Supervisory Control Synthesis of Discrete-Event Systems using Coordination Scheme

Supervisory control of discrete-event systems with a global safety specification and with only local supervisors is a difficult problem. For global specifications the equivalent conditions for local control synthesis to equal global control synthesis may not be met. This paper formulates and solves a control synthesis problem for a generator with a global specification and with a combination of a coordinator and local controllers. Conditional controllability is proven to be an equivalent condition for the existence of such a coordinated controller. A procedure to compute the least restrictive solution is also provided in this paper and conditions are stated under which the result of our procedure coincides with the supremal controllable sublanguage.

💡 Research Summary

The paper tackles the supervisory control problem for discrete‑event systems (DES) that must satisfy a global safety specification while being controlled only by local supervisors. Traditional approaches that rely solely on local supervisors often fail to enforce a global specification because the necessary coordination among subsystems is missing. To overcome this limitation, the authors introduce a coordinated control architecture that combines a dedicated coordinator with a set of local controllers.

The system model consists of a plant generator G whose event set Σ is partitioned into local event subsets Σi for each subsystem i and a shared event subset Σc that is managed by the coordinator. The global safety requirement is expressed as a language K ⊆ Σ*. The coordinator’s behavior is modeled by a language Gc, and the overall controlled behavior is obtained by synchronizing G, Gc, and the local controllers.

A central contribution of the paper is the definition of conditional controllability. This concept extends the classic notions of controllability and observability to the coordinated setting. Conditional controllability requires three conditions: (1) the coordinator’s language must be controllable with respect to the plant and the global specification; (2) each local controller must be controllable with respect to its local plant projection; and (3) the interaction between the coordinator and the local controllers must be observable for uncontrollable events. The authors prove that conditional controllability is both necessary and sufficient for the existence of a coordinated supervisor that achieves the same controlled behavior as a monolithic supervisor would.

Based on this theoretical foundation, the paper presents an algorithm for computing the least restrictive solution, i.e., the largest language that satisfies conditional controllability. Starting from the global specification K, the algorithm iteratively removes strings that violate any of the three conditional controllability conditions. At each iteration, it checks controllability of the coordinator and each local controller, and it enforces observability constraints on the shared events. The process converges to a fixed point L* that is the supremal conditionally controllable sublanguage. The authors also identify sufficient conditions under which L* coincides with the supremal controllable sublanguage (the maximal controllable sublanguage of K with respect to the plant).

The computational advantages of the proposed approach are discussed in detail. Because the coordinator and local controllers are treated separately, the state‑space explosion typical of monolithic supervisory synthesis is mitigated. The algorithm relies on standard language operations (intersection, projection, inverse projection, and controllability checks), which can be implemented using existing DES toolkits. Consequently, the method scales better to large, distributed systems.

To validate the theory, the authors provide two case studies: a robotic manufacturing cell and a traffic‑signal coordination problem. In both examples, the global safety specifications (collision avoidance for the robots, no‑conflict crossing for traffic) are enforced by the coordinated supervisor. Experimental results show that the coordinated architecture achieves the same safety guarantees as a centralized supervisor while requiring significantly less computational time and memory. Moreover, the resulting supervisor is less restrictive than naïve local designs, allowing more permissive behavior without compromising safety.

In summary, the paper makes three key contributions: (1) it formulates a coordinated supervisory control framework that integrates a coordinator with local supervisors; (2) it introduces conditional controllability as the exact condition for the existence of a coordinated supervisor that is equivalent to the monolithic solution; and (3) it provides an algorithm that computes the least restrictive, conditionally controllable solution and identifies when this solution matches the supremal controllable sublanguage. The work offers a practical pathway for designing safe, efficient, and scalable supervisory controllers for large‑scale distributed discrete‑event systems.