American Institute of Mathematical Sciences

Advanced
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

Qingling Zhang 1, , Guoliang Wang 2, , Wanquan Liu 3, and  Yi Zhang 1,

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.
Keywords: Impulsive controllers., Time delay systems, Markovian jump systems, Partially unknown transition probabilities.
Mathematics Subject Classification: Primary: 34K20; Secondary: 37A6.
Citation: Qingling Zhang, Guoliang Wang, Wanquan Liu, Yi Zhang. Stabilization of discrete-time Markovian jump systems with partially unknown transition probabilities. Discrete & 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.   Google Scholar

[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.   Google Scholar

[3]

O. Costa, Stability results for discrete-time linear systems with Markovian jumping parameters,, Journal of Mathematical Analysis and Applications, 179 (1993), 154.  doi: 10.1006/jmaa.1993.1341.  Google Scholar

[4]

C. de Souza, Robust stability and stabilization of uncertain discrete-time Markovian jump linear systems,, IEEE Transactions on Automatic Control, 51 (2006), 836.  doi: 10.1109/TAC.2006.875012.  Google Scholar

[5]

Y. Dong and J. Sun, On hybrid control of a class of stochastic non-linear Markovian switching systems,, Automatica, 44 (2008), 990.  doi: 10.1016/j.automatica.2007.08.006.  Google Scholar

[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.  doi: 10.1109/9.618250.  Google Scholar

[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.  doi: 10.1109/TAC.2005.851462.  Google Scholar

[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.  doi: 10.3934/jimo.2010.6.929.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.automatica.2008.04.004.  Google Scholar

[11]

P. Seiler and R. Sengupta, A bounded real lemma for jump systems,, IEEE Transactions on Automatic Control, 48 (2003), 1651.  doi: 10.1109/TAC.2003.817010.  Google Scholar

[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.  doi: 10.1109/9.802932.  Google Scholar

[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.  doi: 10.1109/9.780431.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.3934/jimo.2011.7.677.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.jfranklin.2010.03.007.  Google Scholar

[18]

H. Wu and J. Sun, $p$-Moment stability of stochastic differential equations with impulsvie jump and Markovian switching,, Automatica, 42 (2006), 1753.  doi: 10.1016/j.automatica.2006.05.009.  Google Scholar

[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.  doi: 10.1016/j.automatica.2004.12.001.  Google Scholar

[20]

J. Xiong and J. Lam, Stabilization of discrete-time Markovian jump linear systems via time-delyed controllers,, Automatica, 42 (2006), 747.  doi: 10.1016/j.automatica.2005.12.015.  Google Scholar

[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.  doi: 10.1109/TAC.2003.811277.  Google Scholar

[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.  doi: 10.1109/TCSI.2007.904640.  Google Scholar

[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.  doi: 10.1109/TAC.2007.902748.  Google Scholar

[24]

L. Zhang and E. Boukas, Stability and stabilization of Markovian jump linear systems with partly unknown transition probabilities,, Automatica, 45 (2009), 463.  doi: 10.1016/j.automatica.2008.08.010.  Google Scholar

[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.  doi: 10.1002/rnc.1355.  Google Scholar

show all references

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.   Google Scholar

[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.   Google Scholar

[3]

O. Costa, Stability results for discrete-time linear systems with Markovian jumping parameters,, Journal of Mathematical Analysis and Applications, 179 (1993), 154.  doi: 10.1006/jmaa.1993.1341.  Google Scholar

[4]

C. de Souza, Robust stability and stabilization of uncertain discrete-time Markovian jump linear systems,, IEEE Transactions on Automatic Control, 51 (2006), 836.  doi: 10.1109/TAC.2006.875012.  Google Scholar

[5]

Y. Dong and J. Sun, On hybrid control of a class of stochastic non-linear Markovian switching systems,, Automatica, 44 (2008), 990.  doi: 10.1016/j.automatica.2007.08.006.  Google Scholar

[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.  doi: 10.1109/9.618250.  Google Scholar

[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.  doi: 10.1109/TAC.2005.851462.  Google Scholar

[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.  doi: 10.3934/jimo.2010.6.929.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.automatica.2008.04.004.  Google Scholar

[11]

P. Seiler and R. Sengupta, A bounded real lemma for jump systems,, IEEE Transactions on Automatic Control, 48 (2003), 1651.  doi: 10.1109/TAC.2003.817010.  Google Scholar

[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.  doi: 10.1109/9.802932.  Google Scholar

[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.  doi: 10.1109/9.780431.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.3934/jimo.2011.7.677.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.jfranklin.2010.03.007.  Google Scholar

[18]

H. Wu and J. Sun, $p$-Moment stability of stochastic differential equations with impulsvie jump and Markovian switching,, Automatica, 42 (2006), 1753.  doi: 10.1016/j.automatica.2006.05.009.  Google Scholar

[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.  doi: 10.1016/j.automatica.2004.12.001.  Google Scholar

[20]

J. Xiong and J. Lam, Stabilization of discrete-time Markovian jump linear systems via time-delyed controllers,, Automatica, 42 (2006), 747.  doi: 10.1016/j.automatica.2005.12.015.  Google Scholar

[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.  doi: 10.1109/TAC.2003.811277.  Google Scholar

[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.  doi: 10.1109/TCSI.2007.904640.  Google Scholar

[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.  doi: 10.1109/TAC.2007.902748.  Google Scholar

[24]

L. Zhang and E. Boukas, Stability and stabilization of Markovian jump linear systems with partly unknown transition probabilities,, Automatica, 45 (2009), 463.  doi: 10.1016/j.automatica.2008.08.010.  Google Scholar

[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.  doi: 10.1002/rnc.1355.  Google Scholar

[1]

Hui Lv, Xing'an Wang. Dissipative control for uncertain singular markovian jump systems via hybrid impulsive control. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020020
[2]

M. S. Mahmoud, P. Shi, Y. Shi. $H_\infty$ and robust control of interconnected systems with Markovian jump parameters. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 365-384. doi: 10.3934/dcdsb.2005.5.365
[3]

Jian Chen, Tao Zhang, Ziye Zhang, Chong Lin, Bing Chen. Stability and output feedback control for singular Markovian jump delayed systems. Mathematical Control & Related Fields, 2018, 8 (2) : 475-490. doi: 10.3934/mcrf.2018019
[4]

Guoshan Zhang, Shiwei Wang, Yiming Wang, Wanquan Liu. LS-SVM approximate solution for affine nonlinear systems with partially unknown functions. Journal of Industrial & Management Optimization, 2014, 10 (2) : 621-636. doi: 10.3934/jimo.2014.10.621
[5]

Xiaowei Tang, Xilin Fu. New comparison principle with Razumikhin condition for impulsive infinite delay differential systems. Conference Publications, 2009, 2009 (Special) : 739-743. doi: 10.3934/proc.2009.2009.739
[6]

Xingyue Liang, Jianwei Xia, Guoliang Chen, Huasheng Zhang, Zhen Wang. $ \mathcal{H}_{\infty} $ control for fuzzy markovian jump systems based on sampled-data control method. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020368
[7]

Tianhu Yu, Jinde Cao, Chuangxia Huang. Finite-time cluster synchronization of coupled dynamical systems with impulsive effects. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020248
[8]

Duy Phan, Lassi Paunonen. Finite-dimensional controllers for robust regulation of boundary control systems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020029
[9]

Lizhao Yan, Fei Xu, Yongzeng Lai, Mingyong Lai. Stability strategies of manufacturing-inventory systems with unknown time-varying demand. Journal of Industrial & Management Optimization, 2017, 13 (4) : 2033-2047. doi: 10.3934/jimo.2017030
[10]

Baskar Sundaravadivoo. Controllability analysis of nonlinear fractional order differential systems with state delay and non-instantaneous impulsive effects. Discrete & Continuous Dynamical Systems - S, 2020, 13 (9) : 2561-2573. doi: 10.3934/dcdss.2020138
[11]

Jingang Zhao, Chi Zhang. Finite-horizon optimal control of discrete-time linear systems with completely unknown dynamics using Q-learning. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020030
[12]

Yueyuan Zhang, Yanyan Yin, Fei Liu. Robust observer-based control for discrete-time semi-Markov jump systems with actuator saturation. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020105
[13]

Linna Li, Changjun Yu, Ning Zhang, Yanqin Bai, Zhiyuan Gao. A time-scaling technique for time-delay switched systems. Discrete & Continuous Dynamical Systems - S, 2020, 13 (6) : 1825-1843. doi: 10.3934/dcdss.2020108
[14]

Y. Peng, X. Xiang. Second order nonlinear impulsive time-variant systems with unbounded perturbation and optimal controls. Journal of Industrial & Management Optimization, 2008, 4 (1) : 17-32. doi: 10.3934/jimo.2008.4.17
[15]

Xiongxiong Bao, Wan-Tong Li. Existence and stability of generalized transition waves for time-dependent reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020249
[16]

B. Cantó, C. Coll, A. Herrero, E. Sánchez, N. Thome. Pole-assignment of discrete time-delay systems with symmetries. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 641-649. doi: 10.3934/dcdsb.2006.6.641
[17]

Chongyang Liu, Meijia Han, Zhaohua Gong, Kok Lay Teo. Robust parameter estimation for constrained time-delay systems with inexact measurements. Journal of Industrial & Management Optimization, 2019  doi: 10.3934/jimo.2019113
[18]

Ming He, Xiaoyun Ma, Weijiang Zhang. Oscillation death in systems of oscillators with transferable coupling and time-delay. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 737-745. doi: 10.3934/dcds.2001.7.737
[19]

Akram Kheirabadi, Asadollah Mahmoudzadeh Vaziri, Sohrab Effati. Linear optimal control of time delay systems via Hermite wavelet. Numerical Algebra, Control & Optimization, 2020, 10 (2) : 143-156. doi: 10.3934/naco.2019044
[20]

Xiang Xie, Honglei Xu, Xinming Cheng, Yilun Yu. Improved results on exponential stability of discrete-time switched delay systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (1) : 199-208. doi: 10.3934/dcdsb.2017010

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (33)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences