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Stability of singular jump-linear systems with a large state space : a two-time-scale approach

Nguyen, Dung Tien and Mao, Xuerong and Yin, G. and Yuan, Chenggui (2012) Stability of singular jump-linear systems with a large state space : a two-time-scale approach. The Australian and New Zealand Industrial and Applied Mathematics Journal, 52 (4). pp. 372-390.

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Abstract

This paper considers singular systems that involve both continuous dynamics and discrete events with the coefficients being modulated by a continuous-time Markov chain. The underlying systems have two distinct characteristics. First, the systems are singular, that is, characterized by a singular coefficient matrix. Second, the Markov chain of the modulating force has a large state space. We focus on stability of such hybrid singular systems. To carry out the analysis, we use a two-time-scale formulation, which is based on the rationale that, in a large-scale system, not all components or subsystems change at the same speed. To highlight the different rates of variation, we introduce a small parameter ε>0. Under suitable conditions, the system has a limit. We then use a perturbed Lyapunov function argument to show that if the limit system is stable then so is the original system in a suitable sense for ε small enough. This result presents a perspective on reduction of complexity from a stability point of view.