A graph-theoretic optimal control problem for terminating discrete event processes

Abstract

Most of the results to date in discrete event supervisory control assume a “zero-or-infinity” structure for the cost of controlling a discrete event system, in the sense that it costs nothing to disable controllable events while uncontrollable events cannot be disabled (i.e., their disablement entails infinite cost). In several applications however, a more refined structure of the control cost becomes necessary in order to quantify the tradeoffs between candidate supervisors. In this paper, we formulate and solve a new optimal control problem for a class of discrete event systems. We assume that the system can be modeled as a finite acylic directed graph, i.e., the system process has a finite set of event trajectories and thus is “terminating.” The optimal control problem explicitly considers the cost of control in the objective function. In general terms, this problem involves a tradeoff between the cost of system evolution, which is quantified in terms of a path cost on the event trajectories generated by the system, and the cost of impacting on the external environment, which is quantified as a dynamic cost on control. We also seek a least restrictive solution. An algorithm based on dynamic programming is developed for the solution of this problem. This algorithm is based on a graph-theoretic formulation of the problem. The use of dynamic programming allows for the efficient construction of an “optimal subgraph” (i.e., optimal supervisor) of the given graph (i.e., discrete event system) with respect to the cost structure imposed. We show that this algorithm is of polynomial complexity in the number of vertices of the graph of the system.