Nonlinear Control Sy.. - Free

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A.2. CHAPTER 4 313 Lemma 3.5: The positive limit set N of a solution x(t, xo, to) of the autonomous system is invariant with respect to (A.6). x = f(x) (A.6) Proof: We need to show that if xo E N, then x(t, xo, to) E N for all t > to. Let p c N. Then there exists a sequence {tn} such that to -+ oc and x(tn, xo, to) -> p as n -> oo. Since the solution of (A.6) is continuous with respect to initial conditions, we have Also, lim x[t, x(tn, xo, to), to] = x(t,p, to). n-+oo x[t, x(tn, x0, to), to] = x[t + tn, xo, to). It follows that lim x[t + tn, xo, to] = x(t, P, to) (A.7) n ,oo and since x(t, xo, to) approaches N as t - oo, (A.7) shows that x(t, p, to) is in NVt. A.2 Chapter 4 Theorem 4.6: Consider the system i = A(t)x. The equilibrium state x = 0 is exponentially stable if and only if for any given symmetric, positive definite continuous and bounded matrix Q(t), there exist a symmetric, positive definite continuously differentiable and bounded matrix P(t) such that -Q(t) = P(t)A(t) + AT(t)p(t) + P(t) (A.8) Proof: Sufficiency is straightforward. To prove necessity, let Thus, P(t) = j D'(T, t)Q(T),D(T, t) dr. t 00 xT 4T (T, t)Q(T)4)(T, t)x dT oo r4)T(T, t)x(t)]T`w (T) [4(T, t)x(t)]dr it TOO J x(T)T Q(T)x(T)dT.
314 APPENDIX A. PROOFS Since the equilibrium is globally asymptotically stable, there exist k1, k2 such that (see, for example, Chen [15] pp. 404) kle-k2(t-to) Ix(t)II < Thus xTPx = f t ft 00 1 kle-k2(T-t) Q(T)kle-k2(T-t) dT k2Q(T)e-2k2(T-t) dr. The boundedness of Q(t) implies that there exist M: 11Q(t)II < M. Thus t "0 or 2 xTPx < kM 2 Also, this implies that xTPx < A2IIx1I2 On the other hand, since Q is positive definite, 3a > 0 such that xT Qx > axT x. Also, A bounded implies that I I A(t) I I < N, Vt E R. Thus Thus, or xTPx = J t 1 k2Me-2k2(T-t) dr. 00 xT4T (T, t)Q(T)d)(T, t)x dT x(T)TQ(T)x(T) d7 f' aIIAN)II IIx(T)II2 dT f ' N II x(T)T A(T)x(T) II dT > II f Nx(T)T drx(T) dTII NxTx. xTPx > a NxTx AlIIxII2
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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
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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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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
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