On A Generalization of Robust Supervisory Control of Discrete Event Systems

Abstract

In this paper we consider a generalization of the robust supervisory control problem introduced by Lin (Lin, 1993), improved by Takai (Takai, 2000; Takai, 2002), Park and Lim (Park and Lim, 2000; Park and Lim, 2002) and Cury and Krogh (Cury and Krogh, 1999). Following the formalism in (Lin, 1993), we suppose the plant language L(G) ⊆ Σ* belongs to a finite collection of non-empty, prefix-closed languages L(G) Є {L1, L2,...,Ln}, where Li(≠0) ⊆ Σ*, i Є {1, 2,..., n}. The event-set Σ is partitioned into controllable (Σc) and uncontrollable (Σu) subsets respectively. We assume all events are observable, and the supervisor has no prior knowledge as to the value of L(G) Є {L1, L2,..., Ln}. For each Li ⊆ Σ* we suppose there exists a prefix-closed language Ki ⊆ Li. We present three conditions that are necessary and sufficient for the existence of a supervisor that enforces Ki if the plant language L(G) = Li. It is possible that for a given choice of the sets {L1, L2,..., Ln} and {K1, K2,..., Kn}, the conditions identified in this paper are not satisfied. This calls for finding a {K1,K2,...Kn}, such that ∀iЄ{1,2,...,n},Ki⊆Ki that meets the required conditions, and each K⌢ satisfies some property that we might be interested in. The search for a satisfactory {K1,K2,...,Kn} using the notion of monotone properties is also presented.