On the index of convergence of a class of Boolean matrices with structural properties

Abstract

ABSTRACT Boolean matrices are of prime importance in the study of discrete event systems (DES), which allow us to model systems across a variety of applications. The index of convergence (i.e. the number of distinct powers of a Boolean matrix) is a crucial characteristic in that it assesses the transient behaviour of the system until reaching a periodic course. In this paper, adopting a graph-theoretic approach, we present bounds for the index of convergence of Boolean matrices for a diverse class of systems, with a certain decomposition. The presented bounds are an extension of the bound on irreducible Boolean matrices, and we provide non-trivial bounds that were unknown for classes of systems. Furthermore, the proposed method is able to determine the bounds in polynomial time. Lastly, we illustrate how the new bounds compare with the previously known bounds and we show their effectiveness in cases such as the benchmark IEEE 5-bus power system.