Topics in Synchronization and Motion Control 2013.pdf · control of marine craft using the concepts...

Ehsan Peymani

Topics in Synchronization andMotion Control

Doctoral thesisfor the degree of Philosophiae

Trondheim, July 2013

Norwegian University of Science and TechnologyFaculty of Information Technology, Mathematics andElectrical EngineeringDepartment of Engineering Cybernetics

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NTNU

Norwegian University of Science and Technology

Doctoral thesisfor the degree of Philosophiae

Faculty of Information Technology, Mathematics andElectrical EngineeringDepartment of Engineering Cybernetics

c© 2013 Ehsan Peymani.

ISBN 978-82-471-4525-8 (printed version)ISBN 978-82-471-4524-1 (electronic version)ISSN 1503-8181

Doctoral theses at NTNU, 2013:205

Printed by NTNU-trykk

Preface

This thesis presents the results of my PhD research, carried out at thedepartment of the engineering cybernetics and the center for ships andocean structures, Norwegian university of science and technology (ntnu),in Trondheim, Norway, and at the school of electrical engineering and com-puter science, Washington state university (wsu), in Pullman, Washington,under the guidance of Professor Ali Saberi during a one-year research visit.The research was conducted between August 2009 and June 2013.

The research for this thesis was financially supported by the ResearchCouncil of Norway through the center for ships and ocean structures(CeSOS), the center of excellence, from August 2009 to August 2012.

I gratefully acknowledge the support and generosity of the departmentof engineering cybernetics for the financial support from September 2012to June 2013 by providing an opportunity for me to work as a researchassistant.

I wish to thank Professor Thor I. Fossen, my advisor, for his supporttowards the completion of this thesis. I would like to thank him for helpingme extend the initial contract.

I would like to express my special thanks of gratitude to Professor AliSaberi and Dr. Havard F. Grip for taking my visit to Washington StateUniversity serious, and making it as fruitful and productive as it could be.They gave me the golden opportunity to work on the topic of H∞ almostsynchronization. No doubts, without their help and guidance this part ofthe thesis would not have been completed.

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Summary

This thesis considers three issues: the problem of output synchronizationin homogenous and heterogeneous networks of linear agents, the problemof path following for 3-DOF marine vehicles, and the problem of motioncontrol of marine craft using the concepts from analytical mechanics.

The thesis studies synchronization of multi-agent systems in the pres-ence of external disturbances, and brings forth the notions of “H∞ almostsynchronization”, “H∞ almost regulated synchronization”, and “H∞ almostformation”. In almost synchronization, the objective is to find decentral-ized, distributed, dynamic protocols so that agents reach an agreement ona certain quantity of interest with any arbitrary degree of accuracy, whichshould be defined appropriately. The objective of the “H∞ almost” synchro-nization problem is to find protocols such that the H∞-norm of the transferfunction from disturbance to synchronization errors can be made arbitrarilysmall. Similarly, the problem of H∞ almost regulated synchronization withrespect to a reference, which is generated by an exosystem and is known toa subset of agents, is to find protocols such that the H∞-norm of the trans-fer function from disturbance to regulation errors can be made arbitrarilysmall; thus, any arbitrary degree of accuracy can be obtained for regulation.

The thesis considers multi-agent systems in which communication linksare unidirectional. Agents are assumed to have multiple inputs and multipleoutputs, and are described by linear, time-invariant models. The problemof H∞ almost synchronization is first solved for heterogeneous networksof introspective agents; i.e. networks possess nonidentical agents whichhave access to their own output. From a practical point of view, bothlocal and global sensing are required for such networks. Moreover, theproblem of H∞ almost synchronization is solved for homogeneous networksof non-introspective agents; i.e. networks possess identical agents whichhave no access to their own state or output. Therefore, only local sensingis essential. The protocols are derived based on the structural properties ofagents. The design procedures are given in a step-by-step manner, and the

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proofs are provided in details to reveal the strong mathematical features ofthe methodology.

In addition, the thesis conducts a study of path-following control meth-ods for 3-DOF marine craft. Innovative results are presented for straight-line path-following in which a particular emphasis is forwarded to the role ofspeed. Two approaches are proposed so that the speed becomes dynamicallydependent on the geometric error and its derivative. In one approach, back-stepping is utilized, and a nonlinear dynamic controller is derived. Underthe proposed control law, the vehicle accelerates during transient to movetoward the path with higher speeds than the desired speed. In the othermethod, with the aid of the method of least squares, a nonlinear controllerfor path following is derived. The controller is first designed for fully actu-ated craft, and then applied to a linearized model of underactuated vessels.It provides a less-computationally extensive method to deal with externaldisturbances by inclusion of an integral action to the system. In fact, itallows the marine craft to compensate for external forces by increasing thespeed. This approach lends itself to cooperative path following.

Moreover, the thesis presents a framework for motion control of marinecraft using [the accepted principles of] analytical mechanics. More accu-rately, a motion control problem is portrayed as the modeling problem ofa constrained multi-body system, and the Lagrangian approach to find theequation of motion of constrained systems is taken into consideration inorder to find the forces under which the system moves in a way that theconstraints are satisfied exactly. These forces are considered as the controlforces, and are applied to the system by virtue of the actuators. There-fore, the system is controlled as if Nature has been the control engineer.The novelty of the work is to generalize a recently established framework toincorporate both holonomic and nonholonomic constraints; thus, scenariossuch as path maneuvering or coordinated path following fall within the scopeof the framework. A leader-follower formation is given as an illustration.

Contents

Preface i

Summary iii

1 The Scope of this Thesis 1

1.1 Topics of this Thesis . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Synchronization under External Disturbances . . . . . 1

1.1.2 Guided Motion Control of Marine Craft . . . . . . . . 6

1.1.3 Motion Control using Analytical Mechanics . . . . . . 8

1.2 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Part I Synchronization under External Disturbances 13

2 Preliminary (Part I) 15

2.1 Linear Systems Theory . . . . . . . . . . . . . . . . . . . . . . 15

2.1.1 Left- and Right-invertibility . . . . . . . . . . . . . . . 19

2.1.2 Finite Zeros and Infinite Zeros . . . . . . . . . . . . . 20

2.1.3 Geometric Subspaces . . . . . . . . . . . . . . . . . . . 21

2.2 Special Coordinate Basis . . . . . . . . . . . . . . . . . . . . . 23

2.3 Squaring-down Compensator . . . . . . . . . . . . . . . . . . 28

2.4 H∞ Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.5 Algebraic Graph Theory . . . . . . . . . . . . . . . . . . . . . 35

3 Synchronization in Multi-agent Systems 39

3.1 Multi-agent Systems . . . . . . . . . . . . . . . . . . . . . . . 39

3.2 Synchronization in Multi-agent Systems . . . . . . . . . . . . 41

3.3 Review of Previous Work . . . . . . . . . . . . . . . . . . . . 47

3.3.1 Synchronization in Multi-agent Systems . . . . . . . . 47

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3.3.2 Synchronization under External Disturbances . . . . . 51

4 Heterogeneous Networks of Introspective Agents 534.1 The Topic of The Chapter . . . . . . . . . . . . . . . . . . . . 534.2 Notations and Preliminaries . . . . . . . . . . . . . . . . . . . 544.3 Heterogeneous Multi-Agent Systems . . . . . . . . . . . . . . 554.4 H∞ Almost Synchronization . . . . . . . . . . . . . . . . . . . 56

4.4.1 Special Case . . . . . . . . . . . . . . . . . . . . . . . 584.4.2 Homogenization . . . . . . . . . . . . . . . . . . . . . 594.4.3 Design Method . . . . . . . . . . . . . . . . . . . . . . 61

4.5 H∞ Almost Regulated Synchronization . . . . . . . . . . . . 614.5.1 Homogenization of the Augmented Network . . . . . . 644.5.2 Dynamic Protocol . . . . . . . . . . . . . . . . . . . . 654.5.3 Design Scheme . . . . . . . . . . . . . . . . . . . . . . 65

4.6 H∞ Almost Formation . . . . . . . . . . . . . . . . . . . . . . 664.7 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . 674.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Appendix 4.A Proof of Lemma 4.1 . . . . . . . . . . . . . . . . . . 71Appendix 4.B Proof of Lemma 4.2 . . . . . . . . . . . . . . . . . . 77Appendix 4.C Manipulation of Exo-system . . . . . . . . . . . . . 81Appendix 4.D Proof of Theorem 4.3 . . . . . . . . . . . . . . . . . 83Appendix 4.E Proof of Proposition 4.1 . . . . . . . . . . . . . . . 84Appendix 4.F Simulation Data . . . . . . . . . . . . . . . . . . . 85

5 Homogeneous Networks of Non-introspective Agents 875.1 The Topic of The Chapter . . . . . . . . . . . . . . . . . . . . 875.2 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.3 Homogeneous Multi-Agent Systems . . . . . . . . . . . . . . . 895.4 H∞ Almost Synchronization . . . . . . . . . . . . . . . . . . . 90

5.4.1 Problem Formulation . . . . . . . . . . . . . . . . . . 905.4.2 Preliminaries and Assumptions . . . . . . . . . . . . . 915.4.3 Protocol Development (SISO) . . . . . . . . . . . . . . 925.4.4 Protocol Development (MIMO) . . . . . . . . . . . . . 96

5.5 Other Possible Scalings . . . . . . . . . . . . . . . . . . . . . 1015.6 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 1025.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103Appendix 5.A Special Coordinate Basis . . . . . . . . . . . . . . . 105Appendix 5.B Proof: Theorem 5.1 – SISO Case . . . . . . . . . . 107Appendix 5.C Proof: Theorem 5.2 – MIMO Case . . . . . . . . . 115Appendix 5.D Proof of Theorem 5.3 – MIMO case – Irregular

Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Contents vii

Part II Guided Motion Control of Marine Craft 135

6 Preliminary (Part II) 1376.1 Mathematical Modeling of Ocean Vehicles . . . . . . . . . . . 137

6.1.1 3-DOF Model of Marine Craft . . . . . . . . . . . . . 1416.2 Underactuated Marine Craft . . . . . . . . . . . . . . . . . . 143

6.2.1 Model of Underactuated Marine Craft . . . . . . . . . 1446.2.2 Why to Study Control of Underactuated Marine Craft?1456.2.3 Properties of Underactuated Marine Craft . . . . . . . 146

6.3 Motion Control Scenarios . . . . . . . . . . . . . . . . . . . . 1496.3.1 Path Maneuvering vs. Trajectory Tracking . . . . . . 1546.3.2 Underactuated vs. Fully Actuated Control Problems . 1556.3.3 Guidance-based Path Maneuvering . . . . . . . . . . . 155

6.4 Previous Work on Marine Control Systems . . . . . . . . . . 1576.4.1 Control of Fully Actuated Marine Craft . . . . . . . . 1576.4.2 Control of Underactuated Marine Craft . . . . . . . . 159

6.5 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . 1676.5.1 Lyapunov Stability . . . . . . . . . . . . . . . . . . . . 1676.5.2 Input-to-State Stability . . . . . . . . . . . . . . . . . 1706.5.3 Stability of Cascade Systems . . . . . . . . . . . . . . 1706.5.4 Brockett’s Theorem . . . . . . . . . . . . . . . . . . . 173

7 Speed-varying Path Maneuvering: A Nonlinear Approach 1757.1 Motivation and Objective . . . . . . . . . . . . . . . . . . . . 1757.2 Model of Horizontal Motion . . . . . . . . . . . . . . . . . . . 1777.3 Line-of-sight Guidance System . . . . . . . . . . . . . . . . . 1797.4 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . 181

7.4.1 Problem Formulation . . . . . . . . . . . . . . . . . . 1827.5 Control Design Method . . . . . . . . . . . . . . . . . . . . . 183

7.5.1 Control Development: Backstepping . . . . . . . . . . 1837.5.2 Main Result . . . . . . . . . . . . . . . . . . . . . . . . 1867.5.3 Design Procedure . . . . . . . . . . . . . . . . . . . . . 192

7.6 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 1937.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

8 Path Maneuvering: A Least Squares Approach 1978.1 Introduction & Objective . . . . . . . . . . . . . . . . . . . . 1978.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . 199

8.2.1 Model of 3-DOF Marine Craft . . . . . . . . . . . . . 2008.2.2 Guidance System . . . . . . . . . . . . . . . . . . . . . 2018.2.3 Problem Formulation . . . . . . . . . . . . . . . . . . 202

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8.3 Design Method . . . . . . . . . . . . . . . . . . . . . . . . . . 2038.3.1 Desired Accelerations to Make Cross-track Error zero 2048.3.2 Desired Accelerations to Achieve Heading and Speed

Objectives . . . . . . . . . . . . . . . . . . . . . . . . . 2058.3.3 Accelerations to Achieve All Objectives . . . . . . . . 206

8.4 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2078.4.1 Properties of Proposed Controller . . . . . . . . . . . . 210

8.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 2148.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216Appendix 8.A Required Relations . . . . . . . . . . . . . . . . . . 217Appendix 8.B Proof of Lemma 8.1 . . . . . . . . . . . . . . . . . . 218Appendix 8.C Proof of Theorem 8.2 . . . . . . . . . . . . . . . . . 219

9 Path Maneuvering for UMC: A Least Squares Approach 2239.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2239.2 Path Maneuvering Problem . . . . . . . . . . . . . . . . . . . 225

9.2.1 Model of 3-DOF Marine Craft . . . . . . . . . . . . . 2259.2.2 Guidance System . . . . . . . . . . . . . . . . . . . . . 2269.2.3 Problem Formulation . . . . . . . . . . . . . . . . . . 228

9.3 Control Development . . . . . . . . . . . . . . . . . . . . . . . 2299.4 Simulation Result . . . . . . . . . . . . . . . . . . . . . . . . . 2369.5 Concluding Remark . . . . . . . . . . . . . . . . . . . . . . . 237Appendix 9.A Required Relations . . . . . . . . . . . . . . . . . . 238Appendix 9.B Proof of Lemma 9.1 . . . . . . . . . . . . . . . . . . 238

Part III Motion Control of Lagrangian Systems usingAnalytical Mechanics 243

10 Motion Control using Analytical Mechanics 24510.1 Introduction and Objectives . . . . . . . . . . . . . . . . . . . 24510.2 Lagrangian Approach to Multi-body Systems . . . . . . . . . 247

10.2.1 Equations of Motion . . . . . . . . . . . . . . . . . . . 24810.2.2 Constrained Motion . . . . . . . . . . . . . . . . . . . 24910.2.3 Dealing with Constrained Motion . . . . . . . . . . . . 253

10.3 Control Methodology . . . . . . . . . . . . . . . . . . . . . . . 25510.3.1 Overview of the Control Methodology . . . . . . . . . 25610.3.2 Constraint Stabilization . . . . . . . . . . . . . . . . . 25610.3.3 Virtual Constraints . . . . . . . . . . . . . . . . . . . . 258

10.4 Case Study: Master-Slave Formation . . . . . . . . . . . . . . 26010.4.1 Model of Formation . . . . . . . . . . . . . . . . . . . 260

Contents ix

10.4.2 Virtual Constraints . . . . . . . . . . . . . . . . . . . . 26110.4.3 The Control Law . . . . . . . . . . . . . . . . . . . . . 26410.4.4 Simulation Results . . . . . . . . . . . . . . . . . . . . 266

10.5 The Fundamental Equation of Motion . . . . . . . . . . . . . 26810.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . 274Appendix 10.A Fundamental Equation: Derivation . . . . . . . . . 275

11 Concluding Remarks & Future Work 277

Bibliography 283

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Chapter 1

The Scope of this Thesis

1.1 Topics of this Thesis

The thesis is divided into three parts, each of which deals with a particulartopic. These three parts are discussed in the following sections.

1.1.1 Synchronization under External Disturbances

The problem of synchronization in networks of dynamic systems possessesdiverse applications such as synchronization of coupled oscillators, distributedrobotics such as formation flying, flocking and swarming, distributed sensorfusion in mobile sensor networks, distributed and parallel computing, quan-tum networks, networked economics, biological synchronization, and socialnetworks, to mention but a few.

In general, the problem of synchronization in networks of agents is todesign decentralized controllers so that agents reach an agreement on acertain quantity of interest which depends on the state or output of agents.To be more accurate, a network of agents, which is often referred to as amulti-agent system, is a group of dynamic systems, each of which is usuallydescribed by a state-space model, that communicate with each other bymessage passing according to an information exchange topology. A completedefinition of multi-agent systems and various classifications are presented inSection 3.1.

Having a look at the literature on synchronization in multi-agent sys-tems, one may easily grasp that the problem of synchronization in multi-agent systems has been addressed in a myriad of articles using a variety oftechniques. An overview of the current trends on synchronization is pro-vided in Section 3.3.

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2 The Scope of this Thesis

On the whole, much of attention has been given to networks of identicalagents with linear dynamics. Although networks with bidirectional linkswere initially at the heart of attention, networks with limited communicationlinks have already been dominated. Uniform and non-uniform delay incommunication links are among the hot topics of synchronization. Networksof nonlinear agents and networks of non-identical agents are still activeresearch topics.

A large number of articles consider synchronization when the agents arenot exposed to external disturbances. In fact, the problem of synchroniza-tion in multi-agent systems in the presence of external disturbances has notbeen addressed, particularly as a topic of research, and there are only afew articles that consider synchronization in the presence of external dis-turbances.

Articles in which the problem of synchronization is studied in the pres-ence of external disturbances are reviewed in Section 3.3. Roughly speaking,those papers can be categorized into three categories.

(1) Articles that analyze the disturbance attenuation properties of dis-tributed protocols without proposing any specific design that achievessynchronization with a desired degree of accuracy in terms of an ac-cepted measure of accuracy.

(2) Articles that expand the results of the suboptimal H∞ control problemfor the conventional single-agent case to the case of multi-agent systems.They basically guarantee that the H∞ norm of the transfer functionfrom disturbance to a linear combination of synchronization errors isless than a given γ > 0 if some linear matrix inequalities hold. Theseworks, usually, do not make any argument about the solvability of theproblem for the given γ.

(3) Articles that solve the suboptimal H∞ control problem over a network,where the objective is to make the H∞ norm of the transfer functionfrom disturbance to the output of individual agents smaller than a givenγ > 0. Their main result is that such a H∞ control problem can besolved in a decentralized manner. Their result is limited to identicalagents and undirected graphs (bidirectional links). Their objective hasnothing to do with the accuracy of synchronization, which should bethe primary goal. In fact, the H∞ norm of the chosen transfer functionis not a measure of the accuracy of synchronization, and it shows thelevel disturbance rejection from the output of agents.

1.1. Topics of this Thesis 3

Hence, one observes a distinct lack of systematic approaches to de-sign synchronization protocols for networks of general linear systems, cou-pled through partial state information and under directional communicationlinks, which are capable of synchronizing the agents with arbitrary accuracyin the presence of external disturbances. This thesis initiates a line of re-search to fill this gap. In other words, the thesis intends to make a clearextension from robust control of single-agent systems to robust synchroniza-tion of multi-agent systems.

The central part of robust control has been occupied by the problemof (almost) disturbance decoupling. The basic problem of (almost) distur-bance decoupling is to find a controller accepting the measured output yas its input and generating the control signal u as its output such that theapplication of the control signal to the system results in an internally sta-ble closed-loop system and guarantees that the output z as the output tobe controlled is exactly (or approximately in some sense) decoupled froman exogenous disturbance w. In the milder case when almost disturbancedecoupling is requested, the measure of decoupling between z and w isthe H∞-norm of the closed-loop transfer function from w to z. Then, theproblem of H∞ almost disturbance decoupling as an optimization problemarises, in which the H∞-norm of the transfer function from w to z is to bereduced to any arbitrarily small given value. This is a very important topicin robust control theory which builds the backbone of the H∞ optimizationproblem. This can be seen as a special case of the suboptimal H∞ controlproblem in which γ∗, as the infimum of the H∞ norm of the correspond-ing closed-loop transfer function under all stabilizing controllers, is equal tozero. This topic is reviewed in Section 2.4.

Inspired by this concept, the first goal in analysis and design of proto-cols to make a network of agents synchronize while agents are subject toexternal disturbances is to find protocols that attenuate the disturbance inthe synchronization error to any arbitrarily small value. Then, synchroniza-tion in the presence of external disturbances is achieved with any desiredaccuracy (in some sense). It is referred to as almost synchronization.

The measure of the accuracy of synchronization, in this thesis, is givenby the H∞ norm of the transfer function from disturbances to the synchro-nization errors. Thus, it makes sense to call it the problem of H∞ almostsynchronization, in which the H∞ norm of the transfer function under studycan be reduced to any arbitrarily small value. Introducing this notion, theobjective of the thesis is to address the following question:

? (Q1) Considering networks of homogeneous/heterogeneous agents, whenand how H∞ almost synchronization is doable?


The tendency is to utilize asymptotic methods of time-scale assignment(Ozcetin et al., 1992) which use singular perturbation methodology (Koko-tovic et al., 1986). In these methods (for a single systems), the controlleris parameterized in terms of a parameter ε ∈ (0, 1]. The smaller the valueof ε, the more the disturbance d is decoupled from the controlled output z.The structure of the controller does not depend on the value of ε; therefore,ε is used as a tuning parameter to attain the desired level of disturbance at-tenuation. Since the controller is continuous in ε, tuning can be carried outonline. Even optimization methods can be employed in order to find an op-timal value for ε which minimizes a cost function. It implies that the designis not an iterative design, but it is a one-shot design. In addition, the orderof the controller is fixed for any desired level of disturbance attenuation.

To be able to tackle question (Q1), two projects are defined and solved.The first project is titled

Project (1) H∞ almost synchronization for heterogeneous networks of in-trospective agents under external disturbances.

In this project, agents are non-identical; however, it is assumed that eachagent is partially aware of its own state in addition to a network measure-ment that is a linear combination of its own output relative to the outputof its neighbors.

That the agents have partial access to their own states enables us todesign local pre-compensators and pre-feedbacks so as to make the agentsasymptotically imitate the desired identical dynamics. In fact, the networkof non-identical (or heterogenous) agents is converted to a network of almostidentical agents. This process is called homogenization. The homogeniza-tion process, employed in the thesis, is inspired by the work of Yang et al.(2011b), which suffers from a technical problem disabling the method to beapplicable to all multi-input multi-output (MIMO) systems. This problemis resolved in the thesis and a homogenization process that is applicable toany MIMO system is designed.

Since the output of the system, based on which agents communicate,is not accessible, pre-compensators are only viable in the homogenizationprocess. Therefore, the agents are confined to be right-invertible; as a con-sequence, the output dimension must be less than the input dimension.

The derivation of the decentralized protocol is inspired by the work ofOzcetin et al. (1992). However, a novel method of “fictitious disturbances”is introduced in order to avoid transforming the system into the specialcoordinate basis of Sannuti and Saberi (1987), which can be computation-ally extensive. The solvability conditions are clearly stated in terms of the


dynamics of agents for families of communication network graphs.

In addition to H∞ almost synchronization, which is an unconstrainedsynchronization meaning that the synchronization trajectory is not con-trolled, the notion of “H∞ almost regulated synchronization” is introducedand solved for networks of Project (1). In this problem, a subset of agentsare aware of a reference signal which is generated by an exosystem. Theobjective is to suppress the impact of disturbances on the regulation errorsto any desired value in the sense of the H∞ norm of the transfer functionfrom disturbances to the regulation errors.

The problem of H∞ almost synchronization lends itself to the problemof formation. In this thesis, the problem of “H∞ almost formation” is,moreover, brought forward, in which agents must synchronize while theymaintain their relative outputs as desired.

The other project that is defined to help us realize how to resolve ques-tion (Q1) is as below:

Project (2) H∞ almost synchronization for homogeneous networks of non-introspective agents under external disturbances.

In Project (2), networks of identical agents are taken into consideration;however, agents do not have access to their own state or output. This relax-ation significantly affects the solvability of the H∞ almost synchronizationproblem, and considerably complicates the solution.

It should be noted that, in Project (1) where agents are introspective,using the homogenization process, agents are forced to mimic the dynam-ics of a chain of integrators of a specific order. It means that we shoulddeal with systems which only own infinite-zero dynamics of a particularrank. However, in the absence of self-measurements, this is not possible toshape agents into a desired identical form. Thus, we are confronted with ageneral linear system which may have nonminimum phase dynamics, non-right invertibility dynamics, and non-left invertibility dynamics in additionto infinite zero dynamics with non-identical orders. The design procedureis developed with the aid of the special coordinate basis (scb), intro-duced by Sannuti and Saberi (1987), and relies on the structural propertiesand decomposition of the model of agents.

That the orders of infinite zeros are not identical (i.e. the system is not ofa uniform rank1) complicates the design and requires a multiple time-scaleassignment technique, which has been worked out particularly for Project(2) and presented in Chapter 5. The novel multiple time-scale assignment

1Refer to Section 2.2 for clear definition.


technique can also be used to solve the H∞ almost disturbance decouplingproblem for a single linear system when either the output matrix C or theinput matrix B is uncertain; thus, it can be regarded as an extension to(Ozcetin et al., 1992).

The problem of H∞ almost synchronization is solved for homogeneousnetworks of non-introspective agents with the aid of a distributed observerand under the assumption that agents are capable of exchanging additionalinformation according to the network topology. The necessary and sufficientconditions for the solvability of the problem are given in terms of the con-cepts from the geometric control theory developed by Wonham (1985). It isshown that if the problem of H∞ almost disturbance decoupling is solvablefor a single agent, the problem of H∞ almost synchronization is solvablefor a network of identical agents with any communication topology whosegraph is a member of a specific family of network graphs.

1.1.2 Guided Motion Control of Marine Craft

The second part of this thesis is concerned with guided motion control of ma-rine craft. Guided motion control methods are referred to those controllersthat employ guidance systems to generate reference trajectories that are tobe tracked by a marine vehicle so that the vehicle fulfills the objectives.As an example, one may indicate to guidance-based two-dimensional path-following controllers in which the desired course/heading angle is generatedusing a guidance system so that the vehicle moves towards the path.

The thesis focuses on path maneuvering and coordinated path maneu-vering for marine craft which move horizontally. In the path-maneuveringscenario, given a desired path and a desired speed profile, a marine crafthas to converge to and follow the path with the desired speed profile. InChapter 6, an overview of possible motion control scenarios is presented.The topic of motion control of marine craft has been tackled with a delugeof articles. A broad perspective of related works is given in Section 6.4.

The general methodology for path maneuvering is to decompose it intotwo independent tasks, namely, the geometric task of converging to thepath and the dynamic task of assigning speed to the vehicle. The underlyingassumption is that moving on the path is more important than moving witha desired path. Therefore, path-maneuvering controllers are designed as twodecoupled controllers, namely, speed controllers that use speed informationfor feedback, and heading autopilots that use the geometric information2

2By the geometric information, I mean the distance to a desired point on the pathand the rate of convergence to that point.


and heading information for generating feedback laws.

It implies that the speed of a vehicle is not determined according tothe geometric information; thus, assuming ideal conditions and a perfecttracking controller, the speed is exactly the same as a desired speed evenif the vehicle is far from the path. Almost all published works on path-maneuvering of marine craft possess this feature.

However, in every-day life, a driver/captain may change the speed inorder to move towards the path faster and to reduce the time of convergence,or to avoid skidding and sliding sideways. In other words, the geometricinformation is utilized in the speed control loop. Identifying this missinglink between the speed and the geometric information, this thesis attemptsto address the following query

? (Q2) How to incorporate the geometric information in the speed controlloop such that the stability of the closed-loop system is preservedand the path-maneuvering objectives are achieved?

Two methods are proposed to provide a resolution for question (Q2).The first method is built on the achievements of the work of Fossen et al.(2003) where, using backstepping, a controller is designed for an underac-tuated marine craft for guided straight-line path following. Given a desiredprofile ud(t), the essence of the proposed method in this thesis is to find anappropriate function f(e) where e is the geometric error between the marinecraft and the path such that the controller forces the marine vehicle to movewith the speed of ud(t) + f(e), and to follow the path. The function f(e)should be chosen such that the speed varies so as to help path followingoccur (better in some sense, e.g. faster).

In the proposed design, a general model of underactuated marine craftis taken into account, in which a force acting on the sway dynamics isgenerated due to rotation although sway is not actuated. The proposeddesign does not need a coordinate transformation3, which can be seen asan advantage for the proposed method. Also, motivated by (Fossen et al.,2003), to deal with underactuation, a dynamic equation is injected into thecontrol system. Thus, the controller increases the order of the closed-loopsystem since it is dynamic.

One drawback of (Fossen et al., 2003) is that the proof of the conver-gence to the path is not addressed4; as a result, it was concluded that the

3A common approach is to transform the model into a new coordinate to decouplethe unactuated dynamics from the control forces; see Subsection 6.2.1.

4The control system contains two loops; the outer loop which contains a guidance


convergence to the path was guaranteed for any value of the guidance pa-rameter ∆. In this thesis, a formal proof for convergence is given, whichstates that the origin of the closed-loop system is globally asymptoticallystable for sufficiently large ∆, which depends on the speed and the structureof the marine craft5.

In the second approach, with the aid of the method of least squares, anonlinear controller is derived for straight-line path following of fully actu-ated and underactuated marine craft. It is shown that although path fol-lowing is attainable for any arbitrarily small ∆ and sufficiently large speedgains when the craft is fully actuated, the guidance parameter ∆ cannot bechosen arbitrarily small when the craft is underactuated.

Unlike the first method in which the speed depends only on the dis-tance between the path and the craft, in this approach, the speed becomesdynamically dependent on the distance and its derivative. The inclusionof the derivative allows the control system to predict the future motion ofthe craft. This method offers a less computationally extensive approach tocope with external disturbances. This approach lends itself to formationcontrol problems where the speed becomes a function of the geometric er-rors between vehicles and is adjusted so that the vehicles form a desiredconfiguration.

1.1.3 Motion Control of Lagrangian Systems using Analyti-cal Mechanics

Motion control of Lagrangian systems has been treated using many tech-niques, usually with the help of tools from mathematics. Conducting aninspection, one realizes that a nonlinear model of the system, obtained us-ing the Lagrangian mechanics, is usually taken into account and a stan-dard control methodology is applied to find a stabilizing control law. Thisframework is mathematical and it does not take advantage of physical in-formation, subtly hidden behind the equations of motion. For example,linearization about one operating point destroys the geometric propertiesof the Lagrangian systems. This is while the physical characteristics canbe exploited to improve the performance of the control system. The thesisintends to find a solution for the following conceptual query:

system producing the desired heading, and the inner loop forcing the craft to track thedesired signals. In (Fossen et al., 2003), it was assumed that these two loops are decoupled,and stability of the inner loop is only studied. The analysis relies on the fact that thestability can be established at the kinematic level and the kinetic level separately.

5The result corresponds with the result of Fredriksen and Pettersen (2006)

1.2. Publications 9

? (Q3) How would the control law look like if Nature was the control en-gineer? How may one derive a motion controller for a mechanicalsystem based on the principles of analytical mechanics?

This question has been partly answered by Ihle (2006). In this thesis,inspired by analytical mechanics, a motion control problem for a mechanicalsystem is portrayed as the modeling problem of a constrained multi-bodysystem; then, using the tools from the Lagrangian mechanics, the equationsof motion of the constrained system is derived, in which the impact ofconstraints on motion is viewed as additional forces that are placed on thesystem dynamics so as to guide the system in a way that the constraintsare satisfied. These constraints are considered as the control forces and areapplied to the system by virtue of actuators.

This thesis generalizes the result of Ihle (2006) in order to provide aframework that both position and velocity control requirements can be han-dled. Then, a complete framework is brought in, which can be used to con-trol mechanical systems which lie within the formulation of Lagrangian orNewtonian. The framework is able to solve scenarios which require speed as-signment such as path maneuvering or coordinated path maneuvering. Theapproach is illustrated by solving a coordinated path maneuvering problemincluding two surface marine vehicles.

The focus of the thesis is on the method of Lagrange multipliers to han-dle constrained motion. Nonetheless, another approach based on Gauss’sprinciple, which introduces a method to deal with constrained motion with-out invoking the method of Lagrange multipliers, is given in order to providea deeper insight into the way that Nature acts as a controller engineer. Fur-thermore, it is delineated how optimal Nature would choose control forces.

1.2 Publications

The main results of this thesis have been published in several internationalconferences and journals. The list contains published, accepted and sub-mitted papers as well as the articles that are going to be submitted soon.

J4 E. Peymani, H.F. Grip, A. Saberi, X. Wang, and T.I. Fossen, H∞ almostsynchronization for heterogeneous networks of introspective agents un-der external disturbances, Automatica, 2012. (Provisionally acceptedas a regular paper)

J3 E. Peymani, H.F. Grip, and A. Saberi, Homogeneous networks of non-introspective agents under external disturbances, H∞ almost synchro-


nization, Automatica, 2013. (submitted)

J2 E. Peymani, T.I. Fossen, Path following for autonomous marine vehi-cles: a least-square approach, IEEE Transactions on Control SystemTechnology, 2013. (submitted)

C9 E. Peymani, H.F. Grip, A. Saberi, and T.I. Fossen, H∞ almost syn-chronization for homogeneous networks of SISO and non-introspectiveagents under external disturbances, In proc. 52nd IEEE conference onDecision and Control, Italy, 2013. (to appear)

C8 E. Peymani, T.I. Fossen, Speed-varying path following for underactuatedmarine craft, 9th IFAC conference on Control Application in MarineSystems, Japan, 2013. (submitted)

C7 E. Peymani, T.I. Fossen, 2D path following for marine craft: a least-square approach, In proc. 9th IFAC Symposium on Nonlinear ControlSystems, France, 2013. (to appear)

C6 E. Peymani, H.F. Grip, A. Saberi, X. Wang, and T.I. Fossen, H∞ al-most regulated synchronization and H∞ almost formation for hetero-geneous networks under external disturbances, In proc. 12th biannualEuropean Control Conference, Switzerland, 2013.

C5 E. Peymani, H.F. Grip, A. Saberi, X. Wang, and T.I. Fossen, H∞ al-most synchronization for non-identical introspective multi-agent sys-tems under external disturbances, In proc. American Control Confer-ence, Washington, DC, 2013.

C4 E. Peymani, T.I. Fossen, Leader-follower formation of marine craft us-ing constraint forces and lagrange multipliers, In proc. 51st IEEEAnnual Conference on Decision and Control, Hawaii, USA, 2012.

C3 E. Peymani, T.I. Fossen, Direct inclusion of geometric errors for path-maneuvering of marine craft, In proc. 9th IFAC Conference on Ma-neuvering and Control of Marine Craft, Italy, 2012.

J1 E. Peymani, T.I. Fossen, Path following of underwater robots using La-grange multipliers, Robotics and Autonomous Systems, 2011. (sub-mitted)

C2 E. Peymani, T.I. Fossen, A Lagrangian framework to incorporate po-sitional and velocity constraints to achieve path-following control, Inproc. 50th IEEE Conference on Decision and Control and EuropeanControl Conference, Florida, USA, 2011.

1.3. Organization 11

C1 E. Peymani, T.I. Fossen, Motion control of marine craft using virtualpositional and velocity constraints, In proc. 9th IEEE InternationalConference on Control and Automation, Chile, 2011.

The following papers are under preparation.

• E. Peymani, Cooperative dynamic positioning – nonlinear H∞ almostsynchronization. (In preparation)

• E. Peymani, H∞ almost regulated synchronization for homogeneousnetworks of non-introspective agents. (In preparation)

• E. Peymani, Techniques for speed-varying path following for underac-tuated marine craft. (In preparation)

1.3 Organization

The technical contents of this thesis are based on the articles listed in Sec-tion 1.2. I have tried to prepare the thesis in a way that chapters can beread independently. It requires a degree of overlap between some of thechapters. The thesis is organized in three parts and includes 11 chapters.A perspective of the thesis is provided here.

Part I: Synchronization under External Disturbances

Chapter 2 is the first preliminary chapter that reviews the linear systemstheory, the H∞ control problem, and the graph theory. The specialcoordinate basis which is the base for the thesis developments is alsogiven.

Chapter 3 presents a clear definition of multi-agent systems and synchro-nization, and classifies them. Mathematical machinery that is usuallyused for synchronization is introduced by an example. A literaturereview on synchronization is provided in this chapter.

Chapter 4 presents the result of Project (1), as defined in Subsection 1.1.1.The notion of H∞ almost synchronization is defined clearly, and theprotocol which solves the problem is derived. This chapter containsthe results of J4, C5 and C6.

Chapter 5 presents the result of Project (2), as defined in Subsection 1.1.1.To facilitate understanding the result, the protocol is proposed forSISO agents; then, the general MIMO agents are considered. Thischapter is composed of the results of J3 and C9.


Part II: Motion Control of Marine Craft

Chapter 6 gives a brief overview of modeling of marine craft with an em-phasis on underactuated marine craft. Motion control scenarios areelucidated. A perspective of previous work on motion control of ma-rine systems is provided.

Chapter 7 presents a two-dimensional path-following controller for under-actuated marine craft, where the speed is nonlinearly dependent onthe distance between the vehicle and the path. This chapter is basedon C3.

Chapter 8 introduces a novel path-following controller for fully actuatedmarine craft, where the speed depends on both the distance to thepath and the rate of convergence. This chapter is composed of theresults of J2 and C7.

Chapter 9 extends the result of Chapter 8 to a linear model of under-actuated marine craft. Convergence to the path and stability of theunactuated dynamics are taken into account. This chapter is basedon C8.

Part III: Motion Control Using the Concept of Constraint Forces

Chapter 10 explains how one can design controllers using the Lagrangianmechanics, and introduces a framework for motion control of La-grangian systems. A formation problem is solved using the proposedframework. This chapter includes the results of C1, C2, and J1.

Part I

Synchronization underExternal Disturbances

13

Chapter 2

Preliminary (Part I)

The chapter reviews fundamental concepts of multivariable lineartime-invariant systems that are crucial in the developments of Part I.The special coordinate basis that reveals the structural properties of alinear time-invariant system is presented, which is the base for solvingthe problem of H∞ almost synchronization. In addition, a squaring-down compensator is presented. A review of the H∞ control problemand the graph theory is presented.

2.1 Linear Systems Theory

Consider a continuous linear, time-invariant system described by1

Σ :

x(t) = Ax(t) + Bu(t)

y(t) = Cx(t) + Du(t)

(2.1a)

(2.1b)

where x ∈ Rn is the state, u ∈ Rm is the control, and y ∈ Rp is the output.In addition, A ∈ Rn×n, B ∈ Rn×m, C ∈ Rp×n, and D ∈ Rp×m are constantmatrices. The transfer function of the system Σ is computed using

T(s) = C(sI−A)−1B + D (2.2)

which represents the input-output map of the system in the frequency do-main. The solution of the system Σ, denoted x(t), with an initial conditionx = x(t) is uniquely expressed as

x(t) = eAtx +

∫ t

t

eA(t−τ)Bu(τ)dτ, t ≥ t

1The concepts that are given in this chapter have been widely discussed in many textbooks; see for example (Chen et al., 2004; Hespanha, 2009; Liu et al., 2005; Saberi et al.,2012; Trentelman et al., 2001).

15

16 Preliminary (Part I)

The matrix exponential function is given by

eAt =∞∑k=0

1

k!Aktk

It satisfies

d

dteAt = eAtA = AeAt

Stability of the system Σ is related to the behavior of the state trajecto-ries of the system in the absence of external inputs, u(t). Thus, the system

x = Ax, x(t) = x (2.3)

is said to be marginally stable or stable in the sense of Lyapunov if thesolution of the system for every bounded initial condition x is bounded. Itis said to be asymptotically stable2 if it is stable and the solution satisfies

limt→∞

x(t) = limt→∞

eAtx = 0

It is worth noting that stability is the property of the equilibrium point. Aslinear systems have only one equilibrium point, it is common to say thatthe system is stable instead of stating that the equilibrium point is stable.

The system (2.3) is stable if and only if all the eigenvalues of A are inthe closed left-half complex plane with those on the jω axis having Jordanblocks of size one. It is asymptotically stable if and only if all eigenvaluesof A are in the open left-half complex plane. Such a matrix A is calledHurwitz stable. If A is Hurwitz stable, then there exist positive scalarsα1 > 0 and α2 > 0 that

‖eAt‖ ≤ α1e−α2t

The following result is fundamental in the stability of linear systems.

Theorem 2.1. The origin of the continuous linear time-invariant system(2.3) is asymptotically stable if and only if, for any given symmetric positivedefinite matrix Q ∈ Rn×n, the Lyapunov equation

ATP + PA = −Q (2.4)

has a unique symmetric positive definite solution P ∈ Rn×n.

For the case that all eigenvalues of A has negative real parts, the solutionof the Lyapunov equation for a Q = QT > 0 is

P =

∫ ∞0

eATtQeAtdt

2For linear systems, asymptotic stability implies exponential stability.

2.1. Linear Systems Theory 17

Input-output Relation: For any p ∈ [1,∞), let Lmp denote the linearspace formed by all measurable signals g : [0,∞)→ Rm such that∫ ∞

0|g(t)|dt <∞

The linear space Lm∞ is defined as the space that g(t) ∈ Lm∞ implies

|g(t)| <∞, ∀t ≥ 0

If g(t) ∈ Lmp for p ∈ [1,∞), its Lp-norm exists and is defined as

‖g(t)‖p ,(∫ ∞

0|g(t)|dt

) 1p

For p =∞, the L∞-norm of g(t) ∈ Lm∞ is defined as

‖g(t)‖∞ , supt≥0|g(t)|

Theorem 2.2. Consider the continuous liner time-invariant system (2.1)where A is Hurwitz. Assume the system is strictly proper (i.e. D = 0).

• If u ∈ Lm1 , then y ∈ Lp1 ∩ Lp∞, y ∈ Lp1 and y is absolutely continuousand y(t)→ 0 as time tends to ∞.

• If u ∈ Lm2 , then y ∈ Lp2∩Lp∞, y ∈ Lp2 and y is continuous and y(t)→ 0as time tends to ∞.

• If u ∈ Lm∞, then y ∈ Lp∞, y ∈ Lp∞ and y is uniformly continuous.

• If u ∈ Lm∞ with u(t)→ u∞ ∈ Rm as t→∞, then y(t)→ y∞ ∈ Rp ast→∞, and the convergence is exponential.

• If u ∈ Lmq , 1 < q <∞, then y ∈ Lpq and y ∈ Lpq .

If the system is proper (i.e. D 6= 0): If u ∈ Lmq , 1 < q <∞, then y ∈ Lpq .

Definition 2.1 (Controllability). The system Σ given by (2.1) is said tobe controllable if for any initial condition x and any given x1, there exista finite time t1 and a control signal u(t), t ∈ [0, t1], such that the result-ing trajectory satisfies x(t1) = x1. Otherwise, the system is said to beuncontrollable. J


If the linear system Σ is controllable, we usually say that the pair (A,B)is controllable. The system Σ is controllable if and only if the controllabilitygrammian of Σ defined as

Wc =

∫ ∞0

e−AτBBTe−ATτdτ

is nonsingular for all t ≥ 0. Alternatively, one can show that the system Σis controllable if and only if the controllability matrix defined as

Qc =[

B AB · · · An−1B]

is full rank (i.e. rankQc = n).Represent the eigenvalues of A by λi for i = 1, · · · , n. An eigenvalue of

A that satisfies

rank[λiI−A B] = n (2.5)

is called a controllable mode. Otherwise, the mode is uncontrollable. If allthe eigenvalues of A satisfy (2.5), the system is controllable.

Definition 2.2 (Stabilizability). The system Σ is said to be stabilizable ifall its uncontrollable modes are asymptotically stable. J

Given the system Σ, if the pair (A,B) is stabilizable, there exists a statefeedback gain F ∈ Rm×n such that the state feedback u = Fx makes theclosed-loop system asymptotically stable. In other words, λ(A+BF) ∈ C−.

Definition 2.3 (Observability). The system Σ is said to be observable if,for any unknown initial state x, there exists a finite time t1 such that theknowledge of u(t) and y(t) over [0, t1] suffices to uniquely determine theinitial state x. Otherwise, the system is called unobservable. J

If the linear system Σ is observable, we usually say that the pair (A,C)is observable. The system Σ is observable if and only if the observabilitygrammian of Σ defined as

Wo =

∫ ∞0

e−ATτCTCe−Aτdτ

is nonsingular for all t ≥ 0. Alternatively, one can show that the system Σis observable if and only if the observability matrix defined as

Qo =

C

CA...

CAn−1


is full rank (i.e. rankQo = n). An eigenvalue of A that satisfies

rank[λiI−A

C

] = n (2.6)

is called an observable mode. Otherwise, the mode is unobservable. If allthe eigenvalues of A satisfy (2.6), the system is observable.

Definition 2.4 (Detectability). The system Σ is said to be detectable if allits unobservable modes are asymptotically stable. J

Given the system Σ, if the pair (A,C) is detectable, there exists anoutput injection gain K ∈ Rn×p such that λ(A + KC) ∈ C−.

2.1.1 Left- and Right-invertibility

The concept of system invertibilities is fundamental in linear systems theory;however, it is usually withdrawn from many text books.

Definition 2.5 (Saberi et al. (2012)). Consider the linear system Σ givenby (2.1).

• Let u1(t) and u2(t) be any inputs to the system Σ, and let y1(t) andy2(t) be the corresponding outputs for the same initial conditions.The system Σ is said to be left-invertible if y1(t) = y2(t) for all t ≥ 0implies that u1(t) = u2(t) for all t ≥ 0.

• The system Σ is said to be right-invertible if, for any yref(t) definedon [0,∞), there exist a control signal u(t) and an initial condition xsuch that y(t) = yref(t) for all t ∈ [0,∞).

• The system Σ is said to be invertible if it is both left and right invert-ible.

• The system Σ is said to be degenerate if it is neither left nor rightinvertible. J

The invertibility of systems can be verified using the concept of transferfunction in frequency domain. Consider the system Σ given by (2.1). Thetransfer function of the system Σ is computed using (2.2). Without loss ofgenerality, we assume that both [BT DT] and [C D] are full rank.

• The system Σ is right invertible if and only if its transfer functionmatrix is a surjective rational matrix. That is, there exists a rationalmatrix, function of s, say R(s), such that

T(s)R(s) = Ip


• The system Σ is left invertible if and only if its transfer function matrixis an injective rational matrix. That is, there exists a rational matrix,function of s, say L(s), such that

L(s)T(s) = Im

Clearly, an invertible system has the same number of inputs as the num-ber of outputs (i.e. m = p). The system which satisfies m = p is called asquare system. A square system is not necessarily invertible.

2.1.2 Finite Zeros and Infinite Zeros

For a single-input single-output (SISO) system, the zeros are defined asthose frequencies for which the transfer function is zero. They are foundas the roots of the numerator of the transfer function. However, for multi-input multi-output (MIMO) systems, the concept of finite and infinite zerosis intricate and requires to be clarified. The normal rank of the transferfunction T(s) is found as bellow.

normrankT(s) = maxrankT(λ) | λ ∈ C

The Rosenbrock system matrix for the system Σ is given as

PΣ(s) =

[sI−A −B

C D

]Definition 2.6. Given the linear system Σ, a scalar z ∈ C is said to be aninvariant zero of Σ if

rankPΣ(z) < n+ normrankT(s)

A scalar z ∈ C is called a transmission zero or a blocking zero if

T(z) = 0 J

A blocking zero is an invariant zero; however, the reverse is not neces-sarily true. For SISO systems, they are identical.

An invariant zero implies that there exists an initial condition, an inputsignal at an appropriate direction, and a frequency which make the outputof the system identically zero (while the input is nonzero). Indeed, since therank of PΣ(s) drops at the invariant zero z, there exist nonzero xR ∈ Rnand uR ∈ Rm such that

PΣ(z)

[xR

uR

]= 0


For x = xR and u(t) = uRezt for t ≥ 0, it is proven (see Proposition 3.6.1Chen and Huang, 2004) that

y(t) = 0, x(t) = xRezt, t ≥ 0

For MIMO systems, there are other types of zeros. Input decoupling zerosare uncontrollable modes of the pair (A,B). Output decoupling zeros areunobservable modes of the pair (A,C). Input-output decoupling zeros arethe eigenvalues of A that are both uncontrollable and unobservable. Fi-nite zeros of MIMO systems are the collection of all theses zeros includinginvariant zeros and transmission zeros.

A rational matrix T(s) possesses an infinite zero of order k when T(1z )

has a finite zero of precisely that order at z = 0. The number of zerosat infinity along with their orders defines the infinite zero structure of thesystem Σ. It will be shown later that using the special coordinate basisintroduced by Sannuti and Saberi (1987) all these properties can be foundeasily.

2.1.3 Geometric Subspaces

Geometric subspaces are explained in this subsection since they are used tosolve the problem of H∞ almost synchronization3. A subspace H is calledA-invariant if h ∈ H implies Ah ∈ H. Let x be the initial condition ofthe system (2.1) at the initial time t = t. The (A,B)-invariant subspaceis defined as

V := x ∈ V | ∃u(t) : x(t,x) ∈ V, t ≥ 0

The subspace V is also called controlled invariant. In other words, the sub-space V is controlled invariant if there exists a state feedback u = Fx suchthat, for any initial condition that starts from V, the closed-loop responsestays in V. The (A,B)-controllability subspace is defined as

R := x ∈ R| ∃u(t), T > 0 : x(t,x) ∈ R, ∀t ∈ [0, t] and x(T,x0) = 0

That is, it is possible to steer the state starting from an arbitrary point in Rto the origin within a finite time and without leaving R. It is conspicuousthat R ⊆ V. The largest controllability subspace in Rn, denoted R∗(Rn),is the space that is spanned by the image of the controllability matrix; i.e.

3The contents of this subsection is taken from Grip (2009); Saberi et al. (2012);Trentelman et al. (2001).


R∗(Rn) = ImQc. These concepts can be used for disturbance decoupling.Consider

x(t) = Ax(t) + Bu(t) + Gw(t) (2.7a)

y(t) = Cx(t) (2.7b)

where w(t) is the external disturbance. The objective is to find the con-ditions under which it is possible to design a stabilizing control signal u(t)such that the output is decoupled from the disturbance. What is required isto confine the influence of disturbance w in a subspace contained in Ker C.Therefore, the pair (A,B) is to be controllable in order to provide the largestcontrollability subspace in Rn (denoted R∗(Rn))using a state feedback. Inaddition, the image of G is to be contained in R∗(Rn). Then, any change tothe states due to w does not affect the rest of the state space. If R∗(Rn) iscontained in Ker C, any change in the states due to w is not visible on theoutput. Thus, it is required that Im G ⊂ R∗ ⊂ Ker C. Let e(t) denote theerror between the states of the system (2.1) and the states of the observerwith an appropriately selected output injection gain. The initial conditionof e(t) is represented by e. The (C,A)-invariant subspace is defined as

S := e ∈ S | e(t, e) ∈ S, t ≥ 0

The subspace S is also called conditioned invariant. In other words, aconditioned invariant subspace S owns the property that if we construct anobserver with an appropriately selected output injection gain, the responsefor any initial observation error in S is confined to S.

More generally, we define the following subspaces. Before stating them,define the coset x/S as

x/S = x+ v|v ∈ S, ∀x ∈ Rn

The set of all cosets is defined as Rn/S = x/S|x ∈ Rn. It indicates thattwo elements of Rn/S, say x1/S and x2/S, are equal if x1 − x2 ∈ S.

Definition 2.7. Consider the system Σ given by (2.1).

• The C−-stabilizable weakly unobservable subspace V−(Σ) is definedas the largest subspace of Rn for which a matrix F exists such that thesubspace is (A + BF)-invariant, contained in Ker(C + DF), whereasthe eigenvalues of (A + BF) are contained in C−.

• The C−-detectable strongly controllable subspace S−(Σ) is defined asthe smallest subspace of Rn for which a matrix K exists such that the

2.2. Special Coordinate Basis 23

subspace is (A + KC)-invariant, contains Im(B + KD), and is suchthat the eigenvalues of the map that is induced by (A + KC) on thefactor space Rn/S are contained in C−. J

It can be shown that

S−(Σ) = (V−(dual of Σ))⊥

The subspaces V−(Σ) and S−(Σ) are invariant under state feedback andoutput injection.

2.2 Special Coordinate Basis

This subsection reviews the special coordinate basis (scb) presented by San-nuti and Saberi (1987). This is a structural decomposition of multivariablelinear time-invariant systems which reveals the system’s finite and infinitezero structures and invertibility properties. The scb decomposes the sys-tem into separate but interconnected subsystems which show how the inputsare related to the outputs and demystify the inner workings of the system(Grip, 2009). The scb can be computed using available software, eithernumerically (Liu et al., 2005) or symbolically (Grip and Saberi, 2010).Consider a linear, time-invariant system4 described by

Σ :

x = Ax + Bu

y = Cx

(2.8a)

(2.8b)

where x ∈ Rn is the state, u ∈ Rm is the control, and y ∈ Rp is the output.Also, A ∈ Rn×n, B ∈ Rn×m and C ∈ Rp×n. Without loss of generality,we assume that B and C are full rank. According to (Sannuti and Saberi,1987), for any system Σ characterized by the matrix triple (A,B,C), thereexist

(i) unique coordinate-free non-negative integers na, nb, nc, nd, 1 ≤ r ≤ n,and qj , j = 1, · · · , r.

(ii) nonsingular state, output and input transformations Γx, Γy, and Γu

such that

x = Γxx, y = Γyy, u = Γuu (2.9)

4A strictly proper LTI system is considered in this thesis to keep the result simple,readable and concise. Extension to non-strictly proper systems is performed using inputand output pre-transformations to separate the direct-feedthrough part from the rest andpresented in (e.g. Saberi et al., 2012, Chapter 3).


where

x =

xa

xb

xc

xd

, y =

[yd

yb

], u =

[ud

uc

]

where the states xa, xb, xc, xd have dimensions na, nb, nc, and nd, respec-tively. Also,

ud, yd ∈ Rmd=pd uc ∈ Rmc yb ∈ Rpb

which implies p = pd + pb and m = md +mc. Moreover, xd, ud and yd arepartitioned as

xd =

x1d

x2d...xrd

, yd =

y1d

y2d...yrd

, ud =

u1d

u2d...urd

In fact, one can represent them as

xd = col xjd, yd = col yjd, ud = col ujd

for j = 1, · · · , r. Here, xjd ∈ Rjqj and ujd, yjd ∈ Rqj . Obviously,∑r

j=1 jqj =nd. For every j = 1, · · · , r, define

Ajd =

[0 Iqj(j−1)

0 0

]∈ Rjqj×jqj , Bjd =

[0qj(j−1)

Iqj

], Cjd =

[Iqj 0qj(j−1)

]Clearly, for j = 1, we obtain A1d = 0, B1d = C1d = Iq1 .

The transformations, introduced in (2.9), take Σ into the scb describedby the following set of equations:

xa = Aaxa + Ladyd + Labyb (2.10a)

xb = Abxb + Lbdyd (2.10b)

xc = Acxc + Lcdyd + Lcbyb +Bc(uc + Ecaxa) (2.10c)

and for each j = 1, · · · , r, there are:

xjd = Ajdxjd + Ljdyd +Bjd(ujd + Ejaxa + Ejbxb + Ejcxc + Ejdxd)(2.10d)


for some constant Aa, Ab, Ac, Lad, Lbd, Lcd, Lab, Lcb5, Bc, Eca, Eja, Ejb,

Ejc, Ejd, and Ljd of appropriate dimensions6. Denote Cd = diagCjd forj = 1, · · · , r. The outputs are given by

yjd = Cjdxjd, yd = Cdxd, yb = Cbxb (2.10e)

In addition, one can partition xa as

xa =

x+a

x0a

x−a

where x+

a ∈ Rn+a , x0

a ∈ Rn0a , x−a ∈ Rn

−a , and na = n+

a + n0a + n−a . Then, the

dynamics (2.10a) can be written as

x+a = A+

a x+a + L+

adyd + L+abyb (2.11a)

x0a = A0

ax0a + L0

adyd + L0abyb (2.11b)

x−a = A−a x−a + L−adyd + L−abyb (2.11c)

where it is assumed that λ(A+a ) ∈ C+, λ(A−a ) ∈ C−, and λ(A0

a) ∈ C0 inwhich C−, C+ and C0 show the open right-half complex plane, the openright-half complex plane, and the imaginary axis of the complex plane,respectively. Clearly, we obtain

Aa =

A+a 0 0

0 A0a 0

0 0 A−a

For the case xb is existent, there exists an integer s such that nb = spb;therefore, we have

Ab =

[0 Ipb(s−1)

0 0

]Cb =

[Ipb

0]

The presented scb explicitly reveals the system’s finite and infinite zerostructures and invertibility properties.

5Lcb can be written as Lcb = BcEcb. That is, yb affects the dynamics of xc in therange space of Bc. Equivalently, (2.10c) can be written as

xc = Acxc + Lcdyd +Bc(uc + Ecaxa + Ecbxb)

6Notice that (2.10d) presented here is different from the one given in (e.g. Saberiet al., 2012, Chapter 3). In fact, each subsystem given by (2.10d) represents a chain ofj integrators which has qj inputs and qj outputs. However, in (e.g. Saberi et al., 2012,Chapter 3), each chain of integrators is assumed to be single-input single-output of orderqj . These two different representations are available in literature.


• The invariant zeros of the system Σ are the eigenvalues of Aa. If xa

is nonexistent, the system has no invariant zeros. It is evident that ifthe system is minimum phase, x+

a and x0a are non-existent.

• The system is right-invertible if and only if xb and hence yb are nonex-istent (nb = 0, pb = 0).

• The system is left-invertible if and only if xc and hence uc are nonex-istent (nc = 0, mc = 0).

• The system is invertible if and only if both xc and xb are nonexistent.Such a system is square and p = m = pd = md.

• The xjd subsystems form the infinite zero structure of the system.Thus, Σ has jqj infinite zeros of order j, and has nd zeros at infinity.

Therefore, one can name the subsystems as follows:

• The finite zero dynamics are represented by (2.10a). The state xa isneither controlled by any input nor does it affect directly any output.

• The equation (2.10b) is the non-right-invertibility dynamics which isnot directly affected by any inputs. The pair (Ab, Cb) is observable;thus, xb is observable from yb.

• The equation (2.10c) is the non-left-invertibility dynamics which doesnot influence any outputs. The pair (Ac, Bc) is controllable; thus, thestate xc can be steered to any desired value using the control uc.

• The infinite zero dynamics are described by r subsystems describedby (2.10d). For every j = 1, · · · , r, the pair (Ajd, Bjd) is controllableand the pair (Ajd, Cjd) is observable. Therefore, the state xjd is con-trollable using ujd through a stack of integrators of order j and canbe visible from the output yjd.

The observability and controllability of the system Σ are also determinedusing the scb as bellow.

• The system Σ is observable (detectable) if and only if the pair (Aobs, Cobs)is observable (detectable).

Aobs =

[Aa 0

BcEca Ac

], Cobs =

[BdEda BdEdc

]


• The system Σ is controllable (stabilizable) if and only if the pair(Acnt, Bcnt) is controllable (stabilizable).

Acnt =

[Aa LabCb

0 Ab

], Bcnt =

[Lad

Lbd

]Definition 2.8. The system is of uniform rank nq if all infinite zeros areof order nq. Equivalently, if Markov parameters Mi+1 = CAiB are zero fori = 0, 1, · · · , nq − 2 and Mnq = CAnq−1B is nonsingular. J

An invertible system with p inputs and p outputs is called of uniformrank nq if, in its scb, r = nq and qr = p and q1 = q2 = · · · = qr−1 = 0.In other words, the system contains infinite zero dynamics which includeonly one chain of integrator as well as finite zero dynamics. The systemdynamics represented in the scb are given by

xa = Aaxa + Ladyd

xd = Ardxd +Brd(ud + Edaxa + Eddxd), r = nq, xd ∈ Rnqp

yd = Crdxd

For invertible systems of a uniform rank, output transformation is not re-quired. It is worth noting that, for an invertible system of a uniform rank,the term Ldd = Lrd can be made zero by an appropriate state transforma-tion. Therefore, if the system Σ is invertible and of the uniform rank nqand has no invariant zeros, one can easily conclude that there exists a statetransformation x = T0xd which transforms the system into

xd = Axd +B(Mu + Eddxd), xd ∈ Rnqp

y = Cxd

where A, B, and C are as

A =

[0 Ip(nq−1)

0 0

], B =

[0Ip

], C =

[Ip 0

](2.12)

and R ∈ Rp×pnq and M ∈ Rp×p is non-singular. Note that nqp = n. In fact,the system can be represented as a chain of integrators

xd =

x1

x2...

xnq−1

xnq

,y = x1

x1 = x2

x2 = x3...

xnq−1 = xnqxnq = Eddxd +Mu


where xj ∈ Rp. The transformation matrix for an invertible system of auniform rank which has no invariant zeros is the observability matrix.

According to the scb, the state space is decomposed into several distinctsubspaces. The state space X is decomposed as

X = X−a ⊕X 0a ⊕X+

a ⊕Xb ⊕Xc ⊕Xd

Here X−a is related to the stable invariant zeros, and X 0a ⊕X+

a is related tothe unstable invariant zeros. Similarly, Xb is related to the non-right invert-ibility dynamics, and if the system is right invertible, Xb = 0. Likewise,Xc is related to the non-left invertibility dynamics, and if the system is leftinvertible, Xc = 0. Finally, Xd is related to the infinite zero dynamics.We have the following properties:

• V−(Σ) is equal to X−a ⊕Xc.

• S−(Σ) is equal to X 0a ⊕X+

a ⊕Xc ⊕Xd.

2.3 Squaring-down Compensator

Consider a linear time-invariant system described by

Σ :

x = Ax+Bu

y = Cx

(2.13a)

(2.13b)

which is not invertible and probably has an unequal number of inputs andoutputs. Notice that the given system may have an equal number of inputsand outputs, but it is not invertible.

The task of squaring down is to deign pre- and/or post compensatorssuch that the resulting system is invertible and has an equal number ofinputs and outputs. In case the system is right-invertible, the squaring-down task is to design pre-compensators in order to combine all inputs intoa new set of inputs so as to have the same number of inputs as the numberof outputs as well as to have an invertible system. Likewise, for the casewhere the system is left-invertible, the squaring-down task is to deign postcompensators to combine outputs into a new set of outputs so that theresulting system is square and invertible. For a degenerate system, pre- andpost compensators are required for squaring down. The objective of thissection is to review a method of squaring down proposed by (Saberi andSannuti, 1988). Before that, the following result from observer theory isgiven.

2.3. Squaring-down Compensator 29

Lemma 2.1 (O’Reilly (1983)). Consider a stabilizable and detectable sys-tem, described by (2.13) where x ∈ Rn is the state, u ∈ Rm is the control,and y ∈ Rp is the output. Let u = Fx+ u0 be a state feedback that makes(A+BF ) Hurwitz stable. Here, u0 is a new open-loop control. A minimalorder observer is characterized by matrices

W ∈ R(n−p)×n, N ∈ R(n−p)×(n−p), M ∈ R(n−p)×p

such that

i) WA = NW +MC

ii) rank[WT CT] = n

iii) λ(N) ∈ C−.

Then, the observer-compensator that realizes the state feedback control lawu = −Fx+ u0 is given by

z = (N −WBG)z + (M −WBJ)y +WBu0

u = −Gz − Jy + u0

where [G, J ] is given by

[G, J ]T = [WT CT]−1FT

Left-invertible systems are considered first. Consider the system Σgiven by (2.13) and find the scb. It will be written as

xa = Aaxa + Ladyd + Labyb (2.14a)

xb = Abxb + Lbdyd (2.14b)

yb = Cbxb

and for each j = 1, · · · , r, there are:

xjd = Ajdxjd + Ljdyd +Bjd(ujd + Ejaxa + Ejbxb + Ejdxd) (2.14e)

yd = Cdxd

Choose the static compensator Kpost such that the new output of the systemis a linear combination of yd and yb:

ynew = Kposty = K1yd +K2yb


Let K1 be nonsingular to preserve the structure of the system at infinity.Choose K1 = Ind

. Then, the objective is to choose K2 such that the systemwith the new output ynew = yd +K2yb is invertible.

If one chooses K2 = 0, it means that yb is totally neglected from theoutputs, and ynew = yd. It indicates that xb is the same as xa which is notinfluenced by any inputs and does not affect any output. Thus, λ(Ab) areadditional invariant zeros due to squaring down, which are not necessarilyin C−. Hence, K2 is to be chosen such that the added invariant zeros arein C−. Writing the system with new output in the scb (see Saberi andSannuti, 1988, page 361), one may find that it is required to choose K2 suchthat Ab − LbdK2Cb is Hurwitz stable. In general, it is not always possibleto fulfill this need. Hence, a dynamic compensator is required.

Assume that the system is stabilizable and detectable. Due to stabi-lizability, the pair (Ab, Lbd) is stabilizable. Choose F such that λ(Ab −LbdF ) ∈ C−. Choose N that λ(N) ∈ C−. Using Lemma 2.1 and takingthe triple (Ab, Lbd, Cb) as the system matrix triple (A,B,C), calculate thematrices W,M,G and J . The dynamic compensator is then given by

Kpost :

z = (N −WLbdG)z + (M −WLbdJ)yb +WLbdynew

ynew = Gz + Jyb + yd

(2.15a)

(2.15b)

where z ∈ R(nb−pb). The substitution of (2.15b) in (2.15a) yields

z = Nz +Myb +WLbdyd (2.16)

Thus, the dynamic post compensator Kpost is exponentially stable. There-fore, the compensator does not deteriorate the robustness and performanceof an eventual feedback system. The post compensator that has to be im-plemented is given by

Kpost(s) =(G(sI−N)−1 [WLbd, M ] + [I, J ]

)Γ−1

y

Denote the cascade of Σ and Kpost(s) (i.e. the squared-down system) byΣsq. One can show the following properties:

• Σsq is square and invertible.

• The invariant zeros of Σsq are the invariant zeros of Σ, λ(N) andλ(Ab−LbdF ). The number of invariant zeros has changed from na tona + nb + nb − pb.

• The poles of Σsq are the poles of Σ and λ(N).

2.3. Squaring-down Compensator 31

System

Γu Kpre( )

Γu ,

( )

System Γy ( ) ,

Kpost( )ΓyFigure 2.1: Squaring-down post compensator for left-invertible systems.

System

Γu Kpre( )

Γu ,

( )

System Γy ( ) ,

Kpost( )Γy

Figure 2.2: Squaring-down pre compensator for right-invertible systems.

• The order of the squared-down system is n+ nb − pb.

• The infinite zero structure of the system and that of the squared-downsystem are identical.

Therefore, if the system Σ is stabilizable, detectable, and minimum phase,so is the squared-down system Σsq. See Fig. 2.1.

Right-invertible Systems are considered, where the xb subsystem isnonexistent in the corresponding scb. The squaring-down task for right-invertible systems which needs pre-compensators is algebraically dual of thesquaring-down task for left-invertible systems. The precompensator Kpre isdesigned as follows (See Fig. 2.2):

1. Transpose the given right-invertible system to obtain the left-invertibledual system, denoted Σd.

2. Design a squaring-down post compensator, denoted Kdpost, for Σd.

3. The pre-compensator Kpre is obtained by transposing Kdpost.

Degenerate Systems are considered, where both the xb and xc subsys-tems are existent in the corresponding scb. For such systems, to obtain asquare invertible system, one requires to design a post-compensator Kpost

to reduce the number of outputs. Then, a pre-compensator Kpre is designedby designing a post-compensator for the dual system. The cascade of Kpre,Σ and Kpost is invertible (for the new set of inputs and outputs).


2.4 H∞ Control

The thesis partly discusses synchronization in the presence of external dis-turbances, and uses the concept of H∞ control. Thus, some of the relevantfacts taken from (Chen, 2000; Khalil, 2002; Saberi et al., 1999) are brieflyreviewed.

In many applications, it is required to make the impact of the externaldisturbances on the output small. Assuming that input signals belong toL2, one should design a controller such that the closed-loop input-outputmap becomes finite-gain L2 stable. We seek a controller that minimizes theupper bound on the L2 gain. Consider the linear time-invariant system

x = Ax+Bu

y = Cx+Du

where A is Hurwitz stable. Let G(s) = C(sI − A)−1B + D. It is straight-forward to verify that (Khalil, 2002, Theorem 5.4)

‖y‖22 ≤(

supω∈R‖G(jω)‖2

)2

‖u‖22

which can be used to show that the L2 gain is equal to supω∈R‖G(jω)‖2.In fact, this is the induced 2-norm of the complex matrix G(jω), and it isknown as the H∞ norm of G(jω) when G(jω) is regarded as an element ofthe Hardy space H∞. Thus, define the H∞ norm of the transfer functionG(s) of a proper, asymptotically stable LTI continuous system as

‖G(s)‖∞ , supω∈[−∞,∞)

σmax[G(jω)] = sup‖ω‖2=1

‖y‖2‖u‖2

where σmax[G(jω)] =√λmaxG(jω)TG(jω). For a system which is not nec-

essarily stable, the L∞-norm is defined as

‖G(s)‖L∞ , supω∈[0,∞)

σmax[G(jω)]

Without doubt, the L∞-norm coincides with the H∞-norm if G(s) repre-sents an asymptotically stable system.

The basic problem of H∞ control is to design, if possible, a controller Ksuch that 1) the closed-loop system is internally stable, and 2) for a givenγ > 0, the H∞ norm of the closed-loop transfer function from externaldisturbances to the controlled (performance) output is less than γ. Thisproblem is called the H∞ γ-suboptimal control problem.

2.4. H∞ Control 33

Controller ( ) Plant

Figure 2.3: The control configuration for the H∞ control problem, wherew represents unknown disturbances, and z is the controlled output. Thecontroller uses the measured output y to produce the input u such that theH∞ norm of the closed-loop transfer function from w to z becomes less thanγ > 0.

The infimum γ∗ is the infimum of the H∞-norm of the closed-loop trans-fer matrix over all stabilizing controllers. The H∞ optimal control problemis to find a controller K such that the resulting system is internally stableand the H∞-norm of the closed-loop transfer matrix is equal to γ∗.

H∞ Almost Disturbance Decoupling7 can be seen as a special caseof the general H∞ control problem, where γ∗ = 0. There is a long his-tory behind the disturbance decoupling problem where the goal is to designa controller in order to decouple the controlled output from external dis-turbances8. In almost disturbance decoupling, the objective is to design acontroller such that the controlled output is approximately decoupled fromthe disturbances; in other words, the controller has to be able to attenuatethe disturbances in the controlled output to any arbitrarily small value.

The problem of H∞ almost disturbance decoupling is said to be solvableif, for any γ > 0, there exists a controller such that the closed-loop systemis internally stable, and the H∞ norm of the closed-loop transfer functionfrom disturbances to the controlled output is less than γ. Therefore, theimpact of disturbances on the controlled output can be made arbitrarilysmall in the sense of the H∞-norm of the closed-loop transfer function.

Lemma 2.2. Let γ > 0 and consider

x = f(x, u)

y = g(x, u)

7Refer to (Ozcetin et al., 1992) for a comprehensive overview on available methods.8Notice that the objective is to remove the unwanted impact of disturbances on the

output, and the internal stability is not required to obtain.


where x ∈ Rn is the state, u ∈ Rm is the input and y ∈ Rp is the output.Suppose there exists V : Rn → Rn such that

1. V (x) ≥ 0 for all x and V (0) = 0.

2. V ≤ −yTy + γ2uTu.

Then, the L2-gain of the system is no more than γ.

Lemma 2.2 links the H∞ gain with dissipativity as it states that thesystem is dissipative with the supply function γ2uTu− yTy and the storagefunction V . The sketch of the proof is given. Consider

V (x(T ))− V (x(0)) = V (x(T )) =

∫ T

0V (x(t), u(t))dt

≤∫ T

0(−y(t)Ty(t) + γ2u(t)Tu(t))dt

Thus, since V (x(T )) ≥ 0, one obtains∫ T

0y(t)Ty(t) ≤ γ2

∫ T

0u(t)Tu(t)dt

⇒ ‖y‖2 ≤ γ‖u‖2For linear systems as

x = Ax+Bu

y = Cx

One may choose V = xTPx and show

V = xT(ATP + PA)x+ xTPBu+ uTBTPx ≤ −yTy + γ2uTu

for a P = PT ≥ 0, then the H∞-norm of the transfer function is less thanor equal to γ. It can be written as[

xT uT] [ ATP + PA+ CTC PB

BTP −γ2I

] [xu

]≤ 0

Thus, if there exists a P = PT ≥ 0 such that[ATP + PA+ CTC PB

BTP −γ2I

]≤ 0

Then, the H∞-norm of the transfer function is less than or equal to γ. Bytaking the Schur complement, we obtain

ATP + PA+ CTC − γ−2PBBTP ≤ 0

which is a Riccati-like quadratic matrix inequality. It turns out that forlinear systems this condition is not only sufficient, but also necessary.

2.5. Algebraic Graph Theory 35

2.5 Algebraic Graph Theory

A graph G consists of a finite set of vertices (nodes) V(G) = v1, v2, · · · , vNand edges (arcs) E(G) ⊂ V × V that connect the vertices9. An edge is anordered pair of distinct vertices. We use eij = (vi, vj) ∈ E which shows thereis a directed edge from node vj to node vi. If eij = (vi, vj) ∈ E implies thateji = (vj , vi) ∈ E for all i, j, then the graph is called undirected . Otherwise,the graph is directed. A directed graph is called digraph.

A digraph is weighted if a positive weight (not necessarily 1) is associatedto each edge. Weights are denoted by aij . Thus, if eij = (vi, vj) ∈ E , aij > 0;if εij /∈ E , aij = 0. For undirected graphs, aij = aji. If self-loops are notallowed, eii = (vi, vi) /∈ E and aii = 0; such graphs are termed as acyclic.For every node vi ∈ V(G), the set of neighbors of vi is defined as

Ni = j ∈ 1, · · · , N : eij = (vi, vj) ∈ E or aij 6= 0

In the context of networked agents, the neighbors of node i are those thatsend information to node i.

For graphs that are not weighted, the in-degree of a node v ∈ V(G) is thenumber of edges that have this node as a head. The out-degree of a nodev ∈ V(G) is the number of edges that have this node as a tail. For weightedgraphs, the in-degree is the sum of weights of edges that enter the node andthe out-degree is the sum of the weights of edges that leave the node. Inother words,

degin(vi) =N∑j=1

aij , degout(vi) =N∑j=1

aji

For unweighted graphs, these quantities are found by setting all wightsaij = 1. If the in-degree of a node is equal to its out-degree, the node iscalled a balanced node. If all nodes of a graph are balanced, the graph istermed as a balanced graph10.

A strong path11 is a sequence of q distinct nodes (v1, · · · , vq) such that(vi, vi+1) ∈ E for i = 1, · · · , q − 1. If the first node and the last node areidentical, the path forms a cycle. A weak path is a sequence of q distinct

9This section only provides the fundamentals of the algebraic graph theory that arelater used and is mainly taken from (Biggs, 1993; Godsil and Royle, 2001).

10The word “balanced” is usually given to digraphs. For undirected graphs, the word“regular” is usually used. A regular (undirected) graph is a graph that every node hasthe same number of edges.

11It is usually called “path”. In fact, usually, a weak path is not considered in thecontext of multi-agent systems.


15

4 3 2

3

3

1

4 6

2 5

(a) A balanced digraph.ehsan ehs anehsan eh s a n h s asm n asa

Eigenvalues of Graphs

λ is an eigenvalue of a graph⇔ λ is an eigenvalue of the adjacency matrix⇔ A~x = λ~x for some vector ~x

Adjacency matrix is real, symmetric ⇒real eigenvalues, algebraic and geometric multiplicities are equalminimal polynomial is product of linear factors for distinct eigenvalues

Eigenvalues:

λ = 3 m = 1

λ = 1 m = 5

λ = −2 m = 4

Rob Beezer (U Puget Sound) An Introduction to Algebraic Graph Theory Pacific Math Oct 19 2009 11 / 36(b) A balanced undirected graph. (A reg-ular graph of degree 3)

Figure 2.4: Examples of balanced and regular graphs.2446 H.F. Grip et al. / Automatica 48 (2012) 2444–2453

1 2 3 4

9876

5

10

2

1.5

2.4 2

1.3

3

2.51.7 1.4

2.7

1.5 1

1

0.8

Fig. 1. The depicted digraph contains multiple directed spanning trees, rooted atnodes 2, 3, 4, 8, and 9. One of these, with root node 2, is illustrated by bold arrows.

where ηj ∈ Rp is a variable produced internally by agent j as partof the controller. This variable will be specified as we proceed withthe control design.

2.1. Assumptions

We make the following assumptions about the network topol-ogy and the individual agents.

Assumption 1. The digraph G has a directed spanning tree withroot agent K ∈ 1, . . . ,N, such that for each i ∈ 1, . . . ,N \ K ,

(1) (Ai, Bi) is stabilizable(2) (Ai, Ci) is observable(3) (Ai, Bi, Ci,Di) is right-invertible(4) (Ai, Bi, Ci,Di) has no invariant zeros in the closed right-half

complex plane that coincide with the eigenvalues of AK .

Remark 1. A directed tree is a directed subgraph of G , consisting ofa subset of the nodes and edges, such that every node has exactlyoneparent, except a single root nodewith noparents. Furthermore,theremust exist a directed path from the root to every other agent.A directed spanning tree is a directed tree that contains all the nodesofG . A digraphmay containmanydirected spanning trees, and thusthere may be several choices of root agent K . Fig. 1 illustrates adigraph containing multiple directed spanning trees.

Remark 2. Right-invertibility of a quadruple (Ai, Bi, Ci,Di)meansthat, given a reference output yr(t), there exist an initial conditionxi(0) and an input ui(t) such that yi(t) = yr(t) for all t ≥ 0.For example, every single-input single-output system is right-invertible, unless its transfer function is identically zero.

Let the matrix GK = [gij]i,j=K be defined from G by removingrow and column number K , corresponding to the root of a directedspanning tree of G . We shall need the following result, which isproven in Appendix A.

Lemma 1. All the eigenvalues of GK are in the open right-halfcomplex plane.

3. Control design

In this section we describe the construction of decentralizedcontrollers that achieve output synchronization. Before embarkingon the actual design procedure, however, we shall describe themotivation behind the design.

The main idea is to set the control input of the root agent Kto zero (i.e., uK = 0) and to also set ηK = 0. We then designcontrollers for all the other agents such that their outputs asymp-totically synchronize with the trajectory yK (t). That is, for eachi ∈ 1, . . . ,N\K wewish to achieve limt→∞(yi −yK ) = 0. Equiv-alently, we wish to regulate the synchronization error variable

ei := yi − yK

to zero, where the dynamics of ei is governed byxixK

=

Ai 00 AK

xixK

+

Bi0

ui, (2a)

ei =Ci −CK

xixK

+ Diui. (2b)

The system (2) is in general not stabilizable. If xi and xK wereavailable to agent i as measurements, then the problem of mak-ing ei converge to zero would nevertheless be solvable by standardoutput-regulationmethods (see, e.g., Saberi, Stoorvogel, & Sannuti,2000). But alas, the only information available to agent i is ζi and ζi.To achieve our objective with such limited information, we carryout our design for each agent i ∈ 1, . . . ,N \ K in three steps.

In Step 1 we construct a new state xi, via a transformation of xiand xK , so that the dynamics of the synchronization error variableei can be described by the alternative equations

˙xi = Aixi + Biui :=

Ai Ai12

0 Ai22

xi +

Bi0

ui, (3a)

ei = Cixi + Diui :=Ci −Ci2

xi + Diui. (3b)

The purpose of this state transformation is to reduce the dimensionof themodel underlying ei by removing redundantmodes that haveno effect on ei. In particular, the model (2) may be unobservable,but the model (3) is always observable.

The properties of the model (3) also allow us, in Step 2 ofthe design, to construct a controller that regulates ei to zeroby using state feedback from xi. This controller is not directlyimplementable, however, because xi is not known to agent i. Thisbrings us to Step 3 of the design, where we construct an observerthat makes an estimate of xi available to agent i. This observeris based on the information ζi and ζi received via the network,and it works in a distributed manner together with the observersfor the other agents to achieve convergence. The observer designis based on previous results on distributed observer design forhomogeneous networks. Since our network is heterogeneous, wefirst perform a second state transformation of xi to χi, in order toobtain a dynamical model that is substantially the same as for theother agents. In particular, themodel differences nowoccur only inparticular locations where they can be suppressed by using high-gain observer techniques. By combining the observer estimateswith the state-feedback controller designed in Step 2, we achieveoutput synchronization.

3.1. Design preliminaries

Due to the design strategy of setting uK = 0, the trajectoryyK (t) becomes the unforced response of agent K , consisting of alinear combination of the observable modes of the pair (AK , CK ).Asymptotically stable modes vanish as t → ∞, and they thereforeplay no role asymptotically. For simplicity of presentation, wetherefore assume that all the eigenvalues of AK are in the closedright-half complex plane and that (AK , CK ) is observable. Wemakethis assumption without any loss of generality since, if AK doescontain unobservable or asymptotically stable modes, we canalways create an auxiliary model excluding those modes for thepurpose of control design (see Appendix C for details).

Below we describe the three steps of the design procedure thatmust be carried out for each agent i ∈ 1, . . . ,N \ K . In additionto agent i’s systemmatrices (Ai, Bi, Ci,Di), the information neededto carry out these three steps for agent i is as follows:

• the matrices AK and CK of the root agent

Figure 2.5: A digraph with multiple directed spanning trees. A spanningtree rooted at node 2 is highlighted by bold arrows. Taken from (Grip et al.,2012).

nodes (v1, · · · , vq) such that either (vi, vi+1) ∈ E for i = 1, · · · , q − 1 or(vi, vi−1) ∈ E for i = 2, · · · , q.

A subgraph of a digraph is a directed tree if every node of that hasexactly one incoming edge, except one distinguished node called the rootwith no incoming edge. Clearly, a tree cannot contain cycles since everynode in a tree must have exactly one parent except the root which has noparents. If a directed tree contains all the nodes of the digraph, it is saidto be a spanning directed tree. In other words, if a digraph contains adirected spanning tree, there is a node in the digraph from which all nodesare accessible by a directed (strong) path (i.e. there are directed pathsfrom the root to every other nodes.). See Fig. 2.5 which depicts a digraphcontaining multiple directed spanning trees with roots at nodes 2, 3, 4, 8,and 9. The directed spanning tree rooted at node 2 is illustrated by boldarrows.

2.5. Algebraic Graph Theory 37

If any ordered pair of distinct nodes are joined by a strong path, thedigraph is strongly connected. A digraph is said to be quasi-strongly con-nected if, for every pair of nodes vi and vj , there exists a node vr that canreach both nodes by a strong path. That is, if a digraph contains a directedspanning tree, it is at least quasi-strongly connected. Note that a strongly-connected graph is definitely quasi-strongly connected. A digraph is weaklyconnected if any pair of distinct nodes are joined by a weak path.

The main properties of graphs are usually studied by introducing thefollowing matrices:

• Adjacency matrix A = [aij ] ∈ RN×N . That is, the entry (i, j) is theweight of the edge from node vj to node vi.

• Degree matrix D ∈ RN×N is a diagonal matrix with entry (i, i) equalto the in-degree of node vi.

• Incident matrix B = [bij ] ∈ RN×E where E is the number of edges:

bij = 1 if there is an edge from vj to vi;

bij = −1 if there is an edge from vi to vj ;

bij = 0 otherwise.

• Laplacian L = D −A. Let us denote the matrix as L = [lij ] where

lii =N∑

j=1,j 6=iaij , lij = −aij , i 6= j

The Laplacian is a diagonally dominant matrix. It is straightforward toverify that the sum of each row is zero. It then implies that L1 = 0 where1 , [1, 1, , · · · , 1]T; that is, 1 is a right eigenvector associated to the eigen-value at zero. In addition, it shows that L has at least one eigenvalue atzero. Therefore, rankL ≤ N − 1. It is worth noting that although 1 is aright eigenvector of L, 1T is not necessarily a left eigenvector associated tozero eigenvalues. If 1TL = 0 and L1 = 0, the graph is balanced.

The number of the eigenvalues of the Laplacian at zero is important sinceit is related to the connectivity properties of the graph. The zero eigenvalueof L is simple if and only if the graph contains a directed spanning tree (Renand Beard, 2005) or, equivalently, if the graph is quasi-strongly connected.The number of connected components is equal to the multiplicity of the zeroeigenvalue of the Laplacian.


Therefore, the eigenvalues of the Laplacian of a graph plays an importantrole in determining the features of the graph. For undirected graphs, theLaplacian is symmetric. Since it is a real valued matrix, the eigenvalues ofthe Laplacian are all real, and it is proven that λ(L) ≥ 0. One can sortthem in an incremental order:

λ1 ≤ λ2 ≤ λ3 ≤ · · · ≤ λN

The algebraic connectivity is then defined as the second smallest eigenvaluesof the Laplacian for an undirected graph (Fiedler, 1973, 1989). The eigen-vector associated with the algebraic connectivity is called the Fiedler vector.It is evident from the previous discussion that if the algebraic connectivityof a graph is positive (λ2 > 0), the graph is connected.

The notion of the algebraic connectivity cannot be extended directly todigraphs. A generalization of Fiedler’s notion of the algebraic connectivityto directed graphs is proposed by Wu (2005). For directed graphs, theLaplacian is not always symmetric; thus, the eigenvalues may be complex.However, it is proved that λ(L) ∈ C+ ∩ C0. As discussed, if a digraph hasa directed spanning tree, it has only one eigenvalue at zero and the rest arein C+.

Chapter 3

Synchronization inMulti-agent Systems

The chapter provides a clear definition for multi-agent systems, and

highlights the challenges that show up in synchronization of multi-agent

systems. Furthermore, a concise literature review on synchronization

in multi-agent systems is given, where an emphasis is placed on syn-

chronization in the presence of external disturbances.

3.1 Multi-agent Systems

A multi-agent system is referred to a network of N multi-input multi-outputsystem described by state-space models as

Agent i :

xi = fi(xi, ui) + gi(wi)

yi = hi(xi)

(3.1a)

(3.1b)

in which i ∈ S , 1, 2, · · · , N, xi ∈ Rni is the state, ui ∈ Rmi is theinput, yi ∈ Rp is the output, and wi ∈ Rωi is the external disturbanceswhose power is bounded (i.e. limT→∞

12T

∫ T−T w

Ti widt < ∞). The vector

functions fi, gi, and hi are such that the system is well-defined, and thesolution exists.

Heterogenous and Homogeneous Multi-agent Systems A multi-agent system is said to be homogeneous if all agents have identical dynamics;that is, for a multi-agent systems, fi(·, ·) = F (·, ·), gi(·) = G(·), and hi(·) =H(·) for all i ∈ S. Otherwise, it is termed as heterogenous.

39

40 Synchronization in Multi-agent Systems

Network Measurements Agents are allowed to exchange informationaccording to the network’s communication topology which is described by aweighted graph L, with no self loops, where each node corresponds to anagent. An edge from node j to node i, weighted with a real positive numberaij , shows agent j sends information to agent i.

The communication graph is associated with the adjacency matrix AL =[aij ] and the Laplacian L = [lij ]. The Laplacian of the digraph L is foundas lii =

∑nj=1,j 6=i aij and lij = −aij for i 6= j. Each agent may receive

information about its own state or output relative to that of its neighbors.A common assumption, which has high practical importance, is that eachagent receives a linear combination of its own state or output relative tothat of its neighbors. In particular, agent i has access to the quantity

ζsi =

∑N

j=1aij(xi − xj) (3.2)

If the agents are coupled by network information as (3.2), it is called a multi-agent system with full state coupling. Obviously, in case agents exchangethe quantity

ζoi =

∑N

j=1aij(yi − yj) (3.3)

it is called a multi-agent system with partial state coupling. In terms of theelements of the Laplacian, one can represent the network measurements as

ζsi =

∑N

j=1lijxj , ζo

i =∑N

j=1lijyj (3.4)

It is possible to assume that agents exchange outputs (i.e. yi) instead ofrelative outputs (i.e. yi − yj). However, control of multi-agent systemsexchanging relative information would be more challenging and has practicalimportance because it removes the need for global sensing and relies on localdifferential sensing.

One-directional vs. Bidirectional Links The network communicationgraph may be undirected, indicating that the communication between agentsis mutual. That is, if agent i sends information to agent j, agent j sendsinformation to agent i. The communication topology may be otherwisedirected where agent i communicates with agent j but the reverse does notneed to happen.

Deterministic vs. Stochastic Network Topologies In determinis-tic network topologies, the links between agents are fixed/switching in a

3.2. Synchronization in Multi-agent Systems 41

deterministic setting. That is, the associated adjacency matrix A(t) is de-terministic. Stochastic network topologies are necessary to consider whenrandom communication failures and communication channel instabilities aretaken into account. Then, the adjacency matrix A(t) stochastically evolves.

Introspective vs. Non-introspective Agents In some applications, asubset of agents possesses partial knowledge of its own states; equivalently,one may say that the agent has some local measurements. In particular, thefollowing measurement is available for agent i:

ym,i = Cm,ixi (3.5)

In this case, the multi-agent system is called a network of introspectiveagents1. If no local measurements are available for agents and the onlyavailable measurement for each agent is the one transmitted over the net-work, the multi-agent system is said to be a network of non-introspectiveagents. Clearly, the presence of local measurements gives additional flexi-bility to design whereas lack of them will be a challenge.

3.2 Synchronization in Multi-agent Systems

The objective of synchronization in multi-agent systems is to design proto-cols based on local (if available) and network information such that agentsagree on a certain quantity of interest which depends on the state of agents.Output synchronization is defined as

limt→∞

(yi − yj) = 0 ∀i, j ∈ 1, 2, · · · , N (3.6)

One may define a stronger synchronization as

limt→∞

(xi − xj) = 0 ∀i, j ∈ 1, 2, · · · , N (3.7)

which is termed as state synchronization. In state/output synchronization,the primary objective is to force all agents to agree on a common syn-chronization trajectory while it does not impose any constraints on thesynchronization trajectory.

As far as synchronization in heterogeneous networks is concerned, statesynchronization no longer makes sense because agents may be described by

1Using the pinning idea (Li et al., 2009), where self-loops are allowed for a subset ofnodes in the graph, indicates that a subset of agents are aware of a quantity dependingon their own states. Therefore, the system is a network of introspective agents.


models of different dimensions, and states may have different physical inter-pretations, which makes comparison between states impossible and mean-ingless. However, in output synchronization, agents will agree on a certainquantity of interest which is determined a priori and is comparable. Outputsynchronization has been studied in several papers, such as (Chopra andSpong, 2008; Grip et al., 2012; Kim et al., 2011).

Controlled synchronization2 intends to make all agents synchronize whilethe synchronization trajectory is constrained to a desired trajectory. Let y0

be the desired reference. Regulated output synchronization is achieved if

limt→∞

yi − y0 = 0 ∀i ∈ 1, 2, · · · , N (3.8)

The desired trajectory y0 may be generated by an exo-system or a referencemodel. Formation control can be defined within the framework of synchro-nization. Considering a formation set f1, · · · , fN, formation control isdefined as

limt→∞|(yi − fi)− (yj − fj)| = 0 ∀i, j ∈ 1, 2, · · · , N (3.9)

Similarly, it is viable to define formation control with regulation of outputsynchronization which is a well-known scenario called “virtual leader for-mation control” in the context of multi-robot systems. In the following, asimple example is provided in order to explain mathematical machinery insynchronization of multi-agent systems.

Example 3.1. Consider a homogeneous network of non-introspective linearagents which are described by

xi = Axi +Bui

where xi ∈ Rn is the state, ui ∈ Rm is the control, and the pair (A,B)is controllable. The communication topology is described by a graph Lassociated with the adjacency matrix A = [aij ] and the Laplacian matrixL = [lij ]. Consider full state information coupling:

ζi =

N∑i=1

aij(xi − xj) ⇒ ζi =

N∑i=1

lijxj

The objective is to design decentralized protocols (controllers) for agentssuch that limt→∞(xi − xj) = 0 for all i, j ∈ 1, 2, · · · , N.

2Controlled synchronization may also be called constrained consensus or regulatedsynchronization.


Take the distributed protocol into account

ui = −FN∑i=1

lijxj

where F ∈ Rn×n is the protocol gain which is to be chosen appropriately.The application of the protocol i to agent i yields

xi = Axi −BFN∑i=1

lijxj

Find the collective dynamics:

χ =

x1

x2...xN

, χ = ((IN ⊗A)− (L⊗BF ))χ

where ⊗ symbolizes the Kronecker product3. Let J be the Jordan form ofL: J = U−1LU . By definition, L has at least one eigenvalue at zero and itsatisfies L1 = 0. The matrix U is chosen such that

U =[U1 1

], U−1 =

[U2

rT

], J =

[∆ 00 0

],

Introduce the state transformation

ϕ =

[ϕ1

ϕ0

]= (U−1 ⊗ In)χ ⇒ χ = (U ⊗ In)ϕ

where ϕ1 ∈ R(N−1)n and ϕ2 ∈ Rn. Then, the representation of the collectivedynamics in the new coordinates is given by[

ϕ1

ϕ0

]=

((IN ⊗A)− (

[∆ 00 0

]⊗BF )

)[ϕ1

ϕ0

]3For two matrices R = [rij ] ∈ Rm×n and W ∈ Rq×p, the Kronecker product is

R⊗W = [rijW ] =

r11W · · · r1nW...

. . ....

rm1W · · · rmnW

The following properties are used:

(A⊗B)T = AT ⊗BT, (A⊗B)(C ⊗D) = AC ⊗BD

(A⊗B)−1 = A−1 ⊗B−1, A⊗B +A⊗ C = A⊗ (B + C)


One part of the dynamics is described by

ϕ1 = ((IN−1 ⊗A)− (∆⊗BF ))ϕ1

We would like to study the stability of ϕ1 = 0. The matrix (IN−1 ⊗ A) −(∆⊗BF ) has an upper triangular structure. Therefore, its eigenvalues aredetermined by the eigenvalues of the elements on its main diagonal whichare A− λiBF where λi for i = 1, · · · , N − 1 are the eigenvalues of L exceptone eigenvalue at zero. Thus, if one can make all N − 1 matrices A− λiBFHurwitz stable, ϕ1 = 0 is globally exponentially stable. The first thing isthat L has to have only one eigenvalue at zero. To hold this, the followingassumption is made on the communication graph L: the communicationgraph contains a directed spanning tree.

Under the assumption that λi 6= 0 in A−λiBF for i = 1, · · · , N−1, onemay find a gain F such that A−λiBF is Hurwitz stable for all possible λi. Itis easy to verify that if F is chosen in a way that A−BF is Hurwitz4, thereis a bounded set S such that for λ ∈ S, A − λBF remains Hurwitz stable.However, we are interested in choosing F in a way that an unbounded setS appears such that for any λ ∈ S, the matrix A− λBF is Hurwitz stable.This problem is the standard problem of designing a state feedback for thepair (A, λB) where the pair (A,B) is controllable and the multiplicativeuncertainty λ satisfies Reλ ≥ τ > 0 such that the design is robust withrespect to the multiplicative uncertainty. Choose

F = BTP

where P = PT > 0 is the unique solution of the algebraic Riccati equation

ATP + PAT − 2τPBBTP = −Q

for some Q = QT > 0. It guarantees (A − λiBF ) is Hurwitz stable forall λi : Reλi ≥ τ > 0 since one may show that (A − λiBB

TP )∗P +P (A − λiBBTP ) = −Q − 2(λi − τ)PBBTP < 0. Therefore, for a set ofcommunication graphs, each member of which has a directed spanning tree,and its Laplacian has nonzero eigenvalues that satisfy Reλ ≥ τ > 0, thereexists a feedback gain F such that ϕ1 is vanishing.

The rest of the system is described by ϕ0 = Aϕ0. The stability ofϕ2 = 0 depends on the stability of individual agents, and it cannot bemade asymptotically stable if agents are unstable. Assuming that ϕ1 = 0 is

4It is possible since the pair (A,B) is controllable.


asymptotically stable, the asymptotic behavior of individual agents is found.According to the state transformation, one may write

χ(t) = (U ⊗ In)ϕ1(t) + (1⊗ In)ϕ0(t)t→∞−−−→ χ(t) = (1⊗ In)ϕ0(t)

One the other hand, ϕ0(t) = ϕ0(0)eAt. From ϕ0(t) = (rT ⊗ In)χ(t), itfollows that χ(t) = (1rT ⊗ eAt)χ(0) as time tends to ∞. Hence, the agentsasymptotically synchronize:

limt→∞

x(t) = (rT ⊗ eAt)χ(0) =

N−1∑j=1

rjeAtxj(0)

where rT = [r1, r2, · · · , rN−1]. Thus, the synchronization trajectory is alinear combination of the initial conditions of agents, and depends on thenetwork topology, the dynamics and the initial conditions of every agent.

Synchronization as Robust Stabilization Consider a homogeneousnetwork of non-introspective linear agents which are described by

xi = Axi +Bui

yi = Cxi

where xi ∈ Rn is the state, ui ∈ Rm is the control, yi ∈ Rp is the out-put, the pair (A,B) is controllable, and the pair (A,C) is observable. Thecommunication topology is described by a directed graph L associated withthe adjacency matrix A = [aij ] and the Laplacian matrix L = [lij ]. Thegraph L is fixed and contains a directed spanning tree, which implies thatthe Laplacian matrix L has one eigenvalue at zero with multiplicity one.Consider partial-state (output) information coupling:

ζi =N∑i=1

aij(yi − yj) ⇒ ζi =N∑i=1

lijyj

The objective is to design decentralized protocols (controllers or filters)ui = K(s)ζi such that limt→∞(xi − xj) = 0 for all i, j ∈ 1, 2, · · · , N isachieved. In particular, the filter K(s) is a dynamic system with the internalstate χi of a specific order, which maps ζi to ui. It can be described by

χi = Akχi +Bkζi

ui = Ckχi +Dkζi


where χi ∈ Rq, Ak ∈ Rq×q, Bk ∈ Rq×p, Ck ∈ Rm×q, and Dk ∈ Rm×p.Therefore, K(s) = Ck(sIq −Ak)−1Bk +Dk. Let ξi = col xi, χi ∈ Rn+q forevery i ∈ 1, 2, · · · , N. Then, the dynamics of each agent under feedbackare given by

χi =

[A BCk0 Ak

]χi +

[BDk

Bk

]ζi ,Aχi + Bζi

yi =[C 0

]χi , Cχi

Collect the states of all agents as ϑ = col χi for all i ∈ 1, 2, · · · , N. Thecollective dynamics are then given by

ϑ = (IN ⊗A+ L⊗ BC)ϑ

Moving along the same line of Example 3.1, one is able to find two decoupleddynamics as [

ϕ1

ϕ0

]=

((IN ⊗A) + (

[∆ 00 0

]⊗ BC)

)[ϕ1

ϕ0

]

where

[∆ 00 0

]is the Jordan form of L; that is, ∆ is an upper-triangular

matrix with the nonzero eigenvalues of L, denoted λj for j = 1, 2, · · · , N−1,on its main diagonal. Thus, the stability of ϑ is equivalent to the stabilityof N systems in the form of

ρ1 = Aρ1

ρj = (A+ λjBC)ρj , j = 2, · · · , N

As a result of discussion provided in Example 3.1, synchronization is achievedif ρj = 0 for j = 2, · · · , N is globally asymptotically stable. Also, ρ1 deter-mines the synchronization trajectory, which may be unbounded.

Therefore, one may describe the synchronization problem among Nagents as follow. Define a family of plants, denoted P, whose membersare described by

η = Aη +Bυ

ξ = λjCη, j = 2, · · · , N

The synchronization problem is regarded as the problem of designing a singlefixed controller K(s) such as in the configuration of Fig. 3.1 that stabilizesthe closed-loop system when the plant is any member of the family P.

3.3. Review of Previous Work 47

CONTROLLER PLANT

Figure 3.1: Control system configuration. The control law υ = K(s)ξstabilizes the plant which is any member of the family P.

This problem is termed as simultaneous stabilization (Kwakernaak, 1982;Vidyasagar and Viswanadham, 1982).

A generalization of the problem is to include external disturbances inthe dynamics of agents. In this case, the goal is to design a fixed controllerK(s) such that the control law υ = K(s)ξ guarantees the internal stabilityof the closed-loop system when the plant is any member of the family P.Moreover, in the closed-loop system, the disturbance is to be rejected fromthe output to any arbitrary degree of accuracy. This problem is referred to asthe problem of simultaneous stabilization with almost disturbance decoupling(Weiland and Willems, 1989). In the context of synchronization, it is calledalmost synchronization, which is the topic of this thesis.

Notice that, due to linearity, one can consider that the members of Pare described by

η = Aη + λjBυ

ξ = Cη, j = 2, · · · , N

3.3 Review of Previous Work

The objective of this section is to provide a brief review of the current resulton synchronization of multi-agent systems. This section does not intend toreview all directions in the topic of synchronization; it focuses on articlesthat are more related to this thesis from the author’s view. The recentsurvey articles (Cao et al., 2013; Murray, 2007; Olfati-Saber et al., 2007)provide a great insight into the recent developments in this topic.

3.3.1 Synchronization in Multi-agent Systems

Consensus problem has a long history in computer science where the ideawas to divide a computational problem into several pieces such that eachpart can be solved by independent entities while they communicate with


each other by message sending (DeGroot, 1974). In control systems soci-ety, the work of (Tsitsiklis et al., 1986) is the pioneering work where anasynchronous agreement problem was studied to resolve the problem of dis-tributed decision-making.

Nowadays, the problem of synchronization in networks of dynamic sys-tems possesses diverse applications in various disciplines of science and engi-neering including synchronization of coupled oscillators, distributed roboticssuch as formation flying, flocking and swarming, distributed sensor fusion inmobile sensor networks, quantum networks, networked economics, biologicalsynchronization, and social networks.

The seminal works of (Wu and Chua, 1995a) and (Wu and Chua, 1995b)have substantially contributed in the analysis and design of multi-agentsystems by introducing the application of the graph theory for a formaldescription of communication topologies, and by bringing in the Kroneckerproduct for describing the collective dynamics.

The works of Jadbabaie et al. (2003); Moreau (2005); Olfati-Saber andMurray (2004), and Ren and Beard (2005) have been instrumental in pavingthe way that consensus protocols have been developed. Jadbabaie et al.(2003) offered a mathematical analysis for the model of networked agentsraised by Vicsek et al. (1995). The result states that agents move withidentical heading if these agents are periodically linked with one another.The consensus problem in networks of first order or second order integratordynamics was explored in (Olfati-Saber and Murray, 2004; Ren and Beard,2005). They basically relied on full state information coupling and designedstatic decentralized protocols. Ren (2008) studied consensus in networks ofdouble-integrator systems where a group reference state was available to asubset of the agents.

Olfati-Saber and Murray (2004) studied time delays in communicationlinks and showed that the maximum time delay that can be tolerated in afixed network of single integrators is inversely proportional to the maximumeigenvalue of the Laplacian of the network graph. Ren and Beard (2005)and Moreau (2005) considered dynamic and time-varying networks. Theyproved that consensus is accomplished if the underlying directed graph has adirected spanning tree in terms of a union of its time-varying graph topolo-gies. The topic of (Coogan and Arcak, 2012) is to scaling the formationwhile only a subset of agents are aware of the desired scale.

Synchronization of networks of general linear agents was addressed in(Fax and Murray, 2004; Li et al., 2010; Qu et al., 2008; Seo et al., 2009;Tuna, 2009; Yang et al., 2011c; Zhang et al., 2011a) where partial-stateinformation is given to each agent via the network, and dynamic protocols


are introduced. Seo et al. (2009) proposed a low-gain approach by filteringthe information that each agent receives whereas Fax and Murray (2004)considered self-feedback for all agents.

A significant breakthrough in the design of dynamic protocols was pre-sented by Li et al. (2010) where conventional observers were expanded todistributed observers by allowing every agent to exchange information aboutthe state of its own protocol over the network. The result was extended toLQR-based optimal design by Zhang et al. (2011a). Predictive pinningcontrol was proposed by Zhang et al. (2011b, 2008a,b) for accelerated con-sensus.

The impact of communication delays on the convergence and perfor-mance was addressed in (Ma and Zhang, 2010). The primary result of(Olfati-Saber and Murray, 2004) for networks of single-integrators was ex-panded to networks of double integrators by Lin and Jia (2009), to generallinear systems by Yao et al. (2009b), to rigid bodies by Nuno et al. (2011),to nonlinear systems by Bokharaie et al. (2010, 2011).

Stochastic networks of single integrators were studied by Tahbaz-Salehiand Jadbabaie (2008) where synchronization is achieved almost surely if andonly if the expectation of the network topology has a directed spanning tree.Extension to stochastic networks of double-integrator systems was providedby Zhang and Tian (2009).

The worst-case synchronization rate is determined by the second small-est eigenvalue of the Laplacian matrix for undirected grapghs (Olfati-Saberand Murray, 2004). Considering a network of single-integrators, Kim andMesbahi (2006) intended to maximize the convergence rate by choosing op-timal weights for edges. Extension to double-integrator agents was providedby Carli et al. (2011). Recently, Shafi et al. (2012) have optimized the nodeand edge weightings of undirected graphs so as to impose bounds on theLaplacian spectra.

Synchronization of heterogeneous networks is an active research. Thecommon assumption is that agents are introspective; that is, agents possesssome knowledge about their own states. For networks of nonlinear agents,Zhao et al. (2011) presented criteria for state consensus. Xiang and Chen(2007) introduced the concept of V -stability and assumed that a commonLyapunov function candidate is available to analyze stability with respectto a common equilibrium point. Using pining control, output consensusfor weakly minimum-phase systems of relative degree one was studied in(Chopra and Spong, 2008) where local feedbacks were designed in order toshape agents. To facilitate passivity-based designs, Bai et al. (2011) alsoused pre-feedback using local information.


Kim et al. (2011) proposed a controller for SISO minimum-phase sys-tems. The main idea of Kim et al. (2011) is to embed an identical modelwithin each agent such that the output of each agent tracks the outputof the embedded model. Using the principle of internal model, Wielandet al. (2011) proposed a dynamic controller for general linear systems. Theyshowed that output synchronization is possible if a virtual exosystem exists,which produces a trajectory to which all the agents asymptotically converge.

Yang et al. (2011b), by designing pre-compensators and local outputfeedbacks within each agent, developed a method to represent a hetero-geneous multi-agent system as a network of asymptotically homogeneousagents.

Relaxing the self-knowledge (introspective) assumption, Zhao et al. (2010)proposed a static protocol for networks of passive nonlinear systems. Passiv-ity is a strong assumption that requires the agents to be weakly minimum-phase and of relative degree one. A moderately less restrictive design wasput forward by Grip et al. (2012) for output synchronization in heteroge-neous networks of non-introspective agents.

Networks of nonlinear systems have also been studied. Chopra andSpong (2009) addressed synchronization with exponential rate in networksof Kuramoto oscillators. Networks of complex systems of high order werestudied in (Arcak, 2007; Ihle et al., 2007; Yao et al., 2009a) where passivitytheory is utilized as a tool for synchronization. A new passivity analy-sis that enables delay analysis was presented in (Summers et al., 2013).A discontinuous and time-invariant decentralized feedback was derived byDimarogonas and Kyriakopoulos (2007) for synchronization of networkednonholonomic unicycles. Fully actuated, autonomous rigid-body attitudesynchronization under directed and varying graphs was discussed in (Sar-lette et al., 2009). For heterogeneous networks of introspective Lagrangiansystems, Chung and Slotine (2009) presented a decentralized state feedbackbased on contraction theory. Parametric uncertainty and time-delays werediscussed in (Nuno et al., 2011) for synchronization of networks of noniden-tical Euler-Lagrange Systems.

Many books have been published on synchronization of multi-agentsystems. A thorough coverage of earlier work, including static and dy-namic protocols, the effect of communication delay, and dynamic interactiontopologies may be found in the books of Wu (2007) and Ren and Cao (2011).A broad overview of distributed control of synchronous networks of robots isgiven by Bullo et al. (2009). Passivity-based synchronization of networkeddynamical systems is reviewed in (Bai et al., 2011). The book (Mesbahi andEgerstedt, 2010) provides an introduction to dynamic multi-agent networks


by focusing on graph theoretic methods, and introduces new formalism foranalysis and synthesis of such systems. Synchronization of multi-robot sys-tems where only position variables are available is the topic of (Nijmeijer andRodriguez-Angeles, 2003). (Pettersen et al., 2006) is a collection of articleson synchronization of vehicular systems and formations of marine vehicles.The work of Vicsek and Zafeiris (2012) has provided a broad overview ofcollective motion, and is recommended for reading. It reviews the topicof consensus and collective dynamics in biological systems and manmadesystems, and provides mathematical models for simulation.

The excellent work (Mauroy et al., 2013) makes a comparison betweentwo fundamental models of oscillator synchronization, namely, diffusive syn-chronization and kick synchronization. Diffusive model of synchroniza-tion dominates the mathematical literature on synchronization; however,kick synchronization is usually seen in nature (such as synchronization ofmetronomes) for which only few theoretical results exist.

3.3.2 Synchronization under External Disturbances

Although synchronization of multi-agent systems has been vastly studiedin ideal conditions, relatively few references are concerning synchroniza-tion in the presence of disturbances. They mainly focus on homogeneousmulti-agent systems and utilize linear matrix inequalities to solve the H∞optimization problem over a network of agents. Some only analyze the dis-turbance attenuation properties of consensus protocols and do not presenta specific design procedure to solve the problem of disturbance rejection inmulti-agent systems.

Under bidirectional communication links, according to (Li et al., 2010),the distributed H∞ control problem to achieve a desired H∞ gain for thetransfer function from disturbance to each agent’s output is converted tothe H∞ control problem of a set of independent systems. Using the pinningidea, which implies a subset of agents has access to its own states, Li et al.(2011b) introduced the notions of H2/H∞ performance regions, and pro-posed a static protocol that assures an unbounded H∞ performance region.Dynamic protocols were presented in (Li et al., 2009).

The works of Li et al. (2011b, 2010, 2009) do not explore disturbanceattenuation from synchronization error dynamics, which can be seen as adrawback for their problem definition since the objective should be to rejectthe impact of disturbances on synchronization error dynamics.

For complete and circulant networks, Massioni and Verhaegen (2009)proposed a decomposition approach to solve distributed H∞ control prob-lem for homogenous multi-agent systems. Assuming full-state coupling, for


networks of non-introspective integrators and introspective linear systems,Mo and Jia (2011) and Liu and Jia (2011) presented static controllers whichsolved H∞ control over a network.

For directed networks of first-order integrator systems, Lin et al. (2008)proposed staticH∞ controllers; extensions to networks of introspective high-order multi-agent systems were given in (Lin and Jia, 2010; Liu and Jia,2010). For networks of introspective single-integrators, Hong-Yong et al.(2011) proposed a consensus protocol by designing an observer for distur-bances generated by an exosystem. For strongly connected and balanceddirected graphs, Li et al. (2012) proposed a static protocol for intercon-nected non-introspective agents possessing Lipschitz nonlinear dynamics,which guaranteed a desired H∞ performance. Moreover, Ugrinovskii (2011)and Ugrinovskii and Langbort (2011) sought an H∞ distributed consensusobserver using the concept of vector dissipativity.

Besides LMI-based H∞ consensus protocols, the following articles inves-tigate the disturbance attenuation properties of consensus protocols. Thenotion of leader-to-formation stability (Tanner et al., 2004) was proposedto assess robustness of followers with respect to the leader’s input for non-linear agents. Li and Zhang (2009) solved unbiased mean-square average-consensus by introducing time-varying consensus gains. For first-order dy-namical systems under bidirectional links where network measurements arecorrupted by bounded noise, Bauso et al. (2009) proposed a static controllerwhich guaranteed convergence of all states to a cylinder. Using non-smoothfinite-time consensus algorithms for networks of double-integrators, Li et al.(2011a) and Du et al. (2012) analyzed the disturbance attenuation propertyof the closed-loop system in the presence of external disturbances.

Chapter 4

H∞ Almost Synchronizationfor Heterogeneous Networksof Introspective AgentsUnder External Disturbances

This chapter brings forward the notion of “H∞ almost synchroniza-tion” for heterogeneous multi-agent systems under directed intercon-nection structures. Agents are assumed to be linear, right-invertibleand introspective (i.e. each agent has partial knowledge of its ownstate). The objective is to suppress the impact of disturbances onthe synchronization error dynamics in terms of the H∞ norm of thecorresponding closed-loop transfer function. In addition, the problemof regulating the synchronization trajectories to a reference signal isaddressed. The application of the proposed method to the formationproblem is furthermore elucidated.This chapter is based on Peymani et al. (2012, 2013c,d).

4.1 The Topic of The Chapter

The main objective of this chapter is to deal with heterogeneous networksof introspective, right-invertible linear agents in order to propose a methodto achieve output synchronization with any desired accuracy (in the senseof the H∞-norm of the corresponding transfer function) when the commu-nication links are directional and external disturbances influence agents.

To this end, the notion of “H∞ almost synchronization” is brought forth.The aim is to construct a family of distributed, observer-based, linear time-invariant protocols such that i) synchronization is accomplished in the ab-

53

54 Heterogeneous Networks of Introspective Agents

sence of disturbances, and ii) the impact of disturbances on the synchro-nization error dynamics is attenuated to any arbitrarily small value in thesense of the H∞ norm of the corresponding transfer function.

The proposed method facilitates the regulation of consensus trajectoriesto a given reference and allows for bringing in the notion of “H∞ almostregulated synchronization”. For the sake of completeness, the concept of“H∞ almost formation” is also introduced1.

Methodology Using the time-scale structure assignment techniques(Ozcetin et al., 1992) rooted in the methodology of singular perturbation(Kokotovic et al., 1986), we propose a family of linear time-invariant pro-tocols parameterized in terms of a tuning parameter ε. The structure ofthe protocols is independent of the parameter ε; thus, one may develop thestructure at one stage and tune the parameter ε later so as to obtain thedesired accuracy of synchronization. Due to continuity in ε, tuning may beeven carried out online. Hence, the method is a one-shot design and is notiterative2.

To pursue the objective, first, a local output feedback along with prec-ompensators is designed for each agent so that all agents are shaped into aparticular form, and a network of almost identical agents with non-identicaldisturbance matrices emerges. Shaping, which plays a pivotal role in solv-ability of the problem, is viable because agents are introspective that meansthey have partial knowledge about their own states. Then, assuming thatagents can transmit information about their protocol’s state according tothe network topology that exchange information about their relative out-puts, a family of dynamic protocols is derived to attain the desired H∞ gainfor the system from external disturbances to synchronization errors.

4.2 Notations and Preliminaries

Throughout the paper, In denotes the identity matrix of dimension n, and1n ∈ Rn means a vector whose entries are all one. Matrix A is denotedA = [aij ] where aij represents the element (i,j) of A. Given a matrixA, transpose, complex conjugate transpose, and induced 2-norm of A arerespectively represented by AT, AH and ‖A‖. The Kronecker product of Aand B is symbolized by A⊗B.

1To the best knowledge of the author, these notions have not been introduced andsolved before.

2This can be viewed as an advantage over LMI-based methods. However, the mainadvantage of the method that this chapter introduces is that solvability of the problem isguaranteed under a set of conditions.

4.3. Heterogeneous Multi-Agent Systems 55

Considering matrices Ai’s, diagAi for i = 1, · · · , n indicates a block-diagonal matrix constructed by Ai’s. Also, x = col xi for i = 1, · · · , nis adopted to denote x = [xT

1 , · · · , xTn ]T where xi’s are vectors. The open

left-half and right-half complex planes are represented by C− and C+, re-spectively. The real part of a complex number λ, is represented by Reλ.For a transfer function T (s), the H∞ norm is denoted ‖T (s)‖∞.

Let L be a weighted directed graph with n nodes. If there is an edgefrom node j to node i, aij > 0, aij ∈ R is assigned to the edge; otherwise,aij = 0. If the graph is not allowed to have self-loops, aii = 0. AL = [aij ]is the weighted adjacency matrix of L. The Laplacian of L is denoted byL = [lij ] where lii =

∑nj=1,j 6=i aij and lij = −aij for i 6= j. It implies that 1n

is a right eigenvector of L associated with the eigenvalue at zero. A digraphis said to have a directed spanning tree if there is a node from which adirected path exists to every other nodes. L has a simple eigenvalue at zeroand all the other eigenvalues are in C+, provided that L contains a directedspanning tree (Ren and Beard, 2005).

4.3 Heterogeneous Multi-Agent Systems

A heterogeneous multi-agent system is referred to a network of multiple-input multiple-output agents described by non-identical linear time-invariantmodels

Agent i :

˙xi = Aixi +Biui +Giwi

yi = Cixi

(4.1a)

(4.1b)

in which i ∈ S , 1, · · · , N, xi ∈ Rni , ui ∈ Rmi , yi ∈ Rp, wi ∈ Rωi :

limT→∞1

2T

∫ T−T w

Ti widt < ∞ are the agent’s state, input, output, and ex-

ternal disturbance, respectively. Agents are introspective; that is, each agentpossesses partial knowledge of its own states through the local measurement:

ym,i = Cm,i xi (4.1c)

Furthermore, agents are allowed to exchange information according to thenetwork’s communication topology which is characterized by a directedgraph L, with no self loops, associated with the adjacency matrix AL = [aij ]and the Laplacian matrix L = [lij ]. Accordingly, each agent has access to aweighted linear combination of its own output relative to that of the others.In particular, the network measurement for agent i ∈ S is:

ζi =

N∑j=1

aij(yi − yj) in terms of the Laplacian matrix⇐==================⇒ ζi =

N∑j=1

lijyj (4.1d)


Also, the agents can exchange additional information over the network. Thetransmission of the information conforms with the communication topologyand facilitates the design of a distributed observer for the network. Thus,agent i has access to the following quantity:

ζi =

N∑j=1

lijηj (4.1e)

where ηj ∈ Rp depends on the state of protocol j, and will be specified laterin Section 4.4.1.

Assumption 4.1. We make the following assumptions for agent i ∈ S.

1. (Ai, Bi, Ci) is right-invertible 3;

2. (Ai, Bi) is stabilizable and (Ai, Ci) is detectable;

3. (Ai, Cm,i) is detectable.

It should be pointed out that ym,i can be the same as yi. However, ingeneral, agents may measure an output which is different from the one thatexchange. In other words, ym,i comes from global sensing while yi is the onethat is used in local sensing.

4.4 H∞ Almost Synchronization

This section deals with the synchronization problem for a heterogenous net-work of introspective agents under directional communication links in thepresence of external disturbances. The problem is called “H∞ Almost Syn-chronization” and is precisely stated in Definition 4.1.

Problem Formulation We define the following vectors which are formedby stacking the corresponding vectors of each agent:

w , col wi, u , col ui, ζ , col ζi

for i ∈ S. Let mutual disagreements between a pair of agents be defined as:

ei,j , yi − yj ∀i, j ∈ S, i > j

The stacking column vector of all mutual disagreements is denoted e. Wedefine the following transfer function with the appropriate dimension: e =Twe(s)w.

3The definition of right-invertible systems is given in Chapter 2.

4.4. H∞ Almost Synchronization 57

Dynamic LTI protocol parameterized in terms of

Multi-agent System Heterogeneous & Introspective

Figure 4.1: The block diagram of the multi-agent system.

Definition 4.1 (H∞ almost synchronization). Consider a multi-agent sys-tem, described by (4.1), with a communication topology L. Given a setof network graphs G and any γ > 0, the “H∞ almost synchronization”problem is to find, if possible, a linear time-invariant dynamic protocol suchthat, for any L ∈ G, the closed-loop transfer function from w to e satisfies‖Twe(s)‖∞ < γ. J

Fig. 4.1 depicts the block diagram of the multi-agent system.

Definition 4.2 (Communication network set Gβ). For given β > 0 andinteger N0 ≥ 1, Gβ is the set of directed graphs composed of N nodes whereN ≤ N0 such that every L ∈ Gβ has a directed spanning tree and theeigenvalues of its Laplacian, denoted λ1, · · · , λN , satisfy Reλi > β forλi 6= 0. J

The Laplacian L associated with L ∈ Gβ has a simple eigenvalue at zeroand the rest are located in C+.

Theorem 4.1. Under Assumption 4.1 and for the set Gβ, the H∞ almostsynchronization problem is solvable; specifically, there exists a family oflinear time-invariant dynamic protocols, parameterized in terms of a tuningparameter ε ∈ (0, 1], of the form

χi = Ai(ε)χi + Bi(ε) col ζi, ζi, ym,iui = Ci(ε)χi +Di(ε) col ζi, ζi, ym,i

(4.2a)

(4.2b)

where χi ∈ Rqi and i ∈ S such that

(i) given β > 0, there exists an ε∗1 ∈ (0, 1] such that, for every ε ∈ (0, ε∗1],output synchronization is achieved in the absence of disturbance; i.e.∀ε ∈ (0, ε∗1] when w = 0, ei,j = yi− yj → 0, ∀i, j ∈ S, i > j, as t→∞.

(ii) given γ > 0, there exists an ε∗2 ∈ (0, ε∗1] such that, for every ε ∈ (0, ε∗2],the closed-loop transfer function from w to e satisfies ‖Twe(s)‖∞ < γ.ehsa nn


The proof of Theorem 4.1 is presented in the subsequent sections ina constructive way. It is shown that Theorem 4.1 follows directly fromLemma 4.1 and Lemma 4.2. In the following, first, we present the solutionfor a special class of multi-agent systems; then, we generalize the methodto encircle multi-agent systems in the form of (4.1).

4.4.1 Special Case

Consider a special case of multi-agent system (4.1):

xi = Axi +B(Mui +Rxi) + Eiwi

yi = Cxi

(4.3a)

(4.3b)

where xi ∈ Rpnq , ui, yi ∈ Rp, wi ∈ Rωi , Ei ∈ Rpnq×ωi , and

A =

[0 Ip(nq−1)

0 0

], B =

[0Ip

], C =

[Ip 0

](4.4)

The square matrix M ∈ Rp×p is nonsingular, and R is a p × pnq matrix.nq ≥ 1 is an integer. If nq = 1, A = 0, B = C = Ip. Each agent collectsnetwork measurements (4.1d) and (4.1e). This special class of multi-agentsystems does not require local measurements (4.1c). Let ε ∈ (0, 1] be thetuning parameter, and define the scaling matrix S as

S = diagIp, ε Ip, · · · , εnq−1Ip ∈ Rpnq×pnq (4.5)

For i ∈ S, construct the observer-based protocol

˙xi = A xi +B(Mui +Rxi)− ε−1K(ζi − ζi) (4.6a)

ui = ε−nqM−1FSxi (4.6b)

where xi ∈ Rpnq ; F and K are the gains to be specified. ζi is given by (4.1e)where ηi = Cxi. Consider the following structure for the observer gain:

K =

[K1

KK1

]where

K1 ∈ Rp×pK ∈ Rp(nq−1)×p (4.7)

Before presenting the design procedure, we need to partition the systemmatrices as

A =

[0p×p C1

0p(nq−1)×p A1

], B =

[0p×pB1

], Ei =

[E1,i

E2,i

]where C1 = [Ip, 0, · · · , 0] ∈ Rp×p(nq−1) and E1,i ∈ Rp×ωi . Also partitionR = [R1, R] where R1 ∈ Rp×p.


Design Procedure The gains of (4.6) are chosen as follows:

• Considering the controllable pair (A,B), choose F such that A+BFis Hurwitz stable.

• Considering the observable pair (A1 + B1R, C1), choose K such thatAz , A1 +B1R− KC1 is Hurwitz stable.

• Let K1 = KT1 < 0.

Lemma 4.1. For any given γ > 0 and Gβ, there exists a sufficiently small0 < ε ≤ 1 such that the dynamic protocol (4.6) solves the H∞ almostsynchronization problem for multi-agent systems of the form (4.3) with anycommunication topology L ∈ Gβ.

Proof. See Appendix 4.A.

Remark 4.1. It is worth noting that having non-identical disturbance ma-trices in (4.3) significantly complicates H∞ analysis for the network sinceit does not allow us to use input-output transformation as proposed by Liet al. (2010, 2009). Thus, their result no longer holds. It is also noted that,in this work, directed communication topologies whose Laplacian matricesare not necessarily symmetric are considered. Thus, the eigenvalues arecomplex values while undirected graphs with real eigenvalues are taken intoaccount in the works of Li et al. (2010, 2009).

4.4.2 Homogenization

In this section, we present a method to shape a multi-agent system of theform (4.1) into the form (4.3). The requisite for shaping is the availabilityof local measurements (4.1c). Therefore, the result given in the precedingsection is expanded to the general form (4.1). Indeed, shaping agents intothe desired structure, which is essentially a chain of integrators, is the keyelement for solvability of the H∞ almost synchronization problem.

To achieve the aim, we utilize a modified version of (Yang et al., 2011b)where, with the aid of local measurements (4.1c), a local dynamic compen-sator is designed for each agent. Accordingly, each agent can be describedby

xi = Axi +B(Mui +Rxi) + Ed,iwi + ρi

yi = Cxi

(4.8a)

(4.8b)

where ui, yi ∈ Rp; A, B and C are given by (4.4). The matrix R ∈Rp×p(nq−1) and the non-singular matrix M ∈ Rp×p are chosen arbitrarily.


Agent

,

(a) Agent i with inputs and out-puts.

Agent Local Feedback

,

(b) Homogenized agent i: agent i embedded withdynamic compensators.

Figure 4.2: Homogenization process is to force agents to imitate desireddynamics.

Also, ρi is evolved from an exponentially stable system, which is describedby

˙xi = Hixi + Eo,iwi

ρi = Wixi

(4.9a)

(4.9b)

Ed,i, Eo,i, Hi, and Wi can be found explicitly by following the proceduregiven in the proof of Lemma 4.2. See Fig. 4.2 for a schematic diagram forhomogenization process.

Definition 4.3. nq0 ≥ 1 is the maximum order of the infinite zeros of triples(Ai, Bi, Ci) for i ∈ S. J

Lemma 4.2. Consider a multi-agent system described by (4.1) satisfyingAssumption 4.1. Let nq ≥ nq0. For each agent, there exists a local dynamiccompensator as

˙xi = Ai ˜xi + B1iui + B2iym,i

ui = Ci ˜xi + D1iui + D2iym,i

such that the resulting system can be written as (4.8) and (4.9) whereR ∈ Rp×pnq and nonsingular M ∈ Rp×p are selected arbitrarily while A, Band C are as (4.4).

Proof. See Appendix 4.B.

Lemma 4.2 shows that a heterogeneous network under disturbances canbe converted to an almost homogeneous network. As Hi is Hurwitz stablein (4.9), xi and ρi have the same nature as wi; thus, one can redefineexternal disturbances as wi , col wi, xi. Hence, the model (4.8) is recastas (4.3) where Ei = [Ed,i, Wi]. Redefining disturbance changes the H∞norm; however, as theH∞ norm of (4.9) is constant with respect to ε, it doesnot affect the solvability of the problem and H∞ almost synchronization issolvable for any given Gβ and γ > 0 by an appropriate choice of ε.

4.5. H∞ Almost Regulated Synchronization 61

Remark 4.2. In Lemma 4.2, R and M are chosen arbitrarily to satisfythe designer’s wish. Actually, Lemma 4.2 states that agents can be shapedinto the dynamics of any invertible system with no invariant zeros and ofuniform rank nq. Such a system may be transformed into the form of (4.3)according to Lemma 4.4.

4.4.3 Design Method

To summarize the section and present a road map to design the protocol formulti-agent systems as described by (4.1), the design procedure is stated toachieve H∞ almost synchronization.

Given a multi-agent system (4.1) which satisfies Assumption 4.1 withany L ∈ Gβ, the design is two-stage.

i) Choose nq according to Definition 4.3, and select R and M as statedin Section 4.4.2, and design local compensators, in accordance withLemma 4.2, to homogenize the network.

ii) Construct (4.6) and select K and F according to the design procedure.Find ε∗1 according to Theorem 4.1, and tune ε to obtain the desiredaccuracy of synchronization.

Figure 4.3 shows the overall control system.

4.5 H∞ Almost Regulated Synchronization

Section 4.4 concerns leaderless synchronization where an agreement amongstagents is paramount. Put clearly, the proposed protocol prescribes no re-strictions on the consensus trajectories. However, in many applications, itis desirable to regulate the output of each individual agent to a particulartrajectory while synchronizing them. The section tackles the problem ofregulated output synchronization in the presence of external disturbances.

Problem Formulation Consider reference y0 which is the output of anexosystem in the form

Σ0 :

˙x0 = A0x0

y0 = C0x0

(4.11a)

(4.11b)

where x0 ∈ Rn0 and y0 ∈ Rp; the pair (A0, C0) is observable; all the eigen-values of A0 are in the closed-left half complex plane, and C0 is full rank.


Agent 1 Local Feedback 1

Agent Local Feedback

,

,

Observer 1

Observer

Protocol 1

Protocol

Information Exchange Topology

Information Exchange Topology

Figure 4.3: A heterogeneous network of introspective agents under externaldisturbances. With the aid of local measurements, local compensators areembedded within individual agents so that the network becomes partiallyidentical. Observer-based protocols are designed to achieve H∞ almost syn-chronization.

Define the regulation error for every agent i ∈ S as

ei,0 = yi − y0

Let e0 = col ei,0 for i ∈ S. The problem is posed as “H∞ almost regulatedsynchronization” and stated in Definition 4.4.

Definition 4.4. Consider a multi-agent system described by (4.1) with acommunication topology L. Given a set of network graphs G and any γ > 0,the problem of “H∞ almost regulated synchronization” with respect to areference y0 evolved from (4.11) is to find, if possible, a linear time-invariantdynamic protocol such that, for any L ∈ G, the closed-loop transfer functionfrom w to e0 satisfies ‖Twe0(s)‖∞ < γ. ehsan J

Assumption 4.2. Every node of the network graph L is a member of adirected tree with the root contained in the “root set” π ⊂ S.

Remark 4.3. Note that if the network graph is one connected componentcontaining a spanning tree, the set π may only own one node which is theroot of a spanning tree.

A certain subset of agents must know how far their outputs are fromthe reference y0; otherwise, regulation is not possible. The set π contains


those agents which receive ei,0 = yi−y0 via the network. It implies that thenetwork measurement (4.1d) is altered to

ζi =N∑j=1

aij(yi − yj) + ψi(yi − y0), i ∈ S (4.12)

where ψi > 0 if i ∈ π; otherwise, ψi = 0. Now, the exosystem is regardedas a new node, labeled with ‘0’, and added to the network graph. Theresulting graph is called the augmented network graph, denoted L. TheLaplacian matrix associated with L, is represented by L = [lij ] and termedas augmented Laplacian. Let ψ = col ψi and Ψ = diagψi for i ∈ S.Then, (4.12) is recast in terms of L:

ζi =N∑j=0

lijyj , L =

[0 0−ψ L+ Ψ

](4.13)

Assumption 4.2 ensures that the augmented graph L has a directed spanningtree (see Grip et al., 2012).

Definition 4.5. For given β > 0, an integer N0 ≥ 1, and an index setπ ⊂ S, G∗β,π is the set of directed graphs composed of N nodes whereN ≤ N0 such that every L ∈ G∗β,π satisfies Assumption 4.2 with the root set

π, and the eigenvalues of the Laplacian of the augmented graph L, denotedλ0, λ1, · · · , λN , satisfy Reλi > β for λi 6= 0. J

Likewise, the additional information (4.1e) is adapted4 according to theaugmented network and is given by

ζi =N∑j=0

lijηj , i ∈ S , 0 ∪S (4.14)

where ηj ∈ Rp will be specified later. In fact, ζi = ζi + ψi(ηi − η0) for everyi ∈ S. As agent 0, the exosystem, receives no information from the network,ζ0 = 0 and ζ0 = 0. Note that ζ = 0 means simultaneous achievement ofsynchronization and output regulation; i.e. y1 = · · · = yN = y0.

Theorem 4.2. Under Assumption 4.1 and for the set G∗β,π, the problem ofH∞ almost regulated synchronization is solvable; specifically, there exists a

4Notice that the network graph has not structurally changed and the only change isto add node ‘0’ to the set of nodes, and connect it to the members of the root set π.


family of linear time-invariant protocols, parameterized in terms of a tuningparameter ε ∈ (0, 1], of the form

χi = Ai(ε)χi + Bi(ε) col ζi, ζi, ym,iui = Ci(ε)χi +Di(ε) col ζi, ζi, ym,i

(4.15a)

(4.15b)


(i) given β > 0, there exists an ε∗1 ∈ (0, 1] such that, for every ε ∈ (0, ε∗1],output regulation to the reference y0 is accomplished in the absenceof disturbance; i.e. ∀ε ∈ (0, ε∗1] when w = 0, ei,0 = yi − y0 → 0, i ∈ S,as t→∞.

(ii) given γ > 0, there exists an ε∗2 ∈ (0, ε∗1] such that if ε ∈ (0, ε∗2], theclosed-loop transfer function from w to e0 satisfies ‖Twe0(s)‖∞ < γ.ehsan

The proof is given in a constructive way in the following subsections.We move along the same threads of thought as in Section 4.4; we first makethe augmented network imitate a homogeneous multi-agent system; then,the protocol is designed for the resulting system.

4.5.1 Homogenization of the Augmented Network

Denote the largest observability index of the pair (A0, C0) by n∗q0. Thereexist a series of state manipulations for the exosystem (4.11) and a matrixB0 such that the resulting system characterized by the triple (A0, B0, C0)is invertible, of the uniform rank nq ≥ n∗q0, with no invariant zeros (seeAppendix 4.C). According to Lemma 4.4, the exosystem can be furtherrepresented in the following form:

Σ0 :

x0 = Ax0 +B(M0u0 +R0x0)

y0 = Cx0

(4.16a)

(4.16b)

where x0 ∈ Rpnq and u0 ∈ Rp; A, B and C are as (4.4). The squarematrix M0 ∈ Rp×p is nonsingular and R0 ∈ Rp×pnq . As the exosystem isautonomous and we have no control over it, u0 must be identically zero.

Therefore, for nq ≥ maxnq0, n∗q0, in accordance with Lemma 4.2, thereexist local dynamic compensators that shape agents into (4.3) for particularR0 and M0 obtained from (4.16). Consequently, the augmented network canbe homogenized, and represented as a network of almost identical agents


described by (4.3) for i ∈ 0∪S where M = M0, R = R0 and E0 = 0. Forease of reference, the obtained model is given here

xi = Axi +B(M0ui +R0xi) + Eiwi

yi = Cxi

(4.17a)

(4.17b)

4.5.2 Dynamic Protocol

In view of (4.17) with the particular choice of R0 and M0, for every i ∈ S =0 ∪S, we propose

˙xi = Axi +B(M0ui +R0xi)− ε−1K(ζi − ζi) (4.18a)

ui = ε−nqM−10 FSxi (4.18b)

In (4.18), xi ∈ Rpnq and ε ∈ (0, 1] is the tuning parameter. Let ηj = Cxj in

ζi given by (4.14). The matrix S is as (4.5) and K is partitioned as (4.7).The gains are chosen in the same way as Section 4.4.1. Notice that settingx0(0) = 0 leads to x0(t) = 0 and u0(t) = 0, ∀t ≥ 0.

The proposed method provides a straightforward solution for regulatedsynchronization as a result of shaping because the pair (A,B) is commonbetween the exosystem and agents.

4.5.3 Design Scheme

Given a multi-agent system of the form (4.1) with a communication topologyL ∈ G∗β,π, and an exosystem (4.11). The problem of H∞ almost regulatedsynchronization is solved in the following fashion.

1. Choose nq ≥ maxnq0, n∗q0. Represent the exosystem (4.11) in theform (4.16) to obtain R0 and M0.

2. According to Lemma 4.2, design local compensators within each agentso as to convert the multi-agent system to a network of almost identicalagents (4.3) with R = R0 and M = M0.

3. Use the parameterized protocol (4.18). Find ε∗1 ∈ (0, 1], and tuneε ∈ (0, ε∗1] to achieve a desired accuracy of output regulation.

Remark 4.4. It is worth remarking that the protocol (4.18) is equivalentto

˙xi = Axi +B(M0ui +R0xi)− ε−1K(ζi − ζi)− ε−1Kψi(yi − y0 − Cxi)ui = ε−nqM−1

0 FSxi


Remark 4.5. It is evident that the structure of the protocols is independentof the parameter ε, and one can develop the protocol structure at one stageand tune the parameter ε ∈ (0, ε∗1] in order to reach the desired accuracyof output regulation. Since the structure is continuous in the parameter ε,tuning may be even carried out online.

4.6 H∞ Almost Formation

The proposed synchronization method lends itself to formation control. Information control, the objective is to maintain the relative outputs amongagents as desired. In the presence of external disturbances, the problem offormation with the desired H∞ performance is posed.

Let formation be defined in terms of a set of formation vectors

Sf , f1, · · · , fN, fi ∈ Rp

Define yf,i = yi − fi for i ∈ S. The mutual disagreement for formationproblem is then defined as ei,j = yf,i− yf,j for i, j ∈ S, i > j. Let ef be thestacking column vector of ei,j ’s.

Definition 4.6. Consider a multi-agent system described by (4.1) witha communication topology L. Given a set of network graphs G and anyγ > 0, the problem of “H∞ almost formation” with respect to a formationset Sf is to find, if possible, a linear time-invariant dynamic protocol suchthat, for any L ∈ G, the closed-loop transfer function from w to ef satisfies‖Twef (s)‖∞ < γ. J

Formation is possible when agents exchange yf,i’s. Therefore, the net-work information (4.1e) is to be modified to

ζf,i = ζi −∑N

j=1lijfj =

∑N

j=1lijyf,j (4.20)

The formation controller relies extensively on the fact that homogeniza-tion (i.e. shaping to a desired almost identical dynamics) is viable. Thus,Lemma 4.2 guarantees existence of local feedback laws to shape the systeminto the desired structure as (4.3) for arbitrary R and M (det(M) 6= 0). Inview of (4.3) along with (4.20) and (4.1e), for every i ∈ S, we propose theprotocol

˙xi = Axi +B(Mui +Rxi +R1fi)− ε−1K(ζf,i − ζi) (4.21a)

ui = ε−nqM−1FSxi −M−1R1fi (4.21b)

4.7. Illustrative Example 67

In (4.21), xi ∈ Rpnq and ε ∈ (0, 1] is the tuning parameter. The matrix S isas (4.5) and K is partitioned as (4.7). The gains are chosen in Section 4.4.1.In ζi given by (4.1e), let ηi = Cxi.

Theorem 4.3. Under Assumption 4.1 and for the set Gβ, the problemof H∞ almost formation is solvable; specifically, there exists a family oflinear time-invariant dynamic protocols, parameterized in terms of a tuningparameter ε ∈ (0, 1], of the form

χi = Ai(ε)χi + Bi(ε) col ζf,i, ζi, ym,i, fiui = Ci(ε)χi +Di(ε) col ζf,i, ζi, ym,i, fi

(4.22a)

(4.22b)


(i) given β > 0, there exists an ε∗1 ∈ (0, 1] such that, for every ε ∈ (0, ε∗1],the desired formation is attained in the absence of disturbance; i.e.∀ε ∈ (0, ε∗1], ei,j = yf,i − yf,j → 0 as t→∞, when w = 0.

(ii) given γ > 0, there exists an ε∗2 ∈ (0, ε∗1] such that, for every ε ∈ (0, ε∗2],it is satisfied that ‖Twef (s)‖∞ < γ.

Proof. See Appendix 4.D.

Remark 4.6. It is worth noting that the given protocol imposes no restric-tions on the agreement trajectories; that is, although agents establish thedesired configuration, it is not clear where the whole system heads to. Theformation problem can be combined with the regulation problem, giving riseto the problem of “H∞ almost regulated formation5” in which agents, underthe influence of external disturbances, is to make a desired formation whilethey are asked to track a reference with a desired accuracy.

4.7 Illustrative Example

The result is illustrated for a network consisting of N = 4 right-invertibleagents with p = 1. The interconnection topology of the network is givenby the digraph displayed in Fig. 4.4a. The models of agents are given inAppendix 4.F. Disturbances are chosen w1 = sin(t), w2 = 1, w4 = sin(2t),and ‖w3‖ ≤ 5 which is a random number normally distributed. The orderof the infinite zeros of agent 1 to 4 are respectively 3, 2, 1, and 2. It isobtained that nq0 = 3.

5Equivalently, one may call it “H∞ almost formation with regulation of output syn-chronization”.


Network Homogenization The first step is to design a local outputfeedback for each agent to homogenize the multi-agent system. Requiredinformation is presented in Appendix 4.F, where we have kept the notationsthe same as Appendix 4.B. We choose nq = nq0. Together with the pre-compensators, all the agents have the form of (4.3) with p = 1 and nq = 3.

Agents 1 and 2 are invertible, but agents 3 and 4 need to be squareddown. The pre-compensator Σ1,3 squares down agent 3 and locates theadditional invariant zeros at −2,−2,−2,−2,−1. Placing the additionalinfinite zeros at −2,−2,−1, the interconnection of Σ1,4 and agent 4 isinvertible.

Together with the pre-compensators, all the agents are invertible andsingle-input single-output. Now, we make them of the same relative degreenq. Agents 2, 3, and 4 are compensated by Σ2,2, Σ2,3 and Σ2,4, representedin Appendix 4.F.

After pre-compensation agent 1-4 are of orders 3, 3, 8 and 7, respec-tively. The third step to achieve an almost identical representation is todesign output feedbacks for the pre-compensated agents, which requireslinear observers. After pre-compensation, agent 1 to 4 are of orders 3,3, 8 and 7, respectively. We choose M = 1 and R = [0, −1, 0] to haveλ(A+BR) = 0,±i.

H∞ Almost Synchronization The appropriately shaped agents are nowconsidered and the control law is generated according to (4.6). We selectK1 = −1. F and K are selected such that λ(A+BF ) = −3,−4,−5 andλ(Az) = −2,−3. Fig. 4.5 shows the result for ε = 0.01 and ε = 0.05.The smaller ε is, the smaller ζi is, the more w is rejected from mutualdisagreements.

4 5 5 3

2 2

3 4

1

(a) Primary Network

4 5 5

2 1 2

3 4

0

3

1

(b) Augmented Network

Figure 4.4: The communication topology of the network.

4.7. Illustrative Example 69

H∞ almost synchronizeion - ε = 0.05

ζ i,i

∈S

Time (sec)

H∞ almost synchronizeion - ε = 0.01

ζ i,i

∈S

5 6 7 8 9 10

5 6 7 8 9 10

−5

0

5

−5

0

5

Figure 4.5: H∞ almost output synchronization. A blow-up of the simulationresults.

H∞ Almost Regulated Synchronization The consensus trajectory forthe multi-agent system is desired to be 2 cos(ω0t). The exo-system (4.11) isthen given by

A0 =

[0 1−ω2

0 0

], C0 =

[1 0

], x0(0) =

[20

]Thus, nq = nq0 = 3 is preserved, but the exo-system has to be shaped asdesired. For B0 = [0, 1]T, the resulting system is invertible and one maydesign a rank-equalizing pre-compensator to change the relative degree tonq. Transforming the system using the observability matrix renders the exo-system as (4.16) with R0 = [0, −ω2

0, 0], M0 = 1, and x0 = [2, 0, −2ω20]T.

Agent 1 is the root of one spanning tree in the network graph, and welink agent 0 (i.e. the exosystem) with agent 1 with weight ψ1 = 1 as depictedin Fig. 4.4b. In fact, we choose π = 1. Let ω0 = 0.5.

Fig. 4.6 displays the results, where the outputs of all agents are plotted,for ε = 0.01 and ε = 0.05. As expected, smaller ε leads to an accuratesynchronization. To discern the contrast between the leaderless synchro-nization and the regulated synchronization, the consensus trajectories areplotted in Fig. 4.7. Note that the consensus trajectory for the leaderlesssynchronization is unbounded although λ(A+BR) = 0,±i.


H∞ almost regulation of output consensus - ε = 0.05

yi,i∈S

Time (sec)

H∞ almost regulation of output consensus - ε = 0.01

yi,i∈S

0 5 10 15 20 25 30

0 5 10 15 20 25 30

−6

−4

−2

0

2

4

6

−6

−4

−2

0

2

4

6

Figure 4.6: H∞ almost regulated synchronization. The red dash lines showy0.

H∞ Almost Regulated Formation This part builds up H∞ almostregulated synchronization into the the problem of H∞ almost formationwith output regulation. To be specific, agents have to form a desired con-figuration while they are forced to follow the reference y0. This can beportrayed as virtual reference formation control. Simulation is carried outfor the same network as the previous part. The formation set is selected asSf = 10, 5,−5,−10. Thus, the objective is to have y1 − 10 = y2 − 5 =y3 +5 = y4 +10 = y0

.= 2 cos(0.5t) as t→∞. The result is given in Fig. 4.8.

4.8 Conclusion

The article introduces the notion ofH∞ almost synchronization, and presentsa method to achieve any accuracy of synchronization in heterogeneous net-works of introspective agents. The main focus is on the leaderless synchro-nization in which the aim is to make the outputs of agents synchronousalthough the consensus trajectory is not controlled. Afterwards, the ap-proach is expanded to solve “H∞ almost regulated synchronization” wherethe objective is to design protocols which attenuate the impact of distur-bances on the regulation and synchronization errors to any arbitrarily smallvalue in the sense of the H∞ norm of the transfer function. The method

4.A. Proof of Lemma 4.1 71

Regulation of output consensus ε = 0.01

yi,i∈S

Time (sec)

Leaderless synchronization - ε = 0.01

yi,i∈S

0 5 10 15 20

0 5 10 15 20

−10

−5

0

5

10

−100

0

100

200

300

400

Figure 4.7: Leaderless synchronization (upper) vs. regulated synchroniza-tion (lower).

is, moreover, broadened to encompass “H∞ almost formation”, where theagents are asked to maintain their relative outputs as desired while synchro-nizing.

Appendix 4.A Proof of Lemma 4.1

Before embarking on the proof, we stress that the problem of H∞ almostsynchronization is solved by showing that the proposed protocol approx-imately decouples ζ from w in the sense of H∞ norm of the closed-looptransfer function from w to ζ. We shall show in Lemma 4.3 that a similardecoupling effect is seen between every ei,j and w.

Closed-loop Equations Let x∗i , xi − xi be the observation error foragent i; then, in view of (4.3) and (4.6), the closed-loop equations for agenti can be written as

xi = Axi +BRxi + ε−nqBFS(xi − x∗i ) + Eiwi

x∗i = Ax∗i +BRx∗i + ε−1KC∑N

j=1lijx∗j + Eiwi


H∞ almost formation with output regulation - ε = 0.01

yi−

y0,i∈S

Time (sec)

H∞ almost formation with output regulation - ε = 0.01

yi,i∈S

y3

y0

y1

y1 − y0

y3 − y0

y4 − y0

y2 − y0

y4

y2

y3

y0

y1

y1 − y0

y3 − y0

y4 − y0

y2 − y0

y4

y2

0 5 10 15 20

0 5 10 15 20

−10

−5

0

5

10

−15

−10

−5

0

5

10

15

Figure 4.8: H∞ almost formation with output regulation.

Let R = [R1, R] with R = [R2, · · · , Rnq ] where R1, · · · , Rnq ∈ Rp×p.Consider the following state transformations

ei = Sxi , zi = Sx∗i , S ,

[Ip 0

−εK ε Ip(nq−1)

]The closed-loop equations are then recast as:

ei = ε−1(A+BF )ei +Reei − ε−1BFSS−1zi + SEiwi

zi = Azzi +Rzzi + ε−1CTK1C∑N

j=1lijzj + SEiwi

where Re = B[εnq−1R1, εnq−2R2, · · · , εRnq−1, Rnq ] and

Rz = εB[R1 + RK, ε−1R

]K ′ , A1 − KC1

, Az =

[C1K ε−1C1

εK ′K K ′

]We split zi into z1,i = Czi ∈ Rp and z2,i such that zi = col z1,i, z2,i.Partition

F =[F1 F2 · · · Fnq

], K =

[KT

1 KT2 · · · KT

nq−1

]T

where Fj , Kj ∈ Rp×p for j = 1, · · · , nq. Let FSS−1 = [F ∗1 , F∗2 ] where F ∗1 =

F1+∑nq−1

s=1 εsFs+1Ks and F ∗2 = [F2, εF3, · · · , εnq−3Fnq−1, εnq−2Fnq ]. Then,


one may show the closed-loop equations as

εei = (A+BF )ei + εReei −BF ∗1 z1,i −BF ∗2 z2,i + εSEiwi

εz1,i = εC1Kz1,i + C1z2,i +K1

∑N

j=1lijz1,j + εE1,iwi

z2,i = εEzz1,i + Azz2,i + εE2,iwi

where E2,i , E2,i − KE1,i and Ez , AzK +B1R1. For i ∈ S, consider thefollowing notations

G = diagSEi, G1 = diagE1,i, G2 = diagE2,ie = col ei, z1 = col z1,i, z2 = col z2,i

Considering the network Laplacian L = [lij ] for i, j ∈ S, the closed-loopequations for the network are given by

εe = (IN ⊗ (A+BF )) e+ (IN ⊗ εRe)e− (IN ⊗BF ∗1 )z1 − (IN ⊗BF ∗2 )z2 + εGw

εz1 = (IN ⊗ εC1K + L⊗K1)z1 + (IN ⊗ C1)z2 + εG1w

z2 = (IN ⊗ εEz)z1 + (IN ⊗ Az)z2 + εG2w

ζ = (L⊗ C)e

Reduced-order Dynamics Let 1 ∈ RN be the right eigenvector of L forthe eigenvalue at zero. Let 1L be the associated left eigenvector. Supposethe Jordan form of L is obtained using the matrix U as

U =[U 1

], U−1 =

[UTL

1TL

]⇒ UT

L U = IN , 1TLU = 0

UTL 1 = 0, 1T

L1 = 1

Thus, we denote U−1LU = diag∆, 0 and LU = [L, 0] where L = U∆.Considering the transformation matrices

T1 = (U−1 ⊗ Ipnq), T2 = (U−1 ⊗ Ip), T3 = (U−1 ⊗ Ip)

where p = p(nq − 1), the following state transformations are introduced:[ee∗

]= T1e ,

[z1

z∗1

]= T2z1 ,

[z2

z∗2

]= T3z2

Denoting N = N − 1, the states e, z1 and z2 are vectors of the dimensionsNpnq, Np and N p, respectively. Obviously, the states e∗, z∗1 and z∗2 are


vectors of the dimensions pnq, p and p, respectively. Consider the followingnotations:

Ge = (UTL ⊗ Ipnq)G, G∗e = (1T

L ⊗ Ipnq)G

Gz1 = (UTL ⊗ Ip)G1, G∗z1 = (1T

L ⊗ Ipnq)G1

Gz2 = (UTL ⊗ Ip)G2, G∗z2 = (1T

L ⊗ Ip)G2

As a result, the system dynamics are divided into two subsystems. Onesubsystem is of order 2pnqN , and is given by

ε ˙e = (IN ⊗ (A+BF ))e+ (IN ⊗ εRe)e− (IN ⊗BF ∗1 )z1 − (IN ⊗BF ∗2 )z2 + ε Gew (4.25a)

ε ˙z1 = (IN ⊗ εC1K + ∆⊗K1)z1

+ (IN ⊗ C1)z2 + ε Gz1w (4.25b)

˙z2 = (IN ⊗ εEz)z1 + (IN ⊗ Az)z2 + ε Gz2w (4.25c)

Note that ζ = (L⊗ C)e. The other subsystem is given by

εe∗ = (A+BF + εRe)e∗ −BF ∗1 z∗1 −BF ∗2 z∗2 + εG∗ew

εz∗1 = εC1Kz∗1 + C1z

∗2 + εG∗z1w

z∗2 = εEzz∗1 + Azz

∗2 + εG∗z2w

When all agents have reached an agreement, ζ is zero, which imposes noconstraints on (e∗, z∗1 , z

∗2). In fact, (e∗, z∗1 , z

∗2)-subsystem determines the

consensus trajectories when ζ = 0. Computation of the synchronizationtrajectory in the presence of external disturbances is not trivial and is notas easy as the case when external disturbances are absent.

It enunciates the fact that the consensus trajectories may be unbounded.It is inferred that the objective in synchronization of networked systems isto design a protocol such that the reduced-order dynamics (4.25) vanishesin time.

Disturbance Rejection Properties : We demonstrate that ‖Twζ‖∞shrinks as ε reduces. Let ‖ζ‖ ≤ ρζ‖e‖ for some ρζ independent of ε. AsA+BF is Hurwitz stable, there exists Pc = PT

c > 0 which solves

(IN ⊗ (A+BF ))T Pc + Pc (IN ⊗ (A+BF )) = −2INpnq

Find s0 ≥ ‖Pc(IN ⊗Re)‖, s1 ≥ ‖Pc(IN ⊗BF ∗1 )‖, s2 ≥ ‖Pc(IN ⊗BF ∗2 )‖ and

ρ1 ≥ ‖PcGe‖ for all ε ∈ (0, 1], which are all bounded. Therefore, there exists


an ε11 ∈ (0, 1] such that, for every ε ∈ (0, ε11], the inequality (1− 2εs0) > 0holds. Let Wc = εeTPce. Differentiation yields

Wc ≤−‖e‖2+2ρ2

√‖z1‖2+‖z2‖2‖e‖+2ερ1‖e‖‖w‖

where ρ2 ≥√

2 maxs1, s2. As L ∈ Gβ, λ(∆) ∈ C+. Therefore, sinceK1 < 0, ∆⊗K1 is Hurwitz stable. Thus, for any Gβ, there exists a P1 > 0such that the following holds

(∆⊗K1)HP1 + P1(∆⊗K1) = −Q1 < 0 (4.26)

Proposition 4.1. Given Gβ, replacing P1 = −diagp1, · · · , pN ⊗K−11 ,

where pN = 1 and pi = β2

9 pi+1, in (4.26) gives rise to Q1 > 4qI where q > 0depends on β. It implies ‖P1‖ is bounded.

Proof. See Appendix 4.E.

It is worth noting that P1 is found for a set of networks, say Gβ, notfor a given network L. Choose W1 = qεzT

1 P1z1. An upper bound for W1 isthen given by

W1 ≤ −2q2‖z1‖2 − 2q‖z1‖2(q− εs3)

+ 2qs4‖z1‖‖z2‖+ 2qερ3‖z1‖‖w‖

where s3 ≥ ‖P1(IN ⊗C1K)‖, s4 ≥ ‖P1(IN ⊗C1)‖, and ρ3 ≥ ‖P1Gz1‖ for allε ∈ (0, 1]. Since Az is Hurwitz stable, the equation

(IN ⊗ Az)TP2 + P2(IN ⊗ Az) = −(2q + q−1s24)IN p

has the solution P2 = PT2 > 0. The derivative of W2 = q zT

2 P2z2 along thetrajectories of (4.25c) is bounded by

W2 ≤ −(2q2 + s24)‖z2‖2 + 2qεs5‖z2‖‖z1‖+ 2qερ4‖z2‖‖w‖

in which s5 ≥ ‖P2(IN ⊗ Ez)‖ and ρ4 ≥ ‖P2Gz2‖. Consider Wo = W1 +W2

and differentiate it in time. One may find an upper bound for Wo as

Wo ≤ −q2‖z1‖2 + 2qs4‖z1‖‖z2‖ − s24‖z2‖2

− 2q‖z1‖2(q− εs3) + 2qεs5‖z2‖‖z1‖ − q2‖z2‖2

− q2‖z1‖2 − q2‖z2‖2 + 2qερ3‖z1‖‖w‖+ 2qερ4‖z2‖‖w‖The first line is non-positive. There exists an ε22 ∈ (0, 1] such that, forevery, 0 < ε ≤ ε22, [

2(q− εs3) −εs5

−εs5 q

]> 0


Then, for every ε ∈ (0, ε22], it turns out that

Wo ≤ −q2‖z1‖2 − q2‖z2‖2 + 2qερ5

√‖z1‖2 + ‖z2‖2‖w‖

where ρ5 ≥√

2 maxρ3, ρ4. Choose V = (2+ρ2ζ)Wc+(1+(2+ρ2

ζ)2ρ2

2 q−2)Wo

as the Lyapunov function candidate for the system (4.25). Let ε∗1 = minε11, ε22.For every ε ∈ (0, ε∗1], the time derivative of V satisfies

V ≤− ‖e‖2 + 2(2 + ρ2ζ)ρ2

√‖z1‖2 + ‖z2‖2‖e‖

− (2 + ρ2ζ)

2ρ22(‖z1‖2 + ‖z2‖2)

− ρ2ζ‖e‖2 − (q2‖z1‖2 + q2‖z2‖2 + ‖e‖2)

+ 2ερ6

√q2‖z1‖2 + q2‖z2‖2 + ‖e‖2‖w‖ (4.27)

where ρ6 ≥√

2 maxρ1(2 + ρ2ζ), ρ5(1 + (2 + ρ2

ζ)2ρ2

2 q−2). The first two lines

form a non-positive term. Completing the square using the last two lines,we arrive at

V + ‖ζ‖2 − (ερ6)2‖w‖2 ≤ 0 (4.28)

From Kalman-Yakubovich-Popov Lemma (see e.g. Zhou and Doyle, 1998),it follows that ‖Twζ‖∞ ≤ ερ6. Obviously, from (4.27), it is observed that(e, z1, z2) is globally exponentially stable at the origin when w = 0.

We point out that the problem is solved for Gβ, not for a given network L.The proposed protocols reject w from ζ to any arbitrary level. Lemma 4.3shows that (4.6) has a similar decoupling effect on e; thus, H∞ almostsynchronization is achieved. We define ei,j = T i,jwe (s)w.

Lemma 4.3. Let ‖Twζ‖∞ ≤ εγ. There exists a positive constant γ′ such

that ‖T i,jwe ‖∞ ≤ εγ′.Proof. By an appropriate choice of A and B, the relation between the outputζ and the input w can be described as

ζ = ε(L⊗ C)(sI− A)−1Bw

We pick one agent arbitrarily. Let it be agent N . As L1 = 0, it holds that∑Nj=1 lijyN = 0. Thus, we have ζi =

∑Nj=1 lijyj−

∑Nj=1 lijyN =

∑Nj=1 lijej,N .

Defineσi , ζi − ζN , σ , col σi, eN , col ei,N

for i ∈ S1 where S1 = 1, 2, · · · , N − 1. Then, one may find σi =∑N−1j=1 l∗ijej,N where l∗ij = lij − lNj for j ∈ S. Let L , [l∗ij ] for i ∈ S1,

j ∈ S be obtained by removing the last row of L − 1 lTN where lTk denotesthe k -th row of L.

4.B. Proof of Lemma 4.2 77

Let L∗ , [l∗ij ] for i ∈ S1, j ∈ S1 be the reduced Laplacian which is found

by discarding the last column of L. According to (Yang et al., 2011a), L∗ isnon-singular. It therefore yields σ = (L∗⊗ Ip)eN . On the other hand, fromthe definition of σi, it follows that σ = ε (L⊗ C)(sI− A)−1Bw. Hence,

eN = ε (L∗⊗ Ip)−1(L⊗ C)(sI− A)−1Bw

It shows that ‖T i,jwe ‖∞ is of order ε, and there exists a constant γ′ > 0 suchthat ‖T i,jwe (s)‖∞ < εγ′.

Appendix 4.B Proof of Lemma 4.2

Before starting the proof, we recall the following result from (Sannuti andSaberi, 1987), which has been reviewed in Chapter 2.

Lemma 4.4. Consider an invertible system which has no invariant zerosand is of uniform rank nq (i.e. all infinite zeros have the same order nq).Let it be characterized by the matrix triple (A, B, C) with x ∈ Rpnq asstate and z, u ∈ Rp as output and input. There exists a nonsingular statetransformation, say Γ0, such that x = Γ0 x transforms the system into

x = Ax+B(Mu+Rx)

z = Cx

(4.29a)

(4.29b)

where A, B, and C are as (4.4), and the square matrix M ∈ Rp×p is non-singular and R ∈ Rp×pnq .

Step 1: According to (Saberi and Sannuti, 1988), design a squaring-downpre-compensator for the right-invertible system6. It is given by

Σ1,i :

p1,i = Ap1,ip1,i +Bp1,iup1,i

ui = Cp1,ip1,i +Dp1,iup1,i

(4.30a)

(4.30b)

where up1,i ∈ Rp, and Ap1,i is Hurwitz. The cascade of Σ1,i and agenti is invertible. It is established in (Saberi and Sannuti, 1988) that thesquaring-down pre-compensator may add stable invariant zeros and it pre-serves stabilizability, detectability, and stability of the finite and infinitezero dynamics of the original system.

6The squaring-down compensators are reviewed in Chapter 2


Step 2: Represent the resulting system in the special coordinate basisaccording to (Sannuti and Saberi, 1987). To that end, one finds an integerr ≥ 1 and nonsingular transformations Γ1,i, Γ2,i, and Γ3,i such that[

p1,i

xi

]= Γ1,i

[xa,i

xd,i

], up1,i = Γ2,i ud,i, yi = Γ3,i yd,i

such that

yd,i = col yj,i, ud,i = col uj,i, xd,i = col xj,i for j ∈ Sr , 1, · · · , r

where yj,i, uj,i ∈ R and xj,i ∈ Rrj,i . Let

Γ−11,iGi =

[Ea,i

Ed,i

], where Ed,i = stack Ej,i for j ∈ Sr

The dynamics of the system are described by

xa,i = Aa,ixa,i + La,iyd,i + Ea,iwi

xj,i = Aj,ixj,i + Lj,iyd,i +Bj,i(uj,i +Raj,ixa,i +Rd

j,ixd,i) + Ej,iwi

where Aa,i, La,i, Lj,i, Raj,i and Rd

j,i are constant matrices with appropriatedimensions; Aj,i and Bj,i have the same structure as A and B, given in(4.4), for p = 1 and nq = rj,i. Moreover, yj,i = Cj,ixj,i, which is equal tothe first element of xj,i. That is,

Aj,i =

[0 I(rj,i−1)

0 0

], Bj,i =

[0rj,i−1

1

], Cj,i =

[1 0rj,i−1

]Notice that we generate p1,i; thus, it is measured. It is straightforward toverify that the pair ([

Ap1,i 00 Ai

],

[I 00 Cm,i

])is detectable. An observer is designed to estimate xa,i and xd,i, which aredenoted by xa,i and xd,i = col xj,i, respectively. We also define the esti-mation errors as

xa,i = xa,i − xa,i , xd,i = xd,i − xd,i , xj,i = xj,i − xj,i

Apply the local feedback

uj,i = uj,i −Raj,ixa,i −Rd

j,ixd,i (4.31)


to the resulting system, which gives rise to

xa,i = Aa,ixa,i + La,iyd,i + Ea,iwi

xj,i = Aj,ixj,i + Lj,iyd,i +Bj,iuj,i +Bj,iRaj,ixa,i +Bj,iR

dj,ixd,i + Ej,iwi

yj,i = Cj,ixj,i

in which the estimation error dynamics are described by

˙xi = Hixi + Eo,iwi (4.32)

where Hi is Hurwitz stable and

xi = Γ1,i

[xa,i

xd,i

]The feedback (4.31) decouples the zero-dynamics from the infinite zero dy-namics. It also decouples the chains of integrators xj,i from each other.

Step 3: Consider the following system

xj,i = Aj,ixj,i + Lj,iyd,i +Bj,iuj,i + %j,ixi + Ej,iwi

yj,i = Cj,ixj,i

where %j,i = Bj,i[Raj,i, Rd

j,i]Γ−11,i .

We make the system of the uniform rank nq by adding states to thebottom of each chain and setting their initial values equal to zero. It doesnot change the output of each chain. To that end, apply the followingpre-compensator to each chain

p2j,i =

[0 Inq−rj,i−1

0 0

]p2j,i +

[0nq−rj,i−1

1

]uj,i for nq > rj,i

uj,i =[

1 0 0 · · · 0]p2j,i

which simply implies that uj,i =(

1s

)nq−rj,i uj,i. If rj,i = nq, let uj,i = uj,i.For j ∈ Sr, define

up2,i = col uj,i, p2,i = col p2j,i, ui = col uj,i

Then the rank-equalizing pre-compensator is described by

Σ2,i :

p2,i = Ap2,ip2,i +Bp2,iup2,i

ui = Cp2,ip2,i +Dp2,iup2,i

(4.33a)

(4.33b)


The cascade of Σ2,i, Σ1,i and agent i, along with the output feedback, isinvertible and has the uniform rank nq. Recall xd,i = col xj,i for j =1, · · · , r. Denote

x∗i = col xd,i, p2,i

Thus, the compensated system can be represented as

x∗i = A∗ix∗i +B∗i up2,i + %∗i xi + E∗i wi (4.34a)

yd,i = C∗i x∗i (4.34b)

where A∗i , B∗i , C∗i , %∗i and E∗i can be found by inspection.

We restore the output (yd,i) to the original output (yi) since outputtransformation does not change the system’s invertibility properties. There-fore, the resulting system (4.34) is invertible, of the uniform rank nq withno invariant zeros.

According to Lemma 4.4, there exists a state transformation xi = Tix∗i

such that the system can be described by

xi = Axi +B(Mup2,i + Rxi) + Wixi + Ed,iwi

yi = Cxi

where A, B, and C are as (4.4), M is nonsingular; R, Wi and Ed,i are foundas

A+BR = TiA∗iT−1i

BM = TiB∗i

Wi = Ti%∗i

Ed,i = TiE∗i

Step 4: Given desired R and M of appropriate dimensions, apply

Mup2,i = Mui + (R− R)xi (4.35)

where xi is the estimation of xi and is given by

xi = Ti

[xd,i

p2,i

]Application of (4.35) yields

xi = Axi +B(Mui + (R− R)xi + Rxi) + Wixi + Ed,iwi

= Axi +B(Mui +Rxi) +B(R−R)Ti

[xd,i

0

]+ WiΓ1,i

[xa,i

xd,i

]+ Ed,iwi

4.C. Manipulation of Exo-system 81

Therefore, for some Wi, we can write the system as

xi = Axi +B(Mui +Rxi) +Wixi + Ed,iwi

It is worth noting that yi has not changed.

Appendix 4.C Manipulation of Exo-system

We will show that there exists a state transformation, state manipulationand a matrix B0 for the exosystem (4.11) such that the resulting systemcharacterized by the triple (A0, B0, C0) is invertible, of the uniform ranknq > nq0, with no invariant zeros.

First, we take the procedure from (Wang et al., 2013). According to(Luenberger, 1967), there exists a non-singular state transformation T0 suchthat z0 = T0x0 transforms the system (4.11) into the following canonicalform:

˙z0 = A0z0

y0 = C0z0

(4.36a)

(4.36b)

where

A0 = T0A0T−10 =

A1 0 0 0 · · · 0? ? ? ? · · · ?

0 A2 0 0 · · · 0? ? ? ? · · · ?...

......

......

...

0 0 0 0 0 Ap? ? ? ? · · · ?

,

C0 = C0T−10 =

C1 0 0 0 · · · 0

0 C2 0 0 · · · 0...

......

......

...

0 0 0 0 · · · Cp

in which ? denotes possible nonzero elements (indicates a possible non-zerorow), and

Ai =

0 1 0 · · · 00 0 1 · · · 0...

.... . .

. . ....

0 0 · · · 0 1

,Ci =

[1 0 · · · 0 0

]


Ai’s may not have the same size. However, by adding integrators to the bot-tom of each block of Ai and setting the initial conditions of the added statesto zero, we can extend the dimension of Ai to (nq−1)×nq while the systemgiven by (4.37) still produces the same output. The new representation ofthe system can be given by

˙z0 = A0z0 (4.37a)

y0 = C0z0 (4.37b)

where

A0 =

A∗ 0 0 0 · · · 0? ? ? ? · · · ?0 A∗ 0 0 · · · 0? ? ? ? · · · ?...

......

......

...0 0 0 0 0 A∗

? ? ? ? · · · ?

, A∗ =

[0 Inq−1

]

C0 =

C∗ 0 0 0 · · · 00 C∗ 0 0 · · · 0...

......

......

...0 0 0 0 · · · C∗

, C∗ =[

1 0 0 · · · 0]∈ Rnq

We choose

B0 =

B1 0 0 0 · · · 0 0

0 B2 0 0 · · · 0 0...

......

... · · · ......

0 0 0 0 · · · 0 Bp

, Bi =

00...01

We find that the triple (A0, B0, C0) is invertible, of uniform rank nq, andhas no invariant zero.

There is also possible to show this by another approach. Let Ai ∈ Rki×ki ,for ki ≥ 1, and A∗i ∈ Rnq×nq be as

Ai =

[0 Iki−1

0 0

], A∗i =

[0 Inq−1

0 0

](4.38)

For ki = 1, Ai = 0. It follows from (Chen et al., 2004, Theorem 4.3.1)that there exist nonsingular state and output transformations as x0 = Tsz0

4.D. Proof of Theorem 4.3 83

and y0 = Tyy0 such that the exosystem (4.11) can be described by p-interconnected systems in the form of:

˙zi = Aizi + Liy0 (4.39a)

y0i = Cizi , i = 1, · · · , p (4.39b)

in which y0i ∈ R, y0 = col y0i and zi ∈ Rki where the set k1, · · · , kpis the observability index of the pair (A0, C0) and

∑pj=1 kj = n0. Also,

Ci = [1, 0, · · · , 0] ∈ Rki and Li ∈ Rki×p, i = 1, · · · , p, are some constantmatrices. Notice that the pair (A0, C0) is observable.

We extend the dimension of each subsystem to nq by adding an appro-priate number of integrators to the bottom of zi. The initial values of thenew states are set equal to zero. Then, the systems described by (4.39) arerecast as

˙zi = A∗i zi + L∗i y0

y0i = C∗i zi, zi ∈ Rnq

where C∗i = [1, 0, · · · , 0], L∗i = [LTi , 0]T. Define L∗ = stack L∗i , C∗ =

diagC∗i , and A0 = diagA∗i +L∗C∗. Let B∗i = [0, 0, · · · , 0, 1]T ∈ Rnq andB0 = diagB∗i . The triple (A0, B0, C

∗) is invertible, of the uniform ranknq with no invariant zeros. We restore the output to the original outputby the output transformation Ty. Thus, let C0 = TyC

∗. As invertibility isinvariant under output transformation, the triple (A0, B0, C0) is invertible,of the uniform rank nq, with no invariant zeros.

Appendix 4.D Proof of Theorem 4.3

It follows from Lemma 4.2 that there exists a dynamic compensator thatmakes agent i ∈ S have the dynamics of (4.3) for an arbitrary R andnonsingular M . Partition R as before (R = [R1, R] where R1 ∈ Rp×p).The vector fi ∈ Rpnq for i ∈ S is formed as fi = col fi, 0 such thatfi = Cfi. Let xf,i = xi − fi. In view of Afi = 0, Rfi = R1fi and fi = 0,(4.3) is recast as

xf,i = Axf,i +B(Muf,i +Rxf,i) + Eiwi, yf,i = Cxf,i (4.41)

where Muf,i = Mui + R1fi. In compliance with Section 4.4, the observer-based protocol for (4.41) will have the same structure as (4.6). It yieldsclosed-loop equations similar to those given in Section 4.4.1. The rest of theproof is akin to the proof of Lemma 4.1.


Appendix 4.E Proof of Proposition 4.1

We seek P1 and Q1 which satisfy (4.26). We choose P1 = −(P ⊗K−11 ) in

which P = diagp1, · · · , pN−1 > 0 must be chosen appropriately. Substi-tution of P1 in (4.26) results in

P1(∆⊗K1) + (∆⊗K1)HP1 = −(P∆ + ∆HP )⊗ Ip

Choosing Q1 = Q⊗Ip, the objective is reduced to show that ∆HP+P∆ = Qwhere Q > 0. We intend to find P so that ∀v ∈ RN−1, v 6= 0, vTQv > 0.Since ∆ is in the Jordan form, vTQv can be expressed as

vTQv =∑N−1

i=1Reλipiv2

i + 2∑N−2

i=1ρipivivi+1

in which ρi ∈ 0, 1; ρi = 1 if λi is a repeated eigenvalue of L; otherwise,ρi = 0. Let ρi = 1 for all i = 1, · · · , N − 1; then one may write:

vTQv = 13

∑N−1

i=1Reλipiv2

i

+ 13Reλ1p1v

21 + 1

3ReλN−1pN−1v2N−1

+

N−2∑i=1

√13Reλi+1pi+1vi+1 +

ρipi√13Reλi+1pi+1

vi

2

+∑N−2

i=1

(13Reλipi −

ρip2i

13Reλi+1pi+1

)v2i (4.42)

If ρi = 0, any positive pi satisfies (4.26) for Q > 0. Otherwise, equation(4.42) is always positive if we set pN−1 = 1 and define pi’s recursively

according to pi = β2

9 pi+1 for i = 1, · · · , N − 2. It happens since the firstthree lines of (4.42) are non-negative for pi > 0, and the last line is positivefor this particular choice because Reλi > β and one may show

1

3Reλipi −

p2i

13Reλi+1pi+1

=1

3pi

(Reλi −

β2

Reλi+1

)> 0

Thus, ∃ c > 0 such that ‖P1‖ = max1, (β2/9)N−2‖K−11 ‖ < c. Also, it

turns out that there exists a q > 0, dependent on β, such that Q > 4qIsince vTQv > 1

3

∑N−1i=1 βpiv

2i ⇒ Q > 1

3βP .

4.F. Simulation Data 85

Appendix 4.F Simulation Data

Models of Agents The transfer functions are given.

y1 =1

s3u1, y3 =

(s3 + 1)u1,3 − su2,3

s3

y2 =s+ 2

s2 − 6u2, y4 =

(s3 − 2s− 1)u1,4 + (s+ 1)u2,4

s5 − 2s4 − 2s3 + s2 + 2s+ 1

The transfer functions from disturbances to the output:

y1 =s2 + 2s+ 3

s3w1, y3 =

−s2 − 2s+ 3

s3w3

y2 =3s+ 7

s2 − 6w2, y4 =

3s4 + 2s3 + s2 − 7s− 5

s5 − 2s4 − 2s3 + s2 + 2s+ 1w4

Squaring-Down Pre-compensators There is no need for squaring downagents 1 and 2; thus, Σ1,1 : u1 = up1,1 and Σ1,2 : u2 = up1,2 . Σ1,3 and Σ1,4

are designed for agents 3 & 4:

Σ1,3 : u3 =[−(s2 + 9s+ 16), (16s2 + 55s+ 39)]

s2 + 3s+ 2up1,3

Σ1,4 : u4 =[−(s+ 6), −5(3s+ 2)]

s+ 1up1,4

Rank-Equalizing Pre-compensators Agent 1 does not need rank equal-ization. Thus, Σ2,1 : up1,1 = up2,1 . Since agent 2 and 3 are of relative degree1, the compensators are the same; Σ2,k : up1,k

= 1s2up2,k

for k = 2, 3. For

agent 4: Σ2,4 : up1,4 = 1sup2,4


Chapter 5

H∞ Almost Synchronizationfor Homogeneous Networksof Non-introspective AgentsUnder External Disturbances

This chapter addresses the problem of “H∞ almost synchronization”for homogeneous networks of general linear agents subject to exter-nal disturbances. Agents are presumed to be non-introspective; i.e.the only available information for each agent is the network measure-ment which is a linear combination of relative outputs; in other words,agents are not aware of their own states or outputs. Under a set ofnecessary and sufficient conditions, a family of dynamic protocols isdeveloped such that the impact of disturbances on the synchronizationerror dynamics, expressed in terms of the H∞ norm of the correspond-ing closed-loop transfer function, is reduced to any arbitrarily smallvalue.This chapter is based on Peymani et al. (2013a,b).

5.1 The Topic of The Chapter

In the preceding chapter, we introduced the notion of H∞ almost synchro-nization, in which the impact of external disturbances on the disagreementdynamics is attenuated to ‘any’ arbitrarily small value in the sense of H∞norm of the closed-loop transfer function. Hence, synchronization can beachieved with any arbitrary degree of accuracy. The preceding chapter wasconcerned with heterogeneous networks of introspective agents; i.e. the dy-namics of agents are non-identical, and agents have partial knowledge abouttheir own states in addition to network measurements.

87

88 Homogeneous Networks of Non-introspective Agents

In this chapter, the problem of H∞ almost synchronization is solved forhomogeneous networks of non-introspective agents; i.e. the network consistsof identical agents which are not allowed to have access to their own states oroutputs, and the only available measurements are those that are transmittedover the network.

We stress the fact that the lack of self-measurements does not allowus to shape the agents into an identical desired dynamics, as performedin the previous chapter1. Hence, we are confronted with general linearsystems, where the finite and infinite zero structures as well as invertibilityproperties of agents are explicitly exploited in order to achieve H∞ almostsynchronization.

More specifically, the objective is to design decentralized observer-basedprotocols which reduce the H∞ norm of the closed-loop transfer functionfrom disturbance to synchronization error to any arbitrarily small value forhomogeneous networks of non-introspective and linear agents. In this work,communication links are assumed to be directed.

Furthermore, a novel time-scale structure assignment technique whereinfinite zero dynamics of different orders are scaled with different time-scalesis proposed for the first time. The motivation for such a multiple time-scaleassignment is to make the design robust with respect to a multiplicativeuncertainty arising from the interaction topology (see Chapter 3 for moreinformation on the source of multiplicative uncertainty.).

5.2 Notations

Matrix A is represented by A = [aij ] where the element (i,j) of A is shownby aij . Let KerA and ImA denote, respectively, the kernel and the imageof A. The Kronecker product of matrices A = [aij ] and B = [bij ] is definedas A⊗B = [aijB]. Let ‖A‖ symbolize the induced 2-norm.

A block diagonal matrix constructed by Ai’s is shown by diagAi fori = 1, · · · , n. Also, stack Ai for i = 1, · · · , n indicates [AT

1 , AT2 , · · · , AT

n ]T.Likewise, x = col xi for i = 1, · · · , n is adopted to denote x = [xT

1 , · · · , xTn ]T

where xi’s are vectors.The identity matrix of order n is symbolized by In. Let 1n ∈ Rn be the

vector with all entries equal to one. The real part of a complex number λis represented by Reλ. The open left-half and open right-half complexplanes are represented by C− and C+, respectively. For a transfer func-tion T (s) which belong to the Hardy space H∞, the H∞ norm is denoted

1In fact, in the preceding chapter, using self-measurements, each agent is convertedto a desired system which only owns infinite zero dynamics.

5.3. Homogeneous Multi-Agent Systems 89

‖T (s)‖∞. For a space V, the orthogonal complement is shown by V⊥. wefrequently use the following sets of integers:

Ω = 1, · · · , r, Ω = 1, · · · , rwhere r and r are positive integers.

5.3 Homogeneous Multi-Agent Systems

A homogeneous multi-agent system is referred to a network of multiple-inputmultiple-output agents described by identical linear time-invariant modelsas

Agent i :

xi = Axi + Bui + Gwi

yi = Cxi

(5.1a)

(5.1b)

in which i ∈ S , 1, · · · , N. Also, xi ∈ Rn is the state, ui ∈ Rm is thecontrol, yi ∈ Rp is the output, wi ∈ Rω : limτ→∞

12τ

∫ τ−τ wT

i widt < ∞ isthe external disturbance.

Agents are non-introspective and no self-measurements are available; inother words, agent i does not have access to its own states xi or the outputyi. Thus, the only information which is given to agent i is the networkmeasurement which is transmitted over the network.

The network’s communication topology, based on which agents exchangeinformation, is described by a directed graph G whose nodes correspond withagents in the network. If an edge exists from node k to i, a positive realweight aik is given to the edge. We assume that no self-loops are allowed;i.e. aii = 0. The graph G is associated with the Laplacian matrix G = [gik]where

gik = −aik, i, k ∈ S, i 6= k;

gii =∑N

k=1 aik

It follows that λ = 0 is an eigenvalue of G with a right eigenvector 1N .Thus, the network measurement given to agent i ∈ S is:

ζi =

N∑k=1

aik(yi − yk)

=N∑k=1

gikCxk (5.1d)

In addition, it is assumed that agents are capable of exchanging additionalinformation over the network. The transmission of this additional informa-tion conforms with the network’s communication topology and facilitates


the design of a distributed observer for the network. Thus, agent i hasaccess to the following quantity:

ζi =N∑k=1

gikηk (5.1e)

where ηk ∈ Rp depends on the state of the protocol of agent k ∈ S; it willbe clarified later when a dynamic protocol is introduced.

5.4 H∞ Almost Synchronization

This section tackles the problem of H∞ almost synchronization for a net-work of homogeneous, and non-introspective agents subject to external dis-turbances, as described by (5.1). The goal has to be achieved using networkmeasurements, and no self-measurements are available for agents.

5.4.1 Problem Formulation

We define the following vectors which are formed by stacking the corre-sponding vectors of each agent:

w , col wi, ζ , col ζi (5.2)

for i ∈ S. Let the mutual disagreement between a pair of agents be definedas:

ei,k , yi − yk, for i, k ∈ S, i > k (5.3)

The stacking column vector of all mutual disagreements is denoted e. Obvi-ously, synchronization is achieved if e = 0. We define the following transferfunction with the appropriate dimension:

e = Twe(s)w (5.4)

Fig. 5.1 depicts the block diagram of the multi-agent system. The problemthat we cope with in this chapter is precisely defined below.

Problem 5.1. Consider a multi-agent system described by (5.1) with acommunication topology G. Given a set of network graphs G∗ and anyγ > 0, the “H∞ almost synchronization” problem is to find, if possible,a linear time-invariant dynamic protocol such that, for any G ∈ G∗, theclosed-loop transfer function from w to e satisfies

‖Twe(s)‖∞ < γ J


Dynamic LTI protocol parameterized in terms of

Multi-agent System Homogeneous & Non-introspective

Figure 5.1: The block diagram of the multi-agent system under externaldisturbances. u = col ui for all i ∈ S.

Accordingly, for agent i ∈ S, the protocol which maps ζi and ζi toui has the internal state ξi ∈ Rqi for some integer qi > 0, and takes thefollowing general form

ξi = Ac(ε) ξi + Bc(ε) col ζi, ζiui = Cc(ε) ξi +Dc(ε) col ζi, ζi

(5.5a)

(5.5b)

5.4.2 Preliminaries and Assumptions

The necessary and sufficient conditions under which development of thedesired protocol is viable are given in terms of geometric subspaces2 andan appropriate set of networks. Geometric subspaces and its applicationto exact disturbance decoupling are explained in the books of Wonham(1985) and Trentelman et al. (2001). The application to almost disturbancedecoupling is presented by Weiland and Willems (1989).

Let V∗KerC(A,B,C) be the maximal (A−BF)-invariant subspace of Rncontained in Ker C such that the eigenvalues of (A − BF) belong to C−for some F. The supremal Lp-almost controllability subspace ‘contained’in Ker C is represented by R∗KerC(A,B,C). Let S∗ImB(A,B,C) denote theminimal (A−KC)-invariant subspace of Rn containing Im B such that theeigenvalues of (A −KC) belong to C− for some K. Eventually, we definethe following subspaces of the state space.

Vb,KerC(A,B,C) = V∗KerC(A,B,C)⊕R∗KerC(A,B,C)

Sb,ImB(A,B,C) = (Vb,KerBT(AT,CT,BT))⊥

Assumption 5.1. We make the following assumptions.

1. (A,B) is stabilizable;

2This topic was reviewed in Chapter 2.


2. (C,A) is detectable;

3. Im G ⊂ Vb,KerC(A,B,C);

4. Sb,ImG(A,G,C) ⊂ Vb,KerC(A,B,C);

5. Sb,ImG(A,G,C) ⊂ Ker C;

6. The matrix triples (A,B,C) and (A,G,C) have no invariant zeroson the imaginary axis.

The geometric subspaces can be computed by virtue of the special coor-dinate basis proposed by Sannuti and Saberi (1987) (reviewed in Chapter 2)using available software, either numerically (Liu et al., 2005) or symbolically(Grip and Saberi, 2010).

Definition 5.1. For given β > 0 and integer N0 ≥ 1, Gβ is the set of graphscomposed of N nodes where N ≤ N0 such that every G ∈ Gβ has a directedspanning tree, and every nonzero eigenvalue of its Laplacian, denoted λi fori = 1, · · · , N , satisfies

Reλi > β J

A directed graph G has a directed spanning tree if it has a node fromwhich there are directed paths to every other nodes. According to (Ren andBeard, 2005), the Laplacian matrix G associated with G ∈ Gβ has a simpleeigenvalue at zero, and the rest are located in C+.

5.4.3 Protocol Development (SISO)

To facilitate understanding the method, at first, networks of single-inputsingle-output agents are considered3. Thus, a multi-agent system, wherem = p = 1, is taken into account, and a family of dynamic protocols ispresented, which solves the problem of H∞ almost synchronization as statedin Problem 5.1. More clearly, it is shown that there exists a distributeddynamic protocol parameterized in terms of a tuning parameter ε ∈ (0, 1]in the form of

˙xi = (A−BFcon(ε))xi + Kobs(ε)(ζi − ζi)ui = −Fcon(ε)xi

(5.6a)

(5.6b)

3The design procedure, provided in this section, is also applicable to multiple-inputmultiple-output systems which are invertible and of a uniform rank. Notice that SISOsystems are invertible.


where xi ∈ Rn and ζi is given by (5.1e) where ηj = Cxj . A step-by-step design procedure for determining the gains Fcon(ε) and Kobs(ε), whichdepend on the triple (A,B,C) and a network set G∗, is presented in thissection. It is established that under Assumption 5.1 and for any β > 0,there exists an ε∗1 ∈ (0, 1] such that, for every ε ∈ (0, ε∗1], synchronization isachieved exponentially fast in the absence of external disturbances, providedthat the network graph G belongs to the graph set Gβ. Also, for any givenγ > 0, there exists ε∗2 ∈ (0, ε∗1] such that every ε ∈ (0, ε∗2] makes the closed-loop transfer function from w to ζ fulfill ‖Twe(s)‖∞ < γ.

? Step S-1

Find nonsingular transformations Γx, Γu and Γy in order to represent theSISO system characterized by the matrix triple (A,B,C) into the scb asstated in Appendix 5.A. Therefore, we have

xi = Γxxi, yi = Γyyd,i, ui = Γuud,i (5.7)

where xi = col x−a,i, x+a,i, xd,i; x−a,i ∈ Rn

−a , x+

a,i ∈ Rn+a , xd,i ∈ Rnd , and

n = n−a + n+a + nd. We represent Γ−1

x G = stack G−a , G+a , Gd. The system

dynamics are given by:

x−a,i = A−a x−a,i + L−adyd,i +G−a wi (5.8a)

x+a,i = A+

a x+a,i + L+

adyd,i +G+a wi (5.8b)

xd,i = Adxd,i +Bd(ud,i + E−dax−a,i + E+

dax+a,i + Eddxd,i) +Gdwi (5.8c)

yd,i = Cdxd,i (5.8d)

The dimensions of the variables as well as the size and structure of thesystem matrices conform with the scb stated in Appendix 5.A.

? Step S-2

Select the feedback gains F+a and Fd such that the following matrices become

Hurwitz stable:

A∗d = Ad −BdFd, Ass = A+a − L+

adF+a

Since the pair (Ad, Bd) is controllable and the pair (A+a , L

+ad) is stabilizable

under Assumption 5.1-(1), the existence of F+a and Fd is guaranteed. The

dimensions of the gains F+a and Fd are 1× n+

a and 1× nd, respectively.


? Step S-3

Consider ε ∈ (0, 1], and define S as

S(ε) = diag1, ε, · · · , εnd−2, εnd−1 ∈ Rnd×nd (5.9)

Fdε = ε−ndFdS (5.10)

where ε is the tuning parameter and will be specified later.

? Step S-4

Form Fd = Fdd + Fdε where

Fdd =[E−da E+

da Edd

]Fdε =

[0 FdεC

Td F

+a Fdε

]Find Fcon(ε):

Fcon(ε) = ΓuFd Γ−1x (5.11)

? Step S-5

Similar to Step S-1, find nonsingular transformations Γx, Γw and Γy in orderto represent the SISO system characterized by the matrix triple (A,G,C)into the scb as stated in Appendix 5.A. For simplicity, we keep the notationused in Step 1 unchanged and place bars on the variables, matrices, and theirdimensions. Then choosing

xi = Γxxi, yi = Γyyd,i, wi = Γwwd,i (5.12)

where xi = col x−a,i, x+a,i, xd,i. Let Γ−1

x B = stack B−a , B+a , Bd. The sys-

tem is then given by:

˙x−a,i = A−a x−a,i + L−adyd,i + B−a ui (5.13a)

˙x+a,i = A+

a x+a,i + L+

adyd,i + B+a ui (5.13b)

˙xd,i = Adxd,i + Gd(wd,i + E−dax−a,i + E+

dax+a,i + Eddxd,i) + Bdui (5.13c)

yd,i = Cdxd,i (5.13d)

The dimensions of the variables, the size and structure of the matrices con-form with the scb stated in Appendix 5.A.


? Step S-6

By an appropriate selection of the observer gain K+a , make the following

matrix Hurwitz stable:

Ass = A+a − K+

a E+da (5.14)

Such K+a exists under Assumption 5.1-(2). Find Pd = PT

d > 0 which solvesthe following algebraic Riccati equation:

AdPd + PdATd − 2τPdC

Td CdPd = −Ind

where 0 < τ ≤ β. The existence of such Pd follows from the observability ofthe pair (Cd, Ad). Now, define Kd = PdC

Td . We point out that K+

a ∈ Rn+a

and Kd ∈ Rnd .

? Step S-7

Define

S(ε) = diagε−(nd−1), ε−(nd−2), · · · , ε−2, ε−1, 1 (5.15)

Kdε = ε−ndS−1Kd (5.16)

? Step S-8

Form Kdε ∈ Rn as below:

Kdε =

0K+

a GTd Kdε

Kdε

(5.17)

Now, one may obtain Kobs(ε) using

Kobs(ε) = ΓxKdε Γ−1y (5.18)

Theorem 5.1 formalizes the result.

Theorem 5.1. Under Assumption 5.1 and for the set Gβ, the parameterizedprotocol (5.6), where Fcon(ε) is selected as in (5.11) and Kobs(ε) is selectedas in (5.18), solves Problem 5.1. Precisely, the following hold

1. for any given β > 0, there exists an ε∗1 ∈ (0, 1] such that, for every ε ∈(0, ε∗1], synchronization is accomplished in the absence of disturbance;i.e. ∀ε ∈ (0, ε∗1] when w = 0

ei,j = yi − yj → 0, ∀i, j ∈ S, i > j as t→∞


2. for any given γ > 0, there exists an ε∗2 ∈ (0, ε∗1] such that, for ev-ery ε ∈ (0, ε∗2], the closed-loop transfer function from w to e satisfies‖Twe(s)‖∞ < γ.


5.4.4 Protocol Development (MIMO)

Now, we consider a general linear system. That is, it is assumed that eachagent is a multiple-input multiple-output linear systems as described by(5.1a), and (5.1a). Agents can be degenerate; that is, they are nighterright-invertible nor left-invertible. The family of dynamic protocols whichsolves the problem of H∞ almost synchronization will be the same as theSISO case. For ease of reference, it is rewritten here:

˙xi = (A−BFcon(ε))xi + Kobs(ε)(ζi − ζi)ui = −Fcon(ε)xi

(5.19a)

(5.19b)

where xi ∈ Rn, ε ∈ (0, 1] is the tuning parameter, and ζi is given by (5.1e)where ηj = Cxj . The result for MIMO case is quite the same as the resultfor SISO case. That is, under Assumption 5.1 and for any β > 0, there existsan ε∗1 ∈ (0, 1] such that, for every ε ∈ (0, ε∗1], synchronization is achievedin the absence of external disturbances, provided that the network graphG belongs to graph set Gβ. Moreover, for any given γ > 0, there existsan ε∗2 ∈ (0, ε∗1] such that every ε ∈ (0, ε∗2] makes the closed-loop transferfunction from w to ζ be less than γ. A step-by-step design procedure fordetermining the gains Fcon(ε) and Kobs(ε) is provided4.

• Step M-1

Find nonsingular transformations Γx, Γu and Γy in order to represent thesystem characterized by the matrix triple (A,B,C) into the scb as statedin Appendix 5.A. To that end, let

xi = Γxxi, yi = Γy

[yd,i

yb,i

], ui = Γu

[ud,i

uc,i

](5.20)

4The algorithm makes use of the special coordinate basis for multivariable linearsystems, proposed by Sannuti and Saberi (1987); see Appendix 5.A.


where xi = col x−a,i, x+a,i, xb,i, xc,i, xd,i. The transformations yield the scb

for each agent i ∈ S:

x−a,i = A−a x−a,i + L−adyd,i + L−abyb,i +G−a wi (5.21a)

x+a,i = A+

a x+a,i + L+

adyd,i + L+abyb,i +G+

a wi (5.21b)

xb,i = Abxb,i + Lbdyd,i +Gbwi (5.21c)

xc,i = Acxc,i + Lcdyd,i + Lcbyb,i +Bc(uc,i + E−cax−a,i + E+

cax+a,i) +Gcwi

(5.21d)

and, for every j ∈ Ω , 1, · · · , r, there are:

xjd,i=Ajdxjd,i+ Ljdyd,i+Bjd(ujd,i + Ejxi) +Gjdwi (5.21e)

Note that xd,i = col xjd,i, yd,i = col yjd,i, and ud,i = col ujd,i, ∀j ∈ Ω.In addition,

yjd,i = Cjdxjd,i, yd,i = Cdxd,i, yb,i = Cbxb,i (5.21f)

where Cd = diagCjd, ∀j ∈ Ω. It is also used that

G = Γx stack G−a , G+a , Gb, Gc, Gd

where Gd = stack Gjd, ∀j ∈ Ω, where Gjd ∈ Rjqj×ω. The dimensions ofthe variables as well as the size and structure of the system matrices conformwith the scb stated in Appendix 5.A. Define the following matrices

As =

[A+

a L+abCb

0 Ab

], Lsd =

[L+

ad

Lbd

](5.22)

• Step M-2

Select the feedback gain matrices F+a , Fb, Fc and Fjd for j = 1, · · · , r such

that the following matrices become Hurwitz stable:

Acc = Ac −BcFc

A∗jd = Ajd −BjdFjdAss = As − LsdFs

where Fs = [F+a , Fb]. Since the pairs (Ac, Bc) and (Ajd, Bjd) are control-

lable and the pair (As, Lsd) is stabilizable under Assumption 5.1-(1), theexistence of F+

a , Fb, Fc and Fjd is guaranteed. The dimensions of the gainsF+

a , Fb, Fc and Fjd are pd×n+a , pd×nb, mc×nc, and qj× jqj , respectively.


• Step M-3

For every j ∈ Ω, define a matrix Sj ∈ Rjqj×jqj as

Sj(ε) = diagIqj , εIqj , · · · , εj−1Iqj (5.23)

where ε ∈ (0, 1] is the tuning parameter and will be specified later. Also,for j = 1, · · · , r, define

Fjdε = ε−jFjdSj Fdε = diagFjdε (5.24)

• Step M-4

Form Fc ∈ Rmc×n and Fd ∈ Rmd×n as below.

Fc =[

0 0 0 Fc 0 · · · 0], Fd = Fdd + Fdε

where we define

Fdd = stack Ej, for j = 1, · · · , rFdε =

[0 FdεC

Td F

+a FdεC

Td Fb 0 Fdε

]Now, we are in the position to find Fcon(ε). It is obtained using

Fcon(ε) = Γu

[Fd

Fc

]Γ−1

x (5.25)

• Step M-5

Find nonsingular transformations Γx, Γw and Γy in order to represent thesystem characterized by the matrix triple (A,G,C) into the scb as stated inAppendix 5.A. For simplicity, we keep the notation used in Step 1 unchangedand place bars on the variables, matrices, and their dimensions. To obtainthe scb, let

xi = Γxxi, yi = Γy

[yd,i

yb,i

], wi = Γw

[wd,i

wc,i

](5.26)

where xi = col x−a,i, x+a,i, xb,i, xc,i, xd,i. Transforming the system into new

coordinates yields the scb for each agent i ∈ S:

˙x−a,i = A−a x−a,i + L−adyd,i + L−abyb,i + B−a ui (5.27a)

˙x+a,i = A+

a x+a,i + L+

adyd,i + L+abyb,i + B+

a ui (5.27b)

˙xb,i = Abxb,i + Lbdyd,i + Bbui (5.27c)

˙xc,i = Acxc,i + Lcdyd,i + Lcbyb,i + Gc(wc,i + E−cax−a,i + E+

cax+a,i) + Bcui

(5.27d)


and for each j ∈ Ω , 1, · · · , r, there are:

˙xjd,i=Ajdxjd,i+Ljdyd,i+Gjd(wjd,i+ Ejxi) + Bjdui (5.27e)

Note that xd,i = col xjd,i, yd,i = col yjd,i, and wd,i = col wjd,i, ∀j ∈ Ω.Moreover, the outputs are given as

yjd,i = Cjdxjd,i, yd,i = Cdxd,i, yb,i = Cbxb,i (5.27f)

where Cd = diagCjd, ∀j ∈ Ω. Also, we have used the following notation

B = Γx stack B−a , B+a , Bb, Bc, Bd

where Bd = stack Bjd, ∀j ∈ Ω, where Bjd ∈ Rjqj×m. The dimensions ofvariables and the size and structure of the matrices as well as matrix parti-tioning conform with the scb stated in Appendix 5.A. Define the followingmatrices

As =

[A+

a 0GcE

+ca Ac

], Eds = stack [E+

ja, Ejc] (5.28)

• Step M-6

By an appropriate selection of the observer gain matrices K+a and Kc, make

the following matrix Hurwitz stable:

Ass = As − KsdEds where Ksd =

[K+

a

Kc

]Such Ksd exists under Assumption 5.1-(2). Find Pb = PT

b > 0 and Pjd =PTjd > 0 which solve the following algebraic Riccati equations:

AbPb + PbATb − 2τPbC

Tb CbPb = −Inb

AjdPjd + PjdATjd − 2τPjdC

TjdCjdPjd = −Ijqj

for every j ∈ Ω, and a chosen τ ∈ (0, β]. Then, define

Kb = PbCTb Kjd = PjdC

Tjd

The existence of such Pb and Pjd follows from the observability of the pairs(Cb, Ab) and (Cjd, Ajd). We point out that K+

a , Kb, Kc, and Kjd have thedimensions of n+

a × pd, nb × pb, nc × pd and jqj × qj , respectively.


• Step M-7

For every j ∈ Ω, define a matrix Sj ∈ Rjqj×jqj as

Sj(εj) = diagIqj , εjIqj , · · · , εj−2j Iqj , ε

j−1j Iqj (5.29)

where εj = εrj . Also, for every j ∈ Ω, define

Kjdε = ε−1j S−1

j Kjd, Kdε = diagKjdε (5.30)

• Step M-8

Form Kb ∈ Rn×pb and Kdε ∈ Rn×pd as below:

Kdε =

0

K+a G

Td Kdε

0KcG

Td Kdε

Kdε

, Kb =

00Kb

00

(5.31)

where Gd = diagGjd, ∀j ∈ Ω. Now, one may obtain Kobs(ε) using

Kobs(ε) = Γx

[Kdε Kb

]Γ−1

y (5.32)

Theorem 5.2 formalizes the result.

Theorem 5.2. Under Assumption 5.1 and for the set Gβ, the parameterizedprotocol (5.19), where Fcon(ε) is selected as (5.25) and Kobs(ε) is selectedas (5.32), solves Problem 5.1. Precisely, the following hold

(i) for any given β > 0, there exists an ε∗1 ∈ (0, 1] such that, for every ε ∈(0, ε∗1], synchronization is accomplished in the absence of disturbance;i.e. ∀ε ∈ (0, ε∗1] when w = 0

ei,j = yi − yj → 0, ∀i, j ∈ S as t→∞

(ii) for any given γ > 0, there exists an ε∗2 ∈ (0, ε∗1] such that, for everyε ∈ (0, ε∗2], the closed-loop transfer function from w to e satisfies

‖Twe(s)‖∞ < γ

Proof. See Appendix 5.C.

Remark 5.1. We draw the reader’s attention to this fact that ε∗1 dependson β > 0 or the set of network graphs Gβ. Obviously, ε∗2 < ε∗1 depends on γand indirectly depends on β > 0.

5.5. Other Possible Scalings 101

5.5 Other Possible Scalings

As it is clear from the works that have been provided in the precedingchapter and the current chapter, the scaling consists the heart of high-gaindesign. It shows how states in a chain of integrators should scaled so as tosuppress the effect of disturbances on the controlled output by squeezingthe parameter ε. However, the scaling matrix is not unique. In this section,another multiple-time scale assignment method is proposed to solve theproblem of H∞ almost synchronization. The design procedure is the sameas Section 5.4.4 except Step M-7 which is changed to:

• Step M-7∗

Let Ω = 1, · · · , r and Ω1 = 2, · · · , r. For every j ∈ Ω, define a matrixSj ∈ Rjqj×jqj as

Sj = diagε−(j−1)j Iqj , ε

−(j−2)j Iqj , · · · , Iqj, j ∈ Ω1;

Sj = ε−r Iqj , j = 1.(5.33)

where εj , εr

j−1 for j ∈ Ω1. Moreover, defineKjdε = ε−jj S−1

j Kjd, j ∈ Ω1;

Kjdε = ε−(r+1)Kjd, j = 1.

Accordingly, one may consider

Kdε = diagKjdε for j ∈ Ω (5.34)

The result is the same as before.

Theorem 5.3. Under Assumption 5.1 and for the set Gβ, the parameterizedprotocol (5.19), where Fcon(ε) is selected as in (5.25) and Kobs(ε) is selectedas in (5.32) considering Step M-7∗, solves Problem 5.1.

Proof. See Appendix 5.D.

The scaling which is introduced by (5.33) is termed as irregular scalingwhereas the scaling which is given by (5.29) is called regular scaling.


5.6 Simulation Results

A homogeneous network of four non-introspective agents is considered asdepicted in Fig. 5.2. Each agent is described by the following state-spacemodel:

xi =

−1 1 00 1 10 0 0

xi +

001

ui +

001

wi

yi =

[0 1 00 0 1

]xi

We intend to solve the problem of H∞ almost synchronization for this net-worked dynamical system.

Clearly, each agent is one-input two-output system which satisfies As-sumption 5.1. Notice that the system is already represented in the scb withrespect to either B or G, and both the scb are identical; each system isleft-invertible, minimum phase, and it has one infinite zero of order one.Consequently, it is straightforward to verify that Sb,ImG(A,G,C) is empty.Thus, Assumption 5.1-(4) and (5) hold. Since Im G ⊂ R∗KerC(A,B,C),Assumption 5.1-(3) also holds.

We assume that G ∈ Gβ with β = 3.5. Therefore, Theorem 5.2 en-sures that the problem is solvable using the controller (5.19). Accordingto Fig. 5.2, the eigenvalues of the Laplacian of the communication networkgraph are 0, 7, 4 ± j2.2361. Taking τ = 3.5, we found Kd = 0.378 andKb = 0.5469. We also select Fb = 5 and Fd = 20. Thus, Kobs(ε) andFcon(ε) are given by

Kobs(ε) =

0 00 Kb

1ε2Kd 0

, Fcon(ε) =

01εFdFb

1εFd

T

We assume that wi(t) = i + sin( i2 t + πi ). The result of the simulation is

shown in Fig. 5.3, where we have plotted ζk,i =∑4

j=1 gijyk,j for k = 1, 2.

4 5 5 3

2 2

3 4

1

Figure 5.2: The communication topology of the network.

5.7. Conclusion 103

ζ 2,i

Time(sec)

ζ 1,i

H∞ almost synchronization, ε = 0.03

0 5 10 15 20

0 5 10 15 20

−20

−10

0

10

20

−15

−10

−5

0

5

10

Figure 5.3: H∞ almost synchronization. The upper plot shows ζ1,i and thelower plot shows ζ2,i.

Clearly, ζk,i = 0 means yk,i = yk,j . In Fig. 5.4, a comparison betweenε = 0.01 and ε = 0.03 is presented. Reducing ε results in more accuratesynchronization.

Theorem 5.2 guaranteesH∞ almost synchronization for sufficiently smallε, which amplifies Fcon(ε) and Kobs(ε). In other words, Theorem 5.2 pro-vides lower bounds for Fcon(ε) and Kobs(ε). However, for practical purposeswhere measurements are corrupted with noise, these gains should be tunedappropriately in order to ensure synchronization by choosing them suffi-ciently high, and to achieve the best performance by limiting the magnitudeof the gains.

5.7 Conclusion

In this work, the problem of synchronization for multi-agent systems withidentical linear dynamics under directed communication structures and inthe presence of external disturbances is studied. Under a set of necessary


Time(sec)

ε = 0.01

ζ 2,i

Time(sec)

ε = 0.03

ε = 0.01

ζ 1,i

ε = 0.03

12 14 16 18 2012 14 16 18 20

12 14 16 18 2012 14 16 18 20

−0.05

0

0.05

−0.05

0

0.05

−0.05

0

0.05

−0.05

0

0.05

Figure 5.4: H∞ almost synchronization. A blow-up of the results for ε =0.01 and ε = 0.03. The upper plots show ζ1,i and the lower plots show ζ2,i.

and sufficient conditions, a family of dynamic protocols is proposed, whichguarantees any arbitrarily accurate synchronization in the sense of the H∞norm of the closed-loop transfer function from disturbance to the synchro-nization error.

It is worth noting that if H∞ almost disturbance decoupling is doablefor a single agent, H∞ almost synchronization is doable for a networkedmulti-agent system consisting of identical agents under a directional com-munication topology which has a directed spanning tree. However, thedesign must be robust with respect to a multiplicative uncertainty, whichcomes from the interaction topology. This robustification is carried out withthe aid of algebraic Riccati equations, an appropriate gain structure, andsufficiently small ε.

5.A. Special Coordinate Basis 105

Appendix 5.A Special Coordinate Basis

The protocol development relies extensively on a special coordinate basis(scb) - see e.g. (Saberi et al., 2012, Chapter 3)- originally proposed bySannuti and Saberi (1987). The scb is reviewed in Chapter 2. It is recalledhere for ease of reference. Consider a linear, time-invariant system as

Σ :

x = Ax + Bu

y = Cx

(5.35a)

(5.35b)

where A ∈ Rn×n, B ∈ Rn×m and C ∈ Rp×n. Also, x ∈ Rn is the state,u ∈ Rm is the control, and y ∈ Rp is the output. According to (Sannuti andSaberi, 1987), for any system Σ characterized by the matrix triple (A,B,C),there exist

(i) unique coordinate-free non-negative integers n−a , n+a , nb, nc, nd, 1 ≤

r ≤ n, and qj , j = 1, · · · , r.

(ii) nonsingular state, output and input transformations Γx, Γy, and Γu

such that x = Γxx, y = Γyy and u = Γuu. Partition

x = col x−a , x+a , xb, xc, xd, y = col yd, yb, u = col ud, uc

where the states x−a , x+a , xb, xc, xd have dimensions n−a , n+

a , nb, nc, andnd, respectively. Also,

ud, yd ∈ Rmd=pd uc ∈ Rmc yb ∈ Rpb

which implies p = pd + pb and m = md +mc. Moreover, xd, ud and yd arepartitioned as

xd = col xjd yd = col yjd ud = col ujd

for j = 1, · · · , r. Here, xjd ∈ Rjqj and ujd, yjd ∈ Rqj . for j = 1, · · · , r.Here, xjd ∈ Rjqj and ujd, yjd ∈ Rqj . Obviously,

∑rj=1 jqj = nd. For every

j = 1, · · · , r, define

Ajd =

[0 Iqj(j−1)

0 0

]∈ Rjqj×jqj , Bjd =

[0qj(j−1)

Iqj

], Cjd =

[Iqj 0qj(j−1)

]Clearly, for j = 1, we obtain A1d = 0, B1d = C1d = Iq1 .


The transformations take Σ into the scb described by the following setof equations:

x−a = A−a x−a + L−adyd + L−abyb (5.36a)

x+a = A+

a x+a + L+

adyd + L+abyb (5.36b)

xb = Abxb + Lbdyd (5.36c)

xc = Acxc + Lcdyd + Lcbyb +Bc(uc + E−cax−a + E+

cax+a ) (5.36d)

and for each j ∈ 1, · · · , r, there are:

xjd = Ajdxjd + Ljdyd +Bjd(ujd + Ejx) (5.36e)

where Ej ∈ Rqj×n and is appropriately partitioned as

Ej =[E−ja E+

ja Ejb Ejc Ejd], Ejd =

[Ej1 · · · Ejr

]where Ejk ∈ Rqj×kqk for k = 1, · · · , r such that Ejx = E−jax

−a + E+

jax+a +

Ejbxb + Ejcxc +∑r

k=1Ejk. The outputs are given by

yjd = Cjdxjd, yd = Cdxd, yb = Cbxb (5.36f)

where Cd = diagCjd for j = 1, · · · , r.I have kept the notation in this chapter the same as the notations in

Chapter 2. Three main properties of the scb is reviewed here. To recall allthe properties of the scb, the reader should consult Chapter 2.

The pairs (Cb, Ab) and (Cjd, Ajd) are observable. The system Σ is ob-servable (detectable) if and only if the pair (Cobs, Aobs) is observable (de-tectable), where

Cobs =[E−da E+

da Edc

], Aobs =

A−a 0 00 A+

a 0BcE

−ca BcE

+ca Ac

in which for j = 1, · · · , r

E−da = col E−ja, E+da = col E+

ja, Edc = col Ejc

Moreover, the pairs (Ac, Bc) and (Ajd, Bjd) are controllable. The system Σis controllable (stabilizable) if and only if the pair (Acon, Bcon) is controllable(stabilizable), where

Acon =

A−a 0 L−abCb

0 A+a L+

abCb

0 0 Ab

Bcon =

L−ad

L+ad

Lbd

5.B. Proof: Theorem 5.1 – SISO Case 107

The geometric subspaces can be expressed in terms of appropriate unions ofsubspaces that describe the scb of Σ. According to (Ozcetin et al., 1992),we have the following property which establishes a connection between scband the geometric subspaces.

Property 5.1. Suppose the state space is described by x−a ⊕x+a ⊕xb⊕xc⊕xd.

• x−a ⊕ xc ⊕ xd spans Vb,KerC;

• x+a ⊕ xc spans Sb,ImB.

Appendix 5.B Proof: Theorem 5.1 – SISO Case

The objective is to show that theH∞ norm of the transfer function from dis-turbance w to ζ is made arbitrarily small. In other words, the disturbanceis decoupled from ζ to any arbitrary degree of accuracy.

5.B.1 Estimation Error Dynamics for Agent ‘i’

Define the estimation error as xi = xi− xi, and find the dynamics accordingto (5.1) and (5.6). It gives rise to

˙xi = Axi + Gwi −Kobs(ε)(ζi − ζi) (5.37)

where ζi− ζi =∑N

j=1 gijCxj and Kobs(ε), which is given by (5.18), is foundusing the coordinates corresponding to the scb with respect to the triple(A,G,C). Thus, we introduce the following transformations:

xi = Γxxi, Cxi = Γyyd,i, wi = Γwwd,i

where xi = col x−a,i, x+a,i, xd,i; x−a,i ∈ Rn

−a , x+

a,i ∈ Rn+a , xd,i ∈ Rnd , and

n = n−a + n+a + nd. We define ζi − ζi = Γyζd,i where ζd,i =

∑Nj=1 gij yd,j .

Accordingly, we rewrite (5.37):

˙x−a,i = A−a x−a,i + L−adyd,i

˙x+a,i = A+

a x+a,i + L+

adyd,i − K+a G

Td Kdεζd,i

˙xd,i = Adxd,i + Gd(wd,i + E−dax−a,i + E+

dax+a,i + Eddxd,i)− Kdεζd,i

where we have used (5.18). Define

zs,i = x+a,i − K+

a GTd xd,i (5.38)


and consider the following scalings:

zsε,i = εzs,i, xdε,i = Sxd,i (5.39)

Recalling (5.14), the dynamics of the system are given by:

˙x−a,i = A−a x−a,i + εnd−1L−adCdxdε,i (5.40a)

˙zsε,i = Asszsε,i + εE−sax−a,i + εEsdεxdε,i + εGsswi (5.40b)

ε ˙xdε,i = Adxdε,i + εGd(E−dax−a,i + Eddεxdε,i) + GdE

+dazsε,i + εGdΓ−1

w wi − Kdζ∗d,i

(5.40c)

where ζ∗d,i =∑N

k=1 gikCdxdε,k, and

Esdε = (AssK+a G

Td + L+

adCd − K+a Edd)S−1

Gss = −K+a Γ−1

w

E−sa = −K+a E−da

Eddε = (Edd + E+daK

+a G

Td )S−1

It is easy to verify that ‖εEsdε‖ = O(ε), ‖εE−sa‖ = O(ε), ‖εGss‖ = O(ε),‖εGdE

−da‖ = O(ε), ‖εGdΓ−1

w ‖ = O(ε), and ‖εEddε‖ = O(ε).

5.B.2 Agent ‘i’ under Feedback

The control law is ui = −Fcon(ε)xi = −Fcon(ε)(xi − xi), where Fcon(ε)is selected as (5.11). State-feedback gain Fcon(ε) is calculated using thecoordinates corresponding to the scb with respect to the triple (A,B,C).Therefore, it makes sense to transform the equations into that coordinate.

In Step S-1, using (5.7), we found (5.8). Thus, all we need to do is toexpress ud,i in terms of the coordinates corresponding to (5.8). Representingxi in terms of the coordinates of the scb with respect to the triple (A,B,C),one obtains

xi = Γxˇxi (5.41)

where ˇxi = col ˇx−a,i, ˇx+a,i,

ˇxd,i. It clarifies that there exists a relationbetween estimation errors expressed in these two scb, which is given byˇxi = Γ−1

x Γxxi. That is, the components of one can be expressed as a linearcombination of the other’s components. According to Property 5.1, one canshow

• ˇx−a,i ⊕ ˇxd,i spans Vb,KerC(A,B,C);


• x+a,i spans Sb,ImG(A,G,C).

According to Assumptions 5.1-(4),(5), x+a,i ⊂ (ˇx−a,i ⊕ ˇxd,i) and x+

a,i ⊂ Ker C.

Computing the orthogonal complements, we acquire ˇx+a,i ⊂ (x−a,i ⊕ xd,i) and

(Ker C)⊥ ⊂ (x−a,i ⊕ xd,i). If we show ˇxd,i = col ˇxjd,i for j = 1, · · · , nd, we

can show that Cdˇxd,i = ˇx1d,i ∈ (Ker C)⊥. Therefore, one concludes:

• Assumptions 5.1-(4),(5) imply that ˇx1d,i and ˇx+a,i are expressed in

terms of x−a,i and xd,i.

• Assumption 5.1-(3) implies that G+a = 0 since Im G is expressed in

terms of x−a,i and xd,i.

Define the following variables

sd,i = F+a x

+a,i + yd,i, zd,i = CT

d F+a x

+a,i + xd,i

which implies that sd,i = Cdzd,i. Transforming the state-feedback gain(5.25) into the right coordinate, we obtain ud,i = −Fd(xi − ˇxi), which iswritten as

ud,i = −Fddxi − Fdεzd,i + u1,i + u2,i (5.42)

where u1,i = Fddˇxi and u2,i = FdεC

Td F

+a

ˇx+a,i+Fdε

ˇxd,i. From the assumptionson geometric subspaces, one can represent Bdu1,i as a linear combination ofthe components of xi; i.e., there exist M−da, Mds, and Mdd, constant matriceswith appropriate dimensions and independent of ε, such that

Bdu1,i = M−dax−a,i + ε−1Mdszsε,i +Mddεxdε,i

where we have used (5.38) and (5.39), and Mddε = (Mdd +MdsK+a G

Td )S−1.

Partition Fd = [f1, · · · , fnd] where fk ∈ R for k = 1, · · · , nd. Considering

(5.9), one can show that Bdu2,i is equal to

Bdu2,i = ε−ndBd

(f1F

+a

ˇx+a,i + f1

ˇx1d,i

)+ ε−ndBd

(ε∑nd

k=2εk−2fk ˇxkd,i

)(5.43)

According to Assumptions 5.1-(4),(5), the first line of (5.43) depends onlyon x−a,i and xd,i. Therefore, there exist constant matrices N−da and Ndd withappropriate dimensions and independent of ε such that

Bd

(f1F

+a

ˇx+a,i + f1

ˇx1d,i

)= N−dax

−a,i +Nddεxdε,i


where Nddε = NddS−1. Similarly, each ˇxkd,i for k = 2, · · · , nd can be

expressed as a linear combination of components of xi. Thus, for constantmatrices M−ka, Mks and Mkd, k = 2, · · · , nd, we get

εBd

nd∑k=2

εk−2fk ˇxkd,i = ε

nd∑k=2

εk−2(M−kax−a,i +

1

εMkszsε,i + MkdS

−1xdε,i)

, εM−dax−a,i + Mdszsε,i + εMddxdε,i

where

M−da =

nd∑k=2

εk−2M−ka, Mds =

nd∑k=2

εk−2Mks, Mdd =

nd∑k=2

εk−2MkdS−1

We introduce the following scalings:

x−aε,i = εx−a,i, zdε,i = Szd,i (5.44)

Recalling Step S-1 and Step S-2, write the closed-loop equations:

x−aε,i = A−a x−aε,i − εL−adF

+a x

+a,i + εL−adCdzdε,i + εG−a wi

x+a,i = Assx

+a,i + L+

adCdzdε,i

εzdε,i = A∗dzdε,i + εSL∗ddCdzdε,i + εSL+dax

+a,i + εSGdwi

+N−dax−a,i +Nddεxdε,i +Hdszsε,i + εH−dax

−a,i + εHddxdε,i

where

L∗dd = CTd F

+a L

+ad, Hds = Mds + εnd−1Mds

L+da = CT

d F+a Ass, H−da = M−da + εnd−1M−da

Hdd = Mdd + εnd−1Mddε

It can be confirmed that the norms of εL−adF+a , εL−adCd, εG−a , εSL∗dd, εSL+

da,εSGd, εH−da, and εHdd are of order ε.

5.B.3 Closed-loop Dynamics of Agent ‘i’: Compact Form

Define

zi = col x−a,i, xdε,i, zsε,i, zi = col x+a,i, x

−aε,i, zdε,i

A = diagA−a , Ad, Ass, A = diagAss, A−a , A

∗d

S = diagIn−a , εInd, In+

a, S = diagIn+

a, In−a , εInd


The closed-loop equations are given in the compact form by:

S zi = A zi + L zi + ε E wi +D zi (5.45a)

S ˙zi = A zi + L zi + ε E wi −∑N

k=1gikD zk (5.45b)

where L = εLε + L0, ‖εLε‖ = O(ε) and L = εLε + L0, ‖εLε‖ = O(ε).

Lε =

0 0 0−L−adF

+a 0 L−adCd

SL+da 0 SL∗ddCd

, L0 =

0 0 L+adCd

0 0 00 0 0

Lε =

0 0 0GdE

−da GdEddε 0

E−sa Esdε 0

, E =

0GdΓ−1

w

Gss

L0 =

0 εnd−1L−adCd 00 0 GdE

+da

0 0 0

, E =

0G−aSGd

In (5.45), D = εDε +D0 in which ‖εDε‖ = O(ε) and

Dε =

0 0 00 0 0H−da Hdd 0

, D0 =

0 0 00 0 0N−da Nddε Hds

D =

0 0 00 KdCd 00 0 0

5.B.4 Collective Dynamics for the Multi-agent System

For i ∈ S, collect the states as χ = col zi and χ = col zi. Then, thecollective dynamics are described by

(IN ⊗ S)χ = (IN ⊗A)χ+ (IN ⊗ L)χ+ ε(IN ⊗ E)w + (IN ⊗D)χ (5.46a)

(IN ⊗ S) ˙χ =((IN ⊗ A)− (G⊗ D)

)χ+ (IN ⊗ L)χ+ ε (IN ⊗ E)w (5.46b)

Notice yi = Cxi = ΓyCdxd,i and yd,i = Cdzdε,i−F+a x

+a,i. Then, there exists

a matrix Γ∗x, independent of ε, such that yd,i = Γ∗xzi. Thus,

ζ = (G ⊗ ΓyCdΓ∗x)χ (5.46c)


Let 1 ∈ RN be the right eigenvector of G associated with the eigenvalue atzero. Let 1T

L represent its left eigenvector. Suppose the Jordan form of G isobtained using the matrix U which is chosen as

U =[U 1

]⇒ U−1 =

[UTL

1TL

]Thus, one can find U−1GU = diag∆, 0. It implies that GU = [G, 0] whereG = U∆. We introduce the following state transformations[

ee0

]= (U−1 ⊗ In)χ,

[ee0

]= (U−1 ⊗ In) χ (5.47)

Denote N = N − 1. Then, we find two sets of equations.

(IN ⊗ S)e = (IN ⊗ (A+ L0)) e+ ε(IN ⊗ Lε)e+ (IN ⊗D)e+ ε (UTL ⊗ E)w

(5.48a)

(IN ⊗ S) ˙e =(IN ⊗ (A+ L0)−∆⊗ D

)+ ε(IN ⊗ Lε)e+ ε (UT

L ⊗ E)w(5.48b)

ζ = (G ⊗ ΓyCdΓ∗x)e (5.48c)

The remaining is given by

S e0 = (A+ L) e0 +D e0 + ε (1TL ⊗ E)w (5.49a)

S ˙e0 = (A+ L) e0 + ε (1TL ⊗ E)w (5.49b)

When disagreement among agents disappear, the state (e0, e0) determinesthe agreement trajectories, which may happen if when w = 0 or w is de-coupled from the disagreement dynamics.

5.B.5 H∞ Analysis

The aim is to establish the disturbance decoupling property of the proposedprotocols on the synchronization error dynamics. Take the reduced-ordersystem (5.48) into account, and let ζ be the controlled output.

Find % > 0 that ζTζ ≤ %2 eTe holds.The matrix A is Hurwitz stable because A−a , Ass, and A∗d are Hurwitz

stable. Due to the upper-triangular structure of L0, the matrix (IN ⊗ (A+ L0))is upper triangular and Hurwitz stable. It implies that there exists a sym-metric Q > (%2 + 4)In such that the Lyapunov equation:

(A+ L0)TP + P(A+ L0) = −Q


has a unique solution P = PT > 0 which is block-diagonal with the blocksizes that correspond to the block sizes in S. It guarantees that P and Scommute. Let Vc = eT(IN ⊗ SP)e. Taking derivative gives rise to

Vc ≤ −ζTζ − 3‖e‖2 − (1− 2εµε)‖e‖2+ 2µdε‖e‖‖e‖+ 2εµw‖e‖‖w‖

where µw = ‖(UTL ⊗ PE)‖

µε ≥ maxε∈(0,1]

‖PLε‖, µdε ≥ maxε∈(0,1]

‖PD‖,

Since µε is bounded, there exists a sufficiently small ε11 ∈ (0, 1] such that1− 2εµε > 0 for every ε ∈ (0, ε11]; thus,

Vc ≤ −ζTζ − 3‖e‖2+ 2µdε‖e‖‖e‖+ 2εµw‖e‖‖w‖

In (5.48b), let A∗ = IN ⊗ (A + L0) − ∆ ⊗ D. From the structure of ∆,it is observed that A∗ is upper triangular. Thus, A∗ is Hurwitz stable ifand only if all matrices on the main diagonal are Hurwitz stable. In otherwords, A + L0 − λD must be Hurwitz stable for all λ’s which are nonzeroeigenvalues of the Laplacian G. Since A − λD is a block-diagonal matrixand L0 is upper triangular, the eigenvalues of A∗ are determined by theeigenvalues of A−a , Ass, and A∗d = Ad − λKdCd. The matrix A−a is Hurwitzstable by definition, and Ass was made Hurwitz stable in Step S-6. It canbe confirmed that A∗d is Hurwitz stable. To see that we recall Kd = PdC

Td

and show

(A∗d)Pd + Pd(A∗d)H = AdPd + PdATd − 2ReλPdC

Td CdPd

= AdPd + PdATd − 2τPdC

Td CdPd

− 2(Reλ − τ)PdCTd CdPd ≤ −Inb

Notice since G ∈ Gβ, Reλ > β ≥ τ > 0 if λ 6= 0. It follows that A∗d isHurwitz stable. Hence, A+ L0 − λD and accordingly A∗ is Hurwitz stablefor β ≥ τ . Let λN = 0 and λi ∈ C+ for i ∈ 1, · · · , N − 1. It is easy toshow that, for every i ∈ 1, · · · , N − 1, there exists a Qi = QT

i > 0 suchthat the Lyapunov equation

(A+ L0 − λiD)HPi + Pi(A+ L0 − λiD) = −Qi

has a unique solution Pi = PTi > 0 which is block-diagonal with the block

sizes that correspond to the block sizes in S. It is possible to choose one


identical P for all eigenvalues (i.e. one may choose Pi = P−1d ). In addition,

one may show5 that the block-diagonal matrix P which is constructed as

P = diagδ1P1, · · · , δN−1PN−1 (5.50)

where δi = 1 i = N − 1

δi = δi+1qiqi+1

9η2i

i = 1, · · · , N − 2

(5.51)

(5.52)

in which qi = ‖Qi‖ and ηi = ‖PiD‖, has a bounded norm (i.e. ‖P‖ isbounded) for any β > 0. Such a P commutes with (IN ⊗ S). It is straight-forward to see that P solves the Lyapunov function

(A∗)HP + PA∗ = −Q (5.53)

for a symmetric Q > (3 + µ2dε)INn. Choose

Vo = eT(IN ⊗ S)Pe

taking derivative yields

Vo ≤ −(2 + µ2dε)‖e‖2 − (1− 2ερε)‖e‖2 + 2ερw‖e‖‖w‖

where

ρε ≥ maxε∈(0,1]

‖P(IN ⊗ Lε)‖, ρw ≥ ‖P(UTL ⊗ E)‖

Due to boundedness of P for any β > 0, ρε and ρw are bounded for anyβ > 0. As a consequence, one may find an ε22 ∈ (0, 1] such that 1−2ερε > 0for every ε ∈ (0, ε22]; thus, for every ε ∈ (0, ε22], we obtain

Vo ≤ −(2 + µ2dε)‖e‖2 + 2ρw‖e‖‖w‖

Choose V = Vc + Vo. Let ε∗1 = minε11, ε22. For every ε ∈ (0, ε∗1], an upperbound on V is given by

V ≤ −ζTζ − ‖e‖2 − ‖e‖2 + (εσw)2‖w‖2

where σw ≥√

2 maxµw, ρw. Hence, it follows that

5See Subsection 5.C.5 for a complete discussion and proof.

5.C. Proof: Theorem 5.2 – MIMO Case 115

• the origin is globally exponentially stable in the absence of distur-bance;

• in the presence of disturbance, ‖Twζ‖∞ ≤ εσw; see Section 2.4.

Put alternatively, the impact of w on ζ can be made arbitrarily small inthe sense of the H∞ norm of the transfer function.

To reach the objective, it is required to show that the influence of dis-turbance on every mutual disagreement, ei,j , can be made small to any

arbitrary degree. Define ei,j = T i,jwe (s)w.Then, from Lemma 5.1, it follows that if ‖Twζ‖∞ ≤ εσw, there exists

σ > 0 such that ‖T i,jwe ‖∞ < εσ. Therefore, for any given γ > 0, there existsan ε∗2 ∈ (0, ε∗1] such that every ε ∈ (0, ε∗2] yields ‖T i,jwe ‖∞ < γ. Thus, anydesired accuracy of synchronization can be obtained by reducing ε in therange (0, ε∗1].

Lemma 5.1 (Peymani et al. (2012)). Let ‖Twζ‖∞ ≤ εσw. There exists a

positive constant σ such that ‖T i,jwe ‖∞ ≤ εσ.

Proof. See Chapter 4.

Appendix 5.C Proof: Theorem 5.2 – MIMO Case

First, we show that the H∞ norm of the transfer function from disturbancew to ζ can be made arbitrarily small. Then, we demonstrate that the H∞norm of the transfer function from disturbance to all synchronization errors,as defined in (5.4), can be made arbitrarily small, too.

5.C.1 Estimation Error Dynamics for Agent ‘i’

We start the proof by finding the estimation error dynamics for agent i ∈ S.Define the estimation error as xi = xi− xi, and find the dynamics accordingto (5.1) and (5.19). It gives rise to

˙xi = Axi + Gwi −Kobs(ε)(ζi − ζi) (5.54)

where ζi− ζi =∑N

j=1 gijCxj and Kobs(ε), which is given by (5.32), is foundusing the coordinates corresponding to the scb with respect to the triple(A,G,C). Thus, using the transformation matrices found in Step 5 (seeEq. (5.26)), we rewrite (5.54) in terms of the variables:

xi = Γxxi, Cxi = Γy

[yd,i

yb,i

], wi = Γw

[wd,i

wc,i

]


where xi = col x−a,i, x+a,i, xb,i, xc,i, xd,i in which xd,i = col xjd,i, yd,i =

col yjd,i, and wd,i = col wjd,i for all j ∈ Ω = 1, · · · , r. The dimensionsconform with Appendix 5.A, but we place bars on the variables; for example,xjd,i ∈ Rjqj and wjd,i, yjd,i ∈ Rqj . It is observed that

ζb,i =N∑j=1

gij yb,j , ζjd,i =N∑k=1

gikyjd,k

and ζd,i = col ζjd,i for all j ∈ Ω. Then, in view of the scb given by (5.27)along with (5.30) and (5.31), one can write

˙x−a,i = A−a x−a,i + L−adyd,i + L−abyb,i

˙x+a,i = A+

a x+a,i + L+

adyd,i + L+abyb,i − K+

a GTd Kdεζd,i

˙xb,i = Abxb,i + Lbdyd,i − Kbζb,i

˙xc,i = Acxc,i + Lcdyd,i + Lcbyb,i + Gc(wc,i + E−cax−a,i + E+

cax+a,i)− KcG

Td Kdεζd,i

˙xjd,i = Ajdxjd,i + Ljdyd,i + Gjd(wjd,i + Ejxi)− Kjdεζjd,i

yjd,i = Cjdxjd,i, yd,i = Cdxd,i

yb,i = Cbxb,i

The structure of matrices follows from the scb explained in Appendix 5.A.For all j ∈ Ω, define

E−da = stack E−ja, Edb = stack Ejb, Ldd = stack Ljd

Lsb =

[L+

ab

Lcb

], Lsd =

[L+

ad

Lcd

], Gs =

[0Gc

]Define xs,i = col x+

a,i, xc,i, and

zs,i = xs,i − KsdGTd xd,i (5.55)

where Gd = diagGjd for all j ∈ Ω. In view of Step 6, one may find

˙zs,i = Asszs,i + E−sax−a,i + Esbxb,i + Esdxd,i + Gsswi

where Gss = [−Ksd, Gs]Γ−1w and

E−sa = GsE−ca − KsdE

−da

Esb = LsbCb − KsdEdb

Esd = AssKsdGTd + LsdCd − KsdG

Td LddCd − KsdEdd


Recalling the scaling matrix Sj , in (5.29), we define S = diagSj for all

j ∈ Ω. Considering the matrix Esd, one may demonstrate that εrEsdS−1 =

εEsdε, indicating that ‖εEsdε‖ = O(ε). To show that, we partition Esd as

Esd = [Es1, Es2 , · · · , Esr] where Esk ∈ R(n+a +nc)×kqk ; therefore, it is obtained

εrEsdS−1 = εr

[Es1S

−11 · · · EsrS

−1r

]Since ‖εrEskS

−1k ‖ = O(εrε

−(k−1)k ) = O(εk), one may write εrEskS

−1k = εEskε

for some appropriate Eskε which is uniformly bounded for all ε ∈ (0, 1].Therefore, Esdε = [Es1ε, Es2ε, · · · , Esrε]. Denote E∗jd = EjsKsdG

Td + Ejd.

Then, for every j ∈ Ω, one may show that

(εjSjLjdCd + εrGjdE∗jd)S−1 = εEjdε

for some Ejdε which is uniformly bounded for all ε ∈ (0, 1], and ‖εEjdε‖ =

O(ε). Notice that ‖εjSjLjdCdS−1‖ = O(εj).

Consider the following state transformations

zsε,i = εrzs,i, xjdε,i = Sj xjd,i (5.56)

and define xdε,i = col xjdε,i for all j ∈ Ω. Let Ejs = [E+ja , Ejc]. Conse-

quently, the dynamics of the observation error system are given by:

˙x−a,i = A−a x−a,i + L−adCdxdε,i + L−abyb,i (5.57a)

˙xb,i = Abxb,i + LbdCdxdε,i − Kbζb,i (5.57b)

˙zsε,i = Asszsε,i + εrE−sax−a,i + εrEsbxb,i + εEsdεxdε,i + εrGsswi (5.57c)

S ˙xdε,i = Addxdε,i + ε Eddεxdε,i + εr E−dax−a,i + εr Edbxb,i

+ Edszsε,i + εr Gddwi − Kddζd,i

(5.57d)

where we have used the following notations, for all j ∈ Ω,

Add = diagAjd Eddε = stack EjdεE−da = GdE

−da Edb = GdEdb

Eds = Gd stack Ejs Gdd = [Gd, 0]Γ−1w

S = diagεjIjqj Kdd = diagKjd

Note that ζd,i =∑N

k=1 gikCdxdε,k.


5.C.2 Agent ‘i’ under Feedback

We obtain the dynamics of agent i, (5.1a), under the feedback (5.19b).Clearly, ui = −Fcon(ε)xi = −Fcon(ε)(xi − xi). Indeed, we obtain

xi = (A−BFcon(ε))xi + BFcon(ε)xi + Gwi

= (A−BFcon(ε))xi + BΓu

[Fd

Fc

]Γ−1

x xi + Gwi

State-feedback gain Fcon(ε) is calculated using the coordinates correspond-ing to the scb with respect to the triple (A,B,C). Therefore, it makes senseto transform the equations into that coordinate. In Step 1, using (5.20), wefound (5.21). Thus, we intend to express uc,i and ud,i in terms of the coor-dinates corresponding to the scb with respect to the triple (A,B,C), andapply them to system equations (5.21).

Representing xi in terms of the coordinates of the scb with respect tothe triple (A,B,C), one obtains

xi = Γxˇxi (5.58)

where ˇxi = col ˇx−a,i, ˇx+a,i,

ˇxb,i, ˇxc,i, ˇxd,i in which ˇxd,i = col ˇxjd,i, ∀j ∈Ω = 1, · · · , r, where ˇxjd,i ∈ Rjqj . It clarifies that there exists a relationbetween estimation errors expressed in these two scb, which is given by

ˇxi = Γ−1x Γxxi

That is, the components of one can be expressed as a linear combination ofthe other’s components. According to Property 5.1, one can show

• ˇx−a,i ⊕ ˇxc,i ⊕ ˇxd,i spans Vb,KerC(A,B,C);

• x+a,i ⊕ xc,i spans Sb,ImG(A,G,C).

According to Assumptions 5.1-(4),(5), one may obtain

(x+a,i ⊕ xc,i) ⊂ (ˇx−a,i ⊕ ˇxc,i ⊕ ˇxd,i)

(x+a,i ⊕ xc,i) ⊂ Ker C

Computing the orthogonal complements of the above sub-spaces gives riseto

(ˇx+a,i ⊕ ˇxb,i) ⊂ (x−a,i ⊕ xb,i ⊕ xd,i)

(Ker C)⊥ ⊂ (x−a,i ⊕ xb,i ⊕ xd,i)

It is observed that Cjd ˇxjd,i = ˇxj1d,i ∈ (Ker C)⊥ for all j ∈ Ω. Therefore, forall j ∈ Ω and every i ∈ S, one concludes that


• Assumptions 5.1-(4),(5) imply that ˇxj1d,i, ˇx+a,i and ˇxb,i are expressed

in terms of x−a,i, xb,i and xd,i.

• Assumption 5.1-(3) implies that G+a = Gb = 0 since Im G is spanned

by x−a,i, xc,i, and xd,i.

Before finding uc,i and ud,i, partition F+a and Fb such that

F+a = stack F+

ja Fb = stack Fjb

for all j ∈ Ω, where F+ja ∈ Rqj×n

+a and Fjb ∈ Rqj×nb . Define Fjs = [F+

ja, Fjb]

and Fs = stack Fjs for all j ∈ Ω. Denote xs,i = col x+a,i, xb,i. Define the

following variables

sjd,i = Fjsxs,i + yjd,i, zjd,i = CTjdFjsxs,i + xjd,i (5.59)

implying that sjd,i = Cjdzjd,i. Let sd,i = col sjd,i and zd,i = col zjd,i forall j ∈ Ω. It implies that sd,i = Fsxs,i + yd,i. Therefore, in view of (5.22),one obtains

xs,i = Assxs,i + Lsdsd,i (5.60)

In light of the geometric assumptions and the state-feedback gain (5.25), weobtain

uc,i = −Fc(xc,i − ˇxc,i) (5.61a)

ujd,i = −Ejxi − Fjdεzjd,i + uj1,i + uj2,i (5.61b)

where

uj2,i = Fjdε(CTjdF

+ja

ˇx+a,i + CT

jdFjb ˇxb,i + ˇxjd,i)

uj1,i = Ej ˇxi = E−jaˇx−a,i + E+

jaˇx+

a,i + Ejb ˇxb,i + Ejc ˇxc,i + Ejd ˇxkd,i

From the assumptions on the geometric subspaces, one can representBcFcˇxc,i

as a linear combination of the components of xi; i.e.

BcFcˇxc,i = M−cax

−a,i +Mcbxb,i +Mcszs,i +Mcdxd,i

where zs,i is defined in (5.55), and M−ca, Mcb, Mcs, and Mcd are some con-stant matrices independent of ε. In view of the scaling (5.56), BcFc

ˇxc,i ismodified to


−a,i +Mcbxb,i + ε−rMcszsε,i + ε−(r−1)Mcdεxdε,i


where we have used the fact that McdS−1 = ε−(r−1)Mcdε since ‖McdS

−1‖ =O(ε−(r−1)); Mcdε is uniformly bounded for all ε ∈ (0, 1].

Likewise, there exist some constant matrices M−ja, Mjb, Mjs, and Mjd

for j ∈ Ω that MjdS−1 = ε−(r−1)Mjdε, where Mjdε is uniformly bounded for

all ε ∈ (0, 1], such that

Bjduj1,i = M−jax−a,i +Mjbxb,i + ε−rMjszsε,i + ε−(r−1)Mjdεxdε,i

where ‖ε−rMjs‖ = O(ε−r) and ‖Mjkdε‖ = O(εk). We also need to expressBjduj2,i in terms of xi to be able to close the loop around agent i. For everyj ∈ Ω, partition Fjd = [Fj1d, · · · , Fjjd] where Fjkd ∈ Rqj×qj for k = 1, · · · , j.Then, one can show

Bjduj2,i = ε−jBjdFjd

(CTjdF

+ja

ˇx+a,i + CT

jdFjb ˇxb,i

)+ ε−jBjd

(Fj1d

ˇxj1d,i +

j∑k=2

Fjkdεk−1 ˇxjkd,i

)(5.62)

where we have used the fact that ˇxjd,i = col ˇxjkd,i, for all k = 1, · · · , jand every j ∈ Ω, where ˇxjkd,i ∈ Rqj . According to Assumptions 5.1-(4),(5),the term

ε−jBjdFjd

(CTjdF

+ja

ˇx+a,i + CT

jdFjb ˇxb,i

)+ ε−jBjdFj1d

ˇxj1d,i

depends only on x−a,i, xb,i and xd,i. Therefore, there exist some constant

matrices N−ja, Njb, and Njd for j ∈ Ω, independent of ε such that

ε−jBjdFjd

(CTjdF

+ja

ˇx+a,i + CT

jdFjb ˇxb,i

)+ ε−jBjdFj1d

ˇxj1d,i

= ε−j(N−jax−a,i +Njbxb,i + ε−(r−1)Njdεxdε,i)

where NjdS−1 = ε−(r−1)Njdε, where Njdε is uniformly bounded for all ε ∈

(0, 1]. Similarly, ˇxjkd,i for k = 2, · · · , j and every j ∈ Ω can be expressed asa linear combination of components of xi. Thus, we get

Bjd

j∑k=2

Fjkdεk−1 ˇxjkd,i =

ε

j∑k=2

εk−2(M−jkax−a,i + Mjkbxb,i + ε−rMjkszsε,i + ε−(r−1)Mjkxdε,i)


It is straightforward to verify that Mjk, k = 2, · · · , j and j ∈ Ω, is uniformlybounded for all ε ∈ (0, 1]. Denote

Mjdε =∑j

k=2εk−2Mjk, Mjs =

∑j

k=2εk−2Mjks

M−ja =∑j

k=2εk−2M−jka, Mjb =

∑j

k=2εk−2Mjkb

Therefore,

Bjd

j∑k=2

Fjkdεk−1 ˇxjkd,i =

εM−jax−a,i + εMjbxb,i + ε−(r−1)Mjszsε,i + ε−(r−2)Mjdεxdε,i

Let us find the closed-loop equations. Considering (5.23), introduce thestate transformations:

x−aε,i = εx−a,i, xcε,i = εxc,i, zjdε,i = Sjzjd,i (5.63)

Denote zdε,i = col zjdε,i ∀j ∈ Ω. In light of Step M-2, one may demonstratethe dynamics of the systems as

x−aε,i = A−a x−aε,i + εL−asxs,i + εL−adCdzdε,i + εG−a wi

xs,i = Assxs,i + LsdCdzdε,i

xcε,i = Accxcε,i +BcE−cax−aε,i + εEcsxs,i + εLcdCdzdε,i + εGcwi

+ εM−cax−a,i + εMcbxb,i + ε−(r−1)(Mcszsε,i + εMcdεxdε,i)

εzdε,i = Addzdε,i + εLddzdε,i + εLdsxs,i + εGddwi

+ εM−dax−a,i + εMdbxb,i + ε−(r−1)Mdszsε,i + ε−(r−2)Mddxdε,i

+ ε−(r−1)Nddxdε,i +N−dax−a,i +Ndbxb,i

where L−as = −L−adFs + [0, L−abCb] and Ecs = −LcdFs + [BcE+ca, LcbCb].

Also, for j ∈ Ω, we have defined

L∗jd = CTjdFjsLsd + Ljd Ljs = CT

jdFjsAss − LjdFs

Add = diagA∗jd Gdd = stack SjGjdLdd = stack SjL∗jdCd Lds = stack SjLjsM−da = stack εj−1M−ja + M−jaMdb = stack εj−1Mjb + Mjb N−da = stack N−jaMds = stack εj−1Mjs + Mjs Ndb = stack NjbMdd = stack εj−1Mjdε + Mjdε Ndd = stack Njdε


5.C.3 Closed-loop Equations for Agent i: Compact Form

Now, the closed-loop equations under the proposed protocol are representedin a compact form. Define

zi =

x−a,ixb,i

xdε,i

zsε,i

, zi =

xcε,i

xs,i

x−aε,izdε,i

Then, the closed-loop equations are given by

S zi = A zi + L zi + ε E wi +D zi (5.64a)

S ˙zi = A zi + L zi + εrE wi −∑N

k=1gikD zk (5.64b)

where

A = diagA−a , Ab, Add, Ass, S = diagI, I, S, IA = diagAcc, Ass, A

−a , Add, S = diagI, I, I, εI

In S and S, we have dropped the dimension of I’s. Also, L = εLε + L0,where ‖εLε‖ = O(ε) and

Lε =

0 Ecs 0 LcdCd

0 0 0 00 L−as 0 L−adCd

0 Lds 0 Ldd

L0 =

0 0 BcE

−ca 0

0 0 0 LsdCd

0 0 0 00 0 0 0

Moreover, L = εLε + L0 in which ‖εLε‖ = O(ε) and

Lε =

0 0 0 00 0 0 0

εr−1E−da εr−1Edb Eddε 0εr−1E−sa εr−1Esb Esdε 0

, E =

00

Gdd

Gss

L0 =

0 L−abCb L−adCd 00 0 LbdCd 00 0 0 Eds

0 0 0 0

, E =

Gc

0G−aGdd

In (5.64), D = Dε + ε−(r−1)D0 in which

Dε =

εM−ca εMcb ε−(r−2)Mcdε 0

0 0 0 00 0 0 0

εM−da +N−da εMdb +Ndb ε−(r−2)Mdd 0


D =

0 0 0 00 KbCb 0 00 0 KddCd 00 0 0 0

, D0 =

0 0 0 Mcs

0 0 0 00 0 0 00 0 Ndd Mds

5.C.4 Closed-loop Equations for the Multi-agent System

Here, we find the closed-loop equations for the multi-agent system (thecollective closed-loop dynamics); then, unobservable modes are removedfrom ζ to find the reduced-order dynamics.Collect the states as

χ = col zi, χ = col zi, ∀i ∈ S

Then, the collective dynamics are described by

(IN ⊗ S)χ = (IN ⊗A)χ+ (IN ⊗ L)χ+ ε(IN ⊗ E)w + (IN ⊗D)χ (5.65a)

(IN ⊗ S) ˙χ =((IN ⊗ A)− (G⊗ D)

)χ+ (IN ⊗ L)χ+ εr(IN ⊗ E)w (5.65b)

Recall yd,i = col yjd,i for all j ∈ Ω and every agent i ∈ S. Based on (5.59)and (5.63), we obtain

yjd,i = Cjdzjd,i − Fjsxs,i

= Cjdzjdε,i − Fjsxs,i

In addition, yb,i = [0, Cb]xs,i. It implies that there exists a matrix Γ∗y,independent of ε, such that yi = ΓyΓ∗yzi. Therefore, in view of (5.2),

ζ = (G ⊗ ΓyΓ∗y)χ (5.65c)

Let 1 ∈ RN be the right eigenvector of G associated with the eigenvalue atzero6. Let 1L represent its left eigenvector. Suppose the Jordan form of Gis obtained using the matrix U which is chosen as

U =[U 1

]⇒ U−1 =

[UTL

1TL

]It implies that UT

L U = IN−1, UTL 1 = 1T

LU = 0, and 1TL1 = 1. Thus, one can

find

U−1GU =

[∆ 00 0

], GU =

[G 0

](5.66)

6Without loss of generality, assume that λN = 0 and λi ∈ C+ for i = 1, · · · , N − 1.


where G = U∆. Since G ∈ Gβ, λ(∆) ∈ C+. We introduce the following statetransformations[

ee0

]= (U−1 ⊗ In)χ,

[ee0

]= (U−1 ⊗ In) χ (5.67)

Denote N = N − 1. Then, we find two sets of equations. The first set isgiven as bellow.

(IN ⊗ S)e = (IN ⊗ (A+ L0)) e+ ε(IN ⊗ Lε)e+ ε−(r−1)

(IN ⊗ (εr−1Dε +D0)

)e+ ε(UT

L ⊗ E)w (5.68a)

(IN ⊗ S) ˙e =(IN ⊗ (A+ L0)−∆⊗ D

)e+ ε(IN ⊗ Lε)e+ εr(UT

L ⊗ E)w(5.68b)

ζ = (G ⊗ ΓyΓ∗y)e (5.68c)

The remaining is given by

S e0 = (A+ L) e0 +D e0 + ε(1TL ⊗ E)w (5.69a)

S ˙e0 = (A+ L) e0 + εr(1TL ⊗ E)w (5.69b)

The state (e0, e0) determines the agreement trajectories when ζ = 0. Hence,the objective in synchronization is to make the origin of the system (5.68)globally asymptotically stable in the absence of disturbance, and to reducethe effect ofw on ζ in the presence of external disturbances. In the following,we show that, using the proposed protocol, we can suppress the impactof disturbances on the synchronization error and we obtain any desiredaccuracy of synchronization.

5.C.5 H∞ Analysis

In this subsection, disturbance decoupling property of the proposed pro-tocols on the synchronization error dynamics is studied. To that end, weconsider the reduced-order system (5.68) with the controlled output ζ, whichis perturbed by external disturbances w.

Choose % > 0 such that ζTζ ≤ %2 eTe for all ε ∈ (0, 1].The matrix A is Hurwitz stable because A−a , Ass, Acc, and A∗jd, ∀j ∈ Ω,are Hurwitz stable. Due to the upper-triangular structure of L0, the matrix(IN ⊗ (A+ L0)) is upper triangular and Hurwitz stable. Therefore, thereexists a symmetric Q > (%2 + 4)In such that the Lyapunov equation:

(A+ L0)TP + P(A+ L0) = −Q


has a unique positive definite and symmetric solution P which is block-diagonal with the block sizes that correspond to the block sizes in S7. Itguarantees that P and S commute. We choose the function

Vc = eT(IN ⊗ SP)e

Differentiation along the trajectories of (5.68a) gives rise to

Vc ≤ −ζTζ − 3‖e‖2 − (1− 2εµε)‖e‖2

+ 2ε−(r−1)µdε‖e‖‖e‖+ 2εµw‖e‖‖w‖

where


‖PLε‖ , µdε ≥ maxε∈(0,1]

‖P(εr−1Dε +D0)‖

µw ≥ ‖(UTL ⊗ PE)‖


Vc ≤ −ζTζ − 3‖e‖2

+ 2ε−(r−1)µdε‖e‖‖e‖+ 2εµw‖e‖‖w‖

In (5.74b), let A∗ =(IN ⊗ (A+ L0)−∆⊗ D

). From the structure of ∆,

it is observed that A∗ is upper triangular. Thus, A∗ is Hurwitz stableiff all matrices on the main diagonal are Hurwitz stable. In other words,A+L0−λD must be Hurwitz stable for all λ’s which are nonzero eigenvaluesof the Laplacian matrix G. Notice that since G ∈ Gβ, Reλ > β > 0 ifλ 6= 0. Since A−λD is a block-diagonal matrix and L0 is upper-triangular,the eigenvalues of A∗ are determined by the eigenvalues of A−a , Ass, Abb =Ab − λKbCb, and A∗jd = Ajd − λKjdCjd.

A−a is Hurwitz stable by definition, and Ass was made Hurwitz stable inStep M-6. According to Step M-6, it can be confirmed that Abb and Ajd,∀j ∈ Ω, are Hurwitz stable. To see that, we recall Kb = PbC

Tb and β ≥ τ ;

therefore, we can show

AbbPb + PbAHbb = AbPb + PbA

Tb − 2ReλPbC

Tb CbPb

= AbPb + PbATb − 2τPbC

Tb CbPb

− 2(Reλ − τ)PbCTb CbPb ≤ −Inb

7Notice that P is block diagonal; thus, the Lyapunov function results in a symmetricQ > 0 of some particular form. See Section 5.C.5.


It follows that Abb is Hurwitz stable. Similarly, it is confirmed that A∗jd’s

are Hurwitz stable. Hence, A+L0−λiD and A∗ are Hurwitz stable for everynonzero λi. Accordingly, for λi, i = 1, · · · , N − 1, there exists a symmetricQi > 0 such that the Lyapunov equation

(A+ L0 − λiD)HPi + Pi(A+ L0 − λiD) = −Qi

has a unique solution Pi = PTi > 08 which is block-diagonal with the block

sizes that correspond to the block sizes in S. Denote qi = ‖Qi‖ and ηi =‖PiD‖. Consider a block diagonal matrix

P = diagδ1P1, · · · , δN−1PN−1 (5.70)

where δi > 0 are constants that are defined recursively by δN−1 = 1 and

δi = δi+11

9

qiqi+1

η2i

for i = 1, · · · , N−2, to account for Jordan blocks of orders

larger than one. It is straightforward to see that ‖P‖ is bounded for anyβ > 0 and P solves the Lyapunov function

(A∗)HP + PA∗ = −Q (5.71)

for some symmetric Q > 0. To prove that, we move along the same line of(Peymani et al., 2012, Proposition 1). We show υTQυ > 0 for every nonzerovector υ , col νs, for s = 1, · · · , N − 1, where νs ∈ Rn. We have

−υTQυ = −N−1∑i=1

δiνTi Qiνi + 2

N−2∑i=1

ρiδiνTi PiDνi+1

in which ρi ∈ 0, 1 is the superdiagonal of each Jordan block of ∆. Then

−υTQυ ≤ −N−1∑i=1

δiqi‖νi‖2 + 2N−2∑i=1

δiηi‖νi‖‖νi+1‖

This can be written as

−υTQυ ≤ −1

3

N−1∑i=1

δiqi‖νi‖2 −1

3δN−1qN−1‖νN−1‖2 −

1

3δ1q1‖ν1‖2

− 1

3

N−2∑i=1


3

N−2∑i=1

δi+1qi+1‖νi+1‖2 + 2N−2∑i=1

δiηi‖νi‖‖νi+1‖

8See Section 5.C.5 for a discussion on P.


Completing the square, we will arrive at

−υTQυ ≤ −1

3

N−1∑i=1


3δN−1qN−1‖νN−1‖2 −

1

3δ1q1‖ν1‖2

−N−2∑i=1

√1

3δi+1qi+1‖νi+1‖ −

δiηi√13δi+1qi+1

‖νi‖

2

−N−2∑i=1

(1

3δiqi −

δ2i η

2i

13δi+1qi+1

)‖νi‖2

The last line is equal to zero. Thus, −υTQυ ≤ −13

∑N−1i=1 δiqi‖νi‖2, implying

that Q ≥ 13 maxi=1,··· ,N−1(δiqi)I. From discussion, it follows that there

exists a Q > (3 + µ2dε)INn such that P given by (5.70) solves the Lyapunov

equation (5.76). In view of the fact that P and (IN ⊗ S) commute, choose

Vo = eT(IN ⊗ S)Pe

taking derivative yields

Vo ≤ −(2 + µ2dε)‖e‖2 − (1− 2ερε)‖e‖2 + 2εrρw‖e‖‖w‖

where


‖P(IN ⊗ Lε)‖, ρw ≥ ‖P(UTL ⊗ E)‖

Because P is bounded for all β > 0 (i.e. for all network graphs G ∈ Gβ), ρεand ρw are bounded for all β > 0. Accordingly, there exists an ε22 ∈ (0, 1]such that 1 − 2ερε > 0 for every ε ∈ (0, ε22]; thus, for every ε ∈ (0, ε22], weobtain

Vo ≤ −(2 + µ2dε)‖e‖2 + 2εrρw‖e‖‖w‖

Choose V = Vc + ε−2(r−1)Vo. Let ε∗1 = minε11, ε22. For every ε ∈ (0, ε∗1],an upper bound on V is given by

V ≤ −ζTζ − 3‖e‖2 + 2ε−(r−1)µdε‖e‖‖e‖+ 2εµw‖e‖‖w‖− ε−2(r−1)(2 + µ2

dε)‖e‖2 + 2ε−(r−2)ρw‖e‖‖w‖≤ −ζTζ − 2‖e‖2 − 2ε−2(r−1)‖e‖2 + 2ε(µw‖e‖+ ε−(r−1)ρw‖e‖)‖w‖

− ‖e‖2 + 2ε−(r−1)µdε‖e‖‖e‖ − ε−2(r−1)µ2dε‖e‖2


The third line is equal to −(‖e‖ − ε−(r−1)µdε‖e‖)2 ≤ 0. Denote

σw =√

2 maxµw, ρw

Then, one may write

V ≤ −ζTζ − 2‖e‖2 − 2ε−2(r−1)‖e‖2

+ 2εσw

√‖e‖2 + ε−2(r−1)‖e‖2 ‖w‖

where we have used the fact that ‖x‖ + ‖y‖ ≤√

2√x2 + y2. Completing

the square results in

V ≤ −ζTζ − ‖e‖2 − ε−2(r−1)‖e‖2 + (εσw)2‖w‖2

Hence, from the bounded-real lemma, it follows that ‖Twζ‖∞ ≤ εσw. There-fore, for any given γ′ > 0, there exists an ε2 ∈ (0, ε∗1] such that everyε ∈ (0, ε2] yields ‖Twζ‖∞ ≤ γ′. It also shows that the origin is globallyexponentially stable in the absence of disturbance.

So far, we have shown that the proposed family of protocols can make theinfluence of w on ζ asymptotically zero by squeezing ε. We have to demon-strate that they have a similar decoupling effect on the vector of mutualdisagreements, e. That is, the impact of w on every mutual disagreementei,j can be made arbitrarily small. We define

ei,j = T i,jwe (s)w, i, j ∈ S, i > j

Then, from Lemma 5.1 (given in Subsection 5.B.5), it follows that if ‖Twζ‖∞ ≤εσw, there exists σ > 0 such that ‖T i,jwe ‖∞ < εσ. Therefore, for anygiven γ > 0, there exists an ε∗2 ∈ (0, ε∗1] such that every ε ∈ (0, ε∗2] yields‖T i,jwe ‖∞ < γ. Thus, any desired accuracy of synchronization can be obtainedby reducing ε in the range (0, ε∗1].

Explanatory Discussion on a Block-diagonal P

SupposeA1 andA2 are Hurwitz matrices. Consider the matrixA =

[A1 L0 A2

]for some constant L. Consider P =

[P1 00 P2

]where P1 and P2 are the

solutions of

AT1 P1 + P1A1 = Q1

AT2 P2 + P2A2 = Q2

5.D. Proof of Theorem 5.3 – MIMO case – Irregular Scaling 129

for symmetric and positive definite Q1 and Q2. Then

ATP + PA =

[AT

1 P1 + P1A1 P1LLTP1 AT

2 P2 + P2A2

]=

[−Q1 P1LLTP1 −Q2

], −Q

Choosing Q2 sufficiently large, one may make Q > 0, and even Q > qI forq > 0.

Explanatory Discussion on Pi

The matrix A+ L0−λiD is upper triangular. The objective is to constructa block diagonal Pi for A+ L0 − λiD. We construct P∗i for A − λiD, first.Let

(A−a )TPa + PaA−a < 0

(Ass)TPs + PsAss < 0

Then

P∗i =

Pa 0 0 0

0 P−1b 0 0

0 0 diagP−1jd 0

0 0 0 Ps

Now, one may find the following matrix easily

Pi =

Pa 0 0 00 Pb 0 00 0 Pd 00 0 0 Ps + ‖PdEds‖2

, P

where

Pb = P−1b + ‖PaL

−abCb‖2

Pd = diagP−1jd + ‖PaL

−adCd‖2 + ‖PbLbdCd‖2

In fact, one may choose Pi’s all identical by this choice.

Appendix 5.D Proof of Theorem 5.3 – MIMO case– Irregular Scaling

The proof is quite similar to the proof of Theorem 5.2. The main differenceis due to the irregular scaling (which makes the proof much easier to follow).


5.D.1 Estimation Error Dynamics for Agent i

Akin to Subsection 5.C.1, find the estimation error dynamics accordingto the scb with respect to the triple (A,G,C). The notations are keptunchanged and (5.55) holds, which leads to the same ˙zs,i. Recalling thescaling matrix Sj , in (5.33), consider the following state transformations

zsε,i = εzs,i, xjdε,i = Sj xjd,i (5.72)

Let xdε,i = col xjdε,i and S = diagSj for j ∈ Ω. Since εj = εr

j−1 forj ∈ Ω1 = 2, · · · , r, one may show εj = εεδj where δj = r

j−1 − 1 > 0 for

j ∈ Ω1. Furthermore, we define ε1 , εr+1 and δ1 , 0. Let Ejs = [E+ja , Ejc].

We find the dynamics of xjdε,i:

εj ˙xjdε,i = Ajdxjdε,i − Kjdζ∗jd,i + εεδj Ejεxdε,i

+ εεδj Gjd(E−jax−a,i + Ejbxb,i) + εδj Ejszsε,i + εεδj Gjdwjd,i

where Ejε = εrSjLjdCd+Gjd(EjsKsdGTd +Ejd)S−1; ζ∗jd,i =

∑Nk=1 gikCjdxjdε,k.

Note A1d = 0, G1d = 1, and L1d = 0 by definition, and ‖εEjε‖ = O(ε). Con-sequently, the dynamics of the system are given by:

˙x−a,i = A−a x−a,i + εrL−adCdxdε,i + L−abyb,i (5.73a)

˙xb,i = Abxb,i + εrLbdCdxdε,i − Kbζb,i (5.73b)

˙zsε,i = Asszsε,i + εE−sax−a,i + εEsbxb,i + εEsdεxdε,i + εGsswi (5.73c)

S ˙xdε,i = Addxdε,i + ε Eddεxdε,i + ε E−dax−a,i

+ ε Edbxb,i + Edszsε,i + ε Gddwi − Kddζ∗d,i (5.73d)

where ζ∗d,i =∑N

k=1 gikCdxdε,k and we have used Esdε = EsdS−1 and the

following notations for all j ∈ Ω:

Add = diagAjd, E−da = stack εδj GjdE−jaEddε = stack εδj Ejε, Edb = stack εδj GjdEjbKdd = diagKjd, Gdd = [diagεδj Gjd, 0]Γ−1

w

S = diagεjIjqj, Eds = stack εδj Ejs

Clearly, ‖εEsdε‖ = O(ε), ‖εE−sa‖ = O(ε), ‖εEsb‖ = O(ε), ‖εGss‖ = O(ε),‖εE−da‖ = O(ε), ‖εEdb‖ = O(ε), ‖εGdd‖ = O(ε), and ‖εEddε‖ = O(ε).


5.D.2 Agent i under Feedback

This section follows the same reasoning of Subsection 5.C.2. Use the samenotations and introduce (5.59), (5.60). The feedback laws (5.61) are stillvalid. However, use the following expressions.

From the assumptions on the geometric subspaces, one can representBcFc

ˇxc,i as a linear combination of the components of xi; equivalently,


−a,i +Mcbxb,i +Mcszs,i +Mcdxd,i

for some constant matrices M−ca, Mcb, Mcs, and Mcd with appropriate di-mensions and independent of ε. In view of the scaling (5.72), BcFc

ˇxc,i ismodified to


−a,i +Mcbxb,i + 1

εMcszsε,i +Mcdεxdε,i

where Mcdε = McdS−1 which is uniformly bounded for all ε > 0. Likewise,

there exist some constant matrices M−ja, Mjb, Mjs, and Mjd ∀j ∈ Ω, withappropriate dimensions and independent of ε such that

Bjduj1,i = M−jax−a,i +Mjbxb,i + 1

εMjszsε,i +Mjdεxdε,i

where Mjdε = MjdS−1 which is uniformly bounded in ε > 0. We also need

to express Bjduj2,i, given in (5.62), in terms of xi to be able to close theloop around agent i. Therefore, there exist some constant matrices N−ja,Njb, and Njd , for j ∈ Ω, with appropriate dimensions and independent ofε such that

ε−jBjdFjd

(CTjdF

+ja

ˇx+a,i + CT

jdFjb ˇxb,i

)+ ε−jBjdFj1d

ˇxj1d,i

= ε−j(N−jax−a,i +Njbxb,i +Njdεxdε,i)

where Njdε = NjdS−1, uniformly bounded for all ε > 0. Similarly, ˇxjkd,i for

k = 2, · · · , j for every j ∈ Ω can be expressed as a linear combination ofcomponents of xi. Thus, we get

Bjd

j∑k=2

Fjkdεk−1 ˇxjkd,i = ε

j∑k=2

εk−2(M−jkax−a,i + Mjkbxb,i +

1

εMjkszsε,i + Mjkxd,i)

= εM−jax−a,i + εMjbxb,i + Mjszsε,i + εMjdεxdε,i

where Mjdε =∑j

k=2 εk−2MjkS

−1, and Mjs, M−ja and Mjb are as Subsec-

tion 5.C.2. Use the state transformations introduced by (5.63), and write


the closed-loop equations:

x−aε,i = A−a x−aε,i + εL−asxs,i + εL−adCdzdε,i + εG−a wi

xs,i = Assxs,i + LsdCdzdε,i

xcε,i = Accxcε,i +BcE−cax−aε,i + εEcsxs,i + εLcdCdzdε,i

+ εGcwi + εM−cax−a,i + εMcbxb,i +Mcszsε,i + εMcdεxdε,i

εzdε,i = Addzdε,i + εLddzdε,i + εLdsxs,i + εGddwi

+ εM−dax−a,i + εMdbxb,i +Mdszsε,i + εMddxdε,i

+N−dax−a,i +Ndbxb,i +Nddxdε,i

where all the notation as the same as Subsection 5.C.2.

5.D.3 Closed-loop Equations for Agent i: Compact Form

The compact form of the closed-loop equations under the proposed protocolis found as in Subsection 5.C.3. Here, those matrices that are different fromSubsection 5.C.3 are given.

Lε =

0 Ecs 0 LcdCd

0 0 0 00 L−as 0 L−adCd

0 Lds 0 Ldd

Lε =

0 0 0 00 0 0 0

E−da Edb Eddε 0E−sa Esb Esdε 0

, L0 =

0 L−abCb εr−1L−ad 00 0 εr−1Lbd 00 0 0 Eds

0 0 0 0

In (5.64), D = εDε +D0 in which

Dε =

M−ca Mcb Mcdε 0

0 0 0 00 0 0 0

M−da Mdb Mdd 0

,D0 =

0 0 0 Mcs

0 0 0 00 0 0 0N−da Ndb Ndd Mds

5.D.4 Closed-loop Equations for the Multi-agent System

Without doubt, the closed-loop equations for the multi-agent system willtake a similar form to the ones derived in Subsection 5.C.4. For ease ofreference, the following sets of equations are given here, which are found


by collecting all the equations and using the change of coordinate (5.67).Recall N = N − 1.

(IN ⊗ S)e = (IN ⊗ (A+ L0)) e+ ε(IN ⊗ Lε)e+ (IN ⊗ (εDε +D0))e+ ε (UT

L ⊗ E)w (5.74a)

(IN ⊗ S) ˙e =(IN ⊗ (A+ L0)−∆⊗ D

)e+ ε(IN ⊗ Lε)e+ ε (UT

L ⊗ E)w(5.74b)

ζ = (G ⊗ ΓyC∗Γ∗x)e (5.74c)

and

S e0 = (A+ L) e0 +D e0 + ε (1TL ⊗ E)w (5.75a)

S ˙e0 = (A+ L) e0 + ε (1TL ⊗ E)w (5.75b)

5.D.5 H∞ Analysis

Since the matrix (IN ⊗ (A+ L0)) is Hurwitz stable, there exists a symmetricQ > (%2 + 4)In such that the Lyapunov equation:

(A+ L0)TP + P(A+ L0) = −Q

has a unique positive definite and symmetric solution P which is block-diagonal with the block sizes that correspond to the block sizes in S. Itguarantees that P and S commute. We choose the Lyapunov function

Vc = eT(IN ⊗ SP)e

Differentiation along the trajectories of (5.74) gives rise to

Vc ≤ −ζTζ − 4‖e‖2 − (1− 2εµε)‖e‖2+ 2εµdε‖e‖‖e‖+ 2εµw‖e‖‖w‖

where


‖PLε‖ , µdε ≥ maxε∈(0,1]

‖P(εDε +D0))‖

µw ≥ ‖(UTL ⊗ PE)‖


Vc ≤ −ζTζ − 4‖e‖2+ 2µdε‖e‖‖e‖+ 2εµw‖e‖‖w‖


With an argument akin to Subsection 5.C.5, one can find a positive definiteand symmetric P > 0, which is block-diagonal with block sizes correspondwith (IN⊗ S) and ‖P‖ is bounded for any β > 0, which solves the Lyapunovfunction

(A∗)HP + PA∗ = −Q (5.76)

for some symmetric Q > (3 + µ2dε)INn.Choosing

Vo = eT(IN ⊗ S)Pe

yields

Vo ≤ −(2 + µ2dε)‖e‖2 − (1− 2ερε)‖e‖2 + 2ερw‖e‖‖w‖

where


‖P(IN ⊗ Lε)‖ ρw ≥ ‖P(UTL ⊗ E)‖

Because ρε is bounded for any β > 0, there exists an ε22 ∈ (0, 1] such that1− 2ερε > 0 for every ε ∈ (0, ε22]; thus, for every ε ∈ (0, ε22], we obtain

Vo ≤ −(2 + µ2dε)‖e‖2 + 2ερw‖e‖‖w‖

Choose V = Vc + Vo. Let ε∗1 = minε11, ε22. For every ε ∈ (0, ε∗1], an upperbound on V is given by

V ≤ −ζTζ − 4‖e‖2+ 2µdε‖e‖‖e‖+ 2εµw‖e‖‖w‖− (2 + µ2

dε)‖e‖2 + 2ερw‖e‖‖w‖≤ −ζTζ − 2‖e‖2 − 2‖e‖2 + 2εσw

√‖e‖2 + ‖e‖2‖w‖

where σw ≥√

2 maxµw, ρw. Then, one may write

V ≤ −ζTζ − ‖e‖2 − ‖e‖2 + (εσw)2‖w‖2

Hence, it is established that ‖Twζ‖∞ ≤ εσw. The rest of the proof is thesame as Subsection 5.C.5.

Part II

Guided Motion Control ofMarine Craft

135

Chapter 6

Preliminary (Part II)

The chapter provides an overview of mathematical modeling and ad-vanced motion control of marine craft. A special attention is given tounderactuated marine vehicles. Different motion control scenarios areaccounted. A brief review on previous work on development of motioncontrollers is provided. At the end of this chapter, mathematical toolsthat are utilized in Part II are reviewed.

6.1 Mathematical Modeling of Ocean Vehicles

To analyze the motion of ocean vehicles, the following coordinate frames areconsidered. The Earth-centered inertial (ECI) frame i = (xi, ui, zi) is aninertial frame which is not accelerating. Newton’s laws of motion apply tothe ECI frame. The origin of ECI is located at the center of the the earth.The Earth-centered Earth-fixed (ECEF) reference frame e = (xe, ye, ze)has its origin oe fixed to the center of the Earth; but its axes rotate relativeto the inertial frame ECI. The North-East-Down (NED) coordinate systemn = (xn, yn, zn) with origin on is defined relative to the Earth’s referenceellipsoid. NED is usually defined as the tangent plane on the surface of theEarth moving with the craft. The x-axis and the y-axis point towards trueNorth and East, respectively, and the z-axis points downwards normal tothe surface of the Earth. Figure 6.1a shows different reference frames.

To describe the equations of motions, an inertial frame and a body-fixedreference frame are needed. As the speed of marine vessels is relatively low,ECEF can be considered as the inertial frame. If the marine craft operateslocally, NED can be approximated to be inertial.

The body-fixed reference frame (BODY), denoted b = (xb, yb, zb), withorigin ob is a moving coordinate frame that is fixed to the craft. The originob, usually referred to as CO, is chosen to coincide with a point midships

137

138 Preliminary (Part II)

BODY

ECEF

NED

(a) Earth fixed reference frames (ECI &ECEF) and NED frame.

COCF

LCF

(b) Body-fixed reference frame (BODY).eh s h s h s j d f h j dgjf j

Figure 6.1: Reference frames that are required to described motion.

in the water line. The axes of the BODY frame are chosen to coincide withthe principal axes of inertia. See Figure 6.1b.

A marine craft experiences motion in six degrees of freedom (DOFs).The DOFs are the set of independent displacements and rotations thatuniquely determine the position and orientation of a marine craft. Formarine craft, the DOFs are defined as surge, sway, heave, roll, pitch, andyaw. See Figure 6.2 for a clear demonstration.

The position and orientation of the vehicle are expressed in an inertialframe, and the linear and angular velocities are expressed in the BODYframe. Consider the following vectors according to the notation of SNAME(1950)1.

η = [x, y, z, φ, θ, ψ]T ∈ R3 × S3 expressed in an inertial frame

ν = [u, v, r, p, q, r]T ∈ R6 expressed in the body-fixed frame

τ = [X,Y, Z,K,M,N ]T ∈ R6 Body-fixed forces and moments

In derivation of the equations of motion, we study kinematics, whichdescribes the geometrical aspects of motion, and kinetics, which analyzethe dynamics of the motion due to external forces. The equations of motionfor a marine craft can be written as (Fossen, 2011)

η = J (η)ν (6.1a)

Mbν + Cb(ν)ν +Db(ν)ν + g(η) = τ b (6.1b)

1(See Fossen, 2011, Tabel 2.1).

6.1. Mathematical Modeling of Ocean Vehicles 139

( ℎ) ( ) ( ) ( )

( )

(ℎ )

Figure 6.2: Motion of marine craft in 6 degrees of freedom.

Kinematics Equation (6.1a) relates the body-fixed velocities to the gen-eralized velocities, in which J (η) = J (Θ) where Θ = [φ, θ, ψ]T contains theEuler angles, and determines the orientation of the craft. The matrix J (Θ)is the transformation matrix and has the following block-diagonal form:

J (Θ) =

[R(Θ) 0

0 T (Θ)

]The rotation matrix R(Θ) is given by

R(Θ) =

cos(ψ) cos(θ) − sin(ψ) cos(φ) + sin(φ) sin(θ) cos(ψ)sin(ψ) cos(θ) cos(ψ) cos(φ) + sin(φ) sin(θ) sin(ψ)− sin(θ) sin(φ) cos(θ)

sin(ψ) sin(φ) + cos(φ) sin(θ) cos(ψ)− cos(ψ) sin(φ) + cos(φ) sin(θ) sin(ψ)

cos(φ) cos(θ)

The rotation matrix belongs to the special orthogonal group of order three,denoted SO(3); therefore, the following properties hold for R(Θ).

• R(Θ)R(Θ)T = R(Θ)TR(Θ) = I3 and detR(Θ) = 1.

• R(Θ)−1 = R(Θ)T.

The time derivative of R(Θ) is given by R(Θ) =d

dtR(Θ) = R(Θ)S(ν2)

where ν2 = [p, q, r]T and S(ν2) : R3 → R3×3 and given by

S(ν2) =

0 −r qr 0 −p−q p 0


The matrix S(ν2) is skew-symmetric. The angular velocity transformationT (Θ) is given by

T (Θ) =

1 sin(φ) tan(θ) cos(φ) tan(θ)0 cos(φ) − sin(φ)

0 sin(φ)cos(θ)

cos(φ)cos(θ)

which is singular at θ = ±π

2 . If the region containing θ = ±π2 is required to

be considered, a four-parameter representation based on unit quaternionsshould be utilized.

Kinetics The kinetics, described by (6.1b), are derived from rigid-bodydynamics and hydrodynamics. The procedure to derive the kinetics ofmarine craft is given by Fossen (2011). In (6.1b), Mb = MT

b > 0 andCb = −CT

b ; thus, for any υ ∈ R6, we obtain υTCbυ = 0. Db(ν)ν con-tains hydrodynamic damping forces. One may decompose the hydrody-namic damping matrix into two parts as

Db(ν) = Dln +Dn(ν)

where Dln is the linear damping matrix, and Dn(ν) is the nonlinear dampingmatrix. For low-speed application, linear damping is dominant; however,quadratic damping dominates at high speeds. The damping matrix is non-symmetric, strictly positive for a rigid body moving through an ideal fluid.Restoring forces and moments are denoted by g(η).

An alternative representation for (6.1) is given by

η = J (Θ)ν (6.2a)

Mn(η)η + Cn(ν,η)η +Dn(ν,η)η + gn(η) = J (Θ)−Tτ b (6.2b)

in which

Mn(η) = J (Θ)−TMbJ (Θ)

Cn(ν,η) = J (Θ)−T[Cb(ν)−MbJ (Θ)−1J (Θ)]J (Θ)

Dn(ν,η) = J (Θ)−TDbJ (Θ)

gn(η) = J (Θ)−Tg(η)

6.1. Mathematical Modeling of Ocean Vehicles 141

The fundamental properties for the system matrices are as following:

• Mb is symmetric and Cb is skew-symmetric.

• Mb, Cb, and Db are wave-frequency independent.

• Mn is symmetric and positive definite; i.e. Mn =MTn > 0.

• Db > 0 and generally nonsymmetric.

• The matrix (Mn − 2Cn) is skew-symmetric; i.e. µT(Mn − 2Cn)µ = 0∀µ ∈ R6.

• (Mb − 2Cb) is skew-symmetric and Dn(ν,η) > 0.

6.1.1 3-DOF Model of Marine Craft

The 3-DOF model of marine craft can be used for describing the horizontalmotion of surface and underwater vehicles. In this model, it is assumed thatthe roll, pitch and heave are close to zero and their dynamics are negligi-ble. Thus, the motion is described by the surge, sway and yaw dynamics.Therefore, we can consider

η =

xyψ

, ν =

uvr

, φ = 0,θ = 0,z = 0,

p = 0q = 0w = 0

Assumption The marine craft has xz-plane of symmetry and has homo-geneous mass distribution. The center of gravity coincides with the centerof added mass.

Having the xz-plane of symmetry is reasonable since it means the marinecraft has port/starboard symmetry. The consequence of the assumption isthat Ixy = Iyz = 0. The assumption that is made for the center of gravityand the center of added mass allows us to have simplifiedMA and CA. Also,it results in the fact that surge is decoupled from sway and yaw. Therefore,the rigid-body mass and inertia matrix and the added mass matrix is foundas

MRB =

m 0 00 m mxg0 mxg Iz

, MA =

−Xu 0 00 −Yv −Yr0 −Nr −Nr


Accordingly, the rigid-body and the added mass Coriolis and centripetalmatrices are given by

CRB =

0 0 −m(xgr + v)0 0 mu

m(xgr + v) −mu 0

,CA =

0 0 Yvv − Yrr0 0 −Xuu

−Yvv − Yrr Xuu 0

Damping in surge is coupled from sway and yaw. Thus, linear and nonlineardamping matrices are modeled as

Dln =

−Xu 0 00 −Yv −Yr0 −Nv −Nr

Dn =

−X|u|u|u| 0 0

0 −Y|v|v|v| − Y|r|v|r| −Y|r|v0 −N|v|v|v| −N|r|v|r| −N|r|v|v| −N|r|r|r|

For 3-DOF horizontal motion, the rotation matrix is reduced to one princi-pal rotation about the z-axis. Therefore, J (ψ) is given by

J (ψ) =

cos(ψ) − sin(ψ) 0sin(ψ) cos(ψ) 0

0 0 1

∈ SO(3) (6.3)

The resulting 3-DOF model for a surface vessel becomes:

η = J (ψ)ν (6.4a)

Mbν + Cb(ν)ν +Db(ν)ν = τ b (6.4b)

where the system matrices have the following forms

Mb =

m11 0 00 m22 m23

0 m23 m33

, Db =

d11 0 00 d22 d23

0 d23 d33

(6.5a)

Cb =

0 0 −(m22v +m23r)0 0 m11u

(m22v +m23r) −m11u 0

(6.5b)

where

Mb =MRB +MA, Cb = CRB(ν) + CA(ν), Db = Dln +Dn(ν)

It should be noted that τ = [τu, τv, τr]T, where τu, τv, and τr are the forces

and moments that act on the surge, sway and yaw dynamics, respectively.

6.2. Underactuated Marine Craft 143

Low-speed model for dynamically positioned ships is simpler. In fact, atlow speeds, Coriolis and centripetal forces are negligible, and linear dampingdominates nonlinear damping terms. Therefore, the model for zero-speeddynamically positioned marine systems is approximated as

η = J (ψ)ν (6.6a)

Mbν +Dlnν = τ b (6.6b)

Indeed, the kinetic equation is linear.

6.2 Underactuated Marine Craft

As this thesis partly focuses on underactuated 3-DOF marine craft, we allo-cate this section to review some of the properties of 3-DOF underactuatedmarine craft and challenges in control of such vehicles. The content of thissection has been partially gathered from (Do and Pan, 2009; Oriolo andNakamura, 1991; Pettersen and Egeland, 1996; Reyhanoglu, 1997; Wich-lund et al., 1995a,b).

Underactuated vehicles are referred to the group of vehicles that ownfewer actuators than degrees-of-freedom to be controlled. The following for-mal definition of underactuated mechanical systems is adapted from (Gold-stein et al., 2002), quoted from (Aguiar and Hespanha, 2007).

Definition 6.1. Consider an affine mechanical system described by

q = f(q, q) +G(q)u (6.7)

where q is a vector of independent generalized coordinates, f is a vectorfield that captures the dynamics of the system, G is the input matrix, andu is the vector of generalized inputs.

The system (6.7) is fully actuated if the rank of G is equal to the di-mension of q. Therefore, the generalized inputs can instantaneously set theaccelerations in all directions of the space that is defined by q.

The system (6.7) is underactuated if the rank of G is smaller than thedimension of q. The system (6.7) is overactuated if the rank of G is largerthan the dimension of q.

Underactuated marine craft are those that have a larger number ofdegrees of freedom than the number of independent controls acting on everydegree of freedom. A marine craft that operates in 3-dimensional space has


a maximum of six degrees of freedom. To be fully actuated, it requires to beequipped with six independent actuators. Any actuator configuration lessthan six for a six-DOF marine craft makes the craft underactuated.

6.2.1 Model of Underactuated Marine Craft

The following model describes the horizontal motion of a 3-DOF marinecraft:

η = J (ψ)ν (6.8a)


where τ b is the vector of forces and moments decomposed in the body-fixedreference frame; thus one may represent it as

τ b = [τu, τv, τr]T (6.9)

in which τu and τv are the forces acting on the surge and the sway dynamics,and τr is the moment which causes a rotation about the z-axis of the body-fixed reference frame. If the marine craft is unactuated in sway, one mayassume that there is no control forces acting along the y-axis of the body-fixed reference frame. It implies that

τv = 0 ⇒ τ b = [τu, 0, τr]T (6.10)

However, in practical situations, a marine craft may be equipped with rud-ders that generates a force along the y-axis of the body-fixed reference framein addition to a yaw moment, which indicates that the sway force is nonzero.The sway force can be expressed as

τv = στr

Suppose that τu is generated with the forward thrust, denoted T , and τrdepends on the rudder deflection, denoted δ. Therefore, one may realize thevector of forces and yaw moment as:

τ b = B[Tδ

]where B =

b1 00 b20 b3

(6.11)

Then, it is easy to verify that σ = b2/b3, where b2 = Yδ and b3 = Nδ

according to the notation of SNAME (1950). It happens because the originof the body-fixed reference frame (ob or alternatively CO) does not coincide


with the the center of rotation. Proposed by Do and Pan (2003), one maychange the center of the body-fixed reference frame to a point where rotationdoes not produce a sway yaw. Representation of the system using thefollowing change of coordinates

x = x+ ε cos(ψ)y = y + ε sin(ψ)v = v + εr

where ε =m33b2 −m23b3m23b2 −m22b3

, m23b2 −m22b3 6= 0

one obtains a vector of forces and moments in the form of (6.10) whereτv = 0. That is, the unactuated dynamics and the controls are decoupled,and the effect of control signals are removed from the unactuated dynamics.

It is worth noting that the same argument may apply to 6-DOF under-actuated marine craft, and there exists such a coordinate transformationwhich decouples the unactuated dynamics from the control variables; see(Børhaug et al., 2007). However, this transformation does not bear anyphysical meaning.

In control of underactuated marine craft, in addition to this approach inwhich a coordinate transformation is utilized, another method is also pro-posed by (Peymani and Fossen, 2012) which will be discussed in Chapter 7.

6.2.2 Why to Study Control of Underactuated Marine Craft?

Three-DOF underactuated marine craft are very common in practice. Shipsare typically equipped either with two independent aft trusters or withone aft truster and one rudder. Even if bow, side or tunnel thrusters aremounted, they are not efficient at high speeds. In addition, it is not oftenpractical to fully actuate autonomous vehicles owing to power restriction,weight, complexity, reliability as well as efficiency.

Underwater marine craft are, also, usually underactuated. Assumingthat an underwater vehicle moves in six degrees of freedom, namely, surge,sway, heave, roll, pitch and yaw, it is practically underactuated at highspeeds. The minimum actuator configuration is to actuate surge, pitchand yaw. It means that the underwater vehicles moves like flying objects,and cannot move in the y- and z-axes of the body-fixed reference frameinstantaneously and independently. Also, it cannot turn about the x-axisof the body-fixed reference independently. Fully actuated marine craft areusually used in low speed applications.


Figure 6.3: An underactuated underwater flying vehicle. Courtesy of Supery-acht.

6.2.3 Properties of Underactuated Marine Craft

This is straightforward to verify that underactuation imposes the followingconstraint on the dynamics of a marine vehicle:

hb(η,ν, ν) = 0 (6.12)

Representation of the aforementioned constraint in terms of the generalizedcoordinates and the generalized velocities, one may obtain

ha(η, η, η) = 0 (6.13)

This function depends on the position, velocity and acceleration variables.The first question that comes to mind is whether this function is integrable.In the case of partial integrability, we will obtain a function as

hv(η, η) = 0 (6.14)

If ha(η, η, η) is fully integrable, we can integrate it to obtain a function as

hp(η) = 0 (6.15)

Thus, underactuation may impose a constraint on the motion of the system.If the constraint (6.13) is integrable, underactuation constrains the positionof the vehicle to the manifold hp(η) = 0. In mechanics, such a constraintfunction is called holonomic. Thus, there are some points in the config-uration space that are not accessible since the holonomic constraint mustalways hold. In this case, the system dimension is reduced by eliminatingdependency between the generalized coordinates.


In case the constraint (6.13) is partially integrable, underactuation placesa nonholonomic constraint on the system. It is known that nonholonomicconstraints preclude the system from moving in certain directions. However,if the constraint (6.13) is not integrable, underactuation imposes a nonholo-nomic constraint at the acceleration level; alternatively, it is referred to asa second-order nonholonomic constraint.

Oriolo and Nakamura (1991) study the correlation between underactu-ation and nonholonomy for robot manipulators. They provide sufficientconditions for integrability of the constraints. The results of (Oriolo andNakamura, 1991) cannot directly be applied to vehicles since centripetal andCoriolis matrix in vehicles cannot be parameterized by Christoffel symbols.Moreover, vehicles have different damping terms and kinematic transforma-tion between velocities and positions.

The relation between underactuation and nonholonomy for underactu-ated vehicles is presented by Wichlund et al. (1995a) where the focus is onconstraints represented in the body-fixed reference frame. We present theresult of (Wichlund et al., 1995a) without proof.

Lemma 6.1 (Wichlund et al. (1995a)). Consider an underactuated vehicle,and represent the unactuated dynamics as a constraint in the following form:

Muν + Cu(ν)ν +Du(ν)ν + gu(η) = 0 (6.16)

• The constraint is partially integrable if and only if

1. gu(η) is a constant vector.

2. Cu(ν) +Du(ν) is a constant matrix

3. The distribution ker(Cu(ν) + Du(ν))J (Θ)−1 is completely in-tegrable.

• The constraint is completely integrable if and only if

1. The constraint is partially integrable.

2. Cu(ν) +Du(ν) = 0.

3. The distribution kerMuJ (Θ)−1 is completely integrable.

• If gu = 0, there is no C1 state feedback law that renders the equilibriumpoint asymptotically stable.

Although both (Wichlund et al., 1995a) and (Oriolo and Nakamura,1991) claimed that their presented conditions were necessary and sufficient,Tarn et al. (2003) showed that the conditions were only sufficient conditions;it presented new integrability conditions.


Property 6.1. Underactuated 3-DOF surface marine craft are second-ordernonholonomic systems. That is, the equation that arises from underactua-tion is not integrable.

Underactuated surface marine craft possess a second-order nonholo-nomic constraint (Wichlund et al., 1995a) which is in contrast with wheeledrobots that have a strict nonholonomic constraint at the velocity level forthe lateral motion. It turns out that the orientation of a surface marinecraft is not necessarily tangent to the path that is traveled by the craft. Inother words, considering a vessel moving horizontally, the heading angle ψmay be different from the course angle χ (which is defined as the angle thatthe total speed vector makes in the inertial frame i.)Property 6.2. Underactuated 3-DOF surface marine craft is not asymp-totically stabilizable to a single equilibrium point using a continuously dif-ferentiable state feedback.

It should be noted that underactuated 3-DOF surface marine craft isasymptotically stabilizable to an equilibrium manifold (Wichlund et al.,1995a).

Controllability of an underactuated ocean vehicle is a critical issue sinceit has fewer controls than the number of degrees of freedom that must becontrolled.

Linearizing the system around each equilibrium point gives a linear sys-tem with an uncontrollable eigenvalue at the origin. Thus, the system is notlinearly stabilizable and no smooth feedback can asymptotically stabilize thesystem (Reyhanoglu, 1997).

Property 6.3. No linear smooth state feedback can asymptotically stabilizean underactuated marine craft at an equilibrium point.

The system does not satisfy Brockett’s necessary condition (Brockett,1983). Hence, we have the following property.

Property 6.4. No continuously differentiable state feedback is availableto asymptotically stabilize an underactuated marine craft at an equilibriumpoint. If such a control law is applied, Lyapunov stability cannot be utilizeddirectly.

It indicates that to asymptotically stabilize an underactuated marinecraft to an equilibrium point, one should seek time-varying or discontinuouscontrol laws. According to (Do and Pan, 2009), trajectory tracking forunderactuated marine craft is viable using linear feedback.

6.3. Motion Control Scenarios 149

Figure 6.4: Dynamic positioning of vessels.

Property 6.5 (Pettersen and Nijmeijer (2001)). Underactuated surfacemarine craft is not feedback linearizable nor are they transformable into thechain form.

In fact, the model of underactuated surface marine craft is not drift-less,and methods that are common for control of nonholonomic systems cannotbe employed in order for stabilization.

Property 6.6 (Pettersen and Egeland (1996)). The underactuated surfacemarine craft is locally strongly accessible and small time locally controllablefrom any equilibrium .

It implies that piecewise continuous analytic feedback laws exist, whichasymptotically stabilize the closed-loop system at a given point (Reyhanoglu,1997).

6.3 Motion Control Scenarios

In this section, we classify various motion control scenarios that may comeup in marine control systems.

Point stabilization is referred to the task of forcing a marine craft toreach to a desired pose (i.e. a desired position with a desired orientation)from an initial pose in the space. This task is a point-to-point motion. Froma control point of view, point stabilization is the task of set-point regulationin which the states of the marine craft should take desired constants. Thus,one can call the controller a regulator. It is also termed as posture stabi-lization. Typical examples of point stabilization are dynamic positioning orposition mooring, for accomplishment of which, fully actuated marine craftare usually employed. Figure 6.4 visualizes the dynamic positioning task.


Trajectory-tracking control problem is the task of forcing the statesof the system to track a desired, feasible, predetermined trajectory definedin Cartesian space. A more accurate term for this task can be the full-statetrajectory-tracking problem where every state of the system has to tracka reference trajectory that specifies the time evolution of the state. Math-ematically, full-state trajectory-tracking problem is formulated as follows.Consider the dynamical system

x = f(t, x, u) (6.17)

where x ∈ Rn is the state, u ∈ Rm is the control and t denotes time.Suppose xd(t) ∈ Rn is a feasible trajectory for the given system; i.e. thereexists a control u such that x = xd. The full-state trajectory-trackingcontrol problem is to find a (state or output) control law u such that

limt→∞

(x− xd) = 0

Determining a feasible desired trajectory xd can be cumbersome. In ma-rine applications, xd is usually generated with a suitable virtual craft or adynamical model reference in open loop. See (Fossen, 2011, Section 10.2.1)for a note about reference models for trajectory generation.

Path-tracking control problem is referred to the task of making amarine vehicle follow a geometric path, which is defined in Cartesian spaceand specified with a timing law, from an initial condition which may notbe on the desired path. The path-tracking control problem is alternativelytermed as the partial-state trajectory-tracking control problem. Consideran output for the system defined by (6.17):

y = h(x) (6.18)

where y ∈ Rq, where q < n, is the requested quantity that is to track thereference trajectory yd(t) ∈ Rq. The objective is to find a (state or output)control law u such that all the states are (globally) bounded and

limt→∞

(y − yd) = 0

In maritime applications, the output is usually the vector of position or thevector of position and orientation. That is, one asks a vessel to track adesired path ηd(t), where ηd(t) 6= 0, while no specific constraints are placedon the velocity. In fact, the desired path is not needed to be generated by a


virtual vessel or a dynamic model reference. See Fig. 6.5 for a visualizationof the path-tracking scenario.

The reader’s attention should be drawn to the fact that, in the path-tracking task, the output has to track a trajectory which determines its timeevolution. Thus, the output should be at the given point at the desired time.Examples of partial state trajectory are given in (Aguiar and Hespanha,2007; Berge et al., 1999; Do and Pan, 2005; Godhavn, 1996).

Path-maneuvering control problem2 is the task of forcing a marinecraft to follow a desired geometric path without imposing a timing law. Infact, the path-tracking control problem is a subclass of the path-maneuveringcontrol problem. Therefore, in the path-maneuvering problem, it is not de-termined when the marine craft has to be at a given point. Let us define thepath-maneuvering control problem mathematically. Consider the nonlinearsystem defined in the state space as

x = f(t, x, u) (6.19a)

y = h(x) (6.19b)

where x ∈ Rn is the state, u ∈ Rm is the control, y ∈ Rp is the output, andt denotes time. The system is assumed to be forward complete; thus, forevery initial state x = x(0), the solution, which is denoted x(t), exists fort ≥ 0.

Define a geometric path which is continuously parameterized in terms ofa path parameter θ; it is denoted as yd(θ). The path yd(θ) is supposed to besmooth enough; that is, yd(θ) and its n derivatives are uniformly bounded.The parameter θ can be a function of time and/or the state of the system.The following definition is adopted from (Skjetne et al., 2004).

The output-maneuvering problem is to find a control law u and an updatelaw for θ such that

1. The geometric task of convergence of y to yd(θ) is accomplished forany path parameter. That is

limt→∞

y − yd(θ) = 0, ∀θ (6.20)

2. One or more of the following dynamic tasks are achieved:

2It is worth noting that “path following” and “path maneuvering” are used inter-changeably in literature, and it can cause confusion.


(a) Time assignment which is to make the path parameter θ track aspecified time-dependent signal zt(t); i.e.

limt→∞

θ − zt(t) = 0 (6.21)

(b) Speed assignment which is to make the path parameter speed θtrack a desired speed zs(t, θ); i.e.

limt→∞

θ − zs(t, θ) = 0 (6.22)

(c) Acceleration assignment which is to make the path parameteracceleration θ converge to a desired acceleration za(t, θ, θ); i.e.

limt→∞

θ − za(t, θ, θ) = 0 (6.23)

The signals zt(t), zs(t, θ), and za(t, θ, θ) are sufficiently smooth.

It is worth noting that the path maneuvering problem reduces to pathtracking problem if the path parameter θ is replaced with time t.

Take the speed assignment task into account. Consider the followingnonlinear system defined in the state-space form:

x = f(t, x, θ) (6.24a)

θ = w(t, x, θ) (6.24b)

y = h(x) (6.24c)

which is assumed to be forward complete. Suppose (6.24) solves the outputmaneuvering problem. Therefore, there exists x = ζ(t, θ) such that y =yd(θ); also, there exists θ = zs(t, θ) such that (n − 1) derivatives of y holdalong the solutions of (6.24). Consider a nonlinear function F : Rn → Rn.Define the error signals

e , F (x− ζ(t, θ))

ω , θ − zs(t, θ)

Take t as a state with t = 1 and t(0) = 0; refer to (Skjetne et al., 2004) fordetail on this definition. It is said that the output maneuvering problemwith a speed assignment task is solved if the noncompact error set

M = (e, ω, θ, t) | e = 0, ω = 0

is rendered uniformly globally asymptotically stable. A comparison betweenpath tracking and path maneuvering is drawn in Fig. 6.5.


( ) ( )( )

( )

( ) ( )

Figure 6.5: Path tracking (the upper plot) vs. path maneuvering (the lowerplot). The parameter θ is viewed as an additional degree of freedom todesign the control system.

Other motion control scenarios can be viewed as an extension of thesefour main tasks. In literature, one may see the target-tracking control prob-lem which is the task of forcing a marine craft to track a moving or even sta-tionary target whose current motion is known. In fact, the target-trackingcontrol problem can be viewed as the path-maneuvering control problemwhen there is no information about the future path/trajectory.

Another motion scenario is formation problems which are generally de-fined as accomplishment of a mission using two or more marine vehicles.As an example, one may indicate to underway replenishment3 (Kyrkjebøet al., 2006; Miller and Combs, 1999) where the guide craft has to be placedin a desired distance relative to the approach craft while they are in tran-sit. Formation control problems can be translated into cooperative path-maneuvering, cooperative dynamic positioning, etc.

According to Section 6.2.3, for underactuated marine craft (and all non-holonomic vehicles), the point-stabilization control problem is more com-plicated to be solved than the trajectory-tracking and path-maneuveringcontrol problems.

3The objective of underway replenishment is to transfer cargo, munitions, fuel andpersonnel from one vehicle to another one while they move. See Figure 6.6 for an illus-tration.


Figure 6.6: The fast combat support ship, left, conducts an underway re-plenishment with an aircraft carrier. (U.S. Navy photo - Reference wikime-dia.org)

6.3.1 Path Maneuvering vs. Trajectory Tracking

In trajectory tracking, we have to answer where the vehicle has to be placedat every instant of time. It gives rise to impose stringent timing laws onevery state of the system. However, in path maneuvering, we only need toknow which path the vehicle has to follow and it is not essentially importantwhen the vehicle has to be at a given point.

The underlying assumption in path maneuvering is that the task ofdriving the marine craft to the desired path can be executed separatelyfrom the speed assignment task along the path. This assumption results inthe fact that the path-maneuvering control problem is decomposed into twoindependent subproblems according to (Skjetne et al., 2004):

• a geometric task which is to drive the marine craft to reach the path,and remain on the path;

• a dynamic task which is to satisfy a time, speed or acceleration as-signment along the path.

In contrast to trajectory tracking, in the path-following scenario, smootherconvergence to the path is obtained; also, the control forces are less likelyto cause saturations in actuators (Aguiar et al., 2004). The fundamentaldiscrepancy between path maneuvering and trajectory tracking is that per-formance limitations due to unstable zero-dynamics can be eliminated inthe path-maneuvering problem (Aguiar et al., 2005).


6.3.2 Underactuated vs. Fully Actuated Control Problems

It is important to recognize the difference between an underactuated controlproblem and an underactuated vehicle. To distinguish between these twofacts, we need, first, to define the configuration space and the workspace.

Definition 6.2. A configuration space is the space in which possible posi-tions and orientations that a marine craft may attain is defined.

It should be noted that for a configuration space of dimension n, thereexists an n dimensional vector of generalized coordinates which uniquelydefines the configuration of the marine vehicle.

Definition 6.3. A workspace is the space in which the control objectivesare defined.

Considering a motion control problem, if the dimension of the corre-sponding workspace is less than the number of actuated degrees of freedom,the control problem is underactuated. Solving underactuated control prob-lems is nontrivial.

Notice that an underactuated vehicle is the one that the dimension ofactuated degrees of freedom is less than the dimension of the configurationspace. Also, it should be noted that a motion control problem for an under-actuated marine craft can be considered fully actuated if the dimension ofactuated degrees of freedom is equal to the the dimension of the workspacethat the motion control is defined in.

Two-dimensional path-tracking problem for underactuated surface ma-rine craft is essentially an underactuated control problem if the problem isdefined in Cartesian coordinates. This is due to the fact that the position (xand y) and the orientation (ψ) of the surface marine craft have to track thedesired trajectories while there are only two independent controls, namely,surge speed and yaw moment. The dimension of the configuration space isthree while the dimension of the workspace is two.

6.3.3 Guidance-based Path Maneuvering

The objective of a guidance system is to compute the reference signals (po-sitions, velocities, and accelerations) for the control system of a marine craftso that the craft acts as desired.

Guidance systems can be utilized for target-tracking control problemswhere the future motion is not determined a priori; see (Fossen, 2011, Sec-tion 10.1). Moreover, they can be utilized to generate a reference trajectoryfor trajectory-tracking problems. Alternatively, guidance systems may be


employed to generate a path for path-maneuvering problems. The generatedpath guides the marine vehicle to reach to the desired path.

A guidance system is usually utilized in order to reduce the dimensionof the workspace such that the number of the outputs equals the number ofthe controls. Then, the control problem becomes fully actuated in the newworkspace.

The path-maneuvering problem is defined as being at a desired path de-fined by xp(θ), yp(θ) with the desired orientation arctan(y′p/x

′p) where y′p =

∂yp/∂θ. With this three-dimensional workspace, the path-maneuvering isunderactuated for underactuated surface marine craft. However, for two-dimensional path maneuvering, it is reasonable to control only the forwardspeed and the heading angle in order to catch and follow the path and totravel with a desired speed along the path.

It suggests to convert the desired position to a desired heading angleψd

4, giving a new workspace of dimension two; consequently, the controlproblem becomes a fully actuated control problem for an underactuatedsurface marine craft. In other words, the objective of a guidance system isto map the desired position onto the desired heading angle. It provides atwo-dimensional workspace for the path-maneuvering control problem, andenables the underactuated marine craft to follow the path by using onlytwo controls (which are the surge speed and the yaw moment). Hence, thevessel follows the desired heading by means of rudders in order to reach thepath and to achieve the geometric objective while it acquires the desiredspeed by means of propellers in order to accomplish the speed assignmenttask.

In addition, with aid of a guidance system, the designer is able to achievea smooth convergence to the path. It also facilitates rerouting in order toobstacle avoidance or weather optimal path planning. Using a guidance sys-tem, straight-line path following can be extended to curved path-following5

where the path is approximated by straight-line segments. In fact, in prac-tical situation, the route of a ship is determined using a set of waypoints,which is defined in Cartesian coordinate (xi, yi). The reference path is thenmade up of straight-line segments that connect every two successive way-points. The reader should consult (Fossen, 2011, Ch.10) and (Aguiar andPascoal, 2007; Børhaug and Pettersen, 2005a; Fredriksen and Pettersen,2006; Pettersen and Lefeber, 2001) for details on way-point tracking. Theprinciples of guidance-based path following are reviewed in (Breivik and

4In fact, ψd = arctan(y′px′p

) when the marine craft is on the path.5Curved path following is referred to the case where the path that is supposed to be

followed has nonzero curvature.

6.4. Previous Work on Marine Control Systems 157

Fossen, 2005).

6.4 Previous Work on Marine Control Systems

The objective of this section is to provide a brief overview of the researchon motion control of marine systems. In this section, we review articleswhich have been instrumental in paving the way in trajectory tracking andpath maneuvering of ocean vehicles, with an emphasis on motion control ofunderactuated marine craft.

The books of (Do and Pan, 2009) and (Fossen, 2011) provide a profoundinsight into marine control systems. The former is more concerned withnonlinear motion control of underactuated ships while the latter considersvarious aspects in modeling, guidance, and control of marine craft. Thebook (Perez, 2005) also provides a deeper insight into ship modeling fromhydrodynamic aspects related to control, wave-induced motion modeling,and addresses roll stabilization and constrained control system designs.

6.4.1 Control of Fully Actuated Marine Craft

Traditional motion controllers for fully actuated marine craft are developedby linearizing the kinematic equations around a predefined set of constantheading angles so as to be able to apply linear control theory. Due to thenonlinear nature of the systems, linear controllers, designed based on a lin-earized model, lead to local stability, and there is no guarantee for globalstability. For traditional dynamic positioning systems, the kinematic equa-tions are usually linearized about 36 different yaw angles; i.e. 36 steps of 10

are considered. The first dynamic positioning systems utilized conventionalPID controllers where low-pass and notch filters were designed in order toreduce the influence of wave-induced motion components. The linearizationmakes it viable to apply linear Kalman filter theory and gain-schedulingtechniques. An general overview for early history of ship autopilots, con-tribution of gyroscope in ship control, and progression of PID controllers isgiven by Fossen (2000). Fossen and Perez (2009) discuss the application ofKalman filtering for heading control and positioning of ships and rigs.

It is very restricting if we assume that all states including position,orientation and velocities are measured. Velocity variables are not measureddirectly nor are they measured accurately. On the whole, position andorientation are measured and velocities should be reconstructed by the aidof an observer.

As fully actuated marine craft are employed in low-speed applications


such as position keeping, dynamic positioning and thrusted mooring wherecrabwise movements are required; much of effort has been forwarded tothese applications. For dynamic positioning and position keeping scenarios,velocities are close to zero; thus, both the nonlinear damping forces and theCoriolis and centripetal forces and moments can be neglected. An overviewof dynamic positioning systems can be found in (Sørensen and Strand, 2000).For position mooring control systems, the reader may refer to (Nguyen andSørensen, 2009).

The concept of weather optimal position control is introduced by Fos-sen and Strand (2001) to keep the position of a ship in a fixed place whilethe heading of the ship should be adjusted automatically to the mean en-vironmental disturbances in order to save energy and reduce the amount ofpollution. To derive the method, adaptive backstepping design is utilized.The fundamental feature of the weather optimal position control system isthat it does not need any environmental sensors; thus, no information aboutthe environmental disturbances is required.

For moored and free-floating ships, using backstepping, Strand et al.(1998) and Strand and Fossen (1998) propose controllers that achieve localH∞ optimality and global inverse optimality.

In (Fossen and Grovlen, 1998), a solution for dynamic positioning offully actuated ships is proposed using observer backstepping method. Thecritical assumption is that the ship must be sway-yaw stable, which maynot happen for large tankers; or technically, the ship must be straight-linestable6. The extension to cover unstable sway-yaw dynamics is presentedin (Robertsson and Johansson, 1998) where the observer-based controller isderived under a detectability condition which implies stable surge dynamics,which is natural for ships.

An observer is designed in (Fossen and Strand, 1999) for dynamic po-sitioning which is proven to be passive and globally exponentially stable.In the design procedure, bias, external forces, and wave-induced forces areconsidered. In (Loria et al., 2000), the observer of (Fossen and Strand, 1999)is utilized in order to design an output feedback law where by means of thetheory of cascade interconnected systems a separation principle for the non-linear system is developed. Thus, the observer and the state feedback aredesigned separately, and combined together to achieve an output feedbackcontrol law.

For tracking purposes, the Coriolis and centripetal forces as well as

6Straight-line stability physically implies that a new path of the ship will be a straight-line after an action in yaw. The direction of the new path will usually differ from that ofthe initial path (Fossen, 1994).


nonlinear damping forces should be considered. Inspired by the work ofBerghuis and Nijmeijer (1993), a tracking controller based on passivitywhich guarantees semi-globally exponentially stability of error dynamicsis put forward by Pettersen and Nijmeijer (1999a). The model includes theCoriolis and centripetal forces, but is does not consider nonlinear dampingforces.

An observer-controller design is proposed by Aamo et al. (2001) wherenonlinear damping terms are considered in the model while the Coriolis andcentripetal terms are neglected. The controller renders the error dynamicsglobally uniformly asymptotically stable. The critical assumption is to havemonotonic damping; that is, the nonlinear damping term d(ν) : Rn → Rnsatisfies

(x− y)TP (d(x)− d(y)) ≥ 0, ∀x, y ∈ Rn, P = PT > 0

Monotonic damping assumption does not, in general, hold for marine craft(Wondergem et al., 2011).

An observer-controller scheme for tracking of fully actuated ships is pro-posed in (Wondergem et al., 2011) where both nonlinear damping forcesand Coriolis and centripetal forces are considered in the model. The designproves semi-globally uniformly exponential stability.

Robust output maneuvering problem for fully actuated marine craft isproposed by Skjetne et al. (2004). Extensions to cover parametric uncertain-ties is presented in (Skjetne et al., 2005) where the method is experimentallyverified. A tracking controller is also proposed by Ihle et al. (2006a) wherethe controller is derived using the Lagrangian approach to handle model-ing of mechanical multi-body systems. A nonlinear adaptive path-followingcontroller is proposed for fully actuated vessels in (Almeida et al., 2007) tocope with ocean currents. To deal with modeling uncertainties, Kamineret al. (2005) propose a robust path-following controller for fully actuatedmarine vehicles.

6.4.2 Control of Underactuated Marine Craft

Motion control systems for underactuated mechanical systems are reviewed.Then, an overview of motion control strategies for underactuated marinecraft is provided.

Underactuated Mechanical Systems

A comprehensive discussion on controllability and stabilizability of under-actuated mechanical systems with non-integrable dynamics is provided in


(Reyhanoglu et al., 1999) where it is established that they cannot be asymp-totically stabilized to a point by a continuous, time-invariant feedback law.The result may be regarded as Brockett’s condition for nonholonomic sys-tems. This obstacle may be overcome if time-varying or discontinuous con-trol laws are adopted; see (Kolmanovsky and McClamroch, 1995). Control-lability and the design of motion algorithms for underactuated Lagrangiansystems on Lie groups are presented in (Bullo et al., 2000). Normal forms forunderactuated mechanical systems with symmetry are presented in (Olfati-Saber, 2002), which can be useful for control design.

The classical approach to trajectory tracking of underactuated vehi-cles relies on local linearization and decoupling of dynamics so as to steerthe same number of degrees of freedom as the number of existent controlforces. This technique allows us to utilize standard control methods, butperformance and global stability may be questionable. Linearization of errordynamics about trajectories is alternatively utilized, which leads to lineartime-varying systems; gain-scheduling and linear parameter varying designalgorithms are then used to enlarge the region of attraction

Output feedback linearization (Isidori, 1995) has also been reported inseveral publications such as (Al-Hiddabi and McClamroch, 2002; Koo andSastry, 1998; Lawton et al., 2003); however, application of such methodsmay be tough and complicated as they essentially involve dynamic inversion,or the resulting controller may render an unstable zero dynamics. Lyapunov-based designs have been more promising and effective to circumvent theproblem with nonlinearity and uncertainties; thus, they result in painlessdesigns; a detailed overview can be taken from (Do and Pan, 2009; Doet al., 2002b; Frazzoli et al., 2000; Jiang, 2000; Spong, 1998). In (Dixonet al., 2000; Jiang and Nijmeijer, 1997), tracking control of mobile robots isachieved under some persistently exciting conditions, which implies that thereference trajectory is not a fixed-set point. A dynamic tracking controlleris designed for underactuated hovercraft system in (Sira-Ramirez and C.A.,2000).

In the path-maneuvering problem, which is often called the path-followingproblem, to steer an object to a desired path is of primary importance whilethe speed assignment task along the path may be of secondary importance.In maneuvering problems, the desired path is continuously parameterizedin terms of a path parameter θ; the path parameter θ comes into play as anadditional degree of freedom and treated as a new state to assign a specificdynamic along the path. This problem has been studied in (Al-Hiddabiand McClamroch, 2002; Hauser and Hindman, 1995, 1997; Samson, 1992;Skjetne et al., 2004; Song et al., 2000).


Path following of wheeled mobile robots is presented in (Samson, 1992)which is a pioneering work in the area of path following. Samson (1995)addresses path following problem for a chain of trailers that are pulled bya car. A controller for curved path-following that achieves local asymptoticstability for a chain of n trailers is proposed by Altafini (2002).

Hauser and Hindman (1997) propose a path maneuvering algorithm forflying vehicles. A maneuver regulation controller is put forward in (Hauserand Hindman, 1995) where the path parameter is determined using a nu-merical projection of the current state onto the path. However, this methodis suitable for feedback linearizable and the uniqueness of the solution canbe guaranteed when the vehicle is close to the path. Using the idea ofHauser and Hindman (1997), a solution to output maneuvering problem ispresented by Encarnacao and Pascoal (2001) where backstepping is utilized.Motivated by (Hauser and Hindman, 1997), a robust output maneuveringcontroller for a class of nonlinear systems in vectorial strict feedback form inthe presence of bounded disturbances is proposed by Skjetne et al. (2004),and applied to path tracking of fully actuated marine craft. Al-Hiddabiand McClamroch (2002) attack nonminimum phase underactuated flightsystems.

Underactuated Marine Craft – Point Stabilization

The stabilization problem for an underactuated marine craft has been ad-dressed in (Mazenc et al., 2002; Pettersen and Egeland, 1996, 1999; Pet-tersen and Nijmeijer, 1999b; Reyhanoglu, 1997; Wichlund et al., 1995a). Ithas been shown that underactuated surface marine craft is not feedbacklinearizable nor are they transformable into chain form. In fact, the modelof underactuated surface marine craft is not drift-less, and methods that arecommon for control of nonholonomic systems cannot be employed in orderfor stabilization. A kinematic drift-less model is considered with four con-trols in (Egeland et al., 1996) to design stabilizing controllers. In addition,there exist no continuous time-invariant state-feedback laws that stabilizethe origin asymptotically in the sense of Lyapunov (Pettersen and Egeland,1996; Wichlund et al., 1995a) and the necessary condition of Brockett forstabilization (Brockett, 1983) fails for underactuated marine vehicles. Wich-lund et al. (1995a) propose a continuously differentiable state feedback toasymptotically stabilize a surface vessel to an equilibrium manifold. Rey-hanoglu (1997) proposes a time-invariant discontinuous feedback law whichachieves locally exponential convergence rate.

A time-varying continuous controller for point stabilization of underac-tuated vessel is proposed by Mazenc et al. (2002). A time-varying discon-


tinuous feedback law is found in (Sørdalen and Egeland, 1995) for underac-tuated ships. Motivated by Samson (1991), time-varying control laws whichcontain explicit time-periodic sinusoidal terms is proposed in (Pettersen andEgeland, 1997, 1999) to obtain local exponential regulation; (Pettersen andEgeland, 1997) is concerned with stabilization of position and orientationof surface vehicles with parametric uncertainties and (Pettersen and Ege-land, 1999) is concerned with stabilization of position and orientation ofunderwater vehicles. Experimental result and inclusion of integral actionare presented in (Pettersen and Fossen, 2000, 1998). Bullo (2000) studiesstabilization problem on Riemannian manifolds. Using transforming themodel to skew form, dynamic feedbacks for stabilization of underactuatedships are derived by Olfati-Saber (2006).

Underactuated Marine Craft – Trajectory Tracking

Although trajectory tracking is very well understood for fully actuated ma-rine craft, it is still an active research topic for underactuated marine craft(Aguiar and Hespanha, 2007).

First attempts for tracking control of underactuated marine craft us-ing nonlinear models are reported in (Berge et al., 1999; Godhavn, 1996)where, under the assumption that the forward velocity is positive, global ex-ponential position tracking is given using controllers derived from feedbacklinearization and backstepping. Since they control only two degrees of free-dom (heading is not controlled), full state tracking control is not achieved.A generalization of (Godhavn, 1996) to encompass different types of forcesis provided by Toussaint et al. (2000) where vectorial backstepping is uti-lized. This work is limited to the types of trajectories that the vessel cantrack.

A controller based on linearization and high-gain control method is pro-posed by Sira-Ramirez (1999) which results in global exponential stabilityof the position trajectories. The design is based on the fact that underac-tuated ships are Liouvillian systems. Liouvillian systems are a special classof non-flat systems (i.e. non feedback linearizable); the interested readershould consult with (Chelouah, 1997; Sira-Ramirez, 1999).

First full state tracking controller for underactuated marine craft is pro-posed in (Pettersen and Nijmeijer, 1999b) where global exponential practi-cal stability is achieved. That is, an arbitrarily small neighborhood of thereference trajectory is globally exponentially stable.

Local exponential stability for full-state tracking is reported in (Pet-tersen and Nijmeijer, 2001) where a coordinate transformation is proposedto transform the model into a triangular-like form and a continuous con-


troller is designed using the recursive method of integrator backstepping.The result of Pettersen and Nijmeijer (2001) might be extendable to semi-global exponential stability under restrictive conditions.

Using the theory of cascaded systems, Lefeber et al. (2003) proposes afull-state tracking controller for underactuated marine craft with diagonalinertial and damping matrices which achieves globally exponentially stabil-ity around the desired trajectory if the reference yaw velocity is persistentlyexciting, which is a restrictive assumption from a practical point of view;it implies that the control method cannot be applied to straight-line paths.In addition to local exponential stability for the origin of the tracking errorsystem, the controller achieves global asymptotic stability7.

Inspired by the work of (Lefeber et al., 2003), Jiang (2002) achievesglobal asymptotic tracking of underactuated ships by application of Lya-punov’s direct method and passivity approach, which allows an explicitconstruction of Lyapunov functions and facilitates robust and adaptive de-signs. The proposed design is constructive and exploits the inherent cascadeinterconnected structure of the closed-loop system.

The tracking controllers proposed in (Jiang, 2002; Lefeber et al., 2003;Pettersen and Nijmeijer, 2001) need a desired yaw velocity that is persis-tently exciting in order for exponential tracking. Ti-Chung and Zhong-Ping(2004) propose a global tracker to achieve global K-exponential stabilityof the tracking error dynamics under weaker persistently exciting conditionthan (Lefeber et al., 2003). The proposed controller is able to achieve globalK-exponential convergence in case of a straight-line path.

The restriction due to persistently exciting desired yaw velocity is re-laxed by Do et al. (2002b), where the condition is replaced with nonzerodesired speed or L1 desired yaw velocity. Thus, the reference trajectory isallowed to be a curve including straight line (zero curvature).

Based on this relaxation, the first universal controller that solves stabi-lization and tracking simultaneously is proposed by Do et al. (2002a). Thesystem matrices are assumed to be diagonal and nonlinear damping termsare neglected. Since stabilization of an underactuated ship is not viable us-ing time-invariant feedback laws, the proposed control law is time-varying.

In (Behal et al., 2002), a continuous, time-varying, high-gain controllaw is developed to achieve globally uniformly ultimately bounded regula-tion and tracking for underactuated ships. Actually, the result is globallyexponentially practical stability. Akin to (Do et al., 2002a), a unified frame-work is proposed, in which regulation is treated as a subclass of tracking

7Therefore, it achieves global K-exponential stability for the origin of the trackingerror system according to (Sørdalen and Egeland, 1995).


problem. Extension to adaptive controllers to estimate uncertain hydrody-namic parameters is presented in (Behal et al., 2001). A robust adaptivecontroller for path-tracking of underactuated ships is proposed by Do et al.(2004), where matrices are diagonal.

Underactuated Marine Craft – Path Following

Straight-line path following can be translated into the problem of stabilizinga marine craft on a given linear course while traveling at fixed surge speed(Indiveri et al., 2000). Assuming that the ship has to catch the path whichlies on the x axis of the inertial frame, the objective can be translatedinto stabilizing the heading (course) angle, the sway velocity and the lateraldistance to zero while moving with a desired speed. In general, the objectiveshould be accomplished only using two controls, namely, the surge speed andthe yaw moment.

In maritime applications, a classical method for path following is todefine an error space using the Serret-Frenet frame; e.g. see (Encarnacaoand Pascoal, 2000; Lapierre and Soetanto, 2007). The principles of guidance-based path following are reviewed in (Breivik and Fossen, 2005).

Indiveri et al. (2000) propose a path-following controller for underac-tuated marine craft. Assuming that the vessel has a constant speed, theobjective is to stabilize the underactuated sway-yaw dynamics using onecontrol. They design a robust controller with respect to parametric uncer-tainty and state measurement errors. A robust adaptive design is proposedin (Do et al., 2003b).

Assuming that the forward speed is constant, for a 2-DOF nonlinearmodel with diagonal matrices, Zhang et al. (1998) and Pettersen and Lefeber(2001) design path following controllers. The design of Zhang et al. (1998)is based on sliding model design method. Using feedback linearization andwith aid of the line-of-sight guidance system, Pettersen and Lefeber (2001)propose a straight-line path-following controller that achieves global asymp-totic stability of the error dynamics. In (Pettersen and Lefeber, 2001), itis theoretically shown that the guidance parameters influences the stabil-ity of the closed-loop system, and the guidance system should be adaptedaccording to the parameters of the model and the velocity.

Allowing a time-varying forward speed with bounded derivative, Doet al. (2003a) design state and output controllers for straight-line path fol-lowing of 2-DOF nonlinear models of underactuated marine craft. Theoutput feedback design is valid for straight-line stable ships; thus, it doesnot encompass large tankers and high-speed craft.


2-D path following of straight lines and circles in the presence of constantocean currents is addressed in (Encarnao et al., 2000) where the Serret-Frenet frame is used to define the error dynamics and backstepping is uti-lized to design a controller which achieves local convergence. A currentestimator is also considered to cope with irrotational ocean currents. 3-Dpath following is the topic of (Encarnacao and Pascoal, 2000).

Lapierre and Soetanto (2007) put forward a controller for path followingof 3-DOF underactuated marine craft. The path is parameterized by a pathparameter, and inspired by (Encarnao et al., 2000), the Serret-Frenet frameis used to define the geometric error dynamics. The main contribution ofthe paper is to apply the method that was initially proposed in (Soetantoet al., 2003) for path-following of land robots. This approach relaxes therestrictive assumption that was shown up in following paths with nonzerocurvature. The assumption was “the initial position error must be smallerthan the smallest radius of curvature present in the path”; it is resolvedby adjusting the velocity of the Serret-Frenet frame moving on the path.The proposed controller depends on accelerations, and relies extensively onthe knowledge of the system parameters. An extension to robust adaptivedesigns is presented in (Lapierre and Jouvencel, 2008).

A solution to the problem of global boundedness and convergence of theposition tracking error to an arbitrarily small neighborhood of the origin isproposed by Aguiar and Hespanha (2007) using adaptive switching super-visory control schemes. The algorithm is proposed to cope with large andsudden parametric uncertainty in the model of the vehicle. The result issuccessfully applied to both position trajectory tracking and path following.

Many articles have assumed that the system matrices (i.e. the mass andinertia matrix or the linear and nonlinear damping matrices) are diagonal.In fact, either they assume that the marine system possesses three planesof symmetry (top/bottom, port/starboard and fore/aft symmetry), or theyassume that off-diagonal element are negligible comparing with the diagonalelements. Many marine craft have port/starboard symmetry, but do nothave fore/aft symmetry. Thus, the assumption of diagonal matrices is notrealistic.

A 2-DOF model of underactuated ships with constant speed is taken intoconsideration in (Do and Pan, 2003) where diagonal assumption on massand linear damping matrices is removed. Under a certain set of conditionson system parameters, a straight-line path following controller is designed,which globally asymptotically and locally exponentially stabilizes the errorsystem at the origin.

Considering a 3-DOF model with non-diagonal matrices and a param-


eterized path, where the curvature is allowed to be nonzero, Do and Pan(2005) introduces a controller that achieves globally asymptotically and lo-cally exponentially stability of the origin of the error dynamics.

A two-dimensional guidance-based straight-line path-following controlleris put forward in (Fossen et al., 2003) for 3-DOF model of marine craftwith nonzero off-diagonal terms in the system matrices while Coriolis andcentripetal forces are neglected. The controller is derived using vectorialbackstepping; to tackle underactuation a dynamical equation is injected intothe control system; thus, the order of the closed-loop system increases due tothe dynamic controller. Although it guarantees global ultimate boundednessof the sway velocity, it does not prove convergence to the path formally.

In (Fredriksen and Pettersen, 2006), a 3-DOF model of surface marinecraft with nonzero off-diagonal terms in system matrices, where Coriolis andcentripetal as well as linear damping forces are considered; it is presumedthat the rudder deflection generates sway force in addition to yaw moment.Thus, the unactuated dynamics is perturbed by one control action that actson the unactuated dynamics. Do and Pan (2003) introduce a coordinatetransformation in order to decouple the unactuated dynamics from the con-trols. Then, motivated by the line-of-sight guidance system used in (Fossenet al., 2003) and (Pettersen and Lefeber, 2001), (Fredriksen and Pettersen,2006) proposes PD-based feedback linearizing control laws for the surgeforce and yaw moment, and investigates the condition under which globalasymptotic and local exponential stability of the origin of the error dynam-ics are given. Various extensions of this work has been reported in Børhaug(2008). A method to cope with constant ocean currents in guidance-basedpath-following controllers is proposed by Børhaug et al. (2008), which istermed as integral line-of-sight path following. In addition, Burger et al.(2009a,b) study the problem of path following and straight-line formationfor fully actuated and underactuated marine craft, and utilize the conceptof conditional integrators to cope with path following in the presence ofexternal disturbances.

Pedone et al. (2010) derive the desired yaw rate by computing the desiredlinear velocity components such that the path-following problem is solvedasymptotically. Underactuation is taken into account when the desired ve-locities are derived although the sway dynamics is not analyzed. The surgespeed is commanded manually; thus, a yaw controller is only designed.

6.5. Mathematical Preliminaries 167

6.5 Mathematical Preliminaries

This section presents mathematical tools that are used in the subsequentchapters in the developments of control laws. The contents of this sectionare standard results available in references. The proofs are thus omittedand the reader is referred to references.

6.5.1 Lyapunov Stability

Consider the following ordinary differential equation which is called thesystem

x = f(t, x) (6.25)

where f : [0,∞)×D → Rn is piecewise continuous in t and locally Lipschitzin x. The domain D ⊆ Rn contains the origin x = 0. Locally Lipschitz im-plies that for a fixed t and for each point x ∈ D, there exists a neighborhoodD0 ⊆ D such that

‖f(t, x)− f(t, y)‖ ≤ L‖x− y‖, ∀x, y ∈ D0

where L is called the Lipschitz constant on D0. Denote the solution of(6.25) at time t for the initial time t and the initial state x = x(t) byx(t; t, x). The system is called forward complete if the solution exists forall t ≥ t. The origin x = 0 is the equilibrium point of (6.25) if

f(t, 0) = 0

Stability of the equilibrium point in the sense of Lyapunov is given in thefollowing definition.

Definition 6.4. The equilibrium point x = 0 of (6.25) is

• stable if for each ε > 0, there is δ = δ(ε, t) such that

‖x‖ ≤ δ(ε, t)⇒ ‖x(t)‖ < ε, ∀t ≥ t ≥ 0

• uniformly stable if for each ε > 0, there is δ = δ(ε) independent of tsuch that

‖x‖ ≤ δ(ε)⇒ ‖x(t)‖ < ε, ∀t ≥ t ≥ 0


• asymptotically stable if it is stable and there is a positive constantc = c(t) such that

limt→∞

x(t) = 0, ∀‖x(t)‖ ≤ c

• uniformly asymptotically stable if it is uniformly stable and there is apositive constant c independent of t such that uniformly in t

limt→∞

x(t) = 0, ∀‖x(t)‖ ≤ c

• globally uniformly asymptotically stable if it is uniformly stable, δ(ε)can be chosen to satisfy limε→∞ δ(ε) =∞ and for each pair of positivenumbers η and c, there is T = T (η, c) > 0 such that

‖x‖ < η ∀t ≥ t + T (η, c), ‖x‖ < c

• is uniformly exponentially stable8 if there exist positive constants c, k,and λ, independent of t, such that

‖x(t)‖ ≤ k‖x‖e−λ(t−t), ∀t ≥ t, ‖x‖ < c

• is semi-globally uniformly exponentially stable if for each c > 0, thereexist positive constants k and λ, independent of t, such that

‖x(t)‖ ≤ k‖x‖e−λ(t−t), ∀t ≥ t, ‖x‖ < c

• globally uniformly exponentially stable if it is uniformly exponentiallystable for any initial state x.

We define comparison functions as follows.

Definition 6.5. A continuous function α : R+ → R+ is said to belong toclass K if it is strictly increasing and α(0) = 0. It is said to belong to classK∞ if α(r)→∞ as r →∞.

Definition 6.6. A continuous function β : R+ × R+ → R+ is of class KLif the mapping β(r, s) belongs to class K with respect to r for each fixeds, the mapping β(r, s) is decreasing with respect to s for each fixed r andβ(r, s)→ 0 as s→∞.

8In literature, it is sometimes assumed that exponential stability is uniform withrespect to t. However, here, a difference is recognized and it is clearly assumed that kand λ are independent of t.


Lyapunov stability can be stated using the comparison functions.

Definition 6.7. The equilibrium point x = 0 of (6.25) is

• uniformly stable, if there exists a class K function α and a positiveconstant c, independent of t, such that

‖x(t)‖ ≤ α(‖x‖), ∀t ≥ t ≥ 0 ∀‖x‖ < c

• uniformly asymptotically stable, if there exists a class KL function βand a positive constant c, independent of t, such that

‖x(t)‖ ≤ β(‖x‖, t− t), ∀t ≥ t ≥ 0 ∀‖x‖ < c

• globally uniformly asymptotically stable if it is uniformly asymptoti-cally stable for any initial state x.

• uniformly exponentially stable if it is uniformly asymptotically stablewith β(‖x‖, t − t) = k‖x‖e−(λ(t−t)) for constant positive λ and kindependent of t.

We are interested in uniform stability of non-autonomous systems asit brings us robustness with respect to bounded disturbances. In fact, ifthe vector field f(t, x) is locally Lipschitz in t and the origin of the sys-tem x = f(t, x) is locally uniformly asymptotically stable, it is possible toshow that the system is locally input-to-state stable (ISS). In addition, lo-cal ISS implies total stability, which cannot be implied by weaker forms ofasymptotic stability (Loria and Panteley, 2005).

The following notion is equivalent to having both globally uniformlyasymptotic stability and locally uniformly exponential stability (Lefeber,2000). It is called globally K-exponential stability, introduced by Sørdalenand Egeland (1995). It is also termed as exponential stability in any ball ofinitial conditions (Loria and Panteley, 2005).

Definition 6.8. The equilibrium point x = 0 of system (6.25) is said to beK-exponentially stable iff there exists positive constants c and γ, indepen-dent of t, and a class K function κ such that

‖x(t)‖ ≤ κ(‖x‖)e−λ(t−t), ∀t ≥ t, ‖x‖ < c

If the inequality holds for any initial condition x, the origin is globallyK-exponentially stable.

K-exponential stability is exponential stability if the class K function κis linear; thus, K-exponentially stability corresponds to a weaker form ofstability than the exponential stability.


6.5.2 Input-to-State Stability

Input-to-state (ISS) stability is introduced for evaluation of robustness ofnonlinear systems in the presence of external disturbances.

Definition 6.9. The system

x = f(t, x, u)

where f is piecewise continuous in t and locally Lipschitz in x and u, is saidto be input-to-state stable (ISS) if there exist a class KL function β and aclass K function γ such that, for any x and for any input u(·) continuousand bounded on [0,∞), the solution exists for all t ≥ t ≥ 0 and satisfies

‖x(t)‖ ≤ β(‖x‖, t− t) + γ

(sup

t≤τ≤t‖u(τ)‖

)It implies that if a system is ISS, then the unforced system is glob-

ally uniformly asymptotically stable. However, the other direction is notnecessarily true.

If the unforced system is locally uniformly asymptotically/exponentiallystable, and the vector field is locally Lipschitz, the system is locally ISS forexternal inputs.

If the unforced system is globally exponentially stable, and the vectorfield is C1 and globally Lipschitz in (x, u) uniformly in t, the system isglobally ISS for external inputs.

6.5.3 Stability of Cascade Systems

Cascade systems are common in practical situations. A very interestingapplication is to design observer-based controllers where the tracking errordynamics e is cascaded with the observation error dynamics x; consideringa linear system, the overall system can be written as

e = (A−BK)e+BKx

˙x = (A− LC)x

The asymptotic stability of (e, x) = 0 is given if both matrices (A−BK) andA − LC are Hurwitz stable. This is a direct consequence of the so-calledseparation principles. On the other hand, one may look at the systemas two cascade systems through the term BKx. The origin (e, x) = 0 isglobally exponentially stable since e = (A−BK)e and ˙x = (A− LC)x areglobally exponentially stable, and the interconnection term is exponentially


decaying. However, for nonlinear systems, it is not so easy as linear coupledsystems.

Consider the following system

x1 = f1(t, x1, x2) (6.26a)

x2 = f2(t, x2) (6.26b)

where f1 and f2 are piecewise continuous in t and locally Lipschitz in[xT

1 , xT2 ]T. Assume that the origin is the equilibrium of (6.26). Suppose

that x1 = f1(t, x1, 0) is globally uniformly asymptotically stable at x1 = 0.In addition, suppose that the system (6.26b) is globally uniformly asymp-totically stable at x2 = 0.

Lemma 6.2. (Khalil, 2002, Lemma 4.7) The origin of the cascade (6.26)is globally uniformly asymptotically stable if the system (6.26a), with x2 asinput, is input-to-state stable.

However, ISS property is not always obtainable, and it makes the useof ISS property to assess stability of cascades limited. There have beenpublished many result on stability of cascade interconnected systems. Wefocus on two of them. A nice review on this topic can be found in (Jankovicet al., 1996; Loria and Panteley, 2005).

Consider the cascaded nonlinear system

Σ0 :

x = f(x) + h(x, ζ) f(0) = 0, h(x, 0) = 0

ζ = g(ζ) (x, ζ) ∈ Rn × Rm(6.27a)

(6.27b)

Function h is globally Lipschitz in x for any fixed ζ. Suppose (x, ζ) = (0, 0)is the equilibrium of Σ0. Lemma 6.3 is taken from (Seibert and Suarez,1990, Corollary 4.3).

Lemma 6.3. Suppose the origins of x = f(x) and ζ = g(ζ) are globallyasymptotically stable. The origin of Σ0 is globally asymptotically stable ifall the solutions of (6.27) are bounded.

To guarantee the boundedness of the solutions, we present Lemma 6.4according to (Jankovic et al., 1996, Lemma 1).

Lemma 6.4. The solutions of Σ0 with any initial condition (x, ζ) ∈ Rn×Rmare bounded if the following assumptions are satisfied.

A.6.4.1 The interconnection term has linear growth in x; i.e

‖h(x, ζ)‖ ≤ σ1(‖ζ‖) + σ2(‖ζ‖)‖x‖where σ1 and σ2 are class K functions.


A.6.4.2 There exist positive constants c1 and c2 such that the Lyapunovfunction W1(x) which establishes global stability of x = f(x) withLfW1 ≤ 0 satisfies

‖x‖ > c1 ⇒ ‖∂W1

∂x‖‖x‖ ≤ c2W1(x)

Assumption A.6.4.1 is to prevent a finite escape time of x(t). If W1(x)is a polynomial function of x, A.6.4.2 is satisfied.

For non-autonomous cascade systems, uniformity of the stability is im-portant since it results in robustness with respect to bounded disturbances.Consider a nonlinear time-varying system as

x = f1(t, x) + g(t, x, ζ)

ζ = f2(t, ζ)

(6.28a)

(6.28b)

where x ∈ Rn1 , ζ ∈ Rn2 . The functions f1(·, ·), f2(·, ·) and g(·, ·, ·) arecontinuous in their arguments, locally Lipschitz in x, ζ and [xT, ζT]T re-spectively, uniformly in t. Also, f1(·, ·) is continuously differentiable in botharguments. In addition, there exists a nondecreasing function $(·) suchthat |g(t, x, ζ)| ≤ $(|x|). Suppose (x, ζ) = (0, 0) is the equilibrium of thesystem, and g(t, x, 0) = 0. We make the following assumptions.

A.6.5.1 The system x = f1(t, x) is uniformly globally asymptotically stableat x = 0. It is established with a C1 Lyapunov function V (t, x) :R≥0 × Rn1 → R≥0 such that V (t, x) is positive definite and radiallyunbounded. Also, there exist positive scalars c1, c2 and η that Vsatisfies

|∂V∂x| ≤ c1V (t, x), ∀|x| ≥ η

|∂V∂x| ≤ c2, ∀|x| ≤ η

A.6.5.2 The interconnection term has linear growth in x. In other words,there exists two nondecreasing functions ρ1, ρ2 : R≥0 → R≥0 such that

‖g(t, x, ζ)‖ ≤ ρ1(‖ζ‖) + ρ2(‖ζ‖)‖x‖

A.6.5.3 There exists a class K function µ(·) such that, for all t ≥ 0, thetrajectories of the system ζ = f2(t, ζ) satisfy∫ ∞

t

‖ζ(t; t, ζ(t))‖ dt ≤ µ(‖ζ(t)‖)


Lemma 6.5 is taken from (Loria and Panteley, 2005, Theorem 2.1).

Lemma 6.5. The origin of the cascade (6.28) is uniformly globally asymp-totically stable if assumptions A.6.5.1, A.6.5.2 and A.6.5.3 hold.

If the system ζ = f2(t, ζ) is globally exponentially stable, assumptionA.6.5.3 is always satisfied. If stability of the origin of system x = f(t, x) isestablished using a polynomial Lyapunov function of x, assumption A.6.5.1is satisfied.

The stability theory for cascade interconnected nonlinear systems is aninstrumental and useful tool for analysis and design of nonlinear systems.It allows us to break down a complicated nonlinear system into smallersystems that are connected to one another as cascaded systems; then, eachsubsystem is less involved to analyze. It is also handy in output feedbackdesign of nonlinear systems. As the separation principle is not applicablefor nonlinear systems, the theory of cascade systems provides a methodto invent separation principles for various classes of nonlinear systems; forefforts leading to separation principles for nonlinear systems, see (Atassi andKhalil, 1999) and (Battilotti, 1999) for local stabilization of input–outputlinearizable systems, Lin (1995) for the case of non-affine systems, Loriaand Panteley (1999) for a class of Euler–Lagrange systems, and Loria et al.(2000) for output–feedback design for dynamic positioning of ships.

6.5.4 Brockett’s Theorem

Brockett (1983) presented a necessary condition for existence of control lawsfor asymptotic stabilization of systems at an equilibrium point. Consider

x = f(x, u)

where x is the state with initial condition x and u is the input. Theequilibrium point of the system is x∗. Suppose that f(·, ·) is continuouslydifferentiable in a neighborhood of x∗ for u = 0, and f(x∗, 0) = 0.

Theorem 6.1. A necessary condition for the existence of a continuouslydifferentiable control law that renders x = x∗ locally asymptotically stableis that

1. the linearized system has to have no uncontrollable modes associatedwith eigenvalues whose real part is positive;

2. there exists a neighborhood Ω of (x∗, 0) such that for each ς ∈ Ω thereexists a control uς defined on [0,∞) such that uς steers the solutionof x = f(x, uς) from x(t = 0) = ς to x(∞) = x∗;


3. The mapping φ : (x, u(x)) → f(x, u(x)) should be onto an open setcontaining 0. That is

∀δ > 0, ∃γ = γ(δ) such that Bγ ⊂ φ(Bδ)

Proof. See Brockett (1983).

A special system is presented here. Consider a system in the form of

x =m∑i=1

uigi(x)

where x ∈ Rn is the state, ui ∈ R is the input, and gi(x), for i = 1, · · · ,m,are linearly independent at x∗, which is the equilibrium point. There existsa C1 static state-feedback law to make x∗ locally asymptotically stable ifand only if m = n. Here, one may notice that in case of underactuation (i.e.m < n), there is no such controllers for asymptotic stabilization.

Chapter 7

Speed-varying PathManeuvering:A Nonlinear Approach

This chapter proposes a dynamic controller for two-dimensional pathmaneuvering of underactuated marine craft using the concept of back-stepping. Under the proposed controller, the speed of the marine craftvaries according to the geometric error, which is portrayed either asthe distance to the closest point on the path, or as the distance to amoving point whose speed can be adjusted to avoid singularity in thecontrol system. In other words, the speed is nonlinearly dependent onthe geometric error. To cope with lack of actuator in sway, a dynamicalequation is injected into the control system. The controller renders theorigin of the error system globally uniformly asymptotically and locallyuniformly exponentially stable.This chapter is based on (Peymani and Fossen, 2012).

7.1 Motivation and Objective

In marine applications, path following is referred to the task of forcing amarine vehicle to reach to and follow a geometric path without imposing atiming law. In accordance with (Skjetne et al., 2004), the path-maneuveringscenario1 is characterized as two distinct tasks: (i) a geometric task; (ii) adynamic task. The former is to catch the path, to follow it, and to remainon it; the latter is to assign the desired speed to the craft. The funda-mental assumption is that these two tasks can be carried out separately.Therefore, almost all two-dimensional path-following controllers for marine

1This topic is studied in Section 6.3.

175

176 Speed-varying Path Maneuvering: A Nonlinear Approach

craft consist of one speed controller which is designed independently of aheading autopilot; for example, see (Børhaug and Pettersen, 2005b; Do andPan, 2003; Fossen et al., 2003; Fredriksen and Pettersen, 2006; Lapierre andSoetanto, 2007; Pettersen and Lefeber, 2001). Heading autopilots are usu-ally dependent on the geometric errors (distance to the path) while speedcontrollers only use the speed information.

In addition to this motivation to have decoupled controllers, one maynotice that surge is usually decoupled from sway and yaw in 3-DOF modelsof horizontal motion. This is due to the fact that port/starboard symmetry(xz plane of symmetry) is usually presumed for marine craft. Thus, theelements related to sway and yaw in the surge dynamics in the mass andinertial matrix and the damping matrix are all zero, which makes it moredemanding to design speed controller independently.

This decoupled design method results in the fact that the speed of themarine craft is controlled independently of the distance to the path. How-ever, in practical situations, the vehicle may speed up to catch the pathfaster; also, it may need to slow down in turns to avoid moving sideways.It implies that the speed of the craft should depend on the geometric error.The objective of this section is to propose a method to design a path-following controller that use information about both the speed and the ge-ometric error for the speed controller. Thus, the speed varies in accordancewith the distance between the marine vehicle and the path.

This paper builds on the achievements of the earlier paper by Fossenet al. (2003) and tries to enhance the obtained result. In (Fossen et al.,2003), for a 3-DOF underactuated marine craft, a dynamic controller isdesigned that guarantees the speed error and the heading error are globallyexponentially stable at zero. The unactuated sway velocity is guaranteed tobe globally bounded. It does not consider the Coriolis and centripetal forceswhich are not negligible for high-speed applications. In addition, neither itproves that the marine craft converges to the desired path, nor does it showthe convergence rate2.

The topic of this chapter is to propose a new guidance-based path-following controller that guarantees convergence to a straight-line path witha desired speed profile. It proves that the geometric error is globally asymp-totically and locally exponentially stable at the origin. Specifically, the con-troller forces the marine craft to speed up as long as the craft is not on

2I personally believe that their result is not complete. According to the analysis givenin this chapter, for global stability, the guidance parameter and the desired speed play acrucial role.

7.2. Model of Horizontal Motion 177

the path. Thus, the speed explicitly depends on the geometric error. Therelation can be established using a nonlinear function that possesses cer-tain specifications. That one can choose a nonlinear function to relate thespeed with the geometric error enables us to take account of the actuator’scapability to avoid saturation.

Moreover, a more general model for underactuated marine craft thanthat of (Fossen et al., 2003) is taken into consideration. No coordinatetransformations are required to derive the control laws unlike (Fredriksenand Pettersen, 2006). The stability analysis of the closed-loop system iscarried out using the stability theory of cascade interconnected systems.

Organization. In the following, the model of marine craft under study ispresented. Then, the line-of-sight guidance system is explained. The path-following problem is clearly stated in Section 7.4 and the control objectivesare mathematically formulated. The control development is presented inSection 7.5 where the main result of the section lies. Next, simulationresults are given. This chapter ends with concluding remarks.

7.2 Model of Horizontal Motion

Consider q = [pT, ψ]T as the vehicle pose expressed in the inertial framei3. The position is represented by p = [x, y]T ∈ R2 and the heading isdenoted by ψ ∈ S. Let ν = [u, v, r]T ∈ R3 be the craft velocity vectordecomposed in the body-fixed reference frame b; therefore, u and v arethe linear velocities in surge and sway respectively, and r is the angularvelocity due to rotation about the z-axis of the body-fixed reference frame.

Let J (ψ) = diagR(ψ), 1 be the rotation matrix from b to i where

R(ψ) =

[cos(ψ) − sin(ψ)sin(ψ) cos(ψ)

]∈ SO(2) (7.1)

According to (Fossen, 2011), the dynamic equations of motion are describedby

q = J (ψ)ν (7.2a)


in which Mb = MTb > 0, Mb = 0, Cb(ν) = −Cb(ν)T, and Db > 0.

Homogeneous mass distribution and xz-plane symmetry are presumed, and

3A more detailed discussion on the 3-DOF model of underactuated marine craft ispresented in Section 6.2.


the surge is assumed to be decoupled from the sway-yaw dynamics; thus,the system matrices take the following forms

Mb =

m11 0 00 m22 m23

0 m23 m33

, Db =

d11 0 00 d22 d23

0 d23 d33

(7.2c)

Cb =

0 0 −(m22v +m23r)0 0 m11u

(m22v +m23r) −m11u 0

(7.2d)

where linear damping is assumed4. In (7.2b), τ b , [τu, τv, τr]T represents

the vector of generalized forces acting on the system, expressed in b. τ b

captures forces and moments due to actuators: τu is a function of the for-ward thrust, denoted T , and τr depends on the rudder deflection, denoted δ.Underactuated marine craft lack an independent actuator for sway dynam-ics. Thus, τv may be assumed to be zero. However, the rudder deflectionmay produce force in sway direction and affects the sway dynamics; there-fore, it is realized that:

τ b = B[Tδ

]where B =

b1 00 b20 b3

(7.2e)

Therefore, it indicates that

τv = στr, σ =b2b3

(7.4)

Contrary to wheeled robots that own a strict nonholonomic constraint atthe velocity level for the lateral motion, underactuated surface marine craftpossess a second-order nonholonomic constraint (Wichlund et al., 1995a);it turns out that the orientation of a surface marine craft is not necessarilytangent to the path that is traveled by the craft. In other words, the head-ing angle ψ may be different from the course angle χ (see Fig. 7.1). Thedifference is called the sideslip angle and denoted β. Indeed, it is given by

β = χ− ψ, β = arctan 2(v, u) ∈ S , [−π, π]

Nonzero sideslip angle happens, for instance, in the presence of ocean cur-rents. Thus, the stability of the unactuated dynamics is vital.

4Extension to encompass nonlinear damping forces is possible but this is outside thescope of this work.

7.3. Line-of-sight Guidance System 179

7.3 Line-of-sight Guidance System

Roughly speaking, path following for marine craft is solved using a guidancesystem that produces the desired heading angle to be tracked by the vesselso that the distance between the vessel and the path is minimized. In thisrespect, the guidance system provides a reduced-dimension workspace5 forthe path-following control problem by mapping a point (which is a two-dimensional quantity) to an angle (which is a one-dimensional quantity).Therefore, the guidance-based 2-dimensional path-following control problemis a fully actuated control problem for underactuated marine craft whichmove in the horizontal plane.

In this section, the line-of-sight (LOS) guidance system together withthe lookahead steering method6 is employed according to (Fossen, 2011,Chapter 10). Consider a straight-line path that connects two points pk andpk+1. The slope of the path is represented by ψk ∈ S = [−π, π].

At each instant of time, find the point plos = [xlos, ylos]T which is located

a lookahead distance ∆ > 0 ahead of the direct projection of p onto the path.The vector that starts from p and ends to plos is called the LOS vector. Theorientation of the LOS vector is called the LOS angle and is denoted ψlos.In fact, the LOS angle is the angle that the LOS vector makes with thex-axis of the inertial frame i. See Fig. 7.1.

Locate a path-fixed reference frame at pk; denote the frame by p.The x-axis of the path-fixed reference frame p has been rotated a positiveangle ψk, and lies on the path, directing towards the end point pk+1.

The distance between p and plos is denoted by ε. One can express ε inthe path-fixed reference frame p; it is obtained using

ε = RT(ψk) (p− pk) (7.5)

where R(ψk) is given by (7.1). Let ε , [s, e]T where e(t) is the cross-trackerror, and s(t) is the along-track error.

As mentioned earlier, the goal of the guidance system is to create amapping from a desired point onto a desired angle that guides the craft tohead towards that point. In the LOS guidance system, the desired pointis plos. Therefore, the goal is to guide the marine vehicle to move towardsplos at each instant of time. It implies that the x-axis of the body-fixedreference frame b must be aligned along the LOS vector at each instantof time.

5The workspace is the space in which the control objectives are defined.6An alternative method is the enclosure-based steering which employs the concept of

circle of acceptance. This method is more computationally expensive than the lookahead-based steering. Refer to (Fossen, 2011, PP. 258) for details.


Equivalently, the heading angle of the marine craft must be equal to theLOS angle (ψlos), which is computed using the following equation:

ψlos = ψk + ψr (7.6)

where ψk is the slope of the path, and ψr is the relative angle (approachangle) which is found using:

ψr = arctan(− e

∆

)(7.7)

Hence, the control objective is to find a controller that forces the headingangle ψ to track the LOS angle ψlos. One should notice that the path thatis traveled by a vehicle is determined by the vehicle’s total speed. In otherwords, the total speed vector of a vehicle is always tangent to the path.

On the other hand, since underactuated marine vehicles are second-ordernonholonomic, the total speed vector may not be aligned with the x-axisof the body-fixed reference frame. Therefore, aligning the x-axis of thebody-fixed reference frame along the LOS vector may result in a nonzerocross-track error for the occasions that the total speed is not aligned withthe x-axis of the body-fixed reference frame. Thus, a better alternative isto align the total speed with the LOS vector instead of aligning the x-axisof b with the LOS vector. Since ψ = χ− β where β is the sideslip angle,the control objective is modified to find a law such that

limt→∞

ψ = ψlos − β

In case the path is made up of n waypoints, represented by pi for i =1, · · · , n, the straight line connecting each two successive waypoints isrecognized as the desired path. Suppose pk and pk+1 denote the activewaypoints and sk denotes the along-track error between them expressed inpk; that is

sk = (xk+1 − xk) cos(ψk) + (yk+1 − yk) sin(ψk)

where each point pk is expressed in Cartesian coordinate as pk = [xk, yk]T.

The following switching logic is utilized to change the active waypoints:

if sk − s ≤ L,use pk+1 and pk+2 as the active waypoints.

where L > 0 is a predetermined value.

7.4. Problem Statement 181

iy

ix

kp

1kp

v

u

V

e

iy

ix

s

k

r

LOS vector

losp

p 2kp

Figure 7.1: Geometric representation of the straight-line path-followingproblem.

7.4 Problem Statement

The problem of path following is solved for underactuated marine craftwhich are confined to move on the horizontal plane. Particularly, a controlleris designed such that the marine craft reaches to and follows a desired pathwhile it retains the speed, along the path, as desired. As stated in (Skjetneet al., 2004), the path-following problem can be decomposed into two tasks:

• Geometric task in which the distance between the craft and the pathis reduced.

• Dynamic task in which the speed of the craft tracks the desired speed.

In the path-following problem, moving on the path is of higher impor-tance than moving with the desired speed. Therefore, the geometric tasktakes precedence over the dynamic task. Accordingly, one may call the ge-ometric task the primary task while the dynamic task may be called thesecondary task.

The aim of the chapter is, specifically, to design path-following con-trollers that modify the forward speed of the marine craft according to thedistance to the path. Thus, if the marine vehicle is far from the path, itmoves towards the path with a higher speed than the desired speed. It there-fore leads to the fact that the dynamic task of speed assignment is sacrificedin order to fulfil the geometric task faster. However, in the ideal condition


where there is no external disturbances, the dynamic task must be achievedperfectly when the geometric task has been accomplished.

Since the total velocity of underactuated marine craft is uncontrollable,the dynamic task is changed to make the the forward velocity track thedesired speed profile.

This work focuses on straight-line paths. Using a switching logic, thework can be extended to waypoint tracking where a path is described by aset of points that are connected by straight-line segments.


Consider a straight-line path P characterized by two points pk and pk+1.As stated, the primary objective is to converge to the path and follow it.Convergence to the path, which is referred to as the geometric task, is for-mulated as

limt→∞

e(t) = 0 (7.8a)

It is desired that the marine craft converges to the path as smoothly asdesired; so, the heading angle has to track a desired angle; that is:

limt→∞

(ψ − ψd) = 0 (7.8b)

The desired heading angle is selected as

ψd = ψlos

It implies that the marine craft follows the LOS angle and the x-axis of bis to be aligned with the LOS vector.

The secondary objective is the speed assignment task in which the for-ward velocity is to track a desired profile ud(t) > 0, ∀t. It is stated as:

limt→∞

(u− ud) = 0 (7.8c)

Furthermore, the unactuated sway dynamics must be globally bounded:

|v(t)| ∈ L∞, limt→∞

v = 0 (7.8d)

Now, the path-following problem can be stated clearly.

7.5. Control Design Method 183

Path-Following Problem Consider a 3-DOF underactuated marine craftdescribed by (7.2). Given a straight-line path P and a speed profile ud(t),the path-following control problem is to find stabilizing feedback laws forsurge speed and yaw moment such that the objectives (7.8) are accom-plished. J

The path-following problem is usually solved by designing a speed con-troller decoupled from a heading autopilot; see e.g. (Fossen et al., 2003)and (Fredriksen and Pettersen, 2006). In the speed controller, only, speedinformation is used for feedback. This means than the speed is controlledno matter how far from the path the marine craft is.

However, in this chapter, we intend to find a controller that manipulatesthe speed so as to catch the path faster when the marine vehicle is not onthe path whereas the craft has to move with the desired speed when it is onthe path; therefore, the speed assignment should become a function of thecross-track (geometric) error. In particular, we solve Problem 7.1.

Problem 7.1. Consider a 3-DOF underactuated marine craft described by(7.2). Given a straight-line path P and a speed profile ud(t), i) design afeedback law to generate a yaw moment such that the objectives (7.8a),(7.8b) and (7.8d) are accomplished; ii) find an appropriate desired speedprofile u∗d(t, e) which satisfies lime→0 u

∗d = ud, and design a feedback law to

generate a surge force that limt→∞(u− u∗d) = 0. J

7.5 Control Design Method

Inspired by (Fossen et al., 2003), the integrator backstepping idea is utilizedto develop a control system so that Problem 7.1 is solved. For integratorbackstepping, consult (Krstic et al., 1995).

7.5.1 Control Development: Backstepping

Define the error signal z0 ∈ S and z ∈ R3 according to

z0 , ψ − ψd (7.9a)

z , [z1, z2, z3]T = ν −α (7.9b)

where α is the vector of stabilizing functions as

α = [α1, α2, α3]T


and will be specified later. Let h = [0, 0, 1]T. Therefore, one can write

z0 = r − ψd

= hTz + α3 − ψd (7.10)

where ψd can be found by proper filtering of ψd or analytically. In addition,using the model of the system, one may find the dynamics of z as

Mbz = −Cb(ν)ν −Db(ν)ν + τ b −Mbα (7.11)

Consider the following control Lyapunov function

V =1

2(z2

0 + zTMbz)

and differentiate it in time, which yields:

V = z0z0 + zTMbz

= z0(α3 − ψd) + zT(−Cbν −Dbν + τ b −Mbα+ hz0)

Consider

τ b , [τ1, τ2, τ3] = Cb(ν)ν +Db(ν)ν +Mbα− hz0 −Kz (7.12)

α3 = ψd − k0z0 (7.13)

The application of α3 and τ b into V gives rise to

V = −k0z20 − zTKz (7.14)

which is negative definite (V < 0, ∀z0 6= 0, z 6= 0) if k0 > 0 and K > 0.Suppose that α1 and α2 are chosen appropriately such that their derivativesexist and bounded. Also, suppose that the implementation of τ b is viable;that is, the marine craft is fully actuated. By standard Lyapunov argument,[z0, z

T]T is globally exponentially stable (GES) at the origin; thus, [z0, zT]T

is globally bounded. Hence,

limt→∞

(u− α1) = 0

limt→∞

(v − α2) = 0

As for underactuated marine craft, it is not possible to assign all three ele-ments of τ b. It is only viable to prescribe values for the surge force τu and


the yaw moment τr; that is, one may assign τu = τ1 and τr = τ3. There-fore, for underactuated marine craft, (7.14) is not valid, and the globallyexponential stability of the origin of the error system is not concluded.

Let K = diagk1, k2, k3. Thus, according to (7.12), one may obtain thefollowing surge force and yaw moment:

τ1 = −(m22v +m23r)r + d11u+m11α1 − k1z1 (7.15a)

τ3 = (m22v +m23r)u−m11uv + d32v + d33r

+m32α2 +m33α3 − z0 − k3z3 (7.15b)

Furthermore, the second element of τ b is given by

τ2 = m11ur + d22v + d23r +m23α2 +m23α3 − k2z2 (7.16)

Due to underactuation, this force cannot be applied to the system. Althoughα3 can be found by taking time derivative of (7.13), α1 and α2 have notbeen assigned yet. Let α1 be chosen as

α1 = ud + f(t, e) (7.17)

It implies that

α1 = ud +d

dtf(t, e) +

∂

∂ef(t, e)e

Let umax be the maximum attainable forward speed for the marine craftunder study. We make the following assumption.

Assumption 7.1. The function f(t, e) is continuous and smooth and sat-isfies

• for all e, 0 ≤ f(t, e) ≤ γ where γ is a positive constant and max∀t(ud(t)+γ) ≤ umax.

• for e = 0, f(t, 0) = 0.

The following assumption is also made, which is necessary for controlla-bility of the system.

Assumption 7.2. The desired speed is lower bounded by a positive valueumin; i.e. ud(t) ≥ umin > 0, ∀t.

Underactuation makes it impossible to assign τ2 to τv. Underactuationplaces the following constraint on the dynamics of the system

m22v +m23r + d22v + d23r +m11ur = στr (7.18)


As obvious from (7.15), the obtained control force and moment dependon v; thus, it is required to prove that v, evolved from (7.18), is globallybounded. In fact, we need to analyze that the proposed yaw moment doesnot destabilize the sway dynamics.

The stabilizing function α2 can be regarded as one degree of freedom tocope with the problem of underactuation. According to the model of themarine craft (7.2e) and (7.4), there is a relation between τr and τv, whichis imposed by the physics of the system. The idea is to adapt α2 such thatthe computed force τ2 complies with the physics of the system and be equalto στ3.

To find an update low for α2, let τ2 = στ3. Rewrite it in terms ofthe error signal v = z2 + α2, and substitute τ3 by (7.15b). The followingdynamic equation emerges:

m22α2 = −d22v − d23r − m13ur − m11uv

− m23α3 + k2z2 − σk3z3 − σz0 (7.19)

where

m22 = m22 − σm32, d22 = d22 − σd32

d23 = d23 − σd33, m13 = m11 − σm23

m23 = m23 − σm33, m11 = σ(m11 −m22)

By integrating (7.19), the value of α2 is computed on-line such that τ2 = στ3.Assuming that limt→∞ z2 = 0, one may conclude that limt→∞ v = α2.Therefore, α2, as a state of the controller, plays a key role in the controlsystem. If it is proven that α2 is globally bounded, then, the control laws(7.15) and the sway dynamics are globally bounded. Before stating the mainresult, we make the following assumption.

Assumption 7.3. The craft satisfies m22 > 0 and d22 > 0.

In the sext section, it is established that the proposed guidance-basedcontroller law solves Problem 7.1.

7.5.2 Main Result

Theorem 7.1 states that the proposed controller is able to solve Problem 7.1.In Section 7.5.3, the design procedure is reviewed.

Theorem 7.1. Under Assumption 7.1-7.3, Problem 7.1 is solvable. Specifi-cally, there exists ∆∗ > 0 such that for every ∆ > ∆∗, the application of the


control laws (7.15) together with the update law (7.19) to the underactu-ated marine craft ensures the achievement of the objectives (7.8) while theforward speed, u, tracks ud(t) + f(e). Moreover, e, (u− ud) and v are glob-ally uniformly asymptotically and locally uniformly exponentially stable atzero, and ψ − ψd = 0 is globally exponentially stable.

Proof. Consider the error signals (7.9) which are written here for ease ofreference:

z0 = ψ − ψd, z = [z1, z2, z3]T = ν −α

We find the dynamics of the error system according to (7.13), (7.19) and(7.15):

z0 = −k0z0 − hTz (7.20)

Mbz = −Cb(ν)ν −Db(ν)ν +

τ1

στ3

τ3

−Mbα (7.21)

Notice that α2 is found by integration of the update law (7.19) such thatτ2 = στ3. Let ζ = [z0, z

T]T. Then, the closed-loop equations are given by:

ζ =

[−k0 hT

−M−1b h −M−1

b K

]ζ (7.22)

For k0 > 0 and K > 0, the Lyapunov function V = 12(z2

0 + zTMbz) is pos-itive definite and radially unbounded. Taking derivative along the solutionof (7.22) results in V = −k0z

20 − zTKz which is negative definite. Thus, by

standard Lyapunov argument, ζ is established to be globally exponentiallystable at the origin and globally bounded.

Therefore, the objective (7.8b) is achieved. Moreover, the objective(7.8c) will be accomplished if (7.8a) is obtained since f(t, e) → 0 if e → 0under Assumption 7.1. Therefore, it is required to prove that e is globallyasymptotically stable at zero. In what follows, we will show that (e, α2)→ 0as t → ∞. It will complete the proof since v → α2 as t → ∞; thus, theobjective (7.8d) is then attained.

According to (7.5), ε = R(ψ − ψk)υ where υ = [u, v]T. Thus, thedynamics of the cross-track error can be written as

e = u sin(ψ − ψk) + v cos(ψ − ψk) (7.23)


In view of (7.9a) and (7.6), ψ − ψk = z0 + ψr. We expand sin(ψ − ψk) andcos(ψ − ψk), and write it as:

sin(ψ − ψk) = sin(ψr) + gsin(e, z0)z0

cos(ψ − ψk) = cos(ψr) + gcos(e, z0)z0

where

gsin(e, z0) =sin(z0)

z0cos(ψr) +

cos(z0)− 1

z0sin(ψr)

gcos(e, z0) = −sin(z0)

z0sin(ψr) +

cos(z0)− 1

z0cos(ψr)

Notice that sin(z0)z0

and cos(z0)−1z0

are well-defined functions and globally bounded.According to (7.7), we have

sin(ψr) =−e√

e2 + ∆2, cos(ψr) =

∆√e2 + ∆2

Substituting u = z1 + ud + f(t, e) and v = z2 + α2 in (7.23) yields:

e =−(ud + f(t, e))√

e2 + ∆2e+

∆√e2 + ∆2

α2 + ge(e, α2, ζ)ζ (7.24)

where

ge(e, α2, ζ) =[

(ud + z1)gsin(e, z0) + α2gcos(e, z0), 0, 0, 0]

It is straightforward to verify that ge(e, α2, ζ) has linear growth in [e, α2]T;that is

|ge(e, α2, ζ)| ≤ ρe,1(‖ζ‖) + ρe,2(‖ζ‖)[e, α2]T

where ρe,i(‖ζ‖) : R≥0 → R≥0 for i = 1, 2. Similarly, (7.19) is cast byreplacing u = z1 + α1, v = z2 + α2, r = z3 + α3 and α3 = ψd − k0z0. As

ψd = − ∆

∆2 + e2e (7.25)

the function ψd depends of e, which complicates analysis. To facilitate thesystem under study, we propose the following state transformation

α2 , α2 + εψd where ε =m23

m22(7.26)


Then, we will obtain

m22 ˙α2 = −d22v − d23r − m13ur − m11uv

+ m23k0(z3 − k0z0) + k2z2 − σk3z3 − σz0 (7.27)

It is now required to rewrite (7.24) in terms of α2:

me e =−(ud + f(t, e))√

e2 + ∆2e+

∆√e2 + ∆2

α2 + ge(e, α2, ζ)ζ (7.28a)

where

me = 1− ε ∆

∆2 + e2cos(z0 + ψr)

There exists ∆∗1 > 0 such that for every ∆ > ∆∗1, it is guaranteed thatme > 0 for all t. Denote the maximum and minimum of me as bellow

0 < 1− ε

∆≤ me ≤ 1 +

ε

∆, ∀∆ > ∆∗1 (7.29)

Now, we replace u, v, and r with their equivalents (i.e. u = z1 + α1,v = z2 + α2 − εψd, r = z3 + α3) and recast (7.27) in terms of ζ, e and α2.With this aim in view, we can write

m13ur + m11uv = (ud + f(t, e))(m11α2 + m13ψd) + guvr(e, α2, ζ)ζ

d22v + d23r = d22α2 + d23ψd + gv,r(ζ)ζ

where m13 = m13 − εm11, d23 = d23 − εd22, and

guvr(e, α2, ζ)ζ = (ud + f(t, e))(m13(z3 − k0z0) + m11z2)

+ z1(m13(z3 − k0z0) + m11z2) + z1(m11α2 + m13ψd)

gv,r(e, α2, ζ)ζ = d22z2 + d23(z3 − k0z0)

Finally, (7.27) is recast as

m22 ˙α2 = −mψ∆(ud + f(t, e))

me(∆2 + e2)32

e

−(m11(ud + f(t, e)) + d22 −

mψ∆2

me(∆2 + e2)32

)α2 + gα2(e, α2, ζ)ζ

(7.28b)


where

mψ = d23 + m13(ud + f(t, e))

gα2(e, α2, ζ)ζ = m23k0(z3 − k0z0) + k2z2 − σk3z3 − σz0

−(guvr(e, α2, ζ) + gv,r(ζ)− mψ∆

me(∆2 + e2)ge(e, α2, ζ)

)ζ

It is straightforward to verify that gα2(e, α2, ζ) grows linearly with respectto [e, α2]T. Denote ξ = [e, α2]T, then, according to (7.28), we can show that

ξ = f(t, ξ) + g(t, ξ, ζ)

where f and g are found by inspection. We need to show that the systemwhich is described by (7.22) and (7.28)7 is uniformly globally asymptoticallystable (UGAS). To this end, we intend to deploy the concepts from the the-ory of the cascade-interconnected systems. Therefore, the dynamic systemdescribed by (7.28) is regarded as a nonlinear system which is perturbedby the linear autonomous system autonomous system (7.22) through theinteracting terms gα2(e, α2, ζ)ζ and ge(e, α2, ζ)ζ.

To prove that the origin of the cascade (7.22) and (7.28) is globallyuniformly asymptotically stable, Lemma 6.5 is provoked. The trajectoriesof (7.22) satisfy A.6.5.3, as it is globally exponentially stable. The couplingterm g(t, ξ, ζ) = [ 1

mege(e, α2, ζ), gα2(e, α2, ζ)]T grows linearly with respect

to [e, α2]T; thus, A.6.5.2 holds.To verify A.6.5.1, the system (7.28) is evaluated at ζ = 0. Consider the

following inequality:

mψ ≤ mψ,max (7.31)

where mψ,max , d23 + m13umax. We choose

V =1

2e2 +

κ

2m22α

22

which is positive definite and radially unbounded. Differentiating V in timeyields

V =−(ud + f(t, e))

me

√e2 + ∆2

e2

+ α2e√

e2 + ∆2

∆

me

(1− κmψ(ud + f(t, e))

∆2 + e2

)− κ

(d22 + m11(ud + f(t, e))− mψ∆2

me(∆2 + e2)32

)α2

2

7Notice that (7.28) consists of (7.28a) and (7.28b).


Therefore, it leads to

V ≤ −e2umin∆

me(e2 + ∆2)+|α2||e|√e2 + ∆2

∆

me

(1 + κ

mψ,maxumax

∆2

)− κ

(d22 + m11(ud + f(t, e))− mψ∆2

me(∆2 + e2)32

)α2

2

in which we have used the fact that (ud+f(t,e))

me√e2+∆2

≥ ∆me

umine2+∆2 . Select

κ =∆2

mψ,maxumaxp

in which p > 0, and complete the square including the first two terms of theupper bound of V :

V ≤ − ∆

me

( √umin|e|√e2 + ∆2

− (1 + p)

2√umin

|α2|)2

+∆

me

(1 + p)2

4uminα2

2

− κ(d22 + m11(ud + f(t, e))− mψ∆2

me(∆2 + e2)32

)α2

2 (7.32)

Therefore, the problem of proving globally uniformly asymptotic stabilityreduces to choose ∆ such that

d22 + m11(ud + f(t, e))− mψ∆2

me(∆2 + e2)32

− mψ,max

me∆

umax

umin

(1 + p)2

4p> 0

It requires that

d22 + m11umin >mψ,max

me∆(1 +

umax

umin)

⇒ me∆ >mψ,max

d22 + m11umin(1 +

umax

umin) (7.33)

Denoting the right-hand side by µ (i.e. µ =mψ,max

d22+m11umin(1 + umax

umin)), we will

show that there exits

∆∗ = maxε+ µ,∆∗1 (7.34)

such that, for every ∆ > ∆∗, the inequality (7.33) holds. It implies thatthere exists %1 > 0 such that

V ≤ − ∆

me

( √umin|e|√e2 + ∆2

− (1 + p)

2√umin

|α2|)2

− %1α22 (7.35)


Consequently, the system (7.28) evaluated at ζ = 0 is globally uniformlyasymptotically stable with a Lyapunov function that satisfies A.6.5.1.

It is worth noting that we study the uniform stability because the per-turbed system (that is (7.28) at ζ = 0) is non-autonomous. Actually, theterm me depends on z0 (equivalently, ζ) while cos(z0 + ψr) is seen as atime-varying term cos(z0(t) + ψr) because z0 exists for all the time.

Finally, one concludes that e, α2 → 0 as t→∞. Accordingly, α2 and vare globally asymptotically stable at zero since z2 = 0 is globally exponen-tially stable. Hence, the objectives (7.8) are accomplished.

Also, one may conclude that for every ∆ > ∆∗, and for every ballBδ, δ > 0, if e(t = t) ∈ Bδ, the derivative of the Lyapunov functionproves locally uniformly exponential stability of ξ = 0. It proves globalasymptotic and local exponential stability of the origin of the closed-loopsystem8 according to (Loria and Panteley, 2005, Proposition 2.3).

Hence, z3 is globally exponentially stable while e, v and u − ud areglobally uniformly asymptotically and locally uniformly exponentially stableat zero.

A possible choice for f(t, e) that fulfills Assumption 7.1 is

f(e) = γ1

(tanh

(e

γ2

))2

(7.36)

where γ1 and γ2 are positive real values and can be selected according tothe marine craft capabilities and the designer’s wish.

7.5.3 Design Procedure

Given a straight-line path P and a desired speed profile ud(t) which satisfiesAssumption 7.2, take the following steps to design the control system:

0. Verify that the system parameters satisfy Assumption 7.3. If it holds,continue.

1. Choose a function f(t, e) such that it satisfies Assumption 7.1.

2. Consider α3 = ψd−k0(ψ−ψd) where k0 > 0 and ψd is found by meansof the LOS guidance system.

8It is also called global K-exponential stability.

7.6. Simulation Results 193

3. Construct the update law

m22α2 = −d22v − d23r − m13ur − m11uv

− m23α3 + k2(v − α2)− σk3(r − α3)− σ(ψ − ψd)

where k2, k3 > 0.

4. Construct the control laws

τu = −(m22v +m23r)r + d11u+m11α1 − k1(u− α1)

τr = (m22v +m23r)u−m11uv + d32v + d33r

+m32α2 +m33α3 − (ψ − ψd)− k3z3

where k1 > 0 and α1 = ud(t) + f(t, e).

5. Find ∆∗ according to (7.34). Choose ∆ > ∆∗.


The nonlinear model of 3-DOF AUV according to (Fossen et al., 2003) isused to evaluate the performance of the proposed controller. The AUV isunactuated in sway. It is assumed that the yaw moment generates a purerotational moment and it does not produce a sway force; therefore, σ = 0.The maximum propeller thrust is 10 N which makes the AUV move at aspeed of 4.5 m/s in calm water. The rudder deflection saturates at 30.Actuator dynamics is approximated by a first-order transfer function. Thissystem is described by a simplified model according to (7.2) which is usedfor designing the controller.

The AUV follows a path described by a set of waypoints. The functionf(e) is chosen as (7.36) where γ1 = 1 m/s and γ2 = 2 m. The controller gainsare selected as k0 = 1, k1 = 10, k2 = 5, k3 = 3. ∆ = 31 m as ∆∗ = 30.9.The desired speed is ud = 2.5 m/s. The result of the simulation is shown inFigs. 7.2 and 7.3.

7.7 Conclusions

The paper employs the recursive method of backstepping to derive a path-following controller for underactuated marine craft. In general, the controldesign is divided into design of a speed controller and a heading controller.

By incorporating the geometric errors in the design of the speed con-troller, the speed of the marine craft is modified according to the distance to


Nor

th(m

)

East (m)

The path travelled by AUV

0 50 100 150 200 250 300 350

−100

−50

0

50

100

Figure 7.2: The performance of the controller for waypoint tracking

Time (sec)

[m]

(b) The geometric objective: Cross-track Error eTime (sec)

[m/s

]

The dynamic objective: u− ud

0 50 100 150 200 250

0 50 100 150 200 250

−10

0

10

20

30

−0.5

0

0.5

1

Figure 7.3: The dynamic task in the top plot and the geometric task in thelower plot.

7.7. Conclusions 195

the path. In results in the fact that the dynamic task of path maneuveringis sacrificed so that the geometric task is achieved. The approach can beextended to curved and parameterized paths.

The controller is achieved without a need for coordinate transformationalthough the yaw moment affects the unactuated sway dynamics.

The control system includes an update law in order to deal with un-deractuation in sway dynamics. Therefore, the controller is dynamic andincrease the order of the system.


Chapter 8

Speed-varying PathManeuvering:A Least-Square Approach

This chapter studies the problem of straight-line path maneuveringfor fully actuated marine craft. A controller is proposed, which adjuststhe speed of the marine craft according to the geometric distance andthe rate of convergence to the path. It is demonstrated that it sig-nificantly affects the performance and enhances the robustness of thecontroller to external disturbances. In the derivation of the control law,the method of least squares to approximate overdetermined systems isutilized. The conditions under which the closed-loop error system isglobally asymptotically and locally exponentially stable are derived.The stability proof relies on the theory of cascaded systems. Moreover,a method to eliminate offset in the geometric error in the presence ofocean currents is proposed. The effectiveness of the method is verifiedby performing computer simulations.This chapter is based on (Peymani and Fossen, 2013b,c).

8.1 Introduction & Objective

Path following, in marine applications, is referred to the task of forcing amarine craft to follow a geometric path without imposing a timing law onthat; i.e. it is not specified when the craft has to be at a given point onthe path. Path following of marine craft is required in many operationssuch as cable laying, towing, and dredging, and control systems must bedesigned in a way that they act accurately and cost-effectively. In this sortof operations, fully actuated marine craft are typically employed.

In fact, the common approach in two-dimensional path maneuvering of

197

198 Path Maneuvering: A Least Squares Approach

Figure 8.1: Cable laying. (Courtesy of fiberinc.net)

marine vehicles is to use the geometric distance (the distance between thevehicle and a desired point on the path) in order to find a desired headingangle that the vehicle has to obtain so that it moves towards the desiredpoint while a speed controller is derived independently so as to force thecraft to move with a desired speed. To the best of the knowledge of theauthor, almost all articles on path maneuvering of marine craft use thisapproach; see (Børhaug and Pettersen, 2005b; Do and Pan, 2003; Fossenet al., 2003; Fredriksen and Pettersen, 2006; Lapierre and Soetanto, 2007;Pettersen and Lefeber, 2001). As a result, the controller always tries tomaintain the speed as desired even if the marine craft is far away from thepath.

However, while driving a vehicle, we may change the speed of the vehicleaccording to the distance to the path and the rate of convergence to the path.That is, if we are far away from the path, we exert more thrust on the vehicleto move faster; then, we reduce the thrust while getting closer to the pathin accordance with the rate of convergence. Indeed, using the convergencerate, we anticipate whether the vehicle will stay on the path or will passacross the path. The same may happen in path maneuvering of marinevehicles. It indicates that one should consider the geometric information inaddition to the speed information in the design of speed controllers.

In Chapter 7, a nonlinear controller based on backstepping is derivedsuch that the control system increases the forward velocity of an underactu-ated marine craft according to the distance to the path while it guaranteesthat the craft moves with the desired speed on the path. The controllerincreases the speed based on the geometric error. In the present chapter,an alternative approach, which takes the advantage of the geometric errorand its derivative to properly modify the speed, is proposed.


The main contribution of this chapter is to propose a two-dimensionalpath-maneuvering controller that is capable of manipulating the speed ofthe marine craft when the craft is off the path. In fact, the speed of the craftbecomes a function of the geometric error and its derivative. The controllaw is derived from the method of least squares to map a four-dimensionalspace to a three-dimensional space. It is demonstrated that the applicationof the control law to the marine craft allows all the objectives to be satisfiedglobally asymptotically. In contrast to Chapter 7, the speed of the craft islinearly proportional to the geometric error. In addition, the derivative ofthe geometric error is also considered in order to provide anticipation.

It is also shown that the proposed controller enhances robustness withrespect to external disturbances. A method is, moreover, introduced tomake the craft move on the path, with no offset, in the presence of irrota-tional constant ocean currents; indeed, we propose a method to resolve theinherent drawback of those path-following controllers that are based on theline-of-sight guidance system1. Therefore, unlike (Lapierre and Soetanto,2007)2 which requires the accelerations, this approach only uses the gener-alized coordinates and their derivatives.

Organization. In the following, first, we state the problem clearly. Specif-ically, the 3-DOF model of horizontal motion of marine craft, the line-of-sight guidance system, and the objectives are mentioned. Then, the controllaw is derived. The main result is presented in Section 8.4 where the stabil-ity proof and properties of the controller are given. Section 8.5 is devotedto simulation results. Concluding remarks end the chapter.

8.2 Problem Statement

The chapter deals with the path-maneuvering problem for 3-DOF marinecraft. Particularly, a controller is designed such that a marine vehicle con-verges to and follows a desired path with no timing laws; it imposes a set ofgeometric constraints on the position and orientation of the vehicle; thus,it is called the geometric task. In addition, path maneuvering requires thatthe speed of the vehicle tracks a desired nonzero speed profile along thepath; this task is referred to as the speed-assignment task.

1The problem is seen in the line-of-sight guidance system where the objective is toalign the x-axis of the body-fixed reference frame with the line-of-sight vector.

2The paper of Lapierre and Soetanto (2007) is originally concerned with sway-unactuated 3-DOF marine vehicles; however, extension of the idea to fully actuated marinecraft is straightforward.


As moving on the path is more important than moving with the de-sired speed, the geometric task takes precedence over the speed-assignmenttask. According to (Skjetne et al., 2004), these two tasks can be executedseparately.

8.2.1 Model of 3-DOF Marine Craft

Let p = [x, y]T ∈ R2 be the earth-fixed position and ψ ∈ S be the yawangle3. Denote the body-fixed velocities in surge, sway, and yaw by u, vand r, which are expressed in the body-fixed reference frame, b. Considerthe following vector of the generalized coordinates and velocities:

q =

xyψ

∈ R2 × S, ν =

uvr

∈ R3

Let J (ψ) be the rotation matrix from the body-fixed reference frameb to the inertial reference frame, represented by i. The matrix J (ψ) isgiven by

J (ψ) =

[R(ψ) 0

0 1

]∈ SO(3), R(ψ) =


]∈ SO(2)

(8.1)

According to (Fossen, 2011), the dynamic equations are described by

q = J (ψ)ν (8.2a)


in whichMb =MTb > 0, Mb = 0, is the mass and inertia along with added

mass due to hydrodynamic forces and moments, Cb = −CTb , is the matrix of

Coriolis and centripetal together with its counterpart due to hydrodynamicforces and moments, and Db is the linear damping matrix. Homogeneousmass distribution and xz-plane symmetry are presumed, and surge is as-sumed to be decoupled from sway and yaw.

In (8.2b), τ b , [τu, τv, τr]T represents the vector of generalized forces,

expressed in b, which acts on the system and captures forces and momentsdue to actuators as well as due to external disturbances. It is assumed thatthe marine craft is fully actuated. Thus, there exist independent actuatorsacting on each degree of freedom.

3See Chapter 6 for more details.


The heading angle ψ is not necessarily equal to the course angle χ (whichindicates the direction of the total speed U =

√u2 + v2). Let β = χ−ψ be

called the sideslip angle which is equal to

β = arctan2(v, u) ∈ [−π, π]

8.2.2 Guidance System

A guidance system is required to provide a desired heading so that the vesselmoves toward the path smoothly. In fact, the guidance system maps thedesired position onto the desired heading angle. We employ the line-of-sight(LOS) guidance system (Fossen, 2011, Ch.10)4.

Consider a straight-line path P connecting points pk and pk+1. Theslope of the path is denoted ψk ∈ [−π, π]. Also consider a path-fixed refer-ence frame, denoted pk, that originates at pk. Its x-axis has been rotatedby a positive angle ψk.

Let e(t) denote the cross-track error which is the shortest distance be-tween the marine craft and the path. Also, let s(t) be the along-track errorwhich is the distance between the direct projection of the marine craft onthe path and the start point pk. Defining ε , [s, e]T, one can find

ε = R(ψk)T (p− pk) (8.3)

Denote the point which is located a lookahead distance ∆ > 0 aheadof the direct projection of p onto the path by plos = [xlos, ylos]

T, and referto the vector that starts from p and ends to plos as the line-of-sight (LOS)vector5. See Fig. 8.2 for visualization of the problem. The LOS angle,denoted ψlos, is the angle that the LOS vector makes with the x-axis of theinertial reference frame i. The ψlos is computed by:

ψlos = ψk + ψr (8.4)

where ψk is the slope of the path, and the relative angle (approach angle)ψr is found using:

ψr = arctan(− e

∆

)(8.5)

The objective is to align the x-axis of the body-fixed reference frame bwith the LOS vector.

4The method that is utilized in the thesis is explained in Chapter 7 in details. However,in order for coherence and ease of reference, it is briefly reviewed in this Chapter.

5This is based on the lookahead-based steering method (Fossen, 2011).


iy

ix

kp

1kp

v

u

V

e

iy

ix

s

k

r

LOS vector

losp

p



The geometric task of path maneuvering is to converge to the path P andfollow it. It is regarded as the primary objective and is formulated as

limt→∞

e(t) = 0 (8.6a)

The marine craft is to converge to the path P as smoothly as desired.Therefore, the heading angle has to track a desired angle; that is:

limt→∞

(ψ − ψd) = 0 (8.6b)

We choose the desired heading angle as ψd = ψlos. The secondary objectiveis that the speed of the marine craft, represented by U =

√u2 + v2, is to be

regulated to the desired value ud,>0; it is stated as:

limt→∞

(U − ud) = 0 (8.6c)

By the secondary objective, we mean that the dynamic task of speed assign-ment has less importance than the geometric task, and it can be sacrificedso as to have the main objective perfectly satisfied. Moreover, it is intendedthat the marine craft does not move sideways; that is:

limt→∞

v = 0 (8.6d)

It implies that the forward speed of the marine craft is to track the desiredspeed ud. In addition, the sway velocity is to converge to zero and thesideslip angle β is to be vanishing. The path-following problem for fullyactuated marine craft is clearly stated here.

8.3. Design Method 203

Path-Following Problem Consider a 3-DOF fully actuated marine craftdescribed by (8.2). Given a straight-line path P and a speed profile ud(t),the path-following control problem is to find stabilizing feedback laws forsurge and sway speeds and yaw moment such that the objectives (8.6) areaccomplished. J

The standard solution is to design feedback laws τu = τu(u) and τv =τv(v) to generate surge and sway forces in addition to a feedback law τr =τr(ψ, r, e) to produce a yaw moment.

In this chapter, the goal is to design a path-following controller thatadjusts the speed of the marine craft in accordance with the distance to thepath and the rate of convergence to the path. The idea is to take advantageof the vehicle’s speed in order to fulfill the geometric task. Although thecontrol system modifies the speed while the craft does not move on the path,the dynamic task of speed-assignment is to be satisfied perfectly when thecraft moves on the path and there is no external disturbances. Therefore,the problem that is posed in this chapter is different from the standardpath-following problem. The problem under study is stated in Problem 8.1.

Problem 8.1. Consider a 3-DOF fully actuated marine craft described by(8.2). Given a straight-line path P and a desired speed ud, i) design a feed-back law to generate a yaw moment τr = τr(ψ, r, e) such that the objectives(8.6a) and (8.6b) are accomplished; ii) use the geometric information todesign feedback laws τu = τu(u, e) and τv = τv(v, e) for the surge and swayforces such that the objectives (8.6c) and (8.6d) are achieved. J

Part (ii) of Problem 8.1 can be interpreted as to derive a desired speedu∗d = u∗d(ud, e) such that lime→0 u

∗d = ud, and to design feedback laws for

surge and sway forces such that limt→∞(U − u∗d) = 0 and limt→∞ v = 0.

8.3 Design Method

The control law is designed in two steps. In the first step, the accelerationsthat are required for an exponential convergence to the path (and hold(8.6a)) are derived. The second step is devoted to find accelerations thatsatisfy (8.6b), (8.6c) and (8.6d). Finally, the vector of the control laws isderived based on the method of least squares which is utilized to find thebest approximation for the ‘desired’ accelerations.


8.3.1 Desired Accelerations to Make Cross-track Error zero

We aim to make the cross-track error e(t) converge to zero as time tends toinfinity. Let ρ1(ψ), ρ2(ψ), and ρ3 be as

ρ1(ψ)T = [cos(ψ), − sin(ψ), 0]ρ2(ψ)T = [sin(ψ), cos(ψ), 0]ρT

3 = [0, 0, 1],⇒ J (ψ) =

ρ1(ψ)T

ρ2(ψ)T

ρT3

According to (8.3), the first and second time-derivatives of the cross-trackerror e(t) will be

e = ρ2(γ)Tν (8.7a)

e = ρ2(γ)Tν + ρ2(γ)Tν (8.7b)

in which γ = ψ − ψk. The objective is recast to make X = [e, e]T glob-ally asymptotically/exponentially stable (GAS/GES) at the origin. Thedynamics of X are given by:

X = AX + Bue where A =

[0 10 0

], B =

[01

](8.8)

The virtual control input ue is utilized to stabilize e(t) at the origin. As thepair (A,B) is controllable, there exists a vector Ke = [ke1, ke2] such thatthe state-feedback control law

ue = −KeX (8.9)

renders the equilibrium point (X = 0) GES. In fact, Ke is chosen such thatthere exits Pe = PT

e > 0 which solves

(A− BKe)TPe + Pe(A− BKe) = −Qe

for any Qe = QTe > 0. According to linear control theory, any ke1, ke2 > 0

give rise to a stabilizing control law ue. Therefore, choosing Ve = XTPeXand taking derivative yield Ve < 0 for all X 6= 0. Closing the loop with(8.9) and considering (8.7b), it follows that

e+ ke2e+ ke1e = 0 ⇒ ρ2(γ)Tν − σe = 0 (8.10)

in which

σe = −ρ2(γ)Tν − ke2e− ke1e (8.11)

Eq. (8.10) yields the desired accelerations that make the vehicle convergeto the path with an exponential rate.

8.3. Design Method 205

8.3.2 Desired Accelerations to Achieve Heading and SpeedObjectives

Define z0 , ψ − ψd. Then, z0 = ψ − ψd = ρT3 ν − ψd. Consider V1 = 1

2z20

and differentiate it in time:

V1 = z0z0

= z0(ρT3 ν − ψd) (8.12)

In order to regulate z0 to zero, the velocities ν are chosen as virtual controlinputs; we define ν , z+α where the new state variables z and the vectorof stabilizing functions α are as

z =

z1

z2

z3

, α =

α1

α2

α3

Therefore, (8.12) can be written as

V1 = z0(ρT3 z + α3 − ψd) (8.13)

Choosing α3 = ψd − k0z0 yields

V1 = −k0z20 + zTρ3z0, k0 > 0 (8.14)

Now, the goal is to stabilize z at the origin. Choose α2 = 0 and α1 = ud.It implies that if z → 0, u and v will converge to α1 and α2, respectively.The dynamics of z are given by z = ν − α. Let V2 = V1 + 1

2zTz be the

control Lyapunov function. The time-derivative of V2 along the trajectoryof the system (z0,z) is

V2 = −k0z20 + zTρ3z0 + zTz

= −k0z20 + zT(ρ3z0 + ν − α)

Let σz = α − ρ3z0 − Kz where K , diagk1, k2, k3 > 0. Therefore, if theconstraint

ν − σz = 0 (8.15)

holds, it turns out that

V2 = −k0z20 − zTK z < 0, ∀z0 6= 0,∀z 6= 0 (8.16)


8.3.3 Accelerations to Achieve All Objectives

In view of (8.10) and (8.15), the vector of velocities that fulfills all theobjectives is computed by integrating the following equation:

H(γ)ν = b(γ,φ) (8.17)

in which φ , [e, e, z0, zT]T, and

H(γ) =

[I3

ρ2(γ)T

], b(γ,φ) =

[σz

σe

](8.18)

where Ii is the i× i identity matrix. Define

Hb(γ) , H(γ)TH(γ) = I3 + ρ2(γ)ρ2(γ)T (8.19)

which is non-singular ∀γ. Therefore, H−1b (γ) exists and is given by

H−1b (γ) = I3 −

1

2ρ2(γ)ρ2(γ)T (8.20)

To find ν, both sides of (8.17) are pre-multiplied by H(γ) = H−1b (γ)H(γ)T.

It gives rise to:

ν = H(γ) b(γ,φ) (8.21)

One may perceive H as Moore-Penrose psuedoinverse of H. Substituting(8.21) in the equations of motion (8.2b) gives the control forces that arerequired to make the marine craft have the acquired acceleration (8.21).The control laws are given by

τ pb =MbH(γ)b(γ,φ) + Cb(ν)ν +Db(ν)ν (8.22)

It is worth noting that (8.21) yields the accelerations that satisfy both(8.10) and (8.15) simultaneously if and only if b ∈ im H. Otherwise, (8.21)is not the solution of (8.17), and the substitution of (8.21) in the left-handside of (8.17) does not equal b.

Equation (8.21) yields the best approximation for ν such that the func-tion ‖H(γ)ν−b(γ,φ)‖2 is minimized. Taking V = Ve +V2 as the Lyapunovfunction is, therefore, meaningless, and it cannot be used to establish thestability proof of the closed-loop system. Hence, to investigate the stabilityof the closed-loop system under the derived control law (8.22) is crucial.

8.4. Main Result 207

8.4 Main Result

In this section, we study the stability of the closed-loop system. To facilitateanalysis, we make a change in the control law (8.22). Clearly, ρ2(γ)T =γρ1(γ)T. On the other hand, according to (8.3), ρ1(γ)Tν = s(t) which isthe speed of the craft along the path. We replace ρ1(γ)Tν with ud. Itis reasonable, as the vehicle is supposed to move along the path with thedesired speed. Accordingly, (8.11) is altered to

σ∗e = −γud − ke2e− ke1e (8.23)

Then, we define b∗ = [σTz , σ

∗e ]T and use b∗ instead of b in (8.17). It gives

rise to the following control law

τ ∗b =MbH(γ)b∗(γ,φ) + Cb(ν)ν +Db(ν)ν (8.24)

Theorem 8.1 provides a solution for Problem 8.1.

Theorem 8.1. Let ud and ∆ be positive constants. Apply the control law(8.24) to the system (8.2). The origin (e, z0, z) = 0 is globally asymptoticallyand locally exponentially stable if

T.8.1.1 k0, k3, ke1 > 0 and ke2 >ud

∆;

T.8.1.2 k1 = k2 = k such that k >3

4

k2e2

ke1∆ud

Proof. The proof of the theorem relies on the theory of nonlinear compositesystems (Jankovic et al., 1996). We find the closed-loop equations. In viewof (8.20), one can write

H = (HTH)−1HT =

[I3 − 1

2ρ2ρT2 ,

12ρ2

](8.25)

where we have dropped the argument γ. From ν = H(γ) b∗(γ,φ), it followsthat

ν = (I3 −1

2ρ2(γ)ρT

2 (γ))(α− ρ3z0 −K z)− 1

2ρ2(γ)(γud + ke2e+ ke1e)

One may find γ = z3 + ψd − k0z0. Recalling ψd = ψlos, it is straightforwardto show that ψd = ψr; thus, we obtain

ψd = − ∆

e2 + ∆2e (8.26)


Also, notice that

ρ2(γ)ρ2(γ)T =

sin2(γ) sin(γ) cos(γ) 0sin(γ) cos(γ) cos2(γ) 0

0 0 0

Since α1 = α2 = 0, we obtain ρ2ρ

T2 α = 0. Moreover, one may find that

ρ2ρT2 σz and ρ2σe do not influence z3. Find ρi(ψ) for i = 1, 2 such that

ρi(ψ) = [ρi(ψ)T, 0]T for i = 1, 2 (8.27)

Define z , [z1, z2]T. The closed-loop equations are expressed by (8.28) and(8.29):

Σ2 :

z0 = −k0z0 + z3

z3 = −z0 − k3z3

(8.28a)

(8.28b)

˙z = −(I2 −1

2ρ2(γ)ρ2(γ)T)Kz − 1

2ρ2(ke2e+ ke1e)

+1

2ρ2ud

∆e

e2 + ∆2− 1

2ρ2(z3 − k0z0)ud

(8.29)

It is required to include e since (8.29) depends on it. According to (8.7a),e = u sin(γ) + v cos(γ). Note that γ = ψ − ψk = z0 + ψr. Thus, one canwrite

sin(z0 + ψr) = sin(ψr) + gsin(e, z0)z0 (8.30a)

cos(z0 + ψr) = cos(ψr) + gcos(e, z0)z0 (8.30b)

Functions gsin(e, z0) and gcos(e, z0), given in Appendix 8.A, are globallybounded. According to the guidance system (reflected by (8.5)), we have

sin(ψr) =−e√

e2 + ∆2, cos(ψr) =

∆√e2 + ∆2

(8.31)

Define ζ , [z0, z3]T and ξ , [e, z1, z2]T. Now, we are ready to express thedynamics of the cross-track error and recast (8.29) as described by

e = − ud + z1√e2 + ∆2

e+z2∆√e2 + ∆2

+ ge(ξ, ζ)ζ (8.32a)

˙z = −Kz(ψr)z − 12 ρ2(ψr)ke1e

− 12 ρ2(ψr)Ω1(− ud + z1√

e2 + ∆2e+

z2∆√e2 + ∆2

) + gz(ξ, ζ)ζ (8.32b)


Figure 8.3: A schematic diagram for the closed-loop equations as two sys-tems which are connected in cascade.

where gz(ξ, ζ) and ge(ξ, ζ) are given in Appendix 8.A, and

Kz(ψr) = (I2 −1

2ρ2(ψr)ρ

T2 (ψr))K (8.33)

Ω1 = ke2 −∆

e2 + ∆2ud (8.34)

in which K = diagk1, k2. Hence, the closed-loop system, comprising (8.28)and (8.32), is a nonlinear composite system. Thus, one may write the closed-loop equations as

ξ = f(ξ) + g(ξ, ζ) (8.35a)

ζ = A2 ζ (8.35b)

where f(ξ), g(ξ, ζ) and A2 are found from (8.32) and (8.28). A block-diagram of the cascade system is depicted in Fig. 8.3. In other words, thesystem described by (8.32) is regarded as the nonlinear system ξ = f(ξ)cascaded with the linear system described by (8.28) through the intercon-nection term

g(ξ, ζ) =

[ge(ξ, ζ)gz(ξ, ζ)

]ζ (8.36)

To prove global asymptotic stability of (ξ, ζ) = 0, we invoke Lemmas 6.3and 6.4 (stated in Section 6.5.3).

The perturbing system Σ2 described by (8.28) is globally exponentiallystable if k0, k3 > 0. This is established by choosing a positive definite,radially unbounded Lyapunov function W2 = z2

0 + z23 . Hence, to prove that

the origin of the closed-loop system is asymptotic stability reduces to provethat the origin of ξ = f(ξ) is asymptotic stability and that all the solutionsof the closed-loop system are bounded.

Lemma 8.1 formally expresses the circumstances under which the originof (8.32) when ζ = 0 (i.e. the origin of the system ξ = f(ξ)) is establishedto be globally asymptotically and locally exponentially stable.


Lemma 8.1. Under conditions T.8.1.1 and T.8.1.2, the origin of the sys-tem described by ξ = f(ξ) is globally asymptotically and locally exponen-tially stable. It is established using a quadratic Lyapunov function.


Thus, from Lemma 6.3, it follows that the origin of the closed-loop sys-tem (8.35) is globally asymptotically stable if all the solutions are bounded.To prove boundedness of all the solutions, we verify A.6.4.1 and A.6.4.2.

The interconnection term g(ξ, ζ), given by (8.36), vanishes at ζ = 0 andis globally Lipschitz in ξ for any fixed ζ. It follows from Property 8.1 inAppendix 8.A that g(ξ, ζ) has linear growth in ξ and satisfies AssumptionA.6.4.1. According to Lemma 8.1, a radially unbounded polynomial Lya-punov function is used to show the globally asymptotic stability of the originof the unperturbed system (8.32); therefore, Assumption A.6.4.2 is satisfied.Hence, all the solutions are globally bounded and accordingly (ξ, ζ) = 0 isglobally asymptotically stable.

Similarly, one can prove that the origin of the closed-loop system islocally exponentially stable for every ball of the initial conditions accordingto (Loria and Panteley, 2005, Proposition 2.3).

Before proceeding to the next section, we would like to clarify that ac-cording to Assumptions T.8.1.1 and T.8.1.2, for any choice of the desiredspeed ud > 0 and the lookahead distance ∆ > 0, one may choose thecontroller gains ke1, ke2, k0, k1, k2 and k3 such that the objectives are ac-complished. That is, path following for fully actuated marine craft can beachieved for any arbitrary ∆ > 0 and ud > 0 whereas for underactuatedmarine craft, the guidance parameter ∆ depends on the structural proper-ties of the marine craft and should be chosen appropriately by considerationof the speed of the marine craft.

Also, note that the control performance may degrade for very small ∆.Too small ∆ leads to a large k which indicates that a high-gain controlleris required to stabilize the closed-loop system. It is worth mentioning thatz0 = ψ − ψd is globally exponentially stable at origin.

8.4.1 Properties of Proposed Controller

Now that we have established that the proposed control law accomplishesthe objectives, we elucidate the properties of the controller.


Manipulation of Speed:

The proposed controller (8.24) modifies the speed of the marine craft whenthe cross-track error, e, is nonzero. In other words, when the marine craftdoes not move on the path, the control law alters the linear velocities ac-cording to the magnitude and sign of e and e.

To see how it happens, using (8.25), one may show that the control lawcan be decomposed into two distinct parts; in fact, τ ∗b can be written as

τ ∗b = τ n + τ e (8.37)

where

τ n =Mbσz + (Cb(ν) +Db(ν))ν

τ e =1

2Mbρ2(γ)

(−ρ2(γ)Tσz + σ∗e

)The control law τ n is the control law that one obtains if (8.10) is notconsidered. Much of work on path following of marine craft has introducedsuch controllers; for example, (Fossen et al., 2003; Fredriksen and Pettersen,2006)6. The control force τ n intends to regulate z0 and z.

However, in the proposed path-following controller, τ e makes a differ-ence. The term σ∗e is nonzero when e and e are nonzero. On the otherhand, due to the structure of ρ2(γ), τ e only affects the dynamics of thelinear velocities, and does not influence the heading dynamics. Therefore,the proposed controller adjusts the speed of the vehicle according to the dis-tance to the path and the rate of convergence. Hence, the speed assignmentobjective is sacrificed so as to fulfil the path-following (geometric) task.

Robustness to Ocean Currents

The proposed method makes the geometric task robust with respect to exter-nal disturbances to some extent because the controller changes the vehicle’sspeed when e 6= 0; therefore, it exerts more forces on the craft and resultsin smaller error. More important, it is possible to obtain zero cross-trackerror in the presence of constant disturbances by means of augmentation ofintegral action to the control system. Augment

eI = ude√

e2 + ∆2(8.38)

6These works have mainly focused on underactuated marine craft; however, they canbe adapted easily for fully actuated vehicles.


to the system described by (8.7). In (8.10), replace σe with σe,I which isgiven by

σe,I = −γud − ke2e− ke1e− ke0eI (8.39)

in which, with an argument similar to the previous section, ρ2(γ)Tν hasbeen replaced with γud. Then, form b∗I = [σT

z , σe,I]T and use it to derive

the control law

τ ∗b,I =MbH(γ)b∗I (γ,φ) + Cb(ν)ν +Db(ν)ν (8.40)

Theorem 8.2 states the result formally.

Theorem 8.2. Let ud and ∆ be positive constants. Apply the control law(8.40) to the system (8.2). The origin (eI, e, z0, z) = 0 is guaranteed to beglobally asymptotically stable if

T.8.2.1 k0, k3 > 0;

T.8.2.2 ke1 > ke0 > 0 and ke2 >ud

∆;

T.8.2.3 k1 = k2 = k such that k >3

4

ud

∆

k2e2

ke1 − ke0

Proof. See Appendix 8.C.

Theorem 8.2 states that if external disturbances shift the equilibriumpoint of the cross-track error to a nonzero constant, integration as proposedby (8.38) resolves the problem. It is worth mentioning that we add merelythe integral of the cross-track error to the control system. Therefore, al-though it is assured that the geometric objective is obtained, there is noguarantee that the speed-assignment task is achieved in the presence of ex-ternal disturbances. In other words, the controller modifies the speed of thecraft so as to achieve the geometric objective.

Due to the presence of error integration, care should be taken to avoidintegral wind-up. As |e|/

√e2 + ∆2 < 1, a large initial cross-track error does

not lead to a large eI.It is worth noting that the line-of-sight guidance method suffers from an

inherent drawback when, in the presence of ocean currents, it is used to alignthe x-axis of the body-fixed reference frame b with the LOS vector. As aresult of external disturbances, the marine craft rotates to provide sufficientforce to counteract the component of the current that is perpendicular to thepath. It causes a nonzero steady-state cross-track error, e. In fact, nonzero


e is inevitable since it takes a while for the vessel to rotate and produceforces to counteract disturbances. It leads to the fact that the vessel doesnot move on the path and the equilibrium of the error system is not zero.

After transient, the vessel moves on a path which is parallel to thedesired path, and there is no mechanism in the control system to make thevessel capable of reaching the desired path. As the vessel is moving on apath parallel to the desired path, one may conclude that χ , χ − ψd = 0;i.e. the total speed is aligned with the path. Because χ = ψ + β where βis the sideslip angle, one obtains ψ + β − ψd = ψ + β − ψr − ψk = 0. Thus,ψ − ψk = ψr − β; i.e. the extent that the vessel rotates with respect to thepath is equal to tan−1(−e/∆)− β.

There has been proposed several remedies to cope with this problem.One method is to compensate for β by aligning the total speed with theLOS vector. It means that the desired course angle χd is ψlos, which impliesthat ψd has to track ψlos−β. In the presence of currents, the side slip anglecan be found using β = arctan( v−vcu−uc ), which requires the knowledge of thespeed of ocean currents. Some papers intend to estimate ocean currents andtheir generated forces (Aguiar and Pascoal, 2007; Encarnacao and Pascoal,2001). Also, the control system may require the knowledge of β which leadsto a need for measuring or estimating the accelerations; see (Lapierre andSoetanto, 2007) for such a solution where accelerations are computed byrelying on the system model.

Augmentation of integral action to the yaw control force is proposed in(Fossen, 2011). However, the problem does not arise from an inaccurateheading control. Fossen (2011) also proposes the following control law

τr = τn,r − kde− kpe− ki∫ t

0e(%)d% (8.41)

where τn,r is appropriately designed to tackle nonlinearity of the dynamics.Alternatively, the integral LOS guidance strategy is put forward by Børhauget al. (2008) as

ψr = tan−1

(κe+ σ

∫ t

0eint(%)d%

)(8.42)

in which the relative angle ψr is modified to allow the craft to have a sideslipangle while e = 0. In fact, the method aims to correct the desired headingangle ψd. Inclusion of integral action to the proposed controller yields an-other solution, which intends to provide additional force by modifying thespeed so that the vessel can get back to the path and moves along it.



A ship’s model is chosen according to (8.2) where

Mb =

2376.4 0 00 3949.9 2891.80 2891.8 3349.8

, Db =

354 0 00 346.8 −435.80 686.1 1427.2

The initial conditions are chosen as q(0) = [10, −250, π/4]T and ν(0) =[1, 0, 0]T. The objective is to converge to and follow a straight-line pathwhich is parallel to the y-axis of i, 40 meters to the north. It is as-sumed that there exists a current flow whose speed expressed in i isUc = [+.75, 0, 0]T(m/s). The relative velocity νr = ν − J (ψ)TUc is con-sidered in the simulation model. We choose ud = 2 (m/s) and ∆ = 20 (m).Thus, ke2 > 0.1; we choose ke1 = 2 and ke2 = 1. If integrator is considered,ke0 = 0.5. Then, T.8.2.2 implies that k > 3; we choose k = 10. Also,k0 = 3 and k3 = 1. It is observed that the conditions of the theorems arenot restrictive for practical situations.

We make a comparison between a controller with integral action (labeledby ‘LS with integral action’) and a controller without considering integralaction (represented by ‘LS without integral action’) to discern the distur-bance rejection properties of the control systems. We also run simulationwith the method presented in (Fossen et al., 2003) (labeled as ‘Standardmethod’) but we adapted the method for fully actuated vessels. The resultis shown in Figs. 8.4, 8.5 and 8.6.

As expected, the least-squares approach with augmentation of integralaction results in a zero steady-state cross-track error while the other er-rors are nonzero. It is also realized that the least-square approach leads tofaster convergence with respect to the standard method. As explained be-fore, explicit incorporation of the geometric error in the design of the speedcontroller will lead to more robust response to external disturbances as thesteady-state cross-track error is smaller than that of the standard method.The price to pay is to use control efforts with large amplitude and to havelarge sideslip angles due to large sway velocities.

Figure 8.6 shows the state of the AUV in the two-dimensional space.Augmentation of the integral action which leads to zero cross-track errorsmay cause larger sideslip angles than those that are generated by the stan-dard method.

8.5. Simulation Results 215

Standard method

LS without integral action

LS with integral action

erro

r[m

]

Time (sec)

Cross-Track Error (e)

erro

r[d

eg]

z0 = ψ − ψd

0 50 100 150 200

0 50 100 150 200

−10

0

10

20

30

0

10

20

30

Figure 8.4: The heading error and the cross-track error.

Standard method

LS without integral action

LS with integral action

Time (sec)

erro

r[m

/sec

]

Sway velocity (v)

erro

r[m

/sec

]

Forward Speed Error (u− ud)

0 50 100 150 200

0 50 100 150 200

−5

−4

−3

−2

−1

0

−0.5

0

0.5

Figure 8.5: The speed assignment task.


Nor

th(m

)

East (m)

−250 −240 −230 −220 −210 −200 −190 −180 −170 −160

0

10

20

30

40

50

60

70

(a) LS with integral action

Nor

th(m

)

East (m)

−260 −240 −220 −200 −180 −160 −140 −120

−20

0

20

40

60

80

(b) Standard Method

Figure 8.6: The path that is traveled by the AUV.

8.6 Conclusions

A nonlinear controller is proposed in order to achieve two-dimensional pathmaneuvering for fully actuated marine craft. The controller is able to adjustthe speed of a marine craft according to the geometric error and the rate ofconvergence.

To derive the control law, the method of least squares to approximateoverdetermined systems is utilized. In fact, the control law is derived sothat the marine craft acquires the accelerations that approximately i) forcethe marine craft to reach to the desired path with an exponential rate andii) satisfy the heading and speed objectives. Using the theory of the cascadeinterconnected systems, the stability of the closed-loop system is established.

Having the speed of the craft as a function of the geometric error, also,leads to a method to resolve the inherent drawback of line-of-sight guidancebased path-following controllers. The controller forces the craft to changethe linear velocities in order to counteract the adverse effect of externaldisturbances. Thus, it provides a simple method which is model-free anddoes not need acceleration measurement.

In contrast to the method that was proposed in Chapter 7, the speed ofthe marine vehicle is linearly proportional to the cross-track error and itsderivative.

8.A. Required Relations 217

Appendix 8.A Required Relations

Considering (8.30), we have

gsin(ψr, z0) =sin(z0)

z0cos(ψr) +

cos(z0)− 1

z0sin(ψr)

gcos(ψr, z0) = −sin(z0)

z0sin(ψr) +

cos(z0)− 1

z0cos(ψr)


and cos(z0)−1z0

are well-defined functions and globally bounded.In this regard, one may find

ρ2(γ) = ρ2(ψr) +Rg(ψr, z0)ζ

in which Rg = [ρ2g(ψr, z0), 0] where

ρ2g(ψr, z0) =

[gsin(ψr, z0)gcos(ψr, z0)

]Accordingly, in (8.32a), one may find

ge(ξ, ζ) = [(ud + z1)gsin(ξ, ζ) + z2gcos(ξ, ζ), 0]

The function gz(ξ, ζ) in (8.32b) is equal to

gz(ξ, ζ) = 12 ρ2(γ) ([k0ud, −ud]− Ω1ge(ξ, ζ))− 1

2G∗z(ξ, ζ)

− 12Rg(ke1e+ Ω1(ud + z1) sin(ψr) + Ω1z2 cos(ψr)))

where G∗z(ξ, ζ) = [Gz0(ξ, ζ)Kz, 0] in which

Gz0(ξ, ζ)z0 = ρ2(γ)ρT2 (γ)− ρ2(ψr)ρ

T2 (ψr)

is a 2× 2 matrix. Let gz,ij be entry (i, j) of Gz0(ξ, ζ). Then, one may find

gz,11 = z0g2sin(ψr, z0) + 2gsin(ψr, z0) sin(ψr)

gz,22 = z0g2cos(ψr, z0) + 2gcos(ψr, z0) cos(ψr)

gz,12 = gz,21 = gsin(2ψr, 2z0)

The following property is easily established.

Property 8.1. The functions ge(ξ, ζ) and gz(ξ, ζ) grow linearly in ξ; i.e.

‖gx(ξ, ζ)‖ ≤ σx1(‖ζ‖) + σx2(‖ζ‖)‖ξ‖, x = e, z

where σx1, σx2 : [0,∞)→ [0,∞) are continuous.



Before proceeding to prove, we need to clarify the following points. Ac-cording to T.8.1.1, Ω1, given by (8.34), is always a positive value; i.e.ke2 > Ω1 ≥ Ω∗1 > 0, ∀e.

Property 8.2. The following inequality holds

xT(I2 −1

2ρ2(ψr)ρ

T2 (ψr))x ≥

1

2xTx ∀x =

[xy

]6= 0

Rewrite ρ2(ψr) in view of (8.31). Then, (8.32b) is recast as:

˙z = −Kz(ψr)z −1

2

[−e∆

](ke1e√e2 + ∆2

− Ω1

(e2 + ∆2)((ud + z1)e−∆z2)

)Let k1 = k2 = k > 0. Choose V1 = 1

2 zTKz where K = kI2 > 0. Differentia-

tion yields

V1 = −KKz(ψr)z +k

2

(ke1√e2 + ∆2

− Ω1ud

e2 + ∆2

)(e2z1 − ez2∆)

− kΩ1

2

(e2z2

1

e2 + ∆2− 2ez1z2∆

e2 + ∆2+

∆2z22

e2 + ∆2

)(8.43)

Choose V2 = 14kke1e

2 where ke1 > 0 and take derivative along the solutionof (8.32a):

V2 = −kke1

2

ud√e2 + ∆2

e2 − k

2

ke1√e2 + ∆2

(e2z1 − ez2∆

)(8.44)

Select V = V1 + V2 as a positive definite, radially unbounded Lyapunovfunction candidate, and take derivative with respect to time. In light of thefact that the second line of (8.43) is non-positive, V is bounded by:

V ≤ −1

2k2zTz − kke1

2

ud√e2 + ∆2

e2 +kΩ1ud

2(e2 + ∆2)(e2|z1|+ |e||z2|∆)

As 0 < ∆√e2+∆2

≤ 1 for all e, the next inequalities hold

− 1√e2 + ∆2

≤ − ∆

e2 + ∆2⇒ ∆

e2 + ∆2≤ 1√

e2 + ∆2(8.45)

8.C. Proof of Theorem 8.2 219

Therefore, we obtain a bound on V as

V ≤ −1

2k2zTz − kke1

2

ud∆

e2 + ∆2e2

+kΩ1ud

2

e2

e2 + ∆2|z1|+

kΩ1ud

2

|e|√e2 + ∆2

|z2| (8.46)

One can write it as

V ≤ −1

2k2zTz − kke1

3× 2

ud∆

e2 + ∆2e2 − kke1

3× 2

ud∆

e2 + ∆2e2

+kΩ1ud

2

e2

e2 + ∆2|z1| −

kke1

3× 2

ud∆

e2 + ∆2e2 +

kΩ1ud

2

|e|√e2 + ∆2

|z2|

= −1

2k2zTz − kke1

3× 2

ud∆

e2 + ∆2e2 − kud

2

e2

e2 + ∆2(ke1∆

3− Ω1|z1|)

− kud

2(ke1∆

3

e2

e2 + ∆2− Ω1|z2|

|e|√e2 + ∆2

)

and complete the squares. As 0 ≤ e2

e2+∆2 < 1, we obtain

V ≤ −kke1ud∆

3× 2

e2

e2 + ∆2− k

2(k − 3udΩ2

1

4ke1∆)(z2

1 + z22)

Under assumption T.8.1.2, one can find positive constants γ1, γ2 and γ3

such that

V ≤ −γ1z21 − γ2z

22 − γ3

e2

e2 + ∆2

Thus, V < 0 for nonzero z1, z2 and e, which proves the unforced system(8.32) is globally asymptotically stable at zero.

For every e which in inside the ball with radius δ (i.e. |e| < δ), we obtain

V ≤ −γ1z21 − γ2z

22 − γ∗3(δ)e2

since − e2

e2+∆2 ≤ − e2

δ2+∆2 . Therefore, the system is exponentially stable forany ball. Hence, the system is K-exponential stable because it is globallyasymptotically and locally exponentially stable for any ball.

Appendix 8.C Proof of Theorem 8.2

Let ζ = [z0, z3]T and χ = [eI, e, z1, z2]T. The application of (8.40) to thesystem gives rise to

Σ2 :

z0 = −k0z0 + z3

z3 = −z0 − k3z3

(8.47a)

(8.47b)


eI = ude√

e2 + ∆2(8.48a)

e = − ud + z1√e2 + ∆2

e+z2∆√e2 + ∆2

+ ge(ξ, ζ)ζ (8.48b)

˙z = −Kz(ψr)z − 12 ρ2(ψr)ke0eI − 1

2 ρ2(ψr)ke1e (8.48c)

+ 12 ρ2(ψr)Ω1(

ud + z1√e2 + ∆2

e− z2∆√e2 + ∆2

) + g∗z(χ, ζ)ζ

where according to T.8.2.1, Ω1, given by (8.34), is always a positive value;i.e. ke2 > Ω1 ≥ Ω∗1 > 0, ∀e. The closed-loop equations are composedof (8.47) and (8.48). Therefore, similar to the proof of Theorem 8.1, theclosed-loop system is regarded as a cascade composite system as

χ = f∗(χ) + g∗(χ, ζ) (8.49a)

ζ = A2 ζ (8.49b)

where

f∗(χ) = f(ξ)− 12ke0eI

00

ρ2(ψr)

, g∗(χ, ζ) =

0ge(ξ, ζ)g∗z(χ, ζ)

and g∗z(χ, ζ) = gz(ξ, ζ) − 1

2Rgke0eI. Clearly, g∗z(χ, ζ) has linear growth inχ, so does g∗(χ, ζ) in accordance with Property 8.1.

Obviously, the perturbing system ζ = A2 ζ is globally exponentially sta-ble for any positive k0 and k3, as discussed in the proof of Theorem 8.1. Con-sidering this fact along with the fact that g∗(χ, ζ) has linear growth in χ,in order to show that the origin of the closed-loop system is globally asymp-totically stable, one requires to show that the origin of the unperturbedsystem χ = f∗(χ) is globally asymptotically stable with a Lyapunov func-tion which satisfies Assumption A.6.4.2 of Lemma 6.4. Lemma 8.2 statesthis fact accurately.

Lemma 8.2. Under conditions T.8.2.1 and T.8.2.2,

• There exists a polynomial positive definite, radially unbounded Lya-punov function W (χ) which proves the global stability of χ = f∗(χ)with Lf∗W ≤ 0.

• The origin of the system described by χ = f∗(χ) is globally asymp-totically stable.

8.C. Proof of Theorem 8.2 221

Proof. Let k1 = k2 = k. Choose W1 = 12 z

TKz where K = kI2 > 0. Takederivative along the solutions of (8.48):

W1 = −KKz(ψr)z +kke0

2

eI√e2 + ∆2

(ez1 − z2∆)

+k

2

(ke1√e2 + ∆2

− Ω1ud

e2 + ∆2

)(e2z1 − ez2∆)

− kΩ1

2

(e2z2

1

e2 + ∆2− 2ez1z2∆

e2 + ∆2+

∆2z22

e2 + ∆2

)(8.50)

Consider ke1 > ke0 > 0 so that Ω2 = ke1 − ke0 > 0. Thus, the function

W2 =k

4

[eI e

] [ ke0 ke0

ke0 ke1

] [eI

e

]is positive for e 6= 0 and eI 6= 0. Differentiation yields

W2 = −kudΩ2

2

e2

√e2 + ∆2

− kke0

2

eI√e2 + ∆2

(ez1 − z2∆)

− k

2

ke1√e2 + ∆2

(e2z1 − ez2∆)

The last line of (8.50) is non-positive. An upper bound for the time deriva-tive of W = W1 + W2 as a radially unbounded, positive definite Lyapunovfunction is given by

W ≤ −1

2k2zTz − kΩ2

2

ud√e2 + ∆2

e2 +kΩ1ud

2(e2 + ∆2)(e2|z1|+ |e||z2|∆)

Considering (8.45), one may find

W ≤ −1

2k2zTz − kΩ2

2

ud∆

e2 + ∆2e2

+kΩ1ud

2(e2 + ∆2)e2|z1|+

kΩ1ud

2√e2 + ∆2

|e||z2| (8.51)

We follow the same procedure as taken in the proof of Theorem 8.1 tocomplete squares. It gives rise to

W ≤ −kΩ2ud∆

3× 2

e2

e2 + ∆2− k

2(k − 3udΩ2

1

4Ω2∆)(z2

1 + z22)

Assumption T.8.2.2 ensures the existence of the positive constants γ1, γ2

and γ3 such that

W ≤ −γ1z21 − γ2z

22 − γ3

e2

e2 + ∆2


Thus, W ≤ 0. It proves global stability of χ = 0. It is straightforward toutilize LaSalle’s invariance principle in order to prove that χ = 0 is globallyasymptotically stable.

We point out that the globally asymptotic stability of the origin of theclosed-loop system is established in the absence of disturbances; so, if thedisturbance changes the equilibrium point of e, it is guaranteed that e isenforced to zero7 while there is no guarantee to make z0, z1 and z2 zero inthe presence of disturbances. It completes the proof.

7If it is shown that the system is input-to-state stable, in the presence of externaldisturbances, eI is to be bounded. It implies that e converges to zero at time tends toinfinity.

Chapter 9

Speed-varying PathManeuvering:A Least Squares Approach

(Extension to Underactuated Marine Craft)

This chapter is dedicated to extend the result of Chapter 8 to marinecraft that are unactuated in sway. In particular, the two-dimensionalpath-following problem for sway-unactuated 3-DOF marine craft istaken into account, and a nonlinear control law is proposed such thatthe speed of the marine craft is modified according to the geometricerror and its derivative. The proposed control law is derived using themethod of least squares to find an approximate solution of overdeter-mined systems. We show the circumstances under which the proposedcontroller achieves path maneuvering. The origin of the error dynam-ics is proven to be globally asymptotically and locally exponentiallystable. The properties of the proposed design are also clarified. Theperformance of the controller is evaluated with computer simulations.This chapter is based on (Peymani and Fossen, 2013a).

9.1 Introduction

Advanced control of underactuated marine craft (UMC) (having fewer actu-ators than degrees of freedom to be controlled) has been an active researchfor the past two decades. Underactuated marine craft are widespread inpractical situations. Ships are typically equipped either with two indepen-dent aft thrusters or with one aft thruster and one rudder. Even if bow,side or tunnel thrusters are mounted, they are not efficient at high speeds.

223

224 Path Maneuvering for UMC: A Least Squares Approach

In addition, it is not often practical to fully actuate autonomous vehiclesowing to power restriction, weight, complexity, reliability, and efficiency.

In trajectory tracking, every state of an underactuated marine craft hasto track a reference trajectory that specifies the time evolution of the state.However, in the path-maneuvering problems, an underactuated marine crafthas to obtain a specified position and orientation without imposing anytiming law; therefore, it is not determined when the craft has to be at agiven point1.

The underlying assumption in the design of path-following controllers isthat the speed-assignment task along the path can be done independentlyof the geometric task of convergence to the path. Almost all work on pathfollowing for marine craft design independent speed controllers and headingautopilots2. It implies that the speed of the craft is controlled regardless ofits distance to the path. However, in practice, one may increase/decreasethe speed to reach the path faster or to avoid sliding sideways.

The topic of the chapter is to extend the result of Chapter 8 to marinecraft that do not have an independent actuator acting on the sway dynamics.In Chapter 8, a controller is proposed for straight-line path following of 3-DOF fully actuated marine craft, which modifies the speed of the marinecraft according to the distance to the path and the rate of convergence.To derive the controller, first, we find the linear accelerations that makethe vehicle converge to the path exponentially fast. Second, we obtain theaccelerations that render the speed error dynamics and the heading errordynamics exponentially stable at the origin. Then, using the method ofleast squares, the acceleration vector that approximates the two obtainedacceleration vectors is derived. The control law is found so that the marinecraft has the approximated vector of acceleration in every instant of time.

Now, we extend this concept to underactuated marine craft where onlytwo controls are used to control three DOFs. Therefore, the objective is tofind a path-following controller that forces the marine craft to move towardsthe path as smoothly as desired, and alters the speed of the marine craftaccording to (i) the distance to the path, and (ii) the rate of convergenceto the path. When the craft moves on the path, it has to have the de-sired speed. The controller should guarantee the stability of the unactuated

1A comprehensive discussion on the properties of underactuated marine craft, variousmotion scenarios, and a literature review on motion control of underactuated marine craftwas provided in Chapter 6.

2Heading autopilots use the geometric distance whereas speed controllers only usespeed information.

9.2. Path Maneuvering Problem 225

dynamics. In order to compensate for underactuation, we consider a stabi-lizing surge force. This algorithm proposes a simple method for cooperativepath following and formation control.

Also, it is reminded that, in Chapter 7, a controller that increases thespeed of underactuated marine craft according to the shortest distance tothe path was put forward, and it was shown how the guidance parametershould be selected to guarantee convergence.

9.2 Path Maneuvering Problem

The straight-line path-maneuvering problem for 3-DOF underactuated ma-rine craft is studied. In particular, a controller is designed such that amarine vehicle converges to and follows a desired path with a desired speed.According to Skjetne et al. (2004), the path-maneuvering scenario can bedecomposed into two tasks; namely, geometric task and speed-assignmenttask. The geometric task is to make the position and orientation of themarine craft as desired. The speed-assignment task is to make the speedof the craft acquire the desired value. Moving on the path is more im-portant than moving with the desired speed; therefore, the geometric tasktakes precedence over the speed-assignment task, which indicates that thedynamic task of speed assignment may be sacrificed in order to accomplishthe geometric task better, in some sense. According to (Skjetne et al., 2004),these two tasks can be executed separately.

9.2.1 Model of 3-DOF Marine Craft

Consider the vehicle pose η = [pT, ψ]T where p = [x, y]T ∈ R2 is the earth-fixed position and ψ ∈ S , [0, 2π] is the yaw angle; η is described relativeto the inertial reference frame, represented by i. Consider the body-fixedreference frame, denoted b, as described in Chapter 6. Let ν = [u, v, r]T ∈R3 be the vector of velocities expressed in b, where u and v are the linearvelocities and r is the angular velocity. Let J (ψ) = diagR(ψ), 1 be therotation matrix from b to i. The matrix R(ψ) is given by

R(ψ) =


]∈ SO(2) (9.1)

The linearized maneuvering equations in surge, sway and yaw are given by(Fossen, 2011, Chap. 6)

η = J (ψ)ν (9.2a)

Mbν +Dbν = τ b (9.2b)


where nonlinear Coriolis and centripetal forces are linearized about thecruise speed U and nonlinear damping is approximated by a linear dampingmatrix. This is a reasonable assumption for straight-line path following. Aprevalent assumption is that surge is decoupled from sway-yaw. Thus, thesystem matrices are constant and take the following structures

Mb =

m11 0 00 m22 m23

0 m23 m33

, Db =

d11 0 00 d22 d23

0 d32 d33

(9.2c)

where

m11 = m−Xu, d11 = −Xu

m22 = m− Yv, d22 = −Yvm23 = mxg − Yr, d23 = (m− Yv)U − Yrm32 = mxg −Nv, d32 = −Nv

m33 = Iz −Nr, d33 = (mxg − Yr)U −Nr

We also assume that the unactuated dynamics are open-loop stable; thatis, we assume

d22 > 0 (9.2d)

In (9.2b), τ b , [τu, τv, τr]T represents the vector of generalized forces, ex-

pressed in b, which acts on the system and captures forces and momentsdue to the actuators. If the craft is unactuated in sway, there exist no inde-pendent control forces acting on the sway dynamics; one may consider thatτv = 0. Thus, for underactuated marine craft, the vector of forces is

τ b = [τu, 0, τr]T (9.2e)

In practice, a vessel may be equipped with rudders which produce τv inaddition to τr; one may consider τv = στr for some σ 6= 0. A coordinatetransformation was proposed by Do and Pan (2003) to remove the effect ofτr from the unactuated dynamics. An alternative approach was proposedby Peymani and Fossen (2012). See Chapter 6 for a discussion about thistopic.

9.2.2 Guidance System

A guidance system is required to provide the desired heading so that thevessel moves toward the path as smoothly as desired3. In fact, the guidance

3A more comprehensive study of the line-of-sight guidance method has been given inChapter 7.

9.2. Path Maneuvering Problem 227

iy

ix

kp

1kp

by

V

e

iy

ix

s

k

r

LOS vector

losp

p

bx


system maps the desired position onto the desired heading angle. Therefore,it provides a two dimensional workspace that enables the underactuatedmarine craft to follow the path by using only two controls (which are thesurge speed and the yaw moment). In other words, the vessel obtains thedesired heading by means of rudders in order to reach the path and toachieve the geometric objective while it acquires the desired speed by meansof propellers in order to accomplish the speed assignment task. We employthe line-of-sight (LOS) guidance system (Fossen, 2011, Ch.10).

Consider a straight-line path connecting points pk and pk+1. The slopeof the path is denoted ψk ∈ S. Also, consider a path-fixed reference frame,denoted pk, that originates at pk. Its x-axis has been rotated by a positiveangle ψk with respect to the inertial frame i.

Define ε , [s, e]T where e(t) is the cross-track error which is the shortestdistance between the craft and the path, and s(t) is the along-track errorwhich is the distance between pk and the projection of the craft on the path.One can express ε in pk as

ε = R(ψk)T (p− pk)

Let plos = [xlos, ylos]T be the desired point on the path that the vessel has

to reach at each time instant. To find the point plos, the lookahead-basedsteering method (Fossen, 2011, Chapter 10) is utilized. Accordingly, plos

is a point on the path which is located a lookahead distance ∆ > 0 aheadof the direct projection of p onto the path. See Fig. 1 for a geometricrepresentation of the problem.


The LOS vector is the vector from p to plos. The LOS angle, denotedψlos, is the angle that the LOS vector makes with the x-axis of the inertialreference frame i:

ψlos = ψk + ψr

where ψk is the slope of the path, and ψr is the relative angle (approachangle) which is found using:

ψr = arctan(− e

∆

)(9.4)

The objective of the guidance system is to align the x-axis of b with theLOS vector at each instant of time. Equivalently, the heading (yaw) anglehas to track the LOS angle ψlos.


The primary objective is to converge to the path and follow it. Convergenceto the path, which is referred to as the geometric task, is formulated as

limt→∞

e(t) = 0 (9.5a)

It is desired that the marine craft converges to the path smoothly; so, theheading angle has to track a desired angle; that is:

limt→∞

(ψ − ψd) = 0 (9.5b)

We choose the desired heading angle as ψd = ψlos. The secondary objectiveis the speed assignment task in which the forward velocity is to track adesired profile ud(t) > 0, ∀t. It is stated as:

limt→∞

(u− ud) = 0 (9.5c)

By the secondary objective, we mean that the dynamic task of speed assign-ment has less importance than the geometric task, and it can be sacrificed soas to have the main objective (which is the geometric task) perfectly/rapidlysatisfied. Furthermore, the unactuated sway dynamics must be globallybounded:

|v(t)| ∈ L∞, limt→∞

v = 0 (9.5d)

Path-Following Problem Consider a 3-DOF underactuated marinecraft described by (9.2). Given a path P and a desired speed ud(t), the

9.3. Control Development 229

path-following control problem is to find a stabilizing controller such thatthe objectives (9.5) are accomplished. ehsan J

The standard solution is to design a feedback law τu = τu(u) to gen-erate a surge force and a feedback law τr = τr(ψ, r, e) to produce a yawmoment. That is, the speed controller uses only the speed information andis decoupled from the heading controller; see e.g. (Fossen et al., 2003) and(Fredriksen and Pettersen, 2006).

However, in this work, we intend to find a controller that manipulatesthe speed so that the vehicle catches the path accurately/qucikly whereasthe craft has to move with the desired speed when it is on the path; there-fore, the speed assignment should be carefully dependent on the cross-track(geometric) error. It resembles finding a desired speed profile that dependson the cross-track error and its derivative, denoted by u∗d(ud, e), such thatlime→0|u∗d−ud| = 0. Then, the controller should achieve limt→∞(u−u∗d) = 0in addition to the objectives (9.5). To be accurate, we pose Problem 9.1 asstated below.

Problem 9.1. Consider a 3-DOF underactuated marine craft described by(9.2). Given a straight-line path P and a desired speed ud(t), i) design afeedback law to generate a yaw moment τr = τr(ψ, r, e); ii) use the geometricinformation to design a feedback law τu = τu(u, e) to produce a surge forcesuch that the objectives (9.5a), (9.5b), (9.5c) and (9.5d) are accomplished.e ere e e e J

9.3 Control Development

In this section, we develop the control law and present the main result ofthe article. Let γ = ψ − ψk. Considering the look-ahead distance ∆ ∈ R+

where R+ is the set of positive real numbers, define

me = 1− m23

m22

∆

∆2 + e2cos(γ) (9.6)

Choose ke2, ke1 ∈ R+. Define

σe = −γ ud − ke2e−ke1

mee (9.7)

Let z0 = ψ − ψd and z = ν − α where z ∈ R3 and α ∈ R3 which can beshown as

z =

z1

z2

z3

, α =

ud

0

ψd − k0z0


It implies that z2 = v. Consider K , diagk1, 0, k3 where k1, k3 ∈ R+.Define

σz = α− ρ3z0 −Kz (9.8)

Let ρ1(ψ), ρ2(ψ), and ρ3 be as

ρ1(ψ)T = [cos(ψ), − sin(ψ), 0]ρ2(ψ)T = [sin(ψ), cos(ψ), 0]ρT

3 = [0, 0, 1]⇒ J (ψ) =

ρ1(ψ)T

ρ2(ψ)T

ρT3

Form the matrix H ∈ R4×3 and the vector b ∈ R3 as below

H(γ) =

[I3

ρ2(γ)T

], b(γ,φ) =

[σz

σe

](9.9)

where Ii is the i × i identity matrix, and φ , [e, e, z0, zT]T. We define

H(γ) ∈ R3×4 as

H(γ) =(H(γ)TH(γ)

)−1H(γ)T

H(γ) exists since H(γ)TH(γ) = I3 + ρ2(γ)ρ2(γ)T is non-singular ∀γ, and(H(γ)TH(γ)

)−1= I3 −

1

2ρ2(γ)ρ2(γ)T (9.10)

We construct the following vector of forces, denoted τ ls ∈ R3:

τ ls =MbH(γ)b(γ,φ) +Db(ν)ν (9.11)

We define

w2 = z2 +m23

m22ψd , σ =

d23

d22− m23

m22(9.12)

Let τs1 = σ ∆e2+∆2

1me

w2√e2+∆2

e. Construct the vector of force

τ s =

τs1

00

Then, form

τ = τ ls + τ s, τ =

τb1

τb2

τb3

(9.13)


As τ belongs to R3, one may represent τ = [τb1, τb2, τb3]T. We can onlyapply τb1 and τb3 to an underactuated marine craft. That is, τu = τb1 canbe provided by the thrusters, and τr = τb3 can be provided by the rudders.Since the craft is not actuated in sway, τb2 cannot be applied to the craft.Theorem 9.1 provides the conditions under which the application of τb1 andτb3 accomplishes the objectives (9.5). Before that, denote the maximumand minimum of me with mmax and mmin, respectively; i.e.

mmin ≤ me ≤ mmax

where mmin = 1− m23

m22

1

∆and mmax = 1 +

m23

m22

1

∆. Define

dα , (1− σ ∆

e2 + ∆2

1

me

∆√e2 + ∆2

)

Theorem 9.1. Consider an underactuated marine craft of the form (9.2).Let ud be a positive constant. Apply τb1 and τb3, found in (9.13), as thesurge force and the yaw moment to the marine craft. The origin (e, z0, z) =0 is uniformly globally asymptotically and uniformly locally exponentiallystable if

T.9.1.1 k0, k3 > 0.

T.9.1.2 ke1, ke2 and ∆ are chosen such that

∆3mmax

m2minke1ud

(ke1 + |σ|2ud

∆2)2 +

ke2

mmin< 8(1− σ

mmin∆)

while ke2 ≥ ud∆ and ∆ is sufficiently large that mmin > 0 and 1− σ

mmin∆ > 0.

T.9.1.3 k1 is selected such that k1 >ud

ke1∆

3mmax

8m2min

k2e2

Proof. The proof of the theorem relies on the theory of nonlinear compositesystems (Jankovic et al., 1996) and (Loria and Panteley, 2005). First, wefind the closed-loop equations. To that end, one may take derivative of ε,which gives ε = J (γ)ν; then, it is obtained that

e = ρ2(γ)Tν = (z1 + ud) sin(γ) + z2 cos(γ) (9.14)

Consider the new variable w2, defined in (9.12), where

ψd = − ∆

∆2 + e2e


Therefore, (9.14) is recast as

mee = (z1 + ud) sin(γ) + w2 cos(γ) (9.15)

Note that γ = ψ − ψk = z0 + ψr. Thus, one can write

sin(z0 + ψr) = sin(ψr) + gsin(e, z0)z0 (9.16a)

cos(z0 + ψr) = cos(ψr) + gcos(e, z0)z0 (9.16b)

Functions gsin(e, z0) and gcos(e, z0), given in Appendix 9.A, are globallybounded. According to the guidance law (9.4), we have

sin(ψr) =−e√

e2 + ∆2, cos(ψr) =

∆√e2 + ∆2

(9.17)

Define ζ , [z0, z3]T and ξ , [e, z1, w2]T. Now, we re-write the dynamics ofthe cross-track error (9.15) as:

mee = − ud + z1√e2 + ∆2

e+w2∆√e2 + ∆2

+ ge(ξ, ζ)ζ (9.18a)

where ge(ξ, ζ) is given in Appendix 9.A. The sway dynamics, denoted v = z2,is not actuated; i.e. τv = 0. Therefore, according to the model (9.2), onemay write

m22z2 +m23r + d22z2 + d23r = 0

It is straightforward to verify that r = z3 + ψd − k0z0. The dynamics of w2

are now given by

m22

d22w2 = −dαw2 + σ

∆

e2 + ∆2

1

me×

(− ud√e2 + ∆2

e− z1√e2 + ∆2

e) + gw(ξ, ζ)ζ (9.18b)

where gw(ξ, ζ) is given in Appendix 9.A.We intend to find the dynamics of z0, z1 and z3. Clearly, one may find

z0 = −k0z0 + z3. To derive the dynamics of z1 and z3, we look at thedynamics of z; however, one should notice that z2 which is given in thisvector is not valid since the marine craft is not actuated in sway, and τb2

cannot be applied to the system. Having this point in mind, we apply τ tothe craft (9.2), which yeilds

ν = (I3 −1

2ρ2(γ)ρT

2 (γ))σz +1

2ρ2(γ)σe + τ s

= (α− ρ3z0 −K z)− 1

2ρ2(γ)ρ2(γ)T(α− ρ3z0 −K z)

− 1

2ρ2(γ)(γud + ke2e+

ke1

mee) + τ s


Since ρ2(γ)ρ2(γ)T(α− ρ3z0) = 0, we will obtain

z = −ρ3z0 − (I3 −1

2ρ2(γ)ρ2(γ)T)Kz

− 1

2ρ2(γ)

(ke2e+

ke1

mee+ (z3 + ψd − k0z0)ud

)+ τ s (9.20)

Recalling that K = diagk1, 0, k3, one may find

z1 = −(1− 1

2sin2(ψr))k1z1 +

ke1

2me

e2

√e2 + ∆2

+ke2 − ∆

e2+∆2ud

2me(− ud + z1

e2 + ∆2e2 +

w2∆

e2 + ∆2e)

+ σ∆

e2 + ∆2

1

me

w2√e2 + ∆2

e+ gz1(ξ, ζ)ζ (9.21)

where gz1(ξ, ζ) is given in Appendix 9.A. In (9.21), we have used the factthat

sin2(γ) = sin2(ψr) + z0(z0g2sin(z0, ψr) + 2gsin(z0, ψr) sin(ψr))

Due to the structure of ρ3, ρ1 and τ s, the second line of (9.20) does notinfluence z3; thus, one may notify the following independent subsystem:

Σ2 :

z0 = −k0z0 + z3

z3 = −z0 − k3z3

(9.22a)

(9.22b)

Therefore, the closed-loop equations are given by (9.18), (9.21) and (9.22).The closed-loop equations may be written as

ξ = f(t, ξ) + g(t, ξ, ζ) (9.23a)

ζ = A2 ζ (9.23b)

where f(t, ξ), g(ξ, ζ) and A2 are found by inspection of (9.18) and (9.22).In other words, the system described by ξ = f(t, ξ) is a nonlinear systemcascaded with the linear system described by (9.22) through the intercon-nection term

g(ξ, ζ) =

1mege(ξ, ζ)

gz1(ξ, ζ)d22m22

gw(ξ, ζ)

ζ (9.24)


The reader might wonder why the function f(·) is time-varying. In fact, me,given by (9.6), depends on cos(γ) = cos(z0 + ψr). The state z0 is generatedby the autonomous system Σ2; by an appropriate selection of k0 and k3,the solutions (z0(t), z3(t)) can be continued for all t ≥ 0. Thus, one mayregard cos(z0 + ψr) as a function of a time-varying variable z0(t) instead ofthe system state z0. Therefore,

me := me(t, ξ) = 1− m23

m22

∆

∆2 + e2cos(z0(t) + tan(

e

∆))

To prove global asymptotic stability of (ξ, ζ) = 0, we utilize Lemma 6.5for stability of cascaded time-varying systems, which is briefly reviewed inSection 6.5.3.

It is easy to verify that f(t, ξ) and g(ξ, ζ) are continuous in their ar-guments, locally Lipschitz in ξ and ζ, uniformly in t. In addition, f(t, ξ)is continuously differentiable in t and ξ. We can see that g(t, ξ, 0) = 0.According to Property 9.1 in Appendix 9.A, one may show that there existsa nondecreasing function σg(‖ξ‖, ‖ζ‖) such that ‖g(ξ, ζ)‖ ≤ σg(‖ξ‖, ‖ζ‖).Thus, to establish stability of the cascade, we shall show that assumptionsA.6.5.1, A.6.5.2 and A.6.5.3 hold.

The perturbing system Σ2 described by (9.22) is globally exponentiallystable under condition T.9.1.1. This is established by choosing a positivedefinite, radially unbounded Lyapunov function W2 = z2

0 + z23 . Therefore,

assumption A.6.5.3 is trivially satisfied due to the globally exponential sta-bility of Σ2.

Lemma 9.1 formally expresses the circumstances under which the glob-ally uniformly asymptotic stability of the origin of (9.18) when ζ = 0 (i.e.the origin of the system ξ = f(t, ξ)) is established.

Lemma 9.1. Under conditions T.9.1.2 and T.9.1.3, the origin of the un-perturbed system (9.18) (i.e. when ζ = 0) is globally uniformly asymptot-ically and locally uniformly exponentially stable. It is established using aquadratic Lyapunov function.


From Lemma 9.1, it is observed that assumption A.6.5.1 is fulfilled. Itfollows from Property 9.1 in Appendix 9.A that g(t, ξ, ζ) has linear growthin ξ; thus, assumption A.6.5.2 is also satisfied. Hence, (ξ, ζ) = 0 is glob-ally uniformly asymptotically and locally uniformly exponentially stableaccording to (Loria and Panteley, 2005, Proposition 2.3). In accordance


with (9.12), one can readily verify that z2 = v is globally uniformly asymp-totically and locally uniformly exponentially stable at zero, as well. Theproof is now complete.

Remark 9.1. It is straightforward to study that, for sufficiently large ∆and appropriate choice of ke1 and ke2, the condition T.9.1.2 is satisfied.Roughly speaking mmin,mmax ≈ 1 for large ∆ > 0; thus, the inequalityreduces to

3∆

4udke1 + ke2 < 8ε1

in which ε1 < 1. Therefore, if ∆ is such that ud∆ ≤ ke2 < 8, we require to

choose ke1 = 43ud∆ (8ε1−ke2)ε2, for ε2 ∈ (0, 1), in order to hold the inequality;

it shows the possibility to satisfy the condition. Without doubt, the choiceof ∆, ke1 and ke2 depends on the system characteristics. We should avoidvery large ∆, which leads to very small ke1 and ke2. In this case, the controlsystem will be very sluggish, and convergence to the path is slow. Obviously,for any choice of ke1, ke2 and ∆, one may choose k1 sufficiently large suchthat T.9.1.3 is satisfied. It is worth noting that larger ∆ will allow us toselect smaller k1.

Remark 9.2. The origin is globally asymptotically stable and locally ex-ponentially stable. According to (Lefeber, 2000), the origin can be calledglobally K-exponentially stable (Sørdalen and Egeland, 1995).

Properties of The Controller

The controller modifies the speed of the marine craft whenever e is nonzero.It is a common assumption in modeling of 3-DOF marine craft that the surgesubsystem is decoupled from the sway-yaw subsystem due to port/starboardsymmetry. It has resulted in the fact that since path-following controllersconsist of decoupled speed controllers and heading autopilots, the speed ofthe marine craft is kept as desired regardless how far the marine craft isfrom the path. Therefore, the speed of the craft does not change if the craftis far from the path. This is while one may adapt the speed according to thedistance to the path and the rate of convergence to the path. The proposeddesign method intends to manipulate the speed. It makes the speed of thecraft dependent on the cross-track error and its derivative.

The control law τ ls is derived using an approximation, based on themethod of least squares, for the forces that make the craft converge to thepath with an exponential rate and the forces that regulate z0, z1 and z3.


This chapter can be viewed as an extension of the method which wasproposed in Chapter 8 for fully actuated marine craft to be applicable tounderactuated marine craft. In fact, the controller is designed in, more orless, the same fashion as for fully actuated marine craft, but only the surgeforce and the yaw moment are applied to the system. To compensate forunderactuation and to guarantee stability of the closed-loop system, τ s isderived and applied to the surge dynamics. The properties of the proposedcontroller, thus, are the same as the properties of the one that was proposedin Chapter 8.

The guidance parameter plays a central rule in stability of the closed-loop system, and it should be chosen carefully according to the systemstructure and the forward speed.

The controller possesses more robust performance than conventionalpath-following controllers since it is allowed to manipulate the speed inorder to catch the path in the presence of external disturbances. However,since the heading ψ is to track ψlos, the cross-track error will not be zeroin the presence of constant disturbances. One solution is to compensate forthe sideslip angle and to make the course angle χ track ψlos. The integralLOS method might be another remedy (Børhaug et al., 2008). In (Peymaniand Fossen, 2013c), using the same framework as here, we have proposedintegration of the cross-track error to exert additional forces on the craft toretrieve the zero cross-track error.

We point out that the speed is linearly proportional to e and e, whichmay cause saturation for very large e. However, it is possible to use saturation-like functions (e.g. tan−1(γe)) to resolve this drawback. One may noticethat the controller does not consider the sideslip angle β = arctan( vu), whichis time-varying and nonzero in transient. Since the unactuated dynamics vis globally bounded and vanishing, β is well-behaved.

9.4 Simulation Result

The proposed control method is evaluated using computer simulations. Weconsider a model of small unmanned marine vehicle with the length of 6m.The model is given by (9.2) where

Mb =

309.60 0 00 405.6 12.10 12.1 33.1

, Db =

24 0 00 84 1.20 1.2 6.0

Accordingly, σ = −0.015. The desired forward velocity is ud = 2m/s. Wechoose ke1 = 0.1 and ke2 = 0.2. Then, for ∆ = 20m, the condition T.9.1.2

9.5. Concluding Remark 237

∆

5 10 15 20 25 30 35 40 45 500

2

4

6

8

10

12

14

16

18

20

Figure 9.2: This plot shows the variation of ∆ 3mmax

m2minke1ud

(ke1+|σ|2ud∆2 )2+ ke2

mmin

with respect to ∆. For ∆ > 12, the inequality in T.9.1.2 holds.

holds. Figure 9.2 shows the left-hind side of the condition T.9.1.2 vs. ∆.

Every k1 > 0.15 holds the condition T.9.1.3 ; we choose k1 = 10.We select k0 = 1 and k3 = 2. The initial pose and velocity are given byq = [0,−250, π4 ] and ν = [2.1, 0, 0]. The objective is to follow a desiredpath which lies on x = 40m. The results are shown in Figs 9.3 and 9.4.

9.5 Concluding Remark

The method of Chapter 8 is extended to underactuated marine craft. Theproblem of path maneuvering for underactuated marine craft is studiedand a nonlinear controller is derived. The path-maneuvering scenario isdivided the dynamic task of speed assignment and the geometric task ofpath following. The proposed controller alters the speed of the marinevehicle, during transient to reach to the path, according to the distance tothe path and the rate of convergence to the path. To derive the controllaw, the method of least squares is utilized. The conditions under which thecontroller achieves the objectives are established.


Appendix 9.A Required Relations

Considering (9.16), we have

gsin(ψr, z0) =sin(z0)

z0cos(ψr) +

cos(z0)− 1

z0sin(ψr)

gcos(ψr, z0) = −sin(z0)

z0sin(ψr) +

cos(z0)− 1

z0cos(ψr)


and cos(z0)−1z0

are well-defined functions and are globallybounded. Accordingly, in (9.18a) and (9.18b), we have

ge(ξ, ζ) = [(ud + z1)gsin(ξ, ζ) + w2gcos(ξ, ζ), 0]

gw(ξ, ζ) = σ∆

e2 + ∆2

1

mege(ξ, ζ) +

1

d22gv1(ξ)

where gv1(ξ) = [m23(1− k20)− d23k0, m23(k3 + k0) + d23]. Also

gz1(ξ, ζ)ζ = −ud

2sin(γ)(z3 − k0z0)

+k1

2z1z0(z0g

2sin(z0, ψr) + 2gsin(z0, ψr) sin(ψr))

− 1

2mesin(γ)(ke2 −

∆

e2 + ∆2ud)ge(ξ, ζ)ζ

− ke1

2megsin(ψr, z0)z0e−

(ke2 − ∆ude2+∆2 )

2megsin(ψr, z0)z0×

((ud + z1) sin(ψr) + w2 cos(ψr))

The following property is easily established.

Property 9.1. ge(ξ, ζ), gw(ξ, ζ) and gz1(ξ, ζ) grow linearly in ξ; i.e.

‖gk(ξ, ζ)‖ ≤ σk1(‖ζ‖) + σk2(‖ζ‖)‖ξ‖, k = e, w, z1

where σk1, σk2 : [0,∞)→ [0,∞) are continuous and non-decreasing.


Denote Ω = ke2 − ∆∆2+e2

ud. If we choose ke2 >ud∆ , we obtain ke2 > Ω >

Ω∗ > 0, ∀e. Second, 1− 12 sin2(ψr) = 1

2(1 + cos2(ψr)) ≥ 12 , for all e.


Select V1 = 12k1z

21 , and differentiate along the trajectories of (9.21). It

gives rise to

V1 = −(

1− 1

2sin2(ψr)

)k2

1z21 +

k1ke1

2me

e2z1√e2 + ∆2

+k1Ω

2me

(− ude

2z1

e2 + ∆2− z2

1e2

e2 + ∆2+w2z1e∆

e2 + ∆2

)+ k1σ

∆

e2 + ∆2

1

me

w2√e2 + ∆2

ez1

Choose V2 = 14k1ke1e

2, and take derivative in time. It yields:

V2 =k1ke1

2me

(− ude

2

√e2 + ∆2

− z1e2

√e2 + ∆2

+∆√

e2 + ∆2ew2

)Choose V3 = 1

2k1m22d22

w22, and find the time derivative:

V3 = −k1dαw22 − k1σ

∆

e2 + ∆2

1

me

ud + z1√e2 + ∆2

ew2

Let V = V1 +V2 +V3 be the positive definite, radially unbounded Lyapunovfunction for the system described by (9.18a), (9.18b) and (9.21) when ζ = 0.We find V :

V = −k21

2

(1 + cos2(ψr)

)z2

1 − k1dαw22 −

k1ke1

2me

ud√e2 + ∆2

e2

− k1Ωud

2me

e2z1

e2 + ∆2+k1Ω

2me

(− z2

1e2

e2 + ∆2+w2z1e∆

e2 + ∆2

)+

k1

2me

∆√e2 + ∆2

(ke1 − σ

2ud

e2 + ∆2

)ew2

The upper bound on V is given by

V ≤ −1

2k2

1z21 − k1dαw

22 −

k1ke1

2mmax

ud√e2 + ∆2

e2

+k1Ωud

2mmin

e2|z1|e2 + ∆2

+k1Ω

2me

(− z2

1e2

e2 + ∆2+|w2||z1||e|∆e2 + ∆2

)+

k1∆

2mmin(ke1 + |σ|2ud

∆2)

|e|√e2 + ∆2

|w2|

Since

− z21e

2

e2 + ∆2+|w2||z1||e|∆e2 + ∆2

= −(|z1||e|√e2 + ∆2

− |w2|∆2√e2 + ∆2

)2 +1

4

w22∆2

e2 + ∆2


V is upper bounded by

V ≤ −1

2k2

1z21 − k1dαw

22 −

k1ke1∆

3× 2mmax

ud

e2 + ∆2e2

− k1ke1∆

3× 2mmax

ud

e2 + ∆2e2 +

k1Ωud

2mmin

e2|z1|e2 + ∆2

− k1ke1∆

3× 2mmax

ud

e2 + ∆2e2 +

k1∆(ke1 + |σ|2ud∆2 )

2mmin

|e||w2|√e2 + ∆2

+k1Ω

8mmin

w22∆2

e2 + ∆2

where we have used the fact that − e2√e2+∆2

≤ − e2∆e2+∆2 . Then, we obtain

V1 ≤ −1

2k2

1z21 − k1dαw

22 −

k1ke1∆

3× 2mmax

ud

e2 + ∆2e2

− k1ud

2

e2

e2 + ∆2

(ke1∆

3mmax− Ω

mmin|z1|)

− k1∆

2

(ke1ud

3mmax

e2

e2 + ∆2− (ke1 + |σ|2ud

∆2 )

mmin

|e|√e2 + ∆2

|w2|)

+k1Ω

8mmin

w22∆2

e2 + ∆2

Complete the squares in the second and third lines. Considering ∆2

e2+∆2 ≤ 1

and e2

e2+∆2 ≤ 1, V is then bounded by

V ≤ −1

2k2

1z21 − k1dαw

22 −

k1ke1∆

3× 2mmax

ud

e2 + ∆2e2

+k1ud

2

Ω2

4m2min

3mmax

ke1∆z2

1

+k1

8

(∆

(ke1 + |σ|2ud∆2 )2

m2min

3mmax

ke1ud+

Ω

mmin

)w2

2

There exist positive scalars β1, β2 and β3 such that

V ≤ −β1z21 − β2w

22 − β3

e2

e2 + ∆2(9.25)


if the gains are selected in a way the following inequalities hold

k1 >ud

2

k2e2

4m2min

3mmax

ke1∆(9.26)

8dα > ∆(ke1 + |σ|2ud

∆2 )2

m2min

3mmax

ke1ud+

ke2

mmin(9.27)

We can find sufficiently large k1 for any ke1, ke2 and ∆ such that (9.26) holds.Notice that for a large class of marine craft σ < 0. Nonetheless, we assumethat ∆ is chosen sufficiently large that dα > 1− σ

mmin∆ > 0. It implies that,under condition T.9.1.2, the inequality (9.27) holds. Thus, (e, z1, w2) = 0is uniformly globally asymptotically stable, so is (e, z1, z2) = 0.

Since for each |e| < δ, δ > 0, we have − e2

e2+∆2 ≤ − e2

δ2+∆2 , one can showthat for every ball with radius δ, we have

V ≤ −β1z21 − β2w

22 − β∗3(δ)e2

Thus, the origin is uniformly locally exponentially stable.


travelled

way-points

desired

Nor

th(m

)

East (m)

The path travelled by the Ship

−300 −250 −200 −150 −100−60

−40

−20

0

20

40

60

80

Figure 9.3: Path following of an underactuated marine craft in calm water.

Geo

met

ric

Err

or[m

]

Time (sec)

Cross-track Error

vel

oci

ty[m

/sec

]

Time (sec)

Sway Velocity (v)

erro

r[m

/sec

]

Speed Error (u− ud)

0 10 20 30 40 50 60

0 10 20 30 40 50 60

0 10 20 30 40 50 60

0

20

40

−0.05

0

0.05

−5

0

5

Figure 9.4: Speed assignment task is depicted in upper plot. The swayvelocity converges to zero (middle plot). The cross track error is depictedin the lower plot.

Part III

Motion Control ofLagrangian Systems using

Analytical Mechanics

243

Chapter 10

Motion Control usingAnalytical Mechanics

This chapter proposes a framework for motion control of mechanicalsystems with application to marine craft. Motivated by analytical me-chanics, a control problem is reformulated as the problem of modelingof a constrained multi-body system. Then, techniques for forward dy-namics simulation are employed to obtain valid models. The forces thathold various bodies of the system connected are found using analyticalmechanics, and applied to the system using actuators. To exemplifythe approach, a formation problem are taken into consideration, andthe validity of the theoretical developments is verified by computer sim-ulations.This chapter is based on (Peymani and Fossen, 2011a,b,c).

10.1 Introduction and Objectives

This chapter provides a novel framework for motion control of Lagrangiansystems. Put accurately, it views the problem of motion control of mechan-ical systems from a new angle. The gist of the proposed method is to viewa motion control problem as the modeling problem of a mechanical sys-tem which consists of several bodies subject to constraints. In mechanicalmulti-body systems, with the help of the Lagrangian mechanics (see e.g.Pars, 1965), a mathematical representation to describe the motion of eachbody is individually derived as if the body is allowed to move freely in theselected configuration space. Imposing constraints, the variational principleof analytical mechanics (see e.g. Lanczos, 1986) and the Lagrange multipliermethod, as the most well-known method, are taken into account in orderto find the forces that have to be exerted on each body so that it meetsall the corresponding constraints. In control problems, one may presume

245

246 Motion Control using Analytical Mechanics

that the system is artificially/virtually constrained by means of control ob-jectives; these artificial/virtual1 constraints exert additional forces on thesystem. These forces can be viewed as control forces and are to be appliedto the system by means of actuators. Hence, a framework for motion controlappears that relies on the powerful capabilities of analytical mechanics forhandling constrained multi-body systems instead of relying only on mathe-matical tools.

This bright idea has not been used in maritime applications frequently.The works of Ihle et al. (2006a,b) are actually the only works in this field,where the problems of trajectory tracking and formation are able to betackled for fully actuated marine surface craft. However, this has been oflimited use in practice since marine craft are mainly underactuated, andthere are scenarios which have a speed assignment task. In the presentchapter, this technique is taken one stage further to include both positionand velocity constraints, which allows the motion scenarios, such as pathmaneuvering, in which the speed has to be controlled independently of theposition.

Related Work The concept of virtual constraints was first introduced inMason (1981) in the field of force control of robot manipulators interact-ing with the environment; it was further developed within the task frameformalism by Bruyninckx and De Schutter (1996). It is very popular inthe context of force control to handle robot manipulators in contact withthe environment (see e.g. Siciliano and Villani, 2000). Another thread ofthought of virtually constrained systems was presented by Shiriaev et al.(2010) in which orbital stabilization and periodic motion planning are con-sidered while the concept of constraint forces are not used. Pertinent to(Ihle et al., 2006a,b), a formation controller for spacecraft was reported in(Xing et al., 2009) where virtual holonomic constraints are placed on thesystem. A similar approach was taken by Lee et al. (2005) for distributedmotion planning.

In (Udwadia and Kalaba, 1992), a fundamental equation of motion forconstrained systems was developed based on Gauss’s principle, where with-out utilizing the concept of Lagrange multipliers and with the aid of the

1The word ‘artificial’ is used for something that is not real or natural, but it is made tolook or work like something real or natural. The word ‘virtual’ is used for something thatis made, done or seen on a computer rather than in the real world (according to LongmanDictionary of Contemporary English). I think that both words are appropriate to use.If it is viewed from the angle of the field of simulation of multi-body systems, virtualconstraints are more appropriate. From a control point of view, artificial constraints aresuitable.

10.2. Lagrangian Approach to Multi-body Systems 247

concept of Moore-Penrose generalized inverse, the force of constraints is de-termined. Their equation of motion is valid for constraints that are nonlin-ear in the generalized velocities, and that could be functionally dependent.More interesting, their method can be applied to constraints that do notconform with D’Alemebert’s principle of virtual work.

Subsequently, Udwadia (2003) used the fundamental equation of con-strained motion, presented in (Udwadia and Kalaba, 1992), for trackingcontrol of nonlinear systems. Afterwards, the idea was expanded to trajec-tory tracking of robot manipulators (Peters et al., 2008), synchronizationof gyroscopes (Udwadia and Han, 2008), and formation keeping of satellites(Cho and Udwadia, 2010). They all consider full state trajectory trackingwhich leads to placing holonomic constraints and do not discuss externaldisturbances.

Among the other related work, in (Spry and Hedrick, 2004; Spry, 2002),a general method of specifying the configuration of a group of vehicles interms of a set of generalized coordinates is provided, which allows the groupto be controlled as a single entity. The approach characterizes formationsof vehicles as location, orientation and shap.

Lee and Li (2003) put forward a decomposition to divide the dynamicsof a group of spacecraft into two dynamics: 1) an average system which de-scribes the dynamics of the overall group motion; 2) a shape system whichdescribes the dynamics of the group internal formation. Thus, maneuveringof groups of spacecraft can be achieved by controlling the average systemwhile formation is accomplished by controlling the shape system. The cou-pling between these two subsystems is also shown.

In an similar effort, using the Jacobi shape theory – to describe the shapeand orientation of deformable bodies – Fumin (2010) presented a geometricapproach to decouple the translation dynamics from the orientation andshape dynamics. The method is applied to formations of underwater vehiclesin (Huizhen and Fumin, 2010).

10.2 Lagrangian Approach to Multi-body Systems

Mechanics is the branch of science that studies the interaction between phys-ical bodies, which are subject to forces and displacements, and the environ-ment. Mechanics has two major divisions, namely, classical mechanics andquantum mechanics. Classical mechanics is the ancient branch of mechan-ics while quantum mechanics, which may encompass classical mechanics,concerns with particles at molecular and atomic level.

Classical mechanics, which gives accurate results as long as the domain


of study is limited to sufficiently large objects with speeds not approachingthe speed of light, includes two main formulations. One is the Newtonianmechanics which utilizes the vector of forces in order to analyze the motion;accordingly, it is often called the vectorial mechanics. The other formulationis analytical mechanics, which in contrast to the vectorial mechanics, usestwo scalar quantities of a system to study the motion. These quantities arethe kinetic energy and the potential energy.

Analytical mechanics has two intertwined parts: Lagrangian mechanicsand Hamiltonian mechanics. In the Lagrangian mechanics, which is re-viewed briefly in this chapter, the Lagrangian is defined as a function ofthe generalized coordinates and velocities. In the Hamiltonian mechanics,the Legendre transformation of the Lagrangian replaces the generalized co-ordinates and velocities with the generalized coordinates and generalizedmomenta.

10.2.1 Equations of Motion

The starting point in the derivation of the dynamic equations of motion is tofind the equations of motion for an unconstrained substitute of the system,which is obtained from the original system by discarding all constraints.

The Lagrangian of an unconstrained mechanical system (Lanczos, 1986),with n degrees of freedom, in an n-dimensional configuration space, whosepose is uniquely characterized by n generalized coordinates q = [q1, q2, · · · , qn]T

and whose velocity is described by n generalized velocities, is defined as

L(q, q) , T (q, q)− V(q) (10.1)

in which T is the kinetic energy and V is the potential energy of the sys-tem. The configuration space Q is an n-dimensional smooth manifold, lo-cally diffeomorphic to an open subset of Rn, which contains all possibleconfigurations of the mechanical systems.

Hamilton’s principle relates the trajectories of the system to the La-grangian. In fact, it states that the actual path (trajectory – q(t) –) of aparticle is the path that minimizes the action integral S as below

S =

∫ t2

t1

L(q, q, t)dt (10.2)

The Lagrange-D’Alembert principle states how the Lagrangian of a me-chanical system exposed to the generalized forces τ , [τ1, τ2, · · · , τn]T isconnected with the equations of motion:

d

dt

∂L∂qi− ∂L∂qi

= τi, i = 1, · · · , n (10.3)


which are also known as the Euler-Lagrange equations. If the kinetic energyis yielded by a quadratic type equation as T = 1

2 qTM(q)q where the matrixM(q) ∈ Rn×n,M > 0 is the mass and inertia matrix, the equation of motionis given by

M(q)q + n(q, q) = τ (10.4)

where

n(q, q) =

(M(q)− 1

2qT∂M(q)

∂q

)q +

∂V(q)

∂q

In literature, C(q, q) , M(q)−12 qT ∂M(q)

∂q is called the matrix of Coriolis and

centripetal, and ∂V(q)∂q represents the vector of gravity forces. It is proven

that M − 2C is a skew-symmetric matrix. This is the typical structurefor models of robot manipulators, mobile robots (Murray et al., 1994), andmarine craft (Fossen, 2011).

10.2.2 Constrained Motion

The motion of mechanical systems may be constrained. For a single rigid-body, one may choose a configuration space in which the motion is describedby a set of generalized coordinates subject to a set of constraint functions.For example, consider a pendulum. Its motion can be described by x and yin the Cartesian space, while its motion is confined to a circle x2 + y2 = L2

where L is the length of the rob. Alternatively, the motion can be uniquelydescribed by θ, the angle between the rob and the vertical axis, while itis unconstrained. For multi-body systems, motion of one body may beconfined to the others. In this section, various constraints that may comeup in analysis of mechanical systems are introduced.

The motion of a system may be confined to a function of the form

f(q, q, t) = 0 (10.5)

In general, constraints in the form of (10.5) are termed as kinematic con-straints. A kinematic constraint (10.5) is said to be rheonomic if it explicitlydepends on time; i.e. the constraint is a moving constraint. Otherwise, it issaid to be scleronomic; that is, it is a stationary constraint2.

Bilateral constraints (two-sided constraints) are those that can be ex-pressed in terms of equalities , while unilateral constraints (one-sided) re-quired inequalities.

2In Greek, ‘rheo’ means flowing and ‘sclero’ means rigid.


In mechanics, kinematic constraints (10.5) are called holonomic con-straints3 if and only if they are integrable and reduce to

g(q, t) = 0 (10.6)

which depends only on the generalized coordinates and, possibly, time. Oth-erwise, the kinematic constraints (10.5) are termed as nonholonomic con-straints. A very common class of kinematic constraints are those in whichthe generalized velocity appears linearly:

A(q, t)q +B(q, t) = 0 (10.7)

These constraints are termed as linear-in-velocity or linear constraints. With-out doubt, if they are integrable, they reduce to holonomic constraints (10.6)even if the closed-form of its integral is not known. If (10.7) is integrablebut the integral is not known, it is called semiholonomic.

Constraints (10.7) is driftless if the term B(q, t) does not exist. A drift-less linear-in-velocity constraint that is scheronomic is called Pfaffian:

A(q)q = 0 (10.8)

Holonomic Constraints

As mentioned, holonomic constraints, G(q, t) : Rn × R → Rk, depend onlyon generalized coordinates and take the form of (10.6). When motion of thesystem is subject to such constraints, the position at time t which is q(t)must satisfy G(q, t) = 0. That is, q(t) must lie on the (n− k)-dimensionalmanifold of the configuration space described by G = 0. In other words,the dimensionality of the space of accessible configurations is reduced fromn to n − k. Therefore, the holonomic constraints confine the system geo-metrically; thus, they are often called geometric constraints.

Implicit function theorem implies that holonomic constraints (10.6) canbe solved in terms of n − k generalized coordinates. This eliminates thedependency among the generalized coordinates, and offers a new set of gen-eralized coordinates, based on which the motion is described with no holo-nomic constraints. Elimination may not be always possible or may havelocal validity; also, it may result in algebraic singularities (Yun and Sarkar,1998). It is recommended to seek a new configuration space of dimensionn − k in which the equations of motion are not subject to holonomic con-straints. The motion of this reduced order system is equivalent with that

3Holonomic is derived from a Greek word meaning “whole” and “integer”. (De Lucaand Oriolo, 1995)


372 F.A.C.C. Fontes, D.B.M.M. Fontes, and A.C.D. Caldeira

The control methodology selected is a two-layer control scheme where eachlayer is based on model predictive control (MPC). Control of multi-vehicle for-mations and/or distributed MPC schemes have been proposed in the literature.See the recent works [5,22] and references therein.

The reason why two-layers are used in the control scheme is because there aretwo intrinsically different control problems:

– the trajectory control problem: to devise a trajectory, and correspondingactuator signals, for the formation as a whole.

– maintain the formation: change the actuator signals in each vehicle to com-pensate for small changes around a nominal trajectory and maintain therelative position between vehicles.

These control problems are intrinsically different because, on the one hand, mostvehicles (cars, planes, submarines, wheeled vehicles) are nonholonomic (cannotmove in all directions). On the other hand, while the vehicles are in motion, therelative position between them in a formation can be changed in all directions(as if they were holonomic).

As an example consider a vehicle whose dynamics we are familiar with: a car.Consider the car performing a parking maneuver or performing an overtakingmaneuver. See figures 1 and 2.

Fig. 1. Car in a parking maneuver: cannot move sideways

In the first situation, we are limited by the fact that the car cannot movesideways: it is nonholonomic. It is a known result that we need a nonlinearcontroller, allowing discontinuous feedback laws, to stabilize this system in thissituation [23,3].

In the second, the vehicle is in motion and small changes around the nomi-nal trajectory can be carried out in all directions of the space. In an overtakingmaneuver we can move in all directions relative to the other vehicle. This factsimplifies considerably the controller design. In fact, a linear controller is an ap-propriate controller to deal with small changes in every spatial direction arounda determined operating point.

Figure 10.1: Parking of a car as a nonholonomic vehicle which cannot movesideways. It is nonlinearly controllable. Courtesy of Fernando A.C.C. Fonte.

of the original one. Note that all points of the configuration space of thereduced system is accessible.

The source of holonomic constraints are interconnections between vari-ous bodies of the system. Differentiation of (10.6) gives

∂

∂qg(q, t)q +

d

dtg(q, t) = 0

which is in the form of linear-in-velocity constraints, which confine the set ofattainable generalized velocities. In fact, although velocities do not directlyappear in holonomic constraints, constraints are placed on the velocitiessince holonomic constraints must hold at all time.

Nonholonomic Constraints

Nonholonomic constraints in the form of (10.5) confine the motion by re-stricting the generalized velocity to an (n−m)-dimensional subspace of thetangent space Tq(Q). Notice that all the configurations that are accessible inthe absence of nonholonomic constraints remain accessible in their presence;i.e. the configuration space Q is not changed if nonholonomic constraintsare placed on the system. In fact, nonholonomic constraints preclude thesystem from moving in certain directions. Automobiles are typical examplesof the systems which own nonholonomic constraints; see Fig. 10.1.

Considering m < n Pfaffian constraints (10.8), one can find the matrixH(q) ∈ Rn×(n−m) that describes the null-space of A(q); i.e., A(q)H(q) = 0.Thus, the admissible q is expressed in terms of the columns of H(q); thatis

q = H(q)w, w ∈ Rn−m (10.9)


Equation (10.9), which describes a nonlinear and affine system, is calledthe kinematic model of a system subject to nonholonomic constraints, andw may be interpreted as the virtual input of the system. Notice that thesystem is underactuated from a control viewpoint, implying a close relationbetween nonholonomy and underactuation. The dependency of q is removedif (10.9) is differentiated and substituted in (10.4), which results in

HTMHw +HT(n+MHw) = HTτ (10.10)

Equation (10.10) is usually called the dynamic model of a nonholonomicsystem (Bloch et al., 1992), leading to a canonical state-space form in con-junction with (10.9).

Remark 10.1. Note that although the dynamic model is always attainablefor a naturally constrained system, it may be unfeasible for virtually con-strained systems. This will be clarified later.

The question which may arise is that: how can one realize that a kine-matic constraint (10.5) is essentially a holonomic constraint? The integra-bility of kinematic constraints is directly linked to the controllability of theassociated kinematic model; see (De Luca and Oriolo, 1995) for details.

It is well-known that the major source of holonomic constraints is in-terconnections between bodies of a mechanical systems. For nonholonomicconstraints, three sources are usually mentioned (De Luca and Oriolo, 1995):

• Rolling contacts without slipping such as wheeled mobile robots, rollingdisk, unicycle, car-like robot, N -trailer robot;

• Conservation of angular momentum, which usually happens in systemsallowed to float freely without a fixed body. It takes place in spaceapplications.

• Robotic devices with a particular control operation. This group fitsinto the underactuated systems which have fewer independent con-trols than degrees of freedom. This category includes nonholonomicconstraints at the acceleration level.

Nonholonomic constraints at the acceleration level may arise dueto underactuation. In general, a constraint may be placed on the systemas h(q, q,q, t) = 0. The integrability of such systems have been studiedin several papers (see e.g. Oriolo and Nakamura, 1991; Tarn et al., 2003;Wichlund et al., 1995a). This topic was briefly reviewed in Chapter 6.


10.2.3 Dealing with Constrained Motion

The Lagrangian framework deals with constraints in an appealing way thatis briefly explained. In the Lagrangian mechanics, the equations of motionfor a constrained system is found in this fashion. Find the equations ofmotion for the unconstrained system which is obtained by removing theconstraints. The constraints exert additional forces on the system. Thus,the equations of motion must be rectified by considering the force of con-straints. In the following, it is expressed how the force of constraints canbe derived.

Assume that the system is subject to k holonomic constraints in thevector form:

G(q, t) = 0, G ∈ Rk (10.11)

It has been shown (Lanczos, 1986) that Hamilton’s generalized principlehandles such constraints by modifying the Lagrangian to

L∗(q, q, t) = L(q, q) + λTGG(q, t), λG ∈ Rk (10.12)

Then applying the Euler-Lagrange equations to the augmented LagrangianL∗ over the extended set of free coordinates q,λG gives the equations ofmotion. This is the well-known Lagrange multiplier rule, which leads to

M(q)q + n(q, q) = τ + τ h (10.13)

where

τ h = −WTG(q, t)λG, WG = ∂G/∂q ∈ Rk×n (10.14)

That is, the effect of holonomic constraints are seen as the additional forceτ h. Dealing with nonholonomic constraints is harder than holonomic ones.This is due to the confusion over the generalization of Hamilton’s princi-ple and the use of the Lagrange multipliers for the general nonholonomicconstraints (10.5). The confusion misled Ray (1966b) and Goldstein et al.(2002) into generalizing Hamilton’s principle for (10.5) akin to (10.12), butthis generalization resulted in a wrong outcome for the Pfaffian constraints,as explained by Ray (1966a) and Murray et al. (1994). It is while Moreschiand Castellano (2006) encounter the same problem and believe the La-grange multiplier method is applicable for the general case (10.5) by somealteration. Still, Olga (2009) and Li and Berakdar (2008) acknowledge thefallacy of the Lagrange multiplier method for the general case and proposea multiplier-free approach to tackle nonholonomy.


Flannery (2005) in an article entitled “the enigma of nonholonomic con-straints” neatly demystifies the point of confusion, and proposes a convinc-ing answer to the bewildering problem of nonholonomic constraints. Toput it simply, general nonholonomic constraints (10.5) are completely out-side of the scope of any principle based on D’Alembert’s principle, andany generalization according to this principle is baseless. It is valuable tonote that D’Alembert’s principle of virtual work consists in the fact thatconstraint forces do no work under any virtual displacements; it is the fun-damental principle in analytical mechanics that Hamilton’s principle canbe extracted from. Flannery (2005) also states that D’Alembert’s gener-alized principle can be applied to holonomic constraints, and undoubtedlythe Lagrange multiplier technique is workable, which is not an extraordi-nary result, though. The Lagrange multiplier method is surprisingly nolonger valid for the nonholonomic linear-in-velocity constraints (10.7) whilethey can be treated by D’Alembert’s basic principle. Consequently, for amechanical system expressed by the Lagrangian mechanics and subject tolinear-in-velocity constraints (10.7), the additional forces that are placed onthe system are given by

τ nh = −AT(q, t)λK , λK ∈ Rm (10.15)

The force of the linear-in-velocity constraints (10.7) takes the same form asthat of the holonomic constraints does even though the way they are derivedare different.

It is worthwhile mentioning that constraints (10.6) and (10.7) satisfythe principle of virtual work whereas the general case (10.5) may not. Con-straints are termed ideal if the work done by the force of constraints underany virtual displacement is zero. Those constraints whose respective forcesproduce work under virtual displacements are said to be nonideal. It hasbeen shown (Liu and Huston, 2008; Udwadia and Kalaba, 2002) that incase the constraint (10.5) is nonideal and its first derivative is linear in ac-celeration (i.e. A(q, q, t)q + b(q, q, t) = 0), the constraint force (10.15) willchange to

τ nh = −AT(q, t)λK +U(q, q, t), U ∈ Rn (10.16)

where the vector U specifies the work done by nonideal constraints and hasto be chosen based on the analyst’s knowledge for the specific system. Inthis thesis, ideal constraints are only taken into consideration.

Holonomically and Nonholonomically Constrained Systems If asystem is subject to m nonholonomic linear-in-velocity constraints K and

10.3. Control Methodology 255

k holonomic constraints G, the equation of motion is modified to

M(q)q + n(q, q) = τ + τ c (10.17)

where τ c is the force4 that secures the satisfaction of the constraints, andis given by

τ c = τ h + τ nh , −WTλ (10.18)

in which W =

[AWG

]∈ Rn×(m+k) is the Jacobian matrix and λ =[

λKλG

]∈ Rm+k is the multiplier vector. Notice that it is assumed that

q(0) and q(0) satisfy the equations (10.6) and (10.7).Therefore, the Lagrangian approach to deal with constrained motion is

to exert additional forces on the equation of motion so that the accelerationssatisfy the constraints. The total sum of the work that is done by theseadditional forces under any virtual displacement must be zero.

Remark 10.2. The methods that involve with elimination of dependencybetween position variables or velocities are presented in (Pars, 1965) and(Bloch et al., 1992), respectively. However, we are interested in keeping theconstraints and considering the effect of constraints as the forces acting onthe system. These forces guide the system to move so that the constraintshold at each instant of time.

Remark 10.3. In mathematics, the method of multiplier is applicable fordifferential equations subject to general nonholonomic constraints (Brunt,2004).

10.3 Control Methodology

In the previous section, naturally constrained motion and the concept of re-action forces were explained. In this section, the notion of artifitial/virtualconstraints are given, and artificially/virtually constrained motion whicharises from control needs is explained. Akin to naturally/mechanically con-strained systems, the reaction forces that guide the system to behave suchthat the constraints hold are derived. These forces are conceived as thecontrol inputs and applied to the system by virtue of actuators. In this

4The force of constraints are defined in the space that the generalized coordinatesspan. To see how the force of constraints is found when the equation of motion of theunconstrained system is obtained using quasi-velocities refer to (Udwadia and Phohomsiri,2007b).


Marine Craft

Constrained System

Constraint Unification

Reaction Forces

Constraint Stabilization

External Forces

Model

Velocity Constraints

Positional Constraints

Figure 10.2: Flow chart showing the proposed method for control of virtuallyconstrained mechanical systems.

section, we go through the procedure of designing controllers based on theconcept of virtual constraints.

10.3.1 Overview of the Control Methodology

Fig. 10.2 illustrates the overall perspective of the method. For this controlstrategy, the control objectives are formulated in the forms of (10.11) and(10.7), and regarded as virtual constraints imposed on the system. The sys-tem along with the virtual confinements comprises the “constrained system”block, which was the topic of Section 10.2. The other block, namely “con-straint stabilization”, is to resolve inconsistency to the initial conditions,external disturbances, as well as to solve the drift problem. This block isinspired by the stabilization method of Baumgarte (Baumgarte, 1983) fornumerical resolutions of constrained systems. From the control viewpoint,it makes the control system robust to external disturbances, measurementnoise, and modeling uncertainties by virtue of a feedback loop. In this sec-tion, the concept of constraint stabilization is enunciated. At the end of thesection, we can grasp the physical interpretation of the control methodology.

10.3.2 Constraint Stabilization

The goal of the constraint stabilization is to make the constraint manifoldinvariant and attractive so that if an initial condition lies in the manifold,it stays in the manifold for all time, and if the constraints are violatedfor any reason, whether due to inconsistent initial conditions or due tounknown perturbations, they are properly enforced. To this end, a two-step


procedure is presented. In the first step, holonomic functions are stabilizedat the velocity level in order to be merged with nonholonomic constraints.As a result, a unified constraint function is defined. In the next step, theunified constraint function is stabilized, and the vector of reaction forces isderived.

Step 1: Constraint Unification As k holonomic constraints (10.11)must hold always, the corresponding manifold is made invariant using G =u2G. Picking u2G = −PGG where PG ∈ Rk×k is positive definite, G = 0 isglobally exponentially stable and

G+ PGG = 0⇒WGq +Gt + PGG = 0 (10.19)

in whichWG = ∂G/∂q ∈ Rk×n, and Gt = ∂G/∂t ∈ Rk. Eq. (10.19) is iden-tical to the nonholonomic linear-in-velocity constraints (10.7) in structure.Putting (10.19) and (10.7) into a unified representation, one may define

Φ(q, q, t) ,W(q, t)q + a(q, t) (10.20)

a(q, t) =

[B(q, t)

Gt(q, t) + PGG(q, t)

](10.21)

where W is the constraint Jacobian matrix defined in (10.18).

Step 2: Φ ∈ Rm+k is called the unified constraint function, and it mustbe always zero. With this aim in view, the system Φ = uΦ is made globallyexponentially stable by an appropriate uΦ. Considering the positive definitematrix PΦ ∈ R(m+k)×(m+k), uΦ = −PΦΦ gives rise to

Φ + PΦΦ = 0⇒Wq + Wq + a+ PΦΦ = 0 (10.22)

Reaction Forces To derive the force of constraints, one should solve themodel (10.4) for acceleration; i.e.

q =M(q)−1 (τ + τ c − n(q, q)) (10.23)

where τ is the impressed force vector and τ c is given by (10.18). Substitut-ing (10.23) into (10.22) yields

WM−1WTλ =WM−1(τ − n) + Wq + a+ PΦΦ (10.24)

In (10.24), if W is full row rank, the product WM−1WT is a nonsingularsquare matrix sinceM is nonsingular. After computing the vector of multi-pliers λ, the reaction forces are found using (10.18) at each instant of time.


In case m+ k > n but W has n linearly independent columns, the productWM−1WT is not invertible. To solve this problem, the constraint force vec-tor can be directly computed by utilizing the Moore-Penrose (MP) pseudo-inverse for WM−1 which is full column rank. Therefore, the MP pseudo-inverse, denoted (WM−1)†, is equal toMW† in whichW† = (WTW)−1WT.Consequently, the vector of constraint forces emerges as

τ c = −(τ − n)−MW†(Wq + a+ PΦΦ

)(10.25)

In this case, the control law minimizes the cost function ‖WM−1τ c + Y ‖2,where Y =WM−1(τ − n) + Wq + a+ PΦΦ is minimized.

Remark 10.4. Those constraints that conflict one another are said to beinconsistent or conflicting. Inconsistent constraints must be avoided. More-over, it is required that the constraints are independent, meaning that theyare not redundant or unnecessary (Udwadia and Kalaba, 1996).

The derived control law follows from first principles and is physically in-terpretable. Viewing the control objectives as virtual constraints along witha stabilization process resembles placing generalized dampers and springsbetween bodies. Violation of constraints is tantamount to the fact thatdampers and springs leave their equilibrium, and consequently they exertforces on the bodies. These imaginary objects can be seen in the derivedequations for the force of constraints (10.24) or (10.25). On the other hand,although the control laws are derived from physical first principles, the pro-cedure bears superficial resemblance to the recursive method of backstepping(Krstic et al., 1995) and sliding-mode control (Slotine and Li, 1991).

It is worth mentioning that there exist many choices for uG and uΦ, andthey can be found such that the overall system becomes robust with respectto uncertainties and external disturbances. As demonstrated in (Ihle et al.,2006a) and (Ihle, 2006, Ch. 5), the design can be made robust with respectto external disturbances using backstepping.

10.3.3 Virtual Constraints

Control objectives in different strategies can be viewed as virtual mechani-cal constraints. Let η and η denote the generalized coordinates (the pose)and the generalized velocities of a vehicle5. Point stabilization is to forcethe vehicle to acquire a desired pose η∗d for all the time. It places a set of

5In marine literature, the generalized coordinates and the generalized velocities areshown by η and η whereas, in analytical mechanics, people use q and q.


holonomic constraints on the system dynamics: Hps = η − η∗. Trajectorytracking is the problem of forcing the state of the system to asymptoticallytrack a desired time-varying signal, ηd(t). This scenario leads to a set ofholonomic constraints: Htt(η, t) = η − ηd. Path following is the problemof forcing a craft to asymptotically converge to and follow a given pathwith a desired speed profile. As discussed in the preceding chapters, it usu-ally imposes two nonholonomic constraints and one holonomic constraint:Hpf(η, η) = [u(t)− ud(t), v(t)− vd(t), ψ − ψd(t)]T in which ud(t), vd(t) andψd(t) are the desired signals, and u and v are the velocity components ex-pressed in the body-fixed reference frame.

Formation control is the problem of forcing a group of vehicles to forma desired geometric configuration. Let ηi and ηi represent the generalizedcoordinates and the generalized velocities of vehicle i. Also, consider thestacking vectors η = [ηT

1 , · · · ,ηTN ]T. A formation can be achieved by as-

signing desired values rij > 0 to relative distance among members, which

bring in holonomic constraints: H ijf1(η) = ‖ηi − ηj‖ − rij ; alternatively,

a formation can be obtained by considering offsets oij among vehicles as

H ijf2(η) = ηi − ηj − oij . In fact, from the viewpoint of this methodology,

the formation control problem is the problem of modeling of an artificiallyconstrained multi-body system.

Let eij and sij be the cross-track error and the along-track error betweenvehicle i and j when vehicle i is assumed to be the leader. An alternativefor formation can be achieved by imposing the following constraint on thesystem: H f3(q, q, t) = [ui − uj , vi − vj , ψi − ψj , eij − qij , sij − pij ]T wherepij and qij are the desired offsets. This type of constraints is a combinationof holonomic and nonholonomic constraints.

In (Ihle et al., 2006a), formation average position is introduced as aholonomic constraint on the average of the vehicles’ pose:

Hav =1

n

n∑i=1

ηi − ηd(t) , η − ηd(t)

where ηd(t) is a reference trajectory. In addition, formation variance isintroduced as the following holonomic constraint

Hvr =1

n

n∑i=1

(ηi − η)2 − σd

where σd is the desired variance around the average position η.


10.4 Case Study: Master-Slave Formation

In this section, the problem of master-slave formation of two marine craftis portrayed as the modeling problem of a constrained multibody system.Without loss of generality, we focus on a group of two craft. A common ex-ample can be underway replenishment operation (Miller and Combs, 1999),where given a desired path and a desired speed, one marine craft plays therole of master and the other, called slave, has to maintain its position rela-tive to the master while the master follows the desired path with the desiredspeed profile. Therefore, the problem can be decomposed into two tasks:

1. Path maneuvering of the master in which the master follows the pathwhile it retains the speed as desired.

2. Synchronization of the slave with the master such that the desiredformation is achieved.

The objective is to design a controller to accomplish these two tasks fora group of two marine craft under Assumption 10.1.

Assumption 10.1. The following assumptions are made:

• Each marine craft has access to its own pose and velocities.

• The slave receives the information about the master’s pose and veloc-ities through a flawless communication channel.

• The slave has no knowledge about the desired speed and the path thatthe master has to follow.

• The desired path and the desired speed are feasible, known, and smooth.

First, we present the typical model for marine craft according to Fossen(2011); then the formation problem is portrayed as the modeling problemof a constrained multi-body system.

10.4.1 Model of Formation

An individual vehicle is described by

η = J (ψ)ν (10.26a)


in which η = [pT, ψ]T is the vehicle pose where p = [x, y]T ∈ R2 and ψ ∈ Sis the yaw angle; ν = [u, v, r]T ∈ R3, where u and v are the linear velocities

10.4. Case Study: Master-Slave Formation 261

and r is the angular velocity expressed in the body-fixed reference frameb. The matrix J (ψ) = diagR(ψ), 1 is the rotation matrix from b tothe inertial reference frame. The matrix R(ψ) is given by

R(ψ) =


]∈ SO(2) (10.27)

R(ψ) = R(ψ)S(ψ) S(ψ) =

[0 −11 0

]ψ (10.28)

In addition, Mb = MTb > 0, Mb = 0, Cb = −CT

b , and Db > 0 take thefollowing forms

Mb =

m11 0 00 m22 m23

0 m23 m33

, Db =

d11 0 00 d22 d23

0 d23 d33

(10.29a)

Cb =

0 0 −(m22v +m23r)0 0 m11u

(m22v +m23r) −m11u 0

(10.29b)

Using the kinematics (10.26a), the kinetics (10.26b) is represented in theform of (10.4).

For a fleet of marine vehicles, each vehicle together with correspond-ing signals and matrices is labeled with j = m, s; that is, Mj(ηj)ηj =F j(ηj , ηj , t). The vector of generalized coordinates is formed by stacking

ηm and ηs; thus, η = [ηTm,η

Ts ]T and η = [ηT

m, ηTs ]T. Consequently, the

model is given by

M(η)η + n(η, η)η = τ c

where

M(η) = diagMm(ηm),Ms(ηs) (10.30)

n(η, η) = [nm(ηm, ηm)T,ns(ηs, ηs)T]T (10.31)

τ c = [τTc,m, τ

Tc,s]

T (10.32)

10.4.2 Virtual Constraints

Now, we lay out the constraints that are imposed on the multibody systemdue to the control objectives. As mentioned earlier, the formation controlproblem is divided into two tasks. One is path maneuvering which is carriedout by the leader, and the other is synchronization with the leader which isconducted by the follower.


ix

( )pp t

py

px

pfe

pfs

r

( )sp t

mV iy

m

fe

fs LOS vector

( )mp t

Figure 10.3: The geometry of master-slave formation of two surface marinecraft.

Constraints on the Leader The leader must follow a parameterizedpath with the desired speed profile, ud(t) > 0,∀t > 0.

The position of each point on the path is denoted pp(%) = [xp(%), yp(%)]T ∈R2, % ∈ R. Consider a virtual vehicle (VV) moving on the path. Thus, theposition of the VV takes that of the path and is denoted pp($), $ ∈ R.The VV’s heading angle is equal to the slope of the path tangential line,which is computed by ψp = arctan(y

′p/x

′p) where x

′(%) , ∂x

∂% , ψp ∈ [−π, π].The body-fixed reference frame of VV is represented by p. The vectorυp = [up, 0]T represents the linear velocity of VV expressed in p. Toenable the VV to move with the speed of up, the following update law isimposed on the parameter $

$ =up√

(x′p)2 + (y′p)

2(10.33)

The error vector between the leader and the VV, expressed in p, is denotedεpf , [spf , epf ]T and given by

εpf = RT(ψp)(pm − pp

)(10.34)

Fig. 10.3 captures the geometry of the problem in hand.

From the standpoint of constrained multibody systems, the path maneu-vering problem (Skjetne et al., 2005) for the leader is equivalent to restrictingthe motion of the leader to that of the VV so that the leader follows thepath in the desired direction with the commanded speed. Therefore, the


following five constraints are placed on the leader:

Km , RT(ψm)pm − [ud, vd,m]T (10.35)

Gm ,[Gm,1, ε

Tpf

]T, Gm,1 , ψm − ψd (10.36)

The nonholonomic constraints Km are placed on um and vm. For a fullyactuated leader, vd,m can be set equal to zero. For an underactuated leadervd,m acts as an additional degree of freedom for control design. Let usgloss over the reasoning behind choosing this constraint for underactuatedleaders, and only mention that the constraint is neither conflicting nor re-dundant. It will be discussed later.

The holonomic constraints are imposed on εpf and the heading. ψd isrecognized as the desired heading. The leader is asked to follow the desiredheading angle so that it reaches the path as smoothly as desired. Inspiredby the line-of-sight guidance method for straight-line path following, thedesired heading is computed using (see Fig. 10.3)

ψd = ψp + ψr, ψr = arctan(−epf∆

), ∆ > 0 (10.37)

where the relative angle ψr ∈ [−π/2, π/2]. It is possible to assign ψd = ψp;however, it may lead to aggressive behaviors. Using a guidance law, adegree of freedom is injected into the design. On the other hand, it mayresult in inaccuracy in following paths with nonzero curvature since a curveis approximated with a straight-line segment at every instant of time.

The speed of the VV, up, has not been assigned yet. One choice may beup = ud. However, it is recommended to use up = ud + σspf , σ > 0 sincethis choice makes the VV speed up, slow down, or even move backward sothat it can stay at the shortest distance to the leader; in other words, itallows the VV to adjust itself with the leader. Accordingly, it reduces theleader’s control effort and improves the performance.

Constraints on the Follower The follower has to synchronize itself withthe leader such that the desired geometric configuration is made. Therefore,we place the following three constraints on the follower:

Ks , RT(ψs)ps − [um, vd,s]T (10.38)

Gs,1 , ψs − ψm (10.39)

The nonholonomic constraints place restrictions on the velocity componentsof the slave; The first constraint forces the slave to asymptotically track theforward velocity of the master. Similar to the constraints on the Leader’s


speed components, for a fully actuated follower, vd,s can be set equal tovd,m. For an underactuated follower, vd,s cannot be assigned arbitrarily,and will act as an additional degree of freedom to tackle underactuation. Itis demonstrated in the next section that vd,s must be adjusted according tothe dynamics of the system and the controller. The holonomic constraintGs,1 indicates that the follower has to own the leader’s orientation.

The follower must remain in the desired position relative to the leaderwhile it does not have knowledge about the leader’s path; thus, the pathhas to be estimated. One way to estimate the leader’s path is to conceivethat the path is composed of straight-line segments. At time t, estimatethe path that is traveled by the leader with a straight line whose slope isdenoted ψ∗; consider a point on the estimated path, represented by p∗(t).Place a frame on p∗(t) whose x-axis is rotated a positive angle ψ∗. Letεf , [sf , ef ]T denote the distance between the follower and p∗(t) expressedin the defined reference frame, such that:

εf = RT(ψ∗) (ps − p∗) (10.40)

If εd = [sd, ed]T describes the desired formation, the following holonomic

constraints are imposed on the follower:

Gs,2 , εf − εd (10.41)

Thus, Gs = [Gs,1,GTs,2]T. Consequently, the multibody system is subject to

10 constraints expressed by (10.35), (10.36), (10.38), (10.39), and (10.41).

10.4.3 The Control Law

To derive the force of constraints (10.32), the aforementioned stabilizationprocedure is utilized. What is required is to form the constraint JacobianmatrixW, and examine its rank property. To that end, the time derivativesof εpf and εf are computed and given by:

εpf = R(ψp)Tpm + S(ψp)

Tεpf − υpεf = R(ψ∗)Tps + S(ψ∗)Tεf −R(ψ∗)Tp∗

Therefore, one can write the constraint Jacobian matrix for each vehicle as:

Wm =

RT(ψm) 02,1 02,3

01,2 1 01,3

RT(ψp) 02,1 02,3

, Ws =

Ws RT(ψs) 02,1

ws 01,2 102,3 RT(ψ∗) 02,1


where 0i,j denotes a zero matrix of dimension i× j, and

Ws =

[− cos(ψm) − sin(ψm) 0

0 0 0

], ws =

[0 0 −1

]StackingWm andWs yields the constraint Jacobian matrixW. Likewise, wefind a and Φ. SinceW possesses six independent columns and det(WTW) =20, the formula (10.25) gives the vector of constraint forces, which providesforces that carry out the path maneuvering task and the synchronizationtask. These forces have to be applied to the vehicles through the actuators.

For underactuated leader and follower, it is impossible to apply thecorresponding forces (i.e. the second and the fifth elements of τ c denotedby τc,2 and τc,5, respectively) to the vehicles. To circumvent the problem,we take the same line of thinking as (Fossen et al., 2003) and (Peymaniand Fossen, 2011b). The solution is to make τc,2 and τc,5 equal to zeroby utilizing the unassigned signals vd,m and vd,s. Setting τc,2 and τc,5 zeroyields two differential equations as:

m22,mvd,m = −d22,mvd,m + gm(ζ, t) (10.42a)

m22,svd,s = −d22,svd,s + gs(ζ, t) (10.42b)

in which ζ = [KTm,K

Ts , G

Tm, G

Ts ,G

Tm,G

Ts ]T, and gj(ζ, t), j = m, s are

found by inspection. It is possible to show that vd,m and vd,s are globallybounded if the leader and follower are open-loop stable in the unactuateddynamics (sway).

We would like to draw the reader’s attention to the fact that the controllaw derived from this method is based on the method of least squares.That is, the control law provides the best approximation for the forcesthat make the geometric errors [εT

f , εTpf ]T GES in addition to making Km,

Ks, Gm,1, and Gm,2 GES. We can easily show that [Gm,1, Gm,2] is GES.However, to establish uniform global asymptotic stability (UGAS) of ξ =[εTf , ε

Tpf ,K

Tm,K

Ts ]T requires more work. It is viable to show that there

exists a ∆∗ > 0 such that ξ is UGAS if ∆ > ∆∗.

Remark 10.5. The proposed controller is a dynamic controller. Accordingto Wichlund et al. (1995a), underactuated marine craft possess a second-order nonholonomic constraint. In fact, in the proposed approach, the first-order nonholonomic constraints are treated by the stabilization procedurewhereas the second-order nonholonomic constraints are treated by injectingdynamics into the control system.


w

bz

u

v

b

p

q

r

bx

by

p

q

ya

bz

hea

bz

aw (ψ)

ave

pi

sway

itch (θ)

roll (φsurge

bx

by

φ)

Figure 10.4: INFANTE AUV.

10.4.4 Simulation Results

A nonlinear model of INFANTE AUV (Silvestre, 2000) is used for simulatinga realistic environment (see Fig. 10.4). The INFANTE AUV is unactuated insway. The contribution of the rudder is seen in both the surge and the swaydynamics. The maximum propeller thrust is 920 N which makes the AUVmove at a speed of 5 m/s in calm water. The rudder deflection saturates at30. Actuator dynamics is approximated by a first-order transfer function.This system is described by a simplified model according to (10.29) whichis used for designing the controller.

Two identical vehicles are considered. The desired path is described bypp(%) = [−60, %]T. The initial conditions are η0,m = [−50,−90, π/2]T,η0,s =

[−90,−100, π/4]T and ν0,m = ν0,s = [1.5 m/s, 0, 0]T. The controller gainsare selected as PG,m = PG,s = diag1, 1, 1,PΦ,m = diag30, 20, 1, 1, 1,and PΦ,s = diag5, 2, 1, 1, 1. ∆ = 20 m and ud = 2.5 m/s. The desiredformation is given by εd = [0,−10]T for t < 90 s. εd smoothly changes to[0,−20]T for t ≥ 90 s. The examination is performed for two cases.

Ideal Condition: In this case, the simulation is carried out in theabsence of external disturbances. Fig. 10.5 contains two blow-ups of thesimulation result.

In the Presence of Ocean Currents: Now, a current with the speedvector ηc = [−.5 m/s, 0, 0]T perturbs the craft. To make the controllerrobust, integral action is augmented to the stabilization procedure. Fig. 10.6shows the result of the simulation. For the sake of comparison, the resultof the controller without integral action is superimposed in Fig. 10.8.

Discussions The simulation results reveal that the algorithm is successfulto accomplish both path maneuvering and synchronization tasks. It is also


Nor

th(m

)

East (m)

The path travelled by AUVs

Nor

th(m

)

East (m)


Desired position of the follower

Desiredpath

Leader

Follower

120 140 160 180 200 220 240 260 280 300

−120 −100 −80 −60 −40 −20 0 20 40

−100

−90

−80

−70

−60

−50

−100

−90

−80

−70

−60

−50

Figure 10.5: Formation of two marine craft in calm water. The desireddistance changes after t = 90 s.

Nor

th(m

)

East (m)


Nor

th(m

)

East (m)


Ocean Currents

Ocean Currents

Leader

Follower

120 140 160 180 200 220 240 260 280 300

−120 −100 −80 −60 −40 −20 0 20 40

−90

−80

−70

−60

−50

−40

−90

−80

−70

−60

−50

Figure 10.6: Formation of two marine craft in the presence of ocean currents.Integral action is augmented to the controller. Circles show the desiredposition of the follower at each instant.


Time (sec)

[deg

ree]

AUV2: Rudder Deflection (δ2)

[N]

AUV2: Forward Thrust (T2)

Time (sec)

[deg

ree]

AUV1: Rudder Deflection (δ1)

[N]

AUV1: Forward Thrust (T1)

0 50 100 150

0 50 100 150

0 50 100 150

0 50 100 150

−40

−20

0

20

40

0

200

400

600

800

1000

−40

−20

0

20

40

0

200

400

600

800

1000

Figure 10.7: The forward thrust and the rudder deflection for each craft.It shows that the controller performs satisfactorily within the saturationbounds. Ocean currents are present.

demonstrated that in the presence of ocean currents, the augmentation ofintegral action can be a remedy for fragility of the control system. Inte-grators can be placed only for the geometric errors (i.e. εf and εpf ) andthere is no need for adding integrators for the constraints on the sway andyaw dynamics. In fact, constant ocean currents force underactuated marinecraft to rotate so that a component of the forward velocity is provided inorder to counteract the component of ocean currents that is normal to thepath; with integral action, this approach makes ψr and e zero. On the otherhand, if e = 0 then ψ 6= 0 and ψ − ψp 6= 0. The reason is that ψd is foundsuch that the x-axis of b is aligned along the line of sight.

In this paper, p∗ and ψ∗ are chosen as pm and χm, respectively. Actually,the path traveled by the leader is approximated with straight-line segmentswith the slope χm at each instant of time. χm = ψm + βm is termed as thecourse angle, which is the angle that the overall speed makes in i, andβm = arctan(vm/um) is called the sideslip angle.

10.5 The Fundamental Equation of Motion

Firdaus E. Udwadia and Robert E. Kalaba proposed a new perspective onconstrained motion (Udwadia and Kalaba, 1992), and derived the equation

10.5. The Fundamental Equation of Motion 269

Time (sec)

[m]

(d) epf

[m]

(c) spf

Time (sec)

[m]

(b) ef

[m]

(a) sf

0 50 100 150

0 50 100 150

0 50 100 150

0 50 100 150

−15

−10

−5

0

5

−100

−50

0

50

−10

0

10

20

30

−15

−10

−5

0

5

10

Figure 10.8: Geometric errors for path following and synchronization tasks.The red dashed lines represent the controller without integral action. Oceancurrents are available.

Time (sec)

[deg

ree]

ψ1 − ψp

[m/s

]

Sway velocity of AUV 1

[m/s

]

u1 − ud

Time (sec)

[deg

ree]

ψ2 − ψ1

[m/s

]

Sway velocity of AUV 2

[m/s

]

u2 − u1

0 50 100 150

0 50 100 150

0 50 100 150

0 50 100 150

0 50 100 150

0 50 100 150

−50

0

50

−2

0

2

−2

0

2

−200

0

200

−1

0

1

−2

0

2

Figure 10.9: The difference between the forward velocities and headingangles. The sway velocities are also depicted. Ocean currents influence thecraft.


of constrained motion without invoking the notion of Lagrange multipliers.They expound the idea to the systems which may have holonomic and non-holonomic constraints, and the constraint forces may or may not6 satisfyD’Alembert’s principle at each instant of time (Udwadia and Kalaba, 1996,2002). In this section, the essence of their work is presented.

The result is applicable to much wider arena of equality constraints. Infact, the system may be constrained to a general kinematic equation (10.5),no matter if it is holonomic, nonholonomic, rheonomic or scleronomic. Tobe more explicit, all constraints which have a linear-in-acceleration form as

A(q, q, t)q = b(q, q, t) (10.43)

can be handled in this method. It is assumed that the constraint equation(10.43) is consistent; however, it need not be linearly independent. Presumethat the equation of motion of unconstrained system is given by

Mq = F(q, q, t) (10.44)

where M, which may depend on q, is the mass matrix which is positivedefinite and symmetric. The vector field F is the vector of impressed forces.Let a denote the accelerations that the unconstrained system acquires, givenby

a =M−1F(q, q, t) (10.45)

Assuming that the generalized position q and velocity q are known at anytime t, the objective is to find the actual accelerations at time t, as a resultof the given impressed force F and the constraints (10.43).

Gauss’s principle asserts that among all the accessible accelerations thatthe system can have at time t which are compatible with the constraints,those accelerations are actual accelerations that minimize the Gaussian asdefined

G(q) = (q− a)TM(q− a) (10.46)

The Gauss’s principle is applicable to any kind of kinematic constraints.Let ∆q , q − a which is the difference between the accelerations in thepresence of constraints and the accelerations in their absence. The Gaussiancan be written as G(q) = ‖∆q‖M = (M1/2∆q)T(M1/2∆q). Clearly, in theabsence of constraints, q = a and G(q) = 0.

6This means that their approach includes non-ideal constraints that do not satisfythe principle of virtual work. Holonomic constraints and linear-in-velocity nonholonomicconstraints conform with D’Alembert’s principle of virtual work.


The fundamental equation of motion is given by7

q = a +M− 12

(AM− 1

2

)†(b−Aa) (10.47)

in which (AM− 12 )† is the Moore-Penrose inverse of the constraint matrix

AM− 12 . The proof is easy. One needs to verify that (i) (10.47) satisfies

the constraints (10.43); (ii) (10.47) is the unique vector that minimizesthe Gaussian. Then, according to Gauss’s principle, (10.47) is the actualaccelerations that the system has in the presence of constraints.

To show that (10.47) satisfies (10.43), substitute (10.47) in (10.43):

Aq = Aa +AM− 12

(AM− 1

2

)†(b−Aa)

=(

I−AM− 12 (AM− 1

2 )†)Aa +AM− 1

2 (AM− 12 )†b

=(

I−AM− 12 (AM− 1

2 )†)

(AM− 12 )M 1

2 a +AM− 12 (AM− 1

2 )†b

From the main properties of the MP-generalized inverse of AM− 12 , it follows

that the first term is zero. Thus, Aq = AM− 12 (AM− 1

2 )†b. Since the con-

straints are consistent, one may concluded that Aq = AM− 12 (AM− 1

2 )†b =b. In fact, (10.43) can be expressed as

Aq = (AM−1/2)(M1/2q) = b (10.48)

To have M1/2q = (AM−1/2)†b as its solution, with the fact that the con-straints are consistent, it is required to have

(AM−1/2)(AM−1/2)†b = b (10.49)

It means that the consistency of the constraints is equivalent with (10.49).To show that (10.47) satisfying (10.43) is the unique acceleration that min-imizes the Gaussian, let h = q + v, for any v 6= 0, be another accelerationthat satisfies (10.43). The aim is to show that G(h) > G(q) for any v 6= 0.

From Ah = Aq +Av = b, it follows that Av = 0⇒ AM−1/2M1/2v =0. According to (see Udwadia and Kalaba, 1996, Ch.2):

(M1/2v)T(AM−1/2)† = 0 (10.50)

7In (Udwadia and Kalaba, 1996), from various directions, it is shown how this funda-mental equation is derived; see Appendix 10.A for a short note. Here, it is demonstratedthat this equation is valid.


Write G(h):

G(h) = (q + v − a)TM(q + v − a)

= (M− 12 (AM− 1

2 )†(b−Aa) + v)TM(M− 12 (AM− 1

2 )†(b−Aa) + v)

= ((AM− 12 )†(b−Aa) +M 1

2 v)T((AM− 12 )†(b−Aa) +M 1

2 v)

= ((AM− 12 )†(b−Aa))T((AM− 1

2 )†(b−Aa)) + (M 12 v)T(M 1

2 v)

+ ((AM− 12 )†(b−Aa))TM 1

2 v + (M 12 v)T((AM− 1

2 )†(b−Aa))

= G(q) + vTMv

where (10.50) has been used. Therefore, for any v 6= 0 due to positivedefiniteness of M, G(h) > G(q). Since no assumption is made on themagnitude of v, q obtained from (10.47) is the unique acceleration thatminimizes the Gaussian (the problem is a global minimization problem).

A close look at (10.47) reveals that the equation of motion subject tothe constraints in the form of (10.43) will take the following form

Mq = F + Fc, Fc =M 12

(AM− 1

2

)†(b−Aa) (10.51)

In fact, in the presence of constraints, the system is under additional forcesso as to satisfy the constraints. The force of constraints is denoted Fc,and explicitly is found. To get a better feel of the constraint force, lete , b − Aa; in fact, e is the level of dissatisfaction of the constraintsevaluated by the accelerations of the unconstrained system (i.e. a). Then,the force of constraints can be represented as

Fc = K e, K =M 12

(AM− 1

2

)†It implies that the constraints exert force on the system which is propor-tional to the error e (i.e. proportional to the extent to which the accelerationcorresponding to the unconstrained system does not satisfy the constraints;see (Udwadia and Kalaba, 1996)).

The MP-inverse of any matrix Y can be written as Y † = Y T(Y Y T)†. IfY is full row rank, (Y Y T) is nonsingular and (Y Y T)† = (Y Y T)−1. Accord-ingly, the equation of motion is modified to

Mq = F +AT(AM−1AT

)†(b−Aa) (10.52)

where M =MT is assumed. In case the constraints are independent8, thematrix A is full row rank. Then, due to positive definiteness of M, the

8The constraint are consistent and independent.


matrix AM−1AT is full rank and(AM−1AT

)†=(AM−1AT

)−1. In this

case, the equation of motion is given by

Mq = F +AT(AM−1AT

)−1(b−Aa) (10.53)

Relation to Lagrange Multipliers For a system subject to constraintsin the form of A(q, t)q = b(q, t)9, the equation of motion is given by

Mq = F +ATλ (10.54)

where F is the impressed force that enters into the equation of motion ofthe unconstrained system, and λ is the vector of Lagrange multipliers withappropriate dimension. Drawing a comparison between (10.54) and (10.52),

one may notice that ATλ = AT(AM−1AT

)†(b−Aa) , Fc; thus

ATλ = Fc ⇒ λ = (AT)†Fc + (I− (AT)†AT)v

= AA†(AM−1AT

)†(b−Aa) + (I−AA†)v (10.55)

for any arbitrary v. The following facts have been used (AT)† = (A†)T andAA† = (AA†)T. Eq. (10.55) implies that λmay not be unique. However, theequation of motion is unique. If (I−AA†) is identically zero,the multipliersare unique. It occurs if the matrix A is full row rank, or equivalently,the constraints are independent. For a set of independent and consistentconstraints, the Lagrange multiplier vector is given by

λ =(AM−1AT

)−1(b−Aa) (10.56)

For more details, refer to (Udwadia, 2000). To see whether there is an-other generalized inverse which works in the fundamental equation (10.47),consult (Udwadia and Kalaba, 1996, Chapter 8). To deal with systemswith singular mass matrix, refer to (Schutte and Udwadia, 2011; Udwadiaand Phohomsiri, 2006); a similar discussion is presented by Liu and Hus-ton (2008). Their approach has been, so far, utilized to control of robotswith redundant degrees of freedom (Peters et al., 2008), synchronization ofchaotic gyroscopes (Udwadia and Han, 2008), satellite formation-keeping(Cho and Udwadia, 2010, 2013), and optimal tracking control of nonlinearsystems (Udwadia, 2008). An interesting discussion on constrained systems

9Expansion to the constraints of the form of (10.43) is possible by revisiting thedefinition of virtual displacement; i.e. virtual displacement is a nonzero vector v whichsatisfies A(q, q, t)v = 0 and vTFc = 0.


with the help of Poincare equations of motion, which extends Gauss’s prin-ciple of least squares, is given in (Udwadia and Phohomsiri, 2007a), andits application to nonlinear control is presented in (Udwadia and Phohom-siri, 2007b). Poincare equations take Lagrangian equations to first-orderequations and usually are used for obtaining the equation of motion usingquasi-velocities (similar to the equation of motion in a body-fixed referenceframe).

An Inherent Optimality The actual accelerations that satisfy the con-straints are those that minimize the Gaussian (10.46), according to Gauss’sprinciple. It is straightforward to show that the force of constraints Fc

minimizes the following cost function

J(t) = FTc M−1 Fc := ‖Fc‖2M−1 (10.57)

Simply, it is equivalent with J(t) = ‖M−1/2Fc‖2. To see that, consider theconsistent constraints Aq = b that are imposed on a system described by(10.44); thus, the equation of motion will be Mq = F + Fc. Then

AM−1/2M1/2q = b

⇒ AM−1/2M−1/2(F + Fc) = b ⇒ AM−1/2M−1/2Fc = b−AM−1F

According to (Graybill, 2001), the solution forM−1/2Fc subject to the costfunction (10.57) is given by

M−1/2Fc = (AM−1/2)†(b−AM−1F)⇒ Fc =M1/2(AM−1/2)†(b−Aa)

which is equivalent with (10.51). It states that Nature would exert addi-tional force whose magnitude is minimum at every instant of time. In otherwords, “of the entire set of constraint forces that cause a constrained me-chanical system to exactly satisfy the constraints, Nature seems to choosethe force that minimizes (10.57) at each instant of time”(Udwadia, 2008).If this methodology is used to derive a control action (similar to the topic ofthis chapter), one may claim that the control is optimal such that it causesthe system to minimize the cost (10.57) at each instant of time t. The op-timal design with the aid of analytical mechanics is the topic of (Udwadia,2008).

10.6 Concluding Remarks

One circle of thinking for motion control of a mechanical system is to con-ceive the system constrained to some imaginary manifolds/bodies by means

10.A. Fundamental Equation: Derivation 275

of some imaginary springs and dampers. The interconnection between bod-ies may be expressed in terms of constraints depending on the positionand/or velocity variables. The forces coming out of the imaginary springsand dampers would keep the system moving as it should. These are theforces that are to be applied to the system using actuators. To derive theforce of constraints that secure the fulfillment of the constraints, the La-grangian mechanics, more specifically the Lagrange multiplier method, isemployed.

Consequently, a complete framework for motion control of mechanicalsystems is proposed, which generalizes (Ihle et al., 2006a,b) for situationsinvolving speed assignment tasks. The generalization is made in a way thatboth positional and velocity restrictions are covered simultaneously. Themethod is general and systematic, and it provides a closed-form solutionfor control forces. Moreover, control laws by this method can be physicallyinterpretable.

Appendix 10.A Fundamental Equation: Deriva-tion

Consider an unconstrained system whose motion is described by Mq = F.Denote the unconstrained acceleration by a =M−1F. Suppose the systemis subject to A(q, q, t)q = b(q, q, t). Let B = AM−1/2 and r = M1/2q;thus, the constraints are recast as Br = b. It implies that r = B†b + (I −B†B)h for any h. r is known if h is known. Thus, h is determined inaccordance with an accepted principle from analytical mechanics (which is,here, D’Alembert’s principle).

The equation of motion constrained to constraints will be changed toMq = F + Fc where Fc is the additional force due to the constraints. Theprinciple of D’Alembert asserts that the total work done by Fc under anyvirtual displacement is zero. The virtual displacement is a nonzero vectorv such that Av = 0. Therefore, vTFc = 0 or vT(Mq− F) = 0.

We haveAv = Bu = 0 where u =M1/2v. It implies that uTB† = 0. Onthe other hand, from vT(Mq−F) = 0, it follows that uT(r−M−1/2F) = 0.Substitution for r gives

uT(B†b + (I−B†B)h−M−1/2F) = 0⇒ uT(h−M−1/2F) = 0

It implies that h−M−1/2F lies in the range space of B†; that is, there is a


w such that h−M−1/2F = B†w. Thus, h =M−1/2F + B†w. Hence

r =M1/2q = B†b + (I−B†B)(M−1/2F + B†w)

= B†b + (I−B†B)M−1/2F

Hence, we will obtain (10.47).

Chapter 11

Concluding Remarks andFuture Work

The thesis consists of three main subjects, each of which is enunciated inone part of the thesis.

(I) Synchronization in multi-agent systems in the presence of externaldisturbances.

(II) Speed-varying path-following of marine vehicles.

(III) Motion control of marine craft using analytical mechanics.

Part I of the thesis has initiated a novel line of research in synchronizationin multi-agent systems while agents are exposed to external disturbances.The concepts of “H∞ almost synchronization”, “H∞ almost regulated syn-chronization”, and “H∞ almost formation” were introduced for the firsttime. The novel notions are defined accurately, and the solutions have asolid theoretical foundation in the theory of linear systems.

The multi-agent systems that are considered in this work are describedby linear, time-invariant multiple-input multiple-output models. Communi-cation topologies are characterized by directed graphs. Each agent receivesa linear combination of its own output relative to that of its neighbors. Thisdescription for communication topologies removes the strong assumption ofbi-directional links, and allows for local sensing rather than global sensing.Thus, it is of high practical importance.

The essence of the work is to design decentralized, linear, time-invariantprotocols based on a distributed observer such that the impact of distur-bances on the synchronization error dynamics can be made arbitrarily small,

277

278 Concluding Remarks & Future Work

in the sense of the H∞-norm of the corresponding closed-loop transfer func-tion. Hence, output synchronization can be achieved with any arbitrarydegree of accuracy.

The proposed solution is based on the time-scale structure assignmenttechnique, which allows for online adjustment of the accuracy of synchro-nization. Also, the order of the controller is fixed and does not vary forvarious degrees of accuracy. In addition, the problem is solved for a set offixed communication topologies (in contrast to one specific topology); thus,the design is robust with respect to network topologies.

To the best of the author’s knowledge, so far, methods that systemati-cally design protocols that reduce the impact of disturbance on synchroniza-tion have not been published. The available articles can be categorized intothree general groups as (1) those articles that analyze the disturbance at-tenuation properties of distributed protocols without proposing any specificdesign that achieves synchronization with a desired degree of accuracy ; (2)those works that expand the results from single-agent case to multi-agentcase and relate the H∞ norm to the existence of matrices that satisfy linearmatrix inequalities without making any argument about the solvability ofthe problem; (3) those papers that solve sub-optimal H∞ control problemover a network, where the objective is to make the H∞ norm of the transferfunction from disturbance to the output of individual agents smaller than agiven γ > 0, which has nothing to do with the accuracy of synchronization.

Compared to the available articles, the networks considered in this thesisare much more realistic. The problem of H∞ almost synchronization wassolved for two different networks:

(Project 1) Heterogeneous networks of introspective agents.

(Project 2) Homogeneous networks of nonintrospective agents.

The research topic is promising and is potential to various expansions. Thefollowing extensions are required to make the result much more powerful.

• In this thesis, full-order observers are considered. Since high-order sys-tems are challenging in practical situations, reduced-order distributedobservers should be taken into consideration.

• Sets of time-varying/switching communication topologies rather thansets of fixed topologies should be considered.

• This thesis assumes that measurements are not corrupted with noise.However, it is a well-known fact that high gains amplify measurementnoise. Therefore, the effect of measurement noise should be taken into

279

account. It is also very interesting to consider model uncertainties ashigh gains can make the control system robust with respect to modeluncertainties.

• An effort should be made on expanding the result for networks ofnonlinear agents and for nonlinear H∞ almost synchronization.

In the thesis, it is presumed that each agent is capable of exchanging infor-mation about the state of its own protocol relative to that of its neighbors.One should take account of

• the case where the exchange of additional information is not permis-sible. Therefore, a distributed observer in the form that is proposedin this thesis is not doable, and one may require to design decentral-ized dynamic protocols by designing local filters that map the networkinformation to the control signal, which achieves H∞ almost synchro-nization.

The thesis deals with the problem of H∞ almost synchronization, where theobjective is to make the H∞-norm of the transfer function from disturbanceto synchronization errors arbitrary small. For Project 1, the key assump-tion for solvability is local information (which requires global sensing). ForProject 2, a set of assumptions on agent’s dynamics are made to ensuresolvability. The problems that seem interesting to solve are

• to find the infimum of theH∞-norm of the transfer function, called γ∗,for a homogeneous/heterogeneous network of nonintrospective agents.Then, the optimal/suboptimal H∞ problem for networks must besolved for a given γ ≥ γ∗.

• to include the effect of delay in communication links while agents areexposed to external disturbances. Determination of γ∗ is imperativeand should be given in relation to delay. A low-high gain controllermay be proposed to reduce the impact of disturbance on synchroniza-tion error dynamics while guaranteeing synchronization in the pres-ence of delay.

Part II addressed the problem of path following for marine vehicles. Thecontribution of this part is to make the speed of the vehicle dynamicallydependent on the distance to the path and the rate of convergence. Itthen offers accelerated path following, as the control system commands thevehicle to speed up to move faster toward the path or to slower down to


avoid moving sideways while the speed of the marine vehicle must be asdesired when it moves on the path.

Two methods to achieve the goal are proposed. In one approach, usingbackstepping, the speed is made dependent on the geometric error through anonlinear mapping. Contrary to available articles that perform a coordinatetransformation to eliminate the coupling between the actuators and theunactuated dynamics, the proposed dynamic controller is derived using themodel in the original coordinates.

In the other approach, using the method of least squares, the speedlinearly depends on the geometric error and its derivative. For fully actuatedmarine craft, the method is extended to include integration which results ina robust controller with respect to constant external disturbances. Actually,the method can be regarded as a remedy for path-following controllers whichuse the line-of-sight guidance system for aligning the x-axis of the body-fixedreference frame with the line-of-sight vector.

According to the analysis given in the thesis, the guidance parameterplays a central role in the stability of the error dynamics if the vehicle isunderactuated. However, in the case of full actuation, the guidance param-eter does not impact on the stability theoretically although it affects theperformance.

The speed-varying path following using the method of least squares wassolved for a linearized model of underactuated marine craft.

• The extension of the method presented in Chapter 9 to nonlinearmodels of underactuated marine craft is essential. In addition, theinclusion of integral action to remove offset in the geometric error inthe presence of external disturbances, akin to what has been done forfully actuated marine craft, seems indispensable.

Notice that when Coriolis and centripetal forces are considered in the modelof underactuated marine craft, the linear growth condition does not hold forthe interconnection term. Moreover, the growth rate of the interconnectionterm relative to the growth of the drift term does not conform with any well-known results on the stability of cascade interconnected systems. Thus, oneshould try to figure out how to establish the stability of the closed-loopnonlinear system.

In path following using the method of least squares, the thing that canbe regarded as a drawback is that the generated surge and sway forces arelinearly proportional with the geometric error and its derivative. Hence,it may cause saturation in the actuators for large initial geometric errors.Although a remedy is to reroute the path with the aid of the guidance

281

system, one can use nonlinear functions to take care of the capabilities ofthe actuators. A suggestion is to change σe to something like

σe = −ρ2(γ)Tν − ke2f2(e)− ke1f1(e) (11.1)

where, for instance, fi(x) can be chosen as tan−1(γ1x) for i = 1, 2. Anotherline of research is

• to solve the formation problem using the method of least squares. Itcan be done by considering the cross-track error and the along-trackerror between a leader and a follower in the slave’s path-followingcontroller.

Part III introduced a novel framework for motion control of mechanicalsystems, including marine craft, using the principles from analytical me-chanics. The idea is to portray a controlled mechanical system as a mechan-ical system constrained to constraint functions. Then, invoking methodsfrom analytical mechanics, the equation of motion of the constrained sys-tem is derived. In accordance with the Lagrangian framework, constraintsexert additional forces on the system, which are conceived as the controlaction to be applied to the system by means of actuators.

The work presented in this thesis generalizes the work of Ihle (2006)so that the method includes both holonomic and nonholonomic constraintsalthough nonholonomic constraints are limited to those that are linear invelocities due to limitations imposed by the principle of D’Alembert. There-fore, scenarios in which the speed of a marine vehicle is controlled (whilethere is not a specific positional constraint corresponding to the velocityconstraint) can be handled. The application of the method to coordinatedpath following was demonstrated.

This approach shows how the control action would be if Nature was thecontrol engineer. Possible extensions may be

• to show how one-directional links can be taken into consideration whenit comes to formation control among a group of vehicles.

• to include unilateral constraints in the framework. Unilateral con-straints can be used to define a proximity of convergence in formationproblems.


Bibliography

Aamo, O. M., Arcak, M., Fossen, T. I., and Kokotovic, P. V. (2001). Global outputtracking control of a class of euler-lagrange systems with monotonic non-linearities inthe velocities. International Journal of Control, 74(7):649–658.

Aguiar, A. P., Dacic, D. B., Hespanha, J. P., and Kokotovic, P. (2004). Path-following orreference-tracking? an answer based on limits of performance. In 5th IFAC/EURONSymposium on Intelligent Autonomous Vehicles, Lisbon, Portugal.

Aguiar, A. P. and Hespanha, J. P. (2007). Trajectory-tracking and path-following of un-deractuated autonomous vehicles with parametric modeling uncertainty. IEEE Trans-actions on Automatic Control, 52(8):1362–1379.

Aguiar, A. P., Hespanha, J. P., and Kokotovic, P. V. (2005). Path-following for nonmin-imum phase systems removes performance limitations. IEEE Transactions on Auto-matic Control, 50(2):234–239.

Aguiar, A. P. and Pascoal, A. M. (2007). Dynamic positioning and way-point tracking ofunderactuated auvs in the presence of ocean currents. International Journal of Control,80(7):1092 – 1108.

Al-Hiddabi, S. and McClamroch, N. (2002). Tracking and maneuver regulation control fornonlinear nonminimum phase systems: application to flight control. IEEE Transactionson Control Systems Technology, 10(6):780–792.

Almeida, J., Silvestre, C., and Pascoal, A. (2007). Path-following control of fully-actuatedsurface vessels in the presence of ocean currents. In Proc. of 7th IFAC Conference onControl Applications in Marine Systems, volume 7, pages 26–31.

Altafini, C. (2002). Following a path of varying curvature as an output regulation problem.IEEE Transactions on Automatic Control, 47(9):1551–1556.

Arcak, M. (2007). Passivity as a design tool for group coordination. IEEE Transactionson Automatic Control, 52(8):1380–1390.

Atassi, A. and Khalil, H. (1999). A separation principle for the stabilization of a class ofnonlinear systems. IEEE Transactions on Automatic Control, 44(9):1672–1687.

Bai, H., Arcak, M., and Wen, J. (2011). Cooperative Control Design – A Systematic,Passivity-Based Approach. Communications and Control Engineering. Springer.

283

284 Bibliography

Battilotti, S. (1999). Separation results for semiglobal stabilization of nonlinear systemsvia measurement feedback. In Nijmeijer, H. and Fossen, T., editors, New Directionsin nonlinear observer design, volume 244 of Lecture Notes in Control and InformationSciences, pages 183–206. Springer London.

Baumgarte, J. W. (1983). A new method of stabilization for holonomic constraints.Journal of Applied Mechanics, 50(4a):869–870.

Bauso, D., Giarr, L., and Pesenti, R. (2009). Consensus for networks with unknown butbounded disturbances. SIAM Journal on Control and Optimization, 48(3):1756–1770.

Behal, A., Dawson, D., Xian, B., and Setlur, P. (2001). Adaptive tracking control of un-deractuated surface vessels. In Proceedings of the 2001 IEEE International Conferenceon Control Applications, pages 645–650.

Behal, A., Dawson, D. M., Dixon, W. E., and Fang, Y. (2002). Tracking and regula-tion control of an underactuated surface vessel with nonintegrable dynamics. IEEETransactions on Automatic Control, 47(3):495–500.

Berge, S. P., Ohtsu, K., and Fossen, T. I. (1999). Nonlinear control of ships minimizingthe position tracking errors. Modeling, Identification and Control, 20(3):177–187.

Berghuis, H. and Nijmeijer, H. (1993). A passivity approach to controller-observer designfor robots. IEEE Transactions on Robotics and Automation, 9(6):740–754.

Biggs, N. (1993). Algebraic Graph Theory. Cambridge Mathematical Library, secondedition edition.

Bloch, A. M., Reyhanoglu, M., and McClamroch, N. H. (1992). Control and stabiliza-tion of nonholonomic dynamic systems. IEEE Transactions on Automatic Control,37(11):1746–1757.

Bokharaie, V., Mason, O., and Verwoerd, M. (2010). D-stability and delay-independentstability of homogeneous cooperative systems. IEEE Transactions on Automatic Con-trol, 55(12):2882–2885.

Bokharaie, V. S., Mason, O., and Verwoerd, M. (2011). Correction to “d-stability anddelay-independent stability of homogeneous cooperative systems”. IEEE Transactionson Automatic Control, 56(6):1489–1489.

Børhaug, E. (2008). Nonlinear Control and Synchronization of Mechanical Systems. Phdthesis, Department of Engineering Cybernetics, Norwegian University of Science andTechnology, Trondhiem, Norway.

Børhaug, E., Pavlov, A., and Pettersen, K. Y. (2007). Straight line path following forformations of underactuated underwater vehicles. In Decision and Control, 2007 46thIEEE Conference on, pages 2905–2912.

Børhaug, E., Pavlov, A., and Pettersen, K. Y. (2008). Integral LOS control for pathfollowing of underactuated marine surface vessels in the ocean currents. In 47th IEEEConference on Decision and Control, pages 4984–4991.

Bibliography 285

Børhaug, E. and Pettersen, K. Y. (2005a). Adaptive way-point tracking control for un-deractuated autonomous vehicles. In Proceeding of 44th IEEE Conference on Decisionand Control and European Control Conference, pages 4028–4034.

Børhaug, E. and Pettersen, K. Y. (2005b). Cross-track control for underactuated au-tonomous vehicles. In Decision and Control, 2005 and 2005 European Control Confer-ence. CDC-ECC ’05. 44th IEEE Conference on, pages 602–608.

Breivik, M. and Fossen, T. I. (2005). Principles of guidance-based path following in 2dand 3d. In Proceeding of 44th IEEE Conference on Decision and Control and EuropeanControl Conference, pages 627–634.

Brockett, R. W. (1983). Asymptotic stability and feedback stabilization. Differentialgeometric control theory, 27:181–191.

Brunt, B. v. (2004). Holonomic and nonholonomic constraints. In The Calculus of Vari-ations, pages 119–133. Springer.

Bruyninckx, H. and De Schutter, J. (1996). Specification of force-controlled actions inthe task frame formalism-a synthesis. IEEE Transactions on Robotics and Automation,12(4):581–589.

Bullo, F. (2000). Stabilization of relative equilibria for underactuated systems on rieman-nian manifolds. Automatica, 36(12):1819 – 1834.

Bullo, F., Cortes, J., and Martınez, S. (2009). Distributed Control of Robotic Networks.Applied Mathematics Series. Princeton University Press. Electronically available athttp://coordinationbook.info.

Bullo, F., Leonard, N. E., and Lewis, A. D. (2000). Controllability and motion algorithmsfor underactuated lagrangian systems on lie groups. IEEE Transactions on AutomaticControl, 45(8):1437–1454.

Burger, M., Pavlov, A., Borhaug, E., and Pettersen, K. Y. (2009a). Straight line pathfollowing for formations of underactuated surface vessels under influence of constantocean currents. In Proceeding of American Control Conference, pages 3065–3070.

Burger, M., Pavlov, A., and Pettersen, K. Y. (2009b). Conditional integrators for pathfollowing and formation control of marine vessels under constant disturbances. InProceeding of Manoeuvring and Control of Marine Craft, pages 179–184, Brazil.

Cao, Y., Yu, W., Ren, W., and Chen, G. (2013). An overview of recent progress inthe study of distributed multi-agent coordination. IEEE Transactions on IndustrialInformatics, 9(1):427–438.

Carli, R., Chiuso, A., Schenato, L., and Zampieri, S. (2011). Optimal synchronizationfor networks of noisy double integrators. IEEE Transactions on Automatic Control,56(5):1146–1152.

Chelouah, A. (1997). Extensions of differential flat fields and liouvillian systems. InProceedings of the 36th IEEE Conference on Decision and Control, volume 5, pages4268–4273 vol.5.

286 Bibliography

Chen, B. M. (2000). Robust and H∞ Control. Springer.

Chen, B. M., Lin, Z., and Shamash, Y. (2004). Linear Systems Theory: A StructuralDecomposition Approach. Birkhauser.

Chen, J. and Huang, T. (2004). Applying neural networks to on-line updated pid con-trollers for nonlinear process control. Journal of Process Control, 14(2):211–230.

Cho, H. and Udwadia, F. E. (2010). Explicit solution to the full nonlinear problem forsatellite formation-keeping. Acta Astronautica, 67(3-4):369–387.

Cho, H. and Udwadia, F. E. (2013). Explicit control force and torque determination forsatellite formation-keeping with attitude requirements. Journal of Guidance, Control,and Dynamics, 36(2):589–605.

Chopra, N. and Spong, M. (2008). Output synchronization of nonlinear systems withrelative degree one. In Blondel, V., Boyd, S., and Kimura, H., editors, Recent Advancesin Learning and Control, volume 371 of Lecture Notes in Control and InformationSciences, pages 51–64. Springer Berlin / Heidelberg.

Chopra, N. and Spong, M. (2009). On exponential synchronization of kuramoto oscillators.IEEE Transactions on Automatic Control, 54(2):353–357.

Chung, S.-J. and Slotine, J.-J. (2009). Cooperative robot control and concurrent synchro-nization of lagrangian systems. IEEE Transactions on Robotics, 25(3):686–700.

Coogan, S. and Arcak, M. (2012). Scaling the size of a formation using relative positionfeedback. Automatica, 48(10):2677 – 2685.

De Luca, A. and Oriolo, G. (1995). Modelling and control of nonholonomic mechanicalsystems. In Angeles, J. and Kecskemethy, A., editors, Kinematics and Dynamics ofMulti-Body Systems, CISM Courses and Lectures, volume 360, page 277342. Springer-Verlag.

DeGroot, M. H. (1974). Reaching a consensus. Journal of the American Statistical Asso-ciation, 69(345):pp. 118–121.

Dimarogonas, D. and Kyriakopoulos, K. (2007). On the rendezvous problem for multiplenonholonomic agents. IEEE Transactions on Automatic Control, 52(5):916–922.

Dixon, W. E., Dawson, D. M., Zhang, F., and Zergeroglu, E. (2000). Global exponentialtracking control of a mobile robot system via a pe condition. IEEE Transactions onSystems, Man, and Cybernetics, Part B, pages 129–142.

Do, K., Jiang, Z., and Pan, J. (2002a). Universal controllers for stabilization and trackingof underactuated ships. Systems & Control Letters, 47(4):299 – 317.

Do, K. and Pan, J. (2003). Global waypoint tracking control of underactuated ships underrelaxed assumptions. In Proceedings. 42nd IEEE Conference on Decision and Control,volume 2, pages 1244–1249 Vol.2.

Do, K. and Pan, J. (2009). Control of Ships and Underwater Vehicles: Design for Under-actuated and Nonlinear Marine Systems. Springer Verlag.

Bibliography 287

Do, K. D., Jiang, Z. P., and Pan, J. (2002b). Underactuated ship global tracking underrelaxed conditions. IEEE Transactions on Automatic Control, 47(9):1529–1536.

Do, K. D., Jiang, Z. P., and Pan, J. (2003a). Robust global stabilization of underactuatedships on a linear course: State and output feedback. International Journal of Control,76(1):1–17.

Do, K. D., Jiang, Z. P., and Pan, J. (2004). Robust adaptive path following of underac-tuated ships. Automatica, 40(6):929–944.

Do, K. D. and Pan, J. (2005). Global tracking control of underactuated ships with nonzerooff-diagonal terms in their system matrices. Automatica, 41(1):87–95.

Do, K. D., Pan, J., and Jiang, Z. P. (2003b). Robust adaptive control of underactuatedships on a linear course with comfort. Ocean Engineering, 30(17):2201 – 2225.

Du, H., Li, S., and Shi, P. (2012). Robust consensus algorithm for second-order multi-agentsystems with external disturbances. International Journal of Control, 85(12):1913–1928.

Egeland, O., Dalsmo, M., and Srdalen, O. (1996). Feedback control of a nonholonomicunderwater vehicle with a constant desired configuration. The International Journalof Robotics Research, 15(1):24–35.

Encarnacao, P. and Pascoal, A. (2000). 3D path following for autonomous underwatervehicle. In Proceedings of the 39th IEEE Conference on Decision and Control, volume 3,pages 2977–2982 vol.3.

Encarnacao, P. and Pascoal, A. (2001). Combined trajectory tracking and path following:an application to the coordinated control of autonomous marine craft. In Proceedingsof the 40th IEEE Conference on Decision and Control, volume 1, pages 964–969 vol.1.

Encarnao, P., Pascoal, A., and Arcak, A. (2000). Path following for autonomous marinecraft. In Proceedings of the fifth IFAC conference on manoeuvring and control of marinecraft, page 117122, Aalborg, Denmark.

Fax, J. A. and Murray, R. M. (2004). Information flow and cooperative control of vehicleformations. IEEE Transactions on Automatic Control, 49(9):1465–1476.

Fiedler, M. (1973). Algebraic connectivity of graphs. Czechoslovak Mathematical Journal,23:298.

Fiedler, M. (1989). Laplacian of graphs and algebraic connectivity. Combinatorics andGraph Theory, 25:57–70.

Flannery, M. R. (2005). The enigma of nonholonomic constraints. American Journal ofPhysics, 73(3):265–272.

Fossen, T. I. (1994). Guidance and control of ocean vehicles. John Wiley & Sons Ltd.

Fossen, T. I. (2000). A survey on nonlinear ship control: From theory to practice. InProceedings of the IFAC Conference on Maneuveringand Control of Marine Craft.

288 Bibliography

Fossen, T. I. (2011). Handbook of marine craft, hydrodynamics, and motion control. JohnWiley & Sons Ltd.

Fossen, T. I., Breivik, M., and Skjetne, R. (2003). Line-of-sight path following of under-actuated marine craft. In IFAC Conference on Manoeuvring and Control of MarineCraft, pages 244–249.

Fossen, T. I. and Grovlen, A. (1998). Nonlinear output feedback control of dynamicallypositioned ships using vectorial observer backstepping. IEEE Transactions on ControlSystems Technology, 6(1):121–128.

Fossen, T. I. and Perez, T. (2009). Kalman filtering for positioning and heading controlof ships and offshore rigs. Control Systems Magazine, IEEE, 29(6):32–46.

Fossen, T. I. and Strand, J. P. (1999). Passive nonlinear observer design for ships usinglyapunov methods: full-scale experiments with a supply vessel. Automatica, 35(1):3 –16.

Fossen, T. I. and Strand, J. P. (2001). Nonlinear passive weather optimal positioningcontrol (wopc) system for ships and rigs: experimental results. Automatica, 37(5):701–715.

Frazzoli, E., Dahleh, M., and Feron, E. (2000). Trajectory tracking control design forautonomous helicopters using a backstepping algorithm. In Proceedings of the AmericanControl Conference, volume 6, pages 4102–4107 vol.6.

Fredriksen, E. and Pettersen, K. (2006). Global K-exponential way-point maneuvering ofships: Theory and experiment. Automatica, 42(4):677 – 687.

Fumin, Z. (2010). Geometric cooperative control of particle formations. Automatic Con-trol, IEEE Transactions on, 55(3):800–803.

Godhavn, J. (1996). Nonlinear tracking of underactuated surface vessels. In Proceedingsof the 35th IEEE Conference on Decision and Control, volume 1, pages 975–980 vol.1.

Godsil, C. and Royle, G. F. (2001). Algebraic graph theory, volume 8. Springer New York.

Goldstein, H., Poole, C., and Safko, J. (2002). Classical Mechanics. AddisonWesley, NewYork, third edition.

Graybill, F. A. (2001). Matrices with Applications in Statistics. Duxbury PublishingCompany.

Grip, H. F. (August 2009). Structural aspects of linear systems theory - geometry theoryand the special coordinate basis. Technical report, Norwegian University of Scienceand Technology.

Grip, H. F. and Saberi, A. (2010). Structural decomposition of linear multivariable sys-tems using symbolic computations. International Journal of Control, 83(7):1414–1426.

Grip, H. F., Yang, T., Saberi, A., and Stoorvogel, A. A. (2012). Output synchronizationfor heterogeneous networks of non-introspective agents. Automatica, 48(10):2444 –2453.

Bibliography 289

Hauser, J. and Hindman, R. (1995). Maneuver regulation from trajectory tracking: Feed-back linearizable systems. In Proceedings of the IFAC symposium on nonlinear controlsystems design, page 595600, Lake Tahoe, CA, USA.

Hauser, J. and Hindman, R. (1997). Aggressive flight maneuvers. In Proceedings of the36th IEEE Conference on Decision and Control, volume 5, pages 4186–4191 vol.5.

Hespanha, J. P. (2009). Linear Systems Theory. Princeton Press, Princeton, New Jersey.

Hong-Yong, Y., Lei, G., and Chao, H. (2011). Robust consensus of multi-agent sys-tems with uncertain exogenous disturbances. Communications in Theoretical Physics,56(6):1161.

Huizhen, Y. and Fumin, Z. (2010). Geometric formation control for autonomous underwa-ter vehicles. In IEEE International Conference on Robotics and Automation (ICRA),pages 4288–4293.

Ihle, I.-A. F. (2006). Coordinated Control of Marine Craft. PhD thesis, EngineeringCybernetics Department, Norwegian University of Science and Technology, Trondheim,Norway.

Ihle, I.-A. F., Arcak, M., and Fossen, T. I. (2007). Passivity-based designs for synchronizedpath-following. Automatica, 43(9):1508 – 1518.

Ihle, I. A. F., Jouffroy, J., and Fossen, T. I. (2006a). Formation control of marine surfacecraft: A lagrangian approach. Oceanic Engineering, IEEE Journal of, 31(4):922–934.

Ihle, I. A. F., Jouffroy, J., and Fossen, T. I. (2006b). Robust formation control of marinecraft using lagrange multipliers. In Pettersen, K., Gravdahl, J., and Nijmeijer, H.,editors, Group Coordination and Cooperative Control, volume 336 of Lecture Notes inControl and Information Sciences, pages 113–129. Springer Berlin / Heidelberg.

Indiveri, G., Aicardi, M., and Casalino, G. (2000). Robust global stabilization of anunderactuated marine vehicle on a linear course by smooth time-invariant feedback.In Proceedings of the 39th IEEE Conference on Decision and Control, volume 3, pages2156–2161 vol.3.

Isidori, A. (1995). Nonlinear Control Systems. Springer-Verlag New York, Inc.

Jadbabaie, A., Lin, J., and Morse, A. (2003). Coordination of groups of mobile au-tonomous agents using nearest neighbor rules. IEEE Transactions on Automatic Con-trol, 48(6):988 – 1001.

Jankovic, M., Sepulchre, R., and Kokotovic, P. (1996). Constructive lyapunov stabilizationof nonlinear cascade systems. IEEE Transactions on Automatic Control, 41(12):1723–1735.

Jiang, Z.-P. (2000). Lyapunov design of global state and output feedback trackers fornon-holonomic control systems. International Journal of Control, 73(9):744–761.

Jiang, Z.-P. (2002). Global tracking control of underactuated ships by lyapunov’s directmethod. Automatica, 38(2):301–309.

290 Bibliography

Jiang, Z. P. and Nijmeijer, H. (1997). Tracking control of mobile robots: A case study inbackstepping. Automatica, 33(7):1393 – 1399.

Kaminer, I., Pascoal, A., and Yakimenko, O. (2005). Nonlinear path following control offully actuated marine vehicles with parameter uncertainty. In Proceedings of the 16thIFAC World Congres.

Khalil, H. K. (2002). Nonlinear systems. Prentice Hall, Upper Saddle River, N.J., 3rdedition.

Kim, H., Shim, H., and Seo, J. H. (2011). Output consensus of heterogeneous uncertainlinear multi-agent systems. IEEE Transactions on Automatic Control, 56(1):200 –206.

Kim, Y. and Mesbahi, M. (2006). On maximizing the second smallest eigenvalue of astate-dependent graph laplacian. IEEE Transactions on Automatic Control, 51(1):116– 120.

Kokotovic, P. V., O’Reilly, J., and Khalil, H. K. (1986). Singular Perturbation Methodsin Control: Analysis and Design. Academic Press, Inc., Orlando, FL, USA.

Kolmanovsky, I. and McClamroch, N. H. (1995). Developments in nonholonomic controlproblems. Control Systems Magazine, IEEE, 15(6):20–36.

Koo, T. and Sastry, S. (1998). Output tracking control design of a helicopter model basedon approximate linearization. In Proceedings of the 37th IEEE Conference on Decisionand Control, volume 4, pages 3635–3640 vol.4.

Krstic, M., Kokotovic, P., and Kanellakopoulos, I. (1995). Nonlinear and adaptive controldesign. John Wiley & Sons, Inc. New York, NY, USA.

Kwakernaak, H. (1982). A condition for robust stabilizability. Systems & Control Letters,2(1):1 – 5.

Kyrkjebø, E., Panteley, E., Chaillet, A., and Pettersen, K. (2006). A virtual vehicleapproach to underway replenishment. In Pettersen, K., Gravdahl, J., and Nijmeijer,H., editors, Group Coordination and Cooperative Control, volume 336 of Lecture Notesin Control and Information Sciences, pages 171–189. Springer.

Lanczos, C. (1986). The variational principles of mechanics. Dover Publications, NewYork, fourth edition.

Lapierre, L. and Jouvencel, B. (2008). Robust nonlinear path-following control of an auv.IEEE Journal of Oceanic Engineering, 33(2):89–102.

Lapierre, L. and Soetanto, D. (2007). Nonlinear path-following control of an AUV. OceanEngineering, 34(11-12):1734 – 1744.

Lawton, J., Beard, R., and Young, B. (2003). A decentralized approach to formationmaneuvers. IEEE Transactions on Robotics and Automation, 19(6):933–941.

Lee, D. and Li, P. Y. (2003). Formation and maneuver control of multiple spacecraft. InProceedings of the American Control Conference, volume 1, pages 278–283 vol.1.

Bibliography 291

Lee, L.-F., Bhatt, R., and Krovi, V. (2005). Comparison of alternate methods for dis-tributed motion planning of robot collectives within a potential field framework. InProceedings of the IEEE International Conference on Robotics and Automation, pages99–104.

Lefeber, A. (2000). Tracking Control of Nonlinear Mechanical Systems. PhD thesis,Universiteit Twente.

Lefeber, E., Pettersen, K., and Nijmeijer, H. (2003). Tracking control of an underactuatedship. Control Systems Technology, IEEE Transactions on, 11(1):52 – 61.

Li, S., Du, H., and Lin, X. (2011a). Finite-time consensus algorithm for multi-agentsystems with double-integrator dynamics. Automatica, 47(8):1706 – 1712.

Li, S.-M. and Berakdar, J. (2008). A lagrangian formalism based on the velocity-determined virtual displacements for systems with nonholonomic constraints. Interna-tional Journal of Theoretical Physics, 47(3):732–740.

Li, T. and Zhang, J.-F. (2009). Mean square average-consensus under measurement noisesand fixed topologies: Necessary and sufficient conditions. Automatica, 45(8):1929 –1936.

Li, Z., Duan, Z., and Chen, G. (2011b). On H∞ and H2 performance regions of multi-agent systems. Automatica, 47(4):797 – 803.

Li, Z., Duan, Z., Chen, G., and Huang, L. (2010). Consensus of multiagent systemsand synchronization of complex networks: A unified viewpoint. IEEE Transactions onCircuits and Systems I: Regular Papers, 57(1):213 –224.

Li, Z., Duan, Z., and Huang, L. (2009). H∞ control of networked multi-agent systems.Journal of Systems Science and Complexity, 22:35–48.

Li, Z., Liu, X., Fu, M., and Xie, L. (2012). Global H∞ consensus of multi-agent systemswith lipschitz nonlinear dynamics. available at http://arxiv.org/abs/1202.5447.

Lin, P. and Jia, Y. (2009). Consensus of second-order discrete-time multi-agent sys-tems with nonuniform time-delays and dynamically changing topologies. Automatica,45(9):2154 – 2158.

Lin, P. and Jia, Y. (2010). Robust H∞ consensus analysis of a class of second-order multi-agent systems with uncertainty. Control Theory Applications, IET, 4(3):487 –498.

Lin, P., Jia, Y., and Li, L. (2008). Distributed robust H∞ consensus control in directednetworks of agents with time-delay. Systems & Control Letters, 57(8):643 – 653.

Lin, W. (1995). Bounded smooth state feedback and a global separation principle fornon-affine nonlinear systems. Systems & Control Letters, 26(1):41 – 53.

Liu, C. and Huston, R. (2008). Another form of equations of motion for constrainedmultibody systems. Nonlinear Dynamics, 51(3):465–475.

Liu, X., Chen, B., and Lin, Z. (2005). Linear systems toolkit in matlab: structuraldecompositions and their applications. Journal of Control Theory and Applications,3:287–294.

292 Bibliography

Liu, Y. and Jia, Y. (2010). Consensus problem of high-order multi-agent systems withexternal disturbances: An H∞ analysis approach. International Journal of Robust andNonlinear Control, 20(14):1579–1593.

Liu, Y. and Jia, Y. (2011). RobustH∞ consensus control of uncertain multi-agent systemswith time delays. International Journal of Control, Automation and Systems, 9:1086–1094.

Loria, A., Fossen, T. I., and Panteley, E. (2000). A separation principle for dynamic po-sitioning of ships: theoretical and experimental results. IEEE Transactions on ControlSystems Technology, 8(2):332–343.

Loria, A. and Panteley, E. (1999). A separation principle for a class of euler-lagrangesystems. In New Directions in nonlinear observer design, pages 229–247.

Loria, A. and Panteley, E. (2005). Cascaded nonlinear time-varying systems: Analysisand design. In Lamnabhi-Lagarrigue, F., Lora, A., and Panteley, E., editors, AdvancedTopics in Control Systems Theory, volume 311 of Lecture Notes in Control and Infor-mation Sciences, pages 579–579. Springer Berlin / Heidelberg.

Luenberger, D. (1967). Canonical forms for linear multivariable systems. IEEE Transac-tions on Automatic Control, 12(3):290 –293.

Ma, C.-Q. and Zhang, J.-F. (2010). Necessary and sufficient conditions for consen-susability of linear multi-agent systems. IEEE Transactions on Automatic Control,55(5):1263–1268.

Mason, M. T. (1981). Compliance and force control for computer controlled manipulators.IEEE Transactions on Systems, Man and Cybernetics, 11(6):418–432.

Massioni, P. and Verhaegen, M. (2009). Distributed control for identical dynamically cou-pled systems: A decomposition approach. IEEE Transactions on Automatic Control,54(1):124 –135.

Mauroy, A., Sacre, P., and Sepulchre, R. (2013). Kick synchronization versus diffusivesynchronization. online: arXiv:1209.4970v3.

Mazenc, F., Pettersen, K., and Nijmeijer, H. (2002). Global uniform asymptotic stabi-lization of an underactuated surface vessel. IEEE Transactions on Automatic Control,47(10):1759–1762.

Mesbahi, M. and Egerstedt, M. (2010). Graph Theoretic Methods in Multiagent Networks.Princeton University Press.

Miller, M. O. and Combs, J. A. (1999). The next underway replenishment system. NavalEngineers Journal, 111(2):45–55.

Mo, L. and Jia, Y. (2011). H∞ consensus control of a class of high-order multi-agentsystems. Control Theory Applications, IET, pages 247 –253.

Moreau, L. (2005). Stability of multiagent systems with time-dependent communicationlinks. IEEE Transactions on Automatic Control, 50(2):169 – 182.

Bibliography 293

Moreschi, O. M. and Castellano, G. (2006). Geometric approach to non-holonomic prob-lems satisfying hamilton’s principle. Revista de la Unin Matemtica Argentina, 47:125–135.

Murray, R. M. (2007). Recent research in cooperative control of multivehicle systems.Journal of Dynamic Systems, Measurement, and Control, 129:571–583.

Murray, R. M., Sastry, S. S., and Zexiang, L. (1994). A Mathematical Introduction toRobotic Manipulation. CRC Press, Inc.

Nguyen, D. T. and Sørensen, A. J. (2009). Switching control for thruster-assisted positionmooring. Control Engineering Practice, 17(9):985 – 994.

Nijmeijer, H. and Rodriguez-Angeles, A. (2003). Synchronization of Mechanical Systems.World Scientific Publishing Co. Pte Ltd.

Nuno, E., Ortega, R., Basanez, L., and Hill, D. (2011). Synchronization of networksof nonidentical euler-lagrange systems with uncertain parameters and communicationdelays. IEEE Transactions on Automatic Control, 56(4):935–941.

Olfati-Saber, R. (2002). Normal forms for underactuated mechanical systems with sym-metry. IEEE Transactions on Automatic Control, 47(2):305–308.

Olfati-Saber, R. (2006). Flocking for multi-agent dynamic systems: algorithms and theory.IEEE Transactions on Automatic Control, 51(3):401–420.

Olfati-Saber, R., Fax, J., and Murray, R. (2007). Consensus and cooperation in networkedmulti-agent systems. Proceedings of the IEEE, 95(1):215 –233.

Olfati-Saber, R. and Murray, R. (2004). Consensus problems in networks of agentswith switching topology and time-delays. IEEE Transactions on Automatic Control,49(9):1520 – 1533.

Olga, K. (2009). The nonholonomic variational principle. Journal of Physics A: Mathe-matical and Theoretical, 42(18):185201.

O’Reilly, J. (1983). Observer for Linear Systems. New York: Academic.

Oriolo, G. and Nakamura, Y. (1991). Control of mechanical systems with second-ordernonholonomic constraints: underactuated manipulators. In Proceedings of the 30thIEEE Conference on Decision and Control, pages 2398 –2403 vol.3.

Ozcetin, H. K., Saberi, A., and Sannuti, P. (1992). Design for H∞ almost disturbancedecoupling problem with internal stability via state or measurement feedback–singularperturbation approach. International Journal of Control, 55(4):901–944.

Pars, L. A. (1965). A Treatise on Analytical Dynamics. Heinemann, London.

Pedone, P., Zizzari, A. A., and Indiveri, G. (2010). Path following for the dynamic modelof a marine surface vessel without closed-loop control of the surge speed. In 8th IFACConference on Control Applications in Marine Systems (CAMS), Germany.

294 Bibliography

Perez, T. (2005). Ship Motion Control: Course Keeping and Roll Stabilisation UsingRudder and Fins. Springer London.

Peters, J., Mistry, M., Udwadia, F., Nakanishi, J., and Schaal, S. (2008). A unifyingframework for robot control with redundant dofs. Autonomous Robots, 24(1):1–12.

Pettersen, K. and Egeland, O. (1996). Exponential stabilization of an underactuatedsurface vessel. In Proceedings of the 35th IEEE Decision and Control, volume 1, pages967 –972 vol.1.

Pettersen, K. and Egeland, O. (1997). Robust control of an underactuated surface vesselwith thruster dynamics. In Proceedings of the American Control Conference, volume 5,pages 3411–3415 vol.5.

Pettersen, K. and Fossen, T. (2000). Underactuated dynamic positioning of a ship-experimental results. IEEE Transactions on Control Systems Technology, 8(5):856–863.

Pettersen, K., Gravdahl, J. T., and Nijmeijer, H., editors (2006). Group Coordination andCooperative Control. Lecture Notes in Control and Information Sciences. Springer.

Pettersen, K. and Lefeber, E. (2001). Way-point tracking control of ships. In Proceedingsof the 40th IEEE Conference on Decision and Control, volume 1, pages 940 –945 vol.1.

Pettersen, K. and Nijmeijer, H. (1999a). Output feedback tracking control for ships. InNijmeijer, H. and Fossen, T., editors, New Directions in nonlinear observer design,volume 244 of Lecture Notes in Control and Information Sciences, pages 311–334.Springer London.

Pettersen, K. Y. and Egeland, O. (1999). Time-varying exponential stabilization of the po-sition and attitude of an underactuated autonomous underwater vehicle. IEEE Trans-actions on Automatic Control, 44(1):112–115.

Pettersen, K. Y. and Fossen, T. I. (1998). Underactuated ship stabilization using integralcontrol: Experimental results with cybership i. In Proc. 1998 IFAC Symposium onNonlinear Control Systems Design, pages 127–132.

Pettersen, K. Y. and Nijmeijer, H. (1999b). Global Practical Stabilization and Track-ing for an Underactuated Ship - A Combined Averaging and Backstepping Approach.Modeling, Identification and Control, 20(4):189–200.

Pettersen, K. Y. and Nijmeijer, H. (2001). Underactuated ship tracking control: theoryand experiments. International Journal of Control, 74(14):1435 – 1446.

Peymani, E. and Fossen, A. (2013a). Speed-varying path following for underactuatedmarine craft. In 9th IFAC conference on Control Application in Marine Systems, Osaka,Japan. (Submitted).

Peymani, E. and Fossen, T. (2011a). Motion control of marine craft using virtual posi-tional and velocity constraints. In the proceeding of 9th IEEE International Conferenceon Control and Automation (ICCA), pages 410 – 416, Santiago, Chile.

Bibliography 295

Peymani, E. and Fossen, T. (2013b). Path following for autonomous marine vehicles: aleast-square approach. IEEE Transactions on Control System Technology. (Submitted).

Peymani, E. and Fossen, T. I. (2011b). A lagrangian framework to incorporate positionaland velocity constraints to achieve path-following control. In the proceeding of 50thIEEE Conference on Decision and Control and European Control Conference (CDC-ECC), pages 3940 – 3945.

Peymani, E. and Fossen, T. I. (2011c). Path following of underwater robots using lagrangemultipliers. Robotics and Autonomous Systems. (submitted).

Peymani, E. and Fossen, T. I. (2012). Direct inclusion of geometric errors for path-maneuvering of marine craft. In Proc. of the 9th IFAC Conference on Maneuveringand Control of Marine Craft (MCMC), pages 60 – 65, Arenzano, Italy.

Peymani, E. and Fossen, T. I. (2013c). 2D path following for marine craft: A least-squareapproach. In proceeding of the 9th IFAC Symposium on Nonlinear Control Systems,Toulouse, France. (to appear).

Peymani, E., Grip, H. F., and Saberi, A. (2013a). Homogeneous networks of non-introspective agents under external disturbances – H∞ almost synchronization. Auto-matica. (Submitted).

Peymani, E., Grip, H. F., Saberi, A., and Fossen, T. (2013b). H∞ almost synchronizationfor homogeneous networks of SISO and non-introspective agents under external dis-turbances. In proceeding of 52nd IEEE conference on Decision and Control, Florance,Italy. (to appear).

Peymani, E., Grip, H. F., Saberi, A., Wang, X., and Fossen, T. I. (2012). H∞ almostsynchronization for heterogeneous networks of introspective agents under external dis-turbances. Automatica. (provisionally accepted for publication as a regular paper).

Peymani, E., Grip, H. F., Saberi, A., Wang, X., and Fossen, T. I. (2013c). H∞ almostregulated synchronization and H∞ almost formation for heterogeneous networks underexternal disturbances. In proceeding of 12th biannual European Control Conference,pages 1890 – 1895, Zurich, Switzerland.

Peymani, E., Grip, H. F., Saberi, A., Wang, X., and Fossen, T. I. (2013d). H∞ al-most synchronization for non-identical introspective multi-agent systems under exter-nal disturbances. In proceeding of American Control Conference, pages 1900 – 1905,Washington, DC.

Qu, Z., Wang, J., and Hull, R. (2008). Cooperative control of dynamical systems with ap-plication to autonomous vehicles. IEEE Transactions on Automatic Control, 53(4):894–911.

Ray, J. R. (1966a). Erratum: Nonholonomic constraints. American Journal of Physics,34(12):1202–1203.

Ray, J. R. (1966b). Nonholonomic constraints. American Journal of Physics, 34(5):406–408.

296 Bibliography

Ren, W. (2008). On consensus algorithms for double-integrator dynamics. IEEE Trans-actions on Automatic Control, 53(6):1503–1509.

Ren, W. and Beard, R. (2005). Consensus seeking in multiagent systems under dy-namically changing interaction topologies. IEEE Transactions on Automatic Control,50(5):655 – 661.

Ren, W. and Cao, Y. (2011). Distributed Coordination of Multi-agent Networks. Com-munications and Control Engineering, Springer-Verlag.

Reyhanoglu, M. (1997). Exponential stabilization of an underactuated autonomous sur-face vessel. Automatica, 33(12):2249–2254.

Reyhanoglu, M., van der Schaft, A., McClamroch, N., and Kolmanovsky, I. (1999). Dy-namics and control of a class of underactuated mechanical systems. IEEE Transactionson Automatic Control, 44(9):1663–1671.

Robertsson, A. and Johansson, R. (1998). Comments on ”nonlinear output feedbackcontrol of dynamically positioned ships using vectorial observer backstepping”. IEEETransactions on Control Systems Technology, 6(3):439–441.

Saberi, A. and Sannuti, P. (1988). Squaring down by static and dynamic compensators.IEEE Transactions on Automatic Control, 33(4):358 –365.

Saberi, A., Stoorvogel, A., and Sannuti, P. (1999). Output Regulation and Control Prob-lems with Regulation Constraints. Springer-Verlag.

Saberi, A., Stoorvogel, A. A., and Sannuti, P. (2012). Internal and External Stabilizationof Linear Systems with Constraints. Birkhuser Boston.

Samson, C. (1991). Velocity and torque feedback control of a nonholonomic cart. InCanudas de Wit, C., editor, Advanced Robot Control, volume 162 of Lecture Notes inControl and Information Sciences, pages 125–151. Springer Berlin Heidelberg.

Samson, C. (1992). Path-following and time-varying feedback stabilization of a wheeledmobile robot. In Proc. of the ICARCV, page RO13.1.1RO13.1.5, Singapore.

Samson, C. (1995). Control of chained systems application to path following and time-varying point-stabilization of mobile robots. Automatic Control, IEEE Transactionson, 40(1):64–77.

Sannuti, P. and Saberi, A. (1987). Special coordinate basis for multivariable linear-systems- finite and infinite zero structure, squaring down and decoupling. International Journalof Control, 45(5):1655–1704.

Sarlette, A., Sepulchre, R., and Leonard, N. E. (2009). Autonomous rigid body attitudesynchronization. Automatica, 45(2):572 – 577.

Schutte, A. and Udwadia, F. (2011). New approach to the modeling of complex multibodydynamical systems. Journal of Applied Mechanics, 78:021018–1 – 021018–11.

Seibert, P. and Suarez, R. (1990). Global stabilization of nonlinear cascade systems.Systems & Control Letters, 14(4):347 – 352.

Bibliography 297

Seo, J. H., Shim, H., and Back, J. (2009). Consensus of high-order linear systems usingdynamic output feedback compensator: Low gain approach. Automatica, 45(11):2659– 2664.

Shafi, S., Arcak, M., and El Ghaoui, L. (2012). Graph weight allocation to meet laplacianspectral constraints. IEEE Transactions on Automatic Control, 57(7):1872–1877.

Shiriaev, A. S., Freidovich, L. B., and Gusev, S. V. (2010). Transverse linearization forcontrolled mechanical systems with several passive degrees of freedom. IEEE Transac-tions on Automatic Control, 55(4):893–906.

Siciliano, B. and Villani, L. (2000). Robot Force Control, volume 540 of The SpringerInternational Series in Engineering and Computer Science. Springer.

Silvestre, C. J. F. (2000). Multi-objective optimization theory with applications to the in-tegrated design of controllers/plants for autonomous vehicles. PhD thesis, Departmentof Electrical Engineering, Instituto Superior Tecnico (ISR).

Sira-Ramirez, H. (1999). On the control of the underactuated ship: a trajectory plan-ning approach. In Proceedings of the 38th IEEE Conference on Decision and Control,volume 3, pages 2192–2197.

Sira-Ramirez, H. and C.A., I. (2000). On the control of the hovercraft system. Dynamicsand Control, 10(2):151–163.

Skjetne, R., Fossen, T. I., and Kokotovic, P. V. (2004). Robust output maneuvering fora class of nonlinear systems. Automatica, 40(3):373–383.

Skjetne, R., Fossen, T. I., and Kokotovic, P. V. (2005). Adaptive maneuvering, withexperiments, for a model ship in a marine control laboratory. Automatica, 41(2):289–298.

Slotine, J. and Li, W. (1991). Applied nonlinear control. Prentice Hall Englewood Cliffs,NJ.

Soetanto, D., Lapierre, L., and Pascoal, A. (2003). Nonsingular path-following control ofdynamic wheeled robos with parametric modeling uncertanity. In Proceedings of the11th International Conference on Advanced Robotics, Coimbra, Portugal.

Song, M., Tarn, T., and Xi, N. (2000). Integration of task scheduling, action planning, andcontrol in robotic manufacturing systems. Proceedings of the IEEE, 88(7):1097–1107.

Sørdalen, O. and Egeland, O. (1995). Exponential stabilization of nonholonomic chainedsystems. IEEE Transactions on Automatic Control, 40(1):35–49.

Sørensen, A. J. and Strand, J. P. (2000). Positioning of small-waterplane-area marineconstructions with roll and pitch damping. Control Engineering Practice, 8(2):205 –213.

Spong, M. (1998). Underactuated mechanical systems. In Control Problems in Roboticsand Automation, pages 135–150. Springer.

298 Bibliography

Spry, S. and Hedrick, J. K. (2004). Formation control using generalized coordinates. Inthe proceeding of the 43rd IEEE Conference on Decision and Control, volume 3, pages2441–2446 Vol.3.

Spry, S. C. (2002). Modeling and Control of Vehicle Formations. PhD thesis, Universityof California, Berkeley.

Strand, J. P., Ezal, K., Fossen, T. I., and Kokotovic, P. V. (1998). Nonlinear control ofships: A locally optimal design. In Preprints of the IFAC Nonlinear Control Systems.

Strand, J. P. and Fossen, T. I. (1998). Nonlinear output feedback and locally optimalcontrol of dynamically positioned ships: Experimental results. In Proc. of the IFACConference on Control Applications in Marine Systems, pages 89–95, Fukuoka, Japan.

Summers, E., Arcak, M., and Packard, A. (2013). Delay robustness of interconnectedpassive systems: An integral quadratic constraint approach. IEEE Transactions onAutomatic Control, 58(3):712–724.

Tahbaz-Salehi, A. and Jadbabaie, A. (2008). A necessary and sufficient condition forconsensus over random networks. IEEE Transactions on Automatic Control, 53(3):791–795.

Tanner, H., Pappas, G., and Kumar, V. (2004). Leader-to-formation stability. IEEETransactions on Robotics and Automation, 20(3):443 – 455.

Tarn, T.-J., Zhang, M., and Serrani, A. (2003). New integrability conditions for classifyingholonomic and nonholonomic systems. In Directions in Mathematical Systems Theoryand Optimization, volume 286, pages 317–331. Springer Berlin / Heidelberg.

Ti-Chung, L. and Zhong-Ping, J. (2004). New cascade approach for global κ-exponential tracking of underactuated ships. IEEE Transactions on Automatic Control,49(12):2297–2303.

Toussaint, G., Basar, T., and Bullo, F. (2000). Tracking for nonlinear underactuatedsurface vessels with generalized forces. In Proceedings of the 2000 IEEE InternationalConference on Control Applications, pages 355–360.

Trentelman, H. L., Stoorvogel, A., and Hautus, M. (2001). Control Theory for LinearSystems. Springer.

Tsitsiklis, J., Bertsekas, D., and Athans, M. (1986). Distributed asynchronous determinis-tic and stochastic gradient optimization algorithms. IEEE Transactions on AutomaticControl, 31(9):803–812.

Tuna, S. (2009). Conditions for synchronizability in arrays of coupled linear systems.IEEE Transactions on Automatic Control, 54(10):2416–2420.

Udwadia, F. (2000). Fundamental principles of lagrangian dynamics: Mechanical systemswith non-ideal, holonomic, and nonholonomic constraints. Journal of MathematicalAnalysis and Applications, 251:341–355.

Bibliography 299

Udwadia, F. E. (2003). A new perspective on the tracking control of nonlinear struc-tural and mechanical systems. Proceedings of the Royal Society of London. Series A:Mathematical, Physical and Engineering Sciences, 459(2035):1783–1800.

Udwadia, F. E. (2008). Optimal tracking control of nonlinear dynamical systems. Pro-ceedings of the Royal Society A: Mathematical, Physical and Engineering Science,464(2097):2341–2363.

Udwadia, F. E. and Han, B. (2008). Synchronization of multiple chaotic gyroscopes usingthe fundamental equation of mechanics. Journal of Applied Mechanics, 75(2):021011.

Udwadia, F. E. and Kalaba, R. E. (1992). A new perspective on constrained motion.Proceedings: Mathematical and Physical Sciences, 439(1906):pp. 407–410.

Udwadia, F. E. and Kalaba, R. E. (1996). Analytical Dynamics: A New Approach.Cambridge University Press.

Udwadia, F. E. and Kalaba, R. E. (2002). What is the general form of the explicit equa-tions of motion for constrained mechanical systems? Journal of Applied Mechanics,69(3):335–339.

Udwadia, F. E. and Phohomsiri, P. (2006). Explicit equations of motion for constrainedmechanical systems with singular mass matrices and applications to multi-body dy-namics. Proceedings of the Royal Society A: Mathematical, Physical and EngineeringScience, 462(2071):2097–2117.

Udwadia, F. E. and Phohomsiri, P. (2007a). Explicit poincar equations of motion forgeneral constrained systems. part i. analytical results. Proceedings of the Royal SocietyA: Mathematical, Physical and Engineering Science, 463(2082):1421–1434.

Udwadia, F. E. and Phohomsiri, P. (2007b). Explicit poincar equations of motion forgeneral constrained systems. part ii. applications to multi-body dynamics and nonlinearcontrol. Proceedings of the Royal Society A: Mathematical, Physical and EngineeringScience, 463(2082):1435–1446.

Ugrinovskii, V. (2011). Distributed robust filtering with H∞ consensus of estimates.Automatica, 47(1):1 – 13.

Ugrinovskii, V. and Langbort, C. (2011). Distributed H∞ consensus-based estimationof uncertain systems via dissipativity theory. Control Theory Applications, IET,5(12):1458 –1469.

Vicsek, T., Czirok, A., Ben-Jacob, E., Cohen, I., and Shochet, O. (1995). Novel type ofphase transition in a system of self-driven particles. Phys. Rev. Lett., 75:1226–1229.

Vicsek, T. and Zafeiris, A. (2012). Collective motion. Physics Reports, 517:71 – 140.

Vidyasagar, M. and Viswanadham, N. (1982). Algebraic design techniques for reliablestabilization. IEEE Transactions on Automatic Control, 27(5):1085–1095.

Wang, X., Saberi, A., Stoorvogel, A. A., Grip, H. F., and Yang, T. (2013). Synchroniza-tion in a heterogeneous network of introspective right-invertible agents with uniformconstant communication delay. To appear in IEEE Transactions on Automatic Control.

300 Bibliography

Weiland, S. and Willems, J. (1989). Almost disturbance decoupling with internal stability.IEEE Transactions on Automatic Control, 34(3):277 –286.

Wichlund, K., Srdalen, O., and Egeland, O. (1995a). Control of vehicles with second-ordernonholonomic constraints: Underactuated vehicles. In European Control Conference.

Wichlund, K. Y., Sordalen, O., and Egeland, O. (1995b). Control properties of un-deractuated vehicles. In Proceedings IEEE International Conference on Robotics andAutomation, volume 2, pages 2009–2014 vol.2.

Wieland, P., Sepulchre, R., and Allgwer, F. (2011). An internal model principle is neces-sary and sufficient for linear output synchronization. Automatica, 47(5):1068 – 1074.

Wondergem, M., Lefeber, E., Pettersen, K., and Nijmeijer, H. (2011). Output feedbacktracking of ships. IEEE Transactions on Control Systems Technology, 19(2):442–448.

Wonham, W. M. (1985). Linear multivariable control: a geometric approach. Springer;3rd ed. edition.

Wu, C. W. (2005). Algebraic connectivity of directed graphs. Linear and MultilinearAlgebra, 53(3):203–223.

Wu, C. W. (2007). Synchronization in complex networks of nonlinear dynamical systems.World Scientific.

Wu, C. W. and Chua, L. (1995a). Application of graph theory to the synchronization inan array of coupled nonlinear oscillators. IEEE Transactions on Circuits and SystemsI: Fundamental Theory and Applications, 42(8):494 –497.

Wu, C. W. and Chua, L. (1995b). Application of kronecker products to the analysis ofsystems with uniform linear coupling. IEEE Transactions on Circuits and Systems I:Fundamental Theory and Applications, 42(10):775 –778.

Xiang, J. and Chen, G. (2007). On the v-stability of complex dynamical networks. Auto-matica, 43(6):1049 – 1057.

Xing, J., Lei, Y., Cheng, W., and Tang, G. (2009). Nonlinear control of multiple spacecraftformation flying using the constraint forces in lagrangian systems. Science in ChinaSeries E: Technological Sciences, 52(10):2930–2936.

Yang, T., Roy, S., Wan, Y., and Saberi, A. (2011a). Constructing consensus controllersfor networks with identical general linear agents. International Journal of Robust andNonlinear Control, 21(11):1237–1256.

Yang, T., Saberi, A., Stoorvogel, A. A., and Grip, H. F. (2011b). Output consensus fornetworks of non-identical introspective agents. In 50th IEEE Conference on Decisionand Control and European Control Conference, pages 1286 –1292.

Yang, T., Stoorvogel, A., and Saberi, A. (2011c). Consensus for multi-agent systems –synchronization and regulation for complex networks. In American Control Conference,pages 5312 –5317.

Bibliography 301

Yao, J., Guan, Z.-H., and Hill, D. J. (2009a). Passivity-based control and synchronizationof general complex dynamical networks. Automatica, 45(9):2107 – 2113.

Yao, J., Wang, H. O., Guan, Z.-H., and Xu, W. (2009b). Passive stability and synchro-nization of complex spatio-temporal switching networks with time delays. Automatica,45(7):1721 – 1728.

Yun, X. and Sarkar, N. (1998). Unified formulation of robotic systems with holonomic andnonholonomic constraints. IEEE Transactions on Robotics and Automation, 14(4):640–650.

Zhang, H., Lewis, F., and Das, A. (2011a). Optimal design for synchronization of coop-erative systems: State feedback, observer and output feedback. IEEE Transactions onAutomatic Control, 56(8):1948 –1952.

Zhang, H.-T., Chen, M., and Stan, G. B. (2011b). Fast consensus via predictive pinningcontrol. IEEE Transactions on Circuits and Systems I: Regular Papers, 58(9):2247–2258.

Zhang, H.-T., Chen, M., Stan, G.-B., Zhou, T., and Maciejowski, J. (2008a). Collectivebehavior coordination with predictive mechanisms. Circuits and Systems Magazine,IEEE, 8(3):67–85.

Zhang, H.-T., Chen, M. Z., Zhou, T., and Stan, G.-B. (2008b). Ultrafast consensus viapredictive mechanisms. EPL (Europhysics Letters), 83(4):40003.

Zhang, R., Chen, Y., Sun, Z., Sun, F., and Xu, H. (1998). Path control of a surface shipin restricted waters using sliding mode. In Proceedings of the 37th IEEE Conferenceon Decision and Control, volume 3, pages 3195–3200.

Zhang, Y. and Tian, Y.-P. (2009). Consentability and protocol design of multi-agentsystems with stochastic switching topology. Automatica, 45(5):1195 – 1201.

Zhao, J., Hill, D., and Liu, T. (2010). Passivity-based output synchronization of dynamicalnetworks with non-identical nodes. In 49th IEEE Conference on Decision and Control,pages 7351 –7356.

Zhao, J., Hill, D., and Liu, T. (2011). Synchronization of dynamical networks withnonidentical nodes: Criteria and control. IEEE Transactions on Circuits and SystemsI: Regular Papers, 58(3):584 –594.

Zhou, K. and Doyle, J. (1998). Essentials of robust control. Prentice Hall Upper SaddleRiver, NJ.

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