Supervisor Design for Minimal Event Observation in Discrete Event Systems: A Linear Programming Approach

This paper studies the supervisory control of discrete event systems (DESs) from an event observation perspective and addresses the problem of supervisor design with minimal observation. In supervisory control, a supervisor enables or disables controllable events based on its observation of the system trajectory to guarantee controllability and nonblocking behavior with respect to a given specification, while the number of observed events critically affects the implementation complexity and cost of the control logic. Rather than minimizing the state space of the supervisor, which is the focus of classical supervisor reduction, this paper is dedicated to the minimization of observable events. Specifically, it aims to reduce the observation alphabet while preserving control equivalence with the original supremal supervisor. By analyzing the consistency of disabling decisions between event-enabled and event-disabled states, necessary and sufficient distinguishability conditions are derived and represented using Parikh vectors, which enables their formulation as linear separation constraints. In addition, event-enabled circles are introduced to capture intrinsic structural observability requirements induced by cyclic behaviors of the supervisor. These results lead to a mixed-integer linear programming (MILP) formulation that minimizes the observation alphabet while preserving control equivalence with the original supremal supervisor, together with an E-closure-based construction that synthesizes an executable event-minimal supervisor. Illustrative examples demonstrate that the proposed method can significantly reduce observation requirements even when state-minimal supervisors are already available, thereby improving implementation efficiency in resource-constrained DES applications.