Nonlinear Control Sy.. - Free

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2.8. DIFFERENTIABILITY 2.8 Differentiability Definition 2.20 A function f : R -> R is said to be differentiable at x if f is defined in an open interval (a, b) C R and exists. f(x + hh - f(x) f'(x) = l a The limit f'(x) is called the derivative of f at x. The function f is said to be differentiable if it is differentiable at each x in its domain. If the derivative exists, then f(x + h) - f(x) = f'(x)h + r(h) where the "remainder" r(h) is small in the sense that lim r(h) = 0. h,O h Now consider the case of a function f : R" -+ R. Definition 2.21 A function f : R' -4 R' is said to be differentiable at a point x if f is defined in an open set D C R" containing x and the limit exists. lim If(x+h) - f(x) - f'(x)hIl = 0 h-+0 lhll Notice, of course, in Definition 2.21 that h E R'. If IhII is small enough, then x + h E D, since D is open and f (x + h) E JRt. The derivative f'(x) defined in Definition 2.21 is called the differential or the total derivative of f at x, to distinguish it from the partial derivatives that we discuss next. In the following discussion we denote by f2, 1 < i < 7n the components of the function f, and by {el, e2, , en} the standard basis in R" fi (x) f2(x) f(x) = I Lfm(x)I el = fol f°1 fBl ,e2 = 0 0 en = 1 49
50 CHAPTER 2. MATHEMATICAL PRELIMINARIES Definition 2.22 Consider a function f : ]R" -4 IItm and let D be an open set in ]R". For xEDCIR" andl
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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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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
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
Page 46 and 47: 1.10. EXERCISES 29 m2 Figure 1.19:
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 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 76 and 77: 2.12. EXERCISES 59 Denoting t1 = to
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Page 80: 2.12. EXERCISES 63 Notes and Refere
Page 83 and 84: 66 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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