Computation of State Avoidance Control for Innnite State Systems in Assignment Program Framework|the Expanded Version 1; 2

In this paper we study supervisory control of discrete event systems using state variables for representation and speci cation. The motivation is two fold: rstly, a state variable representation allows a concise characterization of systems with in nitely many states, and secondly, state variable speci cation allows characterization of qualitative properties of general nondeterministic systems. An assignment program consisting of state variables and a nite set of conditional assignment statements is used for representing a discrete event system, and a set of forbidden states is used for representing a control speci cation. Although the state avoidance control of in nite state systems has been studied in literature, there is little work on computation of supervisors for general in nite state systems (except for certain classes of Petri nets). This is not unexpected since as we show, the control synthesis problem is undecidable in general. The contribution of this paper is to show that the supervisory control problem reduces to that of solving arithmetic equations for which \semi-automated" solution techniques exist. This, to our knowledge, is a rst attempt towards supervisory computation of general in nite state systems. Moreover, in the special case when the system can be represented as a vector addition system or equivalently, a Petri net, and the set of forbidden states is a right-closed set we provide techniques for maximally permissive supervision. This generalizes the existing results on controlled vector addition systems.