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566 Book reviews / Automatica 38 (2002) 563–567
An introduction to hybrid dynamical systems
A. van der Schaft, and H. Schumacher; LNCIS 251,
Springer, Berlin, 2000, ISBN 1-85233-233-6
In systems and control theory a system is said to be
hybrid when its state has continuous and discrete com-
ponents which interact via the system’s dynamics. As a
result, the hybrid state of such a system evolves both
smoothly in continuous time and undergoes discontinu-

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566
Book reviews/Automatica 38 (2002) 563–567
An introduction to hybrid dynamical systems
A. van der Schaft, and H. Schumacher; LNCIS 251,Springer, Berlin, 2000, ISBN 1-85233-233-6In systems and control theory a system is said to behybrid when its state has continuous and discrete com- ponents which interact via the system’s dynamics. As aresult, the hybrid state of such a system evolves bothsmoothly in continuous time and undergoes discontinu-ous changes in discrete time, that is to say, at some count-able set of time instants (which may be determined bythe system state and the control input).Hybrid systems appear in a vast range of settings.A list of domains where one would nd systems withhybrid features would arguably include at least chem-ical engineering, aerospace, air trac control, auto-motive engineering and robotics. In part this is dueto the widespread use of computers for the regula-tion of physical systems with continuous dynamics.Consequently, as observed by van der Schaft andSchumacher, hybrid systems arise in various guisesin systems and control theory, computer science andsystem simulation. One may add that there are alsomany examples of hybrid systems to be found in nat-ural settings; consider, for instance, systems exhibit-ing hysteresis, state phase changes (e.g. the states of water) and systems possessing local, but not global,stability.The book by van der Schaft and Schumacher is awelcome introduction to this rapidly developing area.The six chapters are entitled respectively: Modellingof Hybrid Systems, Examples of Hybrid DynamicalSystems, Variable-structure Systems, ComplimentaritySystems, Analysis of Hybrid Systems, Hybrid ControlDesign.Chapter 1, Modelling of Hybrid Systems, containsa presentation of the now standard
hybrid system
(
orautomaton
)
model
which has continuous states (more precisely, state components) and discrete locations (i.e.discrete state components), invariants (i.e constraintswhich must be satised by a continuous state in a givenlocation) and guards and jumps. Uncontrolled jumpsare discontinuous changes in the continuous state or thelocation, or both, which must occur when a guard sethas been entered and controlled jumps are those whichmay be chosen to occur in a guard set. While the sys-tem is in a specic location the continuous state evolvesaccording to a corresponding dierential equation withcontrol inputs. The instants at which jumps occur arecalled event times and they are carefully discussed inthe hybrid systems setting specied in the rst chap-ter; in particular, the issue of accumulation points of switching times, the so-called Zeno times, arises in thedescription of possible hybrid system behaviours. Theauthors do not give general existence, uniqueness andcontinuity theorems for hybrid system trajectories (alsocalled executions in the hybrid system literature) andcorrespondingly the geometric and analytic hypothe-ses which would be required for such an analysis areomitted.The authors also give an alternative formulation of hybrid systems which involves the concatenation of
event-ow formulas
. Here the ow formulas describethe evolution of subsets of the continuous state compo-nents in various locations and the event formulas specifylocally the invariants, guards and associated jump transi-tions. An objective of this is to facilitate the specicationof the overall system in terms of the behaviour of theindividual system modules (potentially very large innumber) and to treat the interlinking of the modules viacommunicating variables in a manner analogous to theconstruction of the synchronous product of automata.Chapter 2, Examples of Hybrid Dynamical Systems,contains a very useful set of basic examples; specically,the authors describe systems exhibiting hysteresis andthe various bouncing ball, heater-thermostat, water-levelmonitor, controlled rail-road crossing, power converter and constrained pendulum systems and, nally, a formof the Van der Pol oscillator. In particular, the monitor and rail-road examples are presented via the event-owformalism for each of the components of the system.However, it is not evident that this leads to more com- prehensible descriptions than the standard hybrid systemspecication (via the continuous dynamics in each lo-cation with the system of guards and jumps) which arestraightforward to formulate in both of the cases under consideration.Chapter 3, Variable-structure Systems, relates the sys-tem descriptions introduced in the text to variable struc-turesystemsandessentiallygeneralizesFillipovsolutionswithin the hybrid systems framework.Chapter 4, Complimentarity Systems, deals with atopic to which the second author and his co-workers havemade signicant contributions; this is the class of systems for which there is a complementarity pair-ing between inputs
u
i
and outputs
y
i
, for each
i
,such that both are non-negative and at least oneis zero. (There is also a lexicographical generaliza-tion of this notion.) The signicance of this classof models is established via examples includingmechanics and economics; subject to reasonable condi-tions, theorems are given asserting the well-posednessof linear complementary systems (which means hereexistence and uniqueness of executions up to Zenotimes).Chapter 5, Analysis of Hybrid Systems, briey dis-cusses the question of the stability of hybrid systemsand the occurrence of chaotic phenomena. Chapter 6,Hybrid Control Design, introduces the issues of stabi-lization by switching and set point regulation for hybridsystems.
Book reviews/Automatica 38 (2002) 563–567
567
Many issues are not addressed in this book, such asgeneral well posedness theorems, the general methodswhich have appeared for analysing the stability of hy- brid systems (including Lie algebra techniques) and, in particular, the versions of the Maximum Principle andDynamic Programming theorems which have been ob-tained for the optimal control of hybrid systems. Also,methodologies developed in the computer science andsimulation domains are only briey described. However,the decision to give a limited presentation of the subjectwhich omitted the many technicalities arising in further developments was certainly a sound one. The result isthat “An Introduction to Hybrid Dynamical Systems” isa very accessible and well written basic introduction to arapidly moving eld. Furthermore the spirit of the subjectcomes across clearly and the bibliography is extensive.In conclusion, I denitely recommend this volume to allthose interested in hybrid system theory.
PII: S0005-1098(01)00238-2
Peter E. Caines
Electrical and Computer Engineering Dept.,McConnell Engineering Building 512,3480 University, Montreal, QC, CanadaE-mail address:
peterc@cim.mcgill.ca
About the reviewer
Peter E. Caines
received the BA degree in Mathematics fromOxford University in 1967 and the DIC and Ph.D. degrees inSystems and Control from the Imperial College of Science andTechnology, London, in 1970. Since 1980 he has been with the De- partment of Electrical and Computer Engineering, McGill University,Montreal. Peter Caines is the author or coauthor of more than twohundred journal and conference papers on systems and control theoryand is the author of
Linear Stochastic Systems
published by JohnWiley in 1988; he is a Fellow of the IEEE and the Canadian Institutefor Advanced Research. Peter Caines has been a visiting researcher and professor at several institutions; most recently, in 2000, he wasa William Mong Visiting Research Fellow at the University of HongKong and in 2001 a Lady Davis Visiting Professor at the Technion,Israel. His research interests lie in the areas of stochastic systemsand hierarchical, hybrid and discrete event systems.

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