Nonlinear Control Sy.. - Free

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11.4. LIPSCHITZ SYSTEMS 301 11.4 Lipschitz <strong>Sy</strong>stems The nonlinear observer discussed in the previous section is inspired by the work on feedback linearization, and belongs to the category of what can be called the differential geometric approach. In this section we show that observer design can also be studied using a Lyapunov approach. For simplicity, we restrict attention to the case of Lipschitz systems, defined below. Consider a system of the form: J i = Ax + f (x, u) y=Cx (11.23) where A E lRnxn C E Rlxn and f : Rn x R -a Rn is Lipschitz in x on an open set D C R', i.e., f satisfies the following condition: If(x1,u*) - f(x2,u )lI < 7IIx1 -x211 Vx E D. (11.24) Now consider the following observer structure x=Ax+f(x,u)+L(y-Cx) (11.25) where L E Rnxl. The following theorem shows that, under these assumptions, the estimation error converges to zero as t -+ oo. Theorem 11.3 Given the system (11.23) and the corresponding observer (11.25), if the Lyapunov equation P(A - LC) + (A - LC)T P = -Q (11.26) where p = PT > 0, and Q = QT > 0, is satisfied with y< .min (Q) 2A, (P) then the observer error 2 = x - x is asymptotically stable. Proof: = [Ax + f (x, u)] - [Ax + f (x, u) + L(y - C2)] = (A - LC)2 + f (x, u) - f (x, u). (11.27)
302 CHAPTER 11. NONLINEAR OBSERVERS To see that I has an asymptotically stable equilibrium point at the origin, consider the Lyapunov function candidate: but and V iPi + iPx _ -2TQ2+22TP[f(i+2,u) - f(2,u)] pTQiII Amin(Q)II.t1I2 II2xT P[f [(.i+ x, u) - f(, u)jjj J12±TPII II f p+ x, u) - f p, u) jj < 2y\,, Therefore, V is negative definite, provided that or, equivalently Amin(Q)II:i1I2 > 2yAmax(P)jjill2 Example 11.4 Consider the following system: Setting we have that Solving the Lyapunov equation y )'min (Q) 2A""' (P) LX2J=[12J[x2J+[ y=[10][ 2 A-LC= L=[° [ 0 1 1 2 0 2 I P(A - LC) + (A - LC)T P = -Q 0
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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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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 363 and 364: 346 LIST OF FIGURES 3.1 Stable equi
Page 366 and 367: Index Absolute stability, 179 Asymp
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INDEX discrete-time, 132 nonautonom
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