Nonlinear Control Sy.. - Free

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4.4. PROOF OF THE STABILITY THEOREMS 117 is contained in D. By the assumptions of the theorem there exist class IC functions al, a2, and a3 satisfying a1(IIxII) < W(x,t) < a2(IIXII) Vt,Vx E BR (4.19) a3(114) < -W(x,t) Vt,Vx E BR. (4.20) Given that the at's are strictly increasing and satisfy a,(0) = 0, i = 1, 2, 3, given e > 0, we can find 5i, b2 > 0 such that x2(81) < al(R) a2(52) < min[a1(c),a2(b1)] (4.21) (4.22) where we notice that b1 and b2 are functions of e but are independent of to. Notice also that inequality (4.22) implies that 61 > 62. Now define Conjecture: We claim that T = C" (R) a3(a2) 11x011 < 61 = lx(t')II < b2 for some t = t` in the interval to < t' < to + T. (4.23) To see this, we reason by contradiction and assume that 1xo11 < 61 but llx(t`)f1 > b2 for all t in the interval to < t' < to + T. We have that 0 < a,(52) al is class IC al(b2) b2. Also, since W is a decreasing function of t this implies that But 0< al(a2) < W(x(to +T), to +T) to+T < W(x(to), to) + J W(x(t), t) dt to < W(x(to),to) -Ta3(d2) from (4.20) and (4.23). W(x(to),to) < a2(11x(to)11) a2(5i), since 11xoll < 61, by assumption. by (4.19)
118 CHAPTER 4. LYAPUNOV STABILITY IL NONAUTONOMOUS SYSTEMS Thus, we have 0 < W(x(to),to) - Ta3(82) 0 < a2(81) - al(R) by (4.23) which contradicts (4.21). It then follows that our conjecture is indeed correct. Now assume that t > t'. We have ai(IIx(t)II) < W(x(t),t) oo as IIxII -4 oo. Thus for any a > 0 there exists b > 0 such that If IIxoII < a, then al(b) > a2(a). ai(b) > a2(a) ? W(xo,to) >_ W(x(t),t) >- al(IIxII) Thus, IIxII < a = IIxII < b, and we have that all motions are uniformly bounded. Consider El, 8, and T as follows: a2 (b) < al (c) T = al (b) a3 (a) and using an argument similar to the one used in the proof of Theorem 4.2, we can show that for t > to + T 11X(011 < -E provided that IIxoII < a. This proves that all motions converge uniformly to the origin. Thus, the origin is asymptotically stable in the large.
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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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Page 117 and 118: 100 CHAPTER 3. LYAPUNOV STABILITY I
Page 125 and 126: 108 CHAPTER 4. LYAPUNOV STABILITY H
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Page 143 and 144: 126 CHAPTER 4. LYAPUNOV STABILITY H
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 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
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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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