Stability and performance analysis of stochastic singular jump linear systems

Abstract

Mean square stability, Stochastic stability and Exponential stability are three types of stability concepts associated with discrete-time Markov jump linear systems, that is, dynamical systems that model phenomena whose dynamic is subject to abrupt changes that are modeled by discrete-time Markov chains. A linear dynamical system can be represented by a four-tuple (S,A,B,C), where matrix S is associated with the left hand side of the sate-space of the system; matrix A, with the right hand side state-space; matrix B, with the input signal and matrix C, with the output or response of the system. Jump linear dynamical systems are those in which the four-tuple (S,A,B,C) is indexed by a Markov chain thereby determining the change in the parameters of the system. When S is a non singular matrix, the system is called a nonsingular jump linear system. In this case, it is known that under certain conditions the class of stabilities referred above are equivalent. For the singular case, that is, when the matrix S is singular, a first attempt to establish conditions under which this equivalence holds has been presented recently. One of the main goals of this research is to extend this result for singular systems that are not necessarily driven by Markov chains, but for any finite-state discrete-time stochastic process. (AU)