Extensions to the Theory of Optimal Control of Discrete Event Systems

Abstract

This paper describes new results concerning the optimal control problem (OCP) formulated in [6]. The discrete event system (DES) is modeled as a finite vertex directed graph, denoted by G = < V _G , E _G , υ₀, υ_m >, where V _G is the finite set of vertices of G, E _G the set of directed edges, υ₀ the unique initial vertex, and υ_m the unique terminal vertex; it is also assumed that G has at most one edge between a pair of vertices and that all vertices in G are accessible with respect to υ₀ and co-accessible with respect to υ_m. We call a graph G with such properties an admissible graph. In general we will use the functional notation E _f (.) and V _f (.) to denote the set of edges and the set of vertices respectively of an admissible graph. Any path on this graph represents a possible behavior of the DES. Intuitively, paths starting at υ ₀ and ending at υ _m may be viewed as complete or properly terminating behaviors. Accordingly it is assumed that no non-terminating paths are admissible in the controlled behavior. The admissible paths correspond to the language marked by G, denoted by L _m (G). In [6] the same problem was treated but with many restrictions on G. Principal amongst these was the requirement that G be acyclic. This critical qualification is now dispensed with albeit with some increase in the computational complexity.

Research supported in part by the National Science Foundation under grant ECS-9057967, with additional support from DEC and GE.