Probabilistic Supervisory Control of Probabilistic Discrete Event Systems
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abstract
This thesis considers probabilistic supervisory control of probabilistic discrete
event systems (PDES). PDES are modeled as generators of probabilistic languages.
The probabilistic supervisors employed are a generalization of the
deterministic ones previously employed in the literature. At any state, the supervisor
enables/disables events with certain probabilities. The probabilistic
supervisory control problem (PSCP) that has previously been considered in
the literature is revisited: find, if possible, a supervisor under whose control
the behavior of a plant is identical to a given probabilistic specification. The
existing results are unified, complemented with a solution of a special case and
the computational analysis of synthesis problem and the solution. The central place in the thesis is given to the solution of the optimal
probabilistic supervisory control problem (OPSCP) in the framework: if the
conditions for the existence of probabilistic supervisor for PSCP problem are
not satisfied, find a probabilistic supervisor such that the achievable behaviour
is as close as possible to the desired behaviour. The proximity is measured
using the concept of pseudometric on states of generators. The distance between
two systems is defined as the distance in the pseudometric between the
initial states of the corresponding generators. The pseudometric is adopted from the research in formal methods community
and is defined as the greatest fixed point of a monotone function. Starting
from this definition, we suggest two algorithms for finding the distances
in the pseudometric. Further, we give a logical characterization of the same
pseudometric such that the distance between two systems is measured by a
formula that distinguishes between the systems the most. A trace characterization
of the pseudometric is then derived from the logical characterization by
which the pseudometric measures the difference of (appropriately discounted) probabilities of traces and sets of traces generated by systems, as well as some
more complicated properties of traces. Then, the solution to the optimal probabilistic
supervisory control problem is presented. Further, the solution of the problem of approximation of a given probabilistic
generator with another generator of a prespecified structure is suggested
such that the new model is as close as possible to the original one in
the pseudometric (probabilistic model fitting). The significance of the approximation
is then discussed. While other applications are briefly discussed, a
special attention is given to the use of ideas of probabilistic model fitting in
the solution of a modified optimal probabilistic supervisory control problem.
