Communications and Control Engineering

Series Editors E.D. Sontag M. Thoma A. Isidori J.H. van Schuppen

Published titles include:

Stability and Stabilization of Infinite Dimensional Systems with Applications Zheng-Hua Luo, Bao-Zhu Guo and Omer Morgul

Nonsmooth Mechanics (Second edition) Bernard Brogliato

Nonlinear Control Systems II Alberto Isidori

L2-Gain and Passivity Techniques in Nonlinear Control Arjan van der Schaft

Control of Linear Systems with Regulation and Input Constraints Ali Saberi, Anton A. Stoorvogel and Peddapullaiah Sannuti

Robust and H Control Ben M. Chen

Computer Controlled Systems Efim N. Rosenwasser and Bernhard P. Lampe

Control of Complex and Uncertain Systems Stanislav V. Emelyanov and Sergey K. Korovin

Robust Control Design Using H Methods Ian R. Petersen, Valery A. Ugrinovski and Andrey V. Savkin

Model Reduction for Control System Design Goro Obinata and Brian D.O. Anderson

Control Theory for Linear Systems Harry L. Trentelman, Anton Stoorvogel and Malo Hautus

Functional Adaptive Control Simon G. Fabri and Visakan Kadirkamanathan

Positive 1D and 2D Systems Tadeusz Kaczorek

Identification and Control Using Volterra Models Francis J. Doyle III, Ronald K. Pearson and Bobatunde A. Ogunnaike

Non-linear Control for Underactuated Mechanical Systems Isabelle Fantoni and Rogelio Lozano

Robust Control (Second edition) Jürgen Ackermann

Flow Control by Feedback Ole Morten Aamo and Miroslav Krsti

Learning and Generalization (Second edition) Mathukumalli Vidyasagar

Constrained Control and Estimation Graham C. Goodwin, María M. Seron and José A. De Doná

Randomized Algorithms for Analysis and Control of Uncertain Systems Roberto Tempo, Giuseppe Calafiore and Fabrizio Dabbene

Switched Linear Systems Zhendong Sun and Shuzhi S. Ge

Subspace Methods for System Identification Tohru Katayama

Digital Control Systems Ioan D. Landau and Gianluca Zito

Multivariable Computer-controlled Systems Efim N. Rosenwasser and Bernhard P. Lampe

Dissipative Systems Analysis and Control (2nd Edition) Bernard Brogliato, Rogelio Lozano, Bernhard Maschke and Olav Egeland

Algebraic Methods for Nonlinear Control Systems Giuseppe Conte, Claude H. Moog and Anna M. Perdon

Polynomial and Rational Matrices Tadeusz Kaczorek

Simulation-based Algorithms for Markov Decision Processes Hyeong Soo Chang, Michael C. Fu, Jiaqiao Hu and Steven I. Marcus

Iterative Learning Control Hyo-Sung Ahn, Kevin L. Moore and YangQuan Chen

Distributed Consensus in Multi-vehicle Cooperative Control Wei Ren and Randal W. Beard

Control of Singular Systems with Random Abrupt Changes El-Kébir Boukas

Alessandro Astolfi • Dimitrios Karagiannis Romeo Ortega

Nonlinear and Adaptive Control with Applications

123

ISBN 978-1-84800-065-0 e-ISBN 978-1-84800-066-7

DOI 10.1007/978-1-84800-066-7

Communications and Control Engineering Series ISSN 0178-5354

British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library

Library of Congress Control Number: 2007941070

© 2008 Springer-Verlag London Limited

MATLAB® and Simulink® are registered trademarks of The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098, USA. http://www.mathworks.com

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers.

The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use.

The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made.

Cover design: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig, Germany

Printed on acid-free paper

9 8 7 6 5 4 3 2 1 springer.com

Dimitrios Karagiannis, PhD Department of Electrical and Electronic

Engineering Imperial College London London SW7 2AZ UK

Alessandro Astolfi, PhD Department of Electrical and Electronic

Engineering Imperial College London London SW7 2AZ UK

and

Dipartimento di Informatica, Sistemi e Produzione

Università di Roma “Tor Vergata” 00133 Roma Italy

Romeo Ortega, PhD Centre National de la Recherche Scientifique Laboratoire des Signaux et Systèmes Supélec 91192 Gif-sur-Yvette France

To Elisabetta and the kids (A.A.)

To Lara (D.K.)

To the memory of my father (R.O.)

Preface

In the last few years we have witnessed the appearance of a series of challeng-ing control engineering problems. Two common features of these new controlproblems are that the interesting range of operation of the system is not nec-essarily close to an equilibrium, hence nonlinear effects have to be explicitlytaken into account for a successful controller design, and that, even thoughphysical modelling allows to accurately identify certain well-defined nonlineareffects, the controller has to cope with a high level of uncertainty, mainly dueto lack of knowledge on the system parameters and the inability to measurethe whole system state.

This situation justifies the need for the development of tools for controllerdesign for uncertain nonlinear systems, which is the main topic of this book.

Numerous theoretical control design methodologies for nonlinear sys-tems have emerged over the last two decades. When viewed from a con-ceptual standpoint, they can be broadly classified into analytically-orientedand computationally-oriented. (The qualifiers analytical and computationalare used to distinguish between symbolical analysis and numerical computa-tions.)

The former approach, which is the one adopted in this book, proceeds froman analytical model of the system, and the controller design is the outcome ofa systematic process that guarantees some specific behaviour. Since stability isa sine qua non condition, research following this approach usually runs underthe heading robust stabilisation, and it includes Lyapunov-based methods,gain-assignment methods, and classical robust and adaptive tools.

Computationally-oriented techniques, on the other hand, do not necessar-ily require an analytical model, and they may be developed on the basis of anumerical model of the system to be controlled—obtained, for instance, by col-lecting large amounts of data to approximate its behaviour. Neural networksbased control, fuzzy control and intelligent control are the more conspicuousrepresentatives of this school. Recently, a second class of computationally-oriented techniques, that relies on analytical models of the system, has gainedsome popularity. In an attempt to mimic the developments of linear systems

viii Preface

theory, piecewise linear (or linear parameter-varying) models are proposedto capture nonlinear effects. Typically some optimal control objective is for-mulated and the task of the controller design is to prove that, for the givennumerical values of the system model, the optimisation is feasible, e.g., itcan be translated into linear matrix inequalities, and a control signal can benumerically computed. The optimal control approach suffers from two draw-backs. First, the solutions are fragile with respect to plant uncertainty, e.g.,lack of full state measurement and parametric uncertainty, which is the pre-vailing concern in many, if not all, practical applications. Second, computationof the optimal control law is feasible only for low-dimensional systems, whichputs a serious question mark on the applicability of the method for nonlinearsystems. In addition there is not always a clear reason, besides mathematicalconvenience, to express the desired behaviour of a dynamical system in termsof a scalar criterion to be optimised.

In summary, computationally-oriented approaches, while leveraging off aswiftly growing computer technology, provide solutions to some specific prob-lems, but do not aim at explaining why, how and when these solutions indeedwork. At a more philosophical level, it is the authors’ opinion that controllerdesign should not be reduced to the generation of numerical code that imple-ments a control law, without any attempt to try to understand the underlyingmechanism that makes it work. This information is encoded in the nonlinearsystem dynamics and revealed through a full-fledged nonlinear analysis.

We consider nonlinear control systems subject to various types of uncer-tainty, including lack of knowledge on the parameters, partial measurement ofthe system states and uncertainty on the system order and structure. To dealwith all these situations we follow a common thread encrypted in the wordsimmersion and invariance (I&I).

In the I&I approach we propose to capture the desired behaviour of thesystem to be controlled introducing a target dynamical system. The controlproblem is then reduced to the design of a control law which guarantees thatthe controlled system asymptotically behaves like the target system. Moreprecisely, the I&I methodology relies on finding a manifold in state-space thatcan be rendered invariant and attractive, with internal dynamics a copy of thedesired closed-loop dynamics, and on designing a control law that robustlysteers the state of the system sufficiently close to this manifold.

I&I should be contrasted with the optimal control approach where theobjective is captured by a scalar performance index to optimise. In addition,because of its two-step approach, it is conceptually different from existing(robust) stabilisation methodologies that rely on the use of control Lyapunovfunctions. However, it resembles the procedure used in sliding-mode control,where a given manifold—the sliding surface—is rendered attractive by a dis-continuous control law. The key difference is that, while in sliding-mode con-trol the manifold must be reached by the trajectories, in the proposed ap-proach the manifold need not be reached. (This feature is essential in adaptivecontrol and in output feedback design.)

Preface ix

The book is organised as follows. After a brief introduction where the mainideas of I&I are illustrated by means of examples (Chapter 1), in Chapters 2and 3 we introduce the I&I framework and show how it can be used to solvestabilisation and adaptive control problems.

In Chapter 4 the method is applied to nonlinear systems with parametricuncertainties, where it is assumed that the full state is available for feedback.

In Chapter 5 we show that I&I provides a natural framework for observerdesign for general nonlinear systems.

In Chapter 6 the problem of output feedback stabilisation is solved forclasses of nonlinear systems, which include systems with unstable zero dy-namics and the well-known output feedback form. Furthermore, the methodis extended to allow unstructured uncertainties to enter the system equations.

In Chapter 7 the I&I approach is used to design a class of nonlinearproportional-integral controllers, where the gains of the controller are nonlin-ear functions that are chosen to guarantee stability for systems with unknownparameters and uncertain disturbances.

Chapters 8, 9 and 10 are devoted to applications from electrical, me-chanical and electromechanical systems, including power converters, powersystems, electric machines and autonomous aircraft.

Appendix A provides the basic definitions and recalls briefly results usedthroughout the book. In particular, characterisations of Lyapunov stability,input-to-state stability and a nonlinear version of the small-gain theorem aregiven along with some useful lemmas.

Acknowledgements

This book is the result of extensive research collaborations during the last fiveyears. Some of the results of these collaborations have been reported in thepapers [20, 19, 14, 15, 157, 162, 21, 99, 105, 16, 92, 186, 95, 106, 100, 103, 101,93, 94, 102, 96, 39, 17, 97, 40, 104, 98, 18]. We are grateful to our co-authors,Nikita Barabanov, Daniele Carnevale, Gerardo Escobar, Mickael Hilairet, LiuHsu, Zhong-Ping Jiang, Eduardo Mendes, Mariana Netto, Hugo Rodrıguez,and Aleksandar Stankovic, for several stimulating discussions and for theirhospitality while visiting their institutions. We also thank the research staffof the Laboratoire de Genie Electrique de Paris, for rendering available theirexperimental facilities.

Some of the topics of this book have been taught by the authors in a seriesof one-week graduate control courses offered in Paris for the last four years.These have been organised by the European Commission’s Marie Curie Con-trol Training Site (CTS) and by the European Embedded Control Institute(EECI) in the framework of the European Network of Excellence HYCON.We would like to thank Francoise Lamnabhi-Lagarrigue for giving us the op-purtunity to teach during these schools.

Workshops on the topics presented in this book were organised at theIEEE Conference on Decision and Control, Las Vegas, USA, 2002, and at the

x Preface

XII Latin-American Congress on Automatic Control, Bahia, Brazil, 2006. Amini-tutorial was given at the European Control Conference, Kos, Greece,2007. We have delivered lectures on selected topics of the book in the DISCSummer School, Eindhoven, The Netherlands, 2007, and in numerous researchseminars.

Finally, a large part of this work would not have been possible withoutthe financial support of several institutions. The first author would like tothank the Engineering and Physical Sciences Research Council (EPSRC) andthe Leverhulme Trust. The second author’s work was supported first by theEuropean Commission’s Training and Mobility of Researchers (TMR) pro-gramme through the Nonlinear and Adaptive Control (NACO2) network, thenby BAE Systems and the EPSRC via the FLAVIIR Integrated Programme inAeronautical Engineering, and finally by EPSRC via the Control and PowerPortfolio Partnership. The third author would like to thank the EuropeanNetwork of Excellence HYCON for supporting part of his work.

Rome, London, Paris Alessandro AstolfiApril 2007 Dimitrios Karagiannis

Romeo Ortega

Contents

Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 An I&I Perspective of Stabilisation . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Discussion and Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Applications of I&I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.1 Robustification of Control Laws . . . . . . . . . . . . . . . . . . . . . 61.3.2 Underactuated Mechanical Systems . . . . . . . . . . . . . . . . . . 91.3.3 Systems in Special Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3.4 Adaptive Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3.5 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3.6 Nonlinear PI Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 I&I Stabilisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1 Main Stabilisation Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Systems with Special Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3 Physical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 I&I Adaptive Control: Tools and Examples . . . . . . . . . . . . . . . . 333.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2 An Introductory Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3 Revisiting Classical Adaptive Control . . . . . . . . . . . . . . . . . . . . . . 38

3.3.1 Direct Cancellation with Matching . . . . . . . . . . . . . . . . . . 383.3.2 Direct Cancellation with Extended Matching. . . . . . . . . . 393.3.3 Direct Domination. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.3.4 Indirect Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.4 Linearly Parameterised Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.5 Linearly Parameterised Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.6 Example: Visual Servoing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

xii Contents

4 I&I Adaptive Control: Systems in Special Forms . . . . . . . . . . . 554.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2 Systems in Feedback Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2.1 Adaptive Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.2.2 Asymptotic Properties of Adaptive Controllers . . . . . . . . 594.2.3 Unknown Control Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.2.4 Unmatched Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.3 Lower Triangular Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.3.1 Estimator Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.3.2 Controller Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.3.3 Estimator Design using Dynamic Scaling . . . . . . . . . . . . . 74

4.4 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.4.1 Linear SISO Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.4.2 Linear Multivariable Systems . . . . . . . . . . . . . . . . . . . . . . . 81

5 Nonlinear Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.2 Reduced-order Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.3 Systems with Monotonic Nonlinearities Appearing in the

Output Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.4 Mechanical Systems with Two Degrees of Freedom . . . . . . . . . . . 107

5.4.1 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.4.2 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.4.3 Non-diagonal Inertia Matrix with Bounded Entries . . . . 1105.4.4 Diagonal Inertia Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.5 Example: Ball and Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6 Robust Stabilisation via Output Feedback . . . . . . . . . . . . . . . . . 1156.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1166.2 Linearly Parameterised Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1176.3 Control Design Using a Separation Principle . . . . . . . . . . . . . . . . 1216.4 Systems with Monotonic Nonlinearities Appearing in the

Output Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1246.5 Systems in Output Feedback Form. . . . . . . . . . . . . . . . . . . . . . . . . 1296.6 Robust Output Feedback Stabilisation . . . . . . . . . . . . . . . . . . . . . 133

6.6.1 Robust Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1356.6.2 Stabilisation via a Small-gain Condition . . . . . . . . . . . . . . 1366.6.3 Systems Without Zero Dynamics . . . . . . . . . . . . . . . . . . . . 1406.6.4 Systems with ISS Zero Dynamics . . . . . . . . . . . . . . . . . . . . 1416.6.5 Unperturbed Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1416.6.6 Linear Nonminimum-phase Systems . . . . . . . . . . . . . . . . . 142

6.7 Example: Translational Oscillator/Rotational Actuator . . . . . . . 145

Contents xiii

7 Nonlinear PI Control of Uncertain Systems . . . . . . . . . . . . . . . . 1517.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1517.2 Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

7.2.1 Bounded Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1547.2.2 Comparison with Adaptive Control . . . . . . . . . . . . . . . . . . 1577.2.3 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1577.2.4 Unbounded Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

7.3 Unknown Control Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1617.3.1 State Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1617.3.2 Observer-based Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1647.3.3 Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

7.4 Example: Visual Servoing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

8 Electrical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1738.1 Power Flow Control Using TCSC . . . . . . . . . . . . . . . . . . . . . . . . . . 173

8.1.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1748.1.2 Modified Model of the TCSC . . . . . . . . . . . . . . . . . . . . . . . 1758.1.3 Controller Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1768.1.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

8.2 Partial State Feedback Control of the DC–DC Cuk Converter . 1798.2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1818.2.2 Full-information Controller . . . . . . . . . . . . . . . . . . . . . . . . . 1828.2.3 I&I Adaptive Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1848.2.4 Partial State Feedback Controller . . . . . . . . . . . . . . . . . . . 1868.2.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

8.3 Output Feedback Control of the DC–DC Boost Converter . . . . 1898.3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1928.3.2 Full-information Controller . . . . . . . . . . . . . . . . . . . . . . . . . 1938.3.3 Output Feedback Controller . . . . . . . . . . . . . . . . . . . . . . . . 1948.3.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1978.3.5 A Remark on Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

8.4 Adaptive Control of the Power Factor Precompensator . . . . . . . 2008.4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2018.4.2 Full-information Controllers . . . . . . . . . . . . . . . . . . . . . . . . 2038.4.3 I&I Adaptation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2058.4.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

9 Mechanical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2119.1 Control of Flexible Joints Robots . . . . . . . . . . . . . . . . . . . . . . . . . . 211

9.1.1 Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2139.1.2 A 2-DOF Flexible Joints Robot . . . . . . . . . . . . . . . . . . . . . 2169.1.3 A 3-DOF Flexible Joints Robot . . . . . . . . . . . . . . . . . . . . . 217

9.2 Position-feedback Control of a Two-link Manipulator . . . . . . . . . 2189.2.1 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2199.2.2 State Feedback Controller . . . . . . . . . . . . . . . . . . . . . . . . . . 220

xiv Contents

9.2.3 Output Feedback Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 2229.2.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

9.3 Adaptive Attitude Control of a Rigid Body . . . . . . . . . . . . . . . . . 2269.3.1 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2269.3.2 Controller Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2279.3.3 Estimator Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2289.3.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

9.4 Trajectory Tracking for Autonomous Aerial Vehicles . . . . . . . . . 2299.4.1 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2309.4.2 Controller Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2319.4.3 Airspeed Regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2339.4.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

10 Electromechanical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23710.1 Observer Design for Single-machine Infinite-bus Systems. . . . . . 237

10.1.1 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23810.1.2 Controller Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23910.1.3 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23910.1.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

10.2 Adaptive Control of Current-fed Induction Motors . . . . . . . . . . . 24310.2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24410.2.2 Estimator Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24610.2.3 Rotor Flux Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24710.2.4 Controller Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24810.2.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

10.3 Speed/Flux Tracking for Voltage-fed Induction Motors . . . . . . . 25210.3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25210.3.2 Nonlinear Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 25410.3.3 Adaptive Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 25810.3.4 Simulations and Experimental Results . . . . . . . . . . . . . . . 262

A Background Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269A.1 Lyapunov Stability and Convergence . . . . . . . . . . . . . . . . . . . . . . . 269A.2 Input-to-state Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271A.3 Invariant Manifolds and System Immersion . . . . . . . . . . . . . . . . . 273

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

Notation

We consider dynamical systems described by differential equations of theform1

x = f(x, u, t),

where x ∈ Rn is the system state, u ∈ R

m is the input signal, t denotes time(in seconds), and the overdot “ ˙ ” denotes differentiation with respect to time.The solutions (or trajectories) of the above equation are denoted by x(t), witht ∈ [t0, T ) and t0 < T ≤ ∞, and x(t0) are the initial conditions. For simplicitywe often assume t0 = 0.

The transpose of a matrix A is denoted by A� and I is the identity matrix;ei ∈ R

n denotes a vector whose ith element is 1 and all other elements arezero. A diagonal n×n matrix is also written as diag(a1, . . . , an), where ai arethe diagonal elements. A column vector consisting of subvectors x1, . . . , xn isalso written as col(x1, . . . , xn).

The absolute value of a scalar y is denoted by |y|. The p-norm of a vectorx = [x1, . . . , xn]� ∈ R

n, for 1 ≤ p <∞, is defined as

|x|p � (|x1|p + · · · + |xn|p)1/p.

Note that if n = 1 then |x|p = |x|, for all 1 ≤ p <∞. In this book we mainlyuse the 2-norm (or Euclidean norm) in which case the subscript p is oftendropped, i.e., |x| � |x|2. The induced p-norm of a matrix A, for 1 ≤ p < ∞,is defined as

|A|p � supx �=0

|Ax|p|x|p

,

where sup denotes the supremum, i.e., the least upper bound. For p = 2 theabove definition yields |A| � |A|2 =

(λmax(A�A)

)1/2, where λmax(A�A) is themaximum eigenvalue of A�A.

1This class obviously includes time-invariant systems and systems that are affinein the control.

xvi Notation

The Lp-norm of a (vector) signal x(t), defined for all t ≥ 0, for 1 ≤ p <∞,is defined as

‖x‖p �(∫ ∞

0

(|x(t)|p)pdt)1/p

.

Lp denotes the space of signals x : R≥0 → Rn such that ‖x‖p exists and is

finite; L∞ denotes the space of signals x : R≥0 → Rn that are bounded, i.e.,

supt≥0 |x(t)| exists and is finite. (Note that R≥0 denotes the set of nonnegativereal numbers.)

We denote by Cn the class of functions or mappings that are n times differ-entiable, i.e., their partial derivatives up to order n exist and are continuous.Throughout the book we also use the term smooth to indicate functions ormappings that are Ck, for some large k such that all required derivatives arewell-defined.

For any C1 function V : Rn → R and any vector field f : R

n×Rm×R → R

n,LfV (x) denotes the Lie derivative of V along f at x, i.e.,

LfV (x) � ∂V

∂xf(x, u, t).

A list of acronyms used in the text is given in the table below.

BIBS Bounded-input bounded-stateI&I Immersion and invarianceIFOC Indirect field-oriented controllerIOS Input-to-output stableISS Input-to-state stableLTI Linear time-invariantMIMO Multi-input multi-outputMRAC Model-reference adaptive controlPDE Partial differential equationPFP Power factor precompensatorPI Proportional-integralPWM Pulse-width modulationRMS Root mean squareSISO Single-input single-outputSMIB Single-machine infinite-busTCSC Thyristor-controlled series capacitorTORA Translational oscillator with rotational actuator