A Game-Theoretical Approach for Optimal Supervisory Control of Discrete Event Systems for Cyclic Tasks

In this paper, we investigate the problem of optimal supervisory control for cyclic tasks in the context of discrete-event systems (DES). We consider the completion of each single task as the visit of a marked state, and overall control objective is to complete tasks cyclically in the sense that marked states are visited infinitely often. Following the standard optimal supervisory control framework, two types of costs, disable cost and occurrence cost, are considered. However, instead of considering the standard accumulated total cost or the average cost per event, we propose a new measure for the control performance using the average cost per task. We show that such an optimality measure is more suitable for tasks that need to be completed cyclically. Our goal is to design a live and non-blocking supervisor such that the average cost per task in the worst-case is minimized. To solve the problem, we propose a game-theoretical approach by converting the optimal control problem as a two-player graph game. The constructed game is then solved in two stages: one focuses on the optimal execution within each single task cycle and the other focuses on the scheduling strategy among different tasks. Illustrative examples are provided to demonstrate the proposed algorithm.