Use of a Metric in Supervisory Control of Probabilistic Discrete Event Systems

Abstract

This work represents a natural extension of our work on optimal probabilistic supervisory control of probabilistic discrete event systems (PDESs). In that work, a pseudometric on the initial states of two probabilistic automata that represent probabilistic systems is used to measure the distance between two systems. The pseudometric is given a fixed point characterization. This paper gives a logical characterization of the same pseudometric that justifies the intuition that two systems are close if they satisfy similar properties. A trace characterization of the pseudometric is then derived from the logical characterization. Further, the solution of the problem of approximation of a given probabilistic automaton with another automaton is suggested such that the new model is as close as possible to the original one in the pseudometric. The significance of the approximation is then discussed.

Optimal Supervisory Control of Probabilistic Discrete Event Systems

“…The work of [24] is closely related to the well-established work on pseudometrics presented in [29]- [35]. In [36], [37], we further characterize our pseudometric by giving it logical and trace characterizations. The logical characterization measures the distance between two systems by a real-valued formula that distinguishes between the systems the most.…”

“…Furthermore, as the pseudometric is suggested for a large class of systems, it allows for an extension of our work to, for example, nondeterministic systems. Further, as presented in [36], [37], the pseudometric has both logical and trace characterizations. The logical characterization measures the distance between two systems by a -valued formula that distinguishes between the systems the most, while the trace characterization describes the similarity between the probabilistic traces of similar systems.…”

“…Note that the aforementioned results hold for . Also, as previously mentioned, a modification of the presented algorithm can be used to solve the control problem presented in Section III with requirement 2) changed so that the distance between the controlled plant and the unmodified requirement is minimized (see [36], [37]).…”

A Framework for Supervisory Control of Probabilistic Discrete Event Systems

“…The pseudometric is sensitive to all differences between corresponding transition probabilities, as opposed to e.g., the pseudometric of Giacalone et al (1990) that, roughly speaking, considers only the maximum of the differences between the corresponding probabilities. The logical and trace characterization of the pseudometric were given in Pantelic and Lawford (2010), Pantelic and Lawford (2012b). In logical characterization, the pseudometric is characterized via a real-valued logic: the distance in the pseudometric between two systems is measured by a formula that distinguishes between the systems the most.…”