14.1.   HYBRID SYSTEM MODELS

A hybrid control system is a system in which the behavior of interest is
determined by interacting processes of distinct characteristics - in particular,
interacting continuous and discrete dynamics. Hybrid systems typically generate
mixed signals that consist of combinations of continuous- and discretevalued
signals. Some of these signals take values from a continuous set (e.g.,
the set of real numbers) and others take values from a discrete, typically
finite set (e.g., the set of symbols {a, b, c}). Furthermore, these continuous- or
discrete-valued signals depend on independent variables such as time, which
may also be continuous- or discrete-valued. Another distinction that can be
made is that some of the signals can be time-driven, while others can be
event-driven in an asynchronous manner.

The dynamic behavior of such hybrid systems is captured in hybrid models.
In a manufacturing process, for example, parts may be processed in a
particular machine, but only the arrival of a part triggers the process; that is,
the manufacturing process is composed of the event-driven dynamics of the
parts moving among different machines and the time-driven dynamics of
the processes within particular machines. Frequently in hybrid systems in the
past, the event-driven dynamics were studied separately from the time-driven
dynamics, the former via automata or Petri net models (also via PLC, logic
expressions, etc.) and the latter via differential or difference equations. To
understand fully the system’s behavior and meet high-performance specifications,
one needs to model all dynamics together with their interactions. Only
then may problems such as optimization of the whole manufacturing process
be addressed in a meaningful manner. There are, of course, cases where the
time-driven and event-driven dynamics are not tightly coupled or the demands
on the system performance are not difficult to meet, and in those
cases, considering simpler separate models for the distinct phenomena may
be adequate. However, hybrid models must be used when there is significant
interaction between the continuous and the discrete parts and high-performance
specifications are to be met by the system

Hybrid models may be used to significant advantage, for example, in
automotive engine control, where there is a need for control algorithms with
guaranteed properties, implemented via embedded controllers, that can substantially
reduce emissions and gas consumption while maintaining the performance
of the car. Note that an accurate model of a four-stroke gasoline
engine has a natural hybrid representation, because from the engine control
point of view, on one hand, the power train and air dynamics are
continuous-time processes, while, on the other hand, the pistons have four
modes of operation that correspond to the stroke each is in and so their
behavior is represented as a discrete-event process described, say, via a
finite-state machine model.