Nonlinear Control Sy.. - Free

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itrJ A.3. CHAPTER 6 V 315 Thus, f If P = (DT(T,t)Q(T)a(D(T,t) dr+J 4T(T,t)Q(T)41(T,t) dT - Q(t) J 4pT (T, t)Q(T)4D (T, t) dTl - Q(t) -l A(t)-AT(t) 00 which implies that A.3 Chapter 6 P = -P(t)A(t) - AT(t)p(t) - Q(t). Theorem 6.1 Consider a function P(s) E R(s). Then F(s) E A if and only if (i) P(s) is proper, and (ii) all poles of F(s) lie in the left half of the complex plane. Proof: Assume first that P(s) satisfies (i) and (ii), and let n(s)andd(s) E R[s] respectively, be the numerator and denominator polynomials of F(s). Dividing n(s) by d(s), we can express F(s) as: F(s) = k + d s) = k + G(s) (A.9) where k E 1R and d(s) is strictly proper. Expanding G(s) in partial fractions and antitransforming (A.9), we have n f (t) = G-1{F(s)} = kb(t) + E g2(t) where n is the degree of d(s) and each g2 has one of the following forms [u(t) represents the unit step function : (i) ktii-1e)tu(t), if A < 0 is a real pole of multiplicity m; and (ii) ke(°+.3.+)tu(t), a < 0, if A = (v +,yw) is a complex pole. It follows that f (t) E A, and so F(s) E A. To prove the converse, notice that if (i) does not hold, then f (-) contains derivatives of impulses and so does not belong to A._Also, if (ii) does not hold, then, clearly, for some i, g2 ¢ G1 and then f (t) V A and F(s) il A, which completes the proof. Theorem 6.2 Consider a linear time-invariant system H, and let represent its impulse response. Then H is L stable if and only if hoo(t) + ha,(t) E A and moreover, if H is Gp stable, then JjHxjjr < jh11A11x11p. 2=1 J
316 APPENDIX A. PROOFS Proof: The necessity is obvious. To prove sufficiency, assume that E A and consider the output of H to an input u E Cp . We have We analyze both terms separately. Clearly For the second term we have that h(t) * u(t) = hou(t) + J halt t - T)u(T) dT = 91+92 0 IIg1TIIc, = Ilhou(t)IIc, = IhoIIu(t)IIc,. 192(t)I=I I tha(t-T)u(T)dTI < ftlha(t-T)I Iu(T)I dr 0 and choosing p and q E 1R+ such that 1/p + 1/q = 1, it follows that (A.10) t t I92(t)I
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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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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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