Towards Supervisory Control of Interactive Markov Chains: Controllability

We propose a model-based systems engineering framework for supervisory control of stochastic discrete-event systems with unrestricted nondeterminism. We intend to develop the proposed framework in four phases outlined in this paper. Here, we study in detail the first step which comprises investigation of the underlying model and development of a corresponding notion of controllability. The model of choice is termed Interactive Markov Chains, which is a natural semantic model for stochastic variants of process calculi and Petri nets, and it requires a process-theoretic treatment of supervisory control theory. To this end, we define a new behavioral preorder, termed Markovian partial bisimulation, that captures the notion of controllability while preserving correct stochastic behavior. We provide a sound and ground-complete axiomatic characterization of the preorder and, based on it, we define two notions of controllability. The first notion conforms to the traditional way of reasoning about supervision and control requirements, whereas in the second proposal we abstract from the stochastic behavior of the system. For the latter, we intend to separate the concerns regarding synthesis of an optimal supervisor. The control requirements cater only for controllability, whereas we ensure that the stochastic behavior of the supervised plant meets the performance specification by extracting directive optimal supervisors.