Nonlinear Control Sy.. - Free

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3.3. STABILITY THEOREMS 73 In other words, the theorem says that asymptotic stability is achieved if the conditions of Theorem 3.1 are strengthened by requiring V (x) to be negative definite, rather than semi definite. The discussion above is important since it elaborates on the ideas and motivation behind all the Lyapunov stability theorems. We now provide a proof of Theorems 3.1 and 3.2. These proofs will clarify certain technicalities used later on to distinguish between local and global stabilities, and also in the discussion of region of attraction. Proof of theorem 3.1: Choose r > 0 such that the closed ball Br={xE III": IxMI 0, by the fact that V (x) > 0 E D). IxJJ=r Now choose Q E (0, a) and denote 0,3={xEBr:V(x) 0 1x(O)11 < b => Ix(t)II < T < f Vt > 0 which means that the equilibrium x = 0 is stable. ED Proof of theorem 3.2: Under the assumptions of the theorem, V (x) actually decreases along the trajectories of f (x). Using the same argument used in the proof of Theorem 3.1, for every real number a > 0 we can find b > 0 such that Ib C Ba, and from Theorem 3.1 whenever the initial condition is inside f2b, the solution will remain inside 52b. Therefore, to prove asymptotic stability, all we need to show is that 1b reduces to 0 in the limit. In
74 CHAPTER 3. LYAPUNOV STABILITY I. AUTONOMOUS SYSTEMS Figure 3.4: Pendulum without friction. other words, 52b shrinks to a single point as t -> oo. However, this is straightforward, since by assumption f7(x) < 0 in D. Thus, V (x) tends steadily to zero along the solutions of f (x). This completes the proof. Remarks: The first step when studying the stability properties of an equilibrium point consists of choosing a positive definite function Finding a positive definite function is fairly easy; what is rather tricky is to select a whose derivative along the trajectories near the equilibrium point is either negative definite, or semi definite. The reason, of course, is that is independent of the dynamics of the differential equation under study, while V depends on this dynamics in an essential manner. For this reason, when a function is proposed as possible candidate to prove any form of stability, such a is said to be a Lyapunov function candidate. If in addition happens to be negative definite, then V is said to be a Lyapunov function for that particular equilibrium point. 3.4 Examples Example 3.4 (Pendulum Without Friction) Using Newton's second law of motion we have, ma = -mg sin 9 a = la = l9
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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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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
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Page 56 and 57: 2.4. MATRICES 39 2.4 Matrices We as
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Page 62 and 63: 2.6. SEQUENCES 45 Compact set: A se
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Page 76 and 77: 2.12. EXERCISES 59 Denoting t1 = to
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Page 80: 2.12. EXERCISES 63 Notes and Refere
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Page 117 and 118: 100 CHAPTER 3. LYAPUNOV STABILITY I
Page 125 and 126: 108 CHAPTER 4. LYAPUNOV STABILITY H
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124 CHAPTER 4. LYAPUNOV STABILITY I
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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
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6.3. LINEAR TIME-INVARIANT SYSTEMS
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6.4. L GAINS FOR LTI SYSTEMS where
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6.5. CLOSED-LOOP INPUT-OUTPUT STABI
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6.6. THE SMALL GAIN THEOREM 171 In
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6.6. THE SMALL GAIN THEOREM 173 N(x
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6.7. LOOP TRANSFORMATIONS 175 Figur
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6.7. LOOP TRANSFORMATIONS 177 U l M
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6.8. THE CIRCLE CRITERION 179 (i) g
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6.9. EXERCISES 181 (6.3) Prove the
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Chapter 7 Input-to-State Stability
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7.2. DEFINITIONS 185 Setting u = 0,
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7.3. INPUT-TO-STATE STABILITY (ISS)
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7.3. INPUT-TO-STATE STABILITY (ISS)
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7.4. INPUT-TO-STATE STABILITY REVIS
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7.4. INPUT-TO-STATE STABILITY REVIS
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7.5. CASCADE-CONNECTED SYSTEMS U E2
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7.5. CASCADE-CONNECTED SYSTEMS 197
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7.6. EXERCISES 199 (i) Is it locall
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202 CHAPTER 8. PASSIVITY v i Figure
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204 CHAPTER 8. PASSIVITY Moreover,
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206 CHAPTER 8. PASSIVITY Definition
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208 CHAPTER 8. PASSIVITY Thus (UT,
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210 CHAPTER 8. PASSIVITY Thus, H f
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212 CHAPTER 8. PASSIVITY Under thes
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214 CHAPTER 8. PASSIVITY Under thes
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216 CHAPTER 8. PASSIVITY (i) H is p
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218 CHAPTER 8. PASSIVITY Definition
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220 CHAPTER 8. PASSIVITY Example 8.
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Chapter 9 Dissipativity In Chapter
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9.2. DIFFERENTIABLE STORAGE FUNCTIO
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9.3. QSR DISSIPATIVITY 227 Definiti
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9.4. EXAMPLES 229 5- Very strictly-
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9.5. AVAILABLE STORAGE 9.4.2 Mass-S
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9.6. ALGEBRAIC CONDITION FOR DISSIP
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9.6. ALGEBRAIC CONDITION FOR DISSIP
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9.7. STABILITY OF DISSIPATIVE SYSTE
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9.8. FEEDBACK INTERCONNECTIONS 239
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9.8. FEEDBACK INTERCONNECTIONS 241
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9.9. NONLINEAR G2 GAIN 9.9 Nonlinea
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9.9. NONLINEAR L2 GAIN 245 (iii) IH
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9.10. SOME REMARKS ABOUT CONTROL DE
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9.10. SOME REMARKS ABOUT CONTROL DE
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9.11. NONLINEAR L2-GAIN CONTROL 251
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9.12. EXERCISES 253 9.12 Exercises
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Chapter 10 Feedback Linearization I
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10.1. MATHEMATICAL TOOLS 257 Lfh(x)
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10.1. MATHEMATICAL TOOLS 259 10.1.3
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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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