Nonlinear Control Sy.. - Free

More documents

Recommendations

Info

A.5. CHAPTER 8 327 H1 :.......................................................... : ........ - .............................................. H2 14 :.......................................................... : Figure A.4: The feedback system S. Hi = H1(I - Yi H2 = H2+e eHl)-1
328 APPENDIX A. PROOFS It follows that HZ is strictly passive for any e > 0. We now argue that Hi is SPR for sufficiently small E. To see this, notice that, denoting 7(H1), then by the small gain theorem, Hi is stable for any e < '. Also, since H1 is SPR, there exist positive definite matrices P and L, a real matrix Q and µ sufficiently small such that (8.29)-(8.30) are satisfied. But then it follows that, given P, L, Q, µ, P(A + eBC) + (A + eBC)T P = _QQT - µL + 2CCT C and provided that 0 < e < e' = p)tmjn[L]/(2A,,,ny[CTC]), the matrixµL-2eCTC is positive definite. We conclude that for all e E (0, min(c*, -y- 1)), Hi is strong SPR, hence passive, and HZ is strictly passive. Thus, the result follows from Theorem 8.4. A.6 Chapter 9 Theorem 9.2: The nonlinear system Eli given by (9.13) is QSR-dissipative [i.e., dissipative with supply rate given by (9.6)] if there exists a differentiable function 0 : ]Rn -, R and function L : Rn -> Rq and W : 1Rn Rgxm satisfying gT fi(x) > 0, 0(0) = 0 (A.58) 81- 31-f (x) = ax (O)T hT (x)Qh(x) - LT(x)L(x) ST h(x) - WT L(x) (A.59) (A.60) R = WTW (A.61) Proof: Sufficiency was proved in Chapter 9. To prove necessity we show that the available storage defined in Definition 9.5 is a solution of equations (A.61)(A.61) for appropriate functions L and W. Notice first that for any state x0 there exist a control input u E U that takes the state from any initial state x(t_1) at time t = t-1 to xo at t = 0. Since the system is dissipative, we have that In particular / t' t T w(t) dt = o / , fo T t, t , w(t) dt = O(x(tl)) - 4(x(to)) > 0. w(t) dt + w(t) dt = cb(x(T)) - O(x(t_1)) > 0
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 8 and 9:
CONTENTS ix 3.8 The Invariance Prin
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 30 and 31:
Page 32 and 33:
Page 34 and 35:
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: 10.5. INPUT-OUTPUT LINEARIZATION 27
Page 296 and 297: 10.5. INPUT-OUTPUT LINEARIZATION 27
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 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: 326 APPENDIX A. PROOFS and substitu
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

Create successful ePaper yourself

Delete template?

Save as template?