REGULATION IN SWITCHED BIMODAL LINEAR …...ii REGULATION IN SWITCHED BIMODAL LINEAR SYSTEMS...

REGULATION IN SWITCHED BIMODAL LINEAR SYSTEMS

by

Zhizheng Wu

A thesis submitted in conformity with the requirements

for the degree of Doctor of Philosophy

Department of Mechanical and Industrial Engineering

University of Toronto

© Copyright by Zhizheng Wu 2009

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REGULATION IN SWITCHED BIMODAL LINEAR SYSTEMS

Zhizheng Wu

Ph.D. Thesis, 2009

Department of Mechanical and Industrial Engineering

University of Toronto

ABSTRACT

In the past few decades, significant progress has been made in addressing control

problems for a variety of engineering systems having smooth dynamics. In practice, one often

encounters also non-smooth systems in various branches of science and engineering, such as

for example mechanical systems subject to impact. Motivated by the read/write head flying

height regulation problem in hard disk drives, where the close proximity of the read/write

head to the disk surface results in intermittent contact between the two and a bimodal system

behavior, this thesis studies the output regulation problem in switched bimodal linear systems

against known and unknown exogenous input signals.

The regulation problems in bimodal systems presented in this thesis are solved within sets

of Q -parameterized controllers, in which the Q parameters are designed to yield internal

stability and exact output regulation in the closed loop switched system. The proposed

parameterized controllers are constructed mainly in two steps. The first step is based on

constructing a switched observer-based state feedback central controller for the switched

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linear system. The second step involves augmenting the switched central controller with

additional dynamics (i.e. Q parameter) to construct a parameterized set of switched

controllers. Based on the proposed sets of Q -parameterized controllers, four main regulation

problems are addressed and corresponding regulator synthesis algorithms are proposed. The

first problem concerns regulation against known deterministic exogenous inputs, where no

stability or structural constraints are imposed on the Q parameter. The second problem is

similar to the first, except that the Q parameter is constrained to be a linear combination of

basis functions. This structure of the Q parameter is considered in the rest of the thesis. The

third problem involves regulation against exogenous inputs involving known deterministic

components and unknown random components, and where the regulator is designed subject to

an 2H performance constraint. The last problem involves the development of adaptive

regulators against unknown sinusoidal exogenous inputs. The different regulator synthesis

algorithms are developed based on solving sets of linear matrix inequalities or bilinear matrix

inequalities. The last two proposed regulation methods are successfully evaluated on an

experimental setup motivated by the flying height regulation problem in hard disk drives, and

involving a mechanical system with switched dynamics.

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ACKNOWLEDGEMENTS

I would like to thank my academic supervisor, Professor Foued Ben Amara, for his guidance,

encouragement, helpful discussions and support throughout the course of the work. My thanks

also go to Professor James K. Mills, Professor Beno Benhabib and Professor Yu Sun, who have

served as members of my thesis examination committee.

Furthermore, I would also like to express my sincere gratitude to all my colleagues in the

Mechanical and Industrial Engineering Department at the University of Toronto, including

Mr. Maurizio Ficocelli, Mr. Azhar Iqbal, Mr. Jason Li and Mr. Edward Lee who provided

invaluable help and advice in completing this thesis. I gratefully acknowledge the financial

support from the Ontario Graduate Scholarship (OGS), the Ontario Graduate Scholarship in

Science and Technology (OGSST), and the University of Toronto Fellowship. Finally, I

would like to thank my family for all their support and encouragement throughout my studies.

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TABLE OF CONTENTS

ABSTRACT ........................................................................................................................................................II

ACKNOWLEDGEMENTS............................................................................................................................ IV

LIST OF FIGURES...................................................................................................................................... VIII

LIST OF TABLES..........................................................................................................................................XII

CHAPTERS

1 INTRODUCTION.................................................................................................................................... 1

1.1 MOTIVATION................................................................................................................. 1 1.2 LITERATURE REVIEW ON SWITCHED SYSTEMS ............................................................... 3

1.2.1 Stability of switched systems ........................................................................................... 4 1.2.2 Output regulation in switched systems .......................................................................... 9 1.2.3 Adaptive regulation in linear and nonlinear systems...............................................12

1.3 PROBLEM STATEMENT................................................................................................. 14 1.4 CONTRIBUTIONS OF THE THESIS .................................................................................. 16 1.5 ORGANIZATION OF THE THESIS.................................................................................... 19

2 PARAMETERIZED REGULATOR SYNTHESIS FOR SWITCHED BIMODAL SYSTEMS BASED ON BILINEAR MATRIX INEQUALITIES.........................................22

2.1 INTRODUCTION............................................................................................................ 22 2.2 REGULATION PROBLEM FOR SWITCHED BIMODAL SYSTEMS ........................................ 24 2.3 PARAMETERIZATION OF A SET OF CONTROLLERS......................................................... 26

2.3.1 Observer-based state feedback controller ...................................................................27 2.3.2 Parameterized output feedback controller...................................................................27 2.3.3 Stability of the parameterized switched closed loop system .................................30

2.4 REGULATION CONDITIONS FOR THE SWITCHED SYSTEM .............................................. 32 2.4.1 Regulation conditions for 1

clΣ and 2clΣ .....................................................................33

2.4.2 Equivalent impulsive switched closed loop system model....................................36 2.4.3 Regulation conditions for the switched system.........................................................38

2.5 REGULATOR SYNTHESIS FOR THE SWITCHED SYSTEM.................................................. 43 2.6 NUMERICAL EXAMPLE................................................................................................. 48 2.7 CONCLUSION............................................................................................................... 53

3 REGULATION IN SWITCHED BIMODAL SYSTEMS BASED ON LINEAR MATRIX INEQUALITIES....................................................................................................................................56

3.1 INTRODUCTION............................................................................................................ 56 3.2 THE REGULATION PROBLEM FOR SWITCHED BIMODAL SYSTEMS ................................. 58

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3.3 PARAMETERIZATION OF A SET OF STABILIZING CONTROLLERS .................................... 60 3.3.1 Q -parameterized controller ...........................................................................................60 3.3.2 Internal stability of the Q -parameterized switched closed loop system ..........62

3.4 REGULATION OF BIMODAL SYSTEMS: A COMMON LYAPUNOV FUNCTION APPROACH... 67 3.4.1 Input-output stability.........................................................................................................67 3.4.2 State space regulation conditions for 1

clΣ and 2clΣ ................................................73

3.4.3 Regulation conditions for the switched system.........................................................76 3.4.4 Regulator synthesis for the switched system .............................................................88 3.4.5 Numerical example............................................................................................................91

3.5 REGULATION OF BIMODAL SYSTEMS: A MULTIPLE LYAPUNOV FUNCTION APPROACH . 95 3.5.1 Regulation conditions for 1

clΣ and 2clΣ in the frequency domain ....................98

3.5.2 Regulation condition and regulator synthesis for the switched system............104 3.5.3 Numerical example..........................................................................................................112

3.6 CONCLUSION............................................................................................................. 114

4 REGULATION IN SWITCHED BIMODAL SYSTEMS WITH AN 2H

PERFORMANCE CONSTRAINT................................................................................................117

4.1 INTRODUCTION.......................................................................................................... 117 4.2 THE REGULATION PROBLEM FOR THE SWITCHED BIMODAL SYSTEM .......................... 119 4.3 PARAMETERIZATION OF A SET OF STABILIZING CONTROLLERS .................................. 120

4.3.1 Q -parameterized controller .........................................................................................120 4.3.2 Stability of the Q -parameterized switched closed loop system........................123

4.4 REGULATION CONDITIONS FOR SWITCHED SYSTEM ................................................... 127 4.4.1 Regulation conditions for 1Σ and 2Σ .....................................................................128

4.4.2 Regulation condition for the switched system.........................................................134 4.4.3 Regulation condition for the switched closed loop system cl

rΣ with an 2H performance constraint ..................................................................................................137

4.5 REGULATOR SYNTHESIS ............................................................................................ 139 4.6 NUMERICAL EXAMPLE............................................................................................... 144 4.7 EXPERIMENTAL EVALUATION.................................................................................... 149

4.7.1 Description of the experimental setup .......................................................................150 4.7.2 Simulation of the external disturbances and of the contact force ......................151 4.7.3 Identification of the switched system model............................................................155 4.7.4 Experimental results........................................................................................................158

4.8 CONCLUSION............................................................................................................. 162

5 ADAPTIVE REGULATION IN SWITCHED BIMODAL LINEAR SYSTEMS ...........167

5.1 INTRODUCTION.......................................................................................................... 167 5.2 THE ADAPTIVE REGULATION PROBLEM FOR BIMODAL SYSTEMS................................ 169 5.3 PARAMETERIZATION OF A SET OF STABILIZING CONTROLLERS .................................. 170

5.3.1 Q -parameterized controller .........................................................................................170 5.3.2 Stability of the Q -parameterized switched closed loop system........................173

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5.4 REGULATION CONDITIONS FOR THE SWITCHED SYSTEM ............................................ 174 5.5 ADAPTIVE REGULATION IN THE SWITCHED CLOSED LOOP SYSTEM ............................ 185 5.6 NUMERICAL EXAMPLE............................................................................................... 193 5.7 EXPERIMENTAL EVALUATION.................................................................................... 197 5.8 CONCLUSION............................................................................................................. 200

6 CONCLUSION .....................................................................................................................................207

6.1 CONTRIBUTIONS OF THE THESIS ................................................................................ 207 6.1.1 Development of a regulator synthesis procedure in bimodal linear systems

against known deterministic exogenous inputs......................................................208 6.1.2 Development of a regulator synthesis procedure in bimodal linear systems

with an 2H performance constraint. .......................................................................209

6.1.3 Development of an adaptive regulation approach in bimodal linear systems against unknown sinusoidal exogenous inputs.......................................................210

6.2 FUTURE WORK .......................................................................................................... 211

APPENDIX ...................................................................................................................................................214

BIBLIOGRAPHY .......................................................................................................................................228

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LIST OF FIGURES

Figure 2.1. Closed loop system with Q -parameterized controller. ....................................... 27 Figure 2.2. State trajectories of 1

clΣ with initial conditions on the switching surface inside the set Bγ for case 1. ................................................................................................ 43

Figure 2.3. State trajectories of 2

clΣ with initial conditions on the switching surface inside the set Bγ for case 2. ................................................................................................ 43

Figure 2.4. Diagram of a mechanical system with switched dynamics. ................................. 48 Figure 2.5. Simulation results for the case of 30ev = − micrometers showing the

performance variable e and the switching component (1,1)rd in the disturbance

rd . ........................ ………………………………………………………………54 Figure 2.6. Simulation results for the case of 30ev = − micrometers showing the regulated

mass height v and the contact surface displacement sv . .................................. 54 Figure 3.1. State trajectories of 1

clΣ with initial conditions on the switching surface inside the set xB for case 1…....………………… ……………………………………… 86

Figure 3.2. State trajectories of 2

clΣ with initial conditions on the switching surface inside the set xB for case.. ................................................................................................. 86

Figure 3.3. Simulation results for the case of 30ev = − micrometers showing the

performance variable e and the switching component (1,1)rd in the disturbance rd ...………………………………………………………………..93

Figure 3.4. Simulation results for the case of 30ev = − micrometers showing the regulated

mass height v and the contact surface displacement sv . ................................. 93

Figure 3.5. State space partitions where the vector Tex exC C= . ............................................ 105

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Figure 3.6. Simulation results for the case of 30ev = − micrometers showing the performance variable e and the switching component (1,1)rd in the disturbance rd . . ………………………………………………………………115

Figure 3.7. Simulation results for the case of 30ev = − micrometers showing the regulated

mass height v and the contact surface displacement sv . ............................... 115 Figure 4.1. Closed loop system with a Q -parameterized controller. ……………………121 Figure 4.2. Simulation results showing the performance variable e obtained using the

controller designed without accounting for the 2H performance constraint. .. 146 Figure 4.3. Simulation results showing the switching component (1,1)rd in the known input

rd and the random disturbance wd . ................................................................. 146 Figure 4.4. Simulation results showing the displacement of the mass obtained using the

controller designed without accounting for the 2H performance constraint. .. 147 Figure 4.5. Simulation results showing the performance variable e obtained using the

controller designed based on the 2H performance constraint.......................... 147 Figure 4.6. Simulation results showing the displacement of the mass obtained using the

controller designed based on the 2H performance constraint.......................... 148 Figure 4.7. Schematic diagram of the experimental setup.................................................... 151 Figure 4.8. The experimental setup: (a) General view, (b) Close-up view of the suspension

beam with the actuators...................................................................................... 153 Figure 4.9. The frequency response of the identified analytical model (solid line) and the real

system based on experimental data (dashed line) in the non-contact mode: (a) transfer function from u to vy , (b) transfer function from d to vy . .......... 158

Figure 4.10. The frequency response of the identified analytical model (solid line) and the real

system based on experimental data (dashed line) in the contact mode: (a) Transfer function from u to vy , (b) transfer function from d to vy ........ 159

Figure 4.11. Experimental results showing the performance variable e and the position y

for the case of 0dw = and obtained using the controller designed without accounting for the 2H performance constraint. ............................................. 163

x

Figure 4.12. Experimental results showing the random disturbance dw and the total current

in the coil for the case of 0dw = and obtained using the controller designed without accounting for the 2H performance constraint. ................................ 163


in the presence of dw and obtained using the controller designed without accounting for the 2H performance constraint. ............................................. 164


in the coil obtained using the controller designed without accounting for the 2H performance constraint. .................................................................................... 164


in the presence of dw and obtained using the controller designed by accounting for the 2H performance constraint................................................................. 165


in the coil obtained using the controller designed by accounting for the 2H performance constraint………………………………………………………...165

Figure 5.1. Diagram of the switching sequence for the closed loop system. ……………….186 Figure 5.2. Simulation results showing the performance variable e and the switching

component (1,1)id in the input id ................................................................... 195

Figure 5.3. Simulation results showing estimated parameter vector 2θ and nominal parameter vector 0θ ........................................................................................... 195

Figure 5.4. Experimental results showing the performance variable e for the case where the

nominal parameter vector 0θ is used. .............................................................. 201 Figure 5.5. Experimental results showing the tip position ty of the suspension beam and the

contact surface cs for the case where the nominal parameter vector 0θ is used............................................................................................................................. 201

Figure 5.6. Experimental results showing the current i in the coil used to simulate the

disturbance force and the contact force for the case where the nominal parameter vector 0θ is used............................................................................................... 202

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Figure 5.7. Experimental results showing the control input signal u driving the piezoelectric actuator for the case where the nominal parameter vector 0θ is used.............. 202

Figure 5.8. Experimental results showing the performance variable e for the case where the

adaptive regulator isused.………………………………………………………203 Figure 5.9. Experimental results showing the tip position ty of the suspension beam and the

contact surface cs for the case where the adaptive regulator is used............... 203 Figure 5.10. Experimental results showing the current i in the coil used to simulate the

disturbance force and the contact force for the case where the adaptive regulator is used. .............................................................................................................. 204

Figure 5. 11. Experimental results showing the control input signal u driving the

piezoelectric actuator for the case where the adaptive regulator is used. ........ 204 Figure 5.12. Experimental results showing the desired parameter vector 0θ (dashed line) and

the estimated parameter vector θ (solid line) for the case where the adaptive regulator is used. .............................................................................................. 205

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LIST OF TABLES

Table 1.1. Summary of the contributions................................................................................ 21 Table 6.1. Summary of advantages and limitations of the proposed algorithms .................. 211

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CHAPTER 1

Introduction

The main theme of the research presented in this dissertation is the development and analysis

of regulation techniques against exogenous input signals in switched bimodal linear systems.

The exogenous input signals could represent reference signals to be tracked or disturbance

signals to be rejected. Two kinds of regulator design problems are addressed in this thesis.

The first problem is the design of regulators for switched bimodal systems to reject known

exogenous input signals. The second problem is the development of an adaptive regulation

method in switched bimodal systems to reject unknown exogenous sinusoidal input signals.

The rest of this chapter is organized as follows. A motivation for the regulation problem

treated in this thesis is presented in Section 1.1. A literature review on the stability analysis

and regulation of switched systems is then presented in section 1.2. Section 1.3 outlines the

problems addressed in this thesis, followed by a summary of contributions in Section 1.4.

The organization of the rest of the thesis is presented in Section 1.5.

1.1 Motivation

To this day, Hard disk drives (HDDs) remain the major archival mass data storage medium in

computing devices and are also playing an increasingly significant role in consumer

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electronics devices. Over the past decade, HDD data storage capacity has been increasing at a

rate of 60% to 100% per year. To accommodate the demand for more storage capacity in the

same volume, the magnetic medium has to store the information at ever higher areal densities,

which requires that the read/write (R/W) head be held at a very small constant flying height

above the moving magnetic medium (disk surface). Most of the existing modern hard-disk

drives operate in the flying mode [1-4], where the R/W head attached on the slider is lifted by

the air flow beneath and flies above the disk surface when the hard disk rotates. The flying

height will vary with time due to a number of unknown factors, including spindle vibrations,

hard disk surface microwaviness, and also airflow disturbances generated by the disk rotation.

Therefore, it is highly desirable to design a closed-loop feedback system to actively suppress

the flying height variations.

One of the major issues associated with the design of a flying height regulation system is

the fact that, for small flying heights, the slider/disk interface becomes a system with

switched dynamics due to the intermittent contact between the R/W head and the disk surface.

Moreover, the various causes of flying height variation (microwaviness of the disk surface,

aerodynamic disturbances, spindle vibrations, etc.) are not well characterized and are not

known a priori. It is well known that the techniques developed to address traditional

regulation problems in linear and smooth nonlinear systems cannot be applied to switched

systems directly. At present time, there are only limited results on the regulation problem for

switched systems. In this thesis, motivated directly by the flying height regulation problem in

hard disk drives, some original results related to the exact output regulation problem in

switched bimodal systems against both known and unknown exogenous input signals is

presented.

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1.2 Literature Review on Switched Systems

The purpose of this section is to summarize the main results presented in the literature related

to the control of switched systems. Recently, switched control systems have attracted much

attention in the control community since they present problems that are not only

academically challenging, but also of practical importance. Switched systems are hybrid

systems that combine various types of dynamic behavior, including continuous-time

dynamics, discrete-time dynamics, switching and state variable jumps [5-6]. There are three

main factors that motivate the study of switched systems. First, many of the natural non-

smooth dynamic systems can be modeled as switched systems. Examples of such systems

include mechanical systems with impact, hopping robots, linear systems with static

nonlinearities such as relays and saturations, network communication systems with delay, etc.

[2,7,8,9]. Second, switched systems offer much better approximations of nonlinear or hybrid

systems than linear systems. This is due to the fact that linear systems only approximate the

true dynamics in small regions around equilibrium points and do not capture important

properties of nonlinear systems such as limit cycle oscillations [10]. Lastly, the study of the

switched systems is also motivated by the widespread applications of switched controllers in

systems with increasing performance requirements, especially in the presence of large

uncertainties or disturbances. A good example of such systems is the so-called supervised

control design that involves switching between different controllers. It is well understood

that switched controller systems, as a special type of switched systems, can provide control

solutions that guarantee a level of stability and good performance that no single-controller

system can provide [11-14]. For example, many complex nonlinear systems that are not

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stabilizable by a single controller can be stabilized by switching between a finite number of

controllers [15-17].

1.2.1 Stability of switched systems

The stability issues in switched systems, especially switched linear systems, have been of

increasing interest for researchers in the past decade, and include several interesting

phenomena. For example, even when all the subsystems in a switched system are

exponentially stable, the switched system may have divergent trajectories for certain

switching signals [5,18,19]. Another remarkable fact is that one may carefully switch

between unstable subsystems to make the switched system exponentially stable [20,21]. As

these examples suggest, the stability of switched systems depends not only on the dynamics

of each subsystem in the switched system, but also on the properties of switching signals.

The stability analysis of switched systems involves two main types of problems, namely, the

stability analysis for switched systems under arbitrary switching and the stability analysis for

switched systems under constrained switching.

Stability of switched systems under arbitrary switching

Among the large variety of stability analysis problems encountered in switched systems,

one can assume that the switching sequence is not known a priori and look for stability

results under arbitrary switching sequences. A well known approach to deal with this

problem is based on the existence of common quadratic Lyapunov functions (CQLF) to

check the asymptotic stability for the switched system under arbitrary switching. The

existence of a common quadratic Lyapunov function for all subsystems in the switched

system assures the quadratic stability of the switched system. However, the conditions for

the existence of a common quadratic Lyapunov function are not easy to determine. In fact,

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there have been various attempts to derive conditions for the existence of a common

quadratic Lyapunov function. Ooba, et al. [22] and Shorten, et al. [23] presented some

easily verifiable algebraic conditions for proving the existence of the common quadratic

Lyapunov function for general switched linear systems. Dayawansa and Martin [24]

proved that uniform exponential stability is equivalent to the existence of a smooth

positive definite function, which is a common Lyapunov function for the switched system,

but cannot guarantee the existence of a common quadratic Lyapunov function. If a

common quadratic Lyapunov function does not exist or cannot be found, then the multiple

Lyapunov function method can be used for stability analysis. Branicky [25,26] proposed

conditions for multiple Lyapunov functions to ensure the stability of the switched system.

Numerical algorithms to compute multiple Lyapunov functions were developed by

Brayton and Tong [27] and Polanski [28]. However, the computation of the multiple

Lyapunov functions is not a tractable problem in practice. For discrete-time switched

systems, Daafouz, et al. [29] developed a more practical scheme to check the stability of

the system using multiple quadratic Lyapunov functions. The main idea is to associate a

quadratic Lyapunov function with each subsystem in the switched system, and then check

the asymptotic stability using the switched quadratic Lyapunov functions which are

designed to decrease at each of the switching instances. The existence of such Lyapunov

functions can be reduced to the compatibility of a set of linear matrix inequalities (LMIs).

There also have been other results in the literature that propose constructive synthesis and

computation methods to find multiple Lyapunov functions using linear matrix inequalities

or bilinear matrix inequalities [30,31,32].

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Stability of switched systems under constrained switching

In this class of problems, one important case of stability studies involves a finite number

of linear systems together with a set of switching rules defined by switching surfaces

(state dependent switching). In this case, the state space is partitioned into cells, and the

dynamics of the system are described using different linear models corresponding to

operation in the different cells. Johansson and Rantzer [33] presented the first rigorous

result on the analysis of general switched systems where switching is based on space

bounds (switching surfaces). This method relaxes the problem to the search for piecewise

quadratic Lyapunov functions for the switched system, which can be formulated by

solving a set of linear matrix inequalities. If the piecewise quadratic Lyapunov functions

cannot be found with the given natural partitioning of the state space, then a refinement of

the partitions can be applied to improve the flexibility of finding piecewise quadratic

Lyapunov functions. For high order systems, although it is very hard to obtain a

refinement of the proper partitions in the state space to efficiently analyze the stability

using piecewise quadratic Lyapunov functions, the idea of this approach still appears to be

very powerful and can be furthered in a large number of directions, including performance

analysis, global linearization, controller optimization, and model reduction. Hassibi and

Boyd [34] addressed the question of stability and control of piecewise linear time-

invariant systems using linear matrix inequalities. Using Lyapunov theory, they derived

sufficient conditions for stability and performance that can be checked by solving convex

optimization problems with LMI constraints along the switching surface. The approach is

to search among special classes of functions for a Lyapunov function that proves stability

or performance for the piecewise linear system. They demonstrated applications of the

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proposed methods by considering controller synthesis of a simple mechanical system

subject to input saturation, and stability analysis of an electrical circuit with multiple

equilibrium points. Goncalves, Mgretski, and Dahleh [35] developed a stability analysis

methodology for switched linear systems using impact maps and surface Lyapunov

functions. This methodology solely studies the system behaviour at the switching surface

and constructs a surface Lyapunov function to show that impact maps associated with the

switched linear system are contracting. This representation of impact maps allows the

search for surface Lyapunov functions to be done by simply solving a semidefinite

program, allowing global asymptotic stability, robustness, and performance of limit cycles

and equilibrium points of piecewise linear system to be efficiently checked. This analysis

methodology has been successfully applied to relay feedback, on/off and saturation

systems. Rodrigues and How [36] proposed a synthesis method for both state and dynamic

output feedback control of a class of piecewise affine systems, where the state space

related to each subsystem is partitioned as polytopic cells. The synthesis procedure

delivers stabilizing controllers that do not necessarily have measurements of the switching

parameters and enables switching in the controller based on the estimated states rather

than on measured parameters. The proposed technique formulates the search for a

piecewise quadratic control Lyapunov function and a piecewise affine control law as a bi-

convex optimization program subject to some linear constraints and a bilinear matrix

inequality, which can be solved effectively using iterative algorithms.

Another type of switched systems with constrained switching corresponds to switched

systems where the switching is time dependent. One well known approach to guarantee

stability in this case is to find a lower bound on the dwell time or average dwell time in

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the switched system in any given mode of operation of the switched system [5]. The basic

idea is that a switched system is stable if all individual subsystems are stable and the

switching is sufficiently slow as to allow the transient effects to dissipate after each switch.

Recently, the stability of switched systems under random switching, in which the

switching times are unknown but can characterized by stochastic processes, has also been

studied in the literature. For example, the stochastic stabilizability for discrete-time linear

switching systems is investigated by Boukas and Benzaouia [37], where the switching

signal is a finite-state Markovian jump process. A sufficient condition for stochastic

stability of the system is deduced by using a non-quadratic Lyapunov function, and the

necessary and sufficient conditions allowing the constrained control law to always be

admissible despite the stochastic character of the system are also presented. Similar

problems are also addressed in the papers of Fang and Loparo [38], and Costa and Tuesta

[39]. In other cases, where the switching signal can be controlled and designed, one then

can study the existence of specific switching rules that ensure stability of the switched

system. In these cases, both the switching signal and the control input are assumed to be

design variables. As in the case of linear systems, the concepts of controllability play a

fundamental role in the design and synthesis for switching systems if the switching signal

can be designed. Loparo, Aslanis, and Hajek [40] addressed the issues of controllability

and reachability for continuous-time planar switched systems. Sufficient conditions and

necessary conditions for controllability and reachability were reported by Ezzine and

Haddad [41], Szigeti [42], Sun and Ge [43], Xie, et al. [44], Yang [45], and Cheng, et al.

[46].

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In addition to the results mentioned before, numerous other results are published in recent

years and are summarized in books [47-49] and the references therein.

1.2.2 Output regulation in switched systems

Till now, most of the literature on switched systems focused on the stability analysis for such

systems. However, as mentioned in section 1.1, in practice, in additional to stability

requirements, there is a need to find controllers that would achieve output regulation. Output

regulation is the problem of finding a control law by which the output of the system of

interest asymptotically tracks a reference signal and rejects undesired disturbance inputs,

both of which are deterministic signals generated by an exosystem. The control law must also

asymptotically stabilize the system whenever the exosignal is absent. The output regulation

problem for linear time-invariant systems has been solved by Francis, Wonham and Davison

in [50-52]. The main approach is based on the Internal Model Principle, which states that

asymptotic disturbance rejection can be achieved if a model of the exosystem generating the

disturbance is included in the stable closed loop system. For output regulation in nonlinear

systems, Isidori and Byrnes [53] and Byrnes et al. [54] presented necessary and sufficient

conditions to solve this problem. In general, the solvability of the problem is based on the

center manifold theory. Isidori and Byrnes found that it is possible to use a set of mixed

nonlinear partial differential and algebraic equations, called regulator equations, to

characterize the steady state of the system. The solution of the regulator equations provided a

feedward control to cancel the steady state tracking error. Based on the solution of the

regulator equations, both state feedback and error feedback control laws can be readily

synthesized to achieve asymptotic tracking and disturbance rejection for an exactly known

plant. Additional results have also been reported in the application of the theory to practical

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design problems [55-60].

Although significant progress has been made in addressing the regulation problem for

both linear and nonlinear systems, the existing results cannot be applied to switched systems

since the vector fields of switched systems are discontinuous and non-smooth. The literature

on regulation for switched systems covers two main problems: exact output regulation and

optimal output regulation.

1) Exact output regulation: In this problem, it is desired to design a controller such that the

output of the resulting closed loop switched system asymptotically tracks a reference

signal and/or rejects disturbance inputs. The work of Devasia [61,62] and Sakurama [63]

are closely related to this problem. Devasia et al. studied the exact-output tracking

problem for linear switching systems, and necessary and sufficient conditions for the

existence of exact output-tracking bounded state trajectories are established. It is

assumed that the reference trajectories are generated by an exosystem, where either the

dynamics and initial conditions of the exosystem are completely known, or the dynamics

of the exosystem is known but the initial conditions are not known. Moreover, the

switching times are assumed known a priori, so the necessary and sufficient regulation

conditions are derived only for the elimination of switching induced output transients.

Therefore, the developed results cannot be applied to the case when switching is signal

driven and the switching times cannot be set a priori. Sakurama et al. recently discussed

the trajectory tracking problem for bimodal switched linear systems. It is assumed that

the reference trajectory for the switched system is known and the switching times are

measurable. An error variable and an error system are introduced, based on which the

tracking controller is designed using a Lyapunov-like function. However, the conditions

11

for existence of the desired regulator are sometimes very difficult to check. van de

Wouw and Pavlov [64] also studied the tracking problem for a class of piecewise affine

systems, where the desired system state trajectory is assumed to be known and

predefined. An observer-based putput-feedback control design method is proposed,

which consists of a feedforward, a piecewise affine feedback law and a model-based

observer. The synthesis conditions are formulated in terms of bilinear matrix inequalities.

2) Optimal output regulation: In this case, the controller is designed so that the resulting

switched closed loop system achieves a pre-specified optimal performance. To date, only

few results related to this topic have been reported in the literature based on 2l gain

analysis and H∞ closed loop system performance. Hespanha [65] studied the root-mean-

square (RMS) gain of switched linear systems and a bisection algorithm is proposed to

compute the slow switching RMS gain of the switched system in terms of the solutions

to a collection of Riccati equations. Lee and Dullerud [66] studied the disturbance

attenuation problem for discrete-time switched and markovian jump linear systems. An

exact condition for uniform stabilization and disturbance attenuation is given via the

union of an increasing family of linear matrix inequalities. A semidefinite programming

based controller synthesis method was proposed. Lee and Khargonekar [67] studied the

infinite-horizon suboptimal LQG control problem in discrete-time switched systems,

which aims at maintaining the average output variance below a given level subject to the

uniform exponential stability of the closed-loop system. The solution is given in the form

of a dynamic linear output feedback controller which not only observes the present mode

of the system but also recalls a finite number of past modes. These results are also

extended to Markovian jump linear systems. Rodrigues and Boukas [68] studied the

12

optimal inventory control problem in switched production systems, where the inventory

control problem is formulated as a switched H∞ control problem and the synthesis of a

state feedback controller that quadratically stabilizes the production dynamics and at the

same time rejects the external demand fluctuations is cast as a set of linear matrix

inequalities. Lin and Antsaklis [69] studied the asymptotic stability with a finite 2l

induced gain for a class of discrete-time switched linear control systems, where

sufficient conditions for the co-design of continuous variable controllers and discrete

event switching logic are proposed. Geromel et al. [70] investigated the stability

conditions and the output feedback regulator design for switched linear systems in both

the continuous and discrete-time domains, where optimal switching control strategy are

derived in terms of Lyapunov-Metzler inequalities. Some other results concerning 2l

gain analysis and H∞ control for certain classes of hybrid systems or hybrid control are

also available in [71-75].

1.2.3 Adaptive regulation in linear and nonlinear systems

When the disturbances or reference signals are deterministic signal with unknown properties,

adaptive regulation approaches are usually used. Most of the proposed approaches are based

on the internal model principle, where an internal model-based controller is designed online

to accommodate uncertainties in the exogenous signals. Marino and Tomei [76] designed an

adaptive internal model using an adaptive observer which provides estimates of the

observable states and of the exosystem unknown parameters. Ben Amara et al. [77,78]

proposed an adaptive regulation approach within a set of parameterized stabilizing controllers,

where the parameter in the stabilizing controller is tuned online to asymptotically reject

13

sinusoidal disturbances. A similar approach has been considered by Landau, et al. [79] to

adaptively reject narrow-band unknown disturbances in active suspension systems. Another

approach, based on the external model principle, involves the use of a model of the

disturbance outside the feedback loop to estimate the values of the disturbance input. Bodson

and Douglas [80] developed an adaptive feedforward compensator, where the feedforward

controller outside the feedback loop is designed to offset the steady state error induced by the

exogenous disturbance. The parameters in the feedforward controller are updated online

using the regulated error. Tomizuka et al. [81] developed an adaptive regulator, where a

stabilizing controller is computed first, then a model of the disturbance outside the feedback

loop is used to estimate the values of the disturbance input. The latter are then used in a

feedforward disturbance rejection algorithm to provide cancellation of the disturbance.

Serrani [82] developed a hybrid adaptive external model-based controller to reject harmonic

disturbances occurring at the input of a linear system. The adaptive unit is outside the stable

loop and possesses a self-tuning mechanism to achieve regulation even in the absence of

persistency of excitation.

Significant progress has also been reported in the literature in addressing the adaptive

output regulation problem for nonlinear systems. The internal model principle is still best

suited to this problem. Serrani et al. [83] studied the problem of output regulation for

nonlinear systems driven by a linear and neutrally stable exosystem whose frequencies are

not known a priori. The solution is realized by the parallel connection of a robust stabilizer

and a parameterized family of internal models which is adaptively tuned to reproduce the

signals generated by the exosytem and the nonlinearities of the plant. This adaptive design

methodology was also recognized as a useful way to cope with parameter uncertainties in the

14

exosystem. For instance, it was used in [84] for the solution of a global decentralized

regulation problem for a class of nonlinear systems with a neutrally stable uncertain

exosystem. Priscoli et al. [85] extended the existing theory of adaptive output regulation for

linear systems to nonlinear systems by allowing nonlinear internal models based on the

theory of adaptive observers for nonlinear systems [86,87] and the general nonequilibrium

theory [88]. Xi and Ding [89] discussed the adaptive output regulation problem for a class of

nonlinear systems with nonlinear exosystems. The nonlinear internal model is constructed

based on the circle criterion to produce the desired feedforward control term and

accommodate the nonlinearities in the exosystems. Ding [90] also studied the global

disturbance rejection problem in nonlinear systems, where the disturbance is assumed to be

sinusoidal with completely unknown phases, amplitude and frequencies, but the number of

distinct frequencies or the order of the corresponding unknown linear exosystem is known.

Despite the progress mentioned above, to the best of the author’s knowledge, no results

exist in the literature on the exact output regulation for switched systems in the presence of

unknown exogenous input signals.

1.3 Problem Statement

A formal statement of the main focus of this dissertation is given as follows. Consider a

switched bimodal linear system given by

:

,,

, r

xr r r ry yr r re er r r

x A x B u D wy C x D w

e C x D w

Σ

⎧Δ = + +⎪

= +⎨⎪ = +⎩

(1.1)

15

where xΔ denotes dxdt

in the continuous-time case and ( 1)x k + in the discrete-time case,

nx∈R is the state vector, u∈R is the control input, y∈R is the measurement signal to be

fed to the controller, hrw = R is the exogenous input signal, e∈R is the performance

variable to be regulated and is assumed to be measurable, and { }1, 2r∈ is the index of the

system rΣ under consideration at time t or k . The switching between the systems 1Σ and

2Σ is performed according to the value of the performance variable e as

1 if ,2 if ,

e er r re er r r

e C x D wr

e C x D wδδ

⎧ = + ≤= ⎨

= + >⎩ (1.2)

where δ is constant and 0δ ≠ . For the plant (1.1) subject to the exogenous input rw , it is

desired to design a controller that yields internal stability and asymptotic exact output

regulation, namely as time goes to infinity, the value of the performance variable e should

go to zero. This thesis concerns the development of regulator synthesis methods for the

above regulation problem in switched systems (1.1) under the switching law defined by (1.2).

More specifically, this thesis aims mainly at:

1) The development of a regulator design method to reject known deterministic exogenous

inputs in the switched system (1.1).

2) The development of adaptive regulator design method to reject sinusoidal exogenous

inputs with unknown properties in the switched system (1.1).

16

1.4 Contributions of the Thesis

The regulation problems in bimodal linear systems presented in this thesis are solved within

sets of Q -parameterized controllers, where the parameterizing dynamics are designed not

only to make the switched closed loop system stable, but also to achieve exact output

regulation in the switched system. The proposed sets of parameterized controllers are

constructed in two steps. The first step is based on constructing a switched observer-based

state feedback central controller for the switched linear system. The second step involves

augmenting the switched central controller with additional dynamics, referred to as the Q

parameter, to construct a parameterized set of switched controllers. Then, based on the sets of

Q -parameterized controllers, four regulator synthesis methods are developed to deal with

different cases. The original contributions presented in this thesis are summarized as follows:

1) Development of a regulator synthesis procedure for bimodal systems based on bilinear

matrix inequalities. A parameterized regulator design method based on solving bilinear

matrix inequalities is proposed for continuous-time switched bimodal linear systems

subject to known deterministic exogenous inputs. A set of Q -parameterized controllers

is considered, where, unlike the case of controller parameterization in linear systems, no

stability or structural constraints are imposed on the Q parameter. Sufficient regulation

conditions are derived for the resulting parameterized switched closed loop system and a

corresponding regulator synthesis approach is proposed based on solving properly

formulated bilinear matrix inequalities. The advantage of considering Q -parameterized

controllers where no stability or structural constraints are imposed on the Q parameter is

that it offers more flexibility in the design of the desired regulators. However, the

17

synthesis of the regulators based on solving bilinear matrix inequalities results only in

local optimal solutions in the search algorithm. Moreover, a general structure for the Q

parameter is not convenient to work with when it comes to tuning the Q parameter in the

case of adaptive regulation against unknown exogenous sinusoidal inputs.

2) Development of a regulator synthesis procedure in bimodal systems based on linear

matrix inequalities. Parameterized regulator design methods based on solving properly

formulated linear matrix inequalities are proposed for continuous-time switched bimodal

linear systems subject to known deterministic sinusoidal exogenous inputs. The design

of the proposed regulators involves two main steps. First, a set of observer-based Q -

parameterized stabilizing controllers for the switched system is constructed. The Q

parameters are represented as linear combination of stable basis functions. The stability

of the resulting closed loop switched system with the stabilizing Q -parameterized

controllers is analyzed. Second, regulation conditions for each of the two subsystems in

the resulting bimodal switched closed loop system are presented in both the state space

domain and the frequency domain. Then, regulation conditions and regulator synthesis

methods for the switched closed loop system are proposed using a common Lyapunov

function approach and a multiple Lyapunov function approach. There are two main

advantages in considering Q parameters that are expressed as linear combinations of

stable basis functions. First, the regulator synthesis procedure for the case of known

exogenous inputs can be developed based on solving a set of linear matrix inequalities.

Second, an adaptive online tuning algorithm for the free Q parameters can be

conveniently developed in the case of unknown sinusoidal exogenous inputs.

18

3) Development of the regulator synthesis procedure in bimodal systems within a class of

2H controller. A parameterized regulator synthesis method is proposed for switched

discrete-time bimodal linear systems where it is desired to achieve regulation

simultaneously against known deterministic sinusoidal exogenous inputs and unknown

random exogenous inputs. First, a Q -parameterized set of stabilizing controllers for the

discrete-time switched system is constructed and that yield regulation against the known

deterministic sinusoidal exogenous inputs. A subset of 2H controllers is then identified

within the already constructed set of regulators to effectively deal with the random

exogenous input signals. A corresponding regulator synthesis method is developed by

solving properly formulated linear matrix inequalities. A switched mechanical system

experimental setup is used to successfully validate the proposed regulator design

approach.

4) Development of an adaptive regulation approach in bimodal systems. An adaptive

regulation approach is proposed for discrete-time switched bimodal linear systems,

where it is desired to achieve regulation against unknown deterministic sinusoidal

exogenous input signals. First, assuming the properties of the exogenous input signal are

known, a set of observer-based Q -parameterized stabilizing controllers for the switched

system is constructed, and a sufficient regulation condition for the resulting switched

closed loop system is presented. To deal with the case of exogenous sinusoidal inputs

with unknown properties, an adaptation algorithm is developed to tune the Q parameter

in the expression of the parameterized set of controllers. The tuning is such that the

19

tuned Q parameter converges to the desired Q parameter that guarantees regulation in

the switched system. The proposed adaptation approach is successfully validated on an

experimental setup involving a switched mechanical system.

The four contributions are each treated in Chapters 2, 3, 4 and 5, respectively, and are

summarized in Table 1.1.

1.5 Organization of the Thesis

The remainder of this thesis is organized as follows. Chapter 2 presents a parameterized

regulator synthesis method for continuous-time switched bimodal linear systems against

known sinusoidal exogenous inputs, where no stability or structural constraints are imposed

on the Q parameter. A regulator synthesis algorithm is developed based on solving bilinear

matrix inequalities. Chapter 3 presents a regulator synthesis approach along the same lines as

that outlined in chapter 2, but where the Q parameter is constrained to be a linear

combination of stable basis functions. A regulator synthesis algorithm is developed based on

solving properly formulated linear matrix inequalities. Two regulator synthesis approaches

are proposed. The first approach is based on considering a common Lyapunov function for

the closed loop switched system, whereas the second is based on considering multiple

Lyapunov functions for the switched closed loop system. In chapter 4, the problem of output

regulation in discrete-time switched bimodal systems against known deterministic sinusoidal

exogenous inputs and unknown random exogenous inputs is solved within a class of 2H

controllers. Chapter 5 presents an adaptive regulation method for discrete-time switched

bimodal linear systems against unknown sinusoidal exogenous inputs. Chapters 4 and 5 also

20

include experimental results that validate the regulation approaches presented in those two

chapters. The experimental results are obtained using a switched mechanical system

experimental setup that is motivated by the flying height regulation problem in hard disk

drives.

21

Contributions

1st 2nd 3rd 4th Chapter

2 3 4 5 Known √ √ √ √ Continuous-time √ √ Plant model Discrete-time √ √ Deterministic known √ √ √ Deterministic unknown √

PLA

NT

Exogenous inputs Random √ Unconstrained √

Q Parameter Linear combination of stable basis functions √ √ √ Common Lyapunov function √ √ √

Closed loop system Lyapunov function Multiple Lyapunov functions √ √ Derived based on frequency domain regulation conditions (interpolation conditions) for each of the subsystems in the switched closed loop system

√ √ √

Regulation conditions Derived based on state space domain regulation conditions for each of the subsystems in the switched closed loop system

√ √

BMIs √ √

CO

NT

RO

LL

ER

Regulator synthesis algorithm LMIs √ √ √

EXPERIMENTAL RESULTS √ √

Table 1.1 Summary of the contributions.

22

CHAPTER 2

Parameterized Regulator Synthesis for

Switched Bimodal Systems Based on Bilinear

Matrix Inequalities

2.1. Introduction

In this chapter, a controller design approach is proposed for switched bimodal linear systems

where it is desired to reject known disturbance signals and track known reference inputs

simultaneously. Switching in the bimodal system is defined by a time-varying switching

surface. A regulator synthesis method based on solving bilinear matrix inequalities (BMIs) is

presented. The proposed regulator design approach consists of three steps. In the first step, a

switched observer-based state-feedback central controller is constructed for the switched

linear system. The second step involves augmenting the switched central-controller with

additional dynamics to construct a parameterized set of switched controllers. Stability

analysis of the resulting switched closed loop system is then presented. In the third step, two

sufficient regulation conditions are derived for the switched closed loop system. The first

sufficient condition is derived based on the input-output stability property of the switched

23

closed loop system. The second sufficient condition for regulation is derived by transforming

the forced switched closed loop system into an unforced impulsive switched system. As such,

the regulation problem is transformed into a stability analysis problem for the impulsive

switched system. Based on the parameterized controller structure and the derived regulation

conditions, proper bilinear matrix inequalities are formulated and a regulator synthesis

method is proposed. The main advantage of the proposed regulator synthesis approach is that

it offers a numerical procedure that can be implemented and used to develop the desired

switched regulator.

The rest of the chapter is organized as follows. In section 2.2, the general regulation

problem for switched bimodal linear systems is presented. In section 2.3, the construction of

a parameterized set of switched controllers for the switched system is discussed and the

stability properties of the resulting closed loop switched system are analyzed. Regulation

conditions for the switched system are presented in section 2.4 and a regulator synthesis

algorithm for switched system is proposed in section 2.5. The controller design method is

illustrated in section 2.6 using a numerical example, followed by the conclusion in section

2.7.

Notation: In the following, the state space representations :x Ax Buy Cx Du= +⎧

Σ ⎨ = +⎩ and

:A BC D⎡ ⎤

Σ ⎢ ⎥⎣ ⎦

will be used interchangeably for the sake of brevity. For a given 1 N× matrix X ,

X denotes the 1N × vector with the same entries as those of TX .

24

2.2. Regulation problem for switched bimodal systems

Consider the continuous-time switched system given by the state space representation:

0

0

, (0) , , (0) ,

, :

,

1 if ,

2 if ,

xr r r r

r r r r ry yr r r

r e er r

e er r

e er r

x A x B u D w x xw H w w w

y C x D we C x D w

e C x D wr

e C x D wδδ

= + + == =

= +∑

= +

⎧ = + ≤= ⎨

= + >⎩

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

(2.1)

where nx∈R is the state vector, u∈R is the control input, y∈R is the measurement signal

to be fed to the controller, e∈R is the performance variable to be regulated and is assumed

to be measurable, hrw ∈R is the state vector of the exogenous systems generating the signal

rw , h hrH ×∈R have simple eigenvalues on the imaginary axis, { }1,2r∈ is the index of the

system rΣ under consideration at time t , and δ is a constant satisfying 0δ > . The

switching between the systems 1Σ and 2Σ is performed according to the value of the

performance variable e , and is determined based on the location of x with respect to a

switching surface S given by:

{ }: n e er rS x e C x D w δ= ∈ = + =R . (2.2)

The switching surface S is not fixed but changes with time given that the term er rD w is in

general a time-varying term. The switching between the two modes takes place as follows: if

1r = and e becomes strictly greater than δ , then the mode switches to 2r = ; and if 2r =

and e becomes less than or equal to δ , then the mode switches to 1r = . In the following, it

is assumed that for any given 0t ≥ , the system must operate in only one of the two modes

25

corresponding to { }1,2r∈ . This assumption is motivated by physical considerations in some

applications of interest, such as the example presented at the end of the chapter or the system

treated in [2,3]. For the switched system (2.1), it is desired to construct an output feedback

controller to regulate the performance variable e of the switched system against the external

input signal rw . Given the switching nature of the plant, the output feedback controller is

also chosen to be a switching feedback controller r , { }1, 2r∈ , with a state space

representation given by:

,:

,

c cc r c r

r cr c

x A x B y

u C x

⎧ = +⎪⎨

=⎪⎩ (2.3)

where cncx ∈R , and where the switching among controllers is to obey the same rule given in

(2.1) for switching between the two plant models. Therefore, the resulting closed loop system

is given by:

,

: 0 ,

1 if ,

2 if .

c xr r r r

rc y c c yc cr r r r r

cl e er r r

c

e er r

e er r

x xA B C Dw

x xB C A B D

xe C D w

x

e C x D wr

e C x D wδδ

⎧ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤= +⎪ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦⎪⎪ ⎡ ⎤⎪ ⎡ ⎤∑ = +⎨ ⎢ ⎥⎣ ⎦

⎣ ⎦⎪⎪ ⎧ = + ≤⎪ = ⎨⎪ = + >⎩⎩

(2.4)

The output feedback regulation problem for the switched system (2.1) can be stated as

follows.

Output feedback regulation problem:

Given the switched system (2.1), where switching is performed according to the rule given in

(2.1), find a switched output feedback controller of the form (2.3) such that the resulting

closed loop system satisfies the following conditions:

26

1) Internal stability: With (0) 0rw = , the equilibrium point 0TT T

cx x⎡ ⎤ =⎣ ⎦ of the unforced

switched closed loop system is asymptotically stable.

2) Output regulation: For any (0) (0) cT nT T n

cx x⎡ ⎤ ∈ ×⎣ ⎦ R R and (0) hrw ∈R , the response ( )e t

of the closed loop system involving the switched system (2.1) and the switched controller

(2.3) satisfies:

lim ( ) 0t

e t→∞

= .

The following section presents a framework within which the design of an output

feedback regulator for the switched system (2.1), including the development of regulation

conditions and a regulator synthesis procedure, will be conducted.

2.3. Parameterization of a set of controllers

The controller design approach presented in this chapter relies on the construction of a

parameterized set of output feedback controllers for the switched system (2.1). In this section,

the construction of such a set is first discussed, followed by an analysis of the stability

properties of the resulting switched closed loop system. The construction of a parameterized

set of switched controllers involves two steps. The first step consists of designing a central

controller in the form of an observer-based state feedback controller. The second step

involves augmenting the central controller with additional dynamics to construct a

parameterized set of controllers.

27

2.3.1 Observer-based state feedback controller

Consider the following observer-based state feedback controller for the switched system (2.1):

( ) 0ˆ ˆ ˆ ˆ ˆ, (0) ,:

ˆ,o r r rr

r

x A x B u L y y x xu K x

⎧ = + + − =⎪⎨

=⎪⎩ (2.5)

where ˆ nx∈R is the estimate of the plant state vector x and ˆ ˆyry C x= is an estimate of the

plant output y . The mode { }1, 2r∈ is determined according to the rule given in (2.4).

Moreover, it is assumed that there are no impulsive changes in the controller states at the

switching times.

2.3.2 Parameterized output feedback controller

The construction of a parameterized set of controllers for the switched system is based on

considering, for each { }1, 2r∈ , a linear fractional transformation involving a fixed system

rJ and a proper system rQ as shown in Figure 2.1. The proposed controllers are similar to

the parameterized stabilizing controllers for linear systems [91-93], but where no stability or

structural assumptions are placed on the system rQ . The state space representation of the

system rJ is given by:

0

1

ˆ ˆ ˆ ˆ ( ) , (0) ,( )

1 0ˆ: ,

ˆ ˆ ,

y y xr r r r r r Qr r r r

qr r Q

yr

x A B K L C x L y y x xB L D D

J u K x y

y y C x y×

⎧ = + + − + =⎡ ⎤+⎣ ⎦⎪⎪ = + ⎡ ⎤⎨ ⎣ ⎦⎪

− = −⎪⎩

(2.6)

and the system rQ is given by:

( ) 0ˆ , (0) ,

:,

r r

r

Q Q Q Q Q Q

r QQ Q

q q

x A x B y y x x

Q Cy x

I ×

⎧ = + − =⎪⎪

⎡ ⎤⎨= ⎢ ⎥⎪⎢ ⎥⎪ ⎣ ⎦⎩

(2.7)

28

Figure 2.1. Closed loop system with Q -parameterized controller.

where qQx ∈ and 1q

Qy +∈ . In particular, throughout the rest of the chapter, the system rQ

is such that the matrices rQA ,

rQB and rQC also change according to r .

Remark 2.1

The state space representation of the system rJ given in (2.6) differs from that traditionally

used in the construction of parameterized sets of stabilizing controllers for linear systems in

that the state equation in (2.6) contains the additional term ( )y xr r r QL D D x+ . The presence of

this term makes it possible to derive the sufficient conditions for regulation presented in

Theorem 2.1.

By combining the systems rJ and rQ in (2.6) and (2.7), the state space representation of the

regulator is then given by (2.3), where

rQ

ˆy y− Qy

rJ

Plant r∑

rw e

u y

29

,

ˆ, ,

, , 0.

r

r r

rr

y y xr r r r r r Q r r rc

c r yQ Q r Q

rc cr r r Q r c

Q

A B K L C B C L D Dxx A

x B C A

LB C K C D

B

⎡ ⎤+ + + +⎡ ⎤= = ⎢ ⎥⎢ ⎥

⎢ ⎥⎣ ⎦ ⎣ ⎦−⎡ ⎤

⎡ ⎤= = =⎢ ⎥ ⎣ ⎦−⎣ ⎦

(2.8)

Let ˆx x x= − denote the state estimation error. It follows that the resulting closed loop

system involving the plant (2.1) and the regulator given by (2.3) and (2.8) can be written as

follows:

0 ,0

:

[ 0 0] .

r

r r r

xr r r r r r Q r

y x y x yr r r r r r r r r r

y yQ Q r Q Q Q rcl

r

e er r

Q

A B K B K B Cx x Dx A L C D L D x D L D wx B C A x B D

xe C x D w

x

⎧ +⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤⎪ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= + + + − −⎪ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎪ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ −⎪⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦∑ ⎨

⎡ ⎤⎪⎢ ⎥⎪ = +⎢ ⎥⎪⎢ ⎥⎪ ⎣ ⎦⎩

(2.9)

Let 1

TT T TQ N

x x xχ×

⎡ ⎤= ⎣ ⎦ denote the state vector of the resulting closed loop system with

2N n q= + , and let

ˆ 00

r

r r

r r r r r r Q

y x yr r r r r r r

yQ r Q

A B K B K B C

A A L C D L DB C A

+⎡ ⎤⎢ ⎥

= + +⎢ ⎥⎢ ⎥⎣ ⎦

,

( ) ( ) ( )r

TTx x y T y Tr r r r r Q rE D D L D B D⎡ ⎤= − + −⎢ ⎥⎣ ⎦

,

0 0ex eC C⎡ ⎤= ⎣ ⎦ .

The resulting parameterized switched closed loop system dynamics can then be expressed as:

ˆ , ,

:1 if ,

2 if .

r r rex e

r rclr ex e

r rex e

r r

A E we C D w

e C D wr

e C D w

χ χχ

χ δχ δ

⎧ = +⎪

= +⎪∑ ⎨⎧ = + ≤⎪ = ⎨⎪ = + >⎩⎩

(2.10)

30

In the following, both the internal stability as well as the input-output stability properties of

the system (2.10) are analyzed.

2.3.3. Stability of the parameterized switched closed loop system

In this section, it is desired to study the stability properties of the system (2.10). The stability

analysis involves two steps, namely internal stability analysis and input-output stability

analysis. In the first step, the internal stability of the closed loop system is studied by

considering the system (2.10) in the absence of the signal rw , and studying the stability

properties of the origin for the resulting unforced switched system. The second step builds on

the internal stability results and presents input-output stability results for the system (2.10).

Consider first the system (2.10) in the absence of the signal rw . The state equation for the

resulting system is given by:

ˆrAχ χ= . (2.11)

Note that there are no impulsive changes in the state variables of the above switched system

at the switching times. The internal stability of the system (2.11) is then given by the

following well known result.

Lemma 2.1 [48]

If there exists a matrix 0TP P= > and a constant α such that the following inequalities hold:

{ }ˆ ˆ 2 0, 0, 1,2 ,Tr rA P PA P rα α+ + < > ∈ (2.12)

then the origin is an exponentially stable equilibrium point for the switched system (3.8) with

arbitrary switching, and:

( ) (0)tt ce αχ χ−< , (2.13)

31

where max

min

( )( )PcP

λλ

= .

Remark 2.2

The matrix inequality (2.12) can be solved using numerical algorithms. Here, we search for a

common Lyapunov function for the switched systems (2.11) to guarantee the stability of the

origin for the system (2.11) under arbitrary switching, since the switching surface

{ } : n e er rS x e C x D w δ= ∈ = + =R is not a fixed function of the state vector x and changes

with er rD w .

In the following, we will consider the input-output stability properties of the system (2.10)

with the external input signal rw . Let { }m 1,20

max ( )r rrt

w E w t∈≥

= . Then we have the following

stability result.

Lemma 2.2

Assume the origin for the switched system (2.11) is an exponentially stable equilibrium point.

Then the state vector in (2.10) is bounded, and the states will ultimately evolve inside the

bounded set Bβ given by:

{ }: ,NBβ χ χ β= ∈ ≤R (2.14)

where mcwβ εα

= + , and where ε is an arbitrary small positive number, i.e. 0 1ε< << .

32

Proof: Let 0( , )t tφ be the state transition matrix of the switched system (2.11) [49]. Without

loss of generality, let 0 0t = . The state trajectory of the system (2.10) is given by:

0( ) ( ,0) (0) ( , ) ( )

t

r rt t t E w dχ φ χ φ τ τ τ= + ∫ .

Based on (2.13), we have

( ,0) (0) (0) tt c e αφ χ χ −≤ .

It follows that:

0

( )m0

m

( ) ( ,0) (0) ( , ) ( )

(0)

1 (0) .

t

r r

tt t

tt

t t t E w d

c e ce w d

ec e cw

α α τ

αα

χ φ χ φ τ τ τ

χ τ

χα

− − −

−−

≤ +

≤ +

−≤ +

∫

∫

Hence, ( )tχ is ultimately bounded and we have:

mlim ( )t

cwtχ β εα→∞

≤ = − . (2.15)

Therefore, after a long enough finite time, the state trajectory will enter into the bounded set

Bβ given by (2.14) and will continue to evolve in Bβ thereafter.

The input-output stability of the closed loop system follows immediately from the above

result.

2.4. Regulation conditions for the switched system

The purpose of introducing the parameterized controllers is to find, for each { }1, 2r∈ ,

appropriate rQA ,

rQB and rQC in (2.7) to solve the output feedback regulation problem for the

33

switched system (2.10). The solution to the regulation problem for the switched system (2.10)

is presented in two steps. In the first step, regulation conditions for each of the individual

closed loop systems clrΣ , { }1, 2r∈ , are presented. In the second step, two sufficient

conditions for regulation in the switched closed loop systems clrΣ are derived. To derive the

sufficient conditions for regulation, a coordinate transformation is defined first, allowing the

forced switched closed loop system to be transformed into an unforced impulsive switched

system. Hence, the regulation problem is transformed into a stability analysis problem for the

origin of the resulting impulsive switched system. Conditions for achieving asymptotic

stability in the new impulsive switched system are then presented, which is equivalent to

achieving regulation in the original switched closed loop system.

2.4.1. Regulation conditions for 1clΣ and 2

clΣ

Let c

r r rr c y c

r r r

A B CA

B C A⎡ ⎤

= ⎢ ⎥⎣ ⎦

and xr

r c yr r

DE

B D⎡ ⎤

= ⎢ ⎥⎣ ⎦

in (2.4). For each of the systems rΣ , { }1, 2r∈ ,

and using a controller of the form (2.3), regulation conditions are given by the following

lemma.

Lemma 2.3 [52]:

For each { }1, 2r∈ , consider the system rΣ in (2.1) and a controller r in (2.3), and assume

that rA is a stability matrix. Then, for each { }1, 2r∈ , clrΣ in (2.4) achieves regulation if and

only if there exists a matrix rΠ that solves the linear matrix equations:

,

0 .r r r r r

ex er r

H A E

C D

Π = Π +

= Π + (2.16)

34

Similarly, using the parameterized feedback regulator r given in (2.3) and (2.8), regulation

conditions for each of the closed loop systems clrΣ , { }1, 2r∈ , in (2.10) can be derived and are

presented in the following Theorem.

Theorem 2.1

Assume that for each { }1, 2r∈ , the closed loop system clrΣ given in (2.10) is exponentially

stable. Then each of the systems 1clΣ and 2

clΣ achieves regulation only if, for each { }1, 2r∈ ,

there exists a pair of matrices ( ),r rΩ Ψ which satisfy the following equations:

,

0 .

xr r r r r r r

e er r

H A B D

C D

Ω = Ω + Ψ +

= Ω + (2.17)

Furthermore, if the linear matrix equations in (2.17) admit a solution ( ),r rΩ Ψ , and the

matrices rQA and

rQC in (2.6) are taken to be of the form

,

,r r

r

yQ r Q r

Q r r r

A H B D

C K

= +

= Ψ − Ω (2.18)

then, for each { }1, 2r∈ , clrΣ in (2.10) achieves regulation.

Proof: Equation (2.16) can be rewritten as

,

0 0 .

c xr r r r

r r rc y c c yr r r r r

e er r

A B C DH

B C A B D

C D

⎡ ⎤ ⎡ ⎤Π = Π +⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦⎡ ⎤= Π +⎣ ⎦

Partition rΠ as rr

r

Ω⎡ ⎤Π = ⎢ ⎥Φ⎣ ⎦

. Then (2.16) is the same as

35

( ),

,

0 .

c xr r r r r r r r

c c y yr r r r r r r r

e er r

H A B C D

H A B C D

C D

Ω = Ω + Φ +

Φ = Φ + Ω +

= Ω +

(2.19)

Letting cr r rCΨ = Φ , equation (2.17) follows immediately.

To prove the second part of the theorem, note that by assumption, for each { }1, 2r∈ , ˆrA

in (2.10) is a stability matrix. Moreover, since rA is related to ˆrA by a similarity

transformation, then rA is also a stability matrix. Now suppose rΩ and rΨ satisfy (2.17)

and that rQA and

rQC satisfy (2.18). To show that the closed loop system achieves regulation,

let rr I

Ω⎡ ⎤Φ = ⎢ ⎥

⎣ ⎦. Using the expression for c

rA in (2.8) and substituting in the expression for

rQA in (2.18) yields:

,

.00

r

r r

rr

y y xr r r r r r Q r r rc

r y yQ r r Q r

xrr y yr r

r Q r rQr

A B K L C B C L D DA

B C H B D

LBA DK C C D

BH

⎡ ⎤+ + + += ⎢ ⎥+⎢ ⎥⎣ ⎦

⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤= + + ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎣ ⎦⎣ ⎦⎣ ⎦ ⎣ ⎦

It follows that:

( )

( ) ( )

,0

.

r

r

r

xrr r r r Qc y yr r r

r r r r rQr

xr r r r r Q r c y y

r r r r

r

LB K B CA DA C D

BH

A B K C DB C D

H

Ω + ⎡ ⎤⎡ ⎤Ω + ⎡ ⎤Φ = + + Ω +⎢ ⎥⎢ ⎥ ⎢ ⎥

⎣ ⎦⎣ ⎦ ⎣ ⎦⎡ ⎤Ω + Ω + +

= − Ω +⎢ ⎥⎢ ⎥⎣ ⎦

Given that rr Q r rC KΨ = + Ω from (2.18), then based on (2.17) we have:

( )

( )

,

.

xc c y yr r r r rr r r r r r

r

c y yr r r r r r

A B DA B C D

H

H B C D

⎡ ⎤Ω + Ψ +Φ = − Ω +⎢ ⎥

⎣ ⎦

= Φ − Ω +

36

Therefore, equation (2.19) is satisfied, which implies (2.16) is satisfied and regulation can be

achieved.

The results presented above represent regulation conditions for each of the closed loop

systems in (2.10), individually. In the following sections, regulation conditions for the system

(2.10) subject to switching are discussed.

2.4.2. Equivalent impulsive switched closed loop system model

Based on Lemma 2.3, and using a properly defined coordinate transformation, the forced

switched closed loop system (2.10) can be transformed into an unforced impulsive switched

system. To introduce the appropriate coordinate transformation, note that the original

regulation condition (2.16) can be rewritten as:

ˆˆ ˆ ,ˆ 0 ,

r r r r r

ex er r

H A E

C D

Π = Π +

= Π + (2.20)

where

0 0ˆ 0

0 0r r

II I

I

⎡ ⎤⎢ ⎥Π = − Π⎢ ⎥⎢ ⎥⎣ ⎦

. (2.21)

Consider now the following coordinate transformation:

ˆ .r rwχ χ= −Π (2.22)

Using (2.20), the forced closed loop switched system (2.10) can be transformed into an

unforced switched system given by:

37

ˆ ,,

:1 if ,2 if .

rex

clr ex

ex

Ae C

e Cr

e C

χ χχ

χ δχ δ

⎧ =⎪

=⎪∑ ⎨⎧ = ≤⎪ = ⎨⎪ = >⎩⎩

(2.23)

It should be noted from (2.22) that the coordinate transformation varies depending on the

value of { }1,2r∈ . Consequently, the states ( )tχ in the new system (2.23) undergo impulsive

changes at the switching times. Let sT be the set of switching times kτ , 1, 2,k = … .

Therefore, for a given switching time k sτ ∈T , we have that for kt τ= :

( ) ( ) ( )t t tχ χ χ+ −= + Δ ,

where ( ) ( ) ( ) ( )

ˆ ˆ( ) ( ) ( )r t r t r t r t

t w t w tχ − − + +Δ = Π −Π . Hence, the impulsive switched closed loop

system is given by:

ˆ ,

( ) ( ) ( ), , : ,

1 if ,

2 if .

r

scl exr

ex

ex

A

t t t te C

e Cr

e C

χ χ

χ χ χχ

χ δχ δ

+ −

⎧ =⎪

= + Δ ∈⎪⎪∑ =⎨⎪ ⎧ = ≤⎪ = ⎨⎪ = >⎩⎩

T (2.24)

In the new coordinate system, the switching surface S given in (2.2) is fixed and is

expressed as follows:

{ }: .N exS e Cχ χ δ= ∈ = =R (2.25)

Using the coordinate transformation in (2.22), the original output feedback regulation

problem for the switched system (2.10) is transformed into an asymptotic stability analysis

problem for the origin of the unforced impulsive switched system (2.24).

38

2.4.3. Regulation conditions for the switched system

First, let 0δ denote the distance between the switching surface S and the origin. Then we

have 0 exCδ δ= . Define:

{ }m 1,20

ˆmax ( )r rrt

w tβ∈≥

Δ = Π . (2.26)

Let mγ β β ε= + Δ + , where 1ε , and define the set Bγ as follows:

{ }:NBγ χ χ γ= ∈ ≤R . (2.27)

Using the definition for χ in (2.22), and the fact that the original state vector χ will

ultimately evolve inside a bounded set Bβ given by (2.14), it follows that there exists a finite

time Tγ at which χ will enter the set Bγ and continue to evolve in Bγ thereafter. A

sufficient condition for regulation is then given as follows:

Theorem 2.2

Assume that the switched closed loop system clrΣ given in (2.10) is internally stable under

arbitrary switching, and that for each { }1, 2r∈ , clrΣ given in (2.10) achieves output

regulation. If

0 ,δ γ> (2.28)

then the origin is an asymptotically stable equilibrium point for the impulsive switched

system (2.24), implying that the switched system (2.10) achieves regulation.

Proof: If 0δ γ> , then the switching surface S given by (2.25) does not intersect the bounded

set Bγ . The trajectory of the state vector χ will enter the set Bγ at t Tγ= and continue to

39

evolve inside Bγ thereafter. Consequently, there will be no more switching. Since the system

clrΣ that is active in the half-space containing Bγ is such that ˆ

rA is a stability matrix, and

since 0 Bγ∈ , the trajectory of the state vector χ asymptotically converges to the origin,

implying that regulation in the original switched closed loop system (2.10) is achieved.

If condition (2.28) is not satisfied, i.e. 0δ γ≤ , then the switching surface S must intersect

with the bounded set Bγ . To present regulation conditions in this case, define the matrices

11

ˆ

ex

s ex

CA

C A

⎡ ⎤= ⎢ ⎥⎢ ⎥⎣ ⎦

, 22

ˆ

ex

s ex

CA

C A

⎡ ⎤= ⎢ ⎥⎢ ⎥⎣ ⎦

and 0

bδ⎡ ⎤

= ⎢ ⎥⎣ ⎦

. Sufficient conditions for regulation are presented

in the following theorem.

Theorem 2.3

Assume the switched closed loop system clrΣ given in (2.10) is internally stable under

arbitrary switching and that, for each { }1, 2r∈ , clrΣ given in (2.10) achieves output

regulation. Moreover, assume that 0δ γ≤ , 1 2( ) ( ) 2s srank A rank A= = , and that the direction

of vector exC is not parallel to that of 1êxC A and 2

êxC A . If the following conditions are

satisfied,

( )( ) 1 2

ˆ 0,

,

1 if 0,2 if 0,

Tex exr

T Tsr sr

C C A

b A A b

rr

γ

δδ

−

<

>

= >⎧⎨ = <⎩

(2.29)

or

40

( )( ) 1 2

ˆ 0,

,

2 if 0,1 if 0,

Tex ex

r

T Tsr sr

C C A

b A A b

rr

γ

δδ

−

>

>

= >⎧⎨ = <⎩

(2.30)

then the origin is an asymptotically stable equilibrium point for the impulsive switched

system (2.24), implying that the switched system (2.10) achieves regulation.

Proof: The following proof is presented for the case of 0δ > . The case of 0δ < is treated

using the same ideas. Let region 1 and region 2 denote the half-spaces where the systems 1clΣ

and 2clΣ are active, respectively (See Figures 2.2 and 2.3). Hence, the origin is in region 1

where the system 1clΣ is active. Define the hypersurfaces { }1 1

ˆ: 0N exvS C Aχ χ= ∈ =R and

{ }2 2ˆ: 0N ex

vS C Aχ χ= ∈ =R . Then for all points in 1vS , 1ˆ 0exe C A χ= = and for all points in

2vS , 2ˆ 0exe C A χ= = . Since, by assumption, 0δ γ≤ , the switching surface S intersects with

the bounded set Bγ . Let { }: and N exS Cγ χ χ δ χ γ= ∈ = ≤R be the intersection of the

switching surface S with Bγ . By assumption, the direction of vector exC is not parallel to

that of 1êxC A and 2

êxC A . Therefore, the switching surface S must intersect with either of the

hypersurfaces 1vS or 2vS . Define the intersection of S with the hypersurfaces 1vS and 2vS as

{ }1 1ˆ: 0 and N ex exL C A Cχ χ χ δ= ∈ = =R

and

{ }2 2ˆ: 0 and N ex exL C A Cχ χ χ δ= ∈ = =R .

41

For each { }1, 2r∈ , let minr

r Ll

χχ

∈= denote the distance between the origin and the set rL .

Then, rl can be obtained by solving the problem of minimizing χ subject to the constraint

srA bχ = . Since, by assumption, 1 2( ) ( ) 2s srank A rank A= = , the solution to the above

problem is ( ) 1T Tr sr srl b A A b

−= , { }1, 2r∈ . If ( ) 12 2T T

r sr srl b A A b γ−

= > , then the sets rL ,

{ }1, 2r∈ , are outside the bounded set Bγ . In this situation, the set Sγ does not intersect with

the sets 1vS or 2vS . If ( )1ˆ 0

Tex exC C A < and ( ) 12 2

1 1 1T T

s sl b A A b γ−

= > , then the set Sγ will be

located in one of the two half-spaces defined by the hypersurface 1vS and where all the points

satisfy 1ˆ 0exe C A χ= < . Similarly, if ( )2

ˆ 0T

ex exC C A > and ( ) 12 22 2 2

T Ts sl b A A b γ

−= > , then the

set Sγ will be located in one of the two half-spaces defined by the hypersurface 2vS and

where all the points satisfy 2ˆ 0exe C A χ= > . By deriving conditions on e , and using the

stability properties of 1clΣ and 2

clΣ , it is possible to conclude about the asymptotic stability of

the origin of the impulsive switched closed loop system (2.24). In the following, two cases

are considered in detail.

• Case 1: ( )1ˆ 0

Tex exC C A < and ( ) 1 2

1 1T T

s sb A A b γ−

> (See Figure 2.2). In this case, all the

state trajectories leaving the set Sγ and entering region 1 have 0e < . Therefore, if the

state trajectory leaves the set Sγ and enters region 1, then it will not hit it again. Two

possible cases can be considered here depending on whether the state trajectory enters the

bounded set Bγ from region 1 or region 2 at time Tγ . In the case the state trajectory enters

the set Bγ from region 1, then the state trajectory cannot hit the switching surface S since

42

0e < for all states on the switching surface within the set Bγ . Therefore, the state

trajectory is confined to evolve in region 1 inside the set Bγ . Since 1clΣ is a stable system,

the state trajectory will converge to the origin asymptotically, which implies that

regulation for the switched system (2.10) is achieved. Consider now the case where the

state trajectory enters the set Bγ from region 2. Since the state trajectory is confined to

evolve inside the set Bγ and since the system 2clΣ is asymptotically stable, the state

trajectory must hit the switching surface and switching must take place. Following

switching, and based on the dynamics of the impulsive switched closed system as

presented in (2.24), the state vector immediately following switching will be in region 1.

Since 0e < for all states on the switching surface within the set Bγ for the system 1clΣ , the

state trajectory cannot cross the switching surface again and will continue to evolve in

region 1 thereafter. The state trajectory will approach the origin asymptotically, which

implies that regulation will be achieved.

• Case 2: ( )2ˆ 0

Tex exC C A > and ( ) 1 2

2 2T T

s sb A A b γ−

> (See Figure 2.3). In this case, all the

state trajectories leaving the set Sγ and entering region 2 have 0e > . Assume the state

trajectory enters the set Bγ from region 2 at the time Tγ . Given that the system 2clΣ is

asymptotically stable, and that the trajectories are confined to evolve inside Bγ , the state

trajectory must cross the switching surface to approach the origin. However, this is not

possible since, with respect to the system 2clΣ , 0e > for all Sγχ ∈ . Therefore, it is not

possible to have the state trajectory enter the set Bγ from region 2 at time Tγ . Consider

now the case where the state trajectory enters the set Bγ from region 1. If the state

43

trajectory crosses the switching surface to enter into region 2, then that will result in a

contradiction similar to that discussed for the case where the state trajectory enters the set

Bγ from region 2. Therefore, once the state trajectory enters the set Bγ from region 1, it

will continue to evolve in region 1 forever and will never cross the switching surface to

enter into region 2 for t Tγ> . Since 1clΣ is a stable system, then the state trajectory will

converge to the origin asymptotically, which implies that regulation for the switched

system (2.10) is achieved.

Therefore, based on the above analysis, if the conditions for Case 1 given by (2.29) or the

conditions for Case 2 given by (2.30) are satisfied, then regulation in the switched system

(2.10) can be achieved.

2.5. Regulator synthesis for the switched system

Based on the regulation conditions for the switched system proposed in the previous section,

a regulator synthesis approach is presented in this section. The proposed synthesis approach

is based on solving a set of properly formulated bilinear matrix inequalities. The main idea

behind the regulator synthesis approach is as follows. Consider an output feedback controller

r as given in (2.3) and (2.8), where r r

yQ r Q rA H B D= + and

rQ r r rC K= Ψ − Ω . Since

( ),r rΩ Ψ is the solution to the Sylvester equation (2.17), r is only parameterized in the

unknown matrices rK , rL and rQB . Assume r is such that the resulting closed system

satisfies (2.12). Then, based on Lemma 2.1 and Theorem 2.1, the switched closed loop

system clrΣ given in (2.10) is internally stable under arbitrary switching and, for each

{ }1, 2r∈ , clrΣ given in (2.10) also achieves output regulation. In this case, if any of (2.28),

44

Figure 2.2. State trajectories of 1clΣ with initial conditions on the switching surface inside the

set Bγ for case 1.


set Bγ for case 2.

1vS

Bounded set Bγ

1êxC A

Region 2: 2clΣ

Region 1: 1clΣ

S

S

2vS

exC

Region 2: 2clΣ

Region 1: 1clΣ

Bounded set Bγ

2êxC A

exC

45

(2.29) and (2.30) is satisfied, then based on Theorems 2.2 and 2.3, regulation in the switched

system (2.10) is achieved. The set of conditions in Lemma 2.1 and Theorems 2.2 and 2.3

yield bilinear matrix inequalities in the unknown parameters rK , rL and rQB . In the

following, a solution procedure for the formulated bilinear matrix inequalities is proposed to

determine the unknown parameters rK , rL and rQB in r .

The sufficient regulation condition (2.28) given in Theorem 2.2 is equivalent to:

0 mβ δ β ε< −Δ − . (2.31)

Using mcwβα

= and letting 0

m

m

wδ β εκ −Δ −

= , condition (2.28) is equivalent to:

c κα< . (2.32)

If (2.32) cannot be satisfied, the switching surface S intersects the set Bγ and conditions

(2.29) and (2.30) in Theorem 2.3 need to be verified. For the switched closed loop system

(2.10), the parameters c and α can be found using the following matrix inequalities [94]:

{ }2

ˆ ˆ 2 0, 0, 1, 2 ,

, 0.

Tr rA P PA P r

I P c I c

α α+ + < > ∈

< < > (2.33)

If the matrices rK , rL , rQB and

rQC in ˆrA are unknown, then the above equations define a

bilinear matrix inequality. Combining (2.32) and (2.33), the synthesis procedure for the

controller r can be realized by solving the following BMIs:

0 00 0

r r r r

T

r r r r r r r r r r r r r r r r r r r ry x y y x y

r r r r r r r r r r r ry y y y

Q r r Q r Q r r Q r

A B K B K B B K A B K B K B B KA L C D L D P P A L C D L D

B C H B D B C H B D

⎡ ⎤ ⎡ ⎤+ Ψ − Ω + Ψ − Ω⎢ ⎥ ⎢ ⎥+ + + + +⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥+ +⎣ ⎦ ⎣ ⎦

{ } 2 0, 0, 1,2 ,P rα α+ < > ∈

(2.34)

46

2 , 0,I P c I c< < > (2.35)

,c κα< (2.36)

where the unknown parameters are rK , rL , rQB , P , α and c . Once rK , rL ,

rQB are

determined based on the above BMIs, the matrices rQA and

rQC can be calculated using (2.18)

as given in Theorem 2.1.

In this chapter, a regulator synthesis algorithm, referred to as the P - iteration algorithm,

will be used to find a parameterized regulator that satisfies either one of the regulation

conditions (2.28), (2.29), or (2.30). The P - algorithm iteratively solves for the Lyapunov

matrix P and the parameters in the controller r given in (2.3). The basic idea of the P -

iteration is that a BMI can be converted into an LMI when some of the parameters in the

BMI are fixed. The approach for solving BMI problems is to alternate between two

optimization problems subject to LMIs, which are related to the matrix P in (2.34) and (2.35)

and the parameters in the controller r in (2.3), respectively. In the algorithm, the input data

is represented by rA , rB , yrC , eC , x

rD , yrD , e

rD , rH , δ and mw , whereas the unknown

variables to be determined are rK , rL , rQB ,

rQA , rQC P , α and c . The algorithm is

summarized below, where irK , i

rL , r

iQB , iP , iα and ic denote the solutions rK , rL ,

rQB , P ,

α and c obtained at the thi iteration of the algorithm.

47

1) Calculating κ : Determine ( ),r rΩ Ψ by solving the Sylvester equation (2.17) and let

rr

r

Ω⎡ ⎤Π = ⎢ ⎥Φ⎣ ⎦

with rr I

Ω⎡ ⎤Φ = ⎢ ⎥

⎣ ⎦. Let ˆ

rΠ be as in (2.21). Determine mβΔ using (2.26),

then calculate 0 m

mwδ β εκ −Δ −

= .

2) Initializing r : Initialize the controller parameters 0rK , 0

rL and 0rQB to make the

switched closed loop system (2.10) internally stable, which can be realized by solving

the following two LMIs separately for the unknown matrices rK , rL , rQB , KP and LP

with preset constants 0Kα ≥ and 0Lα ≥ .

{ }[ ] [ ] 0, 1, 2 ,Tr r r K K r r r K KA B K P P A B K P rα+ + + + < ∈ (2.37)

{ }0, 1, 2 .r r r r

Ty x y y x yr r r r r r r r r r r r

L L L Ly y y yQ r r Q r Q r r Q r

A L C D L D A L C D L DP P P r

B C H B D B C H B Dα

⎡ ⎤ ⎡ ⎤+ + + ++ + < ∈⎢ ⎥ ⎢ ⎥+ +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

(2.38)

3) Initializing 0α : Based on the initial controller parameters 0rK , 0

rL and 0rQB from step

2, initialize the maximum decay rate 0α by solving the following optimization problem:

max α

subject to (2.34)

4) P -Step: At the thi iteration, 1,2,i = , given the matrices 1irK − , 1i

rL − and 1r

iQB − and

the scalar 1iα − , solve the following optimization problem for iP and ic :

min c

subject to (2.34) and (2.35)

5) -Step: At the thi iteration, 1,2,i = , given the matrix iP and ic from step 4, solve

the following optimization problem for irK , i

rL , r

iQB and iα :

48

max α

subject to (2.34) and (2.35)

6) Verification: Verify the constraints (2.36), (2.29) and (2.30), and if any of the three

conditions is satisfied, compute rQA and

rQC using (2.18), then stop the algorithm. If

none of (2.36), (2.29) and (2.30) is satisfied, then go to step 4.

The iterative loop is repeatedly executed until a solution is found, or there is no major

reduction in /c α relative to the previous iteration. The algorithm converges to a local

solution of /c α , since at each thi iteration of the algorithm, we have 1i ic c −≤ and 1i iα α −≥ .

Therefore, if no solution can satisfy the regulation conditions (2.36), (2.29) or (2.30) by the

iterative procedure described above, then the initial parameters 0rK , 0

rL and 0rQB can be

adjusted by changing the decay rates Kα and Lα in (2.37) and (2.38), and restarting the

iterative procedure again. If the above algorithm yields a solution, then, according to

Theorems 2.2 or 2.3, the switched closed loop system will achieve regulation.

2.6. Numerical example

In this section, the regulator synthesis method proposed in this chapter will be used to design

a controller that cancels the contact vibrations in a mechanical system. Figure 2.4 shows the

diagram of such a system consisting of a mass m , a contact surface cS , and their respective

coordinates. The mass m is attached to a spring with stiffness k and a damper with damping

coefficient c . The mass m moves only in the vertical direction whereas the contact surface

underneath it moves to the left. The mass m may enter into intermittent contact with the

49

Figure 2.4. Diagram of a mechanical system with switched dynamics.

surface cS , resulting in contact vibrations. When the mass m enters into contact with the

surface cS , the contact characteristics are represented by a spring with stiffness ck and a

damper with damping coefficient cc . The force F represents the external force used to

control the mass m whereas aF represents a disturbance force. This model can be found in

many applications, such as the interface between the read/write head and the disk surface in

hard disk drive systems. In the following, and with respect to the system shown in Figure 2.4,

the control objective is for the mass m to follow the displacement of the contact surface cS

while maintaining a desired constant separation in the vertical direction. Let v be the

deviation of the mass from its equilibrium position. Then the equations of motion of the mass

can be written as:

( )( )

when noncontact mode

when contact modea s

a c s

mv cv kv F F v v

mv cv kv F F F v v

+ + = − + >

+ + = − + + ≤

where cF is the contact force expressed as:

ck

Contact surface cSmaFv

ev sv

ckcc

F

Direction of surface motion

50

( ) ( )c c s c sF c v v k v v= − + − ,

and where sv is the displacement of the contact surface. The wavy contact surface profile is

expressed as a linear combination of sinusoidal functions:

( )coss k k k ek

v c t vω φ= + +∑ , (2.39)

with amplitudes kc , frequencies kω , phases kφ , 1, 2,k = … , and constant offset ev . Therefore,

the distance between the mass m and contact surface cS is sv v− . The control signal is

defined as /u F m= − for the non-contact situation and ( ) /c eu F k v m= − − when contact

takes place. Let ev− denote the desired distance between the mass m and contact surface cS .

The output y , to be fed to the controller, is defined as

y v= .

The performance variable e is defined to be the difference between the actual distance sv v−

and the desired distance ev− :

s ee v v v= − + .

Therefore, the system will switch between the contact and non-contact modes according to

the value of the performance variable e . If ee vδ> = , the system will operate in the non-

contact mode, and if ee vδ≤ = , the system will operate in the contact mode. Let 1x v= and

2x v= , then the switched system model is given by:

51

[ ]

[ ] [ ]

1 1

2 2

1

2

1

2

0 0 01 1 0

1 0:

1 0 0 1

1 if ,2 if ,

r r

r

r

e

e

x xA u d

x x

xy

x

xe d

x

e vr

e v

⎧⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤= + +⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦⎪⎪ ⎡ ⎤⎪ = ⎢ ⎥⎪ ⎣ ⎦∑ ⎨

⎡ ⎤⎪ = + −⎢ ⎥⎪ ⎣ ⎦⎪⎪ ≤⎧

= ⎨⎪ >⎩⎩

(2.40)

where:

1

0 1( ) ( )c c

A k k c cm m

⎡ ⎤⎢ ⎥= + +⎢ ⎥− −⎣ ⎦

, ( )1

1c s c s c e a

s e

c v k v k v Fd m

v v

⎡ ⎤+ − +⎢ ⎥=⎢ ⎥

−⎢ ⎥⎣ ⎦

2

0 1A k c

m m

⎡ ⎤⎢ ⎥=⎢ ⎥− −⎣ ⎦

, 2

/a

s e

F md

v v⎡ ⎤

= ⎢ ⎥−⎣ ⎦.

The surface profile is given by ( )( )6cos 20 10 ms ev t vπ −= × + , and the force

( ) 51.5sin(20 ) 1.5cos(20 ) 10 NaF t tπ π −= + × . Let m , k , c , ck , cc and ev be 200mg, 2 N/m,

21 10−× N/m/Sec, 20 N/m, and 31 10−× N/m/Sec, and 630 10 m−− × , respectively. In the

following, a regulator for the switched system is designed using the synthesis procedure

described in the previous section. Simulation results will illustrate the performance of the

proposed regulator in maintaining the desired system output despite the presence of

switching.

In order to find a feasible solution for the BMIs, we introduce a coordinate transformation

as 1 1

2 2

x xT

x x′⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥′⎣ ⎦ ⎣ ⎦

, where 5

1 00 1 10

T −

⎡ ⎤= ⎢ ⎥×⎣ ⎦

. The motivation of this coordinate transformation

52

is to make the weight of state 2x less than 1x in χ since the performance variable e is only

defined based on the state 1x . Rewriting (2.40) in the state space form (2.1) results in:

5

10 1 101.1 10

A⎡ ⎤×

= ⎢ ⎥− −⎣ ⎦,

5

20 1 100.1 5

A⎡ ⎤×

= ⎢ ⎥− −⎣ ⎦,

1 2

0 2020 0

H Hπ

π⎡ ⎤

= = ⎢ ⎥−⎣ ⎦, 10 20 6

01 10

w w −

⎡ ⎤= = ⎢ ⎥×⎣ ⎦

,

1 2 5

01 10

B B −

⎡ ⎤= = ⎢ ⎥×⎣ ⎦

,

1

0 00.7469 1.75

xD⎡ ⎤

= ⎢ ⎥⎣ ⎦

, 2

0 00.75 0.75

xD⎡ ⎤

= ⎢ ⎥⎣ ⎦

,

[ ]1 2 1 0y y eC C C= = = , 1 2 0y yD D= = , [ ]1 2 0 1e eD D= = − .

It is desired to design a regulator that can reject the disturbance in the switched system.

Based on Theorem 2.1, solving the Sylvester equation (2.17) yields:

[ ]1 75314.1592653590 68947.8417604358Ψ = − − ,

[ ]2 75314.1592653590 68947.8417604357Ψ = − − ,

1 2 4

0 16.2831 10 0−

⎡ ⎤Ω = Ω = ⎢ ⎥×⎣ ⎦

, 1

1 2 1

2 2I ×

Ω⎡ ⎤⎢ ⎥Π = Π = Ω⎢ ⎥⎢ ⎥⎣ ⎦

.

Based on (2.26), we obtain 62 10mβ−Δ = × . Since 630 10δ −= − × and 6

m 2.09 10w −= × , and

with 0.01ε = , we have 13.3950κ = in (2.32). Using the P - algorithm proposed above,

we obtain:

3 91 123.583428456762 10 8.54599379771142 10K ⎡ ⎤= − × − ×⎣ ⎦ ,

3 92 123.582172573746 10 8.54575846443053 10K ⎡ ⎤= − × − ×⎣ ⎦ ,

53

3

1 3

409.395433365613 10581.725609783952 10

L−⎡ ⎤− ×

= ⎢ ⎥− ×⎣ ⎦,

3

2 3

409.395415548061 10581.726610025785 10

L−⎡ ⎤− ×

= ⎢ ⎥− ×⎣ ⎦,

1

3128.47291192622610

194.074510473731QB−⎡ ⎤

= ×⎢ ⎥−⎣ ⎦,

2

3128.47291197947110

194.074510543232QB−⎡ ⎤

×⎢ ⎥−⎣ ⎦,

4.9728c = , 0.3808α = , 13.0601cα= .

Then, based on (2.18), we have

[ ]1

35444.92042576818 54.6355866963264 10QC = − × ,

[ ]2

35444.77256150691 54.6343308133107 10QC = − × .

Since 13.0601< 13.3950c κα= = , condition (2.36) is satisfied. Therefore, regulation can be

achieved in the switched closed loop system using the above designed controller.

The simulation results of the response of the closed loop system under switching are

illustrated in Figures 2.5 and 2.6. It can be seen that if the performance variable is smaller

than -30 micrometers, the mass enters into contact with the contact surface cS and the model

of the system switches. It can also be seen that the disturbance rd changes at the switching

times. But even in the presence of switching, the switched system performance variable e

still converges to zero, which means the mass m asymptotically follows the contact surface

cS at the desired separation 30 micrometers.

2.7. Conclusion

The problem of regulation in switched bimodal systems against known disturbance or

reference signals is discussed. A regulator design approach based on the parameterization of

54

a set of controllers that can achieve regulation for the switched system is presented. The

forced switched closed loop system involving the parameterized controller is first

transformed into an unforced impulsive switched system. Consequently, the original

regulation problem is transformed into a stability analysis problem for the origin of the

impulsive switched system. Then sufficient conditions for regulation in switched systems are

derived and a regulator synthesis method based on solving a set of bilinear matrix

inequalities is presented. A simulation example involving a mechanical system with switched

dynamics is used to illustrate the effectiveness of the proposed regulator designed method.

The parameterization of the regulators considered in this paper was performed without

imposing any stability constraints on the rQ parameters. Although this advantage offers more

flexibility in the design of the desired rQ parameters, the derived structure of the rQ

parameters needed for regulation is constrained in terms of the matrices used in the state

space representation of the rQ parameters. This structure is not desirable when considering

the adaptive regulation problem to be discussed later in the thesis, since it does not allow for

the rQ parameters to be suitably parameterized for the purpose of tuning them online. In the

following chapters, the rQ parameters are represented as linear combination of stable basis

functions, which offers significant advantages when considering the adaptive regulation

problem.

55

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-50

0

50

e (m

icro

met

ers)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-0.5

0

0.5

1

d r(1,1

) (m

/sec

2 )

Time(sec)

Figure 2.5. Simulation results for the case of 30ev = − micrometers showing the performance variable e and the switching component (1,1)rd in the disturbance rd .

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

-40

-30

-20

-10

0

10

20

30

40

v an

d v s (m

icro

met

ers)

Time(sec)

mass height vcontact surface vs

Figure 2.6. Simulation results for the case of 30ev = − micrometers showing the regulated mass height v and the contact surface displacement sv .

56

CHAPTER 3

Regulation in Switched Bimodal Systems

Based on Linear Matrix Inequalities

3.1. Introduction

In this chapter, a Q -parameterized regulator design approach is proposed for switched

bimodal systems where it is desired to reject known disturbance signals and track known

reference inputs simultaneously. The proposed method consists of three steps. First, a set of

Q -parameterized observer-based stabilizing controllers for the switched system is

constructed and the stability of the resulting closed loop switched system is analyzed. The Q

parameters are selected to be a linear combination of stable basis functions. Second, for each

subsystem in the bimodal system, a set of Q -parameterized regulators is derived. In the third

step, regulation conditions for the switched closed loop system are developed using the two

approaches summarized in the following.

• In the first approach, sufficient regulation conditions are derived based on the closed loop

system input-output properties using a common Lyapunov function. Then, using a set of

regulation conditions and appropriate coordinate transformations, the forced switched

57

closed loop system is transformed into an unforced impulsive switched system. Hence,

the regulation problem for the original switched closed loop system is transformed into a

stability analysis problem for the origin of an impulsive switched system. Sufficient

conditions for the stability of the origin of the impulsive switched system are presented,

which are equivalent to regulation in the original switched closed loop system. A

regulator synthesis method is consequently outlined based on solving properly formulated

linear matrix inequalities (LMIs).

• In the second approach, based on the properties of the time-varying switching surface, the

whole closed loop system state space is partitioned into three adjacent regions. Based on

the resulting state space partitions, a regulation condition for the switched closed loop

system is formulated using a multiple Lyapunov function approach, then with the

regulation interpolation conditions for each subsystem derived in the frequency domain,

the corresponding regulator synthesis method is proposed using an iterative LMI

algorithm.

The main reason for considering Q parameters that are linear combinations of stable basis

functions is to simplify the controller synthesis procedure and reduce the number of

unknowns to be determined in the desired Q parameters. Moreover, such representation is

convenient to work with in the adaptive regulation case, as outlined later in the thesis. With

respect to the two regulator synthesis approaches summarized above, the main reason for

considering the second approach involving multiple Lyapunov functions is to attempt to

reduce the conservatism associated with the first approach based on a common Lyapunov

function.

58

The rest of the chapter is organized as follows. Section 3.2 introduces the switched

bimodal system model and presents the general regulation problem for the switched system.

In section 3.3, the construction of a Q -parameterized set of stabilizing controllers for the

switched system is discussed and the stability properties of the resulting closed loop switched

system are analyzed. In section 3.4, regulation conditions and a regulator synthesis method

for the switched system are proposed using a common Lyapunov function approach. In

section 3.5, regulation conditions and a regulator synthesis method for the switched system

are proposed using a multiple Lyapunov function approach, followed by the conclusion in

section 3.6.

3.2. The regulation problem for switched bimodal systems

This section considers switched bimodal systems subject to external inputs representing

disturbance and/or reference signals. Both the system dynamics as well as the external input

signals are assumed to switch according to a switching law defined by a switching surface.

Consider the switched system given by the following state space representation:

0

0

, (0) , , (0) ,

, :

,

1 if ,

2 if ,

xr r r r

r r r r ry yr r r

r e er r

e er r

e er r

x A x B u D w x xw H w w w

y C x D we C x D w

e C x D wr

e C x D wδδ

= + + == =

= +∑

= +

⎧ = + ≤= ⎨

= + >⎩

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

(3.1)



to be measurable, hrw ∈R is the state vector of the exogenous systems generating the signal

59

rw , h hrH ×∈R have simple eigenvalues on the imaginary axis, { }1,2r∈ is the index of the

system rΣ under consideration at time t , and δ is a constant satisfying 0δ > . The

switching between the systems 1Σ and 2Σ is performed according to the value of the

performance variable e , and is determined based on the location of x with respect to a

switching surface S given by:

{ }: n e er rS x e C x D w δ= ∈ = + =R . (3.2)

In the following, it is assumed that for any given 0t ≥ , the system must operate in only one

of the two modes corresponding to { }1,2r∈ . For the switched system (3.1), it is desired to

construct an output feedback controller to regulate the performance variable e of the

switched system against the external input signal rw . Given the switching nature of the plant,

the output feedback controller is also chosen to be a switching feedback controller r ,

{ }1, 2r∈ , with a state space representation given by:

,:

,

c cc r c r

r c cr c r

x A x B y

u C x D y

⎧ = +⎪⎨

= +⎪⎩ (3.3)

where cc

nx ∈R , and where the switching among controllers is to obey the same rule given in

(3.1) and decided by the performance variable e as for switching between the two plant

models. In the following section, a framework of designing an output feedback regulator for

the switched system (3.1) is presented, within which regulation conditions and regulator

synthesis procedure are proposed.

60

3.3 Parameterization of a set of stabilizing controllers

The controller design approach presented in this chapter relies on the construction of a Q -

parameterized set of output feedback stabilizing controllers for the switched system. In this

section, the construction of sets of Q -parameterized output feedback stabilizing controllers

for the switched system (3.1) is first discussed. The stability of the resulting switched closed

loop system is then analyzed. As in the case of linear time invariant systems, the construction

of a Q -parameterized set of stabilizing controllers involves two steps. The first step involves

the construction of a central controller in the form of a switched observer-based state

feedback controller. The second step involves augmenting the central controller with stable

dynamics to construct a Q -parameterized set of stabilizing controllers.

3.3.1. Q - parameterized controller

Consider the following observer-based state feedback controller for the switched system (3.1):

( ) 0ˆ ˆ ˆ ˆ ˆ, (0) ,:

ˆ,o r r rr

r

x A x B u L y y x xu K x

⎧ = + + − =⎪⎨

=⎪⎩ (3.4)

where x is an estimate of the plant state vector x and ˆ ˆry C x= , { }1, 2r∈ is an estimate of

the plant output y . The state feedback gains rK , { }1, 2r∈ , and the observer gains rL ,

{ }1, 2r∈ , are assumed to switch according to the rule given in (3.1). Moreover, it is assumed

that there are no impulsive changes in the controller states at the switching times. The

construction of a Q -parameterized set of stabilizing controllers for the switched system

(3.1) proceeds along the same lines as in the case of linear time-invariant systems [91-93].

Each controller is expressed as a linear fractional transformation involving a fixed system rJ

61

and a proper asymptotically stable system rQ that could be chosen as desired (See Figure

2.1). The state space representation of the system rJ is given by:

0ˆ ˆ ˆ ˆ( ) , (0) ,ˆ: ,

ˆ ˆ.

r r r r r r r Q

r r Q

r

x A L C B K x L y B y x x

J u K x y

y y y C x

⎧ = + + − + =⎪⎪ = +⎨⎪ − = −⎪⎩

(3.5)

The rQ parameter is given by:

( ) 0ˆ , (0) ,:

,r

Q Q Q Q Q Qr

Q Q Q

x A x B y y x xQ

y C x

⎧ = + − =⎪⎨

=⎪⎩ (3.6)

where qnQx ∈ . In particular, throughout the rest of the paper, the rQ parameter is such that

QA is a fixed stability matrix, QB is a fixed matrix, and the matrix rQC changes with

{ }1, 2r∈ .

Remark 3.1

For a given { }1,2r∈ , the controller formulated in (3.5) and (3.6) is a parameterized

stabilizing controller for the system rΣ , where the system rQ represents a free design

parameter that can be chosen as desired as long as it is an asymptotically stable system. A

suitable choice for rQ can be a parameterization of the form 1

( ) ( )qn

ir r i

i

Q s sθ ψ=

= ∑ , where

( )i sψ are stable basis functions. Such representation can be used to approximate any proper

stable real rational transfer function. With such representation for rQ , a corresponding

realization is as given in (3.6), where QA and QB are fixed matrices, and where the

62

parameters , 1ir qi nθ = … , appear in

rQC . Here the rQC matrices are selected to achieve

regulation according to the regulation conditions to be discussed in sections 3.4 and 3.5.

Let ˆx x x= − denote the state estimation error and 1

TT T TQ N

x x xχ×

⎡ ⎤= ⎣ ⎦ denote the state

vector for the resulting closed loop system with 2 qN n n= + . The resulting closed loop

system is given by the following state space representation:

, ,

:1 if ,2 if ,

r r rex e

r rclr ex e

r rex e

r r

A E we C D w

e C D wr

e C D w

χ χχ

χ δχ δ

⎧ = +⎪ = +⎪∑ ⎨

⎧ = + ≤⎪ = ⎨⎪ = + >⎩⎩

(3.7)

where 00 0

rr r r r Q r r

r Q Q r

r r r

A B K B C B K

A A B CA L C

+⎡ ⎤⎢ ⎥

= −⎢ ⎥⎢ ⎥+⎣ ⎦

,

( ) ( ) ( )TTTx y T x y

r r Q r r r rE D B D D L D⎡ ⎤= − +⎢ ⎥⎣ ⎦, 0 0ex eC C⎡ ⎤= ⎣ ⎦ .

3.3.2. Internal stability of the Q -parameterized switched closed loop system

In the following, it is desired to study the stability properties of the system (3.7). The stability

analysis involves two steps, namely internal stability analysis and input-output stability

analysis. In the first step, the internal stability of the closed loop system is studied by

considering the system (3.7) in the absence of the signal rw , and studying the stability

properties of the origin for the resulting unforced switched system. The second step builds on

the internal stability results and presents input-output stability results for the system (3.7).

63

Consider first the system (3.7) in the absence of the signal rw . The state equation for the

resulting system is given by:

0 .0 0

rr r r r Q r r

Q Q Q r Q

r r r

A B K B C B Kx xx A B C xx A L C x

+⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥= −⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥+⎣ ⎦ ⎣ ⎦⎣ ⎦

(3.8)

Consider now the three switched subsystems given by the following state equations:

( ) ,r r rx A B K x= + (3.9)

( ) ,r r rx A L C x= + (3.10)

.Q Q Qx A x= (3.11)

Then, the internal stability of the switched closed loop system (3.8) is given by the following

result.

Lemma 3.1

If the nominal switched systems (3.9) and (3.10) are each asymptotically stable under

arbitrary switching and QA is a stability matrix, then the switched system (3.8) is also

asymptotically stable under arbitrary switching.

Proof: In the following, assume the initial time 0 0t = . If the switched system (3.10) is

asymptotically stable, then according to the definition and lemma 3.1, we have:

0( ) ( , ) (0) (0) tx t t t x c x e λφ −= ≤ , (3.12)

64

for some , 0c λ > , where 0( , )t tφ is the state transition matrix of the switched system (3.10).

Then, in (3.12), we have 0x → as t →∞ for any (0)x . Therefore, for the state variable x in

(3.8), we obtain

lim ( ) 0t

x t→∞

= , (0) nx∀ ∈R . (3.13)

Similarly, if the switched system (3.11) is asymptotically stable, then we have that:

0( ) ( , ) (0) (0) QtQ Q Q Q Qx t t t x c x e λφ −= ≤ ,

for some , 0Q Qc λ > , and where 0( , )Q t tφ is the state transition matrix for the switched system

(3.11). The response of the state variable of Qx in (3.8) is then given by:

00( ) ( , ) (0) ( , ) ( )

t yQ Q Q Q Q rt

x t t t x t B C x dφ φ ξ ξ ξ= − ∫ .

It follows that

00( ) ( , ) (0) ( , ) ( )

t yQ Q Q Q Q rt

x t t t x t B C x dφ φ ξ ξ ξ≤ + ∫

( )

max0( ) (0) (0)Q Q

tt t yQ Q Q Q Q rx t c x e c e B C c x e dλ λ ξ λξ ξ− − − −⇒ ≤ + ∫

( )

max 0( ) (0) (0)Q Q

tt tyQ Q Q Q Q rx t c x e c c B C x e e dλ λ ξ λξ ξ− − − −⇒ ≤ + ∫

max

max

2 (0)( ) (0) if

( ) (0) (0) if

m

m

yQ Q r t

Q Q Q QQ

tyQ Q Q Q Q r Q

c c B C xx t c x e

x t c x c ct B C x e

λ

λ

λ λλ λ

λ λ

−

−

⎧ ⎡ ⎤⎪ ⎢ ⎥≤ + ≠⎪ ⎢ ⎥−⇒ ⎨ ⎣ ⎦⎪

⎡ ⎤≤ + =⎪ ⎣ ⎦⎩

where min( , )m Qλ λ λ= . Therefore, for the state variable Qx in (3.8), we have

lim 0Qtx

→∞= , (0) Qn

Qx∀ ∈R . (3.14)

Similarly, if the switched system (3.9) is asymptotically stable, then

65

0( ) ( , ) (0) (0) tx t t t x c x e λφ −= ≤ ,

for some , 0c λ > , where 0( , )t tφ is the state transition matrix for the switched system (3.9).

The response of the state variable of x in (3.8) is given by:

0 00( ) ( , ) (0) ( , ) ( ) ( , ) ( )

t tQr r Q r rt t

x t t t x t B C x d t B K x dφ φ ξ ξ ξ φ ξ ξ ξ= + +∫ ∫

Let max

max

2 (0)(0) if

(0) (0) if

yQ Q r

Q Q QQm

yQ Q Q Q r Q

c c B C xc x

c

c x c ct B C x

λ λλ λ

λ λ

⎧⎪ + ≠⎪ −= ⎨⎪

+ =⎪⎩

. Then we obtain:

0 00( ) ( , ) (0) ( , ) ( ) ( , ) ( )

t tQr r Q r rt t

x t t t x t B C x d t B K x dφ φ ξ ξ ξ φ ξ ξ ξ≤ + +∫ ∫

( ) ( )

max0 0( ) (0) (0)m

t tt t Q tr r m r rx t c x e ce B C c e d ce B K ce x dλ ξλ λ ξ λ ξ λξξ ξ−− − − − − −⇒ ≤ + +∫ ∫

( ) ( )

max 0 0( ) (0) (0)m

t tt Q t tm r r r rx t c x e cc B C e e d cc B K x e e dλ ξλ λ ξ λ ξ λξξ ξ−− − − − − −⇒ ≤ + +∫ ∫

maxm

mmax

mmax

2 2 (0)(0) if

2 (0)( ) (0) if

(0) (0) if

n

n

n

Qm r r r r t

m

r r tQm r r

tQm r r r r

cc B C cc B K xc x e

cc B K xx t c x cc t B C e

c x cc t B C cct B K x e

λ

λ

λ

λ λ λλ λ λ λ

λ λ λλ λ

λ λ λ

−

−

−

⎧⎡ ⎤⎪⎢ ⎥+ + ≠ ≠

−⎪⎢ ⎥−⎣ ⎦⎪⎪ ⎡ ⎤⎪ ⎢ ⎥⇒ ≤ + + = ≠⎨⎢ ⎥−⎪ ⎣ ⎦⎡ ⎤+ + = =⎣ ⎦

⎩

⎪⎪⎪⎪

(3.15)

where nλ is the minimum value of λ , Qλ and λ . It is easy to see that 0x → as t →∞ for

any (0)x in (3.15). Therefore, for the state variable x in (3.8), we have

lim ( ) 0t

x t→∞

= , (0) nx∀ ∈R . (3.16)

Then based on equalities (3.13), (3.14) and (3.16), it shows that the switched system (3.8) is

asymptotically stable.

66

Based on Lemma 3.1, a parameterized set of stabilizing controllers can be designed by

separately designing the individual switched systems (3.9), (3.10) and the system (3.11). First,

using the following Lemma, the gains rK and rL , { }1, 2r∈ , are chosen to make the switched

systems (3.9) and (3.10) asymptotically stable.

Lemma 3.2 [94]

The origin is an asymptotically stable equilibrium point for the switched systems (3.9) and

(3.10) under arbitrary switching if there exist matrices 0TK KP P= > , 0T

L LP P= > and

matrices rK , rL , { }1, 2r∈ , such that:

( ) ( ) { }( ) ( ) { }

0, 1, 2 ,

0, 1,2 .

Tr r r K K r r r

Tr r r L L r r r

A B K P P A B K r

A L C P P A L C r

+ + + < ∈

+ + + < ∈ (3.17)

By a simple change of variables, the matrix inequalities (3.17) can be transformed to LMIs

and solved using numerical algorithms. Here, we search for a common Lyapunov function

for each of the switched systems (3.9) and (3.10) to guarantee the stability of the systems (3.9)

and (3.10) under arbitrary switching. Moreover, based on the above choice for the matrix QA

in (3.6), the origin for the system (3.11) is an asymptotically stable equilibrium point.

Therefore, if the conditions of Lemma 3.2 are satisfied, then based on the results of Lemma

3.1, the origin is an asymptotically stable equilibrium point for the closed loop switched

system (3.8) under arbitrary switching.

67

3.4 Regulation of bimodal systems: a common Lyapunov function

approach

In this section, a regulator design method for bimodal systems is proposed using a common

Lyapunov function approach. First, the input-output stability property of the switched system

is presented based on a common quadratic Lyapunov function and the state space regulation

conditions for each closed loop subsystem 1clΣ and 2

clΣ are derived, based on which two

sufficient regulation conditions for the switched system are presented. Then a regulator

synthesis procedure is proposed using the properly formulated LMIs and a numerical

example is used to illustrate the effectiveness of the proposed regulator design method.

3.4.1 Input-output stability

In the following, we will consider the input-output stability properties of the switched system

(3.7) in the presence of an external bounded input signal rw based on a common Lyapunov

function. Since system (3.11) is asymptotically stable, there exists a positive definite matrix

QP such that:

Q

TQ Q Q Q nA P P A I+ = − (3.18)

Then based on KP , LP , and QP , we have the following result.

Proposition 3.1

Consider the switched system (3.8), if there exist matrices 0KP > , 0LP > , and 0QP > such

that (3.17) and (3.18) are satisfied, then there must exist a common quadratic Lyapunov

68

function for the switched system (3.8), for example, a common quadratic Lyapunov function

matrix can be blockdiag( , , )K L QP P P Pμ η= , where 0μ > and 0η > .

Proof: Consider the quadratic function ( ) ( ) ( )TV t P tχ χ χ= , where

blockdiag( , , )K L QP P P Pμ η= , and μ and η being positive parameters, we define the

following matrix

( )x Tr r rW A P PA= − +

Based on (3.17), we can have

{ }{ }

[ ] [ ] , 1,2

[ ] [ ] , 1, 2

T Kr r r K K r r r r

T Lr r L L r r r

A B K P P A B K W r

A L C P P A L C W r

+ + + < − ∈

+ + + < − ∈

where KrW and L

rW , { }1, 2r∈ , are positive definite matrices. In view of the block diagonal

form of P , then we have xrW as:

( )( )

1 2

13

23

Q

Kr r r

Txr r n

T T Lr r

W R R

W R I R

R R W

μ μ

μ η

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

where 1rR , 2

rR and 3R are the following matrices

1 [ ] [ ]Q T Qr r r K K r rR B C P P B C= +

2 [ ] [ ]Tr r r K K r rR B K P P B K= +

3 [ ] [ ]y T yQ r Q Q Q rR B C P P B C= − + −

Then, let { }

11 1,2

max rrRψ

∈= ,

{ }2

2 1,2max rr

Rψ∈

= , { }1 min1,2

min ( )Krr

Wλ∈

⎡ ⎤ϒ = ⎣ ⎦ , { }2 min1,2

min ( )Lrr

Wλ∈

⎡ ⎤ϒ = ⎣ ⎦ , and

3 3Rψ = . We have

69

( )( )

1 2

13

23

( )

Q

T xr

kr r r

TT T TQ r n Q

T T Lr r

V W

W R R xx x x R I R x

xR R W

χ χ χ

μ μ

μ η

= −

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎡ ⎤= − ⎣ ⎦ ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ ⎣ ⎦

⎢ ⎥⎣ ⎦

( ) ( )1 2 1 23 3r Q

T TT T T T T T T T T T Lk Q r r r Q n r Q r Q

xx Q x R x R x R x I x R x R x R x W x

xμ μ μ η

⎡ ⎤⎢ ⎥⎡ ⎤= − + + + + + + ⎢ ⎥⎢ ⎥⎣ ⎦⎢ ⎥⎣ ⎦

( ) ( )1 2 1 23 3r Q

T TT T T T T T T T T T Lk Q r r r Q Q n Q Q r Q rx Q x x R x x R x x R x x I x x R x x R x x R x x W xμ μ μ η⎡ ⎤= − + + + + + + + +⎢ ⎥⎣ ⎦

22 21 2 1 2 32 2 2Q Q Qx x x x x x x x xμ η ψ ψ μ ψ⎡ ⎤≤ − ϒ + + ϒ − − −⎢ ⎥⎣ ⎦

1 1 2

1 3

2 3 2

Q Q

x

x x x x

x

ψ ψψ μ μψψ μψ η

⎡ ⎤ϒ − −⎡ ⎤⎢ ⎥⎢ ⎥⎡ ⎤= − − − ⎢ ⎥⎣ ⎦ ⎢ ⎥⎢ ⎥⎢ ⎥− − ϒ⎣ ⎦ ⎣ ⎦

ˆ T

Q Qx x x Q x x x⎡ ⎤ ⎡ ⎤= − ⎣ ⎦ ⎣ ⎦ .

where

1 1 2

1 3

2 3 2

Qψ ψ

ψ μ μψψ μψ η

ϒ − −⎡ ⎤⎢ ⎥= − −⎢ ⎥⎢ ⎥− − ϒ⎣ ⎦

.

Clearly, we need to find proper μ and η to make Q positive definite. The three leading

principal minors of Q are:

1ϒ , 21 1μ ψϒ − ,

2 2 21 2 3 1 1 2 2 2 1 3 2( ) ( ) ( )μη μ ψ ψ η ψ ψ ψ ψ ψ μϒ ϒ − + ϒ ϒ − − + .

Therefore, by selecting

21

1

ψμ >ϒ

, 2 2 2

1 3 1 2 3 2 1 2

1 2 1 2 1

( )ψ μ ψ ψ ψ ψ μ ψ ψημ ψ

ϒ + + +>

ϒ ϒ + ϒ ϒ,

the matrix Q is positive definite.

70

If there exists a common Lyapunov function for the switched system (3.8), then we have the

following input-output stability property for the switched closed loop system (3.7).

Theorem 3.1

Consider the closed loop system (3.7) subject to a bounded input rw . Let { }1,20

max ( ) 0rrt

w tκ∈≥

= ≠ ,

0α > a preset constant and 0 1ε< . If there exist 0TP P= > and positive scalars μ and

β such that the following matrix inequalities are satisfied for all { }1, 2r∈ :

( )2

01

Tr r r

Tr

A P PA P PE

E P I

αε μακ

⎡ ⎤+ +⎢ ⎥ ≤−⎢ ⎥−⎢ ⎥⎣ ⎦

, (3.19)

( )

( )

2 0

0 0

Tex

Ter

ex er

P C

I D

C D I

κ

β μκ

βκ

⎡ ⎤⎢ ⎥⎢ ⎥

⎛ ⎞⎢ ⎥− ≥⎜ ⎟⎢ ⎥⎝ ⎠⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

, (3.20)

then there exits a finite time BT at which the states enter into and continue to evolve inside

the bounded set xB given by:

{ }: ,N TxB Pχ χ χ μ= ∈ ≤R (3.21)

and the output ( )e t in (3.7) satisfies:

( ) , Be t t Tβ≤ ∀ ≥ . (3.22)

71

Proof: Consider the state space representation (3.7) and the quadratic function ( ) TV Pχ χ χ= .

Inequality (3.19) implies that for any nonzero vector N h

rwχ +⎡ ⎤

∈⎢ ⎥⎣ ⎦

R , we have:

( )2

01

TT r r r

Tr rr

A P PA P PE

w wE P I

αχ χε μακ

⎡ ⎤+ +⎡ ⎤ ⎡ ⎤⎢ ⎥ ≤−⎢ ⎥ ⎢ ⎥⎢ ⎥−⎣ ⎦ ⎣ ⎦⎢ ⎥⎣ ⎦

.

It follows from the above inequality that:

( )2

1( ( )) ( ( )) ( ) ( ) 0T

r rd V t V t w t w tdt

ε μαχ α χ

κ−

+ − ≤ . (3.23)

Let ( )2

1( ) ( ) ( )T

r rp t w t w tε μακ−

= , then ( )( ) 1p t ε μα≤ − . Assume the initial time is 0 and let

0( (0))V Vχ = . Using the comparison principle, ( ( ))V tχ then satisfies:

( ) ( )

( ) ( )

( ) ( )

0 0

0 0

0

( ( ))

1

1 1 .

t tt

t tt

t t

V t V e e p d

V e e d

V e e

α τα

α τα

α α

χ τ τ

ε μα τ

ε μ

− −−

− −−

− −

≤ +

≤ + −

≤ + − −

∫∫

Consequently, we have:

( )lim ( ( )) 1t

V tχ ε μ→∞

≤ − .

Therefore, there exists a finite time BT such that

( ( )) , BV t t Tχ μ≤ ∀ ≥ . (3.24)

Based on inequality (3.20), we have:

( )( )

2 00

0

Tex

ex erTe

r

P CC D

I D

κκ

β βμκ

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎡ ⎤− ≥⎛ ⎞ ⎣ ⎦⎢ ⎥ ⎢ ⎥−⎜ ⎟⎢ ⎥ ⎢ ⎥⎣ ⎦⎝ ⎠⎣ ⎦

. (3.25)

72

Inequality (3.20) also implies that 0β μκ− ≥ . Multiplying (3.25) from the left by

T

rwχ⎡ ⎤

⎢ ⎥⎣ ⎦

and

from the right by rwχ⎡ ⎤

⎢ ⎥⎣ ⎦

yields:

( )

( )

2 2

2 2

( ) ( ) ( ) ( )

( ) .

Tr re t V t w t w t

V t

β βκ χ μκ κ

β βκ χ μ κκ κ

⎛ ⎞⎛ ⎞≤ + −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

⎛ ⎞⎛ ⎞≤ + −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

Based on (3.24), it follows that:

2 2 2

2

( )

, .B

e t

t T

β βμκ μ κκ κ

β

⎛ ⎞⎛ ⎞≤ + −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

= ∀ ≥

(3.26)

Since 0β > , ( )e t is ultimately bounded and we have ( ) , Be t t Tβ≤ ∀ ≥ .

Remark 3.3

The advantage of considering a common P matrix in (3.19) and (3.20) is that the set xB is

defined independently of the closed loop system that would be in operation, Bt T∀ ≥ . This

property will be very helpful in deriving the regulation conditions presented in section 3.5.

In the following sections the matrices rQC , { }1, 2r∈ , will be designed to achieve regulation

for the switched closed loop system (3.7).

73

3.4.2 State space regulation conditions for 1clΣ and 2

clΣ

The purpose of introducing the Q -parameterized controllers is to find appropriate matrices

QA , QB and rQC , { }1, 2r∈ , to solve the output feedback regulation problem for the switched

system (3.7). In this section, for a given closed loop system clrΣ , { }1, 2r∈ , a set of rQ

parameters is defined such that the corresponding closed loop system clrΣ achieves regulation.

These sets of rQ parameters will be used in the synthesis of regulators for the original

switched system. The construction of the sets of rQ parameters is achieved by solving a set

of linear matrix equations corresponding to standard regulation conditions for linear systems.

In the following, regulation conditions for each of the closed loop systems are presented and

the construction of the sets of rQ parameters is discussed in detail.

The plant rΣ and the block rJ shown in Figure 2.1 can be combined into a single system

rT . The dynamics of the system rT , { }1,2r∈ , can be represented as follows:

[ ]

,0 0

: 0 0 ,

ˆ 0 0 .

xrr r r r r r r

x yQr r r r r r

re er r

Q

ryr r

Q

wA B K B Kx x D ByA L Cx x D L D

wxT e C D

yx

wxy y C D

yx

⎧ + ⎡ ⎤⎡ ⎤⎡ ⎤⎡ ⎤ ⎡ ⎤= +⎪ ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥+ − −⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎪

⎪ ⎡ ⎤⎡ ⎤⎪ ⎡ ⎤ ⎡ ⎤= +⎨ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦⎪⎪ ⎡ ⎤⎡ ⎤⎪ ⎡ ⎤− = − + ⎢ ⎥⎢ ⎥ ⎣ ⎦⎪ ⎣ ⎦ ⎣ ⎦⎩

(3.27)

Therefore, we have that:

ˆr

rQ

weT

yy y⎡ ⎤⎡ ⎤

= ⎢ ⎥⎢ ⎥−⎣ ⎦ ⎣ ⎦, (3.28)

where

74

11 12

21 22r r

rr r

T TT

T T⎡ ⎤

= ⎢ ⎥⎣ ⎦

,

and where, based on (3.27), the state space representations for 11rT , 12

rT , 21rT and 22

rT are as

follows:

11 1111

11 11: 00

xr r r r r r

r r x yr r r r r r r

r r e er

A B K B K DA E

T A L C D L DC D

C D

⎡ ⎤+⎡ ⎤ ⎢ ⎥

= + − −⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎢ ⎥

⎣ ⎦

,

12 1212

12 12: 0r r r rr r

err r

A B K BA ET CC D

+⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎣ ⎦⎣ ⎦,

21 2121

21 21:x y

r r r r r r r rr y

r r r r

A E A L C D L DT

C D C D⎡ ⎤ ⎡ ⎤+ − −

=⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦, 22 0.rT =

Let ( )rW s and ( )E s denote the Laplace transforms of the inputs rw and the performance

variable e , respectively. Let , ( )r rT QF s denote the closed loop system transfer function

relating ( )rW s to ( )E s . Based on (3.28), the closed-loop system is given by:

,( ) ( ) ( ),r rT Q rE s F s W s= (3.29)

where, for each { }1, 2r∈ ,

11 12 21,

11 12 21

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ).r rT Q r r r r

r r r r

F s T s T s Q s T s

T s Q s T s T s

= +

= + (3.30)

Based on the definition of 11rT , 12

rT and 21rT in (3.28), a realization of the closed loop system

is given as follows:

75

11 11

21 21

12 21 12 12 21

12

11 11

0 0 00 0 0ˆ ˆ0 0

ˆ ˆ0 0 0

0 0r

r r

r rr r

r r r r r

r rQ r Q

r Q r

A EA E

A EE C A E D

C D B C AC C D

⎡ ⎤⎢ ⎥⎢ ⎥⎡ ⎤⎢ ⎥⎢ ⎥ =⎢ ⎥⎢ ⎥⎣ ⎦ ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

. (3.31)

Standard regulation conditions for the system (3.31) are then presented as follows.

Lemma 3.3 [52]

For each { }1,2r∈ , assume the closed loop system clr∑ in (3.7) is asymptotically stable. Then,

for each { }1,2r∈ , i.e. in the absence of switching, clr∑ in (3.7) achieves regulation if and

only if there exists a matrix ˆrΠ which solves the linear matrix equations:

ˆ ˆˆ ˆ ,ˆ ˆˆ 0 .

r r r r r

r r r

H A E

C D

Π = Π +

= Π + (3.32)

Remark 3.4

The main idea behind considering the state space realization of (3.31) instead of (3.7) in

Lemma 3.3 is to express the regulation condition (3.32) as a linear matrix constraint. First

the parameters rK and rL in ˆrA can be determined by solving (3.17), then ˆ

rΠ can be

determined by solving the first equation in (3.32). The second equation in (3.32) becomes a

linear matrix equation in rQC , which appears in ˆ

rC .

76

3.4.3 Regulation conditions for the switched system

For a given { }1, 2r∈ , if the closed loop system clrΣ in (3.7) is asymptotically stable and (3.32)

is satisfied, then regulation in the closed loop system clrΣ can be achieved. However, that

does not guarantee achieving regulation in the switched closed loop system. In this section,

regulation conditions for the switched closed loop system are developed using two

approaches. In the first approach, sufficient regulation conditions are derived based on the

closed loop system input-output properties. In the second approach, the forced switched

closed loop system is transformed into an unforced impulsive switched system using an

appropriate coordinate transformation. Hence, the regulation problem for the switched closed

loop system is transformed into a stability analysis problem for the origin of an impulsive

switched system. A regulator synthesis method based on solving some linear matrix

inequalities is then proposed.

Firstly, the input-output stability properties of the switched system given in Theorem 3.1

are used to provide the following sufficient condition for regulation.

Theorem 3.2

Assume that the switched closed loop system (3.7) is internally stable under arbitrary

switching and that the conditions of Theorem 3.1 are satisfied. If for { }1, 2r∈ , (3.32) is

satisfied and

,β δ< (3.33)

then the switched closed loop system (3.7) achieves regulation.

Proof : Based on Theorem 1, we have that:

77

( ) , Be t t Tβ≤ ∀ ≥ .

If β δ< , then for all Bt T≥ , the output ( )e t will always be such that ( )e t δ< and there

will be no more switching. Since the systems (3.7) is asymptotically stable and since (3.32) is

satisfied, then regulation can be achieved.

The physical meaning of the above theorem is obvious. However, condition (3.33) cannot be

easily satisfied if δ is arbitrarily small. Another alternative to determine regulation

conditions for the system (3.7) is by examining the behavior of the state vector χ . To study

the regulation properties of the closed loop system, an appropriate coordinate transformation

will be used to transform the forced switched closed loop system (3.7) into an unforced

impulsive switched system. As a result, the regulation problem for the switched closed loop

system is transformed into a stability analysis problem for the origin of an impulsive

switched system.

For each { }1,2r∈ , if the closed loop system clr∑ in (3.7) is asymptotically stable and

(3.32) is satisfied, then there must exist a matrix rΠ which solves the following linear matrix

equations:

,

0 .r r r r r

ex er r

H A E

C D

Π = Π +

= Π + (3.34)

Now consider the following coordinate transformation:

r rwχ χ= −Π , (3.35)

Substituting the expression for χ from (3.35) into (3.7), and using (3.34), the forced closed-

loop switched system (3.7) can be transformed into an unforced switched system given by:

78

, ,

:1 if ,2 if .

rex

clr ex

ex

Ae C

e Cr

e C

χ χχ

χ δχ δ

⎧ =⎪

=⎪∑ ⎨⎧ = ≤⎪ = ⎨⎪ = >⎩⎩

Let τS be the set of switching times kτ , 1, 2,k = … . Since for each subsystem in (3.7), a

different coordinate transformation is used, then at a switching time kt ττ= ∈S , we have

( ) ( ) ( ),t t tχ χ χ+ −= + Δ

where ( ) ( ) ( ) ( )

( ) ( ) ( )r t r t r t r t

t w t w tχ − − + +Δ = Π −Π . Hence, the resulting impulsive switched closed

loop system is given by:

,

( ) ( ) ( ), : ,

1 if ,2 if .

r

kcl exr

ex

ex

A

t t t te C

e Cr

e C

τ

χ χ

χ χ χ τχ

χ δχ δ

+ −

⎧ =⎪

= +Δ = ∈⎪⎪∑ =⎨⎪ ⎧ = ≤⎪ = ⎨⎪ = >⎩⎩

S (3.36)

In the new coordinate system, the switching surface S given in (3.2) is fixed and can be

expressed as follows:

{ }: .N exS e Cχ χ δ= ∈ = =R (3.37)

Using the coordinate transformation (3.35), the original output feedback regulation problem

for the switched system (3.7) is transformed into an asymptotic stability analysis problem for

the origin of system (3.36). Note that under the conditions of Theorem 3.1, the original state

vector χ will ultimately evolve inside a bounded set xB given by (3.21). Define

{ }1,20

max ( )r rrt

w tη∈≥

= Π and max ex

BC

ηχγ χ

∈= with { }: NBη χ χ η= ∈ ≤R . A sufficient condition

for regulation is then given as follows.

79

Theorem 3.3


switching and that the conditions of Theorem 3.1 are satisfied. If for { }1, 2r∈ , (3.32) is

satisfied, δ γ> and

( )( )2 01

Tex

ex

P C

C δ γμ

⎡ ⎤⎢ ⎥

>⎢ ⎥−⎢ ⎥

⎣ ⎦

, (3.38)


Proof : Adding ( )exr rC w tΠ to both sides of (3.37) yields

( )( ) ( ) ( )ex exr r r rC t w t C w tχ δ+Π = + Π .

Therefore, the equation for the switching surface S can be rewritten as:

( ) ( )ex exr rC t C w tχ δ= + Π . (3.39)

The term ( )exr rC w tΠ is time-varying and satisfies ( )ex

r rC w t γΠ ≤ . If δ γ> , then the

switching surface S would be closest to the origin when

( ) ( )exC t signχ δ δ γ= − . (3.40)

The switching surface (3.40) does not intersect with xB if an only if

( ){ }min = T exP C signχ χ χ δ δ γ μ− > . (3.41)

The solution to the left hand side of the above inequality is

( )( ) ( )( )1 21 Tex exC P C signδ δ γ−

− − .

80

It follows that (3.41) is equivalent to

( )( ) ( )( )1 21 Tex exC P C signδ δ γ μ−

− − > .

Since ( ) ( )22( )signδ δ γ δ γ− = − , the above inequality becomes

( ) ( )21 1Tex exC P C δ γμ

− < − ,

which is equivalent to (3.38). Then if (3.38) is satisfied, the switching surface S in (3.39)

will never intersect the bounded set xB for all possible ( )exr rC w tΠ , 0t ≥ . Since the

conditions in Theorem 3.1 are satisfied, then for Bt T≥ , the trajectory of the state vector χ

will enter the set xB and continue to evolve inside xB thereafter. Consequently, if (3.38) is

satisfied, there will be no more switching in the set xB . Since the system (3.7) is

asymptotically stable and since (3.32) is satisfied, then regulation will be achieved.

For xBχ ∈ , χ satisfies:

( ) ( )( ) ( )T

r r r rw t P w tχ χ μ+Π +Π ≤ .

It follows that if the conditions of Theorem 3.1 are satisfied, then for Bt T≥ , χ will evolve

inside the bounded set xB given by:

( ) ( ){ }: , TNxB c P c c Bηχ χ χ μ= ∈ + + ≤ ∈R .

Remark 3.5

The bounded set xB includes the union of all ellipsoids with center c− , c Bη∈ . The set xB is

a convex set. This can be verified by considering 1χ and 2χ in xB and 1c and 2c in Bη such

81

that ( ) ( )1 1 1 1Tc P cχ χ μ+ + ≤ and ( ) ( )2 2 2 2

Tc P cχ χ μ+ + ≤ . It can then be shown that

( )1 21χ θχ θ χ= + − , 0 1θ≤ ≤ , is in the set xB since ( ) ( )Tc P cχ χ μ+ + ≤ where

( )1 21c c cθ θ= + − is an element of Bη given that Bη is a convex set.

If condition (3.38) is not satisfied, then the switching surface S in (3.37) intersects with the

bounded set xB . The proof of this statement can be presented by considering the following

arguments. The switching surface S in (3.37) does not intersect with the bounded set xB if

and only if ( ) ( )21 1Tex ex exC P C C cδμ

− < + , for all c Bη∈ . This result can be derived by

considering ( )cχ + as a new coordinate and rewriting the expression for the switching

surface (3.37) in the new coordinate. Given that min ex

c BC c

η

δ δ γ∈

+ ≤ − , if (3.38) is not

satisfied, which means ( ) ( )21 1Tex exC P C δ γμ

− ≥ − , then there must be a point c Bη∈ such

that ( ) ( )21 1Tex ex exC P C C cδμ

− ≥ + . The latter inequality implies that the switching surface S

in (3.37) intersects with the bounded set xB . To present regulation conditions in this case,

define the matrices 11

ex

s ex

CA

C A⎡ ⎤

= ⎢ ⎥⎣ ⎦

, 22

ex

s ex

CA

C A⎡ ⎤

= ⎢ ⎥⎣ ⎦

and 0

bδ⎡ ⎤

= ⎢ ⎥⎣ ⎦

. Sufficient conditions for

regulation are then presented in the following theorem.

Theorem 3.4

Assume that the switched closed loop system clrΣ given in (3.7) is internally stable under

arbitrary switching, that the conditions of Theorem 3.1 are satisfied, and that, for each

82

{ }1, 2r∈ , (3.32) is satisfied. Moreover, assume that (3.38) is not satisfied and that

1 2( ) ( ) 2s srank A rank A= = . If the conditions:

( ) ( )( ) ( ) ( ){ }

11

111 1 1 1

0,

min ,

Tex ex

T Ts s s sc

C A P C

b A c A P A b A cη

δ

μ

−

−−

≤

<

+ + > (3.42)

or

( ) ( )( ) ( ) ( ){ }

12

112 2 2 2

0,

min ,

Tex ex

T Ts s s sc

C A P C

b A c A P A b A cη

δ

μ

−

−−

≤

>

+ + > (3.43)

are satisfied, then the origin is an asymptotically stable equilibrium point for the impulsive

switched system (3.36), implying that the switched system (3.7) achieves regulation.

Proof: In the following, we assume 0δ > . The proof for the case of 0δ < is similar. The

proof consists of two parts. The first part presents condition (3.42) so that 0e < on the

switching surface inside xB for system 1Σ and condition (3.43) so that 0e > on the

switching surface inside xB for system 2Σ . In the second part, and based on the sign of e on

the switching surface as given by conditions (3.42) and (3.43), the asymptotic stability of the

origin for the system (3.36), which is equivalent to achieving regulation in the system (3.7),

is studied.

Part 1: Conditions for a uniform sign of e on the switching surface inside xB .

Let region 1 and region 2 denote the half-spaces where the systems 1clΣ and 2

clΣ are active,

respectively. In Figure 3.1 and Figure 3.2, we assume 0δ > , so that the origin is in region 1

where the system 1clΣ is active. Define two hypersurfaces as

83

{ }1 1: 0N exvS C Aχ χ= ∈ =R ,

{ }2 2: 0N exvS C Aχ χ= ∈ =R .

Then for all points in 1vS , 1 0exe C A χ= = and for all points in 2vS , 2 0exe C A χ= = . By

assumption, the switching surface intersects the bounded set xB . Let

( ) ( ){ }: and , TN exxS C c P c c Bηχ χ δ χ χ μ= ∈ = + + ≤ ∈R

denote the convex set representing the intersection of the switching surface S with the

bounded set xB . If the direction of the vector exC is not parallel to that of 1exC A or 2

exC A ,

then the switching surface S must intersect with the hypersurface 1vS or 2vS . Let

{ }1 1: 0 and N ex exL C A Cχ χ χ δ= ∈ = =R

and

{ }2 2: 0 and N ex exL C A Cχ χ χ δ= ∈ = =R

denote the intersection of S with the two hypersurfaces 1vS and 2vS , respectively. For each

{ }1, 2r∈ and for a given c Bη∈ , let ( ) ( )( ) minr

Tr L

l c c P cχ

χ χ∈

= + + . Then ( )rl c can be

obtained by solving the problem of minimizing ( ) ( )Tc P cχ χ+ + subject to srA bχ = . Since,

by assumption, 1 2( ) ( ) 2s srank A rank A= = , the above problem can be solved analytically and

the point r rLχ∗ ∈ providing the minimum value of ( ) ( )Tc P cχ χ+ + is given by:

( ) ( )11 1T Tr sr sr sr srP A A P A b A c cχ

−∗ − −= + − .

It follows that

( ) ( ) ( )11( ) T Tr sr sr sr srl c b A c A P A b A c

−−= + + , { }1, 2r∈ .

84

Therefore, for all c Bη∈ , a lower bound on the value of rl can be obtained by solving the

following quadratic optimization problem:

( ) ( ) ( ){ }11min ( ) min T Tr sr sr sr src B c

l c b A c A P A b A cη η

−−

∈ ≤= + + .

The above problem can be solved numerically. If

( ) ( ) ( ){ }11min T Tsr sr sr src

b A c A P A b A cη

μ−−

≤+ + > , { }1, 2r∈ ,

then the sets rL , { }1, 2r∈ , are outside the bounded set xB . In this situation, the convex set

xS does not intersect with the sets 1vS or 2vS . Let xSχ′∈ be the intersection point of xS and

an ellipse T Pχ χ ρ= , 0ρ > , tangent to S . Such point is given by

( ) ( )( ) 11 1T Tex ex

exP C C P Cχ δ−

− −′ = . Therefore, if ( )1 0exC A χ′ < , which is equivalent to

( ) ( )11 0

Tex exC A P C δ− < , then we have 0e < in xS for system 1Σ . If ( )2 0exC A χ′ > , which

is equivalent to ( ) ( )12 0

Tex exC A P C δ− > , then we have 0e > in xS for the system 2Σ . In the

following, the above two cases are discussed and it is shown that if any of the conditions

(3.42) or (3.43) is satisfied, then regulation for the switched system can be achieved.

Part 2: Regulation results for the case where either (3.42) or (3.43) is satisfied.

In the rest of the paper, the time BT ′ represents the time at which the χ state trajectory

enters the set xB and continues to evolve in xB thereafter. It is easy to see that B BT T′ ≤ since

x xB B⊂ .

• Case 1: In this case, condition (3.42) is assumed to be satisfied (See Figure 3.1). The set

xS is such that all the state trajectories leaving the set xS and entering region 1 have

85

0e < . Therefore, if the state trajectory leaves the set xS and enters region 1, it will not hit

the set xS again. Two possible cases can be considered here depending on whether the

state trajectory enters the bounded set xB from region 1 or region 2 at time Bt T ′= . In the

case the state trajectory enters the set xB from region 1, then the state trajectory cannot

hit the switching surface S since 0e < for all states on the switching surface within the

set xB . Therefore, the state trajectory is confined to evolve in region 1 inside the set xB .

Since 1clΣ is an asymptotically stable system, the state trajectory will converge to the

origin asymptotically, which implies that regulation in the switched system (3.7) is

achieved. Consider now the case where the state trajectory enters the set xB from region

2. Since the state trajectory is confined to evolve inside the set xB and since the system

2clΣ is asymptotically stable, the state trajectory must hit the switching surface and

switching must take place. Following switching, and based on the dynamics of the

impulsive switched closed system as presented in (3.36), the state vector immediately

following switching will be in region 1. Since 0e < for all states on the switching

surface within the set xB for the system 1Σ , the state trajectory cannot cross the

switching surface again and will continue to evolve in region 1 thereafter. The state

trajectory will approach the origin asymptotically, which implies that regulation will be

achieved.

• Case 2: In this case, condition (3.43) is assumed to be satisfied (See Figure 3.2). The set

xS is such that all the state trajectories leaving the set xS and entering region 2 have

0e > . Assume the state trajectory enters the set xB from region 2 at time Bt T ′= . Given

86

that the system 2clΣ is asymptotically stable, and that the trajectories are confined to

evolve inside xB , the state trajectory must cross the switching surface to approach the

origin. However, this is not possible since, with respect to the system 2clΣ , 0e > for all

xSχ ∈ . Therefore, it is not possible to have the state trajectory enter the set xB from

region 2 at time Bt T ′= . Consider now the case where the state trajectory enters the set xB

from region 1. If the state trajectory crosses the switching surface to enter into region 2,

then that will result in a contradiction similar to that discussed for the case where the state

trajectory enters the set xB from region 2. Therefore, once the state trajectory enters the

set xB from region 1 at time Bt T ′= , it will continue to evolve in region 1 forever and will

never cross the switching surface to enter into region 2. Since 1clΣ is an asymptotically

stable system, then the state trajectory will converge to the origin asymptotically, which

implies that regulation for the switched system (3.7) is achieved.

Therefore, based on the above analysis, if either the conditions for case 1 expressed in (3.42)

or the conditions for case 2 expressed in (3.43) are satisfied, then regulation for the switched

system (3.7) can be achieved.

Remark 3.6

The solution to the regulation problem presented above depends on the maximum magnitude

of the signal rw . This is expected since although switching in the closed loop system takes

place between two linear systems, the resulting switched system is nonlinear. Therefore both

the shape and maximum magnitude of the external inputs are expected in general to affect the

regulator design procedure.

87


set xB for case 1.


set xB for case 2.

S

2vSBounded set xB

ex2C A

exCRegion 2: 2

clΣ

Region 1: 1clΣ

χ′

2L

2χ∗

S

1vS

Bounded set xB

ex1C A

exCRegion 2: 2

clΣ

Region 1: 1clΣ

χ′

1L

1χ∗

88

3.4.4. Regulator synthesis for the switched system

Based on the regulation conditions presented above for the switched system (3.7), a regulator

synthesis method is presented in this section. The proposed method aims at finding a

regulator such that the sufficient regulation conditions of either Theorem 3.2, Theorem 3.3 or

Theorem 3.4 are satisfied. The first step in the synthesis algorithm is to design the gains rK

and rL using (3.17) such that the switched systems (3.9) and (3.10) are asymptotically stable.

Since QA is a fixed stability matrix, it follows from Lemma 3.1 that the switched closed loop

system (3.7) is internally stable under arbitrary switching. With the gains rK and rL already

designed using (3.17), the only remaining unknown parameter in (3.31) is rQC , which

appears in ˆrC . Based on the first equation of (3.32), ˆ

rΠ can be determined first. Therefore, if

rQC is such that

ˆ0 0 0r

e eQ r rC C D⎡ ⎤Π + =⎣ ⎦… , (3.44)

then (3.32) will be satisfied. To avoid having an empty set of solutions rQC in (3.32), we

should have qn h≥ . To determine the unknown variable rQC in the controller so that

regulation is achieved in the switched closed loop system (3.7), conditions (3.19) and (3.20)

of Theorem 3.1 and either one of the sufficient conditions (3.33), (3.38), (3.42) or (3.43) for

regulation given in Theorems 3.2, 3.3 and 3.4 need to be satisfied. Since condition (3.19) is a

bilinear matrix inequality that is generally very hard to solve, a congruence transformation

will be used in the following to transform (3.19) into a linear matrix inequality first.

89

We partition P in (3.19) as [ ]1 2

2 3

n nT

P P

P P×

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

and define 11 1P−Ω = , 1

2 1 2P P−Ω = ,

13 3 2 1 2

TP P P P−Ω = − and 1

2

0T I

Ω⎡ ⎤Ω = ⎢ ⎥−Ω⎣ ⎦

. Then we have:

121

32

121 2

133 2 1 2

111

133 2 1 2

0,

0

,00

00,

00

T

T

TT

IP

I

II P PP

P P P P

PP

P P P P

−

−

−

−

−

ΩΩ ⎡ ⎤⎡ ⎤= ⎢ ⎥⎢ ⎥ Ω−Ω⎣ ⎦ ⎣ ⎦

Ω⎡ ⎤ ⎡ ⎤Ω = =⎢ ⎥ ⎢ ⎥Ω− ⎣ ⎦⎣ ⎦

Ω⎡ ⎤ ⎡ ⎤Ω Ω = =⎢ ⎥ ⎢ ⎥Ω− ⎣ ⎦⎣ ⎦

(3.45)

and

[ ] [ ]

( )

1 2 2

3

1 2 3

0,

00

, , , .

r

r

Q Q rr r r r r r r Q r r

r r rTr

Q Q r

r r r

r Q

A B CA B K A B K B C B K

A L CPA

A B CA L C

M C

⎡ ⎤−⎡ ⎤ ⎡ ⎤+ Ω Ω − + Ω +⎢ ⎥⎢ ⎥ ⎣ ⎦+⎣ ⎦⎢ ⎥Ω Ω = ⎢ ⎥−⎡ ⎤⎢ ⎥Ω ⎢ ⎥+⎢ ⎥⎣ ⎦⎣ ⎦

Ω Ω Ω

It is obvious that if 1 0Ω > and 3 0Ω > , then 0P > . Multiplying (3.19) from the left side by

( ),diag IΩ and from the right side by ( ),Tdiag IΩ yields:

( )

( ) ( )( )

2

1 21 2 3 1 2 3

3 3

22 3

1

0, , , , , ,

0 00.

0 1

r r

T T T Tr r r

T Tr

Tr Q r Q r

Tr T

A P PA P PE

E P I

IM C M C E

IE I

αε μακ

α

ε μακ

⎡ ⎤Ω Ω +Ω Ω + Ω Ω Ω⎢ ⎥

−⎢ ⎥Ω −⎢ ⎥⎣ ⎦⎡ ⎤Ω Ω⎡ ⎤ ⎡ ⎤

Ω Ω Ω + Ω Ω Ω +⎢ ⎥⎢ ⎥ ⎢ ⎥Ω Ω⎣ ⎦ ⎣ ⎦⎢ ⎥= ≤⎢ ⎥−⎡ ⎤⎢ ⎥−⎢ ⎥Ω Ω⎢ ⎥⎣ ⎦⎣ ⎦

(3.46)

90

Similarly, multiplying (3.20) from the left side by ( ), ,diag I IΩ and from the right side

by ( ), ,Tdiag I IΩ yields:

( )

( )

1 12

3 2

1 2

0 00

0

0 0

0

TexT

Ter

ex er

CI

I D

C D II

κ

β μκ

βκ

⎡ ⎤Ω Ω⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥Ω −Ω⎣ ⎦⎣ ⎦⎢ ⎥⎢ ⎥⎛ ⎞⎢ ⎥− ≥⎜ ⎟

⎝ ⎠⎢ ⎥⎢ ⎥Ω −Ω⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦

(3.47)

The main steps in the proposed regulator synthesis procedure can be summarized below.

1) Design of an internally stabilizing controller: Design the controller parameters

rK and rL to make the switched closed loop system (3.7) asymptotically stable, by

solving the LMIs (3.17) for the unknown matrices 0TK KP P= > , 0T

L LP P= > and

matrices rK , rL , { }1, 2r∈ .

2) Design of rQ : Determine ˆrΠ by solving the first equation in (3.32), then design the

controller parameter rQC by solving the following optimization problem in the

unknown scalars 0β > , 0μ > , and matrices 1 0Ω > , 3 0Ω > , 2Ω and rQC , { }1, 2r∈ ,

based on a preset positive scalar α and 0 1ε< :

Minimize β

Subject to (3.46) and (3.47),

ˆ0 0 0.r

ee Q r rC C D

⎧⎪⎨⎡ ⎤Π + =⎪⎣ ⎦⎩ …

(3.48)

3) Verification: If β δ< , then based on condition (3.33) in Theorem 3.2, the

resulting controller given by (3.5) and (3.6) achieves regulation. If β δ≥ , the

condition (3.33) cannot be satisfied and conditions (3.38), (3.42) and (3.43) need to

91

be further checked. Based on the resulting closed loop system, solve for P using

(3.45) and solve for Π using (3.34), then determine the values of η and γ . If any of

the regulation conditions (3.38), (3.42) and (3.43) are satisfied, the designed

controller yields regulation.

Note that (3.46) is linear in the unknown parameters β , μ , 1Ω , 2Ω , 3Ω and rQC only if α

is prefixed. Let mα denote an upper bound on the value of α such that

( ) ( ) 11 2 3 1 2 3

3

0, , , , , , 0

0r r

Tr Q r QM C M C α

Ω⎡ ⎤Ω Ω Ω + Ω Ω Ω + ≤⎢ ⎥Ω⎣ ⎦

,

with unknown 1 0Ω > , 3 0Ω > , 2Ω and rQC . The problem to solve mα is not linear, but it is

easily seen to be quasi-convex. Hence mα can be estimated using the bisection algorithm. A

search for a solution to the LMIs in step 2 and 3 can be performed iteratively by considering

values of α in the interval ( )0, mα α∈ .

In the following section, a numerical example is used to illustrate the regulator synthesis

procedure proposed in this paper.

3.4.5. Numerical example

In this section, the regulator synthesis method proposed above will be used to design a

controller that cancels the contact vibrations in the mechanical system as shown in section

2.6. First, rewriting (2.40) in the state space form given in (3.1) results in:

1

0 1110000 10

A⎡ ⎤

= ⎢ ⎥− −⎣ ⎦, 2

0 110000 5

A⎡ ⎤

= ⎢ ⎥− −⎣ ⎦,

1 2

0 2020 0

H Hπ

π⎡ ⎤

= = ⎢ ⎥−⎣ ⎦, 10 20

0100

w w⎡ ⎤

= = ⎢ ⎥⎣ ⎦

, 1 2

01

B B⎡ ⎤

= = ⎢ ⎥⎣ ⎦

,

92

[ ]1 2 1 0eC C C= = = , 31

0 010

0.8 1.7xD −⎡ ⎤= ×⎢ ⎥⎣ ⎦

, 32

0 010

0.75 0.75xD −⎡ ⎤= ×⎢ ⎥⎣ ⎦

,

[ ] 81 2 1 2 0 1 10y y e eD D D D −= = = = − × .

To construct a parameterized set of stabilizing controllers, the gains rK , rL , { }1, 2r∈ , are

designed based on (3.17) with

[ ] 71 2 1.5081 0.0001 10K K= = − − × , [ ] 5

1 2 0.0018 1186.81 10L L= = − × .

Since 3 3rH ×∈R , the base stabilizing controller is augmented with a parameter rQ with

4 3qn h= > = and here given by ,1 ,2 ,3 ,42 3 4( 1) ( 1) ( 1) ( 1)

r r r rrQ

s s s sθ θ θ θ

= + + ++ + + +

. The corresponding

state space representation for rQ is:

1 0 0 01 1 0 00 1 1 00 0 1 1

QA

−⎡ ⎤⎢ ⎥−⎢ ⎥=⎢ ⎥−⎢ ⎥−⎣ ⎦

,

1000

QB

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

,

,1 ,2 ,3 ,4rQ r r r rC θ θ θ θ⎡ ⎤= ⎣ ⎦ . (3.49)

By solving the first Sylvester equation in (3.32), we have:

10 8

7 9

11 8

7 8

11 8

7 81

18 18

16

1.1374 10 1.0102 106.3478 10 7.1466 106.3469 10 1.0058 10

6.2055 10 1.4305 106.3469 10 1.0058 10

ˆ 6.2055 10 1.4305 10 4.1911 10 3.7735 10

2.3710 10 2

− −

− −

− −

− −

− −

− −

− −

−

× ×− × ×− × − ×

× − ×− × − ×

Π = × − ×× ×

− × 16

20 20

21 22

23 23

.6334 10 6.1103 10 6.5731 10

1.0304 10 9.8888 101.5995 10 1.6145 10

−

− −

− −

− −

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥×⎢ ⎥× − ×⎢ ⎥⎢ ⎥− × − ×⎢ ⎥− × ×⎣ ⎦

,

10 8

7 9

11 8

7 8

11 8

7 82

18 18

16

1.1408 10 1.0104 106.3485 10 7.1684 106.3479 10 1.0058 10

6.2058 10 1.4393 106.3479 10 1.0058 10

ˆ 6.2058 10 1.4393 10 4.2196 10 3.8310 10

2.4071 10 2

− −

− −

− −

− −

− −

− −

− −

−

× ×− × ×− × − ×

× − ×− × − ×

Π = × − ×× ×

− × 16

20 20

21 21

23 23

.6513 10 6.2025 10 6.6170 10

1.0372 10 1.0037 101.6233 10 1.6249 10

−

− −

− −

− −

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥×⎢ ⎥× − ×⎢ ⎥⎢ ⎥− × − ×⎢ ⎥− × ×⎣ ⎦

93

First calculate { }1,20

max ( ) 100rrt

w tκ∈≥

= = . Then, solving the LMIs in (3.46)-(3.48) for the

minimum value of β yields:

[ ]1

3740.5811 796.4368 134.3644 10QC = − × ,

[ ]2

31656.8 647.0492 210.3280 10QC = − − × ,

75.5253 10μ −= × , 632.1251 10β −= × .

Since 6 632.1251 10 30 10β δ− −= × > = × , regulation condition (3.33) cannot be satisfied.

Solving the Sylvester equation (3.34) and evaluating η results in 68.8398 10η −= × .

Condition (3.38) is then verified but cannot be satisfied. Verifying condition (3.43), we have:

( ) ( )( ) ( ) ( ){ }

1 72

11 7 72 2 2 2

4.8559 10 0,

min 6.3438 10 5.5253 10 ,

Tex ex

T Ts s s sc

C A P C

b A c A P A b A cη

δ

μ

− −

−− − −

≤

= × >

+ + = × > = ×

which means that (3.43) is satisfied. Then, based on Theorem 3.4, the designed regulator can

achieve regulation for the switched system (2.40).

The simulation results of the response for the closed loop system under switching are

illustrated in Figures 3.3-3.4. It can be seen that if the performance variable is smaller than

630 10 m−− × , the mass enters into contact with the contact surface cS and the model of the

system switches. It can also be seen that the disturbance rd changes at the switching times.

But even in the presence of switching, the switched system performance variable e still

converges to zero, which means the mass m asymptotically follows the contact surface cS at

the desired separation 30 micrometers.

94

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-100

-50

0

50

100

e (m

icro

met

ers)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-0.2

-0.1

0

0.1

0.2

d r(1,1

) (m

/sec

2 )

Time(sec)


0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-40

-30

-20

-10

0

10

20

v an

d v s (m

icro

met

ers)

Time(sec)



95

3.5 Regulation of bimodal systems: a multiple Lyapunov function

approach

In the previous section, a common Lyapunov function approach was used in the analysis of

the input-output stability of the switched system and a regulator synthesis method was also

presented. The approach based on a common Lyapunov function is a conservative approach

and may not yield the desired regulator. In this section, a regulator synthesis approach based

on multiple Lyapunov functions is proposed. Based on the properties of the time-varying

switching surface S in (3.2), the whole closed loop system state space is partitioned into

three adjacent regions. Using the resulting state space partitions, a regulation condition for

the switched closed loop system is formulated using a multiple Lyapunov function approach

and a corresponding regulator synthesis method is then proposed using an iterative LMI

algorithm. Furthermore, the regulation interpolation conditions for 1clΣ and 2

clΣ will be

derived in the frequency domain and are expressed in affine linear equalities with smaller

dimension as compared with the one in (3.44) and, therefore, are easier for the LMI software

tools to solve.

The switched bimodal system is given by:

0, (0) , , , :

1 if ,2 if ,

dr r r ry ydr r re ed

r rre ed

r re ed

r r

x A x B u E d x xy C x D de C x D d

e C x D dr

e C x D dδδ

⎧ = + + =⎪ = +⎪⎪ = +Σ ⎨⎪ ⎧ = + ≤⎪ = ⎨⎪ = + >⎩⎩

(3.50)



to be measurable, { }1,2r∈ is the index of the system rΣ under consideration at time t , and

96

δ is a constant satisfying 0δ > . Switching between 1Σ and 2Σ is performed according to

the performance variable e which defines a time-varying switching surface S given by:

{ }: n e edr rS x e C x D d δ= ∈ = + =R . (3.51)

The external signal hrd ∈R , ,1 ,[ ]T

r r r hd d d= , representing disturbance and/or reference

signals, is also assumed to switch according to the rule given in (3.50), and where each ,r jd

is of the form:

( ) { }, , , ,, 1

1( ) cos , 1, 2 , 1, ,

j

j

kr j r j r j r j

r j i i i ki

d t c t c r j hω φ +=

= + + ∈ =∑ , (3.52)

with known amplitudes ,r jic , frequencies ,r j

iω , and phases ,r jiφ , 1, , ji k= … ; 1, ,j h= … . The

model in (3.50) can be expressed as in (3.1) easily. The signals ,r jd , { }1, 2r∈ ; 1, ,j h= ;

can be considered to be the outputs of exosystems given by:

0, , , , ,

,, ,

, (0) , :

,r j r j r j r j r jes

r jr j j r j

w H w w w

d C w

⎧ = =⎪∑ ⎨=⎪⎩

where 2,

jkr jw ∈ , and

( )

( )

2,1

,

2,

0 1[0] [0]

0

[0] [0] [0]0 1

[0]0

[0] [0] 0j

r j

r j

r jk

H

ω

ω

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥−⎢ ⎥⎢ ⎥⎣ ⎦⎢ ⎥

⎢ ⎥= ⎢ ⎥⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥−⎢ ⎥⎣ ⎦⎢ ⎥⎣ ⎦

…

…

97

( )( )

( )( )

, ,1 1

, , ,1 1 1

, ,,

, , ,

,1

sin

cos

(0) sin

cos

j j

j j j

j

r j r j

r j r j r j

r j r jr jk k

r j r j r jk k k

r jk

c

c

w c

c

c

φ

ω φ

φ

ω φ

+

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦⎢ ⎥⎢ ⎥

= ⎢ ⎥⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦⎢ ⎥⎢ ⎥⎣ ⎦

[ ] [ ]1 (2 1)

1 0 1 0 1j

j kC

× +⎡ ⎤= ⎣ ⎦

Define the state vector ,1 , 1[ ]T T Tr r r h mw w w ×= , where

1

(2 1)h

jj

m k=

= +∑ . The exosystem esr∑

generating the signal rd is then given by:

0, (0): r r r r res

rr w r

w H w w wd C w

= =⎧∑ ⎨ =⎩

(3.53)

where

,1

,

0 00 00 0

r

r

r h

HH

H

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

, 1 0 0

0 00 0

w

h

CC

C

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

, and

0,1

00,

r

r

r h

ww

w

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

.

It worth noting that there are many ways to obtain matrices rH and wC in (3.53). The model

in (3.50) is considered in this section since it is more suitable to work with to derive

regulation conditions in the frequency domain.

In the following, an output feedback controller will be constructed to regulate the

performance variable e against the external input signal rd such that lim ( ) 0t

e t→∞

= . In this

section, Q parameterized stabilizing controllers, as presented in the case of system (3.1), will

be used with the switched system (3.50) to achieve regulation. The resulting closed loop

system is given by the following state space representation:

98

, ,

:1 if ,2 if ,

r r rex ed

r rclr ex ed

r rex ed

r r

A E de C D d

e C D dr

e C D d

χ χχ

χ δχ δ

⎧ = +⎪

= +⎪∑ ⎨⎧ = + ≤⎪ = ⎨⎪ = + >⎩⎩

(3.54)

where 00 0

rr r r r Q r r

yr Q Q r

yr r r

A B K B C B K

A A B CA L C

+⎡ ⎤⎢ ⎥

= −⎢ ⎥⎢ ⎥+⎣ ⎦

, ( ) ( ) ( )TTTd yd T d yd

r r Q r r r rE E B D E L D⎡ ⎤= − +⎢ ⎥⎣ ⎦,

0 0ex eC C⎡ ⎤= ⎣ ⎦ .

The purpose of introducing the Q -parameterized controllers is to find appropriate

matrices QA , QB and rQC , { }1, 2r∈ , to solve the output feedback regulation problem for the

switched system (3.54). In the following, frequency domain regulation conditions for each of

the closed loop systems are presented first, then a regulation condition for the switched

closed loop system and a corresponding regulator synthesis algorithm are discussed.

3.5.1 Regulation conditions for 1clΣ and 2

clΣ in the frequency domain

Combining the plant rΣ in (3.1) and the block rJ in (3.6) into a single system rT , we have:

11 12

21 22ˆr

rr r

Qr r

T

de T Tyy y T T

⎡ ⎤ ⎡ ⎤⎡ ⎤= ⎢ ⎥ ⎢ ⎥⎢ ⎥−⎣ ⎦ ⎣ ⎦ ⎣ ⎦

, (3.55)

where 11rT , 12

rT , 21rT , 22

rT , { }1, 2r∈ , are as in presented in section 3.4.2. Let ( )rD s , , ( )r jD s

and ( )E s denote the Laplace transforms of the inputs rd , ,r jd and the performance variable

e , respectively. Let ,d jrE , ,ed j

rD , and ,yd jrD denote the thj columns of the matrices d

rE , edrD ,

and ydrD , respectively. Let 1

, , ,( ) ( ) ( )r r r r r r

hT Q T Q T QF s F s F s⎡ ⎤= ⎣ ⎦ denote the closed loop

99

system transfer function relating ( )rD s to ( )E s . Then based on (3.28) and (3.6), for each

{ }1, 2r∈ , the transfer function , ( )r rT QF s can be written as:

11 12 21, ( ) ( ) ( ) ( ) ( )

r rT Q r r r rF s T s T s Q s T s= + . (3.56)

Therefore, we have

, ,1

( ) ( ) ( ),r r

hj

T Q r jj

E s F s D s=

=∑

where

11, 12 21,, ( ) ( ) ( ) ( ) ( )

r r

j j jT Q r r r rF s T s T s Q s T s= + ,

and

,

11, , ,

,

: 00

d jr r r r r r

j y d j dy jr r r r r r r

e ed jr

A B K B K ET A L C E L D

C D

⎡ ⎤+⎢ ⎥

+ − −⎢ ⎥⎢ ⎥⎣ ⎦

, 12 12

1212 12: 0

r r r rr rer

r r

A B K BA ET CC D

+⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎣ ⎦⎣ ⎦

, ,21,

, :y d j yd j

r r r r r rjr y yd j

r r

A L C E L DT

C D⎡ ⎤+ − −⎢ ⎥−⎢ ⎥⎣ ⎦

, 1, ,j h= .

According to the properties of ,r jd in (3.52), all the poles of , ( )r jD s , 1, ,j h= , are simple

poles located on the imaginary axis of the complex plane. For a given { }1, 2r∈ , some of the

disturbances , ( )r jD s , 1, ,j h= , may have common poles. Let prn denote the total number

of the different poles in the expression for ( )rD s . Let rp , 1, , ,prn= denote such poles, rS

denote the set of disturbance indices j such that rp is a pole of , ( )r jD s , and , ( )r j rR p

denote the residue of , ( )r jD s at the pole rp . Then we can state the following interpolation

conditions for regulation.

100

Theorem 3.5

Assume that, for each { }1, 2r∈ , the closed loop system clrΣ in (3.54) is internally stable.

Consider the closed loop system transfer function , ( )r rT QF s given in (3.56). For any given

{ }1,2r∈ , regulation in the resulting closed loop system (3.54) is achieved if and only if the

following interpolation conditions are satisfied

( )11, 12 21,, ( ) ( ) ( ) ( ) ( ) 0, 1, , .

r

j j pr j r r r r r r r r r r

j

R p T p T p Q p T p n∈

+ = =∑S

(3.57)

Proof: The Laplace transform of the closed loop system is given by

, ,1

( ) ( ) ( )r r

hj

T Q r jj

E s F s D s=

= ∑ .

According to (3.52), the poles of , ( )r jD s are all simple poles located on the imaginary axis.

Using partial fraction expansion, the response ( )e t can be expressed as follows:

1, , 0

1

1( ) ( ) ( ) ( )pr

r r

r

nj

r j r T Q rjr

e t L R p F p e ts p

−

= ∈

⎡ ⎤⎛ ⎞= +⎢ ⎥⎜ ⎟⎜ ⎟−⎢ ⎥⎝ ⎠⎣ ⎦

∑ ∑S

, (3.58)

where 1L− is the inverse Laplace transform operator, 0 ( )e t denotes the sum of responses

corresponding to partial fractions with , ( )r rT QF s poles and the response to non-zero initial

conditions. Since for each { }1, 2r∈ the closed loop system is asymptotically stable, the

response to nonzero initial conditions converges asymptotically to zero. Also, since

, ( )r rT QF s RH∞∈ , the space of real rational stable transfer matrices, then the responses

corresponding to partial fractions with , ( )r rT QF s poles asymptotically converge to zero.

101

Therefore, the asymptotic properties of the disturbance response are determined by the first

term on the RHS of (3.58):

1, ,

1

1( ) ( ) ( )pr

r r

r

nj

r j r T Q rjr

e t L R p F ps p

−

= ∈

⎡ ⎤⎛ ⎞′ = ⎢ ⎥⎜ ⎟⎜ ⎟−⎢ ⎥⎝ ⎠⎣ ⎦

∑ ∑S

.

Necessity: Since all the poles rp are on the imaginary axis and are distinct, if lim ( ) 0t

e t→∞

′ = ,

then:

, ,( ) ( ) 0, 1,r r

r

j pr j r T Q r r

j

R p F p n∈

= =∑S

.

It follows that:

( )11, 12 21,, ( ) ( ) ( ) ( ) ( ) 0, 1, ,

r

j j pr j r r r r r r r r r r

j


+ = =∑S

,

which means that the interpolation conditions (3.57) must be satisfied.

Sufficiency: Based on the above discussion, satisfying the interpolation conditions (3.57)

implies that ( )e t in (3.58) asymptotically decays to zero.

Let ,1

( ) ( )qn

r r i ii

Q s sθ ψ=

=∑ , where ( ) ( ) ii s s pψ −= + , 1, , qi n= , and 0p > . Then, the state

space representation of rQ is as given in (3.6) with:

0 01

00 1

q q

Q

n n

pp

A

p×

−⎡ ⎤⎢ ⎥−⎢ ⎥=⎢ ⎥⎢ ⎥−⎣ ⎦

,

1

10

0q

Q

n

B

×

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

,

,1 ,2 , 1.

r qq

Q r r r n nC θ θ θ

×⎡ ⎤= ⎣ ⎦ (3.59)

102

Assume ( )rD s has 0k distinct pairs of complex conjugate poles on the imaginary axis and

one pole at zero (the pole related to the disturbance offset in (3.52)), so that the total number

of the different poles is 02 1prn k= + . Let rp + , 01, ,k= , denote the 0k distinct complex

poles in the top half of the complex plane. Then we have the following result.

Theorem 3.6

Assume, for each { }1, 2r∈ , the closed loop system clrΣ in (3.7) is internally stable. For any

given { }1, 2r∈ , the set rQS of

rQC matrices in (3.49) which yield regulation in the closed

loop system clrΣ in (3.7) can be defined by the following affine constraint on

rQC :

{ }: 0q

r r r r r

nT TQ Q QC A C Bθ θ= ∈ + =S , (3.60)

where the matrix r

Aθ and the vector r

Bθ are given by:

0 0

0 0

1 1 1, ,

1 1 1, ,

1, ,

1, ,

1

( ) ( )

( ) ( )

,( ) ( )

( ) ( )

(0) (0)

q

q

r q

q

q

nr re r r re r

nr im r r im r

nk kr re r r re r

nk kr im r r im r

nr r

V p V p

V p V p

AV p V p

V p V p

V V

θ

+ +

+ +

+ +

+ +

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

0

0

0 1,0 1,

0,0,

0

( )( )

,( )( )(0)

r

r re r

r im r

kr re r

kr im r

r

V pV p

BV pV p

V

θ

+

+

+

+

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(3.61)

and where , ( )ir re rV p + and , ( )i

r im rV p + denote the real and imaginary parts of ( )ir rV p + given by:

12 21,, 0( ) ( ) ( ) ( ) ( ) , 1, , , 1, , ,

r

i jr r i r r j r r r r r q

j

V p p R p T p T p k i nψ+ + + + +

∈

⎛ ⎞= = =⎜ ⎟⎜ ⎟

⎝ ⎠∑S

0 11,, 0( ) ( ) ( ), 1, , .

r

jr r r j r r r

j

V p R p T p k+ + +

∈

= =∑S

103

Proof: (Necessity) Using the expression for rQC given in (3.49), the interpolation conditions

can be rewritten as:

11, 12 21,, ,

1( ) ( ) ( ) ( ) ( ) 0, 1, ,

q

r

nj j p

r j r r r r i r r i r r r rij

R p T p T p p T p nθ ψ=∈

⎛ ⎞+ = =⎜ ⎟⎜ ⎟

⎝ ⎠∑ ∑S

. (3.62)

Define:

0 11,,( ) ( ) ( ), 1, ,

r

j pr r r j r r r r

j

V p R p T p n∈

= =∑S

,

12 21,,( ) ( ) ( ) ( ) ( ), 1, , 1, ,

r

i j pr r r j r r r i r r r r q

j

V p R p T p p T p n i nψ∈

= = =∑S

.

Then (3.62) can be written as

0,

1( ) ( ) 0, 1, , 1, ,

qni p

r r r i r r r qi

V p V p n i nθ=

+ = = =∑ .

Since we have ( ) ( )i ir rV s V s= , 1, , qi n= , where s is the complex conjugate of s , the

interpolation conditions corresponding to the 02k complex conjugate poles are equivalent to

the 0k interpolation conditions corresponding to the 0k poles rp + , 01, ,k= . For any

complex number s , let , ( )ir reV s and , ( )i

r imV s denote, respectively, the real and imaginary parts

of the functions ( )irV s , 1, , qi n= . The interpolation condition evaluated at a pole rp + ,

01, , k= , on the imaginary axis can be written as:

0 0, , , , , ,

1 1

( ) ( ) ( ) ( ) 0q qn n

i ir re r r i r re r r im r r i r im r

i i

V p V p j V p V pθ θ+ + + +

= =

⎡ ⎤ ⎡ ⎤+ + + =⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦∑ ∑ .

Hence, for each such pole, the above equation yields two linear equations of the form:

104

0, , ,

1

0, , ,

1

( ) ( ) 0,

( ) ( ) 0.

q

q

ni

r re r i r rei

ni

r im r i r imi

V s V s

V s V s

θ

θ

=

=

⎡ ⎤+ =⎢ ⎥

⎣ ⎦⎡ ⎤

+ =⎢ ⎥⎣ ⎦

∑

∑ (3.63)

Therefore, considering the 0k poles rp + selected as mentioned above yields 02k linear

equations in the unknowns ,r iθ , 1, , qi n= . On the other hand, the interpolation condition

evaluated at the pole at 0 can be written in the form:

0,

1(0) (0) 0

qni

r r i ri

V Vθ=

⎡ ⎤+ =⎢ ⎥

⎣ ⎦∑ .

Therefore, the interpolation conditions (3.57) are equivalent to the following affine constraint

on the matrix rQC :

0r r r

TQA C Bθ θ+ = ,

where rQC is given by (3.49), and where the p

r qn n× matrix r

Aθ and the 1prn × vector

rBθ are

given in (3.61).

(Sufficiency) For a given ,r rT QF and according to the expressions given above for r

Aθ and r

Bθ ,

satisfying (3.60) implies that (3.62) is satisfied and that regulation can be achieved.

3.5.2 Regulation condition and regulator synthesis for the switched system

The previous section presented the construction of a set rQS of

rQC matrices that yield

regulation for each of the closed loop systems clrΣ , { }1, 2r∈ . If qn is such that p

q rn n≥ , then

the set rQS will not be empty. However, simply selecting

rQC , { }1, 2r∈ , in the sets rQS does

not guarantee achieving regulation in the switched closed loop system. In this section,

regulation conditions for the switched closed loop system and a corresponding regulator

105

synthesis procedure based on piecewise quadratic Lyapunov functions will be presented. Let

{ }1,20

max ( )edr rr

t

D d tσ∈≥

= , δ δ σ− = − and δ δ σ+ = + . Based on the switching condition in (3.54),

the whole closed loop system state space can be partitioned into three adjacent regions 1ℜ ,

2ℜ and 3ℜ (as shown in Figure 3.5), where

{ }1N

exCχ χ δ−ℜ = ∈ ≤R ,

{ }2N

exCχ χ δ+ℜ = ∈ >R ,

3exN

ex

CC

δχ χ

δ−

+

⎧ − ⎫−⎡ ⎤ ⎡ ⎤⎪ ⎪ℜ = ∈ ≤⎨ ⎬⎢ ⎥ ⎢ ⎥⎪ ⎪⎣ ⎦⎣ ⎦⎩ ⎭

R .

It is easy to see that only the system 1clΣ can be active in the region 1ℜ and that only the

system 2clΣ can be active in the region 2ℜ , while any one of the two systems 1

clΣ and 2clΣ can

be active in the region 3ℜ . Furthermore, the boundaries between the three regions are the

two hypersurfaces

{ }13N

exS Cχ χ δ−= ∈ =R

and

{ }23N

exS Cχ χ δ+= ∈ =R .

Let 1 1, , , , 0ex j N NC c c c ×⎡ ⎤= ≠⎣ ⎦ and 1, , , ,T

j Nx x xχ ⎡ ⎤= ⎣ ⎦ , then there must exist an index

{ }1, ,j N∈ … such that 0jc ≠ . Consequently, for 13Sχ ∈ or 23Sχ ∈ , jx can be expressed as

106

Figure 3.5. State space partitions where the vector Tex exC C= .

j exj

x Ccδχ −= + or j ex

j

x Ccδχ += + , respectively, where 1 1 1, , , , ,

T

j j Nx x x xχ − +⎡ ⎤= ⎣ ⎦ and

1 1 11 , , , , ,ex j j N

j

C c c c cc − +⎡ ⎤= − ⎣ ⎦ . Therefore, the hypersurfaces 13S and 23S can be expressed

as

{ }113 1, NS H fχ χ −= + ∈R ,

{ }123 2 , NS H fχ χ −= + ∈R ,

where ( ) ( )

( ) ( )

1 1j j

ex

N j N j

I

H CI

− × −

− × −

⎡ ⎤⎢ ⎥

= ⎢ ⎥⎢ ⎥⎣ ⎦

,

( )

( )

1 1

1

1

0

0

j

j

N j

fcδ− ×

−

− ×

⎡ ⎤⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

and

( )

( )

1 1

2

1

0

0

j

j

N j

fcδ− ×

+

− ×

⎡ ⎤⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

.

We define a piecewise quadratic Lyapunov function ( ) 2 0T Ti i iV P q gχ χ χ χ= + + > for

each iχ ∈ℜ , { }1, 2,3i∈ , where Ti iP P= . For ( )V χ to be continuous on the hypersurfaces

13S and 23S , we should have

2ℜ

1ℜ3ℜ

exC

exC χ δ+=

exC χ δ−=

exC χ δ=

107

( ) ( ) ( )( ) ( ) ( )1 1 1 1 1 1

1 3 1 3 1 3

2

2 ,

T T

T T

H f P H f q H f g

H f P H f q H f g

χ χ χ

χ χ χ

+ + + + +

= + + + + +

( ) ( ) ( )( ) ( ) ( )

2 2 2 2 2 2

2 3 2 3 2 3

2

2 ,

T T

T T

H f P H f q H f g

H f P H f q H f g

χ χ χ

χ χ χ

+ + + + +

= + + + + +

which result in the following constraints, for { }1, 2r∈ ,

( )( ) ( )( ) ( ) ( )

3

3 3

3 3 3

0,

0,

2 0.

Tr

T Tr r r

TTr r r r r r

H P P H

H P P f H q q

f P P f q q f g g

− =

− + − =

− + − + − =

(3.64)

Let 1 exC C= , 2 exC C= − , 3ex

ex

CC

C−⎡ ⎤

= ⎢ ⎥⎣ ⎦

, 1δ δ−= , 2δ δ+= − and 3

δδ

δ−

+

−⎡ ⎤= ⎢ ⎥⎣ ⎦

. We then have the

following result.

Theorem 3.7

Consider the closed loop system (3.54) subject to a bounded input rd . Let

{ }1,20

max ( ) 0rrt

d tγ∈≥

= ≠ . If there exist matrices 1 1TP P= , 2 2

TP P= , 3 3TP P= , 1 2 1 2, , , 0η η ζ ζ ≥ ∈R

and 23 3, 0η ζ ≥ ∈R , and positive scalars α , κ and β such that the following matrix

inequalities are satisfied:

2 0,

T T Tr r r r r r r r r r r r

T T Tr r r r r r r r r r

T Tr r r r

A P P A P A q q C P Eq A q C g q E

E P E q I

α α ζα ζ α ζ δ

κ

⎡ ⎤+ + + −⎢ ⎥

+ − + ≤⎢ ⎥⎢ ⎥−⎣ ⎦

{ }1, 2r∈ (3.65)

3 3 3 3 3 3 3 3

3 3 3 3 3 3 3 3

3 3

2 0,

T T Tr r r r

T T T T Tr r

T Tr r


E P E q I

α α ζα ζ α ζ δ

κ

⎡ ⎤+ + + −⎢ ⎥

+ − + ≤⎢ ⎥⎢ ⎥−⎣ ⎦

{ }1, 2r∈ (3.66)

108

02 0 0

00 0

0

T Ti i i i ex

T T Ti i i i i i

Ted

ex ed

P q C Cq C g

I D

C D I

α α ηα η α η δ

β κγ

βγ

⎡ ⎤+⎢ ⎥+ −⎢ ⎥⎢ ⎥⎛ ⎞ >⎢ ⎥−⎜ ⎟⎢ ⎥⎝ ⎠⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

, { }1, 2,3i∈ (3.67)

1 if 0,0,

2 if 0.r r r

TQA C B rθ θ

δδ>⎧

+ = = ⎨ <⎩ (3.68)

(3.64) and ,β δ≤ (3.69)


Proof: First, (3.67) implies

02

Ti i i i

T T Ti i i i i i

P q Cq C gα α η

α η α η δ⎡ ⎤+

>⎢ ⎥+ −⎣ ⎦, { }1, 2,3i∈ .

Multiplying from the left by 1Tχ⎡ ⎤⎣ ⎦ and from the right by 1TTχ⎡ ⎤⎣ ⎦ yields,

( ) ( )( )( ) 2 ( ) 0Ti i iV t C tα χ η χ δ+ − > , iχ ∈ℜ .

Based on the definition of iη , iC and iδ , we have

( )( ) 0Ti i iC tη χ δ− ≤ , iχ ∈ℜ .

It follows that ( )( ) 0V tχ > for each iχ ∈ℜ .

Now consider the state space representation (3.54). Inequalities (3.65) and (3.66) imply

that for any vector 1TT T

rdχ⎡ ⎤⎣ ⎦ , we have

1 2 1 0

T T T Tr r r r r r r r r r r r

T T Tr r r r r r r r r r

T Tr r r r r r


d E P E q I d

χ α α ζ χα ζ α ζ δ

κ

⎡ ⎤+ + + −⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥+ − + ≤⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦⎣ ⎦

109

and

3 3 3 3 3 3 3 3

3 3 3 3 3 3 3 3

3 3

1 2 1 0

T T T Tr r r r

T T T T Tr r

T Tr r r r


d E P E q I d

χ α α ζ χα ζ α ζ δ

κ

⎡ ⎤+ + + −⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥+ − + ≤⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦⎣ ⎦

, { }1, 2r∈ .

It follows that, for each iχ ∈ℜ , { }1, 2,3i∈ , we have

( )( )( ( )) ( ( )) 2 ( ) ( ) ( ) 0T Ti i i r r

d V t V t C t d t d tdt

χ α χ ζ χ δ κ+ − − − ≤

and

( )( ) 0Ti i iC tζ χ δ− ≤ .

Condition (3.64) implies that the Lyapunov function ( ( ))V tχ is continuous on the boundary

hypersurfaces 13S and 23S . Hence, ( ( )) 0d V tdt

χ ≤ holds whenever

2( ( )) ( ) ( )Tr rV t d t d tα χ κγ κ≥ ≥ in the whole state space. Consequently, ( ( ))V tχ cannot

ultimately exceed the value 2κ γα

, namely,

2limsup ( ( ))t

V t κχ γα→∞

≤ .

Inequality (3.67) is equivalent to

( )[ ]

02 0 0 0 0

0 0

T Ti i i i ex

T T Ti i i i i i ex ed

Ted

P q C Cq C g C D

I D

α α ηγα η α η δβ

β γ κ

⎡ ⎤ ⎡ ⎤+⎢ ⎥ ⎢ ⎥+ − − >⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦

.

Multiplying from the left by 1T Trdχ⎡ ⎤⎣ ⎦ and from the right by 1

TT Trdχ⎡ ⎤⎣ ⎦ yields, for

each iχ ∈ℜ ,

110

( ) ( )( )

( )

2

2

( ) ( ) 2 ( ) ( ) ( )

( ) .

T Ti i i r re t V t C t d t d t

V t

β βα χ η χ δ κγ γ

β βα χ κ γγ γ

⎛ ⎞⎛ ⎞< + − + −⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞

≤ + −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

Since 2limsup ( ( ))t

V t κχ γα→∞

≤ as shown before, we have

2 2 2 2limsup ( )t

e t β βκγ κ γ βγ γ→∞

⎛ ⎞⎛ ⎞< + − =⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠, iχ ∈ℜ .

Hence, we have

limsup ( )t

e t β→∞

< .

If β δ≤ as in (3.69), then after a long enough time, the output ( )e t will always be such that

( )e t β δ< ≤ and there will be no more switching. For the case of 0δ > , the system 1cl∑ is

the active closed loop system once switching stops. Since the systems 1cl∑ is asymptotically

stable and since 1 1Q QC ∈S based on (3.68), then regulation in the switched closed loop system

is achieved. The analysis is similar for the case of 0δ < .

In the following, a regulator synthesis method based on Theorem 3.7 and that makes use

of an iterative LMI algorithm will be presented. Since QA and QB in (3.6) are fixed, then the

synthesis method aims at finding controller gains rK , rL and matrices r rQ QC ∈S , { }1, 2r∈ ,

so that conditions (3.65)-(3.69) are satisfied. Since (3.65) includes the unknown parameters

rK , rL , rP , rq , rg , rη , rζ , α , κ and β , (3.65) is nonlinear in the unknown variables.

Therefore, the gains rK and rL will be designed first to yield an internally stable switched

111

closed loop system (3.54) under arbitrary switching. Using the common Lyapunov function

method, the gains rK and rL can be designed as follows:

[ ] [ ] { }

{ }

0, 0, 0, 1, 2 ,

0, 0, 0, 1, 2 .

Tr r r K K r r r k k k K

Ty yr r r L L r r r L L L L

A B K P P A B K P P r

A L C P P A L C P P r

α α

α α

+ + + + < > > ∈

⎡ ⎤ ⎡ ⎤+ + + + < > > ∈⎣ ⎦ ⎣ ⎦ (3.70)

Less conservative methods based on piecewise quadratic Lyapunov functions or polyhedral

Lyapunov functions can also be used [32,33,34]. Once the gains rK and rL are designed,

then the regulator synthesis is completed by finding matrices r rQ QC ∈S , { }1,2r∈ , that satisfy

the regulation conditions in Theorem 3.7. Let ,i nP , ,i nq , ,i ng , ,i nη , ,i nζ , nα , nκ , nβ , 1

nQC and

2

nQC denote the values of iP , iq , ig , iη , iζ , α , κ , β ,

1QC and 2QC in the thn step in the

proposed iterative synthesis algorithm, where { }1, 2,3i∈ . The main steps in the proposed

regulator synthesis procedure are summarized below.

1) Design an internally stabilizing controller: Design the controller gains rK and rL

to make the switched closed loop system (3.54) internally stable by solving (3.70) for

the unknown matrices 0TK KP P= > , 0T

L LP P= > and the gains rK , rL , { }1, 2r∈ .

2) Initialize rQ : For the case of 0δ > , first find the set 1QS by computing

1Aθ and

1Bθ

in (3.61), then initialize 1QC as

1 1

0Q QC ∈S and initialize

2QC as 2

10 qnQC ×∈R . For

example, we can set 1

0QC by solving

1min QC subject to

1 1 10T

QA C Bθ θ+ = and set

2

010

qQ nC ×= . For the case of 0δ < , follow a similar procedure.

3) Initialize α : One way to select an initial value 0 0α > for α can be as follows. First,

find an upper bound mα on the value of α by solving maxα subject to

112

0T Tr r r

T Tr

A P PA P A q qq A q g

α αα α

⎡ ⎤+ + +<⎢ ⎥

+⎣ ⎦, 0T

P qq g⎡ ⎤

>⎢ ⎥⎣ ⎦

, { }1, 2r∈ , with the unknowns TP P= ,

q and g ; and with known matrices rK , rL , and 0rQC , which is a generalized

eigenvalue problem. Then select an initial value 0α in the interval ( )0 0, mα α∈ .

4) (start the thn step). Minimize nβ subject to (3.64)-(3.67) with the unknowns ,i nP ,

,i nq , ,i ng , ,i nη , ,i nζ , nκ , and nβ ; and with known 1nα − , 1

1nQC − and

2

1nQC − .

5) Minimize nβ subject to (3.65)-(3.68) with the unknowns nα , 1

nQC ,

2

nQC , ,i nη , ,i nζ , nκ

and nβ ; and with known ,i nP , ,i nq , ,i ng .

6) If nβ δ≤ or 1n n ββ β σ− − < , where βσ is a prescribed tolerance, stop the algorithm.

Else set 1n n= + , then go to step 4.

For a given set of initial values for rK , rL , 0rQC and 0α , the above algorithm converges to a

local solution since β is guaranteed to decrease or stay the same in every iteration. Therefore,

if there is no feasible solution that satisfies nβ δ≤ , the initial values can be modified and

the algorithm can be restarted again. In the following, a numerical example is presented to

show the effectiveness of the proposed regulator synthesis method.

3.5.3 Numerical example

In this section, the regulator synthesis method will be used to design a controller that cancels

the contact vibrations in the same mechanical system as in section 2.6.

To construct a parameterized set of stabilizing controllers, the gains rK , rL , { }1, 2r∈ , are

first designed based on (3.70), which results in

113

7 21 2 1.5088 10 7.5351 10K K ⎡ ⎤= = − × − ×⎣ ⎦ ,

2 71 2 1.7969 10 1.1874 10

TL L ⎡ ⎤= = − × − ×⎣ ⎦ .

Based on the analysis of the poles of rd , we have 0 1k = and 02 2prn k= = . A rQ parameter

as in (3.49) with 4qn = is selected and is given by

,1 ,2 ,3 ,42 3 4( )

10 ( 10) ( 10) ( 10)r r r r

rQ ss s s sθ θ θ θ

= + + ++ + + +

.

Therefore, we have

10 0 0 01 10 0 00 1 10 00 0 1 10

QA

−⎡ ⎤⎢ ⎥−⎢ ⎥=⎢ ⎥−⎢ ⎥−⎣ ⎦

and

1000

QB

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

,

Since 0δ < , condition (3.60) needs to be satisfied for 2r = and we consider only the set 2QS

given by (3.60) where

2

19 21 23 25

20 21 23 24

2.6002 10 1.4806 10 5.6920 10 6.4702 105.4004 10 3.9026 10 3.2624 10 8.0294 10

Aθ

− − − −

− − − −

⎡ ⎤− × − × × ×= ⎢ ⎥− × × × − ×⎣ ⎦ ,

2

11

10

2.2489 104.5740 10

Bθ

−

−

⎡ ⎤− ×= ⎢ ⎥×⎣ ⎦

.

The initial values are set to

0 0.3α = , [ ]1

0 0 0 0 0QC = ,

2

0 8 11 12 118.0689 10 1.1509 10 1.0859 10 1.1404 10QC ⎡ ⎤= × − × × ×⎣ ⎦ .

Using the iterative algorithm in the regulator synthesis procedure, after 16 iterations, one

solution is found as

0.2637α = ,

114

1

6 11 11 113.3746 10 7.9973 10 8.3783 10 7.3212 10QC ⎡ ⎤= × − × − × ×⎣ ⎦ ,

2

8 11 12 117.2437 10 2.3512 10 1.0859 10 1.3756 10QC ⎡ ⎤= × − × × ×⎣ ⎦ ,

and 6 628.6357 10 30 10β δ− −= × < = × . Since the regulation conditions (3.19)-(3.33) are

satisfied, then the designed regulator can achieve regulation. The simulation results of the

response of the closed loop system under switching are illustrated in Figure 3.6 and 3.7,

which shows the performance variable e converging to zero even in the presence of

switching. Hence, the position v of the mass m asymptotically follows the contact surface

sv with the desired separation d ev v− .

3.6 Conclusion

The problem of regulation in bimodal switched systems against known disturbance or

reference signals is discussed. Regulator design approaches using the parameterization of a

set of stabilizing controllers for the switched system are presented using properly formulated

linear matrix inequalities. Two regulator synthesis approaches are considered based on

different regulation conditions in this chapter. The first approach is based on working with a

common Lyapunov function for the closed loop switched system. In this case, the forced

switched closed loop system involving a parameterized controller is transformed into an

unforced impulsive switched system. Consequently, the original regulation problem is

transformed into a stability analysis problem for the origin of the impulsive switched system.

Then, three sufficient conditions for regulation are presented based on a common Lyapunov

function and a regulator synthesis method is proposed by solving properly formulated LMIs.

The second approach is based on considering multiple Lyapunov functions for the closed

115

loop switched system. In this case, the regulation interpolation conditions for each subsystem

are first derived in the frequency domain. Then, based on the property of the switching

surface in the switched closed loop system, a sufficient regulation condition for the switched

closed loop system is proposed using multiple Lyapunov functions. A regulator synthesis

algorithm is finally developed by iteratively solving LMIs. A simulation example of a

mechanical system subject to contact vibrations is used to illustrate the effectiveness of the

both proposed regulator synthesis methods.

116

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-50

0

50

e (m

icro

met

ers)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-0.2

-0.1

0

0.1

0.2

d r(1,1

) (m

/sec

2 )

Time(sec)


0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-40

-30

-20

-10

0

10

20

v an

d v s (m

icro

met

ers)

Time(sec)



117

CHAPTER 4

Regulation in Switched Bimodal Systems with

an 2H Performance Constraint

4.1. Introduction

This chapter considers a regulation problem for discrete-time switched bimodal linear

systems where it is desired to achieve output regulation against known deterministic

disturbance or reference signals in the presence of unknown random disturbances. The

development of the proposed regulation method involves two main steps. In the first step, the

regulation problem for the switched system against the known deterministic disturbance or

reference signals is approached by constructing a set of observer-based Q -parameterized

stabilizing controllers that satisfy a sufficient regulation condition for the switched system. In

the second step, an 2H performance constraint is added to identify, from among the already

constructed regulators, those that provide the best 2H performance against the unknown

random disturbances. The proposed regulator is successfully evaluated on a bimodal

switched mechanical system experimental setup, hence demonstrating the effectiveness of

118

the proposed regulation approach in dealing simultaneously with known deterministic and

unknown random exogenous inputs.


problem for the switched bimodal system subject to known deterministic exogenous inputs

and unknown random exogenous inputs is presented. In section 4.3, the construction of a Q -

parameterized stabilizing set of switched controllers for the switched system is discussed. In

section 4.4, regulation conditions for each subsystem in the plant model and regulation

conditions for the resulting switched closed loop system are presented. Then, a set of

parameterized 2H controllers is constructed to deal with the unknown random exogenous

inputs. In section 4.5, a regulator synthesis method is proposed by solving some properly

formulated linear matrix inequalities. In section 4.6, a numerical example is presented to

show the effectiveness of the proposed designed method. In section 4.7, the proposed

regulator is evaluated on an experimental setup motivated by the flying height regulation

problem in hard disk drives. In the experimental setup, the tip of a flexible beam is supposed

to maintain a constant separation with respect to a surface with a known profile, while also

being subject to disturbance forces having a known deterministic component and an

unknown random component. The system exhibits a switching behavior depending on

whether contact takes place between the surface to be tracked and the tip of the beam. The

experimental results successfully demonstrate the effectiveness of the proposed approach in

achieving exact output regulation against known sinusoidal exogenous inputs and in

minimizing the effects of the random disturbance input in the presence of switching in the

system dynamics. Concluding remarks are presented in section 4.8.

119

4.2. The regulation problem for the switched bimodal system

Consider the discrete-time switched system given by the following state space representation:

0( 1) ( ) ( ) ( ) ( ), (0) , ( ) ( ) ( ) ( ), ( ) ( ) ( ) ( ), :1 if

x xr r r r r wy y yr r r r we e er r r r wr

x k A x k B u k D d k F d k x xy k C x k D d k F d ke k C x k D d k F d k

er

+ = + + + == + += + +Σ

=( ) ( ) ( ) ( ) ,

2 if ( ) ( ) ( ) ( ) ,

e e er r r r we e er r r r w

k C x k D d k F d ke k C x k D d k F d k

δδ

⎧⎪⎪⎪⎨⎪ ⎧ = + + ≤⎪ ⎨⎪ = + + >⎩⎩

(4.1)



to be measurable, { }1,2r∈ is the index of the system rΣ under consideration at time k , and

δ is a constant satisfying 0δ > . The external signal lwd ∈R represents unknown signals

such as measurement noise and/or random disturbances. The external signal hrd ∈R ,

representing the known disturbance and/or reference signals, is also assumed to switch

according to the rule given in (4.1). The signal rd is given by:

1[ ]h Tr r rd d d= , (4.2)

where each jrd is of the form:

( ) { }, , , ,1

1( ) ( ) cos ( ) ( ) , 1, 2 , 1, ,

j

j

kj r j r j r j r j

r i i i ki

d k c k k k k c r j hω φ +=

= + + ∈ =∑ , (4.3)

with known amplitudes ,r jic , frequencies ,r j

iω , and phases ,r jiφ , 1, , ji k= … ; 1, ,j h= … . In

the following, it is assumed that for any given 0k ≥ , the system must operate in only one of

the two modes corresponding to { }1,2r∈ . For the switched system (4.1), it is desired to

120

construct an output feedback controller to regulate the performance variable e of the

switched system against the external input signals rd and wd .

4.3. Parameterization of a set of stabilizing controllers

The controller design approach presented in this chapter relies on the construction of a Q -

parameterized set of output feedback stabilizing controllers for the switched system. In this

section, the construction of sets of Q -parameterized output feedback stabilizing controllers

for the switched system (4.1) is first discussed. The stability of the resulting switched closed

loop system is then analyzed. As in the case of linear time invariant systems, the construction

of a Q -parameterized set of stabilizing controllers involves two steps. The first step involves

the construction of a central controller in the form of a switched observer-based state

feedback controller. The second step involves augmenting the central controller with stable

dynamics to construct a Q -parameterized set of stabilizing controllers.

4.3.1. Q -parameterized controller

Consider following observer-based state feedback controller for the switched system (4.1):

( ) 0ˆ ˆ ˆ ˆ ˆ( 1) ( ) ( ) ( ) ( ) , (0) ,:

ˆ( ) ( ),r r ro

rr

x k A x k B u k L y k y k x xu k K x k⎧ + = + + − =⎪⎨

=⎪⎩ (4.4)

where ˆ( )x k is an estimate of the plant state vector ( )x k and ˆ ˆ( ) ( )yry k C x k= is an estimate of

the plant output ( )y k . The state feedback gains rK , { }1, 2r∈ , and the observer gains rL ,

{ }1, 2r∈ , are assumed to switch according to the rule given in (4.1). The construction of a

Q -parameterized set of stabilizing controllers for the switched system (4.1) proceeds along

121

the same lines as in the case of linear time-invariant systems as shown in chapter 3. Each

controller is expressed as a linear fractional transformation involving a fixed system rJ , and

a proper stable parameter rQ that could be chosen as desired (See Figure 4.1). The state

space representation of the system rJ is given by:

0ˆ ˆ ˆ ˆ( 1) ( ) ( ) ( ) ( ), (0) ,ˆ: ( ) ( ) ( ),

ˆ ˆ( ) ( ) ( ) ( ).

yr r r r r r r Q

r r Q

yr

x k A L C B K x k L y k B y k x x

J u k K x k y k

y k y k y k C x k

⎧ + = + + − + =⎪

= +⎨⎪

− = −⎩

(4.5)

The parameter rQ is given by:

( ) 0ˆ( 1) ( ) ( ) ( ) , (0) ,:

( ) ( ),r

Q Q Q Q Q Qr

Q Q Q

x k A x k B y k y k x xQ

y k C x k

⎧ + = + − =⎪⎨

=⎪⎩ (4.6)

where QnQx ∈ . In particular, throughout the rest of the paper, the rQ parameter is such that

QA is a fixed stability matrix, QB is a fixed matrix, and the matrix rQC changes with

{ }1, 2r∈ . For a given { }1,2r∈ , the controller formulated in (4.5) and (4.6) is a

parameterized stabilizing controller for the system rΣ , where the system rQ represents a free

design parameter that can be chosen as desired as long as it is an asymptotically stable

system. A suitable choice for rQ can be a parameterization of the form ( )1

( )qn

ir r i

i

Q z zθ ψ=

=∑ ,

where ( )i zψ are stable basis functions. Such representation can be used to approximate any

proper stable real rational transfer function. With such representation for rQ , a corresponding

realization is as given in (4.6), where QA and QB are fixed matrices, and where the

parameters , 1ir qi nθ = … , appear in

rQC . Here the rQC matrices are selected to achieve

122

Figure 4.1. Closed loop system with a Q -parameterized controller.

regulation according to regulation conditions and an 2H performance criterion to be

discussed later.

Let ˆ( ) ( ) ( )x k x k x k= − denote the state estimation error and 1

TT T TQ N

x x xχ×

⎡ ⎤= ⎣ ⎦

denote the state vector for the resulting closed loop system with 2 QN n n= + . The resulting

closed loop system is given by the following state space representation:

( 1) ( ) ( ) ( ), ( ) ( ) ( ) ( ),

:1 if ( ) ( ) ( ) ( ) ,2 if ( ) ( ) ( ) ( ) ,

d wr r r r w

ex e er r r r wcl

r ex e er r r r wex e er r r r w

k A k E d k E d ke k C k D d k F d k

e k C k D d k F d kr

e k C k D d k F d k

χ χχ

χ δχ δ

⎧ + = + +⎪

= + +⎪∑ ⎨⎧ = + + ≤⎪ = ⎨⎪ = + + >⎩⎩

(4.7)

where 00 0

rr r r r Q r r

yr Q Q r

yr r r

A B K B C B K

A A B CA L C

+⎡ ⎤⎢ ⎥

= −⎢ ⎥⎢ ⎥+⎣ ⎦

,

( ) ( ) ( )TTTd x y T x y

r r Q r r r rE D B D D L D⎡ ⎤= − +⎢ ⎥⎣ ⎦, ( ) ( ) ( )

TTTw x y T x yr r Q r r r rE F B F F L F⎡ ⎤= − +⎢ ⎥⎣ ⎦

,

0 0ex er rC C⎡ ⎤= ⎣ ⎦ .

Plant r∑

rQ

ˆy y−Qy

rJ

rd e

u y

wd

123

4.3.2 Stability of the Q - parameterized switched closed loop system

In this section, the internal stability of the closed loop system is studied by considering the

system (4.1) in the absence of the signals rd and wd , and studying the stability properties of

the origin for the resulting unforced switched system. Consider first the system (4.1) in the

absence of the signal rd and wd . The state equation for the resulting system is given by:

( 1) ( )( 1) 0 ( ) .( 1) 0 0 ( )

rr r r r Q r r

yQ Q Q r Q

yr r r

A B K B C B Kx k x kx k A B C x kx k A L C x k

+⎡ ⎤+⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥+ = −⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥+ +⎣ ⎦ ⎣ ⎦⎣ ⎦

(4.8)

Based on (4.8), define the following three subsystems:

( 1) [ ] ( ),r r rx k A B K x k+ = + (4.9)

( 1) [ ] ( ),yr r rx k A L C x k+ = + (4.10)

( 1) ( ).Q Q Qx k A x k+ = (4.11)


result.

Lemma 4.1

If the switched systems (4.9) and (4.10) are each asymptotically stable under arbitrary

switching and QA is a stability matrix, then the switched system (4.8) is also asymptotically

stable under arbitrary switching.

Proof: If the switched system (4.10) is asymptotically stable under arbitrary switching signal,

then we have:

( ) ( , 0) (0) (0) kx k k x c x aφ= ≤ ,

124

for some 0c > and (0,1)a∈ where ( ,0)kφ is the state transition matrix of the switched

system (4.10). Therefore, 0x → as k →∞ for any (0)x . Similarly, if the system (4.11) is

asymptotically stable, then we have:

( ) ( ,0) (0) (0) kQ Q Q Q Q Qx k k x c x aφ= ≤

for some 0Qc > and (0,1)Qa ∈ , and where ( ,0)Q kφ is the state transition matrix of the

system (4.11). The expression of the state variable Qx in (4.8) is given by:

1

0( ) ( ,0) (0) ( , ) ( )

ky

Q Q Q Q Q rj

x k k x k j B C x jφ φ−

=

= −∑ .

It follows

1

0

1

max0

( ) ( ,0) (0) ( , ) ( )

(0) (0)

ky

Q Q Q Q Q rj

kk y k j j

Q Q Q Q Q r Qj

x k k x k j B C x j

c x a c c x B C a a

φ φ−

=

−−

=

≤ +

≤ +

∑

∑.

Hence,

max

max

(0)( ) (0) ( ), if

1( ) (0) (0) , if

yQ Q Q rk k k

Q Q Q Q Q Q

k y kQ Q Q Q Q Q r Q Q

c ca x B Cx k c x a a a a a

ax k c x a c c x B C ka a a

⎧⎪ ≤ + − ≠⎪

−⎨⎪ ≤ + =⎪⎩

,

which yields

max2 (0)

( ) (0)1

yQ Q r k

Q Q Q m

c c x B Cx k c x a

a

⎡ ⎤⎢ ⎥≤ +

−⎢ ⎥⎣ ⎦, Qa a≠ ,

where max

yQ rB C is the maximum value of y

Q rB C and max( , )m Qa a a= . Therefore, we

have 0Qx → as k →∞ for any (0)Qx . Similarly, we have that 0x → as k →∞ for any

(0)x in (4.8).

125

Consider now the closed loop system (4.7), then we have the following result.

Lemma 4.2

Assume the system (4.8) is asymptotically stable under arbitrary switching. If the exogenous

inputs rd and wd are bounded, then the state vector χ of the system (4.7) is also bounded.

Proof: If the system (4.8) is asymptotically stable under arbitrary switching, then we have:

( ) ( ,0) (0) (0) kx x xk k c aχ φ χ χ= ≤ ,

for some 0xc > , (0,1)xa ∈ , and where ( ,0)x kφ is the state transition matrix of the switched

system (4.8). Then the state vector ( )kχ of system (4.7) is given by

1

0

( )( ) ( ,0) (0) ( , )

( )

krd w

x x r rj w

d kk k k j E E

d kχ φ χ φ

−

=

⎡ ⎤⎡ ⎤= + ⎢ ⎥⎣ ⎦

⎣ ⎦∑ .

Therefore,

1

max 0

( ) (0)k

rk d w k jx x x r r Q

j w

dk c a c E E a

dχ χ

−−

=

⎡ ⎤⎡ ⎤≤ + ⎢ ⎥⎣ ⎦

⎣ ⎦∑ ,

where max

d wr rE E⎡ ⎤⎣ ⎦ are the maximum value of d w

r rE E⎡ ⎤⎣ ⎦ , { }1, 2r∈ . From this we can

prove that ( )kχ is bounded if rd and wd are bounded.

The switched system (4.8) can be designed to be asymptotically stable using the following

result.

126

Theorem 4.1

The origin is an asymptotically stable equilibrium point for the switched system (4.8) under

arbitrary switching if QA is a stability matrix and there exist matrices 0TK KP P= > ,

0TL LP P= > , and matrices rK , rL , { }1, 2r∈ , such that:

{ }{ }

[ ] [ ] 0, , 1, 2 ,

[ ] [ ] 0, , 1, 2 .

Tr r r K r r r K

y T yr r r L r r r L

A B K P A B K P i j

A L C P A L C P i j

+ + − < ∈

+ + − < ∈ (4.12)

Proof: Condition (4.12) implies that each of the switched systems (4.9) and (4.10) admits a

common Lyapunov function. Hence, each of those two systems is asymptotically stable

under arbitrary switching. Based on Lemma 4.1, it follows that the switched system (4.8) is

asymptotically stable under arbitrary switching.

Remark 4.1

The main reason for considering common Lyapunov functions as opposed to multiple

Lyapunov functions in (4.12) is a follows. The existence of a common Lyapunov function for

each of the switched systems (4.9) and (4.10) implies that there exists a common Lyapunov

function for the system (4.8). As outlined in Section 4.4, the regulation conditions are

developed based on a common Lyapunov function for the switched closed loop system. The

use of a common Lyapunov function allows the regulation conditions to be transformed into

linear matrix inequalities (LMIs) in the regulator synthesis algorithm using an appropriate

congruence transformation, while considering multiple Lyapunov functions would have

made such transformation very difficult or impossible.

127

Since the matrix QA in (4.6) is chosen to be a stability matrix, then a parameterized set of

stabilizing controllers can be designed by solving (4.12) numerically. In the following, the

parameters in rQC in (4.6) will be chosen to satisfy the regulation conditions and an 2H

performance criterion simultaneously.

4.4. Regulation Conditions for switched system

This section presents regulation conditions for the switched closed loop system (4.7) against

the external input signals rd and wd . Due to the presence of the unknown random signal wd ,

it is not possible to achieve exact output regulation. Consequently, the goal behind the

regulator design is to achieve exact output regulation against the known deterministic

exogenous input rd and minimize the 2H norm of the closed loop system relating the input

wd to the performance variable e . It is proposed in this section to achieve these goals by

exploiting the flexibility in the selection of the Q parameter in the parameterized set of

stabilizing controllers for the switched system. More specifically, by properly selecting the

matrices rQC , { }1, 2r∈ , in (4.6), it will be possible to develop regulation conditions

corresponding to the above stated goals. The development of the regulation conditions is

performed in three steps. First, regulation conditions for each of the closed loop systems clrΣ ,

{ }1, 2r∈ , against the known deterministic input rd and in the absence of the signal wd are

reviewed. The second step determines conditions on the switched closed loop system such

that, with the controllers designed in step 1, switching in the closed loop system would stop

in a finite time in the presence of the both rd and wd . The conditions obtained in step two

128

would guarantee exact regulation in the closed loop system against rd . In the third step, the

conditions from step two are augmented with additional constraints so that the closed loop

system obtained once switching ends satisfies an 2H norm constraint. Each of the three steps

is discussed in detail in the following.

4.4.1 Regulation Conditions for 1Σ and 2Σ

This section provides the regulation conditions for each of the closed loop systems 1clΣ and

2clΣ in the presence of the deterministic signal rd and with 0wd = . The derivations presented

in the following parallel those for the continuous-time case and are included here for

completeness.

The plant rΣ and the block rJ shown in Figure 4.1 can be combined into a single system

rT . In the absence of wd , the dynamics of the system rT , { }1, 2r∈ , can be represented as

follows:

( )( 1) ( ),

( )0( 1) ( ) 0

( )( ): ( ) 0 0 ,

( )( )

( )ˆ( ) ( ) 0 0

( )

xrr r r r r r r

y x yQr r r r r r

re er r r

Q

y yr r

d kA B K B Kx k x k D By kA L Cx k x k D L D

d kx kT e k C D

y kx k

x ky k y k C D

x k

++ ⎡ ⎤⎡ ⎤⎡ ⎤⎡ ⎤ ⎡ ⎤= + ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥++ − −⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦

⎡ ⎤⎡ ⎤⎡ ⎤ ⎡ ⎤= + ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦

⎡ ⎤⎡ ⎤ ⎡− = − +⎢ ⎥⎣ ⎦ ⎣⎣ ⎦

( ).

( )r

Q

d ky k

⎧⎪⎪⎪⎪⎨⎪⎪ ⎡ ⎤⎪ ⎤ ⎢ ⎥⎦⎪ ⎣ ⎦⎩

(4.13)

Therefore, we have that:

( )( )( )ˆ( ) ( )

rr

Q

d ke kT

y ky k y k⎡ ⎤⎡ ⎤

= ⎢ ⎥⎢ ⎥−⎣ ⎦ ⎣ ⎦, (4.14)

where

129

11 12

21 22r r

rr r

T TT

T T⎡ ⎤

= ⎢ ⎥⎣ ⎦

, (4.15)

and where, based on (4.13), the state space representations for 11rT , 12

rT , 21rT and 22

rT are as

11 1111

11 11: 00

xr r r r r r

r r y x yr r r r r r r

r r e er r

A B K B K DA E

T A L C D L DC D

C D

⎡ ⎤+⎡ ⎤ ⎢ ⎥

= + − −⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎢ ⎥

⎣ ⎦

,

12 1212

12 12: 0r r r rr r

errr r

A B K BA ET CC D

+⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎣ ⎦⎣ ⎦,

21 2121

21 21:y x y

r r r r r r r rr y y

r r r r

A E A L C D L DT

C D C D⎡ ⎤ ⎡ ⎤+ − −

=⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦,

22 2222

22 22

0: .

0

yr r r r r

r yr r r

A E A L CT

C D C⎡ ⎤ ⎡ ⎤+

=⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

Let ( )rD z , ( )jrD z and ( )E z denote the Z transforms of the inputs rd , j

rd and the

performance variable e , respectively. Let ,x jrD , ,e j

rD , and ,y jrD denote the thj columns of

the matrices xrD , e

rD , and yrD respectively. Let 1

, , ,( ) ( ), , ( )r r r r r r

hT Q T Q T QF z F z F z⎡ ⎤= ⎣ ⎦ denote

the closed loop system transfer function relating ( )rD z to ( )E z . It follows that:

,

,1

( ) ( ) ( ),

( ) ( ),

r r

r r

T Q r

hj j

T Q rj

E z F z D z

F z D z=

=

=∑ (4.16)

where 11, 12 21,, ( ) ( ) ( ) ( ) ( )

r r

j j jT Q r r r rF z T z T z Q z T z= + , and where the corresponding state space

representations for 11, jrT , 12 ( )rT z and 21, j

rT are as given similarly in section 3.4.2. Consider

the following representation of ( )rQ z :

1( ) ( )

qni

r r ii

Q z zθ ψ=

=∑ ,

130

where ( ) ( )1 ii z z F zψ −= , 1, qi n= , are stable basis functions, and where

11

11

( )m

mm m

m

b z bF zz a z a

−

−

+=

+ + is a stable function used to adjust the dynamic properties of ( )rQ z .

Let 1, , qTn

r r rθ θ θ⎡ ⎤= ⎣ ⎦… and 1Q qn m n= + − . A realization of ( )rQ z can then be given as

follows:

1

2

1

0 0 0 0 0 01 0 0 0 0 0

0 1 0 0 0 00 0 1 0 0 00 0 0 1 0 0 00 0 0 0 0 00 0 0 0 0 1 0

Q Q

m

m

Q

n n

aa

aA

a

−

×

−⎡ ⎤⎢ ⎥−⎢ ⎥⎢ ⎥⎢ ⎥−⎢ ⎥= ⎢ ⎥−⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

,

1

2

1

1

0

0Q

m

m

Q

n

bb

bB

b

−

×

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

,

( )1 10Q Tr rmC θ× −

⎡ ⎤= ⎣ ⎦ . (4.17)

According to the properties of jrd in (4.3), all the poles of ( )j

rD z , 1, ,j h= , are located on

the unit circle. For a given { }1, 2r∈ , some of the disturbances ( )jrD z , 1, ,j h= , may have

common poles. Let prn denote the total number of the different poles in the expression for

( )rD z . Let rp , 1, , ,prn= denote such poles. Let rS denote the set of disturbance indices

j such that rp is a pole of ( )jrD z . Assume the function ( )rD z has 0k distinct pairs of

complex conjugate poles on the unit circle and one pole at 1 (the pole related to the

disturbance offset in (4.3)), so that the total number of the different poles is 02 1prn k= + . Let

rp + , 01, ,k= , denote the 0k distinct complex poles in the top half of the complex plane.

131

Denote ( )jr rR p the residue of ( )j

rD z at the pole rp . Then we have the following

interpolation conditions for regulation.

Theorem 4.2

Assume, for each { }1, 2r∈ , the closed loop system clrΣ in (4.7) is internally stable. For any

given { }1, 2r∈ , the closed loop system clrΣ in (4.7) achieves regulation if and only if the

parameter vector rθ in rQC satisfies the following affine constraint:

0r rrA Bθ θθ + = , (4.18)

where the matrix r

Aθ and the vector r

Bθ are given by:

00 0

0 0

1 1 10 1, ,,

1 1 1 0 1, , ,

01,, ,

1,, ,

1

( ) ( ) ( )( ) ( ) ( )

, ( )( ) ( )

( ) ( )

(1) (1)

q

q

r rq

q

q

nr re r r re r

r re rn

r im r r im r r im r

n kk kr re rr re r r re r

nk kr ir im r r im r

nr r

V p V p V pV p V p V p

A BV pV p V pVV p V p

V V

θ θ

+ ++

+ + +

++ +

+ +

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥= =⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

00

0

,

( )(1)

km r

r

pV

+

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

(4.19)

and where , ( )ir re rV p + and , ( )i

r im rV p + denote, respectively, the real and imaginary parts of

( )ir rV p + given by:

12 21,0( ) ( ) ( ) ( ) ( ), 1, , , 1, , ,

r

i j jr r r r r r i r r r q

j

V p R p T p p T p k i nψ+ + + + +

∈

= = =∑S

0 11,0( ) ( ) ( ), 1, , .

r

j jr r r r r r

j

V p R p T p k+ + +

∈

= =∑S

Proof: The Z transform of the closed loop system is given by

132

,1

( ) ( ) ( )r r

hj j

T Q rj

E z F z D z=

= ∑ .

According to (4.3), the poles of ( )jrD z are all located on the unit circle. Using partial fraction

expansion, the response ( )e k can be expressed as follows:

1, 0

1

1( ) ( ) ( ) ( )pr

r r

r

nj j

r r T Q rjr

e k Z R p F p e kz p

−

= ∈

⎡ ⎤⎛ ⎞= +⎢ ⎥⎜ ⎟⎜ ⎟−⎢ ⎥⎝ ⎠⎣ ⎦

∑ ∑S

, (4.20)

where 1Z − is the inverse Z transform operator, 0 ( )e k denotes the sum of responses

corresponding to partial fractions with , ( )r rT QF z poles and the response to non-zero initial

conditions. Since for each { }1, 2r∈ the closed loop system is asymptotically stable, the

response to nonzero initial conditions converges asymptotically to zero. Also, since

, ( )r rT QF z RH∞∈ , the space of real rational stable transfer matrices, then the responses

corresponding to partial fractions with , ( )r rT QF z poles asymptotically converge to zero.

Therefore, the asymptotic properties of the disturbance response are determined by the first

term on the right hand side of (4.20):

1,

1

1( ) ( ) ( )pr

r r

r

nj j

r r T Q rjr

e k Z R p F pz p

−

= ∈

⎡ ⎤⎛ ⎞′ = ⎢ ⎥⎜ ⎟⎜ ⎟−⎢ ⎥⎝ ⎠⎣ ⎦

∑ ∑S

.

Necessity: Since all the poles rp are on the unit circle and are distinct, if lim ( ) 0k

e k→∞

′ = , then

it follows that:

,( ) ( ) 0, 1,r r

r

j j pr r T Q r r

j

R p F p n∈

= =∑S

,

or equivalently

( )11, 12 21,( ) ( ) ( ) ( ) ( ) 0, 1, ,r

j j j pr r r r r r r r r r r

j


+ = =∑S

,

133

which means that the following interpolation conditions must be satisfied.

( ) { }11, 12 21,( ) ( ) ( ) ( ) ( ) 0, 1, , , 1, 2r

j j j pr r r r r r r r r r r

j

R p T p T p Q p T p n r∈

+ = = ∈∑S

, (4.21)

Using the expression for rQ given in (4.21), the interpolation conditions can be rewritten as:

11, 12 21,

1( ) ( ) ( ) ( ) ( ) 0, 1, ,

q

r

nj j i j p

r r r r r r r i r r r rij

R p T p T p p T p nθ ψ=∈

⎛ ⎞+ = =⎜ ⎟⎜ ⎟

⎝ ⎠∑ ∑S

. (4.22)

Define:

0 11,( ) ( ) ( ), 1, ,r

j j pr r r r r r r

j

V p R p T p n∈

= =∑S

,

12 21,( ) ( ) ( ) ( ) ( ), 1, , , 1, ,r

i j j pr r r r r r i r r r r q

j

V p R p T p p T p n i nψ∈

= = =∑S

.

Then (4.22) can be written as

0

1( ) ( ) 0, 1, , 1, ,

qni i p

r r r r r r qi

V p V p n i nθ=

+ = = =∑ .

Since we have ( ) ( )i ir rV z V z= , 1, , qi n= , where z is the complex conjugate of z , the

interpolation conditions corresponding to the 02k complex conjugate poles are equivalent to

the 0k interpolation conditions corresponding to the 0k poles rp + , 01, ,k= . For any

complex number z , let , ( )ir reV z and , ( )i

r imV z denote, respectively, the real and imaginary parts

of the functions ( )irV z , 1, , qi n= . The interpolation condition evaluated at a pole rp + ,

01, , k= , can be written as:

0 0, , , ,

1 1( ) ( ) ( ) ( ) 0

q qn ni i i i

r re r r r re r r im r r r im ri i

V p V p j V p V pθ θ+ + + +

= =

⎡ ⎤ ⎡ ⎤+ + + =⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦∑ ∑ .

Hence, for each such pole, the above equation yields two linear equations of the form:

134

0, ,

1

0, ,

1

( ) ( ) 0,

( ) ( ) 0.

q

q

ni i

r re r r r re ri

ni i

r im r r r im ri

V p V p

V p V p

θ

θ

+ +

=

+ +

=

⎡ ⎤+ =⎢ ⎥

⎣ ⎦⎡ ⎤

+ =⎢ ⎥⎣ ⎦

∑

∑ (4.23)

Therefore, considering the 0k poles rp + selected as mentioned above yields 02k linear

equations in the unknowns irθ , 1, , qi n= . On the other hand, the interpolation condition

evaluated at the pole at 1 can be written in the form:

0

1(1) (1) 0

qni i

r r ri

V Vθ=

⎡ ⎤+ =⎢ ⎥

⎣ ⎦∑ .

Therefore, the interpolation conditions (4.21) are equivalent to the following affine constraint

on the parameter vector rθ :

0r rrA Bθ θθ + = ,

where the pr qn n× matrix

rAθ and the 1p

rn × vector r

Bθ are given in (4.19).

Sufficiency: For a given ,r rT QF and according to the expressions given above for r

Aθ and r

Bθ ,

satisfying (4.18) implies that (4.20) is satisfied and that regulation can be achieved.

4.4.2 Regulation condition for the switched system

The previous section presented the construction of rQC matrices that yield regulation in each

of the closed loop systems clrΣ , { }1, 2r∈ , against the deterministic exogenous input rd .

However, such rQC matrices do not guarantee achieving regulation in the switched closed

loop system, even when 0wd = . In this section, based on the regulation condition (4.18) for

135

each of the two subsystems in (4.7), regulation conditions for the switched closed loop

system (4.7) are presented in the following result.

Theorem 4.3

Consider the closed loop system (4.7) subject to bounded inputs rd and wd . Let

{ } ( )2 2

1,20

max ( ) ( ) 0r wrk

d k d kκ∈≥

= + ≠ , and 0α > a preset constant and 0 1ε< . If for

{ }1, 2r∈ , there exist a matrix 0P > and positive scalars μ and β such that the following

matrix inequalities are satisfied,

( ) ( )( ) 0,1

T T d wr r r r r

T Td w d w d wr r r r r r r

A PA P P A P E E

E E PA E E P E E I

α

ε μκ

⎡ ⎤⎡ ⎤− + ⎣ ⎦⎢ ⎥<⎢ ⎥−

⎢ ⎥⎡ ⎤ ⎡ ⎤ ⎡ ⎤ −⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦

(4.24)

( )( ) ( )

( )

0

0 0

Texr

Ter

Ter

ex e er r r

P C

DI

F

C D F I

α

β μκ

β

⎡ ⎤⎢ ⎥⎢ ⎥⎡ ⎤−⎢ ⎥⎢ ⎥ >⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦⎢ ⎥

⎡ ⎤⎢ ⎥⎣ ⎦⎣ ⎦

(4.25)

1 if 0,0,

2 if 0,r rrA B rθ θ

δθ

δ>⎧

+ = = ⎨ <⎩ (4.26)

β δ≤ . (4.27)

then the switched closed loop system (4.7) achieves regulation against rd .

136

Proof: Consider the state space representation (4.7) and the quadratic function ( ) TV Pχ χ χ= .

Then for any nonzero vector 2 1

( )( )( )

N hr

w

kd kd k

χ+ +

⎡ ⎤⎢ ⎥∈⎢ ⎥⎢ ⎥⎣ ⎦

R , we have:

( ) ( )( )

( ) ( )( ) ( ) 01( ) ( )

T T T d wr r r r r

r rT Td w d w d wr r r r r r rw w

k kA PA P P A P E Ed k d k

E E PA E E P E E Id k d k

χ χα

ε μκ

⎡ ⎤⎡ ⎤ ⎡ ⎤⎡ ⎤− + ⎣ ⎦⎢ ⎥⎢ ⎥ ⎢ ⎥ <⎢ ⎥⎢ ⎥ ⎢ ⎥−⎢ ⎥⎡ ⎤ ⎡ ⎤ ⎡ ⎤ −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦⎣ ⎦

.

It follows from the above inequality that for switched system (4.7):

( ) ( ) ( )1( ( 1)) ( ( )) ( ( ))

( ) ( )

Tr r

w w

d k d kV k V k V k

d k d kε μ

χ χ α χκ− ⎡ ⎤ ⎡ ⎤

+ − < −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

. (4.28)

Hence, ( ( 1)) ( ( )) 0V k V kχ χ+ − < holds whenever

( ) ( ) ( )2 21( ( )) 1 ( ) ( )r wV k d k d k

ε μα χ ε μ

κ−

≥ − ≥ + .

Consequently, ( ( ))V kχ cannot ultimately exceed the value ( )1 ε μα−

, and we have:

( )1lim ( ( ))t

V kε μ

χα→∞

−≤ .

Therefore, there exists a finite time bk such that

( ( )) , bV k k kμχα

≤ ∀ ≥ (4.29)

Using Schur complement formula, inequality (4.25) is equivalent to:

( )

( )( )( )

01 0

0

Texr

T ex e eer r rr

Ter

CPC D FD

IF

αβ μ βκ

⎡ ⎤⎢ ⎥⎡ ⎤⎢ ⎥⎢ ⎥ ⎡ ⎤− >− ⎣ ⎦⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦⎢ ⎥⎣ ⎦

.

137

Multiplying from the left by ( )( )( )

T

r

w

kd kd k

χ⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

and from the right by ( )( )( )

r

w

kd kd k

χ⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

yields:

( ) ( )

( )( )

2 ( ) ( )( ) ( )

( ) ( )

< ( ) .

Tr r

w w

d k d ke k V k

d k d k

V k

β μβ α χ

κ

β α χ β μ

⎛ ⎞− ⎡ ⎤ ⎡ ⎤⎜ ⎟< + ⎢ ⎥ ⎢ ⎥⎜ ⎟⎣ ⎦ ⎣ ⎦⎝ ⎠

+ −

Then, based on (4.29), we have:

2 2( ) , be k k kβ< ∀ ≥

Hence, ( )e k is ultimately bounded and we have:

( )e k β< , bk k∀ ≥ (4.30)

If (4.27) is satisfied, then after a long enough time bk , the output ( )e k will always be such

that ( )e k β δ< ≤ and there will be no more switching. For the case of 0δ > , the origin is

in the region where the system 1cl∑ is active. Since the systems 1

cl∑ is asymptotically stable

and since the parameters 1θ satisfy the regulation interpolation condition in (4.26), then

regulation for the switched system is achieved. The analysis is similar for the case of 0δ < .

4.4.3 Regulation condition for the switched closed loop system clrΣ with an 2H

performance constraint

The previous section presented conditions on the switched closed loop system so that the

states and the performance variable remain bounded and exact output regulation is achieved

against rd . However, the closed loop system may not achieve optimal performance against

wd . If the conditions of Theorem 4.3 are satisfied, then after a long enough time bk , there

138

will be no more switching in the closed loop system, and only one of the two systems 1cl∑ or

2cl∑ will be active for bk k≥ . Consequently, in order to minimize the effects of the unknown

random disturbance wd on the performance variable e , it is proposed to add an 2H

performance constraint in the design of the regulator for the closed loop system that is active

for bk k≥ .

In the following, some standard results on the 2H norm of systems are reviewed [94,95].

Consider in the closed loop system (4.7). Let wT denote the transfer function from wd to e

for 1cl∑ for 0δ > or 2

cl∑ for 0δ < . The 2H norm of wT is defined by

( )22

2 0

1 Tr ( ) ( )2

j jw w wT T e T e d

π ω ω ωπ

∗≡ ∫ . (4.31)

It is well known that 2wT γ< if and only if ( ) ( )( ) 2Tr

T Te e w wr r r rF F E XE γ+ < , where 0X >

satisfies

( ) 0TT ex ex

r r r rA XA X C C− + < , 1 if 0,2 if 0.

rδδ>⎧

= ⎨ <⎩

Using the Schur complement formula, we have the following result to calculate the 2H norm.

Lemma 4.3

The 2H norm of wT is lower than γ if and only if there exist real symmetric matrices 0X >

and 0S > such that

( )0

0

0

rTT ex

r r

exr

X XA

A X X C

C I

⎡ ⎤−⎢ ⎥⎢ ⎥− <⎢ ⎥

−⎢ ⎥⎣ ⎦

, 1 if 0,2 if 0,

rδδ>⎧

= ⎨ <⎩ (4.32)

139

( ) ( )

00 0

wr

er

T Tw er r

X XEI F

E X F S

⎡ ⎤⎢ ⎥⎢ ⎥ >⎢ ⎥⎢ ⎥⎣ ⎦

, (4.33)

( ) 2Tr S γ< . (4.34)

The 2H performance conditions (4.32)-(4.34) are then used in the following result to

summarize the performance of the closed loop switched system.

Theorem 4.4

Consider the closed loop system (4.7) subject to bounded inputs rd and wd . Assume the

conditions of Theorem 4.3 are satisfied. Moreover, assume there exist real symmetric

matrices 0X > and 0S > such that the 2H performance conditions (4.32)-(4.34) are

satisfied. Then the switched closed loop system achieves exact output regulation against the

known deterministic input rd and satisfies 2wT γ< with respect to the unknown random

input wd .

4.5. Regulator synthesis

Based on the above regulation conditions and 2H performance criterion, a regulator

synthesis method for the switched system is presented in the following. The synthesis method

aims at finding the proper rK , rL and rQC such that that conditions (4.24)-(4.27) and (4.32)-

(4.34) are satisfied. Since rA includes all the unknown parameters rK , rL and rQC , (4.24) is

nonlinear in the unknown variables. This poses a major difficulty in designing the regulator

using rA directly. But it is worth pointing out that, based on Theorem 4.1, a parameterized

set of stabilizing controllers can be designed by separately designing the individual systems

140

(4.9), (4.10) and (4.11). Therefore, the gains rK and rL can be designed first based on (4.12).

Since QA and QB in (4.24) are fixed, then it is only necessary to find the matrices rQC such

that the regulation conditions (4.24)-(4.27) and (4.33) are satisfied. Since condition (4.24)

and (4.33) are still bilinear matrix inequalities in the unknown variable rQC , a congruence

transformation will be used first to transform (4.24) and (4.33) into a linear matrix inequality.

First, using Schur complement formula, (4.24) is equivalent to

( )( )

0

10 0

T

r

Td wr r

d wr r r

P P A P

I E E P

PA P E E P

α

ε μκ

⎡ ⎤− +⎢ ⎥⎢ ⎥−⎢ ⎥⎡ ⎤− <⎣ ⎦⎢ ⎥⎢ ⎥⎡ ⎤ −⎣ ⎦⎢ ⎥⎣ ⎦

(4.35)

We partition P in (4.35) as [ ]1 2

2 3

n nT

P P

P P×

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

and define 11 1P−Ω = , 1

2 1 2P P−Ω = ,

13 3 2 1 2

TP P P P−Ω = − and 1

2

0T I

Ω⎡ ⎤Ω = ⎢ ⎥−Ω⎣ ⎦

. Then we have:

121

32

121 2

133 2 1 2

111

133 2 1 2

0,

0

,00

00,

00

T

T

TT

IP

I

II P PP

P P P P

PP

P P P P

−

−

−

−

−

ΩΩ ⎡ ⎤⎡ ⎤= ⎢ ⎥⎢ ⎥ Ω−Ω⎣ ⎦ ⎣ ⎦

Ω⎡ ⎤ ⎡ ⎤Ω = =⎢ ⎥ ⎢ ⎥Ω− ⎣ ⎦⎣ ⎦

Ω⎡ ⎤ ⎡ ⎤Ω Ω = =⎢ ⎥ ⎢ ⎥Ω− ⎣ ⎦⎣ ⎦

(4.36)

and

141

[ ] [ ]

( )

1 2 2

3

1 2 3

0,

00

, , , .

r

r

Q Q rr r r r r r r Q r r

r r rTr

Q Q r

r r r

r Q

A B CA B K A B K B C B K

A L CPA

A B CA L C

M C

⎡ ⎤−⎡ ⎤ ⎡ ⎤+ Ω Ω − + Ω +⎢ ⎥⎢ ⎥ ⎣ ⎦+⎣ ⎦⎢ ⎥Ω Ω = ⎢ ⎥−⎡ ⎤⎢ ⎥Ω ⎢ ⎥+⎢ ⎥⎣ ⎦⎣ ⎦

Ω Ω Ω

It is obvious that if 1 0Ω > and 3 0Ω > , then 0P > . Multiplying (4.35) from the left side by

( ), ,diag IΩ Ω and from the right side by ( ), , Tdiag IΩ Ω yields:

( ) ( )( )

( )

11 2 3

3

2 3

2 11 2 3

3 3

01 0 , , ,

0

010 0

0, , ,

0 0

r

r

Tr Q

Td wr r T

d wr Q r r

M C

II E E

IM C E E

α

ε μκ

⎡ ⎤Ω⎡ ⎤− Ω Ω Ω⎢ ⎥⎢ ⎥Ω⎣ ⎦⎢ ⎥

⎢ ⎥− ⎡ ⎤⎢ ⎥⎡ ⎤− <⎢ ⎥⎣ ⎦ Ω Ω⎢ ⎥⎣ ⎦⎢ ⎥

Ω Ω⎡ ⎤ ⎡ ⎤⎢ ⎥⎡ ⎤Ω Ω Ω −⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥Ω Ω⎣ ⎦ ⎣ ⎦⎣ ⎦

(4.37)

Multiplying (4.25) from the left side by ( ), ,diag I IΩ and from the right side by

( ), ,Tdiag I IΩ yields:

( )

( ) ( )( )

1 1

3 2

1 2

0 00

0

0 0

0

TexrT

Ter

Ter

ex e er r r

CI

DI

F

C D F II

α

β μκ

β

⎡ ⎤Ω Ω⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥Ω −Ω⎣ ⎦⎣ ⎦⎢ ⎥⎢ ⎥⎡ ⎤⎢ ⎥− ⎢ ⎥ >⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦⎢ ⎥⎢ Ω −Ω ⎥⎡ ⎤

⎡ ⎤⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦⎣ ⎦

(4.38)

142

Similarly, we partition X in (4.33) as [ ]1 2

2 3

n nT

X X

X X×

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

and define 11 1X −Ψ = , 1

2 1 2X X−Ψ = ,

13 3 2 1 2

TX X X X−Ψ = − and 1

2

0T I

Ψ⎡ ⎤Ψ = ⎢ ⎥−Ψ⎣ ⎦

. Multiplying (4.32) from the left side by

( ), ,diag IΨ Ψ and from the right side by ( ), ,T Tdiag IΨ Ψ yields:

( )

( ) ( )

11 2 3

3

1 11 2 3

3 2

1 2

, , ,

, , , 0

r

r

r Q

TT exr Q rT

exr

M C

M C CI

C II

⎡ ⎤Ψ⎡ ⎤− Ψ Ψ Ψ⎢ ⎥⎢ ⎥Ψ⎣ ⎦⎢ ⎥

⎢ ⎥Ψ Ψ⎡ ⎤ ⎡ ⎤⎢ ⎥Ψ Ψ Ψ − <⎢ ⎥ ⎢ ⎥Ψ −Ψ⎢ ⎥⎣ ⎦⎣ ⎦⎢ ⎥

Ψ −Ψ⎡ ⎤⎢ ⎥−⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦

00

0

0 00

00

,1 if 0,2 if 0.

rδδ>⎧

= ⎨ <⎩

(4.39)

Multiplying (4.33) from the left side by ( ), ,diag I IΨ and from the right side by

( ), ,Tdiag I IΨ yields:

( ) ( )

1 2

3 3

2 3

00

0

wr

we

T Tw wr eT

IE

I DI

E D S

⎡ Ψ Ψ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥Ψ Ψ⎣ ⎦ ⎣ ⎦⎢ ⎥⎢ ⎥ >⎢ ⎥

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥Ψ Ψ⎣ ⎦⎣ ⎦

00

00 , (4.40)

( ) 2Tr S γ< . (4.41)

Note that (4.37) and (4.38) are linear in the unknown parameters β , μ , 1Ω , 2Ω , 3Ω and

rQC only if α is prefixed. In the following, a regulator synthesis algorithm is presented to

obtain rQC matrices that simultaneously minimize the 2H performance γ and satisfy the

regulation conditions in Theorem 4.3. The iterative algorithm converges to a local solution

143

related to an initial value of α . Let nμ , 1,nΩ , 2,nΩ , 3,nΩ , 1,nΨ , 2,nΨ , 2,nΨ , nS , ,Qr nC and nα

denote the values of μ , 1Ω , 2Ω , 3Ω , 1Ψ , 2Ψ , 2Ψ , S , rQC and α in the thn iteration of the

algorithm. The main steps in the proposed regulator synthesis procedure are summarized

below.

1) Design of an internally stabilizing controller: Design the controller parameters

rK and rL to make the switched closed loop system (4.7) internally stable by

solving the LMIs (4.12) for the unknown matrices 0TK KP P= > , 0T

L LP P= > and

matrices rK , rL , { }1, 2r∈ .

2) Initialize α : Find an upper bound mα on the value of α in (4.37) with unknown

1 0Ω > , 2Ω , 3 0Ω > ,and rQC , which is a generalized eigenvalue problem and can be

solved efficiently in Matlab LMI toolbox. Select an initial value 0α for α in the

interval ( )0, mα α∈ .

3) Substitute β by δ in (4.38).

4) (start the thn iteration for nS ). Minimize ( )nTr S subject to (4.26), (4.27),

(4.37)， (4.38), (4.39) and (4.40) in the unknown nμ , 1,nΩ , 2,nΩ , 3,nΩ , 1,nΨ , 2,nΨ ,

2,nΨ , nS , and ,Qr nC with known 1nα − .

5) Minimize ( )nTr S subject to (4.26), (4.27), (4.37), (4.38), (4.39) and (4.40) in the

unknown nμ , 1,nΩ , 2,nΩ , 3,nΩ , 1,nΨ , 2,nΨ , 2,nΨ , nS , ,Qr nC and nα with known 1,nΩ

and 3,nΩ .

6) If ( ) ( )1 2n nTr S Tr S σ−− < , a prescribed tolerance, stop the algorithm.

144

For a given initial value 0α of α , the above algorithm converges to a local solution since

( )nTr S is guaranteed to decrease or stay the same in every iteration. Therefore, if there is no

proper feasible solution that can be found, the initial value 0α can be reset and the algorithm

can be restarted again. In the following, a numerical example is presented to show the

effectiveness of the proposed regulator synthesis method.


In this section, the regulator synthesis method proposed in this chapter will be used to design

a controller that cancels the contact vibrations in the mechanical system discussed in section

2.6. It is assumed that the system is subject to both deterministic and random disturbances.

First, this system is discretized to obtain a zero-order-hold equivalent model with sampling

period 0.001sT s= as in (4.1) with

3 6

1 3

945.6832 10 976.8748 10107.4562 935.9144 10

A− −

−

⎡ ⎤× ×= ⎢ ⎥− ×⎣ ⎦

, 3 6

2 3

995.0125 10 995.8425 109.9584 990.0333 10

A− −

−

⎡ ⎤× ×= ⎢ ⎥− ×⎣ ⎦

,

9

1 6

493.7891 10976.8748 10

B−

−

⎡ ⎤×= ⎢ ⎥×⎣ ⎦

, 9

1 6

498.7520 10995.8425 10

B−

−

⎡ ⎤×= ⎢ ⎥×⎣ ⎦

,

9

1 6

493.7891 10 0976.8748 10 0

xD−

−

⎡ ⎤×= ⎢ ⎥×⎣ ⎦

, 9

2 6

498.7520 10 0995.8425 10 0

xD−

−

⎡ ⎤×= ⎢ ⎥×⎣ ⎦

,

10

1 7

493.7891 10976.8748 10

xF−

−

⎡ ⎤×= ⎢ ⎥×⎣ ⎦

, 10

2 7

498.7520 10995.8425 10

xF−

−

⎡ ⎤×= ⎢ ⎥×⎣ ⎦

,

[ ]1 2 1 2 1 0e e y yC C C C= = = = , 51 2 1 2 0 1 10e e y yD D D D −⎡ ⎤= = = = − ×⎣ ⎦ , [ ]1 2 1 2 0e e y yF F F F= = = = ,

1

0.0747sin(20 ) 0.175cos(20 )0.1cos(20 )

k kd

kπ π

π+⎡ ⎤

= ⎢ ⎥⎣ ⎦

, 2

0.075sin(20 ) 0.075cos(20 )0.1 cos(20 )

k kd

kπ π

π+⎡ ⎤

= ⎢ ⎥×⎣ ⎦.

145

The switching rule is defined as 6

6

1 if 50 102 if 50 10

er

e

−

−

⎧ ≤ − ×= ⎨

> − ×⎩ and wd is a unknown random

disturbance signal with 2max 0.2m / secwd ≤ . To construct a parameterized set of

stabilizing controllers, the gains rK , rL , { }1, 2r∈ , are designed based on (4.12) resulted in:

[ ] 31 42.7610 1.1425 10K = − × , [ ] 3

2 56.2912 1.1810 10K = − − × ,

[ ]1 1.6118 22.8644 TL = − , [ ]2 1.6405 76.0713 TL = − − .

Based on the analysis of the poles of rd , we have 0 1k = and 2prn = . Therefore, the base

stabilizing controller is augmented with a parameter rQ with dimension 3qn = and given by

1 2 3

2 3r r r

rQz z zθ θ θ

= + + . The parameters in the matrices rQC , { }1, 2r∈ , needed to achieve

regulation for each subsystem are computed based on the regulation condition (4.18). Since

0δ < , condition (4.18) needs to consider only the system 2Σ given by:

2

122.6190 2.7779 2.925710

2.6117 2.4421 2.2628Aθ

−− − −⎡ ⎤= ×⎢ ⎥− − −⎣ ⎦

, 2

65.055110

5.6455Bθ

−−⎡ ⎤= ×⎢ ⎥−⎣ ⎦

.

Based on the disturbance signals, we have 0.64κ = . First, without considering the 2H

performance criterion in the regulator synthesis procedure, the regulator is designed to only

satisfy the regulation conditions (4.26), (4.37) and (4.38). We have

1

6 3 62.7121 10 450.4848 10 1.2831 10QC ⎡ ⎤= − × − × ×⎣ ⎦ ,

2

6 3 62.6866 10 547.9511 10 1.1974 10QC ⎡ ⎤= − × − × ×⎣ ⎦ .

Based on the above designed controllers, the simulation results are shown in Figures 4.2-4.4.

Now consider the 2H performance criterion in the regulator synthesis procedure, then using

the iterative algorithm with an initial value 0 0.03α = , and after 8 iterations, we obtain

146

1

6 3 33.3810 10 709.5338 10 543.1366 10QC ⎡ ⎤= − × × ×⎣ ⎦ ,

2

6 3 33.3022 10 679.4347 10 583.2225 10QC ⎡ ⎤= − × × ×⎣ ⎦ ,

0.031α = , 61.5413 10γ −= × .

Based on the above designed optimal parameterized controllers, the simulation results are

shown in Figures 4.5 and 4.6. Compared with the results in Figure 4.2 and 4.4, it can be seen

that the noise attenuation in the performance variable e has been improved by using the

proposed optimal parameterized regulator within a class of 2H controllers.

147

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-60

-50

-40

-30

-20

-10

0

10

20

e (m

icro

met

ers)

Time(sec)

Figure 4.2. Simulation results showing the performance variable e obtained using the controller designed without accounting for the 2H performance constraint.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-0.2

-0.1

0

0.1

0.2

d r(1,1

) (m

/sec

2 )

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-0.2

-0.1

0

0.1

0.2

dw (m

/sec

2 )

Time(sec)

Figure 4.3. Simulation results showing the switching component (1,1)rd in the known input

rd and the random disturbance wd .

148

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-60

-50

-40

-30

-20

-10

0

10

20

v an

d v s (m

icro

met

ers)

Time(sec)


Figure 4.4. Simulation results showing the displacement of the mass obtained using the controller designed without accounting for the 2H performance constraint.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-60

-50

-40

-30

-20

-10

0

10

20

e (m

icro

met

ers)

Time(sec)

Figure 4.5. Simulation results showing the performance variable e obtained using the controller designed based on the 2H performance constraint.

149

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-60

-50

-40

-30

-20

-10

0

10

20

v an

d v s (m

icro

met

ers)

Time(sec)


Figure 4.6. Simulation results showing the displacement of the mass obtained using the controller designed based on the 2H performance constraint.

4.7. Experimental evaluation

In this section, a bimodal switched system experimental setup is used to evaluate the

performance of the proposed adaptive regulator. The experimental setup simulates a

mechanical system subject to intermittent contact with a contact surface, and is motivated by

the flying height regulation problem in hard disk drives. The experimental setup is first

described in the following section. The identification of the switched system model is then

discussed followed by a presentation of the regulation experimental results.

150

4.7.1. Description of the experimental setup

A schematic diagram of the experimental setup is shown in Figure 4.7. The experimental

setup consists of a flexible beam corresponding to a suspension beam taken from a hard disk

drive. The tip of the suspension beam is supposed to track, at a fixed distance, an unknown

surface profile, while being subject to an unknown external disturbance force and to an

intermittent force representing a contact force with the surface being tracked. A multilayer

Lead Zirconate Titanate (PZT) piezoelectric actuator (Model PL 122.11, Physic Instrumente)

is attached to the suspension beam and serves to adjust the position of the beam tip. The PZT

actuator is driven by a power amplifier (E-650 LVPZT amplifier, Physic Instrumente). The

real-time measurement of the absolute tip position as well as the tip velocity is performed

using a Laser Doppler Vibrometer (LDV) (Polytec OFV-072, OFV-552, OFV-5000). A small

size permanent magnet is attached to the tip of the suspension beam. A magnetic coil placed

underneath the beam tip is used to apply external forces on the beam tip. These external

forces simulate the intermittent contact force between the beam tip and a hypothetical contact

surface, as well as other external disturbance forces. The details on how the contact force is

calculated are discussed in the following section. A voltage-to-current converter is used to

generate the current flowing in the coil. A PCI 6040e input-output card from National

Instruments and a personal computer are used to implement the controller and to interface it

with the rest of the system. The adaptive regulation algorithm is implemented using Labview

real-time module and Labview simulation interface toolkit with Matlab Simulink. The

suspension beam, electromagnet, and Laser Doppler Vibrometer are all mounted on a

vibration isolation optical table to reduce the effects of external vibrations. A picture of the

151

actual experimental setup is shown in Figure 4.8(a), and a close-up view of the suspension

beam with the PZT actuator and the electromagnetic actuator is shown in Figure 4.8(b).

4.7.2. Simulation of the external disturbances and of the contact force

This section outlines how the external disturbances and the intermittent contact force applied

to the beam tip are generated using the magnetic coil and a tiny permanent magnet. When a

current i flows in the coil, the axial magnetic flux density B generated by the coil is given

by [96]:

( ) ( ) ( ) ( )( )

2 2 2 22 2 2 20

2 2 222 1 1 11 1

ln ln2

z L r r z r rNiz z L zr r z r rz L r r

μ ⎡ ⎤⎛ ⎞ ⎛ ⎞+ + + + +⎢ ⎥⎜ ⎟ ⎜ ⎟= + −⎢ ⎥⎜ ⎟ ⎜ ⎟− + ++ + + ⎝ ⎠⎢ ⎥⎝ ⎠⎣ ⎦

B , (4.42)

where 0μ is the permeability of free space, 2r and 1r are the inner and outer radii of the

magnetic coil, respectively, z is the distance measured from the top of the coil along its

longitudinal axis, N is the number of turns of wire per unit length in the coil and L is the

length of the coil. The direction of the magnetic field B is parallel to the longitudinal axis of

the coil. Assume that at a distance z h= above the coil there is a cylindrical magnet of radius

R and thickness l and having a fixed and uniform magnetization

sM= −M ,

where the direction of M is also along the longitudinal axis of the coil. The equivalent

volume current density mρ and the equivalent surface current density mσ of the magnet are

given by

0mρ = −∇⋅ =M ,

sm

s

M z hM z h l

σ=⎧

= ⎨− = +⎩ (4.43)

152

Figure 4.7. Schematic diagram of the experimental setup.

The electromagnetic force on the permanent magnet with magnetization M due to an

external magnetic field B is given by [96]:

m mV SF dV dSρ σ= +∫ ∫B B , (4.44)

where V and S are the volume and surface of the permanent magnet, respectively. From

equations (4.42), (4.43) and (4.44), we obtain the vertical force on the permanent magnet

with magnetization sM due to the external magnetic field B generated by the coil as

( ) ( )( )

( ) ( ) ( )( )

( ) ( )( )

( ) ( )( )

2

2 22 22 2 2 22 0

2 22 22 1 1 1 1 1

2 22 22 22 2

2 2 2 21 1 1 1

ln ln2

ln ln .

mS

s

s

F dS

M R h h l

h L r r h l r rNiM R h L h lr r h L r r h l r r

h l L r rh r rh h l L

h r r h l L r r

σ

π

μπ

=

= − +

⎡ ⎛ ⎞ ⎛ ⎞+ + + + + +⎢ ⎜ ⎟ ⎜ ⎟= + + +⎢ ⎜ ⎟ ⎜ ⎟− + + + + + +⎢ ⎝ ⎠ ⎝ ⎠⎣

⎤⎛ ⎞⎛ ⎞ + + + ++ + ⎥⎜ ⎟⎜ ⎟− − + +⎥⎜ ⎟⎜ ⎟+ + + + + +⎝ ⎠ ⎥⎝ ⎠⎦

∫ B

B B

(4.45)

Suspension beam

PZT actuator

Magnetic coil

Permanent magnet

PC Based Real

Time Controller

Laser Doppler

Vibrometer

PZT Amplifier

Laser beam

ty

Voltage-to-current converter

, v vy y

i

u v

153

In the experimental setup, we have 30.5 10R m−= × , 31.5 10l m−= × , 31 3 10r m−= × ,

32 8 10r m−= × , 310 10L m−= × and 0h h hδ= + , where 3

0 1 10h m−= − × and hδ is the

deviation of the position of the magnet from 0h when current flows in the coil. Therefore,

based on (4.45), if hδ is in the range of few tens of micrometers, the force F can be

approximated as a linear function of the coil current i , namely

iF k i≅ .

The coil current is produced by a voltage-to-current converter circuit such that vi k v= where

v is the input voltage to the circuit and 210 /vk A volt−= . In the following, let ty denote the

displacement of the suspension beam tip, and assume the contact surface profile is given by

( )( ) 6( ) 5sin 240 50 10cs t t mπ −= − × .

The electromagnet will be used to apply a force that simulates the contact force cF at the

beam tip as follows:

0 if ( ) ( ) if ,

c t c

c c c t c c t t c

F y sF k s y c s y y s

= >⎧⎨ = − + − ≤⎩

(4.46)

where ck and cc are the stiffness and damping coefficient associated with the contact surface.

Here we set 310c cc k −= × [1]. It follows that the input voltage cv needed to simulate the

contact force is given by:

3

0 if

( ) ( ) 10 if .

c t c

cc c t c t t c

i v

v y skv s y s y y s

k k−

= >⎧⎪⎨ ⎡ ⎤= − + − × ≤⎣ ⎦⎪⎩

154

(a)

(b)

Figure 4.8. The experimental setup: (a) General view, (b) Close-up view of the suspension beam with the actuators.

Suspension Beam

Coil

Permanent Magnet

PZT ActuatorPI PL112.11

Microscope AdapterPolytec OFV-072

NI Labview Real Time Module

NI Labview with Simulation Interface Toolkit

PZT AmplifierPI E-650.00

Vibrometer ControllerPolytec OFV-5000

Fiber Vibrometer Polytec OFV-552

PZT Actuated Suspension Beam

155

In the experiment, we set 48 10 /c

i v

k A mk k

= × . The disturbance force at the tip of the flexible

beam is also simulated by applying a voltage to the voltage-to-current convert circuit as

follows:

( )2cos(240 3 / 4) volts if 2cos(240 3 / 4) 4 volts if .

a t c

ac t c

v t y sv t y s

π ππ π

= + >⎧⎨ = + + ≤⎩

(4.47)

Therefore, in the contact mode, the total voltage needed to simulate both the external

disturbances and the contact force is given by 3( ) ( ) 10cc ac c t c t ac

i v

kv v s y s y vk k

−⎡ ⎤+ = − + − × +⎣ ⎦ .

Hence, the total voltage input v to the voltage-to-current converter corresponding to the two

modes of operation is then

( ) ( )3 3

if

10 10 if .

a c

c cc c ac t t c

i v i v

v v y sk kv s s v y y y s

k k k k− −

= >⎧⎪⎨ = + × + − + × ≤⎪⎩

4.7.3. Identification of the switched system model

In the following, the two subsystem models corresponding to the non-contact mode and to

the contact mode will be obtained and verified. In the experimental setup, the measurement is

a voltage signal vy provided by the Laser Doppler Vibrometer and related to the beam tip

displacement ty by v y ty k y= where 0.1v/yk um= . A second order low pass Butterworth

filter is used to process the measurement vy and minimize the effects of the spillover

associated with the use of a reduced-order model of the PZT actuated suspension beam [97-

99]. The filter is also used to attenuate any measurement noise from the laser interferometer

or from external high frequency disturbances from the environment. A low pass Butterworth

156

filter with input vy and output fy , and having a cut-off frequency of 240Hz, is used and is

given by the following transfer function

( )1 2

11 2

1 23.64 0.268 0.6282f

q qH qq q

− −−

− −

+ +=

− +.

The models developed using system identification relate the measured signal vy to two

inputs, namely the PZT control voltage u and an exogenous input signal d defined as

follows:

( )3

if

10 if .

a c

cc c ac c

i v

d v y skd s s v y s

k k−

= >⎧⎪⎨ = + × + ≤⎪⎩

Hence, the signal d v= in the non-contact mode, and is associated with the external

disturbance defined by av . However, in the contact mode, we have

( ) ( )3 310 10c ct t c c ac

i v i v

k kd v y y s s vk k k k

− −= + + × = + × + ,

which is therefore different from v . The reason why the term ( )310ct t

i v

k y yk k

−− + × is not

part of the exogenous input signal d in this case is that it is incorporated as part of the plant

dynamics corresponding to the contact mode, and is responsible for the change to the model

dynamics, as illustrated in the simulation example in section 2.6.

First, the transfer function for the non-contact mode is identified. The transfer function

relates the piezoelectric actuator control voltage u and the signal d to the measurement vy

of the tip vertical displacement. The excitation signals applied to the PZT actuator and to the

voltage-to-current circuit for system identification are selected as ( ) 10.25sin(2 )u t f t voltsπ=

and ( ) 22sin(2 )d t f t voltsπ= , where 1f and 2f are swept from 1 to 350Hz, respectively. The

157

sampling period is set to 1KHz. Using the Matlab System Identification Toolbox, the system

is identified as a second order system

( ) ( )( ) ( ) ( )

( ) ( )1 1

1 2

1 1

b bv

a a

H q H qy k u k d k

H q H q

− −

− −= + , (4.48)

where

( )1 1 21 0.2459 0.9313aH q q q− − −= + + ,

( )1 1 21 1.004 2.1601bH q q q− − −= + , ( )1 1 2

2 0.2187 0.2267bH q q q− − −= + .

Figure 4.9 illustrates the accuracy of the identified model, where the dashed line shows the

frequency response of the real system under the exciting input signals and the solid line

shows the bode plot of the analytical model. It can be seen that the identified model matches

the real system well in the frequency range from 1 to 300Hz.

To identify the contact mode model, the excitation signals are selected as

( ) 10.25sin(2 )u t f t voltsπ= and ( )23sin 2d f t voltsπ= , where 1f and 2f are swept from 1 to

350Hz. The model is identified as

( ) ( )( ) ( ) ( )

( ) ( )1 1

1 2

1 1

c cb b

v c ca a

H q H qy k u k d k

H q H q

− −

− −= + , (4.49)

where

( )1 1 21 0.5645 0.8735caH q q q− − −= + + ,

( )1 1 21 0.9125 1.805c

bH q q q− − −= + , ( )1 1 22 0.201 0.2021c

bH q q q− − −= + .

Figure 4.10 illustrates the accuracy of the identified model.

158

4.7.4. Experimental results

In the following, based on the models in non-contact and contact mode, respectively, a

regulator is designed so that the tip of the suspension beam tracks a contact surface

( )( ) 65sin 20 60 10cs t mπ −= − × at a desired separation height of 660 10 m−× . The output y , to

be fed to the controller, is defined to be the measurement voltage corresponding to the

difference between the actual separation t cy s− and the desired separation of 660 10 m−× :

( )660 10

= 0.5sin(240 ) volts.y t c

v

y k y s

y kπ

−= − − ×

−

Define the performance variable e to be the same as the output y :

0.5sin(240 ) voltsve y kπ= − .

First, based on the transfer functions (4.48) and (4.49), the state space representation as in

(4.1) is given as

1

0 0.87351 0.5645

A−⎡ ⎤

= ⎢ ⎥−⎣ ⎦, 2

0 0.93131 0.2459

A−⎡ ⎤

= ⎢ ⎥−⎣ ⎦, 1

1.80450.9125

B ⎡ ⎤= ⎢ ⎥⎣ ⎦

, 2

2.16011.0036

B ⎡ ⎤= ⎢ ⎥⎣ ⎦

,

1

0.2021 00.2010 0

xD ⎡ ⎤= ⎢ ⎥⎣ ⎦

, 1

0.20210.2010

xF ⎡ ⎤= ⎢ ⎥⎣ ⎦

, 2

0.2267 00.2187 0

xD ⎡ ⎤= ⎢ ⎥⎣ ⎦

, 2

0.22670.2187

xF ⎡ ⎤= ⎢ ⎥⎣ ⎦

,

[ ]1 1 0 1e yC C= = , [ ]2 2 0 1e yC C= = ,

[ ]1 2 1 2 0 1y y e eD D D D= = = = , [ ]1 2 1 2 0y y e eF F F F= = = = .

1

30.5cos(240 ) 0.4sin(240 ) 0.3cos(240 )4

0.5sin(240 )

k k kd volts

k

ππ π π

π

⎡ ⎤+ + +⎢ ⎥=⎢ ⎥⎣ ⎦

, 2

30.5cos(240 )4

0.5sin(240 )

kd volts

k

ππ

π

⎡ ⎤+⎢ ⎥=⎢ ⎥⎣ ⎦

.

159

101

102

103

104

10-1

100

101

102

Am

plitu

de

101

102

103

104

-500

-400

-300

-200

-100

0

100

Pha

se (d

egre

es)

Frequency (rad/sec) (a)

101

102

103

104

10-2

100

Am

plitu

de

101

102

103

104

-400

-300

-200

-100

0

100

Pha

se (d

egre

es)

Frequency (rad/sec) (b)

Figure 4.9. The frequency response of the identified analytical model (solid line) and the real system based on experimental data (dashed line) in the non-contact mode: (a) transfer function from u to vy , (b) transfer function from d to vy .

160

101

102

103

104

100

102

Am

plitu

de

101

102

103

104

-800

-600

-400

-200

0

Pha

se (d

egre

es)

Frequency (rad/sec) (a)

101

102

103

104

10-2

100

102

Am

plitu

de

101

102

103

104

-600

-400

-200

0

Pha

se (d

egre

es)

Frequency (rad/sec) (b)

Figure 4.10. The frequency response of the identified analytical model (solid line) and the real system based on experimental data (dashed line) in the contact mode: (a) Transfer function from u to vy , (b) transfer function from d to vy .

161

Let wd denote the unknown random disturbance with max 0.5wd volts≤ . The switching

rule is defined as 1 if 62 if 6

e voltsr

e volts≤ −⎧

= ⎨ > −⎩. To construct a parameterized set of stabilizing

controllers, the gains rK , rL , { }1, 2r∈ , designed based on (4.12) are given by :

[ ] 31 -159.1746 504.2805 10K −= × , [ ] 3

2 -126.6187 407.4727 10K −= × ,

[ ] 31 873.4887 564.5477 10TL −= × , [ ] 3

2 931.2618 245.8963 10TL −= × .

Based on the analysis of the poles of rd , we have 0 1k = and 2prn = . Therefore, the base

stabilizing controller is augmented with a parameter rQ with dimension 4qn = and given by

1 2 3 43

2 2 3 4

0.46 0.45 101.4 0.9

r r r rr

zQz z z z z z

θ θ θ θ⎛ ⎞+= + + + ×⎜ ⎟− + ⎝ ⎠

. Since 0δ < , condition (4.18) needs to

consider only the system 2Σ given by:

2

30.8619 8.8250 12.0044 8.676610

11.9739 8.1385 0.1084 8.2966Aθ

⎡ ⎤= ×⎢ ⎥− −⎣ ⎦

, 2

3 305.197110

261.1491Bθ

−⎡ ⎤= ×⎢ ⎥−⎣ ⎦

.

First, without considering the 2H performance constraint in the regulator synthesis procedure,

the regulator is designed based only on the regulation conditions (4.26), (4.27), (4.37) and

(4.38). We have

[ ]1

30 1.0448 0.4263 0.0905 0.0975 10QC −= − − × ,

[ ]2

30 0.5360 0.5121 1.3472 1.3250 10QC −= − − − × .

The experimental results corresponding to the above designed controller re shown in Figures

4.11-4.14, which include the two cases of with and without the unknown random disturbance

dw . It can be seen that the resulting system with the designed controller can exactly reject

the known disturbance in the presence of switching, however it cannot attenuate the

162

disturbance dw properly. By considering the 2H constraint in the regulator synthesis

procedure, we obtain

[ ]1

30 0.1376 0.2452 0.1665 0.0208 10QC −= − × ,

[ ]2

30 0.1129 0.2334 0.2105 0.1002 10QC −= − − × ,

2 0.118γ = .

The experimental results are shown in Figures 4.15 and 4.16, which indicate that the

disturbance dw has been attenuated effectively and the tip of the suspension beam tracks the

contact surface while maintaining the desired separation between the tip and the contact

surface.

4.8. Conclusion

The problem of regulation in bimodal switched systems against known deterministic

exogenous inputs and unknown random inputs is treated. A regulator design approach based

on the parameterization of a set of stabilizing controllers for the switched system is presented.

First, regulation conditions for the switched system are derived and are used to construct a set

of Q parameters needed to achieve regulation against the known deterministic exogenous

inputs. A set of 2H controllers from within the already constructed set of controllers is then

considered to achieve regulation also against the unknown random exogenous input. A

regulator synthesis algorithm is developed based on solving a set of linear matrix inequalities.

Finally, a switched bimodal mechanical system experimental setup involving a flexible beam

subject to contact vibrations is used to successfully demonstrate the performance of the

proposed regulator.

163

The results presented so far in the thesis assume that the deterministic exogenous inputs

are known. However, in some cases, the properties of the deterministic signals maybe

unknown or time-varying. To deal with these types of inputs, an adaptive regulation method

will be developed for discrete-time switched bimodal linear systems in the next chapter.

164

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-100

-50

0

50

100

Per

form

ance

var

iabl

e (m

icro

met

ers)

Time(sec)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-100

-50

0

50

100

Pos

ition

(mic

rom

eter

s)

Time(sec)

Tip positionContact surface

Figure 4.11. Experimental results showing the performance variable e and the position y for the case of 0dw = and obtained using the controller designed without accounting for the 2H performance constraint.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-0.5

0

0.5

Ran

dom

dis

turb

ance

dw

(V)

Time(sec)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

0

50

100

Cur

rent

in th

e co

il (m

A)

Time(sec)

Figure 4.12. Experimental results showing the random disturbance dw and the total current in the coil for the case of 0dw = and obtained using the controller designed without accounting for the 2H performance constraint.

165

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-100

-50

0

50

100

Per

form

ance

var

iabl

e (m

icro

met

ers)

Time(sec)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-100

-50

0

50

100

Pos

ition

(mic

rom

eter

s)

Time(sec)


Figure 4.13. Experimental results showing the performance variable e and the position y in the presence of dw and obtained using the controller designed without accounting for the

2H performance constraint.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-0.5

0

0.5

Ran

dom

dis

turb

ance

dw

(V)

Time(sec)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

0

50

100

Cur

rent

in th

e co

il (m

A)

Time(sec)

Figure 4.14. Experimental results showing the random disturbance dw and the total current in the coil obtained using the controller designed without accounting for the 2H performance constraint.

166

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-100

-50

0

50

100

Per

form

ance

var

iabl

e (m

icro

met

ers)

Time(sec)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-100

-50

0

50

100

Pos

ition

(mic

rom

eter

s)

Time(sec)


Figure 4.15. Experimental results showing the performance variable e and the position y in the presence of dw and obtained using the controller designed by accounting for the 2H performance constraint.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-0.5

0

0.5

Ran

dom

dis

turb

ance

dw

(V)

Time(sec)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

0

50

100

Cur

rent

in th

e co

il (m

A)

Time(sec)

Figure 4.16. Experimental results showing the random disturbance dw and the total current in the coil obtained using the controller designed by accounting for the 2H performance constraint.

167

CHAPTER 5

Adaptive Regulation in Switched Bimodal

Linear Systems

5.1. Introduction

In this chapter, an adaptive regulation method is presented for discrete-time switched

bimodal systems where it is desired to reject unknown sinusoidal disturbance signals and/or

track unknown sinusoidal reference inputs. The primary practical motivation for this problem

came from considering the problem of flying height regulation in hard disk drives. The

switching nature of the system dynamics coupled with the unknown disk surface profile

could be addressed using an adaptive regulation approach. The regulator design approach

involves two main steps. In the first step, a set of switched Q -parameterized observer-based

state-feedback controllers for the switched linear system is constructed. Switching in the

controller is performed according to the same switching rule as in the plant. Assuming the

reference or disturbance signals are known, regulation conditions for the switched closed

loop system are presented in the form of properly formulated linear matrix inequalities

(LMIs). The regulation conditions are used to construct a set of Q parameters needed to

achieve regulation in the switched closed loop system. In the second step, an adaptation

168

algorithm is developed to tune, online, the Q parameter in the expression of the

parameterized controller for the case of unknown reference or disturbance signals. The

tuning is such that the Q parameter converges to the desired Q parameter that guarantees

regulation in the switched closed loop system.


problem for the switched bimodal system is presented. In section 5.3, the construction of a

Q -parameterized set of switched controllers for the switched system is discussed and

stability properties of the resulting switched closed loop system are analyzed. In section 5.4,

regulation conditions for the switched closed loop system are presented using properly

formulated LMIs. In section 5.5, an adaptation algorithm is developed to tune the Q

parameter in the controller and drive it to the desired Q parameter needed to achieve

regulation in the switched closed loop system. In section 5.6, numerical simulation results for

a simple switched mechanical system are used to illustrate the performance of the proposed

regulator. In section 5.7, the proposed adaptive regulator is evaluated on an experimental

setup, where the tip of a flexible beam is supposed to maintain a constant separation with

respect to a surface with an unknown profile, while also being subject to an unknown

disturbance force. The experimental results successfully demonstrate the effectiveness of the

proposed approach in achieving exact output regulation against unknown sinusoidal

exogenous inputs and in the presence of switching in the system dynamics. Concluding

remarks are presented in section 5.8.

169

5.2. The adaptive regulation problem for bimodal systems

This chapter considers discrete-time switched systems given by the following state space

representation:

0( 1) ( ) ( ) ( ), (0) , ( ) ( ) ( ), ( ) ( ) ( ), :1 if ( ) ( ) ( ) ,2 if

xi i i iy y

i i ie ei i ii

e ei i i

x k A x k B u k D d k x xy k C x k D d ke k C x k D d k

e k C x k D d ki

eδ

+ = + + == += +Σ

= + ≤=

( ) ( ) ( ) ,e ei i ik C x k D d k δ

⎧⎪⎪⎪⎨⎪ ⎧⎪ ⎨⎪ = + >⎩⎩

(5.1)



to be measurable, δ is a constant satisfying 0δ > , and { }1,2i∈ is the index of the system

iΣ under consideration at time k .

The external signal hid ∈R , representing disturbance and/or reference signals, is also

assumed to switch according to the rule given in (5.1). The signal id is given by:

1( ) [ ( ), , ( )]h Ti i id k d k d k= … , (5.2)

where

( ) 1,, , ,

1

( ) cos , 1, ,s

s

kk ss s s s

i i i i id k c k c s hν ν ν

ν

ω φ +

=

= + + =∑ , (5.3)

with unknown amplitudes ,sicν , frequencies ,s

iνω , and phases ,s

iνφ , 1, , skν = … ; 1, ,s h= … .

For the switched system (5.1), it is desired to construct an output feedback controller to

regulate the performance variable e . More specifically, a Q -parameterized adaptive output

feedback controller will be developed in this chapter to asymptotically drive the performance

variable e to zero, i.e. lim ( ) 0k

e k→∞

= , in the presence of the unknown external input signal id .

170

Given the switching nature of the plant, the parameterized output feedback controller is also

chosen to be a switching feedback controller and where the switching between the two

controllers is to obey the same rule given in (5.1) as for switching between the two plant

models.

5.3. Parameterization of a set of stabilizing controllers

The development of the proposed adaptive regulator is based on considering a Q -

parameterized set of output feedback stabilizing controllers for the switched system. A

summary of the construction of such a set of controllers is presented followed by an internal

stability analysis for the resulting bimodal closed loop system.

5.3.1. Q -parameterized controller

In the following, a Q -parameterized output feedback controller for the switched system (5.1)

is used, where the controller is expressed as a linear fractional transformation involving a

fixed system iJ , and a proper stable parameter iQ that could be chosen as desired (See

Figure 2.1). The state space representation of the system iJ is given by:

0ˆ ˆ ˆ ˆ( 1) ( ) ( ) ( ) ( ), (0) ,ˆ: ( ) ( ) ( ),

ˆ ˆ( ) ( ) ( ) ( ).

yi i i i i i i Q

i i Q

yi

x k A LC B K x k L y k B y k x x

J u k K x k y k

y k y k y k C x k

⎧ + = + + − + =⎪

= +⎨⎪

− = −⎩

(5.4)

where ˆ( )x k is an estimate of the plant state vector ( )x k and ˆ ˆ( ) ( )yiy k C x k= is an estimate of

the plant output ( )y k . The state feedback gains iK , { }1, 2i∈ , and the observer gains iL ,

171

{ }1, 2i∈ , are assumed to switch according to the rule given in (5.1). The system iQ is given

by:

( ) 0ˆ( 1) ( ) ( ) ( ) , (0) ,:

( ) ( ),Q Q Q Q Q Q

i QQ i Q

x k A x k B y k y k x xQ

y k C x k

⎧ + = + − =⎪⎨

=⎪⎩ (5.5)

where QnQx ∈ . In particular, throughout the rest of the paper, the iQ parameter is chosen to

be of the form as

( )1

( )qn

qi i q

q

Q z zθ ψ=

=∑ , (5.6)

where ( ) ( )1 qq z z F zψ −= , 1, qq n= , are stable basis functions, and where

11

11

( )m

mm m

m

b z bF zz a z a

−

−

+=

+ + is a stable function used to adjust the dynamic properties of ( )iQ z .

Such representation can be used to approximate any proper stable real rational transfer

function. Let 1, , qTn

i i iθ θ θ⎡ ⎤= ⎣ ⎦… and 1Q qn m n= + − . A realization of ( )iQ z can then be given

as follows:

1

2

1

0 0 0 0 0 01 0 0 0 0 0

0 1 0 0 0 00 0 1 0 0 00 0 0 1 0 0 00 0 0 0 0 00 0 0 0 0 1 0

Q Q

m

m

Q

n n

aa

aA

a

−

×

−⎡ ⎤⎢ ⎥−⎢ ⎥⎢ ⎥⎢ ⎥−⎢ ⎥= ⎢ ⎥−⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

,

1

2

1

1

0

0Q

m

m

Q

n

bb

bB

b

−

×

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

,

( )1 10Q Ti imC θ× −

⎡ ⎤= ⎣ ⎦ .

172

It follows that QA is a fixed stability matrix, QB is a fixed matrix, and the matrix QiC

changes with the switching signal { }1, 2i∈ . If the signal id is known, the parameter vectors

iθ , { }1, 2i∈ , can be designed according to the regulation conditions to be discussed in

sections 5.4 and that guarantee the switched closed loop system achieves regulation. In the

case where the properties of the input signal id are unknown, an adaptation algorithm will be

proposed to adjust and drive the parameter vectors iθ in the QiC matrices to the desired

parameter vectors that yield regulation in the adaptive switched closed loop system against

the unknown signal id .

Let ˆ( ) ( ) ( )x k x k x k= − denote the state estimation error and 1

TT T TQ N

x x xχ×

⎡ ⎤= ⎣ ⎦

denote the state vector for the resulting closed loop system with 2 QN n n= + . The resulting

closed loop system is given by the following state space representation:

( 1) ( ) ( ), ( ) ( ) ( ),

:1 if ( ) ( ) ( ) ,2 if ( ) ( ) ( ) ,

i i iex ei i icl

i ex ei i iex ei i i

k A k E d ke k C k D d k

e k C k D d ki

e k C k D d k

χ χχ

χ δχ δ

⎧ + = +⎪

= +⎪∑ ⎨⎧ = + ≤⎪ = ⎨⎪ = + >⎩⎩

(5.7)

where 00 0

Qi i i i i i i

yi Q Q i

yi i i

A B K B C B KA A B C

A L C

⎡ ⎤+⎢ ⎥= −⎢ ⎥⎢ ⎥+⎣ ⎦

, ( ) ( ) ( )TTTx y T x y

i i Q i i i iE D B D D L D⎡ ⎤= − +⎢ ⎥⎣ ⎦,

0 0ex ei iC C⎡ ⎤= ⎣ ⎦ .

173

5.3.2 Stability of the Q -parameterized switched closed loop system

In this section, the internal stability of the closed loop system is studied by considering the

system (5.7) in the absence of the signal id . With 0id = , the origin 0χ = of the unforced

switched closed loop system should be asymptotically stable. In the absence of the signal id ,

the state equation for the resulting closed loop system is given by:

( 1) ( )( 1) 0 ( ) .( 1) 0 0 ( )

Qi i i i i i i

yQ Q Q i Q

yi i i

x k A B K BC B K x kx k A B C x kx k A LC x k

⎡ ⎤+ +⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥+ = −⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥+ +⎣ ⎦ ⎣ ⎦⎣ ⎦

(5.8)


result.

Theorem 5.1

The origin is an asymptotically stable equilibrium point for the switched system (5.8) under

arbitrary switching if QA is a stability matrix and there exist matrices 0kiP > , 0l

iP > and

matrices iK , iL , { }1, 2i∈ , such that:

[ ] [ ] { }0, , 1, 2T k ki i i j i i i iA B K P A B K P i j+ + − < ∈ , (5.9)

{ } 0, , 1, 2Ty l y l

i i i j i i i iA L C P A L C P i j⎡ ⎤ ⎡ ⎤+ + − < ∈⎣ ⎦ ⎣ ⎦ . (5.10)

Proof: Based on (5.8), define the following three subsystems for { }1, 2i∈ :

[ ]( 1) ( ),i i ix k A B K x k+ = + (5.11)

( 1) ( ),yi i ix k A LC x k⎡ ⎤+ = +⎣ ⎦ (5.12)

174

( 1) ( ).Q Q Qx k A x k+ = (5.13)

Systems (5.11) and (5.12) are switched systems. Define the piecewise quadratic Lyapunov

functions 2

1

( ( )) ( ) ( ) ( )T kx i i

i

V x k x k k P x kζ=

⎛ ⎞= ⎜ ⎟

⎝ ⎠∑ for system (5.11) and

2

1

( ( )) ( ) ( ) ( )T lx i i

i

V x k x k k P x kζ=

⎛ ⎞= ⎜ ⎟

⎝ ⎠∑ for system (5.12), respectively, where

1, when the subsystem is active at time ( )

0, otherwise

th

ii k

kζ⎧

= ⎨⎩

, { }1, 2i∈ . If (5.9) and (5.10) are satisfied,

the switched systems (5.11) and (5.12) admit continuously decreasing piecewise quadratic

Lyapunov functions xV and xV , which means that the switched systems (5.11) and (5.12) are

asymptotically stable under arbitrary switching. Since QA is a stability matrix, system (5.13)

is also asymptotically stable. Then, as in the proof for Lemma 4.1, it is easy to have that the

switched system (5.1) is asymptotically stable under arbitrary switching.

5.4. Regulation conditions for the switched system

In this section, regulation conditions for the switched closed loop system (5.7) are presented

under the assumption that the properties of the external input in (5.2) and (5.3), namely

frequencies, amplitudes and phases, are all known.

First, the performance variable ( )e k in the switched closed loop system cliΣ in (5.7) can

be written as:

( )11 12 210( ) ( ) ( )i i i i ie k T T QT d k e k= + + , (5.14)

175

where 12 220( ) ( )i i i Qe k T Q T y k= . Based on the state space realization of the system 22

iT given in

(4.15), the output of the system 22iT is not affected by the input ( )Qy k . Hence, 0( )e k

represents the response of the system 12 22i i iT Q T to its initial conditions, and will be shown

later in this section to be bounded by an exponentially decaying signal. First, the closed-loop

system transfer function can be written as:

( ),

11 12 21

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ).i iT Q i

i i i i i

E z F z D z

T z T z Q z T z D z

=

= + (5.15)

Let 1( )e k denote the response of the switched system 11iT to the input ( )id k , 2 ( )e k denote

the response of the switched system 12iT to an arbitrary input 12 ( )id k and 3( )e k denote the

response of the switched system 21i iQT to the input ( )id k . Based on the definitions of 11

iT ,

12iT and 21

iT in (4.15), we have the following results.

Lemma 5.3

Consider the switched system 11iT subject to a bounded input id . Let

{ }1,20

max ( ) 0iik

d kγ∈≥

= ≠ and

10 1ε< . If there exist matrices 111 0H > , 11

2 0H > and positive scalars 1α , 1μ and 1β such

that the following matrix inequalities are satisfied:

( )( ) ( )

11 11 11 111

1 1 11 112

11 11 11 11 11

0

10 0,

T

i i i j

T

i j

j i j i j

H H A H

I E H

H A H E H

α

ε μγ

⎡ ⎤− +⎢ ⎥⎢ ⎥−⎢ ⎥− ≤⎢ ⎥⎢ ⎥−⎢ ⎥⎣ ⎦

( ) { } { }, 1, 2 1,2i j∀ ∈ × , (5.16)

176

( )( ) ( )

11 111

1 1 112

11 111

0

0 0

T

i i

T

i

i i

H C

I D

C D I

α

β μγ

β

⎡ ⎤⎢ ⎥⎢ ⎥−⎢ ⎥ ≥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

, { }1, 2i∀ ∈ , (5.17)

then there exists a finite time 1k such that 1( )e k satisfies:

1 1 1( ) , e k k kβ< ∀ ≥ . (5.18)

Proof: Consider the state space representation of 11iT and define the piecewise quadratic

Lyapunov function

211

11 11 11 111

( ( )) ( ) ( ) ( )TV k k k H kχ χ ζ χ=

⎛ ⎞= ⎜ ⎟

⎝ ⎠∑ ,

where 1, when the subsystem is active at time ( )

0, otherwise

th kkζ

⎧= ⎨⎩

, { }1, 2∈ , and 211( ) nkχ ∈R . First,

using the Schur complement formula, inequality (5.16) is equivalent to:

( ) ( )

( ) ( ) ( )

11 11 11 11 11 11 11 111

1 111 11 11 11 11 112

0,1

T T

i j i i i i j i

T T

i j i i j i

A H A H H A H E

E H A E H E I

α

ε μγ

⎡ ⎤− +⎢ ⎥

≤⎢ ⎥−⎢ ⎥−⎢ ⎥⎣ ⎦

(5.19)

Inequality (5.19) implies that for any nonzero vector 11 2( )( )

n h

i

kd kχ +⎡ ⎤

∈⎢ ⎥⎣ ⎦

R , we have:

( ) ( )

( ) ( ) ( )

11 11 11 11 11 11 11 111

11 11

1 111 11 11 11 11 112

( ) ( )01( ) ( )

T TT i j i i i i j i

T Ti ii j i i j i

A H A H H A H Ek k

d k d kE H A E H E I

αχ χ

ε μγ

⎡ ⎤− +⎢ ⎥⎡ ⎤ ⎡ ⎤

≤⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦⎢ ⎥−

⎢ ⎥⎣ ⎦

.

It follows that, for all ( ) { } { }, 1, 2 1, 2i j ∈ × ,

177

( ) ( )1 111 1111 11 1 11 11 2

1( 1) ( 1) 1 ( ) ( ) ( ) ( )T T T

j i i ik H k k H k d k d kε μ

χ χ α χ χγ−

+ + − − ≤ . (5.20)

Let { }1, 2i∈ denote the mode of the system that is active at time k and { }1, 2j∈ denote the

mode of the system that is active at time 1k + . Then, for any 0k ≥ , the above inequality

implies that

( ) ( )2 21 111 11

11 11 1 11 11 21 1

1( 1) ( 1) ( 1) 1 ( ) ( ) ( ) ( ) ( )T T T

i ik k H k k k H k d k d kε μ

χ ζ χ α χ ζ χγ= =

−⎛ ⎞ ⎛ ⎞+ + + − − ≤⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠∑ ∑ ,

which is equivalent to

( )1 111 11 11 11 1 11 112

1( ( 1)) ( ( )) ( ) ( ) ( ( ))T

i iV k V k d k d k V kε μ

χ χ α χγ−

+ − ≤ − .

Hence, 11 11 11 11( ( 1)) ( ( )) 0V k V kχ χ+ − ≤ holds whenever

( ) ( )1 1 1 111 11 2

1 1

1 1( ( )) ( ) ( )T

i iV k d k d kε μ ε μ

χα α γ− −

≥ ≥ .

Consequently, 11 11( ( ))V kχ cannot ultimately exceed the value ( )1 1

1

1 ε μα−

, and we have:

( )1 111 11

1

1lim ( ( ))k

V kε μ

χα→∞

−≤ . (5.21)

Therefore, there exists a finite time 1k such that

111 11 1

1

( ( )) , V k k kμχα

< ∀ ≥ . (5.22)

Inequality (5.17) is equivalent to:

( )( )( )

11111

11 111 1 1112

01 0

0

Ti

i

i iT

i

H CC D

I D

αβ μ

βγ

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎡ ⎤− ≥− ⎣ ⎦⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦

. (5.23)

178

Let { }1, 2i∈ denote the mode of the system that is active at time k . Then (5.23) can be

rewritten as:

( )( )( )

211

1111 11 11

1111 12

( ) 01 0

0

T

i

i iT

i

k H CC D

DI

α ζ

ββ μγ

=

⎡ ⎤⎛ ⎞⎡ ⎤⎢ ⎥⎜ ⎟

⎝ ⎠ ⎢ ⎥⎢ ⎥ ⎡ ⎤− ≥⎣ ⎦⎢ ⎥⎢ ⎥−⎢ ⎥⎢ ⎥ ⎣ ⎦

⎢ ⎥⎣ ⎦

∑.

Multiplying from the left by 11( )( )

T

i

kd kχ⎡ ⎤⎢ ⎥⎣ ⎦

and from the right by 11( )( )i

kd kχ⎡ ⎤⎢ ⎥⎣ ⎦

yields:

( ) ( )

( )( )

1 121 1 1 11 11 2

1 1 11 11 1 1

( ) ( ) ( ) ( )

( ) .

Ti ie k V k d k d k

V k

β μβ α χ

γ

β α χ β μ

⎛ ⎞−≤ +⎜ ⎟

⎝ ⎠≤ + −

(5.24)

Then based on (5.22), we have:

1 1 1( ) , e k k kβ< ∀ ≥ .

Lemma 5.4

Consider the switched system 12iT subject to a bounded input 12

id . Let { }

12

1,20

max ( ) 1iik

d k∈≥

= and

20 1ε< . If there exist matrices 121 0H > , 12

2 0H > and positive scalars 2α and 2β such that

the following matrix inequalities are satisfied:

( )( ) ( )

12 12 12 122

12 122

12 12 12 12 12

0

0 1 0,

T

i i i j

T

i j

j i j i j

H H A H

I E H

H A H E H

α

ε

⎡ ⎤− +⎢ ⎥⎢ ⎥− − ≤⎢ ⎥⎢ ⎥−⎢ ⎥⎣ ⎦

( ) { } { }, 1, 2 1,2i j∀ ∈ × , (5.25)

( )12 122

12 22

0T

i i

i

H C

C

α

β

⎡ ⎤⎢ ⎥ ≥⎢ ⎥⎣ ⎦

, { }1, 2i∀ ∈ , (5.26)

179

then there exists a finite time 2k such that 2 ( )e k satisfies:

2 2 2( ) , e k k kβ< ∀ ≥ . (5.27)

Proof: Consider the state space representation of 12iT and define the piecewise quadratic

Lyapunov function 2

1212 12 12 12

1

( ( )) ( ) ( ) ( )TV k k k H kχ χ ζ χ=

⎛ ⎞= ⎜ ⎟

⎝ ⎠∑ where 12 ( ) nkχ ∈R . Proceeding

as in the proof of Lemma 5.3 and based on inequality (5.25), there exists 2 0k > such that

12 12 22

1( ( )) , V k k kχα

< ∀ ≥ . (5.28)

Inequality (5.26) is equivalent to:

( )12 12 122 2

2

1 0T

i i iH C Cαβ

− ≥ , (5.29)

Let { }1, 2i∈ denote the mode of the system that is active at time k . Then (5.29) can be

rewritten as:

( )2

12 12 122 2

1 2

1( ) 0T

i ik H C Cα ζβ=

⎛ ⎞− ≥⎜ ⎟

⎝ ⎠∑ .

Multiplying from the left by 12 ( )T kχ and from the right by 12 ( )kχ yields:

( ) ( )212 212 2 2 12 12( ) ( )iC k V kχ α β χ≤ . (5.30)

Based on (5.28), it follows that:

2 2 2( ) , e k k kβ< ∀ ≥ .

180

Similarly, let 21

21

0iQi

Q i Q

AB C A⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

A and 21

21iQ

iQ i

EB D⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

E . Then we have the following result.

Lemma 5.5

Consider the switched system 21i iQ T subject to a bounded input id . Let

{ }1,20

max ( ) 0iik

d kγ∈≥

= ≠ ,

3 0α > be a preset constant and 30 1ε< . If there exist matrices 1 0QH > , 2 0QH > and a

positive scalar 3β such that the following matrix inequalities are satisfied:

( )( ) ( )

3

3 32

0

10 0,

TQ Q Q Qi i i j

TQ Qi j

Q Q Q Q Qj i j i j

H H H

I H

H H H

α

ε αγ

⎡ ⎤− +⎢ ⎥⎢ ⎥−⎢ ⎥− ≤⎢ ⎥⎢ ⎥−⎢ ⎥⎣ ⎦

A

E

A E

( ) { } { }, 1, 2 1,2i j∀ ∈ × , (5.31)

1

21 3

00

0

TQ Qi n i

Qn i

H C

C β

×

×

⎡ ⎤⎡ ⎤⎣ ⎦⎢ ⎥ ≥⎢ ⎥⎡ ⎤⎣ ⎦⎣ ⎦

, { }1, 2i∀ ∈ , (5.32)

then there exists a finite time 3k such that 3 ( )e k satisfies:

3 3 3( ) , e k k kβ< ∀ ≥ . (5.33)

Remark 5.2

The proof for the above result is similar to the proof of Lemma 5.4. It should also be noted

that Lemma 5.5 holds even if the matrix QiC is time-varying as long as (5.32) is satisfied

with the time-varying QiC matrix. This property will be used in the development and analysis

of the adaptation algorithm presented later.

181

Consider now the following assumption.

Assumption 5.1

There exists a unique parameter vector 0θ satisfying the interpolation condition (4.18) for

1i = in the case of 0δ > and 2i = in the case of 0δ < , which means the matrix iAθ in (4.19)

is square and nonsingular. Furthermore, the parameter vector 0θ satisfies 0l uθ θ θ≤ ≤ and

0θ η≤ , where lθ , uθ and η are known constants and are such that any qnθ ∈R satisfying

l uθ θ θ≤ ≤ also satisfies θ η≤ .

For assumption 5.1 to be satisfied, we should have 02 1qn k= + . It is also worth noting that by

properly selecting the functions ( )q zψ in (5.6), we can have η in any pre-specified desired

range. Let Q ql n n n= + − . Lemma 5.3 implies that the system 11iT is an exponentially stable

switched system. It follows that the system 22iT is also an exponentially stable system.

Moreover, given that QA is a stability matrix, then iQ is an exponentially stable switched

system. Hence, 0( )e k in (5.14) is bounded by an exponentially decaying signal. It follows

that for any arbitrary small positive number 00 1ε< , there exists a finite time 0k such that

0 ( )e k satisfies 0 0( )e k ε< , 0 k k∀ ≥ . Let Q ql n n n= + − . Based on Lemmas 5.3-5.5, a

sufficient regulation condition for the switched closed loop system (5.7) can be given as

follows.

182

Theorem 5.2

Assume the switched closed loop system (5.7) is internally stable under arbitrary switching

and that the conditions of Lemmas 5.3 and 5.4 are satisfied. Let 3 0α > be a preset constant

and 30 1ε< . If Assumption 5.1 is satisfied and 1β δ< , and if there exist matrices

1 0QH > , 2 0QH > such that:

2

22

1 0

(5.31)

0 0 1, if 0,

0 2, if 0q

q q q

l l l nQi

n l n n

iH

I iδβ

η δδ β ε× ×

× ×

⎧⎪⎪ ⎡ ⎤⎛ ⎞ = >⎧⎨ ≥ ⎢ ⎥⎜ ⎟ ⎨⎪ ⎜ ⎟ = <− − ⎢ ⎥ ⎩⎝ ⎠⎪ ⎣ ⎦⎩

(5.34)

then, with

( )

( )

1 0 2 11 1

1 1 2 01 1

0 0 , 0

0 0 , 0

Q

Q

Q T Qnm

Q Q Tn m

C and C if

C and C if

θ δ

θ δ

×× −

× × −

⎧ ⎡ ⎤= = >⎪ ⎣ ⎦⎨

⎡ ⎤= = <⎪ ⎣ ⎦⎩

(5.35)

the switched closed loop system (5.7) achieves regulation.

Proof: Since 0θ η≤ , we have 2

0 0q q

Tn nIη θ θ× ≥ . It follows from (5.34) that

2

2

0 01 0

0 0

0q

q

l l l nQi T

n l

H βθ θδ β ε

× ×

×

⎡ ⎤⎛ ⎞≥ ⎢ ⎥⎜ ⎟⎜ ⎟− − ⎢ ⎥⎝ ⎠ ⎣ ⎦

, 1, if 02, if 0

ii

δδ

= >⎧⎨ = <⎩

.

Hence, 2

121 0

01 0

00 0lQ T

i lH β θθδ β ε×

×

⎛ ⎞ ⎡ ⎤⎡ ⎤− ≥⎜ ⎟ ⎢ ⎥ ⎣ ⎦⎜ ⎟− − ⎣ ⎦⎝ ⎠

, which is equivalent to

1 0

21 0

1 02

0

00

TQ Ti l

Tl

H θ

δ β εθ

β

×

×

⎡ ⎤⎡ ⎤⎣ ⎦⎢ ⎥⎢ ⎥ ≥⎛ ⎞− −⎢ ⎥⎡ ⎤ ⎜ ⎟⎣ ⎦⎢ ⎥⎝ ⎠⎣ ⎦

, 1, if 02, if 0

ii

δδ

= >⎧⎨ = <⎩

. (5.36)

183

When QiC are designed as in (5.35), and if (5.36) is satisfied, then it is easy to see that (5.32)

is also satisfied with 1 03

2

δ β εβ

β− −

= . It follows that

1 03 3

2

( ) , e k k kδ β ε

β− −

< ∀ ≥ .

Let { }0 1 2 3max , ,k k k k k= + , then based on the definition of ( )e k in (5.15) and the results in

Lemmas 5.3-5.5 , we have 1 01 2 0

2

( )e kδ β ε

β β ε δβ

⎛ ⎞− −< + + <⎜ ⎟

⎝ ⎠, k k∀ ≥ . Hence, after

time k k= , there will be no more switching and the closed loop system will evolve in only

one of the two modes { }1, 2i∈ . For the case of 0δ > , the system 1clΣ is active for k k≥ .

Since regulation condition (4.18) is satisfied for the system 1clΣ with 1

QC chosen as in (5.35),

then regulation for the switched closed loop system (5.7) is achieved. The analysis is similar

for the case of 0δ < .

It can be seen in the conditions of Theorem 5.2 that the smaller the term 2

1 0

βδ β ε− −

in

(5.34) is, the easier it is for (5.34) to be satisfied. In Appendices A and B, an approach to the

design of the stabilizing controllers is presented and that will make it easier for (5.34) to be

satisfied. The approach is based on first designing the gains iK by solving the problem of

minimizing 2β subject to (5.25) and (5.26). Then, based on the known iK , the gains iL will

be designed by solving the problem of minimizing 1β subject to (5.16) and (5.17).

Furthermore, (5.16) is equivalent to (5.19) which implies that (5.9) and (5.10) can be

satisfied. Therefore the proposed method to design iK and iL yields an internally stable

184

switched closed loop system (5.7). Solving the two optimization problems mentioned above

is difficult given that (5.16) and (5.25) are nonlinear matrix inequalities. In Appendices A

and B, two algorithms are developed to find local solutions for the two optimization

problems based on iterative LMIs. The resulting gains iK and iL , { }1, 2i∈ , yield the

minimum value 1β∗ of 1β such that (5.16) and (5.17) are satisfied and the minimum value

2β∗ of 2β such that (5.25) and (5.26) are satisfied.

Consider now the following assumption which will be used in the development of the

adaptation algorithm in the next section for the case of exogenous inputs with unknown

properties.

Assumption 5.2

Consider the parameterized stabilizing controllers obtained using the gains iK and iL ,

{ }1, 2i∈ , provided by Algorithms 1 and 2 in Appendices A and B, respectively, and the

parameters iQ , { }1, 2i∈ , with 0θ η≤ . Condition (5.34) in Theorem 5.2 is satisfied with

1 1β β ∗= and 2 2β β ∗= .

Remark 5.3

If Assumption 5.2 is satisfied, then based on Theorem 5.2, as long as QiC is set as given in

(5.35), the switched closed loop system (5.7) can achieve regulation. Based on the matrices

QiH , { }1, 2i∈ , obtained in (5.34), define the following set:

185

1

2

1 01

2

01, if 0

= : 0, 2, if 00

q

TQ Ti l

n

Tl

Hiiθ

θδ

θ δ β ε δθβ

×

∗

× ∗

⎧ ⎫⎡ ⎤⎡ ⎤⎣ ⎦⎪ ⎪⎢ ⎥ = >⎧⎪ ⎪⎢ ⎥∈ ≥⎨ ⎨ ⎬⎛ ⎞− − = <⎢ ⎥ ⎩⎡ ⎤⎪ ⎪⎜ ⎟⎣ ⎦ ⎜ ⎟⎢ ⎥⎪ ⎪⎝ ⎠⎣ ⎦⎩ ⎭

S R . (5.37)

Equation (5.36) implies that 0θ is an element of the set θS . Furthermore, if the parameter

vector iθ , where 1i = in the case of 0δ > and 2i = in the case of 0δ < , is always selected

in θS , regardless of whether iθ changes as a function of time or not, and 10qj nθ ×= , where

2j = in the case of 0δ > and 1j = in the case of 0δ < , then we have ( )e k δ< , k k∀ ≥ ,

where { }0 1 2 3max , ,k k k k k= + . This result follows from the fact that if the first LMI in (5.34)

and the LMI in (5.37) are satisfied, then Lemma 5.5 implies that 1 03

2

( )e kδ β ε

β

∗

∗

− −≤ ,

k k∀ ≥ . Therefore, ( )e k δ< , k k∀ ≥ . Moreover, it should be noted that the set θS is a

convex set, a property that will be exploited in the development of the adaptation algorithm

for the switched closed loop system.

5.5. Adaptive regulation in the switched closed loop system

In the previous sections, it was assumed that the disturbance properties are known. Using the

proposed Q -parameterized controllers given by (5.4) and (5.5), conditions for regulation in

the resulting switched closed loop system are presented. In the case where the disturbance

input properties are unknown and possibly time-varying, it is desired to introduce adaptation

in the controller design process. After obtaining the controller parameters iK , iL and the

corresponding 1β∗ and 2β

∗ values in the previous sections, and if Assumption 5.2 is satisfied,

186

then adaptation can be introduced in the design of the proposed Q -parameterized controllers

given by (5.4) and (5.5). The aim of the adaptation is to tune the parameter vector iθ in QiC

so that it converges to the desired parameter vector 0θ as in (5.35) that yields regulation.

This section presents the development of an adaptation algorithm for the switched closed

loop system and presents the properties of the resulting adaptive switched closed loop system.

Based on the regulation condition in Theorem 4.2, 0θ is defined by only one set of

interpolation conditions in (4.18) corresponding to 1i = in the case of 0δ > and 2i = in the

case of 0δ < . Therefore, at any time k , when the system cliΣ is active, where 1i = in the

case of 0δ > and 2i = in the case of 0δ < , a parameter adaptation algorithm is turned on

to provide an estimate iθ of the parameter vector 0θ . Otherwise, the adaptation algorithm is

turned off and the estimated parameter vector iθ is frozen. In the rest of this section it is

assumed that 10qj nθ ×= where 2j = in the case of 0δ > and 1j = in the case of 0δ < .

Assume that the system cliΣ is active at 0k = . Let b

mk and emk , 1, 2,m = , denote the

beginning time and the end time of the time period during which cliΣ is active before the thm

switching from cliΣ to cl

jΣ (See Figure 5.1). Let e bm m mk kτ = − denote the corresponding dwell

time for cliΣ . Now let lz− denote the l time step delay operator and

1 1,

1

ˆ ( ) ( )qn

q qi k i

q

Q k z F zθ − −

=

= ∑ . Let 22 122( ) ( ) ( )i Qe k T z y k−= , which is an exponentially decaying

signal representing the response of the system 22iT to its initial conditions, regardless of what

( )Qy k is. The performance variable ( )e k in the adaptive closed loop system cliΣ is given by:

187

Figure 5.1. Diagram of the switching sequence for the closed loop system.

( )11 1 12 1 21 1, 22

11 1 12 1,

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ),i i i i k i i

i i i i k

e k T z d k T z Q T z d k e k

T z d k T z Q d k

− − −

− −

= + +

= +

(5.38)

where,

21 122 ˆ( ) ( ) ( ) ( ) ( ) ( ).i id k T z d k e k y k y k−= + = −

Define a modified performance variable

12 1 12 1, ,( ) ( ) ( ) ( ) ( ) ( )i k i i i ke k e k Q T z d k T z Q d k− −= + − . (5.39)

Then we have

11 1 12 1,

11 1 1 1 12 1

1

( ) ( ) ( ) ( ) ( )

ˆ( ) ( ) ( ) ( ) ( ) ( ).q

i i i k i

nq q

i i i iq

e k T z d k Q T z d k

T z d k k z F z T z d kθ

− −

− − − −

=

= +

⎛ ⎞= + ⎜ ⎟

⎝ ⎠∑

(5.40)

Let

11 10

1 12 11

( ) ( ) ( ),

( ) ( ) ( ) ( ).i i

i

v k T z d k

v k F z T z d k

−

− −

=

= (5.41)

Then

10 1

1

ˆ( ) ( ) ( ) ( ),qn

q qi

q

e k v k k z v kθ −

=

= +∑

which can be rewritten as:

bmk e

mk 1bmk + 1

emk +

mτ 1mτ +

k

cliΣ is active

cljΣ is active

188

0ˆ( ) ( ) ( ) ( ),Tie k v k k kφ θ= − (5.42)

where

1 1

1

( ) ( ), , ( 1) ,

ˆ ˆ ˆ( ) ( ), , ( ) .q

T

q

Tni i i

k v k v k n

k k k

φ

θ θ θ

⎡ ⎤= − − − +⎣ ⎦

⎡ ⎤= ⎣ ⎦

(5.43)

It should be noted that the signal 1( )v k can be computed at each time step k since the signal

( )d k in (5.41) represents the second output of the iJ block in Figure 2.1, and is therefore

accessible. Similarly, the modified performance variable ( )e k in (5.39) can be computed

from ( )e k at each time step k .

Let 0θ be the nominal parameter vector satisfying the regulation interpolation conditions

(4.18) and 1 10 0

1

( )qn

q q

q

Q z F zθ − −

=

=∑ . Then define

( )11 1 12 1 1 21 10 22

11 1 1 1 12 10

1

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ).q

i i i i i

nq q

i i iq

e k T z d k T z Q z T z d k e k

T z d k z F z T z d kθ

− − − −∗

− − − −

=

= + +

= +∑ (5.44)

It follows that

0 0( ) ( ) ( ) .Te k v k kφ θ∗ = −

It should be noted that ( )e k∗ is bounded by an exponentially decaying signal. The modified

performance variable can be expressed as follows

( ) ( ) ( ) ( )

( ) ( 1) ( ),T

e k e k e k e k

k k e kφ θ∗ ∗

∗

= − +

= − + (5.45)

where

0ˆ( ) ( )ik kθ θ θ= − .

189

The modified error ( )e k is linearly parameterized in the parameter estimation error vector θ ,

and therefore, will be used instead of performance error ( )e k in the development and

analysis of the proposed adaptation algorithm.

The main idea behind the adaptation algorithm is as follows. If Assumption 5.2 is satisfied,

then we have 0 θθ ∈S , where θS defined in (5.37). If at every time step the estimate ( )i kθ

satisfies ˆ ( )i k θθ ∈S , and 10qj nθ ×= , then based on Remark 5.3, we have ( ) , e k k kδ< ∀ ≥ .

Hence for k k≥ , the system cliΣ is always active. Consequently, if the parameter estimate

ˆ ( )i kθ can be driven to 0θ asymptotically, then based on Theorem 5.2, the adaptive switched

closed loop system cliΣ achieves regulation. In the following, an adaptation algorithm based

on the recursive least squares algorithm with time-varying forgetting factor will be used to

asymptotically drive the parameter estimate ˆ ( )iθ ⋅ to 0θ . However, the recursive least squares

adaptation algorithm cannot guarantee ( )i θθ ⋅ ∈S for all 0k ≥ . Therefore, to deal with the

case where ( )i θθ ⋅ ∉S , it is proposed to use the recursive least squares algorithm with

projection onto the boundary of the set θS whenever ( )i θθ ⋅ ∉S . For [ , ]b em mk k k∈ , 1, 2,m = … ,

the algorithm is given as follows:

( ) ( 1)ˆ ˆ( 1) Proj ( ) ( 1) , 1 ( 1) ( ) ( 1)

1 ( 1) ( ) ( ) ( 1) ( ) ( 1) . ( ) 1 ( ) ( 1) ( )

i i T

T

T

P k kk k e kk P k k

P k k k P kP k P kk k P k k

φθ θφ φ

φ φρ φ φ

⎧ ⎫++ = + +⎨ ⎬+ + +⎩ ⎭

⎡ ⎤− −= − −⎢ ⎥+ −⎣ ⎦

(5.46)

190

where 1ˆ ˆ( ) ( )b ei m i mk kθ θ −= , 1( ) ( )b e

m mP k P k −= , 1 0( ) 0bP k P= > , ( )kρ is a time-varying forgetting

factor satisfying min max0 ( ) 1kρ ρ ρ< ≤ ≤ < , and where ( ){ }ˆProj iθ ⋅ denotes the orthogonal

projection of ( )iθ ⋅ onto the set θS and is computed according to the following steps:

1) If ( )i k θθ ∈S , then

{ }ˆ ˆProj ( ) ( )i ik kθ θ= . (5.47)

2) If ( )i k θθ ∉S , then

(a) Transform the coordinate basis for the parameter space by defining

1 2ˆ ˆ( ) ( ) ( )i ik P k kϑ θ−= ,

and denote by θS the image of θS under the linear transformation 1 2( )P k − .

(b) Orthogonally project the image ˆ ( )i kϑ of ˆ ( )i kθ onto the boundary of θS to yield

ˆ ( )i kϑ∗ by solving:

ˆ ˆmin ( ) ( )i ik kϑ ϑ∗ −

( )

( )

1 21

2

1 01 21

2

ˆ0 ( ) ( )

0ˆ0 ( ) ( )

TTQi l i

T

l i

H P k k

P k k

ϑ

δ β εϑ

β

∗×

∗∗

× ∗

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦⎢ ⎥ ≥⎢ ⎥⎛ ⎞− −⎡ ⎤⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎣ ⎦⎢ ⎥⎝ ⎠⎣ ⎦

(c) Compute the estimated parameter vector as follows:

{ } 1 2ˆ ˆProj ( ) ( ) ( )i ik P k kθ ϑ∗= . (5.48)

As a first step in the adaptation process, an initial estimate ˆ (0)iθ in θS of the unknown

parameter vector is determined by solving the following LMI in the unknown ˆ (0)iθ ,

191

1

2

1 01

2

ˆ0 (0)

0ˆ0 (0)

TQi l i

l i

H θ

δ β εθ

β

×

∗

× ∗

⎡ ⎤⎡ ⎤⎣ ⎦⎢ ⎥

⎢ ⎥ ≥⎛ ⎞− −⎢ ⎥⎡ ⎤ ⎜ ⎟⎢ ⎥⎣ ⎦ ⎜ ⎟⎝ ⎠⎣ ⎦

, (5.49)

where QiH is obtained by solving (5.34). In the following, it will be shown that using the

adaptation algorithm in (5.46)-(5.48), and starting with an initial value ˆ (0)iθ that satisfies

(5.49), the estimated parameter vector ˆ ( )iθ ⋅ converges asymptotically to 0θ .

Based on Assumption 5.1, there exists a parameter vector 0θ satisfying the interpolation

conditions corresponding to the disturbance input properties. In order to analyze the

convergence of ( )kθ as k →∞ , we introduce a bound on the magnitude of the response e∗

in (5.44). Let 0 1β< < denote an upper bound on the rate of decay of the response e∗ and α

denote the maximum magnitude of the response e to the exogenous input id . Then, we have

( ) ( ) , [ , ], 1, 2,bmk k b e

m me k e k k k k mαβ −∗ ∗< = ∈ = … (5.50)

In the following, we will assume the constants α and β in (5.50) are known a priori. The

following assumption is used to derive some properties of the covariance matrix ( )P ⋅ in

(5.46).

Assumption 5.3

The signal ( )1v ⋅ in (5.41) is persistently exciting of order qn .

192

With 02 1qn k= + , and since 22 ( )e k is an exponentially decaying signal, then Assumption 5.3

is valid as long as no zeros of 12 21( ) ( ) ( )i iT z F z T z coincide with poles of ( )iD z . The following

Theorem presents the convergence properties of the adaptation algorithm.

Theorem 5.3


switching. If Assumptions 5.2 and 5.3 are satisfied, then updating the parameters iθ in QiC

using the algorithm given by (5.46)-(5.48), where 1i = for 0δ > and 2i = for 0δ < , and

setting 10qj nθ ×= , j i≠ , we have:

( )0ˆlim ( ) 0ik

kθ θ→∞

− = , (5.51)

and the adaptive switched closed loop system achieves regulation.

Proof: The proof is given in Appendix C.

Hence, the adaptation algorithm is capable of driving the parameter estimates to the desired

parameter vector 0θ needed to achieve regulation for the switched system. In the following

section, a numerical example is used to illustrate the effectiveness of the proposed adaptation

algorithm.

193


In this section, the adaptive regulation method proposed in this chapter will be used to design

a controller that cancels the contact vibrations in the mechanical system discussed in section

2.6. First, the system is discretized to obtain a zero-order-hold equivalent model with

sampling period 0.001sT s= as in (5.1) with

3 6

1 3

945.6832 10 976.8748 10107.4562 935.9144 10

A− −

−

⎡ ⎤× ×= ⎢ ⎥− ×⎣ ⎦

, 3 6

2 3

995.0125 10 995.8425 109.9584 990.0333 10

A− −

−

⎡ ⎤× ×= ⎢ ⎥− ×⎣ ⎦

,

9

1 6

493.7891 10976.8748 10

B−

−

⎡ ⎤×= ⎢ ⎥×⎣ ⎦

, 9

1 6

498.7520 10995.8425 10

B−

−

⎡ ⎤×= ⎢ ⎥×⎣ ⎦

,

9

1 6

493.7891 10 0976.8748 10 0

xD−

−

⎡ ⎤×= ⎢ ⎥×⎣ ⎦

, 9

2 6

498.7520 10 0995.8425 10 0

xD−

−

⎡ ⎤×= ⎢ ⎥×⎣ ⎦

,

[ ]1 2 1 2 1 0e e y yC C C C= = = = , 51 2 1 2 0 1 10e e y yD D D D −⎡ ⎤= = = = − ×⎣ ⎦ ,

1

0.0747sin(20 ) 0.175cos(20 )0.1cos(20 )

k kd

kπ π

π+⎡ ⎤

= ⎢ ⎥⎣ ⎦

, 2

0.075sin(20 ) 0.075cos(20 )0.1 cos(20 )

k kd

kπ π

π+⎡ ⎤

= ⎢ ⎥×⎣ ⎦.

The switching rule is defined as 6

6

1 if 50 102 if 50 10

er

e

−

−

⎧ ≤ − ×= ⎨

> − ×⎩. To construct a parameterized set

of stabilizing controllers, the gains iK , { }1, 2i∈ , are designed with an initial value

2,0 0.02α = . After 10 iterations using Algorithm 1, we obtain

62 1.0838 10β ∗ −= × , 2 0.0256α = ,

6 31 1.9740 10 1.9487 10K ⎡ ⎤= − × − ×⎣ ⎦ , 6 3

2 2.0937 10 1.9816 10K ⎡ ⎤= − × − ×⎣ ⎦ .

194

Then, the gains iL , { }1, 2i∈ , are designed with initial values 1,0 0.02α = ,

[ ]1,0 0.9473 102.7668 TL = and [ ]2,0 0.9951 9.7338 TL = − . After 8 iterations using

Algorithm 2, we obtain

61 21.4066 10β ∗ −= × , 1 0.0249α = , [ ]1 0.9549 100.2073 TL = − , [ ]2 0.8834 1.3441 TL = − .

Based on the analysis of the poles of ( )iD z , we have 0 1k = , and 02 2pin k= = since the

offset in the external input signal is 0. The parameter iQ is selected to be of the form

( )1 2 1 32

0.5 0.5 102 1i i i

qQ qq q

θ θ − −+= + ×

− +. Since 650 10 0δ −= − × < , then according to Theorem 5.3

only the system 2Σ is considered to compute the nominal parameter vector 0θ needed to

achieve regulation. Using regulation condition (4.18), it follows that:

0

37.7747 38.5927

θ−⎡ ⎤

= ⎢ ⎥⎣ ⎦

, 72

5.0431 5.008010

0.4006 0.7165Aθ −− −⎡ ⎤

= ×⎢ ⎥⎣ ⎦

, 62

0.018710

1.2395Bθ −−⎡ ⎤

= ×⎢ ⎥−⎣ ⎦.

The corresponding state space realization of iQ as given in (5.6) is:

0 1 01 2 00 1 0

QA−⎡ ⎤

⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

, 3

0.50.5 100

QB −

⎡ ⎤⎢ ⎥= ×⎢ ⎥⎢ ⎥⎣ ⎦

,

1 20Qi i iC θ θ⎡ ⎤= ⎣ ⎦ .

Assuming 70η = , and solving the LMIs in (5.34) with 1β∗ and 2β

∗ yields

3 7 11 7

3 5 9 14 9

7 9 52

7 14 5

4.3293 4.9191 10 6.9764 10 8.1933 10 6.9755 104.9191 10 1.2463 10 3.1686 10 8.3198 10 3.1685 106.9764 10 3.1686 10 3.8281 2.6668 10 3.82818.1933 10 8.3198 10 2.6668 10 2.

QH

− − − −

− − − − −

− − −

− − −

− × − × − × − ×− × × × × ×

= − × × ×− × × × 7 5

7 9 5

8788 10 2.6362 106.9755 10 3.1685 10 3.8281 2.6362 10 3.8280

− −

− − −

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

× ×⎢ ⎥⎢ ⎥− × × ×⎣ ⎦

.

Therefore, the convex set θS for the adaptive algorithm is formulated as

195

2 1 32

1 3

0= : 0

0 696.0531

TQ T

T

Hθ

θθ

θ

×

×

⎧ ⎫⎡ ⎤⎡ ⎤⎪ ⎪⎣ ⎦⎢ ⎥∈ ≥⎨ ⎬⎢ ⎥⎡ ⎤⎪ ⎪⎣ ⎦⎣ ⎦⎩ ⎭

S R .

Since the external input signal id is assumed to have unknown amplitude, frequency, and

phase, it is desired to search online for the desired parameter vector 0θ that yields the desired

regulator. The forgetting factor in the algorithm is a constant 0.9ρ = . The initial conditions

of the algorithm are ˆ(0) [0 0]Tθ = and 4(0) 10P I= where I is 2 2× identity matrix. In the

adaptation algorithm, the UD factorization algorithm is used in order to improve the

numerical properties of the algorithm in (5.46). The performance of the closed loop adaptive

system is illustrated in Figures 5.2-5.3. It can be seen that the parameter vector ˆ( )θ ⋅

converges to the nominal parameter 0θ , and that the performance variable e is driven to zero

asymptotically.

196

0 0.05 0.1 0.15 0.2 0.25 0.3-100

-50

0

50

e (m

icro

met

ers)

0 0.05 0.1 0.15 0.2 0.25 0.3-0.2

-0.1

0

0.1

0.2

d r(1,1

) (m

/sec

2 )

Time(sec)

Figure 5.2. Simulation results showing the performance variable e and the switching component (1,1)id in the input id .

0 0.05 0.1 0.15 0.2 0.25 0.3-100

-50

0

50

100

150

Time(sec)

Thet

a

estimated parameternominal parameter

Figure 5.3. Simulation results showing estimated parameter vector θ and nominal parameter vector 0θ .

197

5.7. Experimental evaluation

In this section, the experimental setup presented in section 4.7 will be used to evaluate the

proposed adaptive regulation method. In the following, based on the obtained models (4.48)

for the non-contact and (4.49) for the contact modes, an adaptive regulator will be designed

to maintain a constant separation of 650 10 m−× between the tip of the suspension beam and a

hypothetical contact surface ( )( ) 65sin 20 50 10cs t mπ −= − × . The output y , to be fed to the

controller, is defined to be the measurement voltage corresponding to the difference between

the actual separation t cy s− and the desired separation of 650 10 m−× :

( )650 10= 0.5sin(240 ) volts.

y t c

v

y k y sy kπ

−= − − ×

−

Define the performance variable e to be the same as the output y :

0.5sin(240 ) voltsve y kπ= − .

Then based on the transfer functions (4.48) and (4.49), the state space representation for the

switched bimodal system as in (5.1) is given by

1

0 0.87351 0.5645

A−⎡ ⎤

= ⎢ ⎥−⎣ ⎦, 2

0 0.93131 0.2459

A−⎡ ⎤

= ⎢ ⎥−⎣ ⎦,

1

1.80450.9125

B ⎡ ⎤= ⎢ ⎥⎣ ⎦

, 2

2.16011.0036

B ⎡ ⎤= ⎢ ⎥⎣ ⎦

,

1

0.2021 00.2010 0

xD ⎡ ⎤= ⎢ ⎥⎣ ⎦

, 2

0.2267 00.2187 0

xD ⎡ ⎤= ⎢ ⎥⎣ ⎦

,

[ ]1 1 0 1e yC C= = , [ ]2 2 0 1e yC C= = ,

198

[ ]1 2 1 2 0 1e e y yD D D D= = = = ,

111

( )( )

0.5sin(240 ) voltsd k

d kkπ

⎡ ⎤= ⎢ ⎥⎣ ⎦

, 212

( )( )

0.5sin(240 ) voltsd k

d kkπ

⎡ ⎤= ⎢ ⎥⎣ ⎦

,

where

( )311

310 2cos(240 ) 0.4sin(240 ) 0.3cos (240 ) volts4

cc c ac

i v

kd s s v k k kk k

ππ π π− ⎛ ⎞= + × + = + + +⎜ ⎟⎝ ⎠

,

2132cos (240 ) volts4ad v k ππ= = + .

Since it is desired to maintain the tip of the flexible beam at a distance of 650 10 m−× above

the contact surface, the switching rule is defined as 1 if 5 volts2 if 5 volts

er

e≤ −⎧

= ⎨ > −⎩. The parameter

iQ is selected to be of the form ( )1 2 1 32

4.8 4.8 101.5 1i i iqQ q

q qθ θ − −+

= + ×− +

. Since 5 volts 0δ = − < ,

then according to Theorem 5.3 only the system 2Σ is considered to compute the nominal

parameter vector 0θ needed to achieve regulation. The parameterized set of stabilizing

controllers is first designed so that conditions (5.16)-(5.26) are satisfied with

1 3.1753β = , 2 4.7455β = ,

[ ] 31 222.7372 509.4074 10K −= − × , [ ] 3

2 178.5739 397.7601 10K −= − × .

[ ] 31 843.0621 549.3770 10TL −= × , [ ] 3

2 931.2618 245.8963 10TL −= × .

Using regulation condition (4.18), it follows that:

30

4.260510

3.1377θ −⎡ ⎤= ×⎢ ⎥−⎣ ⎦

, 2

54.8736 50.209914.9131 26.6924

Aθ ⎡ ⎤= ⎢ ⎥−⎣ ⎦

, 32

76.245410

147.2916Bθ −−⎡ ⎤

= ×⎢ ⎥−⎣ ⎦.

199

Furthermore, the bounds on the nominal parameter vector 0θ outlined in Assumption 5.1 are

assumed to be given by 3 30

10 1010 10

10 10l uθ θ θ− −−⎡ ⎤ ⎡ ⎤× = ≤ ≤ = ×⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦

and 314.1421 10η −= × .

Therefore, conditions (5.31) and (5.34) are satisfied with

3 3 3 9 3

3 3 3 9 3

3 3 3 3 32

9

10.2086 10 3.8842 10 3.6034 10 7.4146 10 3.6031 103.8842 10 5.2234 10 202.3389 10 325.4124 10 202.3379 10

3.6034 10 202.3389 10 750.1231 10 1.3598 10 750.1203 107.4146 10 325.

QH

− − − − −

− − − − −

− − −

−

× − × × − × ×− × × × × ×

= × × × − × ×− × 9 3 3 3

3 3 3 3 3

4124 10 1.3598 10 10.2271 10 1.3673 103.6031 10 202.3379 10 750.1203 10 1.3673 10 750.1175 10

− − − −

− − −

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

× − × × − ×⎢ ⎥⎢ ⎥× × × − × ×⎣ ⎦

.

In the experiment, the properties of contact surface cs and of the external disturbance are

assumed unknown. The proposed adaptation algorithm is used to tune the parameter iQ in

the controller to yield regulation. The projection algorithm ( ){ }ˆProj iθ ⋅ in the adaptive

algorithm is modified according to the following steps in order to simplify the calculations in

the real-time implementation:

1) If ( )ˆl i ukθ θ θ≤ ≤ , then

{ }ˆ ˆProj ( ) ( )i ik kθ θ= . (5.52)

2) Otherwise

(a) Transform the coordinate basis for the parameter space by defining

1 2ˆ ˆ( ) ( ) ( )i ik P k kϑ θ−= ,

(b) Solve the following optimization problem with the unknown variable ˆ ( )i kϑ∗ :

ˆ ˆmin ( ) ( )i ik kϑ ϑ∗ −

1 2 ˆ( ) ( )l i uP k kθ ϑ θ∗≤ ≤ .

(c) Compute the estimated parameter vector as follows:

200

{ } 1 2ˆ ˆProj ( ) ( ) ( )i ik P k kθ ϑ∗= . (5.53)

First, assume the external disturbance is known, the experimental results with the nominal

parameter 0θ are verified and the results are shown in Figure 5.4-5.7. Then the external

disturbances are assumed unknown and the adaptive regulation method is applied. A branch

and bound type optimization algorithm proposed [100,101] is used to find a solution for the

projection operation in the adaptation algorithm. The forgetting factor in the adaptation

algorithm is a constant 0.93ρ = . The initial conditions for the adaptation algorithm are

ˆ(0) [0 0]Tθ = and 6(0) 10P I= where I is 2 2× identity matrix. In the adaptation algorithm,

the UD factorization algorithm is used in order to improve the numerical properties of the

algorithm. The experimental results are presented in Figures 5.8-5.12. It can be seen that

despite the unknown properties of the contact surface cs and of the external disturbance, and

despite the presence of switching in the plant dynamics, the parameter vector ˆ( )θ ⋅ still

converges to the desired parameter vector 0θ and that the performance variable e is driven to

zero asymptotically using the proposed adaptation algorithm.

5.8. Conclusion

The problem of regulation in bimodal switched systems against unknown disturbance or

reference signals is discussed. An adaptive regulator design approach based on the

parameterization of a set of stabilizing controllers for the switched system is presented. First,

regulation conditions for the switched system are derived and are used to construct a set of

Q parameters needed to achieve regulation for the case of known disturbance or reference

201

signals. For the case of unknown disturbance or reference signals, an adaptation algorithm is

developed to tune online the Q parameter in the expression of the parameterized controller.

The tuning is such that the tuned Q parameter converges to the desired Q parameter that

guarantees regulation for the switched closed loop system. A numerical example of a

mechanical system subject to contact vibrations is used to illustrate the proposed adaptive

regulator design method. The performance of the proposed adaptive regulation system is also

successfully validated on a switched mechanical system experimental setup.

202

0 0.05 0.1 0.15 0.2 0.25 0.3

-100

-80

-60

-40

-20

0

20

40

60

80

100

Per

form

ance

erro

r (m

icro

met

ers)

Time(sec)

Figure 5.4. Experimental results showing the performance variable e for the case where the nominal parameter vector 0θ is used.

0 0.05 0.1 0.15 0.2 0.25 0.3

-100

-80

-60

-40

-20

0

20

40

60

80

100

Pos

ition

(mic

rom

eter

s)

Time(sec)


Figure 5.5. Experimental results showing the tip position ty of the suspension beam and the contact surface cs for the case where the nominal parameter vector 0θ is used.

203

0 0.05 0.1 0.15 0.2 0.25 0.3

-20

0

20

40

60

80

100

120

140

Cur

rent

in th

e co

il (m

A)

Time(sec)

Figure 5.6. Experimental results showing the current i in the coil used to simulate the disturbance force and the contact force for the case where the nominal parameter vector 0θ is used.

0 0.05 0.1 0.15 0.2 0.25 0.3-5

-4

-3

-2

-1

0

1

2

3

4

5

Con

trol i

nput

(vol

ts)

Time(sec)

Figure 5.7. Experimental results showing the control input signal u driving the piezoelectric actuator for the case where the nominal parameter vector 0θ is used.

204

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

-100

-80

-60

-40

-20

0

20

40

60

80

100

Per

form

ance

var

iabl

e (m

icro

met

ers)

Time(sec)

Figure 5.8. Experimental results showing the performance variable e for the case where the adaptive regulator is used.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

-100

-80

-60

-40

-20

0

20

40

60

80

100

Pos

ition

(mic

rom

eter

s)

Time(sec)


Figure 5.9. Experimental results showing the tip position ty of the suspension beam and the contact surface cs for the case where the adaptive regulator is used.

205

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

-20

0

20

40

60

80

100

120

140

Cur

rent

in th

e co

il (m

A)

Time(sec)

Figure 5.10. Experimental results showing the current i in the coil used to simulate the disturbance force and the contact force for the case where the adaptive regulator is used.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-5

-4

-3

-2

-1

0

1

2

3

4

5

Con

trol i

nput

(vol

ts)

Time(sec)

Figure 5.11. Experimental results showing the control input signal u driving the piezoelectric actuator for the case where the adaptive regulator is used.

206

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-0.015

-0.01

-0.005

0

0.005

0.01

0.015

Time(sec)

Thet

a

estimated parameternominal parameter

Figure 5.12. Experimental results showing the desired parameter vector 0θ (dashed line) and

the estimated parameter vector θ (solid line) for the case where the adaptive regulator is used.

207

CHAPTER 6

Conclusion

Motivated by the flying height regulation problem of the read/write head in hard disk drives,

this thesis addressed the regulation problem in switched bimodal linear systems. Three main

contributions are developed with respect to the different cases of the exogenous input signals

and are summarized in the following section. The last section in this chapter addresses future

work based on the research results presented in the thesis.

6.1. Contributions of the thesis

The main approach used in this thesis to solve the regulation problem in bimodal linear

systems is based on designing Q -parameterized output feedback controllers for the switched

system. The free Q parameter in the parameterized controller is selected properly to yield

regulation in the switched bimodal system. A summary of the three main contributions in the

thesis are summarized in the following.

208

6.1.1 Development of a regulator synthesis procedure in bimodal linear systems against

known deterministic exogenous inputs

Regulation methods for switched continuous-time bimodal linear systems against known

disturbance and reference signals are developed. Two main methods are proposed for this

case.

1) Regulator synthesis based on bilinear matrix inequalities: The regulator design

approach consists of two main steps. The first step is based on constructing a switched

observer-based state feedback central controller for the switched linear system, which is

augmented with additional dynamics to construct a Q -parameterized set of switched

controllers. The proposed Q -parameterized controller is different from the traditional Q -

parameterized controllers for linear systems in that the Q parameters are not necessarily

asymptotically stable systems. In the second step, two sufficient regulation conditions

are derived for the resulting parameterized switched closed loop system. The regulation

conditions present guidelines for the selection of parameters in the controller to yield the

desired regulator for the switched bimodal systems. A corresponding regulator synthesis

approach is proposed based on solving properly formulated bilinear matrix inequalities.

2) Regulator synthesis based on linear matrix inequalities: In this approach, the

construction of a set of parameterized controllers is performed using asymptotically

stable Q parameters that are expressed as linear combinations of stable basis functions.

The stability for the resulting closed loop switched system with the Q -parameterized

controllers is analyzed, including the internal stability and input-output stability in the

closed loop system. Regulation conditions for each of the two subsystems in the

resulting bimodal switched closed loop system are derived in both the state space domain

209

and the frequency domain. Regulation conditions for the switched closed loop system are

then proposed. Two approaches are used to develop regulation conditions for the

switched closed loop system, namely, the common Lyapunov function approach and the

multiple Lypunov function approach. In the common Lyapunov function approach, the

input-output stability of the switched system is developed using a common Lyapunov

function, then the corresponding regulator synthesis method is formulated using properly

formulated linear matrix inequalities. In the multiple Lyapunov function approach, the

input-output stability property of the switched closed loop system is analyzed using a

multiple Lyapunov function approach based on the properties of the switching surface. A

corresponding regulator synthesis method is proposed using an iterative linear matrix

inequalities algorithm.

6.1.2 Development of a regulator synthesis procedure in bimodal linear systems with an

2H performance constraint.

A regulator synthesis method for switched discrete-time bimodal linear systems subject to

known disturbance or reference signals and unknown random input signals is developed. The

design of the proposed regulation method involves two main steps. In the first step, a set of

parameterized discrete time controllers that yield exact output regulation against the known

deterministic exogenous inputs in the switched bimodal system is constructed. In the second

step, additional constraints are added to identify 2H controllers within the already

constructed set of parameterized controllers and that can yield optimal output regulation

210

against the unknown random exogenous input signal. A corresponding regulator synthesis

method is developed by solving properly formulated linear matrix inequalities.

6.1.3 Development of an adaptive regulation approach in bimodal linear systems against

unknown sinusoidal exogenous inputs

An adaptive regulation problem for switched discrete-time bimodal linear systems, where it

is desired to achieve exact output regulation against unknown deterministic disturbance or

reference signals, is addressed. Assuming the properties of the deterministic exogenous input

signals are known, a sufficient regulation condition for the resulting switched closed loop

system involving Q -parameterized controllers is presented. The switched Q -parameterized

stabilizing controller is designed by iteratively solving linear matrix inequalities. In the case

of exogenous inputs with unknown properties, an adaptation algorithm is developed to tune

the Q parameter in the expression of the parameterized controller and make it converge to

the desired Q parameter that guarantees regulation for the switched bimodal system.

A summary of the advantages and limitations of various algorithms is given in the table 6.1.

211

rw Regulator Advantages Limitations

With unconstrained Q parameter

-- More flexibility -- BMI synthesis algorithm -- Local solutions -- Q structure not suitable for

adaptation Known deterministic

With constrained Q parameter

-- LMI synthesis algorithm-- Suitable for adaptation

-- Less flexibility in Q selection

Known deterministic and unknown random

With constrained Q parameter and added

2H performance constraints

-- Address both deterministic and random inputs

-- LMI synthesis algorithm

-- Less flexibility in Q selection-- Performance with respect to

steady state system

Unknown deterministic

With constrained Q parameter

-- Estimated Q converge to the desired Q

-- Need to know the number of sinusoids

-- Need to know the convex set θS containing the desired

parameter vector

Table 6.1 Summary of advantages and limitations of the proposed algorithms

6.2. Future work

Based on the research results presented in this thesis, recommendations for future work

include:

• Regulation in more General Switched Bimodal Systems

In this thesis, the switching rule between the two plant models as well as between the

disturbance and reference signals is defined according to the performance variable,

which is expressed as a time-varying switching surface with 0δ ≠ . A possible

extension of this work may include the case of 0δ = . Another possible extension is

the case where the switching times are not known. Future research should also aim at

reducing the conservativeness associated with the proposed synthesis approaches,

212

since the current LMI solvers are not able to deal with high order systems and with

cases involving very small values of δ .

• Regulation in Switched Linear Systems Involving more than Two Subsystems

The regulation methods developed in this thesis are associated with switched bimodal

systems. A more general theoretical analysis is needed for switched systems with

arbitrary numbers of subsystems. Regulation conditions for switching systems should

also be developed for cases where regulation implies that switching will always take

place, i.e. regulation is associated with an infinite number of switching instants. The

cases of known and unknown switching times (i.e. whether, at any given instant in

time, the mode of the plant is known or unknown) should be treated.

• Regulation in Switched Nonlinear Systems of Practical Interest

The flying height regulation problem that motivated this research is actually

associated with a nonlinear plant model due to the nonlinear intermolecular forces

that exist between the read/write head and the disk surface at very small flying

heights. Future research can extend the proposed regulation approach to this and other

types of nonlinear switched systems of practical interest.

• Application to the Flying Height Regulation in Hard Disk Drives

In this thesis, an experimental setup is used to evaluate the performance of the

proposed adaptive regulation method. The experimental setup involves the control of

the tip position of a hard disk drive suspension beam that is actuated using a

piezoelectric actuator. For this particular application, future research may focus on: (1)

The development of in situ real-time flying height measurement method. (2) The

development of PZT thin-film based micro actuators. (3) The development of less

213

conservative regulation approaches involving very small values of δ (i.e. very

small separation between the read/write head and the disk surface).

214

APPENDIX

A. Algorithm to compute iK , { }1, 2i∈ , and 2β∗

First, we introduce the following result.

Lemma A.1

Consider the switched system 12iT and a preset constant 2 0α > . If there exist matrices

121 0U > , 12

2 0U > , 121V and 12

2V , and a positive scalar 2β , such that the following

optimization problem

2min β

subject to

( )( ) ( )

( )

12 12 12 122

2

12 12 12

0

0 1 0,

T

i i i i i i

Ti

i i i i i j

U U AU BV

I B

AU BV B U

α

ε

⎡ ⎤− + +⎢ ⎥⎢ ⎥− − ≤⎢ ⎥⎢ ⎥+ −⎢ ⎥⎣ ⎦

( ) { } { }, 1, 2 1,2i j∀ ∈ × , (A.1)

( )12 12 122

12 12 22

0,T

i i i

i i

U C U

C U

α

β

⎡ ⎤⎢ ⎥ ≥⎢ ⎥⎣ ⎦

{ }1, 2i∀ ∈ , (A.2)

admits an optimal solution 2β∗ , then 2β

∗ is also the optimal solution of (5.25) and (5.26) with

feedback gains iK given by

215

( ) 112 12i i iK V U

−= , { }1, 2i∈ . (A.3)

Proof: Letting ( ) 112 12i iU H

−= , from (A.3), we have ( ) 112 12

i i iV K H−

= . Substituting 12iU and

12iV into (A.1), we have

( ) ( ) ( ) ( )

( ) ( )( )( ) ( )

1 1 112 12 122

2 21 112 12

0

0 1 0.

Ti i i i i i

Ti

i i i i i j

H H H A B K

I B

A B K H B H

α

ε α

− − −

− −

⎡ ⎤− + +⎢ ⎥⎢ ⎥− − ≤⎢ ⎥⎢ ⎥+ −⎢ ⎥⎣ ⎦

(A.4)

Multiplying (A.4) from the left side and the right side by ( )12 12, ,i jdiag H I H , respectively,

and substituting the expressions for 12iA and 12

iE , we immediately have (5.25). Similarly

multiplying (A.2) from the left side and the right side by ( )( )112 ,idiag U I−

, we have (5.26).

Note that (A.1) and (A.2) are linear in the unknowns 12iU , 12

jU , 12iV , and 2β only if 2α is

fixed. In the following, an algorithm is presented to obtain a local solution to the

optimization problem in Lemma A.1. Let 12,i nU , 12

,j nU , 12,i nV , 2,nβ and 2,nα denote the values of

12iU , 12

jU , 12iV , 2β and 2α in the thn iteration of the algorithm.

Algorithm 1: Design the controller gains iK , { }1, 2i∈

Step 1. Find an upper bound 2mα on the value of 2α in (A.1) with unknown 12iU , 12

jU ,

and 12iV , which is a generalized eigenvalue problem and can be solved efficiently using

the Matlab LMI toolbox. Select an initial value 2,0α for 2α in the interval ( )2 20, mα α∈ .

216

Step 2. (start the thn iteration). Minimize 2,nβ subject to (A.1) and (A.2) in the

unknowns 12,i nU , 12

,j nU , 12,i nV and 2,nβ ; and with known 2, 1nα − .

Step 3. Minimize 2,nβ subject to (A.1) and (A.2) in the unknowns 2,nα , 12,i nV and 2,nβ ;

and with known 12,i nU and 12

,j nU .

Step 4. If 2, 2, 1 2n nβ β σ−− < , a prescribed tolerance, go to step 5. Else set 1n n= + , then

go to step 2.

Step 5. Set 2 2,nβ β∗ = and ( ) 112 12, ,i i n i nK V U

−= .

For a given initial value 2,0α of 2α , the above algorithm converges to a local solution since

2β is guaranteed to decrease or stay the same in every iteration.

217

B. Algorithm to compute iL , { }1, 2i∈ , and 1β∗

Based on the designed gains iK , { }1, 2i∈ , in the above algorithm, we can design the gains

iL , { }1,2i∈ , by solving the optimization problem 1min β subject to (5.16) and (5.17).

However, matrix inequalities (5.16) and (5.17) are nonlinear in the unknown parameters 111H ,

112H , 1L , 2L , 1μ , 1β , and 1α . In the following, an algorithm is presented to obtain a local

solution based on preset initial values of 1L , 2L , and 1α . First we introduce the following

result to compute an initial values of iL , { }1, 2i∈ , to make the switched system 11iT

asymptotically stable. Due to the block triangular structure of the 11iA , the switched system

11iT is asymptotically stable under arbitrary switching if and only if the switched systems

(5.11) and (5.12) are asymptotically stable under arbitrary switching. Since (A.1) is satisfied,

it follows that (5.25) and (5.9) are satisfied, and the feedback gains iK , { }1, 2i∈ , provided

by Algorithm 1 yield an asymptotically stable switched system (5.11). However, the observer

gains iL , { }1, 2i∈ , cannot be computed directly based on (5.10) since it is a nonlinear matrix

inequality. Therefore, the following result will be used to find observer gains iL , { }1, 2i∈ ,

such that the switched system (5.12) is asymptotically stable. The computed gains iL ,

{ }1, 2i∈ , will serve as initial values for the observer gains in the algorithm to solve the

optimization problem mentioned above.

218

Lemma B.1

Consider the switched system (5.12), and let 0α ≥ be a preset constant. If there exist

matrices 1 0U > , 2 0U > and iV , and scalars iS , { }1,2i∈ , such that the following matrix

inequalities

( ) 0Ty

i i i i i i

yi i i i j

U U AU V C

AU V C U

α⎡ ⎤− + +⎢ ⎥ <⎢ ⎥+ −⎣ ⎦

, ( ) { } { }, 1, 2 1, 2i j∀ ∈ × , (B.1)

y yi i i iS C C U= , { }1, 2i∀ ∈ , (B.2)

are satisfied, then the observer gains iL given by

1i i

i

L VS

= , { }1,2i∈ , (B.3)

yield an asymptotically stable switched system (5.12).

Proof: From (B.2) and (B.3), we have

y yi i i i iV C L C U= , { }1,2i∈ .

Substituting in (B.1) the above expression for yi iV C and multiplying (B.1) from the left side

and the right side by ( )1 1,i jdiag U U− − , we have

( )( )

1 1 1

1 10

Tyi i i i i j

yj i i i j

U U A L C U

U A L C U

α− − −

− −

⎡ ⎤− + +⎢ ⎥ <⎢ ⎥+ −⎣ ⎦

,

which is equivalent to

1 1 1 0Ty y

i i i j i i i i iA L C U A L C U Uα− − −⎡ ⎤ ⎡ ⎤+ + − + <⎣ ⎦ ⎣ ⎦ , { }, 1, 2i j∈ .

219

Let 1li iP U −= and 1l

j jP U −= , then from the above inequality, we obtain (5.10). Therefore, the

switched system (5.12) is asymptotically stable.

Starting with the initial values for iL , { }1, 2i∈ , computed above and with an initial value for

1α , the following algorithm is proposed to find a local solution to the optimization problem

1min β subject to (5.16) and (5.17). Let 11,i nH , 11

,j nH , 1,nμ , 1,nβ , 1,nα and ,i nL denote the values

of 11iH , 11

jH , 1μ , 1β , 1α and iL in the thn iteration of the algorithm.

Algorithm 2: Design of the controller gains iL , { }1, 2i∈ .

Step 1. Set the initial values ,0iL of iL as ,01

i ii

L VS

= by solving the matrix inequalities

(B.1) and (B.2) in the unknowns 0iU > , iV and iS with a preset constant 0α ≥ .

Step 2. Find an upper bound 1mα on the value of 1α in (5.16) with unknown 11iH , 11

jH ,

and 1μ , which is a generalized eigenvalue problem. Select an initial value 1,0α for

1α in the interval ( )1 10, mα α∈ .

Step 3. (Start the thn iteration). Minimize 1,nβ subject to (5.16) and (5.17) with

unknown 11,i nH , 11

,j nH , 1,nμ and 1,nβ , and with known 1, 1nα − and , 1i nL − .

Step 4. Minimize 1,nβ subject to (5.16) and (5.17) in the unknowns 1,nα , ,i nL , 1,nμ and

1,nβ with known 11,i nH and 11

,j nH .

Step 5. If 1, 1, 1 1n nβ β σ−− < , a prescribed tolerance, go to step 6. Else set 1n n= + , then

go to step 3.

220

Step 6. Set 1 1,nβ β∗ = and ,i i nL L= , { }1, 2i∈ .

For given initial values ,0iL of iL and 1,0α of 1α , the above algorithm converges to a local

solution since 1β is guaranteed to decrease or stay the same in every iteration.

221

C. Proof of Theorem 5.3

First, consider the following Lyapunov function for convergence analysis:

1( ) ( ) ( ) ( )TV k k P k kθ θ−= .

The coordinate transformation 1 2ˆ ˆ( ) ( ) ( )i ik P k kϑ θ−= in the projection algorithm has the effect

of yielding

10 0 0 0

ˆ ˆ ˆ ˆ( ) ( ) ( ) ( ) ( ) ( )T T

i i i iV k k P k k k kθ θ θ θ ϑ ϑ ϑ ϑ−⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤= − − = − −⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ,

where 1 20 0( )P k θϑ θ−= ∈S . Thus, since ˆ ( )i kϑ∗ is an orthogonal projection of ˆ ( )i kϑ onto θS

and since 0 θϑ ∈S , then we have 2 2

0 0ˆ ˆ( ) ( )i ik kϑ ϑ ϑ ϑ∗− ≤ − , which means

{ } { }1 10 0 0 0

ˆ ˆ ˆ ˆProj ( ) ( ) Proj ( ) ( ) ( ) ( )T T

i i i ik P k k k P k kθ θ θ θ θ θ θ θ− −⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤− − ≤ − −⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ . (C.1)

As indicated in [102], it follows from (C.1) that convergence analysis results based on the

function 1( ) ( ) ( ) ( )TV k k P k kθ θ−= for the recursive least squares algorithm without projection

apply also to the algorithm with projection.

The recursive least squares algorithm is given by

ˆ ˆ( 1) ( ) ( 1) ( 1) ( 1) ( 1),i ik k k P k k e kθ θ ρ φ+ = + + + + + (C.2)

1 1 ( 1) ( 1) ( ) ( 1) ( 1) . TP k k P k k kρ φ φ− −⎡ ⎤+ = + + + +⎣ ⎦ (C.3)

Using equation (C.2) and the expression for e in (5.45), we have:

( 1) ( 1) ( 1) ( 1) ( 1) ( ) ( 1) ( 1) ( 1) ( 1).Tk I k P k k k k k P k k e kθ ρ φ φ θ ρ φ ∗⎡ ⎤+ = − + + + + − + + + +⎣ ⎦

From equation (C.3), we have:

222

1( 1) ( 1) ( 1) ( 1) ( 1) ( 1) ( ).TI k P k k k k P k P kρ φ φ ρ −− + + + + = + +

Therefore, for [ , ], 1, 2,b em mk k k m∈ = … , the dynamics of the parameter estimation error

vector is given by:

1( 1) ( 1) ( 1) ( ) ( ) ( 1) ( 1) ( 1) ( 1),k k P k P k k k P k k e kθ ρ θ ρ φ−∗+ = + + − + + + + (C.4)

where 1( ) ( )b em mk kθ θ −= , 1( ) ( )b e

m mP k P k −= . It follows that:

1 1( 1) ( 1) ( 1) ( ) ( ) ( 1) ( 1) ( 1)P k k k P k k k k e kθ ρ θ ρ φ− −∗+ + = + − + + + . (C.5)

From (C.3) we also have

1 11( ) ( 1) ( 1) ( 1)( 1)

TP k P k k kk

φ φρ

− −= + − + ++

. (C.6)

Using (C.4) with (C.6) we obtain

( 1) ( ) ( 1) ( 1) ( 1) ( 1) ( ) ( 1) ( 1) ( 1) ( 1)

Tk k k P k k k kk P k k e k

θ θ ρ φ φ θρ φ ∗

+ = − + + + +− + + + +

. (C.7)

Based on the Lyapunov function 1( 1) ( 1) ( 1) ( 1)TV k k P k kθ θ−+ = + + + and the projection

property of (C.1), using (C.5) and (C.7), we have that

{

}

1

1

2 1

2

( 1) ( 1) ( 1) ( 1)

( 1) ( ) ( 1) ( ) ( 1)

( 1) ( ) ( 1) ( 1) ( 1) ( 1)

( 1) ( 1) ( 1) ( ) ( ) ( 1)

( 1) ( ) ( 1) ( 1) ( 1) ( ) ( )( 1

T

T

T T

T

T T

V k k P k k

k V k k k k

k k k k P k k

k k P k P k k e k

k k k k P k P k kk

θ θ

ρ ρ θ φ

ρ θ φ φ φ

ρ φ θ

ρ θ φ φ θ

ρ

−

−∗

−

+ = + + +

≤ + + + − +

+ + + + + +

− + + + +

− + + + +

+ + 2) ( 1) ( 1) ( 1) ( 1).T k P k k e kφ φ ∗+ + + +

Using the second equation in (5.46), the following identities can be easily derived

( ) ( 1) 1 ( 1) ( 1) ( 1) ( 1)

( ) ( 1) 1 ( 1) ( ) ( 1)

T T

T

T

k k k k P k k

k kk P k k

θ φ ρ φ φ

θ φφ φ

⎡ ⎤+ − + + + + +⎣ ⎦+

= −+ + +

,

223

1( 1) ( 1) ( 1) ( ) ( )

( ) ( 1) 1 ( 1) ( ) ( 1)

T

T

T

k k P k P k k

k kk P k k

ρ φ θ

θ φφ φ

−− + + +

+= −

+ + +

,

( )

1

2

( 1) ( ) ( 1) ( 1) ( 1) ( ) ( )

( ) ( 1)

1 ( 1) ( ) ( 1)

T T

T

T

k k k k P k P k k

k k

k P k k

ρ θ φ φ θ

θ φ

φ φ

−+ + + +

+=

+ + +

,

( 1) ( ) ( 1)( 1) ( 1) ( 1) ( 1)1 ( 1) ( ) ( 1)

TT

T

k P k kk k P k kk P k k

φ φρ φ φφ φ

+ ++ + + + =

+ + +.

Using the above identities in the expression for ( 1)V k + , we obtain

( )2

2

( 1) ( 1) ( )( 1) ( ) ( 1) 2 ( ) ( 1) ( 1)

1 ( 1) ( ) ( 1)( 1) ( ) ( 1) ( 1) ( 1).

1 ( 1) ( ) ( 1)

T TT

T

T

V k k V kk k k k k e k

k P k kk P k kk e k

k P k k

ρρ θ φ θ φ

φ φφ φρφ φ

∗

∗

+ ≤ ++ ⎡ ⎤− + + + +⎢ ⎥⎣ ⎦+ + +

+ ++ + +

+ + +

Completing the square in the above equation we get

2

2

2

( 1)( 1) ( 1) ( ) ( ) ( 1) ( 1)1 ( 1) ( ) ( 1)

( 1) ( 1)1 ( 1) ( ) ( 1)

( 1) ( ) ( 1) ( 1) ( 1)1 ( 1) ( ) ( 1)

(

TT

T

T

T

kV k k V k k k e kk P k k

k e kk P k k

k P k kk e kk P k k

k

ρρ θ φφ φ

ρφ φ

φ φρφ φ

ρ

∗

∗

∗

+ ⎡ ⎤+ ≤ + − + + +⎣ ⎦+ + ++

+ ++ + +

+ ++ + +

+ + +

≤ + 2

2

1) ( ) ( 1) ( 1)

( 1) ( ) ( 1) ( 1).

V k k e k

k V k k e k

ρ

ρ ρ∗

∗

+ + +

≤ + + + +

(C.8)

Using the above equation (C.8) iteratively, it follows that at time emk , we have

224

12

1

1 2 2max max

1

1 2 2max max

1

212

max maxmax

( ) ( 1) ( ) ( ) ( )

( ) ( )

( ) ( )

( )

e eem mm

bbmm

em e b

m m m

bm

mm m

m m

k kke bm m

j ii ki k

kk i i kb

mi k

jb jm

j

bm

V k i V k j e i

V k

V k

V k

τ

ττ τ

τ τ

ρ ρ

ρ ρ α β

ρ ρ α β

βρ α ρρ

−

∗== +=

+ − −

= +

+ −

=

+

⎧ ⎫⎡ ⎤ ⎡ ⎤⎪ ⎪≤ + +⎢ ⎥ ⎢ ⎥⎨ ⎬⎢ ⎥ ⎢ ⎥⎪ ⎪⎣ ⎦ ⎣ ⎦⎩ ⎭

≤ +

≤ +

⎡≤ +

∑∏ ∏

∑

∑

1

1 12 2

max max12max max 2

max

2( 1)22 max

max max 2max

22max2

max max 2max

( )1

( )

( ) .

m

m

m m

m mm

m m

m

j

j

bm

bm

bm

V k

V k

V k

τ

τ

τ τ

τ ττ

τ ττ

β βρ ρ

ρ α ρβρ

β ρ βρ α ρ

ρ β

β ρ βρ α ρ

ρ β

=

+

+

+

⎤⎢ ⎥⎣ ⎦

⎛ ⎞⎡ ⎤ ⎡ ⎤⎜ ⎟−⎢ ⎥ ⎢ ⎥⎜ ⎟⎣ ⎦ ⎣ ⎦⎝ ⎠≤ +

⎡ ⎤− ⎢ ⎥⎣ ⎦

−≤ +

−

⎡ ⎤−⎣ ⎦≤ +−

∑

According to the adaptation algorithm, we have

1ˆ ˆ( ) ( )b e

m mk kθ θ −= ,

1( ) ( )b em mP k P k −= ,

which yields

22max2

max 1 max 2max

( ) ( )m m

me em mV k V k

τ ττ

β ρ βρ α ρ

ρ β−

⎡ ⎤−⎣ ⎦≤ +−

. (C.9)

Let em denote the index for the last switching sequence. It follows that e

bmk k≤ and that

emτ →∞ . Hence,

22max2

max 1 max 2max

lim ( ) lim ( )m me e

me

e ee em me e

e em m

k kV k V k

τ τ

τβ ρ β

ρ α ρρ β−

→∞ →∞

⎛ ⎞⎡ ⎤−⎣ ⎦⎜ ⎟≤ +⎜ ⎟−⎝ ⎠. (C.10)

Using the inequality (C.9) iteratively, we have

225

1

1

1 1

2212max2

1 max 1 max max 21 1 max

22max2

max 2max

( ) ( )

mem meem

jm

e

m me e

mme bm

m j m

V k V kτ ττ

τ

τ τ

β ρ βρ ρ α ρ

ρ β

β ρ βα ρ

ρ β

−

=

− −

−−

−= = +

⎧ ⎫⎡ ⎤⎡ ⎤⎡ ⎤−∑ ⎪ ⎪⎣ ⎦⎢ ⎥⎢ ⎥≤ + ⎨ ⎬−⎢ ⎥⎢ ⎥⎪ ⎪⎣ ⎦⎣ ⎦⎩ ⎭

⎡ ⎤−⎣ ⎦+−

∑ ∏. (C.11)

Since k is finite, then for all em m< , the dwell time mτ for the system cliΣ is finite also. Let

minτ denote the minimum dwell time of the system cliΣ . It follows from inequality (C.11) that

1

1 min

1 1

1

1 min

222max( 1) 2

1 max 1 max max 21 max

22max2

max 2max

(max 1 max

( ) ( )

( )

mem mem

m e

e

m me e

me

mm e

mm me b

mm

mb

V k V k

V k

τ τττ

τ τ

ττ

β ρ βρ ρ α ρ

ρ β

β ρ βα ρ

ρ β

ρ ρ

−

=

− −

−

=

−− −

−=

⎧ ⎫⎡ ⎤⎡ ⎤−∑ ⎪ ⎪⎣ ⎦⎢ ⎥≤ + ⎨ ⎬−⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭⎡ ⎤−⎣ ⎦+

−

∑≤ +

∑

min min

min

min min

1min min

1 min

222max1) 2

max max 21 max

22max2

max 2max

22max( 1)2

max 1 max maxm

( )

e

me

mm e

mm

m

mbV k

τ ττ

τ τ

τ τττ

β ρ βρ α ρ

ρ β

β ρ βα ρ

ρ β

β ρ βρ α ρ ρ

ρ

−

=

−− −

=

−

⎧ ⎫⎡ ⎤⎡ ⎤+⎪ ⎪⎣ ⎦⎢ ⎥⎨ ⎬−⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭

⎡ ⎤+⎣ ⎦+−

⎡ ⎤+∑ ⎣ ⎦≤ +

∑

min

min min

2

21 maxax

22max2

max 2max

1

emm

mτ

τ τ

ρβ

β ρ βα ρ

ρ β

−

=

⎡ ⎤ ⎧ ⎫⎡ ⎤⎪ ⎪⎢ ⎥ ⎨ ⎬⎢ ⎥−⎢ ⎥ ⎣ ⎦⎪ ⎪⎣ ⎦ ⎩ ⎭⎡ ⎤+⎣ ⎦+

−

∑

1min min min min

1 min

min

min min

1

22max( 1) max max2

max 1 max max 2max

max

22max2

max 2max

1 1

( )11

e

me

mm e

m

mbV kτ τ τ ττ

τ

τ

τ τ

β ρ β ρ ρρ α ρ ρ

ρ βρ

β ρ βα ρ

ρ β

−

=

−

−

⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥−⎢ ⎥ ⎢ ⎥⎡ ⎤⎡ ⎤+∑ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎢ ⎥≤ + ⎢ ⎥⎡ ⎤−⎢ ⎥⎣ ⎦ ⎢ ⎥− ⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦

⎡ ⎤+⎣ ⎦+−

226

1min min min min

1

min

min min

22 ( 1)max2 max max

max 1 max 2maxmax

22max2

max 2max

( )1

.

meem

m

mbV k

τ τ τ ττ

τ

τ τ

β ρ β ρ ρρ α ρρρ β

β ρ βα ρ

ρ β

−

=

−⎡ ⎤⎡ ⎤+∑ ⎡ ⎤−⎣ ⎦⎢ ⎥≤ + ⎢ ⎥−−⎢ ⎥ ⎣ ⎦⎣ ⎦⎡ ⎤+⎣ ⎦+

−

It follows from the above that 1( )e

emV k − is bounded. Hence, using inequality (C.10) and the

fact that emτ → ∞ , we get

( )21minlim [ ( )] ( ) lim ( ) 0

e ee e em me e

e e em m m

k kP k k V kλ θ−

→∞ →∞≤ ≤ .

It follows that

21minlim [ ( )] lim ( ) 0

e ee em me e

e em m

k kP k kλ θ−

→∞ →∞≤ .

Based on Assumption 5.3, there exists an integer 0N > and 0 m M< < < ∞ such that, for all

k k≥ , we have

1

( ) ( )k N

T

i k

mI i i MIφ φ+

= +

≤ ≤∑ . (C.12)

Consequently, the covariance matrix ( )P ⋅ satisfies [103]:

11 max

max maxmax min

1[ ( )] ( ) , 1

NN N

MP k I P k I k k Nm

ρρ λρ ρ

−

−⎡ ⎤+ ≤ ≤ ∀ ≥ +⎢ ⎥−⎣ ⎦

.

It follows from the above that

1 1 maxmin max max

max

( ) [ ( )] , 1

N NN

Mm I P k P k I k k Nρρ ρ λρ

− −⎡ ⎤≤ ≤ + ∀ ≥ +⎢ ⎥−⎣ ⎦

,

which implies that

1min min[ ( )] , NP k m k k Nλ ρ− ≥ ∀ > + (C.13)

227

1 1 maxmax max max

max

[ ( )] [ ( )] , .1

NN

MP k P k k k Nρλ ρ λρ

− −⎡ ⎤≤ + ∀ > +⎢ ⎥−⎣ ⎦

(C.14)

Therefore, 1minlim [ ( )]

e eme

em

kP kλ −

→∞ is bounded away from zero as indicated in (C.13) which yields

2lim ( ) 0e eme

em

kkθ

→∞= . Hence, the adaptation algorithm is capable of driving the parameter

estimates iθ to the desired parameter vector 0θ needed to achieve regulation.

228

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