Copyright © 2012 Qiang Ling and Haojiang Deng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper studies the second moment stability of a discrete-time jump linear system with real states and the system matrix switching in a Markovian fashion. A sufficient stability condition was proposed by Fang and Loparo (2002), which only needs to check the eigenvalues of a deterministic matrix and is much more computationally efficient than other equivalent conditions. The proof to the necessity of that condition, however, is a challenging problem. In the paper by Costa and Fragoso (2004), a proof was given by extending the state domain to the complex space. This paper proposes an alternative necessity proof, which does not need to extend the state domain. The proof in this paper demonstrates well the essential properties of the Markov jump systems and achieves the desired result in the real state space.