Hybrid Systems with Constraints E-Book

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 386

Veröffentlichungsjahr: 2013

Beschreibung

Control theory is the main subject of this title, in particular analysis and control design for hybrid dynamic systems. The notion of hybrid systems offers a strong theoretical and unified framework to cope with the modeling, analysis and control design of systems where both continuous and discrete dynamics interact. The theory of hybrid systems has been the subject of intensive research over the last decade and a large number of diverse and challenging problems have been investigated. Nevertheless, many important mathematical problems remain open. This book is dedicated mainly to hybrid systems with constraints; taking constraints into account in a dynamic system description has always been a critical issue in control. New tools are provided here for stability analysis and control design for hybrid systems with operating constraints and performance specifications. Contents 1. Positive Systems: Discretization with Positivity and Constraints, Patrizio Colaneri, Marcello Farina, Stephen Kirkland, Riccardo Scattolini and Robert Shorten. 2. Advanced Lyapunov Functions for Lur'e Systems, Carlos A. Gonzaga, Marc Jungers and Jamal Daafouz. 3. Stability of Switched DAEs, Stephan Trenn. 4. Stabilization of Persistently Excited Linear Systems, Yacine Chitour, Guilherme Mazanti and Mario Sigalotti. 5. Hybrid Coordination of Flow Networks, Claudio De Persis, Paolo Frasca. 6. Control of Hybrid Systems: An Overview of Recent Advances, Ricardo G. Sanfelice. 7. Exponential Stability for Hybrid Systems with Saturations, Mirko Fiacchini, Sophie Tarbouriech, Christophe Prieur. 8. Reference Mirroring for Control with Impacts, Fulvio Forni, Andrew R. Teel, Luca Zaccarian. About the Authors Jamal Daafouz is an expert in the area of switched and polytopic systems and has published several major results in leading journals (IEEE TAC, Automatica, Systems and Control Letters, etc.). He serves as an Associate Editor for the key journal IEEE TAC and is a member of the Editorial Board of the IEEE CSS society. Sophie Tarbouriech is an expert in the area of nonlinear systems with constraints and has published several major results in leading journals (IEEE TAC, Automatica, Systems and Control Letters, etc.) and books. She is a member of the Editorial Board of the IEEE CSS society and has also served as an Associate Editor for the key journal IEEE TAC. Mario Sigalotti is an expert in applied mathematics and switched systems and has published several results in leading journals (IEEE TAC, Automatica, Systems and Control Letters, etc.). He heads the INRIA team GECO and is a member of the IFAC Technical Committee on Distributed Parameter Systems.

Leseprobe

Table of Contents

Preface

Chapter 1 Positive Systems: Discretization with Positivity and Constraints

1.1. Introduction and statement of the problem

1.2. Discretization of switched positive systems via Padé transformations

1.3. Discretization of positive switched systems with sparsity constraints

1.4. Conclusions

1.5. Bibliography

Chapter 2 Advanced Lyapunov Functions for Lur’e Systems

2.1. Introduction

2.2. Motivating example

2.3. A new Lyapunov Lur’e-type function for discrete-time Lur’e systems

2.4. Switched discrete-time Lur’e system with arbitrary switching law

2.5. Switched discrete-time Lur’e system controlled by the switching law

2.6. Conclusion

2.7. Bibliography

Chapter 3 Stability of Switched DAEs

3.1. Introduction

3.2. Preliminaries

3.3. Stability results

3.4. Conclusion

3.5. Acknowledgments

3.6. Bibliography

Chapter 4 Stabilization of Persistently Excited Linear Systems

4.1. Introduction

4.2. Finite-dimensional systems

4.3. Infinite-dimensional systems

4.4. Further discussion and open problems

4.5. Bibliography

Chapter 5 Hybrid Coordination of Flow Networks

5.1. Introduction

5.2. Flow network model and problem statement

5.3. Self-triggered gossiping control of flow networks

5.4. Practical load balancing

5.5. Load balancing with delayed actuation and skewed clocks

5.6. Asymptotical load balancing

5.7. Conclusions

5.8. Acknowledgments

5.9. Bibliography

Chapter 6 Control of Hybrid Systems: An Overview of Recent Advances

6.1. Introduction

6.2. Preliminaries

6.3. Stabilization of hybrid systems

6.4. Static state feedback stabilizers

6.5. Passivity-based control

6.6. Tracking control

6.7. Conclusions

6.8. Acknowledgments

6.9. Bibliography

Chapter 7 Exponential Stability for Hybrid Systems with Saturations

7.1. Introduction

7.2. Problem statement

7.3. Set theory and invariance for nonlinear systems: brief overview

7.4. Quadratic stability for saturated hybrid systems

7.5. Computational issues

7.6. Numerical examples

7.7. Conclusions

7.8. Bibliography

Chapter 8 Reference Mirroring for Control with Impacts

8.1. Introduction

8.2. Hammering a surface

8.3. Global tracking of a Newton’s cradle

8.4. Global tracking in planar triangles

8.5. Global state estimation on n-dimensional convex polyhedra

8.6. Proof of the main theorems

8.7. Conclusions

8.8. Acknowledgments

8.9. Bibliography

List of Authors

Index

First published 2013 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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© ISTE Ltd 2013

The rights of Jamal Daafouz, Sophie Tarbouriech and Mario Sigalotti to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2013934427

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN: 978-1-84821-527-6

Preface

Nonlinear control systems have undergone tremendous advances in the last two decades at the levels of theory and applications. Among these, a class of particular interest is the one resulting from the interaction of a control system with a system governed by the dynamics of a different nature. This class of systems lies in the hybrid and nonlinear control systems field. In the last decade, the study of such hybrid systems, whose behavior can be mathematically described using a mixture of logic-based switching and difference/differential linear or nonlinear equations, has attracted important research efforts. The fact that many physical systems are controlled or supervised by controllers with such mixed dynamics constitutes a great motivation for such studies. We can cite many applications (such as automotive, networked control systems, energy management and biology) in which analysis and design methods for systems evolving both continuous and discontinuous components are then needed. Furthermore, among many important problems formulated in the context of hybrid systems, switched control systems have been attracting much attention in recent years. Nevertheless, many important mathematical problems remain open. These include analysis and control of hybrid systems with a periodic behavior, control of systems with actuator constraints and hybrid control design with prescribed performance. These open problems are mainly motivated by their practical impact. Hybrid systems with periodic behavior cover an important class of embedded systems. Available approaches are mainly dedicated to specific applications of these devices and today there is a serious lack of rigorous tools for analyzing and synthesizing control algorithms for such systems. To improve their performance, the objective is to go beyond the classical simplified modeling that does not capture the hetrogeneous nature of these systems.

This book deals with control theory and, in particular, discusses the problems of analysis and control design in the context of hybrid dynamical systems. This book is mainly focused on hybrid systems with constraints. Taking into account the constraints in a dynamical system, description has always been a critical issue in control theory. The book provides new tools for stability analysis and control design for hybrid systems with operating constraints and performance specifications. Hence, it is important to underline that there is no book that focuses on constraints for the analysis and control of hybrid systems. This book proposes new approaches for open problems with practical impact. We focus on the presence of constraints in hybrid systems considered as a critical issue in control theory. This includes discontinuities arising from non-smooth impacts, saturations and nested saturations on signals, positivity and interconnection structure, algebraic equations, etc. To provide a coherent panel, the book is structured into eight chapters organized in two main parts related to the kind of systems handled: switched systems (which include Chapters 1–4) and hybrid systems (which include Chapters 5–8). Chapter 6 provides, in particular, a nice overview of recent theoretical results and challenging problems.

We think that this book constitutes an add-in overview of results and techniques with respect to the recent literature. We hope that it will be a useful reference for researchers, practitioners, and graduate students in systems and control theory. We hope that readers will appreciate the open problems discussed in this book and methods that take into account various types of constraints such as positivity constraints (Chapter 1), sector nonlinearity (Chapter 2), algebraic constraints (Chapter 3), persistent excitation constraints (Chapter 4), coordination constraints (Chapter 5), actuator constraints (Chapter 7) and discontinuities issued from impacts (Chapter 8).

Chapter 1

Positive Systems: Discretization with Positivity and Constraints

In this chapter, we discuss the problem of preservation of two properties pertaining continuous-time systems under discretization, namely the properties of positivity and sparsity. In the first part of the chapter, the action of diagonal Padé transformations is studied together with the preservation of copositive quadratic and copositive linear Lyapunov functions. A variation of the scaling and squaring method is then introduced and shown to be able to preserve such Lyapunov functions and positivity for small sampling times. In the second part, besides positivity, the problem of preservation of the structure (sparseness) of the continuous-time system under discretization is analyzed. The action of the standard forward Euler discretization method is discussed and a new approximation method – mixed Euler – ZOH (mE-ZOH) is introduced that preserves copositive Lyapunov functions, the sparseness structure and the positivity property for all sampling times.

1.1. Introduction and statement of the problem

This chapter is devoted to the study of the effects of discretization in the preservation of two properties pertaining linear systems, namely (1) positivity and (2) structure. The first property characterizes systems whose inputs, state and outputs take non-negative values in forward time. As part of the more general class of monotonic systems [ANG 03], such systems characterize the dynamic behavior of processes frequently encountered in engineering and in social, economic and biological sciences. A few monographs are now available where both the mathematical properties and the application interest of such systems are underlined [BER 94, FAR 00].

The important problem of obtaining reliable discrete-time approximations to a given continuous-time system arises in many circumstances: in simulation issues, in control system design, in certain optimization problems and in model order reduction [ANT 05, FAL 08]. While a complete understanding of this problem exists for linear time-invariant (LTI) systems [WES 01], and some results are available for switched linear systems [ROS 09, SAJ 11], the analogous problems for positive systems are more challenging since discretization methods must preserve not only the stability properties of the original continuous-time system, but also physical properties, such as state positivity. To the best of our knowledge, this is a relatively new problem in the literature, with only a few recent works on this topic [BAU 10]. In particular, we stress the importance of this issue in the framework of switched positive systems, a research field still in its infancy, but with growing importance in telecommunications, biological networks and cloud computing (see [SHO 07, SHO 06, BAR 89, HAR 02]). Generally speaking, we are interested in the evolution of the system:

[1.1]

where Aσ ∈ n are Hurwitz stable Metzler matrices, xc(t) ∈ n×1 and m ≥ 1. We are interested in obtaining from this continuous-time positive system, a discrete-time representation:

[1.2]

where h > 0 is the sampling interval. The first objective of this chapter is to study diagonal Padé approximations to the matrix exponential. Such a study is well motivated, as diagonal Padé approximations are methods used by control engineers. Following [ZAP 12], we deal with two fundamental questions. First, under what conditions are certain types of stability of the original positive switched system inherited by the discrete-time approximation? Second, we also ask if and when positivity itself is inherited by the discrete-time system. We give sufficient conditions under which the Padé approximation is positivity preserving, and identify a new approximation method that is guaranteed to preserve both stability and positivity.

The second objective of this chapter arises from the need of discretizing large-scale systems. In this context, we are often interested in discretization methods that preserve the structure of a dynamic system. We aim to find efficient discretization methods which preserve, for the elements of (), the same zero/non-zero pattern of . The attention here is focused on positive switched systems only, along the lines traced in [COL 12]. First, we analyze the properties of the forward Euler transformation, which intrinsically preserve the zero pattern of the off-diagonal entries of the dynamic matrix. However, it is well known that the forward Euler transformation can easily lead to a loss of stability even for short sampling times. We then propose a novel discretization method that preserves the structure independently of the sampling time, with improved performance in terms of stability preservation.