Nonlinear Control Sy.. - Free

More documents

Recommendations

Info

9.2. DIFFERENTIABLE STORAGE FUNCTIONS 225 the storage function; O(x(t`)) represents the "energy" stored by the system V) at time t'. fto' w(t) dt: represents the energy externally supplied to the system zG during the interval [to, t1]. Thus, according to (9.2), the stored energy l4(xi) at time t1 > to is, at most, equal to the sum of the energy O(xo) initially stored at time to, plus the total energy externally supplied during the interval [to, t1]. In this way there is no internal "creation" of energy. It is important to notice that if a motion is such that it takes the system V) from a particular state to the same terminal state along a certain trajectory in the state space, then we have (since x1 = xo) 0(Xo) < 0(Xo) + i w(t) dt = J w(t)dt > 0 (9.3) where f indicates a closed trajectory with identical initial and final states. Inequality (9.3) states that in order to complete a closed trajectory, a dissipative system requires external energy. 9.2 Differentiable Storage Functions In general, the storage function 0 of a dissipative system, defined in Definition 9.1 need not be differentiable. Throughout the rest of this chapter, however, we will see that many important results can be obtained by strengthening the conditions imposed on 0. First we notice that if 0 is continuously differentiable, then dividing (9.2) by (t1 - to), and denoting O(xi) the value of O(x) when t = ti, we can write but and thus (9.4) is satisfied if and only if _(xl) - 0(Xo) < 1 tl - to tl - to w(t) dt lim O(x1) - 0(Xo) = dO(x) = 00(x) f(x, u) tl-*to t1 - to dt 8x 0a(x) f (x, u) < w(t) = w(u, y) = w(u, h(x, u)) Vx, it. (9.5)
226 CHAPTER 9. DISSIPATWITY Inequality (9.5) is called the differential dissipation inequality, and constitutes perhaps the most widely used form of the dissipation inequality. Assuming that 0 is differentiable, we can restate definition 9.1 as follows. Definition 9.2 (Dissipativity re-stated) A dynamical system Eli is said to be dissipative with respect to the supply rate w(t) = w(u, y) if there exist a continuously differentiable function 0 : X --> IR+, called the storage function, that satisfies the following properties: (i) There exist class 1C,, functions al and a2 such that .90 8xf al(IIxII)
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 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

Create successful ePaper yourself

Delete template?

Save as template?