Dissipative Systems Analysis And Control

by Brogliato, Bernard; Lozano, Rogelio; Maschke, Bernhard; Egeland, Olav

• ISBN13: 9781846285165
• ISBN10: 184628516X
• Edition: 2nd
• Format: Hardcover
• Copyright: 2007-01-30
• Publisher: Springer Verlag

Summary

Many important physical systems have input-output properties related to the conservation, dissipation and transport of energy. The theory surrounding such "dissipative properties" may be used as a framework for the design and analysis of control systems. The consideration of dissipativity is useful ''‚¬" and may be indispensable ''‚¬" for control applications like robotics, active vibration damping and circuit theory and for some control techniques themselves: adaptive, nonlinear-H-infinity , and inverse-optimal control among them.Dissipative Systems Analysis and Control (second edition) presents a fully revised and expanded treatment of dissipative systems theory, constituting a self-contained, advanced introduction for graduate students, researchers and practising engineers. It examines linear and nonlinear systems with examples of both in each chapter; some infinite-dimensional and nonsmooth examples are also included. Throughout, emphasis is placed on the use of the dissipative properties of a system for the design of stable feedback control laws. The theory is consistently substantiated by experimental results and by reference to its application in illustrative physical cases (Lagrangian and Hamiltonian systems and passivity-based and adaptive controllers are covered thoroughly).The second edition is substantially reorganized both to accommodate new material and to enhance its pedagogical properties. Some of the changes introduced are:''‚¬Â¢ Complete proofs of the main theorems and lemmas.''‚¬Â¢ The Kalman''‚¬"Yakubovich''‚¬"Popov Lemma for non-minimal realizations, singular systems, and discrete-time systems (linear and nonlinear).''‚¬Â¢ Passivity of nonsmooth systems (differential inclusions, variational inequalities, Lagrangian systems with complementarity conditions).''‚¬Â¢ Sections on optimal control and H-infinity theory.''‚¬Â¢ An enlarged bibliography with more than 550 references, and an augmented index with more than 500 entries.''‚¬Â¢ An improved appendix with introductions to viscosity solutions, Riccati equations and some useful matrix algebra.

Author Biography

Rogelio Lozano has worked in a number of institutions with a high reputation for control engineering - the University of Newcastle in Australia, NASA's Langley Research Center and now as CNRS Research Director at the University of Compi+¿gne. He is a very experienced author in the control field having been an associate editor of Automatica and now of International Journal of Adaptive Control and Signal Processing. He has published 26 refereed journal articles in the last five years and he is the co-author of 3 previous titles for Springer (not including the first edition of the present title) in the Communications and Control Engineering and Advances in Industrial Control series:Landau, I.D., Lozano, R. and M'Saad, M. Adaptive Control (3-540-76187-X, 1997)Fantoni, I. and Lozano, R. Non-linear Control for Underactuated Mechanical Systems (1-85233-423-1, 2001)Castillo, P., Lozano, R. and Dzul, A., Modelling and Control of Mini-Flying Machines (1-85233-957-8, 2005)In addition to having served (1991 - 2001) as Charg+¬ de Recherche at CNRS, and as, now, Directeur de Recherche at INRIA, Bernard Brogliato is an Associate Editor for Automatica (since 2001) a reviewer for Mathematical Reviews and writes book reviews for ASME Applied Mechanics Reviews. He has served on the organising and other committees of various European and international conferences sponsored by an assortment of organizations, most prominently, the IEEE. He has been responsible for examining the PhD and Habilitation theses of 16 students and takes an active part in lecturing at summer schools in several European countries. Doctor Brogliato is the director of SICONOS (a European project concerned with Modelling, Simulation and Control of Nonsmooth Dynamical Systems) which carries funding of Gé¼2 million.Olav Egeland is Professor at the Norwegian University of Science and Technology (NTNU). He graduated as siv.ing. (1984) and dr.ing. (1987) from the Department of Engineering Cybernetics, NTNU, and has been a professor at the department since 1989. In the academic year 88/89 he was at the German Aerospace Center in Oberpfaffenhofen outside of Munich. In the period 1996-1998 he was Head of Department of Engineering Cybernetics, Vice Dean of Faculty of Electrical Engineering and Telecommunications, and member of the Research Committee for Science and Technology at NTNU. He was Associate Editor of the IEEE Transactions on Automatic Control 1996-1999 and of the European Journal of Control 1998-2000. He received the Automatica Prize Paper Award in 1996, and the 2000 IEEE Transactions on Control Systems Technology Outstanding Paper Award. He has supervised the graduation of 75 siv.ing. and 19 dr.ing., and was Program Manager of the Strategic University Program in Marine Cybernetics at NTNU. Currently he is coordinator of the control activity of the Centre of Ships and Ocean Structures. He has wide experience as a consultant for industry, and is co-founder of Marine Cybernetics, which is a company at the NTNU incubator. His research interests are within modeling, simulation and control of mechanical systems with applications to robotics and marine systems.

Table of Contents

Introduction	p. 1
Example 1: System with Mass Spring and Damper	p. 2
Example 2: RLC Circuit	p. 3
Example 3: A Mass with a PD Controller	p. 5
Example 4: Adaptive Control	p. 6
Positive Real Systems	p. 9
Dynamical System State-space Representation	p. 10
Definitions	p. 11
Interconnections of Passive Systems	p. 14
Linear Systems	p. 15
Passivity of the PID Controllers	p. 24
Stability of a Passive Feedback Interconnection	p. 24
Mechanical Analogs for PD Controllers	p. 25
Multivariable Linear Systems	p. 27
The Scattering Formulation	p. 28
Impedance Matching	p. 31
Feedback Loop	p. 34
Bounded Real and Positive Real Transfer Functions	p. 36
Examples	p. 47
Mechanical Resonances	p. 47
Systems with Several Resonances	p. 50
Two Motors Driving an Elastic Load	p. 51
Strictly Positive Real (SPR) Systems	p. 53
Frequency Domain Conditions for a Transfer Function to be SPR	p. 54
Necessary Conditions for H(s) to be PR (SPR)	p. 56
Tests for SPRness	p. 57
Interconnection of Positive Real Systems	p. 57
Special Cases of Positive Real Systems	p. 58
Applications	p. 62
SPR and Adaptive Control	p. 62
Adaptive Output Feedback	p. 64
Design of SPR Systems	p. 65
Kalman-Yakubovich-Popov Lemma	p. 69
The Positive Real Lemma	p. 70
PR Transfer Functions	p. 70
A Digression to Optimal Control	p. 76
Duality	p. 78
Positive Real Lemma for SPR Systems	p. 79
Descriptor Variable Systems	p. 91
Weakly SPR Systems and the KYP Lemma	p. 95
KYP Lemma for Non-minimal Systems	p. 100
Spectral Factors	p. 102
Sign-controllability	p. 104
State Space Decomposition	p. 106
A Relaxed KYP Lemma for SPR Functions with Stabilizable Realization	p. 107
SPR Problem with Observers	p. 113
The Feedback KYP Lemma	p. 113
Time-varying Systems	p. 115
Interconnection of PR Systems	p. 116
Positive Realness and Optimal Control	p. 119
General Considerations	p. 119
Least Squares Optimal Control	p. 120
The Popov Function and the KYP Lemma LMI	p. 125
A Recapitulating Theorem	p. 129
On the Design of Passive LQG Controllers	p. 130
Summary	p. 133
A Digression on Semidefinite Programming Problems	p. 134
The Lur'e Problem (Absolute Stability)	p. 135
Introduction	p. 135
Well-posedness of ODEs	p. 137
Aizerman's and Kalman's Conjectures	p. 140
Multivalued Nonlinearities	p. 142
Dissipative Evolution Variational Inequalities	p. 152
The Circle Criterion	p. 160
Loop Transformations	p. 162
The Popov Criterion	p. 166
Discrete-time Systems	p. 170
The KYP Lemma	p. 170
The Tsypkin Criterion	p. 173
Discretization of PR Systems	p. 175
Dissipative Systems	p. 177
Normed Spaces	p. 178
[Characters not Reproducible]p Norms	p. 178
Relationships Between [Characters not Reproducible]1, [Characters not Reproducible]2 and [Characters not Reproducible][Infin] Spaces	p. 180
Review of Some Properties of [Characters not Reproducible]p Signals	p. 180
Example of Applications of the Properties of [Characters not Reproducible]p Functions in Adaptive Control	p. 186
Linear Maps	p. 188
Induced Norms	p. 188
Properties of Induced Norms	p. 188
Extended Spaces	p. 190
Gain of an Operator	p. 190
Small Gain Theorem	p. 191
Dissipative Systems	p. 193
Definitions	p. 193
The Signification of [Beta]	p. 197
Storage Functions (Available, Required Supply)	p. 201
Examples	p. 211
Regularity of the Storage Functions	p. 217
Nonlinear KYP Lemma	p. 222
A Particular Case	p. 222
Nonlinear KYP Lemma in the General Case	p. 223
Time-varying Systems	p. 229
Nonlinear-in-the-input Systems	p. 230
Dissipative Systems and Partial Differential Inequalities	p. 231
The linear invariant case	p. 231
The Nonli ear Case y = h(x)	p. 235
The Nonlinear Case y = h(x) + j(x)u	p. 238
Recapitulation	p. 243
Inverse Optimal Control	p. 243
Nonlinear Discrete-time Systems	p. 247
PR tangent system and dissipativity	p. 249
Infinite-dimensional Systems	p. 252
An Extension of the KYP Lemma	p. 252
The Wave Equation	p. 253
The Heat Equation	p. 255
Further Results	p. 255
Stability of Dissipative Systems	p. 257
Passivity Theorems	p. 257
One-channel Results	p. 257
Two-channel Results	p. 259
Lossless and WSPR Blocks Interconnection	p. 263
Large-scale Systems	p. 264
Positive Definiteness of Storage Functions	p. 266
WSPR Does not Imply OSP	p. 270
Stabilization by Output Feedback	p. 272
Autonomous Systems	p. 272
Time-varying Nonlinear Systems	p. 273
Evolution Variational Inequalities	p. 274
Equivalence to a Passive System	p. 276
Cascaded Systems	p. 281
Input-to-State Stability (ISS) and Dissipativity	p. 282
Passivity of Linear Delay Systems	p. 288
Systems with State Delay	p. 288
Interconnection of Passive Systems	p. 290
Extension to a System with Distributed State Delay	p. 291
Absolute Stability	p. 294
Nonlinear H[Infin] Control	p. 295
Introduction	p. 295
Closed-loop Synthesis: Static State Feedback	p. 300
Closed-loop Synthesis: PR Dynamic Feedback	p. 302
Nonlinear H[Infin]	p. 305
More on Finite-power-gain Systems	p. 307
Popov's Hyperstability	p. 310
Dissipative Physical Systems	p. 315
Lagrangian Control Systems	p. 315
Definition and Properties	p. 316
Simple Mechanical Systems	p. 324
Hamiltonian Control Systems	p. 326
Input-output Hamiltonian Systems	p. 326
Port Controlled Hamiltonian Systems	p. 331
Rigid Joint-Rigid Link Manipulators	p. 340
The Available Storage	p. 341
The Required Supply	p. 342
Flexible Joint-Rigid Link Manipulators	p. 343
The Available Storage	p. 346
The Required Supply	p. 346
A Bouncing System	p. 347
Including Actuator Dynamics	p. 349
Armature-controlled DC Motors	p. 349
Field-controlled DC Motors	p. 354
Passive Environment	p. 358
Systems with Holonomic Constraints	p. 358
Compliant Environment	p. 361
Nonsmooth Lagrangian Systems	p. 363
Systems with C[superscript 0] Solutions	p. 363
Systems with BV Solutions	p. 365
Passivity-based Control	p. 373
Brief Historical Survey	p. 373
The Lagrange-Dirichlet Theorem	p. 375
Lyapunov Stability	p. 375
Asymptotic Lyapunov Stability	p. 376
Invertibility of the Lagrange-Dirichlet Theorem	p. 378
The Lagrange-Dirichlet Theorem for Nonsmooth Lagrangian Systems (BV Solutions)	p. 379
The Lagrange-Dirichlet Theorem for Nonsmooth Lagrangian Systems (C[superscript 0] Solutions)	p. 384
Conclusion	p. 385
Rigid Joint-Rigid Link Systems: State Feedback	p. 386
PD Control	p. 386
PID Control	p. 391
More about Lyapunov Functions and the Passivity Theorem	p. 393
Extensions of the PD Controller for the Tracking Case	p. 398
Other Types of State Feedback Controllers	p. 405
Rigid Joint-Rigid Link: Position Feedback	p. 408
P + Observer Control	p. 408
The Paden and Panja + Observer Controller	p. 410
The Slotine and Li + Observer Controller	p. 412
Flexible Joint-Rigid Link: State Feedback	p. 414
Passivity-based Controller: The Lozano and Brogliato Scheme	p. 414
Other Globally Tracking Feedback Controllers	p. 418
Flexible Joint-Rigid Link: Output Feedback	p. 422
PD Control	p. 422
Motor Position Feedback	p. 424
Including Actuator Dynamics	p. 426
Armature-controlled DC Motors	p. 426
Field-controlled DC Motors	p. 428
Constrained Mechanical Systems	p. 428
Regulation with a Position PD Controller	p. 429
Holonomic Constraints	p. 430
Nonsmooth Lagrangian Systems	p. 431
Controlled Lagrangians	p. 432
Adaptive Control	p. 435
Lagrangian Systems	p. 436
Rigid Joint-Rigid Link Manipulators	p. 436
Flexible Joint-Rigid Link Manipulators: The Adaptive Lozano and Brogliato Algorithm	p. 442
Flexible Joint-Rigid Link Manipulators: The Backstepping Algorithm	p. 452
Linear Invariant Systems	p. 456
A Scalar Example	p. 456
Systems with Relative Degree r = 1	p. 457
Systems with Relative Degree r = 2	p. 460
Systems with Relative Degree r [GreaterEqual] 3	p. 461
Experimental Results	p. 467
Flexible Joint Manipulators	p. 467
Introduction	p. 467
Controller Design	p. 468
The Experimental Devices	p. 469
Experimental Results	p. 473
Conclusions	p. 483
Stabilization of the Inverted Pendulum	p. 496
Introduction	p. 496
System's Dynamics	p. 497
Stabilizing Control Law	p. 500
Simulation Results	p. 503
Experimental Results	p. 503
Conclusions	p. 504
Background Material	p. 507
Lyapunov Stability	p. 507
Autonomous systems	p. 507
Non-autonomous Systems	p. 511
Differential Geometry Theory	p. 515
Normal Form	p. 417
Feedback Linearization	p. 518
Stabilization of Feedback Linearizable Systems	p. 519
Further Reading	p. 520
Viscosity Solutions	p. 520
Algebraic Riccati Equations	p. 523
Reduced Riccati Equation for WSPR Systems	p. 525
Some Useful Matrix Algebra	p. 531
Results Useful for the KYP Lemma LMI	p. 531
Inverse of Matrices	p. 533
Jordan Chain	p. 534
Auxiliary Lemmas for the KYP Lemma Proof	p. 534
Well-posedness Results for State Delay Systems	p. 537
References	p. 539
Index	p. 571
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