Stability and Stabilization

Abstract

This chapter is concerned with the stability and stabilization problems of a class of continuous-time and discrete-time Markov jump linear system (MJLS) with partially unknown transition probabilities (TPs). It will be proved that the system under consideration is more general, which covers the systems with completely known and completely unknown TPs as two special cases, the latter is hereby the switching linear systems under arbitrary switching. Moreover, in contrast with the uncertain TPs, the concept of partially unknown TPs proposed in this chapter does not require any knowledge of the unknown elements. Firstly, the sufficient conditions for stochastic stability and stabilization of the underlying systems are derived via linear matrix inequality (LMI) formulation, and the relationship between the stability criteria currently obtained for the usual MJLS and switching linear systems under arbitrary switching is exposed by the proposed class of hybrid systems. Further, the necessary and sufficient criteria are obtained by fully considering the properties of the transition rates matrices (TRMs) and transition probabilities matrices (TPMs), and the convexity of the uncertain domains. We will show by comparison the less conservatism of the methodologies for obtaining the necessary and sufficient conditions, but note that in the next chapters of Part I, we prefer the ones in the sufficient stability conditions to carry out other studies. The extensions to less conservative results are relatively straightforward and we leave them to readers who are interested.