Nonlinear and Adaptive Control of Complex Systems Mathematics and ItsApplications ManagingEditor: M. HAZEWINKEL CentreJorMathematicsand ComputerScience, Amsterdam,TheNetherlands Volume 491 Nonlinear and Adaptive Control of Complex Systems by Alexander L. Fradkov Institutefor Problems ofMechanical Engineering of theRussianAcademyofSeiences. St Petersburg,Russia Iliya V.Miroshnik St Petersburg,State InstituteofFineMechanicsandOptics, St Petersburg,Russia and Vladimir O.Nikiforov St Petersburg,StateInstituteof Fine MechanicsandOptics, St Petersburg,Russia Springer-Science+Business Media, B.V. AC.I.P.Cataloguerecord for this book isavailablefrom the Library ofCongress. Printed onacid-free paper All Rights Reserved ISBN978-90-481-5294-0 ISBN978-94-015-9261-1(eBook) DOI10.1007/978-94-015-9261-1 ©1999SpringerScience+BusinessMediaDordrecht OriginallypublishedbyKluwerAcademicPublishersin1999. 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Table of Contents Preface xi Notations and Definitions xv 1 FACES OF COMPLEXITY 1 1.1 Complexity and Dccomposition 1 1.2 Multivariable Cont.rol and Geometry 5 1.2.1 Coordinating control . . .. 6 1.2.2 Oscillation svnchronization 8 1.2.3 Spatial motion control 10 1.2.4 Terminal control . . . 15 1.2.5 State tracking systems 16 1.3 Uncertaintv and Adaptation 17 2 NONLINEAR SYSTEMS: ANALYSIS AND DESIGN TOOLS 25 2.1 Stabilitv 01' Nonlinear Systems . . . . . . . 25 2.1.1 Corupleteness and stability 25 2.1.2 Lyapunov Iunctions and their applications 32 2.1.3 Partial sta.bility . . . . . . . . . . . . 37 2.2 Equivalent Models and Coordinate Changes . . 44 2.2.1 Autonornous svsterns . . . . . . . . . . . 45 2.2.2 Single-input svsterns and controllability 47 2.2.:1 Canonical Iorms 49 2.3 Basic Canonical Form and Linearization Techniques 52 2.3.1 Basic canonical form . 52 2.3.2 Exact linearization . . . . . . 56 2.3.:3 Linear approxirnation 57 2.4 Equivalcnco 01' Multi-Input Systems 60 2.5 Input-Output Ca.nouical Forms and Stahilization with Re- spect to Output . . . . . . . . . . . . . . . . . . . . . 63 2.5.1 Main transformatiou and linearized dynamics 64 2.5.2 System zero dvnamics. . . . . . . . . . . . . . 67 v VI 2.5.3 Normal form and local stabilization 68 2.6 Control of Triangular Systems . 71 2.7 Passivity and Passification . . . . 76 2.7.1 Passivity and stability . . 76 2.7.2 Passivity arid dissipativity 78 2.7.3 Passivity and Kalman-Yakubovich lemma 80 2.7.4 Passificatiou and Feedback Kalman-Yakubovich lemma 84 3 SPEED-GRADIENT METHOD AND PARTIAL STABILIZATION 91 3.1 Goal-Oriented Contral Problem Statement. . . . . . . . . . 91 3.2 Design of Speed-Gradient Algorithms . . . . . . . . . . . . . 94 3.2.1 Speed-Gradient algorithms for local objective func- tionals . . . . . . . . . . . . . . . . . . . . . . . . .. 94 3.2.2 Speed-Gradient algorithms for integralobjectivefunc- tionals . . . . . . . . . . . . . . . . . . . . 95 3.2.3 Speed-Gradient algorithms in finite form. 96 3.2.4 Combined algorithms 97 3.3 Convergence of the Speed-Gradient Algorithms 97 3.3.1 Regulation and tracking . . . . . . . . . 97 3.3.2 Partial stabilization 105 3.4 Identifying Properties of Speed-Gradient Algorithms 111 3.5 Robustness of Speed-Gradient Algorithms with Respect to Disturbances 113 3.6 Speed-Gradient Cont.rol of Harniltonian Systems 118 3.6.1 Control of energy . . . . 118 3.6.2 Control of Iirst Integrals . . . . . . . . . . 122 4 NONLINEAR CONTROL OF MULTIVARIABLE SYSTEMS 127 4.1 Multivariable Control and Geometrie approach 127 4.2 Equilibrium Stabilization 129 4.2.1 State regulation problerns . . . . . . . 129 4.2.2 Stabilization via exact linearization . . 131 4.2.3 Stabilization via linear approximation 133 4.3 Attracting Sets . . . . . . . . . . . 135 4.3.1 Attractivity and invariance . . . . . . 135 4.3.2~eighborhood propertics , . . . . . . . 140 4.3.:3 Equivalent dynamics 01" autonomaus systems 146 4.3.4 Couditious 01" at.tractivity . . . . . . . . . . 148 4.4 Set Stabilization 1.53 4.4.1 Coutrol problerus and invariance conditions 153 VII 4.4.2 Problem decornposition and invariant contral 157 4.4.3 Equivalent dvnarnics 163 4.4.4 System st.abilization 167 4.5 State Tracking Control .. 173 4.5.1 Tracking problern . . 173 4.5.2 Invariant. cont.rol .. 175 4.5.3 System st.ahilization 177 5 NONLINEAR CONTROL OF MIMO SYSTEMS 183 5.1 Problems 01' Out.put. Control and Coordination of MIMO Systems . . . . . . . . . . . 183 5.2 Output Rpgulatioll . . . . . . 185 5.2.1 Regulat.ion problern 185 5.2.2 :vIain transforurations 186 5.2.3 Systems without zera dyuamics 191 5.2.4 Zero dvnamics submanifold . . 193 5.2.5 System equivalent dynamics . . 194 5.2.6 Isstres 01' equivalence and stability 199 5.3 Output Coordination . . . . . . . . . . . 212 5.3.1 Coordinatiou rondit.ions . . . . . 212 5.3.2 Problems 01" coordinating contral 216 5.3.3 Transformalion to normal form . 219 5.3.4 Transformalion to task-oriented form. 223 5.4 Coordinating Cont.rol . . . . . . . . . . . . . . 230 5.4.1 Cont.rol prohlerns in state space . . . . 230 5.4.2 Basic cont.rol law and partial decoupling 231 5.4.3 Coni.rol bv using irnplicit models 232 5.4.4 Coutrol by using referonce model 237 5.5 Spatial Motion Coutrol . . . . . . . . . . 242 5.5.1 Spl stabilization in output space 243 5.5.2 Cauonical rr-prcsentat.ions and differential forms . 246 5.5.:3 System cquivalont dynarnics and contral design 257 6 ADAPTIVE AND ROBUST CONTROL DESIGN 265 6.1 State-Fecdback Coutrol 266 6.1.1 Cert.ainty cquivalence design 266 6.1.2 Recu rsive <lPsign procedures . 283 6.2 Output-Feedback C'ont.rol . . . . . . 297 6.2.1 Coni.rolofstrict.ly passive Clnd strictly minimum phase systern-, . . . . . . . . . . . . . 298 6.2.2 AuglTlellted crror based design 303 6.2.3 High-order t.unor based design . 309 viii 6.3 Output-Feedback Adaptive Systems with Implicit Reference Model . . . . . . . . . . . . . . . . . 320 6.3.1 Design 01' adaptive controller . . . . 320 6.3.2 Shunting mct.hod . . . . . . . . . . . . . . . . . . . . 323 6.4 Output-Feedback Coutrol 01' Uncertain Linear Plants under Ideal Conditions 328 6.4.1 Problem statement . . . . . . 329 6.4.2 Plant model parametrizations 329 6.4.3 Cert.ainty equivalence design 337 6.4.4 Dynamic certainty equivalence design 342 6.4.5 Nonlinear adaptive design 1'01' linear plants . 349 6.4.6 Discussion ... ... ...... ....... 369 6.5 Output-feedback Control of Uncertain Linear Plants in the Presence 01' External Disturbances . . . . . . . . 373 6.5.1 Plallt model parametrizations . . . . . . . . . . . 373 6.5.2 Robust cont.rollers with high-erder tuners . . . . 375 6.5.3 Robust controller with nonlinear damping terms 381 7 DECOMPOSITION OF ADAPTIVE SYSTEMS 391 7.1 Separation of Metions in Adaptive Systems . . . . . . . . . 392 7.1.1 TIH' first sehetue of motion separation for continuous time systerns . . . . . . . . . . . . . . . . . . . . . . 392 7.1.2 Tho first sehetue of rnotion separation for discrete- time systcms . . . . . . . . . . . . . . . . . . . . . . 399 7.1.3 Tlte secend scheme of motion separation . . . . . . . 401 7.2 Conditions of Applicability and Estinration of Accuracy of the Motion Separation Schemes in Adaptive Systems . . . . 403 7.2.1 Applicabilitv ofthe Speed-Gradient algorithms 1.0sin gularly peri.urbed systems . . . . . . . . . . . . . . . 405 7.2.2 Discretization of the Speed-Gradient algorithms. . . 417 7.3 Adaptive Deccntralizcd Control of Interconnected Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . 419 7.3.1 Problem statement and control algorithm 419 7.3.2 Properfies of tbe control systern .. . 421 8 CONTROL OF MECHANICAL SYSTEMS 429 8.1 Spatial Motion Control of Rigid Body . . . . 429 8.1.1 Dynamics and kinematical properties . 429 8.1.2 Mass-point control 433 8.1.3 Control of rotation. . . . . . . . . . . 436 8.2 Robot Motion Control . . . . . . . . . . . . . 438 8.2.1 Robot modcl and the problern statement. 439 ix 8.2.2 Differential relations and the main approach . 442 8.2.3 Trajectory cont.rol strategv 446 8.3 Mobile Robot Cont.rol . . . . . . . 449 8.3.1 Analysis 01' robot dynamics 449 8.3.2 Cont.rol problcm statcment 453 8.3.3 Control law design . . . . . 455 8.4 Contral 01' Oscillatory Systems .. 459 8.4.1 Energy cont1'01 of a pendulum with contralIed pivot 459 8.4.2 \Vheeling a cal'out of a ditch . . . . . . . . . . . . . 464 9 PHYSICS AND CONTROL 469 9.1 Feedback Hesonanc<' in a Nonlinear Oscillator 469 9.2 "Superopt.imal" Escape frorn Potential Wells 473 9.3 Stabilization 01' Uust.able Modcs . 475 9.4 Feedback Spcrtroscopv . 476 9.5 Excitability Index . 477 9.6 Speed-Gradient Laws of Dvnamics 479 9.7 Onzagger Equatious 482 9.8 Discussion . 483 A APPENDIX 485 References 493 Index 507