Nonlinear Control Sy.. - Free

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8.4. STABILITY OF FEEDBACK INTERCONNECTIONS 213 Figure 8.6: The Feedback <strong>Sy</strong>stem S. Proof: We prove the theorem assuming that H1 is passive and H2 is strictly passive with finite gain. The opposite case (i.e., H2 is passive and H1 is strictly passive with finite gain) is entirely similar. Proceeding as in Theorem 8.4 (equation (8.20), we obtain Iy1TIIX < b-1IIU1TIIX (8.21) so that y1 E X whenever u1 E X. Also Y2 = H2y1. Then IIy2TIIX = II(H2y1)TIIx, and since H2 has finite gain, we obtain Ily2zTlix < ry(H2)IIy1TIIx
214 CHAPTER 8. PASSIVITY Under these conditions, if both systems are passive and one of them is (i) strictly passive and (ii) has finite gain, then el, e2, yl, and y2 are in X whenever xl, x2 E X. Proof: Omitted. 8.5 Passivity of Linear Time-Invariant <strong>Sy</strong>stems In this section, we examine in some detail the implications of passivity and strict passivity in the context of linear time-invariant systems. Throughout this section, we will restrict attention to the space G2. Theorem 8.7 Consider a linear time-invariant system H : C2e -> G2e defined by Hx = h* x, where h E A, x CGee. We have (i) H is passive if and only if te[H(yw)] > 0 Vw E R. (ii) H is strictly passive if and only if 36 > 0 such that Ste[H(yw)] > b Vw E R. Proof: The elements of A have a Laplace transform that is free of poles in the closed right half plane. Thus, points of the form s = ,yw belong to the region of convergence of H(s), and H(3w) is the Fourier transform of h(t). We also recall two properties of the Fourier transform: (a) If f is real-valued, then the real and imaginary parts of P(p), denoted We[F(3w)] and `Zm[F(yw)], respectively, are even and odd functions on w, respectively. In other words We[F'(7w)] = te[F(-yw)]+ ` m[F'(7w)] _ -`3m[F(-yw)] (b) (Parseval's relation) f f (t)g(t) dt - P(31)15(31))' dw 2a _C0 where G(yw)` represents the complex conjugate of G(yw). Also, by assumption, h c A, which implies that H is causal and stable. Thus, according to Theorem 8.1, H is passive (strictly passive) if and only if it is positive (strictly positive), and we can drop all truncations in the passivity definition. With this in mind, we have
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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)
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
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Page 196 and 197: 6.8. THE CIRCLE CRITERION 179 (i) g
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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)
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Page 208 and 209: 7.4. INPUT-TO-STATE STABILITY REVIS
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Page 214 and 215: 7.5. CASCADE-CONNECTED SYSTEMS 197
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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
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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
Page 250 and 251: 9.6. ALGEBRAIC CONDITION FOR DISSIP
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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 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
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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
Page 272 and 273: Chapter 10 Feedback Linearization I
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
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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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