Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-13T06:02:10.270Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic stability in the distribution of nonlinear stochastic systems with semi-Markovian switching

Published online by Cambridge University Press:  17 February 2009

Zhenting Hou ,
Hailing Dong  and
Peng Shi
Zhenting Hou
Affiliation:
School of Mathematics Central South University, Changsha 410075 Hunan China; email: hailing_f1y@mail.csu.edu.cn.
Hailing Dong
Affiliation:
School of Mathematics Central South University, Changsha 410075 Hunan China; email: hailing_f1y@mail.csu.edu.cn.
Peng Shi
Affiliation:
Faculty of Advanced Technology University of Glamorgan, Pontypridd CF37 1DL UK
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.

In this paper, finite phase semi-Markov processes are introduced. By introducing variables and a simple transformation, every finite phase semi-Markov process can be transformed to a finite Markov chain which is called its associated Markov chain. A consequence of this is that every phase semi-Markovian switching system may be equivalently expressed as its associated Markovian switching system. Existing results for Markovian switching systems may then be applied to analyze phase semi-Markovian switching systems. In the following, we obtain asymptotic stability for the distribution of nonlinear stochastic systems with semi-Markovian switching. The results can also be extended to general semi-Markovian switching systems. Finally, an example is given to illustrate the feasibility and effectiveness of the theoretical results obtained.

Keywords

phase distributionsemi-Markovian switchingassociated Markov chainasymptotic stability in distribution.
Type
Articles
Information
The ANZIAM Journal , Volume 49 , Issue 2 , October 2007 , pp. 231 - 241
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Costa, O. L. V. and Fragoso, M. D., “Stability results for discrete-time linear systems with Markovian jumping parameters”, J Math Anal Appl. 179 (1993) 154–178.CrossRefGoogle Scholar
[2]Davis, MMarkov models and optimization (Chapman and Hall, London, 1992).Google Scholar
[3]Elliott, R. J. and Sworder, D. D., “Control of a hybrid conditionally linear Gaussian processes”, J Optim Theory Appl. 74 (1992) 75–85.Google Scholar
[4]Feng, XLoparo, K. A., Ji, Y and Chizeck, H. J., “Stochastic stability properties of jump linear systems”, IEEE Trans Automat Control 37 (1992) 38–53.CrossRefGoogle Scholar
[5]Fleming, WSethi, S and Soner, M “An optimal stochastic production planning problem with randomly fluctuating demand”, SIAMJ Control Optim. 25 (1987) 1494–1502.CrossRefGoogle Scholar
[6]Jensen, AA distribution model applicable to economics (Munksgaard, Copenhagen, 1954).Google Scholar
[7]Ji, Y and Chizeck, H. J., “Controllability, stabilizability and continuous-time Markovian jump linear-quadratic control”, IEEE Trans Automat Control 35 (1990) 777–788.CrossRefGoogle Scholar
[8]Ji, Y and Chizeck, H. J., “Jump linear quadratic Gaussian control: steady-state solution and testable conditions”, Control Theory Adv Tech. 6 (1990) 289–319.Google Scholar
[9]Ji, YChizeck, H. J., Feng, X and Loparo, K. A., “Stability and control of discrete time jump linear systems”, Control Theory and Advanced Technology 7 (1991) 247–270.Google Scholar
[10]Kushner, HStochastic stability and control (Academic Press, New York-London, 1967).Google Scholar
[11]Neuts, M. F., “Probability distributions of phase type”, in Liber Amicorum Prof Belgium Univ of Louvain, (1975)173–206.Google Scholar
[12]Neuts, M. F., Matrix-geometric solutions in stochastic models (John Hopkins Univ. Press, Baltimore, Md., 1981).Google Scholar
[13]Shi, P and Boukas, E. K., “H∞, control for Markovian jumping linear systems with parametric uncertainty”, J Optim Theory Appl. 95 (1997) 75–99.Google Scholar
[14]Shi, PBoukas, E. K. and Agarwal, R. K., “Control of Markovian jump discrete-time systems with norm bounded uncertainty and unknown delays”, IEEE Trans Automat Control 44 (1999) 2139–2144.Google Scholar
[15]Shi, PBoukas, E. K. and Agarwal, R. K., “Kalman filtering for continuous-time uncertain systems with Markovian jumping parameters”, lEEETrans Automat Controls (1999) 1592–1597.Google Scholar
[16]Shi, PBoukas, E. K., Nguang, S. K. and Guo, X “Robust disturbance attenuation for discrete-time active fault tolerant control systems with uncertainties”, Optimal Control Appl Methods 24 (2003) 85–101.CrossRefGoogle Scholar
[17]Skorohod, A. V., Asymptotic methods in the theory of stochastic differential equations (American Mathematical Society, Providence, RI, 1989).Google Scholar
[18]Yuan, C and Mao, X “Asymptotic stability in distribution of stochastic differential equations with Markovian switching”, Stochastic Process Appl. 103 (2003) 277–291.Google Scholar
[19]Yuan, CZhou, J and Mao, X “Stability in distribution of stochastic differential delay equations with Markovian switching”, System Control Lett. 50 (2003) 277–291.Google Scholar