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8.5. PASSIVITY OF LINEAR TIME-INVARIANT SYSTEMS (x, Hx) (x,h*x) J 27r f 1 te[H(7w)] lX (7w)12 dw + 27r x(t)[h(t) * x(t)] dt 00 X (7w)[H(?w)X (7w)] du) 00 1 IX(.7w)I2 H(7w)' dw 27r f00 00 f 00 3 2rrm[H(7w)] 00 and sincem[H(yw)] is an odd function of w, the second integral is zero. It follows that and noticing that we have that (x, Hx) _ 1 27r 00 J Re[H(34 dw (x, x) I I -`f (7w) I2 dw 27r (x,Hx) > inf te[H(yw)] W from where the sufficiency of conditions (i) and (ii) follows immediately. To prove necessity, assume that 2e[H(3w)] < 0 at some frequency w = w'. By the continuity of the Fourier transform as a function of w, it must be true that te[H(tw)] < 0 Vu ;E 1w - w' I 0. We can now construct 9(y,) as follows: X(pw) > M OW) < m VwE 1w-w*I
216 CHAPTER 8. PASSIVITY (i) H is passive if and only if [H(,w)] + H(3w)*] > 0 Vw E R. (ii) H is strictly passive if and only if 3b > 0 such that Amin[1 (3w)] + H(3w)'] > b Vw E ]R It is important to notice that Theorem 8.7 was proved for systems whose impulse response is in the algebra A. In particular, for finite-dimensional systems, this algebra consists of systems with all of their poles in the open left half of the complex plane. Thus, Theorem 8.7 says nothing about whether a system with transfer function as H(s) = w2 a> 0, w o > 0 (8.24) s2 0 is passive. This transfer function, in turn, is the building block of a very important class of system to be described later. Our next theorem shows that the class of systems with a transfer function of the form (8.24) is indeed passive, regardless of the particular value of a and w. Theorem 8.9 Consider the system H : Gee -> G2e defined by its transfer function H(s) = s2 Under these conditions, H is passive. as w20 Proof: By Theorem 8.3, H is passive if and only if but for the given H(s) S(s) _ SOW) _ 1-H a > 0, w0 > 0. 1+H 00
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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
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)
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Page 208 and 209: 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 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
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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 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
Page 280 and 281: 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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