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

Dung Tien Nguyen, Xuerong Mao, George Yin, Chenggui Yuan

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 $\epsilon>0$. 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 $\epsilon$ small enough. This result presents a perspective on reduction of complexity from a stability point of view.

Keywords


singular system; stability; two-time-scale approach



DOI: http://dx.doi.org/10.21914/anziamj.v52i0.3665



Remember, for most actions you have to record/upload into this online system
and then inform the editor/author via clicking on an email icon or Completion button.
ANZIAM Journal, ISSN 1446-8735, copyright Australian Mathematical Society.