Equivalent Transformation Of Nonlinear Constraint And Linear Constraint

Supervisory control theory has been widely used in modeling and controling discrete event systems(DES).In DES,constraints related to reachable state are a typical and important control specification in supervisory control theory.Petri nets are widely used as an important tool for analyzing,modeling and controlling DES.In the theory of Petri nets,the supervisory control is usually expressed as a set of control places that keep only legal behavior.Control specifications are usually represented as linear constraints on the state of a system.A set of linear constraints can be easily enforced by a set of control places.In general,there are two types of constraints based on Petri nets: linear constraints and nonlinear constraints.The former is also called generalized mutual exclusion constraints(GMEC).Linear constraints can be enforced by adding a control place into the Petri net to construct a place invariant(PI).However,not all specifications can be represented by GMECs.In some cases these specifications are given in the form of nonlinear constraints.There are two approaches to enforce nonlinear constraints.One is to design a Petri net controller by separating a transition to a set of transitions.The main disadvantage of the approach is that the controller structure is too complicated.The other approach is to find a set of GMECs to allow the same behavior as the nonlinear constraint.This paper mainly proposes an equivalent transformation method for nonlinear constraints and linear constraints based on feasible state space and border forbidden markings(BFM)space.The main contributions are as follows:1.In the previous work,if the feasible state space of a nonlinear constraint is convex,then we can equivalently transform it to be a set of conjunctive linear constraints.On the other hand,if the BFM space of a nonlinear constraint is convex,we can equivalently transform it to be a set of disjunctive linear constraints.However,there exist some nonlinear constraints,whose feasible state space and BFM spaces are possible to be non-convex.Hence,this work presents an approach to identify the convexity and concavity of the feasible state space of a nonlinear constraint.If a nonlinear constraint is a continuous function,we can obtain a Hesse matrix.Then,the positivity and negativity of the main sub-form can be used to discriminate the concavity and convexity of the nonlinear constraint.2.Previous work cannot be applied to the case that both feasible state space and BFM spaces of a nonlinear constraint are non-convex.In this case,this thesis focuses on find a set of disjunctive and conjunctive linear constraints.By formulating an integer linear programming problem(ILPP),we can find a convex subset of feasible markings.By using an iterative approaches,we can separate the feasible markings into serval convex subsets.For each convex subset of feasible markings,we can find a set of conjunctive linear constraints.Then,the union of all subsets of feasible markings are the set of all feasible markings.Finally,we can equivalently transform a given nonlinear constraint to be a set of conjunctive and disjunctive constraints,which can deal with the case that both feasible state space and BFM spaces of a nonlinear constraint are non-convex.