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STABILITY OF SINGULAR JUMP-LINEAR SYSTEMS WITH A LARGE STATE SPACE: A TWO-TIME-SCALE APPROACH

Part of: Stochastic systems and control Asymptotic theory Markov processes

Published online by Cambridge University Press:  20 March 2012

DUNG TIEN NGUYEN ,
XUERONG MAO ,
G. YIN  and
CHENGGUI YUAN
DUNG TIEN NGUYEN
Affiliation:
Department of Electrical and Computer Engineering, University of British Columbia, Vancouver V6T 1Z4, Canada (email: dungnt@ece.ubc.ca)
XUERONG MAO
Affiliation:
Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH, UK (email: x.mao@strath.ac.uk)
G. YIN*
Affiliation:
Department of Mathematics, Wayne State University, Detroit, Michigan 48202, USA (email: gyin@math.wayne.edu)
CHENGGUI YUAN
Affiliation:
Department of Mathematics, University of Wales Swansea, Swansea SA2 8PP, UK (email: C.Yuan@swansea.ac.uk)
*
✉For correspondence; e-mail: dungnt@wayne.edu
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Abstract

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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.

Keywords

singular systemstabilitytwo-time-scale approach

MSC classification

Secondary: 93E15: Stochastic stability 60J27: Continuous-time Markov processes on discrete state spaces 60J60: Diffusion processes 34E15: Singular perturbations, general theory
Type
Research Article
Information
The ANZIAM Journal , Volume 52 , Issue 4 , April 2011 , pp. 372 - 390
Copyright
Copyright © Australian Mathematical Society 2012

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