E E Verification and Control of Hybrid Systems

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1 E E Verification and Control of Hybrid Systems

2 Paulo Tabuada Verification and Control of Hybrid Systems A Symbolic Approach Foreword by Rajeev Alur

3 Paulo Tabuada Department of Electrical Engineering University of California, Los Angeles F, Engineering Bldg. IV Los Angeles, CA USA ISBN e-isbn DOI / Springer Dordrecht Heidelberg London New York Library of Congress Control Number: Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (

4 This book is dedicated to my father. Stricken again, by the bitterness of life, while this book was being written.

5 Foreword It is my great pleasure to write this foreword to this remarkable monograph on hybrid systems by Paulo Tabuada. A hybrid system combines the statemachine models of discrete control with differential-equation models of continuous dynamics. Traditionally, state machines have been used extensively for modeling and analysis in computer science, and dynamical systems are studied by researchers in control theory. The embedded applications, consisting of discrete software reacting to continuous environment, have motivated the study of the problems at the boundary of the two fields, leading to the growing and exciting field of hybrid systems. This research discipline brings together researchers from software engineering and formal methods, control theory, and applications such as robotics and systems biology. The results at the computer science end were motivated by a desire to generalize the symbolic methods that had proved to be so successful in modeling and analyzing digital hardware. Early results focused on models such as timed automata and linear hybrid automata that augmented finite-state control with simple dynamics. At the other spectrum, control theorists developed ways for generalizing analysis techniques for linear and non-linear systems to switched counterparts. Over the years, a wealth of results have emerged at the intersection, but more importantly, the cross-fertilization of ideas have led to new insights and analysis techniques. For example, the theory of propertypreserving abstractions, studied extensively in the formal methods literature, has led to computational techniques for reducing the dimensionality of linear control systems; the concepts of barrier certificates and Lyapunov functions from dynamical systems theory have led to effective tools for verification of safety and liveness properties of hybrid systems. The rapidly growing and interdisciplinary research demands that researchers must understand concepts and tools from both computer science and control theory. This is undoubtedly a daunting task, and this is where this book can come to the rescue. It covers a wide range of foundational topics in verification and control of hybrid systems. More impressively, it thematically weaves them together using the recurring theme of simulation relations.

6 VIII Foreword The notion of a simulation or a bisimulation relation between two descriptions of a system, possibly at different levels of detail, has proved to be an important concept, both in theory and practice, for modeling and verification of discrete systems. Using richer concepts such as alternating relations and approximate relations, this book develops the foundations for hybrid control systems. Paulo is ideally suited for the challenge of explaining the foundations of hybrid systems. Having interacted with him for many years, I am well aware of the breadth of his technical expertise that spans across topics in formal verification and control theory. He has made numerous important contributions to the field, particularly to foundations and tools for property-preserving abstractions of hybrid control systems. The exposition is technically rigorous, and uses a number of well-chosen illustrations to explain the concepts. There are a number of conferences on hybrid systems. It is also an integral component of the newly emerging and broader discipline of cyber-physical systems. There are new challenges, and research opportunities, related to, for instance, robustness in presence of uncertainties and design of networked control systems. I recommend this book highly to students and researchers who want to tackle these challenges. Studying this book will be an excellent way to understand the mathematical foundations of this vibrant field. University of Pennsylvania Rajeev Alur

7 Preface Hybrid systems arose more than 15 years ago in a bold attempt to yoke together computer science and control theory in the context of, what are now called, cyber-physical systems. Although it is still early to give an unified account of hybrid systems research, certain conceptual similarities, between different results developed by different researchers, have recently come into view. This book aims at highlighting these similarities by providing a systematic exposition of several key verification and control synthesis results. The guiding concept used in this book is the notion of bisimulation. To understand how it winds through hybrid systems research, a digression into digital control and timed automata is in order. Before hybrid systems, the existing paradigm for computer controlled systems was digital control. Under this paradigm, a computer interfaces with the physical world through an Analog-to-Digital (A/D) converter, transforming measured physical quantities into digital format, and through a Digital-to-Analog (D/A) converter, transforming digital control commands into analog signals. The enabling result of this paradigm states that when the continuous dynamics is described by a controlled linear differential equation, the combination of the continuous dynamics with certain classes of D/A and A/D converters can be represented by a discrete-time linear control system. Consequently, discrete-time linear control systems were widely used as the abstraction of choice for computer controlled physical systems. Although powerful, the digital control paradigm provides no support to study the interaction of software with the physical world. The first models for hybrid systems extended the digital control paradigm by modeling the software as a finite-state machine interfacing with the physical world, described by a controlled differential equation, through A/D and D/A converters. This model for hybrid systems later evolved into hybrid automata but no enabling representation result, such as the one for digital control, was known. The desired representation result had already been partially given by Alur and Dill s work on timed automata. In the 1990 paper [AD90], Alur and Dill showed that timed automata, a special class of hybrid automata, could be represented by a finite-state symbolic model, i.e., a model with finitely

8 X Preface many states where each state or symbol represents infinitely many states of the timed automaton. Essential to their results was the observation that the notion of bisimulation was versatile enough to establish a formal equivalence between timed automata and finite-state symbolic models. Bisimulation, originally introduced by Park [Par81] and Milner [Mil89] as a notion of equivalence between software processes, provided the motto for extensions of Alur and Dill s pioneering work to other classes of hybrid systems. Although the initial effort was on representation results for the verification of hybrid systems, later results showed that symbolic models could also be constructed for control design. A further twist in this research stream occurred recently when it was recognized that bisimulation could be generalized to approximate bisimulation with the purpose of further enlarging the class of hybrid systems admitting symbolic models. The excursion into hybrid systems research, offered in this book, is divided into four parts. The first part presents basic concepts centered around a notion of system that is general enough to describe finite-state machines, differential equations, and hybrid systems. However, a system, by itself, is not very interesting. More interesting are the ways in which systems relate to other systems. Two such relationships are presented in Part II: behavioral inclusion/equivalence and simulation/bisimulation. These relationships are then used to study verification and control synthesis problems for finite-state systems. Only a flavor of the existing results is provided since the focus of the book are the infinite-state (hybrid) systems discussed in Part III and Part IV. By drawing inspiration from timed automata, several classes of hybrid systems with richer continuous dynamics are shown to be related to finite-state symbolic systems in Part III. Once such (bi)simulation relations are established, verification and control synthesis problems can be immediately solved by resorting to the techniques described in Part II for finite-state systems. The same strategy is followed in Part IV by generalizing (bi)simulation relationships to approximate (bi)simulation relationships that can be used for wider classes of hybrid systems. The choice of results presented in this book is admittedly biased by my view of hybrid systems and my own research interests. Moreover, I confess that the guiding concept of bisimulation warrants a longer route visiting many other important results. The decision not to include such topics was difficult to make, but including them would have required more time than I could give to this project at this stage. In addition to the choice of topics, I faced another challenge: to make the book accessible and interesting to both computer scientists and control theorists. On the one hand, the computer scientist will certainly find Part II to be a very narrow account of formal verification and Part III to treat timed automata very superficially. On the other hand, the control theorist will be intrigued with the notion of system used in this book, yet disappointed with the relegation of nonlinear systems to several cursory sections. Nonetheless, I hope the readers, independently of their technical

9 Preface XI background, will find the results interesting enough to consult the specialized literature for the missing details. It is a great pleasure to acknowledge the influence of Rajeev Alur and George J. Pappas that helped shape my view of hybrid systems. Some of the results in Part IV were developed in collaboration with Giordano Pola and Antoine Girard and for that, I am very grateful. My students and postdocs, especially Manuel Mazo Jr. and Ramkrishna Pasumarthy, also deserve a special word of acknowledgement for carefully reading different versions of this book and providing me with valuable feedback. This book would not have been possible without the scholarly environment created by my colleagues at the Electrical Engineering Department of the University of California at Los Angles, and without the support of Dr. Helen Gill and Dr. Radhakishan Baheti from the National Science Foundation. My final words of acknowledgment go to my brother, who showed me how to persevere and prosper among brilliant minds, to my parents, for being a constant source of support, and to my wife, to whom I now have to repay the countless nights and weekends spent in the writing of this book. Los Angeles and Lisboa Paulo Tabuada

10 Contents Part I Basic concepts 1 Systems System definition System behavior Examples Composing systems Notes Verification problems S a = Sb S a S b Control problems S c I S a = Sb S c I S a S b Part II Finite systems 4 Exact system relationships Behavioral relationships Similarity relationships Alternating similarity relationships Notes Verification Behavioral relations Similarity relations Notes

11 XIV Contents 6 Control Feedback composition Safety games Reachability games Behavioral games Similarity games Notes Part III Infinite Systems: Exact symbolic models 7 Exact symbolic models for verification Dynamical and hybrid dynamical systems as systems Timed automata Order minimal hybrid dynamical systems Sign based abstractions Barrier certificates Computation of reachable sets Advanced topics Notes Exact symbolic models for control Control systems as systems Controller refinement Discrete-time linear control systems Continuous-time multi-affine control systems Notes Part IV Infinite Systems: Approximate symbolic models 9 Approximate system relationships Approximate similarity relationships Approximate alternating similarity relationships Notes Approximate symbolic models for verification Stability of linear dynamical systems Dynamical systems as systems Symbolic models for affine dynamical systems Advanced topics Notes

12 Contents XV 11 Approximate symbolic models for control Stability of linear control systems Control and switched systems as systems Approximate feedback composition and controller refinement Symbolic models for affine control systems Symbolic models for switched affine systems Advanced topics Notes Appendix A.1 Lattice theory A.2 Fixed-points References Index

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