Nonlinear Control Sy.. - Free

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List of Figures 1.1 mass-spring system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 The system i = cos x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 The system ± = r + x2, r < 0 .. . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Vector field diagram form the system of Example 1.6 . . . . . . . . . . . . . 10 tem ......................................... 12 1.5 System trajectories of Example 1.7: (a) original system, (b) uncoupled sys- 1.6 System trajectories of Example 1.8 (a) uncoupled system, (b) original system 13 1.7 System trajectories of Example 1.9 (a) uncoupled system, (b) original system 14 1.8 System trajectories for the system of Example 1.10 . . . . . . . . . . . . . . 15 1.9 Trajectories for the system of Example 1.11 . . . . . . . . . . . . . . . . . . 16 1.10 Trajectories for the system of Example 1.12 . . .......... ...... 17 1.11 Stable limit cycle: (a) vector field diagram; (b) the closed orbit.. . . . . . . 19 1.12 Nonlinear RLC-circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.13 Unstable limit cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.14 (a) Three-dimensional view of the trajectories of Lorenz' chaotic system; (b) two-dimensional projection of the trajectory of Lorenz' system. . . . . . . . 22 1.15 Magnetic suspension system . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.16 Pendulum-on-a-cart experiment . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.17 Free-body diagrams of the pendulum-on-a-cart system .. . . . . . . . . . . . 26 1.18 Ball-and-beam experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.19 Double-pendulum of Exercise (1.5) .. . . . . . . ... . . . .......... 29 345
346 LIST OF FIGURES 3.1 Stable equilibrium point .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.2 Asymptotically stable equilibrium point . . . . . . . . . . . . . . . . . . . . . 67 3.3 Mass-spring system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.4 Pendulum without friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.5 The curves V(x) =,3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.6 System trajectories in Example 3.21 . . . . . . . . . . . . . . . . . . . . . . . 95 4.1 (a) Continuous-time system E; (b) discrete-time system Ed. . . . . . . . . . 126 4.2 Discrete-time system Ed .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.3 Action of the hold device H . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 5.1 (a) The system (5.7)-(5.8); (b) modified system after introducing -O(x); (c) "backstepping" of -O(x); (d) the final system after the change of variables. 142 5.2 6.1 Current-driven magnetic suspension system .. . . . . . . . . . . . . . . . . . 151 The system H . .. . . . . .. ..... . .. . ... . .. ........... 155 6.2 Experiment 1: input u(t) applied to system H. Experiment 2: input u(t) = uT(t) applied to system H . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 6.3 Causal systems: (a) input u(t); (b) the response y(t) = Hu(t); (c) truncation of the response y(t). Notice that this figure corresponds to the left-hand side of equation (6.8); (d) truncation of the function u(t); (e) response of the system when the input is the truncated input uT(t); (f) truncation of the system response in part (e). Notice that this figure corresponds to the righthand side of equation (6.8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 6.4 Static nonlinearity N(.) . . ..... .......... ............. 163 6.5 The systems Hlu = u2, and H2u = e1" 1 . . . . . . . . . . . . . . . . . . . . . 163 6.6 Bode plot of JH(yw)j, indicating the IIHIIoo norm of H...... ...... . 169 6.7 The Feedback System S .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 6.8 The nonlinearity N(.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 6.9 The Feedback System S .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 6.10 The Feedback System SK ......................... .... 176 6.11 The Feedback System 5M .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
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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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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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Page 325 and 326: 308 APPENDIX A. PROOFS (ii) It is c
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Page 331 and 332: 314 APPENDIX A. PROOFS Since the eq
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Page 335 and 336: 318 APPENDIX A. PROOFS We have x Fi
Page 337 and 338: 320 APPENDIX A. PROOFS (b) 0 = a ,3
Page 339 and 340: 322 APPENDIX A. PROOFS To this end
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Page 343 and 344: 326 APPENDIX A. PROOFS and substitu
Page 345 and 346: 328 APPENDIX A. PROOFS It follows t
Page 347 and 348: 330 APPENDIX A. PROOFS A.7 Chapter
Page 349 and 350: 332 APPENDIX A. PROOFS Assume now t
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 366 and 367: Index Absolute stability, 179 Asymp
Page 368 and 369: INDEX discrete-time, 132 nonautonom
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