Supervisory control of probabilistic discrete event systems

Abstract

This paper considers supervisory control of probabilistic discrete event systems (PDES). PDESs are modeled as generators of probabilistic languages. The supervisory control problem considered is to find, if possible, a supervisor under whose control the behaviour of a plant is identical to a given probabilistic specification. The probabilistic supervisors we employ are a generalization of the deterministic ones previously employed in the literature. At any state, the supervisor enables/disables events with certain probabilities. Necessary and sufficient conditions for the existence of such a supervisor, and an algorithm for its computation are presented.

Use of a Metric in Supervisory Control of Probabilistic Discrete Event Systems

“…The probabilistic DES (PDES) can be modeled as a probabilistic generator G = (Q, Σ, δ, q 0 , p) (Lawford and Wonham (1993)), where Q is the nonempty finite set of states, Σ is a finite alphabet whose elements we will refer to as event labels, δ : Q × Σ → Q is the (partial) transition function, q 0 ∈ Q is the initial state, and p : Q × Σ → [0, 1] is the statewise event probability distribution. The results to be presented are for prefix closed probabilistic specification languages; hence the lack of marking states in the definition of a probabilistic generator.…”

“…The probability that the system terminates at state q is 1 − σ∈Σ p(q, σ). Throughout the sequel, we will mostly consider nonterminating generators (if a plant is terminating, it can easily be transformed into a nonterminating one using the technique described in Lawford and Wonham (1993)). …”

“…A supervisory control framework of PDESs was proposed in Lawford and Wonham (1993). PDESs are modeled as probabilistic generators from Garg (1992a,b).…”

“…Further, deterministic supervisors for DES are generalized to probabilistic supervisors: after observing a string s, the probabilistic supervisor enables an event σ with a certain probability. The supervisory control problem considered in Lawford and Wonham (1993) is to find, if possible, a supervisor under whose control the behaviour of a plant is identical to a given probabilistic specification. Further, Lawford and Wonham (1993) show that a plant under probabilistic control can generate a much larger class of probabilistic languages than deterministic control, and give the necessary and sufficient conditions for the existence of a supervisor for a class of PDESs.…”

“…The supervisory control problem considered in Lawford and Wonham (1993) is to find, if possible, a supervisor under whose control the behaviour of a plant is identical to a given probabilistic specification. Further, Lawford and Wonham (1993) show that a plant under probabilistic control can generate a much larger class of probabilistic languages than deterministic control, and give the necessary and sufficient conditions for the existence of a supervisor for a class of PDESs. A formal proof of the necessity and sufficiency of the conditions and an algorithm for the calculation of the supervisor, if it exists, are presented in Postma and Lawford (2004), .…”