Dissipative Systems Analysis and Control (eBook)

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 Compiègne. 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 Chargé 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 €2 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.

Preface 6
Contents 7
Notation 13
1 Introduction 15
1.1 Example 1: System with Mass Spring and Damper 16
1.2 Example 2: RLC Circuit 17
1.3 Example 3: A Mass with a PD Controller 19
1.4 Example 4: Adaptive Control 20
2 Positive Real Systems 23
2.1 Dynamical System State-space Representation 24
2.2 Definitions 25
2.3 Interconnections of Passive Systems 28
2.4 Linear Systems 29
2.5 Passivity of the PID Controllers 38
2.6 Stability of a Passive Feedback Interconnection 38
2.7 Mechanical Analogs for PD Controllers 39
2.8 Multivariable Linear Systems 41
2.9 The Scattering Formulation 42
2.10 Impedance Matching 45
2.11 Feedback Loop 48
2.12 Bounded Real and Positive Real Transfer Functions 50
2.13 Examples 61
2.14 Strictly Positive Real (SPR) Systems 67
2.15 Applications 76
3 Kalman-Yakubovich-Popov Lemma 83
3.1 The Positive Real Lemma 84
3.2 Weakly SPR Systems and the KYP Lemma 109
3.3 KYP Lemma for Non-minimal Systems 114
3.4 SPR Problem with Observers 127
3.5 The Feedback KYP Lemma 127
3.6 Time-varying Systems 129
3.7 Interconnection of PR Systems 130
3.8 Positive Realness and Optimal Control 133
3.9 The Lur’e Problem (Absolute Stability) 149
3.10 The Circle Criterion 174
3.11 The Popov Criterion 180
3.12 Discrete-time Systems 184
4 Dissipative Systems 191
4.1 Normed Spaces 192
4.2 Lp Norms 192
4.3 Review of Some Properties of Lp Signals 194
4.4 Dissipative Systems 207
4.5 Nonlinear KYP Lemma 236
4.6 Dissipative Systems and Partial Differential Inequalities 245
4.7 Nonlinear Discrete-time Systems 261
4.8 PR tangent system and dissipativity 263
4.9 Infinite-dimensional Systems 266
4.10 Further Results 269
5 Stability of Dissipative Systems 271
5.1 Passivity Theorems 271
5.2 Positive Deffniteness of Storage Functions 280
5.3 WSPR Does not Imply OSP 284
5.4 Stabilization by Output Feedback 286
5.5 Equivalence to a Passive System 290
5.6 Cascaded Systems 295
5.7 Input-to-State Stability (ISS) and Dissipativity 296
5.8 Passivity of Linear Delay Systems 302
5.9 Nonlinear H Control 309
5.10 Popov’s Hyperstability 324
6 Dissipative Physical Systems 329
6.1 Lagrangian Control Systems 329
6.2 Hamiltonian Control Systems 340
6.3 Rigid Joint–Rigid Link Manipulators 354
6.4 Flexible Joint–Rigid Link Manipulators 357
6.5 A Bouncing System 361
6.6 Including Actuator Dynamics 363
6.7 Passive Environment 372
6.8 Nonsmooth Lagrangian Systems 377
7 Passivity-based Control 387
7.1 Brief Historical Survey 387
7.2 The Lagrange-Dirichlet Theorem 389
7.3 Rigid Joint–Rigid Link Systems: State Feedback 400
7.4 Rigid Joint–Rigid Link: Position Feedback 422
7.5 Flexible Joint–Rigid Link: State Feedback 428
7.6 Flexible Joint–Rigid Link: Output Feedback 436
7.7 Including Actuator Dynamics 440
7.8 Constrained Mechanical Systems 442
7.9 Controlled Lagrangians 446
8 Adaptive Control 449
8.1 Lagrangian Systems 450
8.2 Linear Invariant Systems 470
9 Experimental Results 481
9.1 Flexible Joint Manipulators 481
9.2 Stabilization of the Inverted Pendulum 510
9.3 Conclusions 518
A Background Material 521
A.1 Lyapunov Stability 521
A.2 Differential Geometry Theory 529
A.3 Viscosity Solutions 534
A.4 Algebraic Riccati Equations 537
A.5 Some Useful Matrix Algebra 545
A.6 Well-posedness Results for State Delay Systems 551
References 553
Index 583

1 Introduction (P. 1)

Dissipativity theory gives a framework for the design and analysis of control systems using an input-output description based on energy-related considerations. Dissipativity is a notion which can be used in many areas of science, and it allows the control engineer to relate a set of efficient mathematical tools to well known physical phenomena. The insight gained in this way is very useful for a wide range of control problems.

In particular the input-output description allows for a modular approach to control systems design and analysis. The main idea behind this is that many important physical systems have certain input-output properties related to the conservation, dissipation and transport of energy.

Before introducing precise mathematical de.nitions we will somewhat loosely refer to such input-output properties as dissipative properties, and systems with dissipative properties will be termed dissipative systems.

When modeling dissipative systems it may be useful to develop the state-space or input-output models so that they reffect the dissipativity of the system, and thereby ensure that the dissipativity of the model is invariant with respect to model parameters, and to the mathematical representation used in the model. The aim of this book is to give a comprehensive presentation of how the energy-based notion of dissipativity can be used to establish the input-output properties of models for dissipative systems.

Also it will be shown how these results can be used in controller design. Moreover, it will appear clearly how these results can be generalized to a dissipativity theory where conservation of other physical properties, and even abstract quantities, can be handled. Models for use in controller design and analysis are usually derived from the basic laws of physics (electrical systems, dynamics, thermodynamics).

Then a controller can be designed based on this model. An important problem in controller design is the issue of robustness which relates to how the closed loop system will perform when the physical system di.ers either in structure or in parameters from the design model. For a system where the basic laws of physics imply dissipative properties, it may make sense to define the model so that it possesses the same dissipative properties regardless of the numerical values of the physical parameters.

Then if a controller is designed so that stability relies on the dissipative properties only, the closed-loop system will be stable whatever the values of the physical parameters. Even a change of the system order will be tolerated provided it does not destroy the dissipativity. Parallel interconnections and feedback interconnections of dissipative systems inherit the dissipative properties of the connected subsystems, and this simplifies analysis by allowing for manipulation of block diagrams, and provides guidelines on how to design control systems.

A further indication of the usefulness of dissipativity theory is the fact that the PID controller is a dissipative system, and a fundamental result that will be presented is the fact that the stability of a dissipative system with a PID controller can be established using dissipativity arguments. Note that such arguments rely on the structural properties of the physical system, and are not sensitive to the numerical values used in the design model.