## Abstract and Applied Analysis

### Quantized State-Feedback Stabilization for Delayed Markovian Jump Linear Systems with Generally Incomplete Transition Rates

#### Abstract

This paper is concerned with the robust quantized state-feedback controller design problem for a class of continuous-time Markovian jump linear uncertain systems with general uncertain transition rates and input quantization. The uncertainties under consideration emerge in both system parameters and mode transition rates. This new uncertain model is more general than the existing ones and can be applicable to more practical situations because each transition rate can be completely unknown or only its estimate value is known. Based on linear matrix inequalities, the quantized state-feedback controller is formulated to ensure the closed-loop system is stable in mean square. Finally, a numerical example is presented to verify the validity of the developed theoretical results.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 961925, 9 pages.

Dates
First available in Project Euclid: 2 October 2014

https://projecteuclid.org/euclid.aaa/1412278127

Digital Object Identifier
doi:10.1155/2014/961925

Mathematical Reviews number (MathSciNet)
MR3216084

Zentralblatt MATH identifier
07023406

#### Citation

Li, Yanbo; Zhang, Peng; Kao, Yonggui; Karimi, Hamid Reza. Quantized State-Feedback Stabilization for Delayed Markovian Jump Linear Systems with Generally Incomplete Transition Rates. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 961925, 9 pages. doi:10.1155/2014/961925. https://projecteuclid.org/euclid.aaa/1412278127

#### References

• X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, 2006.
• H. Zhang, Y. Shi, and J. M. Wang, “On energy-to-peak filtering for nonuniformly sampled nonlinear sampled nonlinear systems: a Markovian jump system approach,” IEEE Transactions on Fuzzy Systems, vol. 22, pp. 212–222, 2014.
• Z. Wu, P. Shi, H. Su, and J. Chu, “Stochastic synchronization of Markovian jump neural networks with time-varying delay using sampled-data,” IEEE Transactions on Cybernetics, vol. 43, pp. 1796–1806, 2013.
• Z. D. Wang, Y. R. Liu, L. Yu, and X. H. Liu, “Exponential stability of delayed recurrent neural networks with Markovian jumping parameters,” Physics Letters A, vol. 356, pp. 346–352, 2006.
• Z.-G. Wu, J. H. Park, H. Su, and J. Chu, “Stochastic stability analysis for discrete-time singular Markov jump systems with time-varying delay and piecewise-constant transition probabilities,” Journal of the Franklin Institute. Engineering and Applied Mathematics, vol. 349, no. 9, pp. 2889–2902, 2012.
• Z. Wu, H. Su, and J. Chu, “${H}_{\infty }$ model reduction for discrete singular Markovian jump systems,” Proceedings of the Institution of Mechanical Engineers, vol. 223, no. 7, pp. 1017–1025, 2009.
• M. Liu, P. Shi, L. Zhang, and X. Zhao, “Fault-tolerant control for nonlinear Markovian jump systems via proportional and derivative sliding mode observer technique,” IEEE Transactions on Circuits and Systems I, vol. 58, no. 11, pp. 2755–2764, 2011.
• Y. Kao, C. Wang, and L. Zhang, “Delay-dependent exponential stability of impulsive Markovian jumping cohen-grossberg neural networks with reaction-diffusion and mixed delays,” Neural Processing Letters, vol. 38, no. 3, pp. 321–346, 2013.
• Y.-G. Kao, J.-F. Guo, C.-H. Wang, and X.-Q. Sun, “Delay-dependent robust exponential stability of Markovian jumping reaction-diffusion Cohen-Grossberg neural networks with mixed delays,” Journal of the Franklin Institute. Engineering and Applied Mathematics, vol. 349, no. 6, pp. 1972–1988, 2012.
• Y. Kao, C. Wang, F. Zha, and H. Cao, “Stability in mean of partial variables for stochastic reaction-diffusion systems with Markovian switching,” Journal of the Franklin Institute. Engineering and Applied Mathematics, vol. 351, no. 1, pp. 500–512, 2014.
• H. Zhang, Y. Shi, and J. Wang, “Observer-based tracking controller design for networked predictive control systems with uncertain Markov delays,” International Journal of Control, vol. 86, no. 10, pp. 1824–1836, 2013.
• M. Karan, P. Shi, and C. Y. Kaya, “Transition probability bounds for the stochastic stability robustness of continuous- and discrete-time Markovian jump linear systems,” Automatica, vol. 42, no. 12, pp. 2159–2168, 2006.
• J. Xiong and J. Lam, “Robust ${H}_{2}$ control of Markovian jump systems with uncertain switching probabilities,” International Journal of Systems Science. Principles and Applications of Systems and Integration, vol. 40, no. 3, pp. 255–265, 2009.
• J. Xiong, J. Lam, H. Gao, and D. W. C. Ho, “On robust stabilization of Markovian jump systems with uncertain switching probabilities,” Automatica, vol. 41, no. 5, pp. 897–903, 2005.
• L. Zhang and E.-K. Boukas, “Stability and stabilization of Markovian jump linear systems with partly unknown transition probabilities,” Automatica, vol. 45, no. 2, pp. 463–468, 2009.
• L. Zhang and E.-K. Boukas, “Mode-dependent ${H}_{\infty }$ filtering for discrete-time Markovian jump linear systems with partly unknown transition probabilities,” Automatica, vol. 45, no. 6, pp. 1462–1467, 2009.
• L. Zhang and E.-K. Boukas, “${H}_{\infty }$ control for discrete-time Markovian jump linear systems with partly unknown transition probabilities,” International Journal of Robust and Nonlinear Control, vol. 19, no. 8, pp. 868–883, 2009.
• L. Zhang, E.-K. Boukas, and J. Lam, “Analysis and synthesis of Markov jump linear systems with time-varying delays and partially known transition probabilities,” IEEE Transactions on Automatic Control, vol. 53, no. 10, pp. 2458–2464, 2008.
• H. Zhang, J. Wang, and Y. Shi, “Robust ${H}_{\infty }$ sliding-mode control for Markovian jump systems subject to intermittent observations and partially known transition probabilities,” Systems & Control Letters, vol. 62, no. 12, pp. 1114–1124, 2013.
• Y. Zhang, Y. He, M. Wu, and J. Zhang, “Stabilization for Markovian jump systems with partial information on transition probability based on free-connection weighting matrices,” Automatica, vol. 47, no. 1, pp. 79–84, 2011.
• L. Xiong, J. Tian, and X. Liu, “Stability analysis for neutral Markovian jump systems with partially unknown transition probabilities,” Journal of the Franklin Institute. Engineering and Applied Mathematics, vol. 349, no. 6, pp. 2193–2214, 2012.
• Y. Kao, W. Changhong, X. Jing, and K. Hamid Reza, “Stabilisation of singular Markovian jump systems with generally uncertain transition rates,” IEEE Transactions on Automatic Control, 2014.
• Y. Guo and Z. Wang, “Stability of Markovian jump systems with generally uncertain transition rates,” Journal of the Franklin Institute. Engineering and Applied Mathematics, vol. 350, no. 9, pp. 2826–2836, 2013.
• B.-C. Zheng and G.-H. Yang, “Quantised feedback stabilisation of planar systems via switching-based sliding-mode control,” IET Control Theory & Applications, vol. 6, no. 1, pp. 149–156, 2012.
• D. Liberzon, “Hybrid feedback stabilization of systems with quantized signals,” Automatica, vol. 39, no. 9, pp. 1543–1554, 2003.
• R. W. Brockett and D. Liberzon, “Quantized feedback stabilization of linear systems,” IEEE Transactions on Automatic Control, vol. 45, no. 7, pp. 1279–1289, 2000.
• B.-C. Zheng and G.-H. Yang, “Decentralized sliding mode quantized feedback control for a class of uncertain large-scale systems with dead-zone input,” Nonlinear Dynamics, vol. 71, no. 3, pp. 417–427, 2013.
• M. Fu and L. Xie, “The sector bound approach to quantized feedback control,” IEEE Transactions on Automatic Control, vol. 50, no. 11, pp. 1698–1711, 2005.
• M. Fu and L. Xie, “Quantized feedback control for linear uncertain systems,” International Journal of Robust and Nonlinear Control, vol. 20, no. 8, pp. 843–857, 2010.
• E. Tian, D. Yue, and X. Zhao, “Quantised control design for networked systems,” IET Control Theory and Applications, vol. 1, no. 6, pp. 1693–1699, 2007.
• S. W. Yun, Y. J. Choi, and P. Park, “${H}_{2}$ control of continuous-time uncertain linear systems with input quantization and matched disturbances,” Automatica, vol. 45, no. 10, pp. 2435–2439, 2009.
• W.-W. Che and G.-H. Yang, “Quantised ${H}_{\infty }$ filter design for discrete-time systems,” International Journal of Control, vol. 82, no. 2, pp. 195–206, 2009.
• C. Peng and Y.-C. Tian, “Networked ${H}_{\infty }$ control of linear systems with state quantization,” Information Sciences, vol. 177, no. 24, pp. 5763–5774, 2007.
• B.-C. Zheng and G.-H. Yang, “Quantized output feedback stabilization of uncertain systems with input nonlinearities via sliding mode control,” International Journal of Robust and Nonlinear Control, vol. 24, no. 2, pp. 228–246, 2014.
• B.-C. Zheng and G.-H. Yang, “Robust quantized feedback stabilization of linear systems based on sliding mode control,” Optimal Control Applications & Methods, vol. 34, no. 4, pp. 458–471, 2013.
• B. C. Zheng and G. H. Yang, “${H}_{2}$ control of linear uncertain systems considering input quantization with encoder/decoder mismatch,” ISA Transactions, vol. 52, pp. 557–582, 2013.
• N. Xiao, L. Xie, and M. Fu, “Stabilization of Markov jump linear systems using quantized state feedback,” Automatica, vol. 46, no. 10, pp. 1696–1702, 2010.
• D. Ye, B. C. Zheng, and G. H. Yang, “${H}_{2}$ control of Markov jump linear systems with unknown transition rates and input quantization,” in Proceedings of the 31st Chinese Control Conference, pp. 2673–2678, 2012.
• X.-G. Zhao, D. Ye, and J.-D. Han, “${H}_{2}$ control for the continuous-time Markovian jump linear uncertain systems with partly known transition rates and input quantization,” Mathematical Problems in Engineering, vol. 2013, Article ID 426271, 9 pages, 2013.
• S. Yin, H. Luo, and S. Ding, “Real-time implementation of fault-tolerant control systems with performance optimization,” IEEE Transactions on Industrial Electronics, vol. 64, no. 5, pp. 2402–2411, 2014.
• S. Yin, G. Wang, and H. Karimi, “Data-driven design of robust fault detection system for wind turbines,” Mechatronics, 2013.
• S. Yin, S. X. Ding, A. H. A. Sari, and H. Hao, “Data-driven monitoring for stochastic systems and its application on batch process,” International Journal of Systems Science. Principles and Applications of Systems and Integration, vol. 44, no. 7, pp. 1366–1376, 2013.
• S. Yin, S. Ding, A. Haghani, H. Hao, and P. Zhang, “A comparison study of basic datadriven fault diagnosis and process monitoring methods on the benchmark Tennessee Eastman process,” Journal of Process Control, vol. 22, no. 9, pp. 1567–1581, 2012.
• H. Zhang, X. J. Zhang, and J. M. Wang, “Robust gain-scheduling energy-to-peak control of vehicle lateral dynamics stabilization,” Vehicle System Dynamics, vol. 52, pp. 309–340, 2014.
• H. Zhang, Y. Shi, and B. X. Mu, “Optimal ${H}_{\infty }$-based linear-quadratic regulator tracking control for discrete-time Takagi-Sugeno fuzzy systems with preview actions,” Journal of Dynamic Systems, Measurement and Control, vol. 135, 5 pages, 2013.
• H. Zhang, Y. Shi, and M. X. Liu, “${H}_{\infty }$ step tracking control for networked discrete-time nonlinear systems with integral and predictive actions,” IEEE Transactions on Industrial Informatics, no. 9, pp. 337–345, 2013. \endinput