Design of Optimal Supervisors for the Enforcement of Nonlinear Constraints on Petri Nets

Abstract

This paper proposes an iterative approach to separate a set of admissible markings of a nonlinear constraint into a number of subsets. At each iteration, we find a maximal subset of admissible markings that are separated from inadmissible markings by linear constraints. Then, the union of all the obtained subsets constitutes the set of all admissible markings. For each subset of admissible markings, we obtain a set of conjunctive linear constraints. Accordingly, we can equivalently transform a given nonlinear constraint to be a set of disjunctive/conjunctive linear constraints, which can deal with the case that both admissible and inadmissible marking spaces of a nonlinear constraint cannot be separated by linear constraints from each other. Furthermore, we propose a method to design a Petri net supervisor for a derived set of disjunctive/conjunctive constraints. Some examples are used to demonstrate the proposed approach. Note to Practitioners—Linear constraints on Petri nets have been widely studied in the literature. However, not all control specifications can be represented as linear constraints. This paper proposes to enforce nonlinear constraints on Petri nets. A maximally permissive supervisor is designed to make the controlled net live with all admissible markings with respect to a given nonlinear constraint. Meanwhile, the structural complexity is also considered by compressing the number of control places in the supervisor. Experimental results show that the proposed approach can implement the control specifications represented by nonlinear constraints.