Automatica, March 1999, Volume 35, No. 3

In control engineering, one frequently encounters dynamical systems whose state is
described partly by variables that take values in a continuum and partly by variables that
take only finitely many values. Perhaps the simplest example is that of a thermostat in a
room. The temperature in the room is a continuous variable, and the state of the
thermostat ("on" or "off") is discrete; the continuous and discrete
parts cannot be described independently since clearly there is interaction between the
two. The term *hybrid systems* has recently been used to describe dynamical systems
that have both continuous and discrete state variables. The current issue of *Automatica*
is devoted to hybrid dynamical systems and their use for control purposes.

The consideration of both discrete and continuous aspects in a single system is in itself not new and has occurred many times in control theory as well as in other disciplines. What is new in hybrid systems theory is the aspiration to develop a theory of hybrid systems in a more unified and systematic manner than has been done before. Given the enormous variety of hybrid systems that can be imagined such an endeavor can only be characterized as rather ambitious, and obviously one should be careful not to be overly optimistic about what can be achieved in terms of a unified theory. Nevertheless the challenge is fascinating from an intellectual point of view. At the same time, there are decidedly practical overtones. In view of the widespread use of switching logic in industrial control, part of hybrid control theory can be described as theory catching up with practice - problems are now being made subject to systematic analysis that were previously looked upon as implementation issues. In this way, hybrid systems theory may join other disciplines such as robust control theory in strengthening the ties between control theory and control practice.

Hybrid systems theory has roots in several different fields of science. Even within
control theory *per se* there are several lines of research that have provided
motivation for the study of hybrid systems. Work in nonlinear control theory has led to
the insight that switching control is in some respects fundamentally more powerful than
smooth control. Developments in adaptive control have naturally led to the consideration
of switching control schemes as well. It goes without saying that various types of
switching have traditionally been used in control methods and implementations such as gain
scheduling, sliding mode control, and programmable-logic controllers. Even fuzzy control
could be mentioned here, since fuzzy schemes are typically based on a combination of
different operating regimes.

Outside control theory, computer scientists have been notably active in promoting hybrid systems theory. In particular researchers in software verification have felt the need for a more precise description of the processes in which computers play a role, and so have been motivated to allow continuous variables and differential equations into their domain of study. The interests of computer scientists have brought questions to the fore, for instance concerning safety, that for some time have perhaps received less attention from control theorists. These questions bring a renewed interest in some classical control topics such as reachability analysis.

There are connections of hybrid systems theory with still other fields of research. One of these is mathematical programming. In hybrid systems, changes of continuous states and changes of discrete states are often connected to each other via inequalities in the continuous state variables. Inequalities have been studied extensively in linear and nonlinear programming, and so an influence of this field in hybrid systems theory can be seen; actually this influence also goes in the other direction, by the introduction of differential equations in mathematical programming problems. In dynamical systems research, interaction of continuous dynamics and discrete switching is a subject that goes back at least fifty years. Investigations in this area have in fact been partly motivated by developments in control theory, such as the use of relays in control devices and bang-bang results in optimal control theory. Finally, another area which already has a tradition of dealing with interaction between continuous and discrete dynamics is simulation. The language ACSL is more than twenty years old and has explicit facilities for incorporating continuous as well as discrete parts; it has been followed by many other simulation languages which more or less successfully have answered the challenge of simulating complex hybrid systems. Again this is a development which has close links to control applications.

The multiple backgrounds of hybrid systems theory can be recognized in the papers in
this special issue. In some control problems, mode switching is already inherent in the
system to be controlled. A typical example is that of a car engine in which the cylinders
go periodically through several distinctively different phases. This example is taken up
in the paper by Balluchi *et al*., who present a modeling scheme as well as a
control design for a particular regime. Other examples can be found in the process
industry. In their contribution to the special issue, Bemporad and Morari set up a
modeling framework which is motivated by applications in the process industry, and present
a control design method based on recent developments in mathematical programming. Even in
cases in which the system to be controlled is smooth, there can be good reasons to apply a
switching controller, and then the closed-loop system becomes hybrid. The standard example
of a smooth nonlinear system that can be stabilized by switching control but not by smooth
control is the "nonholonomic integrator". Hespanha and Morse present in their
contribution a time-invariant switching control law which achieves exponential
stabilization of the nonholonomic integrator. In the paper by Chang and Davison, the
motivation for switching comes from the idea of adaptation; an important issue here is
transient behavior, and Chang and Davison present simulations suggesting that the
scheme they propose does quite well in this respect. As noted above, switching is
classically present in many control schemes such as gain scheduling. The paper by Bett and
Lemmon in this issue considers a strategy of the gain scheduling type, to which the hybrid
systems perspective brings some nontraditional elements.

Themes that are classical in control and in dynamic systems in general do of course
return in the context of hybrid systems. Among these themes are stability and
reachability. The contribution of Michel and Hu is devoted to the development of a part of
Lyapunov stability theory for hybrid systems; as the possibility of jumps in the state
variable has to be taken into account, the formulation of conditions for stability is
considered anew. Reachability analysis is carried out in the paper by Lygeros *et al.*
for situations where, in the presence of saturation effects, certain regions of the state
space should be avoided for safety reasons; at the same time the control design aims at
achieving certain performance objectives. Whereas Lygeros *et al.* take a design
approach to safety issues, Kowalewski *et al.* discuss a verification approach: for
a given combination of plant and controller resulting in a hybrid closed-loop system, how
can we check that safety requirements are met? By applying a quantization scheme,
Kowalewski *et al.* are able to use automatic verification software successfully in
a small example; they note however that the method is not yet powerful enough to handle
industry-size problems. This conclusion points in the direction of a complexity problem
which is indeed a main source of concern in hybrid systems theory. The complexity of
stability and controllability verification of hybrid systems is investigated explicitly in
the contribution by Blondel and Tsitsiklis. They find that even quite simple switched
linear systems can behave as badly in this respect as the most general nonlinear systems.
The picture of complexity is filled in further from another point of view by Lunze *et
al.*, who show that only under very strict conditions the quantization of a
deterministic linear system is again deterministic.

Worst-case analysis should not distract us however from noting specific problem classes
for which a successful treatment may be well within reach. The traffic control problem
considered in this special issue by Ball *et al.* belongs to a group of problems
that are usually formalized so as to be treated by methods from stochastic operations
research. Instead, Ball *et al.* use a deterministic control design method and
successfully deal with the inequality constraints that are inherent in the problem.
Switching appears naturally in some physical systems and one may expect that it will be
possible to develop specialized modeling frameworks for such cases. This idea is pursued
in the paper by Escobar *et al.* for the case of power converters. In the
contribution by Lootsma *et al.* it is shown that relay systems can be fitted into
a specific class of hybrid systems whose definition takes inspiration from mathematical
programming, and this fact is used as a basis for a well-posedness investigation.

A. Stephen Morse

Constantinos C. Pantelides

S. Shankar Sastry

J.M. (Hans) Schumacher