Semi-tensor product of matrices approach to stability and stabilization analysis of bounded Petri net systems

Abstract

In this paper, we investigate the problems of stability and stabilization for bounded Petri net systems (BPNSs) by using the semi-tensor product (STP) of matrices. First, a new matrix equation, under the framework of Boolean algebra, is established, and is based on our previous results presented on the marking evolution equation of BPNSs. This equation made it possible to provide the necessary and sufficient condition for the equilibrium point stability of BPNSs. Second, the problem associated with marking feedback stabilizability is solved by introducing the concept and some properties of the marking pre-reachability set of BPNSs, respectively. By resorting to these properties, the necessary and sufficient condition for equilibrium point stabilization is presented. In addition, a design procedure is proposed to find all the optimal marking feedback controllers that implement the minimal length trajectories from each marking to the equilibrium point. The proposed results, in this paper, are based on the matrix form, thus the problems of verifying the stability and stabilization of BPNSs are expressed in the matrix computation. This is very simple and straightforward work by means of the MATLAB toolbox of the STP of matrices. The proposed results are of a very simple form and can conveniently be implemented on a computer. Finally, several examples are presented to illustrate the validity and application of the proposed approaches.