Nonlinear Control Sy.. - Free

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Contents 1 Introduction 1 1.1 Linear Time-Invariant Systems . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Equilibrium Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 First-Order Autonomous Nonlinear Systems . . . . . . . . . . . . . . . . . 5 1.5 Second-Order Systems: Phase-Plane Analysis . . . . . . . . . . . . . . . . . 8 1.6 Phase-Plane Analysis of Linear Time-Invariant Systems . . . . . . . . . . . 10 1.7 Phase-Plane Analysis of Nonlinear Systems . . . . . . . . . . . . . . . . . . 18 1.7.1 Limit Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.8 Higher-Order Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.8.1 Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.9 Examples of Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.9.1 Magnetic Suspension System . . . . . . . . . . . . . . . . . . . . . . 23 1.9.2 Inverted Pendulum on a Cart . . . . . . . . . . . . . . . . . . . . . . 25 1.9.3 The Ball-and-Beam System . . . . . . . . . . . . . . . . . . . . . . . 26 1.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2 Mathematical Preliminaries 31 2.1 Sets ........................................ 31 2.2 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 vii
viii CONTENTS 2.3 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3.1 2.3.2 Linear Independence and Basis . . . . . . . . . . . Subspaces .................... . . . . . . . . . . . . . . ......... 34 36 2.3.3 Normed Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.4 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.4.1 Eigenvalues, Eigenvectors, and Diagonal Forms . . . . . . . . . . . . 40 2.4.2 Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.5 Basic Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.5.1 Basic Topology in ]It" . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.6 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.7 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.7.1 Bounded Linear Operators and Matrix Norms . . . . . . . . . . . . . 48 2.8 Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.8.1 Some Useful Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.9 Lipschitz Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.10 Contraction Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.11 Solution of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . 56 2.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3 Lyapunov Stability I: Autonomous Systems 65 3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.2 Positive Definite Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.3 Stability Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.5 Asymptotic Stability in the Large . . . . . . . . . . . . . . . . . . . . . . . . 77 3.6 Positive Definite Functions Revisited . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Exponential Stability ................ . . ......... 80 82 3.7 Construction of Lyapunov Functions . . . . . . . . . . . . . . . . . . . . . . 82
Page 1 and 2: CONTROi I naiysis ana uesign (4)WIL
Page 3 and 4: Copyright © 2003 by John Wiley & S
Page 8 and 9: CONTENTS ix 3.8 The Invariance Prin
Page 10 and 11: CONTENTS xi 8.1 Power and Energy: P
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Page 18 and 19: Chapter 1 Introduction This first c
Page 20 and 21: 1.2. NONLINEAR SYSTEMS 3 with A=[-o
Page 22 and 23: 1.3. EQUILIBRIUM POINTS 5 1.3 Equil
Page 24 and 25: 1.4. FIRST-ORDER AUTONOMOUS NONLINE
Page 26 and 27: 1.5. SECOND-ORDER SYSTEMS: PHASE-PL
Page 28 and 29: 1.6. PHASE-PLANE ANALYSIS OF LINEAR
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Page 44 and 45: 1.10. EXERCISES 27 Figure 1.18: Bal
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Page 48 and 49: Chapter 2 Mathematical Preliminarie
Page 50 and 51: 2.3. VECTOR SPACES 33 (3) 3 0 E X :
Page 52 and 53: 2.3. VECTOR SPACES 35 where the 1 e
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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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308 APPENDIX A. PROOFS (ii) It is c
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310 APPENDIX A. PROOFS For the conv
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312 APPENDIX A. PROOFS 9V ax-1 xz 9
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314 APPENDIX A. PROOFS Since the eq
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316 APPENDIX A. PROOFS Proof: The n
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318 APPENDIX A. PROOFS We have x Fi
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320 APPENDIX A. PROOFS (b) 0 = a ,3
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322 APPENDIX A. PROOFS To this end
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324 A.5 Chapter 8 APPENDIX A. PROOF
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326 APPENDIX A. PROOFS and substitu
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328 APPENDIX A. PROOFS It follows t
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330 APPENDIX A. PROOFS A.7 Chapter
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332 APPENDIX A. PROOFS Assume now t
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334 APPENDIX A. PROOFS Multiplying
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Bibliography [1] B. D. O. Anderson
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BIBLIOGRAPHY 339 [27] W. Hahn, Stab
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BIBLIOGRAPHY 341 [58] E. Ott, Chaos
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BIBLIOGRAPHY 343 [87] A. van der Sc
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346 LIST OF FIGURES 3.1 Stable equi
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Index Absolute stability, 179 Asymp
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INDEX discrete-time, 132 nonautonom
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