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6.6. THE SMALL GAIN THEOREM 171 In other words, a feedback system is input-output-stable if whenever the inputs u1 and u2 are in the parent space X, the outputs y1 and y2 and errors e1 and e2 are also in X. In the sequel, input-output--stable systems will be referred to simply as "stable" systems. Remarks: Notice that, as with open-loop systems, the definition of boundedness depends strongly on the selection of the space X. To emphasize this dependence, we will sometimes denote X-stable to a system that is bounded in the space X. 6.6 The Small Gain Theorem In this section we study the so-called small gain theorem, one of the most important results in the theory of input-output systems. The main goal of the theorem is to provide open-loop conditions for closed-loop stability. Theorem 6.3 given next, is the most popular and, in some sense, the most important version of the small gain theorem. It says that if the product of the gains of two systems, H1 and H2, is less than 1, then its feedback interconnection is stable, in a sense to be made precise below. See also remark (b) following Theorem 6.3. Theorem 6.3 Consider the feedback interconnection of the systems H1 and H2 : Xe -4 Xe . Then, if y(Hj)7(H2) < 1, the feedback system is input-output-stable. Proof: To simplify our proof, we assume that the bias term 3 in Definition 6.7 is identically zero (see Exercise 6.5). According to Definition 6.14 we must show that u1i u2 E X imply that e1 i e2, y1 and Y2 are also in X. Consider a pair of elements (u1, el), (u2i e2) that belong to the relation E. Then, u1, u2, e1, e2 must satisfy equations (6.23) and (6.24) and, after truncating these equations, we have Thus, e1T = u1T - (H2e2)T (6.28) e2T = u2T + (Hlel)T. IIelTII < IIu1TII + II(H2e2)TII < IIu1TII + 7(H2)IIe2TII Ie2TII Iu2TII + II(Hlel)TII < IIu2TII + 'Y(H1)II e1TIl. Substituting (6.31) in (6.30) we obtain IIe1TII
172 CHAPTER 6. INPUT-OUTPUT STABILITY and since, by assumption, y(Hl)y(H2) < 1, we have that 1 - y(H1)y(H2) # 0 and then Similarly IIe1TII
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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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Page 137 and 138: 120 CHAPTER 4. LYAPUNOV STABILITY I
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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
Page 184 and 185: 6.4. L GAINS FOR LTI SYSTEMS where
Page 186 and 187: 6.5. CLOSED-LOOP INPUT-OUTPUT STABI
Page 190 and 191: 6.6. THE SMALL GAIN THEOREM 173 N(x
Page 192 and 193: 6.7. LOOP TRANSFORMATIONS 175 Figur
Page 194 and 195: 6.7. LOOP TRANSFORMATIONS 177 U l M
Page 196 and 197: 6.8. THE CIRCLE CRITERION 179 (i) g
Page 198 and 199: 6.9. EXERCISES 181 (6.3) Prove the
Page 200 and 201: Chapter 7 Input-to-State Stability
Page 202 and 203: 7.2. DEFINITIONS 185 Setting u = 0,
Page 204 and 205: 7.3. INPUT-TO-STATE STABILITY (ISS)
Page 206 and 207: 7.3. INPUT-TO-STATE STABILITY (ISS)
Page 208 and 209: 7.4. INPUT-TO-STATE STABILITY REVIS
Page 210 and 211: 7.4. INPUT-TO-STATE STABILITY REVIS
Page 212 and 213: 7.5. CASCADE-CONNECTED SYSTEMS U E2
Page 214 and 215: 7.5. CASCADE-CONNECTED SYSTEMS 197
Page 216: 7.6. EXERCISES 199 (i) Is it locall
Page 219 and 220: 202 CHAPTER 8. PASSIVITY v i Figure
Page 221 and 222: 204 CHAPTER 8. PASSIVITY Moreover,
Page 223 and 224: 206 CHAPTER 8. PASSIVITY Definition
Page 225 and 226: 208 CHAPTER 8. PASSIVITY Thus (UT,
Page 227 and 228: 210 CHAPTER 8. PASSIVITY Thus, H f
Page 229 and 230: 212 CHAPTER 8. PASSIVITY Under thes
Page 231 and 232: 214 CHAPTER 8. PASSIVITY Under thes
Page 233 and 234: 216 CHAPTER 8. PASSIVITY (i) H is p
Page 235 and 236: 218 CHAPTER 8. PASSIVITY Definition
Page 237 and 238: 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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