Control using nondeterministic supervisors for partially observed discrete event systems

Abstract

We study the supervisory control of discrete event systems under partial observation using nondeterministic supervisors. We formally define a nondeterministic control policy and also a control and observation compatible nondeterministic state machine and prove their equivalence. We show that when control is exercised using a nondeterministic supervisor, the specification language is required to satisfy a weaker notion of observability, which we define in terms recognizability and achievability. Achievability serves as necessary and sufficient condition for the existence of a nondeterministic supervisor, and it is weaker than controllability and observability combined. When all events are controllable, achievability reduces to recognizability. We show that both existence, and synthesis of nondeterministic supervisors can be determined polynomially. (For deterministic supervisors, only existence can be determined polynomially.) Both achievability and recognizability are preserved under union, also and under intersection (when restricted over prefix-closed languages). Using the intersection closure property we derive a necessary and sufficient condition for the range control problem for the prefix-closed case. Unlike the deterministic supervisory setting where the complexity of existence is exponential, our existence condition is polynomially verifiable, and also a supervisor can be polynomially synthesized.