Nonlinear Control Sy.. - Free

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9.10. SOME REMARKS ABOUT CONTROL DESIGN or, extracting the common factor hT(x)h(x) and choosing k = 1/c we conclude that -ke+212k2+2 k 2_k - 1 9.10 Some Remarks about <strong>Control</strong> Design <strong>Control</strong> systems design is a difficult subject in which designers are exposed to a multiplicity of specifications and design constraints. One of the main reasons for the use of feedback, however, is its ability to reduce the effect of undesirable exogenous signals over the system's output. Therefore, over the years a lot of research has been focused on developing design techniques that employ this principle as the main design criteria. More explicitly, control design can be viewed as solving the following problem: find a control law that 1- Stabilizes the closed-loop system, in some prescribed sense. 2- Reduces the effect of the exogenous input signals over the desired output, also in some specific sense. Many problems can be cast in this form. To discuss control design in some generality, it is customary to replace the actual plant in the feedback configuration with a more general system known as the generalized plant. The standard setup is shown in Figure 9.4, where C= controller to be designed, G= generalized plant, y= measured output, z= output to be regulated (such as tracking errors and actuator signals), u= control signal, and d= "exogenous" signals (such as disturbances and sensor noise). 247
248 CHAPTER 9. DISSIPATIVITY d G u I I y C Figure 9.4: Standard setup. To fix ideas and see that this standard setup includes the more "classical " feedback configuration, consider the following example: Example 9.2 Consider the feedback system shown in Figure 9.5. Suppose that our interest is to design a controller C to reduce the effect of the exogenous disturbance d over the signals Y2 and e2 (representing the input and output of the plant). This problem can indeed be studied using the standard setup of Figure 9.4. To see this, we must identify the inputs and outputs in the generalized plant G. In our case, the several variables in the standard setup of Figure 9.4 correspond to the following variables in Figure 9.5: y = el (the controller input) u = yl (the controller output) d: disturbance (same signal in both Figure 9.5 and the generalized setup of Figure 9.4). Z= L e2 J, i.e., z is a vector whose components are input and output of the plant. Y2 z
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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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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.
Page 240 and 241: Chapter 9 Dissipativity In Chapter
Page 242 and 243: 9.2. DIFFERENTIABLE STORAGE FUNCTIO
Page 244 and 245: 9.3. QSR DISSIPATIVITY 227 Definiti
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Page 250 and 251: 9.6. ALGEBRAIC CONDITION FOR DISSIP
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Page 254 and 255: 9.7. STABILITY OF DISSIPATIVE SYSTE
Page 256 and 257: 9.8. FEEDBACK INTERCONNECTIONS 239
Page 258 and 259: 9.8. FEEDBACK INTERCONNECTIONS 241
Page 260 and 261: 9.9. NONLINEAR G2 GAIN 9.9 Nonlinea
Page 262 and 263: 9.9. NONLINEAR L2 GAIN 245 (iii) IH
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Page 268 and 269: 9.11. NONLINEAR L2-GAIN CONTROL 251
Page 270 and 271: 9.12. EXERCISES 253 9.12 Exercises
Page 272 and 273: Chapter 10 Feedback Linearization I
Page 274 and 275: 10.1. MATHEMATICAL TOOLS 257 Lfh(x)
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Page 278 and 279: 10.1. MATHEMATICAL TOOLS 261 and su
Page 280 and 281: 10.1. MATHEMATICAL TOOLS 263 Defini
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Page 288 and 289: 10.3. EXAMPLES 271 provided that wh
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Page 292 and 293: 10.5. INPUT-OUTPUT LINEARIZATION 27
Page 298 and 299: 10.6. THE ZERO DYNAMICS 281 i=Ax+Bu
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Page 309 and 310: 292 CHAPTER 11. NONLINEAR OBSERVERS
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298 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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