Analysis and Design of Nonlinear Control Systems

Daizhan Cheng Xiaoming Hu Tielong Shen

Analysis and Design of Nonlinear Control Systems

Daizhan Cheng Xiaoming Hu Tielong Shen

Analysis and Design of Nonlinear Control Systems With 63 figures

Authors Daizhan Cheng Academy of Mathematics & Systems Science Chinese Academy of Sciences Beijing, 100190, China Email: [email protected]

Xiaoming Hu Optimization and Systems Theory Royal Institute of Technology 100 44 Stockholm, Sweden Email: [email protected]

Tielong Shen Department of Engineering and Applied Sciences Sophia University Kiocho 7-1, Chiyoda-ku Tokyo, 102-8554 Japan Emai: [email protected]

ISBN 978-7-03-025964-6 Science Press Beijing ISBN 978-3-642-11549-3 e-ISBN 978-3-642-11550-9 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2009942528 © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, 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 protective laws and regulations and therefore free for general use. Cover design: Frido Steinen-Broo, EStudio Calamar, Spain Printed on acid-free paper Springer is a part of Springer Science+Business Media (www.springer.com)

Preface

The purpose of this book is to present a comprehensive introduction to the theory and design technique of nonlinear control systems. It may serve as a standard reference of nonlinear control theory and applications for control scientists and control engineers as well as Ph.D students majoring in Automation or some related elds such as Operational Research, Management, Communication etc. In the book we emphasize on the geometric approach to nonlinear control systems. In fact, we intend to put nonlinear control theory and its design techniques into a geometric framework as much as we can. The main motivation to write this book is to bring readers with basic engineering background promptly to the frontier of the modern geometric approach on the dynamic systems, particularly on the analysis and control design of nonlinear systems. We have made a considerable effort on the following aspects: First of all, we try to visualize the concepts. Certain concepts are dened over local coordinates, but in a coordinate free style. The purpose for this is to make them easily understandable, particularly at the rst reading. Through this way a reader can understand a concept by just considering the case in ℝn . Later on, when the material has been digested, it is easy to lift them to general topological spaces or manifolds. Secondly, we emphasize the numerical or computational aspect. We believe that making things computable is very useful not only for solving engineering problems but also for understanding the concepts and methods. Thirdly, certain proofs have been simplied and some elementary proofs are presented to make the materials more readable for engineers or readers not specializing in mathematics. Finally, the topics which can be found easily in some other standard textbooks or references are briey introduced and the corresponding references are included. Much attention has been put on new topics, new results, and new design techniques. For convenience, a brief survey on linear control theory is included, which can be skipped for readers who are already familiar with the subject. For those who are not majoring in control theory, it provides a tutorial introduction to the eld, which is sufcient for the further study of this book. The other mathematical prerequirements are Calculus, Linear Algebra, Ordinal Differential Equation.

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The materials in the book are selected and organized as self-sustained as possible. Chapter 1 is a general introduction to dynamic (control) systems. First, a brief survey for the theory of linear control systems is given. It provides certain necessary knowledge of systems and control, which will be used in the sequel. Then some special characteristics of nonlinear dynamics are discussed. Finally, some sample nonlinear control systems in practice are presented. Chapter 2 gives an elementary introduction to topological spaces. It focuses primarily on those fundamental concepts of topological space. We emphasize on the second countable Hausdorff space, which is the enduring space of a manifold. Continuity of a mapping, homeomorphism, connectness, and quotient space etc. are also discussed. Chapter 3 investigates differential manifolds. It is the basic tool for the geometric approach to nonlinear control systems. The concept of differential manifold is discussed rst. Fiber bundle is then introduced. Vector elds, their integral curves and Lie brackets etc. are discussed subsequently. Then form, distribution and codistribution are investigated. In fact, smooth function, vector eld and co-vector eld (one-form) are the three fundamental elements in nonlinear geometric control theory. Some important theorems such as Frobenius’ Theorem, Lie series expansions and Chow’s Theorem are presented with proofs. Finally, the tensor eld is introduced, and based on it two important manifolds, namely, Riemanian manifold and symplectic manifold are investigated. Chapter 4 gives a brief review on some basic concepts on abstract algebra, including group, ring and algebra. Then there is a brief introduction to some elementary concepts in algebraic topology, such as fundamental group, covering space etc. The main effort is focused on Lie group and Lie algebra. Particularly, the general linear group, general linear algebra and their sub-groups and sub-algebras are investigated. In brief summary, Chapters 2–4 provide the mathematical foundation for nonlinear control theory that should sufce for the study of the rest of the book. Chapter 5 considers controllability and observability of nonlinear systems. Similar to linear systems, the “controllable” and “observable” sub-manifolds are obtained. Based on them the Kalman decomposition is obtained. Algorithms for some control related distributions are provided. Chapter 6 provides some further discussion for the global controllability of nonlinear systems. It is based on the properties of the integral orbits of a set of vector elds and under a general framework of switched systems. First, a heuristic sufcient condition is provided, which shows the geometric insight of the controllability. Then it is generalized to the case of nested distributions. Finally, it is shown that under certain circumstances the sufcient condition becomes necessary too. Chapter 7 considers stability of nonlinear dynamic systems and feedback stabilization of nonlinear control systems. The Lyapunov theory and its generalization – LaSalle’s invariance principle and their applications are discussed rst. Converse theorems to Lyapunov’s stability theorems are investigated. Stability of invariant set is then considered. Both input-to-output and input-to-state stabilities and stabilizations via control Lyapunov function are also discussed. Finally, for an asymptotically stable equilibrium the region of attraction is investigated. Chapter 8 discusses the decoupling problems. First, as a generalization of (A, B)invariant subspace, the ( f , g)-invariant sub-distribution is dened. It is used to solve

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the disturbance decoupling problem. Then the controlled invariant distribution is introduced and it is used for block decomposition of state space by state coordinate transformation with or without controls. Chapter 9 considers the input-output structure of nonlinear systems. First, the decoupling matrix for afne nonlinear systems is introduced. In the light of it, the input-output decoupling problem (also called the Morgan’s problem), the invertibility of nonlinear systems, and the dynamic feedback decoupling problems are investigated. The normal form is introduced, and then by using the point relative degree (vector), the generalized normal form is presented. Finally, the Fliess functional expansion is introduced to describe the input-output mapping of afne nonlinear systems. Following it, the application of Fliess expansion to tracking problem is discussed. Chapter 10 discusses different linearizations of nonlinear control systems. First, linearization without control, called the Poincar’e linearization, is considered. Then regular state feedback linearization with or without outputs is considered. The results for regular state feedback linearization have a fundamental meaning. Then global linearization is considered. Finally linearization via non-regular state feedback is discussed. Chapters 5–10 may be considered as the kernel of nonlinear control theory. Chapter 11 gives an introduction for the theory of center manifold rst. Then the center manifold is used for the design of stabilizers for nonlinear systems with either minimum or non-minimum phase. For non-minimum phase systems a new tool, called the Lyapunov function with homogeneous derivative is proposed. It is then applied to stabilizing zero dynamics on the center manifold of nonlinear systems with both zero center and oscillatory center separately. The technique developed is applied to stabilization of systems in generalized normal form. Some stabilizer design methods are presented. Chapter 12 is an elementary introduction to output regulation. First, the internal model principle of linear systems is introduced. Then the local output regulation problem is considered in detail. Finally, the results obtained for local output regulation are adopted to solving the robust local output regulation problem. Chapter 13 discusses dissipative systems. First, denition and some useful properties of dissipative systems are introduced. Then, the passivity, as a special case of dissipativity, is investigated, where the main attention is focused on the KalmanYakubovich-Popov lemma. Passivity-based controller design methods are introduced for general nonlinear systems. Finally, some further applications of the design technique are discussed for Lagrange systems and Hamiltonian systems from different physical backgrounds. Chapter 14 considers L2 -gain synthesis problems. Fundamentals of L2 -gain, and Hf norm in linear systems, are discussed. The Hf control problem for linear systems is considered and the extension to nonlinear systems is investigated. Finally, a control system synthesis technique with L2 -gain, mainly disturbance attenuation, is provided using the method of storage function construction. Chapter 15 is about switched systems. Based on the fact that a switched linear system is also a nonlinear system, we consider equally the switched linear and nonlinear systems. Common quadratic Lyapunov functions are considered rst. Based on common Lyapunov functions, the stabilization of switched systems is consid-

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ered. Then the controllability of linear and bilinear switched systems is investigated in detail. A LaSalle’s invariance principle for switched systems has been developed. Finally, the methods developed for switched systems are applied to investigate the consensus problem of different types of multi-agent systems. Chapter 16 provides a tutorial introduction to discontinuous dynamical systems. The Filippov-framework for differential equations with discontinuous right hand side is discussed rstly. Then the non-smooth analysis method, which is motivated by the design problem for non-smooth Lyapunov functions, is reviewed. Stability conditions for the Filippov solutions are discussed and the control design is also investigated for a class of discontinuous systems. Finally, the behavior of sliding mode control systems is revisited in term of Filippov-framework. The issues discussed in Chapters 11–16 are some special systems and special control techniques. Parts of the contents were originally prepared for teaching the course of nonlinear control systems at the Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences. The authors are in debt to their graduate students for their assistance in preparing the materials, pointing out typos and for their suggestions for improving the statements and proofs. Particularly, we thank Dr. Hongsheng Qi, and Dr. Zhiqiang Li for helping in proof-reading and picture redrawing. The authors would like to thank Science Press for their support in editing and publishing this book.

March 1, 2010 Daizhan Cheng Xiaoming Hu Tielong Shen

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Linear Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Controllability, Observability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.2 Invariant Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1.3 Zeros, Poles, Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.1.4 Normal Form and Zero Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2 Nonlinearity vs. Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.2.1 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.2.2 Singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.2.3 Complex Behaviors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.3 Some Examples of Nonlinear Control Systems . . . . . . . . . . . . . . . . . . . . . 20 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2. Topological Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.1 Metric Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.2 Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.3 Continuous Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.4 Quotient Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3. Differentiable Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.1 Structure of Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2 Fiber Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.3 Vector Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.4 One Parameter Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.5 Lie Algebra of Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.6 Co-tangent Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.7 Lie Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.8 Frobenius’ Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.9 Lie Series, Chow’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.10 Tensor Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.11 Riemannian Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.12 Symplectic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

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Algebra, Lie Group and Lie Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.1 Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.2 Ring and Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.3 Homotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.4 Fundamental Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.5 Covering Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.6 Lie Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.7 Lie Algebra of Lie Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.8 Structure of Lie Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5. Controllability and Observability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.1 Controllability of Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.2 Observability of Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.3 Kalman Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 6. Global Controllability of Afne Control Systems . . . . . . . . . . . . . . . . . . . . . 147 6.1 From Linear to Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 6.2 A Sufcient Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 6.3 Multi-hierarchy Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 6.4 Codim(G ) = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 7. Stability and Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 7.1 Stability of Dynamic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 7.2 Stability in the Linear Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 7.3 The Direct Method of Lyapunov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 7.3.1 Positive Denite Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 7.3.2 Critical Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 7.3.3 Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 7.3.4 Asymptotic Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 7.3.5 Total Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 7.3.6 Global Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 7.4 LaSalle’s Invariance Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 7.5 Converse Theorems to Lyapunov’s Stability Theorems . . . . . . . . . . . . . 185 7.5.1 Converse Theorems to Local Asymptotic Stability . . . . . . . . . . . . . . . . . . . 185 7.5.2 Converse Theorem to Global Asymptotic Stability . . . . . . . . . . . . . . . . . . . 187 7.6 Stability of Invariant Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 7.7 Input-Output Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 7.7.1 Stability of Input-Output Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 7.7.2 The Lur’e Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 7.7.3 Control Lyapunov Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 7.8 Region of Attraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 8. Decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 8.1 ( f , g)-invariant Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 8.2 Local Disturbance Decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 8.3 Controlled Invariant Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 8.4 Block Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

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8.5 Feedback Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Input-Output Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 9.1 Decoupling Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 9.2 Morgan’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 9.3 Invertibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 9.4 Decoupling via Dynamic Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 9.5 Normal Form of Nonlinear Control Systems . . . . . . . . . . . . . . . . . . . . . . 253 9.6 Generalized Normal Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 9.7 Fliess Functional Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 9.8 Tracking via Fliess Functional Expansion . . . . . . . . . . . . . . . . . . . . . . . . . 267 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 Linearization of Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 10.1 Poincar´e Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 10.2 Linear Equivalence of Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . 282 10.3 State Feedback Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 10.4 Linearization with Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 10.5 Global Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 10.6 Non-regular Feedback Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 Design of Center Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 11.1 Center Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 11.2 Stabilization of Minimum Phase Systems . . . . . . . . . . . . . . . . . . . . . . . . 317 11.3 Lyapunov Function with Homogeneous Derivative . . . . . . . . . . . . . . . . 319 11.4 Stabilization of Systems with Zero Center . . . . . . . . . . . . . . . . . . . . . . . 328 11.5 Stabilization of Systems with Oscillatory Center . . . . . . . . . . . . . . . . . 335 11.6 Stabilization Using Generalized Normal Form . . . . . . . . . . . . . . . . . . . 341 11.7 Advanced Design Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 Output Regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 12.1 Output Regulation of Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 12.2 Nonlinear Local Output Regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 12.3 Robust Local Output Regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 Dissipative Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 13.1 Dissipative Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 13.2 Passivity Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 13.3 Passivity-based Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 13.4 Lagrange Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 13.5 Hamiltonian Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 L 2 -Gain Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 14.1 Hf Norm and L2 -Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 14.2 Hf Feedback Control Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 14.3 L2 -Gain Feedback Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 14.4 Constructive Design Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 14.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

xii

Contents

15 Switched Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 15.1 Common Quadratic Lyapunov Function . . . . . . . . . . . . . . . . . . . . . . . . . 431 15.2 Quadratic Stabilization of Planar Switched Systems . . . . . . . . . . . . . . 454 15.3 Controllability of Switched Linear Systems . . . . . . . . . . . . . . . . . . . . . . 467 15.4 Controllability of Switched Bilinear Systems . . . . . . . . . . . . . . . . . . . . . 476 15.5 LaSalle’s Invariance Principle for Switched Systems . . . . . . . . . . . . . . 483 15.6 Consensus of Multi-Agent Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 15.6.1 Two Dimensional Agent Model with a Leader . . . . . . . . . . . . . . . . . . . . . . 493 15.6.2 n Dimensional Agent Model without Lead . . . . . . . . . . . . . . . . . . . . . . . . 495 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 16 Discontinuous Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 16.2 Filippov Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510 16.2.1 Filippov Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510 16.2.2 Lyapunov Stability Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 16.3 Feedback Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 16.3.1 Feedback Controller Design: Nominal Case . . . . . . . . . . . . . . . . . . . . . . . 518 16.3.2 Robust Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521 16.4 Design Example of Mechanical Systems . . . . . . . . . . . . . . . . . . . . . . . . . 523 16.4.1 PD Controlled Mechanical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523 16.4.2 Stationary Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524 16.4.3 Application Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 Appendix A Some Useful Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 A.1 Sard’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 A.2 Rank Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 Appendix B Semi-Tensor Product of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 535 B.1 A Generalized Matrix Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535 B.2 Swap Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 B.3 Some Properties of Semi-Tensor Product . . . . . . . . . . . . . . . . . . . . . . . . . 538 B.4 Matrix Form of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541

Symbols

ℂ ℝ ℚ ℤ ℕ := Mm×n Mn V (A) Im(A) Ker(C) C O ⟨A ∣W ⟩ (M, T ) Br (x) A A¯ T (M) T ∗ (M) Cf (M) CZ (M) V (M) V Z (M) etX (x0 ) Lf h ad f g adkf g Lkf h [⋅, ⋅] {⋅ ⋅ ⋅ }LA {⋅, ⋅}

Set of complex numbers Set of real numbers Set of rational numbers Set of integers Set of natural numbers Dened as Set of m × n real matrices Set of n × n real matrices Set of eigenvalues of matrix A Subspace spanned by the columns of A Subspace generated by {X ∣ XC = 0} Controllable subspace Observable subspace Smallest subspace containing W and A invariant Topological space M with topology T Open ball with center x and radius r Interior of A Closure of A Tangent space on manifold M Cotangent space on manifold M Set of Cf functions on M Set of analytic functions on M Set of smooth vector elds on manifold M Set of analytic vector elds on manifold M Integral curve of X with initial value x0 Lie derivative of h (function or form) w.r.t. f Lie derivative of vector eld g w.r.t. f k-th Lie derivative of vector eld g w.r.t. f k-th Lie derivative of h (function or form) w.r.t. f Lie bracket Lie algebra generated by ⋅ ⋅ ⋅ Poisson bracket

xiv

Symbols

Hess(h(x)) H
Hessian matrix of h(x) H is a sub-group of G H is a normal sub-group of G General linear group General linear algebra Special orthogonal linear group Orthogonal linear algebra Symplectic group Symplectic algebra Symmetric group Set of tensors over V with co-variant r and contra-variant s Differential of F Gradient of F Integral manifold of ' passing x0 Involutive closure of a distribution Set of proper stable rational matrices of size n × m Set of proper stable rational functions

Chapter 1

Introduction

In this chapter we give an introduction to control theory and nonlinear control systems. In Section 1.1 we briey review some basic concepts and results for linear control systems. Section 1.2 describes some basic characteristics of nonlinear dynamics. A few typical nonlinear control systems are presented in Section 1.3.

1.1 Linear Control Systems A control system can be described as a black box in Fig. 1.1 (a), with input (or control) u and output y. In this sense, a control system is considered as a mapping from the input space to the output space. Before 1950’s, primarily due to the nature of many electrical and electronic engineering problems then, control problems were largely treated as ltering problems and in the frequency domain, which is particularly suitable for single-input and single-output systems. During 1950’s Rudolf E. Kalman proposed a state space description for control systems. A set of state variables were introduced to describe the box. Intuitively, the black box is split into two parts: the rst part is a set of differential (or difference) equations, which are used to describe the dynamics from control u to state variables x, and then a static equation is used to describe the mapping from state variables x to output y. See Fig. 1.1 (b). There are many different state space descriptions that realize the same inputoutput mapping. These are called the state space realizations of the input-output mapping. A realization is minimum if there is no other realization that has less dimension of the state space. Feedback is perhaps the most fundamental concept in automatic control and has a long history. Feedback means that the control strategy relies on the current status of the system. Depending on what information on the current status is available, it can be classied as state feedback control, see Fig. 1.1 (c), and output feedback control, see Fig. 1.1 (d). A nonlinear control system considered throughout this book is described by { x = F(x, u), x ∈ M, u ∈ U (1.1) y = h(x), y ∈ N, where M, U, and N are manifolds of dimensions n, m, p respectively. F(x, u) and h(x) are smooth mappings (C f mappings unless elsewhere stated). In fact, we conD. Cheng et al., Analysis and Design of Nonlinear Control Systems © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2010

2

1 Introduction

Fig. 1.1

A Control System

sider (1.1) as an expression of the system in a coordinate chart, and under the local coordinates x. In many applications we simply have M = ℝn , U = ℝm , and N = ℝ p . Particularly, when F(x, u) is afne with respect to u, system (1.1) becomes an afne nonlinear system. It is commonly expressed as ⎧ m  ⎨x = f (x) + gi (x)ui := f (x) + g(x)u, x ∈ M, u ∈ U ¦ (1.2) i=1  ⎩y = h(x), y ∈ N. System (1.2) becomes a linear control system if f (x) = Ax, h(x) = Cx are linear and gi (x) are constant. That is

1.1 Linear Control Systems

⎧  ⎨x = Ax +  ⎩y = Cx,

m

¦ biui := Ax + Bu,

x ∈ ℝn , u ∈ ℝm

i=1

3

(1.3)

y ∈ ℝ p.

It is clear that all the denitions and results for (1.1) are also applicable to (1.2) and (1.3). Observing the history of modern control theory, one sees that the theory was rstly developed for linear systems, and then as the theory for linear control systems had become considerably mature, theory for nonlinear control systems started to take off. So in nonlinear control theory, there are many concepts and results that are originated, inherited, developed, or extended from their linear counterparts. For the sake of easy reference, we give a brief survey on linear control theory in the following. Most of the proofs are omitted in the survey. We refer to standard references such as [11, 6, 4] for details of linear control systems.

1.1.1 Controllability, Observability Denition 1.1. System (1.1) is said to be controllable (on a subset V ⊂ M) if for any two points x, z ∈ M (respectively, x, z ∈ V ) there exists a control u such that under this control the trajectory goes from x(0) = x to x(T ) = z for some T > 0 with x(t) ∈ M (respectively, x(t) ∈ V ), t ∈ [0, T ]. Theorem 1.1. For linear system (1.3) the controllable subspace C is C = Span{B, AB, ⋅ ⋅ ⋅ , An−1 B}.

(1.4)

Consequently, the system is controllable, if and only if C = ℝn . Proof. Dene the following matrix

) (t0 ,t) =

∫ t

eA(t−W ) BBT eA

T (t−W )

t0

dW ,

t > t0 .

(1.5)

We claim that C has full rank, if and only if ) (t0 ,t) is nonsingular. If rank(C ) < n, there exists 0 ∕= X ∈ ℝn such that X T Ak B = 0,

k = 0, 1, ⋅ ⋅ ⋅ , n − 1.

It follows from Taylor expansion and Cayley-Hamilton’s theorem that X T eA(t−W ) B = 0. So ) (t0 ,t) is singular. Conversely, if ) (t0 ,t) is singular, then there exists 0 ∕= X ∈ ℝn such that ∫ t ∥X T eA(t−W ) B∥2 dW = 0. t0

Hence,

X T eA(t−W ) B ≡ 0,

t0 ⩽ W ⩽ t.

4

1 Introduction

Differentiate it with respect to W for k times, and set W = t, we have X T Ak B = 0,

k = 0, 1, ⋅ ⋅ ⋅ , n − 1.

The claim is proved. For system (1.3), the trajectory can be expressed as x(t) = eA(t−t0 ) x0 +

∫t t0

eA(t−W ) Bu(W )dW .

(1.6)

Now if rank(C ) < n, and assume the system is controllable. Let 0 ∕= X ∈ ℝn , such that X T Ak B = 0, k = 0, 1, ⋅ ⋅ ⋅ , n − 1. Set x0 = X and x(t) = 0, then by controllability, (1.6) yields ∫ x0 = −

t

t0

eA(t0 −W ) Bu(W )dW .

Multiply both sides by X T yields ∥X∥2 = 0, which is absurd. Next, assume rank(C ) = n. For any x0 and xt , we can choose u as u(W ) = BT eA Then (1.6) yields

T (t−W )

v.

(1.7)

xt = eA(t−t0 ) x0 + ) v.

Using the claim, ) is invertible. So v = ) −1 (xt − eA(t−t0 ) x0 ).

(1.8)

That is, the control (1.7) with v as in (1.8) drives x0 to x1 .

⊔ ⊓

We gave the proof here since the above theorem is fundamental and the proof is constructive. Remark 1.1. 1. Throughout this book we assume the set of admissible controls is the set of measurable functions, unless elsewhere stated. 2. Let x(0) = x0 and u = u(t). Then the solution of (1.3) is x(t) = e x0 + At

∫t 0

eA(t−W ) Bu(W )dW .

(1.9)

3. System (1.3) is controllable, if and only if W (t0 ,t1 ) =

∫ t1 t0

e−AW BBT e−A W dW T

(1.10)

is nonsingular for t1 > t0 . Moreover, in this case the control u, which drives x from x0 at t0 to x1 at t1 is:

1.1 Linear Control Systems

[ ] T u(t) = BT e−A t W −1 (t0 ,t1 ) e−At1 x1 − e−At0 x0 .

5

(1.11)

Denition 1.2. System (1.1) is said to be observable (on a subset V ⊂ M) if for any two points x, z ∈ M (respectively, x, z ∈ V ) there exists a control u such that under this control the output y(t, u, x) ∕= y(t, u, z) for some t > 0. Theorem 1.2. For linear system (1.3) the observable subspace O is O = Span{CT , ATCT , ⋅ ⋅ ⋅ , (AT )n−1CT }.

(1.12)

Consequently, the system is observable, if and only if O = ℝn . Remark 1.2. 1. From Theorem 1.2 one sees that for linear systems the observability is independent of the inputs. However, this is in general not true for nonlinear systems. 2. System (1.3) is observable, if and only if Q(t0 ,t1 ) =

∫ t1 t0

eA W CTCeAW dW T

(1.13)

is nonsingular for t1 > t0 . Spliting the state space into four parts as: C ∩O, C ∩O ⊥ , C ⊥ ∩O, C ⊥ ∩O ⊥ , and choosing coordinates x = (x1 , x2 , x3 , x4 ) in such a way that each block of state variables corresponds to the above four subspaces, then we have the following Kalman decomposition: Theorem 1.3. (Kalman Decomposition) The system (1.3) can be expressed as ⎧⎡ ⎤ ⎡ ⎤ ⎡ 1⎤ ⎡ ⎤  x1 A11 0 A13 0 x B1    ⎢ ⎢ ⎢ ⎢ ⎥ ⎥ ⎥ ⎥ 2 2   ⎢x ⎥ ⎢A21 A22 A23 A24 ⎥ ⎢x ⎥ ⎢B2 ⎥   u = + ⎢ ⎢ ⎢ ⎢ ⎥ ⎥ ⎥  3 3 ⎨⎣x ⎦ ⎣ 0 0 A33 0 ⎦ ⎣x ⎦ ⎣ 0 ⎥ ⎦ (1.14) x4 0 0 A43 A44 x4 0        ] [   ⎩y = C1 0 C3 0 . Moreover, { z = A11 z + B1 u y = C1 z

(1.15)

is a minimum realization of the system (1.3). From the above argument it is clear that a linear system is a minimum realization, if and only if it is controllable and observable. A controllable linear system can be expressed into a canonical form, called the Brunovsky canonical form.

6

1 Introduction

Theorem 1.4. (Brunovsky Canonical Form) Assume system (1.3) is controllable. Then its state equations can be expressed as ⎧  x11 = x12      ..   .       x1n1 −1 = x1n1     m ni m    x1n1 = ¦ ¦ Di1j xij + ¦ Ei1 ui    i=1 j=1 i=1  ⎨. .. (1.16)    m m   x1 = x2    ..    .     m   xm  nm −1 = xnm    m ni m   m m i m   ⎩xnm = ¦ ¦ Di j x j + ¦ Ei ui . i=1 j=1

i=1

Remark 1.3. It is a common assumption that the columns of B are linearly independent, otherwise some input channel(s) will be redundant. In this case, if we use a state feedback ⎛ ⎤⎞ ⎡m n i

⎡

⎤

⎡

E11

u1 ⎢ .. ⎥ ⎢ .. ⎣ . ⎦=⎣ . um E1m

⎤−1 ⎜⎡ ⎤ ⎢ ¦ ¦ Di j x j ⎥⎟ ⎜ v1 ⎥⎟ ⎢i=1 j=1 ⋅ ⋅ ⋅ Em1 ⎜ ⎥⎟ ⎢ .. ⎥⎟ .. ⎥ ⎜⎢ .. ⎥ − ⎢ ⎜ ⎥⎟ , ⎢ . . ⎦ ⎜⎣ . ⎦ ⎢ ⎥⎟ m ⎜ ⎥⎟ ⎢ n m i ⋅ ⋅ ⋅ Em ⎝ vm ⎣ m i D x ¦ ¦ i j j ⎦⎠ 1 i

i=1 j=1

then the system (1.16) has a very neat form, called the feedback Brunovsky canonical form as ⎧  xk1 = xk2      ⎨.. . (1.17)  k k  x  = x  n n −1  k k   ⎩xk = v , k = 1, 2, ⋅ ⋅ ⋅ , m. nk

k

1.1.2 Invariant Subspaces Denition 1.3. 1. For linear system (1.3), a subspace V is called an (A, B)-invariant subspace if there exists an m × n matrix F such that (A + BF)V ⊂ V.

(1.18)

1.1 Linear Control Systems

7

V is called an A-invariant subspace if AV ⊂ V.

(1.19)

2. The matrix F satisfying (1.18) is said to be a friend to V . The set of matrices that are friends to V , is denoted by F (V ) = {F∣(A + BF)V ⊂ V }. Proposition 1.1. (Quaker Lemma) V is an (A, B)-invariant subspace, if and only if AV ⊂ V + Im(B).

(1.20)

If we denote by ⟨A∣W ⟩ the smallest subspace that contains W and is A-invariant, then the controllable subspace of system (1.3) can be expressed as C = ⟨A ∣ Im(B)⟩. Based on this observation, we dene the reachability subspace as follows: A subspace V is called a reachability subspace if there exist matrices F and G with proper dimensions such that V = ⟨A + BF ∣ Im(BG)⟩ . It is obvious that a reachability subspace is an (A, B)-invariant subspace. Proposition 1.2. A subspace V is a reachability subspace, if and only if there exists F such that V = ⟨A + BF ∣ Im(B) ∩V ⟩ .

(1.21)

Now let Z ⊂ ℝn be a subspace. Then we have 1. There exists a maximal (A, B)-invariant subspace S∗ ⊂ Z, denoted by S∗ (Z). That is, for any S ⊂ Z being (A, B)-invariant, S ⊂ S∗ . 2. There exists a maximal reachability subspace R∗ ⊂ Z. 3. Let F ∈ F (S∗ ), then R∗ = ⟨A + BF ∣ Im(B) ∩ S∗ ⟩ .

(1.22)

As an application, we consider the following example. Example 1.1. Consider a system { x = Ax + Bu + Ew y = Cx,

(1.23)

where w is disturbance. The disturbance decoupling problem is to nd a control u = Fx, such that for the closed-loop system the disturbance w does not affect y. It is easy to prove that the disturbance decoupling problem is solvable, if and only if

8

1 Introduction

Im(E) ⊂ S∗ (Ker(C)).

(1.24)

1.1.3 Zeros, Poles, Observers Consider a single-input single-output (SISO) system, namely system (1.3) with m = p = 1. Then W (s) := c(sI − A)−1 b

(1.25)

is called the transfer function of system (1.3). The transfer function can be expressed as a proper rational fraction of polynomials as W (s) =

q(s) , p(s)

where p(s) and q(s) are polynomials with deg(p(s)) > deg(q(s)). The zeros of q(s) and p(s) are called the zeros and the poles of the system respectively. It is easy to see that the poles are the eigenvalues of A. Proposition 1.3. Consider the system (1.3). The followings are equivalent: ∙ It is a minimum realization; ∙ It is controllable and observable; ∙ Its transfer function has no zero/pole cancellation. If there is no zero/pole cancellation, it is the presence of zeros on the transfer function W (s) that makes S ∗ (Ker(C)) nontrivial. Hence for a SISO system it follows that dim(S∗ (Ker(C))) = Number of{zeros of W (s)}.

(1.26)

While the zeros can not be changed by the state feedback control, the poles may. In fact, pole (eigenvalue) assignment is very important for control design. Theorem 1.5. Given a set of n numbers / = {O1 , ⋅ ⋅ ⋅ , On }. 1. Assume (1.3) is controllable (briey, (A, B) is a controllable pair), there exists an F, such that V (A + BF) = / . 2. Assume (1.3) is observable (briey, (C, A) is an observable pair), there exists an L, such that V (A + LC) = / . When the system state is not completely measurable, an observer can be designed to estimate the state. A full-dimensional observer has the following form xˆ = Axˆ + Bu + L(y − Cxˆ). Then the error e := x − xˆ satises

(1.27)

1.1 Linear Control Systems

e = (A − LC)e.

9

(1.28)

As long as the system is observable, Theorem 1.5 assures that there exists L such that A − LC is a Hurwitz matrix. Hence lim e(t) = 0.

t→f

In fact, y is a part of the state that is already available. Thus, it may sufce to construct just a reduced-dimensional observer. Choosing R such that [ ] C T= R is a nonsingular matrix. Let z = T x. Then (1.3) becomes ] [ ] [ ⎧[ ]  ⎨ z1 = TAT −1 z + T Bu := A¯11 A¯12 z + B¯1 u z2 B¯2 A¯21 A¯22  ⎩ y = z1 .

(1.29)

Let v := A¯21 y + B¯2 u,

w := y − A¯11y − B¯1 u,

we have {

z2 = A¯22 z2 + v w = A¯12 z2 .

(1.30)

As long as (C, A) is observable, it is easy to prove that (A¯12 , A¯22 ) is observable [4]. So we can choose an L such that A¯22 − LA¯12 is Hurwitz. Then we can construct an observer for z2 from (1.30) as zˆ2 = (A¯22 − LA¯12 )ˆz2 + Lw + v = (A¯22 − LA¯12 )ˆz2 + L(y − A¯11y − B¯1 u) + (A¯21y + B¯2 u).

(1.31)

This equation contains the derivative of y. It can be eliminated by dening

[ = zˆ2 − Ly.

(1.32)

Then the observer becomes [ ] [ = (A¯22 − LA¯12)[ + (A¯22 − LA¯12 )L + (A¯21 − LA¯11 ) y + (B¯2 − LB¯1 )u.

(1.33)

Finally, the estimator is xˆ = T

−1

[

Ip 0 L In−p

][ ] y . [

(1.34)

10

1 Introduction

1.1.4 Normal Form and Zero Dynamics Next, we consider the zeros and zero dynamics of a linear control system. For a SISO system, assume its transfer function is expressed as W (s) =

D (sm + q1 sm−1 + ⋅ ⋅ ⋅ + qm) q(s) = . p(s) sn + p1sn−1 + ⋅ ⋅ ⋅ + pn

(1.35)

The roots of q(x)–the zeros of the system, are also called the transmission zeros of the system. The relative degree U is dened as

U = deg(p(s)) − deg(q(s)) = n − m.

(1.36)

Assume q(s) and p(s) are coprime, then a minimum state space realization is ⎡ ⎤ ⎡ ⎤ ⎧ 0 0 1 0 ⋅⋅⋅ 0    ⎢ .. ⎥ ⎢ .. .. .. .. ⎥   ⎢ ⎥ ⎢  . . . ⎥  ⎥x + ⎢ . ⎥u ⎨x = ⎢ . ⎣0 ⎦ ⎣ 0 0 0 ⋅⋅⋅ 1 ⎦ (1.37)  −p −p ⋅ ⋅ ⋅ −p D −p n n−1 n−2 1     ] [   ⎩y = q ⋅ ⋅ ⋅ q 1 0 ⋅ ⋅ ⋅ 0 x. m 1 The problem is: Find, if possible, a control u and a set Z ∗ of initial conditions x0 , such that y(t) = 0, ∀t ⩾ 0. If such Z ∗ exists, the restriction of (1.37) on Z ∗ is called the zero dynamics. Calculating the derivatives of the outputs, y(i) , i = 0, 1, ⋅ ⋅ ⋅ , one sees that cAi b = 0, i = 0, 1, ⋅ ⋅ ⋅ , U − 2;

cAU −1 b ∕= 0.

(1.38)

That is y(i−1) = cAi−1 x, y(U )

i = 1, ⋅ ⋅ ⋅ , U ;

= cAU x + cAU −1bu.

(1.39)

It is easy to see that cAi−1 , i = 1, ⋅ ⋅ ⋅ , U are linearly independent. So we can choose a new coordinate frame by letting

[i := cAi−1 x, i = 1, ⋅ ⋅ ⋅ , U ; zi := xi , i = 1, ⋅ ⋅ ⋅ , m.

(1.40)

Then the system becomes ⎧ z = Nz + M[1      [1 = [2     .   ⎨..

[U −1 = [U     [U = Rz + S[ + D u,        ⎩ y = [1 .

(1.41)

1.1 Linear Control Systems

11

where ⎡ ⎢ ⎢ ⎢ N=⎢ ⎢ ⎢ ⎣

0

1

0

.. .

.. .

.. .

0

0

0

⋅⋅⋅

−qm −qm−1 −qm−2

⎤

0

⎥ .. ⎥ . ⎥ ⎥, ⎥ ⋅⋅⋅ 1 ⎥ ⎦ ⋅ ⋅ ⋅ −q1

⎡ ⎤ 0 ⎢ ⎥ ⎢ .. ⎥ ⎢.⎥ ⎥ M=⎢ ⎢ ⎥. ⎢0⎥ ⎣ ⎦ 1

(1.42)

In order to keep y(t) = 0, we must have

[1 = [2 = ⋅ ⋅ ⋅ = [U = 0, and 1 u = − (Rz + S[ ). D It follows that the zero dynamics is dened on the subspace { } Z ∗ = x∣cAi x = 0, i = 0, ⋅ ⋅ ⋅ , U − 1 , and represented by

z = Nz.

The eigenvalues of N are the zeros of q(s). Next, we consider multiple input-multiple output (MIMO) systems. Similar to the SISO case, we dene the transfer function matrix as C(sI − A)−1B. The relative degree vector, denoted by U = (U1 , ⋅ ⋅ ⋅ , U p ), is dened as ci Ak B = 0, k = 0, ⋅ ⋅ ⋅ , Ui − 2; i = 1, ⋅ ⋅ ⋅ , p.

ci AUi −1 B ∕= 0,

(1.43)

Denote the decoupling matrix ⎤ c1 AU1 −1 B ⎥ ⎢ .. D=⎣ ⎦. . ⎡

cp

(1.44)

AU p −1 B

Dene

[ ji = ci A j−1 x,

i = 1, ⋅ ⋅ ⋅ , p, j = 1, ⋅ ⋅ ⋅ , Ui .

It is easy to prove that if D has full row rank (which implies that p ⩽ m), then  L := { ci A j−1  i = 1, ⋅ ⋅ ⋅ , p; j = 1, ⋅ ⋅ ⋅ , Ui } p

are linearly independent. Choosing {zk = pk x∣pk ∈ L⊥ } (there are n − ¦ Ui linearly i=1

independent pk ). Using ([ ji , zk ) as coordinates, the system (1.3) can be converted to

12

1 Introduction

⎧  z = Nz + M [    i  i  [1 = [2   ..    .   ⎨ i [Ui −1 = [Ui i  U −1 i  [U = Ri z + Si [ + ci A i Bu,         y = [1i ,    ⎩ i = 1, 2, ⋅ ⋅ ⋅ , p.

(1.45)

Theorem 1.6. Consider the system (1.3). Assume (i). the relative degree vector is (U1 , ⋅ ⋅ ⋅ , U p ); (ii). the decoupling matrix D has full row rank. Then the zero dynamics under the normal form (1.45) is represented by z = Nz,

(1.46)

which is dened on } {  Z ∗ = x ci A j−1 x = 0, i = 1, ⋅ ⋅ ⋅ , p, j = 1, ⋅ ⋅ ⋅ , Ui . Moreover, the control which makes y(t) = 0, ∀t ⩾ 0 is u = −DT (DDT )−1 (Rz + S[ ). The eigenvalues of N is called the transmission zeros of system (1.3). Dene ] [ sI − A B . P(s) = −C 0 Then we have Proposition 1.4. s is a transmission zero of system (1.3), if and only if rank(P(s)) < n + min(p, m). In most useful cases we assume p ⩽ m because of the aforementioned reason. As an application of the normal form we consider the output tracking problem. Let yd (t) ∈ ℝ p be the reference signal to be tracked. Our goal is to nd a control u(t) = Fx(t)+ G(t) ∈ ℝm , such that y(t)− yd (t) → 0 as t → f and x(t) stay bounded. p Denote yd (t) = [y1d (t), ⋅ ⋅ ⋅ , yd (t)]T and assume (yid (t))(k) , i = 1, ⋅ ⋅ ⋅ , p, k = 0, 1, ⋅ ⋅ ⋅, Ui − 1 are bounded. We dene the (enlarged) tracking errors eij := ci A j−1 x − (yid )( j−1) , Using the normal form (1.45), we have

i = 1, ⋅ ⋅ ⋅ , p, j = 1, ⋅ ⋅ ⋅ , Ui .

1.1 Linear Control Systems

⎧  z = Nz + ME + MYd [    i i   e1 = e2   ..    .   ⎨ i eUi −1 = eiUi  i i (U ) U −1  eUi = Ri z + SiE + SiYd + (yd ) i + ci A i Bu         y = [1i ,    ⎩ i = 1, 2, ⋅ ⋅ ⋅ , p. where

p

13

(1.47)

p

E = [e11 , ⋅ ⋅ ⋅ , e1U1 ; ⋅ ⋅ ⋅ ; e1 , ⋅ ⋅ ⋅ , eU p ]T ; Yd = [y1d , ⋅ ⋅ ⋅ , (y1d )(U1 −1) ; ⋅ ⋅ ⋅ ; ydp , ⋅ ⋅ ⋅ , (ydp )(U p −1) ]T .

Using control where

[ ] U u = −DT (DDT )−1 Rz + SE + SYd + Yd + v , R = [R1 , ⋅ ⋅ ⋅ , R p ]T ; U

S = [S1 , ⋅ ⋅ ⋅ , S p ]T ;

Yd = [(y1d )(U1 ) , ⋅ ⋅ ⋅ , (yd )(U p ) ]T ; v = [v1 , ⋅ ⋅ ⋅ , v p ]. Then the last equation in i-th block becomes p

eiUi = vi ,

i = 1, ⋅ ⋅ ⋅ , p.

So we can choose D ij such that vi = D1i ei1 + ⋅ ⋅ ⋅ + DUi i eiUi ,

i = 1, ⋅ ⋅ ⋅ , p,

which assigns negative eigenvalues to i-th block. Finally, since (yid )(k−1) , i = 1, ⋅ ⋅ ⋅ , p, k = 1, ⋅ ⋅ ⋅ , Ui are bounded, N is Hurwitz, z(t) is bounded. Remark 1.4. For MIMO systems the poles and zeros are dened through its SmithMcMillan form. We give the following example to show this, and refer to [6] for details. Example 1.2. Assume a transfer function matrix is given as ] [ s s(s − 2)2 1 H(s) = , d(s) −s(s − 2)2 2s(3s − 2)(s − 2)2 d(s) =

(1.48)

(s − 2)2 (s + 2)2 .

Then we can nd two unimodular matrices (A unimodular matrix is a polynomial matrix, whose determinant equals 1.) ] ] [ [ 1 0 1 (s − 2)2 , U , U1 (s) = (s) = 2 −(s − 2)2 1 0 1 to convert H(s) into diagonal form (called the Smith-McMillan form) as

14

1 Introduction

[

] s/(s − 2)2 (s + 2)2 0 H(s) = U1 (s) U (s). 0 s2 /(s + 2) 2

(1.49)

From this form, we said that H(s) has three zeros at s = 0, two poles at s = 2, and three poles at s = −2. The aforementioned properties of linear control systems are fundamental. Their counterparts for nonlinear systems will be discussed throughout this book. In this section we skipped some important issues such as output regulation and internal model, linear Hf control etc. It seems better to discuss them together with the nonlinear case.

1.2 Nonlinearity vs. Linearity Many concepts, methods, results or even terminologies in nonlinear control systems have their origins in linear control systems. So we may say that the linear control theory is one of the major foundations of the nonlinear control theory. But nonlinear (control) systems are much more complicated than linear (control) systems. In fact, linear (control) systems are a particular and the simplest class of nonlinear (control) systems. Then one would like to ask what are the fundamental differences between linear (control) systems and general nonlinear (control) systems? In this section we try to answer this question, and for this purpose some special characteristics of nonlinear (control) systems are discussed.

1.2.1 Localization For linear (control) systems most of the properties are global, namely, they are valid for all initial states in ℝn . But for nonlinear (control) systems, a large number of properties are local, namely, valid only for some of the initial states. The rst example of such properties is stability. Example 1.3. Consider a linear system x = Ax,

x ∈ ℝn .

(1.50)

It is well known that the system is asymptotically stable if all the eigenvalues of A have negative real parts. It is unstable if at least one eigenvalue has positive real part. Now consider a nonlinear system x = f (x),

x ∈ ℝn ,

(1.51)

where f (0) = 0. Let A := ww xf ∣x=0 . As will be discussed later on, if all the eigenvalues of A have negative real parts the system is locally asymptotically stable around the origin, and if at least one eigenvalue has positive real part the system is unstable. However, alone from the eigenvalues of A we can not tell anything about the behavior of the system at initial points far from the origin. Moreover, even though we know that there exists a neighborhood U of the origin such that a trajectory starting

1.2 Nonlinearity vs. Linearity

15

from U is bounded and converges to the origin, it is hard to tell in general how large this neighborhood U is. The second example is about controllability. Example 1.4. For linear control system (1.3), it is controllable, if and only if dim(C ) = n. Then for any two points x, y ∈ ℝn , there is an admissible control u, which drives the system from x to y. Now consider the afne nonlinear control system (1.2). Let x0 be an equilibrium point of f (x). Then its linear approximation near x0 is  m w f  x = x + gi (x0 )ui := Ax + Bu. ¦ w x x0 i=1 It is well known that [8] if (A, B) is controllable, then there exists a neighborhood U of x0 such that the system (1.2) is locally controllable on U. That is, for any two points x, y ∈ U, there is an admissible control u that drives the system from x to y. Again, what we do have is a local property. The third example is about the normal form. Example 1.5. Consider system (1.2) again. Assume m = p = 1. Similar to the linear case, we can dene the relative degree at a point x0 as Lg Lkf h(x) = 0, U −1

Lg L f

k < U − 1,

x∈U

h(x0 ) ∕= 0,

where U is a neighborhood of x0 . (One may change the above denition to be global, but it will not help for the normal form.) We can prove that dh(x), dL f h(x), ⋅ ⋅ ⋅ , U −1 dL f h(x) are linearly independent on a neighborhood U of x0 . (We use the same notation U but it could be different in different context.) Then we can choose z(x) = [z1 (x), ⋅ ⋅ ⋅ , zn−U (x)]T such that the set of functions {Lkf h(x), k = 0, ⋅ ⋅ ⋅ , U − 1; z(x)} becomes a local coordinate frame. Under it, a normal form similar to (1.41) can be obtained as ⎧ z = I (z, [ )      [1 = [2     .   ⎨.. (1.52) [U −1 = [U   [ = D (z, [ ) + E (z, [ )u,   U       ⎩ y = [1 , (z, [ ) ∈ U. this normal form is only dened on U, a certain neighborhood of x0 . It is also difcult to tell how large this U is. When analyzing the geometric structures, most of the results for nonlinear (control) systems are local. If we want to obtain global (or even semi-global) results,

16

1 Introduction

usually some special efforts have to be implemented since it is in general difcult to nd straightforward veriable and globally applicable necessary and sufcient conditions.

1.2.2 Singularity For nonlinear (control) systems, there is usually a characteristic number (possibly, more than one) associated with a certain property. For instance, the rank of the decoupling matrix, the dimension of the Lie algebra (as a distribution) etc. If the characteristic quantity has a constant value on a neighborhood of a point x0 , x0 is called a regular point. If such a neighborhood doesn’t exist, it is called a singular point. Roughly speaking, if a point x0 is a regular point , then on a neighborhood U of x0 the system behaves like a linear system as long as U is small enough. Around a singular point the system may behave strange. We investigate singularities through some examples. Example 1.6. Consider Example 1.5 again. Assume at point x0 we have Lg Lkf h(x0 ) = 0, U −1

Lg L f

k < U − 1, (1.53)

h(x0 ) ∕= 0.

Instead of a neighborhood of x0 , we have only zero value of Lie derivatives at x0 . Since we couldn’t nd a neighborhood U of x0 such that Lg Lkf h(x) = 0, x ∈ U, k < U − 1, then x0 becomes a singular point. It is easy to prove that in this case we are not able to get the normal form as (1.52). In a singular case, some special consideration is necessary. For instance, condition (1.53) can be used to dene the so called point relative degree. Using the point relative degree, a generalized normal form, which is different from (1.45) (for SISO case) or (1.47) (for MIMO case) can be obtained. [5] Next, we consider the reachable set of a nonlinear control system. Example 1.7. Consider a nonlinear system ⎡ ⎤ ⎡ ⎤ 0 x ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ [ = ⎢ ⎣y⎦ + ⎣ 0 ⎦ u := f ([ ) + g([ )u, z x2

[ ∈ ℝ3 .

(1.54)

It is easy to see that

' ([ ) := Span{ f ([ ), g([ )} is involutive. According to the generalized Frobenius’ theorem (for analytic distribution) passing each [ ∈ ℝ3 there exists a unique maximal integral manifold of ' . It is easy to see that there are three different cases. Case 1. When [ = O, dim(' (O)) = 0. So the maximal integral manifold of ' passing through O consists of only one point O (where O is the origin).

1.2 Nonlinearity vs. Linearity

17

Case 2. When [ = P ∈ {(x, y, z) ∈ ℝ3 ∖{0}∣x = 0}, dim(' (P)) = 1. The maximal integral manifold of ' passing through P consists of a line passing through O and P (see Fig. 1.2). Taking into consideration the non-negative time, the reachable set of P, denoted by R(P), is the ray starting from P and goes against O. Case 3. When [ = Q ∈ {(x, y, z) ∈ ℝ3 ∣x ∕= 0}, dim(' (Q)) = 2. The maximal integral manifold of ' passing through Q consists of a plane L, satisfying yQx − xQy = 0 (also see Fig. 1.2). Taking into consideration the non-negative time, the reachable set of Q, R(Q), is an open semi-plane (plus {Q}) on the right hand side of Q. In fact, Cases 1 and 2 are singular and Case 3 is regular.

Fig. 1.2

Reachable Sets

Next, we consider singularity of a nonlinear mapping. Consider a mapping S : N → M, where N and M are Cf manifolds of dimension n and m respectively. For the reader who is not familiar with differential geometry, simply set N = ℝn and M = ℝm . Let x0 ∈ N. The rank of S at x0 is the rank of the Jacobian matrix of S at x0 , i.e., [ ] wS rank(S )∣x0 := rank (x0 ) . wx A point x0 is called a regular point if rank(S )x0 = m, otherwise, it is called a singular point. A point y0 ∈ M is called a regular value if S −1 (y0 ), dened as

S −1 (y0 ) := {x ∈ N∣S (x) = y0 }, consists of regular points only. Otherwise, it is called a singular value. For regular value we have

18

1 Introduction

Theorem 1.7. [1] Assume y0 ∈ M is a regular value, then S −1 (y0 ) (if it is not ∅) is a regular sub-manifold of N of dimension n − m. If y0 is a singular value, then we can not draw any conclusion. Fortunately, we have the following Theorem 1.8. (Sard) The set of singular values is nowhere dense in M. The behavior of a nonlinear system at a singular point is different from that at a regular point. Particular attention has to be paid to the singularities of nonlinear systems.

1.2.3 Complex Behaviors A nonlinear dynamic system may have some complex behaviors, which make it completely different from a linear system. We give some examples here. The rst example is the Hopf bifurcation. Example 1.8. (Bifurcation) Consider a dynamic system x = f (x, D ).

(1.55)

The appearance of a topologically nonequivalent phase portrait under variation of parameters is called a bifurcation. In the following we consider a particular system where a bifurcation called Hopf bifurcation appears. { x1 = D x1 − x2 − x1 (x21 + x22 ) (1.56) x2 = x1 + D x2 − x2 (x21 + x22 ). It is not difcult to convert this system into the polar coordinates as in [7] { U = U (D − U 2) M = 1.

(1.57)

Now consider the equilibrium x = 0. If D < 0 the origin is linearly stable (exponentially stable); if D = 0 the origin is nonlinearly stable (not exponentially stable); if D > 0 the origin is linearly unstable (exponentially diverges). We refer to Fig.√1.3 for the phase portrait. Particularly, note that as D > 0 there is a limit circle U = D , which is globally asymptotically stable. This bifurcation is called the Hopf bifurcation. It is also depicted in the phase-parameter space as in Fig. 1.4. The second example is a chaotic system. Example 1.9. (Lorenz System) A dynamic system x = f (x) is a chaotic system if the solution has sensitive dependence on initial conditions [2]. Lorenz system could be the most famous chaotic system. Consider the system ⎧  ⎨x = V (y − x) (1.58) y = rx − y − xz  ⎩ z = xy − bz.

1.2 Nonlinearity vs. Linearity

Fig. 1.3

Hopf Bifurcation

Fig. 1.4

Hopf Bifurcation in Phase-Parameter Space

19

Under some particular chosen parameters the system becomes a chaotic system. Following Lorenz, we choose V = 10, r = 28, and b = 83 . We have four equilibrium √ √ points at (±6 2, ±6 2, 27). The system does not have a global section so the projection onto the x-z plane is shown in Fig. 1.5. It can be seen that the projected orbit

Fig. 1.5

Chaos in Lorenz System

20

1 Introduction

√ √ switches between revolving about (x, z) = (6 2, 27) and (x, z) = (−6 2, 27) in an apparently random way. Moreover, there are two strange attractors as the revolving centers.

1.3 Some Examples of Nonlinear Control Systems In this section some typical nonlinear control systems are presented. They will be discussed in detail somewhere later. The rst example is a car steering system. Example 1.10. (Car Steering System) Consider a car moving in a plane (see Fig. 1.6), where x and y are Cartesian coordinates of the middle point on the rear axle, T is ori-

Fig. 1.6

The Geometry of A Car

entation angle, v is the longitudinal velocity measured at that point, L is the distance of the two axles, and I is the steering angle. In this example we briey review the nonholonomic constraints on such a system. The nonholonomic constraints are basically due to two assumptions on the motion of car: no movement in the direction orthogonal to the front wheels and in the direction orthogonal to the rear wheels. Naturally if dynamical factors such as friction forces acting on the tires are considered, then the situation is much more complicated. That the velocity orthogonal to the rear wheels should be zero implies x sin(T ) − ycos(  T ) = 0.

(1.59)

That the velocity orthogonal to the front wheels should be zero implies d d (x + L cos(T )) sin(T + I ) − (y + L sin(T )) cos(T + I ) = 0. dt dt One can easily verify that the state equations dened by

(1.60)

1.3 Some Examples of Nonlinear Control Systems

⎧  x = v cos(T )   ⎨ y = v sin(T )    ⎩T = v tan I L

21

(1.61)

satisfy the nonholonomic constraints. An alternative way to derive the third equation is to use the facts: LT = v f sin I , v f cos I = v. One can simplify the model a bit by dening the right hand side of the third equation as a new control Z : ⎧  x = v cos(T )   ⎨ (1.62) y = v sin(T )    ⎩ T = Z. By this way we obtain the so-called unicycle model. Example 1.11. (Attitude Control of Spacecraft) The model consists of equations used for describing a rotating rigid body and for describing the effect of control torques. Therefore one can separate the equations into kinematic equations and dynamic equations. Consider a rigid body rotating in the inertial space. Introduce a coordinate system, N, xed in the inertial space and a coordinate system, B, xed in the body. Let the coordinates of an arbitrary point be denoted by [ N if expressed in the N-frame and by [ B if expressed in the B-frame. Then the relation between the two frames is [ B = R[ N , where R is a rotation matrix, that is RT R = I, det R = 1, and

where

R = S(Z )R

(1.63)

⎤ 0 Z3 −Z2 S(Z ) = ⎣ −Z3 0 Z1 ⎦ , Z2 −Z1 0

(1.64)

⎡

Zi are the components of the angular velocity expressed in the B-frame. Alternatively, the angular position can be described by three angles (Euler angles) I , T , \ , consecutive clockwise rotations about the axes in the B-frame. If we use the standard basis in ℝ3 , then ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ 0 Z1 1 0 0 I ⎣ Z2 ⎦ = ⎣ 0 ⎦ + ⎣ 0 cos I sin I ⎦ ⎣ T ⎦ Z3 0 − sin I cos I 0 0 ⎤⎡ ⎤⎡ ⎤ ⎡ cos T 0 − sin T 0 1 0 0 (1.65) + ⎣ 0 cos I sin I ⎦ ⎣ 0 1 0 ⎦ ⎣ 0 ⎦ . sin T 0 cos T 0 − sin I cos I \

22

1 Introduction

Thus, ⎤ ⎡ ⎤⎡ ⎤ I 1 sin I tan T cos I tan T Z1 ⎣ T ⎦ = ⎣ 0 cos I − sin I ⎦ ⎣ Z2 ⎦ . 0 sin I sec T cos I sec T Z3 \ ⎡

(1.66)

Clearly this description is only valid in the region ∣T ∣ ⩽ S2 . The dynamic equations depend on how the craft is controlled. We assume here that m gas jet actuators are used for control. Let J be the inertia matrix of the spacecraft, b1 , ⋅ ⋅ ⋅ , bm be the axes about which the corresponding control torque of magnitude ∥bi ∥ui is applied by means of opposing pairs of gas jets. Then m

J Z = S(Z )J Z + ¦ bi ui .

(1.67)

i=1

The third example is the control of PVTOL aircraft: Example 1.12. (Dynamics of PVTOL) The dynamics of a PVTOL aircraft is described as [3] ⎧  x¨ = − sin(T )u1 + H cos(T )u2   ⎨ y¨ = cos(T )u1 + H sin(T )u2 − 1    ⎩T¨ = u , 2

(1.68)

where x and y are the horizontal and vertical displacements, T is the angle the PVTOL makes with the horizontal line, u1 is the collective input and u2 is the torque. The parameter H is a small coefcient that characterizes the coupling between the rolling moment and the lateral acceleration of the aircraft. The term −1 is the normalized gravitational acceleration. The system is depicted in Fig. 1.7.

Fig. 1.7

PVTOL Aircraft

Using the coordinate transformation:

1.3 Some Examples of Nonlinear Control Systems

z1 = x − H sin(T ) z3 = y + H (cos(T ) − 1) z5 = T , system (1.68) can be expressed in a normal form as ⎧ z1 = z2       z2 − sin(z5 )u1  ⎨z = z 3 4 z4 = cos(z5 )u1 − 1      z5 = z6   ⎩ z6 = u2 .

23

(1.69)

(1.70)

The next example is the control of a robot arm. Example 1.13. (Robot Arm) A robot manipulator consists of a kinematic chain of n + 1 links connected by n joints [10]. It is depicted in Fig. 1.8. Its dynamics can be derived using the general Lagrange equation: d wL wL − = [i , dt w qi w qi

i = 1, 2, ⋅ ⋅ ⋅ , n,

(1.71)

where qi , i = 1, ⋅ ⋅ ⋅ , n are general coordinates for n joints. The Lagrange is L = T −U ,

(1.72)

where T is the kinetic energy and U is the potential energy. The kinetic energy is 1  T = qT B(q)q. 2

(1.73)

Denote the forces generated by the potential energy by gi (q) =

wU . w qi

(1.74)

Putting (1.72)–(1.74) into (1.71) yields B(q)q¨ + C(q, q)  q + g(q) = [ , where 1  q) C(q, q)  q = B(q,  q − 2

(

)T w T (q B(q)q)  wq

(1.75)

(1.76)

is the vector of Coriolis and centrifugal torques. Finally, we give a detailed description on [ .

[i = Wi − W fi − Wei ,

(1.77)

24

1 Introduction

Wi , i = 1, ⋅ ⋅ ⋅ , n are the driving torques at the joints, W f = F q

(1.78)

is the viscous friction. Let f and P be the external end-effector force and the external end-effector moment respectively. Then the ending point torque is [ ] f T We = J (q) , (1.79) P where J is the end-effector geometric Jacobian matrix. Then the transfer matrix from general velocity to ending point velocity becomes:  ve = J(q)q.

(1.80)

Plugging specied [ into (1.75) yields the dynamics of the robot manipulator as B(q)q¨ + C(q, q)  q + F q + g(q) = W − J T (q)h,

(1.81)

[ ] f . P Denote the generalized velocity by p = q,  we can express (1.81) in a 2n dimensional form as (it is reasonable to assume B(q) is nonsingular) { q = p [ ] (1.82) p = [B(q)]−1 −C(q, p)p − F p − g(q) + W − J T (q)h . where

h=

Another example is an electrical power generator system. Example 1.14. (Power Generator) Consider an electrical power generator system. As sketched in Fig. 1.9, the synchronous generator is driven by the mechanical power pm that is provided by the hydraulic turbine in the hydroelectric plant. Suppose that the mechanical power controlled by the turbine governor is constant, and the electric load of the generator can be modeled as a single machine of innity bus system. Then, the dynamics of the power system consists of mechanical and electrical parts. First, the mechanical dynamics of the rotor of generator is represented by { G = Z − Z0 D Z0 Z = − {Z − Z0 } + {pm − pe (Eq′ , G )}, M M

(1.83)

with the active electrical power pe given by pe (Eq′ , G ) =

Eq′ Vs V 2 (x′ − xq6 ) sin G + s ′d 6 sin 2G , ′ xd 6 xd 6 xq6

where the variables G , Z and Eq′ denote the angle, the speed of rotor and the transient EMF in the quadrature axis of the generator respectively. The rest of the letters

1.3 Some Examples of Nonlinear Control Systems

Fig. 1.8

Robot Manipulator

Fig. 1.9

Sketch of Hydroelectric Plant

25

are physical parameters that can be found in [9]. Then, the dynamics of electrical part coordinated by Eq′ , which represents the electrical behavior of the generator including the line and the loads, is describe as 1 1 xd − x′d 1 Vs cos G + (V f + E0 ), E q′ = − ′ Eq′ + ′ Td Td0 x6 Td0

(1.84)

26

1 Introduction

where E0 is the nominal value of the excitation signal that is to establish the equilibrium of the system, and V f denotes the excitation signal for stabilizing the system at the equilibrium. Other physical parameters can be found in [9]. For given constant pm and E0 , the equilibrium of the system, denoted by (G0 , Z0 , ′ ), is given by the algebraic equation Eq0 ⎧ Z0 ′  , G0 )] [pm − pe (Eq0 ⎨0 = M 1 ′ 1 xd − x′d 1  + Vs cos G0 + E0 ⎩ 0 = − ′ Eq0 Td Td0 x′6 Td0

(1.85)

For the sake of simplicity, we summarize the physical parameters and dene ′ . Then we obtain the model of the system in x1 = G − G0 , x2 = Z − Z0 , x3 = Eq′ − Eq0 the following form ⎧  ⎨ x1 = x2 [ ] ′ ,x + G ) (1.86) x2 = −b1 x2 + b2 pm − pe (x3 + Eq0 1 0  ⎩ x3 = b3 [cos(x1 + G0 ) − cos G0 ] − b4x3 + u where the excitation signal u = T1 V f (t) is the control input. d Then, the stabilization problem of the generator system is to nd a state feedback control law such that the closed-loop system is asymptotically stable at the origin (x1 , x2 , x3 ) = (0, 0, 0). The system diagram is shown in Fig. 1.10.

Fig. 1.10

Structure of the Power System

The last example is the control of a dual-ended hydroelectric actuation system. Example 1.15. (Hydroelectric Actuator) The schematic diagram of a double-ended hydraulic actuation system is shown in Fig. 1.11. In the system, the piston is driven by the hydraulic force provided by the load pressure pL . The motion of the piston can be represented as x¨ =

d f A pL − x − sgn(x),  m m m

(1.87)

where m, A, d, f denote the mass of actuator’s moving part, the piston area, friction and the dry friction coefcients respectively. The load pressure is generated by the hydraulic part and its dynamics is described by

References

A cd w pL = − x + √ xsp C C U

√

Ps − sgn(xsp )pL

27

(1.88)

with w the orice area gradient, U the hydraulic uid density, Ps the pump pressure, and cd the orice coefcient of charge. C = Vt /4E is the hydraulic compliance where Vt and E denote the total actuator volume and the effective bulk modulus of the system, respectively. Consider the spool displacement xsp as the control input. Then the positioning problem of this system is to nd a feedback control law for the control input u = xsp such that for any given desired position xd , the closed loop system is stable and x(t) → xd as t → f.

Fig. 1.11

Dual-ended Hydroelectric Actuation System

References 1. Abraham R, Marsden J. Foundations of Mechanics. Reading: Benjamin/Cummings Pub. Com. Inc., 1978. 2. Arrowsmith D, Place C. An Introduction to Dynamic Systems. New York: Cambridge Univ. Press, 1999. 3. Astol A. Nonlinear and Adaptive Control, Tools and Algorithms for the User. London: Imperial College Press, 2006. 4. Chen C. Linear System Theory and Design. New York: CBS College Pub., 1984. 5. Cheng D, Zhang L. Generalized normal form and stabilization of nonlinear systems. Int. J. Contr., 2003, 76(2): 116–128. 6. Kailath T. Linear Systems. New York: Prentice-Hill, Inc., 1980. 7. Kuznetsov Y. Elements of Applied Bifurcation Theory, 2nd edn. New York: Springer, 1995. 8. Nijmeijer H, Van der Schaft A. Nonlinear Dynamical Control Systems. New York: SpringerVerlag, 1990. 9. Shen T, Mei S, Lu Q, et al. Adaptive nonlinear excitation control with L2 distrbance attenuation for power systems. Automatica, 2003, 39(1): 81–89. 10. Siciliano B, Villani L. Robot Force Control. Boston: Kluwer, 1999. 11. Wonham W. Linear Multivariable Control: A Geometric Aproach, 2nd edn. Berlin: SpringerVerlag, 1979.

Chapter 2

Topological Space

The purpose of this chapter is to present some basic topological concepts of point sets. What we discuss here is very elementary, thus should not be considered as a comprehensive introduction to topology. But it sufces for our goal-providing a foundation for further discussion, particularly for the introduction of differential manifold and the geometrical framework for nonlinear control systems. Many standard text books such as [1, 2, 3] can serve as further references. Section 2.1 is about metric space. It is probably the most fundamental and useful topological space since many concepts of general topological spaces can be understood through it. In fact, many of them are motivated and generalized from the corresponding concepts in the metric space. Section 2.2 introduces topological space and its fundamental properties. Some important classes of topological spaces, such as second countable space, Hausdorff space, etc. are discussed. Mappings between topological spaces are considered in Section 2.3. In terms of continuous mappings the relations between topological spaces are discussed. Particular interest is paid to homeomorphic spaces. Section 2.4 considers reduced new topological spaces from given spaces. Subspace, product space and quotient space etc. are discussed.

2.1 Metric Space Denition 2.1. A metric space (M, d) consists of a set M and a mapping, called distance, d : M × M → ℝ, which satises the following: (i). 0 ⩽ d(x, y) < f, ∀ x, y ∈ M; (ii). d(x, y) = 0, if and only if x = y; (iii). d(x, y) = d(y, x); (iv). (Triangle Inequality) d(x, z) ⩽ d(x, y) + d(x, z),

x, y, z ∈ M.

Example 2.1. Usually, ℝn is considered as a metric space with (but not restrict to) one of the following commonly used distances: (i). √ d(x, y) =

n

¦ (xi − yi )2

i=1

D. Cheng et al., Analysis and Design of Nonlinear Control Systems © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2010

(2.1)

30

2 Topological Space

(ii). d(x, y) = max ∣xi − yi ∣

(2.2)

1⩽i⩽n

(iii). n

d(x, y) = ¦ ∣xi − yi ∣

(2.3)

i=1

(iv). d(x, y) =

√ (x − y)TP(x − y),

(2.4)

where P is a given positive denite matrix. We prove that with d dened in (i), (ℝn , d) is a metric space, and leave the others to the reader as an exercise. Requirements (i)–(iii) are obvious. To prove (iv), i.e., √ √ √ n

n

n

i=1

i=1

i=1

¦ (xi − zi )2 ⩽ ¦ (xi − yi)2 + ¦ (yi − zi )2 .

It follows immediately from Schwarz inequality [4]: √ √ n

n

n

i=1

i=1

i=1

¦ ui vi ⩽ ¦ u2i ¦ v2i .

(2.5)

Example 2.2. Consider ℝn again. Dene the distance as { 0, x = y d(x, y) = 1, x ∕= y. It is left to the reader to show that this is a distance. (In fact, in this example the space ℝn can be replaced by any set, S, and dene the distance between two points x, y ∈ S as in the above.) Later, we will show that the topological structures induced by the distances in Example 2.1 are the same. But the topological structure induced by the distance in Example 2.2 is different from the others. Denition 2.2. Let V be a vector space, if there exists a mapping ∣∣ ⋅ ∣∣ : V → ℝ, satisfying (i). ∣∣x∣∣ ⩾ 0, ∀x ∈ V , and ∣∣x∣∣ = 0, if and only if x = 0; (ii). ∣∣rx∣∣ = ∣r∣∣∣x∣∣, r ∈ ℝ, x ∈ V ; (iii). (Triangle Inequality) ∣∣x + y∣∣ ⩽ ∣∣x∣∣ + ∣∣y∣∣,

x, y ∈ V ;

Then (M, ∣∣ ⋅ ∣∣) is called a normed space, and ∣∣x∣∣ is called the norm of x[4].

2.1 Metric Space

31

Example 2.3. Consider ℝn again. With usual addition and scalar multiplication it is a vector space. We can dene different norms as (i). (

)1/2

n

¦ x2i

∣∣x∣∣ =

(2.6)

i=1

(ii). ∣∣x∣∣ = max ∣xi ∣

(2.7)

1⩽i⩽n

(iii). n

∣∣x∣∣ = ¦ ∣xi ∣

(2.8)

i=1

(iv). ∣∣x∣∣ =

√ xT Px,

(2.9)

where P is a given positive denite matrix. We leave as exercise the verication that (2.6)–(2.9) are all norms. Example 2.4. Let L2 [a, b] be the set of (complex) functions which are square integrable, i.e., ∫ b { }  L2 [a, b] = f (x)  f (x) f (x)dx < f . a

It is a vector space over ℂ. Dene √ ∣∣ f (x)∣∣ =

∫ b a

f (x) f (x)dx.

(2.10)

Then it is easy to check that (2.10) denes a norm. Therefore, L2 [a, b] is a normed space. To show this, the only non-trivial thing is to prove the triangle inequality, which simultaneously proves that (2.10) is well dened, i.e., the integral is nite. It can be done easily by use the following Schwarz inequality: √ √ ∫ b a

f (x)g(x)dx ⩽

∫ b a

f (x) f (x)dx

∫ b

g(x)g(x)dx.

(2.11)

a

Denition 2.3. Given a vector space V . If there exists a mapping ⟨⋅, ⋅⟩ : V ×V → ℂ, satises the following requirements: (i). ⟨x, x⟩ ⩾ 0, ∀x ∈ V . Moreover, ⟨x, x⟩ = 0, if and only if x = 0; (ii). ⟨x, y⟩ = ⟨y, x⟩, x, y ∈ V ; (iii). ⟨ax + by, z⟩ = a ⟨x, z⟩ + b ⟨y, z⟩ , x, y, z ∈ V, a, b ∈ ℂ; Then (V, ⟨⋅, ⋅⟩) is called an inner product space, and ⟨x, y⟩ is called the inner product of x, y.

32

2 Topological Space

Note that from (ii) and (iii) we have ⟨x, ay + bz⟩ = a¯⟨x, y⟩ + b¯ ⟨x, z⟩ . Example 2.5. Let l2 be the set of square convergence series, i.e.,  { } f  l2 = {xi }  ¦ x¯i xi < f . i=1 Dene the inner product as f

⟨{xi }, {yi }⟩ = ¦ x¯i yi .

(2.12)

i=1

To see that this is a well dened inner product, the only non-trivial thing is to show that this inner product is well dened, i.e., the LHS (left hand side) of (2.12) converges. Again we can use Schwarz inequality √ √ f

f

f

i=1

i=1

i=1

¦ x¯i yi ⩽ ¦ x¯i xi ¦ y¯i yi .

(2.13)

Proposition 2.1. 1. Let V be an inner product space. Dene a norm on V as √ ∣∣x∣∣ = ⟨x, x⟩.

(2.14)

Then V becomes a normed space. Such a norm is called the norm induced by the inner product. 2. Let M be a normed space. Dene a distance d as d(x, y) = ∣∣x − y∣∣,

(2.15)

Then M becomes a metric space. We leave the verication that (2.14) is a norm and (2.15) is a distance to the reader as exercises. In an inner product space, the vectors X , Y are said to be orthogonal if ⟨X,Y ⟩ = 0. In ℝ2 or ℝ3 it is easy to verify that ⟨X ,Y ⟩ = 0 means X ⊥ Y , i.e., they are perpendicular to each other. Example 2.6. Let X , Y be two vectors in an inner product space H. Vector Z is called the projection of X on Y , if (i). Z = cY , for some c ∈ ℂ. (ii). (X − Z) ⊥ Y .

Fig. 2.1

Projection of Vector Fields

2.1 Metric Space

33

It is easy to check that the projection of X on Y is Z=

⟨X ,Y ⟩ Y. ⟨Y,Y ⟩

Since ⟨X − Z, X − Z⟩ ⩾ 0, using above equality we get √ √ ⟨X ,Y ⟩ ⩽ ⟨X , X ⟩ ⟨Y,Y ⟩

(2.16)

and the equality holds if and only if X − Z = 0, i.e., X = cY for some c ∈ ℂ. (2.16) is a general form of Schwarz inequality. Using it to ℝn , L2 [a, b], l2 , we have Schwarz inequalities (2.5), (2.11), and (2.13) respectively. Based on Proposition 2.1, inner product spaces and normed spaces are always considered as metric spaces in this book. Unless otherwise stated, the distance is always considered as the one induced by the inner product or norm. In a metric space (M, d), let x ∈ M, r > 0. We denote by Br (x) the open ball with radius r and center at x, i.e., Br (x) = {y ∈ X ∣ d(y, x) < r}. Note that the shape of a ball depends on how the distance is dened. For instance, in ℝ2 corresponding to the distances D1–D4 the balls B1 (0) are drawn in Fig. 2.2. For D4 the matrix is chosen as ] [ 10 . P= 02

Fig. 2.2

Shapes of A Ball in Different Distances

In a metric space, M, a sequence {xn } is called a Cauchy sequence if for any H > 0, there exists an N > 0, such that as m, n > N, we have d(xm , xn ) < H .

34

2 Topological Space

Denition 2.4. A metric space M is complete if each Cauchy sequence {xn } converges to a point x ∈ M. Example 2.7. 1. ℝn with a conventional distance is a complete metric space. 2. Let C[a, b] ⊂ L2 [a, b], which is dened in Example 2.4. It is easy to show that C[a, b] is a metric space. But it is not complete. To see this, we dene { nx, 0 ⩽ x ⩽ n1 Fn = 1, otherwise. Then {Fn } is a Cauchy sequence in C[0, 1], but it does not converge to a point in C[0, 1].

2.2 Topological Spaces Denition 2.5. Given a set X and a set of its subsets T . 1. (X, T ) is called a topological space, if T satises the following (i). X ∈ T , ∅ ∈ T ; (ii). If UO ∈ T , ∀O ∈ / ⊂ ℝ, then ∪O ∈/ UO ∈ T ; (iii). If Ui ∈ T , i = 1, ⋅ ⋅ ⋅ , n, then ∩ni=1Ui ∈ T . 2. An element in U ∈ T is called an open set. Its complement, denoted by U c is called a closed set. 3. For a point x ∈ X, a subset N is called a neighborhood of x if there exists an open set U such that x ∈ U ⊂ N. 4. Let B ⊂ T . B is a topological basis if any element U ∈ T can be expressed as a union of some elements in B. 5. A set of neighborhoods N ⊂ T is called a neighborhood basis of x if for any neighborhood N of x, there is an N0 ∈ N , such that N0 ⊂ N. In fact, for a set M, let B be a set of subsets of M, then B can be a topological basis of M for a certain topology if and only if, (i). ∪{V ∣ V ∈ B} = M, and (ii). an intersection of nite elements from B can always be expressed as a union of certain elements from B. The topology, T , on M with B as its topological basis is called the topology generated by B. Example 2.8. At a university the set of all graduate students is denoted by S . Let M be the subset of male students, F be the subset of female students, Ci , i = 1, 2, 3 be the set of i-th year students. Given a set of subsets as B0 = {M, F,C1 }, and the smallest topology containing B0 is denoted by T . Then (i). A topological basis of T is: {A1 = M ∩C1 , A2 = F ∩C1 , A3 = M ∩ (C2 ∪C3 ), A4 = F ∩ (C2 ∪C3 )}; (ii). In (S , J): C1 is an open set, C1 is also a closed set; C2 is neither an open set nor a closed set.

2.2 Topological Spaces

35

(iii). Let x be a male second year graduate student. A neighborhood basis of x consists of a single element set {A3 }. Of course, the neighborhood basis of x is not unique. Say, {A3 , M} is another one. Denition 2.6. 1. A topological space is said to be second countable, if there exists a countable topological basis. 2. A topological space is said to be rst countable, if for each point x ∈ X there exists a countable neighborhood basis. From the denition it is easy to see that second countable implies rst countable. Example 2.9. Given any set X. 1. Dene T = {X , ∅}. It is ready to see that T is a topology. This topology is called the trivial topology. It is the roughest topology. 2. Dene T = {U ∣ U ⊂ X }, namely T consists of all subsets of X. Then it is obvious that T is a topology. This topology is called the discrete topology. It is the nest topology. Example 2.10. In ℝn we consider the set of all open balls: Br (c) = {x ∈ ℝn ∣ ∥x − c∥ < r},

∀c ∈ ℝn , r > 0,

denoted by B. Note that an intersection of nite balls in B can be expressed as a union of some elements in B. Based on this fact, one sees easily that all possible unions of some elements in B form a topology. This topology is the conventional topology in ℝn because the elements in this topology are exactly what we called “open sets” in the usual sense. The topology, T , obtained in the above is called the topology induced by distance, because the balls are dened by distance. By denition, B is a topological basis of ℝn . Consider a subset B r ⊂ B, which consists of all balls with rational center (i.e., the coordinates of the center are rational numbers.) and rational radius. Then B r is a countable set. Moreover, one sees easily that B r is a topological basis of ℝn . Hence ℝn with the conventional topology is a second countable topological space. It is easy to show that in a nite dimensional normed space, all the norms induce the same topology. This fact is commonly expressed as that in a nite dimensional vector space any two norms are equivalent. Example 2.11. Consider a proper polynomial (as a transfer function in control theory) Q(s) qn−1 sn−1 + qn−2sn−2 + ⋅ ⋅ ⋅ + q0 = . P(s) sn + pn−1sn−1 + ⋅ ⋅ ⋅ + p0 To determine if Q(s) and P(s) are co-prime is important in control theory. As we saw in Chapter 1, co-prime Q(s) and P(s) correspond to a minimum realization. Consider the set of coefcients, {qn−1, ⋅ ⋅ ⋅ , q0 , pn−1 , ⋅ ⋅ ⋅ , p0 } as a point in ℝ2n we claim that the set for Q(s) and P(s) to be co-prime is open. Dene a 2n × 2n matrix

36

2 Topological Space

⎡

1 pn−1

⎢ ⎢ 0 1 pn−1 ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ S=⎢ ⎢ 0 qn−1 ⎢ ⎢ ⎢ 0 qn−1 ⎢ ⎢ ⎢ ⎢ ⎣ 0 0

⋅ ⋅ ⋅ p0 ⋅⋅⋅ ..

.

1 pn−1 ⋅ ⋅ ⋅ q0 ⋅⋅⋅ ..

.

0 qn−1

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⋅ ⋅ ⋅ p0 ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎥ q0 ⎥ ⎥ ⎥ ⎥ ⎦ ⋅ ⋅ ⋅ q0 p0

It can be proved by mathematical induction that Q(s) and P(s) are co-prime if and only if det(S) ∕= 0. Then the claim follows. (Later on, we will show that the co-prime set is also dense in ℝ2n .) Denition 2.7. Let A ⊂ M. 1. The closure of A, denoted by A¯ or cl(A), is the smallest closed set containing A. Equivalently, A¯ = ∩{C ∣ C is closed and C ⊃ A}.  is the largest open set contained in A. 2. The interior of A, denoted by int(A) or A, Equivalently, A = ∪{U ∣ U is open and U ⊂ A}. 3. The boundary of A, denoted by bd(A), is bd(A) = A¯ ∩ Ac . 4. A is a dense set of M if A¯ = M. 5. A is nowhere dense in M if A¯ contains no non-empty open set. Example 2.12. Consider ℝ1 with the conventional topology. 1. Let A = [0, 1) ⊂ ℝ1 . Then (i). A¯ = [0, 1]; (ii). A = (0, 1); (iii). bd(A) = {0, 1}. 2. Let ℚ be the set of rational numbers. Then ℚ is dense in ℝ1 . 3. Let ℤ be the set of integers. Then ℤ is nowhere dense in ℝ1 . Proposition 2.2. Let A ⊂ M. Then 1. x ∈ A¯ if and only if for any neighborhood Nx of x, Nx ∩ A ∕= ∅; 2. x ∈ A if and only if there exists an open neighborhood Nx of x, Nx ⊂ A; 3. x ∈ bd(A) if and only if any neighborhood Nx of x, Nx ∩ A ∕= ∅ and Nx ∩ Ac ∕= ∅. Proof. We prove item 1 only by contradiction. Assume x ∈ A¯ and there exists an neighborhood U ∋ x such that U ∩ A = ∅. Without loss of generality we can assume U is open. Then A ⊂ U c . But U c is closed, then A¯ ⊂ U c . Hence x ∈ U c , which is absurd. ¯ then x ∈ (A) ¯ c , which is, therefore, an open neighborConversely, assume x ∕∈ A, c ¯ ¯ hood of x. Obviously, (A) ∩ A = ∅. The proofs of items 2 and 3 are similar and are left for the reader. ⊔ ⊓

2.2 Topological Spaces

37

Denition 2.8. Let M be a topological space. 1. M is said to be separable if there exists a countable subset A that is dense in M. 2. M is said to be of rst category if it can be expressed as a countable union of nowhere dense sets. Otherwise, it is of second category. Theorem 2.1. (Baire’s Category Theorem) If a metric space X ∕= ∅ is complete, it is of second category. Proof. Suppose X=

f

¦ Mk .

k=1

We have only to show that at least one Mk is not nowhere dense. Assume the converse: all Mk are nowhere dense. Choosing p1 and H1 < 1, a ball B1 can be constructed as B1 = BH1 (p1 ) ⊂ M¯1c with H1 < 1. Construct C1 = B0.5H1 (p1 ). Since M2 is nowhere dense and M¯2 does not contain C1 , we can construct an open ball B2 = BH2 (p2 ) ⊂ M¯2c ∩C1 with H2 < 2−1 H1 . Construct C2 = B0.5H2 (p2 ). In general, since Mk+1 is nowhere dense, M¯k+1 does not contain Ck . We can thus nd an open ball Bk+1 = BHk+1 (pk+1 ) ⊂ (M¯k+1 )c ∩Ck with Hk+1 < 2−1 Hk . Construct Ck+1 = B0.5Hk+1 (pk+1 ). Now Bk+1 ⊂ Ck ⊂ Bk . Hence Bk+t ⊂ Ck , and for m > k

t > 0,

d(pk , pm ) < 2−1 Hk ,

which means {pk } is a Cauchy sequence. Since X is complete, pk → p ∈ X, (k → f). Since d(pk , p) ⩽ d(pk , pm ) + d(pm , p) < 2−1 Hk + d(pm, p) < Hk , (m → f) p ∈ Bk ⊂ (M¯k )c . That means p ∕∈ Mk . Then which is a contradiction.

p ∕∈ ∪f k=1 Mk = X , ⊔ ⊓

Denition 2.9. 1. A topological space, (X, T ) is called a T0 space, if for any two points x, y ∈ X there exists either a Ux ∈ T such that x ∈ Ux and y ∕∈ Ux or a Uy ∈ T such that y ∈ Uy and x ∕∈ Uy .

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2 Topological Space

2. A topological space, (X , T ) is called T1 space, if for any two points x, y ∈ X there exist both Ux ∈ T and Uy ∈ T , such that x ∈ Ux and y ∕∈ Ux and y ∈ Uy and x ∕∈ Uy . 3. A topological space, (X , T ) is called T2 space, if for any two points x, y ∈ X there exist both Ux ∈ T and Uy ∈ T , such that x ∈ Ux , y ∈ Uy and Ux ∩Uy = ∅. T2 space is also called Hausdorff space. It follows from denition that T2 ⇒ T1 ⇒ T0 . Example 2.13. Consider ℝ1 . 1. Dene a subset T as the set of semi-lines: T = {(−f, r) ∣ r ∈ ℝ}. Then (ℝ1 , T ) is a topological space. Moreover, it is easy to check that this space is T0 . But it is not T1 . 2. Dene a subset T as T = {U ⊂ ℝ ∣ U c is nite.} Then (ℝ1 , T ) is also a topological space. One can show that this is a T1 space. But it is not T2 . 3. With the convenient topology, it is easy to show that ℝ1 is a T2 space. Denition 2.10. Let (X, T ) be a topological space, S ⊂ X. S is called a topological subspace of X, if the topology of S is dened as TS = {U ∩ S ∣ U ∈ T }. We have to check that TS is really a topology. We leave it as an exercise. Denition 2.11. Let (Xi , Ti ), i = 1, 2 be two topological spaces. Consider the Cartesian product, X1 × X2 = {(x, y) ∣ x ∈ X1 , y ∈ X2 }. Let {U1 × U2 ∣ U1 ∈ T1 , U2 ∈ T2 } be a topological basis, and the topology generated by this basis be denoted by T1 × T2 . Then (X1 × X2 , T1 × T2 ) is a topological space, which is called the product topological space of X1 and X2 . The following theorem is useful: Theorem 2.2. Topological space (X , T ) is Hausdorff, if and only if the diagonal set of the product space (X × X, T × T ), denoted by D = {(x, x) ∣ x ∈ X}, is a closed set in the product space. Proof. (Necessity) Assume X is Hausdorff. Let (x, y) ∈ X × X and (x, y) ∕∈ D, which means x ∕= y. Then there exist open sets Ux ∋ x and Uy ∋ y, such that Ux ∩ Uy = ∅. Now Ux × Uy ⊂ X × X , Ux × Uy is open and Ux × Uy ∩ D = ∅. Hence, Dc is open, i.e., D is closed.

2.3 Continuous Mapping

39

(Sufciency) Assume D is closed. Given any x, y ∈ X and x ∕= y, we know that (x, y) ∈ Dc , which is open. So, there is an open set U ⊂ X × X such that (x, y) ∈ U ⊂ Dc . But {U × V ∣ U,V ∈ T } is the topological basis of T × T , hence there exist Ux ,Vy ∈ T such that (x, y) ∈ Ux × Uy ⊂ U ⊂ Dc . That is, x ∈ Ux and y ∈ Uy and Ux ∩Uy = ∅. Hence X is Hausdorff.

⊔ ⊓

2.3 Continuous Mapping Denition 2.12. Let M, N be two topological spaces. A mapping S : M → N is continuous, if one of the following two equivalent conditions holds: ∙ For any U ⊂ N open, its inverse image

S −1 (U) := {x ∈ M ∣ S (x) ∈ U} is open. ∙ For any C ⊂ N closed, its inverse image S −1 (C) is closed. We need to verify that these two conditions are equivalent. It is straight forward and we leave it to reader. Denition 2.13. Let M, N be two topological spaces. M and N are said to be homeomorphic if there exists a mapping S : M → N, which is one-to-one (injective), onto (surjective) and continuous. Moreover, S −1 is also continuous. S is called a homeomorphism. If a mapping is both injective and surjective it is said to be bijective. Example 2.14. Let S2 be a sphere: S2 = {(x, y, z) ∈ ℝ3 ∣ x2 + y2 + z2 = 1} and N be the north pole, i.e., N = (0, 0, 1). Then S2 ∖N, with the inherited topology from ℝ3 , is homeomorphic to ℝ2 . One of the homeomorphisms is: ⎧ ⎨X = x/(1 − z) ⎩Y = y/(1 − z), where (X,Y ) are the coordinates on ℝ2 . Note that if M and N are two metric spaces and F : M → N is a mapping, we may dene the continuity by H − G language as what we did in Calculus: F is continuous in x0 if for any given H > 0 there exists G > 0 such that d(x, x0 ) < G implies d(F(x), F(x0 )) < H . F is continuous if it is continuous everywhere. It is easily seen that for metric spaces this denition coincides with the general denition for topological spaces as long as the topologies are induced by the metrics.

40

2 Topological Space

Denition 2.14. Given a topological space M. 1. A set U ⊂ M is said to be clopen if it is both closed and open. A topological space, M, is said to be connected if the only two clopen sets are M and ∅. 2. A continuous mapping S : I = [0, 1] → M is called a path on M. M is said to be pathwise (or arcwise) connected if for any two points x, y ∈ M there exists a path, S , such that S (0) = x and S (1) = y. Example 2.15. 1. ℝ is connected. Assume U ∕= ∅ and U ∕= ℝ is a clopen set in ℝ. Choose x ∈ U and y ∈ U c . Say x < y. Consider set V = {z ∈ U ∣ z < y}. Then set V has a lowest upper bound m ⩽ y. Since U is closed m ∈ U, so m < y. Moreover, (m, y] ⊂ U c . Because U is open, U c is closed. Hence (m, y] = (−f, y] ∩ U c is closed. Then m ∈ U c , which is a contradiction. The same argument shows that any interval in ℝ is connected. 2. A pathwise connected space, M, is connected. Assume U ∕= ∅ and U ∕= M is a clopen set in M. Choose x ∈ U and y ∈ U c . Since M is path connected, there is a path S : I → M such that S (0) = x and S (1) = y. By continuity, both S −1 (U) and S −1 (U c ) are clopen non-empty sets on I, which contradicts 1. The following example shows that the converse of the second statement of the above example is incorrect. Example 2.16. On ℝ2 dene a topological ( )subspace as 1 , x > 0} ∪ {(0, y) ∣ y ∈ ℝ}. S = {(x, y) ∈ ℝ2 ∣ y = sin x It is connected, because both Y -axis and the curve can not be open. But it is not pathwise connected, because there is no path connecting such two points, one of which is on Y -axis and the other one on the curve. (Refer to Fig. 2.3)

Fig. 2.3

Connected but not Pathwise Connected

Denition 2.15. A topological space M is said to be locally connected at x ∈ M if every neighborhood Nx of x contains a connected neighborhood Ux , i.e., x ∈ Ux ⊂ Nx . M is said to be locally connected if it is locally connected at each x ∈ M.

2.3 Continuous Mapping

41

The following example shows that local connectedness does not imply connectedness (hence no pathwise connectedness); conversely, pathwise connectedness (connectedness) does not imply local connectedness. Example 2.17. 1. Let ℝ be with its discrete topology. Then it is locally connected, because each point itself is a connected neighborhood. But it is not connected because each point is clopen. 2. Let Bi , i = 1, 2, ⋅ ⋅ ⋅ be closed segments starting from origin and ending at (1, 1i ). B0 is the segment [0, 1] on X-axis. As shown in Fig. 2.4. An innite broom consists of 2 B = ∪f i=0 Bi ⊂ ℝ , which is obviously pathwise connected. But it is not locally connected. For instance, consider a point x = (0.5, 0). The ball B0.2 (x) does not contain a connected neighborhood. (Refer to Fig. 2.4)

Fig. 2.4

Innite Broom

Denition 2.16. Let {UO ∣ O ∈ / } be a set of open sets in M. The set is called an open covering of M if ∪O ∈/ UO ⊃ M. M is said to be a compact space if every open covering has a nite sub-covering, i.e., there exists a nite subset {UOi ∣ i = 1, 2, ⋅ ⋅ ⋅ , k} such that ∪ki=1UOi ⊃ M. From Calculus we know that with the conventional topology, a set, U ⊂ ℝn is compact, if and only if it is bounded and closed. Unfortunately, it is not true for

42

2 Topological Space

general metric spaces. For instance, consider the set of {xk } ∈ l 2 , k = 1, 2, ⋅ ⋅ ⋅ , where (xk )i = Gik , i.e., x1 = (1, 0, 0, ⋅ ⋅ ⋅ ), x2 = (0, 1, 0, ⋅ ⋅ ⋅ ), etc. It is obvious that {{xk }, k = 1, 2, ⋅ ⋅ ⋅ } is a closed and bounded set, but it is not compact. Denition 2.17. In a topological space, M, a sequence {xk } is said to converge to x, if for any neighborhood U ∋ x there exists a positive integer N > 0 such that when n > N, xn ∈ U. To emphasize that the convergence is a topological concept we consider the following simple example: Example 2.18. Consider ℝ1 and let xk = 1k , k = 1, 2, ⋅ ⋅ ⋅ . 1. Under the conventional topology it is obvious that xk → 0 because for any neighborhood U there exists an open neighborhood, ball BH (0) = {x ∣ ∣x∣ < H } such that BH (0) ⊂ U. When N > H1 and n > N, xn ∈ U. So the general denition coincides with that in Calculus. 2. Impose the discrete topology on ℝ1 . i.e., each point is an open set. Then xk converges to nowhere. Because it can never get into any {r}, which is a neighborhood of r. 3. Impose the co-nite topology on ℝ1 . Recall that in this topology U ∕= ∅ is open, if and only if U c is nite. Then xk converges to every point. Let r ∈ ℝ be any point, and U ∋ r be a co-nite neighborhood. Since U c is nite at most nite xk ∈ U c . The conclusion follows. Proposition 2.3. Let M, N be two topological spaces. M is rst countable. f : M → N is continuous, if and only if for each xk → x, f (xk ) → f (x). Proof. (Sufciency) For any open U ∋ f (x), by continuity f −1 (U) ∋ x is open. Since xn → x there is a N > 0 such that xn ∈ f −1 (U), n > N. It follows that f (xn ) ∈ U, n > N. (Necessity) Let U ⊂ N be open. we need to show that f −1 (U) is open. It is enough to show that for each x ∈ f −1 (U) there exists an open V such that x ∈ V ⊂ f −1 (U). If it is not true then there exists at least one point x0 ∈ f −1 (U), such that any neighborhood of it is not contained in f −1 (U). Since M is rst countable, let Uk be its countable neighborhood base, and set Vk = ∩ki=1Ui , we have a sequence of shrinking neighborhoods. Since Vk ∕⊂ f −1 (U), there exists xk ∈ Vk and xk ∕∈ f −1 (U). Now since xk → x0 so f (xk ) → f (x0 ) ∈ U. Then there is an integer N > 0 such that f (xn ) ∈ U, n > N, which means xn ∈ f −1 (U), which is a contradiction. ⊔ ⊓ Denition 2.18. A topological space is called sequentially compact if every sequence contains a convergent subsequence. Theorem 2.3. (Bolzano-Weierstrass) Let M be a rst countable topological space. if M is compact, it is sequentially compact. ¯ We claim that there exists a sequence {ak } ∈ A such Proof. Let A ⊂ M, x ∈ A. that ak → x. Since M is rst countable, using the above argument (in the proof of Proposition 2.3), we see that there exists a set of shrinking neighborhoods Vi ∋ x. Since Vk ∩ A ∕= ∅ we can choose ak ∈ Vk ∩ A for each k. Then ak → x.

2.3 Continuous Mapping

43

Now let {uk } be a sequence and denoted by A = {uk , k = 1, 2, ⋅ ⋅ ⋅ }. If A is not ¯ closed then by the above argument, we can choose x ∈ A∖A, and aki → x. Thus we can assume A is closed. Now we claim that for each ai there exists an open neighborhood Ui such that a j ∕∈ Ui , j ∕= i. If for a point ai such a neighborhood does no exist, then any neighborhood would contain some points from the sequence but itself. Hence there exists a sequence converging to it. Now {Ui , i = 1, 2, ⋅ ⋅ ⋅ } ∪ Ac is an open covering of M, but it does not have any nite sub-covering.

⊔ ⊓

The inverse of the Theorem 2.3 is also true. We prove it for metric spaces. We need some lemmas that themselves are interesting. Lemma 2.1. Let M be a sequentially compact space. Then for any H > 0, there exist x1 , ⋅ ⋅ ⋅ , xn ∈ M such that ∩ni=1 BH (xi ) ⊃ M. Such a structure is called an H -net. Proof. Choose x1 . If BH (x1 ) ⊃ M, we are done. Otherwise, choose x2 ∈ (BH (x1 ))c , and construct BH (x2 ). Keep going in this way and at the k-th step choose xi ∈ c (∪i−1 k=1 BH (xk )) . If at the i-th step we could not nd such xi we are done. Otherwise, we nd a sequence, xi , with d(xi , x j ) ⩾ H . This sequence has no convergent sub-sequence, which is a contradiction. ⊔ ⊓ Lemma 2.2. Let M be a sequentially compact metric space and {UO ∣ O ∈ / } is an open covering. Then there exists a number P > 0, called Lebesgue number such that for any x ∈ M, the ball BP (x) ∈ UO for some UO . Proof. For each x ∈ M choose maximal Px > 0 such that Bx (Px ) ⊂ UOx . It is enough to show that inf Px := P > 0. x∈M

Let {xn } be a sequence that makes Pxn → P . By sequential compactness, there is a sub-sequence xni → x0 . Now for this x0 there exists P0 and UO0 , such that BP0 (x0 ) ∈ UO0 . Once again by the sequential compactness, there is an N > 0 such that xni ∈ B P0 /2 (x0 ). Then BP0 /2 (x0 ) ⊂ UO0 , which implies Pni ⩾ P0 /2. But Pni → P . Hence P ⩾ P0 /2 > 0. ⊔ ⊓ Theorem 2.4. Let M be a sequentially compact metric space. Then M is compact. Proof. Let {UO ∣ O ∈ / } be an open covering. Lemma 2.2 says that there exists a P > 0 such that B P (x) ⊂ UOx . It is trivial to see that {B P (x) ∣ x ∈ M} is an open covering of M. By Lemma 2.1, there exist x1 , ⋅ ⋅ ⋅ , xn such that ∪ni=1 BP (xi ) ⊃ M. A sub-covering is obtained as {UOx ∣ i = 1, 2, ⋅ ⋅ ⋅ , n}. i

⊔ ⊓

The following proposition is simple but very useful. It says that the continuous image of a compact set is compact.

44

2 Topological Space

Proposition 2.4. Let f : M → N be a continuous mapping, and V ⊂ M be a compact set. Then f (V ) is compact. Proof. Let {UO ∣ O ∈ / } be an open covering of f (V ). Then { f −1 (UO ) ∣ O ∈ / } is an open covering of V . Since V is compact, there is a nite sub-covering { f −1 (UOi ), i = 1, ⋅ ⋅ ⋅ , k} of V . It follows immediately that ∪ki=1UOi ⊃ V.

⊔ ⊓

We know that for a continuous mapping the inverse image of an open (or closed) set remains open (closed respectively). But the continuous inverse image of a compact set is not necessarily compact. For example, f (x) = sin(x) : ℝ → ℝ is continuous and I = [−1, 1] is compact, but f −1 (I) = ℝ is not compact. Denition 2.19. Let f : M → N be a continuous mapping. If for any compact set, V ⊂ M, f −1 (V ) is compact, f is called a proper mapping.

2.4 Quotient Spaces Denition 2.20. Let S be any set and ∼ be a relation between two elements of S. ∼ is said to be an equivalent relation if (i). x ∼ x; (ii). If x ∼ y, then y ∼ x; (iii). If x ∼ y and y ∼ z, then x ∼ z. Denition 2.21. Let M be a topological space, “∼” an equivalent relation on M. 1. For each x ∈ M, an equivalent class, denoted by [x], is dened as [x] = {y ∈ M ∣ y ∼ x} 2. The set of equivalent classes is denoted by M/ ∼, and a mapping S : x → [x] is called the canonical projection. 3. Dene a set of subsets on M/ ∼, called the quotient topology, as T∼ = {U ∈ M/ ∼ ∣ S −1(U) is open in M}. Proposition 2.5. Let M/ ∼ and T∼ be as dened above. Then (M/ ∼, T∼ ) is a topological space, called the quotient space of M with respect to ∼. Proof. We have only to show that T∼ is a topology. Obviously, M/ ∼, ∅ ∈ T∼ . Let UO ∈ T∼ , O ∈ / . Since ∪O ∈/ S −1 (UO ) is open in M and ∪O ∈/ S −1 (UO ) = S −1 (∪O ∈/ UO ), we have ∪O ∈/ UO ∈ T∼ . Let Ui ∈ T∼ , i = 1, ⋅ ⋅ ⋅ , k. Since ∩ki=1 S −1 (Ui ) is open in M and

2.4 Quotient Spaces

45

∩ki=1 S −1 (Ui ) = S −1 (∩ki=1Ui ), we have ∩ki=1Ui ∈ T∼ . ⊔ ⊓

Hence, T∼ is a topology.

If in a topological space M, we have an equivalent relation ∼ such that x ∼ y, it is said that these two points are glued together. Various topological spaces are constructed by gluing certain points together. Example 2.19. Let S be a rectangle on ℝ2 S = [0, 1] × [0, 1] ⊂ ℝ2 . Give ℝ2 the convenient topology and consider S ⊂ ℝ2 as a topological subspace with inherited topology.

Fig. 2.5

Gluing Spaces

1. If we glue the right side with the left side of S, i.e., set (0, y) ∼ (1, y),

0 ⩽ y ⩽ 1.

Then the quotient space S/ ∼ is a cylinder. (Fig. 2.5 (a)) 2. If we glue the right side with the left side of S in reverse way, i.e., set (0, y) ∼ (1, 1 − y),

0 ⩽ y ⩽ 1.

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2 Topological Space

Then the quotient space S/ ∼ is a strip, which doesn’t have “inside” and “outside”. Intuitively, an ant can move from one side of the strip to another side without crossing the edge. Such a surface is said to be not orientable. This strip is called Mobius strip. (Fig. 2.5 (b)) 3. If we glue the right side with the upper side of S, i.e., set (1, z) ∼ (z, 1),

0⩽z⩽1

and the left side with the lower side of S, i.e., set (0, z) ∼ (z, 0),

0 ⩽ z ⩽ 1.

Then the quotient space S/ ∼ is a sphere. (Fig. 2.5 (c)) 4. If we glue the right side with the left side of S, i.e., set (0, y) ∼ (1, y),

0⩽y⩽1

and the upper side with the lower side of S, i.e., set (x, 0) ∼ (x, 1),

0 ⩽ x ⩽ 1.

Then the quotient space S/ ∼ is a torus. (Fig. 2.5 (d)) 5. If we glue the right side with the left side of S, i.e., set (0, y) ∼ (1, y),

0⩽y⩽1

and the upper side with the lower side of S in a reverse way, i.e., set (x, 0) ∼ (1 − x, 1),

0 ⩽ x ⩽ 1.

Then the quotient space S/ ∼ is such a sealed “bottle” that an ant can move from inside to outside. It is called Klein bottle. (Fig. 2.5 (e)) In fact such a bottle can not be visualized in ℝ3 .

References 1. Abraham R, Marsden J. Foundations of Mechanics. Reading: Benjamin/Cummings Pub. Com. Inc., 1978 2. Choquet G. Topology. New York: Academic Press, 1966. 3. Janich K. Topology. New York: Springer-Verlag, 1984. Translated by Levy S. 4. Scechter M. Principles of Functional Analysis. New York: Academic Press, 1971.

Chapter 3

Differentiable Manifold

This chapter provides an outline of Differential Geometry. First we describe the fundamental structure of a differentiable manifold and some related basic concepts, including mappings between manifolds, smooth functions, sub-manifolds. The concept of ber bundle is also introduced. Then vector elds, their integral curves, Lie derivatives, distributions are discussed intensively. The dual concepts, namely, covector elds, their Lie derivatives with respect to a vector eld, co-distributions and the relations with the prime ones are also discussed. Finally, some important theorems and formulas, such as Frobenius’ theorem, Lie series expansions and Chow’s theorem etc. are presented. This chapter provides a fundamental tool for the analysis of nonlinear control systems.

3.1 Structure of Manifolds Denition 3.1. Let (M, T ) be a second countable, T2 (Hausdorff) topological space. M is called an n dimensional topological manifold if there exists a subset A = {AO ∣ O ∈ / } ⊂ T , such that (i). ∪O ∈/ AO ⊃ M; (ii). For each U ∈ A there exists a homeomorphism I : U → I (U) ⊂ ℝn , which is called a coordinate chart, denoted by (U, I ). Moreover, if (iii). For two coordinate charts: (U, I ) and (V, \ ), if U ∩ V is not empty, then both \ ∘ I −1 : I (U ∩V ) → \ (U ∩V ) and I ∘ \ −1 : \ (U ∩V ) → I (U ∩V ) are Cr (Cf , CZ ). Such two coordinate charts are said to be consistent. (iv). If a coordinate chart, W , is consistent with all charts in A , then W ∈ A . Then (M, T ) is called a Cr (Cf , analytic, respectively) differentiable manifold. Remark 3.1. In fact, (i)–(iii) are enough to determine a differentiable manifold. Condition (iv) is just for convenience in making certain statements. Example 3.1. 1. Any open subset of ℝn is an n-dimensional manifold. 2. Figure “8” in ℝ2 , dened in polar coordinates as F = {(r, T ) ∣ r = ∣ sin(T )∣, 0 ⩽ T ⩽ 2π} is not a manifold, because around the origin, r = 0, it can not be homeomorphic to an open set of ℝ1 . D. Cheng et al., Analysis and Design of Nonlinear Control Systems © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2010

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3 Differentiable Manifold

Fig. 3.1

Differentiable Manifold

3. S2 is a two dimensional analytic manifold. To see this we may choose six coordinate charts as (Ux+ , Sx+ ), (Ux− , Sx− )(Uy+ , Sy+ ), (Uy− , Sy− ), (Uz+ , Sz+ ), (Uz− , Sz− ), where Ux+ = {(x, y, z) ∈ S2 ∣ x > 0}. ( ) ( ) y Y , (x, y, z)T ∈ Ux+ . = Sx+ (x, y, z) = z Z Then Sx+ (Ux+ ) is an open disk in ℝ2 . Moreover, it is obvious that Sx+ is a homeomorphism and both S and S −1 are analytic. The other ve coordinate charts are dened in a similar way. Recalling with Example 2.19, another set of coordinate charts may be chosen as UN = S2 ∖{N},

US = S2 ∖{S}.

Then dene IN : UN → ℝ2 as {

X = x/(1 − z) Y = y/(1 − z),

and IS : US → ℝ2 as

{

X = x/(1 + z) Y = y/(1 + z)

Therefore, {(UN , IN ), (US , IS )} determines a CZ structure on S2 . Example 3.2. (Grassmann Manifold) Consider the set of k dimensional subspaces in ℝn . We try to “impose” an analytic structure on the set. First, let F(k, n) be a set of bases of the k dimensional subspaces. That is X ∈ F if and only if X = (x1 , ⋅ ⋅ ⋅ , xk ), where xi = (xi1 , ⋅ ⋅ ⋅ , xin )T , i = 1, ⋅ ⋅ ⋅ , k and x1 , ⋅ ⋅ ⋅ , xk are linearly independent. For convenience, we identify each X with an n × k matrix with full column rank. Now it is easily seen that taking xij , i = 1, ⋅ ⋅ ⋅ , k, j = 1, ⋅ ⋅ ⋅ , n as coordinates, F(k, n) can be considered as an open subspace of ℝn×k .

3.1 Structure of Manifolds

49

Let X , Y ∈ F(k, n). It is known that X and Y generate the same subspace if and only if there exists an A ∈ GL(k, ℝ) such that X = YA. Here GL(k, ℝ) is the set of k × k non-singular real matrices. Set an equivalent relation ∼ on F(k, n) as X ∼ Y ⇔ X ∈ Y GL(k, ℝ). Then we can dene the quotient space as G(k, n) = F(k, n)/ ∼ . Then G(k, n) is a well dened topological space. We set a differential structure on G(k, n) in the following way: Since X ∈ F(k, n), it has a k × k non-singular minor, say it consists of the rst k rows. Then ⎞ ⎛ Ik ⎛ 1 ⎞ k ⎟ ⎜ x ⎜ k+1 ⋅ ⋅ ⋅ xk+1 ⎟ (3.1) X ∼ ⎜⎜ . ⎟ . .. ⎠ ⎟ ⎠ ⎝ ⎝ .. x1n ⋅ ⋅ ⋅ xkn Now dene a coordinate chart of G(k, n) as all the equivalent classes [X ] where the rst k rows of X are linearly independent. Then in each equivalent class choose a representation of the form of (3.1). Note that in this chart each [X] has unique representation, we, therefore, can choose I ([X ]) = x1k+1 , x1k+2 , ⋅ ⋅ ⋅ , xkn−1 , xkn as its local coordinates. Choosing k rows from n, we can get totally ( ) n! n N= = k k!(n − k)! such coordinate charts, U i , corresponding to the set of [X] with pre-assigned k linearly independent rows. In this way, we obtain N coordinate charts as (Ui , Ii ). Finally, we have only to show that this is an analytic structure. Let [X] ∈ Ui ∩U j . Then in Ui some required linearly independent rows of X form Ii , that is, I˜i = XA−1 i and the pre-assigned k rows of I˜i forms coordinates Ii , and in U j a certain required linearly independent rows of X form I j , that is, I˜ j = XA−1 j and some other preassigned k rows of I˜ j form coordinates I j . Then the coordinate change: Ii → I j is determined by the transformation mapping

I˜ j = I˜i Ai A−1 j , which is obviously analytic. From the above discussion one sees clearly that the dimension of the Grassmann manifold is dim(G(k, n)) = k(n − k). Denition 3.2. Let M, N be two Cr manifolds with dimensions m, n respectively. F : M → N is called a Cr mapping, if for each x ∈ M and y = F(x) ∈ N there are coordinate charts (U, I ) about x and (V, \ ) about y, such that F˜ = \ ∘ F ∘ I −1

50

3 Differentiable Manifold

is Cr . Moreover, the rank of F at x is dened as the rank of F˜ at I (x). Recall that if a mapping F : ℝm → ℝn is expressed as yi = yi (x1 , ⋅ ⋅ ⋅ , xm ),

i = 1, ⋅ ⋅ ⋅ , n,

then the rank of F at x is dened as ⎛

w y1 w y1 ⋅⋅⋅ w xm ⎜ w x1 ⎜ ⎜ .. rank(F)∣x = rank ⎜ ... . ⎜ ⎝ w yn w yn ⋅⋅⋅ w x1 w xm

⎞   ⎟ ⎟ ⎟ ⎟ . ⎟ ⎠  

(3.2)

x

Later on, in most cases we identify F˜ with F when there is no risk of misleading. Denition 3.3. Let N = ℝ. Then a Cr (Cf , CZ ) mapping from M to N = ℝ is called a Cr mapping (Cf mapping, analytic mapping, respectively), denoted by Cr (M)(Cf (M), CZ (M), respectively). Denition 3.4. Let M, N be two differentiable manifolds, and S : M → N a homeomorphism. If both S and S −1 are Cr (Cf , CZ ), then M, N are said to be Cr (Cf , CZ respectively) diffeomorphic, and S is called a Cr (Cf , CZ , respectively) diffeomorphism. Let x ∈ M. If there exist neighborhoods U ∋ x and V ∋ y such that S : U → V with y = S (x) is a diffeomorphism, then M and N are said to be locally diffeomorphic from x to y, and S is called a local diffeomorphism. Example 3.3. Consider an open disk D = {(x, y) ∈ ℝ2 ∣ x2 + y2 < r2 }, and an elliptic disk

} x2 y2 E = (x, y) ∈ ℝ ∣ 2 + 2 < 1 , a b {

2

they are CZ diffeomorphic. A diffeomorphism, F : D → E, can simply dened as ( ) (a) b F1 (x, y) = x, F2 (x, y) = y. r r Note that the same mapping can also be considered as F : w D → w E. Then one sees that the two boundaries are also diffeomorphic. It can be shown that ℝ2 is diffeomorphic to D. But ℝ can never be diffeomorphic to w D because a diffeomorphism is a homeomorphism, but ℝ and w D can not be homeomorphic. Say they have different fundamental groups. (Z for w D and trivial group {e} for ℝ.) Locally, they are diffeomorphic. In fact, ℝ is the covering space of w D. (Refer to Chapter 4 for fundamental group and covering space.)

3.1 Structure of Manifolds

51

Denition 3.5. Let M, N be two differentiable manifolds with dim(M) = m ⩾ n = dim(N) and F : N → M be a smooth injective mapping. Then F : N → N˜ = F(N) ˜ provides an one-to-one correspondence between N and N. 1. When N˜ is endured with the topology of N, it is called an immersed sub-manifold of M. 2. If N˜ is an immersed sub-manifold, and when it is endured with the inherited topology from M, the F : N → N˜ is a homeomorphism, then N˜ is called an embedded sub-manifold. 3. If N ⊂ M and for every p ∈ N there exists a coordinate chart (U, I ), such that ⎧  ⎨I (p) = 0, (3.3) I (U) = {x ∈ ℝn ∣ ∣xi ∣ < H },  ⎩ I (N ∩U) = {x ∈ I (U) ∣ xn+1 = ⋅ ⋅ ⋅ = xm = 0}, then N is called a regular sub-manifold of M. The coordinate frame (3.3) is called a at coordinate system for N. Example 3.4. 1. Consider a mapping F : ℝ → ℝ2 dened as ( ( π) ( ( π ))) F(t) = 2 cos t − , sin 2 t − . 2 2 The image of F is a gure “8”, F is an immerse because rank(F) = 1. But F is not one-to-one. In fact, as t goes over 2π, F(t) runs an “8”. Thus the image is not an immersed sub-manifold. (cf. Fig. 3.2 (a)) 2. Let g(t) : ℝ → (−π, π) be dened as g(t) = 2tan−1 (t). Dene F : ℝ → ℝ2 as ( ( ( ( π ))) π) , sin 2 g(t) + . F(t) = 2 cos g(t) + 2 2 (cf. Fig. 3.2 (b)) Now F(t) draws only one “8”. This time it is one to one. So F(ℝ) is an immersed sub-manifold. But F(ℝ) is not an embedded sub-manifold because if we give F(ℝ) the subspace topology of ℝ2 , then as t = n → f, F(n) → F(0) = 0. It is obvious that F −1 is not continuous. 3. S2 as a subspace of ℝ3 is a regular sub-manifold. To see this we need to nd a at coordinate chart. It is clear that the natural coordinate frame is not a at chart. Now for any p0 ∈ S2 , there is at least one coordinate of p0 , say the z-coordinate, z p ∕= 0. Assume z p > 0, by continuity, there is a neighborhood U ∋ p0 , such 0

0

that z p ∕= 0, ∀p ∈ U. Dene a mapping I : U → ℝ3 as ⎧  ⎨X = x Y =y √  ⎩ Z = z − 1 − x2 − y2 . It is easily seen that the Jacobian of I at p0 is non-singular, then it is a local diffeomorphism. So as U is small enough, I : U → I (U) ⊂ ℝ3 is a diffeomorphism.

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3 Differentiable Manifold

Then consider (U, I ) as a local coordinate chart, we have S2 ∩U = {p ∈ U ∣ Z(p) = 0}.

Fig. 3.2

Immersion

The essential difference between immersed sub-manifold and embedded submanifold lies on the topology of the subspace. An immersed sub-manifold endures its own topology passed over from the other manifold. Unlike it, an embedded submanifold keeps the subspace topology inherited from its mother’s topology. It is not difcult to see that an immersed sub-manifold is locally an embedded sub-manifold, and an embedded sub-manifold is a regular sub-manifold. [2] Theorem 3.1. Let N, M be two smooth manifolds of dimensions n, m respectively. Suppose a smooth mapping F : N → M has a constant rank, rank(F) = k. Then for any q ∈ F(N), F −1 (q) is an (n − k) dimensional closed regular sub-manifold. Proof. Since the rank of F is k. Let p ∈ N and q = F(p) ∈ M. About p we assume the left upper k × k minor of JF is non-singular. Then we can change the local coordinates about p ∈ N by { X 1 = F 1 (x) X 2 = x2 , dim(X 1 ) = k, dim(X 2 ) = n − k. This is a local coordinate transformation because the assumption assures the Jacobian of the transformation to be locally non-singular. Under this new coordinate frame F becomes { F1 = X1 (3.4) F 2 = F 2 (X 1 ). Note that F 2 depends only on X 1 , otherwise it will violate the rank assumption of F. Now about q = F(p) we change the local coordinates by { Y 1 = y1 (3.5) Y 2 = y2 − F 2 (X 1 (y1 , y2 )), dim(X 1 ) = k, dim(X 2 ) = n − k,

3.2 Fiber Bundle

53

where X 1 = X 1 (y1 , y2 ) is assured by implicit function theorem. Now note that (3.4) yields that on the image of F(N) the F 2 depends on X 1 = y1 only. So

wF2 = O(∥y∥), w y2 which means (3.5) is a local coordinate transformation. Under these new coordinate frames on both sides, we have an expression of F as { Xi , i ⩽ k (3.6) Fi (X1 , ⋅ ⋅ ⋅ , Xn ) = 0, i > k. Now, under new coordinate frames around q and p ∈ F −1 (q) we have F −1 (q) ∩U = {X ∈ U ∣ X1 = ⋅ ⋅ ⋅ = Xk = 0}. Hence F −1 (q) is an (n − k) dimensional regular sub-manifold.

⊔ ⊓

In fact, the expression (3.6) itself is useful. We refer to [9] for detailed constructing process. Example 3.5. Let F : ℝ3 → ℝ be dened as F(x, y, z) =

x2 y2 z2 + + . a2 b 2 c 2

Since JF = (2x, 2y, 2z), for any p ∈ F −1 (1) the rank(JF ) = 1 because x, y, z can not be simultaneously zero. So the ellipsoid F −1 (1) is a regular sub-manifold of ℝ3 of dimension 3 − 1 = 2.

3.2 Fiber Bundle Denition 3.6. A ber bundle consists of three topological spaces: B, M, Y , called the total space, base space, and ber space respectively, and a continuous mapping p : B → M, called the projection, which satisfy the following conditions: (i). p : B → M is surjective (onto). (ii). Yx := p−1 (x), x ∈ M. Then Yx is homeomorphic to Y . Yx is called the bundle over x. (iii). For each point x ∈ M, there is a neighborhood x ∈ U ⊂ M and a homeomorphism h : U × Y → p−1 (U) such that p ∘ h(z, y) = z,

∀z ∈ U, y ∈ Y.

A section is a continuous mapping f : M → B, such that p ∘ f (x) = x. A conventional way to denote a ber bundle is (B, p, M). In fact Y is an interior structure. Example 3.6. 1. Let M and Y be two topological spaces. Dene B = M × Y , and p : M × Y → M is the natural projection. Then (B, p, M) becomes a ber bundle. Such a ber bundle is called a product bundle.

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3 Differentiable Manifold

Fig. 3.3

Bundle over S2

2. Let M = S2 . At each point p ∈ M a tangent space can be attached, which is ℝ2 . Now on a neighborhood p ∈ U ⊂ M, as U is small enough, we can dene a continuous coordinate frame (Xq , Yq ), say a Cartesian frame on the tangent space Tq . Then t ∈ Tq can be expressed by (xt , yt ) as the projection of t on Xq and Yq respectively. Now locally we can dene a continuous mapping I : U × ℝ2 → U × {(p,t) ∣ p ∈ U,t ∈ Tp } as (u, (x, y)) → (u,t) with t ∈ Tu and tx = x, ty = y. In this bundle, the total space is B = {(u,t) ∣ u ∈ S2 ,t ∈ Tu }, base space is M = S2 , bundle space is Y = ℝ2 . B = {(X,t) ∈ ℝ3 × ℝ3 ∣ X12 + X22 + X32 = 1, t1 X1 + t2 X2 + t3 X3 = 0}. Moreover, if the Cartesian coordinates are used, let Pi = (X i ,t i ) ∈ B, i = 1, 2, and X i = (xi1 , xi2 , xi3 ), t i = (t1i ,t2i ,t3i ). Then the topology over B is deduced from d(P1 , P2 ) = d(X 1 , X 2 ) + d(t 1 ,t 2 ). Denition 3.7. Assume two ber bundles (Bi , pi , M) share the same base space. A continuous mapping ) : B1 → B2 is called a bundle mapping if p1 = p2 ∘ ) , i.e., the graph in Fig. 3.4 is commute. A bundle mapping is called a bundle isomorphism if it is a homeomorphism and its inverse ) −1 is also a bundle mapping.

Fig. 3.4

Bundle Isomorphism

3.2 Fiber Bundle

55

Denition 3.8. A ber bundle (B, p, M) is said to be a k dimensional vector bundle, if its ber space is Y = ℝk , and the homeomorphism h : U ×Y → p−1 (U), restricted to each x ∈ U, as hx : {x} × Y → p−1 (x) is a vector space isomorphism, i.e., hx ∈ GL(k, ℝ). A vector bundle is called a trivial bundle if B ≃ M × Y = M × ℝn . Example 3.7. Consider the second example in Example 3.6. The tangent bundle over S2 is obviously a vector bundle. But it is not a trivial bundle. To see this, assume B ≃ S2 × ℝ2 . Let F : B → S2 × ℝ2 be the homeomorphism. We choose 0 ∕= Y ∈ ℝ2 . Then F −1 (S2 × Y ) is a continuous vector eld, which is nowhere zero. It is well known that this is impossible [2]. Denition 3.9. Let (Bi , pi , M), i = 1, 2, be two vector bundles. The bundle mapping ) : B1 → B2 is called a vector bundle mapping if for each x ∈ M −1 ) : p−1 1 (x) → p2 (x)

is linear. Theorem 3.2. Let (Bi , pi , M), i = 1, 2, be two vector bundles. ) : B1 → B2 is a vector bundle mapping. ) is a bundle isomorphism, if and only if, for each x ∈ M −1 ) : p−1 1 (x) → p2 (x)

is a vector space isomorphism. i.e.,

) ∣ p−1 (x) ∈ GL(k, ℝ). 1

Proof. (Sufciency) Let x ∈ U1 ∩U2 where

Ii : Ui × Y → p−1 i (Ui ),

i = 1, 2

are homeomorphisms respectively. Let U = U1 ∩U2 . Since ) : B1 → B2 is a homeomorphism and by commutativity one sees that

) : I1−1 (U) → I2−1 (U) is a homeomorphism. Restricting it to p−1 1 (x), we have

) : {x} × Y → {x} × Y to be bijective. By denition it is linear. Then it is a linear space isomorphism. That is, ) ∣ p−1 (x) ∈ GL(k, ℝ). 1

(Necessity) We prove it by constructing the inverse, ) −1 . For each x ∈ M since ) ∣ p−1 (x) is a linear isomorphism, let < ∣ p−1 (x) be the inverse of ) ∣ p−1 (x) . Then we 1 2 1 can construct the inverse of ) point-wise as < ∣ p−1 (x) ∈ GL(k, ℝ). Thus it is enough 2 to show that < is continuous.

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3 Differentiable Manifold

Using the above notations, we have, on U, that

I2−1 )I1 : (x, y) $→ (x, fx (y)), where fx (y) ∈ GL(k, ℝ) is continuous with respect to x. But I1−1< I2 : (x, y) is the point inverse of I2−1< I1 , it can then be expressed as

I1−1< I2 : (x, y) $→ (x, fx−1 (y)), where fx−1 (y) is the matrix inverse of fx (y). Then it is obvious that I1−1< I2 is continuous, and so is < . ⊔ ⊓ For our purpose, the ber bundle, particularly the vector bundle, is used for investigating a manifold and its tangent space. We may replace all the above topological spaces by Cr manifolds and the mappings by Cr mappings. We refer to [6] for further study.

3.3 Vector Field Let M be an n dimensional smooth manifold, x ∈ M. Denote by Cr (M, x) the set of Cr functions dened around x, in other words, over some neighborhoods of x. Let 0 ∈ (a, b) ⊂ ℝ, and T : (a, b) → M be a smooth mapping such that T (t), t ∈ (a, b) is a curve on M with T (0) = x.

Fig. 3.5

Tangent Vector

( ) For any h ∈ Cr (M, x) dene a mapping T∗ dtd : Cr (M, x) → ℝ as: ( ) d d T∗ h = (h ∘ T )∣t=0 dt dt

(3.7)

Intuitively, it seems that this mapping transfers a derivative operator on ℝ to a derivative operator on M. To make it more clear we express it on a coordinate chart. Now on a coordinate chart about x we express T by its components as

T (t) = (T1 (t), ⋅ ⋅ ⋅ , Tn (t))T and h can be expressed as h = h(x1 , ⋅ ⋅ ⋅ , xn ). Then (3.7) is expressed, by using chain rule, as

3.3 Vector Field

 ( ) n dTi  d h= ¦ T∗   dt i=1 dt

t=0

wh . w xi

It is ready to see that under a local coordinate system the operator T∗  ( ) n dTi  w d = ¦ T∗ .   dt dt w xi i=1

57

(3.8) (d) dt

is (3.9)

t=0

It is also easily seen that (3.8) is ℝ-linear, i.e., ( ) ( ) ( ) d d d (aT1 + bT2 )∗ h = a(T1 )∗ h + b(T2 )∗ h. dt dt dt Hence, under system all the curves, passing x ∈ M, form a vector { a local coordinate } w w space with w x1 , ⋅ ⋅ ⋅ , w xn as a basis. It can be considered as a space of the directional derivatives acting on Cr (M, x). Denition 3.10. For a given point x ∈ M, the vector space of directional derivatives, acting on Cr (M, x), is called the tangent space of M at x and denoted by Tx (M). There are several points that should be claried: First of all, for a given coordinate p p chart (U, I ) we identify a point p ∈ U with I (p) = (x1 , ⋅ ⋅ ⋅ , xn ) ∈ ℝn . For instance, we denote T (t) = (T1 (t), ⋅ ⋅ ⋅ , Tn (t)), in fact it should be I ∘ (T (t)), because only on I (U) ⊂ ℝn the coordinates make sense. This identication, if there is no risk of misleading, is conventional and commonly used. It is known as a local coordinate representation. Similarly, h = h(x1 , ⋅ ⋅ ⋅ , xn ) is a local coordinate representation of h ∘ I −1. Since the denition of tangent space is done under a local coordinate system, to make it uniquely dened we have to prove that it is independent on the choice of the local coordinate frame. i.e., the same T plays the same rule under different coordinate frames. It is easy and left for reader. Under a given coordinate chart, a tangent vector X p ∈ Tp (M) can be expressed as m

X p = ¦ vi i=1

w . w xi

As in Linear Algebra, the basis of vectors may be omitted, so X p can also be briey denoted as a vector: X p = (v1 , ⋅ ⋅ ⋅ , vn )T . Thus, for any h ∈ Cr (M, p), n

X p (h) = ¦ vi i=1

w (h). w xi

Denition 3.11. Let M, N be two smooth manifolds, and F : M → N a smooth mapping. For a point p ∈ M and F(p) ∈ N we dene a mapping F∗ : Tp (M) → TF(p) (N) as

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3 Differentiable Manifold

F∗ (X p )h = X p (h ∘ F),

X p ∈ Tp (M), h ∈ Cr (N, F(p)).

(3.10)

Next, we want to nd the local coordinate representation of F∗ (X p ), which is particularly important in applications. Let x and y be local coordinates of p ∈ M and F(p) ∈ N respectively. Moreover, X p and F∗ (X p ) are presented as X p = a1

w w + ⋅ ⋅ ⋅ + an , w x1 w xn

F∗ (X p ) = b1

w w + ⋅ ⋅ ⋅ + bn , w y1 w yn

and F : M → N, as x $→ y, is expressed as yi = fi (x1 , ⋅ ⋅ ⋅ , xm ),

i = 1, ⋅ ⋅ ⋅ , n.

Then (3.10) can be expressed as m

F∗ (X p )h = X p (h ∘ F) = ¦ ai w (h ∘ F) w xi i=1 ( ) m n w fj w = ¦ ai ¦ h w y j w xi i=1 j=1 ( ) n m w fj = ¦ ¦ ai w h∣F(p) := w x wyj i j=1 i=1 where

w fj ai , i=1 w xi m

bj = ¦

n

¦ b j wwy j h(y)∣F(p),

j=1

j = 1, ⋅ ⋅ ⋅ , n.

Under the matrix form, this relation can be expressed as ⎡

w f1 w f1 ⋅⋅⋅ w xm b1 ⎢ w x1 ⎢ .. ⎥ ⎢ . .. ⎣ . ⎦=⎢ . ⎢ .. ⎣ bn w fn w fn ⋅⋅⋅ w x1 w xm ⎡

⎤

⎤

⎡ ⎤ ⎥ a1 ⎥ ⎢ . ⎥ ⎥ ⎢ . ⎥, ⎥ ⎣ . ⎦ ⎦ am

(3.11)

p

or briey, F∗ (X p ) = JF (p)X p . Since X p is essentially a directional derivative, it has the characteristic of an ordinary derivative. For instance, starting from the denition we can show easily that if f , g ∈ Cr (M, p) and a, b ∈ ℝ, then { X p (a f + bg) = aX p ( f ) + bX p (g) (3.12) X p ( f g) = X p ( f )g + f X p (g).

3.3 Vector Field

59

In fact, Tp (M) is the set of all mappings Cr (M, p) → ℝ, which satisfy (3.12). We refer to [2] for detail. Now we can put all tangent spaces, passing through different points of M together, and denote it as T (M) = ∪x∈M {(x, Tx (M))}. Over each coordinate chart (U, I ) of M we may dene a mapping < : I (U) × ℝn → T (M) as ( ) n w (x1 , ⋅ ⋅ ⋅ , xn )T × (a1 , ⋅ ⋅ ⋅ , an )T $→ q, ¦ ai , i=1 w xi where (x1 , ⋅ ⋅ ⋅ , xn ) = I (q). Since < is a diffeomorphism from an open set of ℝn × ℝn to T (M), we may consider (< (I (U) × ℝn ), < −1 ) as a coordinate chart of T (M). It is ready to verify that (i). such charts form an open covering of T (M); (ii). if M is a Cr manifold then such charts are Cr compatible. We conclude that if M is an n dimensional Cr manifold then with the structure described above, T (M) is a 2n dimensional Cr manifold. Let p : T (M) → M be the natural projection: (p, Tp ) $→ p. Then (T (M), p, M) becomes a vector bundle. It is known as tangent bundle of M. In many cases, we simply use an alternative expression of T (M), which ignores the rst component formally and indicates it implicitly. Then it is expressed as T (M) = ∪x∈M {Tx (M)}. Unless we emphasize some particular bundle properties, this simple form will be used hereafter. Denition 3.12. A vector eld X on M is a section of the tangent bundle T (M), i.e., X : M → T (M). X is called a Cr vector eld if this section is Cr . Under a local coordinate system X can be expressed as n

X = ¦ ai (x) i=1

w . w xi

By denition, that X is Cr means all ai (x), i = 1, ⋅ ⋅ ⋅ , n are Cr . The set of smooth vector elds on manifold M is denoted by V r (M), where r can be any non-negative integer, or f, or Z . In a situation when the degree of smoothness is not concerned, it may simply be denoted by V (M). Recall that if we have two smooth manifolds M and N and a smooth mapping F : M → N, F∗ can map a tangent vector of M to a tangent vector of N. Precisely, let q ∈ M and F(q) ∈ N, Xq ∈ Tq (M), then F∗ (Xq ) ∈ TF(q) (N) as dened in (3.10), and can be calculated in local coordinate frames by (3.11). Now a natural question is: Can F map a vector eld on M to a vector eld on N? The answer is that this is not always true. We discuss it in the following example.

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Example 3.8. Consider a vector eld X = (t 2 + 1) dtd on ℝ. Let F : ℝ → ℝ2 be dened as ) ( 1 1 F : t $→ sin πt, cos πt . 2 2 Choosing any t, say t = 1, then F∗ : T1 (ℝ) → T(1,0) (ℝ2 ) maps X ∣t=1 = 2 dtd to ⎡

dF1 dt ⎣ F∗ (Xt=1 ) = dF2 dt

⎤   ⎦  

) 0 ∈ T(1,0) (ℝ2 ). = −π (

t=1

For any t we can dene F∗ in the same way. But obviously {F∗ (Xt ) ∣ t ∈ ℝ} is not a vector eld on ℝ2 because it is even not dened everywhere. In general, it is easily seen that if F : M → N is a diffeomorphism and X ∈ V (M), then F∗ (X) is a vector eld on N. Moreover, F∗ (X ) = JF X (∘F −1 ).

(3.13)

(3.13) is just a generalization of (3.11). Example 3.9. Let T ∈ GL(n, ℝ), F : ℝn → ℝn be dened as y = T x. Let X = (v1 (x), ⋅ ⋅ ⋅ , vn (x))T . Then

F∗ (X ) = JF X ∣y=T x = T X(T −1 y).

Particularly, we can let X = (sin(x1 + x2 ), ex1 )T and ] [ 10 . T= 11 Then

][ ] 1 0 sin(x1 + x2 )   11 e x1 ] [ [ ] y=T x  sin(x1 + x2 ) sin y2  . = = y1 ex1 + sin(x1 + x2 ) x=T −1 y e + sin y2 [

F∗ (X) =

3.4 One Parameter Group Denition 3.13. Let T : I → M be a curve on M, with the interval, 0 ∈ I ⊂ ℝ. If there exists a vector eld X ∈ V (M), and x0 ∈ M, such that d T (t) = XT (t) , dt

T (0) = x0

then T (t) is called the integral curve of X with initial value T (0) = x0 . As the solution of an ordinary differential equation T (t) has the following properties:

3.4 One Parameter Group

61

1. T (t) ∘ T (s) = T (t + s); 2. T (−t) ∘ T (t) = T (0) = id, where id is the identity mapping. Hence T (t) forms a group with respect to t. (Precisely, only when I = R, T (t) becomes a group.) This group is called the one parameter group generated by X . To emphasize the dependence of the integral curve on the vector eld, one parameter group can also be denoted by etX (x0 ). When this notation is used, we always assume that t is in the largest interval, 0 ∈ I ⊂ ℝ, such that the integral curve exists. Denition 3.14. A vector eld X ∈ V (M) is said to be complete if it is completely integrable, namely, for any x0 ∈ M the integral curve exists for all t ∈ (−f, f). If a vector eld on ℝn satises the Lipschitz condition, i.e., there exists a K > 0 such that ∥X(x) − X(y)∥ ⩽ K∥x − y∥, ∀x, y ∈ ℝn , then it is complete. For a manifold, if there exists a covering of charts such that the Lipschitz condition is satised with the same K, then X is complete. Lemma 3.1. Let etX (x0 ) be the integral curve of X . Assume there exists a neighborhood U ∋ x0 such that etX (x) is dened on U × I. Then for any t0 ∈ I there exists a neighborhood U0 ∋ x0 such that etX0 : U0 → etX0 (U0 ) is a diffeomorphism. Proof. We start from t = 0. Since eX0 is an identity mapping, there exists an open neighborhood t ∈ I0 such that etX has nonsingular Jacobian matrix at x0 . That is, there exists a neighborhood Ux0 ∋ x0 and I0 ∋ 0 such that for any t ∈ I0 , etX : Ux0 → etX (Ux0 ) is a diffeomorphism. A similar argument shows that for each t ∈ I there exists Uxt ∋ etX (x0 ) and an open t ∈ It such that etX : Uxt → etX (Uxt ) is a diffeomorphism. Now for any T ∈ I, {It ∣ 0 ⩽ t ⩽ T } form an open covering of [0, T ]. Hence there exists a nite subcovering {Iti ∣ i = 1, ⋅ ⋅ ⋅ , k}. Now we can construct { U¯k = Uk , U¯i−1 = Ui−1 ∩ (etXi−1 )−1 (U¯i ) , i = k, k − 1, ⋅ ⋅ ⋅ , 1. Then it is ready to see that for U0 = U¯0 the mapping eXT : U0 → eXT (U0 ) is a diffeomorphism. Not that T ∈ I is chosen arbitrarily, the conclusion follows. ⊔ ⊓ The above Lemma can be visualized as follows: for any x0 ∈ M there exists a neighborhood U0 ∋ x0 and an interval 0 ∈ I0 ⊂ ℝ such that U0 “moves” along the time-axis in a “diffeomorphic” way. It seems like a tube of “ow”. So the set of integral curves are called the ow of X. Proposition 3.1. Let X be a complete vector eld. Then for any xed t ∈ ℝ, etX : M → M, dened as: x $→ etX (x), is a diffeomorphism. Proof. We already know that for any vector X , etX is a local diffeomorphism as long as it is dened. Now etX is dened for any x ∈ M and any t ∈ ℝ. We have only to show that it is one-to-one and onto. One-to-one is from the uniqueness of solution of differential equations. Onto is obvious because for any y ∈ M, etX (eX−t (y)) = y. ⊓ ⊔

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3 Differentiable Manifold

Note that if M is a compact manifold, then every X ∈ V (M) is complete [2]. Proposition 3.2. Let F : M → N be a diffeomorphism, X ∈ V (M), and etX (x0 ) the integral curve of X with initial condition x(0) = x0 . Then the integral curve of F∗ (X) is the image of the integral curve of X. Precisely, F (X)

et ∗

(F(x0 )) = F ∘ etX (x0 ).

(3.14)

Proof. We have only to prove it over each coordinate chart. In other words, under a local coordinate representation. Using the chain rule we have d d F ∘ etX (x0 ) = JF [etX (x0 )] = JF X (F −1 ) = F∗ (X). dt dt F (X)

As for the initial condition, it is et ∗

(F(x0 )) = F(x0 ).

⊔ ⊓

3.5 Lie Algebra of Vector Fields Recall that a tangent vector X p is an operator acting on Cr (M, p). Now if X ∈ V (M) and f ∈ Cr (M), then X p ∈ Tp (M) and f ∈ Cr (M, p) at every p ∈ M. Hence we can dene the action of X on f in a natural way as [X ( f )] p := X p ( f ). It is obvious that X( f ) is dened everywhere. Moreover, if we express the above equation on a local coordinate frame, say n

X = ¦ vi (x) i=1

w , w xi

f = f (x1 , ⋅ ⋅ ⋅ , xn ),

the action becomes n

X ( f ) = ¦ vi (x) i=1

wf . w xi

It turns out that X ( f ) ∈ Cr−1 (M). Thus a vector eld X can be considered as a mapping X : Cr (M) → Cr−1 (M). Particularly, if r = f or r = Z , it becomes a mapping on Cf (M) (or CZ (M)). Next, we impose an algebraic structure on V (M). First of all, we introduce the concept of Lie algebra. Denition 3.15. A vector space V over ℝ with the product, denoted by [X,Y ], X,Y ∈ V , is called a Lie algebra if for X,Y, Z ∈ V we have (i). [(aX1 + bX2 ),Y ] = a[X1 ,Y ] + b[X2,Y ], a, b ∈ ℝ; (ii). [X ,Y ] = −[Y, X]; (iii). (Jacobi Identity) [X , [Y, Z]] + [Y, [Z, X ]] + [Z, [X,Y]] = 0. The product, [X,Y ], is called Lie bracket. Now consider the set of vector elds V (M). We dene the Lie bracket as

3.5 Lie Algebra of Vector Fields

[X ,Y ] = XY − Y X.

63

(3.15)

That is, for each p ∈ M and h ∈ Crp (M) [X ,Y ] p (h) = X (Y (h)) − Y (X (h))∣ p . It should be shown rst that this is well dened, namely [X,Y ] p ∈ Tp (M), which implies that we have to prove that [X ,Y ] satises (3.12). This is a straightforward computation, and we leave it to reader. Proposition 3.3. Let X, Y ∈ V (M), and under a local coordinate system X = (a1 (x), ⋅ ⋅ ⋅ , an (x))T ,

Y = (b1 (x), ⋅ ⋅ ⋅ , bn (x))T .

Then [X ,Y ] = where

wY wx

and

wX wx

wY wX X− Y, wx wx

(3.16)

are the Jacobian matrices of X and Y respectively.

Proof. Assume n

[X ,Y ] = ¦ ci (x) i=1

w . w xi

(3.17)

Choosing a particular h = x j , then (3.17) provides [X ,Y ](h) = c j (x). On the other hand, by denition [X,Y ](h) = X (Y (h)) − Y (X (h)) n n = ¦ ai (x) w (b j (x)) − ¦ bi (x) w (a j (x)). w xi w xi i=1 i=1 Thus n

c j (x) = ¦ ai (x) i=1

n w b j (x) w a j (x) − ¦ bi (x) , w xi w xi i=1

j = 1, ⋅ ⋅ ⋅ , n.

It is ready to see that (3.16) is the matrix representation of (3.18).

(3.18) ⊔ ⊓

To understand the geometric meaning of the Lie bracket of two vectors, we consider the following case. Let X,Y ∈ V (M). Consider the compounded integral curve: eY−t eX−t eYt etX (x0 ). We may describe this physically as follows: A particle, starting from x0 , moves along the direction of X. After time t, it changes the direction from X to Y and goes along the direction of Y for time t. Then it moves backward on direction of X by the

64

3 Differentiable Manifold

same time period. Finally, backward on direction of Y for still the same length of time. If X and Y are constant vectors, it is obvious that the particle will back to the starting point x0 . But if they are not constant, the end position, xe , may be different from x0 . What is the direction of the particle really goes? Roughly speaking, this direction from x0 to xe is the direction of [X ,Y ]∣x0 . Precisely, ) 1( Y X Y X e−t e−t et et (x0 ) − x0 = [X ,Y ]x0 . 2 t→0 t lim

(3.19)

(3.19) can be proved by using the Taylor expansion etX (x) = x + X(x)t + O(∣t∣2 ) We encourage our readers to show this, as it was commented in [3] that to understand the meaning of Lie bracket you have to do it at least once in your life time. Proposition 3.4. V (M) with Lie bracket dened above is a Lie algebra. Proof. The rst two conditions of Lie algebra follow from denition (3.15) immediately. We prove only the Jacobi identity. Let h ∈ Cr (M), then ([X , [Y, Z]] + [Y, [Z, X ]] + [Z, [X ,Y]])(h) = X (Y Z(h) − ZY (h)) − (Y Z − ZY )X (h) + Y (ZX(h) − XZ(h)) − (ZX − XZ)Y (h) + Z(XY (h) − Y X(h)) − (XY − Y X)Z(h) = 0, ∀h ∈ Cr (M).

⊔ ⊓

Example 3.10. Consider a control system x = f (x) + g(x)u,

x ∈ ℝn ,

(3.20)

where f (x) and g(x) can be considered as vector elds on ℝn . Then the Lie subalgebra of V (ℝn ), generated by f (x) and g(x) is denoted as C = { f (x), g(x)}LA and is called the accessibility Lie algebra [7]. If f (x) = Ax and g(x) = b, where A is an n × n matrix and b ∈ ℝn . The system becomes a linear system. A simple computation shows [ f , g] = −Ab [ f , [ f , g]] = A2 b .. . [ f , ⋅ ⋅ ⋅ , [ f , g] ⋅ ⋅ ⋅ ] = (−1)k Ak b. Note that if the above multi-fold brackets contain more than one b, the resulting vector eld will be zero. So to get non-zero vector elds, we have only to consider brackets with a single b. Then the controllability Lie bracket becomes C = Span{b, Ab, ⋅ ⋅ ⋅ , An−1 b}.

3.6 Co-tangent Space

65

From linear system theory it is known [11] that a linear system is controllable if and only if rank(C) = n. Proposition 3.5. Let X,Y ∈ V (M), p, q ∈ Cr (M). Then [pX, qY ] = pq[X ,Y ] + pX (q)Y − qY (p)X.

(3.21)

Proof. For any h ∈ Cr (M) [pX, qY ]h = pX(qY (h)) − qY (pX(h)) = pX (q)Y (h) + pqX(Y(h)) − qY (p)X(h) − qpY(X (h)) = pq[X ,Y ](h) + pX(q)Y(h) − qY (p)X(h) = (pq[X ,Y ] + pX(q)Y − qY (p)X )(h). ⊔ ⊓

Since h is arbitrary, the conclusion follows.

3.6 Co-tangent Space For a given manifold M and x ∈ M, the tangent space, Tx (M) at x ∈ M is a nite dimensional vector space. So from linear algebra we know that there is a dual space, denoted by Tx∗ (M). Thus we dene Denition 3.16. 1. The co-tangent space of the manifold M is dened as T ∗ (M) = ∪x∈M Tx∗ (M). 2. A co-vector eld I is a mapping that assigns a co-vector I (x) ∈ Tx∗ (M) at each point x ∈ M. 3. A co-distribution Z is an assignment that assigns a subspace Z (x) ⊂ Tx∗ (M) at each point x ∈ M. Z is said to have a constant dimension r if dim(Z (x)) = r, ∀x ∈ M. Example 3.11. 1. In ℝ3 consider two smooth functions h1 (x) = ex1 + sin(x2 ) + x3 ,

h2 (x) = ex1 + cos(x2 ) − x3 .

Then the differentials of hi (x), i = 1, 2, are dh1 (x) = ex1 dx1 + cos(x2 )dx2 + dx3 , dh2 (x) = ex1 dx1 − sin(x2 )dx2 − dx3. We may consider {dxi ∣ i = 1, 2, 3} as the basis, then dhi (x), i = 1, 2, can simply be expressed as a row vectors { dh1 (x) = (ex1 , cos(x2 ), 1), dh2 (x) = (ex1 , − sin(x2 ), −1).

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3 Differentiable Manifold

Point-wise, dhi (x) are co-vectors in Tx∗ (ℝ2 ). In this way we have dhi ∈ T ∗ (ℝ3 ), i = 1, 2. 2. Using the two functions we can dene a co-distribution Z as

Z = Span{dh1 , dh2 }. Precisely,

Z (x) = Span{dh1 (x), dh2 (x)},

∀x ∈ ℝ3 .

It is easy to show that Z (x) has constant dimension: dim(Z (x)) = 2, ∀x ∈ ℝ2 . Motivated by the above example we choose {dxi , i = 1, ⋅ ⋅ ⋅ , n} as the canonical basis of T ∗ (M) in a coordinate chart. Now if there is a vector eld X(x) = n

n

i=1

i=1

¦ ai (x) wwxi and dh(x) = ¦ wwh(x) xi dxi , then their inner-product is n

⟨dh(x), X(x)⟩ = ¦ ai (x) i=1

w h(x) . w xi

It is exactly the action of X on h(x). A co-vector eld is also called a one form. Using {dxi , i = 1, ⋅ ⋅ ⋅ , n} as the basis of T ∗ (M) in a coordinate chart, a co-vector eld can be expressed as I (x) = n

¦ Ii (x)dxi .

i=1

n

Denition 3.17. A one form I (x) = ¦ Ii (x)dxi is called a closed one form if i=1

w Ii (x) w I j (x) = , wxj w xi

i ∕= j.

(3.22)

I (x) is an exact one form if there exists a smooth function h(x) such that I (x) = dh(x). Theorem 3.3. (Poincare’s Lemma [1]) 1. Locally a closed one form is an exact one form. 2. If M is a simply connected topological space, a closed one form is globally an exact one form.

3.7 Lie Derivatives Denition 3.18. Let F : M → N be a diffeomorphism. 1. For a function f (x) ∈ Cr (N), an induced mapping F ∗ : Cr (N) → Cr (M) is dened by F ∗ ( f ) = f ∘ F ∈ Cr (M).

3.7 Lie Derivatives

67

2. For a vector eld X ∈ V r (M), an induced mapping F∗ : V r (M) → V r (N) is dened by F∗ (X)(h) = X(h ∘ F), ∀h ∈ Cr (N). 3. For a co-vector eld I ∈ V ∗r (N), an induced mapping F ∗ : V ∗r (N) → V ∗r (M) is dened by ⟨F ∗ (I ), X⟩ = ⟨I , F∗ (X)⟩ , ∀X ∈ V r (M). If F is a local diffeomorphism, then the above mappings are locally dened. Denition 3.19. Let X ∈ V (M), f ∈ Cr (M). The Lie derivative of f with respect to X , denoted by LX ( f ), is dened as 1 LX ( f ) = lim [(etX )∗ f (x) − f (x)]. t→0 t

(3.23)

Proposition 3.6. Under the local coordinates Lie derivative (3.23) can be expressed as n

LX ( f ) = ⟨d f , X⟩ = ¦ Xi i=1

wf . w xi

(3.24)

Proof. By denition, (etX )∗ f (x) = f (etX (x)). The Taylor expansion of it with respect to t is ) ( f etX (x) = f (x) + td f ⋅ X (x) + O(t 2). ⊔ ⊓

Plugging it into (3.23) yields (3.24).

Denition 3.20. Let X , Y ∈ V (M). The Lie derivative of Y with respect to X, denoted by adX (Y ), is dened as 1 adX (Y ) = lim [(eX−t )∗Y (etX (x)) − Y (x)]. t→0 t

(3.25)

Proposition 3.7. Under the local coordinates Lie derivative (3.25) can be expressed as adX (Y ) = JY X − JX Y = [X,Y ].

(3.26)

Note that JY denotes the Jacobian of Y , namely, ⎡

w Y1 ⋅ ⋅ ⋅ w Y1 w xn ⎢ w x1 ⎢ .. .. JY = ⎢ . . ⎣ w Yn ⋅ ⋅ ⋅ w Yn w x1 w xn

⎤ ⎥ ⎥ ⎥. ⎦

Proof. Using the Taylor expansion, we have etX (x) = x + (tX) + O(t 2),

(3.27)

Y (etX (x)) = Y (x) + JY (tX) + O(t 2 ).

(3.28)

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3 Differentiable Manifold

Using (3.27), the Jacobian of eX−t becomes JeX = I − tJX + O(t 2 ).

(3.29)

−t

(3.27)–(3.29) yield (eX−t )∗Y (etX (x)) = (I − tJX + O(t 2 ))(Y (x) + JY (tX) + O(t 2 )) = Y (x) + t(JY X − JX Y ) + O(t 2 ). ⊔ ⊓

Putting it into (3.25) yields (3.26). Denition 3.21. Let X ∈ V (M) and Z to X, denoted by LX (Z ), is dened as

∈ V ∗ (M). The Lie derivative of Z

with respect

1 LX (Z ) = lim [(etX )∗ Z (etX (x)) − Z (x)]. t→0 t

(3.30)

Proposition 3.8. Under the local coordinates Lie derivative (3.27) can be expressed as LX (Z ) = (JZ T X )T + Z JX .

(3.31)

Proof. Mimicing the proof of Proposition 3.7, we use the Taylor expansion to get (etX )∗ Z (etX (x)) = (Z (x) + t(JZ T X )T + O(t 2 ))(I + tJX + O(t 2 )) = Z (x) + t(JZ T X )T + t Z (x)JX + O(t 2 ),

where the transpose comes from the convention that under local coordinates a covector eld is expressed as a row. Putting it into (3.30), equation (3.31) follows. ⊓ ⊔ The geometric meaning of Lie derivative adXY is the varying rate of a vector eld Y along the integral curve of X . Fig. 3.6 gives such a description. For a function h and a co-vector eld Z , the corresponding Lie derivatives LX ( f ), and LX (Z ) respectively, have a similar meaning.

Fig. 3.6

Lie Derivative adXY

Some useful formulas are collected in the following:

3.7 Lie Derivatives

69

Proposition 3.9. Let p, q, h ∈ Cr (M), X, Y ∈ V (M), Z ∈ V ∗ (M). Then 1. (Leibnitz Equality) LX ⟨Z ,Y ⟩ = ⟨LX Z ,Y ⟩ + ⟨Z , LX Y ⟩ .

(3.32)

L pX (qZ ) = pqLX (Z ) + pLX (q)Z + q ⟨Z , X⟩ dh.

(3.33)

LX (Z ) = d ⟨ Z , X ⟩ .

(3.34)

2.

3. If Z = dh, then

Proof. 1. First, we prove a formula that itself is useful: d ⟨Z , X⟩ = X T

wX wZ +Z . wx wx

(3.35)

Since n

d ⟨Z , X⟩ = d ¦ Zi Xi i=1

=

n

¦

i=1

(

n

¦ ww Zx ji j=1

)

n

xi + ¦

i=1

(

n

¦ ww Xx ji j=1

)

Zi .

Put RHS into a matrix form. The RHS of (3.35) follows. Using (3.35) to (3.32): ) ( LHS = ⟨d ⟨Z ,Y ⟩ , X ⟩ = Y T w Z + Z w Y X wx wx w w Z Y T X +Z X. =Y wx wx ) ( ( )T w X T wZ Y + Z w Y X − Z w X Y. +Z RHS = X wx wx wx wx It follows that LHS=RHS. 2.

(

w qZ wx

)T

w pX wx [ )T ] ) ( ( = pX T (dq)T Z + w Z q + qZ p w X + X dp wx wx

L pX (qZ ) =

pX T

+ qZ

= pqLX (Z ) + pLX (q)Z + q ⟨Z , X⟩ dp. 3. Using (3.35) again, we have d ⟨Z , X⟩ = d ⟨dh, X⟩ = X T

w (dh) wX wX + dh = X T Hess(h) + dh . wx wx wx

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3 Differentiable Manifold

Since Hess(h) is symmetric, the above becomes

wX = X (Hess(h)) + dh wx T

(

T

w (dh) X wx

)T + dh

wX = LX dh. wx

⊔ ⊓

3.8 Frobenius’ Theory Denition 3.22. 1. A distribution D is involutive if for any two vectors X,Y ∈ D we have [X,Y ] ∈ D,

∀X,Y ∈ D.

(3.36)

2. A sub-manifold S is called the integral manifold of a given distribution D if for each x ∈ S its tangent space is D, i.e., Tx (S) = D(x),

∀x ∈ S.

If a distribution D(x) has its integral manifold, it is said to be integrable. Frobenius’ Theorem is a fundamental tool in the application of Differential Geometry to nonlinear system analysis and control design. To present this theorem, we need some preparations. Lemma 3.2. Let X ∈ V 1 (M) be a C1 vector eld and F : M → M be a diffeomorphism. X is F invariant (i.e., F∗ (X(p)) = X (F(p))), if and only if F(etX (x)) = etX (F(p)).

(3.37)

Proof. (Necessity) Assume X is F invariant. Using the chain rule, we have d X (p)) = J d eX (p) F dt t dt (F ∘ et = JF X (etX (p)) = F∗ (X (F −1 ∘ etX (p)))

(3.38)

= X (etX (p)). But the integral curve of X (etX (p)) is etX (p), the uniqueness of solution of a differential equation assures (3.37). (Sufciency) Assume (3.37). Differentiating both sides yields F∗ (X(etX (p))) = X (etX (F(p))). Setting t = 0 yields

F∗ (X(p)) = X (F(p)).

⊔ ⊓

Lemma 3.3. Let X and Y be two C1 vector elds in M. Then [X,Y ] = 0 if and only if for each point p ∈ M there exists a G p > 0 such that etX ∘ eYs (p) = eYs ∘ etX (p),

∣s∣, ∣t∣ < G p .

(3.39)

3.8 Frobenius’ Theory

71

Proof. (Sufciency) (3.39) means that Y is etX invariant and X is eYt invariant. It follows that there exists a neighborhood V of p such that for q ∈ V , 1 1 [X ,Y ]q = lim ((eX−t )∗ (Y (etX (q)) − Y (q)) = lim (Y (q) − Y (q)) = 0. t→0 t t→0 t (Necessity) Assume [X,Y ] = 0. Set q′ = etX (q) and let

Tq (t) = (eX−t )∗ (Y (etX (q)). Then

Tq (t) = lim 1 [(eX−t+δt (q))∗ (Y (eXδt (q′ )) − (eX−t )∗ (Y (q′ ))] δt→0 δt = (eX−t )∗ ( lim 1 (eX−δt )∗ (Y (eXδt (q′ )) − Y (q′ ))) δt→0 δt = (eX−t )∗ ([X ,Y ]q′ ) = 0.

That is, Tq (t) is constant for t < G and q ∈ V , which means Y is etX invariant. (3.39) follows from Lemma 3.2. ⊔ ⊓ Theorem 3.4. (Frobenius’ Theorem) A distribution D is integrable if and only if it is involutive and has constant dimension. Proof. (Necessity) Assume D is integrable around p, then there exists an integral manifold S of D passing through x0 such that Tp (S) = D(p). Since S is a submanifold, there exists a local coordinate chart (U, x) with p ∈ U such that S = {x ∈ U ∣ xk+1 = ⋅ ⋅ ⋅ = xn = 0}. {

Then it is clear that D = Span

w w ,⋅⋅⋅ , w x1 w xk

} .

The necessity follows. (Sufciency) Assume dim(D) = k. For a given p ∈ M, there exists a neighborhood of p, say V , in which we can nd a local basis: X1 , ⋅ ⋅ ⋅ , Xk such that D(q) = Span{X1 (q), ⋅ ⋅ ⋅ , Xk (q)},

q ∈ V.

Without loss of generality we may assume the upper k × k block of the matrix X (q) = (X1 (q), ⋅ ⋅ ⋅ , Xk (q)), denoted by Q(q), is nonsingular. Then dene −1

Y (q) = X(q)(Q(q))

( ) I := (Y1 (q), ⋅ ⋅ ⋅ ,Yk (q)) = k . ∗

Since D is involutive, [Yi ,Y j ] =

k

¦ cs (q)Ys .

s=1

(3.40)

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3 Differentiable Manifold

Using formula (3.26), we have

( ) 0 . [Yi ,Y j ] = ∗

(3.41)

Putting (3.40) and (3.41) together, it is clear that [Yi ,Y j ] = 0,

0 ⩽ i, j ⩽ k.

That is, (Y1 , ⋅ ⋅ ⋅ ,Yk ) is a commutative basis of D. Now we can choose Yk+1 , ⋅ ⋅ ⋅ ,Yn such that (Y1 (q), ⋅ ⋅ ⋅ ,Yn (q)) is a basis of Tq (M), q ∈ V . Then we can nd a G > 0 and dene the mapping

) (y1 , ⋅ ⋅ ⋅ , yn ) = eYy11 ⋅ ⋅ ⋅ eYynn (p),

∣yi ∣ < G , i = 1, ⋅ ⋅ ⋅ , n.

(3.42)

Since the Jacobian matrix J) (0) = In is nonsingular, so ) denes a local diffeomorphism from a neighborhood of 0 ∈ ℝn to a neighborhood of p ∈ M. By denition

w) = Y1 (eYy11 ⋅ ⋅ ⋅ eYynn (p)). w y1

(3.43)

Y

Since eYyii and ey jj are exchangeable, exchanging eYy11 with eYyii in the right side of (3.42) yields that

w) = Yi (eYy11 ⋅ ⋅ ⋅ eYynn (p)), w yi

i = 1, ⋅ ⋅ ⋅ , k.

(3.44)

That is, in new coordinate frame y, Yi = Gi , i = 1, ⋅ ⋅ ⋅ , k. Therefore S = {y ∈ V ∣ yk+1 = 0, ⋅ ⋅ ⋅ , yn = 0} ⊔ ⊓

is an integral manifold of D. Remark 3.2.

1. Frobenius’ theorem also claims that if the distribution is involutive and with constant dimension, there exists a unique largest integral sub-manifold through each x ∈ M [2]. 2. A generalized version of the Frobenius’ theorem says that if the manifold is analytic and the distribution is also analytic, then the integral sub-manifold exists without the assumption of constant dimension [10].

3.9 Lie Series, Chow’s Theorem In this section all the manifolds, functions, vector elds, and co-vector elds are assumed to be analytic. Proposition 3.10. Let X, Y ∈ V Z (M), h ∈ CZ (M), Z ∈ V ∗ (M). Then 1. d h(eX (x)) = LX h(etX (x)). dt t

(3.45)

3.9 Lie Series, Chow’s Theorem

73

2. (Lie Series of Smooth Functions) h(etX (x)) =

f

tk k ¦ LX h(x). k=0 k!

(3.46)

3. (Lie Series of Vector Fields) f

(eX−t )∗Y (etX (x)) =

tk k ¦ adX Y (x). k=0 k!

(etX )∗ Z (etX (x)) =

¦ k! LkX Z (x).

(3.47)

4. (Lie Series of Forms) f

tk

(3.48)

k=0

Lie series (3.46)–(3.48) are also called the Campbell-Baker-Hausdorff formulas. Proof. 1. Using the chain rule, we have LHS = dh

d X (e (x)) = dhX(etX (x)) = LX h(etX (x)). dt t

2. Using (3.45) recursively, we have dk h(eX (x))∣t=0 = LkX h(etX (x)), dt k t

k ⩾ 0.

(3.49)

Now (3.46) can be obtained by the Taylor expansion with respect to t. 3. We rst work out the derivatives with respect to t. d X X dt (e−t )∗Y (et (x)) 1 [(eX X X X = lim 't −t−'t )∗Y (et+'t (x)) − (e−t )∗Y (et (x))] 't→0

1 [(eX ) (eX ) Y (eX (x)) − (eX ) Y (eX (x))] = lim 't −t ∗ t −'t ∗ −t ∗ t+'t 't→0 X X X X X = (e−t )∗ lim [(e−'t )∗Y (e't et (x)) − Y (et (x))] 't→0

= (eX−t )∗ adX Y (etX (x)). It follows recursively that k−1 dk (eX ) Y (x)∣ X ∗ t=0 = (et )∗ adX Y (x)∣t=0 dt k t = adkX Y (x), k ⩾ 0.

(3.50)

Again a Taylor expansion yields (3.47). 4. The proof for (3.48) is basically the same and can be done in a similar way. Since d X ∗ (e ) Z (etX (x)) = (etX )∗ LX Z (etX (x)), dt t

74

3 Differentiable Manifold

dk X ∗ (e ) Z (etX (x)) = (etX )∗ LkX Z (etX (x)), dt k t  hence  dk X ∗ X (e ) Z (et (x)) = LkX Z (x). dt k t t=0 Using the Taylor expansion yields (3.48). ⊔ ⊓ Corollary 3.1. f

t k+1 k LX (X(x)). k=1 (k + 1)!

etX (x) = x + tX(x) + ¦

(3.51)

Proof. Applying (3.46) for coordinate function xi , one sees f

tk k LX (xi ). k=2 k!

xi (etX (x)) = xi + tLX (xi ) + ¦

Note that LX (xi ) = Xi , which is the i-th component of X . Replacing LkX (xi ), k ⩾ 2 by Lk−1 (Xi ) and arranging n components together, we have (3.51). ⊔ ⊓ Let H be a real subspace of V (M), generated by a set of vector elds over ℝ. A curve V (t), 0 ⩽ t ⩽ 1 is called the integral curve of H if d V (t) ∈ HV (t) , dt

0 ⩽ t ⩽ 1.

Let p ∈ M. The reachable set of H from p, denoted by R(H, p), is the set of points, x ∈ M, for which there exists an integral curve V such that V (0) = p and V (1) = x. Theorem 3.5. (Chow’s Theorem) Let H be a set of completely integrable vector elds, L(H) be the Lie algebra generated by H, and M be path-wise connected. If L(H) = V (M) (under the sense of distribution), then R(H, p) = M. Proof. If [H, H] ⊂ H, we are done. Otherwise there exists at lease a pair of vector elds X,Y ∈ H, such that (eYt )∗ X ∕∈ H for some t. Add (eYt )∗ X into H to get a new vector space, H1 . Replace H by H1 , either [H1 , H1 ] ⊂ H1 , or we can generate H2 . We make three claims: Fact 1: Hk ⊂ L(H). This fact follows from Lie series (3.47). Fact 2: For each x ∈ M there exists a nite k < n such that Hk ∣x = Vx (M). It is obvious because dimVx (M) = n and each time we at least raise the dimension by 1. Fact 3: If X, Y are completely integrable, then so is (etX )∗Y . Therefore, etX is a global diffeomorphism (∀t), and then [(etX )∗Y ]

etX eYs (x) = es

(etX (x)).

(3.52)

Now since M is path-wise connected, there exists a path V (t), 0 ⩽ t ⩽ 1. such that V (0) = p and V (1) = x. According to Fact 2, for each t there exist n vectors X1 , ⋅ ⋅ ⋅ , Xn in Hk such that dimHk (V (t)) = n. Then n

– exp(ti Xi )(V (t)) i=1

3.10 Tensor Field

75

is a local diffeomorphism from a neighborhood of 0 ∈ ℝn to a neighborhood of V (t). Moreover, by (3.52) this diffeomorphism can be realized by the integral curves of the vector elds in H. In other words, R(H, V (t)) contains a neighborhood of V (t). Now V (t), 0 ⩽ t ⩽ 1 is a compact set, it can be covered by nite R(H, V (ti )), i = 1, ⋅ ⋅ ⋅ , m < f. We conclude that x ∈ R(H, p). ⊔ ⊓

3.10 Tensor Field Let V be an n dimensional vector space. A mapping I : V × ⋅ ⋅ ⋅ × V → ℝ is called a / 01 2 k

multi-linear mapping, if it satises (i).

I (X1 , ⋅ ⋅ ⋅ , cXr , ⋅ ⋅ ⋅ , Xk ) = cI (X1 , ⋅ ⋅ ⋅ , Xr , ⋅ ⋅ ⋅ , Xk ); (ii).

I (X1 , ⋅ ⋅ ⋅ ,Yr + Zr , ⋅ ⋅ ⋅ , Xk ) = I (X1 , ⋅ ⋅ ⋅ ,Yr , ⋅ ⋅ ⋅ , Xk ) + I (X1 , ⋅ ⋅ ⋅ , Zr , ⋅ ⋅ ⋅ , Xk ). Let {e1 , ⋅ ⋅ ⋅ , en } be a basis of V . The dual space of V is denoted by V ∗ . A basis of V ∗ , denoted by {d 1 , ⋅ ⋅ ⋅ , d n }, is called the dual basis of {ei }, if it satises { 1, i = j i d (e j ) = 0, i ∕= j. Denition 3.23. A multi-linear mapping I :

I:V × ⋅ ⋅ ⋅ × V ×V ∗ × ⋅ ⋅ ⋅ × V ∗ → ℝ / 01 2 / 01 2 r

s

is called an (r, s)-order tensor on V , where r is called the covariant order, and s is called the contra-variant order. The set of (r, s)-order tensors is denoted by Tsr (V ). Denote by

,ir j1 js J ij11,⋅⋅⋅ ,⋅⋅⋅ , js = I (ei1 , ⋅ ⋅ ⋅ , eir , d , ⋅ ⋅ ⋅ , d ),

where 1 ⩽ i1 , ⋅ ⋅ ⋅ , ir ⩽ r, 1 ⩽ j1 , ⋅ ⋅ ⋅ , js ⩽ s. We construct a matrix, called the structure matrix of I as ⎡ 11⋅⋅⋅1 11⋅⋅⋅n ⋅ ⋅ ⋅ J nn⋅⋅⋅1 ⋅ ⋅ ⋅ J nn⋅⋅⋅n ⎤ J11⋅⋅⋅1 ⋅ ⋅ ⋅ J11⋅⋅⋅1 11⋅⋅⋅1 11⋅⋅⋅1 .. .. .. ⎥ ⎢ .. ⎢ . . . . ⎥ ⎢ 11⋅⋅⋅1 ⎥ 11⋅⋅⋅n nn⋅⋅⋅1 nn⋅⋅⋅n ⎥ ⎢J ⎢ 11⋅⋅⋅n ⋅ ⋅ ⋅ J11⋅⋅⋅n ⋅ ⋅ ⋅ J11⋅⋅⋅n ⋅ ⋅ ⋅ J11⋅⋅⋅n ⎥ ⎢ .. .. .. ⎥ * = ⎢ ... (3.53) . . . ⎥ ⎢ ⎥ 11⋅⋅⋅1 11⋅⋅⋅n nn⋅⋅⋅1 nn⋅⋅⋅n ⎢J ⎥ ⎢ nn⋅⋅⋅1 ⋅ ⋅ ⋅ Jnn⋅⋅⋅1 ⋅ ⋅ ⋅ Jnn⋅⋅⋅1 ⋅ ⋅ ⋅ Jnn⋅⋅⋅1 ⎥ ⎢ . ⎥ . . . .. .. .. ⎦ ⎣ .. 11⋅⋅⋅1 ⋅ ⋅ ⋅ J 11⋅⋅⋅n ⋅ ⋅ ⋅ J nn⋅⋅⋅1 ⋅ ⋅ ⋅ J nn⋅⋅⋅n Jnn⋅⋅⋅n nn⋅⋅⋅n nn⋅⋅⋅n nn⋅⋅⋅n

76

3 Differentiable Manifold n

Let X ∈ V be a column vector. We denote it as X = (a1 , ⋅ ⋅ ⋅ , an )T , i.e., X = ¦ ai ei . Similarly, Z ∈ V ∗ can be denoted as Z = (b1 , ⋅ ⋅ ⋅ , bn ), i.e., Z =

n

i=1

¦ bi d i . Then the

i=1

tensor value can be obtained as

I (X1 , ⋅ ⋅ ⋅ , Xr , Z1 , ⋅ ⋅ ⋅ , Zs ) = Zs ⋉ ⋅ ⋅ ⋅ ⋉ Z1 ⋉ * ⋉ X1 ⋉ ⋅ ⋅ ⋅ ⋉ Xr .

(3.54)

We refer to the Appendix for “ ⋉ ”, which is the symbol for semi-tensor product of matrices. Denition 3.24. Let I ∈ Tsr (V ) and \ ∈ Tqp (V ). Then their tensor product, I ⊗ r+p \ ∈ Ts+q (V ), is dened as

I ⊗ \ (X1 , ⋅ ⋅ ⋅ , Xr+p , Z1 , ⋅ ⋅ ⋅ , Zs+q ) = I (X1 , ⋅ ⋅ ⋅ , Xr , Z1 , ⋅ ⋅ ⋅ , Zs )\ (Xr+1 , ⋅ ⋅ ⋅ , Xr+p , Zs+1 , ⋅ ⋅ ⋅ , Zs+q ). It is easy to prove the following Proposition 3.11. Let I and \ have the structure matrices MI and M\ respectively. Then the structure matrix of I ⊗ \ is MI ⊗\ = MI ⊗ M\ .

(3.55)

Two important special cases are that the covariant order r = 0 and that contravariant order s = 0. They are simply denoted by T r (V ) and Ts (V ) respectively. Next, we consider T r (V ) . Denition 3.25. I ∈ T r (V ) is called a symmetric covariant tensor, if

I (X1 , ⋅ ⋅ ⋅ , Xi , ⋅ ⋅ ⋅ , X j , ⋅ ⋅ ⋅ , Xr ) = I (X1 , ⋅ ⋅ ⋅ , X j , ⋅ ⋅ ⋅ , Xi , ⋅ ⋅ ⋅ , Xr ), X1 , ⋅ ⋅ ⋅ , Xr ∈ V. I is called a skew-symmetric covariant tensor, if

I (X1 , ⋅ ⋅ ⋅ , Xi , ⋅ ⋅ ⋅ , X j , ⋅ ⋅ ⋅ , Xr ) = −I (X1 , ⋅ ⋅ ⋅ , X j , ⋅ ⋅ ⋅ , Xi , ⋅ ⋅ ⋅ , Xr ), X1 , ⋅ ⋅ ⋅ , Xr ∈ V. Denote by S k (V ) ⊂ T k (V ) the set of k-th order symmetric covariant tensors and : k (V ) ⊂ T k (V ) the set of k-th order skew-symmetric covariant tensors. We consider the skew-symmetric case rst. We have

: k (V ) = {0},

k > n.

To prove this, consider a skew-symmetric tensor I , if Xi = X j then

I (X1 , ⋅ ⋅ ⋅ , Xi , ⋅ ⋅ ⋅ , X j , ⋅ ⋅ ⋅ , Xr ) = −I (X1 , ⋅ ⋅ ⋅ , X j , ⋅ ⋅ ⋅ , Xi , ⋅ ⋅ ⋅ , Xr ) = −I (X1 , ⋅ ⋅ ⋅ , Xi , ⋅ ⋅ ⋅ , X j , ⋅ ⋅ ⋅ , Xr ). Therefore, it is zero. Now if r > n, let Xi be a vector from a basis {e1 , ⋅ ⋅ ⋅ , en }, it is obvious that I (ei1 , ⋅ ⋅ ⋅ , eir ) = 0. By multi-linearity I (X1 , ⋅ ⋅ ⋅ , Xr ) is a linear combination of I (ei1 , ⋅ ⋅ ⋅ , eir ), it is zero. Hence we have

3.10 Tensor Field

77

: (V ) = : 0 (V ) ⊕ : 1 (V ) ⊕ ⋅ ⋅ ⋅ ⊕ : n (V ), which is the space of skew-symmetric tensors, where : 0 (V ) := ℝ. Denition 3.26. A symmetric mapping P : T r (V ) → S r (V ) is dened as P(I )(X1 , ⋅ ⋅ ⋅ , Xr ) =

1 (XV (1) , ⋅ ⋅ ⋅ , XV (r) ). r! V¦ ∈Sr

A skew-symmetric mapping A : T r (V ) → : r (V ) is dened as A (I )(X1 , ⋅ ⋅ ⋅ , Xr ) =

1 sgn(V )(XV (1) , ⋅ ⋅ ⋅ , XV (r) ), r! V¦ ∈Sr

where the summation is over the k-th order symmetric group Sk . Proposition 3.12. Let V be an n dimensional vector space. Then the dimension of : (V ) is dim(: (V )) = 2n .

(3.56)

Proof. Note that I ∈ T k (V ) is determined uniquely by

I (ei1 , ⋅ ⋅ ⋅ , eik ),

1 ⩽ i1 < i2 < ⋅ ⋅ ⋅ < ik ⩽ n.

In other words, {

) } ( A d i1 ⊗ ⋅ ⋅ ⋅ ⊗ d ik  1 ⩽ i1 < i2 < ⋅ ⋅ ⋅ < ik ⩽ n

is a basis of T k (V ). Hence dim(: k (V )) =

(k ) n

. The conclusion follows.

⊔ ⊓

The wedge product is particularly important in tensor analysis. Denition 3.27. Let I ∈ : r (V ) and \ ∈ : s (V ). The wedge product, ∧ : I ∧ \ , is dened as

I ∧\ =

(r + s)! A (I ⊗ \ ) ∈ : r+s (V ). r!s!

(3.57)

The following properties are immediate consequences of the denition. Proposition 3.13. 1. (Bi-linearity)

I ∧ (\1 + \2 ) = I ∧ \1 + I ∧ \2 .

(3.58)

I ∧ (\ ∧ K ) = (I ∧ \ ) ∧ K .

(3.59)

2. (Associativity)

78

3 Differentiable Manifold

3. If I ∈ : r (V ) and \ ∈ : s (V ), then

I ∧ \ = (−1)rs \ ∧ I .

(3.60)

4. Let Ii ∈ : ri (V ), i = 1, ⋅ ⋅ ⋅ , k. Then

I1 ∧ ⋅ ⋅ ⋅ ∧ Ir =

(r1 + ⋅ ⋅ ⋅ + rk )! A (I1 ⊗ ⋅ ⋅ ⋅ ⊗ Ik ). r1 !r2 ! ⋅ ⋅ ⋅ rk !

(3.61)

Next, we dene the tensor eld on a manifold. Denition 3.28. On a Ck manifold M, a Ck tensor eld I (x) ∈ Tsr (M) is a rule that assigns for each point x0 ∈ M a tensor I (x0 ) ∈ Tsr (Tx0 (M)), such that for any sets of Ck vectors X1 (x), ⋅ ⋅ ⋅ , Xr (x) and any co-vectors Z1 (x), ⋅ ⋅ ⋅ , Zs (x) the mapping

I (X1 (x), ⋅ ⋅ ⋅ , Xr (x), Z1 (x), ⋅ ⋅ ⋅ , Zs (x)) is a Ck function. Similar to tensors, for tensor elds we can dene symmetry, skew-symmetry, etc. The tensor product and wedge product can also be well dened by using point-wise denition. Locally, in an n dimensional manifold M, let a coordinate chart be (U, x). Then the natural basis of Tx (M) is { wwx ∣i = 1, ⋅ ⋅ ⋅ , n}, and its dual basis is {dxi ∣i = i 1, ⋅ ⋅ ⋅ , n}. Skew-symmetric tensor elds are important both in theory and applications. A natural basis of : k (M) is  { } dxi1 ∧ ⋅ ⋅ ⋅ ∧ dxik  1 ⩽ i1 < ⋅ ⋅ ⋅ < ik ⩽ n . A tensor eld Z ∈ : k (M) is called a k-form. It is obvious that the set of n forms has dim(: n (M)) = 1. An n-form can be used to orient the manifold. Denition 3.29. A manifold M is said to be orientable, if there is a Cr n-form Z , such that Z (p) ∕= 0, ∀p ∈ M. We refer to [2] for more about the orientation of manifolds. Finally, we dene the external differential. It is well known that for a smooth function h ∈ Cr (M) = : 0 (M), its differential is a mapping d : h $→ dh, which is an ℝ linear mapping from : 0 (M) to : 1 (M). In general, we dene an ℝ linear mapping d : Ck (M) → Ck+1 (M) as follows, which is called an external differential: Let Z = a(x)dxi1 ∧ ⋅ ⋅ ⋅ ∧ dxik , and dZ = da(x) ∧ dxi1 ∧ ⋅ ⋅ ⋅ ∧ dxik .

(3.62)

Since d is ℝ linear, the mapping is uniquely determined by (3.62). Theorem 3.6. 1. d2 Z = 0;

(3.63)

3.11 Riemannian Geometry

79

2. Let T ∈ : r (M) and Z ∈ : s (M). Then d(T ∧ Z ) = dT ∧ Z + (−1)r T ∧ dZ .

(3.64)

Proof. Without loss of generality, we assume Z has only one term, i.e., Z = a(x)dxi1 ∧ ⋅ ⋅ ⋅ ∧ dxis . Then d2 Z =

w 2 a(x) k dx ∧ dx j ∧ dxi1 ∧ ⋅ ⋅ ⋅ ∧ dxis . j k j,k∕=i1 ,⋅⋅⋅ ,is w x w x

¦

= wwxka(x) and dxk ∧ dx j = −dx j ∧ dxk , (3.63) follows immediately. To Since wwx ja(x) w xk wxj prove (3.64), let T = a(x)dxi1 ∧ ⋅ ⋅ ⋅ ∧ dxir and Z = b(x)dx j1 ∧ ⋅ ⋅ ⋅ ∧ dx js , assume 2

2

i p ∕= jq ,

p = 1, ⋅ ⋅ ⋅ , r, q = 1, ⋅ ⋅ ⋅ , s.

Otherwise, both sides of (3.64) become zero and the equality is trivial. Then

= =

Z) d(T ∧ (

) b w at + a w bt dxt ∧ dxi1 ∧ ⋅ ⋅ ⋅ ∧ dxir ∧ dx j1 ∧ ⋅ ⋅ ⋅ ∧ dx js wx wx

¦

t∕=i p , j q

¦

t∕=i p , j q

+

w a dxt ∧ dxi1 ∧ ⋅ ⋅ ⋅ ∧ dxir ∧ bdx j1 ∧ ⋅ ⋅ ⋅ ∧ dx js w xt

¦p q a wwxbt dxt ∧ dxi1 ∧ ⋅ ⋅ ⋅ ∧ dxir ∧ dx j1 ∧ ⋅ ⋅ ⋅ ∧ dx js

t∕=i , j

= bdT ∧ Z + (−1)r adxi1 ∧ ⋅ ⋅ ⋅ ∧ dxir

¦

t∕=i p , j q

w b dxt ∧ dx j1 ∧ ⋅ ⋅ ⋅ ∧ dx js w xt ⊔ ⊓

= dT ∧ Z + (−1)r T ∧ dZ .

3.11 Riemannian Geometry When r = 2, the tensor eld I (x) ∈ T 2 (M) is a quadratic form. Under a local coordinate frame it can be expressed as

I (X,Y ) = X T (x)MI (x)Y (x),

X(x),Y (x) ∈ V (M),

where MI (x) is the matrix representation of I , and its entries are ) ( w w i, j . MI (x) = I (x) , w xi w x j The tensor eld I (x) is symmetric (skew-symmetric), if and only if MI (x) is a symmetric (skew-symmetric) matrix. If I (x)(X , X) ⩾ 0, and I (X(x), X (x)) = 0, if and only if X(x) = 0, ∀x ∈ M, then I (x) is positive denite. It is obvious that I (x) is positive denite if and only if MI (x) is a positive denite matrix.

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3 Differentiable Manifold

Denition 3.30. A Cf manifold M with a symmetric tensor eld I ∈ T 2 (M) is called a pseudo-Riemannian manifold. If, in addition, I is positive denite, (M, I ) is called a Riemannian manifold. Let M be a Riemannian manifold. A smooth curve L is a mapping L : [a, b] → M. The length of L is dened as √ ( ) ∫ b dL dL I , dt. (3.65) ∣L∣ = dt dt a The Riemannian distance between two points A = L(a) and B = L(b) is dened as d(A, B) = inf {∣L∣ ∣ L(a) = A, L(b) = B} . L

(3.66)

We leave to readers to check that this is a distance [2]. Therefore, Riemannian manifold is a metric space. Example 3.12. 1. In ℝn assume I is obtained by MI = In . Then it is obvious that this is a Riemannian manifold. In fact, the distance dened by this quadratic form I is the conventional distance on ℝn . 2. In ℝ4 assume I is determined by ⎡ ⎤ c 0 0 0 ⎢ 0 −1 0 0 ⎥ ⎥ MI = ⎢ ⎣ 0 0 −1 0 ⎦ , 0 0 0 −1 where c is the speed of light. Then ℝ4 with this I is a pseudo-Riemannian manifold. It is the frame for time-space continuum in relativity [5]. Next, we consider the integral on Riemannian manifold. First, consider the integral on a manifold. Let M be an oriented manifold, and Z ∈ : n (M) be a nowhere zero n-form, which provides the orientation of M. Z is called a unit volume. Denition 3.31. Let (U, x) be a coordinate chart, i.e., I (U) ⊂ ℝn , under this coordinate, I is expressed as Z (x) = g(x)dx1 ∧ ⋅ ⋅ ⋅ ∧ dxn . Let C ⊂ I (U) be a cub. Then the integral of Z over I −1 (C) is dened as ∫ I −1 (C)

∫

Z=

g(x)dv.

(3.67)

C

Let f (x) be a bounded piece-wise continuous function. Then the integral of f over I −1 (C) is dened as ∫ I −1 (C)

fZ =

∫ C

f (x)g(x)dv.

(3.68)

3.11 Riemannian Geometry

81

To show this integral is well dened, we have to show that it is independent of the choice of the local coordinate chart. Let y = y(x) be another coordinate chart. It is known from calculus that the right hand side of (3.67) is ∫ C′

g(x(y))det(Jx (y))dv′ .

But Z is expressed under y coordinates as

Z = det(Jx (y))dy1 ∧ ⋅ ⋅ ⋅ ∧ dyn . Hence (3.67) remains true. As long as the integral is dened over a small cub, it is dened over any domain, because a domain can be split up into a number of small cubs. It is obvious that the integral over a manifold depends on the choice of unit volume. Fortunately, the following theorem ensures the existence of the unique integral over Riemannian manifold. (We refer to [2] for the proof.) Theorem 3.7. Let M be an oriented Riemannian manifold with Riemannian distance ) . Then there exists a unique tensor eld Z ∈ : n (M) such that for any orthogonal basis e1 , ⋅ ⋅ ⋅ , en , Z (e1 , ⋅ ⋅ ⋅ , en ) = 1. Moreover, Let M) (x) be the matrix form of ) under given coordinate frame x. Then Z can be expressed as

Z=

√

gdx1 ∧ ⋅ ⋅ ⋅ ∧ dxn ,

(3.69)

where g = det(M) ). This Z is called the natural unit volume. When the integral is dened on an oriented Riemannian manifold, the integral is conventionally dened with respect to this Z . Example 3.13. In ℝ3 assume the surface S is dened by z = F(x, y). On S the distance is dened as the distance inherited from ℝ3 . (Shortest curve on the surface ˜ where under the distance of ℝ3 .) Consider the integral over D, D˜ = {(x, y, F(x, y)) ∣ (x, y) ∈ D}. Denote the basis of T ∗ (S) by (E1 , E2 ). Then ⎡ 1 0 [ ] ⎢ 0 1 1 E1 = I −1 ( wwx ) = JI −1 =⎢ ⎣ 0 wF wF wx wy Similarly,

⎡

⎤ 1 ⎢ 0 ⎥ ⎥. E2 = ⎢ ⎦ ⎣ wF wy

Now the matrix of the natural unit volume is M)

⎡ ⎤ 1 [ ] ⎢ 0 ⎥ ⎥ 1 ⎢ ⎥ ⎥. ⎦ 0 =⎣ ⎦ wF wx ⎤

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3 Differentiable Manifold

⎡

⎤ ) ( wF 2 wF wF 1 + ⎢ wx w( x w y) ⎥ ⟨E1 , E1 ⟩ ⟨E1 , E2 ⟩ M) = 2 ⎥. =⎢ ⎣ wF wF ⎦ ⟨E2 , E1 ⟩ ⟨E2 , E2 ⟩ 1+ wF wy wx wy [

]

Hence √ g=

√

(

wF 1+ wx

)2

(

wF + wy

)2 .

Let h(x, y) be a function on S. Its integral over D˜ is √ ( ) ( ) ∫ wF 2 wF 2 h(x, y) 1 + + dx ∧ dy. wx wy D Next, we consider the differential on Riemannian manifolds. It is called connection. Connection is important in both geometry and physics [4]. We use semitensor product of matrices to express some fundamental formulas for the connection. (Please refer to the Appendix for semi-tensor product of matrices.) Denition 3.32. Let f , g ∈ V (M) be two vector elds. A mapping ’ : V (M) × V (M) → V (M), satisfying (i). ’r f sg = rs’ f g,

r, s ∈ ℝ;

(3.70)

(ii). ’h f g = h’ f g,

’ f (hg) = L f (h)g + h’ f g,

h ∈ Cf (M),

(3.71)

is called a connection. Under a coordinate frame x, the action of connection on a basis is expressed as ( ) n w w , ’w = ¦ Jikj wxj w xk w xi k=1 where Jikj is called the Christoffel symbol. Dene Christoffel matrix * as ⎡

1 ⋅⋅⋅ J11 ⎢ .. * =⎣ .

1 ⋅⋅⋅ J1n .. .

1 ⋅⋅⋅ Jn1 .. .

⎤ 1 Jnn .. ⎥ . . ⎦

n ⋅ ⋅ ⋅ Jn ⋅ ⋅ ⋅ Jn ⋅ ⋅ ⋅ Jn J11 nn 1n n1

The connection on two vector elds can be expressed in matrix form as: n

Proposition 3.14. Let f = ¦ fi wwxi and g = i=1

n

¦ g j wwx j . Then

j=1

’ f g = Dg ⋉ f + * ⋉ f ⋉ g.

(3.72)

3.11 Riemannian Geometry

Proof. By denition (3.70) and (3.71), we have [ ] n n n n w w k + ¦ ¦ g j Ji j ’ f g = ¦ fi ¦ L w g j w x j j=1 w xk i=1 j=1 w xi k=1

83

(3.73)

= Dg ⋉ f + * ⋉ f ⋉ g. Note that the vector elds in (3.73) are expressed in vector form. That is, f = n

¦ fi wwxi is simply expressed as f = ( f1 , f2 , ⋅ ⋅ ⋅ , fn )T.

⊔ ⊓

i=1

Let y = y(x) be another coordinate frame. We want to express * in this new coordinate frame. Denote by *˜ and J˜ikj the matrix and its entries under new coordinate frame. Then Lemma 3.4. Under the new coordinate frame y we have ⎞ ⎛ 2 w x1 ⋅ ⋅ ⋅ w 2 x1 ⎛ 1⎞ J˜i j w y j w yn ⎟ ( ⎜ w y j w y1 ) ⎜ .. ⎟ ⎜ w xn T . .. ⎟ ⎟ w x1 ⎜ . ⎝.⎠=⎜ . . ⎟ w yi , ⋅ ⋅ ⋅ , w yi ⎠ ⎝ 2 J˜inj w xn ⋅ ⋅ ⋅ w 2 xn w y j w y1 w y j w yn )T ( ) ( wx wx wx T +* ⋉ w x1 , ⋅ ⋅ ⋅ , n ⋉ w y1 , ⋅ ⋅ ⋅ , w yn . w yi w yi j j Proof. f=

n w w w xs =¦ , w yi s=1 w xs w yi

g=

n w w w xt =¦ . w y j t=1 w xt w y j

(3.74)

Recalling the denition of J , one sees that n

w

¦ J˜ikj w yk = ’ f g.

k=1

⊔ ⊓

Applying (3.72) to the above equation yields (3.74). Theorem 3.8. Under the new coordinate frame y, *˜ has the form

*˜ = D2 xDx + * ⋉ Dx(I ⊗ Dx). Proof. A straightforward computation shows that ⎛ n w 2 x1 w xs ⎜ ¦ w ys w y1 w y1 ⋅ ⋅ ⋅ ⎜ s=1 ⎜ .. D2 x ⋉ Dx = ⎜ . ⎜ ⎜ n ⎝ w 2 xn w xs ⋅ ⋅ ⋅ ¦ w ys w y1 w y1 s=1

n

(3.75)

¦ wwys wx1yn ww yx1s

s=1

2

.. . n w 2 xn w xs ¦ w ys w yn w y1 s=1

84

3 Differentiable Manifold n

2 ⋅ ⋅ ⋅ ¦ w x1 w xs ⋅ ⋅ ⋅ w y s w y1 w yn s=1 .. . n 2 ⋅ ⋅ ⋅ ¦ w xn w xs ⋅ ⋅ ⋅ w ys w y1 w yn s=1

n

2 ¦ wwys wx1yn ww yxns s=1 .. . n 2 ¦ wwys wxnyn ww yxns s=1

⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠

Using (i j) to label its columns, then its (i j)-th column is the rst term of the right hand side of (3.74). Next, denote the i-th column of Dx by Ji , then

* ⋉ Dx = (* ⋉ J1 , * ⋉ J2 , ⋅ ⋅ ⋅ , * ⋉ Jn ). Since I ⊗ Dx = diag(J, ⋅ ⋅ ⋅ , J), we have

* ⋉ Dx ⋉ (I ⊗ Dx) = (* ⋉ J1 ⋉ J1 , ⋅ ⋅ ⋅ , * ⋉ J1 ⋉ Jn , ⋅ ⋅ ⋅ , * ⋉ Jn ⋉ J1, ⋅ ⋅ ⋅ , * ⋉ Jn ⋉ Jn ) . It is clear that the (i j)-column of the above form is the second term of the right hand side of (3.74). ⊔ ⊓ Let M be a Riemannian manifold, with Riemannian distance determined by G = (gi j )n×n . The fundamental theorem of the Riemannian geometry says that there exists a unique Riemannian connection on M [1]. Moreover, its Christoffel symbols are determined by ( ) 1 n ks w gsi w gi j w g js k , (3.76) Ji j = ¦ g − + 2 s=1 wxj w xs w xi where gi j is the (i, j)-th element of G−1 . It is well known that [4] Proposition 3.15. There exists a unique Riemannian connection satisfying (i). [ f , g] = ’ f g − ’g f ,

(3.77)

LX (Z (Y, Z)) = Z (’X Y, Z) + Z (Y, ’X Z).

(3.78)

(ii). for X ,Y, Z ∈ V (M)

The Christoffel matrix is said to be symmetric if

Jikj = J kji ,

∀i, j, k.

(3.79)

In fact, we have Theorem 3.9. If there exists a connection on N that has symmetric Christoffel matrix, then (3.77) holds. Proof. It is easily veried that if the Christoffel matrix * is symmetric, then

3.12 Symplectic Geometry

* ⋉ f ⋉g = * ⋉g⋉ f,

85

∀ f , g ∈ V (N).

Using (3.72) yields ’ f g − ’g f = Dg f − D f g = [ f , g].

⊔ ⊓

Equation (3.76) shows that since for a Riemannian manifold there exists the symmetric Christoffel matrix, we have (3.77). Now Theorem 3.9 says that it is enough if the Christoffel matrix of a manifold is symmetric. A related problem is the geodesic. The equation of a geodesic can also be expressed in matrix form [1]. A curve, r(t), is called a geodesic on M, if and only if it satises the following equation: r¨t = * ⋉ r2 .

(3.80)

In a local coordinate chart, r(t) can be expressed as r(t) = (x1 (t), ⋅ ⋅ ⋅ , xn (t))T . Then the equation (3.80) can be expressed into matrix form as ⎡

⎤ ⎡ ⎤2 x¨1 x1 ⎢ .. ⎥ ⎢ .. ⎥ ⎣ . ⎦=* ⎣ . ⎦ . x¨n

xn

3.12 Symplectic Geometry Denition 3.33. A 2n dimensional Cf manifold M with a 2-form Z is called a symplectic geometry, if Z is skew-symmetric, nonsingular and closed. A k-form Z is said to be closed if dZ = 0. The following method can be used to check whether a 2-form is closed. Proposition 3.16. Let the matrix expression of the Z under a given coordinate chart be MZ = (Vi, j ). Then dZ = 0 if and only if

w w w (V jk ) + (Vki ) + (Vi j ) = 0, w xi wxj w xk

∀i, j, k.

(3.81)

Proof. Since V = ¦ ¦ Vi, j dxi ∧ dx j , then i

j

dV = ¦ ¦ ¦ k

i

j

w Vi, j k dx ∧ dxi ∧ dx j . w xk

For a given index group I, J, K we collect the terms of the form dxI ∧ dxJ ∧ dxK . Let k = I, i = J, j = K. Then we have

w VJK I dx ∧ dxJ ∧ dxK . w xI

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3 Differentiable Manifold

Let k = I, i = K, and j = J. By skew-symmetry, we have

w VKJ I w VJK I w VJK I dx ∧ dxK ∧ dxJ = − dx ∧ dxK ∧ dxJ = dx ∧ dxJ ∧ dxK . w xI w xI w xI Similarly, for k = K, i = I, and j = J (or i = J, and j = I) we have

w VIJ I dx ∧ dxJ ∧ dxK . w xK For k = J, i = K, and j = I (or i = I, and j = K) we have

w VKI I dx ∧ dxJ ∧ dxK . w xJ Collecting the 6 terms yields that the coefcient of dxI ∧ dxJ ∧ dxK is 2

w w w (VJK ) + 2 (VKI ) + 2 (VIJ ) w xI w xJ w xK ⊔ ⊓

which is zero. The conclusion follows.

Example 3.14. In ℝ2n , dene a 2-form Z with its matrix form MZ = J, where ] [ 0 In , (3.82) J= −In 0 then (ℝ2n , Z ) is a symplectic manifold. In fact, equation (3.82) has a general meaning. The following theorem is a fundamental result. We refer to [8] for proof. Theorem 3.10. (Darboux) On a 2n dimensional manifold, a 2-form is a symplectic structure, if and only if for any point p ∈ M there is a local coordinate chart (U, (x, y)) ∋ p such that Z can be expressed locally as n

Z = ¦ dxi ∧ dyi . i=1

Denition 3.34. Let (M, Z ) be a symplectic manifold, and X ∈ V (M) and P ∈ V ∗ (M) be connected by iX (Z ) := Z (X, ⋅) = P . Dene two diffeomorphisms ♯ : V ∗ (M) → V (M) and ♭ : V (M) → V ∗ (M) as X = P ♯,

and P = X ♭ .

(3.83)

If there exists a function H such that X = (dH)♯ , then H is called a Hamiltonian manifold, and X is called the Hamiltonian vector eld deduced by H, denoted by X = XH . Let M and N be two manifolds and F : M → N a diffeomorphism, Z ∈ : k (N). Then a deduced mapping F ∗ : : k (N) → : k (M) is dened by

3.12 Symplectic Geometry

F ∗ (Z )(X1 , ⋅ ⋅ ⋅ , Xk ) = Z (F∗ (X1 ), ⋅ ⋅ ⋅ , F∗ (Xk )),

X1 , ⋅ ⋅ ⋅ , Xk ∈ V (M).

87

(3.84)

The following theorem is sometimes called the Principle of structure invariance. Theorem 3.11. (Liouville’s Theorem) Let (M, Z ) be a symplectic manifold and XH a Hamiltonian vector eld. Then (etXH )∗ (Z (x)) = Z (etXH (x)).

(3.85)

We refer to [1] for the proof. Since symplectic manifolds can only be used for a dynamics with even dimension, a more general geometric structure is necessary. The Poisson manifold is introduced to serve this purpose. Denition 3.35. A Poisson bracket on a manifold M is a smooth mapping: Cr (M) × Cr (M) → Cr (M), satisfying the following (i). (Bi-linearity) {cF + dP, H} = c{F, H} + d{P, H}, {F, cH + dP} = c{F, H} + d{F, P}, where c, d ∈ ℝ; (ii). (Leibniz Rule) {F, GH} = {F, G}H + G{F, H}; (iii). (Skew-symmetry) {F, H} = −{H, F}; (iv). (Jacobi Identity) {F, {G, H}} + {G, {H, F}} + {H, {F, G}} = 0. A manifold M with a Poisson bracket is called a Poisson manifold. The Poisson bracket is said to dene a Poisson structure on M. To show that a Poisson manifold is more general than a symplectic manifold we give the following. Proposition 3.17. Let (M, Z ) be a symplectic manifold. Then there exists a Poisson bracket such that M with this bracket becomes a Poisson manifold. Proof. For any two functions f , g ∈ Cr (M) dene a Poisson bracket as { f , g} = Z (X f , Xg ),

∀ f , g ∈ Cr (M).

It is easy to verify that this is a Poisson bracket.

(3.86) ⊔ ⊓

Denition 3.36. Let M be a Poisson manifold. A smooth real function C : M → ℝ is called a Casimir function, if the Poisson bracket of C with any function is zero. That is, {C, H} = 0, ∀H ∈ Cr (M). To test the stability of a dynamic system, we may search a Hamiltonian function as a candidate for Lyapunov function. In this case Casimir functions provide us more freedom, because for a Casimir function C, we always have XH = XH+C .

88

3 Differentiable Manifold

Denition 3.37. Let M be a Poisson manifold, H : M → ℝ a smooth function. The vector eld VH , generated by H, is the unique vector eld on M satisfying VH (F) = {F, H} = −{H, F},

∀F : M → ℝ.

Proposition 3.18. Let M be a Poisson manifold, F, H ∈ Cr (M) with their generated Hamiltonian vector elds VF , VH , respectively. Then the Hamiltonian vector eld generated by their Poisson bracket {F, H} is V{F,H} = −[VF ,VH ] = [VH ,VF ]. Let x = (x1 , ⋅ ⋅ ⋅ , xm ) be a local coordinate frame on M. The Poisson bracket of basis elements are denoted by J i j (x) = {xi , x j },

i, j = 1, ⋅ ⋅ ⋅ , m.

They are called the structure functions of the Poisson manifold M with respect to the coordinate frame. The structure functions uniquely determine the structure of a Poisson manifold. For convenience, they are arranged as an m × m matrix J(x), called the structure matrix of the manifold. Using the structure matrix, the Poisson bracket can be expressed locally as {F, H} = dFJ(x)’H(x). Proposition 3.19. Let J(x) = (J i j (x)) be an m × m matrix of functions of x = (x1 , ⋅ ⋅ ⋅ , xm ) dened on an open set M ⊂ ℝm . Then J(x) with {F, H} = dFJ’H is a Poisson bracket on M, if and only if it satises (i). (Skew-symmetry) J i j (x) = −J ji (x),

i, j = 1, ⋅ ⋅ ⋅ , m.

(ii). (Jacobi Identity) m

¦ {J il (x)wl J jk (x) + J kl (x)wl J i j (x) + J jl (x)wl J ki (x)} = 0,

l=1

i, j, k = 1, ⋅ ⋅ ⋅ , m, x ∈ M. Proposition 3.20. Let M be a Poisson manifold, x ∈ M. Then there exists a unique linear mapping S = S ∣x : T ∗ M∣x → T M∣x , such that for any real function H ∈ Cr (M),

S (dH(x)) = VH ∣x . Denition 3.38. Let M be a Poisson manifold, x ∈ M. The rank of M at x is dened as the rank of the linear mapping S ∣x : T ∗ M∣x → T M∣x . Proposition 3.21. The rank of a Poisson manifold at any point is even.

References

89

Denition 3.39. Let M and N be two Poisson manifolds. A mapping I : M → N is called a Poisson mapping, if it keeps the bracket unchanged. That is, {I ∗ (F), I ∗ (H)}M = I ∗ {F, H}N ,

F, H ∈ Cr (N).

Proposition 3.22. Let M be a Poisson manifold XH be a Hamiltonian eld. Then for each t, etXH : M → M determined a Poisson mapping from M to itself. Corollary 3.2. Let XH be a Hamiltonian vector eld on a Poisson manifold M. Then the rank of M at etXH (x) for any t ∈ ℝ equals to the rank of M at x. The following is a generalized Darboux’ theorem. Theorem 3.12. ((Generalized) Darboux’ Theorem) Let M be an m dimensional Poisson manifold with constant rank 2n ⩽ m. Then at any point x0 ∈ M there is a local coordinate chart (p, q, z) = (p1 , ⋅ ⋅ ⋅ , pn , q1 , ⋅ ⋅ ⋅ , qn , z1 , ⋅ ⋅ ⋅ , zℓ ),

2n + ℓ = m,

such that the Poisson bracket can be expressed as ) n ( wF wH wF wH {F, H} = ¦ − i i , i i w p wq i=1 w q w p where z1 , ⋅ ⋅ ⋅ , zℓ are Casimir functions.

References 1. Abraham R, Marsden J. Foundations of Mechanics. Reading: Benjamin/Cummings Pub. Com. Inc., 1978. 2. Boothby W. An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd edn. Orlando: Academic Press, 1986. 3. Brockett R. Asymptotic stability and feedback stabilization//Brockett R, Millmann R, Sussmann H. Differential Geometric Control Theory, Progress in Mathematics. Boston: Birkhauser, 1983. 4. Chern S, Chen W. Lectures on Differential Geometry. Beijing: Beijing Univ. Press, 1983. 5. Clarke C. Elementary General Relativity. New York: John Wiley & Sons, 1979. 6. Husemoller D. Fibre Bundles, 2nd edn. New York: Springer-Verlag, 1966. 7. Isidori A. Nonlinear Control Systems, 3rd edn. London: Springer, 1995. 8. Olver P. Applications of Lei Groups to Differential Equations, 2nd edn. New York: SpringerVerlag, 1993. 9. Spivak M. A Comprehensive Introduction to Differential Geometry. Berkeley: Publish or Perish Inc., 1979. 10. Sussmann H. Orbits of families of vector elds and integrability of distributions. Trans. American Math. Soc., 1973, 180: 171–188. 11. Wonham W. Linear Multivariable Control: A Geometric Aproach, 2nd edn. Berlin: Springer, 1979.

Chapter 4

Algebra, Lie Group and Lie Algebra

Geometry, algebra, and analysis are usually called the three main branches of mathematics. This chapter introduces some fundamental results in algebra that are mostly useful in systems and control. In section 4.1 some basic concepts of group and three homomorphism theorems are discussed. Ring and algebra are introduced briey in section 4.2. As a tool, homotopy is investigated in section 4.3. Sections 4.4 and 4.5 contain some primary knowledge about algebraic topology, such as fundamental group, covering space etc. In sections 4.6 and 4.7, Lie group and its Lie algebra are discussed. Section 4.8 considers the structure of Lie algebra.

4.1 Group Denition 4.1. A set G with an operator ⊗ : G × G → G is called a group if the followings are satised: (i). (a ⊗ b) ⊗ c = a ⊗ (b ⊗ c), a, b, c ∈ G. (associative) (ii). There exists an element, e, such that g ⊗ e = e ⊗ g = g,

∀ g ∈ G.

e is called the identity. (iii). For every g ∈ G, there exists an element g−1 , called the inverse of g, such that g ⊗ g−1 = g−1 ⊗ g = e. If in addition we have a ⊗ b = b ⊗ a,

∀ a, b ∈ G,

G is called an Abelian group. Example 4.1. 1. ℂ (set of complex numbers), ℝ (set of real numbers), ℚ (set of rational numbers), ℤ (set of integers) with the conventional addition, “+”, are groups. 2. ℂ∖{0}, ℝ∖{0}, ℚ∖{0} with the conventional multiplication, “×”, are groups. 3. ℤ with the conventional multiplication, “×”, is not a group because ±1 ∕= z ∈ ℤ does not have an inverse in ℤ. 4. Let gl(n, ℂ) (gl(n, ℝ)) be the set of n × n complex (real) matrices. With the conventional matrix addition gl(n, ℂ) (gl(n, ℝ)) is a group. D. Cheng et al., Analysis and Design of Nonlinear Control Systems © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2010

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4 Algebra, Lie Group and Lie Algebra

5. Let GL(n, ℂ) (GL(n, ℝ)) be the set of n × n invertible complex (real) matrices. With the conventional matrix multiplication GL(n, ℂ) (GL(n, ℝ)) is a group. However, this group is not Abelian. Example 4.2. Let ℤn = {0, 1, ⋅ ⋅ ⋅ , n − 1} with operator ⊕ be dened as a ⊕ b := a + b (mod(n)). Then ℤn is a group, called the n-th order circular group. The identity of this group is 0, and the inverse of 0 ∕= z ∈ ℤn is n − z. For instance, in ℤ5 : 3 ⊕ 4 = 2; 4−1 = 1 etc. Example 4.3. Let Gn = {1, 2, ⋅ ⋅ ⋅ , n}. All the permutations on Gn with compound of permutations, as the operator, form a group called the n-th order symmetric group, denoted by Sn . (In fact, it has been used several times before.) For instance, consider G5 : A permutation of it can be expressed as V1 = (1, 2, 3) (4, 5) ∈ Sn , which performs the following permutation: 1 → 2, 2 → 3, 3 → 1, and 4 → 5, 5 → 4. Let another permutation be V2 = (3, 2, 1, 4) ∈ Sn , which performs the following permutation: 3 → 2, 2 → 1, 1 → 4, 4 → 3, and 5 → 5. Now V1 ⊗ V2 is dened as 1 → 2 → 1, 2 → 3 → 2, 3 → 1 → 4, 4 → 5 → 5, 5 → 4 → 3. Putting them together yields V1 ⊗ V2 = (3, 4, 5). This group is not Abelian. Denition 4.2. 1. Let (G, ⊗) be a group, and H ⊂ G. H is called a subgroup of G, denoted by H < G, if (H, ⊗) is also a group. 2. Let H < G, g ∈ G. The set g ⊗ H := {g ⊗ h ∣ h ∈ H} is called the left g coset of H. Similarly H ⊗ g := {h ⊗ g ∣ h ∈ H} is called the right g coset of H. 3. H < G is called a normal subgroup of G, denoted by H ⊲ G, if for any g ∈ G g ⊗ H = H ⊗ g. When there is no risk of confusion, we may omit ⊗. Then a ⊗ b is simply written as ab. Proposition 4.1. ∅ ∕= H ⊂ G is a subgroup of G, if and only if for any two elements h1 , h2 ∈ H, h−1 1 h2 ∈ H. Proof. Necessity is trivial. To prove sufciency, we have to show that H satises (i)–(iii). Choose h ∈ H, then h−1 h = e ∈ H. Hence e ∈ H, which implies (ii). Next, for any h ∈ H, h−1 = h−1 e ∈ H, which is (iii). To see (i) we have only to show that when

4.1 Group

93

−1 −1 h1 , h2 ∈ H, h1 h2 ∈ H . This is true because h1 ∈ H ⇒ h−1 1 ∈ H ⇒ (h1 ) h2 = h1 h2 ∈ H. ⊔ ⊓

Proposition 4.2. Left (right) cosets form a partition, i.e., if g1 H ∩ g2 H ∕= ∅ (Hg1 ∩ Hg2 ∕= ∅) then g1 H = g2 H (Hg1 = Hg2 ). Proof. Assume g1 H ∩ g2 H ∕= ∅, and choose c ∈ g1 H ∩ g2 H. Then c = g1 h1 = g2 h2 for some hi ∈ H, i = 1, 2. For g ∈ g1 H there is a h ∈ H such that g = g1 h. Then −1 g = g1 h = g1 h1 h−1 1 h = g2 h2 h1 h ∈ g2 H.

so g1 H ⊂ g2 H. Similarly, we can prove g2 H ⊂ g1 H. So g1 H = g2 H.

⊔ ⊓

For any group G, it has two trivial subgroups: (i). {e}, a single element group; (ii). G itself, which is called its non-proper subgroup. Both of them are normal. For an Abelian group, every subgroup is normal. Let G be a group, and S ⊂ G. The smallest subgroup H containing S is called the subgroup generated by S. It is easy to see that such a group exists and is unique, because it is the intersection of all subgroups that contain S. −1 Example 4.4. Given a group G. An element g = g−1 1 g2 g1 g2 is called a commutator of G. The group generated by the commutators is called the commutator group, denoted by G′ . It is easy to check that G′ consists of all nite products of the commutators: { }

G′ =

−1 – a−1 i b i ai bi ∣ a i , b i ∈ G

.

i
In fact, G′ ⊲ G. To see this it sufces to show that ( ) g−1

−1 – a−1 i b i a i bi

g ∈ G′ ,

∀g ∈ G.

i
By plugging g−1 g into commutators, one easily sees that it is enough to show g−1 a−1 b−1 abg ∈ G′ . Note that g−1 (a−1 b−1 ab)g = (ag)−1 b−1 (ag)b b−1 g−1 bg ∈ G′ , which completes the proof. Let H ⊲ G. Then gH = Hg for all g ∈ G. Now we dene an operator ⊗ over the cosets as aH ⊗ bH := abH.

(4.1)

Since representation of the cosets is not unique, we have to show that (4.1) is well dened. That is, if a1 H = a2 H and b1 H = b2 H then a1 b1 H = a2 b2 H. Let x = a1 b1 h ∈ a1 b1 H. Since a1 H = a2 H, there exist h1 , h2 ∈ H such that a1 h1 = a2 h2 . i.e., a1 = a2 h2 h−1 1 := a2 h3 for some h3 ∈ H. Similarly, b1 = b2 h4 for some h4 ∈ H. Then x ∈ a2 h3 b2 h4 H. Since H is normal h3 b2 = b2 h5 for some h5 ∈ H.

94

4 Algebra, Lie Group and Lie Algebra

Hence x ∈ a2 b2 h5 h4 H = a2 b2 H. That is, a1 b1 H ⊂ a2 b2 H. By symmetry, we also have a2 b2 H ⊂ a1 b1 H. It is ready to verify that the set of cosets with operator dened in (4.1) form a group. So we have Denition 4.3. Let H ⊲ G. The cosets of H with the operator dened in (4.1) form a group, which is called the quotient group of H in G. Denoted by G/H. Denition 4.4. 1. Let G and Q be two groups. A mapping F : G → Q is called a homomorphism if F(g1 g2 ) = F(g1 )F(g2 ).

(4.2)

The set of homomorphisms from G to Q is denoted by Hom(G, Q). Hence F ∈ Hom(G, Q). 2. If a homomorphism is a bijective mapping, i.e., it is one to one and onto, then it is called an isomorphism. The set of isomorphisms is denoted by Iso(G, Q). 3. If F ∈ Hom(G, G), i.e., a homomorphism from a group G to itself, it is called an endomorphism. The set of endomorphisms is denoted by End(G). 4. If F ∈ Iso(G, G), i.e., an isomorphism from a group G to itself, it is called an automorphism. The set of automorphisms is denoted by Aut(G). Example 4.5. 1. Denote by ℝ+ the set of positive real numbers. With the conventional multiplication, “ ⋅ ”, both G = (ℝ∖{0}, ⋅) and H = (ℝ+ , ⋅) are groups. Moreover, H ⊲ G and G/H = {H, −H}. If we dene F : G → H by x $→ ∣x∣, then F ∈ Hom(G, H). 2. Let W = (ℝ, +), i.e., W is the group of real numbers with the conventional addition, and H is as above. Dene F : H → W by F(x) = log(x). Then F ∈ Iso(H,W ). 3. Consider F : GL(n, ℝ) → G, (refer to Example 4.1 for GL(n, ℝ)) G is as in 1, and F is dened as F(X ) = det(X ). Then F ∈ Hom(GL(n, ℝ), G). 4. Let Z ∈ GL(n, ℝ). Dene F : GL(n, ℝ) → GL(n, ℝ) as F(X) = Z −1 XZ. Then F ∈ Aut(GL(n, ℝ)). All the above statements are easily veriable. Let F ∈ Hom(G, Q), the image of F is dened as im(F) = {F(g) ∣ g ∈ G} ⊂ Q, and the kernel of F is dened as ker(F) = {g ∈ G ∣ F(g) = e ∈ Q}. Then we have Proposition 4.3. 1. im(F) < Q; 2. ker(F) ⊲ G.

4.1 Group

95

Proof. 1. Let x, y ∈ im(F). Then there exist g, h ∈ G such that F(g) = x and F(h) = y. So x−1 y = (F(g))−1 F(h) = F(g−1 )F(h) = F(g−1 h) ∈ im(F). So im(F) < Q. 2. It is easy to see that ker(F) < G. To prove it is normal, let g ∈ G, and k ∈ ker(F). We have only to show that g−1 kg ∈ ker(F). F(g−1 kg) = F(g−1 )F(k)F (g) = F(g−1 )F(g) = F(e) = e, which means g−1 kg ∈ ker(F). Hence, ker(F) ⊲ G.

⊔ ⊓

Theorem 4.1. (First Homomorphism Theorem) 1. Let F ∈ Hom(G, Q). K = ker(F). Then T : G/K → im(F), dened as gK $→ F(g), is an isomorphism, i.e., G/ ker(F) ∼ = im(F). 2. If H ⊲ G, then S : G → G/H dened by g $→ gH is a homomorphism, which is surjective (onto). Moreover, ker(S ) = H. Proof. 1. We rst prove that T is well dened: Assume gK = hK. Then g−1 h ∈ K. Hence

T (hK) = T (gg−1hK) = F(gg−1 h) = F(g)F(g−1 h) = F(g) = T (gK). Next, we have

T (ghK) = F(gh) = F(g)F(h) = T (gK)T (hK). Hence, T is a homomorphism. T is obviously surjective. To see T is an isomorphism we have only to show that T is injective (one to one). Assume T (gK) = F(g) = e. Then g ∈ K, and hence gK = K, which means T is injective. 2. Using the same argument as in the proof of 1, it is easily seen that S is a homomorphism and onto. Now if g ∈ H then S (g) = gH = H, and if g ∕∈ H then S (g) = gH ∕= H. So ker(S ) = H. ⊔ ⊓ Theorem 4.2. (Second Homomorphism Theorem) 1. Let H < G, and N ⊲ G. Then N ∩ H ⊲ H. 2. Set NH = {nh ∣ n ∈ N, h ∈ H}. Then NH < G. 3. For h ∈ H dening S : H/(N ∩ H) → NH/N as h(N ∩ H) $→ hN, then S is an isomorphism, i.e., H/(H ∩ N) ∼ = NH/N.

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Proof. 1. Let h ∈ H and n ∈ N ∩ H. We have only to show that h−1 nh ∈ N ∩ H. It is obvious that h−1 nh ∈ H. Since n ∈ N, which is normal, hn = n′ h for some n′ ∈ N. Hence h−1 nh = n′ ∈ N. 2. Let n1 , n2 ∈ N and h1 , h2 ∈ H, we have only to show that (n1 h1 )−1 n2 h2 ∈ NH. Since −1 −1 −1 (n1 h1 )−1 n2 h2 = h−1 1 n1 n2 h2 = h1 n3 h2 = n4 h1 h2 ∈ NH, −1 −1 where n3 , n4 ∈ N with n3 = n−1 1 n2 and h1 n3 = n4 h1 . (Since N is normal such n4 exists.) 3. First, we show S is a homomorphism:

S (x(N ∩ H)y(N ∩ H)) = S (xy(N ∩ H)) = xyN = (xN)(yN) = S (x(N ∩ H))S (y(N ∩ H)), x, y ∈ H. Next, we show S is surjective. For any x = nh ∈ NH we have xN = nhN = hn′ N = hN, for some n′ ∈ N. Then

S (h(N ∩ H)) = hN = xN. Finally, we prove S is injective: Let x ∈ H and assume

S (x(N ∩ H)) = xN = N. Then x ∈ N, i.e., x ∈ N ∩ H.

⊔ ⊓

Theorem 4.3. (Third Homomorphism Theorem) Assume M ⊲ G, N ⊲ G and N < M. Then 1. M/N ⊲ G/N; 2. (G/N)/(M/N) ∼ = G/M. Proof. 1. M/N < G/N is obvious. We prove it is normal: Let g ∈ G, and m ∈ M. Then (gN)−1 (mN)(gN) = (g−1 mg)N = m′ N ∈ M/N, where m′ ∈ M. So M/N ⊲ G/N. 2. Dene S : (G/N)/(M/N) → G/M by gN(M/N) $→ gM. We rst show that S is well dened. Let g1 N(M/N) = g2 N(M/N). Then (g1 N)−1 (g2 N) ∈ M/N. That −1 is, (g−1 1 g2 )N ∈ M/N, which means g1 g2 ∈ M. Equivalently, g1 M = g2 M. Then

S (g1 N(M/N)) = g1 M = g2 M = S (g2 N(M/N)). Next, we prove S is a homomorphism:

S ((g1 N)(g2 N)(M/N)) = S ((g1 g2 N)(M/N)) = (g1 M)(g2 M) = g1 g2 M = S ((g1 N)(M/N))S ((g2 N)(M/N)).

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97

It is obvious that S is surjective, so we have only to show that S is injective. Assume S (gN(M/N)) = gM = M. Then g ∈ M, i.e., gN ∈ M/N, which means S is injective.

⊔ ⊓

Example 4.6. 1. Let G, Q be two groups. On G × Q dene an operator ⊗ as (g1 , q1 ) ⊗ (g1 , q2 ) := (g1 g2 , q1 q2 ), with g1 , g2 ∈ G and q1 , q2 ∈ Q. Then G× Q becomes a group, which is called the product group of G and Q. If we identify G × {e} with G. Then it is easy to check that G ⊲ G × Q. Similarly, if we identify Q with {e} × Q, Q ⊲ G × Q. 2. The center Z(G) of a given group G is dened as Z(G) = {x ∈ G ∣ xy = yx, ∀y ∈ G}. It is easy to see that Z(G) ⊲ G. 3. Let Q < G. Then for any g ∈ G, g−1 Qg < G. Q and g−1 Qg are called two conjugated subgroups.

4.2 Ring and Algebra Denition 4.5. A set R with two operators : addition, “+”, and multiplication, “⋅”, is called a ring if they satisfy the following conditions: (i). (R, +) is an Abelian group; (ii). The multiplication is associative, i.e., for any a, b, c ∈ ℝ (ab)c = a(bc). (iii). (Distributive Rule) (a + b)c = ac + bc,

a(b + c) = ab + ac.

Example 4.7. 1. The set of integers, ℤ, with the conventional addition and multiplication is a ring. 2. The set of polynomials over a eld, F, i.e., F = ℝ, F = ℂ etc., is a ring, denoted as F(x). 3. Set of n × n matrices with the conventional addition and multiplication is a ring, denoted by Mn . 4. (Boolean Algebra) Consider B = {0, 1}, we dene two operators as: conjunction ∧, and disjunction ∨ as: a ∧ b = min{a, b},

a ∨ b = max{a, b}.

(a) Consider ∨ as the addition + and ∧ as the multiplication, then (B, ∨, ∧) is a ring. (b) Consider ∧ as the addition + and ∨ as the multiplication, then (B, ∧, ∨) is also a ring.

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Denition 4.6. 1. A ring R is called a commutative ring if ab = ba,

∀a, b ∈ R.

2. An element, e ∈ R, is called the identity of ring R, if re = er = r, ∀r ∈ R. If R has identity, it is called a ring with identity. 3. If 0 ∕= a ∈ R and 0 ∕= b ∈ R and ab = 0, then a is called a left zero factor and b is called a right zero factor. 4. A ring with identity and without zero factor is called an integral ring. Example 4.8. In Example 4.7, ℤ and ℝ[x] (or ℂ[x]) are commutative integral rings. Mn is not commutative. It has identity, In . It has zero factors: say, in M2 ] ] [ [ 10 01 . , B= A= 00 00 Both A ∕= 0 and B ∕= 0, but AB = 0. So A is a left zero factor and B is a right zero factor. Denition 4.7. Let S ⊂ R. If S is a ring with the same operators as in R, S is called a subring of R. Proposition 4.4. Let S ⊂ R. S is a subring if (i). For any a, b ∈ S, a + b ∈ S; (ii). For any a, b ∈ S, ab ∈ S. Proof. (i) makes S with addition a group. (ii) makes it closed under multiplication. The conclusion follows. ⊔ ⊓ Denition 4.8. 1. A subring S ⊂ R is called an ideal if rs ∈ S,

sr ∈ S,

∀r ∈ R, s ∈ S.

2. Given an element r ∈ R, the smallest ideal containing r is called a principal ideal, denoted by S = (r). Remark 4.1. It is easy to see that the intersection of any ideals is an ideal. So the principal ideal, (r), is the intersection of all ideals containing r. (r) is also called the ideal generated by r. Example 4.9. 1. In ℤ n dene ⊗ as

a ⊗ b = ab (mod(n)).

Then it is easy to check that ℤn with ⊕ and ⊗ is a commutative ring, with identity 1. If n is a prime number, it is an integral ring.

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99

2. Consider ℤ6 . It is not an integral ring because 2 ⊗ 3 = 0. S = {2, 4, 0} is a subring of it. S is an ideal. In fact, S = (2) is a principal ideal. Denition 4.9. Let R1 , R2 be two rings. F : R1 → R2 is called a ring homomorphism if in addition to a group homomorphism it also satises the following F(ab) = F(a)F(b). A ring homomorphism is a ring isomorphism if it is bijective. Example 4.10. 1. Let F : ℤ → ℤn be dened as z $→ z (mod(n)). It is easy to check that F is a ring homomorphism. 2. Recall the Boolean algebra dened in item 4 of Example 4.7. In Boolean set B dene a mapping ¬ : B → B as ¬(a) = 1 − a. Then it is easy to check that ¬ : (B, ∨, ∧) → (B, ∧, ∨) is a ring isomorphism. Denition 4.10. Let V be a linear space over eld F (In most cases we consider F = ℝ). V is called an algebra if there is a multiplication ⋅ : V × V → V , such that (i). (aX1 + bX2) ⋅Y = aX1 ⋅Y + bX2 ⋅Y , (ii). Y ⋅ (aX1 + bX2 ) = aY ⋅ X1 + bY ⋅ X2 , X1 , X2 ,Y ∈ V and a, b ∈ F. Moreover, if (X ⋅Y ) ⋅ Z = X ⋅ (Y ⋅ Z), ∀X ,Y, Z ∈ V, then it is called an associative algebra. If X ⋅Y = Y ⋅ X,

∀X ,Y ∈ V,

it is called a commutative algebra. Remark 4.2. An associative algebra is obviously a ring. So the structure of an algebra is almost parallel to that of the ring. We outline them briey. Denition 4.11. Given two algebras V, W . 1. A subspace M ⊂ V is called a sub-algebra of V , if with the same multiplication M is an algebra. 2. A sub-algebra M of V is called an ideal of V if XZ ∈ M,

ZY ∈ M,

∀X ,Y ∈ V, Z ∈ M.

3. A linear mapping F : V → W is called an algebra homomorphism if F(X ⋅Y ) = F(X ) ⋅ F(Y ). 4. A homomorphism F : V → W is called an algebra isomorphism if it is bijective.

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Example 4.11. 1. Let Mn be the set of n × n matrices. Under the conventional matrix addition, multiplication and scalar multiplication it is an algebra. This algebra is associative, not commutative, and with identity In . 2. Let Un ⊂ Mn be the set of upper triangular matrices. Then it is easy to check that Un is a subalgebra of Mn . 3. If we consider Mn as a Banach space with a conventional norm, say, √ ∥A∥ = max{O ∈ V (AT A)} then it is called a Banach algebra. 4. If dene on Mn a multiplication, denoted by [⋅, ⋅], as [A, B] = AB − BA. It is easy to check that Mn with this multiplication is an algebra. This algebra is known as Lie algebra. It is not even associative and does not have an identity.

4.3 Homotopy Denition 4.12. Let X , Y be two topological spaces, f , g two continuous mappings from X to Y . 1. f and g are said to be homotopic, denoted by f ≃ g, if there exists a continuous mapping H : X × I → Y , called a homotopy of f and g, such that { H(x, 0) = f (x) (4.3) H(x, 1) = g(x). 2. Let A ⊂ X . f and g are said to be homotopic related to A, denoted by f ≃A g, if in addition to (4.3) we have H(x,t) = f (x) = g(x),

x ∈ A.

(4.4)

3. X, Y are homotopic, denoted by X ≃ Y, if there exist mappings f : X → Y and h : Y → X , such that h ∘ f ≃ idX ;

f ∘ h ≃ idY ,

where idX and idY are the identity mappings on X and Y respectively. 4. X is said to be contractable if it is homotopic to a single point. Example 4.12. 1. If two topological spaces X and Y are homeomorphic then they are homotopic; 2. ℝ2 ∖{0} ≃ S1 ; 3. A star shape set S ⊂ ℝn is such a set that there is a point c ∈ S such that if x ∈ S then the segment (x, c) ∈ S. Particularly, a convex set is a star shape set.

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101

A star shape set on ℝn is contractable. We prove them as follows: 1. Let f : X → Y be the homeomorphism. We can set g = f −1 : Y → X. Then g ∘ f = idX and f ∘ g = idY . Hence the homotopy mappings can be chosen trivially as H(x,t) = idX (respectively H(y,t) = idY ). 2. Using the polar coordinate system, then X := ℝ2 ∖{0} = {(r, T ) ∣ r > 0, 0 ⩽ T < 2π}. Y := S1 = {(1, T ) ∣ 0 ⩽ T < 2π}. Choose f : X → Y as (r, T ) $→ (1, T ), and g : Y → X as (1, T ) $→ (1, T ). Then g ∘ f : (r, T ) → (1, T ), and f ∘ g : (1, T ) → (1, T ). So f ∘ g = idY , and we have only to show that g ∘ f ≃ idX . Set H : M × I → M as H((r, T ),t) = (1 + (r − 1)t, T ). It follows that

{

H((r, T ), 0) = (1, T ) = g ∘ f H((r, T ), 1) = (r, T ) = idX .

3. Let Y = {c}. Dene f : S → Y by f (x) = c, and dene g : Y → S by g(c) = c. Since f ∘ g = idY , we have only to show that g ∘ f ≃ idX . Setting H(x,t) = (1 − t)c + tx, then H(x, 0) = c = g ∘ f , and H(x, 1) = idX . We conclude that S ≃ Y = {c}. Hence, X is contractable. Intuitively, if two mappings are homotopic, then one mapping can be continuously moved to the other. Similarly, two spaces are homotopic if one can be continuously deformed to the other. A space is contractable if it can be continuously shrunk to a point. For instance, ℝ can be shrunk continuously to a point. But a circle can not be shrunk continuously to a point. If one wants to shrink a circle to a point, it would have to be cut at some point x. Then the points on its left can shrink towards the left, and the points on its right towards the right. But at this point the continuity of the shrinking mapping is broken.

4.4 Fundamental Group A way to describe the structural characteristics of a topological space is to investigate a loop on the space. Roughly speaking, on S1 a loop can not shrink to a point. But on ℝ2 , any loop can be shrank to a point. This is the idea behind the fundamental group. We start with paths. Denition 4.13. 1. Let f1 , f2 be two paths. they are equivalent, denoted by f1 ∼ f2 , if they are homotopic related to the two end points. In other words they have the same end points and are homotopic. It can also be denoted as

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f1 ≃({0},{1}) f2 . The equivalence class is denoted as [ f ]. 2. Let f , g be two paths on X , f (0) = x, f (1) = y, g(0) = y, and g(1) = z. We dene the product of these two paths as a path from x to z by { f (2t), 0 ⩽ t ⩽ 1/2 f ∘g = (4.5) g(2t − 1), 1/2 ⩽ t ⩽ 1. Intuitively, the set of paths would form a group by using the product dened by (4.5). The identity element is a single point path, and the inverse element is the inverse path, which goes backward. Unfortunately, the product dened by (4.5) is not associative, i.e., ( f ∘ g) ∘ h ∕= f ∘ (g ∘ h). So we have to consider their equivalent classes. Lemma 4.1. Let f1 , f2 , g1 , g2 be paths on X. If f1 ∼ f2 and g1 ∼ g2 , and f1 (1) = g1 (0), then f1 ∘ g1 ∼ f2 ∘ g2 . Proof. Let H1 (s,t) : I × I be the homotopy of f1 ≃({0},{1}) f2 and H2 (s,t) : I × I the homotopy of g1 ≃({0},{1}) g2 . Dene { H(s,t) =

H1 (2s,t), 0 ⩽ s ⩽ 1/2 H2 (2s − 1,t), 1/2 ⩽ s ⩽ 1.

It is continuous because H1 (1,t) = f1 (1) = g1 (0) = H2 (0,t). From the denition we have H(s, 0) = f1 ∘ g1 ; Moreover {

H(s, 1) = f2 ∘ g2.

H(0,t) = H1 (0,t) = f1 (0) = f2 (0) = f1 ∘ g1 (0) = f1 ∘ g2 (0) H(1,t) = H2 (1,t) = g1 (1) = g2 (1) = f1 ∘ g1 (1) = f2 ∘ g2 (1).

Hence f1 ∘ g1 ∼ f2 ∘ g2 .

⊔ ⊓

The above lemma makes it possible to dene a product over the equivalent classes of paths as [ f ][g] := [ f ∘ g]

(4.6)

Denition 4.14. A path f is called a loop if f (0) = f (1) = x. Point x is called the base point. Next, we want to show that with the multiplication of equivalence classes, dened in (4.6), the equivalence classes of loops form a group. The following three lemmas provide associativity, unit and inverse elements for the group respectively.

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103

Lemma 4.2. Let D (s), E (s) and J (s) be three sequentially connected paths, i.e., D (1) = E (0) and E (1) = J (0). Then ([D ][E ])[J ] = [D ]([E ][J ]). Proof. We have only to show that (D ∘ E ) ∘ J ≃({0},{1}) D ∘ (E ∘ J ). According to the denition, we have ⎧  ⎨D (4s), 0 ⩽ s ⩽ 1/4 (D ∘ E ) ∘ J (s) = E (4s − 1), 1/4 ⩽ s ⩽ 1/2  ⎩ J (2s − 1), 1/2 ⩽ s ⩽ 1, and

⎧  ⎨D (2s), 0 ⩽ s ⩽ 1/2 D ∘ (E ∘ J )(s) = E (4s − 2), 1/2 ⩽ s ⩽ 3/4  ⎩ J (4s − 3), 3/4 ⩽ s ⩽ 1.

Dene a homotopy of the two paths as ) ⎧ ( 4s , 4s − 1 ⩽ t ⩽ 1  D   1+t  ⎨ H(s,t) = E (4s − t − 1), 4s − 2 ⩽ t ⩽ 4s − 1   ) (   ⎩J 4s − t − 2 , 0 ⩽ t ⩽ 4s − 2. 2−t We leave to reader for checking that H(s,t) is continuous and xes the two ending points (for t = 0 and t = 1). ⊔ ⊓ For convenience, we denote by ex the xed loop with base point x, i.e., ex (t) = x, ∀t ∈ [0, 1]. Lemma 4.3. Let D (s) be a path with D (0) = x and D (1) = y. Then [ex ][D ] = [D ] = [D ][ey ]. Proof. We prove the rst equality. The second equality can be proved similarly. We have only to prove that ex ∘ D ≃({0},{1}) D . Construct a homotopy as (Refer to Fig. 4.1) ⎧ ⎨x, 0 ⩽ t ⩽ 1 − 2s ( ) H(s,t) = ⎩D 2s + t − 1 , 1 − 2s ⩽ t ⩽ 1. t +1 {

Then H(s, 0) =

x, s ⩽ 1/2 D (2s − 1), s ⩾ 1/2.

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It is ex ∘ D . Meanwhile,

H(s, 1) = D (s).

In addition, H(0,t) = x; Hence ex ∘ D ∼ D .

Fig. 4.1

H(1,t) = D (1) = y. ⊔ ⊓

Homotopy

When the homotopy is constructed the proof of the above or similar problems is easy. The question is how to construct a homotopy? We explain it briey by using the above proof as example. Refer to Fig. 4.1, for the rectangle, on t = 0 we need two segments: the rst half is x and the second half is D (s). While on t = 1 we have only D . On the left, s = 0 it is x, and on the right s = 1, we have y. One way to construct H is to split the rectangle into two parts: t ⩽ 1 − 2s and t ⩾ 1 − 2s. Then on the left triangle area set H = x, and on the right trapezoid set for each pair (s,t) the value H(s,t) = H(s∗ , 1), where s∗ = 2s+t−1 t+1 . By this way we obtain a H as required. The inverse D −1 (t) of D (t) can simply be dened as D −1 (t) = D (1 − t). It is clear that when D goes from x to y, D −1 goes from y to x. They have to satisfy the following: Lemma 4.4. Let D be a path, with D (0) = x and D (1) = y. Then [D ][D −1 ] = [ex ] and [D −1 ][D ] = [ey ]. Proof. We prove D ∘ D −1 ∼ ex . Using Fig. 4.2, let H(p) = H(p∗ ) and H(q) = H(q∗ ). A homotopy follows as

4.4 Fundamental Group

Fig. 4.2

105

Path of D and D −1

⎧ x, 0 ⩽ s ⩽ t      ⎨ ( 2(s − t) ) 1 , t <s⩽ t+ H(s,t) = D 1 − t 2  ) (     ⎩D 2(1 − s) , t + 1 < s ⩽ 1. 1−t 2 Similarly, we can prove D −1 ∘ D ∼ ey .

⊔ ⊓

Summarizing Lemmas 4.2–4.4 yields a complete group structure. Theorem 4.4. Let x ∈ X. Denote the equivalent classes of the loops with x as its base point by : (X, x). Adopting the multiplication of equivalent classes dened by (4.5) and (4.6), : (X , x) becomes a group. It is called the fundamental group based on x, denoted by S1 (X, x). The following theorem shows that the fundamental group is independent of the choice of base point provided the space X is pathwise connected. Theorem 4.5. Let X be a pathwise connected topological space, and x1 , x2 ∈ X. Then S1 (X , x1 ) ∼ = S1 (X, x2 ). Proof. Let f (s) be a path with ends f (0) = x1 and f (1) = x2 , and D (s) be a loop based on x2 . Then f ∘ D ∘ f −1 is a loop based on x1 . Now we dene a mapping f ∗ : S1 (X, x2 ) → S1 (X, x1 ) as f∗ : [D ] $→ [ f ][D ][ f −1 ].

(4.7)

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It is sufcient to show that f∗ is a group isomorphism. First, we have f∗ ([D ][E ]) = [ f ][D ][E ][ f −1 ] = [ f ][D ][ f −1 ][ f ][E ][ f −1 ] = f∗ ([D ]) f∗ ([E ]). So f∗ is a homomorphism. Next, assume that g is another path connecting x2 to x3 , and [J ] ∈ S1 (X, x3 ). We prove that ( f ∘ g)∗([J ]) = f∗ (g∗ ([J ])). ( f ∘ g)∗ ([J ]) = = = =

(4.8)

[ f ∘ g][J ][( f ∘ g)−1 ] [ f ][g][J ][g−1 ][ f −1 ] [ f ][g][J ][g]−1 [ f ]−1 f∗ (g∗ ([J ]))

which proves (4.8). Using (4.8), we have (ex1 )∗ = ( f ∘ f −1 )∗ = f∗ ∘ f∗−1 . But (ex1 )∗ is an identity mapping, which means f∗ is a bijective mapping. Hence f∗ is an isomorphism. ⊔ ⊓ Based on the previous theorem, when the space X is pathwise connected, we may omit the base point and simply denote the fundamental group S (X, x) by S (X). Example 4.13. 1. Let S ⊂ ℝn be a star shape subset, with x the “center”, and D (s) be a loop with x as the base point. Dene a homotopy H(s,t) = t D (s) + (1 − t)x. It follows that D ∼ ex . Moreover, since S is path connected, the fundamental group is independent of the base point. Hence

S1 (X) = {e}, which is a trivial group containing only one element, identity. Such a fundamental group is called a trivial group. For a pathwise connected topological space, if its fundamental group is trivial, it is said to be simply connected. 2. Let X = S1 . Intuitively, we can imagine that the fundamental group is S1 (S1 ) ∼ = ℤ, where ℤ stands for the addition group of integers. In fact, a path twisted around the circle n times corresponds n ∈ ℤ. When the twisting goes on the opposite direction we have a negative integer, n < 0, to represent the n times twisting on the counter-direction. The detailed argument is tedious. (See i.e., [1].) 3. Let X, Y be two topological spaces, f : X → Y continuous, and f (x) = y. Dene a mapping f∗ : S1 (X, x) → S1 (Y, y) as

4.4 Fundamental Group

f∗ ([D ]) = [ f ∘ D ],

107

(4.9)

which is a group homomorphism. First, we have to show that (4.9) is well dened. Let D ∼ E . Then there exists a homotopy h(s,t) such that h(s, 0) = D (s);

h(s, 1) = E (s).

Then f ∘ h(s, 0) = f ∘ D (s);

f ∘ h(s, 1) = f ∘ E (s),

which means f ∘ D (s) ∼ f ∘ E (s). Hence f∗ is well dened. Next f∗ ([D ][E ]) = f∗ ([D ∘ E ]) = [ f ∘ (D ∘ E )] = [( f ∘ D ) ∘ ( f ∘ E )] = [ f ∘ D ][ f ∘ E ] = f ∗ [D ] f∗ [E ]. Hence, f ∗ is a group homomorphism. Roughly speaking, for product space its fundamental group is the product of the fundamental groups. Proposition 4.5. Let X, Y be two path connected topological spaces. Then

S1 (X × Y ) ∼ = S1 (X ) × S1 (Y ).

(4.10)

Proof. Let x0 ∈ X and y0 ∈ Y , p : X ×Y → X and q : X ×Y → Y the natural projections. Assume D (s) = (x(s), y(s)) be a loop on the product space based on (x0 , y0 ). Then p∗ : S1 (X × Y ) → S1 (X) is a homomorphism, and p∗ ([D ]) = [x(s)]. Similarly, q∗ : S1 (X × Y ) → S1 (Y ) is also a homomorphism, and p∗ ([D ]) = [y(s)]. We claim that (p∗ × q∗ ) : S1 (X × Y ) → S1 (X ) × S1 (Y ) is an isomorphism. For each pair [x(s)] × [y(s)] ∈ S1 (X ) × S1 (Y ) we can construct [x(s), y(s)] ∈ S1 (X × Y ), such that (p∗ × q∗ )([x(s) × y(s)]) = [x(s)] × [y(s)] ∈ S1 (X) × S1 (Y ). Hence, p∗ × q∗ is surjective. To see that it is also injective, let Di = (xi (s), yi (s)), i = 1, 2 be such that (p∗ × q∗ )([D1 ]) = p∗ × q∗ ([D2 ]). Then [x1 ] = [x2 ] and [y1 ] = [y2 ]. That is x1 (s) ∼ x2 (s),

y1 (s) ∼ y2 (s).

Let the homotopies between them be h1 (s,t) and h2 (s,t) respectively. Then we construct H(s,t) = (h1 (s,t), h2 (s,t)). It is easy to verify that this is a homotopy between D1 and D2 . That is [D1 ] = [D2 ]. Hence p∗ × q∗ is injective. ⊔ ⊓

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Example 4.14. Consider a torus (Refer to Example 2.19), denote it by T 2 . It is easy to see that T 2 ∼ = S1 × S1 . (In fact, it can be considered as a particular case of the general denition of T k ∼ ⋅ ⋅ × S21 .) According to the above proposition: = S/ 1 × ⋅01 k

S1 (T ) ∼ = S1 (S1 ) × S1(S1 ) ∼ = ℤ × ℤ = ℤ2 . 2

Next, we will prove an important fact that the fundamental group is invariant under homotopy. We need some preparation. Lemma 4.5. Let g, f be two homotopic continuous mapping from X to Y, H(x,t) a homotopy from g to f . For x ∈ X, dene a path

Z = H(x, s),

s ∈ I = [0, 1].

Then the following graph Fig. 4.3 is commutative. That is, Z∗ ∘ f∗ = g∗ .

Fig. 4.3

Z∗

Proof. Mind that the denition of f∗ , g∗ is different from Z∗ . They have a different meaning. Let [D ] ∈ S1 (X, x). We have to show that g ∘ D ≃({0},{1}) Z ∘ ( f ∘ D ) ∘ Z −1.

(4.11)

Since H is a homotopy from g to f , setting x = D (r) we have a graph as Fig. 4.4 (a). Our purpose is to construct a homotopy such that Fig. 4.4 (b) holds. On two slices t = 4s and t = 4(1 − s), we need Z (t). Then it is easy to see that on the left and right triangle areas we can use Z (4s) and Z (4 − 4s) respectively. For the center trapezoid, we have only to deform Fig. 4.4 (a) by 4s − t . 4 − 2t Summarizing them, we obtain the following homotopy: ⎧ Z (4s), 4s ⩽ t    )) ⎨ ( ( ˜ H(s,t) = H D 4s − t ,t , t ⩽ 4s ⩽ 4 − t  4 − 2t   ⎩ Z (4 − 4s), 4s ⩾ 4 − t. r=

(4.12)

⊔ ⊓

4.5 Covering Space

Fig. 4.4

109

Homotopy of Two Mappings with A Path

Theorem 4.6. Let X, Y be two path connected spaces, f : X ≃ Y a homotopy. Then f∗ : S1 (X ) ∼ = S1 (Y ). Proof. Let x ∈ X and y = f (x) ∈ Y and g : Y → X the homotopy inverse of f . Denote by F : g ∘ f ≃ idX , G : f ∘ g ≃ idY , the corresponding homotopies. Set paths

D (s) := F(x, s), Using Lemma 4.5, we have {

E (s) := G(x, s).

g∗ ∘ f∗ = D∗ (idX )∗ = D∗ f∗ ∘ g∗ = E∗ (idY )∗ = E∗ .

Note that both D∗ and E∗ are isomorphisms, because, say, D −1 = F(x, 1 − s) and hence D∗ D∗−1 = (id)∗ . Therefore, both f∗ and g∗ are isomorphic. ⊔ ⊓

4.5 Covering Space ˜ p) is called a covering space of Denition 4.15. Let X be a topological space. (X, X if (i). X˜ is a topological space, and p : X˜ → X, called the projection, is a continuous surjective mapping; (ii). For each x ∈ X , there exists a neighborhood, U, called a basic neighborhood, such that p−1 (U) consists of disjoint open sets, {VO ∣ O ∈ / }. Moreover, for each O , p : VO → U is a homeomorphism. ˜ The cardinal number of / , ∣/ ∣ is called the multiplicity of X.

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Proposition 4.6. Let (X˜ , p) be a covering space of X, and f : I → X a path with ˜ f (0) = x0 . Suppose a point x˜0 ∈ p−1 (x0 ). Then there exists a unique path f˜ : I → X, such that f˜(0) = x˜0 and p ∘ f˜ = f . Proof. If f (I) is contained in a basic neighborhood, the conclusion is obvious. Otherwise, Let {UO ∣ O ∈ / } be a covering consists of basic neighborhoods. Then { f −1 (UO ) ∣ O ∈ / } is an open covering of I. Let n be large enough such that 1n i is less than the Lebesgue number of the cover. Setting Ii = [ i−1 n , n ], i = 1, 2, ⋅ ⋅ ⋅ , n, then each f (Ii ) is contained in a basic neighborhood. Starting with i = 1 it is easily seen that each f (Ii ) has a unique lift with the initial point as the end point of the previous path lift. ⊔ ⊓ Note that from above proof we can easily see that the set I can be replaced by any compact set in a nite dimensional normed space. Then the following is an immediate consequence: Proposition 4.7. Let (X˜ , p) and X be as in the above, and f : I × I → X a path homotopy, f (0, 0) = x0 , x˜0 ∈ p−1 (x0 ). Then there exists a unique path homotopy ˜ g(0, 0) = x˜0 , such that p ∘ g = f . g : I × I → X, ˜ p) and X be as in the above, x˜0 ∈ p−1 (x0 ). Then p∗ : Proposition 4.8. Let (X, ˜ S (X, x˜0 ) → S (X, x0 ) is a one-to-one homomorphism. Proof. Refer to Example 4.13 to show it is a homomorphism. Here we will only prove it is injective. Let D˜ , E˜ be two loops with x˜0 as their base point. Suppose p∗ ([D˜ ]) = p∗ ([E˜ ]), and denote D = p ∘ D˜ and E = p ∘ E˜ . Then D ∼ E , say, with a homotopy h(s,t). That is, h(s, 0) = D (s), h(s, 1) = E (s), h(0,t) = x0 . ˜ By Proposition 4.7, there is a unique h(s,t) such that ˜ 0) = x˜0 , h(0,

p ∘ h˜ = h.

˜ 0) = D , by Proposition 4.6, Since p ∘ h(s, ˜ 0) = D˜ (s). h(s, Similarly, We conclude that i.e., [D˜ (s)] = [E˜ (s)].

˜ 1) = E˜ (s). h(s,

D˜ (s) ∼ E˜ (s), ⊔ ⊓

Intuitively, we may say that the covering space enlarged the original space but simplied the topological structure, which is one of the implications of the above proposition. The following theorem shows that the lifting paths form the same group at different base points:

4.5 Covering Space

111

Theorem 4.7. Let (X˜ , p) and X be as in the above. Then {p∗ S (X˜ , x˜0 ) ∣ x˜0 ∈ p−1 (x0 )} is a set of conjugate subgroup of S (X , x0 ). Proof. Let p(x˜1 ) = p(x˜2 ) = x0 , J be a path connecting x˜1 to x˜2 . Then p∗ ([J ]) = [p ∘ J ] ∈ S (X , x0 ). ˜ x˜1 ) → S (X, ˜ x˜2 ) as Dene an isomorphism u : S (X, u([D ]) = [J ]−1 [D ][J ] and an isomorphism v : S (X, x0 ) → S (X , x0 ) as v([E ]) = (p∗ [J ])−1 [E ](p∗ [J ]). It is easily seen that the graph Fig. 4.5 is commutative.

Fig. 4.5

Path Homotopy

Namely,

˜ x˜2 ) = [p ∘ J ]−1[p∗ S (X, ˜ x˜1 )][p ∘ J ]. p∗ S (X,

⊔ ⊓

Finally we introduce a theorem for the existence of covering spaces. (Refer to [5] for the proof.) We need a new concept rst. A topological space, X, is said to be semilocally simply connected if each point x ∈ X has a simply connected neighborhood. Theorem 4.8. Let X be a topological space satisfying (i). path connected; (ii). locally connected; (iii). semilocally simply connected. Then there exists a covering space, called the universal covering space (X˜ , p), satisfying (i). If (X˜1 , p1 ) and (X˜2 , p2 ) are two universal covering spaces, then X˜ 1 and X˜2 are homeomorphic. Moreover, the homeomorphism, f , makes the following graphs (Fig. 4.6 (a) and (b)) commutative. (ii). If (X˜ , p) is a universal covering space of X, and (Y, q) is a covering space, ˜ S ) is a covering space of Y . then there exists a mapping S : X˜ → Y , such that (X, Moreover, the following graph (Fig. 4.7) commute.

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Fig. 4.6

University Covering Space

Fig. 4.7

Covering Space

Fig. 4.8

ℝ as the Covering Space of S1

Example 4.15. Let p : ℝ → S1 be dened as p : t $→ (cos(2πt), sin(2πt)). Choose ˆ any point x0 ∈ S1 , say x0 = (1, 0). Consider a neighborhood U = Px 0 Q, which is −1 the right semi-sphere. This is a basic neighborhood because p (U) = {Vk {=}(k − 1 1 2 , k + 2 ) ∣ k ∈ ℤ}. Now the restriction p : Vk → U is a homeomorphism. Hence p is a projection. It results that (ℝ, p) is a covering space of S1 . The multiplicity is F0 . (i.e., countable.) Since S (ℝ) = {0} and S (S1 ) ∼ = ℤ, the induced homomorphism p∗ is p∗ S (ℝ) = {0} < ℤ. (Refer to Fig. 4.8.)

4.6 Lie Group

113

4.6 Lie Group Denition 4.16. A set G is a Lie group, if (i). G is a group; (ii). G is an analytic differentiable manifold; (iii). The group operation ⊗ : G × G → G is analytic; The inverse operation: G → G, as: g $→ g−1 , is analytic. Consider Mn the set of n × n matrices. Set V (A) = (AT1 , ⋅ ⋅ ⋅ , ATn )T ,

A ∈ Mn ,

where Ai is the i-th column of A. Then V is a bijective mapping. Through V and V −1 we can identify Mn with ℝn×n . Later on, we will give Mn the conventional topology of ℝn×n . Example 4.16. Consider GL(n, ℝ) ⊂ Mn . It is an open subset of Mn , and hence it is an n × n analytic manifold. As we have mentioned above, with the matrix multiplication it is also a group. Finally, by considering the elements ai, j as the coordinates of A, one sees that both A × B → AB and A → A−1 are analytic. We conclude that GL(n, ℝ) is a Lie group. In fact, this is a very important Lie group, called the general linear group. Denition 4.17. Given a Lie group G. A subset H ⊂ G is called a Lie subgroup if (i). H is a subgroup of G; (ii). H is a regular sub-manifold of G; (iii). With the subgroup structure and sub-manifold structure, H is a Lie group. Remark 4.3. In some books the Lie subgroup H ⊂ G is dened as an immersed sub-manifold. Hence, H may not be a closed sub-manifold of G. Unless elsewhere stated, we consider only closed Lie subgroup to avoid some complex discussion. Theorem 4.9. Let H ⊂ G be a subgroup of Lie group G. In addition, it is also a regular sub-manifold. Then 1. H itself is a Lie group; 2. H is a closed subgroup of G. Proof. 1. Let gi ∈ H, i = 1, 2, 3. g1 g2 = g3 , and (Ui , Ii ), i = 1, 2, 3 be at coordinate charts around gi , i = 1, 2, 3 respectively. Say, dim(G) = m and dim(H) = n. Under the at coordinates we denote g1 = (x1 , ⋅ ⋅ ⋅ , xn , 0, ⋅ ⋅ ⋅ , 0), g2 = (y1 , ⋅ ⋅ ⋅ , yn , 0, ⋅ ⋅ ⋅ , 0), g3 = (z1 , ⋅ ⋅ ⋅ , zn , 0, ⋅ ⋅ ⋅ , 0). Then the product: U1 ×U2 → U3 (by possibly shrinking U1 and U2 ) can be expressed as zi = zi (x1 , ⋅ ⋅ ⋅ , xm , y1 , ⋅ ⋅ ⋅ , ym ), Restricting it to H, we have

i = 1, ⋅ ⋅ ⋅ , m.

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z˜i = zi (x1 , ⋅ ⋅ ⋅ , xn , 0, ⋅ ⋅ ⋅ , 0, y1 , ⋅ ⋅ ⋅ , yn , 0, ⋅ ⋅ ⋅ , 0),

i = 1, ⋅ ⋅ ⋅ , m,

which is obviously analytic. A similar argument shows that g $→ g−1 is also analytic. Hence H is a Lie group. 2. Let hi ∈ H and hi → h0 , and (U, x) be a at coordinate chart about h0 . Then for some N > 0 when i > N, hi ∈ U. Denote hi = (xi1 , ⋅ ⋅ ⋅ , xin , 0, ⋅ ⋅ ⋅ , 0). Since hi → h0 , xij → xi0 , i → f, j = 1, ⋅ ⋅ ⋅ , m, it is easily seen that h0j = 0, j = n + 1, ⋅ ⋅ ⋅ , m. That is, h0 ∈ H. H is therefore closed. ⊔ ⊓ In fact, the converse of the above 2 is also true. That is: A closed subgroup of a Lie group is a Lie subgroup [6]. Theorem 4.10. Let F : G1 → G2 be a Lie group homomorphism. Then the rank of F is constant. Moreover, the kernel, K = ker(F), is a Lie subgroup of G1 with codimension codim(K) = rank(F). Proof. From Theorem 3.1, if the rank of F is constant, then K = F −1 (e) is a closed sub-manifold of G1 with co-dimension of rank(F). Thus it is a Lie subgroup. So it sufces to prove the constancy of the rank of F. Let a ∈ G1 and b = F(a) ∈ G2 . Then F(x) = F(a)F(a−1 x) = bF(a−1 x) := Lb F(a−1 x), where Lb stands for left-multiplication by b. Now it is obvious that Lb is analytic. Moreover, its inverse, (Lb )−1 = Lb−1 is also analytic. It is therefore a diffeomorphism. We conclude that rank(F(x)) = rank(F(a−1 x)). Since a ∈ G1 is arbitrary, rank (F) is constant everywhere.

⊔ ⊓

Example 4.17. 1. Special linear group: Let SL(n, ℝ) = {A ∈ GL(n, ℝ) ∣ det(A) = 1}. SL(n, ℝ) is called the special linear group, which is a Lie subgroup of GL(n, ℝ). 2. Orthogonal group: Let O(n, ℝ) = {A ∈ GL(n, ℝ) ∣ AT A = I}. O(n, ℝ) is called the orthogonal group, which is a Lie subgroup of GL(n, ℝ). 3. Unitary group: Let U(n, ℂ) = {A ∈ GL(n, ℂ) ∣ A¯T A = I}. U(n, ℝ) is called the unitary group, which is a Lie subgroup of GL(n, ℂ). 4. Symplectic group: Denote ] [ 0 In . J= −In 0

4.7 Lie Algebra of Lie Group

115

Dene Sp(2n, ℝ) = {A ∈ GL(2n, ℝ) ∣ AT JA = J}. Sp(2n, ℝ) is called the symplectic group , which is a Lie subgroup of GL(2n, ℝ). The arguments are more or less similar. We give a detailed discussion for SL(n, ℝ). Consider F : GL(n, ℝ) → (ℝ+ , ×), dened as A → F(A) = det(A). It is obvious that F is a Lie group homomorphism. For any A = (ai j ) ∈ GL(n, ℝ), denote by Ai j the algebraic complement minor of ai j . Since F(A) =

n

¦ (−1)k−1a1k det(A1k ) ∕= 0,

k=1

there exists at least one k such that det(A1k ) ∕= 0. Hence

wF = (−1)k−1 det(A1k ) ∕= 0, w a1k which means rank(F) = 1. Theorem 4.10 tells us that SL(n, ℝ) = ker(F) is a Lie subgroup of GL(n, G) of dimension n2 − 1. For a differentiable manifold M, under Lie bracket [⋅, ⋅], V (M) becomes a Lie algebra. Hence, it is convenient to consider V (M) as a Lie algebra.

4.7 Lie Algebra of Lie Group Consider a Lie group G and let a ∈ G. From the above discussion we know that the left-multiplication (or left-displacement) La : G → G, dened as g $→ ag, is a diffeomorphism. Let X ∈ V (G). X is called a left-invariant vector eld if (La )∗ X = X ,

∀a ∈ G.

Similarly, we can dene a right-invariant vector eld and obtain all parallel results. Now we denote the set of left-invariant vector elds on G by g(G), or simply g, i.e., g = {X ∈ V (G) ∣ (La )∗ X = X , ∀a ∈ G}. g is a Lie sub-algebra of V (M). To see this let X , Y ∈ g and D , E ∈ ℝ, and a ∈ G. Then 1. (La )∗ (D X + E Y ) = D (La )∗ X + E (La )∗Y = D X + E Y ; 2. (La )∗ [X ,Y ] = [(La )∗ X , (La )∗Y ] = [X ,Y ]. Denition 4.18. Given a Lie group G, the set g of the left-invariant vector elds, g = g(G) is called the Lie algebra of the Lie group G. Proposition 4.9. Let X ∈ g. Dene a mapping < : g → Te (G) as X $→ Xe . Then < is a linear isomorphism. Proof. Linearity is trivial. We only need to show that it is bijective. Assuming X ∕= Y , then at a certain point x ∈ G, X(x) ∕= Y (x). Thus

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(Lx−1 )∗ Xx ∕= (Lx−1 )∗Yx . That is: Xe ∕= Ye . So < is one to one. To show that this mapping is a surjective, let Xe ∈ Te (G) and dene a vector eld as X(x) = (Lx )∗ Xe . Since Lx is analytic, X(x) is obviously an analytic vector eld. To see that Xe ∈ g we have only to show that it is left-invariant. For any b ∈ G (Lb )∗ Xx = (Lb )∗ ((La )∗ Xe ) = (Lbx )∗ Xe = Xbx , ⊔ ⊓

which completes the proof.

The above theorem tells that dim(g(G)) = dim(G). It is easily seen by continuity that left-invariant vectors on Lie group are complete. So we can dene the following: Denition 4.19. (Exponential Mapping) Let X ∈ g(G). Dene an exponential mapping exp : g → G as exp(X) = eX1 (e).

(4.13)

Then it is obviously true that exp(tX) = etX (e). Moreover, if G = GL(n, F) or its subgroup, the exponential mapping becomes the conventional matrix exponential mapping. Example 4.18. If we identify a left-invariant vector eld X(x) with its value at identity X(e), we have the following: (The eld F can be either ℝ or ℂ) 1. The Lie algebra of GL(n, F) is gl(n, F). 2. The Lie algebra of SL(n, ℝ) is sl(n, ℝ), where sl(n, ℝ) = {X ∈ gl(n, ℝ) ∣ tr(X) = 0}. 3. The Lie algebra of O(n) is o(n), where o(n) = {X ∈ gl(n, ℝ) ∣ X T = −X}. 4. The Lie algebra of U(n) is u(n), where u(n) = {X ∈ gl(n, ℂ) ∣ X¯T = −X}. 5. The Lie algebra of Sp(2n, ℝ) is sp(2n, ℝ), where sp(2n, ℝ) = {X ∈ gl(2n, ℝ) ∣ X T J + JX = 0}. Example 4.19. Denote SO(n) = {A ∈ O(n) ∣ det(A) = 1},

4.8 Structure of Lie Algebra

117

which is called the special orthogonal group. Consider the Lie algebra of SO(n). Note that the Lie algebra of a Lie group depends only on the property of the Lie group around the identity. Hence if a Lie group as a topological space has several disconnect components, its Lie algebra is determined by the component containing its identity. So O(n) and SO(n) have the same Lie algebra, namely, o(n). In fact, all nite dimensional Lie algebras are basically a sub-algebra of gl(n, F). Theorem 4.11. (Ado’s Theorem) Any nite dimensional Lie algebra g over eld F is isomorphic to a subalgebra of gl(n, F). For a proof we refer to [3]. The following theorem is fundamental for the relationship between Lie group and their Lie algebra. Theorem 4.12. Each nite dimensional Lie algebra, g, corresponds to a unique ˜ Moreover, any other connected Lie connected, simply connected Lie group G. group, G, having g as its Lie algebra is isomorphic to the quotient group of G˜ over a discrete normal subgroup D, i.e., ˜ G ≃ G/D. We refer to [4] for details. In fact, all Lie groups G with the same g as their Lie ˜ All of them are locally isomoralgebra have a common uniform covering space G. phic.

4.8 Structure of Lie Algebra Lie algebra itself is an important concept and useful tool in mathematics as well as in systems and control theory. This section considers Lie algebra independently. We refer to [2] for details. Let L be a given Lie algebra. We dene a sequence of ideals of L , called the derived series, as [ ] L (0) = L , L (k+1) = L (k) , L (k) , k ⩾ 0. Denition 4.20. A Lie algebra, L , is solvable if there exist a k∗ > 0 such that ∗

L (k ) = 0.

(4.14)

For a given Lie algebra, L , we can also dene a descending central series as [ ] L 0 = L , L k+1 = L , L k , k ⩾ 0. Denition 4.21. A Lie algebra, L , is nilpotent if there exist a k∗ > 0 such that ∗

L k = 0.

(4.15)

It follows from the denitions that if a Lie algebra, L , is solvable, it is also nilpotent.

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Denition 4.22. 1. A Lie algebra, L , is said to be simple, if it has no nontrivial idea. That is, L has no idea except {0} and itself. 2. A Lie algebra, L , is said to be semi-simple, if it has no non-zero solvable idea. Denition 4.23. The largest solvable idea of L is called its radical, denoted by r. Theorem 4.13. (Levi Decomposition)[6] A Lie algebra can be decomposed as L = s ⊕ r,

(4.16)

where s is a semi-simple Lie algebra, r is the radical of L . The solvability and the semi-simple property of a Lie algebra can be veried via its representation. We discuss it in the following. Let G be a Lie algebra, and V be an n dimensional vector space, gl(V ) be the Lie algebra consists of all linear transformations on V . (When the basis of V is xed, it becomes gl(n, ℝ).) Denition 4.24. A Lie algebra homomorphism I : G → gl(V ) is called a representation of G on V . The representation of G on itself is called an adjoint representation, and denoted by adG . Example 4.20. sl(2, ℝ) is the set of 2 × 2 real matrices, whose traces equal to zero. It is easy to verify that sl(2, ℝ) is a Lie sub-algebra of gl(n, ℝ). A basis of sl(2, ℝ) is: ] [ [ ] [ ] 1 0 01 00 , e2 = , e3 = . e1 = 0 −1 00 10 Then we can calculate adG . Note that [e1 , e2 ] = e1 e2 − e2 e1 = 2e2 , [e1 , e3 ] = e1 e3 − e3 e1 = −2e3 , [e2 , e3 ] = e2 e3 − e3 e2 = e1 . We also know that [ei , ei ] = 0,

[ei , e j ] = −[e j , ei ],

i, j = 1, 2, 3.

It is clear that corresponding to this set of basis the matrix forms of the adjoint representations of ei are ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 00 0 0 01 0 −1 0 ade1 = ⎣0 2 0 ⎦ , ade2 = ⎣−2 0 0⎦ , ade1 = ⎣0 0 0⎦ . 0 0 −2 0 00 2 0 0 Now let D ∈ sl(2, ℝ), with D = D1 e1 + D2 e2 + D3 e3 . We express it in vector form as D = (D1 , D2 , D3 )T . Let E = adei D = (E1 , E2 , E3 )T . Then in vector form we have

E = adei D .

References

119

Denition 4.25. The Killing form of a Lie algebra G is dened as K(X ,Y ) = tr(adX adY ),

X,Y ∈ G .

(4.17)

Theorem 4.14. [6] A Lie algebra G is solvable, if and only if K(X ,Y ) ≡ 0,

X,Y ∈ G .

(4.18)

Assume G is a nite dimensional Lie algebra with a basis {e1 , ⋅ ⋅ ⋅ , en }. For this basis we can dene an n × n matrix as MK = (K(ei , e j )),

(4.19)

i.e., the (i, j)-th element of the matrix is K(ei , e j ). It is easy to check that the rank of MK is independent of the choice of the basis. Moreover, we have the following. Theorem 4.15. [6] A nite dimensional Lie algebra G is semi-simple, if and only if its Killing matrix is non-singular.

References 1. Croom F. Basic Concepts of Algebraic Topology. New York: Springer-Verlag, 1978. 2. Humphreys J. Introduction to Lie Algebras and Representation Theory, 2nd edn. New York: Springer-Verlag, 1970. 3. Jacobson N. Lie Algerba. New York: Interscience Publishers, 1962. 4. Pontryagin L S. Topological Groups, 2nd edn. New York: Gordon and Breach, 1966. 5. Massey W. Algebraic Topology: An Introduction. New York: Springer-Verlag, 1967. 6. Varadarajan V. Lie Groups, Lie Algebras, and Their Representations. New York: SpringerVerlag, 1984.

Chapter 5

Controllability and Observability

In section 5.1 we consider controllability of nonlinear systems. The materials are mainly based on [11, 10]. We also refer to [5, 9] for related results, to [4] for later developments. Section 5.2 is about observability of nonlinear systems. The observability of nonlinear control systems is closely related to their controllability [1, 3, 8]. The Kalman decomposition of nonlinear systems is investigated in section 5.3. This section is based on [2]. We refer to [6, 7] for decomposition of nonlinear control systems.

5.1 Controllability of Nonlinear Systems Consider a nonlinear system x = f (x, u), x ∈ ℝn , u ∈ ℝm y = h(x), y ∈ ℝ p ,

(5.1)

where x is the state variable, y the output, and u the control. In (5.1), f (x, u) is a smooth mapping. In most cases it is assumed that f (0, 0) = 0. Depending on the context “smooth” could mean Cr , Cf or CZ respectively. Mostly, h(x) is also assumed to be smooth (as smooth as f (x, u)) with h(0) = 0. u can assumed to be piecewise continuous or measurable or smooth as you wish. (In fact, it does not affect the controllability and observability much [11].) When u = u(t), it is called an open-loop control; when u = u(x) (u = u(y)) it is called a state feedback (output feedback) closed-loop control. Generally, the state, control and output spaces may be replaced by n, m, and p dimensional manifolds respectively. A particular form of system (5.1) is afne nonlinear system, which is nonlinear in state x and linear in control u. An afne nonlinear control system is generally described as ⎧ m  ⎨x = f (x) + gi (x)ui := f (x) + g(x)u, x ∈ ℝn , u ∈ ℝm ¦ (5.2) i=1  ⎩y = h(x), y ∈ ℝ p . The afne nonlinear systems are the main object in this book. In fact, it is also the main object of nonlinear control theory. In the geometric approach, the f (x) and gi (x), i = 1, ⋅ ⋅ ⋅ , m in system (5.2) are considered as smooth vector elds on ℝn . D. Cheng et al., Analysis and Design of Nonlinear Control Systems © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2010

122

5 Controllability and Observability

Consider system (5.2). The state space, M, is assumed to be ℝn or any path connected n dimensional differentiable manifold. Let the control, u(t), be measurable functions. For the ease of statement, we also assume that for any feasible control u the vector eld f (x, u) is complete. Although this assumption is not necessary for the following discussion on controllability, It serves the purpose of avoiding discussion of the existence of solution over t. Let x0 ∈ M. If there exists a feasible control u and a moment T > 0 such that the trajectory x(t) satises: x(0) = x0 and x(T ) = x1 , then x1 is said to be in the reachable set of x0 . The reachable set of x0 is denoted by R(x0 ). The reachable set at a particular moment T is denoted by R(x0 , T ). Denition 5.1. Consider system (5.1) and let x0 ∈ M. If the reachable set R(x0 ) = M, the system (5.1) is said to be controllable at x0 . If R(x) = M,

∀x ∈ M,

the system (5.1) is said to be controllable. In many cases it is difcult to get the global properties of a nonlinear system, so we turn to consider the local situation. As for local controllability, let x0 ∈ M and U be a neighborhood of x0 . A point x1 is said to be U-reachable, if x1 ∈ U and there exists a feasible control u and a moment T > 0, such that the trajectory x(t) satises x(0) = x0 , x(T ) = x1 and x(t) ∈ U, 0 ⩽ t ⩽ T. The U-reachable set of x0 is denoted by RU (x0 ). Denition 5.2. Consider system (5.1) and let x0 ∈ M. If for any neighborhood U of x0 the reachable set RU (x0 ) is also a neighborhood of x0 , system (5.1) is said to be locally controllable at x0 . If the system is locally controllable at each x ∈ M, system (5.1) is said to be locally controllable. Neither controllability nor local controllability is symmetric with respect to any two ending points. That is, for two given points x1 , x2 ∈ M, x2 ∈ R(x1 ) doesn’t mean x1 ∈ R(x2 ). Similarly, there exists a neighborhood U of x1 such that x2 ∈ RU (x1 ) does not mean x1 ∈ RU (x2 ). We give a symmetric denition as follows: Denition 5.3. Let x0 , xT ∈ M be given. If there exist x1 , ⋅ ⋅ ⋅ , xk = xT , such that or xi ∈ R(xi−1 ),

or xi−1 ∈ R(xi ),

i = 1, ⋅ ⋅ ⋅ , k,

then xT is said to be weakly reachable from x0 . The weakly reachable set of x0 is denoted by W R(x0 ). Similarly, we can dene a locally weakly reachable set as Denition 5.4. For system (5.1), xT is said to be locally weakly reachable from x0 with respect to a neighborhood, U, if there exist x0 , x1 , ⋅ ⋅ ⋅ , xk = xT , such that the trajectories xs (t) either from xs−1 to xs , or from xs to xs−1 , are contained in U, s = 1, ⋅ ⋅ ⋅ , k. The locally weakly reachable set of x0 with respect to U is denoted as W RU (x0 ).

5.1 Controllability of Nonlinear Systems

123

Denition 5.5. System (5.1) is weakly controllable at x0 ∈ M if the weakly reachable set W R(x0 ) = M. The system is said to be weakly controllable if it is weakly controllable everywhere, i.e., W R(x) = M, ∀x ∈ M. The system is said to be locally weakly controllable at x0 ∈ M, if for any neighborhood U of x0 the locally weakly reachable set W RU (x0 ) is still a neighborhood of x0 . If System (5.1) is locally weakly controllable at every x ∈ M, the system is said to be locally weakly controllable. Proposition 5.1. If system (5.1) is locally weakly controllable, then it is weakly controllable. Proof. For any two points x0 , xT draw a path P = {p(t) ∣ 0 ⩽ t ⩽ 1}, connecting x0 and xT . Using local weak controllability, for each point x = p(t), 0 ⩽ t ⩽ 1, there exists an open neighborhood Vx ⊂ W RU (x). Then ∪x∈PVx ⊃ P, is an open covering of the compact set P. Thus we can nd a nite sub-covering {Vxi ∣ 1 ⩽ i ⩽ k}. By the symmetry of local weak controllability, xi+1 ∈ W R(xi ),

i = 0, ⋅ ⋅ ⋅ , k,

where xk+1 := xT . Therefore, xT ∈ W R(x0 ), which means the system is weakly controllable. ⊔ ⊓ A point xT is said to be reachable from x0 with negative time, denoted as xT ∈ f (x,u) (x0 ) = xT . R− (x0 ) if there exists a feasible u and t < 0 such that et Proposition 5.2. Assume the feasible controls are state feedback controls, u = u(x), which are piecewise constant, then R± (x0 ) = W R(x0 ). f (x,u)

(5.3) f (x,u)

(x0 ). Then x0 = e−t (xT ). Now Proof. Assume xT ∈ R(x0 ). That is, xT = et f (x,u ) ± assume xT ∈ R (x0 ), namely there exist x1 , ⋅ ⋅ ⋅ , xk = xT , ui , Gi , such that eG i (xi ) = i xi+1 , i = 0, ⋅ ⋅ ⋅ , k. If Gi > 0, xk+1 ∈ R(xk ); and if Gi < 0, xk ∈ R(xk+1 ). That is, xT ∈ W R(x0 ). Similarly, xT ∈ W R(x0 ) implies xT ∈ R± (x0 ). ⊔ ⊓ In the following we assume that the feasible controls are piecewise constant. Dene a set of vector elds as F = { f (x, u) ∣ u = const.}. Denition 5.6. The Lie algebra generated by F, F = {F}LA is called the accessibility Lie algebra. Consider F as a distribution. If dim(F )(x0 ) = n, then it is said that the accessibility rank condition of the system (5.1) is satised at x0 . If the accessibility rank condition is satised at every x ∈ M, it is said that for system (5.1) the accessibility rank condition is satised.

124

5 Controllability and Observability

Proposition 5.3. System (5.1) is controllable, if (i). M is simply connected; (ii). all the vector elds in F are complete; (iii). if X ∈ F, then −X ∈ F; and (iv). the accessibility rank condition is satised. Proof. Note that condition (iii) implies that R± (x) = R(x),

∀x ∈ M.

Then the negative t is allowed. Using Chow’s theorem, the conclusion follows.

⊔ ⊓

Example 5.1. For an afne nonlinear system (5.2), if the drifting term f (x) = 0, then condition (iii) in the above Proposition 5.3 is satised automatically. Thus if M is simply connected; gi , i = 1, ⋅ ⋅ ⋅ , m are complete; and dim{g1 , ⋅ ⋅ ⋅ , gm }LA = n, the system is controllable. In general, to verify controllability of a nonlinear control system is a hard work. Thus we consider some weaker types of controllability. Weak controllability and local weak controllability are two of them. We consider the relationship between weak controllability and the accessibility rank condition. Theorem 5.1. Consider system (5.1), if the accessibility rank condition is satised at x0 , the system is locally weakly controllable at x0 . Proof. Since rank(F x0 ) = n, there exists a neighborhood U of x0 such that rank(F x ) = n, x ∈ U. Now if X (x0 ) = 0, ∀X ∈ F, it is obvious that rank(F x ) = 0. Therefore, there exists a f1 ∈ F such that f1 (x) ∕= 0, x ∈ U1 ⊂ U. Then we have an f integral curve, et11 (x0 ), −H1 < t1 < H1 , which is a one dimensional sub-manifold, denoted by L1 . We then claim that there exists a −H1 < t10 < H1 , and f2 ∈ F such that f at t10 , f2 (x1 ) ∕∈ Tx1 (L1 ), where x1 = et 01 (x0 ). Otherwise dim (F (x)) = 1, x ∈ L1 . (We 1

leave the detailed verication to reader.) Now consider the mapping

S2 (t2 ,t1 ) := etf22 etf11 (x0 ). Since the Jacobian matrix JS2 (0,t10 ) = ( f2 (x1 ), f1 (x1 )) is nonsingular, locally this is a diffeomorphism from ∣t2 ∣ < H2 , ∣t1 − t10 ∣ < e1 to the image of S2 . The image is a two dimensional manifold, denoted by L2 . Using the above argument again, we can show that there exists ∣t20 ∣ < H2 and ∣t˜10 − t10 ∣ < e1 , f f such that for a f 3 ∈ F, f3 (x2 ) ∕∈ Tx2 (L2 ), where x2 = et 02 et˜01 (x0 ). For notational ease, we still use t10 for t˜10 and dene

2

1

S3 (t3 ,t2 ,t1 ) := etf33 etf22 etf11 (x0 ). We obtain a three dimensional sub-manifold L3 this way. Continuing this procedure, we can nally nd f1 , ⋅ ⋅ ⋅ , fn ∈ F and construct a local diffeomorphism

5.1 Controllability of Nonlinear Systems

Sn (tn , ⋅ ⋅ ⋅ ,t2 ,t1 ) := S2 (t2 ,t1 ) := etfnn ⋅ ⋅ ⋅ etf22 etf11 (x0 ),

125

(5.4) f

0 from a neighborhood Vn of (0,tn−1 , ⋅ ⋅ ⋅ ,t10 ) ∈ ℝn to a neighborhood Un of e0n ⋅ ⋅ ⋅ f2 f1 et 0 et 0 (x0 ). In addition, we can construct a diffeomorphism 2

1

f

f

f

G(x) = e−t1 0 e−t2 0 ⋅ ⋅ ⋅ e−tn−1 0 (x), 1

2

n−1

x ∈ Un ,

which maps Un back to a neighborhood of x0 . Finally, we dene

S := G ∘ Sn (t),

t ∈ Vn ,

which is a local diffeomorphism from Vn to a neighborhood of x0 . By denition

S (Vn ) ⊂ W RU (x0 ). That is, the system is locally weakly controllable at x0 .

⊔ ⊓

Conversely, if the system is locally weakly controllable or even weakly controllable, we would like to know whether the accessibility rank condition is satised. Proposition 5.4. Assume system (5.1) is locally weakly controllable, then there exists an open dense set D ⊂ M such that the accessibility rank condition is satised on D, i.e., dim{F }∣x = n, x ∈ D. Proof. If the accessibility rank condition is satised at a point x0 , then there is a neighborhood, Ux0 of x0 , such that the accessibility rank condition is satised at all x ∈ Ux0 . So, it is obvious that the set of points, D, where the accessibility condition is satised, is an open set. Now assume D is not dense. Then there is an open set U ∕= ∅, such that dim{F }∣x < n, x ∈ U. Let k be the largest dimension of F on U, i.e., k = max dim {F }∣x < n. x∈U

Then there exists a non-empty open subset V ⊂ U, such that dim{F }∣x = k,

x ∈ V.

According to Frobenius’ theorem, for any x0 ∈ V , there exists an integral submanifold of F , I(F , x0 ), which has dimension k. It is obvious that WAV (x0 ) ⊂ I(F , x0 ) ∩V, which contradicts to the local accessibility of x0 . For analytic case, accessibility implies the accessibility rank condition.

⊔ ⊓

126

5 Controllability and Observability

Proposition 5.5. Assume system (5.1) is analytic and weakly controllable, then the accessibility rank condition is satised, i.e., dim{F }∣x = n,

x ∈ M.

dim{F }∣x < n,

∀x ∈ M.

Proof. Assume Then for any x0 ∈ M the integral sub-manifold has dim(I(F , x0 )) < n. But WA(x0 ) ⊂ I(F , x0 ), which contradicts with weak controllability. Hence there is at least one point x0 ∈ M, such that dim{F }∣x0 = n. Next we claim that if the system is weakly controllable, then for any two points x, y ∈ M, dim{F }∣x = dim{F }∣y . If we can prove the claim then the proof is done. Using the denition of accessibility, we see that there exist x0 = x, x1 , ⋅ ⋅ ⋅ , xk = y, such that xi and xi+1 can be connected by an integral curve of X ∈ F , i.e., xi+1 = etX (xi ). Now for any Y ∈ F , using the Campbell-Baker-Hausdorff formula we have (eX−t )∗Y (xi+1 ) =

f

tk

¦ adkX Y (xi ) k! .

(5.5)

k=0

Since the right hand side of (5.5) is in F (xi ) and the Y ∈ F is arbitrary (eX−t )∗ F ∣xi+1 ⊂ F ∣xi . Since (eX−t )∗ is a diffeomorphism, it is clear that dim{F }∣xi+1 ⩽ dim{F }∣xi . Exchanging xi with xi+1 and using Campbell-Baker-Hausdorff formula again we can show that dim{F }∣xi ⩽ dim{F }∣xi+1 . It follows that dim{F }∣x = dim{F }∣x1 = ⋅ ⋅ ⋅ = dim{F }∣y .

⊔ ⊓

The following corollary is an immediate consequence of Proposition 5.1, Theorem 5.1 and Proposition 5.5. Corollary 5.1. For an analytic nonlinear system of the form of (5.1), local weak controllability is equivalent to the accessibility rank condition.

5.1 Controllability of Nonlinear Systems

127

Next, we consider the real reachable set. Denote the reachable set of x0 ∈ M at time t by R(x0 ,t). It is obvious that R(x0 ) = ∪t⩾0 R(x0 ,t). Denition 5.7. 1. System (5.1) is said to be accessible at x0 ∈ M, if R(x0 ) contains a non-empty open set. The system is said to be accessible if it is accessible at every x ∈ M. 2. System (5.1) is said to be strongly accessible at x0 ∈ M, if for any T > 0, R(x0 , T ) contains a non-empty open set. The system is said to be strongly accessible if it is strongly accessible at every x ∈ M. Theorem 5.2. 1. For system (5.1) assume the accessibility rank condition is satised at x0 ∈ M, i.e., dim{F }∣x0 = n.

(5.6)

Then it is accessible at x0 . Moreover, for any T > 0 the reachable set ∪0⩽t⩽T R(x0 ,t)

(5.7)

has a non-empty interior. 2. If the system is analytic, then the accessibility rank condition (5.6) is necessary and sufcient for the system to be accessible at x0 . 3. Assume the system is analytic and the accessibility rank condition (5.6) holds. Then the set of interiors is dense in the set ∪0⩽t⩽T R(x0 ,t). Proof. 1. Recall the proof of Theorem 5.1, it is clear that in equation (5.4) all the parameters ti , i = 1, ⋅ ⋅ ⋅ , n can be chosen to be positive. Hence it implies that Sn (Vn ) is a non-empty open set, which is contained in R(x0 ). Moreover, all ti can be chosen as small as required. So for any given T > 0, we can assume ∥t∥ < T , that is, (5.7) contains a non-empty set of interiors. 2. We only have to prove the necessity. Assume dim{F }∣x0 = k < n, and p is an interior point in the reachable set of x0 . Then there is a neighborhood of p, denoted by Vp, such that Vp ⊂ R(x0 ). As argued in the proof of Proposition 5.11, dim{F } should be the same over R(x0 ), hence dim{F }∣x = k, ∀x ∈ Vp . Using Frobenius’ theorem we have I(F , p)∩Vp to be a k dimensional sub-manifold. But I(F , p) = I(F , x0 ), and R(x0 ) ⊂ I(F , x0 ), thus

128

5 Controllability and Observability

Vp = R(x0 ) ∩Vp ⊂ I(F , p) ∩Vp ⫋ Vp , which is a contradiction. 3. Assume (5.6). First, we can nd a nite subset D of F such that dim {D}LA ∣x0 = n.

(5.8)

Since dim{F }∣x0 = n, there exist n-vector elds Yi ∈ F , i = 1, ⋅ ⋅ ⋅ , n that are linearly independent at x0 . Now since each Yi can be generated by nite vector elds in F via Lie brackets, there exist nite vector elds D = {X1 , ⋅ ⋅ ⋅ , Xk } ⊂ F, such that (5.8) is satised. Without loss of generality, we assume F = D. Expand F to a symmetric set as ±F = {X1 , ⋅ ⋅ ⋅ , Xk , −X1 , ⋅ ⋅ ⋅ , −Xk }. Choose t = (t1 ,t2 , ⋅ ⋅ ⋅ ,tm ) ∈ ℝm +,

0 ⩽ m < f,

and Xi1 , Xi2 , ⋅ ⋅ ⋅ , Xim ∈ ±F, then for all possible choices of t and i = (i1 , i2 , ⋅ ⋅ ⋅ , im ) we can construct the set Xi

Xi

R∗ (x0 ) = ∪i,t et1 1 et2 2 ⋅ ⋅ ⋅ etmim (x0 ) := ∪i,t Ii (t, x0 ), X

which is the reachable set of ±F, starting from x0 . Since et−X (x0 ) = eX−t (x0 ), we can equally express Xi Xi X Ii (t, x0 ) = et1 1 et2 2 ⋅ ⋅ ⋅ etmim (x0 ), with Xi ∈ F and t ∈ ℝm . According to Chow’s theorem, R∗ (x0 ) = I (F, x0 ). Since the integral sub-manifold I (F, x0 ) has dimension n over a neighborhood of x0 , so R∗ (x0 ) contains a non-empty interior. Next, for any natural number N, we dene R∗i (x0 , N) := ∪∥t∥⩽N Ii (t, x0 ), where

(5.9)

m

∥t∥ = ¦ ∥ti ∥. i=1

Since J := {t ∈ ℝm ∣ ∥t∥ ⩽ N} ⊂ ℝm is a compact set, and R∗i (x0 , N) is the image set of J under the continuous mapping Ii (t, x0 ), R∗i (x0 , N) is also compact in M. It follows by denition that

5.1 Controllability of Nonlinear Systems

R∗ (x0 ) = ∪t,N R∗i (x0 , N).

129

(5.10)

Now Since F is a nite set, its nite subsets (ki , i = 0, 1, 2, ⋅ ⋅ ⋅ ) is countable. So the right hand side of (5.10) is a countable union. Since the left hand side contains a non-empty open set, it is of the second category. By Baire category theorem, at least one R∗i (x0 , N) is not nowhere dense. Say a particular R∗i (x0 , N) is not nowhere dense, that is, its closure R∗i (x0 , N) contains a non-empty open set. But since R∗i (x0 , N) is compact, R∗i (x0 , N) = R∗i (x0 , N) contains a non-empty open set. Say, this particular i we have Xi1 , ⋅ ⋅ ⋅ , Xim ∈ F. Dene a mapping P : ℝm → M as Xi

Xi

X

P(t1 , ⋅ ⋅ ⋅ ,tm ) = et1 1 et2 2 ⋅ ⋅ ⋅ etmim (x0 ). Then R∗i (x0 , N) ⊂ P(ℝm ).

(5.11)

Since the left hand side contains a non-empty open set, using Sard’s theorem (Refer to Appendix A.1) there is at least one point t0 , such that the Jacobian matrix of P, JP has full rank, i.e., rank(JP ∣t0 ) = n. By the analyticity we know that S := {t ∈ ℝm ∣ rank(JP ∣t ) < n} ⊂ ℝm has no interior, otherwise, rank(JP ∣t ) < n, ∀t. Then its complement Sc = ℝm 6 = {t ∈ ℝm ∣ rank(JP ∣t ) = n} is an open dense set. Now let T > 0 and let a point y be y ∈ ∪0⩽t⩽T R(x0 ,t).

(5.12)

we proceed to show that y is in the closure of ∪0⩽t⩽T R(x0 ,t), i.e., y ∈ int(∪0⩽t⩽T R(x0 ,t)). Using (5.12), we have y ∈ ∪0⩽t
130

5 Controllability and Observability

U = Sc ∩ {t ∣ ∥t∥ < T − t0 } ∩ {t ∈ ℝm + }, ¯ Since the rank of P on which is obviously an open set in ℝm . Moreover, 0 ∈ U. U is n, by the Rank theorem (Refer to Appendix A.2), P(U) is an open set in M. Let V = {I j (s, P(t)) ∣ t ∈ U}. Then V is the image of P(U) under a diffeomorphism: z $→ I j (s, z). Hence, V is open. Since ti ⩾ 0, s j ⩾ 0, and ∥t∥ + ∥s∥ < T , then V ⊂ ∪0⩽t⩽T R(x0 ,t), which means that the points in V are the interior of the reachable set (on the right hand side). Now it is enough to show that y ∈ V¯. Choose a sequence {tk } ⊂ U, such that {tk } → 0, then by continuity, lim I j (s, P(tk )) = I j (s, x0 ) = y,

k→f

which completes the proof. ⊔ ⊓ In order to investigate the strong accessibility, we need some preparations. Let L be a Lie algebra, the derived Lie algebra of L is dened as L′ = [L, L] = {[X ,Y ] ∣ X ,Y ∈ L}.

(5.13)

It is obvious that the derived Lie algebra is not only a sub-algebra but also an ideal. Now we dene a sub-algebra F 0 of F as  { }  s s  ′ (5.14) F 0 = ¦ Oi Xi + Y Oi ∈ ℝ, s ⩾ 0, ¦ Oi = 0, Xi ∈ F , Y ∈ F .  i=1 i=1 It is easy to verify that F 0 is a Lie sub-algebra. Denition 5.8. Consider the system (5.1). The Lie algebra F 0 is called the strong accessibility Lie algebra. If for x0 ∈ M dim(F 0 (x0 )) = n, it is said that the strong accessibility rank condition is satised at x0 . If the strong accessibility rank condition is satised at every x ∈ M, it is said that the strong accessibility rank condition is satised. Lemma 5.1. For any point x ∈ M, dim(F (x)) − dim(F 0 (x)) ⩽ 1. Proof. In fact, F = {F, F ′ } and F 0 ⊃ {F ′ }. According to the structures of F and F 0 one sees easily that if there is a vector eld X ∈ F, which is also in F 0 , i.e., X ∈ F 0 , then F ⊂ F 0 . The conclusion follows. ⊔ ⊓

5.1 Controllability of Nonlinear Systems

131

Since time t is taken into consideration in strong accessibility, we consider the time-state product space ℝ × M, and construct a subset of V (ℝ × M) as  } {  w ⊕ X  X ∈ F . (5.15) Fˆ = wt ˆ = {F} ˆ LA , the Lie algebra generated by F. ˆ A simple computation Denote by F shows that its derived Lie algebra is ˆ ′ = {0 ⊕ Y ∣ Y ∈ F ′ }. F ˆ can be expressed as Hence the Lie algebra F  } { ( )  s w  ′ ˆ F = ¦ Oi ⊕ X + 0 ⊕ Y  Oi ∈ ℝ, X ∈ F, Y ∈ F .  wt i=1 Consider an extended system ] [ [ ] 1 t . = fˆ = f (x, u) x

(5.16)

(5.17)

Note that each integral curve of F with initial value x0 corresponds to an integral curve of Fˆ with initial value (0, x0 ). It follows that y ∈ R(x0 , T ), if and only if ˆ (T, y) ∈ R((0, x0 )). ˆ x0 )) has nonLemma 5.2. R(x0 , T ) has non-empty interior in M, if and only if R((0, empty interior in ℝ × M. Proof. (Sufciency) Let V ⊂ R(x0 , T ) be a non-empty open set. Choose an X ∈ F and construct a mapping P : ℝ × V → ℝ × M as P(t, v) = (T + t, etX (v)).

(5.18)

Then the Jacobian matrix of P can be expressed as ] [ 1 0 JP = , ∗ Je where Je is the Jacobian of the diffeomorphism: x $→ etX (x), which is nonsingular. So, JP is nonsingular, and hence the mapping P is an open mapping. According to the construction of P, one sees ˆ P((0, f) × V ) ⊂ R((0, x0 )). Moreover, since P is an open mapping, P((0, f) ×V ) is a non-empty open subset in ˆ R((0, x0 )).

132

5 Controllability and Observability

ˆ (Necessity) Since R((0, x0 )) has a non-empty open subset, according to Theorem 5.2, for any t ∈ (0, T ) ˆ x0 ), s) ∪0⩽s⩽t R((0, contains a non-empty open subset. Let W be a non-empty open set in ℝ and V be a non-empty open set in M, such that ˆ W × V ⊂ ∪0⩽s⩽t R((0, x0 ), s). Then for any t0 ∈ W we have ˆ {t0 } × V ⊂ ∪0⩽s⩽t R((0, x0 ), s), which means V ⊂ R(x0 ,t0 ). Choose any X ∈ F and construct the mapping G : y $→ eXT −t0 (y). Since G is a diffeomorphism, G(V ) is a non-empty open set, and G(V ) ⊂ R(x0 , T ).

⊔ ⊓

From the above Lemma one sees that if there exists a T > 0 such that R(x0 , T ) has non-empty interior, then for any t > 0, R(x0 ,t) has non-empty interior as well, ˆ since the condition that R((0, x0 )) has non-empty interior is independent of time t. Now we are ready to give conditions for the strong accessibility. Theorem 5.3. Consider system (5.1). 1. If it satises the strong accessibility rank condition at x0 ∈ M, i.e., dim{F 0 }∣x0 = n,

(5.19)

then it is strong accessible at x0 . 2. If it is an analytic system and strongly accessible at x0 , it satises the strong accessibility rank condition at x0 . 3. If it either is strongly accessible at x0 or satises the strong accessibility rank condition at x0 , then for any T > 0, the int (R(x0 , T )) is dense in R(x0 , T ). Proof. From Lemma 5.2, the strong accessibility of system (5.1) at x0 is equivalent to the accessibility of system (5.17) at (0, x0 ). According to Theorem 5.14, the accessibility of system (5.17) is assured by dim{F 0 }∣(0,x0 ) = n + 1.

(5.20)

In the analytic case, it is equivalent to (5.20). Hence, to prove items 1 and 2 of the theorem we only have to show that (5.20) is equivalent to (5.19). Assume (5.20). Choosing any vector eld V ∈ Tx0 , we have ( ) 0 ˆ ∣x . ∈F 0 V ˆ , we have Recalling (5.16) for the expression of F

5.1 Controllability of Nonlinear Systems

V=

)  s  O X + Y  , ¦ i i  i=1

133

(

(5.21)

x0

with Xi ∈ F, Y ∈ F ′ , and s

0 = ¦ Oi i=1

w . wt

(5.22)

s

Now (5.22) implies

¦ Oi = 0, and (5.21) implies that V ∈ F 0 ∣x0 . But V ∈ Tx0 (M)

i=1

is chosen arbitrary, which means Tx0 (M) ⊂ F 0 ∣x0 . Hence dim{F 0 }∣x0 = n. Conversely, assume (5.19). Then for V ∈ Tx0 (M) we can express it as )   ¦ Oi Xi + Y  , i=1

( V=

s

x0

where Xi ∈ F, Y ∈ F ′ , and

s

¦ Oi = 0. Then

i=1

( (0 ⊕ V ) =

w ¦ Oi ( w t ⊕ Xi) + 0 ⊕ Y i=1 s

)    

ˆ ∣(0,x ) . ∈F 0

(5.23)

(0,x0 )

Choose any X ∈ F, then by denition, (

)  w ˆ ∣(0,x ) . ˆ (0,x ) ⊂ F ⊕ X  ∈ F∣ 0 0 wt (0,x0 )

(5.24)

Particularly, let V = X , then (5.23) and (5.24) imply that

w ˆ ∣(0,x ) . ⊕0 ∈ F 0 wt

(5.25)

ˆ ∣(0,x ) contains all the vectors in T(0,x ) (ℝ × M), Now (5.23) and (5.24) show that F 0 0 hence ˆ }∣(0,x ) = n + 1. dim{F 0 Next, we prove item 3. Let y ∈ R(x0 , T ), then there exist X1 , ⋅ ⋅ ⋅ , Xm ∈ F, and t = (t1 , ⋅ ⋅ ⋅ ,tm ) ∈ ℝm + , ∥t∥ = T , such that y = etX11 ⋅ ⋅ ⋅ etXmm (x0 ). Choose a sequence {sk } ⊂ (0,tm ), such that

134

5 Controllability and Observability

lim sk = 0.

k→f

Using Theorem 5.2, for each sk we can nd an interior xk in R(x0 , sk ). Set t k = (t1 , ⋅ ⋅ ⋅ ,tm−1 ,tm − sk ), then we can construct a sequence {yk } ⊂ M as yk = etX11 ⋅ ⋅ ⋅ etXmm−sk (xk ) := I (t k , xk ) ∈ R(x0 , T ). Since xk lies in the interior of R(x0 , sk ), and z $→ I (t k , z) is a diffeomorphism, yk is in the interior of R(x0 , T ). Let k → f, then t k → t, and by continuity lim yk = y.

k→f

⊔ ⊓

In summary, we have shown: 1. 2. 3. 4. 5. 6. 7.

Accessibility rank condition (at x0 ) ⇒ Weak controllability (at x0 ); Weak controllability ⇒ Accessibility rank condition on an open dense set; Weak controllability + Analyticity ⇒ Accessibility rank condition; Accessibility rank condition (at x0 ) ⇒ Accessibility (at x0 ); Accessibility (at x0 ) + Analyticity ⇒ Accessibility rank condition (at x0 ); Strong accessibility rank condition (at x0 ) ⇒ Strong accessibility (at x0 ); Strong accessibility (at x0 ) + Analyticity ⇒ Strong accessibility rank condition (at x0 ).

Denition 5.9. Given a vector eld X. A distribution D is said to be X invariant, if adX Y ∈ D,

∀Y ∈ D,

Similarly, a Lie algebra L is said to be X invariant, if adX Y ∈ L,

∀Y ∈ L.

Given two sets of vector elds, F, G, the smallest distribution that contains G and is F invariant is denoted by ⟨F ∣ G⟩. The smallest involutive distribution that contains G and is F invariant is denoted by [F ∣ G]. When Lie algebra is considered as a distribution, the same notations are used. In fact, when the dimension of accessibility or strong accessibility Lie algebra is considered, they are regarded as distributions. Next, we consider an afne nonlinear system (5.2). By denition we have Proposition 5.6. For the afne nonlinear system (5.2) the accessibility Lie algebra is the Lie algebra generated by { f , g1 , ⋅ ⋅ ⋅ , gm }, i.e., F = { f , g1 , ⋅ ⋅ ⋅ , gm }LA . The strong accessibility Lie algebra is the smallest Lie algebra contains {g1 , ⋅ ⋅ ⋅ , gm } and f -invariant, i.e., F 0 = [ f ∣ g1 , ⋅ ⋅ ⋅ , gm ].

5.1 Controllability of Nonlinear Systems

135

Example 5.2. Consider the linear system (1.3). It is ready to verify that the accessibility Lie algebra is F = Span{Ax, B, AB, ⋅ ⋅ ⋅ , An−1 B};

(5.26)

and the strong accessibility Lie algebra is F 0 = Span{B, AB, ⋅ ⋅ ⋅ , An−1 B}.

(5.27)

So for linear systems controllability, weak controllability, accessibility, and strong accessibility are all equivalent. Example 5.3. Consider a time-varying linear system ⎧ m  ⎨x = A(t)x + b (t)u := A(t)x + B(t)u  ⎩y = c(t)x.

¦

i

i

i=1

Consider t as a variable, then system (5.28) can be expressed as ⎧  t = 1   m ⎨ x = A(t)x + B(t)u := f (t, x) + ¦ gi (t)ui   i=1  ⎩ y = c(t)x.

(5.28)

(5.29)

It is easy to obtain that [ [ f , gi ] =

]

0

, w bi (t) − A(t)bi(t) wt

1 ⩽ i ⩽ m;

and [gi , g j ] = 0,

1 ⩽ i, j ⩽ m.

Finally, we dene (

'ck B(t) =

) w − A(t) 'ck−1 B(t), wt

k = 1, 2, ⋅ ⋅ ⋅ ,

where 'c0 B(t) := B(t). Then it is easy to see that the accessibility Lie algebra is F = Span{Bic (t) ∣ i = 0, 1, 2, ⋅ ⋅ ⋅ }.

(5.30)

Example 5.4. Consider a bilinear system m

x = Ax + ¦ ui Bi x, i=1

Note that the Lie bracket of two linear forms is

x ∈ ℝn .

(5.31)

136

5 Controllability and Observability

[Px, Qx] = (QP − PQ)x = [Q, P]x, where the Lie bracket [Q, P] of two n × n matrices is dened as in the Lie algebra gl(n, ℝ). Consider a Lie sub-algebra H ⊂ gl(n, ℝ), generated as H = {A, B1 , ⋅ ⋅ ⋅ , Bm }LA < gl(n, ℝ), then it is clear that for system (5.31) the accessibility Lie algebra is F = {Dx ∣ D ∈ H}.

5.2 Observability of Nonlinear Systems For system (5.1), if in the state space there are two points x1 and x2 such that for any feasible control u the outputs are identically the same, i.e., y(t, x1 , u) ≡ y(t, x2 , u),

t ⩾ 0, ∀u,

then x1 and x2 are said to be in-distinguishable. The set of in-distinguishable points with respect to x0 is denoted by ID(x0 ). Denition 5.10. System (5.1) is said to be observable at x0 if its in-distinguishable set contains only one point, i.e., ID(x0 ) = x0 . Remark 5.1. For a linear system (1.3), observability is independent of the control, since ∫t y = CeA(t−t0 ) x0 + C eA(t−W ) Bu(W )dW . t0

This implies

y2 − y1 = CeA(t−t0 ) (x2 − x1 ),

which is independent of control u. Similar to the case of controllability, global observability is also a difcult topic in the investigation of nonlinear systems. For this reason we are primarily interested in local observability. For a point x0 ∈ M and a neighborhood U of x0 , given a T > 0, a U feasible control u is such a control that the trajectory remains in U, i.e., x(t, x0 , u) ∈ U,

∀t ∈ [0, T ].

If for any T > 0 and any corresponding U feasible control u, y(t, x1 , u) = y(t, x2 , u),

t ∈ [0, T ],

then x1 and x2 are said to be U-in-distinguishable. The set of U-in-distinguishable points of x0 is denoted by IDU (x0 ).

5.2 Observability of Nonlinear Systems

137

Denition 5.11. System (5.1) is said to be locally observable at x0 if for any neighborhood U of x0 the U-in-distinguishable set consists of only one point, i.e., IDU (x0 ) = {x0 },

∀U.

The system is said to be locally weakly observable at x0 if there exists a neighborhood U of x0 such that the U-in-distinguishable set consists only one point, i.e., IDU (x0 ) = {x0 }. It is worth noting that local observability is a very strong property. In fact, it implies observability. So we are mostly interested in local weak observability. To investigate the observability we construct a set of output related functions.  } {  s  i (5.32) H = ¦ Oi LX i ⋅ ⋅ ⋅ LX i (h j ) s, ki < f, 1 ⩽ j ⩽ m, Oi ∈ ℝ, Xk ∈ F . 1 ki  i=1 Using H we dene a co-distribution as H = {dh ∣ h ∈ H}. H is called the observability co-distribution. Denition 5.12. System (5.1) is said to satisfy the observability rank condition at x0 if dim{H }∣x0 = n.

(5.33)

If for every x ∈ M (5.33) is satised, the system is said to satisfy the observability rank condition. Next, we consider the relationship between observability and local weak observability. Theorem 5.4. If system (5.1) satises the observability rank condition at x0 , then it is locally weakly observable at x0 . To prove this theorem we need the following lemma. Lemma 5.3. Let V ⊂ M be an open set. If there exist x1 , x2 ∈ V such that x1 ∈ IDV (x2 ), then for any function c(x) ∈ H, c(x1 ) = c(x2 ). Proof. Choosing any X1 , ⋅ ⋅ ⋅ , Xk ∈ F, we construct X

Ik (x) = etX11 ⋅ ⋅ ⋅ etkk (x).

(5.34)

138

5 Controllability and Observability

when ∥t∥ is small enough, we have h j (Ik (x1 )) = h j (Ik (x2 )),

j = 1, ⋅ ⋅ ⋅ , m.

(5.35)

Differentiating both sides with respect to t1 we have X

X

X

X

LX1 h j (etX11 ⋅ ⋅ ⋅ etkk (x1 )) = LX1 h j (etX11 ⋅ ⋅ ⋅ etkk (x2 )). Setting t1 = 0 yields LX1 h j (etX22 ⋅ ⋅ ⋅ etkk (x1 )) = LX1 h j (etX22 ⋅ ⋅ ⋅ etkk (x2 )). Continuing this procedure with respect to t2 , t3 , ⋅ ⋅ ⋅ , tk , we nally have LXk LXk−1 ⋅ ⋅ ⋅ LX1 h j (x1 ) = LXk LXk−1 ⋅ ⋅ ⋅ LX1 h j (x2 ).

⊔ ⊓

Now we are ready to prove Theorem 5.4. Proof. (Proof of Theorem 5.4) Since dim H = n, we can choose n functions ci (x), i = 1, ⋅ ⋅ ⋅ , n, such that they are linearly independent at x0 . Dene a mapping P(x) = (c1 (x), c2 (x), ⋅ ⋅ ⋅ , cn (x))T . By construction there exists a neighborhood U of x0 such that P : U → P(U) ⊂ ℝn is a diffeomorphism. Now for any x0 ∕= x ∈ U, since P(x) ∕= P(x0 ), x ∕∈ IDU (x0 ). Hence ⊔ ⊓ IDU (x0 ) = x0 . Now a natural question is if the observability rank condition is necessary for local weak observability. We have the following results. Theorem 5.5. If system (5.1) is locally weakly observable, then the observability rank condition is satised on an open dense subset of M. Proof. The set of points where the observability rank condition is satised is obviously an open set. So we have only to prove it is dense. Assume there is a non-empty open set U, such that dim(H (x)) = k < n,

x ∈ U.

(5.36)

In fact, we can assume k = maxx∈U dim(H (x)). Then there exists an open subset V ⊂ U, such that dim(H (x)) = k, x ∈ V . Then we can nd k functions in H such that all the co-vector elds in H can be expressed as a linear combination of the k forms dci (x), i = 1, ⋅ ⋅ ⋅ , k, locally on U. Constructing a local coordinate chart as (U, z), where z = (z1 , z2 ) and z1 = (c1 , ⋅ ⋅ ⋅ , ck )T . Then the system (5.1) can be expressed as ⎧ 1 1  ⎨z = f (z, u) 2 (5.37) z = f 2 (z, u)  ⎩ y j = h j (z), j = 1, ⋅ ⋅ ⋅ , p.

5.2 Observability of Nonlinear Systems

139

Since dh j ∈ H , h j depends only on z1 , i.e., h j (z) = h j (z1 ). Next, we claim that f 1 depends only on z1 too, i.e., f 1 (z, u) = f 1 (z1 , u). Otherwise, say for some 1 ⩽ i ⩽ k, k + 1 ⩽ j ⩽ n

w fi1 (z, u) (x) ∕= 0, w z2j

x ∈ U.

Then the j-th component of L f (z,u) (ci (x)) is  w f 1 (z, u) dL f (z,u) (ci (x)) j = i 2 (x) ∕= 0. wzj Hence for some x ∈ U,

dL f (z,u) (ci (x)) ∕∈ H (x),

which is a contradiction. Now system (5.37) can be locally expressed as ⎧ 1 1 1  ⎨z = f (z , u) 2 z = f 2 (z, u)  ⎩ y j = h j (z1 ), j = 1, ⋅ ⋅ ⋅ , p.

(5.38)

Choosing two points z1 , z2 ∈ U as z1 = (z11 , z21 ),

z2 = (z12 , z22 ),

z11 = z12 ,

z21 ∕= z22 .

Then y(z1 , u(t)) ≡ y(z2 , u(t)), The system is not locally observable at x0 .

z ∈ U. ⊔ ⊓

Note that in the above proof, we assume the feasible controls are piecewise constant. If the feasible controls are state feedback controls, the proof remains true (with some mild modication). Finally we consider the analytic case. Theorem 5.6. Assume system (5.1) is analytic and satises the controllability rank condition. It is locally weakly observable if and only if the observability rank condition is satised. Proof. Since the controllability rank condition is satised, it is locally weakly controllable. That is, for any two points x, y ∈ M, x ∈ W R(y). Using Theorem 5.4, we only have to prove the necessity. Using Theorem 5.5, it sufces to show that H has constant dimension everywhere. Using the local weak controllability, it is enough

140

5 Controllability and Observability

to show dim(H ) is the same for any two points connected by an integral curve of F . Let x2 = etX (x1 ), for some X ∈ F . For any Z ∈ H , we use Campbell-BakerHausdorff formula to get (etX )∗ Z (x2 ) =

f

tk

¦ LkX Z (x1 ) k! .

k=0

Now the left hand side is in H ∣x1 and (etX )∗ is an isomorphism. It follows that dim{H }∣x2 ⩽ dim{H }∣x1 . Using (eX−t )∗ in reverse time, we also have dim{H }∣x1 ⩽ dim{H }∣x2 .

⊔ ⊓

Example 5.5. Consider a linear system (1.3). As we know from the last section that F = Span{Ax, B, AB, ⋅ ⋅ ⋅ , An−1 B}. Then H = Spanℝ {C j x, ⋅ ⋅ ⋅ ,C j An−1 ,C j Ax,C j B, ⋅ ⋅ ⋅ ,C j An−1 B ∣ 1 ⩽ j ⩽ m}, where the subscript ℝ is used to emphasize the span is over ℝ. Then it is ready to verify that the observability co-distribution is H = Span{C,CA, ⋅ ⋅ ⋅ ,CAn−1 }.

(5.39)

So for linear systems observability is equivalent to the observability rank condition. Example 5.6. Consider a time-varying linear system (5.28). Using the extended system (5.29), we can verify that the observability rank condition is equivalent to ⎡ ⎤ c(t) ⎢ ’oC(t) ⎥ ⎢ ⎥ rank ⎢ ’2C(t) ⎥ = n, ⎣ o ⎦ .. . where ’oC(t) = and ’k+1 o C(t) =

d k ’ C(t) + ’koC(t)A(t), dt o

5.3 Kalman Decomposition Consider a dynamic system

d C(t) + C(t)A(t), dt k ⩾ 1.

5.3 Kalman Decomposition

x = f (x),

x ∈ M.

141

(5.40)

Let ' be a distribution, and assume for a given x0 ∈ M there is a neighborhood U of x0 such that ' is involutive over U and with dim(' (x)) = k, ∀x ∈ U. Then by Frobenius’ theorem there exists a local coordinate chart (V, z), x0 ∈ V ⊂ U, such that { } w w ' = Span ,⋅⋅⋅ , . (5.41) w z1 w zk Such a coordinate frame is called a at coordinate frame with respect to ' . Lemma 5.4. Let ' be as above. If ' is f -invariant, then under the at coordinate frame the system (5.40) can be expressed as { z1 = f 1 (z) (5.42) z2 = f 2 (z2 ), z ∈ V, where z1 = (z1 , ⋅ ⋅ ⋅ , zk ). Proof. Assume f 2 (z) ∕= f 2 (z2 ), say

w f 2 (z) w z1i

∕= 0 for some 1 ⩽ i ⩽ k. Then

⎡

⎤ w f 1 (z) ] ⎢ [ 1 ⎥ w ⎢ wz ⎥ f , 1 = ⎢ 2 i ⎥ ∕∈ ' , ⎣ w f (z) ⎦ w zi

w z1i ⊔ ⊓

which is a contradiction.

Next we consider system (5.1). Assume x0 is a regular point with respect to F 0 . That is, there exists a neighborhood U of x0 such that dim(F 0 (x)) = k,

x ∈ U.

Proposition 5.7. Assume x0 is a regular point of F 0 with dim(F 0 ) = k. Then there exists a local coordinate chart (V, z) of x0 , such that system (5.1) can be locally expressed as { z1 = f 1 (z, u) (5.43) z2 = f 2 (z2 ), z ∈ V, where F 0 = Span{ wwz1 , i = 1, 2, ⋅ ⋅ ⋅ , k}. i

Proof. Since F 0 is a Lie algebra, as a distribution it is involutive. Moreover, it is obvious that F 0 is F invariant. According to Lemma 5.4, there exists a local at coordinate frame such that the system can be expressed as { z1 = f 1 (z, u) z2 = f 2 (z2 , u), z ∈ V,

142

5 Controllability and Observability

with F 0 = Span wwz1 . Now we only have to show that f 2 is independent of u. If f 2 i

would depend on u, then we can nd two different constant controls u1 ∕= u2 , such that f 2 (u1 ) ∕= f 2 (u2 ). Then f (u1 ) − f (u2 ) ∕∈ F 0 , which contradicts to the denition of F 0 . ⊔ ⊓ Applying the above proposition to an afne nonlinear system (5.2), we have the following result: Corollary 5.2. For system (5.2), assume x0 is a regular point of F 0 with dim(F 0 ) = k. Then there exists a local coordinate chart (V, z) of x0 , such that the system can be locally expressed as ⎧ m  ⎨z1 = f 1 (z) + g (z)u ¦ i i (5.44) i=1  ⎩z2 = f 2 (z2 ), z ∈ V, where F 0 = Span{ wwz1 , i = 1, 2, ⋅ ⋅ ⋅ , k}. i

Remark 5.2. Replacing the strong accessibility Lie algebra F 0 by the accessibility Lie algebra F , similar decompositions can be obtained. Next, we consider the observability co-distribution H . A vector eld X is said to be perpendicular to H if for any Z ∈ H , ⟨Z , X ⟩ = 0. Now we dene a distribution H ⊥ as H ⊥ = Span{X ∣ ⟨Z , X⟩ = 0, ∀ Z ∈ H }.

(5.45)

Lemma 5.5. The distribution H ⊥ is involutive and F invariant. Proof. Let X ,Y ∈ H

⊥

and Z ∈ H . Then

⟨Z , adX Y ⟩ = LX ⟨Z ,Y ⟩ − ⟨LX Z ,Y ⟩ = 0. Hence [X,Y ] ∈ H ⊥ , which means H ⊥ is involutive. Let X ∈ F and Y ∈ H ⊥ , to show that H ⊥ is F invariant we have to show that adX Y ∈ H ⊥ . Now let Z ∈ H , then ⟨Z ,Y ⟩ = 0, and ⟨Z , adX Y ⟩ = LX ⟨Z ,Y ⟩ − ⟨LX Z ,Y ⟩ = 0. To see that the second term is zero we note that since X ∈ F and Z ∈ H , LX Z ∈H . ⊓ ⊔ Theorem 5.7. Assume x0 is a regular point for the observability co-distribution H . Then there exists a local coordinate chart (V, z) of x0 , such that system (5.1) can be expressed locally as ⎧ 1 1  ⎨z = f (z, u)  2 (5.46) z = f 2 (z2 , u), z ∈ V  ⎩ 2 y = h(z ),

5.3 Kalman Decomposition

143

{

} where the distribution H ⊥ = Span wwz1 . i

Proof. Choose a at local coordinate frame z such that { } w , i = 1, ⋅ ⋅ ⋅ , k . H ⊥ = Span w z1i 〈 〉 Under this coordinate frame, since dh ∈ H , dh, wwz1 = 0. So h(z) = h(z2 ). Since H

⊥

i

is F invariant,

w f 2 (z) = 0, w z1i

i = 1, ⋅ ⋅ ⋅ , k. ⊔ ⊓

(5.46) follows.

Corollary 5.3. For afne nonlinear system (5.2) assume x0 is a regular point for the observability co-distribution H . Then there exists a local coordinate chart (V, z) of x0 , such that the system (5.2) can be expressed locally as ⎧ m  1   z1 = f (z) + ¦ gi (z)ui   i=1 ⎨ m (5.47) 2 2  2 z = f (z ) +  ¦ gi (z2 )ui , z ∈ V    i=1  ⎩ y = h(z2 ), where the distribution H ⊥ = Span{ wwz1 }. i

To obtain a Kalman decomposition for nonlinear control systems, we need the following lemmas. Lemma 5.6. The distributions F 0 ∩H

⊥

and F 0 ∪H ⊥ are involutive.

Proof. The involutivity of F 0 ∩H ⊥ is obvious, because according to the denition it is ready to verify that the intersection of any two involutive distributions is still involutive. As for the involutivity of F 0 ∪H ⊥ , since H ⊥ is invariant with respect to F, it is also invariant with respect to F , particularly to F 0 . ⊔ ⊓ The following lemma is called the generalized Frobenius’ theorem in some literatures. Lemma 5.7. Assume distributions '1 ⊂ ' 2 ⊂ ⋅ ⋅ ⋅ ⊂ 'k are involutive with dimensions d1 < d2 < ⋅ ⋅ ⋅ < dk on a neighborhood, U of x0 . Then there exists a local coordinate frame (V, z), x0 ∈ V ⊂ U, z = {z11 , ⋅ ⋅ ⋅ , z1d1 , z21 , ⋅ ⋅ ⋅ , z2d2 −d1 , ⋅ ⋅ ⋅ , zk1 , ⋅ ⋅ ⋅ , zkdk −dk−1 , zk+1 }, such that {

's =

 } w   i ⩽ s, j = 1, ⋅ ⋅ ⋅ , ds − ds−1 , w zij 

1 ⩽ s ⩽ k.

(5.48)

144

5 Controllability and Observability

Proof. We prove it inductively. Since D1 is involutive, we can nd a at coordinate chart (U, z) such that (5.48) holds for s = 1. Now assume it holds for i ⩽ s. Denote ps := ds − ds−1 , with d0 = 0. Then we can nd ps+1 vector elds {X1 , ⋅ ⋅ ⋅ , X ps+1 } such that ⎫ ⎧⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 I p1 0        ⎜ 0 ⎟ ⎜I p2 ⎟ ⎜ 0 ⎟     ⎜ ⎟ ⎟ ⎜ ⎜  ⎟    ⎜ ⎟ ⎟ ⎜ ⎜  ⎟ . . . ⎬ ⎨⎜ . ⎟ ⎜ . ⎟ ⎜ . ⎟ ( ) . . . ⎟ , ⎜ ⎟ , ⎜ ⎟ , X1 , ⋅ ⋅ ⋅ , X p = 's+1 . Span ⎜ s+1 ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ 0 ⎟ ⎜ 0 ⎟ ⎜ I ps ⎟    ⎜     ⎜ . ⎟ ⎜ . ⎟ ⎜ . ⎟       ⎝ .. ⎠ ⎝ .. ⎠ ⎝ .. ⎠   ⎭ ⎩ 0 0 0 Note that there is at least a ps+1 × ps+1 minor in the last n − ds rows of ) ( X1 , ⋅ ⋅ ⋅ , X ps+1 that is non-singular. Without loss of generality, assume the last ps+1 rows are linearly independent. Denote this minor by T , and dene ( ) ( ) ( ) −1 ∗ Y1 , ⋅ ⋅ ⋅ , Yps+1 = X1 , ⋅ ⋅ ⋅ , X ps+1 T = . I ps+1 Then Y1 , ⋅ ⋅ ⋅ ,Yps+1 are commutative. (Refer to the proof of Frobenius’ theorem.) Let Vi , i = 1, ⋅ ⋅ ⋅ , n be a set of linearly independent vector elds, where the rst ds vector elds are wwzi , i = 1, ⋅ ⋅ ⋅ , s, j = 1, ⋅ ⋅ ⋅ , pi , and j

Vds + j = Y j ,

j = 1, ⋅ ⋅ ⋅ , ps+1 .

Construct a local diffeomorphism from a neighborhood of the origin, W ⊂ ℝn to U as I (t) := eVt11 ⋅ ⋅ ⋅ eVtnn (x0 ). Then (I (W ),t) is a local coordinate chart of x0 . Moreover, on this coordinate chart (5.48) holds for s + 1. (Refer to the proof of Frobenius’ theorem for detailed argument.) Assume x0 ∈ M is a regular point for F 0 , F 0 ∩H ⊥ , F 0 ∪H ⊥ . Using Lemmas 5.6 and 5.7, we can nd a local coordinate chart (V, z), V ⊂ U with z = (z1 , z2 , z3 , z4 ) such that { } F 0 ∩H ⊥ = Span w 2 , wz { } w w , (5.49) , F 0 = Span w z2 w z1 } { H ⊥ = Span w 2 , w 4 . wz wz ⊔ ⊓

References

145

Theorem 5.8. (Kalman Decomposition) Assume for system (5.1) x0 ∈ M is a regular point for F 0 , F 0 ∩H ⊥ , F 0 ∪H ⊥ . Then we can nd a local coordinate chart (V, z) of x0 such that system (5.1) can be locally expressed as ⎧ 1 1 1 3   z = f (z , z , u)    z2 = f 2 (z, u)   ⎨ (5.50) z3 = f 3 (z3 )    4 4 3 4   z = f (z , z )     ⎩y = h(z1 , z3 ), where F 0 = Span{ wwz2 , wwz1 }, H ⊥ = Span{ wwz2 , wwz4 }. Proof. Express the system into the coordinate frame introduced in (5.49). Arguing as in the proofs of Proposition 5.7 and Theorem 5.7, equation (5.50) follows. ⊔ ⊓ Applying Theorem 5.8 to afne nonlinear systems we have Corollary 5.4. (Kalman Decomposition) Assume for system (5.2) x0 ∈ M is a regular point for F 0 , F 0 ∩H ⊥ , F 0 ∪H ⊥ . Then we can nd a local coordinate chart (V, z) of x0 such that system (5.2) can be locally expressed as ⎧⎡ ⎤ ⎡ ⎤ ⎡ 1 1 3 ⎤  f 1 (z1 , z3 ) z1   g (z , z )  ⎢ ⎥ ⎥ ⎢   2⎥ 2 (z) ⎥ ⎢ ⎢ ⎥ ⎢  z  f  ⎥ ⎢ ⎥ ⎢ g2 (z) ⎥ ⎨⎢ ⎥u ⎥+⎢ ⎢ ⎥=⎢ ⎢z3 ⎥ ⎢ f 3 (z3 ) ⎥ ⎣ ⎦ (5.51) 0 ⎣ ⎦ ⎦ ⎣    0   z4 f 4 (z3 , z4 )    ⎩y = h(z1 3 , z ), where F 0 = Span{ wwz2 , wwz1 }, H ⊥ = Span{ wwz2 , wwz4 }.

References 1. Baillieul J. Controllability and observability of polynomial dynamical systems. Nonlin. Anal., Theory, Meth. Appl., 1981, 5(5): 543–552. 2. Cheng D. Canonical structure of nonlinear systems. Proc. 10th IFAC World Congress, 1987: 103–107. 3. Gauthier J, Bornard G. Observability of any u(t) of a class of nonlinear systems. IEEE Trans. Aut. Contr., 1981, 26(4): 922–926. 4. Jurdjevic V. Geometric Control Theory. Cambridge: Cambridge University Press, 1997. 5. Jurdjevic V, Sallet G. Controllability properties of afne systems. SIAM J. Contr. Opt., 1984, 22(3): 501–508. 6. Krener A. A decomposition theory for differentiable systems. SIAM J. Contr. Opt., 1977, 15(5): 289–297. 7. Respondek W. On decomposition of nonlinear control systems. Sys. Contr. Lett., 1982, 1(5): 301–308. 8. Van der Schaft A. Observability and controllability for smooth nonlinear systems. SIAM J. Contr. Opt., 1982, 20(3): 338–354.

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5 Controllability and Observability

9. Stefani G. On the local controllability of a scalar input-control system//Byrnes C, Lindquist A. Theory and Application of Nonlinear Control System Amsterdam: Morth-Holland, 1986: 167–182. 10. Susmann H. A general theorem on local controllability. SIAM J. Contr. Opt., 1987, 25(1): 158–194. 11. Sussmann H. Orbits of families of vector elds and integrability of distributions. Trans. American Math. Soc., 1973, 180: 171–188.

Chapter 6

Global Controllability of Afne Control Systems

This chapter investigates global controllability of general afne control systems, including linear and nonlinear, switched and non-switched ones. The main purpose is to establish criteria for the controllability of nonlinear systems, which is similar to those for linear systems. Since controllability is a property of a set of vector elds, there is no essential difference between switched and non-switched systems as far as controllability is concerned. That is the reason we treat all types of systems together.

6.1 From Linear to Nonlinear Systems Investigation of controllability is one of the most fundamental tasks in control theory. It has been investigated since the very early days of the modern control theory and many signicant results have been obtained. For a linear control system m

x = Ax + ¦ bi ui := Ax + Bu,

x ∈ ℝ n , ui ∈ U .

(6.1)

i=1

Denote by U the set of feasible controls. For convenience, in this chapter we assume U is the set of piecewise continuous functions. It is well known that (6.1) is controllable, if and only if the controllability matrix ] [ (6.2) C = B AB ⋅ ⋅ ⋅ An−1 B has full row rank, i.e., rank(C ) = n. Consider an afne nonlinear control system m

x = f (x) + ¦ gi (x)ui := f (x) + G(x)u,

x ∈ M, u ∈ U m ,

(6.3)

i=1

where M is an n dimensional manifold (with ℝn as a special case); f (x), gi (x) ∈ V f (M), i.e., they are Cf vector elds on M. As we have discussed in the previous chapter, in this case the controllability matrix is generalized to the accessibility Lie algebra La = { f , g1 , ⋅ ⋅ ⋅ , gm }LA , and the strong accessibility Lie algebra D. Cheng et al., Analysis and Design of Nonlinear Control Systems © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2010

(6.4)

148

6 Global Controllability of Afne Control Systems

Lsa = {adkf gi ∣ i = 1, ⋅ ⋅ ⋅ , m; k = 0, 1, ⋅ ⋅ ⋅ }LA .

(6.5)

Note that in the linear case (6.5) is degenerated to (6.3), while La may have one more dimension caused by f (x) if f (x) ∕∈ Lsa (x). In the last chapter, we use R(x0 ) to denote the reachable set from x0 . That is, y ∈ R(x0 ) means there exists u(t) ∈ U , such that the trajectory of the controlled system starting from x0 will reach y at certain time t1 > t0 . We use R(x0 , T ) to denote the reachable set from x0 at a particular moment T > t0 . A big difference between linear and nonlinear systems is that the reachability of linear systems is symmetric, i.e., x ∈ R(y) implies y ∈ R(x) and vise versa. But, it is in general not true for nonlinear systems. Using the (strong) accessibility Lie algebra, we obtained some local control properties of nonlinear systems. But the (strong) accessibility Lie algebra can not determine even the local controllability. Although there are many other controllability results for nonlinear systems, particularly for some nonlinear systems of special forms, such as those in [6, 10], there are few simple, general or global controllability results as in the linear case. This fact shows that controllability of nonlinear systems is a very difcult topic, and global controllability is even more difcult. In [11] Sun and Lei proved a necessary and sufcient condition for planar nonlinear control systems to be globally controllable. Its basic idea is: on each integral curve of g, f has to cross “in” and “out” of the curve at certain points. It will be discussed in Example 6.2 for details. Before our general discussion on nonlinear controllability, we want rst to point out that the controllability rank condition is not a precise description of controllability. It can hardly provide any global picture. The following example shows that it is far from being necessary for global controllability of nonlinear systems. Example 6.1. Consider the following system x = f (y), y = u, (x, y) ∈ ℝ2 , u ∈ U ,

(6.6)

where (0 < G ≪ 1)

f (y) =

⎧ 1 − 1 2 −  (y−1) e (y−1−G )2 ,  ⎨e −

1 (y+1)2

−

1 (y+1+G )2

−e e   ⎩ 0, otherwise.

1 < y < 1+G ,

−1 − G < y < −1

This is a Cf system. Due to the special form of the vector elds, it is easy to see that the system is globally controllable, but the rank of its accessibility Lie algebra is 1 except over two narrow strips, where the rank is 2. Next, if we restrict the system on each strip, say, let M = {(x, y) ∈ ℝ2 ∣ 1 < y < 1 + G }. Then the rank condition is satised but it is not even locally controllable. (Refer to Fig. 6.1.)

6.1 From Linear to Nonlinear Systems

Fig. 6.1

149

Rank Condition vis Controllability

So the (strong) accessibility Lie algebra is not enough to describe controllability. We have to “dig in” into its hierarchy structure. Since controllability is basically the property of a set of vector elds, there is no additional difculties when we consider a switched system. A switched afne nonlinear system is dened as m

V (t)

x = f V (t) + ¦ gi

ui ,

(6.7)

i=1

where V : [0, f) → / is a right continuous function, and the index set / = {1, 2, ⋅ ⋅ ⋅ , N} is for N different smooth models: { f i , gi1 , ⋅ ⋅ ⋅ , gim ∣i = 1, ⋅ ⋅ ⋅ , N}. As a special case, a switched linear system is dened as x = AV (t) x + BV (t)u,

x ∈ ℝn , u ∈ U m .

(6.8)

In this chapter, controllability of afne systems is treated in a unied way, no matter linear or nonlinear, switched or no-switched. In the sequel, only controllability of system (6.7) is treated formally. (6.8) is considered as a particular case of (6.7). Moreover, in our approach N = 1 is allowed. That is, we also consider (6.3) (including (6.1)) as a particular case of (6.7) with only a single switching model. The basic idea in the following approach is rst to construct a sequence of distributions, called the controllability distributions, ' 1 , '2 , ⋅ ⋅ ⋅ , which correspond to Span{B}, Span{B, AB}, Span{B, AB, A2B}, ⋅ ⋅ ⋅ , of the linear case, then to check when the trajectory can be driven from the integral manifold of some lower dimensional distribution to the integral manifold of some higher dimensional distribution. For statement ease, we introduce some notations: ∙ IGk := (−G , G ) × (−G , G ) × ⋅ ⋅ ⋅ × (−G , G ) ⊂ ℝk , where 0 < G ≪ 1. We simply use / 01 2 k

I k when we don’t need to specify how small G > 0 is.

150

6 Global Controllability of Afne Control Systems

∙ F is used to describe the set of drift vector elds of (6.7), i.e., F = ( f 1 , f 2 , ⋅ ⋅ ⋅ , f N ). ∙ Gi is the set of input channels of the i-th switching model, i.e., Gi = (gi1 , ⋅ ⋅ ⋅ , gim ), and G as the union of Gi , i.e., G = (G1 , ⋅ ⋅ ⋅ , GN ). ∙ G : the Lie algebra generated by all input channels, i.e., G = {G}LA . ∙ IG : The integral manifold of G (with IG (x0 ) as the maximal integral manifold of G passing through x0 ). ∙ Let ' be a distribution. '¯ denotes its involutive closure, i.e., the smallest involutive distribution containing ' .

6.2 A Sufcient Condition This section presents a sufcient condition for controllability of system (6.7). As a byproduct we also show how the trajectories can go from leaves (i.e., integral manifolds) of a lower dimensional distribution to leaves of the enlarged distribution. To begin with, we need some new concepts: Denition 6.1. Let S := {XO ∣O ∈ / } ⊂ V (M) be a (nite or innite) set of vector elds. A point x0 ∈ M is called an interior (point) of S, if there exists a nite set {Y1 , ⋅ ⋅ ⋅ ,Ys } ⊂ S such that 0 is an interior point of the convex cone generated by Y1 (x0 ), ⋅ ⋅ ⋅ ,Ys (x0 ). The following proposition tells how to verify the interior of S. Proposition 6.1. x0 is an interior of S, if and only if there exists a nite set {Y1 , ⋅ ⋅ ⋅ , Ys } ⊂ S such that (i). there exist n vectors from {Y1 (x0 ), ⋅ ⋅ ⋅ ,Ys (x0 )} that are linearly independent; (ii). there exist ci > 0, i = 1, ⋅ ⋅ ⋅ , s such that s

¦ ciYi (x0 ) = 0.

(6.9)

i=1

Proof. Necessity of (i) comes from the fact that if (i) fails, it means that Yi (0) are linearly dependent. Then the convex cone generated by them, denoted by con {Y1 (0), ⋅ ⋅ ⋅ ,Ys (0)}, has dimension less than n. Thus it does not have any interior point in the ℝn topology. s

(6.9) with

¦ ci = 1 is a necessary and sufcient condition for 0 to be an interior

i=1

point of the convex cone (with respect to subspace topology, which could be of s

lower dimension). Since the interior is empty, the condition ¦ ci = 1 can obviously i=1

be removed.

6.2 A Sufcient Condition

151

As for the sufciency, condition (i) assures that the cone has dimension n. So as an interior of the cone, 0 is also an interior in the ℝn topology. ⊔ ⊓ Remark 6.1. From Proposition 6.1 it is obvious that to satisfy conditions (i) and (ii) the number of vectors should be s ⩾ n + 1. But it is easy to see that if such a set of vectors exist then at most 2n should be enough (the others are redundant). Throughout this chapter we need the following assumption: A1. G has constant dimension. From Frobenius’ theorem [9] A1 assures that for each x ∈ M the maximal integral manifold IG (x) exists. We now give a rigorous denition for controllability discussed in this chapter. Denition 6.2. System (6.7) is said to be controllable if for any two given points x, y ∈ M, x ∈ R(y). It is controllable on a subset, W ⊂ M, (briey, W is controllable) if for any two given points x, y ∈ W , x ∈ RW (y), which means that there is a trajectory that starts from y, travels within W , and then reaches x. The following lemma is a key for our main result, and is also interesting in its own right: Lemma 6.1. Assume A1 and let x0 ∈ M be an interior of {F, G }. Then there exists a neighborhood W of x0 , which is a controllable subset. To prove Lemma 6.1, we need some preparations: Lemma 6.2. Let Z = {Z1 , ⋅ ⋅ ⋅ , Zs } be a set of vector elds, and G = {Z}LA be of constant dimension m and x0 ∈ IG (x0 ). Then there exist m vector elds {X1 , ⋅ ⋅ ⋅ , Xm } ⊂ G (since m ⩾ s, some Xi may be equal to some Z j ), and t10 , ⋅ ⋅ ⋅ ,tm0 , such that the mapping X1 X1 Xm Xm ) (t1 , ⋅ ⋅ ⋅ ,tm ) := I−t 0 ∘ ⋅ ⋅ ⋅ ∘ I−t 0 ∘ It 0 +t ∘ ⋅ ⋅ ⋅ ∘ It 0 +t (x0 ) m

1

m

m

1

1

(6.10)

is a diffeomorphism from a neighborhood 0 ∈ I m ⊂ ℝm to a neighborhood x0 ∈ W ⊂ IG (x0 ). Proof. Choose any X1 ∈ G such that X1 (x0 ) ∕= 0. Construct a mapping

)1 (t) := ItX1 (x0 ), which is a diffeomorphism from 0 ∈ I 1 ⊂ ℝ1 onto its image. Since the dimension dim (IG (x0 )) = m, there exists a t10 ∈ I 1 and a vector eld X2 ∈ G such that at point p1 = )1 (t10 ), X1 (p1 ) and X2 (p1 ) are linearly independent. (Otherwise, along the integral curve of X1 dim(IG (x0 )) = 1, which is a contradiction.) Next, we construct a mapping )2 (t1 ,t2 ) := ItX2 2 ∘ ItX01+t (x0 ), 1

ℝ2

1

onto its image. A similar argument which is a diffeomorphism from 0 ∈ ⊂ shows that there exists a point p2 = )2 (t10 + δt10 ,t02 ), and a X3 ∈ G, such that X1 (p2 ), X2 (p2 ), and X3 (p2 ) are linearly independent. For notational ease, we still use t10 for t10 + δt10 . Continuing this procedure, we can nally construct a mapping I2

152

6 Global Controllability of Afne Control Systems X

)m (t1 , ⋅ ⋅ ⋅ ,tm ) := ItX0m+t ∘ It 0m−1+t m

m

m−1

m−1

∘ ⋅ ⋅ ⋅ ∘ ItX01+t (x0 ) 1

1

(6.11)

which is a diffeomorphism from 0 ∈ I m ⊂ ℝm to a neighborhood of pm = ItX0m ∘ m

X

It 0m−1 ∘ ⋅ ⋅ ⋅ ∘ ItX01 (x0 ) in IG (x0 ). (Note that according to our construction tm0 = 0.) m−1

Dene

1

X1 Xm ) c (x) := I−t 0 ∘ ⋅ ⋅ ⋅ ∘ I−t 0 (x), m

1

which is a diffeomorphism from a neighborhood of pm back to a neighborhood of x0 . Composing it with )m (t1 , ⋅ ⋅ ⋅ ,tm ) yields

) (t1 , ⋅ ⋅ ⋅ ,tm ) := ) c ∘ )m (t1 , ⋅ ⋅ ⋅ ,tm ) which is a diffeomorphism from 0 ∈ I m ⊂ ℝm to x0 ∈ W ⊂ IG (x0 ).

⊔ ⊓

Remark 6.2. Basically, the above proof is a mimic of the proof of Chow’s Theorem [7]. Lemma 6.3. If there exists a nite set {Y1 , ⋅ ⋅ ⋅ ,Ys } ⊂ S such that conditions (i) and (ii) of Proposition 6.1 are satised, then for any Z ∈ S, we can assume it is a vector in (6.9), i.e., Z = Yi for some i. Proof. Since Y1 , ⋅ ⋅ ⋅ ,Yn are linearly independent, we can express Z as n

Z(x0 ) = ¦ diYi (x0 ).

(6.12)

i=1

Adding (6.12) to (6.9) yields n

Z(x0 ) + ¦ (ci − di)Yi (x0 ) + i=1

s

¦

ciYi (x0 ) = 0.

(6.13)

i=n+1

Now if all the coefcients are positive we are done. Since in (6.9) we can choose ⊔ ⊓ ci > 0 as large as we wish the conclusion follows. Denition 6.3. A mapping ItX (x) : M × ℝ1 → M is called (physically) realizable (by system (6.7)) if  { }  m i i  X ∈ f + ¦ g j u j  i = 1, ⋅ ⋅ ⋅ , N; u j ∈ U  j=1 and t ⩾ 0. Now we are ready to prove Lemma 6.1. Proof. (Proof of Lemma 6.1) According to Proposition 6.1, we can nd Y1 , ⋅ ⋅ ⋅ , Ys ∈ {F, G }, such that (6.9) holds. Moreover, there is a subset of n vector elds, say, Y1 , ⋅ ⋅ ⋅ , Yn , which are linearly independent at x0 (and hence linearly independent over a neighborhood W ∋ x0 ). Construct a mapping

6.2 A Sufcient Condition

4 (t1 , ⋅ ⋅ ⋅ ,tn ) := ItY11 ∘ ItY22 ∘ ⋅ ⋅ ⋅ ∘ ItYnn (x0 ),

153

(6.14)

which is a diffeomorphism from 0 ∈ I n ⊂ ℝn to a neighborhood x0 ∈ W ⊂ M. Note that the region of 4 (t1 , ⋅ ⋅ ⋅ ,tn )(x0 ) contains a neighborhood of x0 . So if each segment of its integral curves is realizable, then the reachable set R(x0 ) of x0 contains a neighborhood of x0 . The following discussion is devoted to modifying 4 to make it realizable. Consider Yi , which should be in one of the following three categories: Case 1. Yi ∈ F: According to Lemma 6.3, we can assume that this Yi is an element in (6.9). Say, it is the Yi in (6.9). Then we construct the following mapping ⎧ Y ⎨It i (x0 ), t ⩾ 0, \i (t) := (6.15) Yi+1 Yn ⎩I Y1c ∘ ⋅ ⋅ ⋅ ∘ I Yi−1 t < 0. ci−1 ∘ I ci+1 ∘ ⋅ ⋅ ⋅ ∘ I cn (x0 ), 1 − t −c t

− c t i

i

We show that

− c t i

ci

 d\i (t)  = Yi (0). dt t=0

(6.16)

 d\i (t)  = Yi (0). dt t=0+

It is obvious that

So we have only to consider the case of t → 0− . Denote

\i− (t) := I Y1c1 ∘ ⋅ ⋅ ⋅ ∘ I −c t i

Yi−1 c − i−1 c t i

Yi+1 c − i+1 c t

∘I

i

∘ ⋅ ⋅ ⋅ ∘ I−Yncn t (x0 ). ci

Using the fact that I0X (p) = p, (I0X )∗ = identity, and an easily proved formula

w X Y I ∘ I (x) = (ItX1 )∗Y (ItX1 ∘ ItY2 (x)), w t2 t1 t2 we can prove that

(6.17)

 s cj w −  \i (t) = − ¦ Y j (x0 ) = Yi (x0 ). wt c t=0 j=1, j∕=i i

The last equality is from (6.9). We hence proved (6.16). Recall 4 of (6.14). If there is an integral segment of Yi ∈ F as its component , we replace ItYi i by \i (ti ). Then we have a new 4 . For notational easy, we still denote it by 4 . We claim that new 4 (t1 , ⋅ ⋅ ⋅ ,tn ) is still a diffeomorphism from 0 ∈ I n ⊂ ℝn to a neighborhood x0 ∈ W ⊂ M. (I n may be shrank if necessary.) Using (6.16), it is easy to prove that we still have the Jacobian matrix of 4 at zero as J4 (0) = [Y1 (x0 ),Y2 (x0 ), ⋅ ⋅ ⋅ ,Yn (x0 )]. The linear independence of {Yi } implies that 4 is a local diffeomorphism. The advantage of new 4 is that any segment of the integral curves in 4 , which involves Yi ∈ F, is now realizable because of its corresponding non-negative time. (Recall (6.15) and note that all ci > 0.)

154

6 Global Controllability of Afne Control Systems

Case 2. Yi ∈ G: Say Yi = gij . Then we have

ItYi i

( f i +gij u)− f i u ti

=I

.

(6.18)

When ti ⩾ 0 choose corresponding control u > 0 and when ti < 0 let u < 0. Then we have ( f i +gij u)− f i

ItYi i = It /u i

.

(6.19)

,

(6.20)

Now we dene ( f i +gij u)

\i (t) := It/u

where t/u ⩾ 0. So \i (t) is also physically realizable. Replacing ItYi i in 4 by this \i (ti ) as Yi ∈ G, we can prove that as ∣u∣ becomes large enough 4 is still a local diffeomorphism. It is because  w \i (t)  = Yi (0) + f i (0)/u. w ti t=0 As ∣u∣ is large enough, similar to Case 1, we can prove that the Jacobian matrix for modied 4 is still non-singular at t = 0. Case 3.Yi ∈ G ∖G: Assume that the region W = 4 (I n ) and 4 : I n → W is a diffeomorphism. We claim that W is a reachable set of x0 . Let x1 ∈ W . It sufces to show that x1 ∈ R(x0 ). After components of Case 1 and Case 2 have been treated, we have only to treat the components of Case 3. Let Yi ∈ G ∖G. Then there exists t 0 = (t10 , ⋅ ⋅ ⋅ ,tn0 ) ∈ I n , such that x1 = 4 (t 0 ) = ItY01 ∘ ItY02 ∘ ⋅ ⋅ ⋅ ∘ ItY0n (x0 ). 1

n

2

(6.21)

Since I n is an open set, we can nd H > 0 such that if t = (t1 , ⋅ ⋅ ⋅ ,tn ) satisfying ∥t∥ < H , then t 0 + t ∈ I n . Dene

4˜ (t) = ItY01+t ∘ ItY02+t ∘ ⋅ ⋅ ⋅ ∘ ItY0n+t (x0 ), 1

1

2

n

2

∥t∥ < H .

n

(6.22)

Checking the Jacobian matrix, it is easy to see that 4˜ is a diffeomorphism from a neighborhood 0 ∈ I n (H ) ⊂ ℝn to a neighborhood of x1 ∈ W1 ⊂ W . Denote by Y

p = It 0i+1 ∘ ⋅ ⋅ ⋅ ∘ ItY0n (x0 ); i+1

n

q = ItY0i (p). i

Since Yi ∈ G , by Chow’s Theorem, we can nd Z1 , ⋅ ⋅ ⋅ , Z j ∈ G and d1 , ⋅ ⋅ ⋅ , d j ∈ ℝ, such that Z q = IdZ11 ∘ IdZ22 ∘ ⋅ ⋅ ⋅ ∘ Id jj (p).

6.2 A Sufcient Condition

155

Using Lemma 6.2, we can construct a ) (W1 , ⋅ ⋅ ⋅ , Wm ) as in (6.10) which is a local diffeomorphism from I m to a neighborhood of q (with respect to the inherited topology from M to the sub-manifold IG (q)). Now dene Z

\i (W1 , ⋅ ⋅ ⋅ , Wm ) := ) (W1 , ⋅ ⋅ ⋅ , Wm ) ∘ IdZ11 ∘ IdZ22 ∘ ⋅ ⋅ ⋅ ∘ Id jj (p). Note that the region of \i (W1 , ⋅ ⋅ ⋅ , Wm )(p) over I m contains a neighborhood of q (under the sub-manifold topology). While ItY0i+t (p), ti ∈ I 1 (H ) contains a special integral i

i

curve of Yi on the sub-manifold. As H > 0 is small enough, we conclude that  } {  ItY0i+t (p)ti ∈ I 1 (H ) ⊂ { \i (W )∣ W ∈ I m } . i

i

Note that since \i (W1 , ⋅ ⋅ ⋅ , Wm ) is dened at q, i.e. \i (W1 , ⋅ ⋅ ⋅ , Wm )(q) is a local diffeomorphism, by continuity of the Jacobian matrix one sees easily that as a neighborhood Wq of q being small enough, the mapping \i (W1 , ⋅ ⋅ ⋅ , Wm )(q′ ), q′ ∈ Wq , is also a diffeomorphism. In our case since I n (H ) can be arbitrary small, \i (W1 , ⋅ ⋅ ⋅ , Wm ) is diffeomorphism for each q′ ∈ Wq . Now we replace ItY0i+t in 4˜ by \i (W1 , ⋅ ⋅ ⋅ , Wm ). Then according to the above ari i gument one sees easily that the region of new 4˜ over I i−1 (H ) × I m × I n−i covers the region of the old 4˜ over I n (H ). Therefore, it covers a neighborhood of x1 . After this modication we can replace all Yi ∈ G ∖G by some Z ∈ G. Then using the technique in Case 2, we can further modify 4˜ to make it realizable. We conclude that W is a reachable set of x0 . Note that unlike Case 1 and Case 2, in Case 3 for each x1 ∈ W we dene 4˜ = 4˜ x1 , which depends on x1 and is a submersion (onto mapping) to a neighborhood of x1 . It is not a diffeomorphism any more. Finally, we have to show that W is a controllable neighborhood of x0 . To see that let x1 ∈ W . Construct 4˜ (t) = 4˜ x1 (t) which starts from x0 and mapped over a neighborhood of x1 . Replace all the constant time variables ti0 by −ti0 to get 4˜ c (t). Then it maps from x1 back to a neighborhood of x0 . Note that in 4˜ (t) there is no vector eld of Case 3. It is so for 4˜ c (t) as well. Using the tricks used in Case 1 and Case 2 for handling negative time, we can show that x0 ∈ R(x1 ), which completes the proof. ⊔ ⊓ The geometric meaning of Lemma 6.1 is obvious. Refer to Fig. 6.2, since x0 is an interior of {F, G }, intuitively it can go wherever within the cone. The disadvantage of Lemma 6.1 is obvious. If the codimension of G is k, then, according to Proposition 6.1, at least k + 1 switching models are required. Thus it can not be applied to non-switched system (6.3). We need to improve it. For this purpose, some new concepts are required: Denition 6.4. 1. Let ' be an involutive distribution on M with constant dimension k. A coordinate frame x is said to be at with respect to ' , if under this coordinate frame } { w w . (6.23) ' = Span ,⋅⋅⋅ , w x1 w xk

156

6 Global Controllability of Afne Control Systems

Fig. 6.2

An Interior Point of {F, G }

2. Let x be a coordinate frame at with respect to an involutive distribution ' , and a vector eld f ∈ V (M) is expressed in this coordinate frame as f (x) = ( f 1 (x), ⋅ ⋅ ⋅ , fn (x))T . The quotient vector eld of f (x) over ' , denoted by f /' , is dened as f (x)/' = f 2 (x), [ where f (x) =

(6.24)

] f 1 (x) and f 2 (x) = ( fk+1 (x), ⋅ ⋅ ⋅ , fn (x))T . f 2 (x)

From Frobenius’ theorem we know that for an involutive distribution with constant dimension, its local at coordinate chart (a chart with at coordinate ame) always exists. Moreover, we would like to emphasize that the quotient vector eld is well dened. Proposition 6.2. [4] Let ' be an involutive distribution with constant dimension, and a vector eld f ∈ V (M) be given. Then the quotient vector eld f /' is uniquely dened. Denition 6.5. 1. Given Y ∈ V (M), a vector X(x0 ) ∈ Tx0 (M) is said to be a G -shifted Y from x1 , denoted by X(x0 ) ∈ G (x0 , x1 )∗Y (x1 ),

(6.25)

if there exist a nite set of vector elds Z1 , ⋅ ⋅ ⋅ , Zs ∈ G, such that x0 = ItZ01 ∘ ⋅ ⋅ ⋅ ∘ ItZ0s (x1 ); s

1

and

( ) ( ) X (x0 ) = ItZ01 ∘ ⋅ ⋅ ⋅ ∘ ItZ0s Y (x1 ). 1

∗

s

∗

2. Let S ⊂ V (M) be a set of vector elds. We may G-shift all vector elds f ∈ S from all points x1 ∈ IG (x0 ) to one point x0 , and denote them by G∗ S(x0 ). That is G∗ S(x0 ) = ∪ { G (x0 , x1 )∗Y (x1 )∣Y ∈ S, x1 ∈ IG (x0 )} .

(6.26)

The following lemma shows that G -shift is “quotient-independent” of the choice of trajectories, which is very important for our purpose.

6.2 A Sufcient Condition

157

Proposition 6.3. Let X(x0 ), Z(x0 ) ∈ G (x0 , x1 )∗Y (x1 ). Then with respect to ' = G we have X(x0 )/' = Z(x0 )/' .

(6.27)

Proof. Assume X (x0 ) is obtained through the mapping: x0 = ItZ01 ∘ ⋅ ⋅ ⋅ ∘ ItZ0s (x1 ); 1

s

Note that Zi ∈ G, so under a at coordinate frame with respect to G (i.e., ' = Span{ wwx , ⋅ ⋅ ⋅ , wwx }), we have 1 k [ 1] Z Zi = i . 0 So the Jacobian matrix of ItZi i has the form ] [ J 0 JI = 11 . 0 I It follows that

(

ItZ01 1

)

[ ] ( ) I (Y 1 (x )) ∘ ⋅ ⋅ ⋅ ∘ ItZ0s Y (x1 ) = ∗ 2 1 Y (x1 ) s ∗ ∗

That is X (x0 )/' = Y 2 (x1 ), which means the quotient vector eld is independent of the choice of Zi .

(6.28) ⊔ ⊓

Note that (6.28) shows that when a vector eld is a G -shifted Y from one point to another, its quotient part is unchanged. Fig. 6.3 depicts this. In Fig. 6.3, [1 , [2 , ⋅ ⋅ ⋅ , K1 , K2 , ⋅ ⋅ ⋅ are vector elds in ' . fa moves along two different paths of ' to b, and results in f ′ and f¯ at b. But they have the same quotient vector fb /' .

Fig. 6.3

Shift f a along '

Now we are ready to present the second key lemma. Lemma 6.4. Assume A1 and let x0 ∈ M. If x0 is an interior of {G∗ F(x0 ); G }, then there exists a neighborhood W of x0 that is a controllable subset.

158

6 Global Controllability of Afne Control Systems

Proof. Exactly following the same procedure proposed in the proof of Lemma 6.1, one sees easily that we can nd an open neighborhood W of x0 and for each x1 ∈ W construct a 4˜ : I ℓ → M, which is composed of a set of integral curves, with the region coving a neighborhood x1 ∈ W1 ⊂ W . Note that ℓ ⩾ n might be a very large integer. Precisely, 4˜ is composed of the integral curves of two categories of vector elds: the rst group of vector elds are as f k + gkj with non-negative time, which is realizable; the second group of vector elds are as f˜i ∈ G (x0 , xi )∗ ( f i ) with non-negative time. That is, there exist Z1 , ⋅ ⋅ ⋅ , Z j ∈ G such that ( ) ( ) ˜f i = I Z01 ∘ ⋅ ⋅ ⋅ ∘ I Z0 j fi (6.29) t t ∗

1

j

∗

Without loss of generality and for notational ease, we assume j = 1, that is ( ) (6.30) f˜i = ItZ0 ∗ f i , ˜i

where Z ∈ G. Now we have to make Itif (x˜0 ) realizable, where x˜0 is a point near x0 (corresponding to the “p” in the proof of Lemma 6.1). A simple computation shows that ˜

(I Z0 )∗ f i

It f0i (x˜0 ) = Iti

t

fi

(x˜0 ) = ItZ0 ∘ Iti

) ( fi Z Z I−t ˜0 ) = ItZ0 ∘ Iti ∘ I−t ˜0 ). 0 (x 0 (x

(6.31)

As is discussed in Case 2 of the proof of Lemma 6.1, we can easily replace the integral of Z ∈ G in (6.31) by some realizable vector elds. Then we have proved that any point x1 in W is reachable from x0 . Note that after the treatment of (6.31), there is no component involving f˜i . Then showing x0 ∈ R(x1 ) is exactly the same as in the proof of Lemma 6.1. ⊔ ⊓ Remark 6.3. The physical meaning of (6.31) is: instead of going along the direction of f˜i , we can go along −Z from x˜0 to x˜1 , then along f from x˜1 to A, and then along Z from A to B. (Refer to Fig. 6.4.) Now let us see the direction of f˜i . In fact, if we set ti = t0 := t and let the trajectory continue to go from B along − f˜i for time t to C, then it is well known [3] that the vector from x˜0 to C is adZ f˜i + O(t 2 ). (Refer to Fig. 6.4.) We conclude that (ItZ )∗ f i ≈ f i + t ⋅ adZ f i .

(6.32)

Similarly, for more than one shifting such as in (6.29) we have j

(ItZ1 1 )∗ ⋅ ⋅ ⋅ (It j j )∗ f i ≈ f i + ¦ tk ⋅ adZk f i . Z

(6.33)

k=1

The advantage of Lemma 6.4 over Lemma 6.1 lies in that the f i (x0 ) in Lemma 6.1 is replaced by f i (xi ). That is, f i ∈ F can be “moved” from all different points of IG (x0 ) to x0 to generate the convex cone. Remark 6.4. It is worthwhile noting that the condition in Lemma 6.4 is a property of the system itself. It can be seen from the following three points:

6.2 A Sufcient Condition

Fig. 6.4

159

Integral Curve of G (x0 , xi )∗ (F)

∙ Condition (i) and (ii) of Proposition 6.1 are independent of the choice of coordinates. So the interior of {F, G } is also coordinate free. ∙ Based on Proposition 6.3 one sees that as ' = G , F/' is invariant under the G (x0 , x1 )∗ transformation. Hence Cx0 := con{G∗ (F)(x0 ); G },

(6.34)

is uniquely and well dened. ∙ It is easy to see that if x0 is an interior of the convex cone of Cx0 , i.e, it satises the condition of Lemma 6.4, then all points x ∈ IG (x0 ) are also interior points of the corresponding convex cones, because shifting is a diffeomorphism. So the condition in Lemma 6.4 is a property of each maximum integral manifold of G . Therefore, instead of {G∗ (F)(x0 ); G }, we can use {G∗ (F); G }. Now we are ready to state the rst main theorem: Theorem 6.1. Assume A1 and that M is pathwise connected. Then system (6.7) is controllable if for each maximum integral manifold IG (x0 ), ∀x0 ∈ M, there exists a point y ∈ IG (x0 ) (equivalently, every point on this integral manifold) that is an interior of Cy = con{G∗ (F); G }. Proof. For any two points x, y ∈ M, connect them by a path c(t), 0 ⩽ t ⩽ 1 (c(0) = x and c(1) = y). According to Lemma 6.4 each point c(t) has a controllable neighborhood, called W (xt ), where xt = c(t). Since c(t) is the continuous image of a compact set [0, 1], it is compact. Since {W (xt )∣0 ⩽ t ⩽ 1} is a covering of c(t), it has a nite sub-covering {W (x0 = x),W (x1 ), ⋅ ⋅ ⋅ ,W (x j = y)}. Without loss of generality, we can assume W (xi ) ∩ W (xi+1 ) ∕= ∅, and pi ∈ W (xi ) ∩ W (xi+1 ). Then x ∈ R(pi ), ∀i, which means x ∈ R(y). Similarly, we have y ∈ R(x). ⊔ ⊓ We give some examples to illustrate this theorem. Example 6.2. 1. Consider the following system ⎧ x1 = xn      x = x3n  ⎨ 2 .. .     xn−1 = x2n−1  n  ⎩ xn = u, x ∈ ℝn , u ∈ U .

(6.35)

160

6 Global Controllability of Afne Control Systems

For notational ease, denote the system as x = f (x) + g(x)u, and x = (x1 , x2 ), where x1 = (x1 , x2 , ⋅ ⋅ ⋅ , xn−1 )T . The integral manifold of G passing x0 is { } IG (x0 ) = x ∈ ℝn ∣ x1 = x10 . Choosing yi = (x10 , ki ), zi = (x10 , −ki ), i = 1, ⋅ ⋅ ⋅ , n − 1, where ki > 0, and ki ∕= k j (i ∕= j). Dene vectors Yi = f (yi ), Zi = f (zi ), i = 1, ⋅ ⋅ ⋅ , n − 1. It is obvious that g∗ (Yi ) = Yi and g∗ (Zi ) = Zi . Choose Yn = (0, ⋅ ⋅ ⋅ , 0, 1)T ∈ G , and Zn = −Yn . Then as a Vandermonde determinant, we have [ ] n−1 det Y1 , ⋅ ⋅ ⋅ ,Yn = – ki i=1

–

(k2j − ki2 ) ∕= 0.

n⩾ j>i⩾1

So, {Yi } are linearly independent. In addition, n

n

i=1

i=1

¦ Yi + ¦ Zi = 0.

Using Proposition 6.1, x0 is an interior point of {G∗ (F); G }. Since x0 is arbitrary, Theorem 6.1 tells us that the system (6.35) is controllable. 2. Consider a planar controllable nonlinear system x = f (x) + g(x)u,

x ∈ ℝ2 .

(6.36)

If g(x) is complete, each integral curve of g(x) splits ℝ2 into two disjoint parts. Now if on each integral curve of g(x) there exist two points, say p and q, where f (p) and f (q) directed to two opposite sides, then the system is globally controllable. This is a particular case of Theorem 6.1. Obviously, it is also necessary [11]. (Please refer to Fig. 6.5.) Particularly, consider a linear system { x1 = x2 (6.37) x2 = ax1 + bx2 + u. It is a special case of system (6.36), thus controllable.

Fig. 6.5

Controllable Planar System

6.2 A Sufcient Condition

161

3. Consider the following system ⎧ x1 = xqn1 + P1 (x)    q2    ⎨x2 = xn + P2 (x) .. .   q   xn−1 = xnn−1 + Pn−1 (x)   ⎩ xn = fn (x) + gn (x)u, x ∈ ℝn .

(6.38)

Assume that (i). gn (x) ∕= 0, ∀x ∈ ℝn ; (ii). qi , i = 1, ⋅ ⋅ ⋅ , n − 1 are distinguished positive odd numbers; (iii) Function Pi (x) has the following property lim

xn →f

Pi (x) = 0, xqni

i = 1, ⋅ ⋅ ⋅ , n − 1.

Then system (6.38) is controllable. The proof is sketched as follows: Using the prefeedback 1 [v − fn (x)] u= gn (x) to simplify f and g rst, we have fn (x) = 0 and gn (x) = 1. Then we consider these simplied f and g. Using similar notations as for system (6.35) and let ki be large enough, since Yi are linearly independent, we can express Zi as n

Zi = f (zi ) = − ¦ ci jY j . j=1

Moreover, as ki → f, it is obvious that cii → 1,

ci j → 0, j ∕= i.

Now we have n−1

n−1 n−1

n−1

n−1

i=1

i=1 j=1

i=1

i=1

¦ Zi + ¦ ¦ ci jY j + Yn + Zn = ¦ Zi + ¦

(6.39)

(

) cii + ¦ ci j Yi + Yn + Zn = 0. j∕=i

The positivity of the coefcients is from (6.39). Example 6.3. Consider the following switched system x = f V (t) (x) + gV (t) (x)u,

x ∈ M, u ∈ U ,

where M = ℝ3 ∖{0} ⊂ ℝ3 , / = {1, 2} and f 1 = (sin(x1 ), 0, 0)T ; f 2 = (0, 0, 0)T ; ⎡ ⎤ ⎡ ⎤ 0 10 0 01 g1 (x) = ⎣−1 0 0⎦ x := B1 x; g2 (x) = ⎣ 0 0 0⎦ x := B2 x. 0 00 −1 0 0 It is easy to verify that as the subset of gl(3, ℝ) we have

(6.40)

162

6 Global Controllability of Afne Control Systems

{B1 , B2 }LA = o(3, ℝ). That is, G = o(3, ℝ)x. So the maximum integral manifolds of G are spheres IG (x0 ) = SO(3, ℝ)x0 = ∥x0 ∥S2 , { } where S2 = x ∈ ℝ3 ∣ ∥x∥ = 1 . Now on each ∥x0 ∥S2 we can nd two points x, y with 0 < x1 < π, x3 > 0 and −π < y1 < 0, y3 > 0 (on upper hemisphere). Then it is easy to see that f1 (x1 ) is going out from and f1 (x2 ) is going in into the sphere. A straightforward verication shows that the condition of Theorem 6.1 is satised. So the system is controllable on M = ℝ3 ∖{0}. Remark 6.5. Motivated by the above example, we have the following observation: Let S ⊂ M be a pathwise connected open subset of M. Then S is a sub-manifold of M with the same dimension. So the controllability result obtained in Theorem 6.1 can be easily extended to the semi-global controllability over S. Example 6.4. [5, 11] Consider the following system [ ] [ ] ] [ − sin(x2 ) cos(x2 ) x1 sin(x2 )e−x1 = u, + x2 cos(x2 )e−x1 sin2 (x2 )

x ∈ ℝ2 .

(6.41)

[5] proved that it is not globally linearizable. [11] proved that it is not globally controllable. We claim that it is semi-globally controllable on each strip Mk = {x ∈ ℝ2 ∣kπ − π/2 < x2 < kπ + π/2},

k ∈ ℤ.

We prove it for k = 0 only. (Argument for other k’s is the same.) A basis of G is ((sin(x2 ), cos(x2 ))T . A straightforward computation shows that the integral manifold passing through x0 ∈ M0 is ⎧ 2 2t 2  ⎨x1 = ln(D e + 1) − t − ln(D + 1) + x1(0), ( t ) 1 + sin(x2 (0))  ⎩x2 = 2 tan−1 D et − 1 , where D = . cos(x1 (0)) De + 1 Now since t → −f, x2 → −π/2 and t → f, x2 → π/2, so there exists t0 such that x2 (t0 ) = 0. It is easy to see that as t > t0 and t < t0 f points to different sides of IG (x0 ). So the system is semi-globally controllable on M0 . Theorem 6.1 is based on G. It was shown in the previous chapter that a switched system without control may still be controllable. In fact, we can prove the following corollary for switched systems in a similar way as before: Corollary 6.1. System (6.7) is controllable if every point in M is an interior point of F.

6.3 Multi-hierarchy Case

163

Proof. Using equation (6.15) in the proof of Lemma 6.1 and the same argument there, and then the same argument in the proof of Theorem 6.1, the conclusion follows. ⊔ ⊓ Example 6.5. Consider a switched system x = fV (t) (x),

x ∈ ℝn ∖{0},

(6.42)

where V ∈ / = {1, 2, ⋅ ⋅ ⋅ , n(n − 1) + 2}, and fi = Bi x,

fn(n−1)/2+i = −Bi x,

i = 1, 2, ⋅ ⋅ ⋅ , n(n − 1)/2,

with {B1 , B2 , ⋅ ⋅ ⋅ , Bn(n−1)/2 } a basis of o(n, ℝ), and the last two are ±In . It is easy to see that at each x ∈ M fi (x), i = 1, ⋅ ⋅ ⋅ , n(n − 1) span a convex cone on the tangent space of Sn−1 at x with x as it interior point (with respect to the sub-space topology of Sn−1). Adding the last two vectors fn(n−1)+1(x) and fn(n−1)+2 (x) as vectors at the ± radius directions make x an interior point of {F, G }. So (6.42) is controllable. This example shows that Corollary 6.1 is not necessary since it is easy to see that modes i = 1, 2, ⋅ ⋅ ⋅ , n(n − 1)/2 plus the last two are enough to make the system controllable.

6.3 Multi-hierarchy Case Before giving general results we want to give some motivations for considering the hierarchical structure of a controllable set. First, consider linear system (6.1). Assume m = 1 and put it into Brunovsky canonical form ⎧ x1 = x2     ⎨.. . (6.43)  xn−1 = xn   ⎩ xn = u, x ∈ ℝn , u ∈ U . Check the condition of Theorem 6.1. Now along IG (x0 ) the vector eld has the form as f = ((x2 )0 , ⋅ ⋅ ⋅ , (xn−1 )0 , xn , 0) . When following the moving style as proposed in Section 6.2 the only direction we can go out of the IG (x0 ) is the xn−1 direction. This motivates us that what is proposed in Section 6.2 is only the “rst step” of the motion. To nd all the ways the system can be driven, a sequence of motions should be followed. For the linear case, it is not difcult to see that the rst motion can reach B, AB. So the second motion will go to A2 B and so on. (Refer to Fig. 6.6(a).) The second motivation is from Lemma 6.1 and Lemma 6.4. In fact, it is obvious that in Lemma 6.1 the movable direction (out of IG (x0 ) ) is f . In the proof of Lemma 6.4, it was shown that, roughly speaking, in addition to the direction of f the additional direction is adkG f (see the Remark after the proof of Lemma 6.4). In fact,

164

6 Global Controllability of Afne Control Systems

( ) the directions the trajectories can really go are ItG ∗ f . Using Baker-CampbellHausdorff formula [1] for any X ∈ G yields ( X) 1 It ∗ f (x0 ) = f (x0 ) + adX f (x0 )t + ad2X f t 2 + ⋅ ⋅ ⋅ . 2!

(6.44)

This equation clearly shows that all the possible directions for the rst step of motion is adkG f , k ⩾ 0. Similar to the linear case, we may consider [g∗ ( f )]∗ f as the next step and keep going on. (Refer to Fig. 6.6(b).)

Fig. 6.6

Controllability Distributions

Motivated by the aforementioned argument, we will construct a sequence of distributions for controllability of system (6.7). To begin with, we need some preparations: 1. Let '1 ⊂ ' 2 ⊂ ⋅ ⋅ ⋅ ⊂ 'k∗ = T (M) be a sequence of nested involutive distributions of constant dimensions, dim(' i ) = ni . Using generalized Frobenious’ theorem (Lemma 5.7), there is a at coordinate chart (U, x), such that locally { } w w 'i = Span ,⋅⋅⋅ , , i = 1, ⋅ ⋅ ⋅ , k∗ . w x1 w xni 2. Let ' be an involutive { distribution}of dimension k, and x be a at coordinate frame, such that ' = wwx , ⋅ ⋅ ⋅ , wwx . Assume a vector eld f (x) is locally ex1

k

pressed as f = ( f1 , ⋅ ⋅ ⋅ , fn )T . Then the projection of f on ' is dened as f ∣' = f − icl∗ ( f /' ), where icl : (I' )⊥ → M is the inclusion mapping. Precisely, let f /' = ( fk+1 (x), ⋅ ⋅ ⋅ , fn (x))T . Then f ∣' = ( f1 (x), ⋅ ⋅ ⋅ , fk (x), 0, ⋅ ⋅ ⋅ , 0)T , and icl∗ ( f /' ) = (0, ⋅ ⋅ ⋅ , 0, fk+1 (x), ⋅ ⋅ ⋅ , fn (x))T . Now we can dene the following sequence of distributions, called the set of controllability distributions, as: { '1 := G  { } (6.45)  'k+1 := '¯k + Span ad j F  j ⩾ 1 , k = 1, 2, ⋅ ⋅ ⋅ . '¯k

6.3 Multi-hierarchy Case

165

Denition 6.6. System (6.7) is said to have a proper set of controllability distributions if the sequence of distributions ' i dened in (6.45) satisfy the following conditions: 1. There exists k∗ such that '¯k∗ = T (M); 2. '¯1 , ⋅ ⋅ ⋅ , '¯k∗ −1 have constant dimensions. It is worth to mention that for linear systems '1 = Span{B}, ' 2 = Span{B, AB}, ⋅⋅⋅. Denition 6.7. Let ' be an involutive distribution on M with constant dimension. A point x ∈ I' (x0 ) is said to be a deep interior point of {(' )∗ (F), ' } if for any r > 0 there exist nite vectors Yi (x0 ) ∈ {(' )∗ (F), ' }, i = 1, ⋅ ⋅ ⋅ , s such that the whole ball Br (0) is in the interior of the convex cone: con{Yi (x0 )∣ i = 1, ⋅ ⋅ ⋅ , s}. Note that here the radius r can be arbitrary large, but assigned before choosing Yi . Now we are ready to state our main result. Theorem 6.2. Consider system (6.7). Assume it has a proper set of controllability distributions dened in (6.45), which satisfy the following conditions (for each x0 ∈ M) : (i). There exists xi ∈ I'¯i (x0 ) (equivalently, for any point x ∈ I'¯i (x0 )), such that { } xi is a deep interior point of ('¯i )∗ (F∣'¯i+1 ), '¯i , with respect to the sub-space topology of I'¯i+1 (x0 ), for i = 1, ⋅ ⋅ ⋅ , k∗ − 2. (ii). There exists xk∗ −1 ∈ I'¯k∗ −1 (x0 ) (equivalently, for any point x ∈ I'¯k∗ −1 (x0 )), { } such that xk∗ −1 is an interior point of ('¯k∗ −1 )∗ (F), '¯k∗ −1 . Then system (6.7) is globally controllable. Proof. First consider '¯k∗ −1 as G , and similar to the proofs of Lemmas 6.1 and 6.4, it is easy to nd for each x0 ∈ M a mapping 4 as in (6.14), such that 4 : I n → W ⊂ M is a diffeomorphism, where W is a neighborhood of x0 . Note that this 4 is composed of integral curves of the vector elds in '¯k∗ −1 or F. Using the same argument as in Case 1 and Case 3 of the proof of Lemma 6.1 and the proof of Lemma 6.4, we can nd, for each x1 ∈ W , Z1 , ⋅ ⋅ ⋅ , Zℓ ∈ 'k∗ −1 ∪ F, such that the modied 4˜ : I ℓ → Wx1 ⊂ W starts from x0 , use the integral curves of Zi , and nally maps onto a neighborhood of x1 . Moreover, for Zi ∈ F the corresponding time is non-negative. When reversing the time and modifying f ∈ F, a new 4˜ , denoted by 4˜ c : I ℓ → Wx0 ⊂ W , goes backward from x1 to a neighborhood of x0 . Roughly speaking, x0 is locally controllable by {F; 'k∗ −1 }. Next, we conne to each leave I'¯k∗ −1 (x0 ), and show that for Z ∈ ' k∗ −1 the corresponding component ItZ in 4˜ can be replaced by the integral curves of vector elds in 'k∗ −2 ∪ F. First, using Baker-Campbell-Hausdorff formula, we can prove easily that } {  (6.46) 'k∗ −1 ⊂ Span ('¯k∗ −2 )∗ F ' ∗ ; '¯k∗ −2 . k −1

166

6 Global Controllability of Afne Control Systems

According to (6.46) and the argument in the{proof of Lemma}2.9, ItZ can be replaced by the integral curves of vector elds in '¯k∗ −2 ∪ F∣'k∗ −1 . Repeating the same argument as in Case 1 and Case 3 of the prove of Lemma 6.1 and the proof of Lemma 6.4, one sees that we can do this. In the resulting mapping we have integral curves of vector elds as Z ∈ '¯k∗ −2 and f ∣'k∗ −1 , f ∈ F. For Z ∈ '¯k∗ −2 we know how to replace it by a set of Zi ∈ 'k∗ −2 as in the rst step. As for f ∣'k∗ −1 , we can formally replace it by its corresponding f . Then the problem caused by this replacement is that the components of the vector eld belong to f ∣'¯k∗ = f have been changed from (0, ⋅ ⋅ ⋅ , 0)T ∗ to f k = ( fnk∗ −1 +1 , fnk∗ −1 +2 , ⋅ ⋅ ⋅ , fn )T . This makes the destination point drift. Fortu∗ nately from Proposition 6.3, f k is coordinate-independent of '¯k∗ −2 . So when the system goes along the trajectories of '¯k∗ −2 and F∣'¯k∗ −1 the part of variables corre∗

sponding to f k drift freely. By the denition of deep interior, the movement along '¯k∗ −2 and f ∣'k∗ −1 can be within a time period as short as we wish. This makes the drift be as small as possible. So it does not affect the property that the region of the corresponding transfer mapping covers a neighborhood of x and vise versa. By induction, we can keep on reducing the subscript of the distributions of {'l } till '1 . Using the argument for Case 2 of the proof of Lemma 6.1, we can easily convert the vector elds in ' 1 to being realizable. ⊔ ⊓ Remark 6.6. Assume (6.7) has a proper set of controllability distributions. We may express (6.7) in a coordinate chart that is at with respect to '¯i . For notational ease, denote {x1 , ⋅ ⋅ ⋅ , xi } = {x1 , x2 , ⋅ ⋅ ⋅ , xni }, i = 1, 2, ⋅ ⋅ ⋅ , k∗ . Then (6.7) can be expressed locally as ⎧ 1 x = f V (t),1 (x) + GV (t),1 (x)u      x2 = f V (t),2 (x)     3 V (t),3 (x2 , x3 , ⋅ ⋅ ⋅ , xk∗ )   ⎨x = f ∗ x4 = f V (t),4 (x3 , x4 , ⋅ ⋅ ⋅ , xk )  .   ..    xk∗ −1 = f V (t),k∗ −1 (xk∗ −2 , xk∗ −1 , xk∗ )    ⎩ k∗ ∗ ∗ ∗ x = f V (t),k (xk −1 , xk ).

(6.47)

We may call it the controllability canonical form of afne control systems. We give some examples. Example 6.6. 1. For linear systems Theorem 6.2 becomes necessary and sufcient. To see this we only have to prove the necessity. Consider a controllable linear system. We prove it for the single-input case. The proof for the multi-input case is exactly the same. Assume the system is already in the canonical form (6.43). First, the set of controllability distributions are easily computed as

6.3 Multi-hierarchy Case

167

'1 = Span{b} = Span{Gn } '2 = Span{b, Ab} = Span{Gn , Gn−1 } .. . 'n = Span{b, Ab, ⋅ ⋅ ⋅ , An−1 b} = T (ℝn ), where Gi is the i-th column of the identity matrix In . Now the restriction of f = Ax on '2 is f ∣'2 = (0, ⋅ ⋅ ⋅ , 0, xn , 0)T and it is obvious that corresponding to the subspace topology of I'2 the f ∣'{2 can approach to} ±f. That is, any point on I'1 is obviously a deep interior of (' 1 )∗ (F∣'2 ), '1 . Continuing the same argument shows that the conditions of Theorem 6.2 are satised. We conclude that the conditions of Theorem 6.2 are necessary and sufcient for linear systems. 2. Consider the following system ⎧ x1 = sin(x2 )       ⎨x2 = x3 .. .     xn−1 = xn   ⎩ xn = u.

(6.48)

Similar to the rst example, one sees easily that the conditions of Theorem 6.2 are satised. So (6.48) is globally controllable. (Note that only at the last step for x ∈ I'n−1 , it is only an interior (not deep interior) of {(' n−1 )∗ (F), 'n−1 }, but it is enough.) Example 6.7. Consider switched linear system (6.8). It is easy to see that

'1 = Span{ Bi ∣ 1 ⩽ i ⩽ N}; '2 = '1 + Span{ Ai1 Bi2 ∣ 1 ⩽ i1 , i2 ⩽ N}; and so on. {As argued in the } linear (non-switched) case, x ∈ I'i is always a deep interior of ('i )∗ (F∣'i+1 ), ' i . (In fact, if a straight line is not in a subspace, we can always nd two points which are on the opposite directions and as far from the subspace as we wish.) So the only condition we have to check is: Whether there exists a k∗ such that 'k∗ = ℝn . This argument leads to the following known result [12]. Proposition 6.4. The switched linear system is controllable, if and only if   'n = Span{Bi1 , Ai2 Bi2 , ⋅ ⋅ ⋅ , Ain1 Ain2 ⋅ ⋅ ⋅ Ainn−1 Binn  1 ⩽ ikj ⩽ N} = ℝn . 1

1

2

Example 6.8. Consider the following system ⎧ x1 = xq21     ⎨.. . q   xn−1 = xnn−1   ⎩ xn = u,

(6.49)

(6.50)

168

6 Global Controllability of Afne Control Systems

where q1 , ⋅ ⋅ ⋅ , qn−1 are positive odd integers. Arguing as for linear systems, it is easy to prove that this system is controllable. q q Moreover, if xi i are replaced by xi i + LOT(xi ), the conclusion remains true. (LOT: lower order terms.) Example 6.9. Consider the following system ⎤ ⎡ ⎤ ⎡ cos(x2 + x3 ) 1 ⎥ ⎢ ⎥ ⎢ x 1 ⎥ + ⎢0⎥ u. x = ⎢ ⎦ ⎣0⎦ ⎣ x34 sin(x2 + x3) 1 Calculate that

⎧⎡ ⎤⎫   1   ⎨ ⎢0⎥⎬ ⎥ ; '1 = G = Span ⎢ ⎣ ⎦   0   ⎩ ⎭ 1

(6.51)

⎧⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎫ 0 0     1 ⎨ ⎢0⎥ ⎢1⎥ ⎢0⎥⎬ ⎥,⎢ ⎥,⎢ ⎥ . '2 = Span ⎢ ⎣ ⎦ ⎣0⎦ ⎣1⎦    0 ⎭ ⎩ 1 0 0

Then it is easy to see that ' 1 is in the deep interior of {(' 1 )∗ f ∣'2 , ' 1 }. Moreover, it is also easy to check '¯3 = T (ℝ4 ). So we only have to check whether ' 2 is in the interior of {(' 2 )∗ f , ' 2 }. At p1 = (0, π/2, 0, 0)T and p2 = 0 we have f (p1 ) = (0, 0, 0, 1)T ,

f (p2 ) = (1, 0, 0, 0)T .

Choosing g1,2 = ±(0, 1, 0, 0)T and g3,4 = ±(0, 0, 1, 0)T , g5 = (1, 0, 0, 1)T and setting Z (p ) = p , we have Z = g1 = (0, 1, 0, 0)T. Iπ/2 2 1 ( ) Z Iπ/2 f (p2 ) = (1, 0, 0, 0)T . ∗

Then we have

( ) Z f (p2 ) + (−g5) + g1 + g2 + g3 + g4 = 0. f (p1 ) + Iπ/2 ∗

Since all ±gi ∈ G , using Proposition 2.2, one sees that ' 2 is in the interior of T (ℝ4 ). We conclude that the system (6.51) is controllable.

6.4 Codim(G ) = 1 This section considers the case when Codim(G ) = 1. We will do two things for this particular case. First, simplify the sufcient condition. Secondly, show that under certain mild restriction on systems the condition is also necessary. Recall that an n dimensional manifold is orientable if it is possible to dene a C f n-form Z on M that is not zero at any point. In this case M is said to be oriented by Z. Denition 6.8. Let f ∈ V (M). It deduces a mapping i f : : k+1 → : k , which is dened for each Z ∈ : k+1 as

6.4 Codim(G ) = 1

i f (Z )(⋅) := Z ( f , ⋅).

169

(6.52)

Now we are ready to give a general result. Theorem 6.3. 1. Assume for each x0 ∈ M there exist V1 ∈ : 1 (M), VG ∈ : n−1 (M), where VG is an orientation of IG (x0 ), and V1 (Z) = 0, ∀Z ∈ G , such that V = V1 ∧ VG is an orientation of M. Moreover, on each IG (x) there are two vector elds f1 , f2 ∈ F and two points x1 , x2 ∈ IG (x0 ) such that i f1 (V )∣G (x1 ) = c1 VG (x1 ),

i f2 (V )∣G (x2 ) = c2 VG (x2 ),

(6.53)

where f1 , f2 ∈ F and c1 , c2 < 0. Then the system (6.7) is globally controllable. 2. If, in addition, each IG (x0 ) separates M, then (6.53) is also necessary. Proof. (Sufciency) According to Theorem 6.1, we only have to prove that G (x2 , x1 )∗ f1 (x1 ) and f2 (x2 ) are on the two sides of IG (x2 ). Since IG (x) is orientable, we can nd a set of coherently oriented coordinate charts (UP , xP ), such that [2] P VG = K P (x)x1P ∧ x2P ∧ ⋅ ⋅ ⋅ ∧ xn−1 ,

where K P (x) > 0, x ∈ UP . Let a canonical local basis of G be chosen as Zi =

w , w xiP

x ∈ UP , i = 1, ⋅ ⋅ ⋅ , n − 1,

which are chart-depending. We have

VG (Z1 , ⋅ ⋅ ⋅ , Zn−1 )(x) > 0,

∀x ∈ M.

(6.54)

P

(Precisely, x ∈ UP and Zi = Zi , which are dened over UP .) Now we have i f1 V (Z1 , ⋅ ⋅ ⋅ , Zn−1 ) = V1 ( f1 )VG (Z1 , ⋅ ⋅ ⋅ , Zn−1 ) = c1 (x)VG (Z1 , ⋅ ⋅ ⋅ , Zn−1 ).

(6.55)

So we have

V1 ( f1 )(x1 ) = c1 (x1 ).

(6.56)

V1 ( f2 )(x2 ) = c2 (x2 ).

(6.57)

Similarly, we have

Recall that G (x2 , x1 )∗ f1 (x1 ) = icl∗ [ f1 (x1 )/G ] + K ,

K ∈ G.

Then using the fact that V1 (Z) = 0, Z ∈ G , we have

V1 [G (x2 , x1 )∗ f1 (x1 )](x2 ) = V1 ( f1 ) = c1 (x1 ). Now since

V [G∗ f1 (x2 , x1 ), Z1 , ⋅ ⋅ ⋅ , Zn−1 ](x2 ) = c1 (x1 )VG (Z1 , ⋅ ⋅ ⋅ , Zn−1 )

(6.58)

170

6 Global Controllability of Afne Control Systems

and

V [ f2 , Z1 , ⋅ ⋅ ⋅ , Zn−1 ](x2 ) = c2 (x2 )VG (Z1 , ⋅ ⋅ ⋅ , Zn−1 ) have opposite signs, [G (x2 , x1 )∗ f1 (x1 )](x2 ) and f2 (x2 ) must be on the different sides of IG (x2 ). To see this dene Y (t) = tY1 + (1 − t)Y2,

t ∈ [0, 1],

where Y1 = [G (x2 , x1 )∗ f1 (x1 )](x2 ) and Y2 = f 2 (x2 ). If both Y1 and Y2 are on the same side of IG (x2 ), then Y (t) is always linearly independent of G (x2 ). But V [Y (t), Z1 , ⋅ ⋅ ⋅ , Zn−1 ](x2 ) has different signs at t = 0 and t = 1, so there exits a 0 < t ∗ < 1, such that V [Y (t ∗ ), Z1 , ⋅ ⋅ ⋅ , Zn−1 ](x2 ) = 0, which is a contradiction. (Because an orientation can never be zero over a set of n linearly independent vector elds.) (Necessity) Assume there is a IG (x) such that (6.53) fails. According to our previous argument we know that for all y, z ∈ IG (x) and f 1 , f2 ∈ F we have V1 ( f1 (y)) and V1 ( f2 (z)) have the same sign. That is, all the f ∈ F point to the same side of IG (x). Since IG (x) separates M, it is easy to see that the half M that is pointed by fi ’s is an invariant set. So the system is not controllable. ⊔ ⊓ Some conditions in Theorem 6.3 are related. We refer to [8] for verication and possible simplication of those conditions. Corollary 6.2. Assume M = ℝn . Let G (x0 ), ∀x0 ∈ M, be an n − 1 hyperplane, and Z1 , ⋅ ⋅ ⋅ , Zn−1 be a basis of G . Then a necessary and sufcient condition for system (6.7) to be globally controllable is that on each IG (x0 ), x0 ∈ M, there are two vector elds, f1 , f2 ∈ F, and two points x1 , x2 ∈ IG (x0 ) (same fi and/or xi are allowed), such that det( f 1 , Z1 , ⋅ ⋅ ⋅ , Zn−1 )(x1 ) det( f2 , Z1 , ⋅ ⋅ ⋅ , Zn−1 )(x2 ) < 0.

(6.59)

Proof. First, (6.59) is equivalent to the following statement: Choose V = dx1 ∧dx2 ∧ ⋅ ⋅ ⋅ ∧ dxn , then

V ( f1 , Z1 , ⋅ ⋅ ⋅ , Zn−1 )(x1 )V ( f2 , Z1 , ⋅ ⋅ ⋅ , Zn−1 )(x2 ) < 0.

(6.60)

Since (6.59) is independent of the choice of orientation V , we only have to prove that there exists a particular orientation such that (6.59) implies (6.53). Choose a particular coordinate frame z = z(x) such that { G = Span

w w ,⋅⋅⋅ , w z2 w zn

} .

Then we simply choose V = dz1 ∧ dz2 ∧ ⋅ ⋅ ⋅ ∧ dzn , and set V1 = dz1 and VG = dz2 ∧ ⋅ ⋅ ⋅ ∧ dzn . Let fi (z) = ( fi1 (z), ⋅ ⋅ ⋅ , fin (z))T , i = 1, 2. Then i f1 V = f11 dz2 ∧ ⋅ ⋅ ⋅ ∧ dzn + dx1 ∧ K ,

where K ∈ : n−2 (M).

References

So

 i f1 V G = f11 dz2 ∧ ⋅ ⋅ ⋅ ∧ dzn .

Similarly,

 i f2 V G = f21 dz2 ∧ ⋅ ⋅ ⋅ ∧ dzn .

171

Note that if Zi ∈ G , then Zi = (0, zi2 , ⋅ ⋅ ⋅ , zin )T . So (6.60) implies that f11 (x1 ) and f21 (x2 ) have different signs, which implies (6.53). ⊔ ⊓ Example 6.10. Consider the following system ⎤ ⎡ ⎤ ⎤ ⎡ ⎡ 2 0 0 x2 + x3 ⎥ ⎢ ⎥ ⎥ ⎢ ⎢ ⎢ sin(x4 ) ⎥ ⎢ f2 ⎥ ⎢0⎥ ⎢ ⎥ ⎥ ⎥ ⎢ ⎢ x = ⎢ ⎥ + ⎢ ⎥ u1 + ⎢ ⎥u . ⎢cos(x4 )⎥ 2 ⎢ f3 ⎥ ⎢0⎥ ⎦ ⎣ ⎦ ⎦ ⎣ ⎣ It is easy to check that

{ G=

A basis of G is

⎡ ⎤ 0 ⎢ ⎥ ⎢0⎥ ⎢ ⎥ Z1 = ⎢ ⎥ ; ⎢0⎥ ⎣ ⎦ 1

⎡

g24

1

f4

0

w w w , , w x2 w x3 w x4

} .

⎤

⎥ ⎢ ⎢ sin(x4 ) ⎥ ⎥ ⎢ Z2 = ⎢ ⎥; ⎢cos(x4 )⎥ ⎦ ⎣

(6.61)

⎡

0

⎤

⎥ ⎢ ⎢ cos(x4 ) ⎥ ⎥ ⎢ Z3 = [Z1 , Z2 ] = ⎢ ⎥. ⎢− sin(x4 )⎥ ⎦ ⎣

0

0

 { } Moreover, IG (x0 ) = x ∈ ℝ4  x4 = x4 (x0 ) , and det( f , Z1 , Z2 , Z3 ) = x22 + x3 . Obviously, (6.59) is satised for some x2 , x3 ∈ IG (x0 ). So system (6.61) is controllable.

References 1. Abraham R, Marsden J, Ratiu T. Manifolds, Tensor Analysis and Applications. New York: Springer-Verlag, 1988. 2. Boothby W. An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd edn. Orlando: Academic Press, 1986. 3. Brockett R. Nonlinear systems and differential geometry. Proceedings of the IEEE, 1976, 64(1): 61–72. 4. Cheng D, Isidori A, Respondic W, et al. On the linearization of nonlinear systems with outputs. Mathenatical Systems Theory, 1988, 21(2): 63–83. 5. Cheng D, Tarn T, Isidori A. Global external linearization of nonlinear systems via feedback. IEEE Trans. Aut. Contr., 1985, 30(8): 808–811. 6. Clarke F, Stern R. Lyapunov and feedback characterizations of state constrained controllability and stabilization. Sys. Contr. Lett., 2005, 54(8): 747–752. 7. Hermann R. Differential Geometry and the Calculus of Variations. New York:Academic Press, 1968.

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6 Global Controllability of Afne Control Systems

8. Hirsch M. Differential Topology. New York: Springer-Verlag, 1976. 9. Olver P. Applications of Lei Groups to Differential Equations, 2nd edn. New York: SpringerVerlag, 1993. 10. Respondek J. Controllability of dynamical systems with constraints. Sys. Contr. Lett., 2005, 54(4): 293–314. 11. Sun Y. Necessary and sufcient condition for global controllability of planar afne nonlinear systems. IEEE Trans. Aut. Contr., 2007, 52(8): 1454–1460. 12. Sun Z, Ge S, Lee T. Controllability and reachability criteria for switched linear systems. Automatica, 2002, 38(5): 775–786.

Chapter 7

Stability and Stabilization

Stability is a fundamental property for dynamic systems. In most engineering projects unstable systems are useless. Therefore in system analysis and control design the stability and stabilization become the rst priority to be consider. This chapter considers the stability of dynamic systems and the stabilization and stabilizer design of nonlinear control systems. In Section 7.1 the concepts about stability of dynamic systems are presented. Section 7.2 considers the stability of nonlinear systems via its linear approximation. The Lapunov direct method is discussed in Section 7.3. Section 7.4 presents the LaSalle’s invariance principle. The converse theory of Lyapunov stability is introduced in Section 7.5. Section 7.6 is about the invariant set. The input-output stability of control systems is discussed in Section 7.7. In section 7.8 the semi-tensor product is used to nd the region of attraction. Many results in this chapter are classical, hence the proofs are omitted.

7.1 Stability of Dynamic Systems Stability theory has had a long history. The concept of stability came originally from mechanics where the equilibrium of a rigid body was studied. The modern stability theory that allows the analysis of a general differential equation was rst developed by Russian mathematician Aleksandr M. Lyapunov (1857–1918). In this section we will introduce stability concepts in the sense of Lyapunov [6, 14]. Consider a time-varying dynamic nonlinear system x = f (x,t),

x ∈ ℝn ,t ⩾ 0.

(7.1)

A point x0 is called an equilibrium if f (x0 ,t) = 0,

t ⩾ 0.

Without loss of generality, we assume x0 = 0. Because if x0 ∕= 0 we may make a coordinate change z = x − x0 . Then, expressing (7.1) into z coordinate frame, z = 0 becomes the equilibrium. For convenience we also assume that the system has a unique solution for each initial condition x0 in a neighborhood N0 of x0 and each t0 , which is denoted by x(x0 ,t,t0 ). We refer to standard ODE textbooks for the existence and uniqueness of the solution of (7.1). In the following we give a simple commonly used result. System (7.1) (or briey, f (t, x)) is said to satisfy Lipschitz condition if there exists a constant D. Cheng et al., Analysis and Design of Nonlinear Control Systems © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2010

174

7 Stability and Stabilization

number L > 0, such that ∥ f (x,t) − f (y,t)∥ ⩽ L∥x − y∥,

x, y ∈ ℝn ,t ⩾ 0.

(7.2)

It is said to satisfy the Lipschitz condition on a region U ⊂ ℝn if (7.2) is satised for any x, y ∈ U. Theorem 7.1. [8] Consider system (7.1). Assume that it satises Lipschitz condition (7.2) for x, y ∈ Br (x0 ) and t ∈ [t0 ,t1 ]. Then there exists a G > 0 such that system (7.1) with initial condition x(t0 ) = x0 has a unique solution on [t0 ,t0 + G ] ⊂ [t0 ,t1 ]. If the time t does not appear explicitly on the right hand side of (7.1), we have x = f (x),

x ∈ ℝn ,

(7.3)

which is called an autonomous system. Autonomous system is mostly important in engineering applications. Because of autonomy we have x(x0 ,t + c,t0 + c) = x(x0 ,t,t0 ), which implies that the solution does not depend on a particular initial time. Thus we can rewrite the solution as x(x0 ,t − t0 ) or simply assume t0 = 0. First, we consider the Lyapunov stability. Denition 7.1. Consider system (7.1) and assume x = 0 is an equilibrium. 1. x = 0 is stable, if for any H > 0 and any t0 ⩾ 0, there exists a G (H ,t0 ) > 0, such that ∥x0 ∥ < G (H ,t0 ) ⇒ ∥x(x0 ,t,t0 )∥ < H ,

∀t ⩾ t0 ⩾ 0.

(7.4)

2. x = 0 is uniformly stable, if it is stable, and in (7.4) the G is independent of t0 , i.e., G (H ,t0 ) = G (H ). 3. x = 0 is unstable, if it is not stable. Next, we consider the convergence of the solutions. Denition 7.2. Consider system (7.1) and assume x = 0 is an equilibrium. 1. x = 0 is attractive, if for each t0 ⩾ 0 there exists a K (t0 ) > 0 such that ∥x0 ∥ < K (t0 ) ⇒ lim x(x0 ,t + t0 ,t0 ) = 0. t→+f

(7.5)

2. x = 0 is uniformly attractive, if there exists an K > 0 such that ∥x0 ∥ < K implies that x(x0 ,t + t0 ,t0 ) converges to zero uniformly in t0 and x0 . Finally we consider the asymptotical stability. Denition 7.3. Consider system (7.1) and assume x = 0 is an equilibrium. 1. x = 0 is asymptotically stable (A.S.), if x = 0 is stable and attractive.

7.2 Stability in the Linear Approximation

175

2. x = 0 is uniformly asymptotically stable (U.A.S.), if x = 0 is uniformly stable and uniformly attractive. We give some simple examples to depict this. Example 7.1. 1. Consider the following system x = −(2 + sin(t))x.

(7.6)

Under the initial condition x(t0 ) = x0 , the solution can be obtained and then it is easy to check that ∥x(t)∥ ⩽ ∥x0 ∥e−(t−t0 ) ,

(7.7)

which implies that the system is uniformly asymptotically stable. 2. Consider the following system x = −

1 x. 1+t

(7.8)

The solution is x=

1 + t0 x0 . 1 + t0 + t

(7.9)

From (7.9) it is clear that the system is asymptotically stable. But it is not uniformly asymptotically stable because it is not uniformly attractive. If the solution of a system satises an exponential relation as (7.7), the system is said to be exponentially stable. We give a rigorous denition. Denition 7.4. System (7.1) is said to be exponentially stable at x0 = 0, if there exist positive numbers a > 0 and b > 0 and a neighborhood N0 of the origin, such that ∥x(x0 ,t,t0 )∥ ⩽ a∥x0 ∥e−b(t−t0 ) ,

t ⩾ t0 ⩾ 0, x0 ∈ N0 .

(7.10)

For autonomous systems, it follows from denitions that the stability and uniform stability are equivalent. In fact, it is also true for periodic system. Autonomy is a special case of periodicity. Lemma 7.1. [6] If f (t, x) is periodic, then uniform (asymptotical) stability is equivalent to (respectively, asymptotical) stability.

7.2 Stability in the Linear Approximation We consider rst a time-invariant linear system: x = Ax,

x ∈ ℝn .

(7.11)

For such a system, we state some well known stability properties as a lemma.

176

7 Stability and Stabilization

Lemma 7.2. Consider system (7.11). 1. x = 0 is asymptotically stable, if and only if all the eigenvalues of A have negative real parts. 2. For system (7.11), asymptotic stability is equivalent to exponential stability. 3. x = 0 is stable, if and only if (i). it does not have eigenvalues with positive real parts; (ii). for imaginary eigenvalues (including 0), their algebraic multiplicity should be equal to their geometric multiplicity. Equivalently, their corresponding Jordan blocks are diagonal. [7] 4. A is asymptotically stable, if and only if for any positive denite matrix Q > 0, there exists a positive denite matrix P > 0 such that PA + ATP = −Q

(7.12)

In other words, if we use such P to construct a quadratic form V (x) = xT Px, then V = −xT Qx < 0. When in (7.1) the coefcient matrix is time-varying, the stability analysis is more difcult [9]. Now let us consider (7.1) and suppose it can be written as: x = A(t)x + g(x,t),

(7.13)

where g(x,t) = O(∥x∥2 ), x ∈ Br (0),t ⩾ t0 ⩾ 0. Then we call z = A(t)x,

(7.14)

the Jacobian linearized system of (7.1). About the exponential stability we have the following: Theorem 7.2. [6] If the equilibrium x = 0 of (7.14) is exponentially stable, the x = 0 of (7.13) is also exponentially stable. We have the following unstable result. Theorem 7.3. [6] Consider system (7.11). If A(t) = A is a constant matrix, and at least one of the eigenvalues of A is located in the open right half plane, then x = 0 of (7.11) is unstable. Remark 7.1. Consider a nonlinear system, it is important to know how much its linear approximation (also called the Jacobian linearization) can tell us. Moreover, the difference between linear time-varying and time-invariant systems is also signicant. We would like to emphasize the followings: 1. The principle of stability in the linear approximation only applies to local stability analysis. 2. In the case where the linearized system is autonomous, and some of the eigenvalues are on the imaginary axis, and the rest are on the left open half plane, (“the critical case”) then one has to consider the nonlinear terms to determine stability.

7.3 The Direct Method of Lyapunov

177

3. In the case of a time-varying linearized system, if z = 0 is only (non-uniformly) asymptotically stable, then one also has to consider the nonlinear terms. Before ending this section, we give two useful results on nonlinear stability. The rst result is for the so-called triangular systems. Proposition 7.1. [13] Consider { z = f (z, y), z ∈ ℝ p y = g(y), y ∈ ℝq .

(7.15)

Suppose y = 0 of y = g(y) and z = 0 of z = f (z, 0) are asymptotically stable, then (z, y) = (0, 0) of (7.15) is asymptotically stable. The second result is for two dimensional systems obtained by Lyapunov, which is quite useful for determining stability in the so-called critical case. Proposition 7.2. [6] Consider a two dimensional system: { m+1 + ⋅ ⋅ ⋅ + x1 (bxn2 + b2 xn+1 + ⋅ ⋅ ⋅) + x21 h(x1 , x2 ) x1 = axm 2 + a2 x2 2 x2 = x1 + g(x1 , x2 ),

(7.16)

where g is at least of second order and h is Lipschitz continuous. The origin is unstable, if one of the following conditions holds: 1. 2. 3. 4. 5.

m is even; m is odd and a > 0; m is odd, a < 0, n even, m ⩾ n + 1 and b > 0; m is odd, a < 0, n odd, and m ⩾ 2n + 2; the equation for x1 contains only x21 h(x1 , x2 ).

The origin is asymptotically stable if 1. m is odd, a < 0, b < 0, n even, and m ⩾ n + 1; m+1 + ⋅ ⋅ ⋅ = 0. 2. n is even, b < 0, and axm 2 + a2 x2 For the rest of the cases, higher order expansions must be considered.

7.3 The Direct Method of Lyapunov The main idea of Lypapunov’s method is that the stability of an equilibrium is decided by the sign of a certain function, without the need to know the explicit solution of the differential equation, which is the reason why this is called the direct method. We note that Lyapunov also used another method, so the direct method is often called Lyapunov’s second method.

7.3.1 Positive Denite Functions Denition 7.5. Let I : ℝ+ → ℝ+ be a continuous function.

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7 Stability and Stabilization

1. I is of class K if it is strictly increasing and I (0) = 0; 2. I is of class Kf if it is of class K and lim I (r) = f; r→f

3. I is of class L if it is continuous, strictly decreasing, I (0) < f and lim I (r) = 0. r→f

Denition 7.6. (Various Function Classes) 1. A function V : ℝ+ × ℝn → ℝ is said to be a locally positive denite function if it is continuous, V (t, 0) = 0, ∀t ⩾ 0 and there exists an r > 0 and a function D ∈ K such that

D (∥x∥) ⩽ V (t, x),

∀t ⩾ 0, ∀ x ∈ Br (0).

(7.17)

2. V is decrescent if there exists a function E of class K such that V (t, x) ⩽ E (∥x∥),

∀x ∈ Br (0), ∀t ⩾ 0.

(7.18)

3. V is positive denite if r = f. 4. V is radially unbounded if there is a smooth function M : ℝ → ℝ, with lim M (r) = r→f f, such that V (t, x) ⩾ M (∥x∥), ∀t > 0. To determine whether a time varying function is (locally) positive denite ((L)PDF), we may compare it with a time invariant (L)PDF function. The following result is straightforward veriable. Proposition 7.3. Consider a continuous function W : ℝn → ℝ, and assume there is a continuous time varying function V (t, x) with V (t, 0) = 0, satisfying V (t, x) ⩾ W (x),

∀t ⩾ 0.

then V (t, x) is a LPDF (PDF) if W (x) is a LPDF (correspondingly, PDF). According to Proposition 7.3, we may judge the (locally) positive deniteness of a time varying function by comparing it with a time invariant (L)PDF. The following results are easily veriable. But they are convenient in use. Proposition 7.4. Let W (x) : U → ℝ be a continuous function, where 0 ∈ U ⊂ ℝn is a neighborhood of the origin. Then W (x) is a LPDF if (i). W (0) = 0; (ii). W (x) > 0, ∀x ∈ Br (0)∖{0}. Remark 7.2. If W (x) ∈ C1 and W (x) is LPDF, then W (x) is also decrescent in Br (0), for certain r > 0. Proposition 7.5. Let W (x) : ℝn → ℝ be a continuous function. Then 1. W (x) is a PDF if in addition to the two conditions in Proposition 7.4, it also satises the following condition: There exists a c > 0, such that inf∥x∥>cW (x) > 0.

7.3 The Direct Method of Lyapunov

179

2. W is radially unbounded if W (x) → f as ∥x∥ → f uniformly in x. Now let us consider some examples. Example 7.2. (Classify Functions) The following functions are some classical examples. We will only point out their types and leave to reader to verify the claims. 1. V (x) = x21 + x22 is positive denite in ℝ2 . 2. V (t, x) = x21 (1 + sin2 (t)) + x22 (1 + cos2 (t)) is positive denite and decrescent. 3. V (t, x) = (t + 1)(x21 + x22 ) is positive denite, but not decrescent. 4. V (x) = x21 + sin2 (x2 ) is LPDF but not PDF. 5. V (x) =

x21 1+x21

+ x22 is positive denite but not radially unbounded.

7.3.2 Critical Stability Consider the system x = f (x,t),

x ∈ ℝn ,

(7.19)

where f ∈ C1 (ℝn ). Suppose x = 0 is the equilibrium of interest, for V (t, x), we dene the total derivative, V as:

wV wV V (t, x) := (t, x) + f (x,t). wt wx

(7.20)

Theorem 7.4. x = 0 of (7.19) is uniformly stable if there exists a C1 , decrescent LPDF V (t, x), x ∈ U, where U is a neighborhood of the origin, such that V ⩽ 0,

∀t ⩾ 0, x ∈ U.

(7.21)

Such a function V (t, x) is called a Lyapunov function. Proof. Since V is decrescent and LPDF, there exist class K functions D and E such that for sufciently small ∥x∥

D (∥x∥) ⩽ V (t, x) ⩽ E (∥x∥). Since V ⩽ 0, then for any solution x(x0 ,t,t0 ) we have

D (x(x0 ,t,t0 )) ⩽ V (x(x0 ,t,t0 ),t) ⩽ V (x0 ,t0 ) ⩽ E (∥x0 ∥). Thus,

∥x(x0 ,t,t0 )∥ ⩽ D −1 (V (t)) ⩽ D −1 (E (∥x0 ∥)).

Obviously, J := D −1 ∘ E is a class K function, then

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7 Stability and Stabilization

∥x(x0 ,t,t0 )∥ ⩽ J (∥x0 ∥), which implies x = 0 is uniformly stable.

⊔ ⊓

Remark 7.3. In fact it is the requirement of V (t, x) being decresent that guarantees uniform stability, otherwise we would only have stability. We note that it is not always possible to nd a time independent Lyapunov function even if the system is autonomous [6]. Example 7.3. {

x1 = a(t)x2n+1 2 x2 = −a(t)x2n+1 . 1

(7.22)

+ x2n+2 ), giving V = 0, and the system is thus uniformly stable. Let V = 12 (x2n+2 1 2 Example 7.4. Consider the motion of a spacecraft along a principal axis ⎧ J2 − J3   Z 1 = Z2 Z3 + u1   J1  ⎨ J3 − J1 Z 1 = Z1 Z3 + u2  J2    J − J2  ⎩Z 1 = 1 Z2 Z1 + u3 . J3

(7.23)

We leave to reader to check that when u1 = u2 = u3 , Z = 0 is uniformly stable.

7.3.3 Instability The following is a standard result on instability [6]. Theorem 7.5. Consider system (7.1) with x = 0 as its equilibrium. x = 0 is unstable if there exists a C1 , decrescent function V (t, x) and t0 ⩾ 0 such that (i). V is a LPDF; (ii). V (t, 0) = 0,∀t ⩾ t0 ; (iii). there exists a sequence {xn ∕= 0} where xn → 0 as n → f such that V (t0 , xn ) ⩾ 0. In particular, if V (t, x) is a LPDF, then V (t, x) automatically satises conditions (ii) and (iii), when (i) is satised.

7.3.4 Asymptotic Stability Theorem 7.6. The equilibrium 0 of (7.1) is uniformly asymptotically stable if there exists a C1 decrescent LPDF V (x,t), such that −V is LPDF. By Theorem 7.4 we know that x = 0 is uniformly stable. So to show that it is uniformly asymptotically stable, we only need to show that it is uniformly attractive. Interested readers can nd proof of the result in [8] or [14]. For autonomous systems, Theorem 7.6 can be restated as:

7.3 The Direct Method of Lyapunov

181

Proposition 7.6. Consider an autonomous system x = f (x).

(7.24)

If there exists a LPDF V (x) such that −V is LPDF, then x = 0 is asymptotically stable. Now an interesting question is what are the initial points from which the solution tends to the origin? Denition 7.7. Consider (7.24). The domain of attraction is dened as D(0) = {x0 ∈ ℝn : x(x0 ,t) → 0 as t → f}. As a matter of fact the domain of attraction is always an open set [6]. Suppose on a domain S, V (x) > 0,

V (x) < 0,

∀0 ∕= x ∈ S.

(7.25)

Does this imply that S is contained in the domain of attraction D(0) and/or S is invariant? Let us look at the following example. Example 7.5. {

x1 = −x1 − x2 + x31 + x1x22 x2 = −x2 + 2x1 + x32 + x21 x2 .

(7.26)

Let V = 12 (2x21 + x22 ), giving V = −2x21 − x22 + (2x21 + x22 )(x21 + x22 ) ⩽ 0,

∀∥x∥ < 1.

But ∥x∥ < 1 is not in D(0). More detailed discussion on domain of atrraction will be discussed later. The next theorem deals with exponential stability. Theorem 7.7. Suppose there exists a LPDF V (t, x) which is bounded by a∥x∥2 ⩽ V (t, x) ⩽ b∥x∥2 ,

∀t ⩾ 0, ∀x ∈ Br (0).

(7.27)

If V (t, x) ⩽ −c∥x∥2 ,

∀t ⩾ 0, ∀x ∈ Br (0),

(7.28)

where a, b, c are positive real numbers, then x = 0 is exponentially stable. Proof. Since −b∥x∥2 ⩽ −V (t, x), we have c V (t, x) ⩽ −c∥x∥2 ⩽ − V (t, x), b and

(7.29)

182

7 Stability and Stabilization c

V (t, x(t)) ⩽ V (t0 , x0 )e− b (t−t0 ) .

(7.30)

Using (7.29) and (7.30), we have 1 ∥x(t)∥2 ⩽ V (t, x(t)) a c 1 ⩽ V (t0 , x0 )e− b (t−t0 ) a c b ⩽ ∥x0 ∥2 e− b (t−t0 ) . a So

√ ∥x(t)∥ ⩽

c b ∥x0 ∥e− 2b (t−t0 ) , a

∀t ⩾ t0 .

(7.31) ⊔ ⊓

7.3.5 Total Stability Suppose system (7.1) is affected by some disturbance, which yields a disturbed system as x = f (x,t) + g(x,t).

(7.32)

Denote the solution of (7.32) by xg (x0 ,t,t0 ). Denition 7.8. x = 0 is called totally stable (stable under persistent disturbances) if for all H > 0 there exists two positive numbers G1 (H ) and G2 (H ) such that ∥xg (x0 ,t,t0 )∥ < H , if

∥x0 ∥ < G1 and ∥g(x,t)∥ < G2 ,

∀t ⩾ t0 ⩾ 0,

(7.33)

x ∈ BH (0), ∀t ⩾ t0 ⩾ 0.

We have the following result for total stability. Theorem 7.8. [6] If x = 0 of (7.32) is uniformly asymptotically stable, it is totally stable.

7.3.6 Global Stability Denition 7.9. Consider system (7.1). x = 0 is globally exponentially stable if there exists constants a, b > 0 such that ∥x(x0 ,t,t0 )∥ ⩽ a∥x0 ∥e−b(t−t0 ) ,

∀t ⩾ t0 ⩾ 0, ∀x0 ∈ ℝn .

(7.34)

Denition 7.10. Consider system (7.2). x = 0 is globally uniformly asymptotically stable, if it is uniformly asymptotically stable and the domain of attraction is ℝn . Theorem 7.9. Consider system (7.1). The equilibrium 0 is globally exponentially stable if there exists a PDF V (t, x) such that

7.4 LaSalle’s Invariance Principle

a∥x∥2 ⩽ V (t, x) ⩽ b∥x∥2 ,

∀t ⩾ 0, ∀x ∈ ℝn ,

183

(7.35)

and V (t, x) ⩽ −c∥x∥2 ,

∀t ⩾ 0, ∀x ∈ ℝn ,

(7.36)

where a, b, c are positive. The proof of this result is quite straight forward, thus omitted here. For the general case, we give the following result without proof. Interested readers can nd proof in for example [8]. Theorem 7.10. Consider system (7.1). x = 0 is globally uniformly asymptotically stable, if there exists a radially unbounded, decrescent PDF V (x,t), such that −V is PDF. We need to point out here that the condition on the Lyapunov function being radially unbounded is very crucial for global stability. Example 7.6. Consider the following system { x1 = −x31 + x22 x2 = −x32 − x1 x2 . Let V = 12 (x21 + x22 ), giving

(7.37)

V = −x41 − x42 ,

which implies x = 0 is globally asymptotically stable (remember: for autonomous systems the uniform condition is automatically satised). Example 7.7. As an illustration of the necessity of radially unboundedness, consider the following system { x1 = −x31 + x22 x31 (7.38) x2 = −x2 . Let V =

x21 1+x21

+ 2x22 , giving V ⩽ −

2x41 − 2x22 . (1 + x21 )2

However, the system is not globally asymptotically stable. This can be seen by solving for x2 and insert it in the rst equation for x1 . Then one can calculate the domain of attraction. We leave this as an exercise for reader.

7.4 LaSalle’s Invariance Principle When we use Lyapunov’s direct method for determining asymptotic stability, it is essential that we can nd a positive denite function whose total derivative is negative denite. However, nding such a Lyapunov function can be difcult in many

184

7 Stability and Stabilization

cases. With Lasalle’s invariance principle, it is possible to determine asymptotic stability even if the total derivative is only negative semi-denite. This section provides some main results about LaSalle’s invariance principle, and we refer to [11] and [8] for more details. Example 7.8. Consider the following system { x1 = x2 , x2 = −x1 − f (x2 ),

(7.39)

where f (0) = 0 and [ f ([ ) > 0 if [ ∕= 0. Let V = 12 (x21 + x22 ), giving V = −x2 f (x2 ). We have V = 0 when x2 = 0. Apparently V is not negative denite. The question is in this case if we can draw any asymptotic stability conclusion? Consider a locally Lipschitz autonomous system x = f (x),

x ∈ ℝn .

(7.40)

A point p ∈ ℝn is called an Z -limit point for the solution x(x0 ,t) of (7.40) if there exists a sequence tn → f such that x(x0 ,tn ) → p as tn → f. A very important property of the Z -limit points is as follows. Proposition 7.7. If a solution x(x0 ,t) of (7.40) is bounded, then the set of its Z -limit points S is compact and invariant. Furthermore, x(x0 ,t) → S as t → f. Now we can state LaSalle’s invariance principle as follows. Theorem 7.11. Consider system (7.40). Let V : ℝn → ℝ be a continuously differentiable function and assume : is a compact set such that V ⩽ 0 in : . Let E be dened as E = {x ∈ : : V (x) = 0}. Then every solution x(x0 ,t) initialized in : approaches S as t → f, where S is the largest invariant set in E. We will not provide proof for either of the results stated above. Interested readers can nd them in, for example, [8] or [11]. It is worthy noting that in the above theorem, V does not need to be positive denite. On the other hand, the choice of : depends on V . In practice, one can always check rst if a level set of V is qualied as : . Since our main interest is to show the equilibrium to be asymptotically stable, we need to make sure that the largest invariant set S is the equilibrium itself. If

7.5 Converse Theorems to Lyapunov’s Stability Theorems

185

we further assume V is positive denite (which makes the choice of : easier), we obtain the following corollary. Corollary 7.1. 1. Let x = 0 be an equilibrium of (7.40). If there exists a C1 LPDF V (x) whose total derivative is non-positive in a neighborhood of the origin N(0), and no trajectory other than x = 0 lies in a region inside N(0) dened by V = 0, then the equilibrium is asymptotically stable. 2. If V (x) is PDF and radially unbounded, and the other conditions in item 1 all hold if N(0) is replaced by ℝn , then x = 0 is globally asymptotically stable. Now let us revisit the Example 7.8. Since we have V = 0 when x2 = 0, we need to check what are the possible trajectories of the form (a(t), 0). Since x2 (t) = 0, ∀t ⩾ 0 implies 0 = x2 = −x1 (t) = −a(t), thus the only trajectory of such kind is x = 0. Therefore, the origin is asymptotically stable. Since our V is radially unbounded and N(0) can be taken as the whole ℝ2 , the origin is globally asymptotically stable.

7.5 Converse Theorems to Lyapunov’s Stability Theorems In this chapter we only consider the autonomous system x = f (x),

x ∈ ℝn ,

(7.41)

where f ∈ C1 (ℝn ) and f (0) = 0. We give a series of converse theorems to Lyapunov’s stability theorems. The proof for most of the results presented in the section can be found in the classical book [6].

7.5.1 Converse Theorems to Local Asymptotic Stability The rst one is the converse to asymptotical stability. Theorem 7.12. Consider system (7.41). Suppose x = 0 is asymptotically stable, then there exists a C1 function V(x) and functions D , E and J of class K such that

D (∥x∥) ⩽ V (x) ⩽ E (∥x∥),

∀x ∈ Br (0);

(7.42)

and V (x) ⩽ −J (∥x∥),

∀x ∈ Br (0).

(7.43)

The second one is the converse to exponential stability. Theorem 7.13. Consider system (7.41). Suppose x = 0 is exponentially stable, then there exists a C1 function V (x) and positive constants a > 0, b > 0, and c > 0 such that a∥x∥2 ⩽ V (x) ⩽ b∥x∥2, and

∀x ∈ Br (0),

(7.44)

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7 Stability and Stabilization

V (x) ⩽ −c∥x∥2 ,

∀x ∈ Br (0).

(7.45)

or equivalently, we have the following (7.46) and (7.47). 9 wV 9 9 9 9 ⩽ P ∥x∥, 9 wx

wV f (x) ⩽ −c∥x∥2 , wx

∀x ∈ Br (0),

(7.46)

∀x ∈ Br (0).

(7.47)

Proof. We prove (7.44) and (7.45) only. (7.46) is the same as (7.45) and refer to [6] for the proof of (7.47). Suppose ∀x0 ∈ Br (0) we have ∥x(x0 ,t)∥ ⩽ D ∥x0 ∥e−J t , Let V (x0 ) = Then V (x0 ) ⩽ On the other hand

∫ f 0

∫ f 0

D , J > 0.

∥x(x0 , 9 )∥2 d9 .

D 2 ∥x0 ∥2 e−2J9 d9 ⩽

D2 ∥x0 ∥2 . 2J

d ( 2) 1 ′ 1 ∥x∥ = x x = x′ f (x), dt 2 2

where x reads x(x0 ,t). Since f ∈ C1 , so f is Lipschitz in Br (0). Therefore    1 ′  x f (x) − 0 ⩽ 1 L ∥x∥2 ,  2 2 or

1 1 1 − L ∥x∥2 ⩽ x′ f (x) ⩽ L ∥x∥2 . 2 2 2

So

d ( 2) 1 ∥x∥ ⩾ − L ∥x∥2 , dt 2

which implies that ∥x∥2 ⩾ ∥x0 ∥2 e− 2 L t . 1

It turns out that V (x0 ) ⩾ Now

2 ∥x0 ∥2 . L ∫

d f dV (x(x0 ,t)) = V (x) = ∥x(x(x0 ,t), 9 )∥2 d9 dt dt 0 ∫ d f = ∥x(x0 ,t + 9 )∥2 d9 . dt 0 Putting r = 9 + t yields d dt

∫ f t

∥x(x0 , r)∥2 dr = −∥x(x0 ,t)∥2 = −∥x∥2 .

⊔ ⊓

7.5 Converse Theorems to Lyapunov’s Stability Theorems

187

7.5.2 Converse Theorem to Global Asymptotic Stability Theorem 7.14. Consider system (7.41). Suppose x = 0 is globally asymptotically stable, then there exists a C1 function V(x) and D , E and J of class Kf such that

D (∥x∥) ⩽ V (x) ⩽ E (∥x∥),

∀x ∈ ℝn ,

(7.48)

and V (x) ⩽ −J (∥x∥),

∀x ∈ ℝn

(7.49)

Remark 7.4. In general we can not show the global version of Theorem 7.13, unless one assumes that f (x) satises a linear growth condition, i.e.,  w f (x)     < K, ∀x ∈ ℝn .  wx Next, we give the following result as an application of the converse theorems. Theorem 7.15. Let f (x) be C2 and A =

wf w x ∣x=0 .

Then x = 0 of

x = f (x)

(7.50)

is exponentially stable if and only if z = 0 of its Jacobian linearized system z = Az

(7.51)

is exponentially stable. Proof. (Sufciency) By the principle of stability in the linear approximation. (Necessity) Suppose x = 0 is exponentially stable, then by Theorem 7.13, there exists V (x) such that

D ∥x∥2 ⩽ V (x) ⩽ E ∥x∥2 ,

and

∀x ∈ Br (0),

(7.52)

V (x) ⩽ −J ∥x∥2 ,

∀x ∈ Br (0),

(7.53)

9 9 9 wV 9 9 9 9 w x 9 ⩽ P ∥x∥,

∀x ∈ Br (0).

(7.54)

Rewrite the equation as x = Ax + F(x), where F(x) = f (x) − Note that then

wf ∣x=0 x = f (x) − Ax. wx

∥F(x)∥ = O(∥x∥2 ),

w V (x) w V (x) (Ax + F(x)) = Ax + O(∥x∥3). V = wx wx

(7.55)

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7 Stability and Stabilization

Let

V (x) = P1 x + x′ P2 x + O(∥x∥3 ).

Using (7.52), it is easy to see that P1 = 0, and P2 > 0. It follows that

w V (x) Ax = (2x′ P2 + O(∥x∥2 ))Ax = x′ (P2 A + A′P2 )x + O(∥x∥3 ). wx Using the fact that V ⩽ −J ∥x∥2 , we have P2 A + A′ P2 is negative denite, which implies that A is stable. ⊔ ⊓

7.6 Stability of Invariant Set Consider x = f (x).

(7.56)

Recall that for a set S ∈ ℝn , we dene, as a convention, the distance for a point x0 to S as d(x0 , S) = inf ∥x0 − y∥. y∈S

Now denote the H -neighborhood of S as UH = {x ∈ ℝn ∣d(x, S) < H }. Denition 7.11. An invariant set M is stable if for each H > 0, there exists a G > 0 such that x(0) ∈ UG ⇒ x(t) ∈ UH ; M is asymptotically stable if it is stable and there exists a G > 0 such that x(0) ∈ UG ⇒ lim d(x(t), M) = 0. t→f

Suppose I (t) is a periodic solution of the system with period T . Then the invariant set J = {I (t)∣0 ⩽ t < T } is called an orbit. It is obvious that an orbit is an invariant set. Denition 7.12. A (nontrivial) periodic solution I (t) is orbitally stable if the orbit J is stable; it is asymptotically orbitally stable if J is asymptotically stable. We note that another possible way to study the stability of a periodic solution I (t) is to consider the error dynamics. Let e(t) = x(t) − I (t), then e = f (e + I (t)) − f (I (t)).

(7.57)

By this way, the problem becomes stability of an equilibrium. However, it is well known [6] that (7.57) can never be asymptotically stable if I (t) is nonconstant. For further investigation, we need a new concept, called the Poincar´e map. A Poincar´e map is a map that maps a local section of a cross section to a periodic orbit J at p onto the section. It can be viewed as a discrete time system.

7.7 Input-Output Stability

189

Let H be a hypersurface transversal to J at a point p. Let U be a neighborhood of p on H, small enough such that J intersects U only once at p. The rst return or Poincar´e map P : U → H is dened for q ∈ U by P(q) = x(q, W ),

(7.58)

where W (q) is the time taken for the ow x(q,t) starting from q to return to H. The map is well dened for U sufciently small. It is not difcult to see that the stability of p for the discrete-time system or map P reects the stability of J . This is summarized as follows. Theorem 7.16. [8] A periodic orbit J is asymptotically stable if the corresponding discrete time system obtained from Poincar´e map is asymptotically stable. In particular, if w P(p) w x has n − 1 eigenvalues of modulus less than 1, then J is asymptotically stable. To apply the above result, we introduce a method, which is commonly called Floquet technique. It is however in general difcult to compute the Poincar´e map. On the other hand, the eigenvalues of the linearization of the Poincar´e map may be computed by linearizing the dynamical system around the limit cycle I (t). This can be done using Floquet technique. The linearization of (7.57) around I (t) is z = Since A(t) = tion map.

w f (I (t)) wx

w f (I (t)) z. wx

(7.59)

is periodic, we have the following property for its state transi-

Proposition 7.8. (Floquet’s Theorem)[11] The transition matrix of the above A(t) can be written as

) (t, 0) = K(t)eBt , where K(t) = K(t + T ) and K(0) = I, and B =

1 T

(7.60)

ln ) (T, 0).

It follows then that the stability of J is determined by the eigenvalues of eBT . These eigenvalues are called Floquet multipliers, and the eigenvalues of B are called the characteristic exponents. If v ∈ ℝn is tangent to I (0), then v is the eigenvector corresponding to the Floquet multiplier 1. The rest of the eigenvalues, if none is on the unity circle, determine the stability of J .

7.7 Input-Output Stability In this section we introduce another type of stability: input-output stability and its relation with Liapunov stability.

190

7 Stability and Stabilization

7.7.1 Stability of Input-Output Mapping A general input-output system can be considered as a black box. An afne nonlinear control system x = f (x) + g(x)u, y = h(x),

x ∈ ℝn , u ∈ ℝm

y ∈ ℝp

(7.61)

is just a special case (though very important) of such systems. Our goal is to nd conditions such that when the input u(t) belongs to some “reasonably bounded” set U , the output is also “reasonably bounded”. Let us now dene the set U as the collection of all “reasonably bounded” u(t). Denition 7.13. A map u : [0, f) → ℝm is said to belong to Lm f if it is measurable and essentially bounded on [0, f). If u ∈ Lm f , we denote ∥u∥f = ess. sup {∣u(t)∣,t ⩾ 0}. Denition 7.14. A map u : [0, f) → ℝm is said to belong to the extension of Lm f, m denoted by u ∈ Lm , if the truncation u of u to any nite interval is in L , i.e., T f,e f ∀T > 0, { u(t), 0 ⩽ t ⩽ T uT (t) := 0, t > T is in Lm f. Example 7.9. Consider f (t) = t. It is easy to check that f (t) = t ∈ Lm f,e , but f (t) ∕∈ Lm . f p Denition 7.15. A map F : Lm f,e →: Lf,e is said to be causal if

(F(u))T = (F(uT ))T ,

∀T ⩾ 0, ∀u ∈ Lm f,e .

(7.62)

From now on, we suppose the systems we considered are causal. Denition 7.16. System (7.61) is said to be Lf (L2 ) stable if every Lf (L2 ) input produces an Lf (L2 ) output. Let us rst consider the linear case: { x = Ax + Bu y = Cx

(7.63)

Theorem 7.17. System (7.63) is Lf (L2 ) stable, that is, every Lf (L2 ) input produces an Lf (L2 ) output, if x = Ax is exponentially stable. The above result is one of the most well known classical results. We leave the verication to reader. The question is if one can draw a similar conclusion for nonlinear systems? We consider the following example, which is classical.

7.7 Input-Output Stability

191

Example 7.10. Consider the following system x = −x + (x2 + 1)u.

(7.64)

Set u = 0, x = 0 is globally exponentially stable. However, if one takes u = 1, we have x = −x + x2 + 1 > 0,

∀x ∈ ℝ.

(7.65)

It is ready to check that even when x(0) = 0, x(t) will diverge to f. Denition 7.17. System (7.61) is small signal Lf stable (uniform BIBO stable) if there exist r1 > 0, r2 > 0, C1 > 0 and C2 > 0 such that when ∣x(0)∣ ⩽ r1 ,

∣u(t)∣ ⩽ r2 ,

∀t ⩾ 0

(7.66)

we have ∥y∥f ⩽ C1 ∥u∥f + C2 ∥x(0)∥.

(7.67)

Theorem 7.18. Consider system (7.61). If x=0 is exponentially stable when u=0, then (7.61) is small signal Lf stable. Proof. By the converse theorem (Theorem 7.13), there exists a V (x) such that

D ∥x∥2 ⩽ V (x) ⩽ E ∥x∥2 , wV f (x) ⩽ −∥x∥2 , wx and

9 9 9 w V (x) 9 9 9 9 w x 9 ⩽ J ∥x∥,

(7.68) (7.69)

∀x ∈ Br (0).

(7.70)

Since x = f (x) + g(x)u, we have

wV wV f (x) + g(x)u ⩽ −∥x∥2 + c∥x∥ ∥u∥f, V = wx wx

∀x ∈ Br (0).

(7.71)

Let us rst assume c∥u∥f ⩽ r. Then if ∥x∥ > c∥u∥f, V < 0. For any ∥x0 ∥ < c∥u∥f ⩽ r, ∃ T > 0, such that V (x(t)) ⩽

max V (x) ⩽ E c2 ∥u∥2f,

∥x∥⩽c∥u∥f

√

It follows that ∥x(t)∥ ⩽ If we let

E c∥u∥f, D

√

∀0 ⩽ t ⩽ T.

∀0 ⩽ t ⩽ T.

E c∥u∥f ⩽ r, D

192

7 Stability and Stabilization

{√

or ∥u∥f ⩽ min

} Dr r , . Ec c

Then ∀∥x0 ∥ < c∥u∥f, we have √ ∥x(t)∥ <

E c∥u∥f , D

∀t ⩾ 0.

(7.72)

On the other hand, if c∥u∥f ⩽ ∥x0 ∥ ⩽ r then V ⩽ 0. Hence V (x(t)) ⩽ V (x0 ) ⩽ E ∥x0 ∥2 ,

∀0 ⩽ t ⩽ T,

(7.73)

or √ ∥x(t)∥ ⩽

E ∥x0 ∥. D

√

If we assume ∥x0 ∥ ⩽

(7.74)

D r, E

then when ∥x0 ∥ ⩾ c∥u∥f √ ∥x(t)∥ ⩽

E ∥x0 ∥. D

Adding (7.72) and (7.75) together, we have that if ∥x0 ∥ ⩽ √ min { DE cr , cr }, √ ∥x(t)∥ ⩽

E c∥u∥f + D

√

E ∥x0 ∥. D

(7.75) √

D E

r, and ∥u(t)∥ ⩽

(7.76)

Since h(x) is C2 , it is locally Lipschitz. That is, there is a k > 0 such that ∣h(x)∣ ⩽ k∥x(t)∥. Therefore, √ √ E E c∥u∥f + k ∥x0 ∥. (7.77) ∥y∥f ⩽ k D D ⊔ ⊓

7.7.2 The Lur’e Problem For many practical feedback control systems there is only one nonlinear block in the model, which for example could be the motor. On the other hand the nonlinearity is so signicant that we can not use a linearized system to study such a system. A. I. Lur’e originally studied this problem in 1951, thus the problem is named after him. In the following we will briey discuss this problem. Consider the following

7.7 Input-Output Stability

193

system {

x = Ax + Bu, x ∈ ℝn , u ∈ ℝm y = Cx + Du, y ∈ ℝm .

(7.78)

The feedback is dened by u = −) (y,t). → ℝm . Then ) is said to belong to the sector Denition 7.18. Suppose ) [a, b] (where a < b) if (i). ) (0,t) = 0, ∀t ⩾ 0 and (ii). : ℝm × ℝ+

() (y,t) − ay)T (by − ) (y,t)) ⩾ 0,

∀t ⩾ 0, ∀y ∈ ℝm .

(7.79)

Next, we consider the following problem, which is commonly called the absolute stability problem: Suppose the pair (A, B) is controllable and the pair (C, A) is observable and let G(s) = C(sI − A)−1 B + D be the transfer function. The problem is to derive conditions involving only the transfer function G(⋅) and the numbers a, b, such that x = 0 is a globally uniformly asymptotically stable equilibrium for every mapping ) belonging to the sector [a, b]. Lemma 7.3. (Kalman-Yakubovich-Popov) Consider system (7.78). Suppose A is Hurwitz and (A, B) controllable, (C, A) observable, and inf {Omin (G(jZ ) + G∗ (jZ )) > 0},

Z ∈ℝ

(7.80)

where Omin denotes the smallest eigenvalue of the matrix. Then there exist a positive denite matrix P ∈ ℝn×n and matrices Q ∈ ℝm×n and W ∈ ℝm×m and H > 0 such that AT P + PA = −H P − QT Q BT P + W T Q = C W T W = D + DT . Then we have the following result about absolute stability. Theorem 7.19. (Passivity) Suppose in system (7.78), A is Hurwitz, the pair (A, B) is controllable, the pair(C, A) is observable, G(s) is strictly positive real, and ) belongs to the sector [0, f), i.e., ) (0,t) = 0 and yT ) (y,t) ⩾ 0,

∀t ⩾ 0, ∀y ∈ ℝm .

(7.81)

Then, the feedback system is globally exponentially stable. Remark 7.5. 1. Condition (7.80) is called the strictly positive real condition. 2. Condition (7.81) is called the passivity property.

7.7.3 Control Lyapunov Function In this subsection we briey discuss an approach for feedback stabilization that is very relevant to Lyapnov’s direct method as we just described.

194

7 Stability and Stabilization

We consider a single input control system x = f (x) + g(x)u.

(7.82)

Denition 7.19. A positive denite, radially unbounded and differentiable function V (x) is called a control Liapunov function (CLF) if for all x ∕= 0, LgV (x) = 0 ⇒ L f V (x) < 0. If the system can be globally asymptotically stabilized by a C1 feedback u(x), then by the converse Lyapunov theorem, there exists a Lyapunov function V (x) such that V = L f V (x) + u(x)LgV (x) < 0, which implies that at any point x0 ∕= 0 such that LgV (x0 ) = 0, L f V (x0 ) < 0. Thus the existence of a CLF is necessary for the existence of a smooth global stabilizing controller. The question is to what extent this condition is also sufcient? It turns out if we are satised with almost smooth feedback controllers, this condition is also sufcient. Denition 7.20. A function D (x) dened on ℝn is called almost smooth if D (0) = 0, D is smooth on ℝn ∖ {0} and at least continuous at x = 0. Theorem 7.20. (Artstein-Sontag) Consider system (7.82). There exists an almost smooth feedback law u = D (x) which globally asymptotically stabilizes the system if and only if there exists a CLF V (x) with the additional property: for each H > 0 there exists a G (H ) > 0 such that for each x ∈ BG (0) ∖ {0}, there exists ux ∈ BH (0) such that L f V (x) + LgV (x)ux < 0. Proof. The necessity is easy. We only show the sufciency. Let ⎧ 0, LgV (x) = 0   ⎨ √ D (x) = L f V (x) + (L f V (x))2 + (LgV (x))4   ⎩− , otherwise. LgV (x)

(7.83)

One can use the condition in the theorem to show it is continuous at 0. With this control, we have √ V = − (L f V (x))2 + (LgV (x))4 < 0, ∀x ∕= 0. Therefore, the closed-loop system is globally asymptotically stable.

⊔ ⊓

7.8 Region of Attraction Consider system (7.2) again, in this section we assume f (x) is an analytic vector eld.

7.8 Region of Attraction

195

Suppose xe is an equilibrium point of (7.2). The stable and unstable sub-manifolds of xe are dened respectively as  { }  W s (xe ) = p ∈ ℝn lim x(t, p) → xe { W u (x

e)

=

t→f

 }  n  p ∈ ℝ  lim x(t, p) → xe . t→−f

(7.84)

Suppose xs is a stable equilibrium point of (7.2). The region of attraction of xs is dened as  { }  (7.85) A(xs ) = p ∈ ℝn lim x(t, p) → xs t→f

The boundary of the region of attraction is denoted by w A(xs ). An equilibrium point xe is hyperbolic if the Jacobian matrix of f at xe , denoted by J f (xe ), has no eigenvalues with zero real part. A hyperbolic equilibrium point is said to be of type-k if J f (xe ) has k positive real part eigenvalues. The problem of determining the region of attraction of power systems has been discussed intensively by many researches. The works of [12, 15, 4, 5] established the theoretical foundations for the geometric structure of the region of attraction. Particularly, [15] and [4] proved that for a stable equilibrium point xs the stable boundary is composed of the stability sub-manifolds of equilibrium points on the boundary of the region of attraction under the assumptions that (i). the equilibrium points on the stability boundary w A(xs ) are hyperbolic; (ii). the stable and unstable sub-manifolds of the equilibrium points on the stability boundary w A(xs ) satisfy the transversality condition; (iii). every trajectory on the stability boundary w A(xs ) approaches one of the equilibrium points as t → f. It is well known that the stability boundary is of dimension n − 1 [4]. Therefore, the stability boundary is composed of the closure of stable sub-manifolds of type-1 equilibrium points on the boundary. Based on this fundamental fact, it is of signicant meaning to calculate or estimate the stable sub-manifold of type-1 equilibrium points. In this section we will give a formula for the Taylor expansion of the stable submanifold. Its rst two terms form the best quadratic approximation. The formula is based on a new matrix product, called the left semi-tensor product of matrices, which is briey introduced in Appendix B. Without loss of generality, we can assume the type-1 equilibrium point is xu = 0. Using Taylor expression, we can express the vector eld f in (7.2) by f

f (x) = ¦ Fi xi = Jx + F2x2 + ⋅ ⋅ ⋅ .

(7.86)

i=1

Where F1 = J = J f (0), and Fi = i!1 Di f (0) are known n × ni matrix. We use A−T for the inverse of AT , which is the transpose of A. Lemma 7.4. Let A be a hyperbolic matrix. Denote by Vs and Vu the stable and unstable sub-spaces of A respectively and by Us and Uu the stable and unstable sub-spaces

196

7 Stability and Stabilization

of A−T respectively. Then Vs⊥ = Uu ,

Vu⊥ = Us .

(7.87)

Proof. Assume A is of type-k, then we can convert A into the Jordan canonical form [ ] as Js 0 −1 Q AQ = , 0 Ju (where J)s and Ju represent the stable and unstable blocks respectively. Split Q = Q1 Q2 , where Q1 and Q2 are the rst n − k and last k columns respectively. Then Vs = Span col{Q1 }, It is easy to see that QT A−T Q−T =

Vu = Span col{Q2 }. ] [ −T 0 Js . 0 Ju−T

) ( Split Q−T = Q˜ 1 Q˜ 2 , where Q˜ 1 and Q˜ 2 are the rst n − k and last k columns of Q−T respectively. Then Us = Span col{Q˜ 1 },

Uu = Span col{Q˜ 2 }.

The conclusion follows from the fact that Q−1 Q = I.

⊔ ⊓

The following is an immediate consequence of the above lemma. Corollary 7.2. Let A be a type-1 matrix and assume its only unstable eigenvalue is P . Let K be the eigenvector of AT with respect to eigenvalue P . Then K is perpendicular to the stable subspace of A. Proof. Since the only unstable eigenvalue of A−T is P1 , denote by K the eigenvector of A−T , then Lemma 7.4 says that Span{K } = Uu = Vs⊥ . Now we have only to show that K is also the eigenvector of AT with respect to P . Since A−T K =

1 K ⇒ AT K = PK , P ⊔ ⊓

the conclusion follows.

Before investigating the main results the following notations are required. In fact, they are from denition immediately. In0 := 1,

)0 := In .

Without loss of generality, we assume hereafter that xu = 0 is a type-1 equilibrium point of system (7.2). The following is a set of necessary and sufcient conditions for the stable submanifold of a type-1 equilibrium point to satisfy. Theorem 7.21. Assume xu = 0 is a type-1 equilibrium point of system (7.2). W s (eu ) = {x ∣ h(x) = 0}.

(7.88)

Then h(x) is uniquely determined by the following necessary and sufcient conditions (7.89)–(7.91).

7.8 Region of Attraction

h(0) = 0,

197

(7.89)

h(x) = K T x + O(∥x∥2 ), L f h(x) = P h(x),

(7.90) (7.91)

where K is an eigenvector of J Tf (0) with respect to its only positive eigenvalue P . Proof. (Necessity) The necessity of (7.89) and (7.90) are obvious. We have only to show the necessity of (7.91). First, note that

wh = K T + O(∥x∥). wx

(7.92)

So locally there exists a neighborhood U of zero, such that rank(h(x)) = 1, Now since W s (eu ) is f -invariant, then { h(x) = 0 L f h(x) = 0,

x ∈ U.

x ∈ W s (eu ).

(7.93)

(7.94)

Since dim(W s (eu )) = n − 1, it follows that ] [ h(x) = 1, rank L f h(x) which means h(x) and L f h(x) should be linearly dependent. Now a straightforward computation shows that L f h(x) = K T J f (0)x + O(∥x∥2) = PK T x + O(∥x∥2 ). Then for x ∈ U, (7.91) follows from the linear dependency of h(x) and L f h(x). Finally, by analyticity of the system, (7.91) is globally true. (Sufciency) First we show that if h(x) satises (7.89)–(7.91), then locally {x ∈ U ∣ h(x) = 0} is the stable sub-manifold in U. According to the rank condition (7.93), we know that (Refer to [1]) V := {x ∈ U ∣ h(x) = 0} is an n − 1 dimensional sub-manifold. Secondly, since L f h(x) = 0, V is locally f -invariant. Finally, (7.90) implies that locally zero is the asymptotically stable equilibrium point of the restriction of f on V . Hence locally, V is the stable sub-manifold of (7.2). But the stable sub-manifold is unique ([2]), so V coincides with W s (eu ). Since the system is analytic, {x ∣ h(x) = 0} coincides globally with W s (eu ). ⊓ ⊔ Next, we investigate the quadratic approximation to the stable sub-manifold of unstable equilibrium points. The advantage of this approach lies on two points:

198

7 Stability and Stabilization

(i). Precise formula is given; (ii). The approach is the unique one, which carrying the error of O(∥x∥3 ). For computational ease, we denote the Taylor expansion of h(x) as 1 h(x) = H1 x + H2 x2 + H3 x3 + ⋅ ⋅ ⋅ = H1 x + xT< x + H3 x3 + ⋅ ⋅ ⋅ . 2

(7.95)

Note that in the above we use two forms for the quadratic term: The semi-tensor product form, H2 x2 , and the quadratic form as 12 xT< x, where < = Hess(h(0)) is the Hessian matrix of h(x) at x = 0, and H2 = VcT ( 12 < ), which is the column stacking form of the matrix 12 < . Lemma 7.5. The quadratic form in the stable sub-manifold equation (7.95) satises

<

(P

) ) (P n I−J + I − J T < = ¦ Ki Hess( fi (0)), 2 2 i=1

(7.96)

where P and K are dened as in Corollary 7.2 for A = F1 = J := J f (0), Hess( fi ) is the Hessian matrix of the i-th component fi of f . Proof. First of all, the linear approximation of h(x) = 0 is H1 x = 0, which should be the tangent space of the stable sub-manifold W s (xu ). Since K is the normal direction of W s (xu ) at xu , it is clear that H1 = K . According to Theorem 7.21, the Lie derivative L f h(x) = 0. Using (B.30)–(B.31), we have Dh(x) = H1 + H2 )1 x + H3 )2 x2 + ⋅ ⋅ ⋅ = H1 + xT< + H3 )2 x2 + ⋅ ⋅ ⋅ . Note that the vector eld f can be expressed as ⎡ T ⎤ x Hess( f1 (0))x 1⎢ ⎥ .. 3 f (x) = Jx + ⎣ ⎦ + O(∥x∥ ). . 2 T x Hess( fn (0))x Calculating L f h yields (

) 1 n Lf h = ¦ Ki Hess( fi (0)) + < J x + O(∥x∥3) 2 i=1 ( ) n 1 = PK T x + xT ¦ Ki Hess( fi (0)) + < J x + O(∥x∥3). 2 i=1

K T Jx + xT

Note that as the invariant sub-manifold of f , we have

(7.97)

7.8 Region of Attraction

{  } W s (eu ) = x  h(x) = 0, L f h(x) = 0 .

199

(7.98)

Using (7.95) and (7.98), we have that for W s (eu ) ( ) ( ) n P 1 xT ¦ Ki Hess( fi (0)) + < J − 2 I x + O(∥x∥3) = 0. 2 i=1 Expressing the quadratic form in a symmetric type, we have (7.96).

(7.99) ⊔ ⊓

Lemma 7.6. Equation (7.96) has unique symmetric solution. Proof. Expressing (7.96) into a conventional linear equation, we have ( ) (A ⊗ In + In ⊗ A)Vc (< ) = Vc

n

¦ Ki Hess( fi (0))

,

(7.100)

i=1

where

P I − JT. 2 The form of (7.100) is obtained from a standard Lyapunov mapping. (see, i.e., [3] for Lyapunov mapping.) It is well known that let Oi ∈ V (A), i = 1, ⋅ ⋅ ⋅ , n be the eigenvalues of A, then eigenvalues of A ⊗ In + In ⊗ A are A=

{Oi + O j ∣ 1 ⩽ i, j ⩽ n, Ot ∈ V (A)}. To see that A ⊗ In + In ⊗ AT is non-singular, we have only to show that all the Oi + O j ∕= 0. let [i ∈ V (J), i = 1, ⋅ ⋅ ⋅ , n be the eigenvalues of J. Then

Oi =

P − [i , 2

i = 1, ⋅ ⋅ ⋅ , n.

Observing the eigenvalues of J, one sees that the only negative eigenvalue of A is − P2 , and all other eigenvalues of A have positive real parts > P2 . Then

Oi + O j ∕= 0,

1 ⩽ i, j ⩽ n.

Hence (7.96) has unique solution. Finally, we show the solution is symmetric. Using Proposition B.2, it is ready to verify that (A ⊗ In + In ⊗ A)W[n] = W[n] (A ⊗ In + In ⊗ A).

(7.101)

Using some properties of semi-tensor product, we have (A ⊗ In + In ⊗ A)Vr (< ) = (A ⊗ In + I( n ⊗ A)W[n]Vc (< ) = W[n] (A ⊗ In + In ⊗ A)Vc(< ) = W[n]Vc ( ( ) = Vr

n

¦ Ki Hess( fi (0))

i=1

= Vc

n

n

¦ Ki Hess( fi (0))

i=1

)

¦ Ki Hess( fi (0))

i=1

)

.

(7.102)

200

7 Stability and Stabilization n

The last equality is from the fact that ¦ [i Hess( fi (0)) is a symmetric matrix, hence i=1

its row stacking form and its column stacking form are the same. Now (7.102) shows that Vr (< ) is another solution of (7.100). But (7.100) has unique solution. Hence we have Vr (< ) = Vc (< ). That is, < is symmetric.

⊔ ⊓

Denote by Vc−1 the inverse mapping of Vc . That is, it recovers a (square) matrix, A, from its column stacking form, Vc (A). Summarizing Lemmas 7.4–7.6, we have the following quadratic form approximation of the stable sub-manifold. Theorem 7.22. The stable sub-manifold of xu , expressed as h(x) = 0, can be expressed as 1 h(x) = H1 x + xT< x + O(∥x∥3 ), 2 where ⎧ T  ⎨H1 = K {  ⎩<

= Vc−1

[( P 2

In − J

T

)

⊗ In + In ⊗

(P 2

In − J

T

)]−1

(7.103)

( Vc

n

)}

¦ Ki Hess( fi (0))

,

i=1

here P and K are dened as in Corollary 7.2 for J = F1 , Hess( fi ) is the Hessian matrix of the i-th component fi of f . Observing (7.99), the following is an immediate consequence, which may simplify some computations. Corollary 7.3. Assume n

¦ Ki Hess( fi (0))

(P

i=1

2

In − J

)−1

is symmetric. Then the quadratic approximation of the stable sub-manifold is (P )−1 n 1 In − J h(x) = K T x + xT ¦ Ki Hess( fi (0)) x = 0. (7.104) 4 i=1 2 Example 7.11. Consider the system { x1 = x1 x2 = −x2 + x21 ,

x ∈ ℝ2 .

Its stable and unstable sub-manifolds are [10] W s (0) = {x ∈ ℝ2 ∣ x1 = 0} } { 1 2 u 2 W (0) = x ∈ ℝ ∣ x2 = x1 . 3

(7.105)

7.8 Region of Attraction

201

We use them to verify the formula (7.104). For (7.105), we have ] [ 1 0 . J= 0 −1 For stable sub-manifold W s (0), it is easy to verify that the stable eigenvalue is P = 1, eigenvector is K = (1 0)T , and [ ] 20 . Hess( f1 (0)) = 0, Hess( f2 (0)) = 00 Then

That is,

)−1 ( 1 2 1 ¦ Ki Hess( fi (0)) 2 I − J = 0. 4 i=1 hs (x) = (1 0)x + 0 + O(∥x∥3) = x1 + O(∥x∥3).

For unstable sub-manifold W u (0), it is easy to verify that the unstable eigenvalue is P = −1, eigenvector is K = (0 1)T . ⎡ ⎤ Then ( )−1 1 2 1 −1 − 0 ¦ Ki Hess( fi (0)) 2 I − J = ⎣ 3 ⎦ . 4 i=1 0 0 That is,

⎡ 1 − hu (x) = (0 1)x + xT ⎣ 3 0

⎤ 0⎦

1 x + O(∥x∥3 ) = x2 − x21 + O(∥x∥3 ). 3 0

Note that we can use the results in the next section to check that in both hs (x) and hu (x) the error terms O(∥x∥3 ) are zero. Finally, we consider the whole Taylor expansion of the stable sub-manifold. That is, we will solve Hk from (7.89)–(7.91). The problem is xk is a redundant basis of the k-th degree homogeneous polynomials, so from (7.89)–(7.91) we can not get unique solution. To overcome this obstacle we consider the natural basis of the k-th homogeneous polynomials. Let S ∈ ℤn+ . The natural basis is dened as  Bkn = { xS  S ∈ ℤn+ , ∣S∣ = k}. Now we arrange the elements in Bkn in alphabetic order. That is, for S1 = (s11 , ⋅ ⋅ ⋅ , s1n ) 1 2 and S2 = (s21 , ⋅ ⋅ ⋅ , s2n ), we sign the order as xS ≺ xS if there exists a t, 1 ⩽ t ⩽ n − 1, such that 1 2 s11 = s21 , ⋅ ⋅ ⋅ , st1 = st2 , st+1 > st+2 . Then we arrange the elements in Bkn as a column and denote it as x(k) . Example 7.12. Let n = 3, k = 2. Then x2 = (x21 , x1 x2 , x1 x3 , x2 x1 , x22 , x2 x3 , x3 x1 , x3 x2 , x23 )T , and x(2) = (x21 , x1 x2 , x1 x3 , x22 , x2 x3 , x23 )T .

202

7 Stability and Stabilization

Next, we consider the dimension of k-th order homogeneous polynomials. The proof of the following formula is an element and interesting exercise. Proposition 7.9. The size of the basis Bkn is ∣Bkn ∣ := d =

(n + k − 1)! , k!(n − 1)!

k ⩾ 0, n ⩾ 1.

(7.106)

Two matrices TN (n, k) ∈ Mnk ×d and TB (n, k) ∈ Md×nk can be dened to convert one basis to the other, that is, xk = TN (n, k)x(k) , and

x(k) = TB (n, k)xk ,

TB (n, k)TN (n, k) = Id .

They are dened in a very natural way, which are described in the item 4 of appendix. We use an example to describe them. Example 7.13. Let n = 2 and k = 3. We construct TB (n, k) as TB (2, 3) = (111) ⎤ ⎡ (112) (121) (122) (211) (212) (221) (222) 1 0 0 0 0 0 0 0 ⎢ 0 1/3 1/3 0 1/3 0 0 0 ⎥ ⎥ ⎢ ⎣0 0 0 1/3 0 1/3 1/3 0 ⎦ 0 0 0 0 0 0 0 1

(111) . (112) (122) (222)

(7.107)

Similarly, we have (111) ⎤ ⎡ (112) (122) (222) 1 0 0 0 ⎢ 0 1 0 0 ⎥ ⎥ ⎢ ⎢ 0 1 0 0 ⎥ ⎥ ⎢ ⎢ 0 0 1 0 ⎥ ⎥ ⎢ TN (2, 3) = ⎢ 0 ⎥ ⎥ ⎢ 0 1 0 ⎢ 0 0 1 0 ⎥ ⎥ ⎢ ⎣ 0 0 1 0 ⎦ 0 0 0 1

(111) (112) (121) (122) . (211) (212) (221) (222)

(7.108)

Recalling (7.95), instead of solving Hk , we may try to solve Gk , where Hk xk = Gk x(k) . Fk is said to be a symmetric coefcient set, provided that if two elements of xk are the same then their coefcients are also the same. We use the following example to describe this. Example 7.14. Let n = 3 and k = 2. Then x2 is as in Example 7.12. For a given k-th degree homogeneous polynomial p(x) = x21 + 2x1 x2 − 3x1x3 + x22 − x23 , we can express it as

7.8 Region of Attraction

203

p(x) = H1 x2 = (1, 2, −3, 0, 1, 0, 0, 0, −1)x2 . Alternatively, we can also express it as ) ( 3 3 p(x) = H2 x2 = 1, 1, − , 1, 1, 0, − , 0, −1 x2 . 2 2 Then H1 is not symmetric and H2 is symmetric. A straightforward computation shows the following: Proposition 7.10. The symmetric coefcient set, Hk is unique. Moreover, Hk = Gk TB (n, k),

Gk = Hk TN (n, k).

(7.109)

Now we consider the higher degree terms of the equation h(x) of the stable submanifold. Denote f (x) = F1 x + F2x2 + ⋅ ⋅ ⋅ ; h(x) = H1 x + H2 x2 + ⋅ ⋅ ⋅ . Note that F1 = J f (0) = J, H1 = K T , and H2 is uniquely determined by (7.103). Proposition 7.11. The coefcients Hk , k ⩾ 2, of h(x) satisfy the following equation. ] [ k

¦ Hi )i−1 (Ini−1 ⊗ Fk−i+1) − P Hk

xk = 0,

k ⩾ 2.

(7.110)

i=1

Proof. Note that since h(x) = 0 is the vector eld f (x) invariant, i.e., the Lie derivative L f h(x) = 0.

(7.111)

Using (B.30) and (B.31), we have Dh(x) = H1 + H2 )1 x + H3 )2 x2 + ⋅ ⋅ ⋅ = H1 + 2xT< + H3 )2 x2 + ⋅ ⋅ ⋅ . Then a straightforward computation shows that L f h(x) = PK T x + [H2 )1 (In ⊗ F1 ) + H1 F2 ] x2 + ⋅ ⋅ ⋅ [ ] +

k

¦ Hi )i−1 (Ini−1 ⊗ Fk+1−i)

xk + ⋅ ⋅ ⋅ .

i=1

Note that the equation of the stable sub-manifold satises { h(x) = 0 L f h(x) = 0.

(7.112)

Subtracting the P times rst equation of (7.112) from the second equation of (7.112), we have, inductively in k, that [ ] k

¦ Hi )i−1 (Ini−1 ⊗ Fk−i+1) − P Hk

i=1

xk + O(∥x∥k+1 ) = 0,

k ⩾ 2,

204

7 Stability and Stabilization

⊔ ⊓

which completes the proof. Observing (7.110), according to Proposition 7.10, it can now be expressed as Gk [P Id − TB(n, k))k−1 (Ink−1 ⊗ F1)TN (n, k)] x(k) [ ] ≡

k−1

¦ Gi TB (n, i))i−1 (Ini−1 ⊗ Fk−i+1)

TN (n, k)x(k) ,

k ⩾ 3.

(7.113)

i=1

The following result, which is a summary of the above arguments, is generically applicable. Theorem 7.23. Assume the matrices Ck := P Id − TB(n, k))k−1 (Ink−1 ⊗ F1 )TN (n, k), are non-singular, then [ Gk =

k−1

k⩾3

(7.114)

]

¦ Gi TB(n, i))i−1 (Ini−1 ⊗ Fk−i+1)

TN (n, k)Ck−1 .

(7.115)

i=1

Example 7.15. [8] A pendulum system can be depicted by the following system: { x1 = x2 (7.116) x2 = − sin x1 − 0.5x2. Fig. 7.1–7.3 give the second order, third order and fth order approximations to the stable manifold.

Fig. 7.1

Second Order Approximation to the Stable Manifold

References

Fig. 7.2

Third Order Approximation to the Stable Manifold

Fig. 7.3

Fifth Order Approximation to the Stable Manifold

205

References 1. Boothby W. An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd edn. Orlando: Academic Press, 1986. 2. Carr J. Applications of Centre Manifold Theory. New York: Springer, 1981. 3. Cheng D. On Lyapunov mapping and its applications. Communications on Information and Systems, 2001, 1(5): 195–212. 4. Chiang H, Hirsch M, Wu F. Stability regions of nonlinear autonomous dynamical systems. IEEE Trans. Aut. Contr., 1988, 33(1): 16–27.

206

7 Stability and Stabilization

5. Chiang H, Wu F. Foundations of the potential energy boundary surface method for power system transient stability analysis. IEEE Trans. Circ. Sys., 1988, 35(6): 712–728. 6. Hahn W. Stability of Motion. Berlin: Springer, 1967. 7. Horn R, Johnson C. Matrix Analysis. New York: Cambbridge Univ. Press, 1985. 8. Khalil H. Nonlinear Systems, 3rd edn. New Jersey: Prentice Hall, 2002. 9. Mu X, Cheng D. On stability and stabilization of time-varying nonlinear control systems. Asian J. Contr., 2005, 7(3): 244–255. 10. Saha S, Fouad A, Kliemamm W, et al. Stability boundary approximation of a power system using the real normal form of vector elds. IEEE Trans. Power Sys., 1997, 12(2): 797–802. 11. Sastry S. Nonlinear Systems. New York: Springer, 1999. 12. Varaiya P, Wu F, Chen R. Direct methods for transient stability analysis of power systems: Recent results. Proceedings of the IEEE, 1985, 73(12): 1703–1715. 13. Vidyasagar M. Decomposition techniques for large-scale systems with nonadditive interactions: Stability and stabilizability. IEEE Trans. Aut. Contr., 1980, 25(4): 773–779. 14. Vidyasagar M. Nonlinear Systems Analysis. New Jersey: Prentice Hall, 1993. 15. Zaborszky J, Huang J, Zheng B, et al. On the phase protraits of a class of large nonlinear dynamic systems such as the power systems. IEEE Trans. Aut. Contr., 1988, 33(1): 4–15.

Chapter 8

Decoupling

Section 8.1 considers the ( f , g)-invariant distribution of a nonlinear system. Quaker lemma is proved, which assures the equivalence between two kinds of ( f , g)invariances. Quaker lemma is the foundation of the feedback decoupling of nonlinear systems [2]. The local disturbance decoupling problem is discussed in Section 8.2. In Section 8.3, the controlled invariant distribution is introduced. The problem of decomposition of the state equations is discussed in Section 8.4 and Section 8.5. In Section 8.4 only a coordinate change is used, while in Section 8.5 a state feedback control is also used. We refer to [3, 5] for feedback decomposition. More details can be found in their books [2, 4].

8.1 ( f , g)-invariant Distribution Consider an afne nonlinear system x = f (x) + g(x)u,

x ∈ ℝn , u ∈ ℝm .

(8.1)

Denition 8.1. An involutive distribution ' (x) is said to be ( f , g)-invariant, if there exists a feedback control u = D (x) + E (x)v with non-singular E (x), such that [ f (x) + g(x)D (x), ' (x)] ⊂ ' (x), [g(x)E (x), ' (x)] ⊂ ' (x).

(8.2)

A distribution ' (x) is said to be weakly ( f , g)-invariant, if [ f (x), ' (x)] ⊂ ' (x) + G(x), [g(x), ' (x)] ⊂ ' (x) + G(x),

(8.3)

where G(x) = Span{g(x)} = Span{g1 (x), ⋅ ⋅ ⋅ , gm (x)}. In most cases we consider the (weak) ( f , g)-invariance locally. ' is said to be (weakly) ( f , g)-invariant at x0 ∈ M, if there exists a neighborhood, U of x0 such that (respectively, (8.3)) (8.2) holds for x ∈ U. D. Cheng et al., Analysis and Design of Nonlinear Control Systems © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2010

208

8 Decoupling

Next, we consider the relationship between the weak ( f , g)-invariance and the ( f , g)-invariance. Since the weak ( f , g)-invariance is easily veriable while the ( f , g)-invariance is very useful in control design, we are looking for the equivalence between them. In fact, under certain non-singularity assumptions they are equivalent. Theorem 8.1. (Quaker Lemma)Consider an involutive distribution ' . Assume ' and ' ∪ G are locally non-singular at x0 , then locally the ( f , g)-invariance is equivalent to weak ( f , g)-invariance. Proof. Assume ' is locally ( f , g)-invariant. Using the fact that [D g, X ] = −LX (D )g + D [g, X ],

X ∈ ',

it is obvious that (8.2) implies (8.3). Conversely, assume ' is locally weakly ( f , g)-invariant. Choosing a at coordinate chart (U, x) of x0 such that { } w w ' = Span ,⋅⋅⋅ , . w x1 w xk Correspondingly, we express f and g as [ 1] [ 1] [ 1] f˜ f g f = 2 , g = 2 , f + gD = ˜2 , f f g

[ 1] g˜ gE = 2 , g˜

where the rst block is of dimension k. Then (8.2) becomes ⎧ w ˜2   f = 0, ⎨ w xi   w g˜2 = 0, i = 1, ⋅ ⋅ ⋅ , k, ⎩ w xi

(8.4)

and (8.3) becomes ⎧ w 2   f = Span{g2 }, ⎨ w xi   w g2 = Span{g2 }, ⎩ w xi

(8.5) i = 1, ⋅ ⋅ ⋅ , k.

First, we assume ' ∩G = {0}. We claim that if the second equation of (8.5) holds then the second equation of (8.4) is equivalent to that there exists an m × (n − k) matrix K, which is independent of xi , i = 1, ⋅ ⋅ ⋅ , k, such that Kg2 is non-singular. Assume the second equation of (8.4) holds. We let K = E T (g2 )T , which is independent of xi . Since ' ∩ G = {0}, rank(g2 ) = k, which implies that (g2 )T g2 > 0 is positive denite. Hence, Kg2 = E T (g2 )T g2 is non-singular. Conversely, from the second equation of (8.5) we can express

w 2 g = g2 *i , w xi

i = 1, ⋅ ⋅ ⋅ , k,

8.1 ( f , g)-invariant Distribution

209

where *i are some m × n matrices of functions. Choosing

E := (Kg2 )−1 , where K is independent of xi , then we have

w 2 w 2 (g E ) = g (Kg2 )−1 w xi w xi w 2 w = (g )(Kg2 )−1 − g2 (Kg2 )−1 (Kg2 )(Kg2 )−1 w xi w xi = g2*i (Kg2 )−1 − g2 (Kg2 )−1 Kg2*i (Kg2 )−1 = 0. Now to show that the second equation of (8.5) implies the second equation of (8.4) it is enough to show that such a matrix K exists. In fact, we can set K = (g2 (x10 , x2 ))T , where x1 is the rst k coordinates of x, and x10 is the rst k coordinates of x0 , which are xed. So K is independent of xi , i = 1, ⋅ ⋅ ⋅ , k. To see Kg2 is non-singular, since it is non-singular at x0 it is non-singular on a neighborhood of x0 . Next, assume dim(' ∪ G) = k + s < k + m. Then we can nd s linearly independent subset of the columns of g2 , say b(x) = (b1 (x), ⋅ ⋅ ⋅ , bs (x)), such that g2 = b(x)E(x),

(8.6)

where E(x) is an s × m matrix. Note that the rank(E(x)) = s. So it is easy to express E(x) via b(x) and vise versa as E(x) = (bT (x)b(x))−1 bT (x)g2 (x),

(8.7)

b(x) = g2 (x)E T (x)(E(x)E T (x))−1 .

(8.8)

and

(8.7) shows that E(x) is smooth. Moreover, since rank(E(x)) = s, we can choose an m × (m − s) matrix, Q(x), with smooth functions such that ( ) rank Q(x), E T (x)(E(x)E T (x))−1 = m. Dene W (x) = Q(x) − E T (x)(E(x)E T (x))−1 E(x)Q(x), L(x) = E T (x)(E(x)E T (x))−1 , E0 (x) = (W (x), L(x)). Then it is easy to see that [

−1

E0 (x) = Q(x) E (x)(E(x)E (x)) T

T

]

[

] 0 Im−s . −E(x)Q(x) Is

(8.9)

210

8 Decoupling

Hence E0 (x) is non-singular. In addition, a straightforward computation shows that ] [ ∗ ∗ , (8.10) g(x)E0 (x) = 0 b(x) where ∗ stands for some no concern elements. Using the last s vector elds of g(x)E0 (x) to span a distribution B(x) as B(x) = Span{(g(x)E0 (x)) j ∣ j = m − s + 1, m − s + 2, ⋅ ⋅ ⋅ , m}. Then we have from (8.3) that {

[ f , ' ] ⊂ ' + B, [B, ' ] ⊂ ' + B.

(8.11)

Note that now B ∩ ' = {0}, we can use the result obtained above to get a E1 (x) such that [B(x)E1 (x), ' ] ⊂ ' (x). [

Set

E (x) = E0 (x)

(8.12)

] Im−s 0 , 0 E1 (x)

[( ) ] ∗ g(x)E (x) = B(x)E1 (x) . 0

then

The rst m − s vector elds are in ' , and the last s vector elds satisfy (8.12), so the second equation of (8.2) follows from (8.12). To prove the rst equation of (8.2), we have to construct D (x). Set D(x) = b(x)E1 (x),

D (x) = −L(z)E1 (x)(DT (x)D(x))−1 DT (x) f 2 (x),

(8.13)

where L(x) is as in (8.9). Then ] [ 1] [ ∗ f . f˜ = f + gD = 2 − b(x)E1 (x)(DT (x)D(x))−1 DT (x) f 2 (x) f Using (8.12), we have

w f˜2 w f 2 w f2 = − D(x)(DT (x)D(x))−1 DT (x) . w xi w xi w xi Using (8.11), there exists *i such that

w f2 = b(x)*i (x), w xi Hence

i = 1, ⋅ ⋅ ⋅ , k.

(8.14)

8.1 ( f , g)-invariant Distribution

211

w f˜2 = b(x)*i − D(x)(DT (x)D(x))−1 DT (x)b(x)*i (x) w xi = b(x)*i − D(x)(DT (x)D(x))−1 DT (x)D(x)E −1 (x)*i (x) = b(x)*i − D(x)E −1 (x)*i (x) = 0. ⊔ ⊓

That proves the rst equation of (8.2). Example 8.1. Given a nonlinear system ⎡ x ⎤ ⎡ ⎤ e 2 (x3 + 1)−1 + ex4 − x1 x2 ex4 + (ex4 − 2)−1 ⎢ ⎥ ⎢ ⎥ ex2 (x3 + 1) − 1 x1 ⎥+⎢ ⎥ x = ⎢ ⎣ ⎦ ⎣(x3 + 1)3 (ex4 − 2)−1 ⎦ u ex2 (x3 + 1)2 e−x4 x2 = f + gu, and a distribution

⎧⎡ x ⎤ ⎡ ⎤⎫ e4 1     ⎨⎢ ⎥ ⎢ ⎬ x3 ⎥ ⎢ 1 ⎥ ⎢ ⎥ ' = Span ⎣ ⎦ , ⎣ . 0 ⎦ 0    ⎩ ⎭ e−x4 1

(8.15)

(8.16)

It is easy to verify that ' is involutive, and ' and ' + G are non-singular around x = 0. Moreover, ' is weakly ( f , g)-invariant. We will nd a suitable feedback law u = D + E v, such that (8.2) holds. To construct a at coordinate chart, we choose four vector elds as X1 = (ex4 , 0, 0, 1)T , X3 = (0, 0, 0, 1)T ,

X2 = (0, 1, 0, 0)T, X4 = (0, 0, 1, 0)T.

They are obviously linearly independent and

' = Span{X1 , X2 }. Dene F(z) = eXz11 eXz22 eXz33 eXz44 (0).

(8.17)

Then, z = F −1 (x) dened a at local coordinate chart. To calculate F we process as X follows: First, solve ez44 (0) by ⎡ ⎡ ⎤ ⎤ x1 (0) x1 ⎢ ⎥ ⎥ d ⎢ ⎢x2 ⎥ = X4 , ⎢x2 (0)⎥ = 0. ⎣ ⎣ ⎦ ⎦ x x (0) dz4 3 3 x4 (0) x4

212

8 Decoupling X

Then solve ez33 eXz44 (0) by ⎡ ⎤ x1 ⎥ d ⎢ x ⎢ 2 ⎥ = X3 , ⎣ dz3 x3 ⎦ x4

⎡ ⎤ x1 (0) ⎢x2 (0)⎥ X4 ⎢ ⎥ ⎣x3 (0)⎦ = ez4 (0). x4 (0)

Continuing this procedure, we nally get that ⎡ ⎤ ⎡ z +z ⎤ x1 e 1 3 − ez3 ⎢x2 ⎥ ⎢ ⎥ z2 ⎢ ⎥ = F(z) = ⎢ ⎥. ⎣x3 ⎦ ⎣ ⎦ z4 x4 z1 + z3 The inverse z = F −1 (x) is ⎡ ⎤ ⎤ ⎡ z1 x4 − ln(ex4 − x1 ) ⎢z2 ⎥ ⎥ ⎢ x2 ⎢ ⎥ = F −1 (z) = ⎢ ⎥ ⎣z3 ⎦ ⎣ ln(ex4 − x1 ) ⎦ . x3 z4 Then

⎡ z −z ⎤ e 2 3 (z4 + 1)−1 + e−z2−z3 − e−z3 + 1 ⎢ ⎥ ez2 (z4 + 1) − 1 ⎥, (F −1 )∗ ( f ) = ⎢ z2 −z3 −1 −z3 ⎣ ⎦ (z4 + 1) + e − 1 −e z 2 2 e (z4 + 1) ⎡ z +z ⎤ (e 1 3 − 2)−1e−z3 + z2 ⎢ ⎥ ez1 +z3 − ez3 ⎥ (F −1 )∗ (g) = ⎢ ⎣ −(ez1 +z3 − 2)−1e−z3 ⎦ . (z4 + 1)3 (ez1 +z3 − 2)−1 ] −e−z3 . b(z) = (z4 + 1)3 [

Now we can choose

Then E0 and E1 are obtained as

E0 = (ez1 +z3 − 2),

E1 = 1.

So E = E0 . Now rewrite f 2 as ] ] [ −z [ z −z e 3 −1 −e 2 3 (z4 + 1)−1 := Y1 + Y2 . f = + 0 ez2 (z4 + 1)2 2

Since Y2 is independent of z1 = (z1 , z2 ), we can replace f 2 by Y1 . Using (8.13), we have D = −E0 (bT b)−1 bTY1 = −(ez1 +z3 − 2)ez2 (z4 + 1)−1 . Finally, for the (D , E ) obtained we have

8.2 Local Disturbance Decoupling

⎡

213

⎤

e−z2 −z3 − e−z3 + 1 − z2 (z4 + 1)−1 ez2 (ez1 +z3 − 2) ⎢e (z4 + 1) − 1 − z2(z4 + 1)−1ez2 (ez1 +z3 − 2)(ez1 +z3 − ez3 )⎥ ⎥, f + gD = ⎢ ⎣ ⎦ e−z3 −1 0 z2

⎡

⎤ e−z2 + z2 (ez1 +z3 − 2) ⎢(ez1 +z3 − 2)(ez1 +z3 − ez3 )⎥ ⎥. gE = ⎢ ⎣ ⎦ e−z3 3 (z4 + 1) It is ready to verify that the equations in (8.2) are satised. In some cases, the non-singularity assumption may be weaken. We refer to [1] for this.

8.2 Local Disturbance Decoupling An immediate application of the structure of ( f , g)-invariance is the local disturbance decoupling. Consider the system ⎧ m s  ⎨x = f (x) + ¦ gi ui + ¦ p j (x)w j := f (x) + g(x)u + p(x)w (8.18) i=1 j=1  ⎩y = h(x), x ∈ ℝn , y ∈ ℝ p , where u are controls and w are disturbances. The local disturbance decoupling problem at an equilibrium point x0 is described as following. Denition 8.2. The local disturbance decoupling problem of system (8.18) at x0 is to nd a neighborhood U of x0 , a regular feedback u = D (x) + E (x)v, where “regular” means E is non-singular, and a local coordinate chart (U, z), z = (z1 , z2 ), such that the system (8.18) can be expressed locally as ⎧ m s  1 = f 1 (z) + 1 (z)v +  z  g p j (z)w j  i ¦ ¦ i   ⎨ i=1 j=1 m (8.19) 2 2 2 z  = f (z ) + g2i (z2 )vi  ¦    i=1  ⎩ y = h(z2 ). From (8.19) one sees clearly that the disturbances w do not affect z2 , and at the same time the outputs y depend on z2 only. Hence, the disturbances do not affect the outputs. That is the physical meaning of the disturbance decoupling. We dene Ker(h) := Span{X ∣ LX h = ⟨dh, X⟩ = 0}.

214

8 Decoupling

Lemma 8.1. The local disturbance decoupling problem is solvable at x0 if there exists a distribution ' , which is non-singular and involutive on a neighborhood of x0 , such that pi (x) ∈ ' ⊂ Ker(h).

(8.20)

Proof. First, we can nd a at coordinate chart (U, z), z = (z1 , z2 ) such that { } w w ' = Span ,⋅⋅⋅ , 1 , w z11 w zk where k = dim(' ) = dim(z1 ). Since ' is ( f , g)-invariant, there exists a control u = D (z) + E (z)v, such that [ f + gD , ' ] ⊂ ' , [gE , ' ] ⊂ ' . Since p ∈ ' , we have

[ p(z) =

] p1 (z) . 0

Using the argument in the last section, we have that [ 1 ] [ 1 ] f (z) g (z) f + g D = 2 2 , gE = 2 2 . f (z ) g (z ) Moreover, since ⟨dh, ' ⟩ = 0, it is obvious that h(z) = h(z2 ). The equation (8.19) follows.

⊔ ⊓

Denition 8.3. Let ' be a given distribution. Its involutive closure, denoted by '¯, is the smallest involutive distribution containing ' . Equivalently, it is the intersection of all involutive distributions containing ' . Lemma 8.2. If ' is a weakly ( f , g)-invariant distribution, then its involutive closure, '¯ is also weakly ( f , g)-invariant. Proof. Denote by ' [k] the s-th fold Lie bracket of vector elds in ' , where s ⩽ k. Then it is clear that [k] '¯ = ∪f k=0 Span{' }. So we have only to show that [gi , X] ∈ '¯ + G,

∀X ∈ ' [k] , k = 0, 1, ⋅ ⋅ ⋅ .

(8.21)

We prove (8.21) inductively. When k = 0 it is trivial. Now assume it is true for k. Then any X ∈ ' [k+1] can be expressed as X ∈ [' [k] , ' [k] ]. So we have only to prove that if Xi ∈ ' [k] , i = 1, 2 satisfy (8.21) that so is [X1 , X2 ]. Assume [gi , X j ] = Z ij + Y ji ,

Z ij ∈ '¯, Y ji ∈ G,

i = 0, 1, ⋅ ⋅ ⋅ , m, j = 1, 2,

8.2 Local Disturbance Decoupling

215

where g0 := f . Then by Jacobi identity [gi , [X1 , X2 ]] = = = ∈

[Z1 , [gi , X2 ]] − [X2 , [gi , X1 ]] [X1 , Z2i + Y2i ] − [X2 , Z1i + Y1i ] [X1 , Z2i ] − [X2, Z1i ] + [X1 ,Y2i ] − [X2,Y1i ] '¯ + '¯ + '¯ + G + '¯ + G = '¯ + G.

Therefore, '¯ is also ( f , g)-invariant.

⊔ ⊓

Theorem 8.2. Let ' be the largest weakly ( f , g)-invariant distribution contained in the Ker(h), and assume ' and ' ∪ G is non-singular at x0 , then the system (6.9) is locally disturbance decouplable at x0 , if and only if pi ∈ ' ,

i = 1, ⋅ ⋅ ⋅ , p.

Proof. (Sufciency) Let ' be the largest weakly involutive distribution contained in Ker(h). We claim that ' is involutive. According to the Lemma 8.2, '¯ is also ( f , g)-invariant. Hence we have only to show that ' ⊂ Ker(h) implies '¯ ⊂ Ker(h). This is true as long as we can prove Ker(h) is involutive. In fact, Ker(h) is involutive, because let X,Y ∈ Ker(h), then ⟨dh, [X,Y ]⟩ = LX LY h − LY LX h = 0. Then the conclusion follows from the Lemma 8.1. (Necessity) From the equation (8.19) one sees that  } { w  i = 1, ⋅ ⋅ ⋅ , k '0 = Span w z1i  is involutive ( f , g)-invariant and pi ∈ ' 0 ⊂ Ker(h). Then for the largest weakly ( f , g)-invariant distribution, ' , contained in Ker(h) we have pi ∈ '0 ⊂ ' ⊂ Ker(h). ⊔ ⊓ Next, we consider how to get the largest weakly ( f , g)-invariant distribution contained in Ker(h). We calculate a sequence of co-distributions as ⎧   ⎨: 0 = Span{dhi ∣ i = 1, ⋅ ⋅ ⋅ , p}, m (8.22) ⊥ ⊥   ⎩: k+1 = :k + L f (G ∩ :k ) + ¦ Lg j (G ∩ :k ), k ⩾ 0. j=1

Theorem 8.3. Assume in the increasing sequence, {:k } of (8.22), if there exists an integer k∗ such that :k∗ +1 = :k∗ , then its annihilate distribution

216

8 Decoupling

' = : k⊥∗ is the largest weak ( f , g)-invariant distribution containing in Ker(h). Proof. We rst prove that ' is weakly ( f , g)-invariant. Setting f := g0 , from (8.22) one sees that Lg j (G⊥ ∩ :k∗ ) ⊂ :k∗ , j = 0, 1, ⋅ ⋅ ⋅ , m. Now let X ∈ ' , and Z ∈ :k∗ ∩ G⊥ . Then 〈 〉 〈 〉 Lg j Z , X = Lg j ⟨Z , X ⟩ − Z , adg j X .

(8.23)

But the left hand and the rst term in right hand are zero. Hence 〈 〉 Z , adg j X = 0, which means adg j X ∈ (:k∗ ∩ G⊥ )⊥ = : k⊥∗ ,

j = 0, 1, ⋅ ⋅ ⋅ , m.

Since X ∈ ' is arbitrary, we have [g j , ' ] ⊂ ' + G,

j = 0, 1, ⋅ ⋅ ⋅ , m.

Next, we prove ' is the largest weakly ( f , g)-invariant distribution containing in Ker(h). Let D be another ( f , g)-invariant distribution containing in Ker(h). Without loss of generality, we can assume D is involutive. Let X ∈ D, Z ∈ D⊥ ∩ G⊥ . Using (8.23), we have 〈 〉 Lgi Z , X = 0. Therefore,

Lgi (D⊥ ∩ G⊥ ) ⊂ D⊥ ,

i = 0, 1, ⋅ ⋅ ⋅ , m.

We claim that the :k dened by (8.22) satisfy

: k ⊂ D⊥ ,

k = 0, 1, ⋅ ⋅ ⋅ .

(8.24)

We prove the claim by induction. By denitions of :0 and D, (8.24) is obviously true for k = 0. Assume (8.24) is true for k, then m

: k+1 = :k + L f (:k ∩ G⊥ ) + ¦ Lgi (:k ∩ G⊥ ) i=1 m

⊂ :k + L f (D⊥ ∩ G⊥ ) + ¦ Lgi (D⊥ ∩ G⊥ ) i=1

⊂ Finally,

D⊥ .

' ⊥ = : k ∗ ⊂ D⊥ ,

which implies D ⊂ '.

8.2 Local Disturbance Decoupling

217

⊔ ⊓ It is obvious that in the above algorithm the key point lies on the existence of the k∗ . Assume x0 is a regular point for :0 , :1 , ⋅ ⋅ ⋅ , :n−1 . Then it is obvious that for this increasing sequence of co-distributions, either there exists a k∗ such that :k∗ = :k∗ +1 , or dim(:n−1 ) = n. In either case ⊥ ' = : n−1 .

(8.25)

In other words, we always have k∗ ⩽ n − 1. When the system is an analytic system, it is easy to see that there exists an open dense subset U of M such that (8.25) holds on U. Example 8.2. Consider the decoupling problem for the following system ⎧  ⎨x = f (x) + g(x)u + p(x)w 1  ⎩y = h(x) = x1 − ex4 + (x3 + 1)−2 2

(8.26)

at a neighborhood of the origin. Where f (x) and g(x) are as in (8.15) of the Example 8.1, and the disturbance w is inputed through p(x) = (ex4 , ex4 , 0, 1)T . We calculate :0 rst. ) ( :0 = Span{dh} = Span{ 1, 0, −(x3 + 1)−3, −ex4 }. It is easy to see that :0 ⊂ G⊥ , hence

: 0 ∩ G⊥ = :0 , : 1 = : 0 + L f :0 + Lg :0 = Span{(1, 0, −(x3 + 1)−3, −ex4 ), (−1, 0, 0, ex4 )}. We have

: 1 ∩ G⊥ = :0 .

Hence :2 = :1 . According to the Theorem 8.3, we have

' = : 1⊥ = Span{(ex4 , x3 , 0, 1)T , (1, 1, 0, e−x4 )T }. This is exactly the distribution in Example 8.1. Since p(x) ∈ ' , the local disturbance coupling problem is solvable. The feedback control, u = D (x) + E (x)v and the at coordinate frame is already obtained in the Example 8.1. A straightforward computation shows that under the at coordinate chart we have p = (1, ez1 +z3 + 1, 0, 0)T, 1 h = −ez3 + (z4 + 1)−2. 2

218

8 Decoupling

Finally, the closed-loop system becomes ⎧ z = ez2 −z3 (z4 + 1)−1 + e−z2 −z3 − e−z3 + 1    1 ( )   + (ez1 +z3 − 2)−1 e−z3 + z2 v + w      ⎨ z2 = ez2 (z4 + 1) − 1 + (ez1 +z3 − ez3 ) v + (ez1 +z3 + 1)w z3 = e−z3 −1 − e−z3 v     z4 = (z4 + 1)3 v,       ⎩ y = −ez3 + 1 (z4 + 1)−2. 2

(8.27)

(8.27) shows clearly that the disturbance is decoupled.

8.3 Controlled Invariant Distribution Given a nite set of vector elds: X = {X1 , ⋅ ⋅ ⋅ , Xt }, and a distribution

'0 ⊂ Span{X1 , ⋅ ⋅ ⋅ , Xt }. The smallest distribution containing ' 0 and X -invariant distribution ' is denoted as

' = ⟨X1 , ⋅ ⋅ ⋅ , Xt ∣ '0 ⟩ . Assume {Y1 , ⋅ ⋅ ⋅ ,Ys } ⊂ Span{X1 , ⋅ ⋅ ⋅ , Xt }, and

'0 = Span{Y1 , ⋅ ⋅ ⋅ ,Ys }. Then ' is also denoted as

' = ⟨X1 , ⋅ ⋅ ⋅ , Xt ∣ Y1 , ⋅ ⋅ ⋅ ,Ys ⟩ . To get ' we propose the following algorithm: ⎧  ⎨' 0 = ' 0 t

 ⎩' k+1 = ' k + ¦ [Xi , ' k ], i=1

Then we have the following Lemma 8.3. 1.

'k ⊂ ' ,

k ⩾ 0;

2. If there exists a k∗ such that

'k∗ +1 = 'k∗ , then ' = 'k∗ .

k ⩾ 0.

(8.28)

8.3 Controlled Invariant Distribution

219

Proof. 1 is trivial. It follows from 1 that

' 0 ⊂ ' k∗ ⊂ ' . But from (8.28) we have t

'k∗ +1 = 'k∗ + ¦ [Xi , 'k∗ ] = 'k∗ . i=1

Therefore, [Xi , ' k∗ ] ⊂ ' k∗ +1 = ' k∗ . ⊔ ⊓ Next, we will answer when such

k∗

exists.

Lemma 8.4. Assume X1 , ⋅ ⋅ ⋅ , Xt ∈ Cf (M). Then for algorithm (8.28) there exists an open dense subset U ⊂ M such that

' = 'n−1 .

(8.29)

Proof. First, we claim that there exists Uk ⊂ M open and dense, such that ' k is nonsingular on Uk . Let O be a non-empty open set and the largest dimension of ' k on O is p. Then by continuity, the set of the points x ∈ O and dim(' k (x)) = p is an open non-empty set. Since O is arbitrary, so the non-singular points form an open dense set Uk . Then U = ∩n−1 i=0 Uk is still open and dense. Now for each x ∈ U, since dim(' k (x)) = const., we have k∗ ⩽ n − 1. (8.29) follows. ⊔ ⊓ In fact, we can prove that ' is involutive. Lemma 8.5. Assume X1 , ⋅ ⋅ ⋅ , Xt ∈ Cf (M). Then ' is involutive. Proof. Let U be the open dense set dened in the Lemma 8.4. Then on U the ' = 'n−1 is a linear combination of adXi1 ⋅ ⋅ ⋅ adXik Y j , k ⩽ n − 1, 1 ⩽ j ⩽ s. We claim that ] [ adXi1 ⋅ ⋅ ⋅ adXik Y j , 'n−1 ⊂ ' n−1 . When k = 0,

(8.30)

[Y j , 'n−1 ] ⊂ 'n = 'n−1 .

Using Jacobi identity, we know that for any two vector elds X, Y and a distribution ' [[X ,Y ], ' ] ⊂ [X , [Y, ' ]] + [Y, [X, ' ]]. Now assume (8.30) is true for k. Then for k + 1 we have [ ] [ ] adXi1 ⋅ ⋅ ⋅ adXik+1 Y j , ' n−1 = [Xi1 , adXi2 ⋅ ⋅ ⋅ adXik+1 Y j ], ' n−1 ] [ ⊂ [Xi1 , ' n−1 ] + [adXi2 ⋅ ⋅ ⋅ adXik+1 Y j ], 'n−1 ⊂ ' n−1 .

(8.31)

220

8 Decoupling

We, therefore, proved (8.30), which implies that 'n−1 = ' is involutive. Now consider any x ∈ M, since U ⊂ M is dense, there exists a sequence {xk } ⊂ U, such that lim xk = x. For any two vector elds X ,Y ∈ ' , we have k→f

[X ,Y ](xk ) ∈ ' (xk ),

k = 1, 2, ⋅ ⋅ ⋅ .

By continuity, we have [X ,Y ](x) ∈ ' (x). That is, ' is involutive.

⊔ ⊓

Consider an afne nonlinear system m

x = f (x) + ¦ gi (x)ui .

(8.32)

i=1

Using a regular feedback u = D (x) + E (x)v, with non-singular E (x), we have that { f˜ = f (x) + g(x)D (x) (8.33) g˜ = g(x)E (x). (8.33) may be locally dened, then the following discussion is a local version. Let / ⊂ {1, 2, ⋅ ⋅ ⋅ , m} be a subset of the indexes. We denote the corresponding input channels as g˜/ = {g˜i ∣ i ∈ / }. Denition 8.4. For system (8.32), assume there exists a regular feedback u = D (x)+ E (x)v, a subset / ⊂ {1, 2, ⋅ ⋅ ⋅ , m}, such that 〉 〈 R(x) = f˜, g˜1 , ⋅ ⋅ ⋅ , g˜m ∣ g˜/ . That is, R(x) is the smallest distribution containing g˜/ and f˜-, g˜1 -, ⋅ ⋅ ⋅ , g˜m - invariant. Then R(x) is called a controlled invariant sub-distribution. From the denition it is obvious that a controlled invariant sub-distribution is an ( f , g)-invariant distribution. Moreover, the Lemma 8.5 tells that it is also involutive. In certain decoupling problems it is important to nd the largest controlled invariant distribution contained in a given distribution ' . Assume ' is given in advance, we give the following algorithm: ⎧  ⎨'0 = ' ∩ G,( ) m (8.34)  ⎩'k+1 = ' ∩ ad f ' k + ¦ adgi 'k + G , k ⩾ 0. i=1

The algorithm (8.34) provides an increasing sequence. In fact, it is clear that '1 ⊃ '0 . Assume 'k ⊃ 'k−1 . Then by mathematical induction, we have ( ) m

'k+1 = ' ∩ ad f 'k + ¦ adgi 'k + G i=1

8.3 Controlled Invariant Distribution

(

221

)

m

⊃ ' ∩ ad f ' k−1 + ¦ adgi 'k−1 + G i=1

= 'k . We want to show that under certain condition the algorithm provides the largest controlled invariant distribution contained in ' . For this goal, we need some preparations. Lemma 8.6. The sequence of distributions, {' k }, generated by the algorithm (8.34) is independent of the feedback (D (x), E (x)). That is, using feedback form (8.33) to (8.34) yields ⎧ ˜  ⎨'0 = ' ∩ G,( ) m  ⎩'˜k+1 = ' ∩ ad ˜ '˜ k + ¦ adg˜ '˜k + G , k ⩾ 0. f

i

i=1

Then '˜ k = 'k , k = 0, 1, 2, ⋅ ⋅ ⋅ . Proof. We rst show that '˜ k ⊂ 'k , k = 0, 1, 2, ⋅ ⋅ ⋅. For k = 0, it is obviously true. Assume '˜k ⊂ 'k . Then ( ) m '˜ k+1 = ' ∩ ad ˜ '˜k + ¦ adg˜ '˜k + G f

(

i

i=1 m

)

⊂ ' ∩ ad f˜ ' k + ¦ adg˜i 'k + G i=1

⊂ ' ∩ ([ f + gD , 'k ] + [gE , ' k ] + G) ⊂ ' ∩ ([ f , ' k ] + [g, ' k ] + G) = ' k+1 . Expressing f = f˜ − gD , and g˜ = g(E )−1 , same argument shows that ' k ⊂ '˜ k , k = 0, 1, ⋅ ⋅ ⋅ . ⊓ ⊔ Lemma 8.7. Let ' be an involutive and ( f , g)-invariant distribution. G ∩ ' is nonsingular. Moreover, for the algorithm (8.34) there exists a k∗ such that ' k∗ = 'k∗ +1 . Then 'k∗ is a controlled invariant distribution contained in ' . Proof. Since ' is ( f , g)-invariant, there exists a feedback (8.33) such that {[ ] f˜, ' ⊂ ' [g, ˜ '] ⊂ '.

(8.35)

Let {g¯1 , ⋅ ⋅ ⋅ , g¯s } be a set of basis of G ∩ ' . Since ' is involutive [g¯i , ' ] ⊂ ' ,

i = 1, ⋅ ⋅ ⋅ , s.

Now we can choose m − s vector elds from {g˜1 , ⋅ ⋅ ⋅ , g˜m }, say {g˜s+1 , ⋅ ⋅ ⋅ , g˜m }, such that {g¯1 , ⋅ ⋅ ⋅ , g¯s , g˜s+1 , ⋅ ⋅ ⋅ , g˜m }

222

8 Decoupling

becomes a basis of G. Moreover, for this group of basis, (8.35) remains true. So, we simply denote g˜i = g¯i , i = 1, ⋅ ⋅ ⋅ , s. Using them, we can construct a sequence of distributions as ⎧  ⎨'¯0 = ' ∩ G, m

¯ ¯ ¯ ¯  ⎩' k+1 = ad f˜ ' k + ¦ adg˜i ' k + '0 ,

k ⩾ 0.

i=1

(8.35) assures that

'¯k ⊂ ' .

And then

'¯k = 'k . Hence, there exists k∗ such that '¯k∗ = '¯k∗ +1 . According to the construction of '¯k one sees easily that 〈 〉 'k∗ = '¯k∗ = f˜, g˜1 , ⋅ ⋅ ⋅ , g˜m ∣ G ∩ ' . ⊔ ⊓

Theorem 8.4. Let ' be an involutive and ( f , g)-invariant distribution. G ∩ ' is nonsingular. Moreover, for the algorithm (8.34) there exists a k∗ such that ' k∗ = 'k∗ +1 . Then 'k∗ is the largest controlled invariant distribution contained in ' . Proof. From Lemma 8.7 we already know that ' k∗ is a controlled invariant subdistribution. So we have only to prove that it is the largest one. Assume there exists another feedback pair (D¯, E¯), f¯= f + gD¯, g¯ = gE¯, such that 〈 〉 D = f¯, g¯1 , ⋅ ⋅ ⋅ , g¯m ∣ g/ , is another controlled invariant sub-distribution. D can be expressed equivalently as 〈 〉 D = f¯, g¯1 , ⋅ ⋅ ⋅ , g¯m ∣ D ∩ G . Construct a sequence of distributions as ⎧  ⎨'¯0 = D ∩ G, m

'¯k+1 = ad f¯'¯k + ¦ adg¯i '¯k + '¯0 , ⎩

k ⩾ 0.

(8.36)

i=1

Since

'¯k ⊂ D ⊂ ' . (

Hence,

)

m

'¯k ⊂ ' ∩ ad f¯'¯k−1 + ¦ adg¯i '¯k−1 + G . i=1

Using it, we can prove that

'¯k ⊂ 'k ,

k = 0, 1, ⋅ ⋅ ⋅ .

(8.37) is obviously true for k = 0. Assume it is true for k. Then ( ) m '¯k+1 ⊂ ' ∩ ad f¯'k + ¦ adg¯ 'k−1 + G = 'k+1 . i

i=1

(8.37)

8.4 Block Decomposition

223

Using (8.36) and (8.37), we have D= ⊂

〈

f  〉 f¯, g¯1 , ⋅ ⋅ ⋅ , g¯m  D ∩ G = ¦ '¯i i=0

f

k∗

i=0

i=0

¦ ' i = ¦ ' i = ' k∗ .

⊔ ⊓

Finally, a natural question is: Given a distribution ' , how to nd the largest controlled invariant distribution contained in ' ? Assume ' is non-singular and involutive. Then we can dene : 0 = ' ⊥ . Using algorithm (8.22), we can get the largest weakly ( f , g)-invariant distribution contained in ' , say 'I . Under non-singularity assumptions, 'I becomes the largest ( f , g)invariant distribution contained in ' . Then (8.34) is implemented to get the largest controlled invariant distribution 'C contained in 'I . Since a controlled invariant distribution is also ( f , g)-invariant, the largest controlled invariant distribution 'C contained in ' I is also the largest controlled invariant distribution contained in ' . Using the continuity and the regularity of the distributions, it can be proved that over an open dense subset U ⊂ M, the above mentioned two algorithms are executable. That is, after nite steps (at most n − 1 for each algorithm) the largest controlled invariant distribution contained in ' is computable locally for the neighborhood of each point x ∈ U.

8.4 Block Decomposition To consider the state equation decomposition problem we have to construct a set of distributions, each one of which represents a block of sub-states. Denition 8.5. Let ' 1 , ⋅ ⋅ ⋅ , ' k be a set of distributions on M. They are said to be simultaneously integrable at x0 ∈ M, if there exists a coordinate neighborhood of x0 , say (U, x), with ( ) x = x01 , ⋅ ⋅ ⋅ , x0n0 , x11 , ⋅ ⋅ ⋅ , x1n1 , ⋅ ⋅ ⋅ , xk1 , ⋅ ⋅ ⋅ , xknk , such that

{

'i = Span

 } w   j = 1, ⋅ ⋅ ⋅ , ni , w xij 

i = 1, ⋅ ⋅ ⋅ , k.

(8.38)

For statement ease, we use some notations. Let I be a subset of indexing set {1, 2, ⋅ ⋅ ⋅ , m}, i.e., I ⊂ {1, 2, ⋅ ⋅ ⋅ , m}. Then

'I := ¦ 'i . i∈I

We also denote

' −i = ¦ ' j . j∕=i

224

8 Decoupling

Theorem 8.5. The following statements are equivalent: 1. '1 , ⋅ ⋅ ⋅ , 'k are simultaneously integrable at x0 ∈ M. 2. '1 , ⋅ ⋅ ⋅ , 'k are linearly independent at x0 ∈ M. Moreover, for any I ⊂ {1, 2, ⋅ ⋅ ⋅ , m}, 'I are non-singular and involutive at x0 ∈ M. 3. '1 , ⋅ ⋅ ⋅ , 'k are linearly independent at x0 ∈ M. Moreover, the distributions

'i + ' j ,

1 ⩽ i, j ⩽ k

are non-singular and involutive. 4. '1 , ⋅ ⋅ ⋅ , 'k are linearly independent at x0 ∈ M. Moreover, the distributions

' −i ,

k

and ' = ¦ ' i

i = 1, ⋅ ⋅ ⋅ , k

i=1

are non-singular and involutive. Proof. 1 ⇒ 2 ⇒ 4 and 2 ⇒ 3 are trivial. We prove (3 ⇒ 2) Let I = {i1 , ⋅ ⋅ ⋅ , is }, and X,Y ∈ ' I . Then X ,Y can be expressed as X = X1 + ⋅ ⋅ ⋅ + Xs , Y = Y1 + ⋅ ⋅ ⋅ + Ys , with X j ,Y j ∈ ' i j . Then s

¦

[X ,Y ] =

[X j ,Yk ] ∈

j,k=1

s

¦ ('i j + 'ik ) = 'I .

j,k=1

(4 ⇒ 1) Since ' is non-singular and involutive, there are n0 := n − dim(' ) linearly independent smooth functions, denoted by a01 , ⋅ ⋅ ⋅ , a0n0 , such that 〈 0 〉 (8.39) da j , ' = 0, j = 1, ⋅ ⋅ ⋅ , n0 . Similarly, since ' −i is non-singular and involutive, there are ni := dim(' i ) smooth functions, denoted by ai1 , ⋅ ⋅ ⋅ , aini , such that 〈 i −i 〉 = 0, j = 1, ⋅ ⋅ ⋅ , ni , i = 1, ⋅ ⋅ ⋅ , k. da j , ' (8.40) We claim that the set of n co-vector elds daij ,

i = 0, 1, ⋅ ⋅ ⋅ , k, j = 1, ⋅ ⋅ ⋅ , ni

are linearly independent at x0 . Let O ji ∈ ℝ1 , and k

ni

Z := ¦ ¦ O ji daij = 0. i=0 j=0

Denote

k

Z = ¦ Zi ,

where Zi :=

i=0

ni

¦ O ji daij .

j=0

Choosing a vector eld X ∈ 'i ⊂ ' − j , By denition, we have

j ∕= i, 1 ⩽ i ⩽ m.

〈 〉 Zi , ' −i = 0.

(8.41)

8.4 Block Decomposition

Moreover, since

⟨Z , X⟩ = ⟨Zi , X ⟩ = 0,

i.e.,

225

X ∈ 'i ,

⟨Zi , ' i ⟩ = 0.

It follows that

⟨Zi , ' ⟩ = 0.

That is,

Zi ∈ ' ⊥ = Span{da0j ∣ j = 1, ⋅ ⋅ ⋅ , n0 }.

Notice that Zi is linearly independent with Span{da0j ∣ j = 1, ⋅ ⋅ ⋅ , n0 }, so

O ji = 0,

i = 1, ⋅ ⋅ ⋅ , k, j = 1, ⋅ ⋅ ⋅ , ni .

Then (8.41) becomes

Z=

n0

¦ O j0 da0j = 0,

j=0

which implies that

O j0 = 0, j = 1, ⋅ ⋅ ⋅ , n0 . The claim is proved. Then we can choose a local coordinate frame as xij = aij ,

i = 0, ⋅ ⋅ ⋅ , k, j = 1, ⋅ ⋅ ⋅ , ni .

According to (8.39) and (8.40), we have s 'i = Span{dx ⋅ , ns } ⊥ { j ∣ s ∕= i, j = 1, 2, ⋅ ⋅} w  ⊃ Span  j = 1, 2, ⋅ ⋅ ⋅ , ni . w xij 

Note that dim(' i ) = ni , then {

'i = Span

 } w   j = 1, 2, ⋅ ⋅ ⋅ , ni , w xij 

i = 1, ⋅ ⋅ ⋅ , k.

⊔ ⊓

In the rest of this section we consider a general nonlinear control system x = f (x, u),

x ∈ M.

(8.42)

Where M = ℝn or n-th dimensional manifold. The problem considered is: When the system has a decomposed form via a coordinate change? Denition 8.6. Let u = (u1 , ⋅ ⋅ ⋅ , uk ) be a partition of u. The system (8.42) is said to have a local parallel decomposition at x0 ∈ M, if there is a local coordinate chart (U, z) of x0 , such that the system (8.42) can be expressed as ⎧ z0 = f 0 (z0 )     ⎨z1 = f 1 (z0 , z1 , u1 ) (8.43) ...    ⎩k z = f k (z0 , zk , uk ).

226

8 Decoupling

The system (8.42) is said to have a local cascade decomposition at x0 ∈ M, if the system (8.42) can be expressed locally as ⎧0 z =      z1 =  ⎨ z1 =  .   ..    ⎩k z =

f 0 (z0 ) f 1 (z0 , z1 , u1 ) f 1 (z0 , z1 , z2 , u1 , u2 )

(8.44)

f k (z, u).

Recall Chapter 5, for system (8.42) we dened a set of vector elds as F = { f (x, u) ∣ u = const .}. The controllability Lie algebra, F , is the Lie algebra generated by F: F = {F}LA . Its derived Lie algebra, F ′ , is F ′ = {[X ,Y ] ∣ X ,Y ∈ F }LA . The strong controllability Lie algebra, F 0 is dened as  } {  p p  ′ F 0 = ¦ Oi Xi + Y  p < f, ¦ Oi = 0, X ∈ F, Y ∈ F .  i=1 i=1 Recall also that an ideal, I, of a Lie algebra F is a sub-algebra of F with the property that [F , I] ⊂ I. Denote u−i = {u1 , ⋅ ⋅ ⋅ , uk }∖{ui}, i.e., the set of u except ui . Dene { } Fi := f (x, ui1 , u−i ) − f (x, ui2 , u−i ) ∣ ui1 = const., ui2 = const ., u−i = const. . Where only the i-th block of controls, ui , can take different constant values. Denote by Ii the smallest idea of F containing Fi . Since Fi ⊂ F 0 ,

i = 1, ⋅ ⋅ ⋅ , k,

and it is easy to see that F 0 is an ideal of F , so Ii ⊂ F 0 ,

i = 1, ⋅ ⋅ ⋅ , k.

(8.45)

That is, Ii is also an ideal of F 0 . Next, we consider the relationship among Ii and F 0 . Denote F0 := { f (x, u1 ) − f (x, u2 ) ∣ u1 = const., u2 = const .} . The structures of F 0 and Ii are described as the follows.

8.4 Block Decomposition

Lemma 8.8. F 0 and Ii can be expressed as  { } F 0 = Span ad f1 ⋅ ⋅ ⋅ ad fs X  s ⩾ 0, fi ∈ F, X ∈ F0 ,  { } Ii = Span ad f1 ⋅ ⋅ ⋅ ad fs X  s ⩾ 0, fi ∈ F, X ∈ Fi ,

i = 1, ⋅ ⋅ ⋅ , k.

Proof. First, we give the structure of F ′ . We claim  { } F ′ = Span ad f1 ⋅ ⋅ ⋅ ad fs fs+1  s ⩾ 1, fi ∈ F .

227

(8.46) (8.47)

(8.48)

To prove (8.48), we rst see that the RHS (right hand side) of (8.48) is contained in the LHS (left hand side) of (8.48). In addition, the RHS is a Lie algebra and it contains {[X,Y ] ∣ X,Y ∈ F}. So it contains the Lie algebra generated by {[X ,Y ] ∣ X,Y ∈ F}, which is the LHS. The equations in (8.47) are obvious, so we prove (8.46) only. Since F 0 is an ideal and F 0 ⊃ F0 , it is obvious that the RHS of (8.46) is a subset of the LHS of (8.46). Next, choose any X ∈ F 0 , then it is expressed as p

X = ¦ Oi Xi + Y,

Xi ∈ F , Y ∈ F ′ .

i=1

p

Since

¦ Oi = 0, then

i=1

p

p−1

i=1

i=1

¦ Oi Xi =

¦ Oi (Xi − Xp) ∈

RHS of (8.46).

So it is enough to show that Y ∈ RHS of (8.46). Using (8.48), we can, without loss of generality, assume Y = ad f1 ⋅ ⋅ ⋅ ad fs fs+1 ,

s ⩾ 1.

Hence, we can rewrite Y as Y = ad f1 ⋅ ⋅ ⋅ ad fs ( fs+1 − fs ) ∈ RHS of (8.46).

⊔ ⊓

Lemma 8.9. Assume F 0 is non-singular at x0 , and Ii , i = 1, ⋅ ⋅ ⋅ , k are linearly independent (as distributions at x0 ). Then Ii are simultaneously integrable at x0 . Proof. We rst claim that Ii are non-singular at x0 . An obvious fact is: a vector eld in F0 can be expressed as a linear combination of the vectors in F1 , ⋅ ⋅ ⋅ , Fk . In fact, f (x, (u)1 ) − f (x, (u)2 ) ) ( = f (x, u11 , ⋅ ⋅ ⋅ , uk1 ) − f (x, u12 , u21 , ⋅ ⋅ ⋅ , uk1 )

228

8 Decoupling

( ) + f (x, u12 , u21 , ⋅ ⋅ ⋅ , uk1 ) − f (x, u12 , u22 , ⋅ ⋅ ⋅ , uk1 ) + ⋅⋅⋅ ) ( k 1 2 k + f (x, u12 , u22 , ⋅ ⋅ ⋅ , uk−1 2 , u1 ) − f (x, u2 , u2 , ⋅ ⋅ ⋅ , u2 ) . According to (8.46)and (8.47), we have F 0 ⊂ I1 + ⋅ ⋅ ⋅ + Ik . Recall (8.45), we have F 0 = I1 + ⋅ ⋅ ⋅ + Ik .

(8.49)

Assume dim(F 0 (x0 )) = s and dim(Ii ) = si , i = 1, ⋅ ⋅ ⋅ , k. Using (8.49) and the fact that Ii , i = 1, ⋅ ⋅ ⋅ , k are linearly independent, we have that k

¦ si = s.

i=1

Now by continuity, there exists a neighborhood U of x0 , such that dim(I i (x)) ⩾ si ,

x ∈ U, i = 1, ⋅ ⋅ ⋅ , k.

Now (8.49) assures that dim(I i (x)) = si ,

x ∈ U, i = 1, ⋅ ⋅ ⋅ , k.

We proved the claim. Choosing X ∈ I i and Y ∈ I j , since both I i and I j are ideals, we have [X ,Y ] ∈ I i ∩ I j ,

1 ⩽ i, j ⩽ k.

This shows that I i + I j is involutive. So, Ii , i = 1, ⋅ ⋅ ⋅ , k are simultaneously integrable. ⊔ ⊓ Using the aforementioned notations, we have Theorem 8.6. Assume F 0 is non-singular at x0 . The local parallel decomposition problem for system (8.42) is solvable at x0 , if and only if there is a partition of the control u, such that the corresponding ideals, Ii , i = 1, ⋅ ⋅ ⋅ , k are linearly independent (as distributions at x0 ). Proof. (Necessity) Dene a sequence of distributions as  } { w  'i = Span  j = 1, ⋅ ⋅ ⋅ , ni , i = 1, ⋅ ⋅ ⋅ , k. w zij  From (8.43) we have Fi ⊂ 'i , For any f ∈ F, we have

i = 1, ⋅ ⋅ ⋅ , k.

8.4 Block Decomposition

[ f,

w w zij

] =−

229

w f ∈ 'i . w zij

That is, 'i is f invariant. Using Jacobi identity, it is easy to see that [X , ' i ] ⊂ 'i ,

∀X ∈ F .

Hence, 'i is an ideal of F . Since Fi ∈ ' i , Ii is the smallest ideal containing Fi , so Ii ⊂ 'i . Note that ' i , i = 1, ⋅ ⋅ ⋅ , k are linearly independent, then so are Ii , i = 1, ⋅ ⋅ ⋅ , k. (Sufciency) Assume Ii , i = 1, ⋅ ⋅ ⋅ , k are linearly independent. Using Lemmas 8.8 and 8.9, there exists a local coordinate chart (U, z) of x0 , such that  { } w  Ii = Span  j = 1, ⋅ ⋅ ⋅ , ni , i = 1, ⋅ ⋅ ⋅ , k. w zij  Under coordinates z, the system (8.42) is expressed as ⎧ z0 = f 0 (z, u)     ⎨z1 = f 1 (z, u) ..   .   ⎩k z = f k (z, u).

(8.50)

Since Ii are f invariant, i.e., [ f , Ii ] ⊂ Ii , it follows that

wfj = 0, w xis

i = 1, ⋅ ⋅ ⋅ , k, s = 1, ⋅ ⋅ ⋅ , ni , j ∕= i.

(8.51)

On a neighborhood V of x0 , we also have

wfj = 0, w uti

i = 1, ⋅ ⋅ ⋅ , k, t = 1, ⋅ ⋅ ⋅ , mi , j ∕= i.

(8.52)

Otherwise, for any given neighborhood V0 , we can nd z ∈ V0 and u = u0 , such that  w f j  ∕= 0. w uti z,u0 So there exists H > 0 small enough such that ˜ − f j (z, u) ∕= 0, f j (z, u) where u˜rj = (u0 )rj ,

(r, j) ∕= (i,t);

u˜ti = (u0 )ti + H .

But according to the denition f (z, u) ˜ − f (z, u0 ) ∈ Ii , which contradicts to (8.53).

(8.53)

230

8 Decoupling

The equations (8.51) and (8.52) assure that (8.50) has the parallel decomposed form (8.43). ⊔ ⊓ Next, we consider the cascade decomposition. Let the partition of u and Fi are dened as before. Let Ji be the smallest ideal of F containing {Fs , s ⩽ i}, i.e., Ji = {Fs , s ⩽ i}I ,

i = 1, ⋅ ⋅ ⋅ , k.

Note that Jk = F 0 . Using the generalized Frobenius’ Theorem and the same argument as in the proof of Theorem 8.6, we have the following result. Theorem 8.7. The cascade decomposition problem for system (8.42) is solvable at x0 , if and only if there exists a partition of u, such that the corresponding nested ideals, Ji , i = 1, ⋅ ⋅ ⋅ , k are non-singular at x0 . A key point to solve the parallel or cascade decomposition problems (without feedback) is to nd the ideals Ii . Ii can be obtained via the following algorithm: { E0i = Fi  } { (8.54) i Es+1 = Esi + Span ad f Esi  f ∈ F , s ⩾ 0. It is easy to see that if there exists a k∗ such that Eki ∗ = Eki ∗ +1 , then Ii = Eki ∗ . Under non-singularity assumption, the existence of k∗ is obvious. Example 8.3. Consider the parallel decomposition of the following system ⎡ ⎤ sin(z2 + z4 + u2 ) ⎡ ⎤ ⎢ ⎥ z1 ⎢ ⎥ 1 1 ⎥ u1 u2 ez1 + (z3 + 1 − ez1 ) ⎢z2 ⎥ ⎢ ⎢ ⎥ 2 2 ⎢ ⎥=⎢ ⎣z3 ⎦ ⎢ez1 sin(z2 + z4 + u2 ) + u3 (z2 − z4 + 1)⎥ ⎥ ⎢ ⎥ z4 ⎣ ⎦ 1 1 z1 z1 u1 u2 e − (z3 + 1 − e ) 2 2 at z0 = 0. To begin with, we calculate Ii , i = 1, 2, 3. ⎧⎛ ⎫ ⎞  0        ⎜   ⎟   ⎜    ⎟ 1 ′ z   1 ⎨ ⎜ (u1 − u1 )u2 e ⎟ ⎬ ⎜2 ⎟ ′ F1 = ⎜ ⎟ u1 , u1 , u2 = const . ⎜   ⎟ 0     ⎟ ⎜     ⎝ ⎠   1    ′ z 1 ⎩ ⎭ (u1 − u1 )u2 e  2 Then

E01 = Span{(0, 1, 0, 1)T } E11 = Span{(0, 1, 0, 1)T } + Span{ad f e10 } = Span{(0, 1, 0, 1)T , (1, 0, ez1 , 0)T }.

8.4 Block Decomposition

231

To calculate E21 , since 1 ad f ((1, 0, ez1 , 0)T ) = (u1 u2 ez1 , 0, u1 u2 ez1 , 0)T ∈ E11 , 2 it follows that E21 = E11 = I1 . Similarly, we can calculate I2 and I3 . Since F2 ∈ I1 , so I2 ⊂ I1 . I3 can be obtained as I3 = Span{(0, 0, 1, 0)T, (0, 1, 0, −1)T }. It is easy to check that I1 and I3 are linearly independent. Choosing X1 = (1, 0, ez1 , 0)T , X2 = (0, 1, 0, 1)T, X3 = (0, 1, 0, −1)T , X4 = (0, 0, 1, 0)T, a at coordinate chart can be obtained by a diffeomorphism: F(x1 , x2 , x3 , x4 ) = eXx11 eXx22 eXx33 eXx44 (0). Then F(x) can be expressed as ⎧ z 1 = x1    ⎨z = x + x 2 2 3 x1  z = e − 1 + x4 3   ⎩ z4 = x2 − x3 . Under the new coordinates x the system becomes ⎧  x1 = sin(2x2 + u2 )     1  x1   ⎨x2 = 2 u1 u2 e 1   x3 = x4    2    ⎩x = u (2x + 1). 4

3

3

It is obvious that the system is decomposed into two sub-systems. First sub-system consists of x1 and x2 and the second sub-system consists of x3 and x4 . Since the system satises the strong accessibility rank condition at the origin, there is no x0 component. Consider an afne nonlinear system m

x = f (x) + ¦ gi (x)ui . i=1

232

Then

8 Decoupling

F = { f , g1 , ⋅ ⋅ ⋅ , gm }LA F 0 = {adif g j ∣ j = 1, ⋅ ⋅ ⋅ , m, i ⩾ 0}LA Fi = {gij ∣ j = 1, ⋅ ⋅ ⋅ , mi }, i = 1, ⋅ ⋅ ⋅ , k.

Then Ii = {adsf gij ∣ gij ∈ Fi , s ⩾ 0}LA ,

i = 1, ⋅ ⋅ ⋅ , k.

And the above decomposition results for general system (8.42) are applicable to the afne nonlinear systems.

8.5 Feedback Decomposition Consider an afne nonlinear system m

x = f (x) + ¦ gi (x)ui ,

x ∈ M.

(8.55)

i=1

To investigate the feedback block decomposition problem, we have to nd a feedback which is suitable for several distributions. Denition 8.7. Let 'i , i = 1, ⋅ ⋅ ⋅ , k be a set of distributions. ' i are said to be (locally) simultaneously ( f , g)-invariant, if there exists a (local) regular feedback control (D (x), E (x)) such that { [ f (x) + g(x)D (x), ' i (x)] ⊂ ' i (x) (8.56) [g(x)E (x), ' i (x)] ⊂ ' i (x), i = 1, ⋅ ⋅ ⋅ , k. For convenience, in this section we assume the system (8.55) satises the strong accessibility rank condition, i.e., rank(F 0 ) = rank({adsf g ∣ s ⩾ 0}LA ) = n.

(8.57)

If (8.57) is not satised, but rank(F 0 ) = const., then we can use standard decomposition to convert the system (8.55) into ⎧  ⎨x1 = f 1 (x1 ) m

2 2  ⎩x = f (x) + ¦ gi ui . i=1

Then consider x1 as a known function and consider the second part of the system, which satises the controllability rank condition. Theorem 8.8. Assume the system (8.55) satised the strong accessibility rank condition at x0 . 'i , i = 1, ⋅ ⋅ ⋅ , k are simultaneously integrable and weakly (f,g)-invariant at x0 . G is regular at x0 , and G = G ∩ ' 1 + G ∩ ' 2 + ⋅ ⋅ ⋅ + G ∩ 'k .

(8.58)

8.5 Feedback Decomposition

233

Then 'i , i = 1, ⋅ ⋅ ⋅ , k are simultaneously ( f , g)-invariant at x0 . Proof. Since 'i , i = 1, ⋅ ⋅ ⋅ , k are simultaneously integrable at x0 , there exists a coordinate neighborhood (U, z) of x0 , z = {zij ∣ i = 0, 1, ⋅ ⋅ ⋅ , k, j = 1, ⋅ ⋅ ⋅ , ni }, {

such that

'i = Span

 } w   j = 1, ⋅ ⋅ ⋅ , ni , w zij 

i = 1, ⋅ ⋅ ⋅ , k.

Let ' = '1 + ⋅ ⋅ ⋅ + ' k . Then ' is also weakly ( f , g)-invariant. We claim that F 0 ⊂ '. Since G ⊂ ' and

[ f,'] ⊂ ' + G = ',

we have {adsf g j ∣ s ⩾ 0, j = 1, ⋅ ⋅ ⋅ , m} ⊂ ' . Since ' is also involutive, F 0 = {adsf g j ∣ s ⩾ 0, j = 1, ⋅ ⋅ ⋅ , m}LA ⊂ ' . Since rank(F 0 ) = n, dim '0 = 0. Using (8.58), we can nd a E0 such that gE0 = (g1 , ⋅ ⋅ ⋅ , gk ),

gi ∈ ' i .

So the feedback system with u = E0 v becomes ⎧ 1 1 1 1   ⎨z = f + g v .. z = f + gE0 v = .   ⎩zk = f k + gk vk . Since G is non-singular, ' i are simultaneously integrable, it is easy to check that G ∩ 'i are also non-singular. Say, dim(Span{gi }) = mi ,

k

¦ mi = m.

i=1

Now under z coordinates we have { [ f , 'i ] ⊂ 'i + G [g, 'i ] ⊂ ' i + G, which are equivalent to

w s f = gs qsij , w zij

i ∕= s,

(8.59)

234

8 Decoupling

w s g = gs Fisj , w zij

i ∕= s,

(8.60)

respectively. Where qsij are the mi × 1 vector functions, and Fisj are the mi × ni matrix functions. Similar to the proof of Quaker Lemma in Section 8.1, we construct ( )−1 Ei = (gs (xs ))T (gs (x)) , and

E = diag(E1 , ⋅ ⋅ ⋅ , Ek ). Then gE0 E = diag(g1 E1 , ⋅ ⋅ ⋅ , gk Ek ) := diag(g˜1 , ⋅ ⋅ ⋅ , g˜k ). Using (8.60) it is ready to prove that

w s g˜ = 0, w zij

i ∕= s.

(8.61)

Let D = diag(g˜1 , ⋅ ⋅ ⋅ , g˜k ). We construct and let

D = − diag(E1 , ⋅ ⋅ ⋅ , Ek )(DT D)−1 DT f , T T f˜ = f + gE0 D = ( f˜1 , ⋅ ⋅ ⋅ , f˜k )T .

Then the (8.59) and (8.61) imply that

w ˜s f = 0, w zij

i ∕= s.

(8.62)

Observing the structure of ' i , (8.61) and (8.62) imply the following two equations: (8.63) and (8.64) respectively: [ ] (8.63) f˜, 'i ⊂ 'i , i = 1, ⋅ ⋅ ⋅ , k, [g˜ j , ' i ] ⊂ ' i ,

j = 1, ⋅ ⋅ ⋅ , m, i = 1, ⋅ ⋅ ⋅ , k.

Then 'i , i = 1, ⋅ ⋅ ⋅ , k are simultaneously ( f , g)-invariant at x0 .

(8.64) ⊔ ⊓

Denition 8.8. Given the system (8.55) and x0 ∈ M. The local feedback block decomposition problem is to nd a regular control u = D (x) + E (x)v and a local coordinate change z = z(x) at x0 , such that the feedback system has a decomposed form as ⎧ 1 1 1 1 1 1  z = f (z ) + g (z )v ⎨ .. (8.65) .   ⎩zk = f k (zk ) + gk (zk )vk .

References

235

For the local feedback block decomposition problem we have the following result. Theorem 8.9. Assume the system (8.55) is strong accessible at x0 ∈ M. The local feedback block decomposition problem is solvable, if and only if there exist k simultaneously integrable weakly (f,g)-invariant distributions, 'i , i = 1, ⋅ ⋅ ⋅ , k, such that G = G ∩ '1 + ⋅ ⋅ ⋅ + G ∩ 'k.

(8.66)

Proof. (Necessity) Dene {

'i = Span

 } w   j = 1, ⋅ ⋅ ⋅ , ni , w zij 

i = 1, ⋅ ⋅ ⋅ , k.

Using equation (8.65), it is obvious that the ' i , i = 1, ⋅ ⋅ ⋅ , k, are simultaneously integrable weakly ( f ,g)-invariant, and (8.66) is satised. (Sufciency) According to the Theorem 8.8, ' i , i = 1, ⋅ ⋅ ⋅ , k, are simultaneously ( f , g)-invariant. So we can nd regular feedback (D (x), E (x)) such that { [ f + gD , ' i ] ⊂ ' i [gE , 'i ] ⊂ ' i Under a simultaneously at coordinate frame of 'i , i = 1, ⋅ ⋅ ⋅ , k, it is easy to see the system (8.55) has the form of (8.65). ⊔ ⊓

References 1. Cheng D, Tarn T. New result on ( f ,g)-invariance. Sys. Contr. Lett., 1989, 12(4): 319–326. 2. Isidori A. Nonlinear Control Systems, 3rd edn. London: Springer, 1995. 3. Isidori A, Krener A, Gori-Giorgi C, et al. Nonlinear decoupling via feedback: A differential geometric approach. IEEE Trans. Aut. Contr., 1981, 26(2): 331–345. 4. Nijmeijer H, Van der Schaft A. Nonlinear Dynamical Control Systems. New York: SpringerVerlag. 5. Nijmeijer H, Schumacher J. The regular local noninteracting control problem for nonlinear control systems. SIAM J. Contr. Opt., 1986, 24(6): 1232–1245.

Chapter 9

Input-Output Structure

This chapter considers the structure of an afne nonlinear systems from a viewpoint of the relation between inputs and outputs. Section 9.1 considers the relative degree and the decoupling matrix. Section 9.2 considers the Morgan’s problem, that is, the input-output decoupling problem. The invertibility of the input-output mapping is discussed in Section 9.3. Section 9.4 provides a dynamic solution to Morgan’s problem. In Section 9.5 the Byrnes-Isidori normal form of nonlinear control systems is introduced. In Section 9.6 a generalization of Byrnes-Isidori normal form is provided. Section 9.7 presents the Fliess functional expansion, which describes the input-output behavior of nonlinear control systems. In Section 9.8 the Fliess functional expansion has been used to missile guide control.

9.1 Decoupling Matrix Consider an afne nonlinear system ⎧  ⎨x = f (x) +

m

¦ gi (x)ui ,

x ∈ ℝn

i=1  ⎩y = h (x), j = 1, ⋅ ⋅ ⋅ , p. j j

(9.1)

Denition 9.1. Consider system (9.1). Assume there exist U j > 0, j = 1, ⋅ ⋅ ⋅ , p such that for a neighborhood U of x0 , ⎧ ⎨Lg Lkf h j (x) = 0, x ∈ U, k < U j − 1 (9.2) ⎩L LU j −1 h (x ) ∕= 0. g f j 0 Then the vector (U1 (x0 ), ⋅ ⋅ ⋅ , U p (x0 )) is called the relative degree vector at x0 . If the relative degree vector is dened and constant at every x ∈ M, then it is simply denoted as (U1 , ⋅ ⋅ ⋅ , U p ). Assume the relative degree vector of system (9.1) is well dened at x0 . Then we can locally dene a p × m matrix D(x) = (di j (x)),

x ∈ U,

with D. Cheng et al., Analysis and Design of Nonlinear Control Systems © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2010

(9.3)

238

9 Input-Output Structure U −1

di j = Lg j L f i

hi (x),

i = 1, ⋅ ⋅ ⋅ , p, j = 1, ⋅ ⋅ ⋅ , m,

and a p dimensional vector b(x) = (bi (x)), with

U

bi = L f i hi (x),

x ∈ U,

(9.4)

i = 1, ⋅ ⋅ ⋅ , p.

D(x) is called the decoupling matrix, and b(x) is called the decoupling vector. It is of fundamental importance to see the change of the decoupling matrix and the decoupling vector under a regular feedback. Proposition 9.1. Let (D (x), E (x)) be a regular feedback and denote f˜(x) = f (x) + g(x)D (x) and g(x) ˜ = g(x)E (x). Correspondingly, we construct D˜ and b˜ by using f˜(x) and g(x) ˜ as U −1 d˜i j = Lg˜ j L f˜i hi (x), U b˜ i = L f˜i hi (x).

Then

{

˜ D(x) = D(x)E (x), ˜b(x) = D(x)D (x) + b(x).

(9.5)

Proof. First, we prove that s ⩽ Ui − 1, x ∈ U.

Lsf˜hi (x) = Lsf hi (x),

(9.6)

It is trivially true for s = 1. Assume it is true for s < Ui − 1. Then Ls+1 hi (x) = L f˜Lsf hi (x) f˜ ( =

m

)

L f + ¦ D j Lg j

Lsf hi (x)

j=1

= Ls+1 f hi (x). Using (9.6), we have that U −1 d˜i j = Lg˜i L f i (x) =

m

U −1

¦ Lgk L f i

Ek j ,

k=1

which implies the rst equation of (9.5). Similarly, the second equation of (9.5) can also be easily obtained from (9.6). ⊓ ⊔ An immediate consequence from Proposition 9.1 and its proof is: Corollary 9.1. The relative degree vector is invariant under a regular feedback. Lemma 9.1. Let X1 , ⋅ ⋅ ⋅ , Xs ∈ F := { f , g1 , ⋅ ⋅ ⋅ , gm }.

9.1 Decoupling Matrix

239

Then the input u j does not affect yi , if and only if for any s > 0 Lg j LX1 ⋅ ⋅ ⋅ LXs hi (x) = 0. Proof. (Necessity) Set

(9.7)

〉 〈 ’ j = f , g 1 , ⋅ ⋅ ⋅ , gm ∣ g j .

Then it is obvious that (9.7) is equivalent to 〈 〉 dhi , ’ j (x) = 0. Now assume it is not true at x0 . Then there exists a neighborhood U of x0 such that 〈 〉 dhi , ’ j (x) ∕= 0, x ∈ U. Since the regular point of ’ j is an open dense set, there exists an x ∈ U such that ’ j has constant dimension around x. Now assume the integral manifold of ’ j , denoted by S, can be locally expressed in a local coordinate chart (V, z), with V ⊂ U, z = (z1 , z2 ), as S = {z ∈ V ∣z2 = 0}. Since the controllability Lie algebra of g j has full 〈rank on 〉S, then the reachable set of1 v j contains an open set of S. But now since dhi , ’ j (x) ∕= 0, hi (z) depends on z . Obviously, u j affects yi . (Sufciency) Dene

:i = Span{ dLX1 ⋅ ⋅ ⋅ LXs hi (x)∣ Xt ∈ F, t = 1, ⋅ ⋅ ⋅ , s, s < f}. Then there exists an open dense set W of M such that : i is regular at x ∈ W . Then locally for each x ∈ W we can nd a local coordinate chart (V, z) such that } { w ⊥ . : i = Span w z1 Now express the system (9.1) locally as ⎧ m  1 = f 1 (z) +  z  g1k (z)uk  ¦   k=1  ⎨ m

z2 = f 2 (z) + ¦ g2k (z)uk     k=1   ⎩ yi = hi , i = 1, ⋅ ⋅ ⋅ , p. Since dhi ∈ :i , we have

yi = hi (z2 ).

Since :i⊥ is f - and gk -invariant, k = 1, ⋅ ⋅ ⋅ , m and g j ∈ :i⊥ , so z2 = f 2 (z2 ) + The system (9.1) becomes

m

¦

k=1,k∕= j

g2k (z2 )uk .

(9.8)

240

9 Input-Output Structure

⎧ m   z1 = f 1 (z) + ¦ g1k uk    k=1    m   ⎨z2 = f 2 (z2 ) + ¦ g2k (z2 )uk k=1,k∕= j    2  yi = hi (z )        ⎩y = h , ℓ ∕= i. ℓ ℓ

(9.9)

It is obvious that u j does not affect yi . Formally we have

w hi = 0, wuj

x ∈ W.

(9.10)

For system (9.1) we can formally write the input-output mapping as hi = hi (x0 , u1 , ⋅ ⋅ ⋅ , um ). Now if u j affects yi , then there exists at least one point p ∈ M such that  w hi  ∕= 0. w u j p Then by continuity, there exists a neighborhood of p such that the partial derivative is non-zero. But this contradicts (9.10). ⊔ ⊓

9.2 Morgan’s Problem The local Morgan’s problem, which is also called the local input-output decoupling problem, is dened as the follows. Denition 9.2. The local Morgan’s (or input-output decoupling) problem of system (9.1) at x0 ∈ M is to nd a neighborhoodU of x0 and a regular feedback (D (x), E (x)), such that for the local feedback system ⎧ m  ⎨x = f˜ + ¦ g˜ j v j , x ∈ U, j=1  ⎩y = h (x), j j

j = 1, ⋅ ⋅ ⋅ , p,

there is a partition v = (v1 , ⋅ ⋅ ⋅ , v p ), such that each vi controls hi and does not affect h j , j ∕= i, as long as x ∈ U. It is obvious that Morgan’s problem has solution only if m ⩾ p. We of course can consider the global Morgan’s problem. But here we restrict ourself on local solution, because we do not know much about the global decoupling problem.

9.2 Morgan’s Problem

241

Lemma 9.2. Assume the relative degree vector is well dened at x0 . Then locally the differentials of following functions are linearly independent: } { U −1 U −1 dh1 , ⋅ ⋅ ⋅ , dL f 1 h1 , ⋅ ⋅ ⋅ , dh p , ⋅ ⋅ ⋅ , dL f p h p . Proof. Let

p Ui −1

Z=¦

¦ Oik dLkf hi .

i=1 k=0

and assume Z (x0 ) = 0. Now ⟨Z , g⟩ = (O1(U1 −1) , ⋅ ⋅ ⋅ , O p(U p −1) )D(x0 ) = 0. Since D(x0 ) has full row rank, Oi(Ui −1) = 0, i = 1, ⋅ ⋅ ⋅ , p. Then we consider 〉 〈 〉 〈 Z , ad f g = L f ⟨Z , g⟩ − L f Z , g = −(O1(U1 −2) , ⋅ ⋅ ⋅ , O p(U p −2) )A(x0 ) = 0. So Oi(Ui −2) = 0, i = 1, ⋅ ⋅ ⋅ , p. Note that if Uℓ = 1, then in the above equation there is no Ot(Uℓ −2) . Correspondingly, the ℓ-th row of D(x0 ) will also be deleted. Since the remaining rows of D(x0 ) are still linearly independent, this set of coefcients are still zero. Continuing in this way, we can nally show that all Oi j = 0, which completes the proof. ⊔ ⊓ Theorem 9.1. Assume the relative degree vector is well dened at x0 . The local input-output decoupling problem at x0 is solvable, if and only if the decoupling matrix has rank p at x0 . Proof. (Necessity) Assume the problem is solved by a regular feedback (D (x), E (x)). Since now the block of control, v j , does not affect yi , j ∕= i, we have that Lg˜ j Lkf hi (x) = 0,

j ∕= i, k ⩾ 0, x ∈ U.

Using it the decoupling matrix with respect to this feedback becomes ⎤ ⎡ D˜ 1 0 ⋅ ⋅ ⋅ 0 ⎢ 0 D˜ 2 ⋅ ⋅ ⋅ 0 ⎥ ⎥ ⎢ ˜ D(x) =⎢ . . .. ⎥ . ⎣ .. .. . ⎦ ˜ 0 0 ⋅ ⋅ ⋅ Dp Since the relative degree vector is well dened, each D˜ i (x0 ) should have rank 1. Hence rank(D(x0 )) = p. (Sufciency) By assumption there exists a neighborhood U of x0 , such that rank(D(x)) = p,

x ∈ U.

Without loss generality, we can express D(x) = (D1 (x), D2 (x)),

242

9 Input-Output Structure

and assume D1 (x) is non-singular. Then by column elimination we can nd a nonsingular E1 (x) such that D(x)E1 (x) = (D1 (x), 0). Let

[ −1 ] D (x) 0 E2 (x) = 1 , 0 Im−p

and

E (x) = E1 (x)E2 (x). Then E (x) is non-singular and D(x)E (x) = (Ir , 0). Set

] (x)b(x) D−1 1 , D (x) = −E1 (x) 0 [

then D(x)D (x) = [D1 (x)

0]

[ ] −D−1 1 (x)b(x) = −b(x). 0

˜ That is, b(x) = 0. Under this feedback control, we have { U −1 Lg˜ j L f˜i hi (x) = Gi j , U

L f˜i hi (x) = 0.

(9.11)

Using (9.11), we claim that Lg˜ j Lkf˜ hi (x) = 0,

k ⩾ 0, j ∕= i.

(9.12)

When k < Ui − 1, it follows from the denition of the relative degree vector. As for k = Ui − 1, it is shown in the rst equation of (9.11). Finally, for k > Ui − 1, it follows from the second equation of (9.11). Using Lemma 9.1, it is obvious that vi does not affect y j , j ∕= i. Finally, we have to prove that ui can control yi . Using Lemma 9.2, we may choose zij = L jf hi , and

i = 1, ⋅ ⋅ ⋅ , p, j = 0, ⋅ ⋅ ⋅ , Ui − 1, p

z0ℓ ,

ℓ = 1, 2, ⋅ ⋅ ⋅ , n − ¦ Ui , i=1

such that {z , z , ⋅ ⋅ ⋅ , z } becomes a local coordinate frame. It is easy to see that under this coordinate frame the feedback system can be expressed as 0

1

p

9.3 Invertibility

⎧⎧  zi1 = zi2        .  ⎨.   .   ⎨ i  zUi −1 = ziUi   ⎩  zi = v , i = 1, ⋅ ⋅ ⋅ , p,  i  Ui   0 = f 0 (z) + J 0 (z)v,   z    ⎩ yi = zi1 , i = 1, ⋅ ⋅ ⋅ , p. From (9.13) it is clear that vi controls yi .

243

(9.13)

⊔ ⊓

For system (9.1), a state feedback control u = D (x) + E (x)v is called non-regular feedback if E (x) is singular, or E (x) is of dimension m × k, k < m. Consider the Morgan’s problem again. If a non-regular feedback is allowed, the full row rank condition in Theorem 9.1 becomes sufcient only. Looking for a necessary and sufcient condition becomes a long standing open problem even if for linear case[2].

9.3 Invertibility Roughly speaking, a control system is said to be invertible if its inputs can be uniquely determined by its outputs. It is obvious that if two sets of inputs are different only over a zero measure set they will cause same outputs. To avoid the problem caused by zero-measure differences, in this section we assume system (9.1) is analytic, and the admissible controls are also analytic. Denition 9.3. 1. A point x0 ∈ M is called an output regular point, if the relative degree vector is well dened at x0 . 2. Assume system (9.1) is analytic. It is said to be invertible at x0 , if for any u1 ∕= u2 y(⋅, u1 , x0 ) ∕= y(⋅, u2 , x0 ). If there exists a neighborhood U of x0 such that Lg j Lkf hi (x) = 0,

x ∈ U, j = 1, ⋅ ⋅ ⋅ , m, k ⩾ 0,

then we denote Ui (x0 ) = f. In this case, x0 is also considered as an output regular point. It is easy to prove that for an analytic control system, the set of output regular points is an open and dense subset of the state space. The following proposition can be veried directly. It is also an immediate consequence of Fliess functional expansion, which will be discussed later. Proposition 9.2. Assume system (9.1) is an analytic system and x0 is an output regular point. Then yi = hi (x) is independent of the inputs, if and only if Ui (x0 ) = f. We rst consider the invertibility of single-input case.

244

9 Input-Output Structure

Theorem 9.2. Assume system (9.1) is analytic with m = 1. It is invertible at x0 , if and only if its strong accessibility Lie algebra satises Lsa (x0 ) ∕⊂ Ker h(x0 ).

(9.14)

Proof. (Necessity) Recall that { } Lsa (x) = adkf g(x)∣k ⩾ 0

LA

According to Proposition 9.2, it sufces to prove that U (x0 ) = f. To see that we provide a formula, called the binomial expansion of higher order Lie derivatives. k

Ladk g h = ¦ (−1)i f

i=0

( ) k k−i L f Lg Lif h. i

(9.15)

Using mathematical induction, (9.15) can be easily veried. Using (9.15), we can prove the equivalence that Ladk g h(x0 ) = 0,

k ⩾ 0,

(9.16)

Lg Lkf h(x0 ) = 0,

k ⩾ 0.

(9.17)

f

if and only if

Note that (9.16) means Lsa (x0 ) ⊂ Ker h(x0 ), and (9.17) means U (x0 ) = f, the conclusion follows. (Sufciency) If (9.14) is true, then U (x0 ) < f. Otherwise, say, U (x0 ) = f. By the output non-singularity at x0 we know that there exists a neighborhood U of x0 such that ⎡ ⎤ Lg Lkf h1 (x) ⎢ ⎥ .. Lg Lkf h(x) = ⎣ ⎦ = 0, x ∈ U, k ⩾ 0. . k Lg L f h p (x) According to (9.15), it is equivalent to Ladk g h(x) = 0, f

x ∈ U, k ⩾ 0,

which means {adkf g∣k ⩾ 0} ⊂ Ker(h). Now since Ker(h) = (Span{dh1 , ⋅ ⋅ ⋅ , dh p })⊥ is involutive, we have Lsa (x) = {adkf g(x)∣k ⩾ 0}LA ⊂ Ker h(x),

x ∈ U.

This contradicts to the assumption. Now we can assume

U (x0 ) = t + 1 < f. By non-singularity of x0 there exists a neighborhood U of x0 such that

9.3 Invertibility

Lg Lkf h(x) = 0,

k < t, x ∈ U.

245

(9.18)

Now assume u1 ∕= u2 . Since they are analytic, there exists (k)

(k)

s = min{k∣u1 (0) ∕= u2 (0)} < f. Next, we compare y(⋅, u1 , x0 ) with y(⋅, u2 , x0 ). We claim that y(k) (x) = Lkf h(x),

k ⩽ t, x ∈ U.

(9.19)

We prove the claim by mathematical induction. It is trivial for k = 0. Assume it is true for k = q < t. Then from (9.18) we have 〈 〉 h + LgL pf hu = L p+1 h, y(p+1) = dL pf h, f + gu = L p+1 f f which proves (9.19). Similarly, we can prove the following equality by mathematical induction: For 0 ⩽ k ⩽ s we have y(t+k+1) (x) = ¦ aℓ (x) ⋅ mℓ (u, u,  ⋅ ⋅ ⋅ , u(k−1) ) + Lg Ltf hu(k)(x),

x ∈ U,

(9.20)

ℓ

where aℓ (x) is an r dimensional analytic vector, mℓ (u, u,  ⋅ ⋅ ⋅ , u(k−1) ) is a monomial (k−1) of u, u,  ⋅⋅⋅, u . Using (9.20), one sees that ( ) (s) (s) y(t+s) (0, u1 , x0 ) − y(t+s) (0, u2 , x0 ) = Lg Ltf h(x0 ) u1 (0) − (u2 (0) ∕= 0. Hence y(0, u1 , x0 ) ∕= y(0, u2 , x0 ).

⊔ ⊓

From the proof of Theorem 9.2 one sees that if x0 is an output regular point and the system is invertible at x0 , then there exists a neighborhood U of x0 such that the system is invertible at every point x ∈ U. Next, we consider multiple input case. x0 is called an output normal point, if it is output regular point, and the largest controllability distribution contained in the kernel of output Ker(y) is also non-singular at x0 . It is also easy to show that for an analytic system the set of output normal points is open dense in the state space. For multiple input case we have the following result. Theorem 9.3. Assume system (9.1) is analytic and x0 is an output normal point. Then (9.1) is invertible at x0 , if and only if there exists a neighborhood U of x0 such that for each x ∈ U the largest controllability distribution contained in Ker(h) is zero. Proof. (Necessity) Assume there exists a neighborhood U of x0 , a feedback control u = D (x) + E (x)v, and ∅ ∕= / ⊂ {1, 2, ⋅ ⋅ ⋅ , m}, such that 〈 〉 f˜, g˜ ∣ g˜/ ⊂ Ker(h), (9.21) where f˜ = f + gD and g˜ = gE . Choose v1 = (v11 , ⋅ ⋅ ⋅ , v1m ) and v2 = (v21 , ⋅ ⋅ ⋅ , v2m ) in such a way that

246

9 Input-Output Structure

v1s = v2s = 0,

s ∕∈ / ;

v1s ∕= v2s ,

s ∈ /.

Then v1 ∕= v2 . Setting ui = D (x) + E (x)vi ,

i = 1, 2,

then u1 ∕= u2 . Corresponding to these two inputs, the feedback system degenerated to ⎧ ⎨x = f˜ + ¦ g˜i vi , i∈/ (9.22) ⎩y = h(x). Now (9.21) implies that for subsystem (9.22) U (x) = f, x ∈ U. Hence y(⋅, u1 , x0 ) = y(⋅, u2 , x0 ). (Sufciency) Assume u1 ∕= u2 , but y(⋅, u1 , x0 ) = y(⋅, u2 , x0 ). Since u1 ∕= u2 , we can assume u1 ∕≡ 0. Then for H > 0 small enough, we dene H1 = {x(t, u1 , x0 ) ∣t ∈ [0, H )}, which is a one dimensional sub-manifold of M isomorphic to [0, H ). Hence, for each t ∈ [0, H ) there exists a q ∈ H1 , such that x(t) = q. Dene a mapping a1i (q) = u1i (t),

i = 1, 2, ⋅ ⋅ ⋅ , m,

which is an analytic mapping: H1 → ℝm . Since H1 is a sub-manifold of M, we can extend a1 = (a11 , ⋅ ⋅ ⋅ , a1m ) analytically to a neighborhood of H1 in M, and denote the extended mapping by D 1 (x). Now we dene m

f˜1 = f + ¦ gi (x)Di1 (x),

(9.23)

i=1

Since

Di1 (x)∣H1 = a1i (q) = u1i (t), H1 is an integral curve of (9.23) through x0 . Next, we consider u2 . Assume u2 ∕≡ 0. Then by a similar way, we can construct H2 , and D 2 , such that H2 is an integral curve of m

f˜2 = f + ¦ gi (x)Di2 (x).

(9.24)

i=1

If u2 ≡ 0, we set D 2 (x) ≡ 0. Finally, we dene a system m

m

i=1

i=1

x = f + ¦ gi (x)Di1 (x) + ¦ gi (x)[Di2 (x) − Di1 (x)]v.

(9.25)

9.4 Decoupling via Dynamic Feedback

247

Case 1. We assume D 1 (x0 ) ∕= D 2 (x0 ). Then we can construct a nonsingular matrix E (x), x ∈ U, which has D 2 (x) − D 1 (x) as its rst column. Denote f˜(x) = f (x) + g(x)D 1 (x),

g(x) ˜ = g(x)E (x),

x ∈ U.

Then equation (9.25) becomes x = f˜ + g˜ 1 v.

(9.26)

Since system (9.26) is not invertible, according to Theorem 9.2 } { Lsa (x) = adkf˜ g˜1 ⊂ Ker(h)(x), x ∈ U. LA

Since Lsa (x) is a weak ( f , g)-invariant distribution and g˜1 ∈ G ∩ Lsa , then the largest controllability distribution contained in Lsa is not zero. Hence the largest controllability distribution contained in Ker(h) is not zero. Case 2. We consider D 1 (x0 ) = D 2 (x0 ). Then u1 (0) = u2 (0). Since u1 ∕= u2 , for any H > 0 there exists t0 ∈ (0, H ) such that u1 (t0 ) ∕= u2 (t0 ). Replacing x0 in Case 1 by x˜ = x(t0 , u1 , x0 ), same arguments as in Case 1 show that on a neighborhood of x˜ there is a non-zero controllability distribution contained in Ker(h). Let H → 0, then x˜ → x0 . Since x0 is an output normal point, then there is a neighborhood of x0 , which has a non-zero controllability distribution. ⊔ ⊓

9.4 Decoupling via Dynamic Feedback This section considers the Morgan’s problem again. Consider system (9.1) again. For statement ease, assume m = p. It is proved in Section 9.2 that the Morgan’s problem is solvable, if and only if the decoupling matrix is non-singular. But when the decoupling matrix is singular, we may try a new method to solve it. This method is called the dynamic feedback. We rst describe this method. For system (9.1), we may rst nd a nonsingular feedback matrix E (x), such that g(x)E (x) = g˜ = (g˜1 , ⋅ ⋅ ⋅ , g˜m ). Then we choose a subset of indexes as / ⊂ {1, 2, ⋅ ⋅ ⋅ , m}. Finally, we add one integrator to each control with the chosen index. Then the dynamic feedback closedloop system becomes ⎧ x = f (x) + ¦ g˜i wi + ¦ g˜i ui   ⎨ i∈/ i∈/ c (9.27) w i = vi , i ∈ /   ⎩ y j = h j (x), j = 1, ⋅ ⋅ ⋅ , m. The sub-system w i = vi ,

i∈/

(9.28)

248

9 Input-Output Structure

is called the dynamic compensator. A general dynamic compensator can be expressed as w = [ (x, w) +

¦ K j (x, w)v j .

(9.29)

j∈/

In this section we use only the special dynamic feedback (9.28). Using it, we have the following result for solving Morgan’s problem via dynamic compensators. Theorem 9.4. Assume system (9.1) is output regular and invertible at x0 , with p = m. Then the Morgan’s problem is solvable by dynamic compensators. The proof of this main result is long. We start from some preparations. Denote by Vi the largest ( f , g)-invariant distribution contained in Ker(hi ) and V the largest ( f , g)-invariant distribution contained in Ker(h). We rst prove some lemmas. Lemma 9.3. Consider system (9.1). Assume rank(D(x0 )) = p. Then the largest ( f , g)-invariant distribution contained in Ker(h) is U −1

i (Span{dLkf hi })⊥ . V = ∩i=1 ∩k=0

p

(9.30)

Proof. First, we prove V is the ( f , g)-invariant distribution contained in Ker(h). Since rank(D(x0 )) = p, in a neighborhood U of x0 there exists a feedback (D (x), E (x)), x ∈ U, such that for the feedback system f˜(x) = f (x) + g(x)D (x), g(x) ˜ = g(x)E (x) the decoupling matrix becomes { ˜ D(x) = D(x)E (x) = (I p , 0) (9.31) ˜ b(x) = D(x)D (x) + b(x) = 0. Hence, we have

{ Lg˜j Lsf˜hi

and

= U

1, 0,

L f˜i hi = 0,

j = i, s = Ui − 1, otherwise, i = 1, ⋅ ⋅ ⋅ , p.

Now choose any X ∈ V , and let k ⩽ Ui . Then 〉 〈 dLkf h, [ f˜, X ] = dL f˜LX Lkf hi − dLX L f˜Lkf hi = 0 − dLX Lk+1 hi = 0. f˜ Similarly,

〈

〉 dLkf h, [g˜ j , X ] = dLg˜ j LX Lkf hi − dLX Lg˜ j Lkf hi = 0.

Above two equations show that [ f˜,V ] ⊂ V and [g˜ j ,V ] ⊂ V , j = 1, ⋅ ⋅ ⋅ , m. Next, we show that V is the largest one. Assume D is another ( f , g)-invariant distribution contained in Ker(h). Then there exists an D (x), x ∈ U, such that f˜(x) = f (x) + g(x)D (x) satises [ f˜, D] ⊂ D. We claim that

9.4 Decoupling via Dynamic Feedback

D ⊂ Ker(Lkf hi ),

k ⩽ Ui − 1, i = 1, ⋅ ⋅ ⋅ , p.

249

(9.32)

We prove the claim by induction. It is obviously true for k = 0. Assume (9.32) is true for k = r < Ui − 1. Choosing any X ∈ D, then 〉 〈 〉 〈 r h , X = L dL h , X dLr+1 i i f f f 〈 〉 〈 〉 = L f dLrf hi , X − dLrf hi , ad f X = 0. That is, X ∈ Ker(Lr+1 f hi ). It follows that U −1

i D ⊂ ∩i=1 ∩k=1 (Span{dLkf hi })⊥ .

p

⊔ ⊓

Corollary 9.2. Assume rank(D(x0 )) = p. Then the largest ( f , g)-invariant distribution contained in Ker(hi ) is U −1

i (Span{dLkf hi })⊥ , Vi = ∩k=0

Proof. Consider the single output system ⎧  ⎨x = f (x) +

i = 1, ⋅ ⋅ ⋅ , p.

(9.33)

m

¦ gi (x)ui

i=1  ⎩y = h (x). j j

(9.34)

Since for the original system rank(D(x0 )) = p, for system (9.34) the decoupling matrix, denoted by D j (x), satises rank(D j (x0 )) = 1. Using Lemma 9.3 to system (9.34), the conclusion follows.

⊔ ⊓

Lemma 9.4. Consider system (9.1). Assume m = p, x0 is an output regular point and system is invertible at x0 . Then the following two conditions are equivalent. (i). V = ∩m i=1Vi , (ii). rank(D(x0 )) = m. Proof. (ii)⇒ (i): It follows from Lemma 9.3 and Coronary 9.2 immediately. (i)⇒ (ii): Denote by G := Span{gi ∣i = 1, ⋅ ⋅ ⋅ , m}. From (i) we have U −1

m k ⊥ i G ∩V = G ∩ ∩m i=1Vi = G ∩ ∩i=1 ∩k=0 (Span{dL f hi }) .

From the denition we know that G ⊂ (Span{dLkf hi })⊥ , It follows that

k < Ui − 1.

250

9 Input-Output Structure U −1

i G ∩V = G ∩m i=1 (Span{dL f

Dene

hi })⊥ .

⎤ U −1 dL f 1 h1 ⎥ ⎢ .. ⎥. H := ⎢ . ⎦ ⎣ U −1 dL f m hm ⎡

We claim that U −1

i G ∩ ∩m i=1 (Span{dL f

hi })⊥ = Span{gD ∣D ∈ Ker(D)}.

(9.35)

To prove it, let X = gD , with D ∈ Ker(D). Then X ∈ G. Moreover, HX = DD = 0. That is,

U −1

i X = gD ∈ ∩m i=1 (Span{dL f

hi })⊥ .

U −1

i Conversely, assume X ∈ G ∩ ∩m hi })⊥ . Then there exists an D such i=1 (Span{dL f that X = gD , and HgD = 0. That is, D ∈ Ker(D). Therefore, (9.35) is proved. Since the system is invertible at x0 , according to Theorem 9.3, the largest controllability sub-manifold contained in Ker(h) is zero. It follows that G ∩ V = {0}. Hence for any gD , D ∈ Ker(D), we have gD = 0. Since rank(G) = m, it follows that D = 0. That is, Ker(D) = 0. We, therefore, have

rank(D(x0 )) = m.

⊔ ⊓

Note that condition (ii) in Lemma 9.4 is a necessary and sufcient condition for the solvability of Morgan’s problem. Now since condition (ii) is equivalent to condition (i), condition (i) is also necessary and sufcient condition for the solvability. This is the key for proving Theorem 9.4. To complete the proof, we need the following algorithm, which provides a constructive procedure for solving Morgan’s problem via dynamic compensators. Algorithm 9.1. Step 1. Compute the relative degree vector and construct the decoupling matrix D(x). If rank(D(x0 )) = m, stop. Step 2. Assume rank(D(x0 )) = s < m. Construct a non-singular matrix E (x), such that on a neighborhood U of x0 we have D(x)E (x) = (D1 (x), 0),

x ∈ U,

where D1 (x) is an m × s matrix with full column rank. It follows that there are r1 , r2 , ⋅ ⋅ ⋅ , rs rows, such that the s × s minor of D1 (x) composed by its r1 , r2 , ⋅ ⋅ ⋅ , rs rows is a nonsingular diagonal matrix. Step 3. Set g(x) ˜ = g(x)E (x). Assume in D1 (x) there are q columns, say, i1 , i2 , ⋅ ⋅ ⋅ , iq , which contain more than one non-zero elements. Denote by / = {i1 , i2 , ⋅ ⋅ ⋅ , iq }, and / c = {1, 2, ⋅ ⋅ ⋅ , m}∖{i1, i2 , ⋅ ⋅ ⋅ , iq }. Construct a dynamically expanded system as

9.4 Decoupling via Dynamic Feedback

⎧ x = f (x) +   ⎨

251

¦ g˜ j w j + ¦c g˜ j v j

j∈/

j∈/

w j = v j , j ∈ / ,   ⎩ yk = hk (x), k = 1, 2, ⋅ ⋅ ⋅ , m.

(9.36)

Note that since s < m, / ∕= ∅. Otherwise, the output yk , k ∕∈ {r1 , r2 , ⋅ ⋅ ⋅ , rs } has Uk = f. That is, it is not affected by inputs. It contradicts to the invertibility. Step 4. Replacing the original system by the system obtained in (9.36), go back to Step 1. Theorem 9.5. Under the conditions of Theorem 9.4, after nite iterations Algorithm 9.1 will be terminated. That is, the dynamically extended system will have nonsingular decoupling matrix. Proof. Consider the dynamically extended system (9.36), where ⎡ ⎤ ([ ] [ ]) f + ¦ g˜ j w j g ˜ 0 j ⎦ , ge = j∈/ , . fe = ⎣ I 0 q j∈/ 0 We calculate the relative degree vector Uie in three cases. Case 1. i = rk , and k ∈ / c . Then by the structure of f e we have dLsf e hi = (dLsf hi , 0),

s = 0, 1, ⋅ ⋅ ⋅ , Ui − 1.

(9.37)

Hence when s < Ui − 1, we have Lge Lsf e hi = 0,

s < Ui − 1.

(9.38)

Since i = rk and k ∈ / c , From the structure of D(x)E (x) we know that U −1

U −1

Lgek L f ie hi = Lg˜k L f i

hi ∕= 0.

Hence Uie = Ui . Recall the proof of Lemma 9.3, it has been proved that if D is an ( f , g)-invariant distribution contained in Ker(hi ) then U −1

i (Span{dLkf hi })⊥ . D ⊂ ∩k=0

(9.39)

Since in the proof of (9.39), we did not use the condition that rank(D(x0 )) = m, (9.39) is independent of rank(D(x0 )). Now for the extended system denote

: ie = Span{dLkf e hi ∣k = 1, 2, ⋅ ⋅ ⋅ , Ui − 1}. Then

Vie ⊂ (: ie )⊥

Moreover, it is easy to see that dim(: ie ) = Uie − 1 = Ui − 1. Case 2. i = rk , k ∈ / . It is easy to check that the equations (9.37) and (9.38) remain true. Next, we calculate the Lie derivative for s = Ui − 1. Using (9.37), we have

252

9 Input-Output Structure

〈 U −1 Lge L f ie hi

U −1 (dL f i hi , 0),

=

U −1

= ((Lg˜ j L f i

([ ] [ ])〉 g˜ j 0 , 0 j∈/ c Iq

hi ) j∈/ c , 0).

Since j ∈ / c , according to the structure of D(x)E (x), at j-th column there is only the ri position which has non-zero element. But now since i = rk , k ∈ / , hence k ∕= j. It follows that U −1

(Lg˜ j L f i

hi ) j∈/ c = 0.

(9.40)

Now we consider s = Ui . We have U

U

L f ie hi = L f i hi + Then

U −1

¦ Lg˜ j L f i

j∈/

hi ⋅ w j . )

( U dL f ie hi

U = (dL f i hi , 0) +

It follows that

¦

j∈/

U −1 U d(Lg˜ j L f i hi ) ⋅ w j , (Lg˜ j L f i hi ) j∈/

〈 U Lge L f ie hi

= =

U (dL f i hi , 0),

([ ] [ ])〉 g˜ j 0 , 0 j∈/ c Iq

.

(9.41)

U −1 (∗, (Lg˜ j L f i hi ) j∈/ ).

Note that since i = rk , and k ∈ / , we have Lg˜k LUf i −1 hi ∕= 0, which is a column in the right half block of (9.41). Hence U

Lge L f ie hi ∕= 0. It follows that Uie = Ui + 1, that is, dim(:ie ) = Uie = Ui + 1. Case 3. i ∕∈ {r1 , ⋅ ⋅ ⋅ , rs }. This case is similar to Case 2. First, because i ∕∈ {r1 , ⋅ ⋅ ⋅ , rs }, so i ∕= ri . Then (9.40) holds. Next, consider (9.41). Since i ∕∈ {r1 , ⋅ ⋅ ⋅ , rs } and in i-th row there is at least one non-zero element. Assume the (i, j) element is non-zero. Then j ∈ / , because in the j-th column the r j -th element is non-zero (i ∕= r j ). Hence the right hand side of (9.41) contains non-zero element. Hence U

Lge L f ie hi ∕= 0. Then

dim(:ie ) = Uie = Ui + 1.

Summarizing the above three cases, one sees that for 1 ⩽ i ⩽ m there are s − q indexes belong to Case 1 and m − s + q indexes belong to Case 2 or 3. It is easy to see that :ie , i = 1, ⋅ ⋅ ⋅ , m are linearly independent. Hence

9.5 Normal Form of Nonlinear Control Systems m

m

i=1

i=1

253

dim ¦ :ie = dim ¦ :i + m − s + q. Calculating its complement yields )⊥ ( m

dim

¦ :ie

e ⊥ = dim ∩m i=1 (:i )

i=1

( = n + q − dim

¦ :ie

i=1

(

(

m

= n + q − dim ( =

m

)

m

)

)

¦ :i

) +m−s+q

i=1

n − ¦ :i + m − s (

i=1

) ⊥ +m−s = dim ∩m i=1 (: i ) Since m − s > 0, we have e ⊥ m ⊥ dim ∩m i=1 (:i ) > dim ∩i=1 (: i ) .

(9.42)

Inequality (9.42) shows that after each iteration (adding dynamic compensators) the e ⊥ dimension of ∩m i=1 (: i ) has been reduced at least by one. Since e m e ⊥ ∩m i=1Vi ⊂ ∩i=1 (: i ) . e Hence as long as the iteration goes on, the dim ∩m i=1Vi will decrease. Though, it may not be reduced in each iteration. Observing the fact that e V e ⊂ ∩m i=1Vi , e e according to Lemma 9.4, as long as V e ∕= ∩m i=1Vi , we have rank(D (x0 )) < m. Hence the iteration is continuing. We, therefore, can nally reach the case that

V e = ∩ri=1Vie . This means rank(De (x0 )) = m. Finally, to apply Lemma 9.4 to the extended system, we have to show that if the original system is invertible at x0 , the extended system is also invertible at x0 . This fact can be seen immediately by the construction of the extended system. ⊔ ⊓ Note that Algorithm 9.1 and Theorem 9.5 form a constructive proof of Theorem 9.4.

9.5 Normal Form of Nonlinear Control Systems In this section we provide a normal form for nonlinear systems. First, we consider the single-input single-output case.

254

9 Input-Output Structure

Proposition 9.3. Consider a single-input single-output system { x = f (x) + g(x)u, x ∈ M y = h(x).

(9.43)

If the relative degree U is dened at x0 ∈ M, then there exists a neighborhood U of x0 , such that the system can be expressed locally as ⎧ z1 = z2      z2 = z3     .   ⎨.. (9.44) zU −1 = zU    zU = b([ ) + d([ )u      w = I ([ )   ⎩ y = z1 , [ ∈ U, where [ = (z, w). (9.44) is called the Byrnes-Isidori normal form. Proof. Using Lemma 9.2, we can nd {zi+1 = Lif h ∣ i = 0, ⋅ ⋅ ⋅ , U − 1}, which are linearly independent. In addition to z1 , ⋅ ⋅ ⋅ , zU we can nd n − U functions, w = {w1 , ⋅ ⋅ ⋅ , wn−U }, such that [ = (z, w) becomes a local coordinate frame on some U. Under [ system (9.43) becomes (9.44) with U

b([ ) = L f h(x),

U −1

d([ ) = Lg L f

h(x).

⊔ ⊓

In general, assume p = m, i.e., the input number and the output number are the same, then we call the following form (9.45) the Byrnes-Isidori normal form. ⎧1 z1 = z12      z12 = z13     ..   .      z1U1 −1 = z1U1    m  1   zU1 = b1 ([ ) + ¦ d1i ([ )ui    i=1    ..   ⎨. m (9.45) zm 1 = z2    m m z2 = z3    .  ..      m  zm Um −1 = zUm   m  m   z  = b ( [ ) + dmi ([ )ui m  ¦ U m   i=1     w = I ([ )   ⎩ yi = zi1 , i = 1, ⋅ ⋅ ⋅ , m, [ ∈ U,

9.5 Normal Form of Nonlinear Control Systems

255

m

where [ = (z, w), and w has dimension n − ¦ Ui . i=1

Proposition 9.4. Consider the nonlinear system (9.1) with p = m. Assume G = Span{g1 , ⋅ ⋅ ⋅ , gm } is non-singular and involutive on a neighborhood of x0 . If the relative degree vector (U1 , ⋅ ⋅ ⋅ , Um ) is well dened at x0 ∈ M and the decoupling matrix A(x0 ) is non-singular, then there exists a neighborhood U of x0 , such that the system can be expressed locally as (9.45). j

Proof. Using Lemma 9.2, we can nd {zij+1 = L f hi ∣ i = 1, ⋅ ⋅ ⋅ , m, j = 0, ⋅ ⋅ ⋅ , Ui − 1}, which are linearly independent. Sine G is non-singular and involutive, locally we can choose a suitable coordinate chart such that {( )} Im G = Span . 0 Now the decoupling matrix is ⎛ ⎜ ⎜ D(x) = ⎜ ⎝

U −1

Lf1

h1

.. . U −1

Lfm

⎞ ⎟ ⎟ ⎟ G := H(x)G = (H 1 (x), H 2 (x))G = H 1 (x). ⎠

hm

where H 1 (x) is the rst m × m block of H(x). It is obvious that we can choose n − m functions, {[i } such that ⟨d[i , G⟩ = 0. In fact, we can simply choose the last n − m m

coordinate functions. Choose ℓ = n − ¦ Ui functions {w j ∣ j = 1, ⋅ ⋅ ⋅ , ℓ} from {[i }, i=1

which are linearly independent with {Lkf hi ∣i = 1, ⋅ ⋅ ⋅ , m, k = 0, ⋅ ⋅ ⋅ , Ui − 2}. Then {Lkf hi ∣i = 1, ⋅ ⋅ ⋅ , m, k = 0, ⋅ ⋅ ⋅ , Ui − 1} and {w j ∣ j = 1, ⋅ ⋅ ⋅ , ℓ} form a local coordinate frame. Under this coordinate frame the system (9.1) can be locally expressed as (9.45). ⊔ ⊓ Remark 9.1. In fact the {di j } in (9.45) form the decoupling matrix, and {bi (x)} form the decoupling vector. That is, D(x) = (di j (x)),

b(x) = (b1 (x), ⋅ ⋅ ⋅ , bm (x))T .

As assumed, D(x) is non-singular. Now if in addition, we assume m

¦ Ui = n.

i=1

Then (i). G is automatically involutive; (ii). The system with a feedback control u = (D(x))−1 [v − b(x)], (9.45) becomes a completely controllable and completely observable linear system.

256

9 Input-Output Structure

9.6 Generalized Normal Form The normal form has been proved to be a very useful tool in the synthesis of nonlinear control systems. For instance, a system in normal form and with minimum phase is very convenient for stabilizer design. Even if in non-minimum phase case, it is also convenient for designing center manifold to stabilize the systems. (We refer to [5–7] for details.) But it has some obvious shortages: ∙ The relative degree (vector) is not always well dened because of its regularity requirement. ∙ It may not provide the largest linearizable sub-system. ∙ The zero dynamics may not have minimum phase just because of the output is “improperly chosen”. We give a simple example to describe this: Example 9.1. Consider the following system ⎧ x1 = x2 + x33    ⎨x = x + x2 x 2 3 2 3 2x  x  = x + x 3 4  3 4  ⎩ x4 = u,

(9.46)

Case 1. Set the output as y = h(x) = sin(x2 + x24 ). Then

(9.47)

Lg h(x) = 2x4 cos(x2 + x24 ).

Since Lg h(0) = 0 and there is not a neighborhood, U, of the origin such that Lg h(x) = 0, x ∈ U. So the relative degree is not dened because of the singularity of the Lie derivative Lg h(x) at the origin. Case 2. Set the output as y = x3 .

(9.48)

Since Lg h(x) = 0, L f (x) = x4 + x23 x4 , and Lg L f h(x) = 1 + x33 ∕= 0, ∥x∥ < 1. The relative degree is 2. So we can get a linearizable sub-system of dimension dim = 2. But this is not the largest one. If we choose y = x2 , the relative degree will be 3, then we can get a linearizable sub-system of dimension dim = 3. (Later on we will see that under the classical denition, y = x2 is the best choice. But the new modied denition will do better.) Moreover, consider the output (9.48) again. According to the above calculation, we can follow a standard procedure by choosing a new coordinate frame as z1 = h(x) = x3 ,

z2 = L f h(x) = x4 (1 + x23 ),

z3 = x1 ,

z4 = x 2 ,

and then convert system (9.46) and (9.48) into the Byrnes-Isidori normal form as follows:

9.6 Generalized Normal Form

⎧  z1 = z2       2z1 z22   ⎨z2 = + (1 + z21 )u 1 + z21  z3 = z4 + z31      z4 = z1 (1 + z24)    ⎩y = z . 1 The zero dynamics is

257

(9.49)

{

z3 = z4 z4 = 0.

(9.50)

Obviously, it is not stable at the origin. So the standard technique for stabilizing minimum phase nonlinear control systems is not applicable. This is obviously a weakness of the method because system (9.46) is stabilizable by linear state feedback. The main purpose of this section is to overcome the rst and the second shortages as much as possible. Roughly speaking, since the restrictions of converting a nonlinear control system into the normal form are rigorous, to apply the normal form analysis to more general systems, we have to generalize the normal form to include as many systems as possible . In Chapter 11 we will see that the generalized normal form is useful in design of the stabilizers for even non-minimum phase cases. We rst dene essential and point relative degrees, which are generalizations of relative degree. We start with SISO systems. Recall the denition of relative degree, it is obvious that the relative degree depends on the output. When we are only interested in the state equations, then for the state equation of (9.1) with m = 1, rewrite it as x = f (x) + g(x)u,

x ∈ ℝn .

(9.51)

We may look for an auxiliary output h(x) such that the relative degree for the state equation with respect to this h(x) can be the maximum one. Denition 9.4. The essential relative degree of (9.51) is the largest relative degree related to an arbitrary chosen smooth output function. It is obvious that the essential relative degree is closely related to the largest feedback linearizable sub-system. Denote by

'i = Span{g, ad f g, ⋅ ⋅ ⋅ , adi−1 f g},

i = 1, 2, ⋅ ⋅ ⋅ .

(9.52)

The following result is from [5] with slightly different statement: Proposition 9.5. [5] The essential relative degree U e is the largest k, such that on some open neighborhood U of 0, { dim {' k (0)}LA = n (9.53) dim {' k−1 (x)}LA = l ⩽ n − 1, x ∈ U,

258

9 Input-Output Structure

where {'i (x)}LA is the Lie-algebra generated by 'i (x). From differential geometry [1] we know that since g(0) ∕= 0, '1 is always nonsingular and involutive locally. That means, the essential relative degree U e of the system (9.51) is always greater than or equal to 1. Consider ( f , g) ∈ V (ℝn ) × V (ℝn ), where V (ℝn ) is the space of Cf vector elds on ℝn . Now if we use Whitney Cf topology to V (ℝn ) [4], it is easy to see that the set of ( f , g) satisfying (9.53) is a zero measure set. That means only a zero measure set of SISO systems have non-trivial essential relative degree. (But we should not be too pessimistic, because many practically useful dynamic systems do have non-trivial essential relative degree.) We give the following denition about the point relative degree. Denition 9.5. 1. For system (9.1) with m = 1, the point relative degree U p at origin is dened as { Lg Lkf h(0) = 0, k < U p − 1, (9.54) U p −1 h(0) ∕= 0. Lg L f 2. For system (9.1) the essential point relative degree is the point relative degree for an auxiliary output h(x) such that the point relative degree for (9.51) with respect to this h(x), denoted by U ep , is the maximum one. We give a simple example to describe these concepts. Example 9.2. Recall system (9.46). Assume the output is y = x4 ex1 .

(9.55)

According to the denitions, a straightforward computation shows that the different relative degrees at the origin are as follows: 1. Since Lg y(0) ∕= 0, the relative degree is U = 1. 2. Since Span{g, ad f g} is non-singular and involutive, and '3 is not involutive and dim{'3 }LA = 4, the essential relative degree is U e = 3. One of the corresponding h is h(x) = x2 . 3. The point relative degree is also U p = 1. In fact, it follows from the denition that if the relative degree is well dened at a point, then the point relative degree should be the same as the relative degree. To tell the difference between the relative degree and the point relative degree, we assume now the output is y = sin(x2 + x24 ).

(9.56)

Then it is ready to verify that with this output the relative degree of the composed system of (9.46) and (9.56) is not dened. But the point relative degree is U p = 3. 4. Now if we choose h(x) = x1 , it is easy to check that the essential point relative degree for the composed system of (9.46) with this output is U ep = 4.

9.6 Generalized Normal Form

259

Next, we consider MIMO case. Denote the state equation of (9.1) as x = f (x) + g(x)u,

x ∈ ℝn , u ∈ ℝm .

(9.57)

Denition 9.6. 1. For system (9.1) the point relative degree vector (U1 , ⋅ ⋅ ⋅ , Um ) is dened as ⎧ ⎨ Lg Lkf hi (0) = 0, k < Ui − 1, (9.58) ⎩ Lg LUi −1 h(0) ∕= 0, i = 1, ⋅ ⋅ ⋅ , m. f 2. The essential relative degree vector, U e = (U1e , ⋅ ⋅ ⋅ , Ume )(the essential point relative degree vector, U ep = (U1ep , ⋅ ⋅ ⋅ , Umep )) , for the state equation (9.57) is dened as the largest one of relative degree (resp. point relative degree), U ∗ , for all possible e ep auxiliary outputs, which makes the decoupling matrix, AU (resp. AU ), nonsingular. That is, m

∥U ∗e ∥ = ¦ Ui∗e = max{∥U e ∥ ∣ AU is nonsingular}, e

(9.59)

i=1

and m

∥U ∗ep ∥ = ¦ Ui∗ep = max{∥U ep∥ ∣ AU is nonsingular}. ep

(9.60)

i=1

Motivated by the equation (9.51), we may ask whether the system (9.51) has the essential point relative degree U ep (relative degree vector U ep = (U1ep , ⋅ ⋅ ⋅ , Umep )), can we always nd a canonical controllable linear part with dimension U ep (or ∥U ep∥ for multi-input case)? The answer is “yes”. In the following we will prove this. Remark 9.2. Later on, we will show that the essential point relative degree (vector) always exists. But in multi-input case we still don’t know whether the essential (point) relative degree vector is unique. Using point relative degree, we can construct generalized normal form of nonlinear systems. Denition 9.7. For system (9.1) the following form is called the generalized normal form: ) ( ⎧  0   zi = Ai zi + biui + + pi (z, w)u, zi ∈ ℝUi    Di (z, w) ⎨ (9.61) i = 1, ⋅ ⋅ ⋅ , m   r  w = q(z, w), w ∈ ℝ    ⎩ yi = zi1 , i = 1, ⋅ ⋅ ⋅ , m, m

where r + ¦ Ui = n, Di (z, w) are scalars, pi (z, w) are Ui × m matrices, q(z, w) is an i=1

r × 1 vector eld, and (Ai , bi ) are Brunovsky canonical form in ℝUi with the form

260

9 Input-Output Structure

⎡ 0 ⎢ .. ⎢ Ai = ⎢ . ⎣0 0

⎤ 0 .. ⎥ .⎥ ⎥, 0 ⋅ ⋅ ⋅ 1⎦ 0 ⋅⋅⋅ 0 1 ⋅⋅⋅ .. .

⎡ ⎤ 0 ⎢ .. ⎥ ⎢ ⎥ bi = ⎢ . ⎥ , ⎣0⎦ 1

and pi (0, 0) = 0. Comparing (9.61) with (9.45), the only difference between them is that in (9.45) gi = (0, ⋅ ⋅ ⋅ , 0, di (x, z))T , i = 1, ⋅ ⋅ ⋅ , m, and in (9.61) there exist pi (z, w) which are higher degree input channels. The following proposition is essential for SISO normal form. Proposition 9.6. If system (9.1) with m = 1 has point relative degree r = U p at x = 0, then there exists a suitable local coordinate change, which converts system (9.1) into system (9.61). Proof. We rst claim that dh(0), ⋅ ⋅ ⋅ , dLr−1 f h(0) are linearly independent. It can be proved in a similar way as the proof of Lemma 9.2. Now we can choose partial coordinate variables (z1 , ⋅ ⋅ ⋅ , zr ) = (h, L f h, ⋅ ⋅ ⋅ , Lr−1 f h). Then we can locally nd n − r functions w1 , ⋅ ⋅ ⋅ , wn−r such that (z, w) is a set of local coordinate variables. Moreover, since Lg Lr−1 f h(0) ∕= 0, w can be chosen in such a way that Lg wi = 0, i = 1, ⋅ ⋅ ⋅ , n − r. Under this coordinate frame we can, through a straightforward computation, convert the original system into ⎧  z1 = z2 + Lg h(z, w)u    ..    .   ⎨ zr−1 = zr + Lg Lr−2 f h(z, w)u (9.62) r−1 r   zr = L f h + LgL f h(z, w)u      w = q(z, w),   ⎩ y = z1 , which is the generalized normal form (9.61) with m = 1.

⊔ ⊓

An immediate consequence is the following: Proposition 9.7. Assume system (9.51) has an essential point relative degree r = U ep at the origin, then it can be expressed as the state equations of (9.61). We will call the state equations of (9.61) the generalized normal state form. For MIMO case we need the following assumptions: A1. D(0) is invertible; A2. g1 (0), ⋅ ⋅ ⋅ , gm (0) are linearly independent and Span{g(x)} is involutive near the origin. Proposition 9.8. Consider system (9.1).

9.6 Generalized Normal Form

261

1. Assume A1 and A2 and that the system has point relative degree vector U p = (U1 , ⋅ ⋅ ⋅ , Um ). Then there exists a local coordinate frame such that the system can be converted into the generalized normal form (9.61). 2. Assume A2 and that system (9.57) has essential point relative degree vector U ep = (U1 , ⋅ ⋅ ⋅ , Um ). Then there exists a local coordinate frame such that the system can be converted into the generalized normal state form as the state equation of (9.61). Proof. We prove 1. Then 2 is an immediate consequence of 1. U −1 Under assumption A1, we can choose hi , L f hi , ⋅ ⋅ ⋅ , L f i hi , i = 1, ⋅ ⋅ ⋅ , m as part of coordinate variables. Recall A2, G := Span{g(x)} is an involutive distribution of dimension m. Hence, there exist n − m functions [1 , ⋅ ⋅ ⋅ , [n−m , such that G⊥ = Span{d[i ∣ i = 1, ⋅ ⋅ ⋅ , n − m}. Denote

: = Span{dL jf hi ∣ i = 1, ⋅ ⋅ ⋅ , m; j = 0, ⋅ ⋅ ⋅ , Ui − 1}, m

and U = ¦ Ui . Since the decoupling matrix can be expressed as i=1

⎡ ⎢ D(x) = ⎢ ⎣

U −1

dL f 1

h1 (x)

.. .

U −1

dL f m

⎤ ⎥ ⎥ g(x), ⎦

hm (x)

and D(0) is non-singular, it is clear than dim(: ∩ G⊥ ) ⩽ U − m. In fact, since U co-vectors in : are linearly independent, the “⩽” should be “=”. We, therefore, are able to choose (n − m) − (U − m) = n − U := r closed one-forms from G⊥ , which are linearly independent with : . Say, they are dw1 , ⋅ ⋅ ⋅ , dwr . j Now we can use L f hi , i = 1, ⋅ ⋅ ⋅ , m, j = 0, ⋅ ⋅ ⋅ , Ui − 1 and wk , k = 1, ⋅ ⋅ ⋅ , r as a complete set of coordinate variables. Then a straightforward computation shows that under this coordinate frame system (9.1) becomes (9.62). ⊔ ⊓ In fact, it is obvious that the essential relative degree (vector) is much better than the relative degree (vector), because it can provide largest linear part. The problem is the relative degree (vector) can provide the required coordinate variables by using output (outputs) and its (their) Lie derivatives. But to get the required coordinate variables, which convert the system into the canonical form with largest linear part, we have to solve a set of partial differential equations. So it is, in general, practically hardly applicable. One of the signicant advantages of the essential point relative degree is that it is easily computable, and then the related generalized normal form, which contains the largest linear part, can be obtained via straightforward computation. We construct it in the rest of this section.

262

9 Input-Output Structure

For SISO systems, we show an analog to the Proposition 9.5 for essential point relative degree. Proposition 9.9. Let 'i be dened as in (9.52). The essential point relative degree U ep is the largest k, such that dim(' i (0)) = i,

i ⩽ k.

(9.63)

Proof. Assume k is the largest one for (9.63) to be true. Then we can nd a local coordinate frame z such that ( ) ( ) I g(0), ⋅ ⋅ ⋅ , adk−1 g(0) = k . f 0 Now choose h(z) = zk . It follows that Lg h(0) = 0, and Lg L f h(0) = Lad f g h(0) − L f Lg h(0). Since Lg h(0) = 0 and f (0) = 0, then L f Lg h(0) = 0. It follows that Lg L f h(0) = 0. Similarly, Lg Lif h(0) = 0, i < k − 1. We also have Lg Lk−1 f h(0) ∕= 0. As an immediate consequence of the denition, k ⩽ U ep . Conversely, assume the system has the essential point relative degree U ep , then the system has the form as (9.62). Through a straightforward computation one sees easily that (9.63) holds for U ep . Hence k ⩾ U ep . We conclude that k = U ep . ⊔ ⊓ For MIMO system (9.1), we denote A=

wf (0), wx

B = (b1 , ⋅ ⋅ ⋅ , bm ) = g(0).

The linear system x = Ax + Bu

(9.64)

is called the Jacobian linearization system of (9.1). Then a set of linear subspaces can be dened as

'i = Span{B, AB, ⋅ ⋅ ⋅ , Ai−1 B},

i = 1, 2, ⋅ ⋅ ⋅ .

Assume 'k is the controllable subspace of (A, B). We can arrange the bases of ' i , i = 1, ⋅ ⋅ ⋅ , k in Table 9.1 (with k1 = k). Table 9.1 The bases of 'i

'1 b1 b2 .. . bm

'2 Ab1 ⋅ ⋅ ⋅ Ab2 ⋅ ⋅ ⋅ Abm ⋅ ⋅ ⋅ Akm −1 bm

Ak2 −1 b2

'k Ak1 −1 b1

9.6 Generalized Normal Form

263

Where the vectors in the table are linearly independent. Moreover, the vectors in the rst i columns form the basis of ' i . It is very easy to get the table by choosing linearly independent vectors column by column. To assure that k1 ⩾ k2 ⩾ ⋅ ⋅ ⋅ ⩾ km , we may need to reorder gi . Proposition 9.10. For system (9.1), (k1 , ⋅ ⋅ ⋅ , km ) is an essential point relative degree vector. Proof. First we choose a linear transformation to convert the linear part of (9.1), say (A, B), into canonical form as ⎧  xi1 = xi2 + D1i (x) + E1i (x)u    ..    .   ⎨ i xki −1 = xiki + Dki i −1 (x) + Ekii −1 (x)u (9.65) i = D i (x) + E i (x)u   x   ki ki ki     i = 1, ⋅ ⋅ ⋅ , m   ⎩ m+1 = p(x) + q(x)u, x w Di

where D ij (0) = 0, j = 1, ⋅ ⋅ ⋅ , ki , w xkj = 0, j = 1, ⋅ ⋅ ⋅ , ki − 1, k = 1, ⋅ ⋅ ⋅ , m, and E ji (0) = 0, j = 1, ⋅ ⋅ ⋅ , ki − 1, i = 1, ⋅ ⋅ ⋅ , m. Choosing yi = zi1 , it is easy to see that the essential point relative degrees are Ui = ki and the decoupling matrix is non-singular . Hence, m

∥U ep ∥ ⩾ ¦ ki . i=1

Conversely, let U ep = (U1 , ⋅ ⋅ ⋅ , Um ) be a point relative degree vector with a nonsingular decoupling matrix, then from Proposition 9.8, system (9.1) can be expressed as (9.61). Then its Jacobian linearization has a controllable subsystem of dimension ∥U p ∥. But k1 + ⋅ ⋅ ⋅ + km is the dimension of the controllable subspace of the Jacobian linearization of system (9.1), then ∥U p∥ ⩽ k1 + ⋅ ⋅ ⋅ + km . We, therefore, have ∥U ep ∥ = k1 + ⋅ ⋅ ⋅ + km. ⊔ ⊓ Next, we show how to get the pseudo-normal form. For (9.63), we choose x11 , ⋅ ⋅ ⋅ , x1m as the auxiliary outputs to generate part of new coordinates z. Precisely speaking, set U −1 1 x1 ,

z11 = x11 , ⋅ ⋅ ⋅ , z1U1 = L f 1

⋅⋅⋅ ,

U −1 m x1 .

m 1 m zm 1 = xm , ⋅ ⋅ ⋅ , zUm = L f

And then, as in the proof of Proposition 9.8, we can choose r = n − U function [i , i = 1, ⋅ ⋅ ⋅ , r, such that on a neighborhood U of origin, Lg j [i (x) = 0,

x ∈ U,

j = 1, ⋅ ⋅ ⋅ , m, i = 1, ⋅ ⋅ ⋅ , r.

Moreover, (z, [ ) is a new local coordinate frame.

264

9 Input-Output Structure

Then we express the system into this new coordinate frame (z, [ ). It becomes the pseudo-normal form as ⎧  zi1 = zi2 + E1i (z, [ )v    .   ..   ⎨i zUi −1 = ziUi + EUi i−1 (z, [ )v (9.66)   ziUi = ci (z, [ ) + di (z, [ )v      zi ∈ ℝUi , i = 1, ⋅ ⋅ ⋅ , m,   ⎩ [ = q(z, [ ), [ ∈ ℝr where ci (0, 0) = 0, E ji (0, 0) = 0, j = 1, ⋅ ⋅ ⋅ , Ui − 1, i = 1, ⋅ ⋅ ⋅ , m, and q(0, 0) = 0.

9.7 Fliess Functional Expansion For a linear system

{

x = Ax + Bu y = Cx,

the input-output response can be expressed as ) ( ∫t A(t−t0 ) A(t−W ) x0 + e Bu(W )dW . y=C e

(9.67)

t0

Now a natural question is: is there a counterpart of (9.67) for afne nonlinear system (9.1)? The answer is “yes”. Such an expression is called a Fliess’ functional expansion. In the following we simply set t0 = 0. To begin with, we need some notations: to make f (x) and gi (x) formally symmetric, we denote g0 = f , u0 = 1. Secondly, we dene a set of multi-integrals inductively as ∫ t ⎧∫ t  ⎨ d[i = ui (W )dW ∫ 0t

 ⎩

0

0

d[ik+1 d[ik ⋅ ⋅ ⋅ d[i1 =

∫ t 0

uik+1 (W )

∫ W 0

(9.68) d[ik ⋅ ⋅ ⋅ d[i1 ,

k ⩾ 1.

Then we have the following expression. Theorem 9.6. The input-output response of system (9.1) can be expressed as the following. f

y j (t) = h j (x0 ) + ¦

m

¦

k=0 ik ,⋅⋅⋅ ,i0 =0

Lgi0 ⋅ ⋅ ⋅ Lgik h j (x0 )

∫ t 0

d[ik ⋅ ⋅ ⋅ d[i0 ,

j = 1, ⋅ ⋅ ⋅ , p. (9.69)

9.7 Fliess Functional Expansion

265

(9.69) is called the Fliess functional expansion of system (9.1). We refer to [5] for the proof. Next, we give an example to show how to calculate it and how to use it for output tracking problem. Example 9.3. Recall Example 1.12 in Chapter 1 for the dynamics of a PVTOL aircraft, where we nally get its dynamics as ⎧ x1 = x2      x2 = sin(x5 )u1    ⎨x = x 3 4 (9.70)  x  = cos(x 5 )u1 − 1  4    x5 = x6   ⎩ x6 = u2 . It is natural to assume that the generalized position coordinates are our concerned outputs. So we set y 1 = x1 y 2 = x3 y 3 = x5 .

(9.71)

{

Now we have

x = f (x) + g1 (x)u1 + g2 (x)u2 y j = h j (x), j = 1, 2, 3,

where

⎤ x2 ⎢0⎥ ⎢ ⎥ ⎢ x4 ⎥ ⎥ f (x) = ⎢ ⎢−1⎥ , ⎢ ⎥ ⎣ x6 ⎦ 0 ⎡

⎤ 0 ⎢− sin(x5 )⎥ ⎥ ⎢ ⎥ ⎢ 0 ⎥, ⎢ g1 (x) = ⎢ ⎥ ⎢ cos(x5 ) ⎥ ⎦ ⎣ 0 0 ⎡

⎡ ⎤ 0 ⎢0⎥ ⎢ ⎥ ⎢0⎥ ⎥ g2 (x) = ⎢ ⎢0⎥ , ⎢ ⎥ ⎣0⎦ 1

and h1 (x) = x1 ,

h2 (x) = x3 ,

h3 (x) = x5 .

Set x0 = 0, g0 (x) = f (x), and consider h1 (x) rst. It is obvious that h1 (0) = 0. For k = 0 we have the rst order Lie derivatives as Lg0 h1 (x) = x2 ,

Lg1 h1 (x) = 0,

Lg2 h1 (x) = 0.

So at x0 = 0 all of them are zero. For k = 1 we consider second order Lie derivatives. Note that if the rst order Lie derivative is constant, its further derivatives are zero. So now we need to consider

266

9 Input-Output Structure

the rst one, i.e., Lg0 h1 (x) only: L2g0 h1 (x) = 0,

Lg1 Lg0 h1 (x) = − sin(x5 ),

Lg2 Lg0 h1 (x) = 0.

At x0 = 0 all of them are still zero. For k = 2 we consider third order Lie derivatives. Lg0 Lg1 Lg0 h1 (x) = −x6 cos(x5 ),

L2g1 Lg0 h1 (x) = 0,

Lg2 Lg1 Lg0 h1 (x) = 0.

They are all zero at x0 = 0. For k = 3 we have 4-th order Lie derivatives as L2g0 Lg1 Lg0 h1 (x) = x26 sin(x5 ),

Lg1 Lg0 Lg1 Lg0 h1 (x) = 0,

Lg2 Lg0 Lg1 Lg0 h1 (x) = − cos(x5 ). Now we have a 4-th order Lie derivative, which is not zero at x0 = 0. It is Lg2 Lg0 Lg1 Lg0 h1 (0) = −1. So if we consider the output y1 (t), up to 4-th order Lie derivative terms, we have y1 (t) = −

∫ t 0

dW1

∫ W1 0

u1 (W2 )dW2

∫ W2 0

dW3

∫ W3 0

u2 (W4 )dW4 + ⋅ ⋅ ⋅ .

(9.72)

Similarly, for h2 (t) we can calculate the up to third order Lie derivatives, and the possible non-zero terms are listed as follows: For k = 0 Lg0 h2 (x) = x4 , Lg1 h2 (x) = 0, Lg2 h2 (x) = 0. At x0 = 0 all of them are zero. For k = 1 L2g0 h2 (x) = −x4 ,

Lg1 Lg0 h2 (x) = x4 cos(x5 ),

Lg2 Lg0 h2 (x) = 0.

At x0 = 0 all of them are still zero. For k = 2 L3g0 h2 (x) = 1,

Lg0 Lg1 Lg0 h2 (x) = − cos(x5 ) − x4 x6 sin(x5 ),

Lg1 L2g0 h2 (x) = − cos(x5 ), L2g1 Lg0 h2 (x) = cos2 (x5 ), Lg2 L2g0 h2 (x) = 0,

Lg2 Lg1 Lg0 h2 (x) = 0.

So the output y2 , up to third Lie derivatives, is y2 (t) =

t3 − 6 +

∫ t 0

∫ t 0

dW 1

dW1

∫ W1

∫ W1 0

0

dW2

∫ W2 0

u1 (W2 )dW2

Finally, we consider h3 (t). Since

u1 (W3 )dW3 −

∫ W2 0

∫ t 0

dW1

∫ W1

u1 (W3 )dW3 + ⋅ ⋅ ⋅ .

0

u(W2 )dW2

∫ W2 0

dW3

9.8 Tracking via Fliess Functional Expansion

Lg0 h3 (x) = x6 ,

Lg1 h3 (x) = 0,

267

Lg2 h3 (x) = 0,

and L2g0 h3 (x) = 0,

Lg1 Lg0 h3 (x) = 0,

Lg2 Lg0 h3 (x) = 1,

all the higher order derivatives are zero. We then have an precise expression as y3 (t) =

∫ t 0

∫ W1

dW1

0

u2 (W2 )dW2 .

9.8 Tracking via Fliess Functional Expansion In this section we consider a tracking problem by using Fliess functional expansion. This section is based on [3]. Recall system (9.1). Assume the relative degree vector U = (U1 , ⋅ ⋅ ⋅ , U p ) is well dened and the decoupling matrix ⎡ ⎤ U −1 U −1 Lg1 L f 1 h1 (x) ⋅ ⋅ ⋅ Lgm L f 1 h1 (x) ⎢ ⎥ .. .. ⎥ , x ∈ U, D(x) = ⎢ (9.73) . . ⎣ ⎦ U p −1 U p −1 Lg1 L f h p (x) ⋅ ⋅ ⋅ Lgm L f h p (x) and the decoupling vector as ⎤ U L f 1 h1 (x) ⎥ ⎢ .. [ (x) = ⎣ ⎦, . Up L f h p (x) ⎡

x ∈ U,

(9.74)

are well dened. Moreover, we assume A3. D(x), x ∈ U has full row rank, i.e., rank(D(x)) = p, x ∈ U. For statement brevity we further assume A4. p = m. Then we consider a feedback control u = −D−1 (x)[ (x) + D−1 (x)v.

(9.75)

Remark 9.3. Assumption A4 is just for statement ease. It can be generalized as follows: Assume m > p and rank(D(x)) = p. Then we can nd a column permutation matrix P such that D(x)P = [D1 (x), D2 (x)], where D1 (x) is non-singular. Dene D+ (x) =

[

] D−1 1 (x) , 0(m−p)×p

and set u = −PD+ (x)[ (x) + PD+ (x)v. All the arguments in the remaining of this section will be applicable.

(9.76)

268

9 Input-Output Structure

Assume A3 and A4 hold. Using control (9.75), and setting f˜ = f − GD−1(x)[ (x),

G˜ = GD−1 (x),

we have the following closed-loop system { ˜ x = f˜ + Gv y = h(x).

(9.77)

Note that the system (9.77) has input-output decomposed form. We would like to give its Fliess functional expansion, and show that it has only nite terms. To do this, we need some preparations. Lemma 9.5. ∫ T 0

dW1

∫ W1 0

dW2 ⋅ ⋅ ⋅

∫ Wk 0

) (Wk+1 )dWk+1 =

∫ T (T − W )k 0

k!

) (W )dW .

(9.78)

Proof. We prove it by mathematical induction. When k = 1, we have (by exchanging the order of integration) ∫ T 0

dW1

∫ W1 0

) (W2 )dW2 = =

∫ T 0

∫ T 0

dW2

∫ T W2

) (W2 )dW1

) (W2 )dW2

∫ T W2

dW 1 =

∫ T 0

(T − W2 )) (W2 )dW2 .

Now assume (9.78) is true for k = s − 1. Then for k = s we have ∫ T 0

= = = = =

∫ T 0

∫ T 0

∫ T 0

∫ T 0

dW1 dW1

∫ W1 0

∫ W1 0

dWs+1

dW 2 ⋅ ⋅ ⋅

∫ Ws 0

) (Ws+1 )

∫ T Ws+1

) (Ws+1 )dWs+1

(W1 − Ws+1 )s−1 dWs+1 (s − 1)!

) (Ws+1 )

) (Ws+1 )dWs+1

(W1 − Ws+1 )s−1 dW1 (s − 1)!

∫ T (W1 − Ws+1 )s−1 Ws+1

(s − 1)!

dW1

 (W1 − Ws+1 )s T ) (Ws+1 ) dWs+1  s! Ws+1

∫ T (T − Ws+1 )s 0

which completes the proof.

s!

) (Ws+1 )dWs+1 , ⊔ ⊓

Lemma 9.6. Consider the closed-loop system (9.77) with the feedback control (9.75). The Lie derivatives of the outputs have the following form:

9.8 Tracking via Fliess Functional Expansion

x ∈ U, k ∕= U j − 1, j = 1, ⋅ ⋅ ⋅ , m, i = 0, ⋅ ⋅ ⋅ , m, { 1, i = j U j −1 , x ∈ U, j = 1, ⋅ ⋅ ⋅ , m. Lg˜i L f˜ h j (x) = 0, i ∕= j

Lg˜i Lkf˜h j (x) = 0,

269

(9.79) (9.80)

Proof. Recall Proposition 9.1, the new decoupling matrix and decoupling vector are respectively ˜ D(x) = D(x)E (x),

(9.81)

[˜ = D(x)D (x) + [ (x).

(9.82)

and

Now under control (9.75) it follows from (9.81) that for f˜(x) and the G˜ the decoupling matrix D˜ = Im . (9.80) follows immediately. Next, for (9.79) with k < U j − 1 it is trivial, because a feedback does not affect the relative degree. To prove it for k > U j − 1 it sufces to show that U

L f˜ j h j (x) = 0.

(9.83)

Using (9.82) we have [˜ = 0. As a component of [˜ , (9.83) follows.

⊔ ⊓

Now we are ready to provide the main result: Theorem 9.7. Under assumptions A3 and A4, the Fliess functional expansion of the closed-loop system (9.77) has the following nite term expression: U j −1

y j (T ) = h j (x0 ) +

Tk

¦ Lkf˜h j (x0 ) k! +

k=1

∫ T 0

v j (W )

j = 1, ⋅ ⋅ ⋅ , m.

(T − W )U j −1 dW , (U j − 1)!

(9.84)

Proof. From (9.83) it is clear that k ⩾ U j.

Lkf˜ h j (x) = 0,

Hence we have only nite control independent terms as U j −1

Tk

¦ Lkf˜h j (0) k! .

k=0

Using (9.79) and (9.80), a straightforward computation shows that the only non-zero control dependent term is U −1

Lg˜ j L f˜ j =

∫ T 0

dW1

h(x0 )

∫ W1 0

∫ T 0

dW2 ⋅ ⋅ ⋅

dW 1

∫ W1 0

∫ WU −1 j 0

dW2 ⋅ ⋅ ⋅

∫ WU −1 j 0

v j (WU j )dWU j .

v j (WU j )dWU j

270

9 Input-Output Structure

Using Lemma 9.5, we have ∫ T 0

dW 1

∫ W1 0

dW2 ⋅ ⋅ ⋅

∫ WU −1 j 0

v j (WU j )dWU j =

∫ T 0

v j (W )

(T − W )U j −1 dW . (U j − 1)!

⊔ ⊓

Let the reference output trajectories be described as yd (t) = (yd1 (t), ⋅ ⋅ ⋅ , ydp (t)).

(9.85)

The general tracking problem is to nd a suitable control u such that the outputs can track the reference outputs. That is, lim ∥y(t) − yd (t)∥ = 0.

t→f

(9.86)

Assume that the controls are piecewise constant with nite possible constant values. Denote by UPC (c1 , ⋅ ⋅ ⋅ , cs ) the set of piecewise constant functions taking values from {c1 , ⋅ ⋅ ⋅ , cs }. Then the control u = (u1 , ⋅ ⋅ ⋅ , um ) satises ui ∈ UPC (c1 , ⋅ ⋅ ⋅ , cs ),

1 ⩽ i ⩽ m.

(9.87)

Choose a time duration T > 0 as the sampling period. Assume within each sampling duration the controls are constant, i.e., ui (t) = uki ,

kT ⩽ t < (k + 1)T,

i = 1, ⋅ ⋅ ⋅ , m,

(9.88)

where uki ∈ {c1 , ⋅ ⋅ ⋅ , cs }. That is, the possible switching (discontinuous) moments are at {kT ∣ k = 1, 2, ⋅ ⋅ ⋅ }, with T > 0 as a period. Denote by xki = xi (kT ),

ykj = y j (kT ),

where i = 1, ⋅ ⋅ ⋅ , n, j = 1, ⋅ ⋅ ⋅ , p, k = 1, 2, ⋅ ⋅ ⋅ . Then the tracking problem considered in this section becomes an optimization problem. We may require the trajectory being as close as possible to the given trajectory in nite steps. We give a step by step algorithm for the optimal u. That is, at each sampling period, say, [kT, (k + 1)T ), when x0 := x(kT ) is known, we want to nd the optimal control u∗ , which minimizes 9 9 9 9 (9.89) min 9y((k + 1)T ) − yd ((k + 1)T )9 . u

For statement ease, we ignore the real time kT by considering it as the initial time, that is, set kT = 0. From (9.88), on [0, T ], u is constant and it follows from (9.75) that v(W ) = D(x(W ))u + [ (x(W )),

W ∈ [0, T ].

(9.90)

Now we are facing a problem of nding v(W ), which depends on x(W ). Getting a precise solution is almost impossible. Fortunately, T is assumed to be small enough, (it is true in missile guiding problem,) we can use linear approximation to calculate x(W ):

9.8 Tracking via Fliess Functional Expansion

x(W ) ≈ x0 + ( f (x0 ) + G(x0 )u)W .

271

(9.91)

In the following calculation we use semi-tensor product of matrices, see appendix for the denitions. Using (9.91), we can get the linear approximation of D(x(W )) as ( ) D(x(W )) ≈ D(x0 ) + L f D(x0 ) + LGD(x0 )u W , (9.92) where the Lie derivative of a matrix with respect to a vector eld is dened as the matrix of Lie derivatives of its elements. That is, ( ) L f A(x) := L f ai j (x) = (dai j (x) f (x)) , and LG A(x) := (Lg1 A(x), Lg2 A(x), ⋅ ⋅ ⋅ , Lgm A(x)) . Similarly, we have the linear approximation of [ (x(W )) as ( ) [ (x(W )) ≈ [ (x0 ) + L f [ (x0 ) + LG [ (x0 )u W .

(9.93)

Plugging (9.92) and (9.93) into (9.90) yields 2 v(W ) ≈ W (L ( G D(x0 )) u ) + D(x0 ) + W (L f D(x0 ) + LG[ (x0 )) u + [ (x0 ) + W L f [ (x0 ).

(9.94)

Denote ⎤ (T − W )U1 −1 ⋅ ⋅ ⋅ 0 ⎥ ⎢ (U − 1)! 1 ⎥ ⎢ ⎥ ⎢ .. .. ⎥ ⎢ ) (W ) = ⎢ . . ⎥. ⎥ ⎢ ⎥ ⎢ ⎣ (T − W )Um−1 ⎦ 0 ⋅⋅⋅ (Um − 1)! ⎡

(9.95)

Then y(T ) = ; + ≈

(∫

∫ T 0

) ) (W )W dW LG D(x0 ) u2

T

0

+ +

(∫

T 0

∫ T 0

) (W )v(W )dW

) (W )dW D(x0 ) +

∫ T 0

) (W )W dW L f [ (x0 ) +

:= Pu2 + Qu + R,

) ) (W )W dW (L f D(x0 ) + LG [ (x0 )) u

∫ T 0

) (W )dW[ (x0 ) + ;

(9.96)

272

9 Input-Output Structure

where ⎧ ∫ T   ) (W )W dW LG D(x0 ) P =   0   ∫ T ⎨ ) ∫T ( ) (W )W dW L f D(x0 ) + LG [ (x0 ) + ) (W )dW D(x0 ). Q=  0 0    ∫ T ∫ T    ⎩R = ) (W )W dW L f [ (x0 ) + ) (W )dW[ (x0 ) + ; 0

(9.97)

0

Note that in the above we use semi-tensor product for all the matrix products. Now the optimal control u∗ satises 9 9 9 9 2 9 d 9 9 9 (9.98) 9y (T ) − Pu∗ − Qu∗ − R9 ⩽ 9yd (T ) − Pu2 − Qu − R9, ∀u. It is easy to nd u∗ from the nite set of controls. In the following we consider the attitude control of a missile. The dynamics of the attitude of missiles is [8] ⎧ dZ x  + (Jz − Jy )Zy Zz = u1 Jx    dt      Jy dZy + (Jx − Jz )Zx Zz = u2   dt      d Zz   + (Jy − Jx )Zx Zy = u3 Jz   dt      dJ  ⎨ = Zx − (Zy cos J − Zz sin J ) tan T dt (9.99)  1 d I    = (Zy cos J − Zz sin J )   dt cos T     dT   = Zy sin J + Zz cos J    dt      y1 = J (t)     y2 = I (t)   ⎩ y3 = T (t), where Jx , Jy , Jz are the inertias with respect to x, y, and z axes of the missile (for simplicity, they are assumed to be constants), Zx , Zy , and Zz are angular velocities with respect to the axes respectively; J , I , and T are three attitude angles. Denoting x = (x1 , x2 , x3 , x4 , x5 , x6 )T , where x1 = Zx , x2 = Zy , x3 = Zz , x4 = J , x5 = I , x6 = T , then the system (9.99) can be converted into a canonical form as ⎧ ⎨x = f (x) + G(x)u ⎩y = h(x),

(9.100)

9.8 Tracking via Fliess Functional Expansion

where

273

⎤ Jy − Jz Zy Zz ⎥ ⎢ Jx ⎥ ⎢ Jz − Jx ⎥ ⎢ Zx Zz ⎥ ⎢ ⎥ ⎢ Jy ⎥ ⎢ Jx − Jy ⎥ ⎢ Z Z ⎥ ⎢ x y f (x) = ⎢ Jz ⎥ ⎢Z − (Z cos J − Z sin J ) tan T ⎥ y z ⎥ ⎢ x ⎥ ⎢ ⎥ ⎢ 1 ⎥ ⎢ ⎣ cos T (Zy cos J − Zz sin J ) ⎦ Zy sin J + Zz cos J ⎡ ⎤ Jy − Jz x2 x3 ⎢ ⎥ Jx ⎢ ⎥ J − J ⎢ ⎥ z x x1 x3 ⎢ ⎥ ⎢ ⎥ Jy ⎢ ⎥ Jx − Jy ⎢ ⎥ x x ⎢ ⎥; 1 2 =⎢ Jz ⎥ ⎢x − (x cos(x ) − x sin(x )) tan(x )⎥ ⎢ 1 2 4 3 4 6 ⎥ ⎢ ⎥ ⎢ ⎥ 1 ⎢ ⎥ ⎣ cos(x ) (x2 cos(x4 ) − x3 sin(x4 )) ⎦ 6 x2 sin(x4 ) + x3 cos(x4 ) ⎤ ⎡1 0 0 ⎥ ⎢ Jx ⎥ ⎢ ⎢0 1 0⎥ ⎥ ⎢ ⎢ Jy ⎥ ⎥ ⎢ G(x) = (g1 , g2 , g3 ) = ⎢ 0 0 1 ⎥ ; ⎢ Jz ⎥ ⎥ ⎢ ⎢0 0 0⎥ ⎥ ⎢ ⎣0 0 0⎦ ⎡

0 0 0 ]T ]T h(x) = J I T = x4 x5 x6 . [

[

Note that we assume ∣T ∣ < π/2 to assure cos(x6 ) ∕= 0. It is easy to calculate the relative degree vector as U1 = U2 = U3 := U = 2. Then we have ⎡ ⎤ Zx − (Zy cos J − Zz sin J ) tan T ⎢ ⎥ 1 Lf h = ⎢ (Zy cos J − Zz sin J ) ⎥ ⎣ ⎦. cos T Zy sin J + Zz cos J The decoupling matrix is ⎤ ⎡ 1 1 1 − cos(x4 ) tan(x6 ) sin(x4 ) tan(x6 ) ⎥ ⎢ Jx Jy Jz ⎥ ⎢ ⎥ ⎢ 1 1 ⎥. 0 cos(x ) sec(x ) − sin(x ) sec(x ) (9.101) D(x) = ⎢ 4 6 4 6 ⎥ ⎢ Jy Jz ⎥ ⎢ ⎦ ⎣ 1 1 0 sin(x4 ) cos(x4 ) Jy Jz

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9 Input-Output Structure

The decoupling vector is

⎡ 2 ⎤ ⎡ ⎤ L f h1 (x) [1 (x) ⎢ 2 ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ [ (x) = ⎢ ⎣L f h2 (x)⎦ := ⎣[2 (x)⎦ , L2f h3 (x) [3 (x)

(9.102)

where

[1 =

Jy − Jz Jz − Jx x2 x3 − x1 x3 cos(x4 ) tan(x6 ) Jx Jy Jx − Jy + x1 x2 sin(x4 ) tan(x6 ) Jz +[x1 − (x2 cos(x4 ) − x3 sin(x4 )) tan(x6 )] ×[x2 sin(x4 ) + x3 cos(x4 )] tan(x6 ) −[x2 sin(x4 ) + x3 cos(x4 )][x2 cos(x4 ) − x3 sin(x4 )] sec2 (x6 ),

[2 =

Jx − Jy Jz − Jx x1 x3 cos(x4 ) sec(x6 ) − x1 x2 sin(x4 ) sec(x6 ) Jy Jz −[x1 − (x2 cos(x4 ) − x3 sin(x4 )) tan(x6 )] ×[x2 sin(x4 ) + x3 cos(x4 )] sec(x6 ) +[x2 sin(x4 ) + x3 cos(x4 )][x2 cos(x4 ) − x3 sin(x4 )] tan(x6 ) sec(x6 ),

[3 =

Jx − Jy Jz − Jx x1 x3 sin(x4 ) + x1 x2 cos(x4 ) Jy Jz +[x1 − (x2 cos(x4 ) − x3 sin(x4 )) tan(x6 )][x2 cos(x4 ) − x3 sin(x4 )].

When the linear approximation of x(W ) is used, the Fliess functional expansion (9.84) for the closed-loop system of (9.99) becomes y(T ) = h(0) + L f˜h(0)T +

∫ T 0

(T − W )I3 v(W )dW := Pu2 + Qu + R.

(9.103)

Using (9.101), it is easy to calculate that LG D(x) = 0, so we have P = 0. Now a straightforward computation shows that ∫ T 0

∫ T 0

1 ) (W )W dW = T 3 I3 , 6 1 ) (W )dW = T 2 I3 . 2

(9.104)

9.8 Tracking via Fliess Functional Expansion

The Lie derivative of D(x) with respect of f (x) is ⎡ ⎤ L f d11 L f d12 L f d13 ⎢ ⎥ ⎥ L f D(x) = ⎢ ⎣L f d21 L f d22 L f d23 ⎦ , L f d31 L f d32 L f d33 where L f d11 = 0, 1 L f d12 = [x1 − (x2 cos(x4 ) − x3 sin(x4 )) tan(x6 )] sin(x4 ) tan(x6 ) Jy 1 − [x2 sin(x4 ) + x3 cos(x4 )] cos(x4 ) sec2 (x6 ), Jy 1 L f d13 = [x1 − (x2 cos(x4 ) − x3 cos(x4 )) tan(x6 )] cos(x4 ) tan(x6 ) Jz 1 + [x2 sin(x4 ) + x3 cos(x4 )] sin(x4 ) sec2 (x6 ), Jz L f d21 = 0, 1 L f d22 = − [x1 − (x2 cos(x4 ) − x3 sin(x4 )) tan(x6 )] sin(x4 ) sec(x6 ) Jy 1 + [x2 sin(x4 ) + x3 cos(x4 )] cos(x4 ) tan(x6 ) sec(x6 ), Jy 1 L f d23 = − [x1 − (x2 cos(x4 ) − x3 cos(x4 )) tan(x6 )] cos(x4 ) sec(x6 ) Jz 1 − [x2 sin(x4 ) + x3 cos(x4 )] sin(x4 ) tan(x6 ) sec(x6 ), Jz L f d31 = 0, 1 L f d32 = [x1 − (x2 cos(x4 ) − x3 sin(x4 )) tan(x6 )] cos(x4 ), Jy 1 L f d33 = − [x1 − (x2 cos(x4 ) − x3 cos(x4 )) tan(x6 )] sin(x4 ). Jz ] [ d[1 = d[11 d[12 d[13 d[14 d[15 d[16 , [ ] d[2 = d[21 d[22 d[23 d[24 d[25 d[26 , ] [ d[3 = d[31 d[32 d[33 d[34 d[35 d[36 ,

Denote

where Jx − Jy Jz − Jx x3 cos(x4 ) tan(x6 ) + x2 sin(x4 ) tan(x6 ) Jy Jz +(x2 sin(x4 ) + x3 cos(x4 )) tan(x6 ),

d[11 = −

d[12 =

Jy − Jz Jx − Jy x3 + x1 sin(x4 ) tan(x6 ) + x1 sin(x4 ) tan(x6 ) Jx Jz −(x2 sin(2x4 ) + x3 cos(2x4 ))(tan2 (x6 ) + sec2 (x6 )),

275

276

9 Input-Output Structure

d[13 =

d[14 =

d[15

Jy − Jz Jz − Jx x2 − x1 cos(x4 ) tan(x6 ) + x1 cos(x4 ) tan(x6 ) Jx Jy −(x2 cos(2x4 ) − x3 sin(2x4 ))(tan2 (x6 ) + sec2 (x6 )), Jx − Jy Jz − Jx x1 x3 sin(x4 ) tan(x6 ) + x1 x2 cos(x4 ) tan(x6 ) Jy Jz +x1 (x2 cos(x4 ) − x3 sin(x4 )) tan(x6 )

−(x22 + x23 )(tan2 (x6 ) + sec2 (x6 )), = 0, Jx − Jy Jz − Jx x1 x3 cos(x4 ) sec2 (x6 ) + x1 x2 sin(x4 ) sec2 (x6 ) Jy Jz +x1 (x2 sin(x4 ) + x3 cos(x4 )) sec2 (x6 )

d[16 = −

−4(x2 cos(x4 ) − x3 sin(x4 ))(x2 sin(x4 ) +x3 cos(x4 )) tan(x6 ) sec2 (x6 ), d[21 =

Jx − Jy Jz − Jx x3 cos(x4 ) sec(x6 ) − x2 sin(x4 ) sec(x6 ) Jy Jz −(x2 sin(x4 ) + x3 cos(x4 )) sec(x6 ), Jx − Jy x1 sin(x4 ) sec(x6 ) − x1 sin(x4 ) sec(x6 ) Jz +2(x2 sin(2x4 ) + x3 cos(2x4 )) tan(x6 ) sec(x6 ),

d[22 = −

d[23 =

Jz − Jx x1 cos(x4 ) sec(x6 ) − x1 cos(x4 ) sec(x6 ) Jy +2(x2 cos(2x4 ) + x3 sin(2x4 )) tan(x6 ) sec(x6 ), Jx − Jy Jz − Jx x1 x3 sin(x4 ) sec(x6 ) − x1 x2 cos(x4 ) sec(x6 ) Jy Jz −x1 (x2 cos(x4 ) − x3 sin(x4 )) sec(x6 ) + 2(x22 + x23 ) tan(x6 ) sec(x6 ),

d[24 = −

d[25 = 0, d[26 =

Jz − Jx x1 x3 cos(x4 ) tan(x6 ) sec(x6 ) Jy Jx − Jy x1 x2 sin(x4 ) tan(x6 ) sec(x6 ) − Jz −x1 (x2 sin(x4 ) + x3 cos(x4 )) tan(x6 ) sec(x6 ) +2(x2 cos(x4 ) − x3 sin(x4 ))(x2 sin(x4 ) +x3 cos(x4 ))(sec3 (x6 ) + tan2 (x6 ) sec(x6 )),

Jx − Jy Jz − Jx x3 sin(x4 ) + x2 cos(x4 ) + x2 cos(x4 ) − x3 sin(x4 ), Jy Jz Jx − Jy = x1 cos(x4 ) + x1 cos(x4 ) − 2(x2 cos(x4 ) Jz

d[31 = d[32

References

277

−x3 sin(x4 )) cos(x4 ) tan(x6 ), d[33 =

Jz − Jx x1 sin(x4 ) − x1 sin(x4 ) + 2(x2 cos(x4 ) Jy −x3 sin(x4 )) sin(x4 ) tan(x6 ),

d[34 =

Jx − Jy Jz − Jx x1 x3 cos(x4 ) − x1 x2 sin(x4 ) Jy Jz −x1 (x2 sin(x4 ) + x3 cos(x4 )) tan(x6 ) +2(x2 sin(x4 ) + x3 cos(x4 ))(x2 cos(x4 ) − x3 sin(x4 )) tan(x6 ),

d[35 = 0, d[36 = −(x2 cos(x4 ) − x3 sin(x4 ))2 sec2 (x6 ). Then the Lie derivative of [ (x) with respect of f (x) is ⎡ ⎤ d[1 f ⎢ ⎥ ⎥ Lf [ = ⎢ ⎣d[2 f ⎦ , d[3 f and the Lie derivative of [ (x) with respect of G is ⎤ ⎡ d[1 g1 d[1 g2 d[1 g3 ⎥ [ ] ⎢ ⎥ ⎢ LG [ = Lg1 [ Lg2 [ Lg3 [ = ⎢d[2 g1 d[2 g2 d[2 g3 ⎥ . ⎦ ⎣ d[3 g1 d[3 g2 d[3 g3 Putting them into (9.97), we have Q and R. Since P = 0, then the optimal solution should be uo = Q−1 (ydT − R). (or uo = Q+ (ydT − R) if Q is singular). It follows that the optimal admissible control u∗ satises ∥uo − u∗ ∥ = min ∥uo − u∥. u∈U

References 1. Boothby W. An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd edn. Orlando: Academic Press, 1986. 2. Cheng D. Semi-tensor product and its application to Morgan’s problem. Science in China Series F: Information Sciences, 2001, 44(3): 195–212. 3. Cheng D, Qiao Y, Yuan Y. Decomposed Fliess expansion and its application to attitute control of missiles. Preprint, 2008 4. Golubitsky M, Guillemin V. Stable Mappings and Their Singularities. New York: SpringerVerlag, 1973 5. Isidori A. Nonlinear Control Systems, 3rd edn. London: Springer, 1995.

278

9 Input-Output Structure

6. Khalil H. Nonlinear Systems, 3rd edn. New Jersey: Prentice Hall, 2002. 7. Nijmeijer H, Van der Schaft A. Nonlinear Dynamical Control Systems. New York: SpringerVerlag, 1990. 8. Yao Y, Yang B, He F, et al. Control of missile via Fliess expansion. IEEE Trans. Control Systems Technology, 2008, 16(5): 959–970.

Chapter 10

Linearization of Nonlinear Systems

Linearization is one of the most powerful tools for dealing with nonlinear systems. Some person says that in fact, what the mathematicians can really deal with is linear problems. Believe it or not, the control theory can treat linear systems perfectly. Hence linearization is an ideal method to deal with nonlinear systems. Section 10.1 considers how to convert a nonlinear dynamics into a linear one, called the Poincar´e’s linearization. Section 10.2 is about the linear equivalence of afne nonlinear systems. It reveals the essential linear structure of a control system. Section 10.3 discusses the state feedback linearization of nonlinear systems. The linearization of nonlinear systems with outputs is investigated in Section 10.4. Section 10.5 considers the global linearization and the linearization algorithms. In Section 10.6 the linearization by non-regular state feedback is investigated.

10.1 Poincar´e Linearization Given a nonlinear dynamics x = f (x),

x ∈ Rn ,

(10.1)

where f (x) is an analytic mapping with f (0) = 0. It is natural to ask such a question: Can we nd a coordinate transformation y = M (x) such that under coordinate frame y the system (10.1) can be expressed as a linear system y = Ay,

y ∈ Rn .

(10.2)

In case the global transformation is not found, we may only be interested in a local version. That is, there exists a neighborhood 0 ∈ U ⊂ Rn , such that (10.1) for x ∈ U is equivalent to (10.2) for y ∈ M (U). We call such a problem the Poincar´e’s linearization. It is based on Poincar´e’s Theorem. We need some preparations. The rst proposition is easily veriable [7]. Proposition 10.1. 1. Let Hnk be the set of k-th degree homogeneous vector elds with x ∈ Rn . Then it is a vector space over R. 2. Let Ax ∈ Hn1 be a linear vector eld. Then the Lie derivative adAx : Hnk → Hnk is a linear mapping. Now since adAx : Hnk → Hnk is a linear mapping, the image adAx (Hnk ) is a sub-space of Hnk . Denote by Dk = adAx (Hnk ) and Gk its complement. Then we have D. Cheng et al., Analysis and Design of Nonlinear Control Systems © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2010

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10 Linearization of Nonlinear Systems

Hnk = Dk + Gk ,

k = 1, 2, · · · .

(10.3)

It is worth noting that for a given Dk , its complement Gk is not unique. Using this decomposition we rst give the following result. Theorem 10.1. Consider the system (10.1), where f (x) is analytic with f (0) = 0. Let L = J f (0)x, where J f is the Jacobian matrix of f . Then there exists a local diffeomorphism x = x(z) such that (10.1) can be expressed in z coordinate frame as z = L + g2 (z) + · · · + gr (z) + Rr (z),

(10.4)

where gi (z) ∈ Gi ,

i = 2, 3, · · · , r;

Rr (z) = O(1z1r+1 ).

Proof. It is enough to show that if we have (10.4), then we can nd a coordinate transformation x = M (z), such that the rst r terms of (10.4) remain unchanged and Rr (z) = gr+1 (z) + Rr+1 (z),

where Rr+1 (z) = O(1z1r+2 ).

Starting from (10.4) and using Taylor expansion, we can express Rr (z) = [r+1 (z) + Rr+1(z),

where Rr+1 (z) = O(1z1r+2 ).

(10.5)

Split [r+1 (z) as

[r+1 (z) = hr+1 (z) + gr+1 (z),

(10.6)

where hr+1 (z) ∈ Dr+1 , gr+1 (z) ∈ Gr+1 . By denition, there exists T (z) ∈ Hnk such that adL (T (z)) = hr+1 (z)

(10.7)

Now dene a coordinate transformation as z = y + T (y). Then y = Q(y)[L + g2 (y + T (y)) + · · · + gr (y + T (y)) + [ (y + T(y)) + Rr+1 (y)], (10.8) where

Q(y) = (I + JT (y))−1 = I − JT (y) + O(1x12r ).

Plugging it into (10.8) and checking the degree of each term carefully, we have y = (I − JT (y))(Ay + AT (y)) + g2 (y) + · · · + gr (y) + [ (y) + O(1y1r+2) = Ay − [JT (y)Az − AT (y)] + [ (y) + g2(y) + · · · + gr (y) + O(1y1r+2) = Ay − adL T (y) + [ (y) + g2(y) + · · · + gr (y) + O(1y1r+2). Since

[ (y) − adL T (y) = [ (y) − hr+1(y) = gr+1 (y) ∈ Gr+1 ,

10.1 Poincar´e Linearization

281

3 2

the conclusion follows.

Remark 10.1. 1. (10.4) is called the normal form of system (10.1) [7]. It has been extended to control systems [4]. 2. In (10.4) there is no special r assigned. So as r → f we have an innite series expression of normal form without Rr (z). We do hope that we could have adL Hnk = Hnk , k = 2, 3, · · · , and get a linear form. Next, we consider when this is true. Denition 10.1. Let A ∈ Mn with eigenvalues V (A) = O = (O1 , · · · , On ). A is a resonant matrix if there exists m = (m1 , · · · , mn ) ∈ Zn+ (i.e., mi  0), with the norm1m1  2, such that for some s

Os = 4m, O 5 . Otherwise it is called a non-resonant matrix. ? > 0 1 is resonant because its eigenvalues are O1 = 1, O2 = For example, A = 2 −1 −2. Taking m = (3, 1), then

O1 = 4(O1 O2 ), m5 . The following proposition provides a sufcient condition. [6] Proposition 10.2. Let V (A) = O = (O1 , · · · , On ) be the eigenvalues of a given Hurwitz matrix A. A is non-resonant if max{ |Re(Oi )| | Oi ∈ V (A)}  2 min{ |Re(Oi )| | Oi ∈ V (A)}.

(10.9)

The following theorem, called the Poincar´e’s theorem [1], provides a sufcient condition for a nonlinear system to be locally equivalent to a linear system. Theorem 10.2. (Poincar´e’s Theorem) Consider the system (10.1) assume f (x) is analytic with f (0) = 0. If the Jacobian matrix  w f  J f (0) := w x x=0 is non-resonant, then there exists an analytic mapping, as a local diffeomorphism, A @ x = y + M (y), where M (y) = O 1y12 , such that under y coordinate frame the system (10.1) can be expressed as y = Ay.

(10.10)

Note that, as the conditions of Theorem 10.2 are satised, the algorithm proposed in the constructive proof of Theorem 10.1 provides an algorithm for Poincar´e’s linearization. However since the algorithm is completed in innity iterations, practically, we can only get an approximately linearized system with an error of the higher degree terms.

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10 Linearization of Nonlinear Systems

10.2 Linear Equivalence of Nonlinear Systems Consider an afne nonlinear system m

x = f (x) + ¦ gi (x)ui := f (x) + g(x)u,

x ∈ Rn , u ∈ Rm ,

(10.11)

i=1

where f (x0 ) = 0. Denition 10.2. The system (10.11) is said to be equivalent to a linear system (locally around an equilibrium point x0 ), if there exists a (local) coordinate chart (U, z) (z(x0 ) = 0) such that the system (10.11) is expressed in z coordinate frame as m

z = Az + ¦ bi ui := Az + Bu,

z ∈ Rn (z ∈ U), u ∈ Rm ,

(10.12)

i=1

where (A, B) is a completely controllable pair. First we consider the local linearization problem. Theorem 10.3. The system (10.11) is said to be equivalent to a linear system locally around an B equilibriumpoint x0 , if and only if C  (i). dim adkf gi (x0 ) 1  i  m, 0  k  n − 1 = n; (ii). there exists a neighborhood U of x0 , such that D s E ad f gi , adtf g j = 0, 1  i, j  m, 0  s,t  n. Proof. (Necessity) Assume the coordinate change, which realizes the linearization, is: T : x 6→ z. Then T∗ ( f ) = Az,

T∗ (gi ) = bi ,

i = 1, · · · , m.

Since the system (10.12) is completely controllable, a simple computation shows that adkT∗ ( f ) T∗ (gi ) = (−1)k Ak bi .

(10.13)

But a diffeomorphism doesn’t affect the Lie bracket, i.e., T∗ [ f , g] = [T∗ ( f ), T∗ (g)]. Hence, (10.13) implies (i) and (ii). (Sufciency) According to (i), there exist n linearly independent vector elds Xik+1 = adkf gi , F

m

¦ ni = n

i = 1, · · · , m, k = 0, 1, · · · , ni − 1,

G which are linearly independent on a neighborhood, U of x0 . Choosing

i=1

a local coordinate frame as

10.2 Linear Equivalence of Nonlinear Systems

283

I H m n z = z11 , · · · , z1n1 , · · · , zm 1 , · · · , z nm ∈ R and constructing a mapping F : z 6→ U as Xn1

X1

Xm

Xm

F(z) = ez11 ◦ · · · ez1 1 · · · ezm1 ◦ · · · ezmnnm (x0 ). n1

1

1

(10.14)

m

Let p = F(z). Using Lemma 3.3, (ii) implies that

wF (p) = Xki (p), w zik

i = 1, · · · , m, k = 1, · · · , ni .

Hence under the coordinates z we have A @ −1 1 F∗ (X1 ), · · ·@, F∗−1 (Xn11 ), · · · , F∗−1 (X1m ), · · ·A, F∗−1 (Xnmm ) = (JF )−1 ◦ X11 , · · · , Xn11 , · · · , X1m , · · · , Xnmm ◦ F = I. For notational ease, we still use f and gi for their new forms under the coordinates z, i.e., F∗−1 ( f ) and F∗−1 (gi ). Moreover, set ni = n1 + n2 + · · · + ni , and denote by Gi ∈ Rn the i-th column of the identity In . Then adkf

gi = G(ni−1 +1) , gi = G(ni−1 +k+1) ,

Denote

(10.15)

i = 1, · · · , m, k = 1, · · · , ni − 1.

@ AT f = f 11 , · · · , fn11 , · · · , f1m , · · · , fnmm .

Using (10.15), a straightforward computation shows that J 1, s = i, and t = j + 1, s, i = 1, · · · , m; w fts = w zij 0, otherwise; t = 1, · · · , ni ; j = 1, · · · , ni − 1. Hence we can express f as AT @ m + Y (z1n1 , · · · , zm f = 0, z11 , · · · , z1n1 −1 , · · · , 0, zm 1 , · · · , znm −1 nm ).

(10.16)

Note that the vector eld Y depends only zini , i = 1, · · · , m. Calculating adnf i gi yields F adnf i gi

=

w Yn1 w Ynm wY m w Y11 , · · · , i 1 , · · · , 1i , · · · , i m i w zni w zni w z ni w z ni

GT .

Note that adnf i gi is commutative with Xki , i = 1, · · · , m, and k = 1, · · · , ni . It follows that

w fts = const. ∀s,t, i. w zini Putting (10.16) and (10.17) together, we have

(10.17)

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10 Linearization of Nonlinear Systems

⎡ 1⎤ ⎡ 1 z A1 · · · ⎢ .. ⎥ ⎢ .. ⎣ . ⎦=⎣ . m

Am 1

z

···

⎧⎡ ⎪ ⎪ 0 ⎪ ⎪ ⎢0 ⎪ ⎪ ⎢ ⎪ ⎪ ⎪ ⎪⎢ ⎢ .. ⎪ ⎪ ⎣. ⎪ ⎪ ⎪ ⎨ 0 Ai j = ⎡ ⎪ 0 ⎪ ⎪ ⎪ ⎢1 ⎪ ⎪ ⎪ ⎪⎢ ⎢. ⎪ ⎪ ⎪ ⎪⎢ ⎣ .. ⎪ ⎪ ⎪ ⎩ 0

where

⎤⎡ 1⎤ ⎡ z b1 · · · A1m .. ⎥ ⎢ .. ⎥ + ⎢ .. . ⎦⎣ . ⎦ ⎣ . Am m

z

⎤ 0 .. ⎥ u, . ⎦ 0 · · · bm

m

⎤ · · · 0 cij1 i · · · 0 c j2 ⎥ ⎥ .. .. ⎥ ⎥, . . ⎦

(10.18)

i := j

in

··· 0 cj i ⎤ 0 · · · 0 cii1 0 · · · 0 cii2 ⎥ ⎥ .. .. ⎥ ⎥, . . ⎦

i = j,

i ni

0 · · · 1 ci

and bi = (1, 0, · · · , 0)T ,

i = 1, · · · , m. 3 2

This is a canonical form of the completely controllable linear systems.

Remark 10.2. From the above argument, one sees easily that if the conditions in Theorem 10.3 are satised globally and the mapping F : Rn → M dened as in (10.14) is a global diffeomorphism, then the system (10.11) is globally equivalent to a completely controllable linear system. Next, we consider an afne nonlinear system with outputs. ⎧ m ⎪ ⎨x = f (x) + g (x)u := f (x) + g(x)u, x ∈ Rn , u ∈ Rm , ⎪ ⎩y = h(x),

¦

i

y∈

R p,

i

i=1

(10.19)

where f (x0 ) = 0, h(x0 ) = 0. Denition 10.3. The system (10.19) is said to be equivalent to a linear system (locally around an equilibrium point x0 ), if there exists a (local) coordinate chart (U, z) (z(x0 ) = 0) such that the system (10.19) is expressed in z coordinate frame as ⎧ m ⎪ ⎨z = Az + bi ui := Az + Bu, z ∈ Rn (z ∈ U), u ∈ Rm ¦ (10.20) i=1 ⎪ ⎩y = Cz, where (A, B) is completely controllable and (C, A) is completely observable. We consider the local case only. Theorem 10.4. System (10.19) is equivalent to a linear system locally around an equilibriumBpoint x0 , if and only if C  (i). dim adkf gi (x0 ) 1  i  m, 0  k  n − 1 = n,

10.2 Linear Equivalence of Nonlinear Systems

285

 C  (ii). dim dLkf h j (x0 ) 1  j  p, 0  k  n − 1 = n, (iii). there exists a neighborhood, U of x0 , such that B

Lgi Lsf Lg j Ltf h(x) = 0,

x ∈ U, 1  i, j  m, s,t  0.

Proof. (Necessity) Let f be any vector eld and h be any smooth function. Then, for any diffeomorphism T , the Lie derivative is invariant in the sense that LT∗ ( f ) ((T −1 )∗ (h)) = (T −1 )T ∗ (L f (h)). Now the necessity of (i), (ii), and (iii) are obvious because they are correct for a linear system. (Sufciency) We claim that there exists a neighborhood U of x0 , such that L[ads gi ,adt g j ] Lkf h(x) = 0, x ∈ U, f f 1  i, j  m, 0  s,t  n − 1, k  0. We need the binomial formula of Lie derivative U V k i k i Lk−i Ladk g h(x) = ¦ (−1) f Lg L f h(x). i f i=0

(10.21)

(10.22)

Using (10.22) and the condition (iii), the claim (10.21) is obvious. Since (10.21) is equivalent to that X W dLkf h(x), [adsf gi , adtf g j ] = 0, the condition (ii) now implies that E D s ad f gi , adtf g j = 0,

1  i, j  m, 0  s,t  n − 1.

According to the Theorem 10.3, the system (10.19) can be locally expressed as J z = Az + Bu y = h(z). Now under z coordinates the vector elds B C {X1 , · · · , Xn } = g1 , · · · , adnf 1 −1 g1 , · · · , gm , · · · , adnf m −1 gm . Then Xi = Gi =

w w zi ,

i = 1, · · · , n. From (iii), It is easy to see that LXi LX j hs (z) = 0,

which means

w 2 hs (z) = 0, w zi w z j

s = 1, · · · , p,

1  i, j  n, s = 1, · · · , m.

That is, hs (z) is only a linear function of z. We also assume h(z(x0 )) = 0, hence

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10 Linearization of Nonlinear Systems

y = Cz.

3 2

Example 10.1. Consider the following system ⎧ ⎪ x1 = x1 + u ⎪ ⎪ ⎪ ⎨ x1 x2 = ⎪ cos(x 2) ⎪ ⎪ ⎪ ⎩ y = h(x) = sin(x2 ).

(10.23)

We consider its local linear equivalence at the origin. It is easy to see that ⎡ ⎤ ⎤ ⎡ 1 x1 > ? 1 ⎢ ⎥ ⎥ ⎢ , and ad f g = ⎣ 1 ⎦ . f = ⎣ x1 ⎦ , g = 0 cos(x2 ) cos(x2 ) The condition (i) is satised. Since dh = (0, cos(x2 )) dLkf h = (1, 0), k  1. The condition (ii) is satised. Moveover, Lg h = 0,

Lg Lkf h = 1,

k  1.

Hence Lg Lsf Lg Ltf h = 0,

s  0, t  0.

The condition (iii) follows. That is, the system (10.23) is equivalent to a linear system around the origin. Next, we look for its linear form. Let ⎡ ⎤ 1 > ? 1 ⎢ ⎥ , X2 = ad f g = ⎣ 1 ⎦ X1 = g = 0 cos(x2 ) Construct the mapping F(z1 , z2 ) = eXz11 ◦ eXz22 (0). First, we solve eXz22 (0), that is, to solve ⎧ dx1 ⎪ ⎪ ⎨ dz = 1 2

dx 1 ⎪ ⎪ ⎩ 2= , dz2 cos(x1 ) with initial condition as

J

x1 (0) = 0 x2 (0) = 0.

10.3 State Feedback Linearization

The solution is x1 = z1 ,

287

x2 = sin−1 (z2 ).

Next, to get F(z1 , z2 ), we solve ⎧ dx1 ⎪ ⎪ =1 ⎨ dz1 ⎪ dx2 ⎪ ⎩ = 0, dz1 with initial condition as

J

x1 (0) = z2 x2 (0) = sin−1 (z2 ).

Hence the mapping F is obtained as J x1 = z1 + z2 x2 = sin−1 (z2 ), with F −1 as

J z1 = x1 − sin(x2 ) z2 = sin(x2 ).

Finally, the system (10.23) can be expressed under coordinate z as ⎧ ⎪ ⎨z1 = u z2 = z1 + z2 ⎪ ⎩ y = z2 .

(10.24)

Linear equivalence means the system is essentially a linear system. This investigation reveals the geometric essence of a linear system. Practically, it is much more useful to use a control to convert a nonlinear system into a linear system. We consider it in the following section.

10.3 State Feedback Linearization The linearization problem asks when a nonlinear control system can be converted into a linear system under a coordinate transformation and a state feedback u(x) = D (x) + E (x)v,

(10.25)

where E (x) ∈ Mm×m is non-singular. Denition 10.4. The system (10.1) is linearizable if there is a local coordinate change z = z(x) with z(0) = 0, and a state feedback control (10.25) with nonsingular E (x) such that the closed-loop system can be expressed as z = Az + Bv,

(10.26)

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10 Linearization of Nonlinear Systems

where (A, B) is a completely controllable linear system. When the coordinate transformation and the feedback are global, the linearization is global. First we dene a sequence of distributions as  I H 'i = Span adsf g j  j = 1, · · · , m; s  i − 1 ,

i = 1, 2, · · · .

The following theorem has essential importance. Theorem 10.5. The system (10.11) is linearizable about x0 , if and only if there is a neighborhood U of x0 , such that (i). 'i (x), x ∈ U, i = 1, · · · , n are non-singular and involutive; (ii). dim(' n )(x) = n, x ∈ U. Proof. (Necessary) An obvious fact is that the distributions ' i , i = 1, · · · , n are feedback invariant. That is, let f˜ = f + gD , then

g˜ = gE ,

 B C  'i = Span adsf˜ g˜ j  j = 1, · · · , m; s  i − 1 .

Denote the diffeomorphism by T , then we know that T∗ ( f˜) = T∗ ( f ) + T∗ (g)D = Az, T∗ (g) ˜ = T∗ (g)E = B. Then it is obvious that (i) and (ii) are satised by T∗ ('i ), and so are by ' i . j (Sufciency) Choosing n linearly independent vector elds as (where Xi := j ad f gi ) ⎤ ⎡ k +1 X10 · · · X1k1 X1k1 +1 · · · X1k2 · · · X1 s−1 · · · X1ks ⎢ . .. .. .. .. .. ⎥ ⎢ .. . . . . . ⎥ ⎥ ⎢ ⎥ ⎢ . .. .. .. ⎥ ks−1 +1 ⎢ . · · · Xmkss ⎥ . . . Xms ⎢ . ⎥ ⎢ . .. .. .. ⎥, ⎢ . (10.27) ⎥ . . . ⎢ . ⎥ ⎢ . .. ⎥ ⎢ . ⎥ ⎢ . . Xmk12+1 · · · Xmk22 ⎥ ⎢ .. ⎥ ⎢ .. ⎦ ⎣ . . k1 0 Xm · · · Xm where the rst t columns of vector elds are the bases of  't = Span{ adsf g j  s < t; j = 1, · · · , m}, t = 1, · · · , ks . Here g1 , · · · , gm may be reordered if necessary. Denoting k0 = 0 and di = ki − ki−1 , i = 1, · · · , s, we dene a set of distributions as  B C  ' ij = Span adkf gt  k < ki−1 + j; j = 1, · · · , m . So ' ij is spanned by the vector elds in the rst i − 1 blocks and the rst j vector elds in the i-th block of (10.27), that is,

10.3 State Feedback Linearization

289

A

@

(' 1 , · · · , 'ks ) = '11 , · · · , 'd11 , · · · , '1s , · · · , 'dss . The perpendicular co-distribution of 'i is denoted by :ks +1−i . Then we have a sequence of co-distributions as :1 , · · · , :ks , where the rst i-th element is the perpendicular co-distribution of the last i-th element in the sequence of {'t }. Corresponding to {' ij }, the co-distribution sequence is re-indexed as (:1 , · · · , :ks ) = (: 11 , · · · , : d1s , · · · , : 12 , · · · , :d2s−1 , · · · , : 1s , · · · , :ds1 ). Y Z⊥ : ij = 'ds+1−i . s+1−i +1− j

Then

Now 'dss is involutive and of co-dimension ms , by Frobenius’s theorem, there exist smooth functions z1 , · · · , zms , such that

: 11 = Span{dz1 , · · · , dzms }. It follows that ⎡

⎤ dz1 [ \ ⎢ ⎥ Ds := ⎣ ... ⎦ adkfs −1 g1 · · · adkfs −1 gms

(10.28)

dzms is non-singular. Now we claim that dLkf z j ,

j = 1, · · · , ms , k = 1, · · · , ks − 1

are linearly independent. Similar to the proof of (10.22), we can prove the following formula: W

U V X k ^ k k−i ] L f Z , adif g . Lkf Z , g = ¦ (−1)i i i=1

Now assume

ks −1 ms

¦ ¦ cij dLif z j = 0.

i=1 j=1

Using (10.29), we have _ ks −1 ms

¦ ¦ cij dLif z j ,

i=1 j=1

[ = c1ks −1

@

`

A

g1 , · · · , g m ⎤ ⎡ \ dz1 [ \ ks −1 ⎢ .. ⎥ adks −1 g · · · adks −1 g = 0. · · · cm ⎦ ⎣ m 1 s f f . s dzms

ks −1 = 0. Then we consider Hence c1ks −1 = · · · = cm s

(10.29)

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10 Linearization of Nonlinear Systems

_

ks −2 ms

¦¦

cij dLif z j ,

`

@

ad f g1 , · · · , ad f gm

A

= 0.

i=1 j=1 ks −2 = 0. Continuing this procedure shows Same argument shows that c1ks −2 = · · · = cm s i i that all c j = 0, that is, dL f z j are linearly independent, i = 1, · · · , ks − 1, j = 1, · · · , ms . In fact, the above also shows that

Lif z j ∈ 'k⊥s −1−i ,

j = 1, · · · , ms .

(10.30)

Next, we can nd ms−1 − ms functions zms +1 , · · · , zms−1 , such that dz1 , · · · , dLdf s z1 , · · · , dzms , · · · , dLdf s zms , dzms +1 , · · · , dzms−1 are linearly independent and they form the basis of Y Z⊥ : 12 = 'ds−1 . s−1 −1 Continuing this procedure, we nally can nd m functions z1 , · · · , zm such that the following n one-forms, Z ij = dLif z j , are linearly independent. ⎡

⎤ Z10 · · · Z1ds Z1ds +1 · · · Z1ds +···+d2 +1 · · · Z1ds +···+d1 ⎢ .. ⎥ .. .. .. .. ⎢ . ⎥ . . . . ⎢ ⎥ ⎢Z 0 · · · Z ds Z ds +1 · · · Z ds +···+d2 +1 · · · Z ds +···+d1 ⎥ ms ms ⎢ ms ⎥ ms ms ⎢ ⎥ .. .. .. ⎢ ⎥. . . . ⎢ ⎥ ⎢ ⎥ d +···+d +1 d +···+d s−1 2 s−1 1⎥ 0 ⎢ Zms−1 · · · Zms−1 · · · Zms−1 ⎢ ⎥ ⎢ ⎥ .. .. ⎣ ⎦ . .

Zm0 1

···

(10.31)

Zmd11

Moreover, the rst t columns of the co-vector elds in (10.31) span the co-distributions :t , t = 1, · · · , km . Denote by r1 = d1 + · · · + ds , r2 = d1 + · · · + ds−1 , · · · , rs = d1 , we can get n linearly independent functions, grouped as ⎡ ⎤ z1 ⎢ .. ⎥ 1 z1 := ⎣ . ⎦ , z12 = L f z11 , · · · , z1r1 Lrf1 −1 z11 ⎡zms ⎤ zms +1 ⎢ .. ⎥ 2 r −1 2 z1 := ⎣ . ⎦ , z2 = L f z21 , · · · , z2r2 = L f2 z21 .. .

zms−1 ⎡

⎤ zm2 +1 ⎢ ⎥ zs1 := ⎣ ... ⎦ , zs2 = L f zs1 , · · · , zsrs = Lrfs −1 zs1 zm1

10.3 State Feedback Linearization

291

Using this set of functions as a local coordinate frame, we have = zi2

zi1 .. .

ziri−1 = ziri ziri = Lrfi zi1 + Lg Lrfi −1 zi1 u, Set

⎡ ⎤ Lg Lrf1 −1 z11 ⎢ r −1 ⎥ ⎢Lg L f2 z21 ⎥ ⎢ ⎥, D=⎢ .. ⎥ ⎣ ⎦ . Lg Lrfs −1 zs1

(10.32) i = 1, · · · , s.

⎡ r1 1 ⎤ L f z1 ⎢Lrf2 z21 ⎥ ⎢ ⎥ F = ⎢ . ⎥. ⎣ .. ⎦ Lrfs zs1

We claim that D is non-singular. Dene an n × m matrix, E as ⎤ ⎡ dz11 ⎥ ⎢ .. ⎥ ⎢ . ⎥ ⎢ r1 −1 1 ⎥ ⎢ _⎢dL f z1 ⎥ ` ⎥ ⎢ .. ⎥,g . E= ⎢ ⎥ ⎢ . ⎥ ⎢ ⎢ dzs1 ⎥ ⎥ ⎢ .. ⎥ ⎢ ⎦ ⎣ . dLrfs −1 zs1

Then rank(E) = m. This is because the rst n forms are linearly independent, and as a convention g1 , · · · , gm are linearly independent. Now the only m non-zero rows of E form D, so rank(D) = m. Using the controls as u = −D−1 F + D−1 v, the system (10.32) becomes zi1 .. .

= zi2

ziri−1 = ziri ziri = vi ,

(10.33) i = 1, · · · , s,

where vi are a partition of v as ⎡ ⎤ ⎡ ⎤ v1 vms +1 ⎢ ⎥ ⎢ ⎥ v1 = ⎣ ... ⎦ , v2 = ⎣ ... ⎦ , vms

··· ,

⎡ ⎤ vm2 +1 ⎢ ⎥ vs = ⎣ ... ⎦ .

vms−1

vm1

The system (10.33) is a completely controllable linear system. Remark 10.3. Note that from the above proof, one sees that as long as dim('i+1 − 'i ) = dim(' i − ' i−1)

3 2

292

10 Linearization of Nonlinear Systems

the involutivaty of 'i is assured by ' i+1 . So the condition of the involutivaties of ' i , i = 1, · · · , n in Theorem 10.5 can be replaced by the following condition: When dim(' i+1 − ' i ) < dim(' i − 'i−1 ),

'i is involutive. ('0 := {0}) Consider a single input system. It is clear that if dim(' n ) = n, then dim(' k ) = k,

k = 1, · · · , n.

Moveover, using the above Remark, if ' n−1 is involutive, then all 'k , k = 1, · · · , n are involutive. So we have the following corollary. Corollary 10.1. The system (10.11) with m = 1 is linearizable about x0 , if and only if there is a neighborhood U of x0 , such that (i). 'n−1 (x), x ∈ U, is involutive; (ii). dim(' n )(x0 ) = n.

10.4 Linearization with Outputs This section considers the state feedback linearization of nonlinear systems with outputs. In certain sense, it is a combination of two previous sections. First, we dene the problem clearly. Denition 10.5. Consider system (10.19), with an equilibrium x0 . (local) linearization with outputs means nding a state feedback control (10.25) and a coordinate transformation z = z(x), such that (locally) the closed-loop system becomes ⎧ m ⎪ ⎨z = Az + b v := Az + Bv, z ∈ Rn (z ∈ U), u ∈ Rm , ¦ ii (10.34) i=1 ⎪ ⎩y = Cz, where (A, B) is completely controllable. We consider only the local linearization problem here, and rst consider the single-input single-output case. Theorem 10.6. Consider system (10.19) with m = p = 1. The system is locally linearizable at x0 with output, if and only if (i). g(x0 ), ad f g(x0 ), · · · , adn−1 g(x0 ) are linearly independent. f (ii). the relative degree is U (x) = U < f, x ∈ U, where U is a neighborhood of x0 . (iii). Set u = D (x) + E (x)v, with ⎧ U L f (x) ⎪ ⎪ ⎪ D (x) = − , ⎪ U −1 ⎨ Lg L f h(x) (10.35) ⎪ 1 ⎪ ⎪ , x ∈ U, ⎪ ⎩E (x) = U −1 Lg L f h(x)

10.4 Linearization with Outputs

and

J

293

f˜(x) = f (x) + g(x)D (x), g(x) ˜ = g(x)E (x), x ∈ U.

Then the n + 1 vector elds g, ˜ ad f˜ g, ˜ · · · , adnf˜ g˜ are commutative. Proof. (Necessity) Since the linearizable system is completely controllable, we can, without loss of generality, assume it has the Brunovsky canonical form. That is, under z = T (x), we have J T∗ ( f ) + T∗ (g)D = (0, z1 , · · · , zn−1 )T T∗ (g)E = (1, 0, · · · , 0)T . Then we have ⎧ T ⎪ ⎨ f˜ := (a(z), z1 , · · · , zn−1 ) g˜ := (b(z), 0, · · · , 0)T ⎪ ⎩˜ h := Cz = (c1 , · · · , cn )z,

(10.36)

where a(z) = −D (z)/E (z), b(z) = 1/E (z). From (10.36) it is ready to check that conditions (i) and (ii) are satised. As for condition (iii), let

D¯ = (T −1 )∗ D ,

E¯ = (T −1 )∗ E .

It can easily be calculated that U ¯ U −1 ¯ D¯ = −L f (h)/L (h) gL f

a 1 =− − (cU +1 z1 + · · · + cn zn−U ), b b(z)cU 1 E¯ = . b(z)cU

(10.37)

Hence under the z coordinate frame we have f˜ = f¯+ g¯ D¯ = (cU +1 z1 , · · · , cn zn−U )T := Az g˜ = g¯ E¯ > ?T 1 = , 0, · · · , 0 := b. cU It follows that adkf˜ g˜ = (−1)k Ak b = const.,

k = 0, 1, · · · , n.

Condition (iii) is satised. (Sufciency) Using Theorem 10.5, the closed-loop system (with respect to f˜, g) ˜ can be expressed as a complete control system:

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10 Linearization of Nonlinear Systems

z = Az + bu.

(10.38)

Now consider the output. It is easy to check that k < U −1

Lg˜ Lkf˜ h = Lg Lkf h = 0,

U −1 U −1 Lg˜ L f˜ h = E Lg L f h = 1

As for k  U , note that U

U

U −1

L f˜ h = L f h + D Lg L f

h = 0.

It follows that Lg˜ Lkf˜ h = 0,

k  U.

Using binomial formula (10.22), it is easy to prove that LXi h = const.,

i = 1, · · · , n,

(10.39)

where g˜ = Ai b, Xi = (−1)i−1 adi−1 f˜

i = 1, · · · , n.

Since h(x0 ) = 0, (10.39) implies that h(z) = cz. 3 2 When p > 1 similar argument shows the following Corollary 10.2. Consider system (10.19) with m = 1. The system is locally linearizable at x0 with outputs, if and only if (i). g(x0 ), ad f g(x0 ), · · · , adn−1 g(x0 ) are linearly independent. f (ii). the relative degree is Ui (x) = Ui < f, i = 1, · · · , p, x ∈ U, where U is a neighborhood of x0 . (iii). Choose any 1  i  p and set u = D (x) + E (x)v, with ⎧ U L f i (x) ⎪ ⎪ ⎪ D (x) = − , ⎪ U −1 ⎨ Lg L f hi (x) ⎪ 1 ⎪ ⎪ , x ∈ U, ⎪ ⎩E (x) = U −1 Lg L f i hi (x) and

J

(10.40)

f˜(x) = f (x) + g(x)D (x), g(x) ˜ = g(x)E (x), x ∈ U.

Then the n + 1 vector elds X1 := g, ˜ X2 := ad f˜ g, ˜ · · · , Xn+1 := adnf˜ g˜ are commutative. (iv). LXi (h j ) = const., i = 1, · · · , n, j = 1, · · · , p.

10.5 Global Linearization

295

When we consider the multi-input multi-output case, it is closely related to inputoutput linearization. We refer to [3] or [9] for a general discussion. The following theorem provides a convenient sufcient condition. In fact, it has been discussed in Chapter 9. Theorem 10.7. Consider system (10.19). Assume (i). its decoupling matrix is of full row rank, i.e. rank(D(x)) = p, x ∈ U, where U is a neighborhood of x0 . (ii).

p

¦ Ui = n.

i=1

Then the system is linearizable with outputs. In fact, it is an immediate consequence of the normal form discussed in Chapter 9. We leave to reader for nding the feedback control.

10.5 Global Linearization This section considers the problem of global linearization. The proof is constructive, so it also provides an efcient algorithm for linearization. Denition 10.6. Consider system m

x = f (x) + ¦ gi (x)ui ,

x ∈ M,

(10.41)

i=1

where M is an n dimensional manifold. The global state feedback linearization means nding the control u(x) = D (x) + E (x)v,

x ∈ M,

(10.42)

and a diffeomorphism T : M → U ⊂ Rn , where U is an open subset of Rn , such that in U the closed-loop system becomes a completely controllable linear system. First, we consider single-input case. Dene

'i = Span{g, ad f g, · · · , adi−1 f g},

i = 1, · · · , n.

Then we have the following necessary conditions and sufcient conditions. Theorem 10.8. If system (10.41) is globally linearizable, then (i). 'i , i = 1, · · · , n are involutive; (ii). there exist Xi ∈ ' i , i = 1, · · · , n, which are linearly independent, and there is an open set V ⊂ Rn , such that a mapping F : V → M dened by F(z1 , · · · , zn ) = eXz11 ◦ · · · ◦ eXznn (x0 )

(10.43)

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10 Linearization of Nonlinear Systems

is a one-to-one and onto mapping. If V is a convex set, conditions (i) and (ii) are also sufcient. Proof. We prove the necessity part rst. Since a completely controllable linear system can be transformed into Bronovsky canonical form, without loss of generality, we can assume that J T∗ (g)E = (1, 0, · · · , 0)T , T∗ ( f ) + T∗ (g)D = (0, z1 , · · · , zn−1 )T . Hence

⎧ VT U D ⎪ ⎪ ¯ ⎨ f := T∗ ( f ) = − , z1 , · · · , zn−1 E U VT ⎪ 1 ⎪ ⎩g¯ := T∗ (g) = , 0, · · · , 0 . E

(10.44)

A straightforward computation shows that k 1 ad f¯g¯ = (×, · · · , ×, (−1)k , 0, · · · , 0)T , / 01 2 E

k

which means under z coordinate frame we have a b w w 'k = Span ,··· , , w z1 w zk Condition (i) is satised. Choose Xi :=

T∗−1

U

V w , w zi

k = 1, · · · , n.

i = 1, · · · , n,

then Xi ∈ 'i , and X1 , · · · , Xn are linearly independent. Dene a mapping F : V → M as F(t1 , · · · ,tn ) := etX11 ◦ · · · ◦ etXnn (T −1 (0)). Using Proposition 3.2, we have w wz

w

T ◦ F = et1 1 ◦ · · · ◦ etwnzn (0), which is an identity mapping on V . Note that T : M → V is a diffeomorphism, so F must be one-to-one and onto. 3 2 To prove the sufciency part, we need some preparations. Lemma 10.1. Assume n vector elds X1 , · · · , Xn ∈ V (M) are linearly independent,

'i = Span{X1 , · · · , Xi },

i = 1, · · · , n

are involutive. F : V → M is a mapping from an open set V ∈ Rn to M, dened by F(t1 , · · · ,tn ) := etX11 ◦ · · · ◦ etXnn (x0 ).

10.5 Global Linearization

297

If F is a one-to-one and onto mapping, the F is a diffeomorphism. Proof. For statement ease, dene X

i+1 ◦ · · · ◦ et j j , e(i, j) := etXi i ◦ eti+1

X

X

X

e−(i, j) := e−tj j ◦ e−tj−1 ◦ · · · ◦ eX−ti i . j−1 To prove F is a diffeomorphism it is enough to show that the Jacobi matrix of F is nonsingular everywhere. It is easy to calculate that > ? wF wF ,··· , , JF = w t1 w tn where

wF = (e1,i−1 )∗ Xi (ei,n (x0 )). w ti We prove the result by contradiction. Assume there are {Oi }, i = 1, · · · , n, which are not identically zero, such that n

¦ Oi (e1,i−1 )∗ Xi (ei,n (x0 )) = 0,

(10.45)

i=1

where (e(1,0) )∗ is understood as the identity mapping. Denote k = max{i | Oi := 0}, then (10.45) becomes k

¦ (Oi H (1,i−1))∗ Xi(ei,n (x0 )) F

i=1

= (e(1,k−1) )∗ ·

k−1

G

¦ Oi (e−(i,k−1))∗ Xi (e(i,n) (x0 )) + OkXk (e(k,n) (x0 ))

= 0.

i=1

Since (e(1,k−1) )∗ is an isomorphism on vector space, we have Xk (H (k,n) (x0 )) =

1 k−1 ¦ Oi (e(i,k−1) )∗ Xi (e(i,n) (x0 )). Ok i=1

(10.46)

Since 'k−1 is involutive, k−1

¦ Oi (e(i,k−1) )∗ Xi (e(i,n)(x0 )) ∈ 'k−1 (e(k,n) (x0)),

i=1

which means (10.46) contradicts the fact that X1 , · · · , Xk are linearly independent. Therefore, JF is invertible. 3 2 Lemma 10.2. Let W ⊂ Rn be a convex set, and H : W → Rn be a Cf mapping. If the Jacobi matrix of H is a non-singular and upper-triangular matrix, then H : W → H(W ) is a diffeomorphism. Moreover, H(W ) ⊂ Rn is an open set. Proof. Denote the Jacobi matrix of H by

298

10 Linearization of Nonlinear Systems

U JH =

 V w hi  i, j = 1, · · · , n . wxj 

Then

w hi = 0, wxj w hi = 0, w xi We show H is one-to-one. Let

i > j, i = 1, · · · , n.

a = (a1 , · · · , an ) := b = (b1 , · · · , bn ). then there is a k := max{i|ai := bi }. Using the mean value theorem, we have  n w hk  hk (b) − hk (a) = ¦  (bi − ai )  i=1 w xi [

= w hk w xk |[ (bk − ak ) := 0,

[ ∈ (a, b).

That is, H(b) := H(a). Since H is one-to-one, and the Jacobi matrix of H is nonsingular, H : W → H(W ) is a diffeomorphism. From rank theorem, H is an open 3 2 mapping. Hence H(W ) ⊂ Rn is open. Now we prove the sufciency part of Theorem 10.8. Proof. (Sufciency part of Theorem 10.8) From Lemma 10.1, F −1 can be chosen as a global coordinate frame. Then we have b a w w , i = 1, · · · , n. 'i = Span ,··· , w z1 w zi Usimg similar argument as in the proof of Theorem 10.5, one sees that ⎧ w fs ⎪ ⎪ = 0, k = 1, · · · , n − 2, s  k + 2, ⎨ w zk ⎪ wf ⎪ ⎩ k+1 := 0, k = 1, · · · , n − 1. w zk Dene a mapping

J R1 :

z1i = fi+1 (z), z1n = zn .

(10.47)

i = 1, · · · , n − 1

Using (10.47), one sees that the Jacobi matrix of R1 is upper triangular with diagonal elements ww zf2 , · · · , wwz f2 , 1. According to Lemma 10.2, R1 : V → R1 (V ) is a 1

n−1

diffeomorphism. Denote f 1 = (R1 )∗ f , and calculate it as ⎧ n ⎪ ⎨ f 1 = ¦ w fi+1 ft , i = 1, · · · , n − 1 i t=1 w zt ⎪ ⎩f1 = f . n

n

(10.48)

10.5 Global Linearization

299

Using (10.47) and (10.48), we can show that f i1 , i = 1, · · · , n as functions of z still satisfy (10.47). Hence we can dene R2 in a similar way. In general, we dene J j−1 (z), i = 1, · · · , n − 1, z j = fi+1 R j : ij j−1 zn = zn . Using mathematical induction, we can prove that ⎧ n w f j−1 ⎪ j−1 ⎨f j = ft , i = 1, · · · , n − 1 ¦ wi+1 i z t t=1 ⎪ ⎩ j fn = fnj−1 ; and

⎧ j ⎪ w fs ⎪ ⎪ = 0, k = 1, · · · , n − 2, s  k + 2 ⎪ ⎪ w zk ⎪ ⎪ ⎪ ⎨ j j−1 j−1 w fk+1 w fk+2 w gk+1 = · > 0, k = 1, · · · , n − 2 ⎪ w zk w zk+1 w zk ⎪ ⎪ ⎪ ⎪ ⎪ w fnj ⎪ w fnj−1 ⎪ ⎩ = > 0. w zn−1 w zn−1

(10.49)

(10.50)

Similar to the proof of Theorem 10.5, we can show that f j = (R j )∗ f j j j = (×, · · · , ×, zn− j , zn− j+1 , · · · , zn−1 )T ,

j = 1, · · · , n − 1.

Finally, we dene T = Rn−1 ◦ F −1 . Then T : M → Rn−1 (V ) is a diffeomorphism. Moreover, under the global coordinate frame z determined by T , we have f = ( f1 (z), z1 , · · · , zn−1 )T . Note that in the coordinate frame determined by F −1 the g has the form as (g1 , 0, · · · , 0)T . Since in each Ri its Jacobi matrix is upper triangular, then in z coordinate frame we have g = (h(z), 0, · · · , 0)T . f (z)

1 , and E (z) = Taking D (z) = − h(z)

1 , h(z)

the global linearized form is obtained.

3 2

Remark 10.4. In Theorem 10.8 the choice of x0 does not affect the conclusion. In fact, if for a certain x0 the required conditions are satised, then it is easy to verify (from the linearized form) that they are also satised for any x<0 ∈ M. Since the proof of Theorem 10.8 is constructive, it provides a linearization algorithm. We summarize it as follows. Algorithm 10.1. Step 1. Choose Xi ∈ ' i , such that X1 , · · · , Xn are linearly independent. Step 2. Construct a mapping F(z1 , · · · , zn ) = eXz11 ◦ · · · ◦ eXznn (x0 ).

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10 Linearization of Nonlinear Systems

Using F −1 as a coordinate frame on M and calculate F∗−1 ( f ), and F∗−1 (g). Step 3. Dene mappings R j recursively as J j−1 z j = fi+1 (z) R j : ij j−1 zn = zn , j = 1, · · · , n − 1. Calculate f j = (R j )∗ f . Step 4. Calculate f n−1 = (Rn−1 )∗ ( f ) = ( f1 (z), z1 , · · · , zn−1 )T gn−1 = (Rn−1 )∗ (g) = (h(z), 0, · · · , 0)T . Step 5. Set u=−

f1 (z) 1 + v h(z) h(z)

to get the linearized system. Next, we give two examples. Example 10.2. Consider the following system ⎤ ⎡ c 1 d > ? x1 + 23 x31 + 15 x51 x1 1+x21 ⎥ ⎢ u, + =⎣ 1 + x21 ⎦ x2 0 2 x2 − x1

(10.51)

where x ∈ R2 . It is easy to calculate that g = ((1 + x21 )−1 , 0)T , ad f g = (−1, (1 + x21)−1 )T . Choose

? > ? > 10 1 + x21 (1 + x21 )2 . = (X1 , X2 ) = (g, ad f g) 01 0 1 + x21

Let F(z1 , z2 ) = eXz11 ◦ eXz22 (0). Then F : R2 → R2 is the identity mapping. The conditions in Theorem 10.8 are satised. Hence the system is globally linearizable. In the following we linearize it. Dene J y1 = f2 = x22 − x1 R: y2 = x2 . Then

? −1 2x2 ; JR = 0 1 ⎡ ⎤ 1 ? > > ? −1 2x2 ⎣ h 2 ⎦ 1 + x1 := F∗ (g) = ; 0 1 0 0 >

10.5 Global Linearization

⎡

301

⎤

> ? x1 + 23 x31 + 15 x51 > ? −1 2x2 ⎢ a ⎥ 2 F∗ ( f ) = 1 + x1 ⎦ := y . 0 1 ⎣ 1 x22 − x1 Here

1 := 0; 1 + (y1 − y22 )2 U V 2 1 a = − (y1 − y22 )2 + (y1 − y22 )3 + (y1 − y22 )5 3 5 2 2 −1 2 ×(1 + (y1 − y2 ) ) + 2y2 − 2(y1 − y22 )y2 . h=−

Taking u = − ah + 1h v, the closed-loop system becomes J y1 = v y2 = y1 . Next, we consider an example, which is not globally linearizable. Example 10.3. Consider the following system ? > ? > ? > −x − sin(x2 ) cos(x2 ) x1 e 1 sin(x2 ) u, = + −x1 e cos(x2 ) x2 sin2 (x2 )

x ∈ R2 .

(10.52)

It was pointed in [2] that this system satises the local linearization condition everywhere over R2 , but it is not globally linearizable. In the following we will show that it does not satisfy the condition (ii) of Theorem 10.8. For system (10.52) we have g = (e−x1 sin(x2 ), e−x1 cos(x2 )) ; T ad f g = (e−x1 cos(x2 ), e−x1 sin(x2 )) . T

Choosing X1 ∈ Span{g} and X2 ∈ Span{g, ad f g} is equivalent to nding a nonsingular and upper-triangular matrix ? > a(x) b(x) , H= 0 c(x) such that

(X1 , X2 ) = (g, ad f g)H.

Choosing any x0 ∈ R2 , and nonsingular upper-triangular matrix H, we construct F : V → R2 as F(y1 , y2 ) = eXy11 ◦ eXy22 (x0 ), where V ⊂ R2 is open. We will show that F can not be onto. Consider the rst integral curve eXy22 (x0 ) : ⎧ dx1 ⎪ ⎪ = be−x1 sin(x2 ) + ce−x1 cos(x2 ) ⎪ ⎪ dy ⎪ 2 ⎨ dx2 = be−x1 cos(x2 ) + ce−x1 sin(x2 ), ⎪ ⎪ dy 2 ⎪ ⎪ ⎪ ⎩ x(0) = x0 .

302

10 Linearization of Nonlinear Systems

Choose an integer k such that x20 < 2kπ+ π/2, where x0 = (x10 , x20 ). We can show that this curve lays under the horizontal line x2 = 2kπ + 3π/2. Since H is nonsingular, c(x) is either positive or negative for all x. Assume c(x) > 0, then as Y2 > 0, the trajectory will not cross with x2 = 2kπ + π/2. Otherwise let y¯2 > 0 be the rst y > 0 such that the trajectory touches the line x2 = 2kπ + π/2. Then at this moment we have d x2 (y¯2 ) = −ce−x1 < 0. dy2 Hence x2 will decrease, and then the trajectory will not cross the line, i.e., it will stay under the line. For the case c(x) < 0, a similar argument shows the same conclusion. Next, we consider the second integral curve eXy11 (x) : ⎧ dx1 ⎪ ⎪ = ae−x1 sin(x2 ) ⎨ dy1 ⎪ ⎪ dx2 = ae−x1 cos(x ). ⎩ 2 dy1 If x2 (0) := mπ + π/2, where m is an integer, then the trajectory will be That is, x1 = − ln | cos(x2 )| + c.

dx1 dx2

= tan(x2 ).

If x2 = mπ + π/2, the trajectory becomes x2 = mπ + π/2. Hence, the trajectory of the second integral stays between x2 = mπ ± π/2. Summarizing the above arguments, we conclude that F(y1 , y2 ) = eXy11 ◦ eXy22 (x0 ) will stay under the line forever. Next, we consider the multi-input case. Consider system m

x = f (x) + ¦ gi (x)ui ,

x ∈ M,

(10.53)

i=1

where M is an n dimensional Cf manifold. Denition 10.7. System (10.53) is said to be globally linearizable, if there exists a diffeomorphism T : M → U, (U ∈ Rn open) and a feedback control u = D (x) + E (x)v, with E (x) nonsingular, such that under coordinate frame z, determined by T , the closed-loop system becomes a completely controllable linear system. Denote

G1 = G = Span{g1 , · · · , gm }, Gi+1 = Gi + ad f Gi , i = 2, 3, · · · .

Then we have the following result. Theorem 10.9. [5] If system (10.53) is globally linearizable, then

10.5 Global Linearization

303

(i). There exist vector elds g¯1 , · · · , g¯m ∈ G1 and integers n1 , · · · , nN , m = n1  n2  · · ·  nN > 0 with

N

¦ ni = n, such that the n vector elds

i=1

B

g¯1 , · · · , adN−1 g¯nN C = g¯1 , · · · , g¯m , · · · , ad f g¯1 , · · · , ad f g¯n2 , · · · , adN−1 f f are linearly independent, and C B Gi = Span adsf g¯js | s  i − 1; js = 1, · · · , ns+1 ,

C

i = 1, · · · , N.

(ii). Let Xi be the i-th vector eld in C . We dene

'i := Span{X1 , · · · , Xi },

i = 1, · · · , n.

Then 'i , i = 1, · · · , n are involutive. (iii). There exist Zi ∈ ' i , i = 1, · · · , n, x0 ∈ M, such that Z1 (z), · · · , Zn (z), z ∈ Rn are linearly independent. Moreover, a mapping \ : V → M, dened as

\ (y1 , · · · , yn ) = eZy11 ◦ eZy22 ◦ · · · ◦ eZynn (x0 ) is bijective, where V ⊂ Rn is open. Conditions (i)–(iii) are also sufcient provided in (iii) the V is convex. Correspondingly, we can have the following algorithm. Algorithm 10.2. Step 1. Choose a set of basis g¯1 , · · · , g¯m of G. Find n linearly independent vector elds B C C = g¯1 , · · · , g¯m , · · · , ad f g¯1 , · · · , ad f g¯n2 , · · · , adN−1 g ¯ , n N f as required in (i) of Theorem 10.9 that the distributions 's , s = 1, 2 · · · , n, which are spanned by the rst s vector elds respectively, are involutive. Step 2. Choose Zs ∈ ' s , s = 1, · · · , n linearly independent, such that F : V → M, dened by F(z1 , · · · , zn ) = eZz11 ◦ eZz22 ◦ · · · ◦ eZznn (x0 ) is one-to-one and onto, where V ⊂ Rn is open. Step 3. Take F −1 as a global coordinate chart of M and calculate F∗−1 ( f ),

F∗−1 (gi ),

i = 1, · · · , m.

Step 4. Denote pi =

i+1

¦ ni ,

i = 1, · · · , N − 1.

s=1

Then dene recursively a set of diffeomorphisms as ⎧ j−1 ⎪ ⎨ f pi+1 +k , s = pi + k, i = 2, · · · , N − 2, j R j : zs = k = 1, · · · , ni+2 ⎪ ⎩ j−1 zs , otherwise,

304

10 Linearization of Nonlinear Systems

where

f 0 = F∗−1 ( f ),

f j−1 = (R j−1 )∗ ◦ F∗−1 ( f ), j  2.

Step 5. Calculate f˜ := (RN−1 )∗ ◦ F∗−1 ( f ) = (h1 , · · · , hm , z1 , · · · , zn2 , z p1 +1 , · · · , z p1 +n3 , · · · , z pN−2 +1 , · · · , z pN−2 +nN )T ; (g˜1 , · · · , g˜m ) := (RN−1 )∗ ◦ F∗−1 (g1 , · · · , gm ) = (W T (z)|0)T , where W T (z) is an m × m nonsingular matrix. Step 6. Setting feedback u = −W (z)−1 (h1 , · · · , hm )T + W (z)−1 v yields the linearized system. We give an example for this. Example 10.4. Consider the following system ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ 1/3 −1 x1 x2 ⎥ ⎣x2 ⎦ = ⎣ ⎦+⎢ 1 0 ⎦ u1 ⎣ 1/2 1/2 1/3 1/2 x3 2x1 x3 + 2x2 x3 −2x2 x3 ⎤ ⎡ 2/3 x1 + x2 − 3x2 ⎥ ⎢ 2/3 +⎣ ⎦ u2 , 3x2 1/2 1/2 −2x1 x3 − 2x2 x3 where

x ∈ M := {x ∈ R3 | x2 > 0; x3 > 0}.

Calculate that

U ad f g1 =

and

(10.54)

2 −2/3 1/2 1 −2/3 1/3 1/2 x , 0, − x2 x3 − 2x2 x3 3 2 3 4/3 1/2

det(g1 , g2 , ad f g1 ) = −6x2 x3

Then we can choose

:= 0,

VT

x ∈ M.

C = {g1 , g2 , ad f g1 }.

Since

−1/3

[g1 , g2 ] = (1 − x2

)g1 ,

the distributions ' 1 , '2 , and ' 3 , spanned by the rst 1, 2, and 3 vector elds of C are involutive. Choosing −1/3

X1 = −x2

1/2

g1 = (−1, 0, 2x3 )T ∈ ' 1 ,

1 1 −2/3 g2 = (−1, 1, 0)T ∈ ' 2 , X2 = − (x1 + x2 )x−1 2 g1 + x2 3 3 X3 = (1, 0, 0)T ∈ ' 3 = T (M),

10.5 Global Linearization

and V = {y ∈ R3 | y1 > −1, y2 > −1}, x0 = (0, 1, 1) ∈ M, we can construct a mapping F : V → M as F(y1 , y2 , y3 ) = eXy11 ◦ eXy12 ◦ eXy33 (x0 ). It is easy to calculate the mapping F as ⎧ ⎪ ⎨x1 = −y1 − y2 + y3 F : x2 = y2 + 1 ⎪ ⎩ x3 = (y1 + 1)2 . So it is ready to check that F is one-to-one and onto. Under the coordinate frame determined by F −1 we have F∗−1 (g1 ) = (−(y2 + 1)1/3, 0, 0)T ; F∗−1 (g2 ) = (y1 − y3 − 1, 3(y2 + 1)2/3, 0)T ; F∗−1 ( f ) = (y3 − y1 + 1, 2(y3 − y1 + 1)(y1 + 1), y3 − y1 + 1)T. Dene R1 : V → R1 (V ) as

⎧ ⎪ ⎨z1 = f3 = y3 − y1 + 1 R 1 : z 2 = y2 ⎪ ⎩ z 3 = y3 ,

where R1 (V ) = {z ∈ R3 | z2 > −1; z3 > z1 − 2}. Set T = R1 ◦ F −1 , then

⎧ ⎪ ⎨z1 = x1 + x2 T : z 2 = x2 − 1 ⎪ ⎩ 1/2 z3 = x1 + x2 + x3 − 2,

which maps M → R1 (V ), is a diffeomorphism. Moreover, under z we have T∗ ( f ) = (0, 1, z1 )T , T∗ (g1 ) = ((z2 + 1)1/3, 0, 0)T , T∗ (g2 ) = (z1 , 3(z2 + 1)2/3, 0)T . Take control u = D (z) + E (z), where ⎡ 1

z1 (z2 + 1)−1

⎤

⎥ ⎢ 3 D =⎣ ⎦, 1 −2/3 − (z2 + 1) 3 > ? 1 3(z2 + 1)2/3 −z1 . E= 0 (z2 + 1)1/3 3(z2 + 1) Then the closed-loop system becomes

305

306

10 Linearization of Nonlinear Systems

⎧ ⎪ ⎨z1 = v1 z2 = v2 ⎪ ⎩ z3 = z1 .

10.6 Non-regular Feedback Linearization Consider system (10.11) again. Throughout this section, it is assumed to be analytic. We give a rigorous denition for non-regular feedback: Denition 10.8. System (10.55) is non-regular state feedback linearizable at the origin, if there exists a feedback control u = D (x) + E (x)v,

(10.55)

with m × k matrix E (x) and a local diffeomorphism z = [ (x) such that, in coordinate frame z, the closed-loop system can be expressed locally as a completely controllable linear system. If E (x) is a square non-singular matrix, it is called the regular state feedback linearization. When k = 1 the linearization is called the single-input linearization. The following lemma can simplify our investigation: Lemma 10.3. (Heymann’s Lemma) [8] Consider a multi-input linear system x = Ax + Bu,

x ∈ Rn , u ∈ Rm .

(10.56)

Assume (A, B) is a completely controllable pair, then there exists a feedback u = Fx + Gv,

v ∈ R1

such that the closed-loop (single-input) system x = (A + BF)x + BGv,

x ∈ Rn , v ∈ R

(10.57)

is also completely controllable. The following lemma, which simplied the study of non-regular state feedback linearization, is an immediate consequence of Heymann’s Lemma. Lemma 10.4. System (10.11) is non-regular state feedback linearizable, if and only if it is single-input linearizable, i.e., linearizable by control (10.55) with m × 1 vector E (x). Based on this lemma, in this section we consider only the single-input linearization problem. The following lemma is useful in the sequel. Lemma 10.5. [10] Consider system (10.11). Let A = J f (0) be the Jacobian matrix of f at the origin, B = g(0). If the system is linearizable, then (A, B) is completely controllable.

10.6 Non-regular Feedback Linearization

307

We rst consider a canonical form of non-regular feedback linear systems. A constant vector, b = (b1 , · · · , bn )T ∈ Rn , is said to be of non-zero component if bi := 0, ∀i. Proposition 10.3. A linear control system m

x = Ax + ¦ bi ui := Ax + Bu,

x ∈ Rn , u ∈ Rm

(10.58)

i=1

is completely controllable, if and only if there exists two matrices F, G such that the closed-loop system x = (A + BF)x + BGv can be converted, by a linear coordinate change, into the following form ⎡ ⎤ ⎡ ⎤ b1 d1 0 · · · 0 ⎢b2 ⎥ ⎢ 0 d2 · · · 0 ⎥ ⎢ ⎥ ⎢ ⎥ z = Az + bv := ⎢ . . .. ⎥ z + ⎢ .. ⎥ v, ⎣.⎦ ⎣ .. .. ⎦ . bn 0 0 · · · dn

(10.59)

where di , i = 1, · · · , n are distinct and b is of non-zero component. Proof. A straightforward computation shows that the controllability matrix C of such a system is a mild variation of Vandermonde’s matrix n

det(C) = – bi –(d j − di ) := 0. i=1

(10.60)

i< j

3 2 From the above proposition we can call (10.59) the reduced single-input feedback A-diagonal (RSIFAD) canonical form. Moreover, we give the following assumption: A1. A is a diagonal matrix with distinct diagonal elements di and is non-resonant. Lemma 10.6. Assume matrix A satises A1 and g is a k-th degree homogeneous vector eld, k  2. Then there exists a k-th degree homogeneous vector eld K such that adAx K = g.

(10.61)

Proof. For a given K , let f = adAx K . Then a straightforward computation shows that fi , the i-th component of f depends only on Ki , the i-th component of K . Now let xr11 · · · xrnn be a term of Ki , then a straightforward computation shows that ⎞ ⎛ ⎞⎤ ⎡ ⎤ ⎡⎛ × × d1 x 1 ⎢⎜ .. ⎟ ⎜ .. ⎟⎥ ⎢ .. ⎥ ⎢⎜ . ⎟ ⎜ . ⎟⎥ ⎢ . ⎥ ⎟ ⎜ r1 r ⎟⎥ ⎢ r1 r ⎥ ⎢⎜ n ⎟⎥ n⎥ ⎟ ⎜ ⎢ ⎜ (10.62) adAx K = ⎢ ⎢⎜ di xi ⎟ , ⎜x1 xn ⎟⎥ = ⎢Pi x1 xn ⎥ , ⎢⎜ . ⎟ ⎜ . ⎟⎥ ⎢ . ⎥ . . . ⎣⎝ . ⎠ ⎝ . ⎠⎦ ⎣ . ⎦ dn xn × × where

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10 Linearization of Nonlinear Systems

Pi = d1 r1 + · · · + dnrn − di ,

(10.63)

since A is non-resonant and Pi := 0. Now for each term xr11 · · · xrnn of gi we can construct a corresponding term P1i xr11 · · · xrnn of Ki such that adAx K = g. 3 2 Since all the vector elds and functions are assumed to be analytic, all the functions and their derivatives have convergent Taylor series expansions. Note that if A satises A1, then for a vector eld g = gk xk + gk+1 xk+1 + · · · ∈ O(1x1k ), applying Lemma 10.6 to each term, we can nd a vector eld K ∈ O(1x1k ) such that adAx K = g. Now let us get back to the linearization. We consider the following system: m

x = Ax + [ (x) + ¦ gi (x)ui ,

(10.64)

i=1

where A satises A1 and [ (x) = O(1x12 ). An immediate result is Proposition 10.4. Consider system (10.64) with A satisfying A1. It is non-regular state feedback linearizable if (i). [ (x) ∈ Span{g1 , · · · , gm }; (ii). There exists a constant vector b of non-zero component, such that b ∈ Span{g1 , · · · , gm }. When either of the conditions of Proposition 10.4 fails, we can use the normal form to further investigate the problem. According to Lemma 10.6, we can always nd a vector eld K (x) such that adAx K (x) = [ (x).

(10.65)

Now we dene a local diffeomorphism as z1 = x − K (x). Then under the coordinate frame z1 system (10.64) can be expressed as m

z1 = Az1 − J0 (x)[ (x) + ¦ g1i (x)ui ,

(10.66)

i=1

where J0 (x) is the Jacobian matrix of K (x) and g1i (x) = (I − J0(x))gi (x). For notational ease, we denote x := z0 , [ (x) := [0 (x), K (x) := K0 (x), and gi (x) := g0i (x). Then we can continue the previous procedure to dene recursively new coordinates as adAx (Kk ) = [k ,

zk+1 = zk − Kk (x),

k  0,

and new vector elds gk+1 (x) = (I − Jk (x)) gki (x), i

1  i  m, k  0,

where Jk (x) is the Jacobian matrix of Kk (x). Then under zk the system is expressed as

10.6 Non-regular Feedback Linearization m

zk = Azk + [k (x) + ¦ gki (x)ui ,

k  1.

309

(10.67)

i=1

As a consequence of the above discussion, we have Corollary 10.3. System (10.64) is non-regular state feedback linearizable if there exists k  0, such that (10.67) satises the conditions (i) and (ii) of Proposition 10.4. Note that according to the recursive algorithm it is easy to see that deg([i ) = ci+1 + 1,

i = 0, 1, · · · ,

where {ci } is the Fibonacci sequence, i.e., (c1 , c2 , · · · ) = (1, 1, 2, 3, 5, 8, · · · ). Hence when k → f, [k (x) → 0, because the convergence is assumed. Then we have the following. Corollary 10.4. System (10.64) is non-regular state feedback linearizable if there exists a constant vector b of non-zero component such that k J f

–(I − Ji(x))g j (x),

b ∈ Span

j = 1, · · · , m .

i=0

Next, we consider the computational realization of the non-regular linearization. We use semi-tensor product of matrices for computation. Please refer to the appendix for this product. To begin with, using Taylor series expression on f (x) with the form of semitensor product, we express system (10.1) as x = Ax + F2 x2 + F3 x3 + · · · ,

(10.68)

where Fk are n × nk constant matrices, and xk are dened in appendix. Next, assume adAx Kk = Fk xk . Using Lemma 10.6, we can easily obtain that

Kk = (*kn = Fk ) xk ,

x ∈ Rn .

(10.69)

Here = is the Hadamard product of matrices [11], i.e., if two m × n matrices A = (ai j ) and B = (bi j ) are given, and C = A = B, then C is of the same size with entries ci j = ai j bi j . *kn can be constructed by (10.63) mechanically as (*kn )i j = U

1 n

¦

Dsj Os

V

,

i = 1, · · · , n, j = 1, · · · , nk ,

(10.70)

− Oi

s=1

where D1j , · · · , Dnj are respectively the powers of x1 , · · · , xn of the j-th component of xk . Now we are ready to present our main result:

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10 Linearization of Nonlinear Systems

Theorem 10.10. Assume A satises A1. Then system (10.68) can be transformed into a linear form z = Az,

(10.71)

by the following coordinate transformation: f

z = x − ¦ Ei x i ,

(10.72)

i=2

where Ei are determined recursively as (with )i as in (B.31)) F2 E2 = *2 = F

G

s−1

Es = *s = Fs − ¦ Ei )i−1 (Ini−1 ⊗ Fs+1−i) ,

s  3.

(10.73)

i=2

Proof. Using (10.72) to system (10.68), we have G G F F f f f w Ei xi i i z = Ax + ¦ Fi x − ¦ Ax + ¦ Fi x i=2 i=2 w x i=2 f f f w Ei xi i i = Az + ¦ Fi x + A ¦ Ei x − ¦ Ax i=2 w x G Fi=2 F i=2 G f f w Ei x i − ¦ Fj x j ¦ i=2 w x j=2 f f U i 2 = Az − ¦ adAx (Ei x ) + F2x + ¦ Fs xs − i=2 s=3 V s−1 w Ei xi Fs+1−i xs+1−i ¦ i=2 w x f

f

i=2

s=2

(10.74)

:= Az − ¦ adAx (Ei xi ) + ¦ Ls , where L2 = F2 x2 s−1 w Ei xi  Fs+1−ixs+1−i Ls = Fs xs − ¦ wx i=2 F G

=

s−1

Fs − ¦ Ei )i−1 (Ini−1 ⊗ Fs+1−i) xs ,

(10.75) s  3.

i=2

Now under assumption A1 we can set Es xs = ad−1 Ax (Ls ), Then (10.68) becomes (10.71).

s = 2, 3, · · · . 3 2

The advantage of this Taylor series expression is that it doesn’t require recursive computation of the intermediate forms of the system under transferring coordinates zi , i = 1, 2, 3, · · ·.

10.6 Non-regular Feedback Linearization

311

Now consider the linearization of system (10.11). Denote A = ww xf |0 , B = g(0), and assume (A, B) is a completely controllable pair. Then we can nd feedback K and a linear coordinate transformation T , such that A˜ = T −1 (A + BK)T satises assumption A1. For the sake of simplicity, we call the above transformation an NRtype transformation. Using the notations and algorithm proposed above the following result is immediate. Theorem 10.11. System (10.11) is single-input linearizable, if and only if there exists an NR-type transformation and a constant vector b of non-zero component such that G JF k b ∈ Span

f

I − ¦ Ei )i−1 xi−1 g j | j = 1, · · · , m .

(10.76)

i=2

Next, we consider approximate linearization. Denition 10.9. System (10.11) is said to be k-th degree non-regular state feedback approximately linearizable if we can nd a state feedback and a local coordinate frame z such that under z the closed-loop system can be expressed as z = Az + O(1z1k+1) + (b + O(1z1k))v,

(10.77)

where (A, b) is a completely controllable pair. For approximate linearization, we may relax the non-resonant constraint. Denition 10.10. Let O = (O1 , · · · , On ) be the eigenvalues of a given matrix A. A is a k-th degree resonant matrix if there exists m = (m1 , · · · , mn ) ∈ Zn+ , and 2  |m|  k, such that for some s, Os = 4m, O 5. From the expression (10.70) it is clear that the following corollary of the Poincar´e’s Lemma is correct. Corollary 10.5. Consider a CZ dynamic system (10.1). If A is k-th degree nonresonant, there exists a formal change of coordinates (10.55), such that system (10.1) can be expressed as z = Az + O(1z1k+1).

(10.78)

If we just consider the k-th degree approximate linearization of system (10.1) with ui = 0, we need to adjust (10.72) as k

z = x − ¦ Ei x i .

(10.79)

i=2

In this case, (10.73) is still valid (here s  k), and (10.71) will become z = Az + O(||x||k+1 ).

(10.80)

We say a transformation is NR-k-type transformation, if it is almost the same as NR-type transformation except the non-resonant condition, which is replaced by the k-th degree non-resonant.

312

10 Linearization of Nonlinear Systems

Theorem 10.12. System (10.11) is k-th degree single-input approximate state feedback linearizable, if and only if there exist an NR-k-type transformation and a constant vector b of non-zero component such that G JF k b ∈ Span

k

I − ¦ Ei )i−1 xi−1 g j | ∀ j

+O(1x1k ).

i=2

(10.81)

We use the following example to demonstrate the linearizing process. Example 10.5. Find a 4-th degree approximate state feedback linearization of the following control system: ⎤ ⎡ ⎤ ⎡ x1 −4 sin x1 − 23 x31 + 5x22 + 6x32 ⎦ ⎣x2 ⎦ = ⎣ −5x2 − 3x23 x3 −6x ⎡ ⎤ 3 ⎡ ⎤ (10.82) 0 1 + ⎣6(1 + x3 )⎦ u1 + ⎣0⎦ u2 . 0 7 Using Taylor series expansion, we express (10.82) as: ⎤ ⎡ 2 ⎤ ⎡ 3⎤ ⎡ 5x2 −4 0 0 6x2 x = ⎣ 0 −5 0 ⎦ x + ⎣−3x23 ⎦ + ⎣ 0 ⎦ 0 0 −6 ⎡ 0⎤ ⎡ ⎤0 0 1 +O(||x||5 ) + ⎣6 + 6x3⎦ u1 + ⎣0⎦ u2 . 0 7

(10.83)

It is easy to calculate that L2 = (5x22 , −3x23 , 0)T , U VT 5 2 3 2 −1 2 E2 x = adAx (L2 ) = − x2 , x3 , 0 ; 6 7 3 2 T L3 = (6x2 − 5x2 x3 , 0, 0) , U VT 6 3 5 −1 2 3 E3 x = adAx (L3 ) = − x2 + x2 x3 , 0, 0 . 11 13 So we get the desired coordinate transformation as follows: ⎤ ⎡ 5 6 5 − x22 − x32 + x2 x23 ⎥ ⎢ 6 11 13 ⎥ ⎢ ⎥, ⎢ 3 z = x−⎢ 2 ⎥ x ⎦ ⎣ 7 3 0 under which system (10.82) is expressed as:

(10.84)

References

⎤ ⎡ −4 0 0 h(x) 1 > ? u z = ⎣ 0 −5 0 ⎦ z + O(||x||5 ) + ⎣ 6 0⎦ 1 , u2 0 0 −6 7 0 ⎡

313

⎤

(10.85)

5 2 70 2 where h(x) = (6 + 6x3)( 53 x2 + 18 11 x2 − 13 x3 ) − 13 x2 x3 . Because ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ 1 h(x) 1 ⎣6⎦ = ⎣ 6 ⎦ × 1 + ⎣0⎦ × (−h(x) + 1), 0 7 7

Theorem 10.12 ensures that the system is 4-th degree approximate state feedback linearizable. Choosing state feedback > ? > ? u1 1 v (10.86) = u2 −h(x) + 1 and substituting it into (10.85), we get ⎡ ⎤ ⎡ ⎤ −4 0 0 1 z = ⎣ 0 −5 0 ⎦ z + O(||x||5 ) + ⎣6⎦ v, 0 0 −6 7

(10.87)

which is the desired 4-th degree approximate state feedback linearization of system (10.82).

References 1. Arnold V. Geometrical Methods on the Theory of Ordinary Differential Equations. New York: Springer, 1983. 2. Boothby W. Some comments on global linearization of nonlinear systems. Sys. Contr. Lett., 1984, 4(3): 143–147. 3. Cheng D, Isidori A, Respondic W, et al. On the linearization of nonlinear systems with outputs. Mathenatical Systems Theory, 1988, 21(2): 63–83. 4. Cheng D, Martin C. Normal form representation of control systems. Int. J. Robust Nonlinear Contr., 2002, 12(5): 409–443. 5. Cheng D, Tarn T, Isidori A. Global external linearization of nonlinear systems via feedback. IEEE Trans. Aut. Contr., 1985, 30(8): 808–811. 6. Devanathan R. Linearization condition through state feedback. IEEE Trans. Aut. Contr., 2001, 46(8): 1257–1260. 7. Guckcnhcimcr J, Ilolmcs P. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Berlin: Springer, 1983. 8. Heymann M. Pole assignment in multi-input linear systems. IEEE Trans. Aut. Contr., 1968, 13(6): 748–749. 9. Isidori A. Nonlinear Control Systems, 3rd edn. London: Springer, 1995. 10. Sun Z, Xia X. On nonregular feedback linearization. Automatica, 1997, 33(7): 1339–1344. 11. Zhang F. Matrix Theory, Basic Results and Techniques. New York: Springer-Verlag, 1999.

Chapter 11

Design of Center Manifold

This chapter provides a systematic technique for designing center manifold of closed loop of nonlinear systems to stabilize the system. The method was rstly presented in [6]. Section 11.1 introduces some fundamental concepts and results about center manifold theory. Section 11.2 considers the case when the zero dynamics has minimum phase. A powerful tool, called the Lyapunov function with homogeneous derivative, is developed in Section 11.3. Section 11.4 and Section 11.5 consider the stabilization of systems with zero center and oscillatory center respectively. The application to generalized normal form is considered in Section 11.6. Section 11.7 is for the stabilization of general control systems.

11.1 Center Manifold To begin with, we give a brief introduction to the center manifold theory without proof. (Refer to [4] or [10] for details.) Consider a dynamic system x = f (x),

x ∈ Rn ,

(11.1)

where f (x) is a smooth, say Cf , vector eld. The fundamental theorem of center manifold theory is the following: Theorem 11.1. [4] Let xe be an equilibrium point of system (11.1). There exist three sub-manifolds S+ , S− , and S0 , such that 1. they are invariant sub-manifolds of (11.1) passing through xe ; 2. Txe (S+ ), Txe (S− ), and Txe (S0 ) are linearly independent and Txe (S+ ) ⊕ Txe (S− ) ⊕ Txe (S0 ) = Txe (Rn ); 3. Restrict (11.1) on S+ (S− , S0 ), then the linear approximation of dynamics on S+ (S− , S0 ) at xe has all positive real part eigenvalues (negative real part eigenvalues, zero real part eigenvalues respectively); 4. Both S + and S− are unique, but S0 may not be unique. S+ , S− , S0 are called the unstable, stable and center manifolds respectively. Center manifold is a local property, so usually we consider it only over a neighborhood U of xe . We give a simple example to describe this: D. Cheng et al., Analysis and Design of Nonlinear Control Systems © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2010

316

11 Design of Center Manifold

Example 11.1. Consider the following system ⎧ x1 = 2x2 − x23 − x4 ⎪ ⎪ ⎪ ⎨x = −2x − 4x + x x 2 1 2 1 3 ⎪ x  = x 3 3 ⎪ ⎪ ⎩ x4 = x24 + 2(x4 − 1)x23 + x43 .

(11.2)

It is easy to check that S+ = {x ∈ R4 | x1 = 0, x2 = 0, x4 + x23 = 0}; S− = {x ∈ R4 | x3 = 0, x4 = 0}; S0 = {x ∈ R4 | x1 = 0, x2 = 0, x3 = 0}; Consider S+ : First, we have to check that it is an invariant set. Since x1 (0) = 0, x2 (0) = 0, so J x1 = 0, x2 = 0. That is: x1 (t) = 0, x2 (t) = 0, ∀t > 0. Let [ = x4 + x23 . Then

[ = [ 2 . As [ (0) = 0, [ (t) = 0, ∀t > 0. We conclude that S+ is invariant. Now we consider the restriction of (11.2) on S+ . It is a one dimensional manifold. We may use x3 to parameterize it. Then we have x3 = x3 . Obviously, it has unstable linear part. We may also use x4 to parameterize it. Then it is easy to check that the dynamics becomes x4 = x24 + 2(x4 − 1)x23 + x43 = x24 + 2(x4 − 1)(−x4 ) + (−x24 ) = 2x4 . We have same conclusion. Similarly, we can check that (a) S− is invariant, and the dynamics on it is J x1 = 2x2 x2 = −2x1 − 4x2, which is stable; (b) S0 is invariant, and the dynamics on it is x4 = x24 , which has zero linear part. The following three theorems are fundamental. Theorem 11.2. Consider the following system J x = Ax + p(x, z), x ∈ Rn , z = Cz + q(x, z), z ∈ Rm ,

(11.3)

11.2 Stabilization of Minimum Phase Systems

317

where ReV (A) < 0, ReV (C) = 0, p(x, z) and q(x, z) vanish at zero with their rst derivatives. Then there exists an m dimensional invariant sub-manifold through the origin, described by S = {(x, z) | x = h(z)},

(11.4)

w h(z) (Cz + q(h(z), z)) − Ah(z) − p(h(z), z) = 0. wz

(11.5)

where h(z) satises

This invariant sub-manifold is the center manifold. Theorem 11.3. The dynamics on the center manifold is z = Cz + q(h(z), z),

z ∈ Rm .

(11.6)

System (11.3) is asymptotically stable (stable, unstable), if and only if system (11.6) is asymptotically stable (stable, unstable). Theorem 11.4. Assume there exists a smooth function I (z), such that under operator M : Cmf (U) → Cnf (U) (dened as follows with U a neighborhood of the origin), we have M(I (z)) :=

w I (z) (Cz + q(I (z), z)) − AI (z) − p(I (z), z) = O(1z1k+1 ). wz

(11.7)

Then 1I (z) − h(z)1 = O(1z1k+1 ).

(11.8)

For convenience, we call Theorem 11.2, Theorem 11.3, and Theorem 11.4 the existence theorem, equivalence theorem, and approximation theorem respectively.

11.2 Stabilization of Minimum Phase Systems The following lemma is an immediate consequence of the above theorems on center manifold. Lemma 11.1. [9] Consider a system J z = Az + p(z, w) w = f (z, w),

(11.9)

where p(0, w) = 0 for all w near 0 and

wp (0, 0) = 0. wz If w = f (0, w) has an asymptotically stable equilibrium at z = 0 and A is Hurwitz, then the system (11.9) has an asymptotically stable equilibrium at (z, w) = (0, 0).

318

11 Design of Center Manifold

This lemma is particularly suitable for stabilization of nonlinear systems with minimum phase zero dynamics. First, we consider the Byrnes-Isidori normal form: ⎧i z1 = zi2 ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ ⎨ziUi −1 = ziUi (11.10) ziUi = ci (z, [ ) + di(z, [ )u ⎪ ⎪ ⎪ ⎪ i = 1, · · · , m ⎪ ⎪ ⎪ ⎪  ⎪ [ = p(z, [ ), [ ∈ Rn−1U 1 ⎪ ⎪ ⎩ yi = zi1 , i = 1, · · · , m, m

where 1U 1 = ¦ Ui , di (z, [ ) = (di1 (z, [ ), · · · , dim (z, [ )), i = 1, · · · , m. i=1

Using Lemma 11.1, a useful stabilization result is the following. Proposition 11.1. [3] If system (11.10) has minimum phase, i.e.

[ = p(0, [ )

(11.11)

is asymptotically stable at the origin, then the system is stabilizable via a pseudolinear control ⎡ U1 ⎤ 1 1 a z ⎡ ⎤−1 ⎡ ⎤ ⎡ ⎤−1 ⎢ ¦ j j ⎥ c1 d1 d1 ⎢ j=1 ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ .. ⎥. (11.12) u = − ⎣ ... ⎦ ⎣ ... ⎦ + ⎣ ... ⎦ ⎢ . ⎢ ⎥ ⎢ Um ⎥ dm cm dm ⎣ ⎦ ¦ amj zmj j=1

Next, we consider the generalized normal form (9.61), we copy it again for convenience. c d ⎧ ⎪ 0 ⎪ i  i ⎪ z = A i z + b i ui + + pi (z, w)u, zi ∈ RUi ⎪ ⎪ ⎪ D (z, w) i ⎨ (11.13) i = 1, · · · , m ⎪ ⎪ ⎪w = q(z, w), w ∈ Rr ⎪ ⎪ ⎪ ⎩ yi = zi1 , i = 1, · · · , m, m

where r + ¦ Ui = n, Di (z, w) are scalars, pi (z, w) are Ui × m matrices, q(z, w) is a i=1

r × 1 vector eld, and (Ai , bi ) are Brunovsky canonical form in RUi with the form ⎡ ⎡ ⎤ ⎤ 0 1 ··· 0 0 ⎢ .. .. ⎢ .. ⎥ .. ⎥ ⎢ ⎢ ⎥ .⎥ Ai = ⎢ . . ⎥ , bi = ⎢ . ⎥ , ⎣0 0 · · · 1⎦ ⎣0⎦ 1 0 0 ··· 0

11.3 Lyapunov Function with Homogeneous Derivative

319

and pi (0, 0) = 0. Using Lemma 11.1, the following stabilization property is obtained, which is a generalization of its counterpart Proposition 11.1 for Byrnes-Isidori normal form. Proposition 11.2. Assume

Di (0, w)pi (0, w) = 0,

i = 1, · · · , m.

(11.14)

Then if the pseudo-zero dynamics w = q(0, w)

(11.15)

is asymptotically stable at zero, then (11.13) is stabilizable by a pseudo-linear state feedback control. Proof. Choosing ui = ai1 zi1 + · · · + aiUi ziUi − Di (z, w),

i = 1, · · · , m,

the closed-loop system becomes ⎧ i i i i i ⎪ ⎨z = Ai z + p (z, w)u(z, w) := Ai z + G (z, w) i = 1, · · · , m ⎪ ⎩ w = q(z, w).

(11.16)

According to (11.14), G i (0, w) = 0, i = 1, · · · , m. Using Lemma 11.1, if we can prove wGi i w z (0, 0) = 0 we are done. Note that Di (0, 0) = 0 and p (0, 0) = 0, the conclusion follows. 3 2 Remark 11.1. An afne nonlinear system can always be converted into the generalized normal form as long as its point relative degree vector exists. So as long as the zero dynamics has minimum phase, the above result is applicable.

11.3 Lyapunov Function with Homogeneous Derivative The main purpose of this chapter is to consider the stabilization of nonlinear systems with non-minimum phase zero dynamics. To get some motivations about how to deal with non-minimum phase, we consider an example. Example 11.2. Consider the following system ⎧ ⎪ x1 = x2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨x2 = x3 , x3 = f (x, z) + g(x, z)u, ⎪ ⎪ ⎪ z = z3 + zx1 ⎪ ⎪ ⎪ ⎩y = x . 1

g(0, 0) := 0

(11.17)

The system (11.17) is in the canonical form of afne non-linear systems. Its zero dynamics is

320

11 Design of Center Manifold

z = z3 ,

(11.18)

which is not stable at the origin. Therefore, a quasi-linear control u=−

1 f (x, z) + (a1 x1 + a2 x2 + a3x3 ) g(x, z) g(x, z)

can not make the origin asymptotically stable, and a non-linear state feedback control should be considered. We may try the following control u=−

1 f (x, z) + (a1 x1 + a2 x2 + a3x3 + bz2 ). g(x, z) g(x, z)

(11.19)

To get a stabilizing control, we can rst choose a1 , a2 , a3 to stabilize the linearly controllable variables x1 , x2 , x3 , and then choose b to stabilize the central variable z. To determine a possible value of b, let ⎡ ⎤ I1 (z) I (z) = ⎣I2 (z)⎦ = O(1z12 ) I3 (z) be used to approximate the center manifold. Using Theorem 11.4, for system (11.17) we have

w I (z) (Cz + q(I (z), z)) − AI (z) − p(I (z), z) wz ⎡ ⎤ I2 (z) ⎦ I3 (z) = DI (z) (q(z)I1 (z)) − ⎣ 2 a1 I1 (z) + a2I2 (z) + a3 I3⎤(z) + bz ⎡ I2 (z) ⎦. I3 (z) = O(1z14 ) − ⎣ 2 a1 I1 (z) + a2 I2 (z) + a3 I3 (z) + bz

MI (z) =

⎧ ⎪ ⎨I1 = − b z2 , a1 ⎪ ⎩I = 0, i = 2, 3,

Choosing

i

According to Theorem 11.4, the center manifold then we have M I (z) = can be expressed as ⎧ ⎪ ⎨x1 = h1 (z) = − b z2 + O(1z14 ) a1 (11.20) ⎪ ⎩x = h (z) = O(1z14 ), i = 2, 3. O(1z14 ).

i

i

The dynamics on the center manifold is U V b 3 z = 1 − z + O(1z14 ). a1 Choose {a1 , a2 , a3 , b} such that the linear part is Hurwitz and

(11.21)

11.3 Lyapunov Function with Homogeneous Derivative

1−

321

b < 0, a1

say a1 = −1, a2 = a3 = −3, b = −2. The feedback control becomes u=−

f (x, z) 1 + (−x1 − 3x2 − 3x3 − 2z2 ). g(x, z) g(x, z)

It follows that system (11.21) is asymptotically stable at origin, and according to Theorem 11.3, so is the closed-loop system. Remark 11.2. Observing this example, one may see some motivating facts as follows: 1. The higher order terms (deg  2) in state feedback do not affect the local stability of the linearly controllable variables but they may affect the center part variables by changing the structure of the center manifold. 2. Higher order terms in feedback can be “injected” into the dynamics on center manifold through the rst variable, x1 , of the integral chain. x1 does not affect the order of approximation of the center manifold. This component of the linear part can be employed to modify the non-linear dynamics. 3. Since the center manifold is approximated up to a certain degree the approximated dynamics of the center manifold should be robust, i.e., approximately stable up to certain degree uncertainties to assure the stability of the original system. Since in general we can only obtain an approximation of the center manifold, it is necessary to have some convenient tools to verify the stability of the dynamics on center manifold through its approximated dynamics. For this purpose a new concept, Lyapunov function with homogeneous derivative (LFHS), is proposed in this section. Consider a dynamical system x = f (x),

x ∈ Rn ,

(11.22)

with f (0) = 0. We use Z+ for the set of non-negative integers. For a multi-index S = (s1 , · · · , sn ) ∈ Zn+ and x = (x1 , · · · , xn ) ∈ Rn , we denote n

|S| := ¦ si , i=1

n

xS := –(xi )si , i=1

n

S! := –(si )! . i=1

Note that 0! = 1, so S! := 0. For a smooth function F(x), we denote

w |S| F(x) w |S| F(x) := s1 s2 . S wx w x1 w x2 · · · w xsnn Then we can give the following denition: Denition 11.1. 1. Let ki be the lowest degree of non-vanishing terms of the Taylor expansion of fi (x), i = 1, · · · , n. A system consisting of only the lowest degree

322

11 Design of Center Manifold

(ki ) terms of system (11.22) is said to be the (lowest degree) approximate system of system (11.22). It can be expressed formally as xi = gi (x) :=

¦

|S|=ki

1 w |S| fi (x) (0)xS , S! w xS

i = 1, · · · , n.

(11.23)

2. System (11.23) is said to be an odd approximation of system (11.22) if all ki are odd. 3. System (11.22) is said to be approximately stable if xi = fi (x) + O(1x1ki +1 ),

i = 1, · · · , n

is locally asymptotically stable at origin. Remark 11.3. 1. In (11.23) gi is a homogeneous polynomial of degree ki . So g = (g1 , · · · , gn )T is a component-wise homogeneous vector eld. 2. When k1 = · · · = kn := k, the approximate stability dened above coincides with the conventional one [8]. Otherwise, it is coordinate-depending. It is clear that approximate stability implies asymptotic stability, but the inverse statement is not true. Denition 11.2. Given a component-wise homogeneous polynomial vector eld g = (g1 , · · · , gn )T , a positive denite polynomial V > 0 is said to be a Lyapunov function with homogeneous derivative (LFHD) along g, if the Lie derivative LgV is homogeneous with U V wV deg(LgV ) = deg + deg(gi ), ∀i = 1, · · · , n. w xi The following example provides two typical LFHDs, which will be used later. Example 11.3. Let g = (g1 , · · · , gn )T be a component-wise homogeneous vector eld with odd degrees, deg(gi ) = ki , i = 1, · · · , n, and m be a given integer satisfying 2m  max{k1 , · · · , kn } + 1. 1. Set 2mi = 2m − ki + 1, i = 1, · · · , n, then n

i V = ¦ pi x2m i

(11.24)

i=1

is a LFHD along g if pi > 0, ∀i. 2. Assume k1 = · · · = kn1 := k1 ; kn1 +1 = · · · = kn1 +n2 := k2 ; · · · ; kn1 +···+nr−1 +1 r

= · · · = kn1 +···+nr := kr , where ki are odd and ¦ ni = n. Denote x = (x1 , · · · , xr ), i=1

with dim(xi ) = ni and set 2mi = 2m − ki + 1, i = 1, · · · , r, then Z Y Z r Y i i i i T V = ¦ (xi1 )m , · · · , (xini )m Pi (xi1 )m , · · · , (xini )m i=1

(11.25)

11.3 Lyapunov Function with Homogeneous Derivative

323

is a LFHD along g if Pi , i = 1, · · · , r are positive denite matrices with dimensions n i × ni . Note that the derivative of V in either (11.24) or (11.25) along g is then a homogeneous polynomial of degree 2m. The following proposition is fundamental for LFHD. Proposition 11.3. System (11.22) is approximately stable at the origin if there exists a LFHD of its approximate system (11.23) such that its derivative along (11.23) is negative denite. Proof. Assume LgV is negative denite, then it should be of even degree, say deg(LgV ) = 2m. We claim that there exists a real number b > 0 such that n

LgV (x)  −b ¦ (xi )2m .

(11.26)

i=1

Since LgV is negative denite on the compact “sphere” k J   n  2m S = z  ¦ (zi ) = 1 , i=1 LgV (x) attains its maximum value −b < 0. That is, LgV (z)  −b < 0,

z ∈ S.

Now any x ∈ Rn can be expressed as x = kz for some z ∈ S. Then n

LgV (x) = LgV (kz) = k2m LgV (z)  −bk2m = −b ¦ (xi )2m , i=1

which proves the claim. Using (11.26), the derivative of the LFHD becomes n

V | f = V |g+O(1x1K+1 )  −b ¦ (xi )2m + O(1x12m+1),

(11.27)

i=1

where g + O(1x1K+1) is a shorthand for ZT Y g(x) = g1 (x) + O(1x1k1 +1 ), · · · , gn (x) + O(1x1kn +1 ) .

3 2

For the homogeneous vector elds an approximate stability result is given in [7] as (with slightly different statement): Theorem 11.5. Assume system (11.22) has k1 = · · · = kn = k and its approximate system (11.23) is asymptotically stable, then system (11.22) is asymptotically stable. The Proposition 11.3 and Theorem 11.5 will be our major tools for testing approximate stability. Next, we investigate some sufcient conditions for testing approximate stability of systems with odd approximate systems. The following inequality plays a fundamental role in later estimation.

324

11 Design of Center Manifold

Lemma 11.2. Let S ∈ Zn+ and x ∈ Rn . The following inequality holds. n

sj

¦ |S| |x j ||S|.

|xS | 

(11.28)

j=1

Proof. Let z1 , · · · , zn > 0 and s1 , · · · , sn  0 with

n

¦ si > 0. Since ln(z) is a convex

i=1

function (concave down), we have ⎛ n

⎜

¦⎜ ⎝

i=1

Equivalently, denote s =

⎞

⎞

⎟ ⎜ ln(zi ) ⎟ ⎠  ln ⎝ ¦ n

si n

i=1

¦ sk

si n

¦ sk

zi ⎟ ⎠.

k=1

k=1 n

¦ sk , we have

k=1

n

ln –

s ( i) zi s

i=1

or

⎛

n

F

G si  ln ¦ zi , i=1 s

s ( i)

– zi s i=1

n

n

si zi . i=1 s

¦

(1) zi s

by |xi |, inequality (11.28) follows. Substituting Note that (11.28) is trivial if some xi are zero.

3 2

Given a component-wise homogeneous polynomial vector eld g = col(g1 , · · · , gn ) with deg(gi ) = ki , i = 1, · · · , n. We express gi as gi (x) = aidi xki i +

¦ aiS xS ,

S:=di

i = 1, · · · , n,

(11.29)

where the index di = ki Gi = (0, · · · , ki , · · · , 0), which indicates the diagonal term. The “diagonal term” is an important concept in the LFHD approach. We explain it as the following: Let V be a LHHD with respect to the component-wise homogeneous polynomial system x = g(x). ⎡

⎤ wV g1 ⎢ w x1 ⎥ ⎢ . ⎥ ⎢ . ⎥ = A  x2m, ⎢ . ⎥ ⎣ wV ⎦ gn w xn i where A ∈ Mn×n2m . Now adi corresponds to the i-th diagonal element in A. Precisely, the i-th diagonal element, which is the (i, (i − 1)n2m−1 + 1)-th element in A, is the product of aidi with the term in ww Vxi of highest degree in xi . It turns out that the term in V with highest degree in xi is obtained from the diagonal term of gi . Then we have

11.3 Lyapunov Function with Homogeneous Derivative

325

Then we have Theorem 11.6. (CRDDP (Cross Row Diagonal Dominating Principle)) The vector eld g, given above, is asymptotically stable at origin, if there exists an integer m with 2m > max{k1 , · · · , kn }, such that V U Y s Z n si + 2m − ki i j i i −adi > + ¦ , (11.30) |aS | ¦ |aS | ¦ 2m 2m j=1, j:=i |S|=k |S|=k ,S:=d i

i

j

where i = 1, · · · , n; S = (s1 , · · · , sn ) ∈ Zn+ . Proof. Choose a LFHD as n

V=¦

U

i=1

V 1 x2m−ki +1 , 2m − ki + 1 i

then we have n

V |g = ¦

¦

i=1 |S|=ki

i aiS xS x2m−k . i

(11.31)

This is a homogeneous polynomial of degree 2m. Now using inequality (11.28) to split each term in (11.30) and collecting terms, inequality (11.30) yields that n

V |g < − ¦ Hi x2m i ,

for some Hi > 0.

i=1

3 2

The conclusion follows immediately.

One obvious improvement for this estimation can be done as the following: Negative semi-denite non-diagonal terms can be eliminated from the estimation. Formally, for each gi dene a set of its terms by their exponents as Qi = {|S| = ki | s j ( j := i) are even and aiS < 0}. Terms with exponents in Qi are negative semi-denite in from inequality (11.30) yields V U si + 2m − ki i i −adi > ¦ |aS | 2m |S|=k ,S∈Q / i

+

wV w xi gi .

Moving such terms

i

n

¦

¦

j=1, j:=i |S|=k j ,S:∈Q j

j |aS |

Y s Z i , 2m

(11.32)

i = 1, · · · , n.

Later on we will simply use inequality (11.32) as CRDDP. Next, we give a simpler form, which deals with each row independently. Corollary 11.1. (DDP (Diagonal Dominating Principle)) Given a polynomial vector eld g as in Theorem 11.6. It is asymptotically stable at origin if

326

11 Design of Center Manifold

−aidi >

¦

|S|=ki ,S∈Q / i

|aiS |,

i = 1, · · · , n.

(11.33)

Proof. Since in inequality (11.32) m can be arbitrary large, let m → f, the right hand side of inequality (11.32) becomes right hand side of inequality (11.33). Hence the strict inequality (11.33) implies inequality (11.32) for large enough m. 3 2 Use inequality (11.28), we can reduce the homogeneous polynomial of deg = 4k into a “dominating” quadratic form with variables x2k i , i = 1, · · · , n. Next, we propose an algorithm for estimating the polynomial form by a quadratic form. Algorithm 11.1. (QFRA (Quadratic Form Reducing Algorithm)) Let g = col(g1 , · · · , gn ) and deg(gi ) = ki , i = 1, · · · , n with odd ki . Step 1. Choose smallest even number m = 2k such that 2m > max{k1 , · · · , kn }. Construct a 2m homogeneous polynomial q(x) as n

i gi . q(x) = ¦ x2m−k i

i=1

Step 2. Find all terms in q(x), for which the index S of xS has component si less than 2k. Split it into two equal exponent groups in the alphabetical order of xi . i.e., for ax21 x2 x53 , we have S = (2, 1, 5), m = 4 and k = 2. It is split as ax21 x2 x53 = ax21 x2 x3 × x43 . For ax21 x52 x3 , we have S = (2, 5, 1), and it is split as ax21 x52 x3 = ax21 x22 × x32 x3 . Step 3. Using inequality (11.28) to convert them into several 2k exponent terms. i.e. x2 x3 × x43 ax21U V V U x4 x4 x4 1 4 4 1 4 4 1 8 x1 x3 + x2 x3 + x3 .  |a| 1 + 2 + 3 x43 = |a| 2 4 4 2 4 4 Replace the original terms in q(x) by their splitting terms. The algorithm produces a quadratic form of x2k i , i = 1, · · · , n. Then the following can be proved by constructing a suitable LFHD: Proposition 11.4. If the resulting quadratic form produced by the above algorithm is negative denite, then q(x) is negative denite. Consequently, g(x) is asymptotically stable at the origin. The following example is used to describe the notations and results in the above Theorem 11.6 through Proposition 11.4. Example 11.4. Find a region for parameter O , such that the following system is asymptotically stable at the origin: ⎧ ⎪ ⎨x1 = sin(x1 ) − x1 cos(2O x2 ), (11.34) x2 = x22 ln(1 − x2 − x3 ) + 0.5x22x3 , ⎪ ⎩ x3 = 2x33 (1 − cosh(x3 − x2 )) − 1.1x53 .

11.3 Lyapunov Function with Homogeneous Derivative

Using Taylor expansion on (11.34), its approximate system is ⎧ 1 ⎪ x1 = − x31 + 2O 2x1 x22 (:= g1 ), ⎪ ⎪ ⎨ 6 1 x2 = −x32 − x22 x3 (:= g2 ), ⎪ ⎪ 2 ⎪ ⎩ x3 = −2.1x53 + 2x43 x2 − x33 x22 (:= g3 ).

327

(11.35)

We gure out all the parameters in Theorem 11.6 and Corollaries 11.1 and Proposition 11.4 as follows: For g1 (x), k1 = 3. Denote by G1 = {S | |S| = k1 = 3}, then G1 = {(300), (210), (201), (120), (111), (102), (030), (021), (012), (003)} := {S1 , S2 , · · · , S10 }. Note that d1 = (300) = S1 , a1d1 = − 16 , a1S4 = 2O 2 , a1Si = 0, (i := 1, and i := 4). It is easy to check that there is no term in Q1 , so Q1 = I . For g2 (x), k3 = 3. Hence G2 = G1 . Then d2 = (030) = S7 , a2d2 = −1, a2S8 = − 12 , 1 aSi = 0 (i := 7, and i := 8). We also have Q2 = I . For g3 (x), k3 = 5. Hence G3 = {(500), (410), (401), · · · , (005)}. Then d3 = (005), a3d3 = −2.1, a3(014) = 2, a3(023) = −1. For the other S ∈ G3 , a3S = 0. Since the last term is in Q3 , so Q3 = {(023)}. Now we are ready to test the negativity of the derivative. We rst check DDP. For the second and third formulas of (11.35), the dominating condition (11.33) is satised. For the rst formula, (11.33) yields 16 > 2O 2 . So |O | < 0.2886751346. Next we check CRDDP. Let m = 3. Then (11.32) yields U V ⎧ 1 ⎪ 2 4 ⎪ > 2O , ⎪ ⎪ 6 ⎪ ⎪ U 6V U V U V ⎨ 1 5 1 2 2 2O + +2 , 1> ⎪ 6 2 6 6 ⎪ U V U V ⎪ ⎪ 5 1 1 ⎪ ⎪ ⎩2.1 > 2 + . 6 2 6 The solution is

|O | < 0.3535533906.

Finally, let’s use QFRA. The smallest even m should be 4. Then 1 1 q(x) = − x81 + 2O 2x61 x22 − x82 − x72 x3 − 2.1x83 + 2x73 x2 − x63 x22 . 6 2 The algorithm produces a quadratic form as ⎤ ⎡ 1 2 1 2 − + O O 0 ⎥ ⎢ 6 2 ⎥ ⎢ ⎢ 1 5 ⎥ 5 2 ⎥. ⎢ − ⎢ 2O 8 16 ⎥ ⎥ ⎢ ⎦ ⎣ 5 −0.6 0 16

328

11 Design of Center Manifold

To make it negative denite we have |O | < 0.3922535218. In fact, we can prove that in general QFRA is stronger than CRDDP and CRDDP is stronger than DDP. But DDP is the easiest one in use, while QFRA is the most difcult one. Later on, according to the problems one or more of these three methods are used for testing the negative deniteness of the derivatives of LFHD.

11.4 Stabilization of Systems with Zero Center Now we are ready to consider the stabilization problem. Consider an afne nonlinear system with the following Byrnes-Isidori canonical form: ⎧ i x1 = xi2 , ⎪ ⎪ ⎪ ⎪ . ⎪ .. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = xini , xi ⎪ ⎪ ⎨ ni i −1 xni = fi ([ ) + gi([ )ui , m ⎪ ⎪ ⎪ ⎪ xi ∈ Rni , i = 1, · · · , m, ¦ ni = n, [ = (x, w, z); ⎪ ⎪ ⎪ i=1 ⎪ ⎪ ⎪ ⎪w = Sw + p([ ), w ∈ Rs , ReV (S) < 0; ⎪ ⎪ ⎩ z = Cz + q([ ), z ∈ Rt , ReV (C) = 0;

(11.36)

where fi (0) = 0, gi (0) := 0, p([ ) and q([ ) vanish at origin with their rst derivatives. Since the rst variables in each integral chain play a particular role, we adopt the following notations: x = (x1 , x1 ), where x1 = (x11 , · · · , xm 1 ),

m x¯1 = (x12 , · · · , x1n1 , · · · , xm 2 , · · · , xnm ).

System (11.36) is said to have zero center if C = 0. Only this case is considered in this section. (r) Let \i (z), r = 2, · · · , h, i = 1, · · · , m, be a set of polynomials of z with degree r. Dene ⎧ h ⎪ ⎪ ⎪ pcj (z) := p j (x(z), 0, z), xi1 = ¦ \i(r) (z), x¯ i1 = 0, ⎪ ⎪ ⎪ r=2 ⎪ ⎪ ⎪ ⎪ i = 1, · · · , m, j = 1, · · · , s ⎪ ⎪ ⎪ h ⎪ ⎪ (r) ⎨qc (z) := q (x(z), 0, z), xi = \ (z), x¯ i = 0, k

k

1

¦

i

1

r=2 ⎪ ⎪ ⎪ i = 1, · · · , m, k = 1, · · · ,t ⎪ ⎪ ⎪ ⎪ h ⎪ ⎪ ⎪q˜c (z) := qk (x(z), w(z), z), xi1 = ¦ \i(r) (z) + E1i (z), ⎪ k ⎪ ⎪ ⎪ r=2 ⎪ ⎩ j = 2, · · · , ni , i = 1, · · · , m,

( x¯ i1 ) j = E ij (z),

11.4 Stabilization of Systems with Zero Center

329

where E ij (z) and w(z) are uncertain functions, and they will be specied later. We denote pc (z) = (pc1 (z), · · · , pcs (z))T , and similarly dene qck (z) and q˜ck (z). The following theorem shows a general design idea. Polynomials of degrees 2 to h are used for the non-linear control design. (r)

Theorem 11.7. Assume C = 0 and there exists a set of polynomials \i (z), r = (r) 2, · · · , h, i = 1, · · · , m, deg(\i (z)) = r, and an integer e  2, such that the following conditions (i)–(iv) hold: (i). pc (z) = O(1z1e+1 ); (ii). qc (z) = O(1z1e ); (iii). If E ij (z) = O(1z1e+1 ), and w(z) = O(1z1e+1 ), then z = q˜c (z) z = qc (z)

· · · (a) · · · (b)

(11.37)

have same approximate system; (iv). z = qc (z) is approximately stable. Then system (11.36) is (locally) asymptotically stabilizable (at origin). Moreover, if conditions (i)–(iv) are satised, a suitable feedback control, which stabilizes system (11.36) is GG F F ni h fi ([ ) 1 (r) i i i ui = − + ¦ a j x j − a1 ¦ \i (z) , i = 1, · · · , m, (11.38) gi ([ ) gi ([ ) j=1 r=2 ni

where O ni − ¦ aij O j−1 is Hurwitz. j=1

Proof. Choose

) (z) =

⎧ ⎪ i ⎪ ⎪ ⎨x1 (z) =

h

(r)

¦ \i

(z),

r=2

⎪ xi1 (z) = 0, ⎪ ⎪ ⎩ w(z) = 0

i = 1, · · · , m,

to approximate the center manifold of the closed-loop system with control (11.38). Using conditions (i) and (ii) and control (11.38) we have ⎡⎛ ⎞ ⎤ (r) h ⎡ ⎤ w \i (z) 0 ⎢⎜ ¦ ⎟ ⎥ , i = 1, · · · , m ⎢⎝ ⎥ c ⎠ M) (z) = ⎢ r=2 w z ⎥ q (z) − ⎣ 0 ⎦ ⎣ ⎦ 0 pc (z) (11.39) 0 = O(1z1e+1 ). The dynamics on the center manifold is z = q(x(z), w(z), z).

(11.40)

According to the approximation theorem, (11.39) ensures that the functions x(z) and w(z) in (11.40) have the following forms:

330

11 Design of Center Manifold

⎧ ⎪ i ⎪ ⎪ ⎨x 1 =

h

(r)

¦ \i

(z) + O(1z1e+1 ),

r=2

⎪ x¯ i1 = O(1z1e+1 ), ⎪ ⎪ ⎩ w = O(1z1e+1 ).

i = 1, · · · , m,

Now (11.40) is of the type of the rst formula of (11.37). So conditions (iii) and (iv) ensure the approximate stability of (11.40). Hence the closed-loop form of system (11.36) is asymptotically stable. 3 2 It is clear from above proof that e + 1 is the degree of the approximation error. It has been pointed out early that the higher degree feedback can be injected into the dynamics on the center manifold through x1 . To distinct different injection types we dene the injection degrees as Denition 11.3. For system (11.36) the injection degree dk is dened as  J k  w |T |+|S| qk  (0) := 0 , k = 1, · · · ,t. dk = min 2|T | + |S|  |T | > 0,  w (x1 )T w zS In fact, dk , k = 1, · · · ,t are the lowest degrees of the non-vanishing terms in the dynamics on center manifold which contains x1 (z). Given system (11.36) the approximation degree e can be estimated from (11.39). Let l j be the lowest degree of the non-vanishing terms in pcj . Then we have e = min{di , i = 1, · · · ,t; l j − 1, j = 1, · · · , s}.

(11.41)

It can be seen intuitively that an even-degree leading system can hardly be homogeneously stable. Our design idea is: When the injection degree, dk , is odd, use it as lowest degree of the resulting system, i.e., for the dynamics on the center manifold, let Lk = dk . Otherwise, choose control to eliminate dk degree terms and turn the lowest degree of the resulting system to odd, i.e., Lk = dk + 1 . In such a way, we nally make the dynamics on the center manifold to have an odd approximate system. Lk will be called the leading degree. Remark 11.4. Even in Theorem 11.7, e + 1 > h is not claimed, it is required implic(r) itly. Otherwise some terms of \i (z) in the designed approximation of the center manifold will be meaningless. Using e, dk , and Lk , conditions (i) and (iii) in Theorem 11.7 are computable. (r)

Proposition 11.5. In Theorem 11.7, for arbitrary chosen \i , i = 1, · · · , m, r = 2, · · · , h, Condition (i) holds, if and only if

w |T |+|S| pk (0) = 0, w (x1 )T w zS

2|T | + |S|  e,

k = 1, · · · , s.

(11.42)

Condition (ii) holds, if

w |T |+|S| qk (0) = 0, w (x1 )T w zS

2|T | + |S|  dk − 1,

k = 1, · · · ,t.

(11.43)

11.4 Stabilization of Systems with Zero Center

331

Condition (iii) holds, if and only if (11.43) holds and when |U| + |V| > 0

w |T |+|S|+|U|+|V | qk (0) = 0, w (x1 )T w zS w (x1 )U w wV 2|T | + |S| + (e + 1)(|U| + |V|)  Lk ,

(11.44) k = 1, · · · ,t.

Proof. In pk ([ ) set x1 = 0 and w = 0, then use Taylor expansion on x1 and z. Note that xk1 (z) = O(1z12 ). Then (11.42) means all terms in pck (z) of degree less than or equal to e are zero. Since e  dk , (11.42) holds for 2|T | + |S|  e − 1, which means all terms in qck (z) of degree less than e are zero. As for condition (iii), note that xk1 (z) = O(1z1e+1 ) and w(z) = O(1z1e+1 ). Then it is easily seen that (11.44) holds, if and only if both x1 and w don’t appear in the approximate system of the dynamics on the center manifold. Hence the two formulas in (11.37) have same approximate system. 3 2 Equation (11.43) is sufcient for condition (ii). But it is necessary for the required leading degrees. So we call (11.42)–(11.44) the degree matching conditions. They are always assumed in the following sections for center manifold design. We use an example to give a detailed description for all the objects in this section. Example 11.5. Consider the following system ⎧ x1 = x2 , ⎪ ⎪ ⎪ ⎪ ⎪ x2 = x3 , ⎪ ⎪ ⎪ ⎨x = u, 3 ⎪ w  = −w + x1 z21 z2 , ⎪ ⎪ ⎪ ⎪ ⎪z1 = az21 + z1 x1 + z2 x2 , ⎪ ⎪ ⎩ z2 = z1 z2 x1 .

(11.45)

For this system m = 1, x1 = (x2 , x3 ), s = 1, t = 2, p(x, w, z) = x1 z21 z2 , q1 (x, w, z) = and q2 (x, w, z) = z1 z2 x1 . Consequently, we have

az21 + z1 x1 + z2 x2 ,

⎧ h ⎪ (r) c (z) = ⎪ p ⎪ ¦ \i (z)z21 z2 , ⎪ ⎪ ⎪ r=2 ⎪ ⎪ ⎨ h (r) c q1 (z) = az21 + z1 ¦ \i (z), ⎪ ⎪ r=2 ⎪ ⎪ ⎪ h ⎪ ⎪ (r) ⎪ ⎩qc2 (z) = z1 z2 ¦ \i (z),

(11.46)

r=2

and

⎧ F G h ⎪ ⎪ (r) c 2 1 1 ⎪ ⎪ ⎨q˜1 (z) = az1 + z1 ¦ \i (z) + E1 (z) + z2 E2 (z), r=2 F G h ⎪ ⎪ (r) ⎪ ⎪q˜c2 (z) = z1 z2 ¦ \i (z) + E12 (z) , ⎩ r=2

(11.47)

332

11 Design of Center Manifold (r)

where \i (z), r = 2, · · · , h will be chosen to design control, E11 , E21 , and E12 are some uncertain terms of O(1z1e+1 ). According to Denition 11.3, the injection degrees are d1 = 3 and d2 = 4. Hence we choose L1 = 3 and L2 = 5. From pc (z) we have l = 5. Then e = min{d1 = 3, d2 = 4, l − 1 = 4} = 3. It is ready to check that (11.42) holds. Consider (11.43). For q1 , when S = (2, 0) and T = (0), |S| + |T |  d1 − 1. But

w |S| q1 w 2 q1 = = 2a. w zS w z21 So (11.43) holds for q1 if and only if a = 0. It is easy to check that (11.43) is true for q2 . For (11.44), only q1 has a term involving x1 and/or w, which is z2 x2 . For this term T = (0), S = (0, 1), U = (1, 0), and V = (0). So 2|T | + |S| + (e + 1)(|U| + |V|) = 5 > L1 = 3. (11.44) is, therefore, satised. We conclude that the degree matching conditions are satised if and only if a = 0. Next, from (11.46) and (11.47) it is clear that the two formulas in (11.37) have same approximate system. Theorem 11.8. Assume system (11.36) with C = 0 has an odd universal injection degree, say dk = L  3, ∀k. The system is state feedback stabilizable, if it satises the degree matching conditions (11.42)–(11.44) with h = 2, 2  e  L and Lk = L, ∀k, and there exists a quadratic homogeneous vector \ (z) = col(\1 , · · · , \m ), such that G F m 1 w |T |+|S| qk zk = ¦ T !S! w (x1 )T w zS (0)zS –(\i )Ti (z) , k = 1, · · · ,t (11.48) i=1 2|T |+|S|=L is asymptotically stable at origin. Moreover, if the above conditions are satised, (2) (11.48) with \i = \i is a suitable feedback control, which stabilizes system (11.36). Proof. Using control (11.38), conditions (11.43) and (11.44) assure the lowest degree of the dynamics of the closed-loop system on the center manifold is L. Note that in this case di = e = L, i = 1, · · · , m. Conditions (11.42)–(11.44) assure the center manifold is described as xi1 = \i (z) + Ri , Ri = O(1z1L+1 ); xi1 = O(1z1L+1 ), i = 1, · · · , m; w = O(1z1L+1 ). Using (11.43)–(11.44), Ri , xi1 , and w will not appear in the degree L terms. Hence the degree L terms of the dynamics are exactly the right side of (11.48). That is, (11.48) is the approximate system of the dynamics on the center manifold. Since

11.4 Stabilization of Systems with Zero Center

333

(11.48) is homogeneous and asymptotically stable at the origin, Theorem 11.5 assures the approximate stability of the dynamics on the center manifold of the closedloop system. Then the asymptotical stability of the closed-loop system follows from Theorem 11.7. 3 2 When dk = 3, k = 1, · · · ,t, it is an interesting case. Now set h = 2, and L = 3. The previous result leads to the following simpler one. Corollary 11.2. System (11.36) with C = 0 is state feedback stabilizable if 2 (i). wwxpi (0) = 0, wwxi wp z (0) = 0, i = 1, · · · , m; 1

1

wp w z (0) = 0,

w2p (0) = 0, w z2 wq w z (0) = 0,

w3p (0) = 0. w z3 2 w q (0) = 0. w z2

(ii). ww qx (0) = 0, (iii). There exists a quadratic homogeneous vector eld

\ (z) = col(\1 , · · · , \m ), such that z = D(z)\ (z) + E(z)

(11.49)

is asymptotically stable. And D(z) and E(z) are t × m and t × 1 matrices with entries as t w 2 qi Di j = ¦ (0)zk , i = 1, · · · ,t, j = 1, · · · , m, j k=1 w x1 w zk Ei =

w 3 qi (0)zS , S!w zS |S|=3

¦

(2)

respectively. Moreover, (11.38) with \i stabilizes system (11.36).

i = 1, · · · ,t,

= \i is a suitable feedback control, which

The following example shows that when the injection degree is 3 we have only to solve a set of algebraic inequalities to obtain the required control. Example 11.6. Consider the following system ⎧ ⎪ ⎪x1 = x2 , ⎪ ⎪ ⎪ ⎪ ⎨x2 = f1 (x, z) + g1 (x, z)u1 , x3 = f2 (x, z) + g2 (x, z)u2 , ⎪ ⎪ ⎪ z1 = q1 (x, z), ⎪ ⎪ ⎪ ⎩z = q (x, z), 2 2

(11.50)

where fi (0) = 0, gi (0) := 0, qi satises the condition (ii) in Corollary 11.2, i = 1, 2. Our goal is to nd a sufcient condition for system (11.50) to be feedback stabilizable. Denote by ckij =

w 2 qk (0), w zi w x j

k = 1, 2,

i = 1, 2,

j = 1, 3;

334

11 Design of Center Manifold

dikj =

w 3 qk (0), i! j!w (z1 )i w (z2 ) j

\1 = az21 + bz1 z2 + cz22 , Then (11.49) leads to the following:

k = 1, 2,

i + j = 3;

\2 = dz21 + ez1 z2 + f z22 . ⎡

3

? ⎢¦ ?> > 1 c11 z1 + c121z2 c113 z1 + c123 z2 \1 (z) ⎢ + ⎢t=0 z = 2 c11 z1 + c221z2 c213 z1 + c223 z2 \2 (z) ⎣3

⎤ dt13−t zt1 z3−t 2 ⎥

⎥ ⎥. 2 t z3−t ⎦ d z ¦ t 3−t 1 2

(11.51)

t=0

Thus the sufcient conditions developed previously may be used to design a stabilizing control. To specify the above general form, we consider the following system as a special case of (11.50). ⎧ ⎪ x1 = x2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨x2 = f1 (x, z) + g1 (x, z)u1 , (11.52) x3 = f2 (x, z) + g2 (x, z)u2 , ⎪ ⎪ 2 z ) − 3 tan(z2 z ), ⎪ z  = sin(x z − z ⎪ 1 1 1 1 2 1 2 ⎪ ⎪ ⎩z = z − z + z3 − z ex1 + z ex3 . 2 1 2 1 2 2 Then the coefcients are computed as c111 = 1,

(11.51) becomes

> >

c113 = 0,

c121 = 0, c123 = 0,

c211 = −1, c213 = 0,

c221 = 0, c223 = 1,

1 = 0, d30

1 = −4, d 1 = 0, d 1 = 0, d21 12 03

2 = 0, d30

2 = 0, d21

z1 0 −z1 z2

2 = 0, d 2 = 1. d12 03

?> ? ? > −4z21 z2 \1 (z) + \2 (z) z32

? az31 + bz21 z2 + cz1 z22 − 4z21 z2 = . −az31 + (d − b)z21 z2 + (e − c)z1z22 + ( f + 1)z32 Using CRDDP with m = 2, we have ⎧ 3 2 3 2 1 ⎪ ⎨−a > |b| + |c| + 3 + |a| + |d − b| + |e − c|, 4 4 4 4 4 ⎪ ⎩− f − 1 > 1 |b| + 2 |c| + 1 + 1 |a| + 2 |d − b| + 3 |e − c|. 4 4 4 4 4

(11.53)

(11.54)

One particular solution of (11.54) is a = −25, b = 4, c = 0, d = 4, e = 0, f = −10. ? > Then −25z21 + 4z1 z2 . \= 4z21 − 10z22 To stabilize linear part, one may choose a11 = −1, a12 = −2, a21 = −1. Then (11.38) yields:

11.5 Stabilization of Systems with Oscillatory Center

335

⎧ 1 f1 (x, z) 2 ⎪ ⎪ ⎨u1 = − g (x, z) + g (x, z) (−x1 − 2x2 − 25z1 + 4z1 z2 ), 1 1 ⎪ 1 f2 (x, z) ⎪ ⎩u2 = − + (−x3 + 4z21 − 10z22). g2 (x, z) g2 (x, z) For more details about the systematic use of center manifold to stabilize nonlinear systems with non-minimum phase zero dynamics we refer to [6, 5].

11.5 Stabilization of Systems with Oscillatory Center Control of an oscillator via center manifold method has been discussed in several articles. i.e. [1, 2]. Usually the system is rst transformed to a normal form. Our purpose is to provide testing criteria and a stabilizing control based on the system’s canonical form. Consider system (11.17) again. When the center is oscillatory the system may be visualized as ⎧ i x1 = xi2 , ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ xini −1 = xini , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨xini = ui , (11.55) i = 1, · · · , m, ⎪ ⎪ ⎪ ⎪ w =dSw + ⎪ c p([ ), d c d c d ⎪ c ⎪ j j j ⎪ ⎪ Z ( [ ) 0 − q z z  j ⎪ 1 1 1 ⎪ , ⎪ j = j + j ⎪ ⎪ q2 ([ ) z2 Zj 0 z2 ⎪ ⎪ ⎩ j = 1, · · · ,t, where x = col(x1 , · · · , xm ) ∈ Rn , dim(xi ) = ni , n1 +· · ·+nm = n; w ∈ Rs ; ReV (S) < 0; z = col(z1 , · · · , zt ), z ∈ R2t ; [ = (x, w, z); Z j := 0; p and q j , j = 1, · · · ,t, vanish with their rst derivative at zero . Some new notations and denitions are needed for further discussion. Let z = (z11 , z12 , · · · , zt1 , zt2 ). We set J ij = {|S| = i | S ∈ Z2t + , sk = 0, for k := 2 j − 1, k := 2 j}, For a quadratic form of z , say I (z) =

¦

1  j  t.

aS zS , I can be expressed as

|S|=2 t

I (z) = ¦ (Di (zi1 )2 + Ei (zi1 )(zi2 ) + Ji (zi2 )2 ) + i=1

¦

2

2

¦ ¦ aOuvP zOu zvP .

1O < P t u=1 v=1

Then we denote

I+ j (z) = D j (z1j )2 + E j (z1j )(z2j ) + J j (z2j )2 , I− j (z) = I (z) − I+ j (z),

336

11 Design of Center Manifold

and dene V t U 1 1I (z)1 = ¦ max(|Di |, |Ji |) + |Ei | ri2 + ¦ 2 i=1 1O < P t where ri =

l

2

2

¦ ¦ |aOuvP |rO rP ,

u=1 v=1

(zi1 )2 + (zi2 )2 , i = 1, · · · , m.

Let ) (z) = (I 1 (z), · · · , I m (z)), we dene @ A )± j (z) = (I 1 )± j (z), · · · , (I m )± j (z) , m

1) (z))1T = – 1I i (z)1Ti . i=1

Let S = (s1 , s2 , · · ·

, s2t ) ∈ Z2t +

and r = (r1 , · · · , rt ) ∈ Rt . We dene s

s

s

(r, r)S = r1s1 r1s2 r23 r2s4 · · · rt 2t−1 rt 2t . We also denote, as in the previous chapter, x1 = (x11 , · · · , xm 1 ). Using above notations we dene a set of polynomials of r as    w 3q j  1   v Hvj = ¦ S!  w zS (0) (r, r)S |S|=3,S∈J / 3j     w 2q j   v + ¦  S (0)  (r, r)S 1) 1T T   w z w (x ) 1 1 |S|=1,S∈J / j

|T |=1 

   w 2q j   v (0)  (r, r)S 1)− j 1T ,  S   w z w (x1 )T

¦

+

S∈J 1j

(11.56)

|T |=1

v = 1, 2, j = 1, · · · ,t.

Next, assume a set of quadratic polynomial of z as I 1 (z), · · · , I m (z) are given. Then we can get that (I i )+ j (z) = D ij (z1j )2 + E ji (z1j )(z2j ) + J ij (z2j )2 ,

j = 1, · · · ,t.

(11.57)

Denote ⎧ ⎪ jk ⎪ ⎪ ⎨csl =

w s+l qkj

(0), k = 1, 2, j = 1, · · · ,t, w (z1j )s w (z2j )l ⎪ w 2 qkj ⎪ jk ⎪ (0), k, s = 1, 2, j, l = 1, · · · ,t. ⎩dsl = w (zsj )w (xl1 ) Using (11.57) and (11.58), we can dene a set of real numbers as

(11.58)

11.5 Stabilization of Systems with Oscillatory Center

Oj =

1 16

F

m

337

m

j1 j1 c30 + 6 ¦ d1tj1 D tj + c12 + 2 ¦ d1tj1 J tj t=1

m

+2 ¦

j1 d2t E tj

t=1 m

+2¦

j2 + c21 + 2

m

¦

t=1

j2 d1t E tj

t=1 m

j2 d2tj2 D tj + c03 +6

t=1

¦

G (11.59)

d2tj2 J tj

t=1

1 j1 j1 j1 + c (c + c02 ) 16Z j 11 20 1 j2 j1 j2 j1 j2 (c j2 (c j2 + c02 ) − c20 c20 + c02 c02 ), − 16Z j 11 20

j = 1, · · · ,t.

Now we are ready to state the following result. Theorem 11.9. The system (11.55) is state feedback stabilizable if the following conditions (i)–(iii) hold. (i).

w |T |+|S| pi (0) = 0, w (x1 )T w zT

2|T | + |S|  3, i = 1, · · · , s.

(11.60)

(ii).

wqj (0) = 0, wx i := j

wqj (0) = 0, wz

and/or k := j,

w 2q j (0) = 0, w zi w zk

(11.61)

i, j, k = 1, · · · ,t.

(iii). There exists a quadratic homogeneous vector ) (z) = col(I 1 , · · · , I m ), such that either one of the following two systems l (11.62) r j = O j r3j + (H1j (r))2 + (H2j (r))2 , j = 1, · · · ,t and r j = O j r3j + H1j (r) + H2j (r),

j = 1, · · · ,t,

(11.63)

is asymptotically stable. Moreover, if conditions (i)–(iii) are satised, a suitable control, which stabilizes system (11.55), is G F ni fi ([ ) 1 i i i i ui = − + ¦ a j x j − a1I , i = 1, · · · , m. gi ([ ) gi ([ ) j=1 1 Proof. Approximate the center manifold equations of x11 , · · · , xm 1 , by I (z), · · · , m I (z), and put them into the equations of center manifold. Using condition (ii) and a Taylor expansion, an approximation of the equations up to degree 3 can be obtained. Then in each pair of equations of z j = (z1j , z2j ) we separate the terms into

338

11 Design of Center Manifold

two categories: the terms involving only xi and the rest. Finally, we can get the equations as: c d > ?c jd c j j d c j d j 0 −Z j z1 f1 (z ) z1 h (z) (11.64) + 1j + O(1z1)4 , j = Z j + f2j (z j ) h2 (z) z2 j 0 z2 where fvj (z j ) =

j

1 w 2 qv (0)zS + S 2 S! w z

¦

S∈J j

+

S∈J j

w 2 qvj (0)zS ()+ j )T , w zS w (x1 )T

¦

S∈J 1j

j

1 w 3 qv (0)zS S 3 S! w z

¦

j = 1, · · · ,t, v = 1, 2.

(11.65)

|T |=1

j

hv (z) =

¦

1 w 3 qvj (0)zS + S S! w z 3

¦

w 2 qvj (0)zS ()− j )T , w zS w (x1 )T

|S|=3,S∈J / j

+

S∈J 1j

w 2 qvj S w z w (x1 )T |S|=1,S∈J / 1j |T |=1

¦

(0)zS ) T (11.66)

j = 1, · · · ,t, v = 1, 2.

|T |=1

We call

c d > ?c d c j j d z1j 0 −Z j z1j f1 (z ) j = Z j + 0 f2j (z j ) z2 z2 j

(11.67)

the reduced sub-systems. Each reduced sub-system can be expressed in normal form via the normal coordinate frame j

j

z˜ j = (˜z1 , z˜2 ),

j = 1, · · · ,t.

From the construction of the normal form, it is easy to show that the Jacobian matrix of the transformation is J=

w (z11 , z12 , · · · , zt1 , zt2 ) = I + O(1˜z1). w (˜z11 , z˜12 , · · · , z˜t1 , z˜t2 )

Consider the penultimate term of (11.64). We may put them together as H(z) = col(h11 (z), h12 (z), · · · , ht2 (z)). Then we have

˜ z) = J −1 H(z(˜z)). H(˜

Since h j (z), j = 1, · · · ,t are third order homogeneous vector eld, and the Jacobian matrix of the transformation has the form as above, it follows that

11.5 Stabilization of Systems with Oscillatory Center

339

˜ z) = H(˜z) + O(1z14) H(˜ which means when we put system (11.64) into a normal coordinate frame and allow O(1z14 ) error, then the last part H(z) needn’t be changed. For notational ease, we still use z for new coordinates, then on the normal coordinate frame system (11.64) becomes c d > ?c j d c j j d j 0 −Z j f˜ (z ) z1 z1 = + 1j j j j Zj 0 z2 z f˜2 (z ) c d 2 j h (z) + 1j + O(1z1)4 , j = 1, · · · ,t. h2 (z) Under polar coordinates, we get normal form with extra terms of H(z) as J j j r j = O j r3j + cos(T j )h1 (z) − sin(T j )h2 (z) + O(1r14 ), T j = \ (r j , T j ), j = 1, · · · ,t.

(11.68)

Note that in this form the f˜vj (z j ), v = 1, 2 have been absorbed into O j . Now we have to estimate cos(T j )h1j (z) − sin(T j )h2j (z) by using (11.66). The estimation is tedious but straightforward. We leave the detailed verication to reader, but give some hints: Using the estimation a(zk1 )2 + b(zk1 )(zk2 ) + c(zk2 )2  |a|(rk cos(Tk ))2 + |b|rk | cos(Tk ) sin(Tk )| + |c|(rk sin(Tk ))2  (max(|a|, |b|) + 12 |b|)rk2 ,

(11.69)

we have |hvj (z)|  Hvj ,

v = 1, 2,

where Hvj are dened in (11.56). Next, using H1j cos(T ) + H2j sin(T )

l  (H1j )2 + (H2j )2 ,

(11.70)

system (11.62) follows. It is clear that the stability of system (11.63) ensures the stability of system (11.62). System (11.63) is a loose but simpler estimation of system (11.62). To verify the expression of O j , j = 1, · · · ,t, in (11.59), using center manifold theorem, a straightforward computation yields (11.59). Note that (11.56) is homogeneous, so asymptotic stability implies approximate stability. The control form is obvious, because our stability analysis is based on the closedloop systems with this control form. 3 2 As with the zero center case, CRDDP and DDP etc. can be used for control design. Example 11.7. Consider the following system

340

11 Design of Center Manifold

⎧ ⎪ ⎪xk = xk+1 , k = 1, · · · , n − 1 ⎪ ⎪ ⎪ xn =df (x)c+ g(x)u ⎪ d c d c ⎪ ⎪ ⎪ ⎨ z11 −z12 sin(x1 z12 ) + z11 z12 = + 1 2 2 z12 z11 z1 (z2 ) + tan(x1 x2 ) ⎪ ⎪ ⎪ d c d c d c ⎪ ⎪ ⎪ z2 2 1 )3 ) − 1 ⎪ −2z exp((z ⎪ 1 2 1 ⎪ + , ⎩ 2 = log(1 + x1 z22 − P x1 z11 ) z2 2z21

(11.71)

where P is a parameter. It is easy to check that conditions (i) and (ii) of Theorem 11.9 are satised. To check condition (iii), let

I (z) = a1 (z11 )2 + b1(z11 )(z12 ) + c1 (z12 )2 +a2 (z21 )2 + b2 (z21 )(z22 ) + c2 (z22 )2 +d11 z11 z21 + d12(z11 )(z22 ) + d21z12 z21 + d22z12 z22 . Using notation (11.58), the parameters can be easily found as ⎧ 11 11 11 11 11 c11 ⎪ c11 30 = c21 = c12 = c03 = 0, c20 = c02 = 0, 11 = 1, ⎪ ⎪ ⎪ ⎪ 12 12 12 12 12 12 12 ⎪ c30 = c12 = c21 = c03 = 0, c20 = c11 = c02 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ 21 21 21 21 21 21 ⎪ c21 ⎪ 30 = c21 = c12 = c03 = 0, c20 = c11 = c02 = 0, ⎪ ⎪ ⎪ ⎪ ⎨ c22 = c22 = c22 = c22 = 0, c22 = c22 = c22 = 0, 30 21 12 03 20 11 02 11 = 0, 11 = 1, ⎪ d21 d11 ⎪ ⎪ ⎪ ⎪ ⎪ 12 12 = d 21 = d 21 = 0, d 22 = 1, ⎪ d = d21 ⎪ 11 21 21 ⎪ ⎪ 11 ⎪ ⎪ 1=a , 1 ⎪ D E ⎪ 1 1 1 = b1 , ⎪ ⎪ ⎪ ⎩ D1 = a , E1 = b , 2

2

2

2

22 = 0, d11

J11 = c1 , J21 = c2 .

Then a straightforward computation shows: ⎧ 1 ⎪ ⎪ O 1 = b1 , ⎪ ⎪ 8 ⎪ ⎪ ⎪ 1 ⎪ ⎪ O2 = (a2 + 3c2 ), ⎪ ⎪ ⎪ 8 ⎪ ⎪ 1 = (r )2 r (|d | + |d | + |d | + |d |), ⎪ H ⎪ 1 2 11 12 21 22 1 ⎪ ⎪ ⎪ ⎪ 1 2 ⎨ H2 = r1 (r2 ) , ⎪ H12 = (r1 )3 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ H22 = (r1 (r2 )2 + | P |(r1 )2 r2 )(|d11 | + |d12| + |d21| ⎪ ⎪ ⎪ Y ⎪ ⎪ ⎪ ⎪ +|d22|) + ((r1 )2 r2 + |P |(r1 )3 ) max(|a1 |, |c1 |) ⎪ ⎪ ⎪ ⎪ Z Y Z ⎪ 1 1 ⎪ ⎩ + |b1 | + (|P |r1 (r2 )2 ) max(|a2 |, |c2 |) + |b2 | . 2 2

(11.72)

11.6 Stabilization Using Generalized Normal Form

341

Set redundant parameters to be zero as: d11 = d12 = d21 = d22 = 0; a1 = c1 = b2 = 0; and let a2 = c2 . Then (11.71) becomes ⎧ 1 ⎪ ⎪ r1 = b1 r13 + r1 r22 , ⎪ ⎪ 8 ⎪ ⎪ ⎨ 1 1 (11.73) r2 = (a2 )r23 + r13 + r12 r2 |b1 | ⎪ 2 2 ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎩ + |P |r13 |b1 | + |P |r1 r22 |a2 |. 2 Using DDP on (11.73) yields that V ⎧ U 1 ⎪ ⎪ ⎨− 8 b1 > 1, U V ⎪ 1 1 1 ⎪ ⎩− a2 > |a2 ||P | + 1 + |b1 ||P | + |b1 |. 2 2 2 It is obvious that the above inequality has solution if and only if |P | < 1/2. For such P we have the following solution: b1 < −8,

a2 < −

|b1||P | + 2 + |b1| . 1 − 2|P |

Particularly, we may choose: b1 = −9, and a2 = −

9| P |+11 1−2| P |

− 1. Then the following:

J n f (x) 1 u=− + Oi x i ¦ g(x) g(x) i=1 U > ?b V A @ 9|P | + 11 −O1 −9z11 z12 + − − 1 (z21 )2 + (z22 )2 1 − 2|P | stabilizes the overall system (11.71).

11.6 Stabilization Using Generalized Normal Form This section considers the system under generalized normal form (11.13) and of non-minimum phase. For non-minimum phase case, we need an assumption: A1. q(z, w) with its rst order derivatives vanish at the origin. In fact, the canonical form (11.13) is not unique, but if we assume there exists another coordinate frame (˜z, w), ˜ such that system (11.13) keeps its structure unchanged. Then it is easy to see that the Jacobian matrix ? > J11 J12 Jz˜,w˜ (0) = . 0 J22 Then one sees that the assumption A1 is independent of the different coordinate frames.

342

11 Design of Center Manifold

The assumption A1 means that the zero dynamics has zero linear part, i.e., the system has zero center. In fact a necessary condition for the stabilizability is that the zero dynamics should have its linear part with zero real part eigenvalues. We dene the injection degrees as  H Oi = min 21K1 + 1T1  1K1 > 0, zK1 wT is in qi (11.74) with non-zero coefcient} , i = 1, · · · r. The leading degrees (L.D.) are dened as J Oi , Oi is odd; Li = Oi + 1, Oi is even,

i = 1, · · · , r.

(11.75)

Next, we give an example to explain the injection degrees. Example 11.8. Consider the following system ⎧ x1 = x2 + w21 u1 ⎪ ⎪ ⎪ ⎪ ⎪ x2 = u1 + w32 u2 ⎪ ⎪ ⎪ ⎨x = x + w sin(x )u 3 4 1 1 1 2 u + ew1 u ⎪ x  = x 4 2 ⎪ 3 1 ⎪ ⎪ ⎪ ⎪w 1 = x1 w2 + w32 ⎪ ⎪ ⎩ w 2 = x4 w1 + x23 w1 w2 + x21 x43 w1 .

(11.76)

This is a generalized normal form. Now z1 = (x1 , x3 ), z¯1 = (x2 , x4 ). In q1 , we have only one term of the form of zK1 wT , that is, z1 w2 . For this term we have K = (1, 0) and T = (0, 1), so the injection degree is

O1 = 21K1 + 1T1 = 3. For q2 , we have two terms with the injection form: For z23 w1 w2 we have K = (0, 2) and T = (1, 1), then 21K1 + 1T 1 = 6. For the second one z21 z43 w1 we have K = (2, 4) and T = (1, 0), so 21K1 + 1T1 = 13. Choosing the smallest one, we have the injection degree O2 = min{6, 13} = 6. Now for the leading degrees, we have L1 = O1 = 3,

L2 = O2 + 1 = 7.

The motivation is to convert all the dynamics on center manifold into an odd leading system. Denition 11.4. A system w i = fi (w),

i = 1, · · · , r

is said to be L = (L1 , · · · , Lr ) approximately asymptotically stable at the origin, if the corresponding uncertain system

11.6 Stabilization Using Generalized Normal Form

w i = fi (w) + O(1w1Li +1 ),

343

i = 1, · · · , r

is asymptotically stable at the origin. Theorem 11.10. For system (11.13), assume there exist m homogenous quadratic functions I (w) = (I1 (w), · · · , Im (w)) and m homogenous cubic functions

\ (w) = (\1 (w), · · · , \m (w)) such that the following hold: (i). There exists an integer s > 3, such that ⎛

⎞

⎜ ⎟ L.D. ⎝Lq(I +\ ,0, · · · , 0,w) (I + \ )⎠  s, / 01 2

(11.77)

1U 1−m

⎛

⎞

⎜ ⎟ L.D. ⎝ p(I + \ , 0, · · · , 0, w)D (I + \ , 0, · · · , 0, w)⎠  s. / 01 2 / 01 2 1U 1−m

(11.78)

1U 1−m

(ii). ⎛

⎞

⎜ ⎟ L.D. ⎝qi (I + \ , 0, · · · , 0, w)⎠ = Li , / 01 2

i = 1, · · · , r.

(11.79)

1U 1−m

(iii). w = q(I + \ , 0, · · · , 0, w) / 01 2

(11.80)

1U 1−m

is L = (L1 , · · · , Lr ) approximately asymptotically stable at the origin. (iv). qi (I + \ + O(1w1s), O(1w1s ), · · · , O(1w1s ), w) / 01 2 1U 1−m

= qi (I + \ , 0, · · · , 0, w) + O(1w1Li +1 ), / 01 2

i = 1, · · · , r.

(11.81)

1U 1−m

Then the overall system (11.13) is state feedback stabilizable. Note that, in the above for notational ease we denote q(z, w) = q(z1 , z¯1 , w) etc. So q(I + \ , 0, · · · , 0, w) means z1 = I + \ and z¯1 = 0. / 01 2 1U 1−m

344

11 Design of Center Manifold

Proof. It is similar to the argument in previous session. We just outline the proof. Choose the control as Y Z ui = −Di (z, w) + ai1 zi1 + · · · + ain ziUi − ai1 (Ii (w) + \i (w)) , i = 1, · · · , m (11.82) and use

J

z1 = I (w) + \ (w), z¯1 (w) = 0

(11.83)

as the approximation of the center manifold of the closed-loop system. Then the approximation error is ⎡ ⎤ Z1 wz ⎢ .. ⎥ q −⎣ . ⎦ w w (z(w),w) Zm wz = q + p(I (w) + \ (w), 0, · · · , 0, w)D (I (w) + \ (w), 0, · · · , 0, w) / 01 2 / 01 2 w w (z(w),w) 1U 1−m

= O(1w1s ), where

1U 1−m

>

? 0 Zi = Ai i ui (z(w), w) + Di (z(w), w) +pi (z(w), w)u(z(w), w), i = 1, · · · , m. zi (w) + b

Note that the rst term is Lq(I +\ ,0,···,0,w) (I + \ ). So (11.77) and (11.78) assure that the approximation error is of O(1w1s ). That is the true center manifold equation can be expressed as J z1 = h1 (w) = I (w) + \ (w) + O(1w1s ) (11.84) z2 = h2 (w) = O(1w1s ). Now according to (11.81), the O(1w1s ) center manifold approximation error does not affect the leading degree terms of each equations of the dynamic equations of the center manifold. Note that the approximately asymptotical stability of (11.80) assures that w = q(I + \ , 0, · · · , 0, w) + O(1w1s ) / 01 2

(11.85)

1U 1−m

is asymptotically stable at the origin. Using (11.84), the true dynamics on the center manifold is w = q(h1 (w), h2 (w), w) = q(I + \ + O(1w1s ), O(1w1s ), w).

(11.86)

Recalling (11.81), one sees that the true dynamics on the center manifold, (11.86), is of the form (11.85), and therefore it is asymptotically stable.

11.6 Stabilization Using Generalized Normal Form

345

Finally, we choose aij such that each linear block is Hurwitz, i.e. Ui

O Ui − ¦ aij O j−1 ,

i = 1, · · · , m

j=1

are Hurwitz polynomials. Then by equivalence theorem of the center manifold, the stability of the dynamics on center manifold assures the asymptotical stability of the overall closed-loop system. 3 2 Remark 11.5. 1. It seems that the condition (11.79) in the Theorem 11.10 is redundant. It is true that this condition is not necessary for the theorem itself. But in practical use, we require that the leading degrees are from the injection degrees and are all odd. In fact, we can only deal with this kind of systems. 2. The control has to be chosen to meet two requirements: First of all, for even injection degree subsystems, the even leading terms have to be eliminated. Secondly, the nal dynamics for the approximate center manifold should be approximately asymptotically stable. 3. The technique of Lyapunov function with homogeneous derivative (LFHD), developed in previous section will be used to design the control to assure the approximately asymptotical stability of the nal dynamics on the center manifold. Example 11.9. Consider a single-input afne nonlinear system ⎧ 2 2 ⎪ ⎪x1 = x2 + x3 + x3 u ⎪ ⎪ x ⎪ ⎪ ⎨x2 = x3 − 2e 3 u x3 = ex3 u ⎪ ⎪ ⎪ w¯1 = −2x1 w¯1 − w¯21 w¯2 ⎪ ⎪ ⎪ ⎩w¯ = −w¯ 3 x − w¯ 3 x . 2 2 1 1 3

(11.87)

The system (11.87) can be approximated by Taylor expansion as following : ⎧ ⎪ x1 = x2 + x23 + x23 u ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎨x2 = x3 − 2(1 + x3)u + O(1x1 )u 2 (11.88) x3 = (1 + x3)u + O(1x1 )u ⎪ ⎪ 2 ⎪  w¯1 = −2x1 w¯1 − w¯ 1 w¯2 ⎪ ⎪ ⎪ ⎩w¯ = −w¯ 3 x − w¯ 3 x . 2 2 1 1 3 Then we take a coordinate change, say ˜ w) ˜ T, (x, w) ¯ T = P(x, where x = (x1 , x2 , x3 ), w¯ = (w¯ 1 , w¯ 2 ), x˜ = (x˜1 , x˜2 , x˜3 ), w˜ = (w˜ 1 , w˜ 2 ), the invertible matrix P is ⎤ ⎡ 1 −2 0 0 0 ⎢0 1 −2 0 0⎥ ⎥ ⎢ ⎥ P=⎢ ⎢0 0 1 0 0⎥ . ⎣0 0 0 1 0⎦ 0 0 0 01

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11 Design of Center Manifold

System (11.88) can be converted into the following form in which the controllable linear part is expressed as a Brunovsky canonical form. ⎧ 2 ⎪ ˜ 2 )u ⎪x˜1 = x˜2 + x˜3 + O(1x1 ⎪ ⎪ ⎪ ⎪ ⎨x˜2 = x˜3 (11.89) ˜ 2 )u x˜3 = u + x˜3 u + O(1x1 ⎪ ⎪ 2 ⎪ w˜ 1 = −2x˜1 w˜ 1 + 4x˜2w˜ 1 − w˜ 1 w˜ 2 ⎪ ⎪ ⎪ ⎩w˜ = −w˜ 3 x˜ + 2w˜ 3x˜ − w˜ 3 x˜ . 2 2 1 2 2 1 3 According to system (11.89), it is easy to nd a function, say h(x) = x˜1 = x1 + 2x2 + 4x3 , and a straightforward computation to show that the essential point relative degree of system (11.87) is U ep = 3, which is equal to the dimension of the controllable linear part of system (11.89). Consequently, we can obtain the generalized normal form by a nonlinear coordinate change. Denote ⎡ 2 ⎤ ⎤ ⎡ x2 + x23 x3 ⎢−2ex3 ⎥ ⎥ ⎢ x 3 ⎢ ⎥ ⎢ ⎥ ⎥ , g = ⎢ ex3 ⎥ 0 f =⎢ ⎢ ⎥ ⎢ ⎥ ⎣ 0 ⎦ ⎣−2x1 w¯ 1 − w¯ 21 w¯ 2 ⎦ 0 −w¯ 32 x1 − w¯ 31 x3 A feasible coordinate change is calculated as (h, L f h, L2f h, w¯1 ,⎤w¯2 )T (z, w)T = ⎡ x1 + 2x2 + 4x3 ⎢ x2 + 2x3 + x2 ⎥ 3⎥ ⎢ ⎥, =⎢ x 3 ⎢ ⎥ ⎣ ⎦ w¯1 w¯2 where z = (z1 , z2 , z3 ), w = (w1 , w2 ), and the Jacobian matrix J(z,w) |(0,0) is nonsingular. Under this coordinate frame system (11.87) can be converted into the generalized normal form as ⎧ 2 3 ⎪ ⎪ ⎪z1 = z2 + z3 u + O(1x1 )u ⎪ 2 3 ⎪ ⎪ ⎨z2 = z3 + 2z3 u + 2z3uO(1x1 )u (11.90) z3 = u + z3u ⎪ ⎪ 2 2 ⎪w 1 = −2z1 w1 + 4z2 w1 − 4z w1 − w w2 ⎪ 3 1 ⎪ ⎪ ⎩w = −w3 z + 2w3 z − 2w3 z2 − w3 z 2 2 1 2 2 2 3 1 3 It is easy to check that this system is of non-minimum phase, we use the technique developed in Section 11.4 to stabilize the dynamics on the designed center manifold, which then also stabilizes the overall system. Note that in system(11.90) the leading degrees are L1 = 3, L2 = 5, and the conditions of Theorem 11.10 hold. According to (11.82), the control is constructed as u = −6z1 − 11z2 − 6z3 + 12w21 + 6w22

11.6 Stabilization Using Generalized Normal Form

347

Then the linear part of (11.90) is Hurwitz. Let ⎤ ⎡ 2 ⎡ ⎤ M1 (w) 2w1 + w22 ⎦ = O(1w12 ) 0 M (w) = ⎣M2 (w)⎦ = ⎣ M3 (w) 0 be used to approximate the center manifold. Then we have M M (w) = DM (w)w = O(1w14 ). According to the approximation theorem, the center manifold can be expressed as ⎧ 2 2 4 ⎪ ⎨z1 = 2w1 + w2 + O(1w1 ) 4 (11.91) z2 = O(1w1 ) ⎪ ⎩ 4 z3 = O(1w1 ) The dynamics on the center manifold becomes J w 1 = −4w31 − 2w1 w22 − w21 w2 + O(1w15 ) w 2 = −w52 − 2w32 w21 + O(1w17 )

(11.92)

For the certain part of (11.92), choose a LFHD as V = 14 w41 + 12 w22 , then V = w31 w1 + w2 w2 = −4w61 − 2w41 w22 − 2w21 w42 − w51 w2 − w62  −4w61 − w51 w2 − w62 5 1  −4w61 − w62 + w61 + w62 6 6 0 V is negative. So system (11.92) is asymptotically stable. By Theorem 11.10, system (11.87) is also asymptotically stable. The second example shows the design of stabilizing control for multi-input case. Example 11.10. Consider the following system ⎧ x1 = x2 + sin(z1 )u1 ⎪ ⎪ ⎪ ⎪x = tan(x ) + ex2 u ⎪ 2 1 1 ⎪ ⎪ ⎪ ⎪ ⎪ x  = x + (x + z )u 4 3 2 1 + z2 u2 ⎨ 3 x4 = x5 + ln(1 + x1 z1 )u2 ⎪ ⎪ ⎪x5 = x3 z2 + u2 ⎪ ⎪ ⎪ ⎪ ⎪ z1 = sin(z1 )x1 + z32 ⎪ ⎪ ⎩ z2 = z2 x1 x3 .

(11.93)

It is obvious that the system is already in a generalized normal form. First, we use two quadratic functions h1 (z) and h2 (z) to approximate the center manifold as

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11 Design of Center Manifold

x1 (z) ∼ h1 (z) = D1 z21 + E1 z1 z2 + J1 z22 x3 (z) ∼ h2 (z) = D2 z21 + E2 z1 z2 + J2 z22 x2 (z) ∼ 0, x4 (z) ∼ 0, x5 (z) ∼ 0.

(11.94)

Then we design the controls as u1 = −e−x2 tan(x1 ) + e−x2 [a11 x1 + a12x2 − a11(D1 z21 + E1 z1 z2 + J1 z22 )] u2 = −x3 z2 + [a21x3 + a22x4 + a23x5 − a21 (D2 z21 + E2 z1 z2 + J2 z22 )].

(11.95)

Now for the closed-loop system we verify the approximation error. ⎡ ⎤ w h1 w h 1 ⎢ w z1 w z2 ⎥ ⎢ ⎥ ⎢ w0 w0 ⎥ ⎤ ⎡ ⎢ ⎥ − sin(z1 ) tan(h1 (z)) ⎢ wz wz ⎥ ⎢ 1 2 ⎥> ⎥ ? ⎢ 0 ⎢ ⎥ ⎥ ⎢ ⎢ w h2 w h2 ⎥ sin(z1 )h1 (z) + z32 3 ⎢ − ⎢−(h2 (z) + z2 ) tan(h1 (z)) ⎥ ⎢ ⎥ ⎥ = O(1z1 ). ⎢ w z1 w z2 ⎥ z2 h1 (z)h2 (z) ⎦ ⎣ − ln(1 + h1(z)z1 )h2 (z)z2 ⎢ ⎥ ⎢ w0 w0 ⎥ 0 ⎢ ⎥ ⎢ w z1 w z2 ⎥ ⎢ ⎥ ⎣ w0 w0 ⎦

w z1 w z2 (11.96) Hence the approximate order is s = 3. Next, we consider the dynamics on the approximated center manifold. We have J z1 = z1 (D1 z21 + E1 z1 z2 + J1 z22 ) + z32 (11.97) z2 = z2 (D1 z21 + E1 z1 z2 + J1 z22 )(D2 z21 + E2 z1 z2 + J2 z22 ). As a convention, we choose L1 = 3 and L2 = 5. The technique developed in previous section can be used to design controls to make (11.97) approximately stable. To be independent, we simply choose a Lyapunov function with homogeneous derivative as V = z41 + z22 , and choose D1 = −1, E1 = 0, J1 = −1, D2 = 0, E2 = 0, J2 = 1. Then V |(11.97) = −4z61 − 4z41 z22 + 4z31 z32 − 2z21z42 − 2z62 , which is negative denite. So (11.97) is approximately stable with homogeneous degrees (3, 5). Finally, we have to check condition (11.80), which means the error doesn’t affect the asymptotical stability. Since z1 (h1 (z) + O(1z13 ) + z32 − (z1 h1 (z) + z32 ) = O(1z14 ) z2 (h1 (z) + O(1z13 ))(h2 (z) + O(1z13 )) − z2 h1 (z)h2 (z) = O(1z16 ), (11.80) follows. To construct the controls we have to choose ai j to make the linear part stable. Say, let a11 = −1, a12 = −2, a21 = −1, a22 = −3, and a23 = −3. Then the controls

11.7 Advanced Design Techniques

349

become u1 = −e−x2 tan(x1 ) + e−x2 [−x1 − 2x2 + (−z21 − z22 )] u2 = −x3 z2 + (−x3 − 3x4 − 3x5 + z22 ).

(11.98)

11.7 Advanced Design Techniques This section considers how to improve the center manifold design techniques. We rst consider some motivate examples. Example 11.11. Consider the stabilization problem of the following system ⎧ x1 = x2 ⎪ ⎪ ⎪ ⎨x = x 2 3 (11.99) ⎪x3 = x1 sin(z) + u ⎪ ⎪ ⎩ z = x1 z. The system is in Byrnes-Isidori normal form. First, we can choose a linear feedback to stabilize the linearly controllable states x. Say, set the eigenvalues as {−1, −2, −3}, to this end we use u = −x1 sin(z) − 6x1 − 11x2 − 6x3 := −x1 sin(z) + a1 x1 + a2 x2 + a3 x3 . Let the center manifold to be designed as x = h(z). We may choose ⎧ 2 ⎪ ⎨x1 = I (z) := −z x2 = 0 ⎪ ⎩ x3 = 0 to approximate the center manifold equation h(z). Then the control can be chosen as u = −x1 sin(z) + a1 x1 + a2 x2 + a3 x3 − a1(I (z)) = −x1 sin(z) − 6x1 − x2 − 7x3 − 6z2 .

(11.100)

It follows that the error degree of the approximation (EDA) is [4]

wx (x1 (z)z) − (Ax(z) + bu(x(z), z)) wz = (−2z)(−z2 z) − 0 = O(1z14 ).

EDA =

So the center manifold can be expressed as ⎡ 2⎤ −z h(z) = ⎣ 0 ⎦ + O(1z14 ). 0

(11.101)

(11.102)

The dynamics on the center manifold of the closed-loop system are given by A @ (11.103) z = h(z)z = −z2 + O(1z14 ) z = −z3 + O(1z15 ).

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11 Design of Center Manifold

Obviously, it is asymptotically stable and so is the closed-loop system. Therefore, the state feedback control (11.100) stabilizes system (11.99). Next, we modify the system a little. Example 11.12. Consider the stabilization problem for the following system ⎧ x1 = x2 + x1 zu ⎪ ⎪ ⎪ ⎨x = x + zu 2 3 (11.104) ⎪ x  = x 3 1 sin(z) + u ⎪ ⎪ ⎩ z = x1 z. The system is in a generalized Byrnes-Isidori normal form. We may still use the same method as in Example 11.11 to stabilize it. The only difference, which has to be veried is the degree of approximation. In fact, the error degree can be calculated ⎤ ⎡ ⎤ ⎡ as dx1 (z) −2z ⎣dx2 (z)⎦ x1 (z)z = ⎣ 0 ⎦ x1 (z)z = O(1z14 ), dx3 (z) 0 ? ? > > and −z2 z(−z2 sin(z)) x1 (z)z(x1 (z) sin(z)) = O(1z14 ). = z(x1 (z) sin(z)) z(−z2 sin(z)) Then (11.102) remains true. We conclude that the control (11.100) can also stabilize system (11.104). The method used above is what is proposed in Section 11.6. Now, we modify system (11.104) to see that the method developed in Section 11.6 may fail. Example 11.13. Consider the stabilization problem of the following system ⎧ x1 = x2 + x1 zu ⎪ ⎪ ⎪ ⎨x = x + zu 2 3 (11.105) ⎪ x  = sin(z) +u 3 ⎪ ⎪ ⎩ z = x1 z. The system is still in a generalized Byrnes-Isidori normal form. We can still use the equations (11.77) and (11.78) to check the error degree of approximation. (11.77) is the same as in Example 11.12. For (11.78) we have > ? ? > 2 x (z) sin(z) −z sin(z) EDA = 1 = O(1z12 ). = z sin(z) z sin(z) So we can only claim that the center manifold has the form of ⎡ 2⎤ −z h(x) = ⎣ 0 ⎦ + O(1z12 ) = O(1z12 ). 0 The dynamics on the center manifold of the closed-loop system is

(11.106)

11.7 Advanced Design Techniques

@

A z = h(z)z = −z2 + O(1z12) z = O(1z13 ).

351

(11.107)

Clearly we can say nothing about the stability of (11.107). It is obvious that the method developed in Section 11.6 fails in Example 11.13. But it doesn’t mean the system (11.105) is not stabilizable by state feedback. In fact, the problem is that the tool is not subtle enough. Let us try to sharpen our tool. Say, choose ⎧ ⎪ ⎨x1 = I1 (z) x2 = I2 (z) ⎪ ⎩ x3 = I3 (z) to approximate the center manifold, and set x1 (z) = I1 (z) = −z2 as before. Then choose x2 , x3 and u in such a way that turns the right hand side of (11.105) to zero. That is ⎧ ⎪ ⎨x2 + x1 zu = 0 x3 + zu = 0 ⎪ ⎩ sin(z) + u = 0 Note that the linear part of u has to be chosen such that the linearly controllable state variables are stable. So the control can be chosen as u = − sin(z) + a1x1 + a2 x2 + a3 x3 − a1 (I1 (z)) − a2 (I2 (z)) − a3 (I3 (z)). (11.108) Accordingly, we have x2 (z) = I2 (z) = −x1 (z)zu(x(z), z) = −z3 sin(z) x3 (z) = I3 (z) = −zu(x(z), z) = −z sin(z). Checking the error degree of approximation, we have ⎤ ⎡ w I1 (z) ⎢ wz ⎥ ⎡ ⎤ ⎥ ⎢ I2 (z) + I1 (z)zu(z, I1 (z), I2 (z), I3 (z)) ⎢ w I2 (z) ⎥ ⎥ ⎣ I3 (z) + zu(z, I1 (z), I2 (z), I3 (z)) ⎦ EDA = ⎢ ⎢ w z ⎥ I1 (z)z − ⎥ ⎢ sin(z) + u(z, I1 (z), I2 (z), I3 (z)) ⎣ w I (z) ⎦ 3 ⎤ ⎡ wz −2z = ⎣−3z2 sin(z) − z3 cos(z)⎦ (−z2 )z = O(1z14 ). − sin(z) − 2 cos(z) Then (11.103) remains true. That is, control (11.108) stabilizes the system (11.105). We would like to emphasize the idea in the above argument: In Section 11.5, we use xi1 = I1i (z) to design the center manifold to assure the stability of the dynamics on center manifold and set xij = 0, j > 1 to assure the required accuracy (degree) of the approximation error. In Section 11.6, the method is used in the context of a generalized Byrnes-Isidori normal form. It was shown in the above examples that unlike in the standard case, xij = 0, j > 1 may not be enough to assure the required accuracy

352

11 Design of Center Manifold

(degree) of the approximation error. But we may also design xij = I ij (z), j > 1 to improve the accuracy (degree) of the approximation error to meet our requirement on the accuracy of dynamics on center manifold. In the rest of this section we will develop this idea into a systematic treatment for the design of an approximate center manifold, xij = I ij (z), j  1. Consider the stabilization problem of the generalized Byrnes-Isidori normal form (11.13). First, assume the linear feedbacks are chosen such that the linearly controllable variables xij , i = 1, · · · , m, j = 1, · · · , Ui are stabilized by ui = ai1 xi1 + · · · + aiUi xiUi ,

i = 1, · · · , m.

Assume I ij (z) are used to approximate the center manifold. Then following the design idea in previous examples, we construct controls as ui (x, z) = −Di (x, z) + ai1 x11 + · · · + aiUi xiUi − ai1 I1i (z) − · · · − aiUi IUi i (z), i = 1, · · · , m.

(11.109)

Now I1i (z) can be chosen freely as a polynomial with lowest degree  2. Then we can solve I ij = xij , j > 1 from the following equations. F1i (x, z) := xi2 + pi1 (x, z)Di (x, z) = 0, .. . FUi i −1 (x, z) := xiUi + piUi −1 (x, z)Di (x, z) = 0,

(11.110) i = 1, · · · , m.

Lemma 11.3. Locally on a neighborhood U of the origin, the xij , i = 1, · · · , m, j = 2, · · · , Ui can be uniquely solved from equation (11.110) as functions of z (denoted by xij = I ij (z), j  2), where xi1 = I1i (z) are known. Proof. Note that since f (0) = 0, D i (0, 0) = 0. Moreover, pij (0, 0) = 0, i = 1, · · · , m, j = 1, · · · , Ui − 1. It follows that the Jacobian matrix  d c  w Fji (x, z)   U − 1 = Im i = 1, · · · , m; j = 1, · · · , J=   i  w xij+1  (0,0)

By the implicit function theorem, xij , j  2 can be solved from (11.110) locally. 2 3 Using the above notations and Lemma 11.3, the following theorem is an immediate consequence of the center manifold theory. Theorem 11.11. Assume there exist I ij (z), i = 1, · · · , m, j = 1, · · · , Ui , such that (i). the error degree of approximation is  d c w I ij (z)  EDA =  i = 1, · · · , m; j = 1, · · · , Ui q(I (z), z) = O(1z1d+1 ). (11.111) wz  (ii). The errors in the dynamics on center manifold, caused by the approximation error, are

References

qk (I (z) + O(1z1d+1 ), z) = qk (I (z), z) + O(1z1tk +1 ),

k = 1, · · · , r.

353

(11.112)

(iii). The approximated dynamics of the center manifold zk = qk (I (z), z),

k = 1, · · · , r,

(11.113)

are (t1 , · · · ,tr )-degree approximately stable, i.e. zk = qk (I (z), z) + O(1z1tk +1 ),

k = 1, · · · , r

(11.114)

are asymptotically stable. Then the system (11.13) is asymptotically stabilizable. An applicable stabilizer is (11.109). Now let [k (z) be an approximation of qk (I (z), z) , consisting of its lowest degree non-vanishing terms, with the degrees of [k (z) being deg([k (z)) = sk , k = 1, · · · , m. Then we can construct the approximate system of (11.113) as zk = [k (z),

k = 1, · · · , r.

(11.115)

Using LFHD, we have Theorem 11.12. Assume (i). sk  tk , k = 1, · · · , r; (ii). there exists a LFHD V > 0 such that V |(11.115) < 0.

(11.116)

Then the system (11.13) is asymptotically stabilizable by control (11.109). Proof. An LFHD V > 0 satisfying (11.116) assures the asymptotical stability of zk = [k (z) + O(1z1sk +1 ),

k = 1, · · · , r.

Now the true dynamics on the center manifold are zk = qk (I (z), z) + O(1z1tk +1 ) = [k (z) + O(1z1sk +1 ) + O(1z1tk +1 ) = [k (z) + O(1z1sk +1 ), k = 1, · · · , r. The conclusion follows.

3 2

References 1. Aeyels D. Stabilization of a class of non-linear systems by a smooth feedback control. Sys. Contr. Lett., 1985, 5(5): 289–294. 2. Behtash S, Dastry D. Stabilization of non-linear systems with uncontrollable linearization. IEEE Trans. Aut. Contr., 1988, 33(6): 585–590. 3. Byrnes C, Isidori A, Willems J. Passivity, feedback equivalence, and the global stabilization ofminimum phase nonlinear systems. IEEE Trans. Aut. Contr., 1991, 36(11): 1228–1240. 4. Carr J. Applications of Centre Manifold Theory. New York: Springer, 1981. 5. Cheng D. Stabilization of a class of nonlinear non-minimum phase systems. Asian J. Contr., 2000, 2(2): 132–139.

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6. Cheng D, Martin C. Stabilization of nonlinear systems via designed center manifold. IEEE Trans. Aut. Contr., 2001, 46(9): 1372–1383. 7. Hahn W. Stability of Motion. Berlin: Springer, 1967. 8. Hermes H. Homogeneous feedback controls for homogeneous systems. Sys. Contr. Lett., 1995, 24(1): 7–11. 9. Isidori A. Nonlinear Control Systems, 3rd edn. London: Springer, 1995. 10. Khalil H. Nonlinear Systems, 3rd edn. New Jersey: Prentice Hall, 2002.

Chapter 12

Output Regulation

Output regulation is the problem of nding a control law by which the output of the concerning plant can asymptotically track a prescribed trajectories and/or asymptotically reject undesired disturbances. Meanwhile, the unforced closed-loop system is required to be asymptotically stable. For linear systems the internal model principle is a powerful tool to solve this problem. It is a central problem in control theory. The purpose of this chapter is to give a brief outline of the basic concepts, some main problems and fundamental results in this eld. Section 12.1 reviews the output regulation of linear control systems. Section 12.2 considers the local output regulation of nonlinear systems. Robust output regulation problem is discussed in Section 12.3.

12.1 Output Regulation of Linear Systems This section considers the output regulation of linear control systems. In certain sense, it is also the foundation of the output regulation of nonlinear systems. The basic solution to the output regulation of linear systems is called the internal model principle. The internal model principle of linear systems was developed in the seventies of last century [7, 16]. We give a prole here. To begin with, we introduce a concept of stability. Denition 12.1. [2, 13] Given a dynamic system x = f (x),

f (0) = 0.

(12.1)

It is said to be Poisson stable at x(0) = p if for any T > 0 and any open neighborhood U of p there exists a t > T such that x(t) ∈ U. The system (12.1) is said to be neutrally stable if the equilibrium p0 = 0 is a stable equilibrium (in the sense of Lyapunov) and each initial state p is stable in the sense of Poisson. We give a simple example to depict this. Example 12.1. Consider the following system J x1 = ax2 x2 = −bx1 , where a > 0 and b > 0. The trajectory is D. Cheng et al., Analysis and Design of Nonlinear Control Systems © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2010

(12.2)

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12 Output Regulation

J √ √ √ √ x1 = k1 a sin( abt) + k2 a cos( abt) √ √ √ √ x2 = k1 b cos( abt) − k2 b sin( abt). It is obvious that 0 is stable and any initial (x0 , y0 ) is Poisson stable. Hence (12.2) is neutrally stable at the origin. Consider a linear system J x = Ax + Bu + Pw, x ∈ Rn , u ∈ Rm , e = Cx + Qw, e ∈ Rm ,

(12.3)

with the exogenous disturbance input w ∈ Rr generated by w = Sw,

(12.4)

which is assumed to be neutrally stable. Denition 12.2. 1. The problem of output regulation is the problem of nding, if possible, a regulator J [ = F [ + Ge (12.5) u = H[ such that the closed-loop system ⎧ ⎪ ⎨x = Ax + BH [ + Pw [ = GCx + F [ + CQw ⎪ ⎩ w = Sw

(12.6)

satises: (i). when w = 0, (12.6) is asymptotically stable; (ii). lim e(t) = 0. t→f

2. If the regulator (12.5), designed for a nominal set of parameters {A0 , B0 ,C0 , P0 , Q0 } is also available for {A, B,C, P, Q} in an open neighborhood of the nominal parameters, the regulator is called a structurally stable regulator. Note that condition (i) is equivalent to that the following matrix J is Hurwitz. ? > A BH . (12.7) J= GC F The following result, commonly called the internal model principle, was established in the works of [7, 16, 5, 6]. A simple proof of the existence of robust linear regulator was presented in [9]. Theorem 12.1. 1. There exists a regulator, if and only if the pair (A, B) is stabilizable, the pair (C, A) is detectable, and the linear matrix equations (called the regulator equations)

12.1 Output Regulation of Linear Systems

J 3 S = A3 + B* + P 0 = C3 + Q

357

(12.8)

have a solution 3 , * . 2. There exists a structurally stable regulator, if and only if the pair (A0 , B0 ) is stabilizable, the pair (C0 , A0 ) is detectable, and the regulator equations J 3 S = A0 3 + B0* + P (12.9) 0 = C0 3 + Q have a solution 3 , * for all P, Q. To prove it we need some preparations. First of all, observing the matrix J in (12.7), we can prove the following: Lemma 12.1. A necessary condition for the matrix J to be Hurwitz is that (A, B) is stabilizable and (C, A) is detectable. Proof. It is obvious that if J in (12.7) is Hurwitz then the pair > ? > ? A 0 B , GC F 0

(12.10)

is stabilizable. Writing (A, B) into a controllable canonical form, which means a suitable linear coordinate change z = T x is used, where > ? T 0 T= 0 . 0 I Then the pair becomes V U V⎤ ⎡U A11 A12 0 ⎢ 0 A22 0 ⎥, ⎣ ⎦ GCT0

F

⎡ ⎤ B ⎣0⎦ . 0

(12.11)

Since overall pair is stabilizable, A22 must be Hurwitz. Now since (A11 , B) is a controllable pair, (A, B) is stabilizable. By duality, (C, A) is detectable. 3 2 Moreover, we have Lemma 12.2. Assume the matrix J in (12.7) is Hurwitz. Then lim e(t) = 0,

t→f

if and only if there exist matrices 3 and 6 satisfying ⎧ ⎪ ⎨3 S = A3 + BH 6 + P 6 S = F6 ⎪ ⎩ 0 = C3 + Q.

(12.12)

(12.13)

358

12 Output Regulation

Proof. This is a particular case of Lemma 12.9.

3 2

Remark 12.1. Lemma 12.2 is very important in whole internal model principle. But (12.13) seems not very natural. We give an intuitive interpretation for its insight [9]. We may assume the steady-state solution of (12.6) is: x = 3 w and [ = 6 w. Then we have ⎧ ⎪ ⎨(3 S − A3 − BH 6 − P)w(t) = 0 (6 S − F 6 )w(t) = 0 ⎪ ⎩ e(t) = (C3 + Q)w(t) → 0, (t → f). For arbitrary w(t), the above equation implies (12.13). Secondly, from Lemmas 12.1, 12.2 it is clear that for structurally stable regulator, the necessity of the condition in 2 of Theorem 12.1 is obvious. In addition to the Lemma 12.2, the only thing we may need to mention is that (12.8) should hold over a neighborhood of the nominal parameters. Particularly, (12.9) should hold around a neighborhood of (P0 , Q0 ). As for linear equation, it is true for all (P, Q). For the sufciency, we will construct the regulator. It is a heavy job. We start with a lemma: Lemma 12.3. The regulator equations (12.9) have a solution 3 , * for all P, Q, if and only if > ? A0 − O I B0 (12.14) C0 0 has independent rows (i.e., nonsingular) for each O ∈ V (S). Proof. Express (12.9) as ?> ? ?> ? > > ? > I 0 3 P A0 B0 3 . − n S=− 0 0 * C0 0 * Q Convert it into a standard linear equation,[4], as following ? U> ?V U ?V U> ?V > > P 3 A B I 0 . = −Vc Ir ⊗ 0 0 − ST ⊗ n Vc C0 0 Q 0 0 *

(12.15)

(12.15) has solution for any (P, Q) is equivalent to that the coefcient matrix has full row rank. Vc (X) is used for the column stacking form of matrix X. Let (O1 , · · · , Or ) = V (S), Converting ST into Jordan canonical form, which affects only P and Q, the coefcient matrix becomes an block upper-triangular form as V ⎤ ⎡U A0 − O1 I B0 ∗ · · · ∗ ⎥ ⎢ C0 0 U ⎥ V ⎢ ⎥ ⎢ − O I B A 0 2 0 ⎥ ⎢ ··· ∗ 0 ⎥ ⎢ C 0 0 ⎥. ⎢ ⎥ ⎢ .. .. .. ⎥ ⎢ . . . ⎢ V⎥ U ⎣ A0 − O s I B 0 ⎦ 0 0 ··· C0 0

12.1 Output Regulation of Linear Systems

The conclusion follows immediately.

359

3 2

We continue the proof of the sufciency of Theorem 12.1, by constructing the regulator step by step. We need some more lemmas. Lemma 12.4. Let S be transformed into a block-diagonal form as ? > ∗ 0 , S= 0 Smin where Smin is a q × q matrix whose characteristic polynomial coincides with the minimal polynomial of S. Construct a matrix ) as

) = Im ⊗ Smin.

(12.16)

Then we can nd a qm × m matrix N and an m × qm matrix 4 , such that () , N) is controllable and (4 , ) ) is observable. Proof. Since S is a cycle matrix, it is similar to ⎤ ⎡ 0 0 · · · 0 a1 ⎢1 0 · · · 0 a 2 ⎥ ⎥ ⎢ ⎢0 1 · · · 0 a 3 ⎥ ⎥. ⎢ ⎢ .. .. .. .. ⎥ ⎣. . . .⎦ 0 0 · · · 1 an @ AT @ A Let b = 1, 0, · · · , 0 and c = 0, · · · , 0, 1 . Then N = Im ⊗ b and 4 = Im ⊗ c meet our requirement. 3 2 Lemma 12.5. 1. (A, B) is stabilizable, if and only if for any O , Re(O )  0 D E rank A − O I B = n, (12.17) where n is the dimension of the state space. 2. (C, A) is detectable, if and only if for any O , Re(O )  0 > ? C rank = n. A−OI

(12.18)

Proof. We prove (12.16) only. (12.17) can be proved by duality. (Necessity) There exists M such that A + BM is stable. Hence D E rank A + BM − O I B = n, which implies (12.17). (Sufciency) Assume (A, B) is not stabilizable, and express the system into controllability canonical form as ? > ? > 0 A11 0 , B= . A= A21 A22 B2

360

12 Output Regulation

Then there exists O0 ∈ V (A11 ) with Re(O0 )  0. Then E D rank A − O0I B < n. 3 2 Lemma 12.6. Suppose (A0 , B0 ) and () , N) are stabilizable, (C0 , A0 ) and (4 , ) ) are detectable and the matrix (12.14) is nonsingular for ∀O ∈ V (S). Then the pair > ? > ? A0 0 B0 (12.19) , NC0 ) 0 is stabilizable and the pair E D C0 0 ,

> ? A 0 B0 4 0 )

(12.20)

is detectable. Proof. Assume (12.19) is not stabilizable. Using Lemma 12.3, there exists O with Re O  0 and x ∈ Rn , w ∈ Rqm such that ⎧ T T ⎪ ⎨x (A0 − O I) = w NC0 x T B0 = 0 ⎪ ⎩ T w () − O I) = 0. Note that w := 0. Otherwise it contradicts to the stabilizability of (A0 , B0 ). Hence O ∈ V () ). Set yT = wT N. Then y := 0. Otherwise, it violates the stabilizability of () , N). Now ? > @ T A A0 − O I B0 x −yT =0 C0 0 A @ for xT −yT := 0 and O ∈ V () ), which is a contradiction. Similar arguments show the detectability of (12.20). 3 2 The following lemma is commonly called the separation principle. Lemma 12.7. (Separation Principle) Assume a linear system J x = Ax + Bu y = Cx

(12.21)

is stabilizable and detectable. Then it is stabilizable by a dynamic output feedback J xˆ = K xˆ + Ly (12.22) u = M x, ˆ with properly chosen K, L, and M. Proof. Assume the dynamic control system is chosen as

12.1 Output Regulation of Linear Systems

361

J

xˆ = Axˆ + L(y − Cx) ˆ + Bu u = M x. ˆ

That is K = A − LC + BM. We have to show the closed-loop system > ? > ? x A BM = LC K xˆ

(12.23)

(12.24)

is asymptotically stable with suitable chosen K, L, M. In fact, we may choose the error e = xˆ − x to replace the state variables x. ˆ Under this set of new coordinates (x, e), the system (12.24) becomes > ? > ? x A + BM BM = . (12.25) e 0 A + LC Let M and L be chosen in such a way that both A + BM and A + LC are stable, we are done. 3 2 Now we are ready to construct the regulator. Recall (12.19) and @ A (12.20). Since feedback does not change the stabilizability, using feedback 0, 4 to (12.19) yields ? > ? > B0 A0 B04 (12.26) , NC0 ) 0 is stabilizable. Similarly, using (12.20), the pair > ? E D A 0 B04 C0 0 , NC0 ) is detectable. Then Lemma 12.7 assures that there exist K, L, M such that A⎤ ⎡U VU V ⎤ ⎡ @ A0 B04 B0 V U AV0 BU0 4 M M ⎣ NC0 ) ⎦=⎣ N 0 ) 0 ⎦ @ A C0 0 K L L C0 0 K

(12.27)

(12.28)

is Hurwitz. Using the matrices ) ∈ Mmq×mq , N ∈ Mqm×m , 4 ∈ Mm×qm , and K ∈ Mn×n , L ∈ Mn×m , M ∈ Mm×n obtained above, we construct the regulator of (12.5) with ? > ? > D E N ) 0 , H= 4 M . , G= (12.29) F= L 0 K We claim that this regulator is a structural stable regulator. To prove it we assume (A, B,C) are close to (A0 , B0 ,C0 ) such that the closed-loop system [ \ ⎧ ⎪ x  = Ax + B [ + Pw 4 M ⎪ ⎪ d c d c c d ⎪ ⎨ ) 0 N N  (12.30) [= Cx + [+ Qw ⎪ 0 K L L ⎪ ⎪ ⎪ ⎩ e = Cx + Qw,

362

12 Output Regulation

makes lim e(t) → 0 as long as the matrix t→f

>

A BH J= GC F

?

is Hurwitz. Using Lemma 12.2, equations (12.13) are satised. Construct a Sylvester equation > ? ?> ? > ? > E A BH 3 3 − S= 1 . GC F 6 6 E2

(12.31)

(12.32)

Its matrix form [4] is (I ⊗ J − ST ⊗ I)X = Vc (E),

where X = Vc

> ? 3 . Since Re V (J) < 0 and Re V (S) = 0, (12.32) has a solution for 6

any E. Setting

? −P , E= −GQ >

we obtain two equations A3 + BH 6 + P = 3 S

(12.33)

F 6 − 6 S = −G(C3 + Q).

(12.34)

(12.34) coincides the rst equation of (12.13). So we have only to show that (12.34) implies the second and the third equations of (12.13). For this purpose, we dene two mappings as F : M(n+mq)×r → M(n+mq)×r and G : Mm×r → M(n+mq)×r , where F(6 ) = F 6 − 6 S,

G(Z) = GZ.

Then (12.34) becomes F(6 ) + G(C3 + Q) = 0.

(12.35)

Now it sufces to prove the following two facts: im(F) ∩ im(G) = {0};

(12.36)

Ker(G) = 0.

(12.37)

F(6 ) = 0;

(12.38)

and

Because (12.35) and (12.36) imply

and

12.1 Output Regulation of Linear Systems

G(C3 + Q) = 0.

363

(12.39)

Then (12.37) and (12.39) imply that C3 + Q = 0.

(12.40)

F6 + 6 S

(12.41)

We need one more lemma. Lemma 12.8. The equation

has at least mr linearly independent solutions 6 S. Proof. Consider the equation Smin X − XS = 0,

(12.42)

with its matrix form [4] (I ⊗ Smin − ST ⊗ I)Vc (X ) = 0. Note that

V (I ⊗ Smin − ST ⊗ I) = {Oi − O j | Oi ∈ V (Smin ), O j ∈ V (S)}, (I ⊗ Smin − ST ⊗ I) has at least r zero eigenvalues as j = 1, · · · , r. Hence (12.42) has at least r linearly independent solutions. Recall that > ? I ⊗ Smin 0 F= m . 0 K Then

⎡ ⎤ X ⎢0⎥ ⎢ ⎥ ⎢ ⎥ 61 = ⎢ ... ⎥ , ⎢ ⎥ ⎣0⎦ 0

⎡ ⎤ 0 ⎢X ⎥ ⎢ ⎥ ⎢ ⎥ 62 = ⎢ ... ⎥ , ⎢ ⎥ ⎣0⎦ 0

··· ,

⎡ ⎤ 0 ⎢0⎥ ⎢ ⎥ ⎢ ⎥ 6m = ⎢ ... ⎥ ⎢ ⎥ ⎣X ⎦ 0

provide mr linearly independent solutions. Using this Lemma we know that dim(im(F))  (n + mq)r − mr.

(12.43)

Note that dim(Z) = mr, so dim(im(G))  mr.

(12.44)

According to the second equation of (12.32), G(C3 ) + F(6 ) = Y has solution for any Y ∈ M(n+mq)×r . That is im(F) + im(G) = M(n+mq)×r .

(12.45)

364

12 Output Regulation

It follows that

dim(F) = (n + mq) × r − mr dim(G) = mr.

The conclusion follows.

3 2

Note that this proves the structural stability, and complete the proof of the second part of Theorem 12.1. Part 1 is a special case of Part 2, so no more proof is necessary. Since the proof is constructive, it provides an algorithm for building the regulator. We give a numerical example to describe the algorithm. Example 12.2. Consider the following system c d c d ⎧ ⎪ ⎨x = −1 0 x + 0 u + Pw := Ax + Bu + Pw 1 1 1 ⎪ ⎩ e = [0 1]x + Qw := Cx + Qw, where P, Q are known matrices, and w satises the following exosystem ? > 0 1 w := Sw. w = −1 0

(12.46)

(12.47)

First, it is easy to check that (A, B) is stabilizable; (C, A) is detectable; and (12.47) is neutrally stable. Checking the condition (12.8) or (12.9), we rewrite (12.8) in a matrix form as ?> ? ?> ? > > > ? AB 3 I2 0 3 P . (12.48) − S=− 0 0 * C0 * Q Converting it into vector form, we have [4] ?V U> ?V ? > > U> ?V U 3 AB I 0 P Vc − ST ⊗ 2 . = −Vc Ir ⊗ 0 0 * C0 Q

(12.49)

Then the coefcient matrix, denoted by L, is U ? > > ?V AB I 0 L = Ir ⊗ − ST ⊗ 2 C0 0 0 ⎤ ⎡ −1 0 0 1 0 0 ⎢ 1 1 1 0 1 0⎥ ⎥ ⎢ ⎢ 0 1 0 0 0 0⎥ ⎥. ⎢ =⎢ ⎥ ⎢−1 0 0 −1 0 0⎥ ⎣ 0 −1 0 1 1 1⎦ 0 0 0 0 10 It is easy to check that L is non-singular. According to Theorem 12.1 the structurally stable output regulation problem is solvable. To get a canonical form of S as suggested in the proof of Lemma 12.4, we let ? > 0 1 w. w˜ = −1 0

12.1 Output Regulation of Linear Systems

Then S becomes

365

? > 0 −1 ˜ , S= 1 0

and P˜ = T −1 P, Q˜ = T −1 Q. Since they do not affect the following design, for nota˜ and S. ˜ ˜ Q, tional ease, we still use P, Q, S for P, According to (12.16), we have ) = S. Following the proof of Lemma 12.4, we choose N = [0 1]T , 4 = [1 0], which make () , N) controllable and (4 , ) ) observable. Next, according to Lemma 12.6 we have ⎤ ⎛⎡ −1 0 0 0 ? > ?V U> ⎜⎢ 1 1 0 0 ⎥ B A 0 ⎥ ⎢ =⎜ , ⎝⎣ 0 0 0 −1⎦ , 0 NC ) 0 11 0

⎡ ⎤⎞ 0 ⎢1⎥⎟ ⎢ ⎥⎟ ⎣0⎦⎠ 0

is stabilizable, and ⎛ U > ?V ⎜D D E A B4 E C0 , 0100 , =⎜ ⎝ 0 )

⎡ −1 0 ⎢1 1 ⎢ ⎣0 0 0 0

⎤⎞ 0 0 ⎟ 1 0⎥ ⎥⎟ ⎦ 0 −1 ⎠ 1 0

is detectable. Now we construct the matrices in (12.26) and (12.27), denoted by (A , B) and (C , A ) respectively, as ⎤ ⎡ −1 0 0 0 > ? ⎢ 1 11 0 ⎥ A B4 ⎥ A = =⎢ ⎣ 0 1 0 −1⎦ ; NC ) 0 01 0 D E D ET B= 0101 ; C = 0101 . Then a standard procedure provides one feasible L as D ET L = 0 −4 −3 −2 , which assigns all the eigenvalues of A + LC to −1. Similarly, we can choose D E M = 0 −4 −2 −1 which assigns all the eigenvalues of A + BM to −1. Finally, ⎤ ⎡ −1 0 0 0 ⎢ 1 1 −1 1 ⎥ ⎥ K = A − LC + BM = ⎢ ⎣ 0 4 0 −1⎦ 0 2 1 0 Then the structural stable regulator (12.30) is constructed easily.

366

12 Output Regulation

12.2 Nonlinear Local Output Regulation Beginning with [11–13] etc., the problem of nonlinear output regulation has been a hot direction in the investigation of nonlinear control. Consider a nonlinear system J x = f (x, u, w) (12.50) e = h(x, w), where state x ∈ Rn , control u ∈ Rm , regulated output e ∈ Rm , and the exogenous disturbance input w ∈ Rr generated by an exosystem w = s(w).

(12.51)

f (x, u, w), h(x, w), and s(w) are assumed to be smooth enough and f (0, 0, 0) = 0, h(0, 0) = 0, s(0) = 0. Denition 12.3. The local output regulation problem is: given a nonlinear system of the form (12.50) with exosystem (12.51), nd, if possible, a controller of the form J [ = K ([ , e), [ ∈ U ⊂ RQ (12.52) u = T ([ ), where U is an open neighborhood of the origin, such that (i). J x = f (x, T ([ ), 0) [ = K ([ , h(x, 0)), has asymptotically stable linear approximation; (ii). the forced closed-loop system ⎧ ⎪ ⎨x = f (x, T ([ ), w) [ = K ([ , h(x, w)) ⎪ ⎩ w = s(w) is such that

(12.53)

(12.54)

lim e(t) = 0

t→f

for (x(0), [ (0), w(0)) in a neighborhood of the origin. Let

⎧ wf wf wh ⎪ ⎪ A= (0, 0, 0), B = (0, 0, 0), C = (0, 0) ⎪ ⎪ ⎪ w x w u wx ⎪ ⎪ ⎨ wf wh ws (0, 0, 0), Q = (0, 0), S = (0) P= ⎪ ww ww ww ⎪ ⎪ ⎪ ⎪ wK wK wT ⎪ ⎪ (0, 0), H = (0, 0), G = (0). ⎩F = w[ we w[

(12.55)

12.2 Nonlinear Local Output Regulation

367

Then condition (i) means the matrix J, as in (12.7), is Hurwitz. Theorem 12.2. Consider the system (12.50) with exosystem (12.51), which is assumed to be neutrally stable. There exists a controller which solves the problem of local output regulation only if there exist mapping S : U0 → Rn and c : U0 → Rm (where 0 ∈ U0 ⊂ U), with S (0) and c(0), such that ⎧ ⎪ ⎨ w S s(w) = f (S (w), c(w), w) ww (12.56) ⎪ ⎩0 = h(S (w), w). We call equations (12.56) the nonlinear regulator equations. To prove this, we need the following lemma, which is a generalization of Lemma 12.2. Lemma 12.9. Assume the matrix J in (12.7) is Hurwitz. Then lim e(t) = 0.

t→f

(12.57)

if and only if there exist smooth mappings S : U0 → Rn and V : U0 → RQ , with S (0) = 0, V (0) = 0, such that ⎧ wS ⎪ ⎪ s(w) = f (S (w), T (V (w)), w) ⎪ ⎪ w ⎪ ⎨ w wV (12.58) s(w) = K (V (w), 0) ⎪ ⎪ w w ⎪ ⎪ ⎪ ⎩ 0 = h(S (w), w). Proof. Consider the closed-loop system (12.54), the Jacobian matrix at the origin of (x, [ , w) = (0, 0, 0) is ⎡ ⎤ > ? A BH ∗ ⎣GC F ∗⎦ = J ∗ . 0S 0 0 S Since J is Hurwitz and Re V (S) = 0, there exists center manifold (x, [ ) = (S (w), V (w)) (see Chapter 11 or [3]), satisfying

wS s(w) = f (S (w), T (V (w)), w), ww wV s(w) = K (V (w), h(S (w), w)). ww

(12.59) (12.60)

The rst equation of (12.58) is fullled. We rst prove the necessity. (Necessity) Let 0 ∈ U ⊂ Rr be small enough such that the center manifold is dened. Let E > 0 be small enough such that 1w0 1 < E implies w(t, w0 ) ∈ U, t  0. This is assured by the neutral stability (hence Lyapunov stability) of (12.4). We claim that lim e(t) = 0 implies t→f

h(S (w), w) = 0.

(12.61)

368

12 Output Regulation

Assume there exists 1w0 1 < E such that 1h(S (w0 ), w0 )1 = 2H > 0. Since lim e(t) = 0, there exists T > 0 such that t→f

1e(t)1 = 1h(S (w(t)), w(t))1 < H ,

t > T.

But by the neutral stability (hence Poisson stability) there exists t < > T such that w(t < ) is so close to w0 that 1h(S (w(t < )), w(t < )) − h(S (w0), w0 )1 < H It is absurd. Now (12.60) and (12.61) imply second equation of (12.58). Next, we show the sufciency. (Sufciency) From third equation of (12.58), we have e(t) = h(x(t), w(t)) − h(S (w(t)), w(t)).

(12.62)

But the center manifold (x, [ ) = (S (w), V (w)) is exponentially attractive, hence there exists M > 0 and a > 0 such that 1x(t) − S (w(t))1  Me−at 1x(0) − S (w(0))1, which implies lim e(t) = 0. t→f

3 2

Remark 12.2. In linear case we can assume the center manifold is J x(w) = 3 w + O(1w12 ) [ (w) = 6 w + O(1w12 ). According to (12.58), it turns out that (3 , 6 ) satises (12.13). As for the sufciency, starting from (12.13), the argument after (12.62) remains effective. Proof. (Proof of Theorem 12.2) Using Lemma 12.9, in the rst equation of (12.58), set T (V (w)) = c(w), we are done.

3 2

To present the necessary and sufcient condition we need the following concept. Denition 12.4. Given two dynamic systems with outputs as J x = f (x), x ∈ X y = h(x), y ∈ Rm ; J z = F(z), z ∈ Z y = H(z), y ∈ Rm ,

(12.63) (12.64)

12.2 Nonlinear Local Output Regulation

369

where 0 ∈ X ⊂ R p , 0 ∈ Z ⊂ Rq , and p  q. System (12.63), denoted by (X, f , h), is said to be immersed into (12.64), (Z, F, H), if there exists a mapping W : X → Z, denoted by z = W (x) satisfying W (0) = 0 such that

wW f (x) = F(W (x)) wx h(x) = H(W (x)). Note that the rst condition implies that

W ()tf (x)) = )tF (W (x)), and the second condition implies that h()tf (x)) = H()tF (W (x))). Theorem 12.3. Consider the system (12.50) with exosystem (12.51), which is assumed to be neutrally stable. There exists a controller which solves the problem of local output regulation, if and only if, there exist mapping S : U0 → Rn and c : U0 → Rm (where 0 ∈ U0 ⊂ U), with S (0) = 0 and c(0) = 0, such that (12.56) holds, and the autonomous system with output J w = s(w) (12.65) u = c(w), is immersed into a system, called an immersion system, J [ = I ([ ) u = J ([ ),

(12.66)

dened on a neighborhood 0 ∈ V ⊂ RP , in which I (0) = 0 and J (0) = 0 and the two matrices wI wJ F= (0), H = (0) w[ w[ are such that the pair ? > ? > B A 0 (12.67) , 0 NC F is stabilizable for some choice of the matrix N, and the pair ? > D E A BH C0 , 0 F

(12.68)

is detectable. Proof. (Necessity) Suppose the following regulator solves the problem of output regulation: J [ = K ([ , e) = F [ + Ge + O(1[ , e12) (12.69) u = T ([ ).

370

12 Output Regulation

Then equations (12.58) are satised. Set c(w) = T (V (w)),

J ([ ) = T ([ ),

I ([ ) = K ([ , 0).

It is clear that S (w) and c(w) satisfy condition (12.56), while I ([ ) and J ([ ) satisfy ⎧ ⎪ ⎨ w V s(w) = I (V (w)) ww (12.70) ⎪ ⎩c(w) = J (V (w)), which shows that {V0 , s, c} is immersed into {U0 , I , J }, where V0 := V (U0 ). Observe that, by denition,

wK wI (0, 0) = (0) = F, w[ w[ By hypothesis,

wT wJ (0) = (0) = H. w[ w[

>

A BH J= GC F

?

? > ? E B D A 0 0H , + 0 GC F ? > ? > B A 0 , 0 GC F

>

is Hurwitz. Since J=

is stabilizable. Similarly, since > ? > ? A BH 0 D E C0 , J= + 0 F G D E C0 ,

? > A BH 0 F

is detectable. (Sufciency) Using (12.67) and (12.68), we have ? > ? > B A BH , 0 NC F is stabilizable and

D E C0 ,

>

A BH NC F

?

is detectable. According to Lemma 12.7 and its proof, there exist K, L, M such that VU V ⎤ ⎡ ⎤ ⎡U A BH B A BH BM M⎦ ⎣ 0 (12.71) D = ⎣ NC = NC F 0 ⎦ @ FA LC 0 K L C0 K is Hurwitz. Next, we construct the regulator as

12.2 Nonlinear Local Output Regulation

⎧  ⎪ ⎨[0 = K [0 + Le [1 = I ([1 ) + Ne ⎪ ⎩ u = M [0 + J ([1 ).

371

(12.72)

Now the Jacobian matrix of (12.53) at (x, [0 , [1 ) = (0, 0, 0) becomes ⎡ ⎤ ⎤ ⎡ A BM BH f (x, M [0 + J ([1 ), 0)  w ⎣ K [0 + Lh(x, 0) ⎦ = ⎣ LC K 0 ⎦ ,  w (x, [0 , [1 )  NC 0 F I ([1 ) + Nh(x, 0) (0,0,0) which is Hurwitz. Moreover, by hypothesis, there exists x = S (x) and u = c(x) such that (12.56) is satised. Finally, there exists an immersing function [1 = W (w) such that

wW s(w) = I (W (w)), ww Setting

c(w) = J (W (w)).

(12.73)

> ? > ? 0 [0 =V = [1 W (w)

and using (12.56) and (12.73), it is ready to verify that (12.58) is satised. The sufciency follows. 3 2 Remark 12.3. Recall the regulator (12.72). Sometimes people split it into two parts as [1] J [0 = K [0 + Le (12.74) u = M [0 , which is called the stabilizer, and J [1 = I ([1 ) + Ne u = J ([1 ),

(12.75)

which is called the Internal Model. Example 12.3. Consider the TORA (transitional oscillator with a rotational actuator) system, which is shown in Fig. 12.1. It was proposed in [15], and also discussed in [8, 10, 14] and other literatures. It is also called the RTAC (rotational/translational actuator) system. The system considers of a cart of mass M, which is attached to a wall with a spring of stiffness k. There is a disturbance force D on the car. An arm of mass m around an axis is locating at the center of the car. The center of the mass, C is  distance from the axis. I is the inertia of a pendulum attached to the arm. The arm is actuated by a torque T . The horizontal displacement of the cart is denoted by e, and the angular displacement of the arm is denoted by T . The torque T is considered as a control, and the purpose for the control is to make the horizontal displacement e tends to zero. The dynamics of the system is described as [15]

372

12 Output Regulation

Fig. 12.1

TORA System

J

(M + m)e¨ + m(T¨ cos T − T 2 sin T ) + ke = D, (I + m2 )T¨ + mecos ¨ T = T.

(12.76)

To get a condensed form we dene a coordinate transformation, which simplied the dynamic equations [14]. Let m ⎧ M+m ⎪ ⎪ x1 = e + K sin T , ⎪ ⎪ I + m2 ⎪ ⎪ ⎪ ⎪ m ⎪ ⎪ ⎪ M + m de dT ⎪ ⎨ x2 = +K cos T , I + m2 dW dW (12.77) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x3 = T , ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ x = dT , 4 dW n where W = k/(M + m)t is a time re-scaling parameter,

K=n

m (M + m)(I + m2 )

,

and the state feedback control T=

E (I + m2 )k D (1 − K 2 cos2 x3 )v − K cos x3 (x1 − (1 + x24)K sin x3 − P F) , M+m (12.78)

where 1 P= k

m

M+m , I + m2

and v is the new control. Then the system (12.76) becomes

12.2 Nonlinear Local Output Regulation

⎧ ⎪ x1 = x2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨x2 = −x1 + K sin x3 + P F x3 = x4 ⎪ ⎪ ⎪x4 = v, ⎪ ⎪ ⎪ ⎩e = Q (x − I sin x ), 1 3 o

where

Q= Setting D =

n

373

(12.79)

I + m2 . M+m

(M + m)/kZ as a new excitation frequency, the exosystem becomes J Z 1 = DZ2 (12.80) Z 2 = −DZ1 ,

and F = Z1 . Now we use Theorem 12.3 to check if the regulation problem is solvable. First, we have to nd a solution for the regulation equation. It is easy to verify directly that the following S (w) and c(w) are a set of solutions of (12.56) [14]. ⎡ ⎤ P w1 − 2 D ⎢ ⎥ ⎢ ⎥ P w2 ⎢ ⎥ − ⎢ ⎥ D ⎢ ⎥ U V ⎥ ⎢ (12.81) S (w) = ⎢ P w1 ⎥ , −1 ⎢ − sin ⎥ 2 ⎢ D K ⎥ ⎢ ⎥ ⎢ ⎥ PD w2 ⎣− l ⎦ 2 D 4 K 2 − P 2 w1

c(w) =

PD 2 w1 (D 2 K 2 − P 2 (w21 + w22 )) . @ A3 D 4 K 2 − P 2 w21

(12.82)

To check the existence of functions I and J in (12.66) we simply set I = s and J = c. Then we check if there exists N such that the matrices in (12.67) are stabilizable pair, and if the matrices in (12.68) are detectable pair. A straightforward computation shows that ⎡ ⎤ ⎤ ⎡ 0 0 100 ⎢0⎥ ⎢−1 0 K 0⎥ ⎢ ⎥ ⎥ A=⎢ ⎣ 0 0 0 1⎦ , B = ⎣0⎦ , 1 0 000 E D C = Q 1 0 −K 0 , ? > D E 0 1 , H = P /K 0 . F= −1 0

374

12 Output Regulation

To specify the computation, we set K = 0.5, and we choose N = [1 1]T . Then we construct the matrices in (12.67) as ⎡ ⎡ ⎤ ⎤ 0 1 0 0 0 0 0 ⎢−1 0 0.5 0 0 0⎥ ⎢0⎥ ⎢ ⎢ ⎥ ⎥ ⎢ ⎥ ⎥ ? ⎢ > ⎢ ⎢ ⎥ ⎥ 0 0 0 1 0 0 A 0 ⎥ , B = ⎢0⎥ . =⎢ A = ⎢ ⎢1⎥ ⎥ NC F ⎢ 0 0 0 0 0 0⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎥ ⎣ 1 0 −0.5 0 0 1⎦ ⎣0⎦ 0 1 0 −0.5 0 −1 0 It is easy to check that (A , B) is a controllable pair. Next, we construct the matrices in (12.68), setting M = 5 (or whatever you like). Then ⎡ ⎤ 0 1 0 0 0 0 ⎢−1 0 0.5 0 0 0⎥ ⎢ ⎥ ⎥ ? ⎢ > ⎢ 0 0 0 1 0 0⎥ D E A BH ⎢ ⎥ , C = 1 0 −0.5 0 0 0 . =⎢ A = ⎥ 0 F ⎢ 0 0 0 0 5 0⎥ ⎢ ⎥ ⎣ 0 0 0 0 0 1⎦ 0 0 0 0 −1 0 Unfortunately, we hang here because it is easy to check that the pair (C , A ) is not detectable. (It is easy to check that the unobservable subspace is of dimension 2 and the eigenvalues of A corresponding to this block are {0, 0}.) Since there is no systematic way to nd I and J , even though we are lucky enough to nd the solution for regulation equation, we still have problem here. Note that in this system (A, B) is stabilizable. (In fact, it is a controllable pair.) In this case [14] proposes the following static controller to solve the regulation problem: • Choose K such that A + BK is Hurwitz; • Construct controller as v = c(w) + K(x − S (w)).

(12.83)

Now as w = 0 the linear approximation of the original system is x = (A + BK)x which is exponentially stable. Now consider z := x − S (w), it is easy to check that z = Az + BKz + h.o.t, where h.o.t represents higher order terms. Hence, locally, x(t) → S (w), which, combined with the second equation of (12.56), assures that e(t) → 0 as t → f. This regulator is called a full information regulator.

12.3 Robust Local Output Regulation Robust output regulation considers the output regulation problem in the presence of parameter uncertainties. We consider the following system

12.3 Robust Local Output Regulation

x = f (x, u, w, Q ), x ∈ Rn , u ∈ Rm , w ∈ Rr e = h(x, w, Q ), e ∈ Rm ,

375

(12.84)

where f and h are assumed to be smooth (Cf ) mappings, Q ∈ R p is a constant unknown parameter vector. The disturbance is generated by the exosystem w = s(w),

w ∈ Rr .

(12.85)

Denition 12.5. Given system (12.84) with exosystem (12.85), the robust local output regulation problem is: nding, if possible, a controller of the form J [ = K ([ , e), [ ∈ U ⊂ RQ (12.86) u = T ([ ), such that, for each Q ∈ I ⊂ R p , with I C 0 as a certain neighborhood of the origin, (i). the equilibrium point (x, [ ) = (0, 0) of the closed-loop system ⎧ ⎨x = f (x, T ([ ), 0, Q ) (12.87) ⎩[ = K ([ , h(x, 0, Q )) has an asymptotically stable linear approximation; (ii). the overall system ⎧ ⎪ x = f (x, T ([ ), w, Q ) ⎪ ⎪ ⎨ [ = K ([ , h(x, w, Q )) ⎪ ⎪ ⎪ ⎩w = s(w)

(12.88)

is such that lim e(t) = 0.

t→f

(12.89)

In fact, the solution of robust out is a straightforward application of the results obtained in the last session. To see this, we may put w and Q together as a new disturbance, which is generated by the extended exosystem c d c d s(w) w e e e e , where w = . (12.90) w = s (w ) = 0 Q In this set up, the system (12.84) is expressed as x = f e (x, u, we ), e = he (x, we ),

x ∈ Rn , u ∈ Rm , we ∈ Rr+p e ∈ Rm ,

(12.91)

which is exactly the same as system (12.50). Now if a controller (12.87) solves the problem of local output regulation problem of system (12.91), then we have

376

12 Output Regulation

(i). J

x = f e (x, T ([ ), 0) [ = K ([ , he (x, 0)),

has asymptotically stable linear approximation; (ii). the forced closed-loop system ⎧ e e ⎪ ⎨x = f (x, T ([ ), w ) [ = K ([ , he (x, we )) ⎪ ⎩ w = s(w)

(12.92)

(12.93)

is such that lim e(t) = 0,

t→f

for (x(0), [ (0), wa (0)) in a neighborhood of the origin. Then for certain neighborhood I of the Q = 0, the requirement of (12.88) is automatically satised. As for (12.87), by continuity, it is obvious that with possible shrinking I , it can also be satised. Conversely, if the controller (12.87) solves the robust local output regulation problem of the system (12.84), it is obvious that it also solve the local output regulation problem of the system (12.91). So, these two problems are equivalent. We, therefore, can use the results in the previous section to produce the necessary and sufcient condition for robust local output regulation. Set > ? > ? > ? w f  w f  w h  A(Q ) = , B(Q ) = , C(Q ) = . w x (0,0,0,Q ) w u (0,0,0,Q ) w x (0,0,Q ) Then we have the following result: Theorem 12.4. Consider the system (12.84) with exosystem (12.85), which is assumed to be neutrally stable. The problem of robust local output regulation is solvable, if and only if there exist mappings x = S e (w, Q ),

u = ce (w, Q ),

with S e (0, Q ) = 0 and ce (0, Q ) = 0, dened on a neighborhood (0 × 0) ∈ U × I ⊂ Rr × R p of the origin, such that ⎧ e ⎪ ⎨ w S s(w) = f (S e (w, Q ), ce (w, Q ), w, Q ) ww (12.94) ⎪ ⎩0 = h(S e (w, Q ), w, Q ), (w, Q ) ∈ U × I . Moreover, the autonomous input-output system ⎧ ⎪ ⎨w = s(w) Q = 0 ⎪ ⎩ u = ce (w, Q )

(12.95)

References

377

is immersed into a system J [ = I ([ ) u = J ([ ),

(12.96)

on a neighborhood 0 ∈ V ⊂ RP , in which I (0) = 0 and J (0) = 0 and the two matrices wI wJ (0), H = (0) F= w[ w[ are such that the pair >

? A(0) 0 , NC(0) F

>

? B(0) 0

is stabilizable for some choice of the matrix N, and the pair ? > D E A(0) B(0)H C(0) 0 , 0 F

(12.97)

(12.98)

is detectable. Proof. It is an immediate consequence of Theorem 12.3. The only thing need to be noticed is: since se (we ) has special form, we have that

w Se e e w S e (w, Q ) s(w). s (w ) = w we ww 3 2

References 1. Byrnes C, Isidori A. Output regulation of nonlinear systems: An overview. Int. J. Robust Nonlinear Contr., 2000, 10(5): 323–337. 2. Byrnes C, Priscoli F, Isidori A. Output Regulation of Uncertain Nonlinear Systems. Boston: Birkhauser, 1997. 3. Carr J. Applications of Centre Manifold Theory. New York: Springer, 1981. 4. Cheng D. Semi-tensor product and its application to Morgan’s problem. Science in China Series F: Information Sciences, 2001, 44(3): 195–212. 5. Davison E. The robust control of a servomechanism problem for linear time-invariant multivariable systems. IEEE Trans. Aut. Contr., 1976, 21(1): 25–34. 6. Francis B. The linear multivariable regulator problem. SIAM J. Contr. Opt., 1977, 14(3): 486–505. 7. Francis B, Wonham W. The internal model principle of control theory. Automatica, 1976, 12(5): 457–465. 8. Huang H, Hu G. Control design for the nonlinear benchmark problem via the output regulation method. Journal of Control Theory and Applications, 2004, 2(1): 11–19. 9. Huang J. A simple proof of the robust linear regulator. Control-Theory and Advanced Technology, 1995, 10(4): 1499–1504. 10. Huang J. Nonlinear Output Regulation, Theory and Application. Philadelphia: SIAM, 2004. 11. Huang J, Rugh W. On a nonlinear multivariable servomechanism problem. Automatica, 1990, 26(6): 963–972. 12. Huang J, Rugh W. Stabilization on zero-error manifold and the nonlinear servomechanism problem. IEEE Trans. Aut. Contr., 1992, 37(7): 1009–1013.

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12 Output Regulation

13. Isidori A, Byrnes C I. Output regulation of nonlinear systems. IEEE Trans. Aut. Contr., 1990, 35(2): 131–140. 14. Pavlov A, van de Wouw N, Nijmeijer H. Uniform Output Regulation of Nonlinear Systems, A Convergent Dynamics Approach. Boston: Birkhauser, 2006. 15. Wan C, Bernstein D, Coppola V. Global stabilization of oscillating eccentric rotor. Proceedings 33th CDC, 1994: 4024–4029. 16. Wonham W. Linear Multivariable Control: A Geometric Aproach, 2nd edn. Berlin: Springer, 1979.

Chapter 13

Dissipative Systems

This chapter discusses the dissipative systems. First, the denition and the properties of two kinds of dissipative systems, mainly passive system and J -dissipative system, are introduced in Section 13.1. Then, the conditions for verifying passivity and dissipativity are presented in Section 13.2. Based on these conditions, the controller design problem is investigated in Section 13.3. Finally, two classes of main dissipative systems, mainly Lagrange systems and Hamiltonian systems are studied in Section 13.4 and Section 13.5 respectively.

13.1 Dissipative Systems Consider a nonlinear system described by the following state-space representation J x = f (x) + g(x)u (13.1) y = h(x), where x ∈ D ⊂ Rn denotes the state variables, u ∈ R and y ∈ R are input and output of the system, respectively. f : D → Rn , g : D → Rn and h : D → R are differentiable. Suppose f (0) = 0, i.e. the origin x = 0 is an equilibrium of the free system (with u = 0). Denition 13.1. For a given function s(u, y), if there exists a C0 non-negative denite function V : D → R such that V (x)  V (x0 ) +

p t 0

s(u(W ), y(W ))dW , ∀t  0

(13.2)

holds for any input u and any initial state x0 = x(0), then the system is said to be dissipative. The function s is called a supply rate, and V (x) is called storage function. Moreover, if there exists a positive denite function Q(x) > 0(Q(0) = 0) such that the following inequality holds V (x) − V (x0 ) 

p t 0

s(u(W ), y(W ))dW −

p T 0

Q(x(W ))dW , ∀t  0,

(13.3)

then, the system is said to be strictly dissipative. Inequality (13.2) has its physical meaning. If we consider the storage function V (x(t)) as the current value of the total energy of the system, and s(u, y) denotes D. Cheng et al., Analysis and Design of Nonlinear Control Systems © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2010

380

13 Dissipative Systems

the energy supplied by external signal u, then dissipativity means that the system is always with energy loss during the dynamical transient from any initial state x0 to the current state x(t), i.e. there is no energy generated from the internal system. If the storage function V ∈ C1 , the dissipative inequality (13.2) is equivalent to the following differential inequality V  s(u(W ), y(W )), ∀t  0.

(13.4)

In nonlinear system analysis and design, the following two kinds of supply rates are usually encountered. As we can see later, dissipativity with respect to these two kinds of supply rates is related to the stability and L2 -gain. Denition 13.2. System (13.1) is said to be passive, if the system is dissipative respect to the following supply rate s(u, y) = uy.

(13.5)

Similarly, the system is said to be strictly passive if it is strictly dissipative respect to this supply rate. Denition 13.3. System (13.1) is said to be J -dissipative, if the system is dissipative respect to the following supply rate s(u, y) =

A 1@ 2 2 J u − y2 . 2

(13.6)

We give some examples of dissipative systems. Example 13.1. Consider the circuit shown in Fig. 13.1. To represent the dynamics of the system, we take the port voltage u as input, and the current through the capacitor C and the voltage across the inductor L as state variables x1 and x2 , respectively. Then, using Kirchhoff’s current law, we obtain the state equations as follows: ⎧ 1 1 ⎪ ⎨ x1 = u − x2 L L (13.7) ⎪ ⎩ x = 1 x − 1 x . 2 1 2 C RC

Fig. 13.1

Passive Loop

If we consider the current x1 as the output of this system, and the following candidate of storage function V (x1 , x2 ) =

L 2 C 2 x + x 2 1 2 2

(13.8)

13.1 Dissipative Systems

381

Then, it is easy to verify the passivity of the system: Since along any trajectory of the system, we have 1 V = x1 u − x22  yu, R

(13.9)

or equivalently, V (x1 (t), x2 (t))  V (x1 (0), x2 (0)) +

p t

yudt.

(13.10)

0

Example 13.2. Consider the mass-damper-spring system shown in Fig. 13.2. In this system, the mass m is moving on a horizonal surface and attached to a vertical surface through the spring k and the damper f . y denotes the displacement of the mass. Let us take the external force u and the velocity y as the input and output of this system, respectively. Then, the dynamics of the system can be represented by my¨ = u − ky − f y or equivalently

(13.11)

⎧ ⎪ ⎨ x1 = x2 k f 1 ⎪ ⎩ x2 = − x1 − x2 + u, M M M

(13.12)

 with the state variables x1 = y, x2 = y.

Fig. 13.2

Spring Damp System

Suppose 2 f > 1 and consider the storage function given by k m V (x1 , x2 ) = x21 + x22 . 2 2 Then, along any trajectories of the system, we have V = x1 u − f x21

V U 1 1 2 y = − y2 + yu − f − 2 2 G2 m U 2 V F u 1 1 1 2 −y − √ u− f − y . = 2 2f −1 2 4f −2

(13.13)

382

13 Dissipative Systems

√ This means that if we take J = 1/ 2 f − 1, then we will have J -dissipative inequality 1 V  (J 2 u2 − y2 ), 2

(13.14)

i.e. the system is J -dissipative with respect to the preassigned value J . If a system is passive with respect to a C1 storage function, then the passivity will lead to Lyapunov stability of the unforced system, since in this case the storage function might play the role of a Lyapunov function. Furthermore, the J -dissipativity guarantees certain L2 -gain of the system. We summarize these facts in the following two theorems. Theorem 13.1. Suppose that there exists a C1 storage function such that the system (13.1) is passive. Then, the unforced system x = f (x) with u = 0 is Lyapunov stable at the equilibrium x = 0. Moreover, if the system is strictly passive, the unforced system is asymptotically stable at x = 0. Proof. Since the storage function V ∈ C1 , the passivity implies that V  uy,

∀t  0

holds for any u. By considering the special case of u = 0, ∀t  0, we have V  0, ∀t  0, which implies the Lyapunov stability by Lyapunov Theorem. Furthermore, if the system is strictly passive, then along any trajectories of the unforced system, we have V  −Q(x),

∀t  0,

(13.15)

with a positive denite function Q(x) (Q(0) = 0). This means the asymptotical stability by the Lyapunov Theorem. 3 2 Denition 13.4. The system (13.1) is said to have L2 -gain less than J , if for any input u the inequality p f 0

y2 dt  J 2

p f

u2 dt

(13.16)

0

holds with zero initial state x(0) = 0. Comparing this denition with the dissipative inequality 1 V  (J 2 u2 − y2 ) 2

(13.17)

yields immediately the following conclusion. Theorem 13.2. For a given J > 0, if the system (13.1) is J -dissipative with respect to a C1 storage function V (x), then the system has L2 -gain less than J .

13.2 Passivity Conditions

383

13.2 Passivity Conditions In this section, we will focus our attention on such kind of dissipative systems that the dissipativity is guaranteed by a C1 storage function. This constraint will loss generality of the presented results, however, it makes the argument simple and it is enough for the design of smooth system in following sections. The rst theorem, so-called KYP lemma, provides a necessary and sufcient condition for the passivity. Another important condition, which is described by HJI inequality, presents a necessary and sufcient condition for the J -dissipativity. It should be noted that if we relax the constriction of the storage function to C0 , the situation becomes complicated. Denition 13.5. The system (13.1) is said to have KYP property, if there exists a positive semi-denite C1 function V (x) with V (0) = 0 such that J L f V (x)  0 ∀x ∈ D. (13.18) LgV (x) = h(x), Theorem 13.3. (Kalman-Yacubovich-Popov) The system (13.1) is passive with respect to a C1 storage function V (x)(V (0) = 0), if and only if the system has KYP property. Proof. (Sufciency) Along any trajectory of the system, it is easy to show that V = L f V (x(t)) + LgV (x(t))u(t)  y(t)u(t)

(13.19)

holds under the condition (13.18). (Necessity) If the system is passive with respect to a C1 function V (x), then the passivity inequality (13.19) holds along any trajectory of the system and any input u. Thus, by considering the special case of u(t) = 0, ∀t  0, we have L f V (x)  0. Moreover, from (13.19) L f V (x(t)) + (LgV (x(t)) − h(x(t))) u(t)  0.

(13.20)

If LgV (x(t)) − h(x(t)) := 0, ∀t, then for the input u(t) =

−L f V (x(t)) + H (t) , LgV (x(t)) − h(x(t))

H (t) > 0,

(13.21)

the left-hand side of (13.20) will become positive, the positivity inequality will fail. Hence, LgV (x) = h(x). 3 2 Observing the proof, it is obvious that if we replace the inequality in the KYP property by strict inequality, then the proof will lead to the condition of strict passivity. Also, the condition can be extended to more general systems with drift term. Consider an afne system with drift term represented as follows. a x = f (x) + g(x)u (13.22) y = h(x) + d(x)u,

384

13 Dissipative Systems

where u ∈ R and y ∈ R are the input and output, respectively. Denition 13.6. The system (13.22) is said to have generalized KYP property, if there exist a C1 function V : D → R with (V (0) = 0), a vector function l : D → Rh and a matrix function W : D → Rh×p such that ⎧ ⎨ L f V (x)  −l T (x)l(x) (13.23) LgV (x) = h(x) − 2l T(x)W (x) ∀x ∈ D. ⎩ d(x) = W T (x)W (x) Theorem 13.4. The system (13.22) is passive with respect to a C1 storage function, if and only if the system has generalized KYP property. Proof. (Sufciency) Using the condition (13.23), the time derivative of V along any trajectory can be calculated as follows V =  = =

L f V (x(t)) + LgV (x(t))u(t) −l T (x)l(x) + (h(x) − 2l T(x)W (x))u yu − l T(x)l(x) − 2l T (x)W (x)u − d(x)u2 yu − (l T(x) + uTW T (x))(l(x) + W (x)u).

It follows the passivity inequality V  uy.

(13.24)

(Necessity) Suppose the system (13.22) is passive with respect to a C1 storage function V , then the following inequality H(x, u, y) = V − uy  0

(13.25)

holds for the storage function V . A straightforward calculation yields H(x, u, y) = L f V (x) + LgV (x)u − (h(x) + ud(x))u = L f V (x) + (LgV (x) − h(x))u − d(x)u2.

(13.26)

Obviously, H is a quadratic function on u. Thus, there exist appropriate functions l(x) and W (x) such that H(x, u, y) = −(l T (x) + uW T (x))(l(x) + W (x)u).

(13.27)

Comparing this representation with (13.26), it is easy to check that (13.23) holds for these l(x) and W (x). 3 2 It is interesting to consider linear systems as a special case. Consider a x = Ax + Bu y = Cx + Du. In this case, we restrict the storage function to be a quadratical form 1 V (x) = xT Px, 2

(13.28)

13.2 Passivity Conditions

385

where P ∈ Rn×n is a positive semi-denite matrix. In fact, if the P satises ⎧ T T ⎪ ⎨A P + PA = −N N (13.29) PB = CT − N T M ⎪ ⎩ D + DT = M T M, then the quadratic function V (x) with functions √ √ 2 2 Nx, W (x) = M l(x) = 2 2 satisfy the generalized KYP property, i.e. a linear system is passive with respect to the storage function V (x). This condition is known as positive real condition [2]. Example 13.3. Consider a second-order nonlinear system given by ⎧ x1 = −x1 (1 + U x2) ⎪ ⎪ ⎪ ⎨ 1 x2 = x21 + u ⎪ U ⎪ ⎪ ⎩ y = x2 ,

(13.30)

where U > 0 is a constant number. Let x = [x1 x2 ]T . Then ⎡ ⎤ ? > 0 −x1 (1 + U x2) , g(x) = ⎣ 1 ⎦ , h(x) = x2 . f (x) = 2 x1 U Choose a positive denite function as 1 V (x) = (x21 + U x22), 2 which satises

> L f V (x) = [x1 U x2 ]

(13.31)

? −x1 (1 + U x2) = −x21  0, x21 ⎡

⎤ 0 LgV (x) = [x1 U x2 ] ⎣ 1 ⎦ = h(x). U This means the system is passive with respect to the storage function V (x). and

Next, we discuss J -dissipative systems. Theorem 13.5. The nonlinear system (13.22) is J -dissipative with respect to a C1 storage function V (x), if and only if (i). {J 2 I − d 2 (x)}  0, ∀x, (ii). there exists a function l(x) such that

386

13 Dissipative Systems

⎧ ⎪ ⎨L f V (x)  − 1 h(x)h(x) − l T(x)l(x) 2 ⎪ ⎩L V (x) = −h(x)d(x) − 2l T(x)W (x)

(13.32)

g

holds, where W (x) is the function satisfying W T (x)W (x) =

A 1@ 2 J I − d 2 (x) . 2

Proof. The proof is similar to Theorem 13.4. (Sufciency) Suppose there exists an l(x) such that the condition holds, then V = L f V (x) + LgV (x)u 1  − h(x)h(x) − l T (x)l(x) − (h(x)d(x) + 2l T(x)W (x))u 2 1 1 = (J 2 u2 − y2 ) − l T l − 2l TWu − u(J 2 I − d 2)u 2 2 1 2 2 2 T T = (J u − y ) − (l − uW )(l − Wu). 2 Hence, the dissipativity inequality 1 V  (J 2 u2 − y2 ), 2

∀u

holds for the storage function V (x). (Necessity) Suppose that there exists a C1 storage function such that the J dissipativity inequality holds, then we have

J 1 H(x, y, u) = V + y2 − u2 2 2 2

1 = L f V (x) + LgV (x)u + h2 (x) + h(x)d(x)u 2 2 J 1 2 + d (x)u2 − u2 2 2 1 2 = L f V (x) + h (x) + (LgV (x) + h(x)d(x))u 2 1 2 2 − u (J I − d 2 (x)) 2  0, ∀u. By the same argument as in the proof of Theorem 13.3, there exist appropriate functions l(x) and W (x) such that 1 H(x, u, y) = L f V (x) + h2 (x) + (LgV (x) + h(x)d(x))u 2 1 − u2 (J 2 I − d 2 (x)) 2 = −(l T (x) + uW T (x))(l(x) + W (x)u)  0.

13.2 Passivity Conditions

387

Therefore, W T (x)W (x) = 12 (J 2 I − d 2(x)), and Z (x) and l(x) satises (13.32).

3 2

If J 2 I − d 2 (x) > 0, we can take √ 1 2 2 (J I − d 2 (x)) 2 > 0, W (x) = 2

∀x ∈ D.

Hence, by (13.32), we obtain 1 1 l(x) = − W − 2 (x)(LgV (x) + h(x)d(x))T . 2

(13.33)

Substituting this l(x) into (13.32) yields an equivalent condition of (13.32) as follows: 1 1 L f V (x) + h2 (x) + (LgV (x) + h(x)d(x)) 2 2 2 2 −1 (J I − d (x)) (LgV (x) + h(x)d(x))T  0.

(13.34)

This is so-called Hamilton-Jacobi-Issacs (HJI) inequality. Theorem 13.6. Suppose J 2 I −d 2 (x) > 0, ∀x ∈ D. The system (13.22) is J -dissipative with respect to a positive semi-denite C1 storage function V (x), if and only if the function V (x) satises the HJI inequality (13.34). As we have seen in the case of passive systems, for the linear system (13.28), we constraint the storage function to quadratic form and the functions l(x) and W (x) to be linear, then it is easy to check the following conditions. Theorem 13.7. Consider the linear system (13.28). Suppose J 2 I − DT D  0. If there exist a positive semi-denite matrix P and a matrix N such that J AT P + PA  −CTC − N T N (13.35) PB = −CT D − N TM, then the system is J -dissipative, where M is constant matrix satisfying J 2 I − DT D = M T M. Furthermore, if J 2 I − DT D = M T M > 0, the matrix P satises the condition (13.35), if and only if the following Riccati inequality holds: AT P + PA + (PB + CTD)(J 2 I − DT D)−1 (PB + CTD)T + CTC  0. Proof. This theorem follows from Theorem 13.6 with √ √ 2 2 1 T V (x) = x Px, l(x) = Nx, W (x) = M. 2 2 2

(13.36)

3 2

Before closing this section, it is worth noting that the denition of dissipativity does not require the storage function to be C1 . An interesting question is that if a system is dissipative with respect to a C0 storage function, is there a function V satisfying the KYP condition? In fact, if the available storage function is well-dened and differentiable, furthermore, the system is reachable, the answer is positive [5].

388

13 Dissipative Systems

13.3 Passivity-based Control As we can see in the previous two sections, the passivity of a system is related to the stability and the J -dissipsativity to the L2 -gain. In the rest of this chapter, we will discuss stabilization problem by applying the verifying conditions presented in Section 13.2. The next section will concern the L2 -gain synthesis problem with the J -dissipativity conditions. Also, it should be noticed that for the storage function of a passive system, in order to play the role of Lyapunov function for the system, the storage function must be positive and radially unbounded. In the follows, we do not indicate the positiveness and the radial unboundedness intentionally. Denition 13.7. 1. System (13.1) is said to be zero-state observable if there is no solution of x = f (x) can stay identically in

:0 = {x | y = h(x) = 0}, other than the zero solution x(t) ≡ 0. In other words, the large invariant subset S ⊂ :0 is 0, i.e., S = 0. 2. System (13.1) is said to be zero-state detectable if for any x0 ∈ S, the solution of x = f (x), x(x0 ,t) → 0, as t → f. Theorem 13.8. Assume the system (13.1) is zero-state observable. 1. Suppose system (13.1) is J -dissipative, then the origin of is x = f (x) asymptotically stable. 2. Suppose system (13.1) is passive with respect to a C1 positive function V (x), then the origin x = 0 is stabilized by static output feedback control u = \ (y),

(13.37)

where \ (y) (\ (0) = 0) is any function satisfying y\ (y) > 0,

∀y := 0.

Proof. 1. By the denition of J -dissipativity, as u = 0 we have 1 1 V  (J 2 u2 − y2 ) = − y2  0. 2 2 So the set for V = 0 is :0 . Using zero-state observability, the asymptotical stability at the origin follows from the LaSalle invariance principle. 2. According to Theorem 13.3, the system has KYP property. That is, the time derivative of V along any trajectory of the closed-loop system saties V = L f V (x) − LgV (x)\ (x)  −y\ (y),

∀t  0.

(13.38)

Thus, V  0, and the set for V = 0 is : 0 . Note that in :0 the feedback system becomes x = f (x) because u = \ (y) = 0. Again the conclusion follows from the zero-state observability and the LaSalle invariance principle. 3 2

13.3 Passivity-based Control

389

Remark 13.1. It is obvious that “zero-state observable” is a particular case of “zerostate detectable”. We would like to show that Theorem 13.8 remains true when the condition “zero-state observable” is replaced by “zero-state detectable”. Note that according to LaSalle invariance principle, a trajectory will converge to S, which is the largest invariant subset contained in :0 . In the case of “zero-state observable” S = 0, we are done. In the case of “zero-state detectable”, we have only to show that each trajectory converges to the origin. Given any H > 0, we have show that each trajectory will eventually enter to BH (0). By the zero-detectability, x(s,t) will enter into BH /2 (0) after t > Ts0 for certain Ts0 > 0. Recall that the solution of a differential equation depends continuously on the initial condition, then for any s ∈ S, there exists a neighborhood Ns , such that for any z ∈ Ns , the trajectory x = (z,t) will enter BH (0) when t > Ts for a xed Ts > 0. Construct N = ∪s∈S Ns . By LaSalle invariance principle, any trajectory will enter N after a nite time. Then as argued above, after one more nite time it will enter BH (0). In the later use of the results in Theorem 13.8, the condition may be replaced by “zero-state detectable”. Applying this result to state feedback stabilization problem, we have the following corollary. Corollary 13.1. Consider a nonlinear system x = f (x) + g(x)u.

(13.39)

If there exists a C 1 positive denite function V (x) such that the system is zero-state observable with respect to the output y = LgV (x), then the system can be stabilized by the state feedback u = −kLgV (x),

(13.40)

where k > 0 is any positive number. The controller (13.40) is well-known as LgV controller. This result shows that if the system is passive with respect to a well-dened output signal, then we can obtain a state feedback stabilizing controller. In fact, even though it is not the case, we can obtain a stabilizing controller by introducing a pre-feedback compensator that renders the system passive and zero-state detectable. Corollary 13.2. For the system (13.39), if there exists a function D (x) and a C1 positive denite function V (x) such that the closed-loop system of (13.39) with u = D (x) + v

(13.41)

is passive and zero-state detectable with respect to the input v and the output y = LgV (x), then the system is stabilized by u = D (x) − \ (LgV (x)).

(13.42)

This corollary is a straightforward consequence of the Theorem 13.8 to the passied system from v to y = LgV (x). This suggests that the passivity-based design of a

390

13 Dissipative Systems

stabilizing controller is to nd a feedback compensation and an output such that the closed-loop system is passive and zero-state detectable. Usually, these two specications can be reached simultaneously by constructing a proper C1 storage function. The next theorem, so-called integral backstepping, provides a design technique for stabilizing controller along this line. Consider the following system J z = f (z, y) (13.43) y = u, where u ∈ R, y ∈ R, z ∈ R. By Hadamard Lemma [1], for the smooth function f (z, y) there exists a function f 1 (z, y) such that f (z, y) = f1 (z, y)y

(13.44)

Theorem 13.9. Suppose, for the unforced z-subsystem z = f (z, 0), there exists a positive denite function W (z) with W (0) = 0 such that L f (z,0)W (z) < 0,

∀z := 0.

(13.45)

u = −L f1 (z,y)W (z) − \ (y),

(13.46)

Then the system (13.43) is stabilized by

where \ (y) is any function satisfying y\ (y) > 0, ∀x := 0. Proof. Denote the control law as u = −L f1 (z,y)W (z) + v, where v = −\ (y). Then the closed-loop system can be represented by x = F(x) + G(x)v with xT = [z y] and

> F(x) =

? f (z, 0) + f1 (z, y)y , −L f1 (z,y)W (z)

(13.47)

G(x) =

> ? 0 . 1

Choose the storage function as 1 V (z, y) = W (z) + y2 , 2

(13.48)

where the output y is chosen by LGV . Then, it is easy to check by noting (13.45) that the closed-loop system with the storage function has KYP property, i.e the system is passive. Furthermore, since the dynamics, with constraint y = 0 and z = f (z, 0), is asymptotically stable, the system is zero-state detectable. From Theorem 13.8, v = −\ (y) yields the state feedback stabilizing controller. 3 2

13.3 Passivity-based Control

391

It is interesting to observe that the system (13.43) is of a cascaded form, that is, the z-subsystem is driven by y-subsystem, where the latter is a passive system, since if we choose the storage function as 1 U(y) = y2 , 2 then U = yu. In fact, this result can be extended to more general case. Consider the cascaded system given by

6 p : x = f (x, y);

(13.49)

and J [ = D ([ ) + E ([ )u 6a : y = h([ ).

(13.50)

Theorem 13.10. Suppose that the cascaded systems (13.49) and (13.50) satisfy the following conditions: (i). For 6 p subsystem, there exists a C1 positive denite function W (x) such that L f (x,0)W =

wW f (x, 0) < 0, wx

∀x := 0.

(ii). There exists a C1 storage function such that 6a subsystem is passive with respect to the output y and zero-state detectable. Then, a stat feedback stabilizing controller is given by u = −L f1 (x,y)W (x) − \ (y),

(13.51)

where f1 is the function satisfying f (z, y) = f1 (z, y)y. Proof. By the assumption and Theorem 13.3, there exists U([ ) such that LD U  −Q([ ),

LE U = h([ ).

(13.52)

Note that the closed-loop system is given by

K = F(K ) + G(K )v, where v = −\ (y) and > ? > > ? ? x f (x, 0) + f1 (x, y)y 0 K= , F(K ) = , G(K ) = . [ D ([ ) + E ([ )c(x) E ([ ) For this closed-loop system, if we choose the storage function as V (x, [ ) = W (x) + U([ ), then, using the condition (13.52), we obtain

(13.53)

392

13 Dissipative Systems

LF V = L f1 (x,0)W + LD U  −Q(x), LGV = LE U = h([ ). This means the closed-loop system is passive, and the zero-state detectability of the whole system is also ensured by the assumptions. Thus, the system can be stabilized 3 2 by output feedback v = −\ (y). For the cascaded system introduced above, the choice of the output to reach the passivity is the output of the driving system. For example, in Theorem 13.10 the passivity is respect to the output of the integrator, since the zero-state detectability with respect to the output can be guaranteed under the assumption. If this assumption is relaxed, then we need to reconstruct the output and the storage function. The next theorem is an extension of Theorem 13.9. Theorem 13.11. Consider the system (13.43). Suppose that for the z-subsystem, there exists a positive denite function W (z) and a smooth function D (z) with D (0) = 0 such that

wW { f (z, D (z))} < 0, wz

∀z := 0.

(13.54)

Then a state feedback stabilizing controller is given by u=

dD (z) w W − f1 (z, y + D (z)) − \ (y − D (z)), dt wz

(13.55)

where f1 is the function satisfying f (z, y) = f1 (z, y)y. Proof. For the whole system, dene a new output signal as y˜ = y − D (z), and change the coordinates of the system as > ? > ? z z →x= y y˜

(13.56)

(13.57)

Then, the system is represented as ⎧ ⎪ ⎨z = f (z, y˜ + D (z)) ⎪ ⎩y˜ = u − dD (z) . dt

(13.58)

Thus, plugging the control law (13.55) into this equation, the closed loop system becomes the following J x = F(x) + G(x)v (13.59) y˜ = y − D (z), ˜ and F and G are dened by where v = \ (y)

13.4 Lagrange Systems

⎡ ⎢ F(x) = ⎣

f (z, y˜ + D (z)) −

dD (z) dt

⎤ ⎥ ⎦,

393

> ? 0 . G(x) = 1

For this system, we choose the C1 storage function as 1 V (x) = W (z) + y˜2 . 2

(13.60)

Then, by straightforward calculation, the KYP property can be validated for the system with the input v and the output y. ˜ Furthermore, the system is zero-state detectable, since y˜ ≡ 0 implies y ≡ D (z) and the dynamics under this constraint is given by z = f (z, D (z)) in which z(t) → 0 as t → f. Hence, by Theorem 13.8, this ˜ 3 2 system is stabilized by v = −\ (y). Example 13.4. Consider a nonlinear system J z = −zy y = u.

(13.61)

Note that using the notation used in Theorem 13.9, f (z, y) = zy. Obviously, when y = D (z) = −z2 , z-subsystem z = −z3 is asymptotically stable at z = 0 and the positive denite function 1 W (z) = z2 2 satises L f (z, D )W (z) = −z4 < 0. Thus, let y˜ = y − D (z), we have

y˜ = u − z(−zy).

Therefore, for the whole system, we choose the candidate Lyapunov function as 1 V (z, y) ˜ = (z2 + y) ˜ 2. 2

(13.62)

Then, according to Theorem 13.8, we obtain the state feedback stabilizing controller as follows: u = x2 − x2 y − \ (y − x2),

(13.63)

where \ (s) is any function satisfying \ (s)s < 0, ∀s := 0.

13.4 Lagrange Systems Mechanical systems such as robot etc. can be modeled as a multi-mass system with constraints. In order to represent the dynamics of this kind of systems, the coordinates of each mass and its velocity are used as state variables, and the dynamical

394

13 Dissipative Systems

equations can be derived by the Lagrange principle

wL d wL − = W j, dt w T j w T j

j = 1, 2, · · · , n,

(13.64)

where T j , j = 1, 2, · · · , n denote the generalized coordinates of each mass, the displacement or the angle of each link in robot, and W j is the active force along the direction T j , respectively. L(T , T ) is so-called the Lagrange function which is dened as L(T , T ) = K(T , T ) − P(T )

(13.65)

with T T = [T1 T2 · · · Tn ] ∈ Rn , W T = [W1 W2 · · · Wn ] ∈ Rn and the total kinetic energy 1 K = T T M(T )T , 2

(13.66)

P = P(T ).

(13.67)

and the total potential energy

Usually, this Lagrange principle leads to the following Lagrange equation M(T )T¨ + C(T , T )T + g(T ) = W , where

(13.68)

> ? @ A  T ) − 1 w T T M(T ) , C(T , T ) = M( 2 wT

and

dP(T ) . dT Example 13.5. Consider a two link rigid manipulator as sketched in Fig. 13.3. Denote the length of the link by li , i = 1, 2, the distance of i-th link mass center from the joint as ri , i = 1, 2, and the mass of i-th link by mi . Ii , i = 1, 2 are the moment of inertia of the i-th link, and Ti represents joint torque input. g(T ) =

Fig. 13.3

Model of Two Link Manipulator

13.4 Lagrange Systems

395

The center of mass m1 and m2 are coordinated as x1 y1 x2 y2

= r1 sin T1 = r1 cos T1 = l1 sin T1 + r2 sin(T1 + T2 ) = l1 cos T1 + r2 cos(T1 + T2 ).

Then, the kinetic energy and the potential energy of mass m1 and m2 are calculated as follows, respectively. 1 1 1 1 K1 = m1 (x21 + y21 ) + I1 T12 + m1 r12 T12 + I1 T12 , 2 2 2 2 1 1 2 2 2 2 K2 = m2 (x2 + y2 ) + I2 (T1 + T2 ) 2 2 1 2 2 = m2 [l1 T1 + 2l1r2 T1 (T1 + T2 ) cos T2 + r22 (T1 + T2 )2 ] 2 1 + I2 (T1 + T2 )2 , 2 P1 = m1 gr1 cos T1 , P2 = m2 g[l1 cos T1 + r2 cos(T1 + T2 )].

(13.69)

(13.70) (13.71) (13.72)

The total kinetic energy can be represented by a matrix form as

with where

1 K = K1 + K2 = T T M(T )T , 2 ? > j1 + j2 + 2 j3 cos T2 j2 + j3 cos T2 , M(T ) = j2 + j3 cos T2 j2 j1 = m1 r12 + m2 l12 + I1, j2 = m2 r22 + I2 , j3 = m2 l1 r2 .

Thus, with the Lagrange function L = K1 + K2 − P1 − P2, we obtain the dynamical equation of the two link manipulators as the form (13.68). The matrix function C and g are given by ? > −T2 −(T1 + T2 )  ; C(T , T ) = j3 sin T2  T1 0 > ? −g1 sin T1 − g2 sin(T1 + T2 ) g(T ) = , −g2 sin(T1 + T2 ) where g1 = (m1 r1 + m2 l1 )g, g2 = m2 r2 g. It is well-known that the Lagrange system has the following physical properties: (i). Positivity. The matrix M(T ) is symmetric positive denite and bounded, i.e. there exist positive numbers Om and OM such that

Om I  M(T )  OM I,

∀T .

(13.73)

396

13 Dissipative Systems

 T ) − 2C(T , T ) is skew symmetric, i.e. (ii). Skew symmetry. The matrix M(  T ) − 2C(T , T ))[ = 0, [ T (M(

∀[ .

(13.74)

(iii). Linearity. If we parameterize all physical parameters as a constant vector E , then there exists an appropriate function ) (T , T , v, a) such that M(T )a + C(T , T )v + g(T ) = ) (T , T , v, a)E .

(13.75)

Due to the properties (i) and (ii), the Lagrange system is passive, if we choose the velocity of the general coordinate T as the output. Theorem 13.12. Let the output of Lagrange system be y = T . Then, the Lagrange system from the input W to the output y is passive with respect to the storage function 1 V (T , T ) = T T M(T )T + P(T ). 2

(13.76)

Proof. Along any trajectory of the system, calculating the time derivative of the storage function, we obtain 1  T )T + T T w P V = T T M(T )T¨ + T T M( 2 wT A 1 T @ T     T )T + T T w P = T W − C(T , T )T − g(T ) + T M( 2 wT A 1 @ = T T W + T T M( T ) − 2C(T , T ) T . 2 Using the physical property (ii) yields V = T T W ,

∀t  0.

The passivity of the system follows.

(13.77) 3 2

However, this physical property is not enough for controller design even though we consider the simple set-point control problem at the origin T = 0, T = 0, since this system is not zero-state detectable with respect to the output y = T . Fortunately, we can introduce a proper pre-feedback compensation that renders the tracking error system for any given desired trajectory passive and zero-state detectable. Let the desired trajectory be given as Td (t), Td (t), T¨d (t). To reach the goal of tracking, we introduce the following pre-compensation

W = u + M(T )T¨d + C(T , T )Td + g(T ),

(13.78)

where u is the new control input, then, the error dynamics is given by M(T )e¨ + C(T , T )e = u,

(13.79)

where the tracking error e = T − Td . Choose the coordinate as > ? > ? e e = x= , e + D1 e y where D1 > 0 is any constant number. Then, the tracking error system with the precompensation can be represented as follows:

13.5 Hamiltonian Systems

J

e = −D1 e + y M(T )y = (D1 M − C)y − D1(D1 M − C)e + u.

397

(13.80)

Theorem 13.13. For the system (13.80), let the control input u be given by u = −e − D1 M(y − D1 e) − D1Ce + v.

(13.81)

Then, the closed-loop system consisting of (13.80) with this controller is passive and zero-state detectable with respect to the output y. Proof. Dene the storage function as 1 1 V (e, y,t) = eT e + yT M(T )y. 2 2

(13.82)

Then, along any trajectory of the closed-loop system, we obtain 1  V = eT e + yTM y + yT My 2 1  = eT e + yT((D1 M − C)e + u) + yT My 2 1 = −D1 eT e + eTy + yT (M − 2C)y 2 +yT (u + D1 M e + D1Ce). Hence, using the property (ii) and substituting the control law, we have V = −D1 eT e + yTv yT v.

(13.83)

The passivity of the closed-loop system follows. Furthermore, by the denition 3 2 of the output y = e + D1 e, the zero-state detectability is clear. By the Theorem 13.8, this result provides a tracking controller by setting v = −\ (y) = \ (e + D1 e). Finally, it should be noticed that the error system (13.80) is actually nonautonomous since the right-hand side of the differential equation depends on the external time varying variables, Td (t), Td (t). Therefore, the denition of passivity must be modied to match this situation. The stability reduced based on Theorem 13.8 should be a kind of uniform stability. In fact, from the physical property (i), it is easy to check that the positive storage function satises 9> ?92 9> ?92 9 9 9 9 ˜Om 9 e 9  V  O˜ M 9 e 9 (13.84) 9 y 9 9 y 9 for some constants Om > 0 and OM > 0. So that, the stability guaranteed by the passivity can be extended to the uniform stability. We leave the exact argument to reader.

13.5 Hamiltonian Systems This section will deal with an alternative class of physical dynamical systems with passivity.

398

13 Dissipative Systems

Denition 13.8. A dynamical system is said to be port-controlled Hamiltonian system with dissipation, if the system can be described as: ⎧ ⎪ ⎨x = [J(x) − R(x)] w H (x) + g(x)u wx (13.85) ⎪ ⎩y = gT (x) w H (x), wx where x ∈ X, an n-dimensional manifold, u, y ∈ Rm are the input and output, respectively; H : Rn → R represents the total stored energy called Hamiltonian function, J(x) is a skew-symmetric structure matrix, and R(x) is a non-negative denite symmetric matrix. Example 13.6. Consider the Lagrange system (13.68). Dene

K=

w L(T , T ) = M(T )T , w T

and the state variable xT = [T T K T ]. Then, the system (13.68) can be transformed into the form (13.85) with the Hamiltonian function given by 1 H(x) = K T M(T )K + P(T ), 2 and the structure matrices

? > ? 0 0 I , R(x) = 0, g(x) = J(x) = I −I 0 >

Usually, Hamiltonian function H(x) is assumed to be lower bounded. Suppose that the Hamiltonian function admits a strict minimum H(x0 ) at x0 , then recon¯ = H(x) − H(x0 ), the system is still of structing the Hamiltonian function as H(x) ¯ Thus, without loss of Hamiltonian form with the non-negative denite function H. generality, we assume the Hamiltonian function H(x) to be nonnegative and to admit a strict minimum at x0 . Theorem 13.14. Let the output of the system (13.85) to be y = gT (x)

wH (x). wx

(13.86)

Then, the port-controlled Hamiltonian system (13.85) is passive with respect to the input u and the output y. Proof. Note that the skew-symmetric structure of J(x) and the semi-positivity of R(x), a straightforward calculation yields ?T > wH wH (x) R(x) (x) + yTu  yT u. H = − (13.87) wx wx 3 2 In order to make use of the passivity of Hamiltonian system in stabilization, the unforced dynamics x = [J(x) − R(x)]

wH (x) wx

(13.88)

13.5 Hamiltonian Systems

399

must have the zero-state detectablity property with respect to the output (13.86), in other words, the dynamics under the constraint y = 0 guarantees x → x0 as t → f. In fact, if the Hamiltonian function H(x) admits a strict minimum at x0 and the x0 is the equilibrium of the unforced system, then the unforced system is Lyapunov stable, since in this case the Hamiltonian function can play the role of Lyapunov function. Furthermore, if R(x) is strict positive, then the unforced system is asymptotically stable. Unfortunately, in many cases, x0 where the Hamiltonian function takes minimum value is not the desired equilibrium. In this case, a natural way to reach a solution of stabilization is to seek feedback control law such that the closedloop system can be formulated as Hamiltonian system with a shaped Hamiltonian function Hc (x), which admits a strict minimum at a desired equilibrium xe , and eventually to inject additional damping R(x) → Rc (x), i.e. [J(x) − R(x)]

wH w Hc (x) + g(x)D (x) = [J(x) − Rc (x)] (x). wx wx

(13.89)

This problem has been investigated by [4]. In the following we will present a design example which demonstrates the design procedure following this idea. Example 13.7. Consider a simplied single-machine innite bus power system with silicon-controlled rectier (SRC) direct excitor as shown in Chapter 1. A model for excitation control of this system can be written as follows [3]: Mechanical equations:

G = Z (t) − Z0, Z = −

D Z0 [Z (t) − Z0 ] + [Pm − Pe (t)]; M M

(13.90)

Generator electrical dynamics: 1 1 xd − x
Eq< (t)Vs x<6

(13.91)

sin G is the active electrical power, G (t) and Z (t) are angle and

speed of the rotor, respectively. Eq< (t) is the transient EMF in the quadrature axis of the generator. V f denotes the control input of the SCR amplier of the generator, and w denotes the unknown disturbance which caused by faults in the line or loads level variation, etc. 1 We only consider the excitation control loop. Hence, we assume that the mechanical input power Pm and the speed of synchronous machine Z0 are constants. Dene the state variable by The notation for the model parameters are as follows: Z0 = 2S f0 : nominal value of the speed; D: damping constant; M: inertia constant; Vs : innite bus voltage; Td0 : the time constant of the excitation circuit; Td< : the direct axis transient short circuit time constant; Pm : the input power of generator; xd : the direct axis reactance of the generator; x<6 = x
400

13 Dissipative Systems

x1 = G , x2 = Z − Z0 , x3 = Eq< .

(13.92)

Then, dynamics of the system can be represented by the following state space model: ⎧ ⎪ ⎨x1 = x2 (13.93) x2 = −DM x2 − bL x3 sin x1 + P ⎪ ⎩ x3 = cL cos x1 − cT x3 + V + w, where the control input V (t) = DM =

1 Td V f (t),

and the parameters are dened by

xd − x
As it is well-known (see i.e. [3] for a recent reference), if we insert a constant excitation input V (t) = u, ¯ then the system with w = 0 has a local equilibrium (x1e , 0, x3e ) satisfying J bL x3e sin x1e = P (13.94) cT x3e − cL cosx1e = u. ¯ Actually, an energy-like function 1 1 bL cT Hc (x) = x22 + bL x3 (cos x1e − cosx1 ) − P(x1 − x1e ) + (x3 − x3e )2 (13.95) 2 2 cL qualies as a Lyapunov function for the forced system (13.93) with the constant input V (t) = u¯ = cT x3e − cL cos x1e ,

(13.96)

where x1e and x3e is a locally unique solution of (13.94). It is easy to check that Hc (x) admits a local strict minimum at (x1e , 0, x3e ), Now, replacing the control input as V (t) = u¯+ v

(13.97)

with auxiliary input v, then the system (13.93) can be represented by the following formulation with the Hamiltonian function (13.95) x = [J(x) − Rc (x)]

w Hc (x) + g(x)(v + w), wx

where g(x) = [0 0 1]T and the structure matrices are dened by ⎤ ⎡ ⎡ ⎤ 0 0 0 0 10 J(x) = ⎣−1 0 0⎦ , Rc (x) = ⎣0 DM 0 ⎦ . 0 0 bcLL 0 00

(13.98)

(13.99)

References

401

References 1. Arnold V I. Ordinary Differential Equations. Tokyo: Springer-Verlag, 1992. 2. Khalil H. Nonlinear Systems, 3rd edn. New Jersey: Prentice Hall, 2002. 3. Lu Q, Sun Y, Mei S. Nonlinear Control Systems and Power System Dynamics. Boston: Kluwer Academic Publishers, 2000. 4. Ortega R, Van der Schaft A J, Maschke B, et al. Stabilization of port-controlled Hamiltonian systems: Passivation and energy-balancing// Stability and Stabilization of Nonlinear Systems. London: Springer, 1999: 239–258. 5. Van der Schaft A. L2 -gain analysis of nonlinear systems and nonlinear state feedback Hf control. IEEE Trans. Aut. Contr., 1992, 37(6): 770–784.

Chapter 14

L2 -Gain Synthesis

This chapter investigates L2 -gain synthesis problem. As L2 -induced norm, L2 -gain is an extension of Hf norm of linear systems. Hence, L2 -gain synthesis problem is usually called the nonlinear Hf control problem. Section 14.1 discusses the Hf norm and L2 -gain. Then, the Hf design problem is formulated in Section 14.2 and the design principle for linear systems is introduced. Section 14.3 focuses on the L2 gain synthesis problem for nonlinear systems. In Section 14.4, a constructive design approach is presented. Finally, some application examples are presented to illustrate the design techniques in Section 14.5.

14.1 H f Norm and L2 -Gain L2 -gain synthesis is also called nonlinear Hf design, since the L2 -gain is an induced norm, and in linear system it is equivalent to the Hf norm of the transfer function. Therefore, we begin this section with linear systems. Consider a linear system with transfer function G(s) = C(sI − 1)−1 B.

(14.1)

Denition 14.1. Let G(s) ∈ RHfn×m. Hf norm of G(s) is dened as follows: 1G(s)1f =

sup

−fZ f

V¯{G(jZ )},

(14.2)

where V¯(·) denotes the largest singular value of the transfer matrix G(s). An important property of Hf norm is that the Hf norm of the transfer function equal to the L2 reduced norm when the transfer function plays the role of operator from the space of input signals to the space of output signals. In other words, if the system is asymptotically stable, i,e, G(s) ∈ RHfn×m , then the output Y (s) = G(s)U(s) ∈ RH2n

(14.3)

for any U(s) ∈ RHfm . Denote y(t) = L −1 {Y (s)} and u(t) = L −1 {U(s)}, and restrict the input signal u(t) to L2 space, then the output y(t) ∈ L2 . We denote the operator describing this behavior of the system as *G : L2 → L2 . Then, we have the following lemma. D. Cheng et al., Analysis and Design of Nonlinear Control Systems © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2010

404

14 L2 -Gain Synthesis

Lemma 14.1. Let G(s) ∈ RHfn×m. Then, 1*G 1 :=

1y12 = 1G(s)1f . 1u1 2 u:=0,u∈L2 sup

(14.4)

Proof. For the sake of simplicity, we consider the scalar case only. By the Parseval identity, we have 1u12 = 1U(s)12 and 1y12 = 1Y (s)12 , then p

1 f 1G(jZ )U(jZ )12 dZ 2π −f p f 1  1U(jZ )12dZ sup V¯2 {G(jZ )} 2π Z −f

1y122 = 1Y (s)122 =

= 1G(jZ )12f 1u122 . Thus, 1y12  1G(s)1f , ∀u ∈ L2 . 1u12

(14.5)

To complete the proof, we show in the following that for any given small H > 0, there exists an input signal uH ∈ L2 such that the corresponding output yH satises 1yH 12  1G(s)1f − H . 1uH 12

(14.6)

Consider the signal given as follows uH (t) =

2 1 e−U t cos Z0 t, 2 1eU t cos Z0 t12

(14.7)

where U > 0 is any number, Z0 is the number satisfying that 1G(jZ0 )1 

sup

−f
1G(jZ )1 − H

(14.8)

for the given small H > 0. It is clear that 1uH 12 = 1. Moreover, V m U Y Z (Z +Z )2 2 1 π − (Z −4ZU0 )2 − 4U 0 +e lim F e−U t cos Z0t = lim e U →f U →f 2 U = π(G (Z − Z0 ) + G (Z + Z0 )), where F (·) denotes the Fourier transformation and G denotes the impulse function. Hence, we have U lim

U →f

1 2π

p f −f

V1/2 |π2 G (Z − Z0 ) + G (Z + Z0 )|2 dZ

and the corresponding output satises

=

√ π,

(14.9)

14.1 Hf Norm and L2 -Gain

lim 1yH 122 = lim

U →f

p 1 f

U →f

2π

−f

1G(jZ )uH (jZ )12 dZ = 1G(jZ0 )12 .

The existence of uH follows.

405

(14.10) 3 2

It should be noticed that the rst equal in (14.4) is the denition of L2 -induced norm, which is not constrained to linear systems. For the nonlinear systems, with the property that the output corresponding to any input signal in L2 belongs to L2 , the denition is still effective. For nonlinear systems, the L2 -induced norm is generally called as L2 -gain. If L2 -induced norm of a system is less than J > 0, then we say the system has L2 -gain less than J . Recalling the Denition 13.4 and Theorem 13.2, it is easy to show the following lemma. Lemma 14.2. A nonlinear system has L2 -gain less than J , if the system is J dissipative with respect to a C1 storage function. For linear systems, Riccati equation or Hamiltonian matrix related to state space realization of the system is used to judge the L2 -gain [3]. We introduce the result here without proof. Theorem 14.1. Let G(s) = {A, B,C, 0}. The following three conditions are equivalent: (i). A is stable and 1G(s)1f < J . (ii). The algebraic Riccati equation 1 AT X + XA + XBBT X + CTC = 0 J has positive semi-denite solution X  0 and A + BBTX is stable. (iii). The following Hamiltonian matrix ⎤ ⎡ 1 T BB A ⎦ H =⎣ J T T −C C −A has no eigenvalues on the imaginary axis. The proof of this Theorem is complicated and can be found in the classical textbook, say [3], and more general version of the conditions for nonlinear systems to have L2 -gain less than J is presented in [5] with appropriate additional conditions. In the following, we introduce a similar proposition that provides equivalent conditions with inequalities which is much convenience to be extended to nonlinear systems. Theorem 14.2. Let G(s) = {A, B,C, 0}. The following three conditions are equivalent: (i). A is stable and 1G(s)1f < J . (ii). The algebraic Riccati inequality AT X + XA + XBBTX + CTC < 0 has a positive denite solution X > 0.

406

14 L2 -Gain Synthesis

(iii). There exists a sufciently small H > 0 such that the following Riccati equation has positive denite solution Z > 0. AT Z + ZA + ZBBTZ + CTC + H I = 0.

(14.11)

Proof. We complete this proof by showing (ii)⇒(i)⇒(iii)⇒(ii). (ii)⇒(i). Let Q = −(AT X + XA + XBBTX +CTC) > 0. Then, from the condition (ii) we have ¯ AT X + XA = −Q,

(14.12)

where Q¯ = XBBTX + CTC + Q > 0. Thus, the stability of matrix A follows by Lyapunov stability theorem. Furthermore, it is easy to check that

* (s) = I − GT(−s)G(s) ¯ − A) ¯ −1 B¯ = I + C(sI ˆ − A) ˆ −1 B, ˆ = I + C(sI ¯ −1 , and ¯ −1 , Bˆ = T B, ¯ Cˆ = CT where Aˆ = T AT > ? > ? 0 −AT CTC ¯ ¯ A= , C¯ = [BT 0], , B= B 0 A

? I X . T= 0 I >

Dene M 2 = Q + XBBTX and N(s) = M −1 XB + M(sI − A)−1 B, then we have

* (s) = N T (−s)N(s) + I − BTXM −2 XB = N T (−s)N(s) + (I + BT XQ−1 XB)−1 This means * (jZ ) is positive denite for all Z , i.e.

* (jZ ) = I − GT(−jZ )G(jZ ) > 0, ∀Z . This inequality implies 1G(s)1f < 1. (i)⇒(iii). Let 1G(s)1f = J0 < 1 and 1(sI − A)−1B1f = U . Then, there exists a sufciently small H0 > 0 such that (1 − H0)I > J02 I  GT (−jZ )G(jZ ), ∀Z . Furthermore, let H > 0 be a sufciently small number satisfying H < H0 /U 2 . Then we have

H0 I  HU 2 I  H BT (−jZ I − AT )−1 (jZ − A)−1 B, ∀Z . Thus, we obtain I > H0 I + GT (jZ )G(jZ ), ∀Z , and by (14.13), I > BT (−jZ I − AT )CHTCH (jZ I − A)−1 B, ∀Z ,

(14.13)

14.1 Hf Norm and L2 -Gain

where CHT = [CT

√

407

H I]. This inequality implies 1CH (sI − A)−1 B1f < 1.

Hence, from Theorem 14.1, there exists a positive semi-denite matrix Z satisfying Riccati equation (14.11). Let S = ZBBT Z + CTC + H I > 0. Then, the Riccati equation can be represented as AT Z + ZA = −S. Therefore, the positivity of the matrix Z follows by Lyapunov stability theorem with the stability of A. (iii)⇒(ii). This is obvious. 3 2 As an extension of Hf norm, for nonlinear systems L2 -gain represent the property of signal transformation from the input space to the output space, and as shown in Lemma 14.1, a system has L2 -gain less than one if the system is J -dissipative with J = 1. Moreover, it has been shown in the previous chapter that the J -dissipativity is guaranteed by Hamilton-Jacobi-Issacs (HJI) inequality. Indeed, the HJI inequality is an extension of Theorem 14.2. For the convenience in citation, we summarize the conclusion in the following theorem. Theorem 14.3. Consider a nonlinear system J x = f (x) + g(x)u y = h(x).

(14.14)

If there exists a positive semi-denite function V (x)  0, ∀x (V (0) = 0) satisfying 1 1 L f V (x) + hT (x)h(x) + LgV LTg V (x)  0, 2 2

(14.15)

then the system has L2 -gain less than 1. Proof. Recalling the HJI inequality (13.34), and noting that d(x) = 0 and J = 1, the theorem follows from Theorem 13.6 and Lemma 14.2. 3 2 In fact, this conclusion is an extension of Theorem 14.1. It is easy to check that if Riccati equation in Theorem 14.1 has positive semi-denite solution X, then the positive semi-denite function 1 V (x) = xT Xx 2 satises HJI inequality (14.15). However, necessity of the HJI inequality for the system to have L2 -gain less than one is not simple. It needs additional condition. Denition 14.2. Consider system (14.14). For a given initial state x(0) = x0 and an input u(t), denote the solution as x(t) = \ (t, x0 , u). Then, the system is said

408

14 L2 -Gain Synthesis

to be reachable at x0 , if for any given x1 , there exist t1  0 and u1 (t) such that x1 = x(t1 ) = \ (t1 , x0 , u1 ). Theorem 14.4. Suppose the system (14.14) is reachable at x0 , and for any given t0  0, the following function is well-dened and differentialble V (x) = − lim

1 2

inf

T →f u∈L2 [t0 ,T ] x(t0 )=x

p T@ t0

A J 2 1u12 − 1y12 dt.

(14.16)

Then the system is J -dissipative, if the system has L2 -gain less that J . Proof. Suppose the system has L2 -gain less than J . Since the system is reachable, for any given x there exist t0 and u0 (t) such that x = \ (t0 , 0, u0 ). Let

J u(t) =

u0 (t),

0  t  t0

v(t),

t0 < t  T,

where v ∈ L2 [t0 , T ] is any signal. Then p T@ 0

A J 2 1u12 − 1y12 dt  0.

That is, p t0 p T @ 2 A @ 2 A J 1u0 12 − 1y12 dt + J 1v12 − 1y12 dt  0, ∀T  t0 . t0

0

Thus, −

1 2

p t0 @ 0

p A A 1 T@ 2 J 2 1u0 12 − 1y12 dt  J 1v12 − 1y12 dt < f. 2 t0

This indicates the function dened by (14.16) is bounded. Furthermore, by the denition, V (x) > 0 and when x0 = 0, t0 can be taken as t0 = 0. Hence, V (0) = 0. On the other hand, if we consider the optimization problem with the following cost function subject to x = f (x) + g(x)u

I (x) =

inf

u∈L2 [t0 ,T ] x(t0 )=x

1 2

p T@ t0

A J 2 1u12 − 1y12 dt,

(14.17)

then, from the optimization theory [1], the optimal cost function I (x) satises b a wI wI 1 2 = − min (J 1u12 − 1y12) + ( f (x) + g(x)u) . (14.18) u wt 2 wx

14.2 Hf Feedback Control Problem

Noticing that I (x) = −V (x) and a

wI wt

409

= 0, we have

b wI 1 2 2 2 min (J 1u1 − 1y1 ) + ( f (x) + g(x)u) u 2 wx U V wV w TV 1 T 1 wV T f (x) + 2 g(x)g (x) + h (x)h(x) =− wx 2J w x wx 2 = 0. Therefore, V (x) satises the HJI inequality.

3 2

14.2 H f Feedback Control Problem Roughly speaking, Hf control problem is to design a feedback controller such that the closed-loop system satises stability condition and a given constraint condition, which is described by forcing Hf norm for linear systems (L2 -gain for nonlinear systems) to be less than a given level. For linear systems, if we consider the state feedback control problem, then the Hf control problem can be described as follows: Consider a linear system J x = Ax + B1w + B2 u (14.19) z = C1 x + D12u, where x is the state and u is the control input. w and z are generalized disturbance and penalty signals, respectively, which are introduced to represent the desired performance via Hf -norm constraint. Hf state feedback design problem can be formulated as follows: for a given scalar J > 0, nd a feedback control law u = Kx,

(14.20)

such that the closed-loop system is asymptotically stable and Hf norm of the transfer function Tzw (s) is less than the given level J , i.e. 1Tzw (s)1f < f,

(14.21)

where Tzw (s) denotes the closed-loop transfer function from w to z, which is given by Tzw (s) = (C1 + D12 K) (sI − (A + B2K))−1 B1 .

(14.22)

By Theorem 14.2, the closed-loop system satises the desired specication, if and only if there exists a positive denite matrix X > 0 satisfying the following Riccati inequality 1 XB1 BT1 X J2 + (C1 + D12 K)T (C1 + D12 K) < 0.

(A + B2 K)T X + X (A + B2K) +

(14.23)

410

14 L2 -Gain Synthesis

Therefore, the design problem can be solved by seeking such a feedback law K that guarantees the existence of X > 0, satisfying the Riccati inequality (14.23). Indeed, a desired K and a positive denite matrix X can be found simultaneously by solving a Riccati inequality. Theorem 14.5. Consider system (14.19). Suppose (A, B2 ) is stabilizable and DT12 [D12 C1 ] = [I 0].

(14.24)

The Hf state feedback design problem is solvable, if and only if there exists a positive denite matrix X , satisfying the following Riccati inequality V U 1 T T T AT X + XA + X B B − B B (14.25) 1 2 1 2 + C1 C1 < 0. J2 Furthermore, a desired static state feedback gain is given by K = −BT2 X .

(14.26)

Proof. It is easy to check that the Riccati inequality (14.23) can be rewritten as V U 1 T T T A X + XA + X B1 B1 − B2 B2 + C1TC1 + RTK RK < 0, (14.27) J2 where

RK = K + BT2X . Hence, if (14.25) has positive denite solution X > 0, then, by taking K = −BT2 X, the Riccati inequality (14.27) holds, i.e. (14.23) holds. On the other hand, if there exists a state feedback control law K such that the closed-loop system satises the design requirements, then Riccati inequality (14.23) has positive denite solution X > 0. Thus, by (14.27), the positive denite matrix X satises (14.25). 3 2

More general case is to consider the state feedback with disturbance forward, i.e. the structure of controller is as follows: u = Ks (s)x + Kw(s)w.

(14.28)

In this case, the design problem is usually called full-information design problem, and the solution is not unique, here we present a typical result. The details of the proof can be found in [3]. Theorem 14.6. Consider system (14.19) with controller (14.28). Suppose the system satises the condition required in Theorem 14.5, and G12 (s) = C1 (sI − A)−1 B2 has no zero on the imaginary axis. Then, for a given positive scalar J > 0, there exist Ks (s) and Kw (s) such that the closed loop system is asymptotically stable and 1Tzw (s)1f < J , if and only if the following Riccati equation V U 1 T T T B1 B1 − B2 B2 + C1TC1 = 0 (14.29) A X + XA + X J2 has a positive semi-denite solution X  0 such that

14.3 L2 -Gain Feedback Synthesis

411

A + (B1 BT1 − B2 BT2 )X is stable. Furthermore, the desired controller is parameterized as follows Ks (s) = −BT2 X − Q(s)BT1 X ,

Kd (s) = Q(s),

(14.30)

where Q(s) ∈ RHf is any function satisfying 1Q(s)1f < 1. Under the assumption that the measurable signal of the system is y = C2 x + D21w.

(14.31)

The Hf output feedback control problem is to nd a feedback controller of the form u = K(s)y,

(14.32)

such that the closed-loop system is stable and the Hf -norm constraint is satised. To nd an Hf output feedback controller, we need to solve two coupled Riccati inequalities. The details can be found in [3]. For nonlinear systems, the Hf design problem can be formulated with L2 -gain constraint as follows: consider a nonlinear system ⎧ ⎪ ⎨x = f (x) + g1 (x)w + g2 (x)u (14.33) z = h1 (x) + d12(x)u ⎪ ⎩ y = h2 (x) + d21(x)w. For a given J > 0, nd a static output feedback controller u = D (y),

(14.34)

or a dynamic output feedback controller u = D ([ , y),

[ = E ([ , y),

(14.35)

such that the closed-loop system satises the following performance specications: (i). When w = 0, the closed-loop system is asymptotically stable; (ii). the closed-loop system from w to z has L2 -gain less than J . In the following sections, we will discuss the L2 -gain design problem.

14.3 L2 -Gain Feedback Synthesis From the presentation in the foregoing section, it follows that under certain condition a nonlinear system has L2 -gain less than J > 0, if and only if the nonlinear system is dissipative, or equivalently, HJI inequality has a non-negative denite solution. Consequently, if a feedback controller can be derived, which can ensure the J -dissipativity of the closed-loop system and guarantee the asymptotical stability of the equilibrium when the external disturbance w = 0, the controller is just a solution to the standard Hf control problem. In this section, we will discuss the standard Hf control problem along this way, provided that the generalized controlled system satises the zero-state detectability.

412

14 L2 -Gain Synthesis

First, consider the state feedback Hf control problem. A generalized controlled system is described as the following form ⎧ ⎨x = f (x) + g1 (x)w + g2 (x)u (14.36) ⎩z = h1 (x) + d12(x)u y = h2 (x) + d21(x)w, and suppose y = x,

i.e. h2 (x) = x,

d21 (x) = 0.

(14.37)

The objective is to nd a state feedback controller u = D (x),

(14.38)

such that the resulting closed-loop system is asymptotically stable when w = 0, and the closed-loop system has L2 -gain less than a given level J > 0. Theorem 14.7. Suppose the system (14.36) satises T d12 (x) [d12 (x) h1 (x)] = [I

0].

(14.39)

Then, for any given J > 0, a state feedback controller (14.38) can render the closedloop system dissipative property with respect to J , if and only if the following HJI inequality U V T 1 wV 1 wV w V T T f (x) + g (x)g (x) − g (x)g (x) 1 2 1 2 wx 2 w x J2 wx (14.40) 1 T + h1 (x)h1 (x)  0, ∀x 2 has a positive semi-denite solution V (x). Furthermore, if there exists a solution V (x)  0 of the HJI inequality (14.40), a state feedback controller, rendering the closed-loop system J -dissipative, is given by u = D (x) = −gT2 (x)

w TV . wx

(14.41)

Proof. (Sufciency) Suppose V (x)  0 is a solution of HJI inequality (14.40). Using this inequality yields wV [ f (x) + g1 (x)w + g2 (x)u] V (x) = wx −

w TV 1 w V w TV 1 wV g1 (x)gT1 (x) + g2 (x)gT2 (x) 2 2J w x wx 2 wx wx

1 wV wV g2 (x)u + g1 (x)w − hT1 (x)h1 (x) wx wx 2 9 9 92 92 T 9 9 19 1 T w V w TV 19 T 9 9 9 = − 9 g1 (x) − J w9 + 9g2 (x) + u9 9 2 J wx 2 wx +

+

J2 1 1w12 − hT1 (x)h1 (x). 2 2

(14.42)

14.3 L2 -Gain Feedback Synthesis

Thus, let the state feedback controller be (14.41). We obtain 9 92 2 9 1 9 1 T w TV 9 + J 1w12 − g (x) J w V (x)  − 9 1 9 9 2 J wx 2 @ A 1 1 − 1u12 − 1D (x)12 + 1h1 (x)1 . 2 2

413

(14.43)

Note that when the condition (14.39) holds, the closed-loop system satises 1z12 = 1D (x)12 + 1h1(x)12 .

(14.44)

Consequently, along the trajectory of the closed-loop system, V (x) satises the following dissipation inequality A 1@ 2 J 1w12 − 1z12 . V (x)  2

(14.45)

(Necessary) Suppose there exists a state feedback controller (14.41) such that the closed-loop system is J -dissipative. With this controller, the closed-loop system can be described as J x = fD (x) + g1 (x)w (14.46) z = hD (x), where fD (x) = f (x) + g2 (x)D (x), hD (x) = h1 (x) + d12 (x)D (x). Thus, according to Theorem 14.3, it follows that there exists a non-negative denite solution V (x) of the inequality 1 wV wV w TV 1 T fD (x) + 2 g1 (x)gT1 (x) + hD (x)hD (x)  0, wx 2J w x wx 2

∀x.

(14.47)

By using the condition (14.39), it is easy to show that (14.47) is equivalent to V T U wV w V 1 T 1 wV 1 T T g1 (x)g1 (x) − g2 (x)g2 (x) f (x) + + h1 (x)h1 (x) wx 2 w x J2 wx 2 92 9 9 9 T w TV 9  0. (x) D (x) (14.48)  −9 g − 9 9 2 wx 3 2 As previously stated, the controller given in Theorem 14.7 ensures the closedloop system to have J -dissipativity, but it can not guarantee the internal stability. To this end, the further condition is needed to impose on the system. Theorem 14.8. Suppose the generalized system (14.36) satises (14.39) and the state feedback controller is given by (14.41). If x = 0 is an equilibrium of the system of (14.36), and ( f (x), h1 (x)) is zero-state detectable, then the closed-loop system is asymptotically stable at x = 0 when w = 0. Proof. From the proof of Theorem 14.7, it follows that for the closed-loop system consisting of (14.36) and (14.41), there exists a non-negative denite solution V (x)

414

14 L2 -Gain Synthesis

such that the dissipation inequality (14.45) holds for any w. Thus, when w = 0, we have 1 (14.49) V (x)  − 1z12  0, ∀t. 2 Consequently, the system is Lyapunov stable at x = 0, and furthermore, V = 0 only if x = 0, or equivalently, h1 (x) = 0, D (x) = 0. Then, clearly, E ⊆ :0 , where

(14.50)

E = {x | V = 0}, :0 = {x | h1 (x) = 0, D (x) = 0},

and in the set :0 , the state trajectory of the system satises x = f (x), h1 (x) = 0.

(14.51)

Therefore, according to the assumption of the zero-state detectability and LaSalle invariant theorem, x(t) → 0. 3 2 From the discussion above, it follows that if the whole state of the generalized controlled system is available for measurement and the assumptions in Theorem 14.7 and Theorem 14.8 are satised, the solution to the standard Hf control problem can be obtained by solving the HJI inequality. For convenience of citation, the above arguments are summarized in the following corollary. Corollary 14.1. Consider the system (14.36). Suppose (14.39) holds and ( f (x), h1 (x)) is zero-state detectable. For any given J > 0, if there exists a non-negative denite solution V (x) to the Hamilton-Jacobi inequality (14.40), the state feedback controller given by (14.41) is a solution to the standard Hf control problem. As a special case of nonlinear systems, consider the following linear system J x = Ax + B1w + B2 u (14.52) z = C1 x + D12u. According to Corollary 14.1, it is easy to obtain the following corollary. Corollary 14.2. Suppose the system (14.52) satises the following conditions: (i). DT12 [D12 C1 ] = [ I 0]; (ii). (C1 , A) is detectable. Moreover for any given J > 0, if there exists a solution P  0 to the Riccati inequality AT P + PA + P(J −2B1 BT1 − B2 BT2 )P + C1TC1  0,

(14.53)

the state feedback controller u = −BT2 Px

(14.54)

can render the resulting closed-loop system asymptotically stable, and furthermore, 1z1T  J 1w1T ,

∀w, ∀T > 0.

(14.55)

14.3 L2 -Gain Feedback Synthesis

415

Proof. Dene a non-negative denite function 1 V (x) = xT Px. 2

(14.56)

Since f (x) = Ax, g1 (x) = B1 , g2 (x) = B2 , h1 (x) = C1 x, h12 (x) = D12 , V (x) satises the Hamilton-Jacobi inequality (14.40). Thus, from Corollary 14.1, this conclusion follows. 3 2 The dynamic output feedback control problem will be handled by the following theorem. Theorem 14.9. Suppose the generalized system (14.36) satises T k12 (x)[k12 (x) h1 (x)] = [I 0],

(14.57)

T k21 (x)[k21 (x)

(14.58)

gT1 (x)]

= [I 0],

and ( f 1 (x), h1 (x)) is zero-state detectable. For any given J > 0, if there exist positive denite functions V (x), W (x, [ ), and a constant matrix G such that HJI inequalities V T U wV w V 1 T 1 wV 1 T T g1 (x)g1 (x) − g2 (x)g2 (x) f (x) + + h1 (x)h1 (x)  0, ∀x. 2 wx 2 wx J wx 2 (14.59) 1 wW wW w TW 1 T fe (xe ) + 2 E(xe )E T (xe ) + he (xe )he (xe )  0, ∀xe w xe 2J w x e w xe 2

(14.60)

hold and the system 1 [ = f ([ ) + 2 g1 ([ )E ([ ) − Gh2([ ) J

(14.61)

is asymptotically stable at [ = 0, then a solution to the standard Hf control problem is given by ⎧ ⎨ u = D ([ ) 1 (14.62) ⎩ [ = f ([ ) + g2 ([ )D ([ ) + 2 g1 ([ )E ([ ) + G(y − h2([ )), J where D (·) is the same as that in Theorem 14.7, and xe = [xT [ T ]T ,

w TV E (x) = g1 (x) (x) wx ⎡ ⎢ fe (xe ) = ⎣

f (x) + g2 (x)D (x) +

1 g1 (x)E (x) J2

⎤

⎥ ⎦, 1 f ([ ) + g2 ([ )D ([ ) + 2 g1 ([ )E ([ ) + G(b2 ([ ) − h2 ([ )) J ? ? > > wW wW wW g1 (x) , . h2 (xe ) = D ([ ) − D (x), E(xe ) = = Gk21 (x) w xe wx w[ Proof. The closed-loop system can be described as

416

14 L2 -Gain Synthesis

J

xe = F(xe ) + E(xe )w z = H(xe ),

(14.63)

⎤ ? > 1 g (x) E (x) g1 (x) 1 2 ⎦ , , E(xe ) = F(xe ) = fe (xe ) − J Gk21 (x) 0 H(xe ) = h1 (x) + k12(x)D (x). It is well known that this system has L2 -gain less than J if it is J -dissipative. Furthermore, according to Theorem 14.7, the system is J -dissipative, if and only if there exists a non-negative denite solution U(xe ) to HJI inequality

where

⎡

⎣−

wU w TU 1 T 1 wU F(xe ) + 2 E(xe )E T (xe ) + H (xe )H(xe )  0, w xe 2J w x e w xe 2

∀xe .

(14.64)

Thus, in the following, to prove the closed-loop system (14.63) has L2 -gain less than J , we will prove that the non-negative denite function U(xe ) = V (x) + W (x, [ )

(14.65)

satises the HJI inequality (14.64). Let V T U wV w V 1 T 1 wV 1 T T g (x)g (x) − g (x)g (x) H1 (x) = f (x) + + h1 (x)h1 (x); 1 2 1 2 wx 2 w x 2J 2 wx 2 (14.66) 1 wW wW w TW 1 T T H2 (x, [ ) = fe (xe ) + 2 E(xe )E (xe ) + he (xe )he (xe ). (14.67) w xe 2J w xe w xe 2 Substituting (14.65) into (14.64) yields

wU w TU 1 T 1 wU F(xe ) + 2 E(xe )E T (xe ) + H (xe )H(xe ) = H1 (x) + H2 (x, [ ). w xe 2J w xe w xe 2 (14.68) This shows that according to the inequalities (14.59) and (14.60), U(xe ) satises the HJI inequality (14.64). In the following, we will show the stability of the system (14.63). In view of the relation between the J -dissipativity and the HJI inequality (see Theorem 14.7), it is clear that U(xe ) satisfying the inequality (14.64) must satisfy the dissipation inequality A @  e )  1 J 2 1w12 − 1z12 . (14.69) U(x 2 Therefore, when w = 0, along the trajectory of the closed-loop system xe = F(xe ),

(14.70)

 e )  − 1 1z12  0. U(x 2

(14.71)

the time derivative of U satises

14.4 Constructive Design Method

417

Consequently, U = 0 only if z = 0, that is h1 (x) = 0, D ([ ) = 0, and the state trajectory of the closed-loop system satises ⎧ ⎨x = f (x) 1 ⎩[ = f ([ ) + 2 g1 ([ )E ([ ) − Gh2 ([ ). J

(14.72)

(14.73)

From the assumptions of the zero-state detectability of ( f (x), h1 (x)) and of the asymptotical stability of the free system (14.61) at the origin, it follows that the states x(t) and [ (t) satisfying the equation (14.73) will tend to zero. Consequently, according to the LaSalle invariant theorem, the closed-loop system (14.63) is asymptotically stable at x = 0, [ = 0. 3 2

14.4 Constructive Design Method Recalling the foregoing proofs for theorems, it can be seen that the solution of the HJI inequality in fact is a storage function ensuring the J -dissipativity of the system, and the L2 -gain condition is derived indirectly by the J -dissipativity. Thus, a possible way to get the solution of HJI indirectly is to construct the storage function directly. Since the J -dissipativity reects the concept of a general storage energy dissipation, for many practical systems, the storage function, ensuring the dissipativity of the system, can be constructed directly by physical meaning, or just, its own energy function. Furthermore, the controller is designed to satisfy the L2 -gain property. This direct design technique by constructing the storage function will be presented in this section. First, a simple mechanical system is considered. Recalling Example 13.2, it can be seen that we can determine that the system has L2 -gain ( 1) by the function V (x, x)  represented the total energy of the system instead by the existence of the solution of the corresponding HJI inequality or Riccati inequality. Consequently, in practice, if this kind of energy function can be found, the controller can be constructed directly. To illustrate this technique, we consider a special system. Suppose a generalized controlled system is given as ⎧ ⎪ ⎨K = f (K ) + f1 (K , y)y (14.74) y = p(K , y)w + u ⎪ ⎩ z = h0 (K , y)y, where (K , y) are the states of the system, u, w, z denote the control input, external disturbance signal and output of the system, respectively. If the subsystem K of (14.74) with y = 0

K = f (K )

(14.75)

is asymptotically stable at K = 0, according to the Lyapunov inverse theorem (see [10]), there exists a positive denite function W (K ) > 0 with W (0) = 0, such that

418

14 L2 -Gain Synthesis

wW f (K ) < 0, wK

∀K := 0.

(14.76)

1 V (K , y) = W (K ) + yT y. 2

(14.77)

Dene a positive denite function as

Then, along the trajectory of the system, the time derivative of V is

wW ( f (K ) + f1 (K , y)y) + yT (u + p(K , y)w) V = wK 9 92 9 A wW 1 T 1@ 2 19 2 2 9 = J 1w1 − 1z1 + f (K ) − 9 p0 (K , y)y − J w9 9 2 wK 2 J U V 1 T wW 1 T T +y u + 2 p(K , y)p (K , y)y + h0 (K , y)h0 (K , y)y + f1 . 2J 2 wK Thus, choosing the state feedback control law u=−

1 1 wW f1 (K , y) − y − 2 p(K , y)pT (K , y)y − hT0 (K , y)h0 (K , y)y, wK 2J 2

(14.78)

yields that the resulting closed-loop system satises the J -dissippation inequality A 1@ 2 J 1w12 − 1z12 . V  2

(14.79)

Moreover, by using (14.79) and (14.76), it can be shown that when w = 0, V satises V < 0, ∀K = : 0, y := 0.

(14.80)

Consequently, the state feedback controller (14.78) is a state feedback Hf controller of the generalized controlled system (14.74). Therefore, for the generalized system (14.74), we have the following result. Theorem 14.10. Consider the generalized system (14.74). For any given J > 0, if there exists a positive function W (K ) such that (14.76) holds, the controller (14.78) renders the closed-loop system an L2 -gain less than J . Moreover, when w = 0, the system is asymptotically stable at the origin. Recall the proof of above theorem, it is clear that we suppose the subsystem K is asymptotically stable at the origin when y = 0, instead of requiring the existence of the solution of the HJI inequality, so that the controller can be obtained for any given J > 0. It implies that the L2 -gain of the closed-loop system can be restrained to be arbitrarily small. In fact, it follows from the fact that the penalty signal z does not contain the weighting term on the control input. This kind of design problem is called the L2 disturbance approximate decoupling problem. The result in Theorem 14.10 can be also extended to the case in which the subsystem K is not asymptotically stable. In fact, it is enough that there exists a smooth function D (K ) (D (0) = 0) such that when y = D (K ), the subsystem K

14.4 Constructive Design Method

K = f0 (K ) = f (K ) + f1 (K , D (K ))D (K )

419

(14.81)

is asymptotically stable at the origin K = 0, namely, there exists a positive function W (K ) such that

wW W = f0 (K ) < 0. wK Then, we can introduce a change of coordinates as > ? > ? K K → , y y˜

(14.82)

(14.83)

where y˜ = y − D (K ), to transform the system (14.74) into the form as a K = fD (K ) + fD 1 (K , y) ˜ y˜ y˜ = p(K , y)w + v,

(14.84)

where v is a new control input, and

wD ( f (K ) + f1 (K , y)y) wK ˜ D (K ) + y) ˜ = fD (K ) + fD 1 (K , y) ˜ y, ˜ f (K ) + f1 (K , D (K ) + y)(

v = u − D = u −

i.e. the system (14.84) is feedback equivalent to the system (14.74). Thus, the desired corresponding controller of the system (14.74) can be obtained provided that we can derive the controller of the system (14.84) that satises the L2 -gain condition and ensures the stability of the closed-loop system. Obviously, the system should satisfy the assumption in Theorem 14.10, namely, when y˜ = 0, the subsystem K is asymptotically stable at the origin. Therefore, providing the penalty signal z satises z = hD (K , y) ˜ y˜ for y, or equivalently, h0 (K , D (K ))D (K ) = 0, the controller can be designed by using Theorem 14.10. In fact, even though z dose not satisfy the condition, the desired controller may be also obtained. This will be shown in the following. Consider nonlinear systems J x = f (x) + g1 (x)w + g2 (x)u (14.85) z = h(x). If for the system (14.85), we can nd an auxiliary output signal y = hD (x),

(14.86)

such that the relative degree of y with respect to the control input u is one, and g1 (x), g2 (x) satisfy the matching condition g1 (x) = g2 (x)gm (x),

∀x.

Moreover, assume there exists a change of coordinates > ? ? > [ T (x) = I (x) = , hD (x) y

(14.87)

(14.88)

420

14 L2 -Gain Synthesis

such that the system (14.85) is feedback equivalent to the system ⎧  ⎪ ⎨[ = f0 ([ , y) y = p([ , y) + v ⎪ ⎩ z = h([ , y),

(14.89)

where v is a new control input, and p([ , y) = Lg2 hD (I −1 (x))gm (I −1 (x)), h([ , y) = h(I −1 (x)). Then we discuss the Hf design problem of the system (14.89). The zero dynamics of the system (14.89) when y = 0 reads [ = f0 ([ , 0) = f∗ ([ ). (14.90) Theorem 14.11. Suppose that for the zero dynamics (14.90) there exist positive definite functions W ([ ) and Q([ ) such that L f∗ W ([ )  −Q([ ),

(14.91)

and 1 T h ([ )h∗ ([ )  Q([ ), 2 ∗

∀[ .

(14.92)

Then, for any given J > 0, the corresponding Hf controller of the system (14.89) is given by u = −y − LTF W ([ ) − H T([ , y) −

1 p([ , y)pT ([ , y)y, 2J 2

(14.93)

where

f0 ([ , y) = f∗ ([ ) + F([ , y)y, h∗ ([ ) = h([ , 0), 1 T 1 h ([ , y)h([ , y) = hT∗ ([ )h∗ ([ ) + H([ , y)y. 2 2 Proof. Dene a positive denite function V as 1 V ([ , y) = W ([ ) + yT y. 2

(14.94)

Then, along the state trajectory of the closed-loop system, the time derivative of V is calculated as V ([ , y) = L f∗ W ([ ) + LF W ([ )y + yT p([ , y)w + yTv A 1 1@ = L f∗ W ([ ) + hT ([ , y)h([ , y) + J 2 1w12 − 1z12 2 2 92 9 V U 9 1 1 T 19 T T T 9 9 − 9 p ([ , y)y − J w9 + y v + LF W ([ ) + 2 pp y 2 J 2J A 1 1@  −Q([ ) + hT∗ ([ )h∗ ([ ) + J 2 1w12 − 1z12 2 2 U V 1 T T T T +y v + LF W ([ ) + H ([ , y) + 2 pp y . 2J

(14.95)

14.4 Constructive Design Method

421

Substituting the feedback controller (14.93) into above inequality yields 1 V  (J 2 1w12 − 1z12). 2

(14.96)

Thus, the closed-loop system is dissipative with respect to J . Consequently, the closed-loop system has L2 -gain less than J . Moreover, when w = 0, repeat the design procedure above for the dissipation inequality to obtain V  −Q([ ) − yTy, ∀t  0.

(14.97)

Then, the asymptotical stability of the closed-loop system at the origin can be shown by Lyapunov stability theory. 3 2 In Theorem 14.11, we require the zero dynamics of the controlled system is asymptotically stable at the origin and the stability margin is large enough to make Q([ ) satisfy the inequality (14.91). In fact, if the zero dynamics is exponentially stable, i.e. there exists a constant H > 0 such that L f∗ ([ )  −H W ([ ).

(14.98)

Then, it can be veried if we choose a positive function dN (T ) + 0T 1 dT

W0 (W ) = W sup

p 2W W

N (T )dT ,

(14.99)

where N (·) is a monotonous nondecreasing function satisfying the condition 1 1h∗ ([ )12  N (W ). 2H

(14.100)

Let the positive function Q([ ) = HN (W ). Then W0 and Q satisfy the assumptions in Theorem 14.11 (see [7, 6]). In the following we consider the case when the system dose not satisfy the matching condition (14.87). In this case, the system (14.85) is feedback equivalent to ⎧  ⎪ ⎨[ = f0 ([ , y) + g11([ , y)w (14.101) y = g12 ([ , y)w + v ⎪ ⎩ z = h([ , y). Comparing with the system (14.89), the difference is that there exists the disturbance signal w in the subsystem [ of (14.91). Theorem 14.12. Consider the system (14.101). Suppose that for its zero dynamics there exists a positive denite function W ([ ) satisfying L f∗ W ([ ) +

1 1 1Lg∗ W ([ )12 + 1h∗ ([ )12 < 0, 2 2J 2

then, the Hf controller is given by

∀[ := 0,

(14.102)

422

14 L2 -Gain Synthesis

v = −y − LTF W ([ ) − H T([ , y) −

1 T M ([ , y), 2J 2

(14.103)

where g∗ ([ ) = g11 ([ , 0), H([ , y) and M([ , y) satisfy 1gT12([ , y) + LTg11 W ([ )12 = 1LTg∗ W ([ )12 + M([ , y)y, and

1 1 1h([ , y)12 = 1h∗ ([ )12 + H([ , y)y. 2 2

Proof. Dene a positive denite function V ([ , y) as in (14.94), then, V ([ , y) = L f∗ W ([ ) + LF W ([ )y + (Lg11W ([ ) + yTg12 ([ , y))w + yT v A 1 1@ = L f∗ W ([ ) + hT ([ , y)h([ , y) + J 2 1w12 − 1z12 2 2 9 92 9 1 91 T T 9 − 9 (g ( [ , y)y + L W ( [ )) − J w 12 g11 9 9 2 J +

9 A @ 1 9 9gT12 ([ , y)y + LTg W ([ )92 + yT v + LTF W ([ ) 2 11 2J

9 @ A 1 9 9LT W ([ )92 + 1 1h∗ ([ )12 + 1 J 2 1w12 − 1z12 2 J 2 g∗ 2 2 U V 1 T T T T (14.104) +y v + LF W ([ ) + H ([ , y) + 2 M ([ , y) . 2J

 L f ∗ W ([ ) +

Thus, substituting the controller (14.103) and the inequality (14.102) into (14.104) yields the dissipation inequality (14.96). When w = 0, the proof for the stability of the closed-loop system is similar to Theorem 14.11. 3 2 From the condition in Theorem 14.12, it follows that the condition for the existence of a positive denite solution of the HJI inequality (14.102) is equivalent to the fact that when w := 0 the zero dynamics J [ = f∗ ([ ) + g∗ ([ )w (14.105) z = h∗ ([ ) is dissipative with respect to J . Thus, the result given in Theorem 14.12 shows that if the relative degree of the system from the output y to the control input u is one and the zero dynamics is J dissipative, then the dissipativity of the closed-loop system can be realized by the state feedback. In fact, similar to the backstepping design approach for the stabilization problem (see [2]), it shows that the controller ensuring the J -dissipativity of the closed-loop system can be also designed by recursively constructing the storage function. For example, if the system (14.85) has the relative degree r > 1, under certain condition, the system is feedback equivalent to the system

14.5 Applications

⎧ [ = f0 ([ , y1 ) + g11([ , y1 )w ⎪ ⎪ ⎪ ⎪ ⎪ y ⎪ ⎨ 1 = y2 + g12([ , y1 , y2 )w .. . ⎪ ⎪ ⎪ ⎪ yr = v + g1(r+1)([ , y1 , y2 , · · · , yr )w ⎪ ⎪ ⎩ z = h([ , y1 , y2 , · · · , yr ).

423

(14.106)

Therefore, if the subsystem [ is dissipative with respect to J , rst, regarding y1 as a virtual control input and apply Theorem 14.12 for the subsystem ([ , y1 ) to derive the virtual control law y2 = D ([ , y1 ). And then, change the coordinates by using the obtained control function D (·), under the new coordinates, y2 is viewed as the virtual control input, the rest may be deduced by analogy, namely, recursively using Theorem 14.12, the Hf controller of the whole system can be obtained. Moreover, if g11 ([ ) = 0, in the rst step of the recursive design above, it is not necessary that the positive function W satises the HJI inequality, it is only required that the subsystem [ is asymptotically stable or exponentially stable at the origin. The reader interested in the recursive design approach can refer to [7, 6].

14.5 Applications In this section, we discuss two design examples using the constructive design approach. It will be shown that for the systems with physical background, the L2 -gain synthesis problem can be solved by directly constructing the storage function. First, consider the Lagrangue system represented by M(T )T¨ + C(T , T )T = W + w,

(14.107)

where T and W denote the generalized coordinates and the torque acting on the direction T . w is the disturbance at torque level. Suppose the desired trajectory is given by Td , Td , T¨d . The problem considered here is to nd a feedback control law such that T and T asymptotically track the desired trajectory when the disturbance w = 0, and the system has L2 -gain from the disturbance to a penalty signal less than a given level J , i.e. p T 0

1z(t)12 dt  J 2

p T 0

1w(t)12 dt,

∀w ∈ L[0, T ],

(14.108)

where z is the penalty signal dened to represent the disturbance rejection performance. To reach this goal, we introduce a feedback compensation

W = u + M(T )T¨d + C(T , T )Td ,

(14.109)

where the tracking error e = T − Td . Then the tracking error dynamics is given by M(T )e¨ + C(T , T )e = u + w.

(14.110)

To achieve the disturbance rejection performance, we dene the following penalty signal

424

14 L2 -Gain Synthesis

>

? q1 e z= , q > 0, q2 > 0, q2 e where q1 > 0 and q2 > 0 are weighting parameters satisfying q1 + q2  1 − H1 for a sufciently small H1 > 0. Then, as discussed in Section 14.4 we can obtain a desired control for the input u by constructing a storage function V satisfying p T 0

1z(t)12 dt  J 2

p T 0

1w(t)12 dt,

∀w ∈ L2 [0, T ].

(14.111)

Theorem 14.13. Consider the error system (14.110). Let the controller be given by u = \ (e, y,t) + 2q22 e − D1 y,

(14.112)

where \ = −e + (M − C)e and y = e + e. Then, for any given J > 0, if the constant gain D1 satises

D1 

1 + q22 + H2 4J 2

(14.113)

for a sufciently small H2 > 0, then the closed-loop system is asymptotically stable when w = 0, and it has L2 -gain less than the given J . Proof. Using y = e + e, the error (14.110) can be rewritten as J e = −e + y M(T )y = (M − C)y − (M − C)e + u + w.

(14.114)

Along any trajectory corresponding to w, dene H(e, y,t, w, z) = V + 1z12 − J 2 1w12 .

(14.115)

Recalling the physical properties as we employed in proof of Theorem 13.12, we have H = −eT e + yT(u − \1 (e, y,t) + w) + 1z12 − J 2 1w12 = −eT e + yT(u−\1+w)+q21 1e12 + q22 1e1  2 − J 2 1w12 9 92 91 9 9 = −(1 − q21 − q22 )1e12 − 9 9 2J y − J w9 1 +yT (u − \1 − 2q22e + ( 2 + q22 )y) 4J U V 1  −(1 − q21 − q22 )1e12 − D2 − ( 2 + q22 ) 1y12 4J  −H12 1e12 − H22 1y12 < 0, ∀e := 0, y := 0.

(14.116)

This leads to the dissipativity inequality 1 V < (J 2 1w12 − 1z12). 2 The stability can be proved by considering the case w = 0 with Lyapunov Theorem. 3 2

14.5 Applications

425

This design technique can be extended to the robust design problem with respect to modeling uncertainty. For example, assume the model is represented by M(T )T¨ + C(T , T )T + g(T ) + ' f (T , T ) = W + w,

(14.117)

where ' f denotes the modeling error. Then, under the feedback compensation (14.109) the tracking error system becomes M(T )e¨ + C(T , T )e + ' f (T , T ) = u + w.

(14.118)

Suppose that ' f can be represented as ˜ Td , Td , e, e) ' f = E(  G˜ (Td , Td , e, e),  ˜ with known function E˜ and unknown function G , which is bounded by a known function U (Td , Td , e, e),  i.e. > ? 1G˜ (Td , Td , e, e)1 e   9> ?9  U ( T , T , e, e),  ∀ := 0. (14.119) d d 9 e 9 e 9 9 9 e 9 Then, we may redene E and G as ˜  = U E, E(Td , Td , e, e)

G (Td , Td , e, e)  =

1˜ G, U

such that

' f = E(Td , Td , e, e)  G (Td , Td , e, e). 

(14.120)

Here ' f is partly bounded as

9> ?9 9 e 9 9   9 1G (Td , Td , e, e)1 9 e 9.

Theorem 14.14. Consider the tracking error system (14.118). Let V U V U O 1 1 e − D1 + u = \1 + 2q22 + + EE T y, O 2O 2

(14.121)

(14.122)

where for any given J > 0, \1 is chosen as in Theorem 14.13, and D1 and O are chosen such that 1 1 D1  2 + q22 + H2 , O > . 4J H1 Then, the closed-loop system is robustly stable when w = 0 and has L2 -gain less than J for all G . Proof. Note that the tracking error system is represented by J e = −e + y M(T )y = (M − C)(y − e) − E G + u + w.

(14.123)

Along the trajectories of the closed-loop system, H(e, y,t, w, z), dened by (14.115), is obtained as

426

14 L2 -Gain Synthesis

9 92 91 9 9 − yT E G H = −(1 − q21 − q22 )1e12 − 9 y − J w 9 2J 9 U > V ? 1 +yT u − \1 − 2q22 e + + q22 y . 4J 2

(14.124)

Furthermore, note that for any O > 0, we have −yT E G 

1 T O T T y EE y + G G. 2 2O

Hence, from (14.121), we obtain

F9> ?9 G 92 1 9 e  2 9 − 1G 1 9 H 2O 9 e 9 9> ?92 9 1 9 O T T 9 e 9 + y EE y + 9 2 2O e 9 > V ? U 1 T 2 2 +y u − \1 − 2q2e + + q2 y 4J 2 U U V > V 1 1 2 2 2 T 2  − 1 − q1 − q 2 − 1e1 + y u − \1 − 2q2 + e O O V ? U O 1 1 + + q22 + + EE T y . 4J 2 2O 2 −(1 − q21 − q22 )1e12 −

Note that D1 

(14.125)

+ q22 + H2 , then substituting (14.122) into this inequality yields U V V? > U 1 1 2 2 2 2 H  − 1 − q1 − q2 − 1e1 − D1 − + q2 1y12 O 4J 2 1 4J 2

 −H02 1e12 − H22 1y12 < 0,

∀e := 0, y := 0,

(14.126)

where H0 > 0 is a sufcient small number. Similar to the proof of Theorem 14.10, this inequality, which holds for all G , ensures the robust stability and L2 -gain constraint. 3 2 As discussed above, for a Lagrange system, the L2 -gain synthesis problem can be solved by using the physical energy function as the storage function to ensure the J -dissipativity. This approach can be also applied to Hamiltonian systems. Consider a Hamiltonian system x = [J(x) − R(x)]

wH (x) + g(x)(u + w), wx

(14.127)

where w ∈ Rm is an essentially bounded unknown disturbance such that the trajectory x(t) remain in D for any initial state x(0) ∈ D. For this system, we dene a penalty signal z = h(x)gT (x)

w Hc (x), wx

(14.128)

14.5 Applications

427

where gT wwHxc is a standard output of Hamiltonian system and h(x) (h(x0 ) = 0) is a weighting matrix. The design object is to nd a feedback control law u = k(x) such that when w = 0, the closed-loop system is asymptotically stable at a given desired equilibrium x0 ∈ Rn , and has L2 -gain less than J . Suppose that for the system (14.127) there exists a feedback u = D (x) that preserves Hamiltonian structure with a modied Hamiltonian function Hc (x) and a symmetric non-negative matrix Rc (x), i.e. (13.89) holds, and Hc admits a strict minimum at the desired equilibrium x0 . A method for seeking such a feedback law D (x) has been proposed by [4]. The following Theorem shows that in this case, the design object can be achieved by damping injection, i.e. the Hamiltonian function Hc can serve as the storage function for the closed-loop system if we insert a proper feedback. Theorem 14.15. Consider the system (14.127) with the penalty signal (14.128). For any given J > 0, let the feedback control law be given by ⎧ ⎨u = D (x) + E (x) (14.129) @ A ⎩E (x) = − 1 J −2 I + hT(x)h(x) gT (x) w Hc (x). 2 wx Then, the closed-loop system is asymptotically stable and has L2 -gain less than J . Proof. Note that, under the feedback (14.129), the closed-loop system with the modied Hamiltonian function Hc can be represented by ⎧ ⎪ ⎨x = [J(x) − Rc (x)] w Hc (x) + g(x)(E (x) + w) wx (14.130) ⎪ ⎩z = h(x)gT (x) w Hc (x). wx Along any trajectory of this system, a straightforward calculation gets

w T Hc w Hc w T Hc (x)Rc (x) (x) + (x)g(x)(E (x) + w) H c = − wx wx wx w T Hc w Hc w T Hc (x)Rc (x) (x) + (x)g(x) =− w x? Uw x > wx V 1 1 w Hc T T I + h (x)h(x) g (x) × E (x) + (x) 2 J2 wx 9 92 1 w Hc 9 19 9 + 1 (J 2 1w12 − 1z12). J w(t) − gT (x) − 9 29 J wx 9 2

(14.131)

Hence, by substituting E (x) into the right-hand side and setting the non-negative denite function w T Hc w Hc (x)Rc (x) (x), Q(x) = wx wx we have A 1@ 2 (14.132) J 1w12 − 1z12 , ∀w. H c + Q(x)  2

428

14 L2 -Gain Synthesis

This means that the Hamiltonian function Hc serves as the storage function for the closed-loop system. 2 3 Example 14.1. Recall Example 13.7. To reject the disturbance w, we introduce the penalty signal dened by (14.128) with the weighting function ⎤ ⎡ q1 (x1 − x1e ) (14.133) h(t) = ⎣q2 (x2 − x2e )⎦ , qi  0, i = 1, 2, 3. q3 (x3 − x3e ) Note that, the standard output is given by U V bL cT w T Hc (x) = bL (cos x1e − cosx1 ) + gT (x) (x3 − x3e ) . wx cL Hence, we have U V2 3 b L cT (x3 − x3e ) ¦ q2i (xi − xie)2 . 1z12 = bL (cos x1e − cosx1 ) + cL i=1

(14.134)

(14.135)

This implies that z(x) = 0, if and only if (x1 , x2 , x3 ) = (x1e , 0, x3e ) or bL (cos x1e − cosx1 ) +

bL cT (x3 − x3e ) = 0. cL

When w(t) vanishes, this equality means x3 = 0 and from the locally uniqueness of equilibrium, x1 = x1e and x3 = x3e . By the equation (13.93), x2 = −DM x2 . This leads, from the positivity of parameter DM , x2 → 0. Therefore, 1z12 represents the perturbation of state from the given equilibrium. The aim of control is the rejection of the state perturbation under the disturbance and the uncertainty. Our goal is as follows: For given desired equilibrium (x1e , 0, x3e ) and any given disturbance rejection level J > 0, nd a feedback control law such that the unforced closed-loop system has a stable equilibrium in xe , and has L2 -gain less that J > 0. As mentioned above, the system (13.93), forced by the constant input D (x) = u, ¯ has the Hamiltonian structure and the total energy function Hc admits a strict minimum at the desired equilibrium. Thus, applying Theorem 14.15 to the system gets the desired feedback law as follows ? > wH 1 1 T I + h (x)h(x) gT (x) u = u¯− (x) 2 J2 wx G F 3 1 1 2 2 + ¦ q i xi = cT x3e − cL cos x1e − 2 J 2 i=1 V U b L cT (x3 − x3e ) , (14.136) × bL (cos x1e − cosx1 ) + cL where w Hc (x) = [ bL x3 sin x1 − P x2 wx

bL (cos x1e − cosx1 ) +

bL cT (x3 − x3e ) ]. cL (14.137)

References

429

References 1. Anderson B, More J. Optimal Control: Linear Quadratic Methods. New Jersey: Prentic-Hall, 1989. 2. Byrnes C, Isidori A, Willems J. Passivity, feedback equivalence, and the global stabilization ofminimum phase nonlinear systems. IEEE Trans. Aut. Contr., 1991, 36(11): 1228–1240. 3. Doyle J, Glover K, Khatgonekar P, et al. State space solution to standard H2 and Hf control problems. IEEE Trans. Aut. Contr., 1989, 34(8): 831–847. 4. Ortega R, Van der Schaft A J, Maschke B, et al. Stabilization of port-controlled Hamiltonian systems: Passivation and energy-balancing// Stability and Stabilization of Nonlinear Systems. London: Springer, 1999. 5. Van der Schaft A. L2 -gain analysis of nonlinear systems and nonlinear state feedback Hf control. IEEE Trans. Aut. Contr., 1992, 37(6): 770–784. 6. Shen T, Tamura K, Nikiforuk P. Robust feedback design of cascaded nonlinear systems with structural uncertainty. Trans. IEE of Japan, 2000, 120-C(5): 692–698. 7. Shen T, Xie L, Tamura K. Constructive design approach to robust control of nonlinear systems with gain bounded uncertainty. Journla of SICE, 2000, 36(3): 242–247.

Chapter 15

Switched Systems

As the simplest hybrid system a switched system has many industrial backgrounds and engineering applications. Theoretically, it is also challenging: Switching adds complexity, and at the same time provides more freedom for control design. This chapter considers switched afne (control) systems. Section 15.1 investigates the problem of common quadratic Lyapunov function. It provides a tool for stability analysis and stabilization of switched linear systems. Section 15.2 gives a necessary and sufcient condition for quadratic stabilization of planar switched linear systems. Controllability of switched linear and bilinear control systems are studied in Sections 15.3 and 15.4 respectively. As an application, Section 15.5 considers the consensus of multi-agent systems.

15.1 Common Quadratic Lyapunov Function A switched (afne) nonlinear system is described as m

x = fV (t) (x) + ¦ giV (t) (x)ui ,

x ∈ Rn , u ∈ R m ,

(15.1)

i=1

where V (t) : [0, +f) → / is a piecewise constant right continuous function, called the switching signal, / is an index set for switching modes. Throughout this chapter we assume / = {1, 2, · · · , N} is a nite set, unless elsewhere stated. fi and gij , i = 1, · · · , N, j = 1, · · · , m, are smooth vector elds. When fi (x) = Ai x and gij (x) = bij , we have the following switched linear control system: m

x = AV (t) x + ¦ biV (t) ui := AV (t) x + BV (t)u,

x ∈ Rn , u ∈ Rm .

(15.2)

i=1

Particularly, as the controls are identically zero, we have a switched linear system as x = AV (t) x,

x ∈ Rn .

(15.3)

This section considers system (15.3) only. It is well known that even if each switching mode of (15.3) is stable, under certain switching law, system (15.3) could be unstable, and vise versa. The following example is from [3]. D. Cheng et al., Analysis and Design of Nonlinear Control Systems © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2010

432

15 Switched Systems

Example 15.1. Consider the following system x = AV (t) x,

x ∈ R2 ,

(15.4)

where V (t) ∈ {1, 2}, and ? −1 10 , −100 −1

> A1 =

? −1 100 . −10 −1

> A2 =

The two switching modes are stable. But if the switching law is as follows: in quadrants 2 and 4, V (t) = 1 and in quadrants 1 and 3, V (t) = 2, then the system is unstable. We refer to Fig. 15.1 for the trajectories. Fig. 15.1 (a) is the trajectory for the rst mode, starting from (1, 0); (b) is the trajectory for the second mode, starting from (0, 1); and (c) is the switching system, starting from (10−6 , 0). Taking −Ai , i = 1, 2 for switching modes, it is easy to see that the same switching law makes the switched system stable, though each mode is unstable.

Fig. 15.1

Unstability Caused by Switching

One way to assure the stability under arbitrary switching laws is a common Lyapunov function. For switched linear system (15.3) we consider a common quadratic Lyapunov function (QLF). Denition 15.1. 1. A matrix A is said to have a QLF, V (x) = xT Px, with P > 0, such that PA + ATP < 0.

(15.5)

P is briey called a QLF of A. If in addition P is diagonal, A is said to have a diagonal QLF. 2. P > 0 is a common QLF of {AO |O ∈ / }, if PAO + (AO )T P < 0,

O ∈ /.

(15.6)

The rst important result is the following: Theorem 15.1. [20] Given a set of stable matrices {A1 , · · · , AN }. Assume they are commutative pairwise. Then they share a common QLF.

15.1 Common Quadratic Lyapunov Function

433

Proof. Choose a positive denite matrix P0 and dene Pi > 0, i = 1, · · · , N, recursively by ⎧ P1 A1 + AT1P1 = −P0 , ⎪ ⎪ ⎪ ⎪ ⎨P2 A2 + ATP2 = −P1 , 2 (15.7) .. ⎪ ⎪ . ⎪ ⎪ ⎩ PN AN + ATN PN = −PN−1 . Then we claim that PN is a common QLF. It can be proved by mathematical induction: It is true for N = 1. Assume Pi,s := Ps Ai + ATi Ps < 0,

s  k, i  s.

If we can prove that Pi,k+1 = Pk+1 Ai + ATi Pk+1 < 0, we are done. Since Pi,k+1 Ak+1 + ATk+1 Pi,k+1 = (Pk+1 Ai + ATi Pk+1 )Ak+1 + ATk+1(Pk+1 Ai + ATi Pk+1 ) = (Pk+1 Ak+1 + ATk+1 Pk+1 )Ai + ATi (Pk+1 Ak+1 + ATk+1 Pk+1 ) = −(Pk Ai + ATi Pk ) > 0, and Ak+1 is stable, it follows that Pi,k+1 < 0,

∀i  k + 1.

3 2

Remark 15.1. One of the advantages of the above result is that its proof is constructive, which provides a way to construct the common QLF. Moreover, PN has a closed form expression as PN =

p f

T

eAN tN

0

p f 0

p f 0

T

eAN−1tN−1 · · ·

T

(15.8)

eA1 t1 P0 eA1t1 dt1 · · · eAN−1 tN−1 dtN−1 eAN tN dtN .

A more general result is the following: Theorem 15.2. [18] Given a set of stable matrices {AO | O ∈ / }. Assume the Lie algebra generated by this set of matrices, {AO | O ∈ / }LA , is solvable. Then they share a common QLF. Proof. It follows from the result in Lie algebra [15]: If a Lie algebra L ⊂ gl(n, R) is solvable, then the elements in L can be converted into upper triangular form simultaneously. Then the conclusion follows immediately. 3 2 Note that in Theorem 15.2, / could be an innite set. Next, we give a numerical approach to common QLF. The following lemma is very simple, yet it is the starting point of our new approach. Lemma 15.1. Assume a set of matrices {AO | O ∈ / } are stable, i.e., ReV (AO ) < 0, and there exists Ha common QLF.  ThenI there exists an orthogonal matrix T ∈ SO(n, R) such that A˜ O = T T AO T  O ∈ / have a common diagonal QLF.

434

15 Switched Systems

Proof. Let P be a common QLF of AO . Then there exists an orthogonal matrix T such that P˜ = T T PT is diagonal. Using it, we have T T (PAO + ATO P)T = P˜A˜ O + A˜ TO P˜ < 0.  I H Hence, A˜ O = T T AO T  O ∈ / have P˜ as their common diagonal QLF. Without loss of generality, we can assume det(T ) = 1, i.e., T ∈ SO(n, R). 3 2 According to Lemma 15.1, instead of searching a common QLF we can search a common diagonal QLF under a common orthogonal transformation on {AO }. Let )n be the set of n × n positive denite matrices, and ;n ⊂ )n be its diagonal subset. Then )n = {T DT T | T ∈ SO(n, R), D ∈ ;n }. Now since in searching a QLF, P ∼ kP, k > 0, (where “∼” stands for equivalence.) we may consider the equivalent classes. That is, we need only to consider the quotient set

0, · · · , xn−1 > 0}. Giving
(15.9)

and then we dene a mapping S : SO(n, R) × Rn−1 + → )n /R+ as (T, x) 6→ T diag(1, x1 , · · · , xn−1 )T T which is obviously a surjective mapping. So we can search the common QLF over SO(n, R) × Rn−1 + . For later discussion, it is not convenient to use the conventional topology of SO(n, R) for searching P. We turn to its Lie algebra. Let si, j ∈ o(n, R) be an element in the orthogonal algebra o(n, R), dened as si j (i, j) = −1;

si j ( j, i) = 1;

si j (p, q) = 0, (p, q) := (i, j) and (p, q) := ( j, i).

Note that the connected Lie group generated by si j , denoted by Si j , is a one dimensional sub-group of SO(n, R), and Si j ∼ = SO(2, R), where ∼ = stands for group isomorphism. Lemma 15.2. There exist L = such that

n(n−1) 2

one dimensional subgroups Si j < SO(n, R),

(S12 S13 · · · S1n )(S23 S24 · · · S2n ) · · · (S(n−1)n) = SO(n, R). Proof. “⊂” is trivial, we prove “⊃” only. Let

(15.10)

15.1 Common Quadratic Lyapunov Function

⎡ a11 ⎢a21 ⎢ : =⎢ . ⎣ ..

a12 · · · a22 · · · .. .

435

⎤

a1n a2n ⎥ ⎥ .. ⎥ ∈ SO(n, R). . ⎦

an1 an2 · · · ann We claim that there exist ei j ∈ Si j such that e(n−1)n e(n−2)n e(n−2)(n−1) · · · e1n e1(n−1) · · · e13 e12 : = I. If this is true, we are done. In fact, we can choose ei j to eliminate a ji of : iteratively. For instance, if a21 := 0, we can choose t ∈ [0, 2π) in such a way that sin(t) = n Let

a21 (a11

)2 + (a

21

)2

,

cos(t) = n

a11 (a11 )2 + (a21 )2

.

(15.11)

⎤ cos(t) sin(t) 0 e12 = exp(−s12t) = ⎣− sin(t) cos(t) 0 ⎦ . 0 0 In−2 ⎡

Then it eliminates a21 . If a21 = 0, we can simply choose e12 = I. Similarly, we can construct e1,i recursively to eliminate ai1 , i = 3, · · · , n. It follows that the resulting matrix is ⎡ ⎤ 1 0 ··· 0 ⎢0 a˜22 · · · a˜2n ⎥ ⎢ ⎥ ⎢ .. .. .. ⎥ . ⎣. . . ⎦ 0 a˜n2 · · · a˜nn Similar procedure converts : to I. Then

: = (E 12 E 13 · · · E 1n )(E 23 · · · E 2n ) · · · E (n−1)n, where E i j = (ei j )−1 ∈ Si j .

(15.12) 3 2

Remark 15.2. 1. Equation (15.10) is an equation in the sense of sets. That is, both sides are the same set. It is not a group isomorphism because o(n, R) is not commutative. 2. For numerical purpose, we need the form of E i j . It is identity out of i-th and j-th rows and columns, and its i − j minor is SO(2, R). So, (assume i < j) ⎡ ⎤ Ii−1 0 0 0 0 ⎢ 0 cos(t) 0 − sin(t) 0 ⎥ ⎢ ⎥ ij 0 I j−i−1 0 0 ⎥ E =⎢ ⎢ 0 ⎥. ⎣ 0 sin(t) 0 cos(t) 0 ⎦ 0 0 0 0 In− j Now we have a natural vector bundle structure induced by the natural projection: n(n−1) L [0, 2π)L × Rn−1 + → [0, 2π) , where L = 2 . Note that the base space of this bundle is a simply connected bounded “cubic” set. Moreover, Lemma 15.1 provides a mapping 3 : [0, 2π)L → SO(n, R) as

436

15 Switched Systems

3 (t1 , · · · ,tL ) = exp(s12t1 ) · · · exp(s1ntn ) · · · exp(s(n−1)ntL ).

(15.13)

It is easy to show that this mapping is surjective. In the rest of this section we will not distinct )n with )n /R+ , unless elsewhere stated. Based on Lemmas 15.1 and 15.2 one sees that we can give the topological structure of [0, 2π)L × Rn−1 + to the set of positive denite matrices (under equivalence). So in the future study we will search the common QLF over this set. Lemma 15.1 and 15.2 provide the foundation of the new technique for searching common QLF. The basic idea is: instead of nding the common QLFs directly, we search the set of diagonal QLFs on the ber space Rn−1 + , along the trajectory of AO , under the group action of SO(n, R). Now for every point on base space, we have to gure out its feasible diagonal QLFs, which are on the ber space. Fortunately, we can give a clear description of the structure of the set of QLFs on the ber space. Denition 15.2. Let x = (x1 , · · · , xn ) be a xed coordinate frame in Rn . A set of limits (L,U), with lower limits L and upper limits U described as: L = {L1 , L2 (x1 ), · · · , Ln (x1 , · · · , xn−1 )}, U = {U1 ,U2 (x1 ), · · · ,Un (x1 , · · · , xn−1 )}. ˜ U) ˜ is deduced as From (L,U), a non-negative set of limits (L, L˜ i (x1 , · · · , xi−1 ) = max{Li (x1 , · · · , xi−1 ), 0}, U˜ i (x1 , · · · , xi−1 ) = max{Ui (x1 , · · · , xi−1 ), 0},

i = 1, · · · , n.

For a given (L,U), the positive cube C(L,U) ⊂ Rn is dened as: x = (x1 , · · · , xn ) ∈ C(L,U), if and only if L˜ 1 < x1 < U˜ 1 , L˜ 2 (x1 ) < x2 < U˜ 2 (x1 ), .. . L˜ n (x1 , · · · , xn−1 ) < xn < U˜ n (x1 , · · · , xn−1 ).

(15.14)

In fact, C(L,U) is a domain of n-dimensional integration. It is the intersection of the domain bounded by (Li ,Ui ), i = 1, · · · , n and the rst quadrat. To describe the set of diagonal QLFs, we need the following fact, stated as a lemma: Lemma 15.3. Assume a matrix A has a diagonal QLF. Then its diagonal elements are all negative, i.e., aii < 0, i = 1, · · · , n. Proof. Let P = diag(p1 , · · · , pn ) be a diagonal Lyapunov function. It is easy to see that the diagonal elements of PA + ATP are {2p1 a11 , · · · , 2pn ann }. The positivity of P and the negativity of PA + ATP imply that aii < 0, i = 1, · · · , n. 3 2 Now we are ready to give a complete characterization of the set of QLF for a given stable matrix.

15.1 Common Quadratic Lyapunov Function

437

According to Lemma 15.1, we look for a T ∈ SO(n, R) such that T T AT has a diagonal QLF as [ = diag(1, x1 , · · · , xn−1 ). From Lemma 15.2 and the above argument we know that T = 3 (t), which is dened in (15.13). Our goal is to nd t and x (with (1, x1 , · · · , xn−1 ) ∈
2a11 (t) a12 (t) + a21(t)x1 .. .

⎤ a12 (t) + a21(t)x1 · · · a1n (t) + an1(t)xn−1 2a22 (t)x1 · · · a2n (t) + an2(t)xn−1 ⎥ ⎥ ⎥, .. .. ⎦ . .

a1n (t) + an1(t)xn−1 a2n (t) + an2(t)xn−1 · · ·

(15.15)

2ann(t)xn−1

with [ = diag(1, x1 , · · · , xn−1 ). To make the Q(t) negative denite, it is enough to show that the determinants of the principal minors of odd orders are negative and those of even orders are positive. We show that these requirements lead to a set of boundary functions of a cub in Rn−1 . According to Lemma 15.3, aii (t), i = 1, · · · , n should be negative. Otherwise we set all Uk (t, x) = Lk (t, x) = 0. Denote the k-th principle minor by Dk . For later use, we solve Uk and Lk carefully by induction. When k = 2, we have ? > a12 (t) + a21(t)x1 2a11 (t) D2 (t, x1 ) = . a12 (t) + a21(t)x1 2a22 (t)x1 Let det(D2 ) > 0. Denoting ' = (2a11(t)a22 (t)− a12(t)a21 (t))2 − (a12(t)a21 (t))2 , we have the following ⎧⎧ ⎨U1 (t) = +f, ⎪ ⎪ ⎪ ⎪ ⎪ a212 (t) ⎪ ⎩ ⎪ ; a21 (t) = 0; L (t) = 1 ⎪ ⎪ 4a11 (t)a22 (t) ⎪ J ⎪ ⎪ ⎪ ⎪ ⎨ U1 (t) = 0, L (t) = 0; a21 (t) := 0, '  0; ⎧ 1 √ ⎪ ⎪ ⎪⎪ (2a11(t)a22 (t) − a12(t)a21 (t)) + ' ⎪ ⎪ ⎪ ⎪ , ⎪ ⎨U1 (t) = ⎪ ⎪ (a21 (t))2 ⎪ ⎪ √ ⎪ ⎪ ⎪ ⎪ (2a11 (t)a22 (t) − a12(t)a21 (t)) − ' ⎪ ⎪ ⎩⎪ ; a21 (t) := 0, ' > 0. ⎩L1 (t) = (a21 (t))2 (15.16) Using the mathematical induction, we assume Uk−1 (t, x1 , · · · , xk−2 ) and Lk−1 (t, x1 , · · · , xk−2 ) have been dened. If there is some D  k − 1, such that

438

15 Switched Systems

{xD | LD (t, x1 , · · · , xt−1 ) < xD < UD (t, x1 , · · · , xt−1 )} is an empty set, then we set Us = 0 and Vs = 0, s  D . Otherwise, we denote Ek = (a1,k+1 (t), a2,k+1 (t), · · · , ak,k+1 (t))T , Fk = (ak+1,1 (t), ak+1,2 (t), · · · , ak+1,k (t))T . >

? Dk (t, x1 , · · · , xk−1 ) Ek + Fk xk Dk+1 (t, x1 , · · · , xk ) = . EkT + FkT xk 2ak+1,k+1 (t)xk A computation with some skill shows that Then

det(Dk+1 > ? ? > (t, x1?, · · · , xk ))> Dk Fk 2 Dk D k Ek Fk + 2 det T x + det T x . = det T Ek 0 Ek ak+1,k+1 (t) k Fk 0 k

(15.17)

Denote (15.17) simply by det(Dk+1 ) = ak (t, x1 , · · · , xk−1 )x2k + 2bk (t, x1 , · · · , xk−1 )xk +ck (t, x1 , · · · , xk−1 ).

(15.18)

We also denote ' k = b2k − ak ck . To get the coefcients of (15.17) we use the following simple fact: Let A be an n × n non-singular matrix and b, c ∈ Rn . Then ? > A b = − det(A)(cT A−1 b) + e det(A). det T c e Applying it to (15.17), a straightforward computation shows that ⎧ T −1 ⎪ ⎨ak (t, x1 , · · · , xk−1 ) = − det(Dk )(Fk Dk Fk ) bk (t, x1 , · · · , xk−1 ) = − det(Dk )EkT D−1 k Fk + det(Dk )ak+1,k+1 (t) ⎪ ⎩ ck (t, x1 , · · · , xk−1 ) = − det(Dk )EkT D−1 k Ek .

(15.19)

Assume k is odd. Then det(Dk ) < 0 and since Dk is negative denite, then ak  0 and ak = 0 if and only if Fk = 0. Note that ck = − det(Dk )EkT D−1 k Ek  0, and when Fk = 0, bk = det(Dk )ak+1,k+1 (t) > 0. Since we require det(Dk+1 ) > 0, then the boundary functions can be expressed as ⎧⎧ ⎪ ⎨Uk (t, x1 , · · · , xk−1 ) = +f, ⎪ ⎪ ⎪ −ck ⎪ ⎪ ⎪ ⎩Lk (t, x1 , · · · , xk−1 ) = ; Fk = 0; ⎪ ⎪ 2bk ⎪ J ⎪ ⎪ ⎪ ⎨ Uk (t, x1 , · · · , xk−1 ) = 0, (15.20) 0, ' k  0; ⎧Lk (t, x1 , · · · , xk−1 ) = 0; Fk := ⎪ √ ⎪ ⎪ −bk − ' k ⎪ ⎪ ⎪ ⎪ ⎨Uk (t, x1 , · · · , xk−1 ) = ⎪ , ⎪ ⎪ ak√ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪Lk (t, x1 , · · · , xk−1 ) = −bk + 'k ; Fk := 0, ' k > 0. ⎩ ⎩ ak

15.1 Common Quadratic Lyapunov Function

439

Assume k is even. Then det(Dk ) > 0 and since Dk is negative denite, then ak  0 and ak = 0 if and only if Fk = 0. Note that now ck  0, and when Fk = 0, bk < 0. Now we require det(Dk+1 ) < 0. Taking this into consideration, a formula can be obtained, which is very similar to (15.20), but with a positive sign before the square roots. Then a general formula, which covers (15.16), (15.20) and the case of even k, can be expressed in a unied form as ⎧⎧ ⎨Uk (t, x1 , · · · , xk−1 ) = +f, ⎪ ⎪ ⎪ ⎪ −ck ⎪ ⎪ ⎩Lk (t, x1 , · · · , xk−1 ) = ; Fk = 0; ⎪ ⎪ 2bk ⎪ J ⎪ ⎪ ⎪ ⎪ U (t, x1 , · · · , xk−1 ) = 0, ⎪ ⎪ k ⎨  0; ⎧Lk (t, x1 , · · · , xk−1 ) = 0; Fk := 0, k' (15.21) √k ⎪ −b + (−1) ' ⎪ ⎪ k k ⎪ ⎪ ⎨Uk (t, x1 , · · · , xk−1 ) = , ⎪ ⎪ ⎪ ak √ ⎪ ⎪ k ⎪ −bk − (−1) ' k ⎪ ⎪ ⎪ ⎪ ⎩Lk (t, x1 , · · · , xk−1 ) = ; Fk := 0, 'k > 0, ⎪ ⎪ ⎪ ak ⎪ ⎩ k = 1, 2, · · · , n − 1. To end the proof, we need only to show that (Lk ,Uk ) is a positive interval. We need only to prove it for the case ' k > 0. Observe that for either odd k or even k we always have (i). −bk /ak > 0, (ii). ak ck  0. It follows that (Lk ,Uk ) ⊂ R+ . Equation (15.21) provides a complete description for admissible t and x. We put the above argument into the following theorem: Theorem 15.3. Let A be a given n × n stable matrix. Then P is a QLF of A, if and only if there exists a set t = (t1 , · · · ,tL ) ∈ [0, 2π)L , a positive cube Ct (L,U) ⊂ )n such that P = T [ T T,

(15.22)

where [ = diag(1, x1 , · · · , xn−1 ), with (x1 , · · · , xn−1 ) ∈ Ct (L,U), T = 3 (t), which is dened by (15.13). The boundary functions (L(t),U(t)) are determined by (15.21). Now for a set of stable matrices {A1 , · · · , AN }, say the boundary functions are obtained as Uki (t, x1 , · · · , xk−1 ) and Lik (t, x1 , · · · , xk−1 ), for i = 1, · · · , N respectively, then we can dene J Uk (t, x1 , · · · , xk−1 ) = min1iN Uki (t, xi , · · · , xk−1 ) (15.23) Lk (t, x1 , · · · , xk−1 ) = max1iN Lik (t, xi , · · · , xk−1 ). Summarizing the above argument, we have the following result: Corollary 15.1. A set of matrices {A1 , · · · , AN } have a common QLF, if and only if there exists a t ∈ [0, 2π)L such that Uk (t, x1 , · · · , xk−1 ) > xk > Lk (t, x1 , · · · , xk−1 ), have a solution x = (x1 , · · · , xn−1 ).

k = 1, · · · , n − 1

(15.24)

440

15 Switched Systems

Since the set of common QLF is an open set (under the inherited topology of [0, 2π)L × Rn−1 + ), if it is not empty it is a set of positive measure. This fact leads to the following necessary and sufcient condition. Theorem 15.4. A set of stable matrices and only if p 2π 0

dt1 · · ·

p 2π 0

dtL

p Vn−2 (t,x1 ,···,xn−3 ) Ln−2 (t,x1 ,··· ,xn−3 )

p V1 (t) L1 (t)

dxn−2

dx1

p V2 (t,x1 ) L2 (t,x1 )

dx2 · · ·

p Vn−1 (t,x1 ,··· ,xn−2 ) Ln−1 (t,x1 ,··· ,xn−2 )

(15.25) dxn−1 > 0,

where Vk (t, x1 , · · · , xk−1 ) = max{Uk (t, x1 , · · · , xk−1 ), Lk (t, x1 , · · · , xk−1 )}, k = 1, · · · , n − 1. Remark 15.3. 1. When Fk := 0 for at least one Ai (1  i  N), k = 1, · · · , n − 1, the integral region is a compact set. It is easy to integrate (15.25) numerically. 2. The integrand of (15.25) is always non-negative and we don’t need the precise value of the integral, so for numerical integration the grid may be chosen very rough. If we do get a positive value, we are done. Otherwise, we may rene the grid and integrate again. To verify the quadratic stability of a switching system it is enough to know the existence of a common QLF for switching modes without guring out a particular QLF. So Theorem 15.4 is enough for this purpose. The formulas obtained above can be used to get the set of QLFs for a given stable matrix or the set of common QLFs for a given nite set of stable matrices. We use the following example to describe the verication and searching procedure. Example 15.2. Consider two stable matrices: ⎤ ⎡ ⎤ ⎡ −3 0.1 0 −1 0 0.2 A = ⎣ 0 −4 2 ⎦ , B = ⎣ 1 −2 1 ⎦ . −1 3 −2 1 −3 1 First, we can answer whether A and B share a common QLF. Using Theorem 15.4, we check p 2π

p 2π

dt1

0

p 2π

dt2

0

0

dt3

p V1 (t) L1 (t)

(V2 (t, x1 ) − L2 (t, x1 ))dx1 .

(15.26)

Using mid-point numerical integration and choosing the step lengths

Gt1 = Gt2 = Gt3 =

2π , 40

Gx1 =

V 1 − L1 , 40

the integral is obtained as p 2π 0

p 2π

dt1

0

p 2π

dt2

0

p V1 (t)

dt3

L1 (t)

(V2 (t, x1 ) − L2 (t, x1 ))dx1 = 1.0811 × 105 > 0.

15.1 Common Quadratic Lyapunov Function

441

Hence, A and B do share a common QLF. Next, we may give a precise description for the set of common QLFs. Since n = 3. A basis of o(3, R) consists of ⎤ ⎤ ⎤ ⎡ ⎡ ⎡ 0 10 0 01 0 0 0 g1 = ⎣−1 0 0⎦ , g2 = ⎣ 0 0 0⎦ , g3 = ⎣0 0 1⎦ . 0 00 −1 0 0 0 −1 0 We describe the set of QLFs in the following procedure: Step 1. For t ∈ [0, 2π)3 , using (15.13) to construct T (t) = exp(g1t1 ) exp(g2t2 ) exp(g3t3 ),

t = (t1 ,t2 ,t3 ) ∈ [0, 2π)3 .

(15.27)

Step 2. Get the trajectory passing through A as A(t) = T (t)T AT (t) = (ai j (t)).

(15.28)

Step 3. Check whether the diagonal elements are negative, i.e., a11 (t) < 0, a22 (t) < 0, a33 (t) < 0? If the answer is ”yes”, continue. Otherwise, go to the next t. Step 4. For k = 1, 2, using (15.19) to get ak , bk , ck , and 'k . Then using them and the formula (15.21) to get L1 , U1 and L2 , U2 respectively, which provide the set of QLFs. Step 5. Using (15.23) to get a common solution for the given set of matrices. A program can be created to search all possible common QLFs for them. We 4π report some results in the following table, where D0 = 0, D1 = 2π 3 , D2 = 3 . If a node t = (t1 ,t2 ,t3 ) (i.e. (D1 , D1 , D1 )) doesn’t appear in the following table, it means the feasible P for this t doesn’t exist. Denote by diag(1, x, y) ∈ <3 . Ix is for interval of feasible x. x0 ∈ Ix is chosen from Ix arbitrarily. Table 15.1

Set of common QLFs for A and B

t1

t2

t3

D1 D1 D1 D1 D2 D2 D2 D2 D2

D2 D2 D2 D2 D1 D1 D1 D1 D1

D2 D0 D0 D0 D2 D2 D0 D0 D0

Ix = (L1 (t),U1 (t)) (8.5086, 21.9110) (4.6019,9.8366) (4.6019,9.8366) (4.6019,9.8366) (2.0327,99.6470) (2.0327,99.6470) (0.4514,8.9252) (0.4514,8.9252) (0.4514,8.9252)

x0 ∈ Ix 10.7423 5.4744 7.2192 8.9641 18.3018 50.8399 1.8637 4.6883 7.5129

Iy (t, x0 ) = (L2 (t, x0 ),U2 (t, x0 )) (0.6085,4.0012) (22.1367,106.9738) (13.6341,139.1501) (11.2268,145.9138) (1.5330,10.9937) (3.0717,11.0326) (4.2069,26.0945) (4.3655,30.7806) (4.6438,15.8890)

Finally, we may check a particular point: t1 = D1 =

2π , 3

t2 = D2 =

4π , 3

t3 = D2 =

4π . 3

From Table 15.1 the lower boundary L1 = 8.5086, the upper boundary U1 = 21.9110, a feasible x is chosen as x0 = 10.7423. For this x0 we have L2 (x0 ) = 0.6085, and U2 (x0 ) = 4.0012. Now we may choose a point y = 2.3049 ∈ Iy . Then the diagonal QLF becomes

442

15 Switched Systems

D = diag(1.0000, 10.7423, 2.3049). The rotating matrix T ∈ SO(3, R) is

⎤ 0.2500 −0.0580 −0.9665 T = exp(t1 g1 ) exp(t2 g2 ) exp(t3 g3 ) = ⎣0.4330 0.8995 0.0580 ⎦ . 0.8660 −0.4330 0.2500 ⎡

Then it produces a common QLF for original systems as ⎤ ⎡ 2.2517 −0.5816 −0.0706 P = T DT T = ⎣−0.5816 8.8872 −3.7757⎦ . −0.0706 −3.7757 2.9082 Finally, it is easy to verify that ⎡

⎤ −4.6446 −0.6563 2.1955 PA + ATP = ⎣−0.6563 −48.4435 20.2607 ⎦ < 0, 2.1955 20.2607 −9.3147 ⎤ ⎡ −14.5323 15.5842 −6.9127 PB + BTP = ⎣ 15.5842 −58.3198 32.7079 ⎦ < 0. −6.9127 32.7079 −19.1845 We claim that Theorem 15.3 remains true for the set of complex matrices. It is used later when we consider the block upper triangular matrices. As pointed in [18], we know that Lemma 15.4. All the leading principal minors of a Hermitian matrix are real. A Hermitian matrix H is positive denite if and only if all its leading principal minors are positive. Corollary 15.2. Theorem 15.3 remains true when the set of matrices are complex matrices. Proof. Now for complex case (15.22) becomes Q(t) := [ At + At∗ [ = ⎡ a11 (t) + a11(t) a12 (t) + a21(t)x1 ⎢a12 (t) + a21(t)x1 (a22 (t) + a22(t))x1 ⎢ ⎢ .. .. ⎣ . . a1n (t) + an1xn−1 a2n (t) + an2(t)xn−1

··· ···

a1n (t) + an1(t)xn−1 a2n (t) + an2(t)xn−1 .. .

···

(ann (t) + ann(t))xn−1

⎤ ⎥ ⎥ ⎥, ⎦ (15.29)

According to Lemma 15.4, right side of (15.18) is a real number with arbitrary xk . It is easy to see the fact that, for a complex polynomial p(x), if for any x ∈ R, p(x) is a real number, then all the coefcients of p(x) are real. Then p(x) itself is a real polynomial. Because let x be large enough, it can bee seen that the highest degree term with imaginary coefcient turns to be real, which leads to a contradiction. We conclude that even for complex case, the right side of (15.18) remains a real quadratic form. So the proof of Theorem 15.3 remains true. 3 2 Next, we will specify our general necessary and sufcient condition to the planar case. Now the orthogonal transformations T ∈ SO(2, R) can be expressed as

15.1 Common Quadratic Lyapunov Function

? cos(t) − sin(t) , Tt = sin(t) cos(t)

443

>

>

Consider a stable matrix A=

0  t < 2π.

(15.30)

? DE . J G

According to Lemma 15.3, we rst consider when A(t) = TtT ATt ,

0t <π

has negative diagonal elements. The reason for considering only half interval, [0, π), will be explained later. For notational ease, set S = sin(t), C = cos(t), S2 = sin(2t), C2 = cos(2t). A straightforward computation shows that ? ?> ?> > C S D E C −S A(t) = −S C J G S C (15.31) ? > 2 D C + (E + J )SC + G S2 E C2 + (G − D )SC − J S2 . = J C2 + (G − D )SC − E S2 G C2 − (E + J )SC + D S2 Using some trigonometric equalities, (15.31) becomes ⎡ ⎤ D +G D −G E +J E −J E +J G −D + + + C S + C S 2 2 2 2⎥ ⎢ 2 2 2 2 2 2 ⎢ ⎥. ⎣ J −E J +E G −D D +G G −D E +J ⎦ + C2 + S2 + C2 − S2 2 2 2 2 2 2 Setting a=

D +G , 2

we have

b= >

A(t) =

D −G , 2

c=

E +J , 2

d=

E −J , 2

? a + bC2 + cS2 d + cC2 − bS2 . −d + cC2 − bS2 a − bC2 − cS2

(15.32)

(15.33)

Lemma 15.3 provides a region for possible solutions. We rst consider the region of t, which assures the diagonal elements being negative. To begin with, we consider a special case. Case 1. Assume b = c = 0: Then > ? a d A(t) = A = . (15.34) −d a Since a = 12 tr(A) < 0, TtT ATt has obviously negative diagonal elements. √ Otherwise, let r = b2 + c2 := 0 and dene P ∈ [0, 2π) by c cos(P ) = , r Then (15.33) becomes

b sin(P ) = , r

0  P < 2π.

444

15 Switched Systems

>

? a + r sin(2t + P ) d + r cos(2t + P ) A(t) = . −d + r cos(2t + P ) a − r sin(2t + P )

(15.35)

Case 2. r < −a: It is clear that any t makes the diagonal elements negative. Note that case 1 is a particular case of case 2. To make expression (15.35) covering case 1, when r = 0 we dene P = 0. Case 3. r  −a: Now to assure the negativity of diagonal elements we need r| sin(2t + P )| < −a.

(15.36)

In fact, (15.36) covers all three cases 1–3. Only in cases 1 and 2 it is satised automatically. Denote by

4 = {t | 0  t < π, r| sin(2t + P )| < −a}. In case 3, (15.36) means U V U V −1 |a| −1 |a| kπ + sin kπ − sin −P −P r r
k ∈ Z.

(15.37)

Note that 0  t < π, it is easy to see that the solution of t consists of 2 or 3 open intervals. Summarizing the above argument we have the following ? > DE . When r  −a, the diagonal Proposition 15.1. Given a stable matrix A = J G elements of A(t) = TtT ATt , 0  t < π are negative, if and only if t satises (15.37). When r < −a, the diagonal elements are always negative. Remark 15.4. From (15.33) one sees that we do not need to consider whole 0  t < 2π. It is enough to consider the problem only for 0  t < π. Note that Proposition 15.1 provides the set of t, which assures that the rotated matrix, A(t), has negative diagonal elements. In fact, this set has a decisive meaning. Later on, we will prove that it is exactly the set for the corresponding rotated matrix to have diagonal QLFs. Now we can start to search the diagonal QLFs P(x) = diag(1, x), where x > 0. For notational ease, let RS = r sin(2t + P ) and RC = r cos(2t + P ). Then P(x)A(t) + AT(t)P(x) ? > 2(a + RS) (−d + RC)x + (d + RC) . = (−d + RC)x + (d + RC) 2(a − RS)x. Dene

(15.38)

D(t, x) = det[P(x)A(t) + AT(t)P(x)].

Now nding a QLF, P, is equivalent to nding t ∈ 4 and x > 0 such that D(t, x) > 0. It is equivalent to

15.1 Common Quadratic Lyapunov Function

−D(t, x) = E(t)x2 + 2F(t)x + G(t) < 0,

445

(15.39)

where E(t) = (RC − d)2 ; F(t) = (RC)2 − d 2 + 2(RS)2 − 2a2; G(t) = (RC + d)2 . Since t ∈ 4 , (15.38) is negative denite, if and only if there exists x > 0 such that (15.39) is satised. Observing (15.39), it is obvious that F(t) < 0 is a necessary condition for the existence of x > 0. Fortunately, for t ∈ 4 this condition can be satised automatically. Lemma 15.5. When t satises (15.36), F(t) = (RC)2 − d 2 + 2(RS)2 − 2a2 < 0. Proof. Since

F(t) = r2 + r2 sin2 (2t + P ) − d 2 − 2a2, then F < 0 is equivalent to sin2 (2t + P ) < We claim that

d 2 + 2a2 − r2 . r2

(15.40)

a2 d 2 + 2a2 − r2 > 2. 2 r r

If the claim is correct, (15.36) implies (15.40). To prove the claim we have only to show that d 2 + a2 − r2 > 0. Recalling the denition of (15.32), we have 1 [(D + G )2 + (E − J )2 − (D − G )2 − (E + J )2 ] 4 = [DG − E J ] = det(A) > 0.

(15.41)

The last inequality is from the fact that A is stable, hence det(A) > 0.

3 2

d 2 + a2 − r 2 =

Next, we look for the feasible x for given t. Start from a special case. Case 1. Assume r = 0: This is a very special case when the rotations do not affect matrix A. It is obvious that (15.34) is satised for any t. In this case we have E = d2;

F = −d 2 − 2a2;

G = d2.

Case 1.1, d = 0: In this case for every t, any x > 0 works. In fact, it is easy to see that now the matrix A = D I, with D < 0. Case 1.2, d := 0: We have d 2 x2 − 2(d 2 + 2a2 )x + d 2 < 0. It leads to the boundary of the solutions as Y a Z2 U 2|a| V mY a Z2 Y a Z2 U 2|a| V mY a Z2 1+2 − +1 < x < 1+2 + + 1. d |d| d d |d| d (15.42) It is easy to see that (15.42) denes a non-empty interval. We have to prove that it is a positive interval. In fact

446

15 Switched Systems

1+2

Y a Z2 d

m U Y a Z2 V Y a Z2 Y a Z2  a    + = 1+ > 2 1+  . d d d d

Hence all the solutions x are positive. Case 2. r := 0. Case 2.1, assume (RC − d) = 0: Then using Lemma 15.5, the feasible x should satisfy x>

(RC + d)2 . −2F

(15.43)

Note that such x is automatically positive. Hence (15.43) is also sufcient. Case 2.2, assume (RC − d) := 0: We rst claim that ' > 0, that is F 2 > ((RC)2 − d 2 )2 .

(15.44)

Recall that RS = r sin(2t + P ) and RC = r cos(2t + P ). Hence (RS)2 + (RC)2 = r2 . Then F 2 − ((RC)2 − d 2 )2 = (F + (RC)2 − d 2 )(F − (RC)2 + d 2 ) = 4(a2 + d 2 − r2 )(a2 − r2 sin2 (2t + P )) > 0. The last inequality comes from (15.41) and (15.36). Now the feasible x satisfying (15.39) should satisfy the following well dened inequalities: n n −F − F 2 − ((RC)2 − d 2 )2 −F + F 2 − ((RC)2 − d 2 )2 <x< . (15.45) (RC − d)2 (RC − d)2 Again, it is easy to see that (15.45) denes a non-empty positive interval. So it is also sufcient.  We ignore the case 1.1, i.e., A = D I, with D < 0, because in this case every P > 0 is its QLF. Looking for a common QLF of a set of matrices, such matrices can simply be ignored. Using (15.42), (15.43) and (15.45), we dene two boundary functions as ⎧ o ⎧ U V2 U V U V2 ⎪ ⎪ ⎪ ⎪ a 2|a| a ⎪ ⎪ ⎪ ⎪ 1+2 − + 1, r = 0, ⎪ ⎪ ⎪ ⎪ d |d| d ⎪ ⎪ ⎨ ⎪ ⎪ 2 ⎪ ⎪ L(t) = (RC + d) , r > 0, RC = d, ⎪ ⎪ ⎪ ⎪ ⎪ −2Fn ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −F − F 2 − ((RC)2 − d 2 )2 ⎪ ⎪ ⎪ ⎩ , r > 0, RC := d; ⎪ ⎨ (RC − d)2 ⎪ ⎪ o ⎧ ⎪ U V2 U V U V2 ⎪ ⎪ ⎪ ⎪ 2|a| a a ⎪ ⎪ ⎪ ⎪ + + 1, r = 0, 1+2 ⎪ ⎪ ⎪ ⎪ ⎨ d |d| d ⎪ ⎪ ⎪ ⎪ U(t) = +f, r > 0, RC = d, ⎪ ⎪ ⎪ ⎪ n ⎪ ⎪ ⎪ ⎪ ⎪ −F + F 2 − ((RC)2 − d 2 )2 ⎪ ⎪ ⎪ ⎩ ⎩ , r > 0, RC := d, t ∈ 4. (RC − d)2 (15.46)

15.1 Common Quadratic Lyapunov Function

447

Summarizing the above argument, the main result for planar systems is obtained as Theorem 15.5. For each t ∈ 4 , there exists an open non-empty interval It = (L(t),U(t)) ⊂ (0, +f), such that P = diag(1, x) is a diagonal QLF of At , if and only if x ∈ It . The L(t) and U(t) are determined by (15.46). Backing to the original coordinate frame, for a given stable 2 × 2 matrix A, the set of QLFs can be described completely as:  b > ? a 1 0 T  (15.47) p > 0, t ∈ 4 , and L(t) < x < U(t) . T P = pTt 0x t  Example 15.3. Let

? > −3 −2 . A= 4 1

We will nd the set of QLFs for it. The parameters are calculated as √ a = −1, b = −2, c = 1, d = −3, r = 5. Then

1 cos(P ) = √ , 5

2 sin(P ) = − √ ⇒ P = 5.1760. 5 U V −1 |a| = 0.4636. T = sin r

To satisfy (15.36), it is easy to get that

4 = {t | 0.3218 < t < 0.7854, or 1.8925 < t < 2.3562}. Now equation (15.46) provides all the corresponding diagonal QLFs over 4 (up to a constant positive coefcient), with the form as P = diag(1, x). Fig. 15.2 shows the set of QLFs of A. Observing the set 4 in Example 15.3, one sees that two component intervals are π/2 displaced. In fact, this is generally true. In the following we simply use (t, x) ∈ [0, π) × (0, +f) to represent a class of QLFs. That is > ?> ?> ? cos(t) − sin(t) 1 0 cos(t) sin(t) (t, x) ∼ . sin(t) cos(t) 0 x − sin(t) cos(t) Proposition 15.2. If (t, x) is a feasible QLF with t < π/2, then (t + π/2, 1/x) is also a feasible QLF. Conversely, if (t, x) is a feasible QLF with t  π/2, then (t − π/2, 1/x) is also a feasible QLF. Proof. We prove the rst half statement. The second part is similar. Since SO(2, R) is Abelian, it is easy to see that

448

15 Switched Systems

Fig. 15.2

Set of QLFs of A

? ?> ?> > cos(t + π/2) sin(t + π/2) cos(t + π/2) − sin(t + π/2) 1 0 0 x − sin(t + π/2) cos(t + π/2) sin(t + π/2) cos(t + π/2) ?> ? ?> > cos(t) − sin(t) cos(π/2) − sin(π/2) 1 0 = 0x sin(π/2) cos(π/2) sin(t) cos(t) ? ?> > cos(t) sin(t) cos(π/2) sin(π/2) × − sin(π/2) cos(π/2) − sin(t) cos(t) ?> ?> ? > cos(t) − sin(t) x 0 cos(t) sin(t) = . sin(t) cos(t) 0 1 − sin(t) cos(t) Now since

the conclusion follows.

>

? > ? x0 1 0 ∼ , 01 0 1/x 3 2

This proposition tells us that to search the common QLFs we have only to search over [0, π/2). Using a, b, c, d, and r as in above, the set 4 can be precisely described as follows. Assume r < |a|. Then 4 = [0, π/2). Otherwise, we rst calculate P as U V ⎧ −1 |b| ⎪ ⎪ sin , b  0, c  0; ⎪ ⎪ r ⎪ ⎪ U V ⎪ ⎪ ⎪ −1 |b| ⎪ ⎪ , b  0, c < 0; π − sin ⎨ r U V (15.48) P= ⎪ ⎪ −1 |b| ⎪ π + sin , b < 0, c < 0; ⎪ ⎪ r ⎪ ⎪ U V ⎪ ⎪ ⎪ |b| ⎪ ⎩2π − sin−1 , b < 0, c  0. r Then we can get the feasible region of the rotations, which assure that the rotated A(t) has diagonal QLFs. Set

15.1 Common Quadratic Lyapunov Function

D1 = sin−1

U

449

V

|a| , r

D2 = π − D1,

D3 = π + D1 ,

D4 = 2π − D1 .

Then

Vb aU Vb ⎧a> D1 − P D2 − P π ⎪ ⎪ ∪ , , P < D1 , 0, ⎪ ⎪ 2 2 2 ⎪ ⎪ aU ⎪ Vb ⎪ ⎪ D2 − P D3 − P ⎪ ⎪ , , D1  P < D2 , ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ a> Vb aU Vb ⎨ D3 − P D4 − P π 0, ∪ , , D2  P < D3 , 4= ⎪ 2 2 2 ⎪ ⎪ aU Vb ⎪ ⎪ D4 − P 2π + D1 − P ⎪ ⎪ , , D3  P < D4 , ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ a> Vb aU Vb ⎪ ⎪ 2π + D1 − P 2π + D2 − P π ⎪ ⎪ ⎩ 0, ∪ , , D4  P < 2π. 2 2 2 (15.49)

Now the problem of common QLF for a set of matrices becomes very simple. To avoid topological complexity, we assume the set / is nite. So {AO } = {A1 , A2 , · · · , AN }. Using (15.49) we can construct the feasible set of t for each matrix, as 4k . Moreover, over each 4k the boundary functions Lk (t) and Uk (t) as in (15.46) are dened. Then we construct the common feasible set as

4 = ∩Nk=14k ⊂ [0, π/2). We know that it consists of only nite intervals. Then for each t ∈ 4 , dene L(t) = max Lk (t), 1kN

U(t) = min Uk (t), 1kN

V (t) = max{U(t), L(t)}.

Summarizing the above argument, we have Theorem 15.6. The set of stable 2 × 2 share a common QLF, if and only if p t∈4

(V (t) − L(t))dt > 0.

(15.50)

Remark 15.5. In fact, 4 , L(t), and V (t) describe the set of common QLFs completely. P is a common QLF, if and only if there exist 4 , x ∈ (L(t),V (t)) such that P ∼ (t, x) or P ∼ (t + π/2, 1/x). Example 15.4. Consider three matrices: ? ? > > −2 3 −4 −1 , , B= A= 1 −2 2 0.1

> C=

? −5 2 . 3 −2

(15.51)

We skip the tedious elementary computation and give the domains of A(t), B(t), and C(t) respectively as

4A = (0.0764, 1.2552), 4B = [0, π/2),

4C = [0, π/2).

450

15 Switched Systems

So

4 = 4A ∩ 4B ∩ 4C = (0.0764, 1.2552). It is easy to calculate the integration as p

t∈4

(V (t) − L(t))dt = 0.2524 > 0.

So A, B, C share a common QLF. Fig. 15.3 shows the set of common QLFs.

Fig. 15.3

Common QLFs for A, B, and C

From Fig. 15.3, it is easy to nd out a common QLF. Say, (t, x) = (0.3π, 0.5) is obviously in the feasible region. Hence, we can choose ? ?> ?> > cos(0.3π) sin(0.3π) cos(0.3π) − sin(0.3π) 1 0 P= 0 0.5 − sin(0.3π) cos(0.3π) sin(0.3π) cos(0.3π) ? > 0.6727 0.2378 . = 0.2378 0.8273 Then it is easy to verify that

? −4.4309 0.0545 < 0, 0.0545 −0.3101 ? > −2.2155 1.8944 < 0, PB + BTP = 1.8944 −1.8824 > ? −5.3009 2.1629 < 0. PC + CT P = 2.1629 −2.3580 Its dual QLF is (0.3π + 0.5π, 1/0.5) = (0.8π, 2), which yields ? > 1.3455 0.4755 . P= 0.4755 1.6545 >

PA + ATP =

It is easy to verify that this P is also a common QLF of A, B, and C.

15.1 Common Quadratic Lyapunov Function

451

Meanwhile, you can easily verify that outside the feasible region P ∼ (t, x) can not be a common QLF. For instance, choose T = 0.2π and x = 0.5. Then ? > 0.8273 0.2378 . P= 0.2378 0.6727 ? > For this P −2.8335 2.2035 T , PB + B P = 2.2035 −1.2644 which is not negative denite. Finally, we show that when a set of matrices have the same block upper-triangular form, the complexity of the verication will be reduced tremendously. The main result is the following Theorem 15.7. Assume a nite set of block triangular (complex) matrices with same diagonal block structure as ⎡ i ⎤ A11 Ai12 · · · Ai1n ⎢ ⎥ ⎢ 0 Ai22 · · · Ai2n ⎥ ⎥ (15.52) Ai = ⎢ ⎢ .. .. .. ⎥ , i = 1, · · · , N, ⎣ . . . ⎦ 0 0 · · · Ainn where the same k-th diagonal blocks Aikk have same dimensions for all i. Then Ai , i = 1, · · · , N, share a common QLF, if and only if, for every k, the diagonal blocks {Aikk | i = 1, · · · , N) share a common QLF. Proof. Without loss of generality, we have only to prove it for n = 2. Then by mathematical induction we can prove it for any n for both necessity and sufciency. (Sufciency) Denote by > ? Xi Yi Ai = , i = 1, · · · , N. (15.53) 0 Zi Assume dim(Xi ) = p and dim(Zi ) = q. Let P and Q be the common QLF of {Xi } and {Zi } respectively. Since Ui := −(PXi + XiT P) > 0, i = 1, · · · , N. There exists a positive real number G > 0, such that all the eigenvalues of Ui are greater than G . Similarly, letting Vi := −(QZi + ZiT Q) > 0, i = 1, · · · , N, all the eigenvalues of Vi are greater then some positive H > 0. We claim that for large enough P > 0, W = diag(P, P Q) is a common QLF of {Ai }. Calculate ? > −Ui PYi T Hi := WAi + Ai W = ∗ . Yi P −P Vi To show Hi < 0, choose [ ∈ C p and K ∈ Cq . Then > ? [ [[ ∗ K ∗ ]Hi = −[ ∗Ui [ + [ ∗ PYi K + K ∗Yi∗ P[ − PK ∗Vi K K 1  −[ ∗Ui [ + G 1[ 12 + K ∗ (Yi∗ P2Yi )K − PK ∗Vi K G> ? 1 = [ ∗ (−Ui + G I p )[ + K ∗ − P Vi + (Yi∗ P2Yi ) K . G

(15.54)

452

15 Switched Systems

Choosing

P>

1 max 1Y ∗ P2Yi 1, G H 1iN i

then it is obvious that (15.54) is less than or equal to zero, and it equals zero, if and only if [ = 0 and K = 0. (Necessity) Assume Ai , i = 1, · · · , N share a common QLF, say P. According to > ? P11 P12 . Then the structure of Ai we split P as P21 P22 > ? P X + Xi∗ P11 × < 0, PAi + A∗i P = 11 i × × where × stands for some uncertain elements. Then P11 Xi + XiT P11 < 0, which means P11 is the common QLF of Xi . > ? H11 H12 Let H = = P−1 > 0. Then H21 H22 H(PAi + A∗i P)H < 0, which leads to Ai H + HA∗i = It is easy to see that

>

? × × < 0. × H22 Zi∗ + Zi H22

−1 −1 H22 Zi + Zi∗ H22 < 0.

−1 is a common QLF of {Zi }. That is, H22

3 2

Denition 15.3. A sequence of nested subspaces of Rn , V1 ⊂ V2 ⊂ · · · ⊂ Vs = Rn , is called a “quasi-ag”. A matrix A is said to be stable with respect to this quasi-ag, if AVi ⊂ Vi , i = 1, · · · , s. When s = n and dim(Vi ) = i, i = 1, · · · , n the quasi-ag is called a ag [15]. The following result is an immediate consequence of the above theorem. Let {A1 , A2 , · · · , AN } be a set of stable matrices. If they are stable with respect to a quasi-ag, V1 ⊂ V2 ⊂ · · · ⊂ Vs = Rn , then we can choose a basis, (e1 , e2 , · · · , en ), of Rn , such that V1 = Span{e1 , · · · , en1 }, V2 = Span{en1 +1 , · · · , en2 }, etc., which is stable with respect to (Ak , k = 1, · · · , N). Consider Ak as the endomorphism on Rn . Then under the basis each Ak can be expressed in a block upper triangular form as ⎤ ⎡ k k A11 A12 · · · Ak1s ⎥ ⎢ ⎢ 0 Ak · · · Ak ⎥ 22 2s ⎥ ⎢ (15.55) Ak = ⎢ . . .. ⎥ , k = 1, · · · , N. . . ⎣ . . . ⎦ 0 0 · · · Akss Corollary 15.3. Assume a set of stable matrices {A1 , A2 , · · · , AN } are stable with respect to a quasi-ag, described as in above. {Ak , k = 1, · · · , N} share a common

15.1 Common Quadratic Lyapunov Function

453

QLF, if and only if each set of {Akj j , k = 1, · · · , N} share a common QLF for j = 1, · · · , s. It is obvious that the result of [18] is a particular case of the Corollary 15.3. The following example shows the application of the above corollary. Example 15.5. Let

⎤ 1 −1 0 A = ⎣16 −7 4⎦ , 7 −2 2 ⎡

⎤ 5 −2 4 B = ⎣ −9 0 −3 ⎦ . −19 5 −12 ⎡

It is easy to nd a common eigenvector as X = (1, 2, −1)T. Then we may choose a linear transformation T with X as its rst column. Say, ⎤ ⎡ 1 0 −1 T = ⎣ 2 2 −1⎦ . −1 1 1 ⎤ ⎤ ⎡ ⎡ Then −1 0 1 −3 1 −1 T −1 AT = ⎣ 0 −4 −3⎦ , T −1 BT = ⎣ 0 −2 3 ⎦ . 0 2 1 0 1 −2 According to Corollary 15.3, if two minors > ? > ? −4 −3 −2 3 C= , D= 2 1 1 −2 have a common QLF then so do A and B. Fig. 15.4 shows that C and D do have a common QLF. Taking parameters (0.85π, 8) (as the dual of (0.35π, 0.125)), we have a common QLF of C and D ? > as 2.4428 2.8316 . P0 = 2.8316 6.5572

Fig. 15.4

Common QLF of C and D

454

15 Switched Systems

According to Theorem 15.7, we can choose a large P to construct a common QLF for A and B. Say, P = 10, then we have ⎤ ⎡ 1 0 0 P1 = ⎣0 24.428 28.316⎦ . 0 2.8316 6.5572 Back to the original coordinate frame, we have ⎤ ⎡ 856.0625 −236.9743 385.1138 P = T −T P1 T −1 = ⎣−236.9743 66.5725 −104.8294⎦ . 385.1138 −104.8294 177.4551 Then we check that

⎤ −479.4604 126.9100 −227.6404 PA + ATP = ⎣ 126.9100 −38.7486 50.4128 ⎦ < 0, −227.6404 50.4128 −128.8147

and

⎡

⎤ −1808.2 421.2 −988.8 PB + BTP = ⎣ 421.2 −100.4 227.4 ⎦ < 0. −988.8 0.227.4 −549 ⎡

Hence P is a common QLF of A and B.

15.2 Quadratic Stabilization of Planar Switched Systems In this section we consider the stabilization of planar switched systems. As a particular case of system (15.2), the system considered is x = AV (t) x + bV (t) u,

x ∈ R2 , u ∈ R,

(15.56)

where V (t) : [0, +f) → / = {1, · · · , N} is a piecewise constant right continuous function. We say that V (t) is observable if it is known at any time. Denition 15.4. System (15.56) is said to be quadratically stabilizable with observable V (t), if there exist a set of state feedback controls uO = KO x such that AO + BO KO , O = 1, · · · , N share a common QLF, xT Px. We need some preparations. First of all, a necessary condition for the system (15.56) to be stabilizable under arbitrary switching is that every mode is stabilizable. So a reasonable assumption is: A1. All the modes are stabilizable. The following lemma shows that without strengthening the assumption we can replace A1 by A2. All the modes are controllable. Lemma 15.6. If a single-input planar linear system (A, b) is stabilizable but not controllable, then for any positive denite matrix, P, there exists a suitable state feedback such that xT Px is a quadratic Lyapunov function of the closed-loop system.

15.2 Quadratic Stabilization of Planar Switched Systems

455

Proof. Without loss of generality, we can assume b = (0, 1). Then the system becomes ? > ? > 0 a 0 u. + x = Ax + bu = 11 a21 a22 1 Since it is stabilizable, a11 < 0. Without loss of generality, we can assume a11 = −1. Then the closed-loop matrix, A˜ = A + bK, is > ? −1 0 ˜ A= , D E where D and E can be chosen arbitrary. > ? 1 p2 Assume P = > 0. We have only to prove that we can choose D and E p2 p3 ˜ T P < 0. A straightforward computation shows such that PA˜ + (A) > ? 2(D p2 − 1) D p3 + (E − 1)p2 ˜ TP = . Q := PA˜ + (A) D p3 + (E − 1)p2 2E p 3 Now since p3 > 0, to get Q < 0 we have only to choose D and E < 0 such that det(Q) > 0. det(Q) is calculated as det(Q) = −(D p3 − E p2)2 + 2D p2 p3 + 2E p22 − p22 − 4E p3. Choosing D = Epp32 yields det(Q) = −4E (p3 − p22 ) − p22 = −4E det(P) − p22. Choosing

E <−

p22 <0 4 det(P)

yields det(Q) > 0, which complete the proof.

3 2

Now if a switching mode, (AO , bO ), is stabilizable but not controllable, we can ignore it. After nding a common QLF for all controllable mode, the stabilizable but not controllable modes can use it as their quadratic Lyapunov function. A2 is assumed hereafter. The following lemma claims that the transformation of a single-input system to its canonical form is unique. It is simple but essential for the later approach and computer coding. Lemma 15.7. Let (A, b) be a single-input linear system. There exists a unique state transformation matrix, T , which converts the system into the Brunovsky canonical form as ⎡ ⎡ ⎤ ⎤ 0 1 0 ··· 0 0 ⎢ 0 0 1 ··· 0 ⎥ ⎢0⎥ ⎢ ⎢ ⎥ ⎥ −1 (15.57) T −1 AT = ⎢ . . . .. ⎥ , T b = ⎢ .. ⎥ . ⎣ .. .. .. ⎣.⎦ .⎦ a 1 a 2 a3 · · · a n Moreover, the parameters ai are

1

456

15 Switched Systems

⎡ ⎤ a1 ⎢a2 ⎥ D E−1 n ⎢ ⎥ A b; ⎢ .. ⎥ = b Ab · · · An−1 b ⎣.⎦ an and the unique state transformation matrix, T = (T1 , T2 , · · · , Tn ), can be inductively determined column by column as J Tn = b, Tk−1 = ATk − ak b, k = n, n − 1, · · · , 2. Proof. We have only to prove the uniqueness of T . From (15.57) it is easy to get the following equation ⎧ AT1 = a1 b ⎪ ⎪ ⎪ ⎪ ⎪ ⎪AT2 = T1 + a2 b ⎨ .. . ⎪ ⎪ ⎪ ⎪ ATn−1 = Tn−2 + an−1 b ⎪ ⎪ ⎩ Ab = Tn−1 + anb. Then a straightforward computation shows the required formulas. 3 2 > ? p p Lemma 15.8. Given a positive denite matrix P = 1 2 > 0, there exists a feedp 2 p3 back u = Kx = (k1 , k2 )x, such that the closed-loop system of the canonical singleinput planar system > ? > ? ? > E 0 1 0 D 0 1 ˜ k k := A = A + bK = + (15.58) a21 a22 DE 1 1 2 has P as its quadratic Lyapunov function, if and only if p2 > 0. Proof. Without loss of generality, we assume p1 = 1. Then for P > 0 to be a ˜ if and only if quadratic Lyapunov function of A, > ?> ? > ?> ? 1 p2 0 1 0 D 1 p2 T ˜ ˜ PA + A P = + < 0, (15.59) p2 p3 D E 1 E p 2 p3 which leads to

? 1 + D p3 + E p2 2 D p2 < 0. Q := 1 + D p3 + E p2 2(p2 + E p3 ) >

(15.60)

Note that E is the trace of the matrix of the closed-loop system, hence E < 0 is a necessary condition for the closed-loop system to be stable. It is assumed in the following discussion. It is obvious that p2 = 0 is not allowed. In fact, D p2 < 0 is necessary. Then we have: Case 1. p2 < 0 and D > 0:

15.2 Quadratic Stabilization of Planar Switched Systems

457

Now for (15.60) to hold, it is necessary and sufcient to have the determinant of the left hand side of (15.60), denoted by D(D , E ), to be positive. After a simple computation, we have D(D , E ) = −(E p2 − D p3 )2 − 2(E p2 + D p3 ) + 4D p22 − 1. Then setting

w D(D , E ) = −2p2 (E p2 − D p3 ) − 2p2 = 0 wE

yields

E p2 = D p3 − 1. So the maximum D(D , E ) is Dmax = −1 − 2(2D p3 − 1) + 4D p22 − 1 = −4D (p3 − p22 ) = −4D det(P) < 0. Hence p2 > 0 is necessary for the matrix Q in (15.60) to be negative denite, which proves the necessity. Case 2. p2 > 0 and D < 0: In this case we can simply choose any D < 0 and let

E=

D p3 − 1 . p2

(15.61)

Then D(D , E ) = −4D (p3 − p22 ) > 0. 3 2

The sufciency is proved.

Based on the previous lemma, we give the following denition. ? > p p Denition 15.5. A matrix M = 1 2 > 0 is said to be canonical-friend (to the p2 p3 canonical controllable form), if p2 > 0. The following lemma shows when a canonical-friend matrix remain canonicalfriend under a coordinate transformation. ? > t t Lemma 15.9. Given a nonsingular matrix T = 11 12 , there exists a canonicalt21 t22 friend matrix P > 0, such that T T PT is canonical-friend, if and only if the following quadratic equation t21t22 x2 + (t12t21 + t11t22 )x + t11t12 > 0 has a positive solution x > 0.

(15.62)

? p1 p2 > 0 with p2 > 0. Calculating Proof. (Sufciency) Consider a matrix M = p2 p 3 T P˜ = T PT shows that >

458

15 Switched Systems

H(p1 , p2 , p3 ) := p˜12 = t21t22 p3 + (t12t21 + t11t22 )p2 + t11t12 p1 . From (15.62), H(1, x, x2 ) > 0. By continuity, there exists H > 0 small enough, such that H(1, x, x2 + H ) > 0. Set p1 = 1, p2 = x and p3 = x2 + H , the matrix P meets our requirement. > ? 1 p2 (Necessity) Without loss of generality, we can assume there exists P = > p2 p3 0 such that both P and T T PT are canonical- friend. That is: p2 > 0 and H(1, p2 , p3 ) > 0. Now, if t21t22 > 0, then it is trivial that (15.62) has a positive solution. As t21t22  0, since p3 > p22 , H(1, p2 , p22 )  H(1, p2 , p3 ) > 0.

(15.63) 3 2

So p2 is a positive solution of (15.62).

Note that for a given state transformation matrix, T , the solution of (15.62) consists of one or two open intervals. It might be empty. We denote the set of solutions as IT . Consider the single-input planar switched system (15.56) and assume / = (1, · · · , N) be a nite set. Then we can get canonical-friend P for each mode with its particular coordinate frame. To get a universal P the coordinate transformations have to be considered. (Refer to Fig. 15.5)

Fig. 15.5

Coordinate Transformations

Now for each switching mode we denote the state transformation matrix, which converts it to the canonical controllable form, by Ci , i = 1, · · · , N. That is, let zi = Ci x,

i = 1, · · · , N.

Then the i-th mode x = Ai x+ bi u, when expressed into zi coordinates, is a Brunovsky canonical form (15.57). According to Lemma 15.7, Ci , i = 1, · · · , N are uniquely determined. −1 Set Ti = C1Ci+1 , i = 1, · · · , N − 1. Ti is the state transformation matrix from zi+1 to z1 . That is −1 z1 = C1 x = C1Ci+1 zi+1 = Ti zi+1 .

15.2 Quadratic Stabilization of Planar Switched Systems

459

Next, we classify Ti = (t ij,k ) into three categories: i i t22 > 0}, S p = {i ∈ / | t21 i t i < 0}, Sn = {i ∈ / | t21 22 i t i = 0}. Sz = {i ∈ / | t21 22

Then / = S p ∪ Sn ∪ Sz . Next, for i ∈ Sz , we dene J i ti + ti ti ci = t12 21 11 22 i ti , i ∈ S . di = t11 z 12 For the other cases, we dene ⎧ i t11 ⎪ ⎪ ⎪ ⎨Di = t i 21 i ⎪ t12 ⎪ ⎪ ⎩E i = i , t22

(15.64)

(15.65) i ∈ S p ∪ Sn .

For each i ∈ Sz , we dene a linear form as Li = ci x + di ,

i ∈ Sz ;

(15.66)

and for each i ∈ S p or i ∈ Sn we dene a quadratic form as Qi = x2 + (Di + Ei )x + Di Ei ,

i ∈ S p ∪ Sn .

(15.67)

According to the Lemma 15.9, we may solve x from ⎧ ⎪ ⎨Qi (x) < 0, i ∈ Sn Qi (x) > 0, i ∈ S p ⎪ ⎩ Li (x) > 0, i ∈ Sz . Noting that Qi in (15.67) can be factorized, the roots are {−Di , −Ei }. So we can simply dene the solution set as follows. If i ∈ S p , we dene an open set consists of two semi-lines as @ A A @ Ii = − f, min(−Di , −Ei ∪ max(−Di , −Ei ), +f ; If i ∈ Sn , we dene an open set as Ii = (min(−Di , −Ei ), max(−Di , −Ei )); If i ∈ Sz , since Ti is non-singular, it follows that ci := 0. Hence we can dene an open set as V ⎧U di ⎪ ⎪ − , f , ci > 0 ⎨ ci V Ii = U ⎪ d ⎪ ⎩ − f, − i , ci < 0. ci

460

15 Switched Systems

As an immediate consequence of Lemma 15.9, we have the following simple conclusion. Theorem 15.8. 1. A sufcient condition for the switched system (15.56) to be quadratically stabilizable is N−1 Ik := ∅. I = ∩k=1

(15.68)

2. If all i ∈ S p , i = 1, · · · , N − 1, then the switched system (15.56) is always stabilizable. 3. If all i ∈ Sn , i = 1, · · · , N − 1, (15.68) is also necessary. Proof. 1. Choose p2 ∈ I and p3 = p22 + H . It is easy to see that when H > 0 small enough, the corresponding matrix > ? 1 p2 P= p 2 p3 + H (in z1 coordinates) becomes the common quadratic Lyapunov function (for suitably chosen controls). 2. Choose p2 > 0 large enough, then p2 ∈ I. 3. From the proof of Lemma 15.9, inequality (15.63) shows that p2 ∈ I. So I := ∅. 3 2 Theorem 15.8 provides a very convenient condition for the existence of a common quadratic Lyapunov function. We may expect that it is also necessary. When N = 2 Lemma 15.9 says that it is. But in general it is not. We have the following counter-example. Example 15.6. Consider a system x = AV (x,t) x + bV (x,t)uV (x,t)

(15.69)

where V (x,t) : R2 × [0, f) → {1, 2, 3, 4}, with four switching modes as: ? ? > > ? > > ? −4 −2 −1 2 −1/3 5 , b1 = ; A2 = , b2 = ; A1 = 9 5 2 −3 0 −6 ? ? > > ? > > ? −7 −6 6 −12 −11 −7 3 3 4 4 , b = ; A = , b = . A = 9.5 8 −7 13 12 8 We investigate whether it is quadratically stabilizable. Suppose z1 , z2 , z3 , and z4 are the canonical coordinates of modes 1, 2, 3, 4 respectively. That is, under zi mode i has Brunovsky canonical form. Let zi = Ci x,

i = 1, 2, 3, 4.

Using Lemma 15.7, we can get Ci as (Note that Ci is T −1 of Lemma 15.7.)

15.2 Quadratic Stabilization of Planar Switched Systems

> C1 =

⎡ 2⎤ −2 −1 ⎢ 3⎥ C2 = ⎣ ⎦; 2 1 3

?

21 ; 11

⎡ 1 1 ⎢ 6 C3 = ⎣ 1 1 3

⎤ 1 ⎥ ⎦; 1

461

⎡

2 1 ⎤ 2 ⎢ 3 3 ⎥ C4 = ⎣ ⎦; 1 2 −1 −1 3 3 2

Then the state transformation matrices between z1 and zi , i = 2, 3, 4, determined by −1 Ti = C1Ci+1 , i = 1, 2, 3, are obtained as > T1 =

? 1 4 ; −1 −1

T2 =

? > −4 5 ; 2 −1

> T3 =

? −3 −6 . 1 1

= −1, E1 = tt21 = −4, so I1 = Now for T1 , since t21t22 = 1, 1 ∈ S p , and D1 = tt11 21 22 t11 = −5, (−f, 1)∪(4, f). For T2 , since t21t22 = −2, 2 ∈ Sn , and D2 = t21 = −2, E2 = tt21 22 t11 t21 so I2 = (2, 5). For T3 , since t21t22 = 1, 3 ∈ S p , and D3 = t21 = −3, E3 = t22 = −6, so I3 = (−f, 3) ∪ (6, f). We conclude that I = I1 ∩ I2 ∩ I3 = ∅. But we may choose the controls as u = Ki x with D E D E K1 = −11.2143 −8.2143 , K2 = −36 −23.3333 , D E D E K4 = 58 46 K3 = −18 −13.5 , and let the quadratic Lyapunov function xT Px be constructed by > ? 31.75 26.25 P= > 0. 26.25 21.75 Then it is ready to verify that ? −246.8929 −205.3929 < 0; P(A + b K1 ) + (A + b K1 ) P = −205.3929 −170.8929 >

1

1

1

1

T

? > −120.5000 −79.5000 < 0; −79.5000 −52.5000 ? > −188.7500 −143.2500 P(A3 + b3K3 ) + (A3 + b3 K3 )T P = < 0, −143.2500 −108.7500 ? > 4 4 4 4 T 3 −1.5005 −1.1955 < 0. P(A + b K4 ) + (A + b K4 ) P = 10 −1.1955 −0.9525 P(A2 + b2 K2 ) + (A2 + b2K2 )T P =

Hence the system (15.69) doesn’t satisfy the condition of Theorem 15.8. But it is quadratically stabilizable with arbitrary but known switching rule. We will go back to this simple condition of Theorem 15.8 again, and also will revisit Example 15.6 to see how to design the controls. Next, we investigate the necessary and sufcient condition for the system (15.56) to be quadratically stabilizable. Carefully going through the proof of Lemma 15.9 and the followed argument, we can nd the following result: Theorem 15.9. Let / = (1, · · · , N). Then system (15.56) is quadratically stabilizable, if and only if there exists a positive x such that

462

15 Switched Systems

⎧ ⎪ ⎨maxi∈Sn Qi (x) < 0 mini∈S p Qi (x) > maxi∈Sn Qi (x) ⎪ ⎩ Li (x) > 0, I ∈ Sz .

(15.70)

Proof. Assume there is a quadratic Lyapunov function in z1 coordinates, which is ? > 1 p2 . According to Lemma 15.8, p2 > 0. It is easy to see from expressed as P1 = p 2 p3 the proof of Lemma 15.9 that P1 is a common quadratic Lyapunov function for the other modes, if and only if Hi (1, p2 , p3 ) > 0, i = 1, 2, · · · , N − 1, which leads to ⎧ ⎪ ⎨ p3 + (Di + Ei )p2 + Di Ei > 0, i ∈ S p (15.71) p3 + (Di + Ei )p2 + Di Ei < 0, i ∈ Sn ⎪ ⎩ ci p2 + di > 0, i ∈ Sz . Since p3 > p22 we may rewrite the rst two equations as J e + p22 + (Di + Ei)p2 + Di Ei > 0, i ∈ S p e + p22 + (Di + Ei)p2 + Di Ei < 0, i ∈ Sn ,

(15.72)

where e > 0. The necessity of (15.70) is obvious. As for the sufciency, assume there is a solution x such that mini∈S p Qi (x) > 0. Then we can choose p2 = x and p3 = x2 + H . As H > 0 small enough, (15.71) is satised. Otherwise, denote w = mini∈S p Qi (x)  0. We choose p2 = x and p 3 = x2 +

U V 1 min Qi (x) − max Qi (x) − w. i∈Sn 2 i∈S p

(15.73)

It is easy to see that for such a choice (15.71) holds. The matrix, therefore, con? > 1 p2 meets the requirement. 3 2 structed as P = p2 p 3 Next, we go back to Theorem 15.8. In fact, it was proved in Theorem 15.8 that if all i ∈ S p , or all i ∈ Sn , i = 1, · · · , N − 1, then (15.68) is also necessary. Since I is simply computable, it is a very convenient condition. Example 15.6 shows that it is not necessary in general. But when N  3 we can prove the following proposition: Corollary 15.4. If N  3, (15.68) is also necessity. Proof. When N = 2 it was proved in Lemma 15.9. We have only to prove it for N = 3. We have T1 and T2 . If both 1 and 2 are in S p (or Sn ), it has been proved in the Theorem 15.8. Without loss of generality we may assume 1 ∈ S p and 2 ∈ Sn . To see the necessity, we assume I1 ∩ I2 = ∅. It turns out that

D1  D2  E2  E1 . Here without loss of generality, we assume D1  E1 and D2  E2 . Now it is obvious that

15.2 Quadratic Stabilization of Planar Switched Systems

Q2 (x)  Q1 (x),

x ∈ (E2 , D2 ).

463

(15.74)

A rigorous proof of (15.74) is tedious. But think about the relative position of the two parabola of the same shape, the conclusion is obvious. Hence the second inequality in (15.70) failed. 2 3 Next, we provide certain numerical details for solving the problem of stabilizing single-input planar systems. First we claim that (15.70) is equivalent to a set of linear inequalities. Consider the rst inequality in (15.70). It is ready to see that this inequality is equivalent to min{D j , E j } < x < max{D j , E j }, j ∈ Sn . For the second inequality, it is equivalent to the fact that each Qi , i ∈ S p is greater than each Q j , j ∈ Sn . Hence, it is equivalent to (Di + Ei − D j − E j )x + Di Ei − D j E j > 0,

i ∈ S p , j ∈ Sn .

Since we are looking for a positive solution, then we have Corollary 15.5. The system (15.56) is quadratically stabilizable, if and only if the following set of linear inequalities have a solution: ⎧ min{Di , E j } < x < max{D j , E j }, j ∈ Sn ⎪ ⎪ ⎪ ⎨(D + E − D − E )x + D E − D E > 0, i ∈ S , j ∈ S i i j j i i j j p n (15.75) ⎪ c x + d > 0, i ∈ S i z ⎪ ⎪ i ⎩ x > 0. Recall the proof of Lemma 15.8, a stabilizing control is easily constructible. We state it formally as: Proposition 15.3. Let (A, b) be a canonical planar system. i.e., ? > ? > 0 0 1 u, + x = 1 a21 a22 > and P =

(15.76)

? p 1 p2 > 0 be canonical-friend. (i.e., p2 > 0.) To make xT Px a quadratic p 2 p3

Lyapunov function of the closed-loop system, a feedback control can be chosen as V? > U O p3 − p 1 (15.77) u = kx = (O − a21) − a22 x, p2 where O < 0 can be any negative real number. Proof. It is from the proof of Lemma 15.2.4. In fact, (15.77) is from (15.61) with an obvious modication for p1 := 1. 3 2 Remark 15.6. Of course (15.77) is only a possible choice. But we always choose (15.77) as our control in later use. Because from the proof of Lemma 15.8 we know

464

15 Switched Systems

that the (D , E ) chosen in this way makes the determine of Q to reach the maximum. In some sense, it makes Q most negative. Next, we give a step by step algorithm for solving the quadratic stabilization problem of the single-input planar systems: Algorithm 15.1. Step 1. Using Lemma 15.7 to nd the state transformation matrices Ci , i = 1, · · · , N, zi = Ci x, such that in coordinate frame zi the i-th switching mode is in Brunovsky canonical form. −1 Step 2. Dene another set of state transformation matrices: Ti = C1Ci+1 ,i= 1, · · · , N − 1, such that z1 = Ti zi+1 ,

i = 1, · · · N − 1.

Step 3. Calculate Di , Ei by (15.65) if i ∈ S p ∪ Sn ; ci , di by (15.64) if i ∈ Sz . Step 4. Construct the system of inequalities (15.75). Find any one solution, x = x0 . (If there is no solution, the quadratic stabilization problem has no solution.) Step 5. Use inequalities (15.72), setting p2 = x0 , to nd a positive solution e > 0. Set p3 = p22 + e. Construct a positive denite matrix ? > 1 p2 > 0, P1 = p 2 p3 which is a common quadratic Lyapunov function for all switching modes with certain feedback controls. (Note that if Step 4 has a solution then there exist solutions for the inequalities of (15.72).) Step 6. Convert P1 to each canonical coordinate frames as: Pi+1 = TiT P1 Ti ,

i = 1, · · · , N − 1.

(15.78)

Convert mode (Ai , bi ) into its canonical coordinate chart as: A˜ i = Ci−1 ACi ,

b˜ i = (0, 1)T ,

i = 1, · · · , N.

(15.79)

Use formula (15.77) to construct the feedback controls: ki , i = 1, · · · , N. Step 7. Back to the original coordinate frame x, the controls should be Ki = (ki )Ci ,

i = 1, · · · , N.

(15.80)

The common quadratic Lyapunov function for all closed-loop switching modes is P = C1T P1C1 .

(15.81)

Example 15.7. Revisiting Example 15.6, we will follow Algorithm 15.1 to construct the controls. Steps 1–3 have already been done in Example 15.6. So we can start from Step 4 to construct the inequalities. Since 2 ∈ Sn , D2 = −2 and E2 = −5, the corresponding inequality is

15.2 Quadratic Stabilization of Planar Switched Systems

465

2 = min{−D2 , −E2 } < x < max{−D2 , −E2 } = 5. Now we consider i = 1 ∈ S p and j = 2 ∈ Sn . Since D1 = −1 and E1 = −4, the corresponding inequality is (D1 + E1 − D2 − E2 )x + D1 E1 − D2 E2 = 2x − 6 > 0. Finally we consider i = 3 ∈ S p and j = 2 ∈ Sn . Since D3 = −3 and E3 = −6, the corresponding inequality is (D3 + E3 − D2 − E2 )x + D3 E3 − D2 E2 = −2x + 8 > 0. Now we have the whole system of the inequalities (15.75) as ⎧ 2<x<5 ⎪ ⎪ ⎪ ⎨2x − 6 > 0 ⎪ −2x + 8 > 0 ⎪ ⎪ ⎩ x > 0. The solution is 3 < x < 4. We know any solution x can be used to construct a common quadratic Lyapunov function. For instance, we may choose x0 = 3.5 and go to Step 5. Then (15.72) becomes ⎧ e + 12.5 + (−1 − 4) × 3.5 + 4 > 0 ⎪ ⎪ ⎪ ⎨e + 12.5 + (−3 − 6) × 3.5 + 18 > 0 ⎪e + 12.5 + (−2 − 5) × 3.5 + 10 < 0 ⎪ ⎪ ⎩ e > 0. The solution is 1 < e < 2. Any solution e can be used to construct a common quadratic Lyapunov function. We may naturally choose e = 1.5. Then setting p2 = x0 , p3 = x20 + e, we have ? > 1 3.5 . P1 = 3.5 14 Now for Step 6–7, converting (A1 , b1 ) to z1 coordinates, we have A˜ 1 = C1 A1C1−1 , > ? > ? which is 01 0 1 1 ˜ ˜ , b = . A = 21 1 Using (15.77), say we choose O = −1 < 0, then > U V? O p3 − p 1 − a22 k˜ 1 = (O − a21) p2 > U V? −1(14) − 1 = (−1 − 2) −1 3.5 ? > 2 = −3 −5 . 7 Then in the original coordinate frame x we have D E K1 = k˜ 1C1 = −11.2857 −8.2857 .

466

15 Switched Systems

Next, for (A2 , b2 ) to z2 coordinates, we have > ? > ? 01 0 2 2 ˜ ˜ , b = . A = 12 1 To get the feedback control law, we need to convert P1 into z2 frame, which is ? > 8.0000 0.5000 . P2 = T1T P1 T1 = 0.5000 2.0000 Then we can get the feedback law as D E k˜ 2 = −2.0000 −22.0000 ;

D E K2 = −18.0000 −11.3333 .

For (A3 , b3 ) to z3 coordinates, we have ? > 0 1 , A˜ 3 = −1 1 P3 = T2T P1 T2 = The feedback law as D E k˜ 3 = 0 −21.0000 ;

> ? 0 b˜ 3 = . 1

? > 16.0000 1.0000 . 1.0000 4.0000

D E K3 = −28.0000 −21.0000 .

Finally, for (A4 , b4 ) to z4 coordinates, we have > ? > ? 01 0 A˜ 4 = , b˜ 4 = . 10 1 ? 2.0000 0.5000 . = 0.5000 8.0000 >

P4 = T3T P1 T3

The feedback law is D E k˜ 4 = −2.0000 −20.0000 ;

D E K4 = 28.0000 22.0000 .

To check whether the result is correct, we do the following verication: Get the quadratic form back to the original coordinates x: ? > 32.0000 26.5000 T . P0 = C1 P1C1 = 26.5000 22.0000 We can check whether it is really a common quadratic Lyapunov function of the four different modes. ? > −253.0000 −211.0000 1 1 1 1 T P0 (A + B K1 ) + (A + B K1 ) P0 = < 0, −211.0000 −176.0000 ? > −67.0000 −44.0000 < 0, P0 (A2 + B2 K2 ) + (A2 + B2K2 )T P0 = −44.0000 −29.0000

15.3 Controllability of Switched Linear Systems

467

? > −308.5000 −233.0000 3 3 3 3 T < 0, P0 (A + B K3 ) + (A + B K3 ) P0 = −233.0000 −176.0000 ? > −751.0000 −596.0000 P0 (A4 + B4 K4 ) + (A4 + B4 K4 )T P0 = < 0. −596.0000 −473.0000 The verication is completed successfully. One of the advantages in this approach is that it provides not only one solution, as in most of the control design problems, but also all the possible quadratic Lyapunov functions. There is no exception at all. This section is based on [4].

15.3 Controllability of Switched Linear Systems This section considers the controllability of the following switched linear system x(t)  = AV (t) x(t) + BV (t) u(t),

x(t) ∈ Rn , u(t) ∈ Rm ,

(15.82)

where V (t) : [0, f) → / is a piecewise constant right continuous mapping and / = {1, 2, · · · , N}. u(t) is a piecewise constant control. From Chapter 6 (or [24]), we have Theorem 15.10. The controllable subspace of system (15.82) is L = 4A1 , · · · , AN | B1 , · · · , BN 5 .

(15.83)

Hence, the system is controllable, if and only if dim(L ) = n. This chapter will consider the case when dim(L ) < n. For system (15.82), the following denition is very natural. In fact, these concepts are adopted from their counterpart of nonlinear systems. Denition 15.6. For system (15.82) the accessibility Lie algebra is dened as La := { Ai x + Bi u | i = 1, · · · , N; u = const.}LA ;

(15.84)

the strong accessibility Lie algebra is dened as a set of constant vectors as Lsa := {[ | [ ∈ 4 A1 , · · · , AN | B1 , · · · , BN 5 } .

(15.85)

Note that (15.85) may be considered as a generalization of (6.2), which can be expressed alteratively as Lsa := 4 f (x)| g(x)5LA . We rst investigate the structure of the accessibility Lie algebra La . Denote A = {A1 , A2 , · · · , AN }. A is said to be symmetric if −Ai ∈ A, ∀i. We consider Ai , i = 1, · · · , N as the elements of the Lie algebra gl(n, R), and denote by A the Lie sub-algebra generated by {A} . That is, A = {A1 , · · · , AN }LA ⊂ gl(n, R). Then we dene the set of linear homogeneous vector elds, denoted by LH(Rn ) = {Ex | E ∈ Mn },

468

15 Switched Systems

where Mn is the set of n × n matrices. A straightforward computation shows that LH(Rn ) ⊂ V Z (Rn ) is a Lie sub-algebra of V Z (Rn ), which is the Lie algebra of the analytic vector elds on Rn . Now we dene a mapping, ) : LH(Rn ) → gl(n, R) by ) (Ex) := −E. It follows that Lemma 15.10. 1. ) : LH(Rn ) → gl(n, R) is a Lie algebra isomorphism. 2. Let L ⊂ gl(n, R) be a Lie sub-algebra. Then its inverse image ) −1 (L ) is a Lie sub-algebra of V Z (Rn ), which can be expressed as

) −1 (L ) = L x.

(15.86)

Proof. 1. A straightforward computation shows that

) ([Ax, Bx]) = ) (BAx − ABx) = −BA + AB, while [) (Ax), ) (Bx)] = [−A, −B] = AB − BA. Hence ) is a Lie algebra homomorphism. It is obvious that ) is a bijective mapping, and so it is an isomorphism. 2. The rst half statement is obvious. (15.86) is from the denition of ) directly with the consideration that a Lie algebra is a vector space, so the negative sign can be omitted. 3 2 Theorem 15.11. For system (15.82) the accessibility Lie algebra can be expressed as La = {Px + Q |P ∈ A , Q ∈ Lsa } .

(15.87)

Proof. Setting P = 0n×n ∈ A , we have {Q | Q ∈ Lsa } = Lsa ⊂ La . Hence “⊃” is obvious. Next, we consider the structure of vector elds in La . It is easy to see that Ai x ∈ La , Bi ∈ La , i = 1, · · · , N. Then since Ai x + Bi u is a linear combination of Ai x and Bi , we have that La = {Ai x, Bi | i = 1, · · · , N}LA .

(15.88)

Consider a bracket-product of Ai x and Bi , which means a term of brackets in any order without ±. For instance, [A1 x, [A3 x, B2 ]],

or [B1 , [B2 , [A3 x, A2 x]]],

etc.

First of all, if a bracket-product contains more than one Bi , it equals zero. This is from the fact that a product containing one Bi is a constant vector eld. Then we have only to consider the terms containing one Bi and the terms containing no Bi .

15.3 Controllability of Switched Linear Systems

469

For terms containing one Bi , using Jacobi identity, it is easy to see that such terms can be expressed as the sum of terms of the form Ai1 · · · Aik Bi ,

(15.89)

which is obviously contained in Lsa . For terms containing no Bi , Lemma 15.10 says that {A1 x, · · · , AN x}LA = A x.

(15.90)

From (15.89) and (15.90), any [ ∈ La can be expressed as [ = Px+Q, where P ∈ A and Q ∈ Lsa , which proves “⊂”. 3 2 Now we are ready to use the Lie algebra approach to investigate the accessibility of switched linear systems. First of all, we can get a result, which is mimic to its nonlinear counterpart. Proposition 15.4. If system (15.82) satises accessibility rank condition at x0 , then the reachable set R(x0 ) contains a non-empty set of interior points. If system (15.82) satises strong accessibility rank condition at x0 , then for any T > 0 the reachable set RT (x0 ) contains a non-empty set of interior points. Proof. Note that each switching mode is an analytic system. Denote L = {Ai x + Bi u | u = const. ∈ Rm }. Then all the vector elds in L is realizable by system (15.82) with suitable constant controls and switching laws. Assume system (15.82) satises accessibility rank condition at x0 . Then there exists a neighborhood U of x0 such that dim(La (x)) = n, x ∈ U. Choose any nonX zero vector eld X1 (x) ∈ L and denote its integral curve as S1 (t1 , x(0)) = et11 (x0 ). If all X ∈ L are linearly dependent with X1 in U, then dim(La (x)) = 1, x ∈ U, which is a contradiction. So there exists a t10 , a vector eld X2 ∈ L, such that at x1 = etX01 (x0 ) 1

the vectors X1 (x1 ) and X2 (x1 ) are linearly independent. Therefore, we can construct a mapping as S2 (t1 ,t2 , x(0)) = etX22 etX11 (x0 ), such that its Jacobian matrix JS2 (t10 , 0) has full rank. So it is a local diffeomorphism. Same argument shows that we can nd (t10 ,t20 ) and X3 ∈ L (t10 may not the same as before), such that S3 (t1 ,t2 ,t3 , x(0)) = X et33 etX22 etX11 (x0 ) is a local diffeomorphism at (t10 ,t20 , 0). Keep going like this, nally we can construct a mapping

Sn (t1 ,t2 , · · · ,tn , x(0)) = etXnn · · · etX22 etX11 (x0 ),

(15.91)

which is a local diffeomorphism at (t10 , · · · ,tn0 ) ∈ Rn . Then there exists an open neighborhood V of (t10 , · · · ,tn0 ), such that Sn−1 (V ) ⊂ R(x0 ) is an open set. The conclusion follows. Next we prove the second part of the Proposition. Assume (15.82) satises strong accessibility rank condition at x0 . Similar to the discussion for nonlinear systems [25], an additional variable xn+1 = t can be added to the original system to get an extended system as J x = AV (t) x + BV (t) u (15.92) xn+1 = 1.

470

15 Switched Systems

It is easy to show that the extended system satises the accessibility rank condition at (x0 , T ) for any T > 0. Recall the above proof for the rst part of this proposition. It is easy to see that in the mapping (15.91) t1 + t2 + · · · + tn > 0 can be arbitrary small. Now consider a moment T0 > 0 and T0 < T . Since (15.92) satised accessibility rank condition at (x0 , T0 ), (15.91) can be constructed for t1 + t2 + · · · + tn < T − T0 . (Precisely, for system (15.92) the mapping should be from (t1 , · · · ,tn+1 ) to Rn , and where tn+1 = T0 + t1 + · · · + tn .) So system (15.92) has non-empty interior at T0 + t1 + · · ·+tn := T1 < T , which implies that for (15.82) RT1 (x0 ) has non-empty interior. Since T1 < T , choosing u(t) and V (t) such that AV (t) + BV (t) u(t) := 0,

T1  t  T,

then RT1 (x0 ) is diffeomorphic to RT (x0 ). The conclusion follows.

3 2

We can have some further results for switched linear systems. Proposition 15.5. Consider system (15.82). The weakly reachable set of any x ∈ Rn , denoted by W R(x), is the largest integral sub-manifold of La passing through x. Proof. First, still use the fact that each vector eld in La is analytic. Hence the generalized Chow’s theorem is applicable. Replacing negative time interval ti by −ti and exchanging two ends of the segment of the integral curve, xi with xi+1 , the conclusion is an immediate consequence of the generalized Chow’s theorem. 3 2 Next, we consider the topological structure of (weak) controllable sub-manifolds for a switched linear system with dim(La ) < n. Let V be the controllable subspace of system (15.82). The following proposition is an immediate consequence of the denition and Theorem 15.10. Proposition 15.6. Consider system (15.82). The controllable subspace is V = I (Lsa )(0), i.e., the integral sub-manifold of the strong accessibility Lie algebra passing through 0. Now the state space can be split as x = (x1 , x2 ), where x1 is the coordinates of V . Since V contains Bi and is Ai invariant for all i, system (15.82) can be expressed as > 1 ? cA11 A12 d > 1 ? > ? V (t) V (t) BV (t) x x = + u. (15.93) x2 x2 0 0 A22 V (t)

The following proposition gives a partition for possible (weakly) controllable sub-manifolds. / V , then Proposition 15.7. Consider system (15.82). For any x0 ∈ Rn , if x0 ∈ W R(x0 ) ∩V = ∅. Proof. Case 1. Assume A x0 ⊂ Lsa . Using analyticity, at any point x ∈ W R(x0 ), dim(La (x)) = dim(Lsa (x)),

x ∈ W R(x0 ).

15.3 Controllability of Switched Linear Systems

471

The weakly reachable set of x0 is the reachable set of x0 , which is x0 + V . Since x0 ∈ / V , (x0 + V ) ∩V = ∅. Case 2. Assume A x0 :⊂ Lsa , then dim(La (x0 )) > dim(Lsa (x0 )). If W R(x0 ) ∩ V := ∅, then x0 is path-wise connected with the origin. Using Lemma 2.7 again, we have dim(La (0)) = dim(Lsa (0)). 3 2

This fact contradicts the assumption.

Remark 15.7. From Proposition 15.7, it is clear that if the initial point x(0) = x0 ∈ /V there is no piecewise constant control, which can drive it to zero. From the decomposed form of (15.93) we denote  I H  A0 := A22 i i = 1, · · · , N LA .

(15.94)

Then we can dene a projection S : A → A0 in a natural way by letting S (Ai ) = A22 i , i = 1, · · · , N. Note that the projection S is dened under a chosen coordinate frame. But it is easy to prove that this project is coordinate independent, because the quotient space Rn /V is Ai invariant. Then the following is obvious. Proposition 15.8. The mapping S is a Lie algebra homomorphism. Moreover, La = A0 x2 ⊕ Lsa ,

(15.95)

where ⊕ is the direct sum (of two subspaces). It is obvious that A0 is uniquely determined by / Ker(S )} . A0 := {Ai | 1  i  N; Ai ∈ Remark 15.8. From Proposition 15.7 and the structure of La the topological structure of the (weak) controllability sub-manifolds is characterized as follows: • System (15.82) is globally controllable if and only if dim(La ) = n. • As dim(La ) < n, system (15.82) can not even be globally weakly controllable. In this case the state space is split into two disjoint parts: subspace V and its complement V c = Rn \V . • Recall (15.93), V = {x ∈ Rn | x2 = 0}, so V c = {x ∈ Rn | x2 := 0}. Assume dim(V ) = n − k. If k > 1, V c is a path-wise connected open set, and if k = 1, V c is composed of two path-wise connected open sets: {x | x2 < 0} and {x | x2 > 0}. Based on the above argument, it is clear that the (weak) controllability of system (15.82) is determined by its (weak) controllability on V c . In later discussion we assume V c is a path-wise connected open set. As for k = 1, it will be replaced by two path-wise connected open sets: {x | x2 < 0} and {x | x2 > 0}. For further discussion we need some auxiliary denitions.

472

15 Switched Systems

Denition 15.7. For a subspace V ⊂ Rn a transformation P is called V -invariant, if PV ⊂ V . Note that in system (15.93) the controllable subspace a> ?b I , V = Span 0 then P is V -invariant, if and only if

> ? P11 P12 P= . 0 P22

Denition 15.8. System (15.93) is V -invariant, feedback block diagonalizable, if there exist a V -invariant transformation P, feedback controls ui = Ki x + vi , i = 1, · · · , N such that the feedback modes have block-diagonal form. Let z = Px, then (15.93) becomes d> ? > > 1 ? cA11 ? V (t) 0 BV (t) z z1 = + v, (15.96) z2 z2 0 0 A22 V (t)

where A, B, used same notations for notational ease, are different from those in (15.93). The following lemma is necessary for proving the main result. Lemma 15.11. Consider a nonlinear control system x = f (x, u),

x ∈ Rn .

(15.97)

Let U be a path-wise connected open set, and dim(La (x)) = n, x ∈ U. Then for any x ∈ U the largest integral sub-manifold of the Lie algebra is I (La )(x) ⊃ U. Proof. If it is not true, there exists a point p ∈ U, which is also on the boundary of I , i.e., p ∈ I¯\I . Denote by I (p) the largest integral sub-manifold of La passing through p. Then I (x) ∪ I (p) is a connected integral manifold of La passing through x, and it is larger than I (x), which is a contradiction. 3 2 The following is the main result about controllability. Theorem 15.12. For system (15.96), assume dim(V ) = n − k, and V c is path-wise connected. 1. V c is weakly controllable, if and only if dim(A0 x) = k,

x ∈ V c.

(15.98)

2. Assume A0 is symmetric, then V c is controllable, if and only if (15.98) holds. Proof. 1. (Necessity) If there exists a x0 ∈ V c with dim(A0 x0 ) = s < k. Then W R(x0 ) ⊂ I (La )(x0 ), and according to the generalized Chow’s theorem, the right is a sub-manifold of dimension n − k + s < n. V c can not be weakly controllable. (Sufciency) Let x0 , x1 ∈ V c . We have to nd a T > 0, a switching law V (t) with switching moments

15.3 Controllability of Switched Linear Systems

473

0 = t0 < t1 < · · · < ts = T, and a switched vector eld X(t) = AV (t) + BV (t) uV (t) , such that

0 < t < T,

x1 = x(T ) = eXdss · · · eXd11 (x0 ),

where Xi = X(t),ti−1  t < ti , di = ti − ti−1 . Denote the starting point and the destination as c 1d c 1d x0 x1 x0 = , x1 = . x20 x21 We design the control in three steps. First, since V is controllable, we can nd switching law and controls such that at a moment T0 we have x1 (T0 ) = 0. Correspondingly, x2 (T0 ) is well known. Now by the controllability [24], there are Xk1 = A11 ik x + Bik uik , k = 1, · · · , s, such that X1

X1

x11 = x1 (T ) = edss · · · ed11 (0).

(15.99)

Let {s1 , · · · , sl } ⊂ {1, 2, · · · , s} be such that the corresponding Ast ∈ A0 ,

t = 1, · · · , l.

Dene Xs2

Xs2

x22 := e−d1s · · · e−dl s (x21 ), 1

(15.100)

l

2 where Xs2i = A22 si x . Then in step 2, we can formally move x2 (T0 ) to x22 (in weak sense). This can be done because of Lemma 15.11. Note that in step 2 the rst part of states x1 remains in the origin by setting all controls vi = 0. Now in step 3, we use the controls in (15.99) to move x1 to the destination x11 . At the same time, according to (15.100), x2 moves from x22 to x21 . 2. Same argument for weak controllability can be used for controllability. The only difference is, now since A0 is symmetric, so, by Chow’s theorem, x22 is physically reachable from x2 (T0 ). 3 2

Next, we dene the local controllability. Denition 15.9. System (15.82) is locally controllable at x0 , if there exists a neighborhood U of x0 such that x ∈ R(x0 ), ∀ x ∈ U. The system is locally weakly controllable at x0 , if there exists a neighborhood U of x0 such that x ∈ W R(x0 ), ∀ x ∈ U.

474

15 Switched Systems

Using the similar argument as in Theorem 15.12, we can get the following local controllability result: Corollary 15.6. For system (15.96) assume dim(V ) = n − k. (i). For a point x0 ∈ V c the system is weakly locally controllable at x0 , if and only if dim(A0 x0 ) = k.

(15.101)

(ii). Assume A0 is symmetric. Then the system is locally controllable at x0 , if and only if (15.101) holds. Proof. Since x0 ∈ V c and V c is open and path-wise connected, so by continuity of the vector elds in A0 , there exists an open neighborhood U ⊂ V c of x0 such that dim(A0 x) = n,

x ∈ U.

(15.102)

Recall the proof of Theorem 15.12. One sees easily that the time durations T0 > 0 and d1 + · · · + ds > 0 can be as small as required by employing high gain controls. Now let T0 > 0 and d1 + · · · + ds > 0 be so small that ? > ? > 0 0 ∈ U, ∈ U. x22 x2 (T0 ) Then the control design technique in the proof of Theorem 15.12 remains available. 3 2 How to verify (15.101) in nite steps is a major question. We give the following proposition for verifying it. Assume A0 ⊂ gl(k, R) is a nite dimensional Lie algebra, it is easy to nd its basis. Let E1 , · · · , Et be a basis of A0 . Then we have the following: Proposition 15.9. Dene t

M(x0 ) := ¦ Ei x0 xT0 EiT .

(15.103)

i=1

Then (15.101) holds, if and only if det(M(x0 )) > 0.

(15.104)

Proof. It is obvious that (15.101) holds if and only if

[ T (E1 x0 , · · · , Et x0 ) = 0 implies [ = 0. Since M(x0 ) = (E1 x0 , · · · , Et x0 ) (E1 x0 , · · · , Et x0 )T , the conclusion follows.

3 2

15.3 Controllability of Switched Linear Systems

475

Finally, we consider the practical stabilization of system (15.93). That is: Given any neighborhood U of the origin, can we nd control and switching law such that the trajectory can enter U at some T > 0. (To get x(t) ∈ U for all t > T , a sequence of {Uk } with Uk+1 ⊂ Uk , k = 1, 2, · · · can be constructed in such a way that ∩Uk = {0}. Then the controls and switches can be constructed such that the trajectory can get into Uk . We omit the details here.) Note that when x2 is small enough, (15.93) can be approximated by (15.96). So the following result is obvious: Corollary 15.7. For system (15.93) assume dim(V ) = n − k, and V c is path-wise connected. Moreover, (15.98) holds and A0 is symmetric. Then system (15.93) is practically stabilizable. We give an example to illustrate the results of controllability. Example 15.8. (i). Consider the following system with n = 4, m = 1 and N = 2. Two modes (Ai , Bi ), i = 1, 2 are ⎛⎡ ⎤ ⎡ ⎤⎞ ⎛⎡ ⎤ ⎡ ⎤⎞ 0000 1 00 0 0 1 ⎜⎢1 0 0 0⎥ ⎢0⎥⎟ ⎜⎢0 1 0 0⎥ ⎢0⎥⎟ ⎜⎢ ⎥ ⎢ ⎥⎟ ⎜⎢ ⎥ ⎢ ⎥⎟ ⎝⎣0 0 1 0⎦ , ⎣0⎦⎠ ; ⎝⎣0 0 0 1⎦ , ⎣0⎦⎠ . 0001 0 0 0 −1 0 0 We skip some routine computation, and show the concerned results directly. The controllable subspace is a> ?b I2 . Lsa = Span 0 Let x = (x1 x2 x3 x4 ) ∈ R4 . Then J La (x) =

Lsa , Tx (R4 ),

x3 = x4 = 0, otherwise,

where Tx (M) stands for the tangent space of a manifold M at x. Dene a subspace V = {x ∈ R4 | x3 = x4 = 0} Then we know that the weakly reachable set of x is J ILsa (0) = V, x ∈ V W R(x) = R4 \V, x∈ / V. (ii). Now, we add two switching modes (Ai , Bi ), i = 3, 4 to (i) as ⎤ ⎡ ⎤⎞ ⎤ ⎡ ⎤⎞ ⎛⎡ ⎛⎡ 1 1 0 −1 0 1 −1 2 0 2 ⎜⎢ 1 2 0 2 ⎥ ⎢1⎥⎟ ⎜⎢3 1 1 0 ⎥ ⎢−1⎥⎟ ⎥ ⎢ ⎥⎟ ⎥ ⎢ ⎥⎟ ⎜⎢ ⎜⎢ ⎝⎣ 0 0 −1 0 ⎦ , ⎣0⎦⎠ ; ⎝⎣0 0 0 −1⎦ , ⎣ 0 ⎦⎠ . 0 00 1 0 0 0 0 0 −1

476

15 Switched Systems

Then one sees easily that A0 is symmetric. So the weak controllability becomes controllability. Precisely, J ILsa (0) = V, x ∈ V R(x) = R4 \V, x∈ / V, Note that we need pre-feedback controls to block diagonalize A3 and A4 rst. This section is based on [6], we refer to [22] for further developments.

15.4 Controllability of Switched Bilinear Systems This section considers the controllability of switched bilinear systems (SBLS) of the form Z m Y x = AV (t) x + ¦ BiV (t) x + ciV (t) ui (15.105) i=1 n m := AV (t) x + BV (t)ux + CV (t) u, x ∈ R , u ∈ R , where V (t) : [0, f) → / is a measurable right continuous mapping and / = {1, 2, · · · , N}. Controls u(t) are piecewise constant functions. The state space Rn can be replaced by any arc-wise connected open submanifold, M, of Rn . So in the sequel, any one of such manifolds may be used without further explanation. m

Note that in (15.105) and in the sequel we use brief notation BO u for ¦ BiO ui . In @ A i=1 fact, this is the semi-tensor product BO  u, where BO = B1O , · · · , Bm O is an n × mn matrix. (See appendix for semi-tensor product.) Three kinds of controllabilities are investigated in this section, namely, weak controllability, approximate controllability and global controllability. As one of the main tools, the accessibility Lie algebra of switched systems is dened and investigated. Roughly speaking, the main result in this paper consists of: (i). it is proved that as the accessibility Lie algebra has full rank the system is weakly controllable; (ii). if in addition, certain symmetric condition is satised, the approximate controllability is obtained; (iii). if the system concerned is practically controllable, some additional local conditions assure the global controllability. The results (ii) and (iii) are applicable to a large class of switched bilinear systems, which satisfy the symmetric condition. To begin with, we give some rigorous denitions for sub-manifolds. Denition 15.10. Consider a SBLS. 1. A sub-manifold, I , of M is called an invariant sub-manifold of a switched control system, if for any piecewise constant controls ui , any switched law V (t), and any x0 ∈ I , the trajectory of the controlled switched system remains in I , i.e., x(x0 , u, V (t),t) ⊂ I , ∀t, ∀u. 2. An invariant sub-manifold I is called a controllable sub-manifold if for any two points x, y ∈ I , x ∈ R(y).

15.4 Controllability of Switched Bilinear Systems

477

3. An invariant sub-manifold I is called a weakly controllable sub-manifold if for any two points x, y ∈ I , x ∈ W R(y). The controllable sub-manifolds are closely related to the Lie algebra generated by the vector elds extracted from the systems. Similar to the classical (non-switching) case, we dene the accessibility Lie algebra for SBLS as Denition 15.11. For system (15.105), the accessibility Lie algebra is dened as  H I La := AO x, BiO x + ciO ,  O ∈ / , i = 1, · · · , m LA .

(15.106)

The following result about weak controllability is a mimic of the corresponding result about general control systems. Proposition 15.10. The system (15.105) is globally weakly controllable, if the accessibility Lie algebra has full rank. That is, rank (La (x)) = n,

∀x ∈ Rn .

(15.107)

If (15.107) is satised, as for non-switching case, it is said that the accessibility rank condition is satised. About the local controllability, we have the following result. Proposition 15.11. 1. Consider a SBLS. For a given point x ∈ Rn if rank(L (x0 )) = k and there exists an open neighborhood U of x0 such that S(x) := {x ∈ U | rank(La (x)) = k} is a k dimensional regular sub-manifold of U, then S(x) is a weak controllable sub-manifold. 2. If La has a k-symmetric generator of the form { fO +gO uO }, S(x) is a controllable sub-manifold. Proof. 1. Note that by the Frobenius’ theorem for analytic manifold, for a bilinear system the k-th degree largest integral manifold passing through x0 always exists. Then for any point y on this manifold, La (y) = k. Now since S(x) is the unique k-th degree regular sub-manifold, it must be contained in the largest integral manifold. 2. Using k-symmetry and the same argument in 1, 2 is obvious. 3 2 Example 15.9. Consider the following switched system x = BV (t) ux, where / = {1, 2} and

> B1 =

It is easy to see that

? 10 , 01

x ∈ R2 \{0}, > B2 =

? 22 . 11

{B1 , B2 }LA = Span{B1 , B2 }.

(15.108)

478

15 Switched Systems

So we have only to consider the rank of the distribution generated by these two vector elds. Observe that A @ det B1 x, B2 x = (x1 + x2 )(x1 − 2x2 ). Then the controllable sub-manifolds consist of 8 components: four angular regions, which are two dimensional sub-manifolds, and four rays, which are one dimensional sub-manifolds. They form a partition of R2 \{0}, which are split by two lines: x1 = −x2 and x1 = 2x2 . Namely, four 2-dimensional controllable sub-manifolds are H I 1 2 1 < x2 < 2 x 1 I , Hx ∈ R2 \{0} | x1 > 0, −x 1 Hx ∈ R2 \{0} | x1 < 0, 2 x1 < x2 < −x1I , Hx ∈ R2 \{0} | x2 > 0, −x2 < x1 < 2x2 I , x ∈ R \{0} | x2 < 0, 2x2 < x1 < −x2 , and four 1-dimensional controllable sub-manifolds are H I 1 2 Hx ∈ R2 \{0} | x1 > 0, x2 = 21 x1 I , Hx ∈ R2 \{0} | x1 < 0, x2 = 2 x1 I, Hx ∈ R2 \{0} | x1 > 0, x2 = −x1 I , x ∈ R \{0} | x1 < 0, x2 = −x1 . Next, we consider the approximate controllability of system (15.105). Approximate controllability means the state can be driven to approach a given destination point. In some journals it is also called eventual controllability. It is practically useful. We give a rigorous denition. Denition 15.12. System (15.105) is said to be approximately controllable at x ∈ M if for any y ∈ M and any given H > 0, there exist suitable controls and switching law such that the spline-trajectories of the switched controlled modes can reach the H neighborhood of y. The system is said to be approximately controllable if it is approximately controllable at every x ∈ M. The constructive nonlinear decomposition technique has been used widely for bilinear systems [16, 17]. For this approach, instead of studying the switched bilinear system (15.105), we consider the following two switched systems: a linear system without control and two switched bilinear homogeneous control systems as follows:

x = BV (t) ux + CV (t) u;

x = AV (t) x;

(15.109)

@ A x = − BV (t) ux + CV (t) u .

(15.110)

Denote by RLH (x0 ) the reachable set of the spline trajectories of (15.109) and (15.110). Then we have the following result, which is due to [17] for BLS. It can be extended to SBLS without any difculties. Lemma 15.12. [17] Consider system (15.105). For every x0 ∈ Rn , denote the reachable set of x0 by R(x0 ), then RLH (x0 ) ⊂ cl{R(x0 )}.

(15.111)

15.4 Controllability of Switched Bilinear Systems

479

Here cl is used to denote the closure of a set. Denote  H I Vc = (BiO x + ciO ) O ∈ / , i = 1, · · · , m LA , which is generated by the vectors of input channels. Using Lemma 15.12, we have the following result immediately: Proposition 15.12. Consider system (15.105). If rank (Vc (x)) = n,

∀x ∈ Rn ,

(15.112)

then the system is approximately controllable. Proof. We claim that the complement of reachable set, denoted by Rc (x0 ), is nowhere dense. Otherwise, there exists a non-empty open set O := ∅, O ⊂ Rc (x0 ). Then Oc ⊃ R(x0 ). Since Oc is a closed set, we have Oc ⊃ cl{R(x0 )}. It follows from (15.111) that O ∩ RLH (x0 ) = ∅. On the other hand, the generator of Vc is from (15.110), which is a symmetric set of vector elds. According to Proposition 2.3 and the symmetry, (15.111) implies that RLH (x0 ) = Rn . This fact leads to a contradiction. Now given y ∈ Rn , for any H > 0, denote its H neighborhood by BH (y). Since c R (x0 ) is nowhere dense, there exits y0 ∈ BH (y) which is also in R(x0 ), which completes the proof. 3 2 To avoid the obstacle of non-symmetry of shift term, we consider a class of systems, which, with properly chosen state feedback on every switching mode, have k-symmetric shift term. Denition 15.13. The system (15.105) is said to have a feedback k-symmetric shift term, if there exist controls u0O , O ∈ / , such that for the new shifts under feedback A˜ O = AO + BO u0O , form a k-symmetric set

O ∈/

(15.113)

I H A˜ := A˜ O |O ∈ / .

Remark 15.9. Consider an afne switched nonlinear system m

x = fV (t) (x) + ¦ giV (t)Ui ,

V (t) ∈ {1, 2, · · · , N}.

(15.114)

i=1

(15.114) has feedback k-symmetric shift term, if and only if for each i there exist a j and a positive real number k > 0, such that  fi + k f j ∈ Span{ gts s = i, j;t = 1, · · · , m}. A sufcient condition for feedback k-symmetry is  @ A dim Span{ gts  s = i, j;t = 1, · · · , m} = n.

(15.115)

480

15 Switched Systems

It is easy to see that a lot of practical systems satisfy (15.115). Similar to Proposition 15.12, we can prove the following: Proposition 15.13. Consider system (15.105). Assume (i). the system has feedback k-symmetric shift term; (ii). dim{La (x)} = n, ∀x ∈ Rn . Then the system is practically controllable. Proof. Using a pre-feedback, we can replace (15.109) by x = A˜ V (t) x.

(15.116)

Then the conditions (i) and (ii) imply that for any x0 ∈ Rn the set RLH (x0 ) corresponding to (15.116)–(15.110) is Rn . The rest argument is the same as the one for Proposition 15.12. 3 2 Remark 15.10. 1. If the feedback k-symmetric shift term forms a generator of La , then the above argument shows that the system is globally controllable. 2. In fact, if there is a subset / < ⊂ / , such that the corresponding feedback vector elds H I A˜ < := A˜ O < |O < ∈ / < form a symmetric set, then we dene  C B j j L < := AO < x, BO x + cO  j = 1, · · · , m; O < ∈ / < , O ∈ /

LA

.

It is clear that the switched bilinear system (15.105) is globally approximately controllable if dim{L < (x)} = n, ∀x ∈ Rn . Example 15.10. Consider a bilinear switched system @ A x = AV (t) x + u BV (t) x + CV (t) ,

(15.117)

where / = {1, 2} and ⎤ 101 A1 = ⎣0 2 0⎦ , 003 ⎡

⎤ ⎡ 0 0 0 B1 = ⎣0 0 1⎦ , 0 −1 0

⎡ ⎤ −2 0 −2 A2 = ⎣ 0 0 0 ⎦ , 0 0 −6

⎡ 00 B2 = ⎣0 1 00

⎤ 0 0⎦ , 0

⎡ ⎤ 1 C1 = ⎣1⎦ ; 1 ⎡ ⎤ 0 C2 = ⎣0⎦ . 0

A straightforward computation shows that rank (La (x)) = 3,

∀x ∈ R3 ,

i.e., the accessibility rank condition is satised. In addition, it is obvious that system (15.117) has control k-symmetric shift term. According to Proposition 15.13, the system is approximately controllable.

15.4 Controllability of Switched Bilinear Systems

481

Finally, we consider the global controllability. Recall a result of local controllability for general control systems rst. Consider a general control system x = f (x, u),

x ∈ Rn ,

(15.118)

where f is a C1 mapping. Let x0 be an equilibrium of the control system with control ue (x), i.e., f (xe , ue (xe )) = 0. Dene     wf wf  (x, ue (x)) (x, ue (x)) E= , D= . (15.119) wx w u x0 ,ue (x0 ) x0 ,ue (x0 ) We have the following sufcient condition for local controllability. Lemma 15.13. [19] Consider system (15.118). Assume there exist x0 ∈ Rn and control ue (x). such that, f (xe , ue (xe )) = 0. Moreover, assume (E, D), dened in (15.119), is completely controllable. Then (15.118) is locally controllable at x0 . That is, there exists an open neighborhood U of x0 , such that for any x, y ∈ U, x ∈ R(y) and y ∈ R(x). Using it and the approximate controllability investigated in last section, we deduce some sufcient conditions for the global controllability. Denition 15.14. Consider a bilinear system x = Ax + Bux + Cu,

x ∈ Rn , u ∈ Rm .

(15.120)

1. A pair (xe , ue ) ∈ Rn × Rm is called an equilibrium pair, if Axe + Bue xe + Cue = 0.

(15.121)

2. An equilibrium pair (xe , ue ) is said to be stable if A + Bue is Hurwitz, and it is said to be anti-stable if −(A + Bue) is Hurwitz. 3. An equilibrium pair (xe , ue ) is said to be controllable, if (A + Bue, B(Im ⊗ xe ) + C) is a controllable pair. Theorem 15.13. Consider system (15.105). Assume (i). it is approximately controllable; (ii). there exist O1 ∈ / and an equilibrium pair (xe , ueO1 ), such that (xe , ueO1 ) is anti-stable for the O1 -th switching mode; (iii). there exist O2 ∈ / (O2 = O1 is allowed) and an equilibrium pairs (xe , ueO ), 2 such that (xe , ueO2 ) is controllable for the O2 -th switching mode. Then the system (15.105) is globally controllable. Proof. Since (xe , ueO2 ) is controllable, by Lemma 15.13 there exists a neighborhood U of xe such that the O2 -th switching mode is controllable over U.

482

15 Switched Systems

Next, we show that for any x, y ∈ Rn we can drive the state from x to y. Since the system is approximately controllable we can rst drive x to a point [ ∈ U. Denote the vector eld of the closed-loop O1 -th switching mode with control ueO1 by V , that is, Z Z Y Y V = AO1 + BO1 ueO1 x + CO1 ueO1 = AO1 + BO1 ueO1 (x − xe ). Since −V is stable, so the integral curve of −V goes from y to xe asymptotically. Hence there is a T > 0 such that e−V T (y) = K ∈ U. Equivalently, y = eVT (K ). To complete the proof we have only to drive the state from [ to K . This can be done by choosing O2 -th switching mode and a suitable control uO2 , because of local controllability of this mode over U. Summarizing the above argument, we can drive x to y in three steps: Step 1. According to approximate controllability, we can drive x to [ ∈ U; Step 2. According to the local controllability, we can drive [ to K ; Step 3. According to the anti-stability, we can drive K to y by feedback vector eld V . 3 2 Example 15.11. Recall Example 15.10. We prove that system (15.117) is globally controllable. Using Theorem 4.3, we have to check conditions (i)–(iii). (i) is proved in Example 15.10. Now we choose a pair as (xe , ue ) = (0, 0). Obviously, it is an equilibrium pair. We then show that for the rst mode it is anti-stable. In fact, A1 + B 1 u e = A 1 , which is anti-stable. So (ii) is satised. Still use this pair to the rst mode. We have (A1 + B1 ue , B1 xe + C) = (A1 , c1 ).

(15.122)

It is easy to check that (15.122) is completely controllable, which implies (iii). The conclusion follows. Following the same train of thought as in the proof of Theorem 4.3, we can have the following result immediately: Proposition 15.14. Consider system (15.105). Assume (i). there exist O1 ∈ / and an equilibrium pair (xe , ueO1 ), such that (xe , ueO1 ) is stable for the O1 -th switching mode; (ii). there exist O2 ∈ / and an equilibrium pair (xe , ueO2 ), such that (xe , ueO2 ) is anti-stable for the O2 -th switching mode; (iii) there exist O3 ∈ / and an equilibrium pairs (xe , ueO ), such that (xe , ueO ) is 3 3 controllable for the O3 -th switching mode. Then the system (15.105) is globally controllable. Remark 15.11. In fact, In Theorem 15.13 xe in condition (ii) (specied as x2e to distinguish it from xe in condition (iii)) can be different from the xe (x3e ) in condition (iii). It is enough that x2e ∈ R(x3e ). Particularly, since R(x3e ) contains a controllable

15.5 LaSalle’s Invariance Principle for Switched Systems

483

open neighborhood, U, of x3e , so it sufces that x2e ∈ U. Similarly, for Proposition 4.5, x3e ∈ R(x1e ) and x2e ∈ R(x3e ) are enough for the global controllability. Particularly, when U is a controllable open neighborhood of x3e , then x1e , x2e ∈ U is enough for the global controllability. This section is based on [5].

15.5 LaSalle’s Invariance Principle for Switched Systems Consider a switched linear system x = AV (t) x, x ∈ Rn ,

(15.123)

where V : [0, +f) → / = {1, 2, · · · , N} is dened as before. For determining the stability of the system (15.123), it is very natural for us to search for a common quadratic Lyapunov function (CQLF), V (x) = xT Px, where P > 0 is a positive denite matrix, such that PAO + ATO P < 0,

∀O ∈ / .

(15.124)

As discussed in section 15.1, nding a CQLF is an interesting and challenging problem. It is worth mentioning that it was shown in [8] that the existence of a common quadratic Lyapunov function is only a sufcient condition for switched linear systems to be asymptotically stable. When the total derivative of the candidate Lyapunov function with respect to each mode is only non-positive, the function is called a weak Lyapunov function. A switching system is said to have a non-vanishing dwell time, if there exists a positive time period W0 > 0, such that the switching instances {Wk | k = 1, 2, · · · } satisfy inf(Wk+1 − Wk )  W0 .

(15.125)

k

Throughout this section we assume that A3. Admissible switching signals have a dwell time W0 > 0. The following is a multi-Lyapunov function approach to LaSalle’s invariance principle for switched linear system. Proposition 15.15. [11] Consider the system (15.123). Suppose that there exist a set {PO |O ∈ / } of n × n symmetric positive denite matrices and a set of n × m matrices {CO |O ∈ / } such that at each switching moment we have xT (t)PV (t) x(t)  xT (t)PV (t − ) x(t),

∀ t > 0,

(15.126)

and PO AO + ATO PO  −COTCO ,

∀ O ∈ /.

(15.127)

Then (15.123) is Lyapunov stable. Moreover, if each pair (CO , AO ) is observable, then (15.123) is asymptotically stable.

484

15 Switched Systems

Note that if (CO , AO ) is observable and there exists PO > 0 such that (15.127) holds, then AO is a Hurwitz matrix ([23]). Motivated by some consensus of multiagent systems, which will be discussed in the following section, we consider an extension of LaShalle’s invariance principle where the asymptotical stability of each mode is not assumed. Since we only require each mode to be Lyapunov stable, in addition to A3, we need to assume certain ergodicity property for switches. A4. There exists a T > 0, such that for any t0  0, the switching signal V (t) satises {t | V (t) = O } ∩ [t0 ,t0 + T ] := ∅,

∀ O ∈ /.

(15.128)

Remark 15.12. If both A3 and A4 hold, then there exists T > 0 (replacing the original T of A4 by T + W0 ) such that |{t | V (t) = O } ∩ [t0,t0 + T ]|  W0 ,

∀ O ∈ / , t0  0,

(15.129)

where | · | denotes the Lebesgue measure. Denition 15.15. Consider system (15.123). 1. If a quadratic function V (x) = xT Px with positive denite P > 0 has the following property PAi + ATi P = −Qi  0,

i = 1, · · · , N,

(15.130)

then V (x) (or briey, P) is called a common weak quadratic Lyapunov function (CWQLF) of system (15.123). 2. A common weak quadratic Lyapunov function of system (15.123) is called a common joint quadratic Lyapunov function (CJQLF) if N

Q := ¦ Qi > 0.

(15.131)

i=1

Remark 15.13. It is easy to prove that for the system (15.123), assume there exists a CWQLF P > 0, then P is a CJQLF if and only if q

Zi = {0},

(15.132)

i∈/

where Zi is the kernal of Qi , i ∈ / . Now, a natural question is: under the assumptions of A3 and A4, is the existence of a CJQLF enough to assure the global asymptotical stability? Unfortunately, the answer is negative and the following is a counter example: Example 15.12. Consider a switched linear system x = AV (t) x,

(15.133)

with the piecewise constant switching law V (t) : [0, +f) → / = {1, 2, 3}. Let

15.5 LaSalle’s Invariance Principle for Switched Systems

⎤

485

⎤ ⎤ ⎡ 0 1 0 0 0 −1 −1 0 0 A1 = ⎣−1 0 0 ⎦ , A2 = ⎣0 −1 0 ⎦ , A3 = ⎣ 0 0 1⎦ . 0 0 −1 1 0 0 0 −1 0 ⎡

⎡

Choosing P = I3 , a straightforward computation shows that Q1 = diag{0, 0, 2}, Q2 = diag{0, 2, 0}, Q3 = diag{2, 0, 0}, and Z1 = Span{(1, 0, 0)T , (0, 1, 0)T }, Z2 = Span{(1, 0, 0)T , (0, 0, 1)T }, Z3 = Span{(0, 1, 0)T, (0, 0, 1)T }. 3

Obviously, Qi  0, i = 1, 2, 3, and Q = ¦ Qi = 2I3 > 0, that is, I3 is a common i=1

joint quadratic Lyapunov function. Choose a switching law, being a periodic switch with period T = 1.5π, as ⎧ ⎪ ⎨1, t ∈ [kT, kT + 0.5π) V (t) = 2, t ∈ [kT + 0.5π, kT + π) ⎪ ⎩ 3, t ∈ [kT + π, (k + 1)T), k = 0, 1, · · · which is ergodic. However, system (15.133) is not asymptotically stable under the above switching law if we take the initial value x(0) = (0, 1, 0)T ∈ Z1 . Indeed, the trajectory (refer to Fig. 15.6) A → B → C → A → · · · , is a closed loop with ||x(t)||≡ 1 for all t  0.

Fig. 15.6

Trajectory of (15.133)

486

15 Switched Systems

It seems that we need to impose certain constrains on system (15.123), or more precisely, on {Ai }. We begin with the following observation. Lemma 15.14. For system (15.123) if there exists a CWQLF, then for each i, Ai is Lyapunov stable, which implies that Ai can only have eigenvalues with non-positive real part. Moreover, the algebraic multiplicity of each eigenvalue on the imaginary axis is equal to its geometric multiplicity. Denition 15.16. Assume P is a CWQLF for system (15.123), then z = P1/2 x is called the normal coordinate frame. Remark 15.14. A straightforward computation shows that under the normal coordi˜ A˜ i and Q˜ i respectively, become nate frame P, Ai and Qi , denoted by P, P˜ = In ;

A˜ i = P1/2 Ai P−1/2,

Q˜ i = P−1/2 Qi P−1/2 ,

i ∈ /.

(15.134)

It is worthwhile noting that since P is identity now, Lyapunov function becomes 1x12 the square of the distance of x to zero. In the following argument we use normal coordinate frame only unless elsewhere stated. We need some preparations rst. Lemma 15.15. Consider a linear system x = Ax,

x ∈ Rn ,

(15.135)

where A is Lyapunov stable. Denote K = Ker(A), and let y ∈ K. Then for any R > 0, there exists r > 0, such that if 1x0 − y1 < r, then 1M (x0 ,t) − y)1 < R,

t  0,

(15.136)

where M (x0 ,t) is the solution of (15.135) with M (x0 , 0) = x0 . Proof. It is easy to see that any y ∈ K is a stable equilibrium of (15.135), and 9 thus 9 the conclusion follows. In particular we can choose r = RL , where L = max 9eAt 9 < t0

+f. Then 1x(t) − y1 = 1eAt (x0 − y)1  L1x0 − y1 < Lr = R,

t  0. 3 2

Lemma 15.16. Consider the switched linear system (15.123). Assume Ai , i ∈ / are r Lyapunov stable matrices. Denote Ki = Ker(Ai ), K = Ki , and let y ∈ K. Assume i∈/

the switch satises A3 and A4, then for any R > 0, there exists r > 0, such that if 1x0 − y1 < r then 1M (x0 ,t) − y1 < R,

0  t  T.

(15.137)

Proof. Dene L = sup ||eAit || < +f, because every Ai is Lyapunov stable. Let t0,i∈/ [ \ k = WT0 + 1. Then there are at most k switches over the duration 0  t  T . Choose

15.5 LaSalle’s Invariance Principle for Switched Systems

r=

487

R . Lk+1

Let O0 , O1 , · · · , Os be the sequential active modes with lasting times t0 ,t1 , · · · ,ts respectively. Then s  k and 9 9 9 9 1M (x0 ,t) − y1 = 9eAOs ts · · · eAO0 t0 x0 − y9 9 9 9 9 A t = 9eAOs ts · · · e O0 0 (x0 − y)9  Lk+1 r = R. 3 2 Lemma 15.17. Ker(Ai ) ⊂ Ker(Qi ).

(15.138)

Proof. Since Ai + ATi = −Qi , letting x0 ∈ Ker(Ai ), then xT0 Qi x0 = −xT0 (Ai + ATi )x0 = 0. So Qi x0 = 0 due to Qi  0. The conclusion follows.

3 2

Denote by M the largest weak invariant subset contained in Z, and let Vi = M ∩ Zi ,

i = 1, · · · , N.

Using Lemma 15.17, one sees easily that Ker(Ai ) itself is an weak invariant set contained in Zi ⊂ Z, hence ker(Ai ) ⊂ Vi . We further assume A5. Ker(Ai ) = Vi , i ∈ / . Then we have Theorem 15.14. Consider the system (15.123). Assume A3 and A4 hold, and there exists a CJQLF such that A5 holds, then the system (15.123) is globally asymptotically stable. Proof. Let x(t) = M (x0 ,t) be any trajectory of the system (15.123) starting at M (x0 , 0) = x0 . We only need to prove the convergence. Since 1M (x0 ,t)1 is decreasing monotonically, the Lyapunov stability is obvious. Moreover lim 1M (x0 ,t)1 = c.

t→f

If c = 0 we are done. So we assume c > 0 and will draw a contradiction. Since x(t) is bounded, then there exists an innite sequence {tk } such that xk := x(tk ) → y,

k → f.

Moreover 1y1 = c > 0. Now since y is a point in the Z -limit set (see [1]), we have y ∈ M ⊂ Z.

488

15 Switched Systems

Split / into two disjoint subsets, I ⊂ / and J = / \ I, satisfying y ∈ Zi , ∀i ∈ I,

y∈ / Z j , ∀ j ∈ J.

(Refer to Fig. 15.7) Since y ∈ M, thus I := ∅ and y ∈ Vi , ∀i ∈ I. According to (15.132), J := ∅.

Fig. 15.7

Evolution of the Trajectory

Denote d = min d(y, Z j ) > 0.

(15.139)

j∈J

We can choose 0 < R < d/2 and dene a ball BR (y). Then we have d(x, Z j ) > R,

x ∈ BR (y),

∀ j ∈ J.

(15.140)

For each x ∈ Rn , we can decompose it with respect to each i ∈ / by x = x1 + x2 , where x1 ∈ Zi and x2 ∈ Zi⊥ . If mode i is active, then d 1x(t)12 = −xT2 Qi x2 , dt and

Vm xT2 x2  xT2 Qi x2  VM xT2 x2 , where Vm and VM are the smallest and largest nonzero (equivalently, positive) eigenvalues of Qi , and 1x2 1 is the distance of x from Zi . Let Pmin = min{Okj | Okj : nonzero eigenvalues of Q j } > 0. j∈J

(15.141)

Then it is clear that if a mode j ∈ J is active, then d 1x(t)12 = −xT (t)Q j x(t)  −Pmin ||x2 ||2  −Pmin R2 , dt

∀x ∈ BR (y).

(15.142)

15.5 LaSalle’s Invariance Principle for Switched Systems

489

The proof will be completed in two steps: Step 1, assume t0 is a moment when (i). a mode j ∈ J becomes active; (ii). x(t0 ) ∈ BR1 (y), where R1 < R. We will show that as R1 small enough we have a contradiction. First, choose a positive number N0 I 1 and let R01 = R/N0 . Then we can nd an D > 0 such that 9 (N0 − 1)R01 9 max 9eA j t − I 9  , 0  t  D. j∈J 1y1 + R01

(15.143)

Because when D = 0 the left hand side of (15.143) is zero, (15.143) follows from the continuity. Now if x(t0 ) ∈ BR0 (y), then we have 1

1x(t) − y1  1x(t) − x(t0 )1 + 1x(t0 ) − y1 

(N0 − 1)R01 1x(t0 )1 + R01  R, 1y1 + R01

t0  t  t0 + D .

(15.144)

Without loss of generality we can assume D  W0 . Denote

G = Pmin R2 D > 0. From (15.143) and the fact that a mode j ∈ J becomes active at t0 , we can use (15.142) to get 1x(t0 + D )12 < 1x(t0 )12 − G

U

G 2 1x(t0 )1 V2 U G = 1x(t0 )1 − . 2 1x(t0 )1

V2

< 1x(t0 )12 − G +

Hence

(15.145)

G . 2 1x(t0 )1 Since x(t0 ) ∈ BR0 (y), then 1x(t0 )1 < 1y1 + R01 < 1y1 + R, and hence 1x(t0 + D )1  1x(t0 )1 −

1

1x(t0 + D )1 < 1x(t0 )1 −

G . 2(1y1 + R)

(15.146)

b G , 2(1y1 + R) and letting x(t0 ) ∈ BR1 (y) ⊂ BR0 (y), then (15.146) yields

Now choosing

a R1 = min R01 , 1

1x(t0 + D )1 < 1y1 + R1 −

G  1y1, 2(1y1 + R)

(15.147)

which is a contradiction. Step 2, using Lemma 15.16 associated with Assumption A5, we can nd 0 < r < R1 , such that when x(t ∗ ) ∈ Br (y) and only modes i ∈ I are active,

M (x(t ∗ ),t) ∈ BR1 (y),

t ∗  t  t ∗ + T.

(15.148)

490

15 Switched Systems

Since y belongs to the Z -limit set, at certain moment t ∗ we have the trajectory x(t ∗ ) ∈ Br (y). Recalling Assumption A4 a j ∈ J mode will be active at some moment te  t ∗ + T . Using (15.148) we conclude that the trajectory M (x(t ∗ ),t), (t ∗  t  te ) remains in BR1 (y). Then as a mode j ∈ J starts to run, according to (15.147) after time D we will have 1x(te + D )1 < 1y1, which is a contradiction. 3 2 In general, it is not straightforward to verify A5. We thus give a sufcient condition here. Proposition 15.16. If rank(Ai ) = rank(Ai + ATi ), equivalently, in the original coordinate frame: rank(Ai ) = rank(PAi + ATi P), i = 1, · · · , N, then A5 is satised. Proof. Lemma 15.17 says that Ker(Ai ) ⊂ Ker(Qi ). Now since rank(Ai ) = rank(Qi ), we have dim(Ker(Ai )) = dim(Ker(Qi )). Therefore, Ker(Ai ) = ker(Qi ) = Zi . Then Vi ⊂ Ker(Ai ). Meanwhile, since Ker(Ai ) is Ai invariant and contained in Ker(Qi ), so, Ker(Ai ) ⊂ Vi . We conclude that Vi = Ker(Ai ). 3 2 A particular interesting case is the following: Corollary 15.8. If Ai , equivalently in the original coordinate frame: P1/2 Ai P−1/2 , i ∈ / , are symmetric, then A5 is satised automatically. Proof. From the proof of Proposition 15.16, one sees that as long as Ker(Ai ) = Ker(Qi ) assumption A5 of Theorem 15.14 is satised automatically. Since now Qi = −(Ai + ATi ) = −2Ai , the conclusion follows. 3 2 We give an example to describe the results obtained above. Example 15.13. Consider the following switched system x = AV (t) x,

x ∈ R4 ,

where V (t) ∈ / = {1, 2, 3}, ⎤ ⎤ ⎡ ⎡ −6 −12 12 −6 −7 −18 19 −10 ⎢ 5 10 −10 5 ⎥ ⎢ 23 59 −63 31 ⎥ ⎥ ⎥ A2 = ⎢ A1 = ⎢ ⎣ 2 4 −4 2 ⎦ ; ⎣ 16 41 −44 21 ⎦ ; −1 −2 2 −1 −7 −18 19 −10 ⎡ ⎤ 26 65 −68 33 ⎢−52 −130 134 −70⎥ ⎥ A3 = ⎢ ⎣−32 −80 82 −44⎦ . 14 35 −36 19 Choosing

⎤ 10 25 −25 16 ⎢ 25 63 −62 43 ⎥ ⎥ P=⎢ ⎣−25 −62 66 −30⎦ > 0, 16 43 −30 60 ⎡

(15.149)

15.5 LaSalle’s Invariance Principle for Switched Systems

491

we convert the system into the normal coordinate frame with P = I4 . Setting y = P1/2 x, we have (by using the same notations) ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ −1 0 0 0 00 0 0 0 0 0 0 ⎢ 0 0 0 0⎥ ⎢0 0 0 0 ⎥ ⎢0 −2 1 0⎥ ⎥ ⎢ ⎥ ⎢ ⎥ A1 = ⎢ ⎣ 0 0 0 0⎦ ; A2 = ⎣0 0 −1 0 ⎦ ; A3 = ⎣0 0 −1 0⎦ . 0 000 0 0 1 −1 0 0 0 0 Then we simply have Qi = Ai + ATi , which are ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ −2 0 0 0 00 0 0 0 0 0 0 ⎢ 0 0 0 0⎥ ⎢0 0 0 0 ⎥ ⎢0 −4 1 0⎥ ⎥ ⎢ ⎥ ⎢ ⎥ Q1 = ⎢ ⎣ 0 0 0 0⎦ ; Q2 = ⎣0 0 −2 1 ⎦ ; Q3 = ⎣0 1 −2 0⎦ . 0 000 0 0 1 −2 0 0 0 0 Then

⎤ −2 0 0 0 ⎢ 0 −4 1 0 ⎥ ⎥ Q = Q1 + Q2 + Q3 = ⎢ ⎣ 0 1 −4 1 ⎦ < 0. 0 0 1 −2 ⎡

We conclude by Theorem 15.14 that system (15.149) is globally asymptotically stable if the switching signal satises A3 and A4. Finally, in order to apply our results to the multi-agent consensus problem we need to modify (in fact weaken) Assumption A4 to the following A4< . There exists a T > 0, such that for any t  0

¦

Q j > 0,

(15.150)

j∈J[t,t+T ]

where J[t,t + T ] := {V (s) | s ∈ [t,t + T ]}. Remark 15.15. In the study of consensus problems a common assumption is that there exist a T0 > 0 and a sequence 0 < t1 < t2 < · · · → f, ti+1 − ti  T0 , ∀i, such that the graphs over [ti ,ti+1 ] are jointly connected. This requires that

¦

Q j > 0,

∀i.

(15.151)

j∈J[ti ,ti+1 ]

In fact (15.150) and (15.151) are equivalent. (15.150) ⇒ (15.151) is obvious. To prove (15.151) ⇒ (15.150) it is enough to set T = 2T0 . Corollary 15.9. Theorem 15.14 remains true when Assumption A4 is replaced by A4< . Proof. According to (15.132), A4< assures that on each time duration [t,t +T ], ∀ t  0, certain j ∈ J mode must be active. So the argument of the proof in Theorem 15.14 remains true. 3 2 This section is based on [7].

492

15 Switched Systems

15.6 Consensus of Multi-Agent Systems In coordination problems for multiple agents, tools from the algebraic graph theory are used frequently. Thus we rst introduce some basic concepts from the graph theory. More details can be found in [10, 14]. Let G = (V , E , A) be a weighted undirected graph of order n with the set of vertices V = {1, 2, · · · , n}, the set of unordered edges E = {(i, j) : i, j ∈ V } ⊂ V × V , and the weighted adjacency matrix A = [ai j ] ∈ Rn×n , where aii = 0 and ai j = a ji = 1. ai j = 1 if and only if there is an edge between agent i and agent j (i.e. ai j = a ji = 1 ⇔ (i, j) ∈ E ) and the two agents are called adjacent (or they are neighbors). The set of neighbors of vertex i at time t is denoted by Ni (t) = { j ∈ V : (i, j) ∈ H , j := i}. The Laplacian matrix of a graph G is dened as the following symmetric matrix L = D − A, where D = diag{d1 , · · · , dn } ∈ Rn×n is the degree matrix of G with diagonal elements di = ¦ ai j for i = 1, · · · , n. By denition, every row sum of L is zero, that j∈Ni

is, L1 = 0, where 1 = (1, · · · , 1)T . In the graph, if in addition to n agents, there is a special agent called a leader. The adjacency graph of n agents is denoted by G ; adding the leader, we have G¯. “The graph G¯ is connected” means that at least one agent in each component of G is connected to the leader. In addition, we dene B = diag{b1 , · · · , bn } as the leader adjacency matrix associated with G¯, where bi = 1 if agent i is adjacent to the leader and otherwise bi = 0. We give an example to depict these. Example 15.14. Consider graph (a) in Fig. 15.8. In this graph we have

Fig. 15.8

Graphs

V = {1, 2, 3, 4}, E = {(1, 2), (2, 3), (3, 4)}. ⎡ ⎤ ⎡ ⎤ 0100 1000 ⎢1 0 1 0⎥ ⎢0 2 0 0⎥ ⎥ ⎢ ⎥ A=⎢ ⎣0 1 0 1⎦ ; D = ⎣0 0 2 0⎦ . 0010 0001 Consider graph (b). It is obtained from (a) by adding a lead, agent “0”. Now in addition to A and D, we need an additional matrix B to describe the adjacency of other agents to the leader. It is easy to see that

15.6 Consensus of Multi-Agent Systems

⎡

000 ⎢0 0 0 B=⎢ ⎣0 0 1 000

493

⎤ 0 0⎥ ⎥. 0⎦ 1

Denition 15.17. The union of a collection of simple graphs, {G1 , G2 , · · · , Gm }, each with vertex set V , is dened as a simple graph, denoted by ∪ni=1 Gi , with vertex set V and edge set equaling the union of the edge sets of all the graphs in the collection. In this section we consider two cases: (i). two dimensional agent model with a leader; (ii). n dimensional agent model without a leader.

15.6.1 Two Dimensional Agent Model with a Leader Consider the following double integrator system of n agents: J qi = pi , pi = ui ,

(15.152)

where qi , pi , ui ∈ R, i = 1, 2, · · · , n, denote the position (or angle), velocity (or angular velocity) and control input of agent i respectively. There does not seem to exist an obvious way to adopt the matrix analysis method directly to the second order dynamics. Thus we will use instead the LaSalle invariance approach to solve the consensus problem. In this subsection we consider the consensus problem for leader-following multi-agent systems. The dynamics of the leader is expressed as follows: q0 = p0 ,

(15.153)

where p0 ∈ R is the desired constant velocity known to all agents. Our control aim is that all the agents follow the leader asymptotically and the desired velocity of all the agents converges to p0 , namely, qi → q0 , pi → p0 , ∀i, as t → f. We use the following neighbor-based feedback control law: ui = −

¦

ai j (t)(qi −q j )−bi (t)(qi −q0 )−k(pi − p0 ), i = 1, 2, · · · , n, (15.154)

j∈Ni (t)

where k > 1 is the control parameter. Let xi = qi − q0 , vi = pi − p0 , i = 1, 2, · · · , n. Then (15.154) can be rewritten as: ui = −

¦

ai j (t)(xi − x j ) − bi(t)xi − kvi , i = 1, 2, · · · , n.

(15.155)

j∈Ni (t)

Denote

⎡ ⎤ H1 ⎢ .. ⎥ H = ⎣ . ⎦ ∈ R2n ,

> ? x Hi = i ∈ R2 , vi

⎡ ⎤ u1 ⎢ .. ⎥ u = ⎣ . ⎦ ∈ Rn .

un Hn With (15.152) and (15.155), the closed-loop system for the followers can be expressed as:

494

15 Switched Systems

J x = v v = −(LV (t) + BV (t) )x − kv,

(15.156)

where LV (t) is the Laplacian matrix and BV (t) = diag(b1 (t), · · · , bn (t)) is the leader adjacency matrix. We use V (t) to emphasize that they change values only when the switchings occur. Putting the variables together, we have

H = FV H , where FV = In ⊗

(15.157)

? ? > > 0 1 0 0 + HV ⊗ ∈ R2n × R2n, 0 −k −1 0

HV = LV + BV .

The next lemma is from [13], which will be used in the sequel. Lemma 15.18. If graph G¯p (p ∈ / ) is connected, then the symmetric matrix H p associated with G¯p is positive denite. An easy consequence is Corollary 15.10. [12] Let the matrices Hi1 , · · · , Him be associated with the graphs G¯i1 , · · · , G¯im , respectively. If these graphs are jointly connected, then

m

¦ Hi j is posi-

j=1

tive denite. The following theorem shows the consensus can be obtained. Theorem 15.15. Consider the system (15.157). Assume the switching signal V (t) satises A3, and there exists a T > 0 such that for any t  0, the collection of the interaction graphs across each interval [t,t + T ] is jointly connected, then the consensus is reached asymptotically, namely, lim H (t) = 0.

(15.158)

t→f

Proof. Take a positive denite matrix > ? k1 ∈ R2n × R2n , P = In ⊗ 11

k>1

as a Lyapunov candidate. Then Q p :=

−(FpT P + PFp)

? > > ? 0 0 21 + Hp ⊗ , = In ⊗ 0 2(k − 1) 10

p ∈ /.

Since H p (p ∈ / ) are positive semi-denite, there exist orthogonal matrices U p (p ∈ > ? / ) such that / 0 U pT H pU p = p , 0 0 where / p = diag{O1 (H p ), · · · , Or p (H p )} with Oi (H p ), i = 1, · · · , r p , the nonzero eigenvalues of H p . r p > 0 is the rank of H p . At the same time, ? ? > ? > > 0 0 / 0 0 1 , (15.159) ⊗ + p F¯p := (U p ⊗ I2)T Fp (U p ⊗ I2 ) = In ⊗ −1 0 0 0 0 −k

15.6 Consensus of Multi-Agent Systems

? > > ? > ? 0 0 21 /p 0 T ¯ Q p := (U p ⊗ I2 ) Q p (U p ⊗ I2 ) = In ⊗ + . ⊗ 0 2(k − 1) 10 0 0

495

(15.160)

Choose 1 (15.161) max{Oi (H p ), p ∈ / , i = 1, 2, · · · , r p } + 1, 4 > ? then 2Oi (H p ) Oi (H p ) > 0, Oi (H p ) 2(k − 1) which implies that Q p  0, p ∈ / . Obviously, P is a CJQLF. In fact, since {G¯p , p ∈ / } contain all possible simple graphs of n + 1 agents, there exists at least one p0 ∈ / such that G¯p0 is connected, which implies Q p0 > 0. Therefore, ¦ Q p > 0. k>

p∈/

Denote J[t,t + T ] = {V (s) | s ∈ [t,t + T ]}, |J[t,t + T ]| = mt . On each interval [t,t + T ], since {G¯p , p ∈ J[t,t + T ]} is jointly connected, by Corollary 15.10, ¦ Hp > 0, ∀ t  0. p∈J[t,t+T ]

F

Dene

O = max{Oi ∈ V

G

¦

H p , ∀ t  0},

p∈J[t,t+T ]

which is well dened since {J[t,t + T ], ∀ t  0} ⊂ 2/ is a nite set. Take k > 1 4 max{O , Oi (H p ) | p ∈ / , i = 1, 2, · · · , r p } + 1, then > > ? ? 0 0 21 Q = I ⊗ H ) ⊗ + ( ¦ p n 0 2mt (k − 1) ¦ p 1 0 > 0. p∈J[t,t+T ]

p∈J[t,t+T ]

A4<

Therefore, the assumption in last session holds. Furthermore, by (15.159) and (15.160), we can easily obtain that rank(F¯p ) = rank(Q¯p ), p ∈ / , which implies A5 of Theorem 15.14 is satised by observing Proposition 15.16. Therefore, by Corollary 15.9, (15.158) is obtained. 3 2

15.6.2 n Dimensional Agent Model without Lead Consider a system with N agents. The dynamics of each agent is xi = Axi + Bui,

xi ∈ Rn , ui ∈ Rm ,

i = 1, · · · , N,

(15.162)

where we assume rank(B) = m. Let xi = (xi1 , xi2 , · · · , xin )T ∈ Rn denote the state of agent i. Dene zi =

¦ (xi − x j ),

i = 1, · · · , N,

(15.163)

j∈Ni

where Ni denotes the set of neighbors of agent i. zi is considered as the local information available for agent i. Denition 15.18. Consider system (15.162). The consensus is said to be achieved using local information if there is a local state error feedback control

496

15 Switched Systems

ui = Kzi ,

i = 1, · · · , N,

(15.164)

such that lim 1xi − x j 1 = 0,

t→f

i, j = 1, · · · , N.

(15.165)

We rst need some preparations. According to the denition, the Laplacian matrix LG of a graph G is ⎧ ⎪ ⎨|Ni |, i = j, (15.166) LG = [li j ]N×N , where li j = −1, j ∈ Ni , ⎪ ⎩ 0, otherwise. By the denition, every row sum of L is zero. We also use 1N = (1, 1, · · · , 1)T ∈ RN ; || · || denotes the Euclidean norm. The following lemma [21, 10] shows some basic properties of the Laplacian L. Lemma 15.19. Let L be the Laplacian of an undirected graph G with N vertices, O1  · · ·  ON be the eigenvalues of L. Then 1. 0 is an eigenvalue of L and 1N is the associated eigenvector, that is, L1N = 0; 2. If G is connected, then 0 is the algebraically simple eigenvalue of L and

O2 =

min

[ :=0,[ ⊥1N

[ T L[ > 0, [ T[

which is called the algebraic connectivity of G ; 3. If 0 is the simple eigenvalue of L, then it is an n multiplicity eigenvalue of L ⊗ In and the corresponding eigenvectors are 1N ⊗ Gi , i = 1, · · · , n. Go back to the consensus problem. The following observations are basically from [9] with some trivial modication. Tentatively, we assume the topology is xed, then we can drop the subscript G of LG . Denote by x and z the concatenations of vectors {x1 , · · · , xN } and {z1 , · · · , zN }, respectively. From (15.163), we have z = (L ⊗ In )x.

(15.167)

Then the closed-loop system of (15.162) with control (15.164) becomes x = [IN ⊗ A + (IN ⊗ BK)(L ⊗ In )] x.

(15.168)

Since L is symmetric, there is an orthogonal matrix T such that T LT T = D = diag(O1 , O2 , · · · , ON ) is diagonal, where {Oi } = V (L) is the spectrum of L. Now let x˜ = (T ⊗ In)x, then (15.168) becomes

(15.169)

15.6 Consensus of Multi-Agent Systems

x˜i = [A + OiBK] x˜i ,

i = 1, · · · , N.

497

(15.170)

First, we consider the case when the adjacent topology is xed. We want to nd a common K that stabilizes the subsystems in (15.168) with i = 2, · · · , N. We need Lemma 15.20. [2] Let n be a positive integer and let P(s) be a stable polynomial of degree n − 1: P(s) = p0 + p1s + · · · + pn−1 sn−1

with all pi > 0.

Then there exists an D > 0 such that Q(s) = P(s) + pn sn is stable if and only if pn ∈ [0, D ). Lemma 15.21. Consider a nite set of linear systems xi = Axi + OiBui ,

i = 1, · · · , k,

(15.171)

where x ∈ Rn , u ∈ Rm , (A, B) is completely controllable, rank(B) = m, and Oi > 0, i = 1, · · · , k. Then there exists a K which simultaneously assign the poles of k systems as negative as possible. Precisely, for any M > 0 there exist ui = Kxi ,

i = 1, · · · , k,

such that ReV (A + OiBK) < −M,

i = 1, · · · , k.

(15.172)

Proof. Without loss of generality, we can assume the pair (A, B) is in Brunovsky canonical form. We prove it in the following two cases: Case 1. Assume m = 1. Then the characteristic polynomials for A + Oi BK are Pi (s) = sn − Oikn−1 sn−1 − · · · − Oi k1 s − Oi k0 − pa (s),

i = 1, · · · , k,

(15.173)

where pa (s) = an−1 sn−1 + · · · + a1 s + a0 . Let Pn−1 (s) = dn−1 sn−1 + · · · + d1s + d0 be any Hurwitz polynomial. Using Lemma 15.17, there exist a > 0 such that when dn < a, sn + d1n Pn−1 (s) is also Hurwitz. Let O ∗ = min Oi . Then 1ik

Pni (s) := sn + Oi

2 Pn−1 (s) aO ∗

is Hurwitz. Denote the roots of Pni (s) by {−si1 , · · · , −sin }, then Re(sij ) > 0. Dene

498

15 Switched Systems

U

Pni (P , s)

V 2 = s + Oi P ∗ dn−1 sn−1 + · · · aO U V U V 2 2 +Oi P n−1 ∗ d1 s + Oi P n ∗ d0 , aO aO n

(15.174) i = 1, · · · , k.

Choose k∗j = −P n− j

2 d j, aO ∗

j = 0, · · · , n − 1.

(15.175)

It is easy to see, by singular perturbation analysis, that when P is sufciently large the effect of pa (s) on the roots of Pi (s) is negligible. Thus when P is sufciently large and M , P> min{Re(sij )|i = 1, · · · , k; j = 1, · · · , n} inequality (15.172) is satised. Case 2. We consider the multi-input case. When m > 1, the characteristic polynomials for A + OiBK are Pi (s) = =

m

– Pji (s) + Qi (s) j=1 m

–(sr j − Oikr j −1 sr j −1 − · · · − Oik1 s − Oik0 ) + Qi(s), j

j

j

j=1

i = 1, · · · , k,

where r j , j = 1, · · · , m are controllability indices,

m

¦ r j = n and

j=1

Qi (s) = pn−1 (Oi , K)sn−1 + · · · + p1 (Oi , K)s + p0 (Oi , K). Let

klj∗ = P r j −l klj ,

j = 1, · · · , m, l = 0, · · · , r j − 1.

Then by Leibniz formula one can easily see that lim

P →f

pl (Oi , K) = 0, P n−l

l = 0, · · · , n − 1.

For each Pji (s), repeat the process of Case 1, we can nd P j large enough, such that the real parts of the roots of Pji (s) are as negative as possible. Choose

P > max{P j , j = 1, · · · , m}. It is easy to see, by singular perturbation analysis, that when P is sufciently large m

the effect of Qi (s) on the roots of – Pji (s) is negligible. Thus when P is sufciently j=1

large, inequality (15.172) is satised.

3 2

Corollary 15.11. Assume (A, B) is a controllable pair, Oi > 0, i = 1, · · · , k. Then for any W > 0 and H > 0 there exists a K such that

15.6 Consensus of Multi-Agent Systems

9 9 9 (A+Oi BK)W 9 9e 9 < H,

i = 1, · · · , k.

499

(15.176)

Proof. Using {k∗ } as dened in (15.175), we know that A + Oi BK has eigenvalues as H I V (A + OiBK) = − P sij | j = 1, · · · , n , i = 1, · · · , k, where Re(−sij )  −V < 0. It is well known that 1e(A+OiBK)W 1  Q(P )e−V PW ,

(15.177)

where Q(P ) is a polynomial of P (see for example [26]). Since lim Q(P )e−V PW = 0,

P →f

3 2

the result follows.

Now we are ready to consider the consensus problem. First, if the adjacent topology is connected and xed, the result is obvious. Proposition 15.17. Consider system (15.162). Assume the adjacent topology is connected and xed. If (A, B) is controllable, then the consensus is achieved via local state error feedback (15.164). Proof. Since the graph is connected, we have 0 = O1 < O 2  O3  · · ·  ON . Using (15.167), when z(t) → 0, t → f, x → 1N ⊗ s, for some s ∈ Rn . So the consensus is obtained. (More precise argument can be found in the second part of this section.) Now, by the denition of z, it is clear that the consensus is achieved, if and only if z(t) → 0, t → f. Using (15.167) and (15.169), we have ⎡ ⎤ 0 ⎢ O2 x˜2 ⎥ ⎢ ⎥ (T ⊗ In )z = (D ⊗ In )x˜ = ⎢ . ⎥ . (15.178) ⎣ .. ⎦

ON x˜N From Lemma 15.21 it is clear that we can nd the feedback law K which simulta3 2 neously stabilizes x˜i , i = 2, · · · , N, and hence z(t) → 0, t → f. Next, we consider the case when the adjacent topology is time-varying and connected. When the adjacent graph is switching, we dene a switching signal V (t) : [0, +f) → {1, 2, · · · , m} which is a piecewise constant right continuous function. A switching system is said to have a non-vanishing dwell time, if there is a positive time period W ∗ > 0, such that the switching moments 0 < t1 < · · · < tk < · · · satisfy inf(tk+1 − tk ) = W ∗ . In the following we assume k

A6. Admissible switching signals have a dwell time W ∗ > 0.

500

15 Switched Systems

Let / be the set of all possible graphs and /c ⊂ / the set of connected graphs. The following is the rst result for varying topology. Theorem 15.16. Consider system (15.162) with varying topology and Assumption A6 holds. Assume (A, B) is controllable and its adjacent graph is connected, then the consensus can be achieved by local state error feedback (15.164). Proof. Note that in this theorem the neighbor set of an agent i, denoted by Ni (t), is time-varying. So we consider a non-switched duration [D , E ), and assume its graph is G p with the Laplacian L p , where p ∈ /c . Denote by Tp L p TpT = D p , where D p = p p diag(0, O2 , · · · , ON ). Using (15.178), we have ⎤ ⎤ ⎡ ⎡ 0 0 ⎢ O p x˜2 (t) ⎥ ⎢ O p x˜2 (D ) ⎥ ⎥ ⎥ ⎢ 2 ⎢ 2 −1 −1 z p (t) = (Tp ⊗ In ) ⎢ ⎥ = (Tp ⊗ In )E p (t) ⎢ ⎥ .. .. (15.179) ⎦ ⎦ ⎣ ⎣ . . p N p N ON x˜ (t) ON x˜ (D ) = (Tp−1 ⊗ In )E p (t)(Tp ⊗ In )z p (D ), t ∈ [D , E ), where ⎤ ⎡ 0 0 ··· 0 A @ ⎥ ⎢0 exp (A + O pBK)(t − D ) · · · 0 2 ⎥. E p (t) = ⎢ ⎦ ⎣0 0 0 A @ 0 0 · · · exp (A + ONp BK)(t − D ) Using Corollary 15.11, we can conclude from (15.176) the following two facts: • For any given H > 0 there exists a K such that 9 9 −1 9(Tp ⊗ In )E p (t)(Tp ⊗ In )9 < H , t  D + W ∗ , ∀ p ∈ /c .

(15.180)

This is due to the fact that the cardinality |/c | < f. • As long as K is chosen there is a boundary M(K) < f for overshoot. That is, 9I H9 −1 9(Tp ⊗ In)E p (t)(Tp ⊗ In )9 < M(K). (15.181) max sup p∈/c D tD +W ∗

This is due to the continuity. Dene the synchronization manifold S := {x ∈ RnN : x1 = · · · = xN } = {1N ⊗ s|s ∈ Rn }. Our aim is to show that x will converge to S . For any x ∈ RnN , we can decompose it with respect to each p ∈ / by x = S + K, where S = 1N ⊗ s ∈ S , K ∈ S ⊥ , the orthogonal complement of S . Noting that (L p ⊗ In)S = (L p ⊗ In )(1N ⊗ s) = (L p 1N ) ⊗ (In s) = 0,

15.6 Consensus of Multi-Agent Systems

501

we have z p = (L p ⊗ In )x = (L p ⊗ In )K ,

(15.182)

and the distance of x to S satises d(x, S ) = ||K ||. Since the graph G p is connected, by Lemma 15.19, 0 is the algebraically simple eigenvalue of L p , and is also the eigenvalue of L p ⊗ In with multiplicity n. All the other eigenvalues of L p are positive. The n linearly independent eigenvectors associated with the eigenvalue 0 of L p ⊗ In are 1N ⊗ Gi , i = 1, · · · , n. Since K ∈ S ⊥ , then K ⊥(1N ⊗ Gi ), i = 1, · · · , n. We have 9 9 Omp 1K 1  9(L p ⊗ In )K 9  OMp 1K 1 , where Omp and OMp are the second smallest and the largest eigenvalues of L p , respectively. Set H I Om = min {Omp } > 0, OM = max OMp , p∈/c

p∈/c

then we have

Om ||K ||  ||(L p ⊗ In )K ||  OM ||K ||,

∀ p ∈ /c .

(15.183)

Let {tk }f k=1 be the switching moments and assume tk → f as k → f. Assume on [tk−1 ,tk ), the mode pk ∈ / is active. Choosing

H=

Om G, OM

where 0 < G < 1,

equations (15.179), (15.180) and Assumption A6 yield that there exists a K such that on each interval [tk−1 ,tk ), ||z pk (tk− )||  H ||z pk (tk−1 )||. Using both (15.182) and (15.183), we get

Om ||K (tk )||  ||(L pk ⊗ In )K (tk )|| = ||z pk (tk− )||  H ||z pk (tk−1 )|| = H ||(L pk ⊗ In )K (tk−1 )||  HOM ||K (tk−1 )||.

(15.184)

It follows that ||K (tk )||  G ||K (tk−1 )||,

k = 1, 2, · · · .

Then we have lim 1K (tk )1 = 0.

k→f

Note that the feedback K is universal. Inequality (15.181) and the fact that 1K (t)1  imply

1 1z(t)1, Om

∀t  0

(15.185)

502

15 Switched Systems

1K (t)1 

M(K)OM 1K (tk−1 )1, Om

tk−1 < t < tk−1 + W ∗ .

Hence lim 1K (t)1 = 0,

t→f

3 2

which means the consensus is achieved.

Finally, we consider the case when the graph has frequently connected topology. Denition 15.19. System (15.162) is said to have frequently connected topology with time period T , if there exists a T > 0, for any t > 0, there exists a t ∗ ∈ [t,t + T ) such that the graph G (t ∗ ) is connected. Throughout this section we assume A7. System (15.162) has a frequently connected topology with time period T . Under Assumption A7 we can nd an alternating connect-disconnect sequence of time segments. Namely, there is a time sequence W1 < t1 < W2 < t2 < · · · → f (refer to Fig. 15.9) such that • • • •

tk − Wk  W ∗ (refer to Assumption A6 for W ∗ ); W ∗  Wk+1 − tk < T ; G (t) is connected, ∀ t ∈ [Wk ,tk ); G (t) is not connected, ∀ t ∈ [tk−1 , Wk ).

Fig. 15.9

Frequent Connection

From the proof of Theorem 15.16 we have the following: Lemma 15.22. Let Assumption A6 hold. Then for a given 0 < G < 1, there exists a set of decentralized controls of the form (15.164) with an universal K such that 1K (tk )1  G 1K (Wk )1,

k = 1, 2, · · · .

(15.186)

So the problem is to investigate what happens during the time period [tk−1 , Wk ) when the graph is not connected. Now consider a non-switching duration [D , E ) ⊂ [tk−1 , Wk ). Assume {G P (t), P = 1, · · · , s} are connected components of G (t), t ∈ [D , E ), and the associated vertex sets are V P , P = 1, · · · , s. Denote the cardinality (size) of the P -th component by NP = |V P | ; the center of the P -th component by x¯P =

¦

i∈V P

NP

xi .

Similarly, we can dene the center of all agents, denoted by x. ¯ Then we have

15.6 Consensus of Multi-Agent Systems

503

Lemma 15.23. Assume t ∈ [D , E ), which is a non-switching duration. Then for the closed-loop system with local state error feedback control, the center of each connected component G P satises the following free drift equation: x¯P = Ax¯P ,

P = 1, · · · , s.

(15.187)

Proof. Since for each connected component we have

¦P zi = ¦P ¦ (xi − x j ) = 0,

i∈V

i∈V

j∈Ni

3 2

the conclusion follows immediately. Particularly, we have Corollary 15.12. The overall center x¯ satises (15.187) for all t  0.

Now let Pi ∈ V P , i = 1, · · · , NP , where NP = |V P |. Denote xP = ((xP1 )T , · · · , ∈ RnNP . Similar to the argument in Section 15.3, we split

(x PNP )T )T

xP = S P + K P , where S P ∈ S P , K P ∈ [S P ]⊥ , and S P is dened as I H S P := xPi = xP j | ∀ Pi , P j ∈ V P . The following lemma gives a precise expression of K P . Lemma 15.24.

K P = xP − 1NP ⊗ x¯P .

(15.188)

Proof. First of all, it is easy to see that  I H S P = 1NP ⊗ [  [ ∈ Rn . Then we have

1NP ⊗ x¯P ∈ S P .

Now a straightforward computation shows that ^ ] P x − 1NP ⊗ x¯P , 1NP ⊗ [ = 0,

∀[ ∈ Rn . 3 2

The conclusion follows. The same argument shows that

K = x − 1N ⊗ x. ¯ Let D  t < E . Then we have 9 9 1xPi (t) − x¯P (t)1  9xP (t) − 1NP ⊗ x¯P (t)9 = 1K P (t)1,

(15.189)

i = 1, · · · , NP . (15.190)

504

15 Switched Systems

Similarly, 9 9 i 9x (t) − x(t) ¯ 9  1K (t)1.

(15.191)

It follows from (15.191) that 1x¯P (t) − x(t)1 ¯  1K (t)1,

∀P .

(15.192)

Now assume there are two agents, belonging to two different connected compo< nents, say, Pi ∈ V P and P <j ∈ V P . We are ready to see how much they can diverge. 9 9 9 9 Pi P< 9x (t) − x j (t)9

9 < 9 9 9 < < 9 9 9 9  1xPi (t) − x¯P (t)1 + 9xP j (t) − x¯P (t)9 + 9x¯P (t) − x¯P (t)9 9 9 9 9 < 9 < 9 9 9  1K P (t)1 + 9K P (t)9 + 9x¯P (t) − x¯P (t)9 .

(15.193)

From the proof of Theorem 15.16 and by using the arguments to each connected component, one sees that as long as t − D  W ∗ we have 1K P (t)1  1K P (D )1 ,

t  D + W ∗,

∀P .

(15.194)

Moreover, note that S is a subspace of S P , and the distance to the whole space is always smaller than or equal to the distance to any of its subspaces. Hence, 1K P (t)1  1K (t)1. We conclude that

9 < 9 9 9 1K P (t)1 + 9K P (t)9  2 1K (D )1 ,

(15.195)

t  D + W ∗.

(15.196)

Using Lemma 15.23 and equations (15.191) and (15.192), we also have 9 9 9 9 99 < < 9 P 9 9 9 99 9x¯ (t) − x¯P (t)9  9eA(t−D ) 9 9x¯P (D ) − x¯P (D )9 9 [9 9 9 < 9\ 9 9 9 9 9 9 9 ¯ D )9 + 9x¯P (D ) − x( ¯ D )9  9eA(t−D ) 9 9x¯P (D ) − x( 9 9 9 9  2 9eA(t−D ) 9 1K (D )1 . (15.197) Plugging (15.196) and (15.197) into (15.193), we have 9 9 9\ 9 [ 9 9 9 9 Pi P< 9x (t) − x j (t)9  2 1 + 9eA(t−D ) 9 1K (D )1 , D + W ∗  t  E .

(15.198)

Using (15.189), we have N 9 9 1K (t)1  ¦ 9xi (t) − x(t) ¯ 9. i=1

9 9 Note that if 9xi − x j 9  H , ∀i, j, then

(15.199)

15.6 Consensus of Multi-Agent Systems

505

9 i 9 9x − x¯ 9  H . Using this fact and equations (15.198) and (15.199), we have 9 9\ [ 9 9 1K (t)1  2N 1 + 9eA(t−D ) 9 1K (D )1 , t  D + W ∗ .

(15.200)

Note that the estimation (15.200) is independent of the choice of the control, which assures that we can design control to drive the system to be consensus. Now we are ready to present our main result: Theorem 15.17. Assume A6, A7 hold. Then the consensus of system (15.162) can be achieved by using local state error feedback control (15.164) with suitably chosen coefcients K. Proof. Recall that the system has frequently connected topology with time period T . Since T  W ∗ , there exists an unique n0 ∈ Z+ such that (n0 − 1)W ∗ < T  n0 W ∗ . Now in the duration [tk−1 , Wk ) there are at most n0 times switching. Of course that the dwell time for each mode is less than or equal to T . Using (15.200) we have [ Y Z\n0 1K (Wk )1  2N 1 + e1AT 1 1K (tk−1 )1 . (15.201) Recall Lemma 15.22. We can choose a suitable set of coefcients, K, in the feedback control, such that the G in (15.176) satises

GD

G0 AEn , 2N 1 + e1AT 1 0 @

where 0 < G0 < 1. It follows that 1K (tk )1  G0 1K (tk−1 )1 ,

k = 1, 2, · · · .

The similar argument for the overshoots between {tk } as in the proof of Theorem 15.16 completes the proof. 3 2 Before giving an illustrative example, we begin with exploring a little bit more about the set of adjacent graphs, which depend only on the number of agents only. So they have a universal meaning. In general, we have |/ | = 2N(N−1)/2 different graphs. For directed graph, it would be |/ | = 2N(N−1) . In this section, we consider only undirected graphs. First, we want to order the graphs in / . They are one-one corresponding to their Laplacians. So we consider the Laplacians. Note that (l12 , · · · , l1N , l23 , · · · , l2N , · · · , l(N−1)N ) are independent elements in a Laplacian, which can be used to describe Laplacians. So we can simply use a binary number

506

15 Switched Systems

|l12 | · · · |l1N | |l23 | · · · |l2N | · · · |l(N−1)N | = |l12 | × 2 := k − 1,

N(N−1) −1 2

+ |l13 | × 2

N(N−1) −2 2

+ · · · + |l(N−2)N | × 2 + |l(N−1)N |

to index L’s as {Lk }, k = 1, 2, · · · , 2N(N−1)/2 . Now let N = 2. Then |/ | = 2. We have ? > ? > 00 1 −1 , ; L2 = L1 = −1 1 00 and G2 is connected. Let N = 3. Then |/ | = 8. We have ⎤ ⎤ ⎡ ⎡ 000 0 0 0 L1 = ⎣0 0 0⎦ ; L2 = ⎣0 1 −1⎦ ; 000 0 −1 1

⎤ 2 −1 −1 L8 = ⎣−1 2 −1⎦ . −1 −1 2 ⎡

··· ;

It is easy to see that |/c | = 4 and /c = {G4 , G6 , G7 , G8 } are connected, and the distinct positive eigenvalues O ’s are {1, 3}. Let N = 4. Then |/ | = 64. We have ⎤ ⎤ ⎡ ⎡ 0000 00 0 0 ⎢0 0 0 0⎥ ⎢0 0 0 0 ⎥ ⎥ ⎥ ⎢ L1 = ⎢ ⎣0 0 0 0⎦ ; L2 = ⎣0 0 1 −1⎦ ; 0000 0 0 −1 1 ⎤ ⎤ ⎡ ⎡ 3 −1 −1 −1 3 −1 −1 −1 ⎢−1 3 −1 −1⎥ ⎢−1 3 −1 −1⎥ ⎥ ⎥ ⎢ · · · ; L63 = ⎢ ⎣−1 −1 2 0 ⎦ ; L64 = ⎣−1 −1 3 −1⎦ . −1 −1 0 2 −1 −1 −1 3 Using Matlab, it is easy to calculate that there are |/c | = 38 connected graphs, which are H  I /c = G p  p ∈ P , (15.202) where

⎫ ⎧ ⎨ 12 14 15 16 20 22 23 24 27 28 29 30 31 ⎬ P = 32 36 38 39 40 42 44 45 46 47 48 50 51 . ⎭ ⎩ 52 54 55 56 57 58 59 60 61 62 63 64

There are 6 different positive O ’s, which are

O = {1.0000, 4.0000, 2.0000, 0.5858, 3.4142, 3.0000}.

(15.203)

When N = 5, |/ | = 1024, |/c | = 628. We list the rst and last 5 indexes p such that G p ∈ /c : ⎫ ⎧ ⎨ 76 78 79 80 84 ⎬ . P = ··· ⎭ ⎩ 1020 1021 1022 1023 1024 There are 20 different positive O ’s, which are

15.6 Consensus of Multi-Agent Systems

507

⎫ ⎧ ⎨ 1.0000 5.0000 2.3111 0.5188 3.0000 0.3820 2.6180 ⎬ O = 1.3820 4.3028 0.6972 4.1701 3.6180 2.0000 0.8299 . ⎭ ⎩ 2.6889 4.0000 4.4812 4.6180 4.4142 2.3820 This information is useful in control design. Example 15.15. Consider a system with 4 agents, satisfying xi = Axi + bui ,

xi ∈ R2 , i = 1, 2, 3, 4,

? 0 1 , −1 0

>

where A=

b=

(15.204)

> ? 0 . 1

First, we design controls K = [k10 , k20 ] to make A + OibK,

i = 1, · · · , 6

(15.205)

stable, where {Oi } are shown in (15.203). Choose K = [P 2 k10 , P k20 ], and initial values as > ? 6 1 , x (0) = 2

with k10 = −3, k20 = −2, P = 10

? −3 x (0) = , 5 >

2

? −4 x (0) = , 3 >

3

> ? 4 x (0) = . 5 4

Three cases are considered. Fig. 15.10 shows the consensus with switching frequently connected topology, in which over a time period 3T , two disconnected modes are active on the rst 2T duration and then one connected mode is active. The modes are also randomly chosen at switching moments. The four curves in each gure denote the trajectories of four agents. In all three cases, the trajectories of four agents will converge to a common circle which is the trajectory of the center x. ¯ This section is based on [7, 27].

Fig. 15.10

Consensus with Frequently Connected Topology

508

References

References 1. Bacciotti A, Mazzi L. An invariance principle for nonlinear switched systems. Sys. Contr. Lett., 2005, 54(11): 1109–1119. 2. Bhattacharyya S, Chapellat H, Keel L. Robust Control: The Parametric Approach. New Jersey: Prentice Hall, 1995. 3. Branicky M. Multiple Lyapunov functions and other analysis tools for switchedand hybrid systems. IEEE Trans. Aut. Contr., 1998, 43(4): 475–482. 4. Cheng D. Stabilization of planar switched systems. Sys. Contr. Lett., 2004, 51(2): 79–88. 5. Cheng D. Controllability of switched bilinear systems. IEEE Trans. Aut. Contr., 2005, 50(4): 511–515 6. Cheng D, Lin Y, Wang Y. Accessibility of switched linear systems. IEEE Trans. Aut. Contr., 2006, 51(9): 1486–1491. 7. Cheng D, Wang J, Hu X. An extension of LaSalle’s invariance principle and its application to multi-agent consensus. IEEE Trans. Aut. Contr., 2008, 53(7): 1765–1770. 8. Dayawansa W, Martin C. A converse Lyapunov theorem for a class of dynamic systems which undergo switching. IEEE Trans. Aut. Contr., 1999, 44(4): 751–760. 9. Fax A, Murray R. Information ow and cooperative control of vehicle formations. IEEE Trans. Aut. Contr., 2004, 40(9): 1453–1464. 10. Godsil C, Royle G. Algebraic Graph Theory. New York: Springer-Verlag, 2001. 11. Hespanha J. Uniform stability of switched linear systems: Extensions of LaSalle’s Invariance Principle. IEEE Trans. Aut. Contr., 2004, 49(4): 470–482. 12. Hong Y, Gao L, Cheng D, et al. Coordination of multi-agent systems with varying interconnection topology using common Lyapunov function. IEEE Trans. Aut. Contr., 2007, 52(5): 943–948. 13. Hong Y, Hu J, Gao L. Tracking control for multi-agent consensus with an active leader and variable topology. Automatica, 2006, 42(7): 1177–1182. 14. Horn R, Johnson C. Matrix Analysis. New York: Cambbridge Univ. Press, 1985. 15. Humphreys J. Introduction to Lie Algebras and Representation Theory, 2nd edn. New York: Springer-Verlag, 1970. 16. Jurdjevic V, Sallet G. Controllability properties of afne systems. SIAM J. Contr. Opt., 1984, 22(3): 501–508. 17. Khalil H. Nonlinear Systems, 3rd edn. New Jersey: Prentice Hall, 2002. 18. Liberzon D, Hespanha J, Morse A. Stability of switched systems: Lie-algebraic condition. Sys. Contr. Lett., 1999, 37(3): 117–122. 19. Mohler R. Nonlinear Systems, Volume II, Applications to Bilinear Control. New Jersey: Prentice Hall, 1991. 20. Narendra K, Balakrishnan J. A common Lyapunov function for stable LTI systems with commuting A-matrices. IEEE Trans. Aut. Contr., 1994, 39(12): 2469–2471. 21. Olfati-Saber R, Murray R. Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans. Aut. Contr., 2004, 49(9): 1520–1533. 22. Qiao Y, Cheng D. On partitioned controllability of switched linear systems Automatica, 2009, 41(1): 225–229. 23. Rugh W. Linear Systems Theory. New Jersey: Prentice-Hall, 1993. 24. Sun Z, Ge S, Lee T. Controllability and reachability criteria for switched linear systems. Automatica, 2002, 38(5): 775–786. 25. Susmann H. A general theorem on local controllability. SIAM J. Contr. Opt., 1987, 25(1): 158–194. 26. Vidyasagar M. Nonlinear Systems Analysis. New Jersey: Prentice Hall, 1993. 27. Wang J, Cheng D, Hu X. Consensus of multi-agent linear dynamic systems. Asian J. Contr., 2008, 10(2): 144–155.

Chapter 16

Discontinuous Dynamical Systems

This chapter discusses analysis and design problems of discontinuous dynamic systems represented by differential equations with discontinuous right-hand side. After a brief review of discontinuous dynamic systems, which is summarized in Section 16.1, Filippov framework including solution and some analyzing tools for control system design is explained in Section 16.2. Furthermore, a feedback design problem is investigated in Section 16.3 for a class of nonlinear systems which have cascaded structure. Section 16.4 demonstrates an application example of stability analysis for PD controlled mechanical systems.

16.1 Introduction System dynamics is usually represented by the differential equation of the state variables, i.e. x = f (t, x), (16.1) where x ∈ Rn is the vector of the state variables and f : R× Rn → Rn is a vector eld. As shown in the previous chapters, most theoretical results on analysis and control of dynamic systems are established under the assumption on the smoothness of the right-hand side of the differential equation. So that, the existence and uniqueness of the solution can be obtained from the conventional denition of the solution of the differential equation. However, for many applications, the dynamics of the system involves discontinuity. In other words, the right-hand side of the differential equation, which is used to describe the model of plant or the dynamics of the closed-loop with feedback control, is discontinuous. For example, for switching control systems [8, 16] or variable structure control systems [23, 24], the right-hand side is always discontinuous at the switching time in the time domain or at the switching surfaces in the state space. Furthermore, in engineering practice, a large number of phenomena can be represented by discontinuous functions such as the static friction in mechanical systems [3] and the traction force of vehicles [18, 20], the stroke changes in internal combustion engines, the contact force in biped robots etc. For such systems, the discontinuities arise in the vector eld that determines the system dynamics. From a theoretical point of view, for the differential equations with discontinuous right-hand side, a natural but fundamental question is whether the conventional solution still works or even exists? If not, what is the new framework of the solution, the existence and uniqueness etc? It should be warned that we have dealt with D. Cheng et al., Analysis and Design of Nonlinear Control Systems © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2010

510

16 Discontinuous Dynamical Systems

the switching systems in many cases without concerning these questions. In fact, it is possible thanks to the fact that the solution is dened locally and can consist of several local pieces along the time axis. Hence, we can analyze the behavior of solution even though the vector eld is discontinuous at the switching point or even without denition. However, it is not the case when the switching behaviors depend on the state variable, such as the switching surface. As it will be shown in the following section, the behavior of the solution on the surface might not be handled in the conventional framework. Indeed, in the eld of mathematics, the differential equation with discontinuous right-hand side is not a new topic. For instance, more than eight kinds of notions of solution have been proposed for the discontinuous dynamical systems in the past half century [11, 15, 7, 14, 9, 12, 13]. Correspondingly, the fundamental issues, such as uniqueness, continuous dependence on initial condition, stability and convergence, have been investigated by many literatures in the eld of mathematics (see [11] and the references therein). Moreover, several textbooks have been published in this area in the past decade [5, 10, 26]. However, in the control community, a few studies were reported on the discontinuous dynamical systems. During early stage, the notions of Filippov and Caratheodory solutions were addressed from the viewpoint of control system theory [21, 17, 21, 9], and the notions with stability criteria were proposed to explain some physical phenomena of mechanical systems [1, 25, 6, 19]. We will start this chapter with a brief review on the Filippov framework for differential equations with discontinuous right-hand side, mainly, the notion of Filippov solution and the stability results with some counter examples. The interested reader may refer to [11] and the aforementioned textbooks to further explore these topics. With these fundamental results, we will present a design technique to the stabilization of a class of nonlinear systems with discontinuity. Finally, an application example with physical background will be demonstrated.

16.2 Filippov Framework 16.2.1 Filippov Solution For the sake of simplicity, we only focus on autonomous systems. Consider the differential equation x = f (x), x(t0 ) = x0 , (16.2) where the vector eld f : Rn → Rn is measurable and essentially locally bounded. As is well-known, a solution of the differential equation (16.2) over an interval [t0 ,t1 ] is a continuous function x(t) : [t0 ,t1 ] → Rn such that x(t) is differentiable and satises x(t)  = f (x(t)) for all t ∈ [t0 ,t1 ]. The existence and uniqueness of the solution can be ensured under the condition that the vector eld f is Lipschitz continuous. However, if f is discontinuous at a switching condition related to the state variables, the uniqueness of the solution may be lost. At the worst case, one can not dene a solution in conventional sense due to the discontinuity. Filippov solution is

16.2 Filippov Framework

511

one of new notions to deal with the discontinuity that can determine the solution on the discontinuous points with a set-valued map associated to f . The Filippov set-valued map at x ∈ Rn is dened as follows q q @ A co f BG (x)\N . (16.3) K[ f ](x) ≡ G >0 P (N)=0

Where P is the Lebesgue measure. Equivalently [17] B C  K[ f ](x) = co lim f (xi )  xi → x, xi :∈ S f ∪ S , i→f

(16.4)

where S ⊂ Rn is any set of measure zero, and S f ⊂ Rn is the set of measure zero where f (x) is not dened. Denition 16.1. (Filippov Solution)[12] A function x(t) is called a Filippov solution of the differential equation (16.2) over [t0 ,t1 ], if x(t) is absolutely continuous and for almost all t ∈ [t0 ,t1 ] x(t)  ∈ K[ f ](x(t)). (16.5) In other words, a Filippov solution of the differential equation (16.2) dened on  ∈ K[ f ](x(t)) [t0 ,t1 ] is an absolutely continuous map x(t) : [t0 ,t1 ] → Rn such that x(t) for almost all t ∈ [t0 ,t1 ]. For the existence of Filippov solution, we have the following conclusion. Theorem 16.1. (Existence Condition)[9] If f (x) is measurable and locally essentially bounded, then there exists at least one Filippov solution of the differential equation (16.2) starting from any initial condition. Generally, Filippov solution of a discontinuous differential equation is not necessary unique. For example, the differential equation x = sgn(x),

x(0) = 0

(16.6)

is dened by the vector eld f (x) = sgn(x). Since the Filippov set-valued map K[ f ](x) can be calculated as ⎧ x>0 ⎨ 1, K[ f ](x) = [ − 1, 1], x = 0 ⎩ −1, x<0 it is easy to conrm that there are three maximal solutions starting from x(0) = 0 x1 (t) = −t, x2 (t) = 0, x3 (t) = t Indeed, Filippov solution of the differential equation (16.2) is a solution of differential inclusion associated with the Filippov set-valued map (16.5). Hence, using the uniqueness condition on solution of differential inclusion, we have the following proposition on the Filippov solution [11, 7, 5]. Theorem 16.2. (Uniqueness Condition) Let f (x) be measurable and locally essentially bounded. Assume that for all x ∈ Rn , there exist Jx , H ∈ (0, +f) such that for

512

16 Discontinuous Dynamical Systems

almost every x1 , x2 ∈ BH (x), ([1 − [2 )T (x1 − x2 )  Jx 1x1 − x2 122

(16.7)

holds for all [1 ∈ K[ f ](x1 ) and [2 ∈ K[ f ](x2 ). Then, for any x0 ∈ Rn , the differential equation (16.2) has a unique Filippov solution with the initial condition x(t0 ) = x0 . In a large number of applications, the vector elds dening dynamics of discontinuous systems have discontinuity at some surface only, i.e. the vector eld f : Rn → Rn is continuous everywhere except at a switching surface of the state space. In this case, the local solutions in each continuous domain are dened by continuous differential equations. Thus, the interested problem is the behavior of the solutions in a neighborhood of the surface. As shown in Fig. 16.1, if the vector eld is discontinuous at the surface S , and G + , G − are the continuous regions, then there are two possible situations at the points of the surface. One possible situation is that the Filippov solution moves away from the points, i.e. the solution will cross the surface, or there are two solutions that start from the points. The other situation is that the solution remains on the surface. The latter is so-called sliding mode motion [24] when the solution moves along the surface, or so-called stationary solution when the point is a stationary solution [26].

Fig. 16.1

Sketch of Discontinuous Vector Field

Let S be a smooth surface, and the domain G − and G + be separated by S . Suppose that the vector eld f (x) is bounded and its limiting vectors f − (x) and f + (x) exist when x ∈ S is approached from G − and G + . Let fN− and fN+ be the projection of the vectors f − (x) and f + (x) on the normal direction to the surface S directed from G − to G + . Theorem 16.3. (Existence of Sliding Motion)[12] Let the vector function x(t) be absolutely continuous and x(t) ∈ S , ∀t ∈ [t1 ,t2 ]. Suppose that fN− (x)  0, fN+ (x)  0, fN− (x) − fN+ (x) > 0 for ∀t ∈ [t1 ,t2 ]. Then, x(t) is a Filippov solution of (16.2) if and only if for almost all t ∈ [t1 ,t2 ],

16.2 Filippov Framework

x = f ◦ (x), where

f ◦ (x) = D f + (x) + (1 − D ) f −(x),

Fig. 16.2

f N+ , fN− and f ◦(x)

Fig. 16.3

G + , G − and f + (x)

Fig. 16.4

Filippov Solution x(t)

513

(16.8)

D = fN− /( fN− − fN+ ).

Theorem 16.4. (No Crossing)[12] If fN+  0 and x(t) ∈ S0 ⊂ S , ∀t ∈ [t1 ,t2 ], then for t1  t  t2 no solution can go out of S0 into G + . Theorem 16.5. (Crossing)[12] If fN+ > 0 and fN− > 0 at all points of S0 ⊂ S , then in the domain G + + S0 + G − we have uniqueness and continuous dependence of the Filippov solution on the initial conditions. Furthermore, the solution goes from G − to G + and has only one point in common with S0 .

16.2.2 Lyapunov Stability Criteria The key of Lyapunov stability criteria is to establish the monotonically decreasing evaluation of Lyapunov function along the trajectories of the solutions. To determine

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16 Discontinuous Dynamical Systems

the boundedness and the convergence of Filippov solution, Lyapunov’s method is still effective, i.e. we can also establish (asymptotical) stability if we can conrm the (strictly) decreasing evaluation of a Lyapunov-like function along all Filippov solutions. Hence, the remained problem is to establish a feasible evaluating tool for a given candidate Lyapunov function along all Filippov solutions. Denition 16.2. [9] A function V : Rn → R is said to be decreasing along the system (16.2), if for@ eachAsolution @ x(t) A of Filippov’s differential inclusion (16.5), we have t1  t2 ⇒ V x(t2 )  V x(t1 ) . The decrease property of a function V (x) along a solution x(t) can be determined by the time derivative of V . A candidate Lyapunov function should be differentiable almost everywhere. Since the function V ◦ I : R → R is an absolutely continuous function under the condition that V : Rn → R is locally Lipschitz continuous function and I : R → Rn is absolutely continuous (Filippov solution must be an absolute continuous function), if we choose the candidate Lyapunov function V (x) to be locally Lipschitz continuous, the time derivative of V exists almost everywhere. To deal with non-differentiable functions, we introduce the following results: Consider a continuous function V : Rn → R. Suppose that V (x) is not differentiable. Denition 16.3. (Clarke’s Generalized Gradient) [10] The generalized gradient of V (x) at x is dened by  H I w V (x) = ] ∈ Rn  V ◦ (x, v)  4v, ] 5, ∀v ∈ Rn , where V ◦ is a generalized directional derivative of V at x for given v V ◦ (x, v) = lim sup

y→x O →0

V (y + O v) − V(y) . O

Lemma 16.1. [10] Let V (x) be Lipschitz near x, and S be any set of Lebesgue measure 0 in Rn . Then, C B  (16.9) w V (x) = co lim KV (xi )  xi → x, xi :∈ S , xi :∈ :V , i→f

where :V is the set of points at which V (x) is not differentiable. Denition 16.4. (Set-Valued Derivative of V (x)) [9] The set valued derivative V (x) with respect to x ∈ K[ f ](x) is dened by B C  V (x) = a ∈ R  ∃v ∈ K[ f ], s.t. 4[ , v5 = a, ∀[ ∈ w V . Theorem 16.6. [9] Let V (x) be a Lipschtitz function. For each xed x ∈ Rn the set V (x) is closed and bounded interval, possibly empty. Moreover if V is differentiable B C V (x) = 4’V (x), v5 | v ∈ K[ f ](x) . With these analysis tools, we have the following conclusion.

16.2 Filippov Framework

515

Theorem 16.7. [9] Let V : Rn → R be a locally Lipschitz continuous function @ A and x(t) be absolutely continuous. Then, for almost all t, there exists [ ∈ w V x(t) such that, A d @ V x(t) = [ T x(t),  (16.10) dt where w V (x) denotes the generalized gradient of V . For applications, we can evaluate V along all Filippov solutions by the set-valued derivative V (x) with respect to K[ f ](x), and more practically, by the generalized time derivative V˜ (x), since we have [9, 21] A A a.e. @ d @ V x(t) ∈ V x(t) dt

(16.11)

and V (x) ⊂ V˜ (x), where the generalized time derivative of V along the differential inclusion (16.5) is given by [21] V˜ (x) =

q

[ T K[ f ](x).

(16.12)

[ ∈w V (x)

Therefore, we have the Lyapunov stability theorem for the Filippov solutions (see [9, 21]). Theorem 16.8. Consider the system (16.2). Let f : Rn → Rn be essentially locally bounded in a region Q ({x ∈ Rn | 1x1 < r} ⊂ Q) and 0 ∈ K[ f ](0). Furthermore, let V (x) be a regular function satisfying V (0) = 0 and V1 (1x1)  V (x)  V2 (1x1), where V1 ,V2 ∈ class K . Then, (i). V˜ (x)  0 in Q implies x(t) ≡ 0 is a uniformly stable solution; (ii). In addition, if there exists a class K function w(·) in Q with the property V˜ (x)  −w(x) < 0,

x := 0,

(16.13)

then the solution x(t) ≡ 0 is uniformly asymptotically stable † . Theorem 16.9. (LaSalle Invariance Principle) Let Z be a compact set such that every Filippov solution of the system (16.2) starting in Z is unique and remains in : for all t  t0 . Let V (x) be a regular function such that V˜ (x)  0.

(16.14)

S = {x ∈ : | 0 ∈ V˜ }.

(16.15)

Dene S by Then, every trajectory in Z converges to the largest invariant set M in the closure of S. Example 16.1. [9] Consider the system described by †

V˜ (x) < 0 implies that v  0, ∀v ∈ V˜ (x), since V˜ (x) is a set valued function.

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16 Discontinuous Dynamical Systems

>

? −sgn(x2 ) x = f (x) = . sgn(x1 )

(16.16)

The Filippov set-valued map K[ f ](x) can be calculated as follows (see Fig. 16.5). ⎧ {−sgn(x2 )} × {sgn(x1 )}, x1 := 0, x2 := 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ [−1, 1] × {sgn(x1 )}, x1 := 0, x2 = 0 ⎪ ⎨ K[ f ] = {−sgn(x )} × [−1, 1], (16.17) x1 = 0, x2 := 0 ⎪ 2 ⎪ ⎪ ⎪ ⎪ a> ? > ? > ? > ?b ⎪ ⎪ 1 −1 −1 1 ⎪ ⎪ , , , , x1 = 0, x2 = 0. ⎩ co 1 1 −1 −1

Fig. 16.5

Vector Field (16.16)

From Theorem 16.1 and 16.2, the system (16.16) has a unique Filippov-solution. Consider a candidate Lyapunov function V (x) =| x1 | + | x2 |. Then, the generalized gradient can be calculated as ⎧ {sgn(x1 )} × {sgn(x2 )}, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ {sgn(x1 )} × {[−1, 1]}, ⎪ ⎨ w V = {[−1, 1]} × {sgn(x )}, ⎪ 2 ⎪ ⎪ ⎪ ⎪ a> ? > ? > ? > ?b ⎪ ⎪ 1 1 −1 −1 ⎪ ⎪ , , , , ⎩ co 1 −1 1 −1

x1 := 0, x2 := 0 x1 := 0, x2 = 0 x1 = 0, x2 := 0 x1 = 0, x2 = 0.

Thus, we have ⎧ {0}, ⎪ ⎪ ⎨ ∅, V (x) = ∅, ⎪ ⎪ ⎩ {0},

x1 := 0 x1 := 0 x1 = 0 x1 = 0

x2 := 0 x2 = 0 x2 := 0 x2 = 0.

16.3 Feedback Stabilization

517

For example, at a given point x0 = [1, 0]T , since a> ? b U1 , U1 ∈ [−1, 1] , K[ f ](x0 ) = 1 a> ? b 1 , U2 ∈ [−1, 1] . w V (x0 ) = U2 There exists no v ∈ K[ f ](x0 ) such that a = 4v, [ 5, ∀[ ∈ w V (x0 ). It follows V (x0 ) = ∅. In fact, we can see V˜ (x)  0 for any x and 0 ∈ V˜ (0). Therefore, from Theorem 16.8, the system is stable at x = 0.

Fig. 16.6

w V (x)

16.3 Feedback Stabilization In this section, we focus our attention on the feedback stabilization problem for a class of cascaded nonlinear systems with piecewise discontinuity. The key of feedback design is to handle the set-valued map that maps each state in the discontinuous surface to the set generated by the allowable control inputs. Then, the stabilization will be achieved by rendering the generalized derivative of candidate of Lyapunov function negative along the set-valued map. Consider a nonlinear system: ⎧ z = f0 (z) + f1 (z, s)s ⎪ ⎪ ⎪ ⎨s = a(z, s) + b(z, s) sgn(s) + y (16.18) ⎪ [ = f2 ([ ) + g([ )u ⎪ ⎪ ⎩ y = h([ ), where fi (·) (i = 0, 1, 2), g(·), a(·), b(·) are nonlinear smooth functions with appropriate dimensions according to the dimension of z ∈ Rn−1 , [ ∈ Rm , and s, y, u ∈ R. The function sgn(s) is dened by J s/|s|, s := 0 (16.19) sgn(s) = U, s = 0,

518

16 Discontinuous Dynamical Systems

where U ∈ [−1, 1] is unknown constant. Obviously, the system (16.18) has discontinuity and cascaded structure, i.e. the subsystem coordinated by (z, s) is driven by the scalar signal y which is the output of the driving subsystem coordinated by [ , and the control input for the whole system is u. In order to simplify the design problem, we assume that the [ -subsystem with respect to the output y has relative degree 1, and the zero dynamics driven by y is input-to-state stable. Then, under these assumptions, the [ -subsystem is feedback equivalent to the following system J K = d0 (K , y) (16.20) y = v, where d0 is a smooth function and v is the new control input. Therefore, the stabilization problem of the system (16.18) is equivalent to stabilization of the system coordinated by (z, s, K , y). Moreover, the dynamics coordinated by K is driven by y only. Thus, with the input-to-state stability condition, to provide a solution for the stabilization problem of the whole systems, we focus our attention on the following system: ⎧ ⎪ ⎨z = f0 (z) + f1 (z, s)s (16.21) s = a(z, s) + b(z, s) sgn(s) + y ⎪ ⎩ y = v.

16.3.1 Feedback Controller Design: Nominal Case First, we consider the case where all functions in the differential equations are exactly known. For the system (16.21), dene e = 3 (z, s, y) = y + a(z, s) + pL f1W (z), where p > 0 is any constant. Then, the system (16.21) becomes ⎧ ⎪ z = f 0 (z) + f1 (z, s)s ⎪ ⎨ s = −pL f1 W (z) + b(z, s) sgn(s) + e ⎪ ⎪ ⎩e = w 3 b(z, s) sgn(s) + R1 (z, s) + v, ws

(16.22)

(16.23)

where

w3 w3 { f0 (z) + f1 (z, s)s} − {pL f1 W (z) − e}. wz ws The next Theorem provides a desired feedback controller. R1 (z, s) =

Theorem 16.10. Suppose the system (16.21) satises the following condition: (i). For the dynamics z = f0 (z), there exists a positive denite function W (z) such that L f0 W (z)  0, ∀z. (ii). b(z, s) < 0, ∀s, z and |b(z, s)|  V (s), where V (s) is a class K function. For any given p > 0, let the feedback law be given by U V w3 w3 v = −R1 (z, s) − s − ke + J b(z, s) sgn e (16.24) ws ws

16.3 Feedback Stabilization

519

with any given constants J > 1 and k > 0. Then, the closed-loop system is Lyapunov stable and every Filippov solution converges to the bounded set M = {(z, s, e) | W (z)  H1 , s = 0, e = 0} where

(16.25)

b  V (0)  , f10 (z) = f1 (z, 0). H1 = min W (z), Q1 = z |L f 0 W (z)|  1 z∈Q1 p a

Proof. Dene z∗ = [zT , s, e]T , then the closed-loop system is represented as ⎤ ⎡ f0 + f1 s ⎥ ⎢ −pL f1 W + b sgn(s) + e ⎥ ⎢ z∗ =F(z∗ )=⎢ ⎥. U V ⎣w3 w3 w3 ⎦ b sgn(s) − s − ke + J b sgn e ws ws ws According to the Filippov framework, the solution of this discontinuous differential equation satises the following differential inclusion: z∗ ∈ K[F](z∗ ),

(16.26)

where K[·] is a set-valued function in the discontinuous point z∗ . In this case, on the surfaces s = 0 or ww3s e = 0 the function K takes set-value. Choose the candidate Lyapunov function as follows: V (z∗ ) = W (z) +

1 2 1 s + e2 . 2p 2p

(16.27)

Then, along the trajectories of the closed-loop system, the time derivative of V¯(z∗ ) can be calculated as 1 1 w3 V¯ =L f0 W (z) + b(z, s)|s| + e b(z, s) sgn(s) p p ws U V w3 w3 k 2 J sgn e − e + b(z, s)e p p ws ws    w3  1 k 2 1   L f0 W (z)+ b(z, s)|s|− e + (J−1)b(z, s)e p p p ws 

(16.28)

From (i) and (ii), this inequality concludes that V¯  0, ∀t  0 and V¯ < 0, ∀s := 0 or/and e ww3s := 0. It should be noted that V¯ is also a set valued and V¯  0 means that all elements of V¯ are negative. Hence, with the positive denite and proper function V (z∗), we have the following conclusion: z(t)∈Lf , s(t)∈Lf , e(t)∈Lf , i.e. the closedloop system is Lyapunov stable in the sense of Filippov solution. Moreover, by applying the Barbalat lemma, it is clear that s(t) → 0, e(t) → 0 as t → f. Thus, the remained task is to investigate the behavior of the Filippov solution on the surfaces Ss (z∗ )=s=0 and Se (z∗ )= ww3s e=0. With the help of LaSalle invariance principle as shown in Theorem 16.9, we can reach the conclusion. In fact, from (16.28) it is clear that the set ZV ={z∗ | 0 ∈V¯(z∗ )} is given by ZV ={ z∗ | s =0, e=0}. If the set M is the largest positive invariant set in ZV , then by the LaSalle invariance

520

16 Discontinuous Dynamical Systems

principle every Filippov solution of the closed-loop system will approach to M as t → f. In order to make M clear, we need to investigate the behavior of every solution included in ZV , which satises the differential inclusion (16.26). However, we do not need to discuss the solution on the surface ww3s e=0, since the surface and the set ZV do not have common part except for the origin s = 0. Therefore, we focus our attention on the behavior of the Filippov solution on the surface s=0 only. Note that any bounded region G of the space Rn+1 is divided by the smooth ∗ + ∗ surface Ss (z∗ ) = 0 into the domains G− s = {z | s < 0} and Gs = {z | s > 0}. On the surface the differential inclusion is described by ⎤ ⎡ f0 ⎥ ⎢ −pL f 0 W + b(z, 0)K1 ⎥, 1 K[F](z∗ ) = ⎢ (16.29) ⎦ ⎣w3 w3 b(z, 0)K1 + JK2 b(z, 0) ws ws where −1  K1  1, −1  K2  1. Moreover, the normal vector of the surface Ss (z∗ ) is given by Ns = [0 1 0]T . In addition, the limit vectors in ZV are given by ⎤ ⎡ f0 ⎥ ⎢ −pL f 0 W + b(z, 0) ⎥, f + (z∗ ) = lim F(z∗ ) = ⎢ 1 ⎦ ⎣ + w3 w3 s→0 ∗ b(z, 0) + JK2 b(z, 0) z ∈ZV ws ⎤ ⎡ ws f0 ⎥ ⎢ −pL f 0 W − b(z, 0) ⎥. 1 f − (z∗ ) = lim F(z∗ ) = ⎢ ⎦ ⎣ − w 3 w 3 s→0 − b(z, 0) + JK2 b(z, 0) z∗ ∈ZV ws ws Thus, we have

J

f N+s = NsT f + (z∗ ) = −pL f 0 W (z) + b(z, 0) 1 fN−s = NsT f − (z∗ ) = −pL f 0 W (z) − b(z, 0),

(16.30)

1

and

J

f N+s  0, if L f 0 W (z)  b(z, 0)/p

(16.31)

1

fN−s  0, if L f 0 W (z)  −b(z, 0)/p. 1

Furthermore, fN−s − fN+s  0, if |L f 0 W (z)|  V (0)/p. By Theorem 16.3, on the sur1 ∗ ∗ face  0, any absolutely continuous function z (t) that satises s = 0, e =  Ss (z ) =   0, L f 0 W (z)  V (0)/p, and 1

⎡ ⎢ z∗=D f ++(1−D)f −=⎣

D=

pL f 0 W+b(z, 0) 1

2b(z, 0)

f0 (z) 0

⎤

⎥ Z⎦ , w3 Y pL f 0 W+JK2 b(z,0) 1 ws

16.3 Feedback Stabilization

521

is a solution of (16.26). This implies that in the set ZV (⊂ Ss ), any solution is presented by z = f0 (z), and from condition (i), the solution is stable, i.e. any solution initialized at t0 in M dened by (16.25) will stay in M for all t  t0 . Hence, M is an invariant set. On the other hand, we have J − fNs > 0, fN+s > 0, if L f 0 W (z)  b(z, 0)/p 1 (16.32) fN−s < 0, fN+s < 0, if L f 0 W (z)  −b(z, 0)/p. 1

By Theorem 16.5, it implies that there is no solution will stay on the surface Ss(z∗) = 0, if the system is initialized in ZV \ M . Therefore, we can conclude that the set M is the largest invariant set in ZV . 3 2 It is interesting to observe that the bound of the invariant set, where the trajectory of the closed loop system will ultimately stay, depends on the switching surface of the plant, but not the switching surface of the controller. Also, the bound of the invariant set is determined by the design parameter p, i.e. to obtain a small invariant M , we can choose a lager p, however, it will make the feedback control gain higher. The other design parameters J and k do not affect the invariant set, but from (16.28), it is clear that larger J and k make rapid decreasing of V¯. Consequently, they render the convergence rate of the state faster. However, larger J and k will cause directly higher feedback gain. In practical application, a trade-off between the performance and the high gain is necessary.

16.3.2 Robust Stabilization In this section, we consider the system (16.21) with the unknown function a(z, s) under the following assumptions: (iii). For the dynamics z = f 0 (z), there exists a positive denite function W (z) such that L f0 W (z)  −1z12 . (iv). There exists a known function N(z, s) (N(0, 0) = 0) such that |a(z, s)|  T N(z, s) where T is an unknown constant. Note that, under the condition (iv) and Hadamard Lemma [4], the function N(z, s) can be decomposed by N(z, s) = N1 (z, s)z + N2 (z, s)s with appropriate functions N1 (z, s) and N2 (z, s). Let U V 1 ˆ2 2 e = D2 (z, s, y, Tˆ ) = y + T N1 (z, s)s + Tˆ 2 N22 (z, s)s + L f1 W (z) + s . 2 Then, the system (16.21) becomes as follows: ⎧ ⎪ ⎪ U V ⎪z = f0 (z) + f1 (z, s)s ⎪ ⎨ 1 ˆ2 2 T N1 (z, s)s + Tˆ 2 N22 (z, s)s + L f1 W (z) + s s = a(z, s) + b(z, s) sgn(s) + e − 2 ⎪ ⎪ ⎪ ⎪ ⎩e = w D2 a(z, s) + w D2 b(z, s) sgn(s) + R3 (z, s, Tˆ ) + v, ws ws (16.33) where

522

16 Discontinuous Dynamical Systems

w D2 ˆ w D2 ( f0 (z) + f1 (z, s)s) R3 (z, s, Tˆ ) = T+ w Tˆ U w z \V 1[ w D2 e − Tˆ 2 N12 (z, s)s + Tˆ 2 N22 (z, s)s + L f1 W (z) + s + ws 2 Theorem 16.11. Suppose the system (16.21) satises (i), (ii) and (iv). Let the adaptive control law be U V @ A w D2 w D2 ˆ ˆ v = −R3 (z, s, T ) − 2s + J b(z, s) − T N(z, s) sgn e − ke ws ws    w D2   N(z, s), (16.34) Tˆ = −2Ja |s|N(z, s) − Ja e ws  with any given constants J > 1, Ja > 0 and k > 0. Then the closed-loop system with the adaptive control (16.34) is Lyapunov stable, and any Filippov solution converges to the bounded set M = { (z, s, e, T˜ ) | z = 0, s = 0, e = 0, |T˜ |  U },

(16.35)

where U is a sufciently large constant. Proof. The closed-loop system is given by > ∗? z = F(z∗ , T˜ ), T˜ with

(16.36)

⎡

⎤ f0 + f1 s I⎥ ⎢ 1 H ˆ2 2 ⎢a + b sgn(s) + e − T (N1 + N22 )s + L f1 W + s ⎥ ⎢ ⎥ 2 ⎢ ⎥ a Vb U ⎢ ⎥ @ A w D2 w D ∗ ˆ 2 ⎢ F(z , T ) = ⎢ a + b sgn(s)+ J b−Tˆ N sgn e −2s−ke⎥ ⎥. ws ⎢ ws ⎥ ⎢ ⎥   ⎢ ⎥  w D2  ⎣ ⎦   −2Ja |s|N − Ja e N ws 

Choose V (z∗ , Tˆ ) as 1 1 V (z∗ , Tˆ ) = W (z) + s2 + e2 + (T − Tˆ )2 . 2 2 Ja Then, from (ii) and (iv), we have V¯ = L f0 W (z) + 2sa(z, s) + 2b(z, s)|s| − Tˆ 2N12 (z, s)s2 −Tˆ 2 N22 (z, s)s2 − s2 − ke2   @ A  w D2   + e w D2 a(z, s) + J b(z, s) − Tˆ N(z, s) e ws  ws

(16.37)

16.4 Design Example of Mechanical Systems

 V U  w D2  w D2   N(z, s) ˆ +e b(z, s) sgn(s)−(T −T ) 2|s|+ e ws ws  A2 @ A2 @  − 1z1 − |s|Tˆ N1 (z, s) − s − |s|Tˆ N2 (z, s)    w D2  . −ke2 + 2b(z, s)|s| + (J − 1)b(z, s) e ws 

523

(16.38) (16.39)

Clearly, from (ii), this inequality ensures that V¯  0, ∀t  0 and V¯ < 0, ∀s := 0 or/and e wwDs2 := 0. Hence, in a similar way to the proof of Theorem 16.10, the Lyapunov stability and the convergence of z, s and e can be shown by Barbalat lemma with the inequality. Also, it is clear that ZV ={(z∗ , Tˆ )|0∈V¯(z∗ , Tˆ )} is given by ZV ={(z∗ , Tˆ )|z=0, s= 0, e=0} and on the surface Ss (z∗ , Tˆ ) ∈ ZV , K[F](z∗,Tˆ)=0. Hence, we can conclude that every Filippov solution converges to the bounded set M ={(z, s, e, T˜ )|z=0, s= 3 2 0, e=0, |T˜ |< U}.

16.4 Design Example of Mechanical Systems Positioning and tracking a desired trajectory are two main issues of motion control for mechanical systems. However, due to unknown friction force, how to improve the accuracy of positioning and tracking is still far from being solved satisfactorily for general mechanical systems. As indicated in [22, 2], the discontinuous static mapping with respect to the relative velocity is a good approximation to represent the uncertain static friction of each joint. If we adopt the switching function to describe the static friction, the system turns into a discontinuous dynamic system. On the other hand, a trade-off between the complexity of the algorithm and the performance requirement is usually made to construct a feasible controller in engineering. As the simplest controller, PD controller has been widely implemented, since the asymptotically stability of the PD controlled mechanical systems can be easily obtained under the ideal condition. However, there exists a nite steady-state position error for the PD-controlled mechanical systems acted by static friction force, which has been presented in [22, 2] for a one-degree-of-freedom (1DOF) mass system. In this section, we refer to the stationary set analysis for PD controlled mechanical systems which is acted by discontinuous static friction force.

16.4.1 PD Controlled Mechanical Systems The mechanical plants of interest is modeled in the form of Euler-Lagrange equation J(q)q¨ + C(q, q)  q + O(q) = Wd ,

(16.40)

where q ∈ Rn includes the joint coordinates, Wd ∈ Rn is the applied torque inputs, J(q) ∈ Rn×n is the inertia matrix, C(q, q)  ∈ Rn×n is the matrix of centripetal and n Coriolis forces, and O(q) ∈ R is the vector of gravitational torque. Let qd ∈ Rn be the desired position, dene the position error x1 = [x11 x12 · · · x1n ]T = q − qd , x2 = [x21 x22 · · · x2n ]T = x1 , and C(x1,2 ) = C(x1 , x2 ), then the error equa-

524

16 Discontinuous Dynamical Systems

tion can be written as

J

x1 = x2 J(x1 )x2 = −C(x1,2 )x2 − O(x1 ) + Wd .

(16.41)

It is well-known that system (16.41) can be stabilized by a PD controller,

Wd = O(x1 ) − K p x1 − Kd x2 ,

(16.42)

where K p = diag{K pi } and Kd = diag{Kdi } with Kpi , Kdi > 0, i = 1, · · · , n. This result can be checked by introducing the Lyapunov function 1 1 (16.43) V0 = xT1 K p x1 + xT2 J(x1 )x2 . 2 2 The following two properties of Euler-Lagrange systems are also useful to calculate the time derivative of V0 . (i). J is positive denite, and there exists two positive numbers m1 , m2 such that m1 In < J < m2 In . (ii). C and J satisfy the skew-symmetric relationship [ T (J− 2C)[ = 0, ∀[ . In order to describe a class of discontinuous uncertainties, such as static friction [22], the model considered here corresponds to the n-dimensional discontinuous map of the relative speed denoted by DSgn (x2 ), where D = diag{di }, di > 0, i = 1, · · · , n, and Sgn (x2 ) is a vector denoted by Sgn (x2 ) = [sgn(x21 ) sgn(x22 ) · · · sgn(x2n )]T . The closed-loop system can be rewritten as J x1 = x2 J(x1 )x2 = −C(x1,2 )x2 − K p x1 − Kd x2 − DSgn (x2 ),

(16.44)

(16.45)

It’s obvious that the differential equation of the closed-loop system has a discontinuous right-hand side. Filippov solution is given by the differential inclusion ? > > ? x2 x1 , (16.46) ∈K x2 Hx2−G(K p x1+Kd x2 )+ LSgn (x2 ) where K[·] is the notation of Filippov’s inclusion dened as Section 16.1. The parameters G, L, H denote J −1 (x1 ), −GD and −GC(x1,2 ), respectively. Let G = (gi j (x1 ))n×n , H = (hi j (x1 , x2 ))n×n , briey G = (gi j ) and H = (hi j ). Obviously, gi j and hi j , i, j = 1, · · · , n, are continuous functions respect to x1 and x2 .

16.4.2 Stationary Set The stationary set includes all the equilibrium points of (16.46) which are nothing but static solutions in the sense of Filippov, i.e. if x¯ = [x¯T1 x¯T2 ]T is an equilibrium point of (16.46), x¯ satises 0 ∈ K[ f (x)]. ¯ The following theorem shows the equilibrium point of (16.46) is not unique but a bounded set including the origin. Theorem 16.12. For the system (16.46), let x = [xT1 xT2 ]T . Then there exits a nonempty set : such that any x¯ ∈ : is an equilibrium point, where

16.4 Design Example of Mechanical Systems

a b di 2n : = x ∈ R | |x1i |  , i = 1, · · · , n, x2 = 0 . K pi Proof. Rewrite (16.46) as

? x2 . x∈K[  f (x)]=K (H − GKd )x2 − GK px1 + LSgn (x2 ) The right-hand side of this inclusion is set-valued, if and only if

525

(16.47)

>

x∈S =

n v

Si ,

(16.48)

(16.49)

i=1

where Si = {x ∈ R2n | x2i = 0}, i = 1, · · · , n. Obviously, for any x ∈ / S , K[ f (x)] is single-valued and 0 ∈ / K[ f (x)]. In other words, the equilibrium point of (16.46) must be in S . Hence, we focus on the set S only. Due to Si ⊂ S , i = 1, · · · , n, we can nd the stationary set by searching the points x satisfying 0 ∈ K[ f (x)] in each Si , i = 1, · · · , n. The whole space R2n is divided into two regions by any Si , Si+= {x ∈ R2n | x2i> 0},

Si−= {x ∈ R2n | x2i< 0}.

(16.50)

We can obtain the following limit values of f (x) under approach to Si from Si+ and Si− , respectively. ⎧ x21 ⎪ ⎪ ⎪ .. ⎪ ⎪ . ⎪ ⎨ x2n (16.51) fi+ (x) = 61 − g1i K pi x1i − g1idi ⎪ ⎪ ⎪ .. ⎪ ⎪ ⎪ . ⎩ 6n − gniKpi x1i − gni di , ⎧ x21 ⎪ ⎪ ⎪ .. ⎪ ⎪ . ⎪ ⎨ x − 2n fi (x) = (16.52) 61 − g1i K pi x1i + g1idi ⎪ ⎪ ⎪ .. ⎪ ⎪ ⎪ . ⎩ 6n − gniKpi x1i + gni di , where for r = 1, · · · , n,

6r =

n

¦

[hr j x2 j − gr j (Kd j x2 j + K p j x1 j + d j [ j )],

j=1, j:=i

and [r ∈ [−1, 1], r = 1, · · · , n, r := i, are unknown constants. Since the normal of the switching surface Si is Ni = [01×n 01×(i−1) 1 01×(n−i)]T , the projections of the vectors fi+ (x) and fi− (x) onto the normal direction of the surface Si , denoted by fN+i (x) and fN−i (x), can be calculated as follows fN+i (x) = 6i − gii (K pi x1i + di ),

(16.53)

fN−i (x)

(16.54)

= 6 i − gii (K pi x1i − di ).

Thus, from Theorem 16.3, the differential inclusion (16.48) has a solution x for t ∈ [t0 ,t1 ] on the surface Si , if and only if x ∈ ' i for all t ∈ [t0 ,t1 ], where 'i is represented by

526

16 Discontinuous Dynamical Systems

'i = {x ∈ Si | fN+i  0, fN−i  0, fN−i − fN+i > 0}.

(16.55)

Furthermore, the solution x(t),t ∈ [t0 ,t1 ] satises x = fi◦ (x) =

fN−i

fi+ (x) +

fN+i

fN−i − fN+i fN−i − fN+i ⎧ x21 ⎪ ⎪ ⎪ .. ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ x ⎪ 2(i−1) ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ x ⎪ 2(i+1) ⎪ ⎪ ⎪ .. ⎪ ⎪ . ⎪ ⎪ ⎨ x2n = 61 − g1i/gii 6i ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ 6 − g i−1 (i−1)i/gii 6 i ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ 6 − g i+1 (i+1)i/gii 6 i ⎪ ⎪ ⎪ .. ⎪ ⎪ ⎪ . ⎪ ⎩ 6n − gni /gii 6i .

fi− (x)

(16.56)

This leads to the following conclusion: x ∈ ' i is an equilibrium in the Filippov’s sense, if and only if fi◦ (x) = 0 holds at the point x, i.e. a x2 j = 0 , j = 1, · · · , n, j := i. (16.57) K p j x1 j + d j [ j = 0 Due to [i ∈ [−1, 1], the above equations imply that the largest range of x1 j , j = 1, · · · , n, j := i is given by > ? dj dj , x1 j ∈ − , j = 1, · · · , n, j := i. (16.58) Kp j Kp j Moveover, from the condition (16.55), we can obtain the range of x1i , that is > ? di di , . (16.59) x1i ∈ − K pi K pi Hence, the static solution set on the switching surface Si is > ? dj dj : i={x ∈ Si |x1 j∈ − , , j = 1, · · · , n, x2 = 0}. Kp j Kp j

(16.60)

Then we can conclude that the equilibrium point set of the closed-loop system is a b n v di : = :i= x∈ R2n | |x1i | , i = 1, · · ·, n, x2 =0 (16.61) K pi i=1 3 2 Theorem 16.12 also implies that the trajectories of the generalized system are not asymptotically stable to the origin but may converge to any equilibrium point in : . The following theorem shows the fact that the stationary set is stable.

16.4 Design Example of Mechanical Systems

527

Theorem 16.13. Consider the system (16.46), the stationary set : is stable for any K p and Kd . Proof. Let (16.43) be a Lyapunov function candidate. In order to show the Lyapunov stability in the Filippov’s sense, we should calculate the generalized time derivative of V0 which is dened as V˜ 0 =

q

] T · K[ f (x)],

(16.62)

] ∈w V0

where w V0 is the Clarke’s generalized gradient. Let V˜0 |x∈D express the value-set of V˜0 at x, at the same time, we use V˜0 to denote the elements of the value-set for simplicity. In this case, since V0 is a continuous function respect to the state x1 and x2 , w V0 is equal to the normal gradient w V0 /w x. Furthermore, it is clear that K[ f (x)] involves single element calculated by the right-hand side with a denite value of Sgn (x2 ) whenever x ∈ / S , i.e. in such region, we have n

2 V˜0 |x∈S / = − ¦ (Kdi x2i + di |x2i |) < 0.

(16.63)

i=1

Moreover, on any switching surface Si , i = 1, · · · , n, K[ f (x)] can be represented by

K[ f (x)] = { fi◦ (x)},

(16.64)

where fi◦ (x) is given in (16.56). Hence, the generalized time derivative of V0 can be calculated by n V˜ | =− (K x2 + d |x |)  0. (16.65) 0 x∈Si

¦

j=1, j:=i

dj 2j

j

2j

For the whole state space, we can conclude that there always exists D ⊂ R2n such that K  0 for any K ∈ V˜0 |∀x∈D and K = 0, if and only if x ∈ S1 ∩ · · · ∩ Sn which is a subset of D. Based on the extended LaSalle’s invariant set principle (Theorem 16.9), any trajectory starting in D will approach to the largest invariant set in the set S 1 ∩ · · · ∩ Sn . Obviously, the largest invariant set is equal to the solution set in which any solution is given by x = f1◦ (x) = · · · = fn◦ (x), (16.66) where fi◦ (x), i = 1, · · · , n are set-valued function dened as (16.64). The equivalent solution set is the equilibrium point set : in Theorem 16.12. Then, the theorem follows. 3 2 This result guarantees that the trajectories from any place in the state space will approach to the compact set : . Therefore, for any given accuracy requirement |xi | < Hi of each freedom, we just regulate the design parameter Kp satisfying Kpi > di /Hi , then the accuracy requirement is achieved. However, we can not get the nal position in : . That shows a basic design ideal of PD control.

528

16 Discontinuous Dynamical Systems

16.4.3 Application Example In this section, we discuss an example applying the results to a 3-axis rotation table. As shown in Fig.16.7, since the table used in the experiment is designed axisymmetric and normal to all the axes, we can ignore the gravitational torque. If we consider the rotational angles as the coordinates of the table system, i.e. x1 = T = [T1 T2 T3 ]T , the physical parameters of the plant can be easily obtained as follows ⎤ ⎡ J1 0 J1 cos T2 ⎦ 0 J2 0 J(T ) = ⎣ 2 J1 cos T2 0 J4 + J3 cos T2 and

⎡

0

−J1 T3 sin T2

0

⎤

1  ⎥ ⎢ 0 J1 T3 sin T2 J3 T3 sin 2T2 ⎥ , C(T , T )=⎢ 2 ⎦ ⎣ 1 1 −J1 T2 sin T2 − J3 T3 sin 2T2 − J3 T2 sin 2T2 2 2 where Ji , i = 1, 2, 3, 4 are 1.2kg · m2, 2.9kg · m2, 0.3kg · m2, and 19.9kg · m2, respectively.

Fig. 16.7

Three-Axis Rotation Table

The controller is implemented in the industrial computer. The control outputs are delivered to the motor driver through a D/A board. Position of each joint are measured by three optical absolute encoders. For the sake of simplicity, we ignore the dynamics of the motor and calculate the rotational speed by the time derivative of the position feedback. And a low-pass lter is also designed to reduce the noise of the velocity signal. In this case, the maximum static friction forces for each axis are 1.2Nm, 1.8Nm, 2.2Nm corresponding to rolling part, pitching part and yawing part, respectively. In order to show the relation between the boundary of the static error and the design parameters, we make a comparison by choosing the different parameters of the PD controllers in the experiments. In experiment A, the parameters of the feedback law (16.42) are chosen as

16.4 Design Example of Mechanical Systems

529

K p1 = 10, K p2 = 15, Kp3 = 20, Kd1 = 5, Kd2 = 5, Kd3 = 5. Then, in experiment B, they are changed to K p1 = 40, K p2 = 60, Kp3 = 80, Kd1 = 5, Kd2 = 5, Kd3 = 5. The position responses and control signals of the experiment A and B are shown in Fig. 16.8 and Fig. 16.9, respectively. The subgures (a), (b) and (c) in Fig.16.8 and Fig. 16.9 are position responses, which corresponds to the rolling, the pitching and the yawing part of the table, respectively. The subgures (d), (e) and (f) are the corresponding control signals. The dashed in subgures (a), (b) and (c) are reference signals. Two parallel dashed lines in subgures (a), (b) and (c) illustrate the boundary of the theoretical positioning error. The experimental results indicate that the servo error is constant at the steady state. The static error is also reduced with respect to the parameter K p . However, the high gain feedback may cause saturation of the actuator when large error occurs during transient process, for example the control signal in Fig.16.9(f). The positioning error in the experiments agrees

Fig. 16.8

Position Responses (a), (b), (c) and Control Signals (d), (e), (f) of Experiment A.

530

16 Discontinuous Dynamical Systems

well with the theoretical analysis of the range of the static error, though we can not determine the exact nial position before the end of the experiment.

Fig. 16.9

Position Responses (a), (b), (c) and Control Signals (d), (e), (f) of Experiment B.

References

Fig. 16.10

531

Position Responses (a), (b), (c) and Control Signals (d), (e), (f) of Experiment C.

References 1. Adly S, Goeleven D. A stability theory for second order nonsmooth dynamical systems with application to friction problems. Journal of Math. Pures Appl., 2005, 83(1): 17–51. 2. Armstrong B, Amin B. PID control in the presence of static friction: A comparison of algebraic and describing function analysis. Automatica, 1996, 32(5): 679–692. 3. Armstrong-Helouvry B, Dupont P, Canudas de Wit C. A survey of models, analyasis tools and compensation methods for the control of machines with friction. Automatica, 1994, 30(7): 1083–1138. 4. Arnold V I. Ordinary Differential Equations. Tokyo: Springer-Verlag, 1992. 5. Aubin J, Cellina A. Differential Inclusion. New York: Springer-Verlag, 1994. 6. Azhmyakov V. Stability of differential inclusions: A computation approach. Mathematical Problems in Engineering, 2006, ID17837: 1–15. 7. Bacciotti A, Rosier L. Lyapunov Functions and Stability in Control Theory. London: Springer, 2005. 8. Casagrande D, Astol A, Parisini T. A stabilizing time-switching control strategy for the rolling sphere. Proceedings of CDC-ECC Conference, 2005: 3297–3302. 9. Ceragioli F. Discontinuous ordinary differential equations and stabilization[Ph.D. dissertation] Firenze: Universita degli Studi di Firenze - Dipartimento di Matematica, 1999. 10. Clarke F. Optimization and Nonsmooth Analysis. New York: SIAM, 1990. 11. Cortes J. Discontinuous dynamical systems: A tutorial on notions of solutions, nonsmooth analysis, and stability. IEEE Control Systems Magazine, 2007. 12. Filippov A. Differential equations with discontinuous right-hand sides. American Mathematical Society Translations, 1964, 42: 199–231. 13. Filippov A. Differential Equations with Discontinuous Right-hand Sides. Netherlands: Kluwer Academic Publishersm, 1988. 14. Hajek O. Discontinuous differential equations I. Journal of Differential Equations, 1979, 32(2): 149–170. 15. Hermes H. Discontinous vector elds and feedback control// Differential Equation and Dynamical Systems. New York: Academic Press, 1967: 155–165. 16. Liberzon D. Switchings in Systems and Control. Boston: Birkhauser, 2003. 17. Paden B, Sastry S. A calculus for computing Filippove differential inclusion with application to the variable structure control of robot manipulators. IEEE Trans. Circuits and Systems, 1987, 34(1): 73–82. 18. Ray L. Nonlinear tire force estimation and road friction identication: Simulation and experiments. Automatica, 1997, 33(10): 1819–1833. 19. Sekhavat P, Wu Q, Sepehri N. Lyapunov-based friction compensation for accurate positioning of a hydraulic actuator//Proceedings of the 2004 American Control Conference, Boston, 2004: 418–423. 20. Shen T. Robust nonlinear traction control with direct torque controlled induction motor// Prceedings of Asian Electric Vehicle Conference, Seoul, 2004. 21. Shevitz D, Parden B. Lyapunov stability theory of nonsmooth systems. IEEE Trans. Aut. Contr., 1994, 39: 1910–1914.

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22. Southward S, Radcliffe C, MacCluer C. Robust nonlinear stick-slip friction compensation. ASME Journal of Dynamic Systems, Measurement and Control, 1991, 113: 639–645. 23. Utkin V. Variable structure systems with sliding mode. IEEE Trans. Aut. Contr., 1997, 22(2): 212–222. 24. Utkin V. Sliding mode in control and optimization//Communication and Control Engineering. New York: Springer-Verkag, 1998. 25. van de Wouw N, Leine R. Attractivity of equilibrium sets of systems with dry friction. Nonlinear Dynamics, 2004, 35(1): 19–39. 26. Yakubovich V, Leonov G, Gelig A K. Statility of Stationary Sets in Control Systems with Discontinuous Nonlinearities. New Jersey: World Scientic, 2004.

Appendix A

Some Useful Theorems

A.1 Sard’s Theorem Denition A.1. Let M and N be two differentiable manifolds. F : M → N is a C1 mapping. Then 1. corank(dF) p = min{dim(M), dim(N)} − rank(dF) p . 2. A point p ∈ M is a critical point of F, if corank(dF)| p > 0. The set of critical points is denoted by C[ f ]. 3. A point q ∈ N is a critical value of F, if q ∈ F(C[F]). Theorem A.1. (Sard’s Theorem)[2] The set of critical values of F has a measure zero in N.

A.2 Rank Theorem Theorem A.2. (Rank Theorem)[1] Let M and N be two differentiable manifolds of dimensions m and n respectively. F : M → N is a C1 mapping. Assume rank(F) = k. Then there exist local coordinate charts, such that F can be locally expressed as F(x1 , x2 , · · · , xm ) = (x1 , · · · , xk , 0, · · · , 0).

References 1. Boothby W. An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd edn. Orlando: Academic Press, 1986. 2. Golubitsky M, Guillemin V. Stable Mappings and Their Singularities. New York: SpringerVerlag, 1973.

Appendix B

Semi-Tensor Product of Matrices

B.1 A Generalized Matrix Product First, we dene the semi-tensor product for two vectors. Denition B.1. 1. Let X be an np dimensional row vector, Y a p dimensional column vector. Splitting X into p equal blocks as X 1 , · · · , X p , each block is a 1 × n row vector. The semi-tensor product of X and Y , denoted by X Y , is dened as p

X  Y = ¦ X i yi ∈ Rn .

(B.1)

i=1

2. Let X be a p dimensional row vector, Y an np dimensional column vector. Splitting Y into p equal blocks as Y 1 , · · · ,Y p , each block is an n × 1 column vector. The semi-tensor product of X and Y is dened as p

X  Y = ¦ xiY i ∈ Rn .

(B.2)

i=1

D E D E Example B.1. 1. Let X = 1 3 −1 2 , Y = 2 − 2 . Then D E D E D E X  Y = 2 1 3 − 2 −1 2 = 4 2 . D E D ET 2. Let X = 2 −3 , Y = 2 1 −3 5 . Then ? > ? > > ? 13 −3 2 . = −3 X Y = 2 −13 5 1 Next, we dene the semi-tensor product for two matrices. Denition B.2. Let M ∈ Mm×n , N ∈ M p×q . If either n is a factor of p or p is a factor of n, then the semi-tensor product C = M  N is dened as follows: C consists of m × q blocks, i.e., C = (Ci j ), and Ci j = M i  N j ,

i = 1, · · · , m, j = 1, · · · , q,

where M i is the i-th row of M and N j is the j-th column of N. Remark B.1. 1. When n = p, according to the denition of semi-tensor product, it coincides with the convenient matrix product. So the semi-tensor product is a

536

Appendix B Semi-Tensor Product of Matrices

generalization of convenient matrix product. Hence, the symbol  can be omitted. We omit it unless we want to emphasize it is different from the convenient product. 2. We dene the semi-tensor product only for multi-dimensional case (i.e., n is a multiplier of p or vise versa). We refer to, i.e., [1], for general case. 3. Let A ∈ Mm,n and B ∈ M p,q . If n = t p we denote A Mt B. If tn = p we denote A ≺t B. Example B.2. Let

Then

⎤ ⎡ 2 1 −1 3 X = ⎣0 1 2 −1⎦ , 2 −1 1 1

> Y=

? −1 2 . 3 2

⎤ (2 1) × (−1) + (−1 3) × 3 (2 1) × 2 + (−1 3) × 2 X  Y = ⎣(0 1) × (−1) + (2 − 1) × 3 (0 1) × 2 + (2 − 1) × 2⎦ (2 − 1) × (−1) + (1 1) × 3 (2 − 1) × 2 + (1 1) × 2 ⎡

⎡ −5 8 2 = ⎣ 6 −4 4 1 4 6

⎤ 8 0⎦ . 0

Denition B.3. Let A ∈ M p×q . Assuming p is a factor of q or vise versa, we dene An , n > 0 inductively as J A1 = A, (B.3) Ak+1 = Ak  A, k = 1, 2, · · · . Remark B.2. 1. If X and Y are two row (column) vectors, then X Y is well dened. 2. If X is a row or column vectors, then X k is well dened. Some basic properties of semi-tensor product are introduced as follows. Theorem B.1. As long as the semi-tensor product is well dened (i.e., the factor matrices have proper dimensions), we have 1. (Distributive Rule) J F  (aG ± bH) = aF  G ± bF  H, (aF ± bG)  H = aF  H ± bG  H,

a, b ∈ R.

(B.4)

2. (Associative Rule) (F  G)  H = F  (G  H). Block multiplication rule is also true. Proposition B.1. Let A Mt B (or A ≺t B). Splitting A and B as ⎡ 11 ⎤ ⎡ 11 ⎤ A · · · A1s B · · · B1t ⎢ .. ⎥ , B = ⎢ .. .. ⎥ . A = ⎣ ... ⎣ . . ⎦ . ⎦ Ar1 · · · Ars

Bs1 · · · Bst

(B.5)

B.2 Swap Matrix

If Aik Mt Bk j , ∀ i, j, k (respectively, Aik ≺t Bk j , ∀ i, j, k), then ⎤ ⎡ 11 C · · · C1t ⎢ .. ⎥ , A  B = ⎣ ... . ⎦ Cr1 where

Ci j =

···

537

(B.6)

Crt

s

¦ Aik  Bk j .

k=1

B.2 Swap Matrix Swap matrix W[m,n] is an mn × mn matrix, dened as follows: Label its columns by double index (i, j) as {11, 12, · · · , 1n, 21, 22, · · · , 2n, · · · , m1, m2, · · · , mn} Label its rows by {11, 21, · · · , m1, 12, 22, · · · , m2, · · · , 1n, 2n, · · · , mn} Then set the element at position [(I, J), (i, j)] as J 1, I = i and J = j, I,J w(IJ),(i j) = Gi, j = 0, otherwise.

(B.7)

Example B.3. 1. Let m = 2, n = 3, W[2,3] is constructed as follows:

W[2,3]

(11) ⎤ ⎡ (12) (13) (21) (22) (23) 1 0 0 0 0 0 ⎢0 0 0 1 0 0⎥ ⎥ ⎢ ⎢0 1 0 0 0 0⎥ ⎥ ⎢ = ⎢ ⎥ ⎢0 0 0 0 1 0⎥ ⎣0 0 1 0 0 0⎦ 0 0 0 0 0 1

(11) (21) (12) . (22) (13) (23)

(11) ⎤ ⎡ (12) (21) (22) (31) (32) 1 0 0 0 0 0 ⎢0 0 1 0 0 0⎥ ⎥ ⎢ ⎢0 0 0 0 1 0⎥ ⎥ = ⎢ ⎢0 1 0 0 0 0⎥ ⎥ ⎢ ⎣0 0 0 1 0 0⎦ 0 0 0 0 0 1

(11) (21) (31) . (12) (22) (32)

2. Consider W[3,2] , it is

W[3,2]

It is easy to check the follows: Proposition B.2. 1.

538

Appendix B Semi-Tensor Product of Matrices T −1 W[m,n] = W[m,n] = W[n,m] .

(B.8)

2. When m = n, (B.8) becomes T −1 = W[n,n] . W[n,n] = W[n,n]

(B.9)

3. W[1,n] = W[n,1] = In . When m = n we denote

(B.10)

W[n] := W[n,n] .

An alternative construction of W[m,n] is as follows. Let Gin be the i-th column of In . Then we have Proposition B.3. D E W[m,n] = G1n  G1m · · · Gnn  G1m · · · G1n  Gmm · · · Gnn  Gmm .

(B.11)

The most useful property of the swap matrix is Proposition B.4. 1. Let X ∈ Rm and Y ∈ Rn be two column vectors. Then W[m,n] XY = Y X.

(B.12)

2. Let X ∈ Rm and Y ∈ Rn be two row vectors. Then XYW[m,n] = Y X.

(B.13)

B.3 Some Properties of Semi-Tensor Product Proposition B.5. Let A and B be two matrices with proper dimensions. Then (A  B)T = BT  AT .

(B.14)

Proposition B.6. Let A and B be two square matrices with proper dimensions. Then 1.

V (A  B) = V (B  A).

(B.15)

2. tr(A  B) = tr(B  A). 3. If A and B are upper triangular (lower triangular, diagonal, orthogonal) matrices, then so is A  B. 4. If both A and B are invertible, then A  B is invertible and (A  B)−1 = B−1  A−1.

(B.16)

Proposition B.7. 1. Let M ∈ Mm×pn . Then M  In = M.

(B.17)

B.4 Matrix Form of Polynomials

539

2. Let M ∈ Mm×n . Then M  I pn = M ⊗ I p .

(B.18)

I p  M = M.

(B.19)

I pm  M = M ⊗ I p.

(B.20)

3. Let M ∈ M pm×n . Then

4. Let M ∈ Mm×n . Then

Proposition B.8. 1. If A ∈ Mm×np , B ∈ M p×q , then A  B = A(B ⊗ In).

(B.21)

A  B = (A ⊗ I p )B.

(B.22)

2. If A ∈ Mm×n , B ∈ Mnp×q , then

Proposition B.9. Given A ∈ Mm×n . 1. Let [ ∈ Rt be a row vector. Then A[ = [ (It ⊗ A).

(B.23)

2. Let Z ∈ Rt be a column vector. Then ZA = (It ⊗ A)Z.

(B.24)

B.4 Matrix Form of Polynomials Let x = (x1 , · · · , xn )T be the set of coordinate variables in Rn , where x can also be considered as a column vector. We use Pnk (x) for the set of k-th degree homogeneous polynomials in Rm and k Hn the set of k-th degree homogeneous vector elds in Rn . Then for a polynomial p(x) ∈ Pnk (x), we can nd a 1 × nk coefcient C such that p(x) = Cxk .

(B.25)

Similarly, if a vector eld f (x) ∈ Hnk , we can nd an n × nk matrix F, such that f (x) = Fxk .

(B.26)

Note that both C in (B.25) and F in (B.26) are not unique, because xk is a redundant basis. Denition B.4. Consider a matrix M(x) ∈ Mp×q with entries as smooth functions of x ∈ Rn . The differential of M(x), DM(x) ∈ M p×nq , is dened by replacing each element mi j (x) of M(x) by its differential

540

Appendix B Semi-Tensor Product of Matrices

U

That is,

V w mi j (x) w mi j (x) ,··· , . w x1 w xn

⎡

w m11 (x) w m11 (x) w m1n (x) w m1n (x) ⎤ ··· ··· ··· ⎢ w x1 w xn w x1 w xn ⎥ ⎢ ⎥ . . . .. ⎢ ⎥. . . . DM(x) = ⎢ . . . . ⎥ ⎣ w m (x) w mn1 (x) w Mnn (x) w mnn (x) ⎦ n1 ··· ··· ··· w x1 w xn w x1 w xn

(B.27)

Higher degree differentials can be dened inductively by Dk+1 M = D(Dk M) ∈ Mp×nk+1 q ,

k  1.

(B.28)

Using the semi-tensor product and the differential form dened above, the Taylor expansion of multi-variable functions (or mappings) has a neat form as the one for single-variable functions (or mappings). Theorem B.2. (Taylor Expansion) Let F : Rm → Rn be a smooth (Cf ) mapping. Then its Taylor expansion can be expressed as f

1 k D F(x0 )(x − x0 )k . k! k=1

F(x) = F(x0 ) + ¦

(B.29)

The following formula is fundamental in differential calculations. Theorem B.3. Let x ∈ Rn . Then D(xk+1 ) = )kn xk ,

k  0,

(B.30)

where k

¦ Ins ⊗ W[nk−s,n] .

)kn =

(B.31)

s=0

Formula (B.30) can be used for the differential of any analytic functions (or mappings). Example B.4. Let F(x) : Rn → Rm be an analytic mapping. Using Taylor expansion, we have f

F(x) = ¦ Di F(x0 )(x − x0 )i .

(B.32)

i=0

Differentiating it term by term, we have f

DF(x) = ¦ Di+1 F(x0 ))in (x − x0 )i .

(B.33)

i=0

References 1. Cheng D. Semi-tensor product of matrices and its applications: A survey. Proceedings of ICCM, 2007, 3: 641–668.

Index

(A, B)-invariant subspace, 6 ( f , g)-invariant, 207 A-invariant subspace, 7 Cr mapping, 50 Cf mapping, 50 CZ mapping, 50 Hf norm, 403 L2 disturbance approximate decoupling, 418 L2 stable, 190 L2 -gain, 405 Lf stable, 190, 191 T0 space, 37 T1 space, 38 T2 space, 38 U-in-distinguishable, 136 U-reachable, 122 X-invariant, 134 H -net, 43 J -dissipative, 380 Z -limit point, 184 k-form, 78 Abelian group, 91 absolute stability problem, 193 accessibility Lie algebra, 64, 123 accessibility rank condition, 123 accessible, 127 adjoint representation, 118 afne nonlinear control system, 2 afne nonlinear system, 121 algebra, 99 algebra isomorphism, 99 algebraic connectivity, 496 algebraic multiplicity, 176 approximate linearization, 311 approximate system, 322 approximately stable, 322 approximation degree, 330 approximation theorem, 317 associative algebra, 99

asymptotical stability, 174 attractive, 174 automorphism, 94 autonomous system, 174 Baire’s Category Theorem, 37 Banach algebra, 100 base space, 53 bifurcation, 18 binomial expansion, 244 Boolean algebra, 97 boundary, 36 Brunovsky canonical form, 5 bundle isomorphism, 54 bundle mapping, 54 Byrnes-Isidori normal form, 254 Campbell-Baker-Hausdorff formula, 73 cascade decomposition, 226 Casimir function, 87 causal, 190 center, 97 center manifold, 315 Chow’s Theorem, 74 Christoffel matrix, 82 Christoffel symbol, 82 circular group, 92 Clarke’s generalized gradient, 527 class K function, 178 class Kf , 178 class L function, 178 clopen, 40 closed one form, 66 closed set, 34 closed-loop control, 121 closure, 36 co-distribution, 65 co-tangent space, 65 co-vector eld, 65

541

542

Index

common joint quadratic Lyapunov function, 484 common Lyapunov function, 432 common quadratic Lyapunov function(QLF), 432 commutative algebra, 99 commutative ring, 98 commutator, 93 commutator group, 93 compact space, 41 complete metric space, 34 complete vector eld, 61 component-wise homogeneous vector eld, 322 conjugated subgroups, 97 connected, 40 connection, 82 continuous mapping, 39 contractable, 100 control Liapunov function, 194 controllability canonical form, 166 controllable, 3, 122 controlled invariant sub-distribution, 220 converge, 42 converse theorem to Lyapunov’s stability theorem, 185 covering space, 50, 109 Cross Row Diagonal Dominating Principle (CRDDP), 325 decoupling matrix, 11, 238 decoupling vector, 238 decrescent function, 178 degree matching conditions, 331 dense set, 36 derived Lie algebra, 130 Diagonal Dominating Principle (DDP), 325 diagonal term, 324 differential of matrix, 539 differential on Riemannian manifold, 82 dissipative, 379 disturbance decoupling, 7, 213 dwell time, 483 dynamic compensator, 248 dynamic feedback, 247 embedded sub-manifold, 51 endomorphism, 94 equilibrium, 173 equivalence theorem, 317 equivalent class, 44 equivalent relation, 44 Euler-Lagrange equation, 523 existence theorem, 317 exosystm, 366 exponential mapping, 116 exponential stability, 175

external differential, 78 feedback Brunovsky canonical form, 6 ber bundle, 53 ber space, 53 Filippov solution, 511 rst category, 37 rst countable, 35 First Homomorphism Theorem, 95 at coordinate frame, 141 at coordinate system, 51 Fliess’ functional expansion, 264 Floquet technique, 189 friend to V , 7 Frobenius’ Theorem, 71 full-dimensional observer, 8 fundamental group, 50, 105 general linear group, 113 generalized Darboux’ Theorem, 89 generalized directional derivative, 514 generalized Frobenius’ theorem, 143 generalized Frobenius’ theory, 16 generalized gradient, 514 geodesic, 85 geometric multiplicity, 176 globally uniformly asymptotically stable, 182 glued together, 45 Grassmann manifold, 48 group, 91 Hadamard product, 309 Hamilton-Jacobi-Issacs, 407 Hamiltonian function, 398, 399, 427 Hamiltonian system, 399, 426 Hausdorff space, 38 Heymann’s lemma, 306 HJI, 407 HJI inequality, 387, 407, 414, 416–418 homeomorphic, 39 homeomorphism, 39 homomorphism, 94 homotopic, 100 homotopic related to A, 100 homotopy, 100 hyperbolic, 195 ideal, 98, 99 ideal generated by r, 98 identity, 98 immersed sub-manifold, 51 immersion system, 369 in-distinguishable, 136 injection degree, 330 inner product, 31 input-output linearization, 295 integrable, 70

Index integral backstepping, 390 integral curve, 60 integral manifold, 70 integral on Riemannian manifold, 80 integral ring, 98 interior, 36 internal model, 371 internal model principle, 355, 356 involutive, 70 involutive closure, 150, 214 isomorphism, 94 Jacobi identity, 62 Jacobian linearization, 176, 262 Kalman decomposition, 5 Killing form, 119 Klein bottle, 46 KYP property, 383 Lagrange function, 394 Lagrange principle, 394 Lagrange system, 395 Laplacian matrix, 492 LaSalle’s invariance principle, 184 leading degree, 330 Lebesgue number, 43 left g coset, 92 left zero factor, 98 Levi decomposition, 118 LFHS, 321 Lie algebra, 62, 100 Lie algebra of a Lie group, 115 Lie bracket, 62 Lie group, 113 Lie series of forms, 73 Lie series of smooth function, 73 Lie series of vector eld, 73 Lie subgroup, 113 linear control system, 2 linearization, 287 linearization with outputs, 292 Liouville’s Theorem, 87 Lipschitz condition, 61, 173 local input-output decoupling, 240 local output regulation, 366 locally connected, 40 locally controllable, 15, 122 locally positive denite function, 178 loop, 102 Lorenz system, 18 Lyapunov function, 179 Lyapunov function with homogeneous derivative, 321 metric space, 29 minimum realization, 1, 5

Mobius strip, 46 Morgan’s problem, 240 multi-linear mapping, 75 multi-Lyapunov function, 483 neighborhood, 34 neighborhood basis, 34 neutrally stable, 355 nilpotent Lie algebra, 117 non-regular feedback, 243 non-resonant matrix, 281 nonlinear control system, 1 norm, 30 norm induced by inner product, 32 normal form, 281 normal subgroup, 92 normed space, 30 nowhere dense, 36 NR-type transformation, 311 observability co-distribution, 137 observability rank condition, 137 observable, 5 odd approximation, 322 one form, 66 one parameter group, 61 open covering, 41 open set, 34 open-loop control, 121 orbit, 188 orientable, 78 orthogonal group, 114 oscillatory center, 335 output feedback control, 1 output normal point, 245 output regular point, 243 output regulation, 356 output tracking, 12 parallel decomposition, 225 passive, 380 passivity property, 193 path, 40 pathwise connected, 40 penalty signals, 409 Poincar´e map, 188 Poincar´e’s linearization, 279 Poincar´e’s Theorem, 281 Poincare’s Lemma, 66 point relative degree, 16, 258 Poisson bracket, 87 Poisson manifold, 87 Poisson mapping, 89 Poisson stable, 355 pole assignment, 8 pole of linear system, 8

543

544

Index

port-controlled Hamiltonian system with dissipation, 398 positive denite function, 178 principal ideal, 98 product bundle, 53 product group, 97 product topological space, 38 projection, 32, 53 proper mapping, 44 pseudo-Riemannian manifold, 80 Quadratic Form Reducing Algorithm (QFRA), 326 quadratically stabilizable, 454 Quaker lemma, 208 quotient group, 94 quotient topology, 44 radially unbounded, 178 rank of F, 50 Rank theorem, 533 reachable set, 17, 122 realization, 1 reduced-dimensional observer, 9 region of attraction, 195 regular point, 16, 17, 141 regular sub-manifold, 51 regular value, 17 regulator equations, 356 relative degree, 10 relative degree vector, 11, 237 Riccati inequality, 387, 409, 417 Riemannian manifold, 80 right g coset, 92 right zero factor, 98 right-invariant vector eld, 115 ring, 97 ring homomorphism, 99 ring isomorphism, 99 ring with identity, 98 robust local output regulation, 374 RSIFAD canonical form, 307 RTAC system, 371 Sard’s theorem, 533 Schwarz inequality, 30–33 second category, 37 second countable, 35 Second Homomorphism Theorem, 95 section, 53 semi-simple Lie algebra, 118 semi-tensor product, 535 separable, 37 separation principle, 360 sequential compact, 42 set-valued, 511 Set-valued derivative, 514

simple Lie algebra, 118 simply connected, 106 simultaneously integrable, 223 singular point, 16, 17 singular value, 17 skew-symmetric covariant tensor, 76 sliding mode motion, 512 Smith-McMillan form, 13 solvable Lie algebra, 117 special linear group, 114 special orthogonal group, 117 stable, 174 stable manifold, 315 stable sub-manifold, 195 star shape, 100 state feedback control, 1 state feedback linearization, 288 stationary solution, 512 storage function, 379 strange attractor, 20 strictly dissipative, 379 strictly passive, 380 strictly positive real condition, 193 strong accessibility rank condition, 130 strong accessibility Lie algebra, 130 strong accessible, 127 structure function, 88 structure matrix, 75, 88 subgroup, 92 subgroup generated by S, 93 subring, 98 supply rate, 379 swap matrix, 537 switched system, 431 switching mode, 431 switching signal, 431 Sylvester equation, 362 symmetric covariant tensor, 76 symplectic geometry, 85 symplectic group, 115 tangent space, 57 Taylor expansion, 540 tensor, 75 Third Homomorphism Theorem, 96 topological basis, 34 topological manifold, 47 topological space, 34 topological subspace, 38 TORA system, 371 torus, 46 total space, 53 total stability, 182 totally stable, 182 transfer function, 8 transfer function matrix, 11 transmission zero, 10, 12

Index triangular system, 177 trivial bundle, 55 trivial group, 106 type-k equilibrium, 195 unicycle Model, 21 uniform BIBO stable, 191 uniformly attractive, 174 uniformly stable, 174 unit volume, 80 unitary group, 114 universal covering space, 111 unstable manifold, 315

unstable sub-manifold, 195 vector bundle, 55 vector bundle mapping, 55 vector eld, 59 weak Lyapunov function, 483 weakly ( f , g)-invariant, 207 weakly reachable, 122 zero center, 328 zero dynamics, 10 zero of linear system, 8

545