Hardware Reference
In-Depth Information
Chapter 15
Supervisory Control
15.1
Supervisory Control
Supervisory control is an important area of discrete control theory that received
a growing attention since the seminal work of Ramadge and Wonham (see, for
example [8, 25, 77, 78, 119]). In this chapter we apply the techniques for language
equation solving to supervisory control problems, taking into account that methods
for language equation solving cannot be directly used in supervisory control, and
vice versa. The reason is that from one side the topology addressed in supervisory
control is a special case of the general topology addressed by language equation
solving; from another side in supervisory control one is required to model also
partial controllability and partial observability, which are formalized in a different
way when solving language equations.
Methods for solving equations over regular languages have an exponential
complexity in the worst-case, because of the complementation operator that requires
determinization, but they can handle arbitrary topologies. The methods for supervi-
sory control under partial controllability with regular languages have a polynomial
complexity, due to the simplified topology. Comparing the two approaches helps in
characterizing the topologies for equation solving whose worst-case computational
complexity is lower than in the general case. An investigation of such topologies
started already in [12, 148].
15.2
Supervisory Control with Full Controllability
and Observability
We start with the general controller topology (see Fig. 15.1 ). Given a plant and a
specification as languages
, the problem of supervisory
control is to find a controller with a prefix-closed language
P
and
S
over the alphabet
˙
C
over
˙
such that
P \ C D S
. Such controller is also called a supervisor or supervisory controller.
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