Nonlinear Control Sy.. - Free

More documents

Recommendations

Info

CONTENTS ix 3.8 The Invariance Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.9 Region of Attraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.10 Analysis of Linear Time-Invariant Systems . . . . . . . . . . . . . . . . . . . 96 3.10.1 Linearization of Nonlinear Systems . . . . . . . . . . . . . . . . . . . 99 3.11 Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4 Lyapunov Stability II: Nonautonomous Systems 107 4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.2 Positive Definite Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.3 Stability Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.4 Proof of the Stability Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.5 Analysis of Linear Time-Varying Systems . . . . . . . . . . . . . . . . . . . 119 4.5.1 The Linearization Principle . . . . . . . . . . . . . . . . . . . . . . . 120 4.6 Perturbation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.7 Converse Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.8 Discrete-Time Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.9 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 4.10 Stability of Discrete-Time Systems . . . . . . . . . . . . . . . . . . . . . . . 130 4.10.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4.10.2 Discrete-Time Positive Definite Functions . . . . . . . . . . . . . . . 131 4.10.3 Stability Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 4.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 5 Feedback Systems 137 5.1 Basic Feedback Stabilization . . . . . . . . . . . ....... . .... .... 138 5.2 Integrator Backstepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.3 Backstepping: More General Cases . . . . . . . . . . . . . . . . . . . . . . . 145
x CONTENTS 5.3.1 Chain of Integrators . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 5.3.2 Strict Feedback Systems . . . . . . . . . . . . . . . . . . . . . . . . . 148 5.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 6 Input-Output Stability 155 6.1 Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 6.1.1 Extended Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 6.2 Input-Output Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 6.3 Linear Time-Invariant Systems . . . . . . . . . . . . . . . . . . . . . . . . . 164 6.4 Lp Gains for LTI Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 6.4.1 L. Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 6.4.2 G2 Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 6.5 Closed-Loop Input-Output Stability . . . . . . . . . . . . . . . . . . . . . . 168 6.6 The Small Gain Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 6.7 Loop Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 6.8 The Circle Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 6.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 7 Input-to-State Stability 183 7.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 7.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 7.3 Input-to-State Stability (ISS) Theorems . . . . . . . . . . . . . . . . . . . . 186 7.3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 7.4 Input-to-State Stability Revisited . . . . . . . . . . . . . . . . . . . . . . . . 191 7.5 Cascade-Connected Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 8 Passivity 201
Page 1 and 2: CONTROi I naiysis ana uesign (4)WIL
Page 3 and 4: Copyright © 2003 by John Wiley & S
Page 6 and 7: Contents 1 Introduction 1 1.1 Linea
Page 10 and 11: CONTENTS xi 8.1 Power and Energy: P
Page 12: CONTENTS xiii A Proofs 307 A.1 Chap
Page 15 and 16: xvi 8, along with some of the most
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 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 76 and 77:
2.12. EXERCISES 59 Denoting t1 = to
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 85 and 86:
Page 87 and 88:
Page 89 and 90:
72 CHAPTER 3. LYAPUNOV STABILITY I:
Page 91 and 92:
Page 93 and 94:
Page 95 and 96:
Page 97 and 98:
Page 99 and 100:
Page 101 and 102:
Page 103 and 104:
Page 105 and 106:
Page 107 and 108:
Page 109 and 110:
Page 111 and 112:
Page 113 and 114:
Page 115 and 116:
Page 117 and 118:
100 CHAPTER 3. LYAPUNOV STABILITY I
Page 119 and 120:
Page 121 and 122:
Page 123 and 124:
Page 125 and 126:
108 CHAPTER 4. LYAPUNOV STABILITY H
Page 127 and 128:
Page 129 and 130:
Page 131 and 132:
Page 133 and 134:
Page 135 and 136:
Page 137 and 138:
Page 139 and 140:
Page 141 and 142:
Page 143 and 144:
126 CHAPTER 4. LYAPUNOV STABILITY H
Page 145 and 146:
Page 147 and 148:
Page 149 and 150:
Page 151 and 152:
Page 154 and 155:
Chapter 5 Feedback Systems So far,
Page 156 and 157:
5.1. BASIC FEEDBACK STABILIZATION 1
Page 158 and 159:
5.2. INTEGRATOR BACKSTEPPING 141 5.
Page 160 and 161:
5.2. INTEGRATOR BACKSTEPPING 143 De
Page 162 and 163:
5.3. BACKSTEPPING: MORE GENERAL CAS
Page 164 and 165:
Page 166 and 167:
Page 168 and 169:
5.4. EXAMPLE 151 5.4 Example 8 Figu
Page 170 and 171:
5.5. EXERCISES 153 [S - 0(x)] = x1
Page 172 and 173:
Chapter 6 Input-Output Stability So
Page 174 and 175:
6.1. FUNCTION SPACES 157 Both L2 an
Page 176 and 177:
6.2. INPUT-OUTPUT STABILITY 159 Thu
Page 178 and 179:
6.2. INPUT-OUTPUT STABILITY 161 (a)
Page 180 and 181:
6.2. INPUT-OUTPUT STABILITY 163 1.2
Page 182 and 183:
6.3. LINEAR TIME-INVARIANT SYSTEMS
Page 184 and 185:
6.4. L GAINS FOR LTI SYSTEMS where
Page 186 and 187:
6.5. CLOSED-LOOP INPUT-OUTPUT STABI
Page 188 and 189:
6.6. THE SMALL GAIN THEOREM 171 In
Page 190 and 191:
6.6. THE SMALL GAIN THEOREM 173 N(x
Page 192 and 193:
6.7. LOOP TRANSFORMATIONS 175 Figur
Page 194 and 195:
6.7. LOOP TRANSFORMATIONS 177 U l M
Page 196 and 197:
6.8. THE CIRCLE CRITERION 179 (i) g
Page 198 and 199:
6.9. EXERCISES 181 (6.3) Prove the
Page 200 and 201:
Chapter 7 Input-to-State Stability
Page 202 and 203:
7.2. DEFINITIONS 185 Setting u = 0,
Page 204 and 205:
7.3. INPUT-TO-STATE STABILITY (ISS)
Page 206 and 207:
7.3. INPUT-TO-STATE STABILITY (ISS)
Page 208 and 209:
7.4. INPUT-TO-STATE STABILITY REVIS
Page 210 and 211:
7.4. INPUT-TO-STATE STABILITY REVIS
Page 212 and 213:
7.5. CASCADE-CONNECTED SYSTEMS U E2
Page 214 and 215:
7.5. CASCADE-CONNECTED SYSTEMS 197
Page 216:
7.6. EXERCISES 199 (i) Is it locall
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
Page 246 and 247:
9.4. EXAMPLES 229 5- Very strictly-
Page 248 and 249:
9.5. AVAILABLE STORAGE 9.4.2 Mass-S
Page 250 and 251:
9.6. ALGEBRAIC CONDITION FOR DISSIP
Page 252 and 253:
9.6. ALGEBRAIC CONDITION FOR DISSIP
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 260 and 261:
9.9. NONLINEAR G2 GAIN 9.9 Nonlinea
Page 262 and 263:
9.9. NONLINEAR L2 GAIN 245 (iii) IH
Page 264 and 265:
9.10. SOME REMARKS ABOUT CONTROL DE
Page 266 and 267:
9.10. SOME REMARKS ABOUT CONTROL DE
Page 268 and 269:
9.11. NONLINEAR L2-GAIN CONTROL 251
Page 270 and 271:
9.12. EXERCISES 253 9.12 Exercises
Page 272 and 273:
Chapter 10 Feedback Linearization I
Page 274 and 275:
10.1. MATHEMATICAL TOOLS 257 Lfh(x)
Page 276 and 277:
10.1. MATHEMATICAL TOOLS 259 10.1.3
Page 278 and 279:
10.1. MATHEMATICAL TOOLS 261 and su
Page 280 and 281:
10.1. MATHEMATICAL TOOLS 263 Defini
Page 282 and 283:
10.2. INPUT-STATE LINEARIZATION 265
Page 284 and 285:
10.2. INPUT-STATE LINEARIZATION 267
Page 286 and 287:
10.2. INPUT-STATE LINEARIZATION and
Page 288 and 289:
10.3. EXAMPLES 271 provided that wh
Page 290 and 291:
10.4. CONDITIONS FOR INPUT-STATE LI
Page 292 and 293:
10.5. INPUT-OUTPUT LINEARIZATION 27
Page 294 and 295:
Page 296 and 297:
Page 298 and 299:
10.6. THE ZERO DYNAMICS 281 i=Ax+Bu
Page 300 and 301:
10.6. THE ZERO DYNAMICS 283 that th
Page 302 and 303:
10.6. THE ZERO DYNAMICS 285 Definit
Page 304 and 305:
10.7. CONDITIONS FOR INPUT-OUTPUT L
Page 306:
10.8. EXERCISES 289 (10.8) Consider
Page 309 and 310:
292 CHAPTER 11. NONLINEAR OBSERVERS
Page 311 and 312:
Page 313 and 314:
Page 315 and 316:
Page 317 and 318:
Page 319 and 320:
Page 321 and 322:
Page 323 and 324:
Page 325 and 326:
308 APPENDIX A. PROOFS (ii) It is c
Page 327 and 328:
310 APPENDIX A. PROOFS For the conv
Page 329 and 330:
312 APPENDIX A. PROOFS 9V ax-1 xz 9
Page 331 and 332:
314 APPENDIX A. PROOFS Since the eq
Page 333 and 334:
316 APPENDIX A. PROOFS Proof: The n
Page 335 and 336:
318 APPENDIX A. PROOFS We have x Fi
Page 337 and 338:
320 APPENDIX A. PROOFS (b) 0 = a ,3
Page 339 and 340:
322 APPENDIX A. PROOFS To this end
Page 341 and 342:
324 A.5 Chapter 8 APPENDIX A. PROOF
Page 343 and 344:
326 APPENDIX A. PROOFS and substitu
Page 345 and 346:
328 APPENDIX A. PROOFS It follows t
Page 347 and 348:
330 APPENDIX A. PROOFS A.7 Chapter
Page 349 and 350:
332 APPENDIX A. PROOFS Assume now t
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 366 and 367:
Index Absolute stability, 179 Asymp
Page 368 and 369:
INDEX discrete-time, 132 nonautonom
show all

Nonlinear Control Sy.. - Free

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?