Supervisory Control of Probabilistic Discrete Event Systems Under Partial Observation

Abstract

The supervisory control of probabilistic discrete event systems (PDESs) is investigated under the assumptions that the supervisory controller (supervisor) is probabilistic and has a partial observation. The probabilistic P-supervisor is defined, which specifies a probability distribution on the control patterns for each observation. The notions of the probabilistic controllability and observability are proposed and demonstrated to be a necessary and sufficient conditions for the existence of the probabilistic P-supervisors. Moreover, the polynomial verification algorithms for the probabilistic controllability and observability are put forward. In addition, the infimal probabilistic controllable and observable superlanguage is introduced and computed as the solution of the optimal control problem of PDESs. Several examples are presented to illustrate the results obtained. 20] investigated the optimal control problem of PDESs. The optimal control aims to synthesize a supervisor that minimizes the distance between the uncontrollable specification and its controllable approximation. In order to measure the distance, Pantelic et al. [21] proposed the notion of the pseudometric, and its calculating algorithms.Chattopadhyay et. al [22] also considered the optimal control issue of PDESs. However, different from [20], the optimal objective is maximizing the renormalized language measure vector for the controlled plant. Based on the measurement, Chattopadhyay et. al also formulated a theory for the optimal control of PDESs.It should be pointed out that the supervisors defined in [16]-[22] are all supposed to have a full observation to the events, which are not always satisfied in practical engineering systems. In this paper, we focus on the supervisory control problem of PDESs with the assumptions that the supervisor is probabilistic and has a partial observation to the events.Different from the full-observation supervisors defined in [16]-[22], we define a partial-observation probabilistic supervisor, called as the probabilistic P-supervisor, which specifies a probabilistic distribution on the control patterns to each observation. Intuitively, for each observation, the probabilistic Psupervisor makes a special roulette. The roulette issues several outcomes with the pre-specified probabilities. Before making a control decision, the supervisor will "roll" the corresponding roulette, and then adopt the jth control pattern if the jth outcome is issued. In addition, we demonstrate the equivalence between the probabilistic P-supervisor and the scaling-factor function.We then present the notions of the probabilistic controllability and observability, and their polynomial verification algorithms. We also demonstrate that the probabilistic controllability and observability are the necessary and sufficient conditions for the existence of the probabilistic P-supervisors, and provide the design method of the probabilistic P-supervisor. Moreover, we consider the optimal control problem of PDESs. Different from [20][21][22], the infim...

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