Set-Theoretic Methods in Control

by Franco Blanchini, Stefano Miani

Overview

This self-contained monograph describes basic set-theoretic methods for control and provides a discussion of their links to fundamental problems in Lyapunov stability analysis and stabilization, optimal control, control under constraints, persistent disturbance rejection, and uncertain systems analysis and synthesis. New computer technology has catalyzed a resurgence of research in this area, particularly in the development of set-theoretic techniques, many of which are computationally demanding.

The work presents several established and potentially new applications, along with numerical examples and case studies. A key theme of the presentation is the trade-off between exact (but computationally intensive) and approximate (but conservative) solutions to problems. Mathematical language is kept to the minimum necessary for the adequate formulation and statement of main concepts. Numerical algorithms for the solution of the proposed problems are described in detail.

Set-Theoretic Methods in Control is accessible to readers familiar with the basics of systems and control theory; prerequisites such as convexity theory are included. The text provides a solid foundation of mathematical techniques and applications and also features avenues for further theoretical study. Aimed primarily at graduate students and researchers in applied mathematics and engineering, the book will also appeal to practitioners since it contains extensive references to the literature and supplies many recipes for solving significant control problems.

Product Details

ISBN-13:	9783319179322
Publisher:	Springer International Publishing
Publication date:	07/04/2015
Series:	Systems & Control: Foundations & Applications
Edition description:	2nd ed. 2015
Pages:	630
Product dimensions:	6.10(w) x 9.25(h) x 0.05(d)

About the Author

Francio Blanchini is a Professor in the Department of Mathematics and Computer Science at the University of Udine, Italy.

Stefano Miani is an Associate Professor in the Department of Electrical, Management, and Mechanical Engineering at the University of Udine, Italy.

Table of Contents

Preface     vii
Introduction     1
Notation     1
Acronyms     3
Basic ideas and motivations     3
The spirit of the book     3
Solving a problem     5
Conservative or intractable?     6
How to avoid reading this book     8
How to benefit from reading this book     9
Past work referencing     9
Outline of the book     9
The link with Lyapunov's theory     10
Uncertain systems     13
Constrained control     18
Required background     25
Related topics and reading     26
Lyapunov and Lyapunov-like functions     27
State space models     27
Differential inclusions     29
Model absorbing     30
The pitfall of equilibrium drift     32
Lyapunov derivative     34
Solution of a system of differential equations     34
The upper-right Dini derivative     35
Derivative along the solution of a differential equation     36
Special cases of directional derivatives     37
Lyapunov functions and stability     39
Global stability     40
Local stability and ultimate boundedness     43
Control Lyapunov Functions     45
Associating a control law with a Control Lyapunov Function: state feedback     46
Associating a control law with a Control Lyapunov Function: output feedback     53
Finding a Control Lyapunov Function     54
Polytopic systems     54
The convexity issue     57
Fake Control Lyapunov Functions     57
Lyapunov-like functions     60
Discrete-time systems     62
Converse Lyapunov theorems     68
Literature review     69
Exercises     70
Convex sets and their representation     73
Convex functions and sets     73
Operations between sets     76
Minkowski function     79
The normal and the tangent cones     81
Ellipsoidal sets     83
Polyhedral sets     86
Other families of convex sets     94
Exercises     96
Invariant sets     99
Basic definitions     99
Nagumo's theorem     101
Proof of Nagumo's theorem for practical sets and regular f     104
Generalizations of Nagumo's theorem     106
An example of application of Nagumo's theorem     108
Discrete-time systems     110
Positive invariance and fixed point theorem     112
Convex invariant sets and linear systems     114
Ellipsoidal invariant sets     120
Ellipsoidal invariant sets for continuous-time systems     120
Ellipsoidal invariant sets for discrete-time systems     124
Polyhedral invariant sets     125
Contractive polyhedral sets for continuous-time systems     126
Contractive sets for discrete-time systems     135
Associating a control with a polyhedral control Lyapunov function and smoothing     138
Existence of positively invariant polyhedral C-sets     142
The positive description     143
Other classes of invariant sets and historical notes     144
Exercises     146
Dynamic programming     149
Infinite-time reachability set     149
Linear systems with linear constraints     156
State in a tube: time-varying and periodic case     164
Historical notes and comments     167
Backward computation of Lyapunov functions     168
The largest controlled invariant set     171
The uncontrolled case: the largest invariant set     179
Comments on the results     184
Exercises     188
Set-theoretic analysis of dynamic systems     191
Set propagation     191
Reachable and controllable sets     191
Computation of set propagation under polytopic uncertainty     194
Propagation of uncertainties via ellipsoids     197
0-Reachable sets with bounded inputs     198
Reachable sets with pointwise-bounded noise     198
Infinite-time reachability and l[subscript 1] norm     207
Reachable sets with energy-bounded noise     209
Historical notes and comments     212
Stability and convergence analysis of polytopic systems     212
Quadratic stability     213
Joint spectral radius     213
Polyhedral stability     215
The robust stability radius     217
Best transient estimate     218
Performance analysis of dynamical systems     220
Peak-to-peak norm evaluation     221
Step response evaluation     226
Impulse and frequency response evaluation     228
Norm evaluation via LMIs     229
Periodic system analysis      231
Exercises     233
Control of parameter-varying systems     235
Robust and gain-scheduling control     237
Stabilization of LPV systems via quadratic Lyapunov functions     241
Quadratic stability     242
Quadratic stabilizability     242
Quadratic Lyapunov functions: the discrete-time case     244
Quadratic stability and H[infinity] norm     245
Limits of quadratic functions and linear controllers     246
Notes about quadratic stabilizability     251
Polyhedral Lyapunov functions     251
Polyhedral stabilizability     251
Universality of polyhedral Lyapunov functions (and their drawbacks)     256
Smoothed Lyapunov functions     261
Gain-scheduling linear controllers and duality     263
Duality in a quadratic framework     267
Exercises     268
Control with time-domain constraints     271
Input constraints     274
Construction of a constrained control law and its associated domain of attraction     278
The stable-unstable decomposition     283
Systems with one or two unstable eigenvalues     284
Region with bounded complexity for constrained input control      291
Domain of attraction for input-saturated systems     295
State constraints     299
A case study     301
Assigning an invariant (and admissible) set     306
Control with rate constraints     312
The rate-bounding operator     314
Output feedback with constraints     315
The tracking problem     317
Reference management device     319
The tracking domain of attraction     324
Examples of tracking problems     330
Exercises     333
(Sub-)Optimal control     337
Minimum-time control     337
Worst-case controllability     337
Time optimal controllers for linear discrete-time systems     341
Time optimal controllers for uncertain systems     342
Optimal peak-to-peak disturbance rejection     347
Constrained receding-horizon control     352
Receding-horizon: the main idea     352
Recursive feasibility and stability     355
Receding horizon control in the presence of disturbances     360
Relatively optimal control     365
The linear dynamic solution     369
The nonlinear static solution     377
Exercises      386
Set-theoretic estimation     389
Worst-case estimation     390
Set membership estimation for linear systems with linear constraints     396
Approximate solutions     403
Bounding ellipsoids     408
Energy-bounded disturbances     408
Including observer errors in the control design     410
Literature review     412
Exercises     412
Related topics     415
Adaptive control     415
A surge control problem     420
The domain of attraction     425
Systems with constraints     426
Hybrid and switching systems     430
Switching and switched systems     432
Switching among controllers     436
Relay systems     441
Planar systems     447
Exercises     449
Appendix     451
Remarkable properties of the Euler auxiliary system     451
MAXIS-G: a software for the computation of invariant sets for constrained LPV systems     456
Software availability     458
Web addresses     458
References     459
Index     477

Editorial Reviews

From the reviews:

"The book is an outstanding monograph of a recent research trend in control. It reflects the vast experience of the authors as well as their noticeable contributions to the development of this field...At the conceptual level, the book was meant as a self contained volume for readers with the usual background in systems and control. The text is very accessible (the style is attractive; the theoretical concepts and results are accompanied by relevant examples so as to facilitate the deep understanding of the discussed topics; both analytical and numerical approaches are presented; the final section of each chapter includes exercises). The book is highly recommended to PhD students and researchers working in control engineering or applied mathematics. The material can also be used for graduate courses in these areas." [Octavian Pastravanu, Zentralblatt MATH]

"This is an excellent book, full of new ideas and collecting a lot of diverse material related to set-theoretic methods. It can be recommended to a wide control community audience." (B. T. Polyak, Mathematical Reviews, Issue 2009 f)

From the Publisher