- Email: [email protected]
Robust model-based fault diagnosis for dynamic systems
Automatica 38 (2002) 1089 – 1094
www.elsevier.com/locate/automatica
Book reviews Automotive control systems U. Kiencke and L. Nielsen; Springer, Berlin, 2000, ISBN-3-540-66922-1 The authors wrote a very good book in control of ground vehicles. They managed to write the book from the synergetic concept integrating many important topics, problems, and issues. The book has nine chapters and appendices. The introduction is given in Chapter 1. Chapter 2 is devoted to the analysis of the internal combustion engine thermodynamics. Engine management systems are covered in Chapter 3, while Chapter 4 reports numerous engine control systems. Drivetrain control is reported in Chapter 5. Chapter 6 is dedicated to introduce nonlinear vehicle modeling. Vehicle state variables and parameters of interest are reported in Chapter 7. Chapter 8 studies di6erent control systems used in the ground vehicles. Finally, road and driver mathematical models are documented in Chapter 9. This book is an excellent reference source and carrying-on book for the practicing engineers and scientists. Wide automotive control systems are thoroughly covered and analyzed by the authors. The experimental data presented helps one to understand and illustrate the need to research particular topics reported from realistic perspectives. Most importantly, the authors depart from linear concepts in modeling, analysis, and design of automotive systems. It is evident that comprehensive nonlinear analysis will improve the ground vehicle performance, and the authors illustrate this issue. In Europe, many automotive departments and programs o6er the courses in control of automotive systems and ground vehicles. These courses are an important part of senior undergraduate and graduate program curricula. The book is oriented to teach the automotive control course using numerous illustrative examples. Therefore, the book is very attractive due to their completeness and breadth. This book with corresponding supplementing lecture notes is suitable for students who already have the feedback control
systems background. The book by Professors Kiencke and Nielsen is a good introductory textbook which is methodically organized to overcome possible students’ weakness in the automotive applications of control by introducing students to many important topics. However, to attain the completeness, more numerical results and examples should be introduced and integrated in the book. The chapters and materials, in general, are well written and presented. The material is based on the nonlinear and linear modeling and control of automotive systems. The authors de?nitely dedicated their e6orts to guarantee full descriptions and comprehensive coverage with valuable examples. While acknowledging the tremendous authors’ e6orts, the authors must eliminate confusions in terminology and weaknesses in technical language, which probably can be done in the second book edition which I hope the authors target. Sergey Edward Lyshevski Department of Electrical and Computer Engineering, Purdue University Indianapolis, 723 West Michigan Street, SL160 Indianapolis, IN 46202–5132, USA E-mail address: [email protected] About the reviewer Sergey Edward Lyshevski was born in Kiev, Ukraine in 1957. He received the M.S. (1980) and Ph.D. (1987) degrees from Kiev Polytechnic Institute, all in Electrical Engineering. From 1980 to 1989 he held faculty positions at the Department of Electrical Engineering at Kiev Polytechnic Institute. From 1989 to 1992 he was a Electromechanical and Microelectronic Systems Division Head at the Academy of Sciences of the Ukraine. Since 1993 he has been with Purdue University at Indianapolis, where he is currently an Associate Professor at the Department of Electrical and Computer Engineering. He is the author of over 200 technical papers and 8 books. His current teaching and research interests include nanoand micro-electromechanical systems, high-performance electromechanical systems and mechatronics, systems theory, intelligence and CAD. He has been active in the design of advanced high-performance aerospace, automotive, electromechanical, and mechanical systems.
PII: S 0 0 0 5 - 1 0 9 8 ( 0 1 ) 0 0 2 9 1 - 6
Robust model-based fault diagnosis for dynamic systems Jie Chen, and Ron J. Patton; Kluwer Academic Publishers, Boston=Dordrecht=London, 1999, ISBN 0-7923-8411-3 The content of the book ?ts its title precisely.The book is essentially a research monograph focusing on several di6erent residual generation schemes for model-based fault diagnosis of dynamic systems. The emphasis has been
placed particularly on the robustness against disturbances and=or modelling uncertainties. The book is well written in the sense that it uni?es several di6erent approaches to fault diagnosis under a central theme of robust residual generation, which not only helps the authors to present the material in a systematic manner, but also provides readers with a broad view of the subject matter. This arrangement is especially helpful for newcomers to the ?eld, as
1090
Book reviews / Automatica 38 (2002) 1089–1094
well as for experienced researchers who would probably just like to use this book as a quick reference. It should be made clear that this is not a book that intends to cover the entire ?eld of model-based fault diagnosis for dynamic systems. Even though this book consists of nine chapters, the ?rst two chapters are tutorial in nature. From Chapter 3 onwards, each chapter concentrates on a speci?c and unique aspect of robust model-based fault diagnosis. Such organization makes the inter-dependence among di6erent chapters minimal, which is excellent from a pedagogical point of view at the post-graduate level because an instructor can cover Chapters 1 and 2 ?rst with several additional technical papers as supplementary handouts, and then he=she can select some materials from the rest of the chapters to focus on speci?c methods of fault diagnosis in detail. The amount of materials and speci?c chapters to adopt will certainly depend on the interest and the emphasis of the instructor and the amount of time available. This book can be a suitable text for a graduate level course on ‘Selected Topics in Fault Diagnosis in Dynamic Systems.’ To increase the e6ectiveness of the book in a classroom setting, the instructor may need to supply some exercise problems. The book contains 354 pages with 107 illustrations and close to 500 references. The majority of the book deals with linear time-invariant systems that can be represented by ?nite dimensional ordinary di6erential equations, though some aspects of nonlinear systems are touched upon in Chapter 9. The style of the presentation in each chapter follows the pattern of an introduction, mathematical analysis, numerical examples, and ?nally a chapter summary. Despite a few cases of practical systems that have been used as illustrated examples, no attempts have been made to address any practical issues in applying fault diagnosis techniques to these systems in practice. The book is theoretical in nature with some added practical Gavour. Chapter 1 presents a very brief overview of fault diagnosis for dynamic systems with motivations and de?nitions of several related terminologies. This chapter summarizes a few historical signi?cant developments in the area of model-based fault diagnosis from 1970s by referring to appropriate references. Chapter 2 sets the stage for the rest of the book by introducing the concept of model-based fault diagnosis, from which the principle of residual generation is presented. A general structure of residual generation schemes is illustrated and di6erent variations are brieGy reviewed. Finally, robustness in residual generation against disturbances has been brought up from the viewpoint of de-coupling. However, it is felt that the representation on the robustness to modelling errors in FDI is not very convincing. Certainly, the idea of an adaptive threshold can be a very e6ective way to deal with modelling uncertainties in practical FDI schemes. The principle of unknown input observer (UIO) has been brieGy examined in Chapter 3, and its application to fault detection and isolation has been illustrated through several
simple examples. The underlying principle of such an approach is to make the state estimation error independent of the disturbances by treating them as unknown inputs to the system. Recognizing the fact that a robust residual generation can be achieved as long as the residual signal, not the estimation error, is independent of the disturbances, a new design scheme based on eigenstructure assignment of the residual generating observer is introduced in Chapter 4. The authors have done an excellent job in presenting the basic idea, algorithms, as well as some illuminating examples, it is still felt that the presentation would have been much more interesting if some e6orts were made to examine similarities and di6erences between the unknown input observer and the eigenstructure assignment approaches when applying to the same problem. One of the main reasons why we can achieve disturbance de-coupling in residual generation is due to the assumption that the way in which the disturbance enters the system is known through the so-called disturbance distribution matrix. For a given practical system, such an assumption needs to be substantiated and the disturbance distribution matrix needs to be determined. Chapter 5 addresses such issues by proposing two methods for estimating this disturbance distribution matrix. A jet engine example has been used to illustrate this approach. The next two chapters deviate from the previous setting by formulating the problem of robust residual generation as optimization problems. In Chapter 6, the conGicting design objectives in residual generation, particularly for incipient faults, have been formulated in a multi-objective optimization framework, which can then be solved by genetic algorithms, while Chapter 7 exploits the use of optimal parity relations between system inputs and outputs to deal with bounded parameter variations and unknown disturbances. The problem of robust model-based residual generation has been formulated in frequency domain in Chapter 8. Three di6erent formulations have been examined: factorization; H∞ ?ltering; and linear matrix inequality (LMI). However, no attempts have been made to examine the relationship among them. Chapter 9 describes several general approaches to fault diagnosis for nonlinear systems. Various methods have been skimmed through, including non-linear observers, arti?cial neural networks, as well as neuro-fuzzy approaches. Even though this chapter provides some very interesting complements to the materials in the earlier chapters, the contents do not appear to ?t with the central theme of the book, i.e. robust model-based fault diagnosis. Nevertheless, this chapter adds a very interesting and di6erent Gavor to the entire book. Five appendices have been included at the end of the book to provide additional terminology in the ?eld of robust model-based fault diagnosis, a dynamic model for an inverted pendulum, as well as a few matrix formulations pertaining to the proofs in the previous chapters.
Book reviews / Automatica 38 (2002) 1089–1094
As a research monograph, the authors have certainly achieved their objectives by presenting a large body of their own research contributions from an interesting perspective with respect to other developments in the ?eld. The book could also serve as a valuable textbook if the instructors use it as a base textbook to introduce students to a wider area of model-based fault diagnosis with the help of some supplementary technical publications. From a research monograph point of view, this book is certainly a welcome addition to several recently published books in this important ?eld of fault-tolerant control systems. This reviewer would like to congratulate both authors for the excellent job presented in this book.
1091
Jin Jiang Department of Electrical and Computer Engineering, Faculty of Engineering Science, University of Western Ontario, London, Ont. N6A 5B9 Canada E-mail address: [email protected] About the reviewer Prof. Jin Jiang received his Ph.D. from the Department of Electrical Engineering, The University of New Brunswick, Fredericton, New Brunswick, Canada in 1989. He was a lecturer in Marine Institute in St. Johns, Newfoundland, and an Assistant Professor in Lakehead University, Thunder Bay, Ontario, before joining the Department of Electrical and Computer Engineering, The University of Western Ontario, London, Ontario, in 1991, where he is currently a full Professor. His research interests include: fault-tolerant control systems, energy and electrical power systems, and biomedical applications of advanced signal processing techniques.
PII: S 0 0 0 5 - 1 0 9 8 ( 0 1 ) 0 0 2 9 0 - 4
Output regulation of uncertain nonlinear systems C.I. Byrnes, F.D. Priscoli, and A. Isidori, Copyright 1997, BirkhIauser, Boston, ISBN: 3-7643-3997-7 One of the fundamental tools in feedback control theory is the use of an internal model of a class of disturbances and references in order to achieve asymptotic tracking and disturbance rejection for all signals in the speci?ed class. For constant signals, this idea reduces to the well established technique of integral control. The regulation, or servomechanism, theory was developed for linear systems in the 1970s by Davison, Francis, and Wonham. The essence of the theory is that the controller consists of two components, called the servocompensator and the stabilizing compensator. The servocompensator’s role is to generate control inputs needed to impose the prescribed asymptotic behavior, which is achieved by duplicating the model of an exosystem that generates the exogenous signals. The stabilizing compensator stabilizes the overall closed-loop system. A key property of the controller is its robustness in the sense that zero steady-state error is achieved for all parameter perturbations for which the closed-loop system remains asymptotically stable. During the 1990s, Isidori and Byrnes led the development of the regulation theory for nonlinear systems in a seminal work in which they used the center manifold theory to show that achieving zero steady-state error for a nonlinear system is equivalent to creating an invariant manifold on which the error is zero and then stabilizing the manifold so that it becomes asymptotically attractive. Further contributions by Byrnes, Isidori, and other researchers, including Huang, Khalil, Krener, Priscoli, and Rugh, have shaped the nonlinear regulation theory, which is beautifully presented in the book under review. The book consists of four chapters: Introduction, Output regulation of nonlinear systems, Existence conditions for regulator equations, and Robust output regulation. It ends with informative bibliographical notes and a list of 50 references, which cover the key references on linear regulation theory, all the essential contributions to
the nonlinear regulation theory, and references on geometric methods and nonlinear feedback stabilization, which are key ingredients of the nonlinear regulation theory. Chapter 1 starts, in Section 1:1, with an overview of the basic ingredients of the asymptotic output regulation problem. Section 1:2 introduces the concept of a steady-state response of a nonlinear system, which can be described as follows: Consider the nonlinear system x˙ = f(x; u); where f(0; 0)=0 and the eigenvalues of the Jacobian matrix [@f=@x](0; 0) have negative real parts. Suppose the input u is generated by the system w˙ = s(w);
u = d(w);
where s(0) = 0, d(0) = 0, and w˙ = s(w) is neutrally stable; that is, it has a stable equilibrium point at the origin, both in forward time and reverse time; hence, the Jacobian matrix S = [@s=@w](0) has all eigenvalues on the imaginary axis. Suppose the functions f and s are suNciently smooth. Then the composite system x˙ = f(x; d(w)); w˙ = s(w) has a (locally de?ned) center manifold x = (w), where satis?es the partial di6erential equation @ s(w) = f( (w); d(w)); @w
(0) = 0:
For suNciently small initial conditions on the manifold; that is, x(0) = (w(0)), the steady-state solution xss (t) = (w(t)) is de?ned for all t, and for suNciently small x(0) = (w(0)), the response x(t) is de?ned for all t and lim x(t) − xss (t) = 0:
t→∞
It is shown how to compute the steady-state response for certain classes of nonlinear systems. The chapter ends,
www.elsevier.com/locate/automatica
Book reviews Automotive control systems U. Kiencke and L. Nielsen; Springer, Berlin, 2000, ISBN-3-540-66922-1 The authors wrote a very good book in control of ground vehicles. They managed to write the book from the synergetic concept integrating many important topics, problems, and issues. The book has nine chapters and appendices. The introduction is given in Chapter 1. Chapter 2 is devoted to the analysis of the internal combustion engine thermodynamics. Engine management systems are covered in Chapter 3, while Chapter 4 reports numerous engine control systems. Drivetrain control is reported in Chapter 5. Chapter 6 is dedicated to introduce nonlinear vehicle modeling. Vehicle state variables and parameters of interest are reported in Chapter 7. Chapter 8 studies di6erent control systems used in the ground vehicles. Finally, road and driver mathematical models are documented in Chapter 9. This book is an excellent reference source and carrying-on book for the practicing engineers and scientists. Wide automotive control systems are thoroughly covered and analyzed by the authors. The experimental data presented helps one to understand and illustrate the need to research particular topics reported from realistic perspectives. Most importantly, the authors depart from linear concepts in modeling, analysis, and design of automotive systems. It is evident that comprehensive nonlinear analysis will improve the ground vehicle performance, and the authors illustrate this issue. In Europe, many automotive departments and programs o6er the courses in control of automotive systems and ground vehicles. These courses are an important part of senior undergraduate and graduate program curricula. The book is oriented to teach the automotive control course using numerous illustrative examples. Therefore, the book is very attractive due to their completeness and breadth. This book with corresponding supplementing lecture notes is suitable for students who already have the feedback control
systems background. The book by Professors Kiencke and Nielsen is a good introductory textbook which is methodically organized to overcome possible students’ weakness in the automotive applications of control by introducing students to many important topics. However, to attain the completeness, more numerical results and examples should be introduced and integrated in the book. The chapters and materials, in general, are well written and presented. The material is based on the nonlinear and linear modeling and control of automotive systems. The authors de?nitely dedicated their e6orts to guarantee full descriptions and comprehensive coverage with valuable examples. While acknowledging the tremendous authors’ e6orts, the authors must eliminate confusions in terminology and weaknesses in technical language, which probably can be done in the second book edition which I hope the authors target. Sergey Edward Lyshevski Department of Electrical and Computer Engineering, Purdue University Indianapolis, 723 West Michigan Street, SL160 Indianapolis, IN 46202–5132, USA E-mail address: [email protected] About the reviewer Sergey Edward Lyshevski was born in Kiev, Ukraine in 1957. He received the M.S. (1980) and Ph.D. (1987) degrees from Kiev Polytechnic Institute, all in Electrical Engineering. From 1980 to 1989 he held faculty positions at the Department of Electrical Engineering at Kiev Polytechnic Institute. From 1989 to 1992 he was a Electromechanical and Microelectronic Systems Division Head at the Academy of Sciences of the Ukraine. Since 1993 he has been with Purdue University at Indianapolis, where he is currently an Associate Professor at the Department of Electrical and Computer Engineering. He is the author of over 200 technical papers and 8 books. His current teaching and research interests include nanoand micro-electromechanical systems, high-performance electromechanical systems and mechatronics, systems theory, intelligence and CAD. He has been active in the design of advanced high-performance aerospace, automotive, electromechanical, and mechanical systems.
PII: S 0 0 0 5 - 1 0 9 8 ( 0 1 ) 0 0 2 9 1 - 6
Robust model-based fault diagnosis for dynamic systems Jie Chen, and Ron J. Patton; Kluwer Academic Publishers, Boston=Dordrecht=London, 1999, ISBN 0-7923-8411-3 The content of the book ?ts its title precisely.The book is essentially a research monograph focusing on several di6erent residual generation schemes for model-based fault diagnosis of dynamic systems. The emphasis has been
placed particularly on the robustness against disturbances and=or modelling uncertainties. The book is well written in the sense that it uni?es several di6erent approaches to fault diagnosis under a central theme of robust residual generation, which not only helps the authors to present the material in a systematic manner, but also provides readers with a broad view of the subject matter. This arrangement is especially helpful for newcomers to the ?eld, as
1090
Book reviews / Automatica 38 (2002) 1089–1094
well as for experienced researchers who would probably just like to use this book as a quick reference. It should be made clear that this is not a book that intends to cover the entire ?eld of model-based fault diagnosis for dynamic systems. Even though this book consists of nine chapters, the ?rst two chapters are tutorial in nature. From Chapter 3 onwards, each chapter concentrates on a speci?c and unique aspect of robust model-based fault diagnosis. Such organization makes the inter-dependence among di6erent chapters minimal, which is excellent from a pedagogical point of view at the post-graduate level because an instructor can cover Chapters 1 and 2 ?rst with several additional technical papers as supplementary handouts, and then he=she can select some materials from the rest of the chapters to focus on speci?c methods of fault diagnosis in detail. The amount of materials and speci?c chapters to adopt will certainly depend on the interest and the emphasis of the instructor and the amount of time available. This book can be a suitable text for a graduate level course on ‘Selected Topics in Fault Diagnosis in Dynamic Systems.’ To increase the e6ectiveness of the book in a classroom setting, the instructor may need to supply some exercise problems. The book contains 354 pages with 107 illustrations and close to 500 references. The majority of the book deals with linear time-invariant systems that can be represented by ?nite dimensional ordinary di6erential equations, though some aspects of nonlinear systems are touched upon in Chapter 9. The style of the presentation in each chapter follows the pattern of an introduction, mathematical analysis, numerical examples, and ?nally a chapter summary. Despite a few cases of practical systems that have been used as illustrated examples, no attempts have been made to address any practical issues in applying fault diagnosis techniques to these systems in practice. The book is theoretical in nature with some added practical Gavour. Chapter 1 presents a very brief overview of fault diagnosis for dynamic systems with motivations and de?nitions of several related terminologies. This chapter summarizes a few historical signi?cant developments in the area of model-based fault diagnosis from 1970s by referring to appropriate references. Chapter 2 sets the stage for the rest of the book by introducing the concept of model-based fault diagnosis, from which the principle of residual generation is presented. A general structure of residual generation schemes is illustrated and di6erent variations are brieGy reviewed. Finally, robustness in residual generation against disturbances has been brought up from the viewpoint of de-coupling. However, it is felt that the representation on the robustness to modelling errors in FDI is not very convincing. Certainly, the idea of an adaptive threshold can be a very e6ective way to deal with modelling uncertainties in practical FDI schemes. The principle of unknown input observer (UIO) has been brieGy examined in Chapter 3, and its application to fault detection and isolation has been illustrated through several
simple examples. The underlying principle of such an approach is to make the state estimation error independent of the disturbances by treating them as unknown inputs to the system. Recognizing the fact that a robust residual generation can be achieved as long as the residual signal, not the estimation error, is independent of the disturbances, a new design scheme based on eigenstructure assignment of the residual generating observer is introduced in Chapter 4. The authors have done an excellent job in presenting the basic idea, algorithms, as well as some illuminating examples, it is still felt that the presentation would have been much more interesting if some e6orts were made to examine similarities and di6erences between the unknown input observer and the eigenstructure assignment approaches when applying to the same problem. One of the main reasons why we can achieve disturbance de-coupling in residual generation is due to the assumption that the way in which the disturbance enters the system is known through the so-called disturbance distribution matrix. For a given practical system, such an assumption needs to be substantiated and the disturbance distribution matrix needs to be determined. Chapter 5 addresses such issues by proposing two methods for estimating this disturbance distribution matrix. A jet engine example has been used to illustrate this approach. The next two chapters deviate from the previous setting by formulating the problem of robust residual generation as optimization problems. In Chapter 6, the conGicting design objectives in residual generation, particularly for incipient faults, have been formulated in a multi-objective optimization framework, which can then be solved by genetic algorithms, while Chapter 7 exploits the use of optimal parity relations between system inputs and outputs to deal with bounded parameter variations and unknown disturbances. The problem of robust model-based residual generation has been formulated in frequency domain in Chapter 8. Three di6erent formulations have been examined: factorization; H∞ ?ltering; and linear matrix inequality (LMI). However, no attempts have been made to examine the relationship among them. Chapter 9 describes several general approaches to fault diagnosis for nonlinear systems. Various methods have been skimmed through, including non-linear observers, arti?cial neural networks, as well as neuro-fuzzy approaches. Even though this chapter provides some very interesting complements to the materials in the earlier chapters, the contents do not appear to ?t with the central theme of the book, i.e. robust model-based fault diagnosis. Nevertheless, this chapter adds a very interesting and di6erent Gavor to the entire book. Five appendices have been included at the end of the book to provide additional terminology in the ?eld of robust model-based fault diagnosis, a dynamic model for an inverted pendulum, as well as a few matrix formulations pertaining to the proofs in the previous chapters.
Book reviews / Automatica 38 (2002) 1089–1094
As a research monograph, the authors have certainly achieved their objectives by presenting a large body of their own research contributions from an interesting perspective with respect to other developments in the ?eld. The book could also serve as a valuable textbook if the instructors use it as a base textbook to introduce students to a wider area of model-based fault diagnosis with the help of some supplementary technical publications. From a research monograph point of view, this book is certainly a welcome addition to several recently published books in this important ?eld of fault-tolerant control systems. This reviewer would like to congratulate both authors for the excellent job presented in this book.
1091
Jin Jiang Department of Electrical and Computer Engineering, Faculty of Engineering Science, University of Western Ontario, London, Ont. N6A 5B9 Canada E-mail address: [email protected] About the reviewer Prof. Jin Jiang received his Ph.D. from the Department of Electrical Engineering, The University of New Brunswick, Fredericton, New Brunswick, Canada in 1989. He was a lecturer in Marine Institute in St. Johns, Newfoundland, and an Assistant Professor in Lakehead University, Thunder Bay, Ontario, before joining the Department of Electrical and Computer Engineering, The University of Western Ontario, London, Ontario, in 1991, where he is currently a full Professor. His research interests include: fault-tolerant control systems, energy and electrical power systems, and biomedical applications of advanced signal processing techniques.
PII: S 0 0 0 5 - 1 0 9 8 ( 0 1 ) 0 0 2 9 0 - 4
Output regulation of uncertain nonlinear systems C.I. Byrnes, F.D. Priscoli, and A. Isidori, Copyright 1997, BirkhIauser, Boston, ISBN: 3-7643-3997-7 One of the fundamental tools in feedback control theory is the use of an internal model of a class of disturbances and references in order to achieve asymptotic tracking and disturbance rejection for all signals in the speci?ed class. For constant signals, this idea reduces to the well established technique of integral control. The regulation, or servomechanism, theory was developed for linear systems in the 1970s by Davison, Francis, and Wonham. The essence of the theory is that the controller consists of two components, called the servocompensator and the stabilizing compensator. The servocompensator’s role is to generate control inputs needed to impose the prescribed asymptotic behavior, which is achieved by duplicating the model of an exosystem that generates the exogenous signals. The stabilizing compensator stabilizes the overall closed-loop system. A key property of the controller is its robustness in the sense that zero steady-state error is achieved for all parameter perturbations for which the closed-loop system remains asymptotically stable. During the 1990s, Isidori and Byrnes led the development of the regulation theory for nonlinear systems in a seminal work in which they used the center manifold theory to show that achieving zero steady-state error for a nonlinear system is equivalent to creating an invariant manifold on which the error is zero and then stabilizing the manifold so that it becomes asymptotically attractive. Further contributions by Byrnes, Isidori, and other researchers, including Huang, Khalil, Krener, Priscoli, and Rugh, have shaped the nonlinear regulation theory, which is beautifully presented in the book under review. The book consists of four chapters: Introduction, Output regulation of nonlinear systems, Existence conditions for regulator equations, and Robust output regulation. It ends with informative bibliographical notes and a list of 50 references, which cover the key references on linear regulation theory, all the essential contributions to
the nonlinear regulation theory, and references on geometric methods and nonlinear feedback stabilization, which are key ingredients of the nonlinear regulation theory. Chapter 1 starts, in Section 1:1, with an overview of the basic ingredients of the asymptotic output regulation problem. Section 1:2 introduces the concept of a steady-state response of a nonlinear system, which can be described as follows: Consider the nonlinear system x˙ = f(x; u); where f(0; 0)=0 and the eigenvalues of the Jacobian matrix [@f=@x](0; 0) have negative real parts. Suppose the input u is generated by the system w˙ = s(w);
u = d(w);
where s(0) = 0, d(0) = 0, and w˙ = s(w) is neutrally stable; that is, it has a stable equilibrium point at the origin, both in forward time and reverse time; hence, the Jacobian matrix S = [@s=@w](0) has all eigenvalues on the imaginary axis. Suppose the functions f and s are suNciently smooth. Then the composite system x˙ = f(x; d(w)); w˙ = s(w) has a (locally de?ned) center manifold x = (w), where satis?es the partial di6erential equation @ s(w) = f( (w); d(w)); @w
(0) = 0:
For suNciently small initial conditions on the manifold; that is, x(0) = (w(0)), the steady-state solution xss (t) = (w(t)) is de?ned for all t, and for suNciently small x(0) = (w(0)), the response x(t) is de?ned for all t and lim x(t) − xss (t) = 0:
t→∞
It is shown how to compute the steady-state response for certain classes of nonlinear systems. The chapter ends,