# American Institute of Mathematical Sciences

November  2011, 16(4): 1197-1211. doi: 10.3934/dcdsb.2011.16.1197

## Stabilization of discrete-time Markovian jump systems with partially unknown transition probabilities

 1 Institute of Systems Science, Northeastern University, Shenyang, Liaoning, 110819, China, China 2 School of Information and Control Engineering, Liaoning Shihua University, Fushun, Liaoning 113001, China 3 Department of Computing, Curtin University of Technology, Perth, WA 6102, Australia

Received  August 2010 Revised  March 2011 Published  August 2011

In this paper the stabilization problem for a class of discrete-time Markovian jump system with partially unknown transition probabilities is investigated via using the time-delayed and impulsive controllers. As some elements in transition matrix are unknown, a new approach is proposed to estimate the unknown elements, in which an impulsive stabilizing controller depending on time delays and system mode is presented in terms of linear matrix inequalities (LMIs) with equality constraints. Especially, if there are no time delays and impulsive effects in the controller, it is derived that the conditions for the existence of $H_\infty$ controller can be expressed by LMIs without equality constraints. Finally, illustrative examples are presented to show the benefits and the validity of the proposed approaches.
Citation: Qingling Zhang, Guoliang Wang, Wanquan Liu, Yi Zhang. Stabilization of discrete-time Markovian jump systems with partially unknown transition probabilities. Discrete and Continuous Dynamical Systems - B, 2011, 16 (4) : 1197-1211. doi: 10.3934/dcdsb.2011.16.1197
##### References:
 [1] E. Boukas and Z. Liu, Robust $H_\infty$ control of discrete-time Markovian jump linear systems with mode-dependent time-delays, IEEE Transactions on Automatic Control, 46 (2001), 1918-1924. [2] W. Chen, Z. Guan and P. Yu, Delay-dependent stability and $H_\infty$ control of uncertain discrete-time Markovian jump systems with mode-dependent time delays, Systems and Control Letters, 52 (2004), 361-376. [3] O. Costa, Stability results for discrete-time linear systems with Markovian jumping parameters, Journal of Mathematical Analysis and Applications, 179 (1993), 154-178. doi: 10.1006/jmaa.1993.1341. [4] C. de Souza, Robust stability and stabilization of uncertain discrete-time Markovian jump linear systems, IEEE Transactions on Automatic Control, 51 (2006), 836-841. doi: 10.1109/TAC.2006.875012. [5] Y. Dong and J. Sun, On hybrid control of a class of stochastic non-linear Markovian switching systems, Automatica, 44 (2008), 990-995. doi: 10.1016/j.automatica.2007.08.006. [6] L. Ghaoui, F. Oustry and M. AitRami, A cone complementarity linearization algorithm for static output-feedback and related problems, IEEE Transactions on Automatic Control, 42 (1997), 1171-1176. doi: 10.1109/9.618250. [7] Z. Guan, D. Hill and X. Shen, On hybrid impulsive and switching systems and application to nonlinear control, IEEE Transactions on Automatic Control, 50 (2005), 1058-1062. doi: 10.1109/TAC.2005.851462. [8] P. V. Laxmi and O. M. Yesuf, Analysis of a finite buffer general input queue with Markovian service process and accessible and non-accessible batch service, Journal of Industrial and Management Optimization, 6 (2010), 929-944. doi: 10.3934/jimo.2010.6.929. [9] F. Leibfritz, An LMI-based algorithm for designing suboptimal static $H_2$/$H_\infty$ output feedback controllers, SIAM Journal on Control and Optimization, 39 (2001), 1171-1735. [10] S. Pan, J. Sun and S. Zhao, Stabilization of discrete-time Markovian jump linear systems via time-delayed and impulsive controllers, Automatica, 44 (2008), 2954-2958. doi: 10.1016/j.automatica.2008.04.004. [11] P. Seiler and R. Sengupta, A bounded real lemma for jump systems, IEEE Transactions on Automatic Control, 48 (2003), 1651-1654. doi: 10.1109/TAC.2003.817010. [12] P. Shi, E. Boukas and R. Agarwal, Control of Markovian jump discrete-time systems with norm bounded uncertainty and unknown delay, IEEE Transactions on Automatic Control, 44 (1999), 2139-2144. doi: 10.1109/9.802932. [13] P. Shi, E. Boukas and R. Agarwal, Kalman filtering for continuous-time uncertain systems with Markovian jumping parameters, IEEE Transactions on Automatic Control, 44 (1999), 1592-1597. doi: 10.1109/9.780431. [14] Y. Shi, P. Shi and M. S. Mahmoud, $H_\infty$ and robust control of interconnected systems with Markovian jump parameters, Discrete and Continuous Dynamical Systems-Series B, 5 (2005), 365-384. [15] M. Telek and Z. Saffer, Analysis of globally gated Markovian limited cyclic polling model and its application to uplink traffic in the IEEE 802.16 network, Journal of Industrial and Management Optimization, 7 (2011), 677-697. doi: 10.3934/jimo.2011.7.677. [16] G. Wang, Q. Zhang and V. Sreeram, Robust delay-range-dependent stabilization for Markovian jump systems with mode-dependent time delays and nonlinearities, Optimal Control Applications and Methods, 31 (2010), 249-264. [17] G. Wang, Q. Zhang and V. Sreeram, $H_\infty$ control for discrete-time singularly perturbed systems with two Markov processes, Journal of The Franklin Institute, 347 (2010), 836-847. doi: 10.1016/j.jfranklin.2010.03.007. [18] H. Wu and J. Sun, $p$-Moment stability of stochastic differential equations with impulsvie jump and Markovian switching, Automatica, 42 (2006), 1753-1759. doi: 10.1016/j.automatica.2006.05.009. [19] J. Xiong, J. Lam, H. Gao and D. Ho, On robust stabilization of Markovian jump systems with uncertain switching probabilities, Automatica, 41 (2005), 897-903. doi: 10.1016/j.automatica.2004.12.001. [20] J. Xiong and J. Lam, Stabilization of discrete-time Markovian jump linear systems via time-delyed controllers, Automatica, 42 (2006), 747-753. doi: 10.1016/j.automatica.2005.12.015. [21] S. Xu, T. Chen and J. Lam, Robust $H_\infty$ filtering for uncertain Markovian jump systems with mode-dependent time delays, IEEE Transactions on Automatic Control, 48 (2003), 900-907. doi: 10.1109/TAC.2003.811277. [22] S. Xu and J. Lam, Delay-dependent $H_\infty$ control and filtering for uncertain Markovian jump systems with time-varying delays, IEEE Transactions on Circuits and Systems, 54 (2007), 2070-2077. doi: 10.1109/TCSI.2007.904640. [23] Z. Yang and D. Xu, Stability analysis and design of impulsive control systems with time delay, IEEE Transactions on Automatic Control, 52 (2007), 1448-1454. doi: 10.1109/TAC.2007.902748. [24] L. Zhang and E. Boukas, Stability and stabilization of Markovian jump linear systems with partly unknown transition probabilities, Automatica, 45 (2009), 463-468. doi: 10.1016/j.automatica.2008.08.010. [25] L. Zhang and E. Boukas, $H_\infty$ control for discrete-time Markovian jump linear systems with partly unknown transition probabilities, International Journal of Robust and Nonlinear Control, 19 (2009), 868-883. doi: 10.1002/rnc.1355.

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##### References:
 [1] E. Boukas and Z. Liu, Robust $H_\infty$ control of discrete-time Markovian jump linear systems with mode-dependent time-delays, IEEE Transactions on Automatic Control, 46 (2001), 1918-1924. [2] W. Chen, Z. Guan and P. Yu, Delay-dependent stability and $H_\infty$ control of uncertain discrete-time Markovian jump systems with mode-dependent time delays, Systems and Control Letters, 52 (2004), 361-376. [3] O. Costa, Stability results for discrete-time linear systems with Markovian jumping parameters, Journal of Mathematical Analysis and Applications, 179 (1993), 154-178. doi: 10.1006/jmaa.1993.1341. [4] C. de Souza, Robust stability and stabilization of uncertain discrete-time Markovian jump linear systems, IEEE Transactions on Automatic Control, 51 (2006), 836-841. doi: 10.1109/TAC.2006.875012. [5] Y. Dong and J. Sun, On hybrid control of a class of stochastic non-linear Markovian switching systems, Automatica, 44 (2008), 990-995. doi: 10.1016/j.automatica.2007.08.006. [6] L. Ghaoui, F. Oustry and M. AitRami, A cone complementarity linearization algorithm for static output-feedback and related problems, IEEE Transactions on Automatic Control, 42 (1997), 1171-1176. doi: 10.1109/9.618250. [7] Z. Guan, D. Hill and X. Shen, On hybrid impulsive and switching systems and application to nonlinear control, IEEE Transactions on Automatic Control, 50 (2005), 1058-1062. doi: 10.1109/TAC.2005.851462. [8] P. V. Laxmi and O. M. Yesuf, Analysis of a finite buffer general input queue with Markovian service process and accessible and non-accessible batch service, Journal of Industrial and Management Optimization, 6 (2010), 929-944. doi: 10.3934/jimo.2010.6.929. [9] F. Leibfritz, An LMI-based algorithm for designing suboptimal static $H_2$/$H_\infty$ output feedback controllers, SIAM Journal on Control and Optimization, 39 (2001), 1171-1735. [10] S. Pan, J. Sun and S. Zhao, Stabilization of discrete-time Markovian jump linear systems via time-delayed and impulsive controllers, Automatica, 44 (2008), 2954-2958. doi: 10.1016/j.automatica.2008.04.004. [11] P. Seiler and R. Sengupta, A bounded real lemma for jump systems, IEEE Transactions on Automatic Control, 48 (2003), 1651-1654. doi: 10.1109/TAC.2003.817010. [12] P. Shi, E. Boukas and R. Agarwal, Control of Markovian jump discrete-time systems with norm bounded uncertainty and unknown delay, IEEE Transactions on Automatic Control, 44 (1999), 2139-2144. doi: 10.1109/9.802932. [13] P. Shi, E. Boukas and R. Agarwal, Kalman filtering for continuous-time uncertain systems with Markovian jumping parameters, IEEE Transactions on Automatic Control, 44 (1999), 1592-1597. doi: 10.1109/9.780431. [14] Y. Shi, P. Shi and M. S. Mahmoud, $H_\infty$ and robust control of interconnected systems with Markovian jump parameters, Discrete and Continuous Dynamical Systems-Series B, 5 (2005), 365-384. [15] M. Telek and Z. Saffer, Analysis of globally gated Markovian limited cyclic polling model and its application to uplink traffic in the IEEE 802.16 network, Journal of Industrial and Management Optimization, 7 (2011), 677-697. doi: 10.3934/jimo.2011.7.677. [16] G. Wang, Q. Zhang and V. Sreeram, Robust delay-range-dependent stabilization for Markovian jump systems with mode-dependent time delays and nonlinearities, Optimal Control Applications and Methods, 31 (2010), 249-264. [17] G. Wang, Q. Zhang and V. Sreeram, $H_\infty$ control for discrete-time singularly perturbed systems with two Markov processes, Journal of The Franklin Institute, 347 (2010), 836-847. doi: 10.1016/j.jfranklin.2010.03.007. [18] H. Wu and J. Sun, $p$-Moment stability of stochastic differential equations with impulsvie jump and Markovian switching, Automatica, 42 (2006), 1753-1759. doi: 10.1016/j.automatica.2006.05.009. [19] J. Xiong, J. Lam, H. Gao and D. Ho, On robust stabilization of Markovian jump systems with uncertain switching probabilities, Automatica, 41 (2005), 897-903. doi: 10.1016/j.automatica.2004.12.001. [20] J. Xiong and J. Lam, Stabilization of discrete-time Markovian jump linear systems via time-delyed controllers, Automatica, 42 (2006), 747-753. doi: 10.1016/j.automatica.2005.12.015. [21] S. Xu, T. Chen and J. Lam, Robust $H_\infty$ filtering for uncertain Markovian jump systems with mode-dependent time delays, IEEE Transactions on Automatic Control, 48 (2003), 900-907. doi: 10.1109/TAC.2003.811277. [22] S. Xu and J. Lam, Delay-dependent $H_\infty$ control and filtering for uncertain Markovian jump systems with time-varying delays, IEEE Transactions on Circuits and Systems, 54 (2007), 2070-2077. doi: 10.1109/TCSI.2007.904640. [23] Z. Yang and D. Xu, Stability analysis and design of impulsive control systems with time delay, IEEE Transactions on Automatic Control, 52 (2007), 1448-1454. doi: 10.1109/TAC.2007.902748. [24] L. Zhang and E. Boukas, Stability and stabilization of Markovian jump linear systems with partly unknown transition probabilities, Automatica, 45 (2009), 463-468. doi: 10.1016/j.automatica.2008.08.010. [25] L. Zhang and E. Boukas, $H_\infty$ control for discrete-time Markovian jump linear systems with partly unknown transition probabilities, International Journal of Robust and Nonlinear Control, 19 (2009), 868-883. doi: 10.1002/rnc.1355.
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