Stability and Stabilization: An Introduction - Rilegato

Terrell, William J.

Rilegato

ISBN 10: 0691134448 ISBN 13: 9780691134444

Casa editrice: Princeton Univ Pr, 2009

Riassunto

Stability and Stabilization is the first intermediate-level textbook that covers stability and stabilization of equilibria for both linear and nonlinear time-invariant systems of ordinary differential equations. Designed for advanced undergraduates and beginning graduate students in the sciences, engineering, and mathematics, the book takes a unique modern approach that bridges the gap between linear and nonlinear systems.


Presenting stability and stabilization of equilibria as a core problem of mathematical control theory, the book emphasizes the subject's mathematical coherence and unity, and it introduces and develops many of the core concepts of systems and control theory. There are five chapters on linear systems and nine chapters on nonlinear systems; an introductory chapter; a mathematical background chapter; a short final chapter on further reading; and appendixes on basic analysis, ordinary differential equations, manifolds and the Frobenius theorem, and comparison functions and their use in differential equations. The introduction to linear system theory presents the full framework of basic state-space theory, providing just enough detail to prepare students for the material on nonlinear systems.

• Focuses on stability and feedback stabilization


• Bridges the gap between linear and nonlinear systems for advanced undergraduates and beginning graduate students


• Balances coverage of linear and nonlinear systems


• Covers cascade systems


• Includes many examples and exercises

Su questo libro

Informazioni sull?autore

William J. Terrell is associate professor of mathematics and applied mathematics at Virginia Commonwealth University. In 2000, he received a Lester R. Ford Award for excellence in expository writing from the Mathematical Association of America.

Dalla quarta di copertina

"This book is a pleasant surprise. William Terrell selects and presents the field's key results in a fresh and unbiased way. He is enthusiastic about the material and his goal of setting forth linear and nonlinear stabilization in a unified format."--Miroslav Krstic, University of California, San Diego

"This textbook has very positive features. The arguments are complete; it does not shy away from making correct proofs one of its main goals; it strikes an unusually good balance between linear and nonlinear systems; and it has many examples and exercises. It is also mathematically sophisticated for an introductory text, and it covers very recent material."--Jan Willems, coauthor ofIntroduction to Mathematical Systems Theory

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Stability and Stabilization

An Introduction

By William J. Terrell

PRINCETON UNIVERSITY PRESS

Copyright © 2009 Princeton University Press
All rights reserved.

ISBN: 978-0-691-13444-4

Contents

List of Figures, xi, 
Preface, xiii, 
1 Introduction, 1, 
2 Mathematical Background, 12, 
3 Linear Systems and Stability, 49, 
4 Controllability of Linear Time Invariant Systems, 82, 
5 Observability and Duality, 109, 
6 Stabilizability of LTI Systems, 124, 
7 Detectability and Duality, 138, 
8 Stability Theory, 161, 
9 Cascade Systems, 189, 
10 Center Manifold Theory, 212, 
11 Zero Dynamics, 233, 
12 Feedback Linearization of Single-Input Nonlinear Systems, 268, 
13 An Introduction to Damping Control, 289, 
14 Passivity, 302, 
15 Partially Linear Cascade Systems, 331, 
16 Input-to-State Stability, 359, 
17 Some Further Reading, 378, 
Appendix A Notation: A Brief Key, 381, 
Appendix B Analysis in R and Rn, 383, 
Appendix C Ordinary Differential Equations, 393, 
Appendix D Manifolds and the Preimage Theorem; Distributions and the Frobenius Theorem, 403, 
Appendix E Comparison Functions and a Comparison Lemma, 420, 
Appendix F Hints and Solutions for Selected Exercises, 430, 
Bibliography, 443, 
Index, 451, 

CHAPTER 1

Introduction

In this short introductory chapter, we introduce the main problem of stability and stabilization of equilibria, and indicate briefly the central role it plays in mathematical control theory. The presentation here is mostly informal. Precise definitions are given later. The chapter serves to give some perspective while stating the primary theme of the text.

We start with a discussion of simple equations from an elementary differential equations course in order to contrast open loop control and feedback control. These examples lead us to a statement of the main problem considered in the book, followed by an indication of the central importance of stability and stabilization in mathematical control theory. We then note a few important omissions. A separate section gives a complete chapter-by-chapter description of the book. The final section of the chapter is a list of suggested collateral reading.

1.1 OPEN LOOP CONTROL

Students of elementary differential equations already have experience with open loop controls. These controls appear as a given time-dependent forcing term in the second order linear equations that are covered in the first course on the subject. A couple of simple examples will serve to illustrate the notion of open loop control and allow us to set the stage for a discussion of feedback control in the next section.

The Forced Harmonic Oscillator. Consider the nonhomogeneous linear mass-spring equation with unit mass and unit spring constant,

[??] + y = u(t).

We use [??] and [??] to denote the first and second derivatives of y(t) with respect to time. The equation involves a known right-hand side, which can be viewed as a preprogrammed, or open loop, control defined by u(t). The general real-valued solution for such equations is considered in differential equations courses, and it takes the form

y(t) = yh(t) + yp(t),

where yp(t) is any particular solution of the nonhomogeneous equation and yh(t) denotes the general solution of the homogeneous equation, [??] + y = 0. For this mass-spring equation, we have

yh(t) = c1 cos t + c2 sin t,

where the constants c1 and c2are uniquely determined by initial conditions for y(0) and [??](0).

Suppose the input signal is u(t) = sin t. This would not be an effective control, for example, if our purpose is to damp out the motion asymptotically or to regulate the motion to track a specified position or velocity trajectory. Since the frequency of the input signal equals the natural frequency of the unforced harmonic oscillator, [??] + y = 0, the sine input creates a resonance that produces unbounded motion of the mass.

On the other hand, the decaying input u(t) = e-t yields a particular solution given by yp(t) = 1/2 e-t. In this case, every solution approaches a periodic response as t -> ∞, given by yh(t), which depends on the initial conditions y(0) and [??](0), but not on the input signal.

Suppose we wanted to apply a continuous input signal which would guarantee that all solutions approach the origin defined by zero position and zero velocity. It is not difficult to see that we cannot do this with a continuous open loop control. The theory for second-order linear equations implies that there is no continuous open loop control u(t) such that each solution of [??] + y = u(t) approaches the origin as t -> ∞, independently of initial conditions.

The Double Integrator. An even simpler equation is [??] = u(t). The general solution has the form y(t) = c1 + c2t + yp(t), where yp(t) is a particular solution that depends on u(t). Again, there is no continuous control u(t) that will guarantee that the solutions will approach the origin defined by zero position and zero velocity, independently of initial conditions.

Open loop, or preprogrammed, control does not respond to the state of the system it controls during operation. A standard feature of engineering design involves the idea of injecting a signal into a system to determine the response to an impulse, step, or ramp input signal. Recent work on the active approach to the design of signals for failure detection uses open loop controls as test signals to detect abnormal behavior; an understanding of such open loop controls may enable more autonomous operation of equipment and condition-based maintenance, resulting in less costly or safer operation.

The main focus of this book is on principles of stability and feedback stabilization of an equilibrium of a dynamical system. The next section explains this terminology and gives a general statement of this core problem of dynamics and control.

1.2 THE FEEDBACK STABILIZATION PROBLEM

The main theme of stability and stabilization is focused by an emphasis on time invariant (autonomous) systems of the form

[??] = f(x),

where f : D [subset] Rn ->Rn is a continuously differentiable mapping (a smooth vector field on an open set D [subset] Rn) and [??] := dx/dt. If f is continuously differentiable, then f satisfies a local Lipschitz continuity condition in a neighborhood of each point in its domain. From the theory of ordinary differential equations, the condition of local Lipschitz continuity of f guarantees the existence and uniqueness of solutions of initial value problems

[??] = f(x), x(0) = x0,

where x0 is a given point of D.

The state of the system at time t is described by the vector x. Assuming that f(0) = 0, so that the origin is an equilibrium (constant) solution of the system, the core problem is to determine the stability properties of the equilibrium. The main emphasis is on conditions for asymptotic stability of the equilibrium. A precise definition of the term asymptotic stability of x = 0 is given later. For the moment, we simply state its intuitive meaning: Solutions x(t) with initial condition close to the origin are defined for all forward time t ≥ 0 and remain close to x = 0 for all t ≥ 0; moreover, initial conditions sufficiently close to the equilibrium yield solutions that approach the equilibrium asymptotically as t -> ∞.

We can now discuss the meaning of feedback stabilization of an equilibrium. Let f : Rn × Rm ->Rn be a continuously differentiable function of (x, u) [member of] Rn × Rm. The introduction of a feedback control models the more complicated process of actually measuring the system state and employing some mechanism to feed the measured state back into the system as a real time control on system operation. The feedback stabilization problems in this book involve autonomous systems with control u, given by

[??} = f(x, u).

In this framework, the introduction of a smooth (continuously differentiable) state feedback control u = k(x) results in the closed loop system

[??} = f(x, k(x)),

which is autonomous as well. If f(0, 0) = 0, then the origin x0 = 0 is an equilibrium of the unforced system, [??] = f(x, 0). If the feedback satisfies k(0) = 0, then it preserves the equilibrium; that is, the closed loop system also has an equilibrium at the origin.

We apply stability theory in several different settings to study the question of existence of a continuously differentiable feedback u = k(x) such that the origin x0 = 0 is an asymptotically stable equilibrium of the closed loop system. For certain system classes and conditions, explicit stabilizing feedback controls are constructed. The system classes we consider are not chosen arbitrarily; they are motivated by (i) their relevance in the research activity on stabilization of recent decades, and (ii) their accessibility in an introductory text.

Feedback in the Harmonic Oscillator and Double Integrator Systems. The system corresponding to the undamped and unforced harmonic oscillator, obtained by writing x1 = y and x2 = [??], and setting u = 0, is given by

[??]1 = x2, [??]2 = -x1.

This system does not have an asymptotically stable equilibrium at the origin (x1, x2) = (0, 0). If we had both state components available for feedback, we could define a feedback control of the form u = k1x1 + k2x2, producing the closed loop system

[??]1 = x2, [??]2 =(k1 - 1)x1 + k2x2.

If we can measure only the position variable x1 and use it for feedback, say u = k1x1, then we are not able to make the origin (0, 0) asymptotically stable, no matter what the value of the real coefficient k1 may be. However, using only feedback from the velocity, if available, say u = k2x2, it is possible to make the origin an asymptotically stable equilibrium of the closed loop system. Verification of these facts is straightforward, and to accomplish it, we can even use the second order form for the closed loop system; for position feedback only, [??] + (1-k1)y = 0; for velocity feedback only, [??] - k2[??] + y = 0. For position feedback, the characteristic equation is r2 + (1 - k1) = 0, and the general real-valued solution for t ≥ 0 is (i) periodic for k1< 1, (ii) the sum of an increasing exponential term and a decreasing exponential term for k1 > 1, and (iii) a constant plus an unbounded linear term for k1 = 1. For velocity feedback, choosing k2< 0 ensures that all solutions that start close to the origin at time t = 0 remain close to the origin for all t ≥ 0, and also satisfy (x1(t), x2(t)) -> (0, 0) as t -> ∞.

For the simpler double integrator equation, [??] = u(t), or its equivalent system,

[??]1 = x2, [??]2 = u,

one can check that neither position feedback, u = k1x1, nor velocity feedback, u = k2x2, can make all solutions approach the origin as t -> ∞. However, feedback using both position and velocity, u = k1x1 + k2x2, will accomplish this if k1 > 0 and k2 > 0.

The study of stability and stabilization of equilibria for ordinary differential equations (ODEs) is a vast area of applications-oriented mathematics. The restriction to smooth feedback still leaves a huge area of results. This area will be explored in selected directions in the pages of this introductory text.

The restriction to smooth feedback avoids some technical issues that arise with discontinuous feedback, or even with merely continuous feedback. Discontinuous feedback is mathematically interesting and relevant in many applications. For example, the solutions of many optimal control problems (not discussed in this book) involve discontinuous feedback. However, a systematic study of such feedback requires a reconsideration of the type of system under study and the meaning of solution. These questions fall essentially outside the scope of the present book.

Although we consider primarily smooth feedback controls, at several places in the text the admissible open loop controls are piecewise continuous, or at least integrable on any finite interval, that is, locally integrable.

The Importance of the Subject

Stability theory provides core techniques for the analysis of dynamical systems, and it has done so for well over a hundred years, at least since the 1892 work of A. M. Lyapunov; see [73]. An earlier feedback control study of a steam engine governor, by J. Clerk Maxwell, was probably the first modern analysis of a control system and its stability. Stability concepts have always been a central concern in the study of dynamical control systems and their applications. The problem of feedback stabilization of equilibria is a core problem of mathematical control theory. Possibly the most important point to make here is that many other issues and problems of control theory depend on concepts and techniques of stability and stabilization for their mathematical foundation and expression. Some of these areas are indicated in the end-of-chapter Notes and References sections.

Some Important Omissions

There are many important topics of stability, stabilization, and, more generally, mathematical control theory which are not addressed in this book.

In particular, as mentioned in the Preface, there is no discussion of transfer function analysis for linear time invariant systems, and transfer functions are not used in the text. Also, there is no systematic coverage of optimal control beyond the single section on the algebraic Riccati equation. Since there is no coverage of numerical computation issues in this text, readers interested specifically in numerical methods should be aware of the text by B. N. Datta, Numerical Methods for Linear Control Systems, Elsevier Academic Press, London, 2004.

The end-of-chapter Notes and References sections have resources for a few other areas not covered in the text.

1.3 CHAPTER AND APPENDIX DESCRIPTIONS

In general, the chapters follow a natural progression. It may be helpful to mention that readers with a background in the state space framework of linear system theory and a primary interest in nonlinear systems might proceed with Chapter 8 (Stability Theory) after the introductory material of Chapter 2 (Mathematical Background) and Chapter 3 (Linear Systems and Stability). Definitions and examples of stability and instability appear in Chapter 3. For such readers, Chapters 4–7 could be used for reference as needed.

Chapter 2. The Mathematical Background chapter includes material mainly from linear algebra and differential equations. For basic analysis we reference Appendix B or the text. The section on linear and matrix algebra includes some basic notation, linear independence and rank, similarity of matrices, invariant subspaces, and the primary decomposition theorem. The section on matrix analysis surveys ifferentiation and integration of matrix functions, inner products and norms, sequences and series of functions, and quadratic forms. A section on ordinary differential equations states the existence and uniqueness theorem for locally Lipschitz vector fields and defines the Jacobian linearization of a system at a point. The final section has examples of linear and nonlinear mass-spring systems, pendulum systems, circuits, and population dynamics in the phase plane. These examples are familiar from a first course in differential equations. The intention of the chapter is to present only enough to push ahead to the first chapter on linear systems.

Chapter 3. This chapter develops the basic facts for linear systems of ordinary differential equations. It includes existence and uniqueness of solutions for linear systems, the stability definitions that apply throughout the book, stability results for linear systems, and some theory of Lyapunov equations. Jordan forms are introduced as a source of examples and insight into the structure of linear systems. The chapter also includes the Cayley-Hamilton theorem. A few basic facts on linear time varying systems are included as well.

The next four chapters, Chapters 4–7, provide an introduction to the four fundamental structural concepts of linear system theory: controllability, observability, stabilizability, and detectability. We discuss the invariance (or preservation) of these properties under linear coordinate change and certain feedback transformations. All four properties are related to the study of stability and stabilization throughout these four chapters. While there is some focus on single-input single-output (SISO) systems in the examples, we include basic results for multi-input multi-output (MIMO) systems as well. Throughout Chapters 4–7, Jordan form systems are used as examples to help develop insight into each of the four fundamental concepts.

(Continues...)

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Dettagli dell'edizione

• Editore: Princeton Univ Pr
• Data di pubblicazione: 2009
• Lingua: Inglese
• ISBN 10: 0691134448
• ISBN 13: 9780691134444
• Rilegatura: Copertina rigida
• Numero di pagine: 457
• Contatto del produttore: non disponibile
• Persona responsabile: non disponibile