Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-13T05:44:49.886Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Asymptotic theory Stochastic systems and control 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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

[1]Boukas, E. K., “Stabilization of stochastic singular nonlinear hybrid systems”, Nonlinear Anal. Theory Methods Appl. 64 (2006) 217–228; doi:10.1016/j.na.2005.05.066.CrossRefGoogle Scholar
[2]Boukas, E. K., Xu, S. and Lam, J., “On stability and stabilizability of singular stochastic systems with delays”, J. Optim. Theory Appl. 127 (2005) 249–262; doi:10.1007/s10957-005-6538-5.Google Scholar
[3]Brunner, A. D., Testing for structural breaks in US post-war inflation data (Board of Governors of the Federal Reserve System, Washington, DC, 1991).Google Scholar
[4]Cai, J., “A Markov model of switching-regime ARCH”, J. Bus. Econom. Statist. 12 (1994) 309–316; doi:10.2307/1392087.Google Scholar
[5]Campbell, S. L., Singular systems of differential equations I (Pitman, San Francisco, 1980).Google Scholar
[6]Campbell, S. L., Singular systems of differential equations II (Pitman, San Francisco, 1982).Google Scholar
[7]Cheng, Z. L., Hong, H. M. and Zhang, J. F., “The optimal regulation of generalized state-space systems with quadratic cost”, Automatica 24 (1988) 707–710; doi:10.1016/0005-1098(88)90120-3.Google Scholar
[8]Dai, L., Singular control systems, Volume 118 of Lecture Notes in Control and Information Sciences (Springer, Berlin, 1989).Google Scholar
[9]Du, Z., Zhang, Q. and Liu, L., “Delay-dependent stabilization of uncertain singular systems with multiple state delays”, Int. J. Innov. Comput. Inform. Control 5 (2009) 1655–1664.Google Scholar
[10]Hale, J. K., Ordinary differential equations, 2nd edn (R. E. Krieger, Malabar, FL, 1980).Google Scholar
[11]Hamilton, J. D. and Susmel, R., “Autoregressive conditional heteroskedasticity and changes in regime”, J. Econometrics 64 (1994) 307–333; doi:10.1016/0304-4076(94)90067-1.CrossRefGoogle Scholar
[12]Hansen, B. E., “The likelihood ratio test under nonstandard conditions: testing the Markov switching model of GNP”, J. Appl. Econometrics 7 (1992) S61–S82; doi:10.1002/jae.3950070506.CrossRefGoogle Scholar
[13]Huang, L. and Mao, X., “Stability of singular stochastic systems with Markovian switching”, IEEE Trans. Automat. Control 56 (2011) 424–429; doi:10.1109/TAC.2010.2088850.CrossRefGoogle Scholar
[14]Lewis, F. L., “A survey of linear singular systems”, Circuits Systems Signal Process. 8 (1986) 3–36; doi:10.1007/BF01600184.CrossRefGoogle Scholar
[15]Wu, Z. G. and Zhou, W. N., “Delay-dependent robust stabilization of uncertain singular systems with state delay”, Acta Automat. Sinica 33 (2007) 714–718; doi:10.1360/aas-007-0714.CrossRefGoogle Scholar
[16]Wu, Z. G., Su, H. Y. and Chu, J., “Delay-dependent robust exponential stability of uncertain singular systems with time delays”, Int. J. Innov. Comput. Inform. Control 6 (2010) 2275–2284.Google Scholar
[17]Yin, G. and Zhang, J. F., “Hybrid singular systems of differential equations”, Sci. China Ser. F: Inform. Sci. 45 (2002) 241–258; doi:10.1360/02yf9022.CrossRefGoogle Scholar
[18]Yin, G. and Zhang, Q., Continuous-time Markov chains and applications: a singular perturbation approach (Springer, New York, 1998).CrossRefGoogle Scholar