Nonlinear Control Sy.. - Free

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9.9. NONLINEAR G2 GAIN 9.9 <strong>Nonlinear</strong> G2 Gain Consider a system V) of the form i = f(x) + g(x)u y = h(x) 243 (9.31) As discussed in Section 9.2, this system is finite-gain-stable with gain y, if it is dissipative with supply rate w = 2(y2IIu112 - IIy112) Assuming that the storage function corresponding to this supply rate is differentiable, the differential dissipation inequality (9.5) implies that fi(t) = aa( )x 1 < w(t) s 272IIu112 - 2IIyl12 < 0. Thus, substituting (9.31) into (9.32) we have that a0(x) f(x) + ao x)g(x)u < w(t)
244 CHAPTER 9. DISSIPATWITY This result is important. According to (9.35), if the system V) given by (9.31) is finite-gainstable with gain ry, then it must satisfy the so-called Hamilton-Jacobi inequality: (ax )T 1 H def a-f(x) + 2,y2 a--gg + 2 IyI 2< 0 (9.36) Finding a function that maximizes f and satisfies inequality (9.36) is, at best, very difficult. Often, we will be content with "estimating" an upper bound for y. This can be done by "guessing" a function 0 and then finding an approximate value for -y; a process that resembles that of finding a suitable Lyapunov function. The true L2 gain of 0, denoted 'y', is bounded above by -y, i.e., 0
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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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Page 212 and 213: 7.5. CASCADE-CONNECTED SYSTEMS U E2
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Page 219 and 220: 202 CHAPTER 8. PASSIVITY v i Figure
Page 221 and 222: 204 CHAPTER 8. PASSIVITY Moreover,
Page 223 and 224: 206 CHAPTER 8. PASSIVITY Definition
Page 225 and 226: 208 CHAPTER 8. PASSIVITY Thus (UT,
Page 227 and 228: 210 CHAPTER 8. PASSIVITY Thus, H f
Page 229 and 230: 212 CHAPTER 8. PASSIVITY Under thes
Page 231 and 232: 214 CHAPTER 8. PASSIVITY Under thes
Page 233 and 234: 216 CHAPTER 8. PASSIVITY (i) H is p
Page 235 and 236: 218 CHAPTER 8. PASSIVITY Definition
Page 237 and 238: 220 CHAPTER 8. PASSIVITY Example 8.
Page 240 and 241: Chapter 9 Dissipativity In Chapter
Page 242 and 243: 9.2. DIFFERENTIABLE STORAGE FUNCTIO
Page 244 and 245: 9.3. QSR DISSIPATIVITY 227 Definiti
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Page 248 and 249: 9.5. AVAILABLE STORAGE 9.4.2 Mass-S
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Page 254 and 255: 9.7. STABILITY OF DISSIPATIVE SYSTE
Page 256 and 257: 9.8. FEEDBACK INTERCONNECTIONS 239
Page 258 and 259: 9.8. FEEDBACK INTERCONNECTIONS 241
Page 262 and 263: 9.9. NONLINEAR L2 GAIN 245 (iii) IH
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Page 268 and 269: 9.11. NONLINEAR L2-GAIN CONTROL 251
Page 270 and 271: 9.12. EXERCISES 253 9.12 Exercises
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Page 274 and 275: 10.1. MATHEMATICAL TOOLS 257 Lfh(x)
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Page 278 and 279: 10.1. MATHEMATICAL TOOLS 261 and su
Page 280 and 281: 10.1. MATHEMATICAL TOOLS 263 Defini
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Page 288 and 289: 10.3. EXAMPLES 271 provided that wh
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Page 309 and 310: 292 CHAPTER 11. NONLINEAR OBSERVERS
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294 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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