Nonlinear Control Sy.. - Free

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2.12. EXERCISES 59 Denoting t1 = to + p, we have that if u is such that r µ Lr+ then the solution x(t) is confined to B. This completes the proof. It is important to notice that Theorem 2.13 provides a sufficient but not necessary condition for the existence and uniqueness of the solution of the differential equation (2.30). According to the theorem, the solution is guaranteed to exist only locally, i.e. in the interval [to, to+s]. For completeness, below we state (but do not prove) a slightly modified version of this theorem that provides a condition for the global existence and uniqueness of the solution of the same differential equation. The price paid for this generalization is a stronger and much more conservative condition imposed on the differential equation. Theorem 2.14 (Global Existence and Uniqueness) Consider again the nonlinear differential equation (2.30) and assume that f (x, t) is piecewise continuous in t and satisfies Il f (x1, t) - f (x2, t) II
60 CHAPTER 2. N ATHEMATICAL PRELIMINARIES (2.4) Prove Theorem 2.1. (2.5) For each of the norms 11x111,11x112, and IlXlloo in lR , sketch the "open unit ball" centered at 0 = [0,0]T, namely, the sets IIxiii < 1, lIx112 < 1, and IIxII,,, < 1. (2.6) Show that the vector norms II 111, II - 112, and 11 - Iloo satisfy the following: IIxl12 < 1141 :5 v1 11x112 Ilxll- 5 1x111 IIx1Io
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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
Page 26 and 27: 1.5. SECOND-ORDER SYSTEMS: PHASE-PL
Page 28 and 29: 1.6. PHASE-PLANE ANALYSIS OF LINEAR
Page 36 and 37: 1.7. PHASE-PLANE ANALYSIS OF NONLIN
Page 38 and 39: 1.8. HIGHER-ORDER SYSTEMS 1.8.1 Cha
Page 40 and 41: 1.9. EXAMPLES OF NONLINEAR SYSTEMS
Page 42 and 43: 1.9. EXAMPLES OF NONLINEAR SYSTEMS
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
Page 54 and 55: 2.3. VECTOR SPACES 37 Proof: First
Page 56 and 57: 2.4. MATRICES 39 2.4 Matrices We as
Page 58 and 59: 2.4. MATRICES 41 Proof: By definiti
Page 60 and 61: 2.4. MATRICES 43 (i) A is positive
Page 62 and 63: 2.6. SEQUENCES 45 Compact set: A se
Page 64 and 65: 2.7. FUNCTIONS 47 If f is a functio
Page 66 and 67: 2.8. DIFFERENTIABILITY 2.8 Differen
Page 68 and 69: 2.8. DIFFERENTIABILITY 51 Theorem 2
Page 70 and 71: 2.9. LIPSCHITZ CONTINUITY 53 Notice
Page 72 and 73: 2.10. CONTRACTION MAPPING 55 and th
Page 74 and 75: 2.11. SOLUTION OF DIFFERENTIAL EQUA
Page 78 and 79: 2.12. EXERCISES 61 (2.10) Show that
Page 80: 2.12. EXERCISES 63 Notes and Refere
Page 83 and 84: 66 CHAPTER 3. LYAPUNOV STABILITY I.
Page 89 and 90: 72 CHAPTER 3. LYAPUNOV STABILITY I:
Page 117 and 118: 100 CHAPTER 3. LYAPUNOV STABILITY I
Page 125 and 126: 108 CHAPTER 4. LYAPUNOV STABILITY H
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110 CHAPTER 4. LYAPUNOV STABILITY I
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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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