Nonlinear Control Sy.. - Free

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Index Absolute stability, 179 Asymptotic stability in the large, 77 Autonomous system, 4 Available storage, 231 Backstepping chain of integrators, 145 integrator backstepping, 141 strict feedback systems, 148 Ball-and-beam system, 26 Basis, 34 Bounded linear operators, 48 Bounded set, 44 Cartesian product, 46 Cauchy sequence, 45 Causality, 159 Center, 16 Chaos, 21 Chetaev's theorem, 100 Circle criterion, 178 Class )CL function, 81 Class TC,,. function, 80 Class JC function, 80 Closed-loop input-output stability, 168 Compact set, 45 Complement (of a set), 44 Complete integrability, 263 Contraction mapping, 54 theorem, 54 Converse theorems, 125 Convex set, 45 349 Coordinate transformations, 259 Covector field, 256 Diagonal form of a matrix, 40 Diffeomorphism, 259 Differentiability, 49 Differentiable function, 49 Differential, 49 Discrete-time systems, 126 stability, 130 Discretization, 127 Dissipation inequality, 224 for input-to-state stability, 193 Dissipative system, 224 Dissipativity, 223 QSR, 226 algebraic condition for, 233 Distinguishable states, 296 Distribution involutive, 263 nonsingular, 262 regular point, 262 singular point, 262 variable dimension, 262 Distributions, 261 Eigenvalues, 40 Eigenvectors, 40 Equilibrium point, 5 asymptotically stable, 67 convergent, 66 exponentially stable, 67
350 globally uniformly asymptotically stable, 109 stable, 66 uniformly asymptotically stable, 109 uniformly convergent, 109 uniformly stable, 109 unstable, 66 Euclidean norm, 38 Exponential stability, 82 Extended spaces, 157 Feedback interconnections, 239 Feedback linearization, 255 Feedback systems, 137 basic feedback stabilization, Finite-gain-stable system, 228 Fixed-point theorem, 54 Focus stable, 16 unstable, 16 Fractal, 21 Frobenius theorem, 264 Function spaces, 156 ,C2, 156 L,, 156 Functions, 46 bijective, 46 continuous, 47 domain, 46 injective, 46 range, 46 restriction, 47 surjective, 46 uniformly continuous, 47 Gradient, 51 Hilbert space, 204 Holder's inequality, 38 Inner product, 204 138 Inner product space, 204 Input-output linearization, 275 Input-output stability, 160 Input-output systems, 155 Input-State linearization, 265 Input-output stability, 159 Input-to-state stability, 183, 226 Instability, 100 Interior point, 44 Internal dynamics, 282 Invariance principle, 85 Invariant set, 86 Inverse function theorem, 52 Inverted pendulum on a cart, 25 ISS Lyapunov function, 186 Jacobian matrix, 50 Kalman-Yakubovich lemma, 218 INDEX Lagrange stable equilibrium point, 106 LaSalle's Theorem, 90 Lie bracket, 257 Lie derivative, 256 Limit cycles, 18 Limit point, 44 Limit set, 86 Linear independence, 34 Linear map, operator, transformation, 48 Linear time-invariant systems, 1 £ stability, 164 analysis, 96 Linearization input-state, 265 Linearization principle, 120 Lipschitz continuity, 52 Lipschitz systems, 301 Loop transformations, 174 Lyapunov function, 74 Lyapunov stability autonomous systems, 65, 71
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CONTROi I naiysis ana uesign (4)WIL
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Copyright © 2003 by John Wiley & S
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Contents 1 Introduction 1 1.1 Linea
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CONTENTS ix 3.8 The Invariance Prin
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CONTENTS xi 8.1 Power and Energy: P
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CONTENTS xiii A Proofs 307 A.1 Chap
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xvi 8, along with some of the most
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Chapter 1 Introduction This first c
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1.2. NONLINEAR SYSTEMS 3 with A=[-o
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1.3. EQUILIBRIUM POINTS 5 1.3 Equil
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1.4. FIRST-ORDER AUTONOMOUS NONLINE
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1.5. SECOND-ORDER SYSTEMS: PHASE-PL
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1.6. PHASE-PLANE ANALYSIS OF LINEAR
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1.7. PHASE-PLANE ANALYSIS OF NONLIN
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1.8. HIGHER-ORDER SYSTEMS 1.8.1 Cha
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1.9. EXAMPLES OF NONLINEAR SYSTEMS
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1.9. EXAMPLES OF NONLINEAR SYSTEMS
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1.10. EXERCISES 27 Figure 1.18: Bal
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1.10. EXERCISES 29 m2 Figure 1.19:
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Chapter 2 Mathematical Preliminarie
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2.3. VECTOR SPACES 33 (3) 3 0 E X :
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2.3. VECTOR SPACES 35 where the 1 e
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2.3. VECTOR SPACES 37 Proof: First
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2.4. MATRICES 39 2.4 Matrices We as
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2.4. MATRICES 41 Proof: By definiti
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2.4. MATRICES 43 (i) A is positive
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2.6. SEQUENCES 45 Compact set: A se
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2.7. FUNCTIONS 47 If f is a functio
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2.8. DIFFERENTIABILITY 2.8 Differen
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2.8. DIFFERENTIABILITY 51 Theorem 2
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2.9. LIPSCHITZ CONTINUITY 53 Notice
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2.10. CONTRACTION MAPPING 55 and th
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2.11. SOLUTION OF DIFFERENTIAL EQUA
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2.12. EXERCISES 59 Denoting t1 = to
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2.12. EXERCISES 61 (2.10) Show that
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2.12. EXERCISES 63 Notes and Refere
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66 CHAPTER 3. LYAPUNOV STABILITY I.
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72 CHAPTER 3. LYAPUNOV STABILITY I:
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100 CHAPTER 3. LYAPUNOV STABILITY I
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108 CHAPTER 4. LYAPUNOV STABILITY H
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126 CHAPTER 4. LYAPUNOV STABILITY H
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Chapter 5 Feedback Systems So far,
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5.1. BASIC FEEDBACK STABILIZATION 1
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5.2. INTEGRATOR BACKSTEPPING 141 5.
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5.2. INTEGRATOR BACKSTEPPING 143 De
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5.3. BACKSTEPPING: MORE GENERAL CAS
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5.4. EXAMPLE 151 5.4 Example 8 Figu
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5.5. EXERCISES 153 [S - 0(x)] = x1
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Chapter 6 Input-Output Stability So
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6.1. FUNCTION SPACES 157 Both L2 an
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6.2. INPUT-OUTPUT STABILITY 159 Thu
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6.2. INPUT-OUTPUT STABILITY 161 (a)
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6.2. INPUT-OUTPUT STABILITY 163 1.2
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6.3. LINEAR TIME-INVARIANT SYSTEMS
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6.4. L GAINS FOR LTI SYSTEMS where
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6.5. CLOSED-LOOP INPUT-OUTPUT STABI
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6.6. THE SMALL GAIN THEOREM 171 In
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6.6. THE SMALL GAIN THEOREM 173 N(x
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6.7. LOOP TRANSFORMATIONS 175 Figur
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6.7. LOOP TRANSFORMATIONS 177 U l M
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6.8. THE CIRCLE CRITERION 179 (i) g
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6.9. EXERCISES 181 (6.3) Prove the
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Chapter 7 Input-to-State Stability
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7.2. DEFINITIONS 185 Setting u = 0,
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7.3. INPUT-TO-STATE STABILITY (ISS)
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7.3. INPUT-TO-STATE STABILITY (ISS)
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7.4. INPUT-TO-STATE STABILITY REVIS
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7.4. INPUT-TO-STATE STABILITY REVIS
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7.5. CASCADE-CONNECTED SYSTEMS U E2
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7.5. CASCADE-CONNECTED SYSTEMS 197
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7.6. EXERCISES 199 (i) Is it locall
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202 CHAPTER 8. PASSIVITY v i Figure
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204 CHAPTER 8. PASSIVITY Moreover,
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206 CHAPTER 8. PASSIVITY Definition
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208 CHAPTER 8. PASSIVITY Thus (UT,
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210 CHAPTER 8. PASSIVITY Thus, H f
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212 CHAPTER 8. PASSIVITY Under thes
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214 CHAPTER 8. PASSIVITY Under thes
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216 CHAPTER 8. PASSIVITY (i) H is p
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218 CHAPTER 8. PASSIVITY Definition
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220 CHAPTER 8. PASSIVITY Example 8.
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Chapter 9 Dissipativity In Chapter
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9.2. DIFFERENTIABLE STORAGE FUNCTIO
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9.3. QSR DISSIPATIVITY 227 Definiti
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9.4. EXAMPLES 229 5- Very strictly-
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9.5. AVAILABLE STORAGE 9.4.2 Mass-S
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9.6. ALGEBRAIC CONDITION FOR DISSIP
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9.6. ALGEBRAIC CONDITION FOR DISSIP
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9.7. STABILITY OF DISSIPATIVE SYSTE
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9.8. FEEDBACK INTERCONNECTIONS 239
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9.8. FEEDBACK INTERCONNECTIONS 241
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9.9. NONLINEAR G2 GAIN 9.9 Nonlinea
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9.9. NONLINEAR L2 GAIN 245 (iii) IH
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9.10. SOME REMARKS ABOUT CONTROL DE
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9.10. SOME REMARKS ABOUT CONTROL DE
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9.11. NONLINEAR L2-GAIN CONTROL 251
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9.12. EXERCISES 253 9.12 Exercises
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Chapter 10 Feedback Linearization I
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10.1. MATHEMATICAL TOOLS 257 Lfh(x)
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10.1. MATHEMATICAL TOOLS 259 10.1.3
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10.1. MATHEMATICAL TOOLS 261 and su
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10.1. MATHEMATICAL TOOLS 263 Defini
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10.2. INPUT-STATE LINEARIZATION 265
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10.2. INPUT-STATE LINEARIZATION 267
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10.2. INPUT-STATE LINEARIZATION and
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10.3. EXAMPLES 271 provided that wh
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10.4. CONDITIONS FOR INPUT-STATE LI
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10.5. INPUT-OUTPUT LINEARIZATION 27
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10.6. THE ZERO DYNAMICS 281 i=Ax+Bu
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10.6. THE ZERO DYNAMICS 283 that th
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10.6. THE ZERO DYNAMICS 285 Definit
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10.7. CONDITIONS FOR INPUT-OUTPUT L
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10.8. EXERCISES 289 (10.8) Consider
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292 CHAPTER 11. NONLINEAR OBSERVERS
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Page 315 and 316: 298 CHAPTER 11. NONLINEAR OBSERVERS
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Page 337 and 338: 320 APPENDIX A. PROOFS (b) 0 = a ,3
Page 339 and 340: 322 APPENDIX A. PROOFS To this end
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Page 351 and 352: 334 APPENDIX A. PROOFS Multiplying
Page 354 and 355: Bibliography [1] B. D. O. Anderson
Page 356 and 357: BIBLIOGRAPHY 339 [27] W. Hahn, Stab
Page 358 and 359: BIBLIOGRAPHY 341 [58] E. Ott, Chaos
Page 360: BIBLIOGRAPHY 343 [87] A. van der Sc
Page 363 and 364: 346 LIST OF FIGURES 3.1 Stable equi
Page 368 and 369: INDEX discrete-time, 132 nonautonom
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