Abstract
The paper is concerned with the problem of H ∞ control for stochastic time-delayed Markovian switching systems with partly known transition rates and input saturation. By employing more appropriate Lyapunov- Krasovskii functional, a state feedback controller is designed to guarantee stochastic stability of the corresponding closed-loop system with H ∞ performance. A linear matrix inequality approach is employed to obtain the controller gain matrix. Two illustrative examples are provided to show the potential of the proposed techniques.
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L. J. Shen and U. Buscher, “Solving the serial batching problem in job shop manufacturing systems,” European Journal of Operational Research, vol. 221, no. 1, pp. 14–26, 2012. [click]
L. G. Wu, P. Shi and H. J. Gao, “State estimation and sliding-mode control of Markovian jump singular sys-tems,” IEEE Transactions on Automatic Control, vol. 55, no. 5, pp. 1213–1219, 2010. [click]
Z. Y. Fei, H. J. Gao, and P. Shi, “New results on stabiliza-tion of Markovian jump systems with time delay,” Auto-matica, vol. 45, no. 10, pp. 2300–2306, 2009. [click]
J. W. Xia, C. Y. Sun, X. Teng, and H. B. Zhang, “Delay-segment-dependent robust stability for uncertain discrete stochastic Markovian jumping systems with interval time delay,” International Journal of Systems Science, vol. 45, no. 3, pp. 271–282, 2014. [click]
X. R. Mao, “Stability of stochastic differential equations with Markovian switching,” Stochastic analysis and appli-cations, vol. 79, no. 1, pp. 45–67, 1999. [click]
J. X. Dong and G. H. Yang, “Robust H2 control of continuous-time Markov jump linear systems,” Automat-ica, vol. 44, no. 5, PP. 1431–1436, 2008. [click]
Z. G. Wu, P. Shi, H. Y. Su, and J. Chi, “Asynchronous l 2 - l ∞ filtering for discrete-time stochastic Markov jump systems with randomly occurred sensor nonlinearities,” Automatica, vol. 50, no. 1, pp. 180–186, 2014. [click]
Z. G. Wu, P. Shi, H. Y. Su, and J. Chu, “Stochastic synchro-nization of Markovian jump neural networks with time-varying delay using sampled-data,” IEEE Transactions on Cybernetics, vol. 43, no. 6, pp. 1796–1806, 2013. [click]
J. X. Dong and G. H. Yang, “Fuzzy controller design for Markovian jump nonlinear systems,” International Journal of Control, Automation, and Systems, vol. 5, no. 6, pp. 712–717, 2007.
M. Q. Shen and D. Ye, “Improved fuzzy control design for nonlinear Markovian-jump systems with incomplete tran-sition descriptions,” Fuzzy Sets and Systems, vol. 217, no. 16, pp. 80–95, 2013. [click]
L. X. Zhang and E. K. Boukas, “Stability and stabilization for Markovian jump linear systems with partly unknown transition probabilities,” Automatica, vol. 45, no. 2, pp. 463–468, 2009. [click]
Y. Zhang, Y. He, M. Wu, and J. Zhang, “Stabilization for Markovian jump systems with partially information on probability based on free-connection weighting matrices,” Automatica, vol. 47, no. 1, pp. 79–84, 2011. [click]
H. Zhang, J. M. Wang, and Y. Shi, “Robust H ∞ sliding-mode control for Markovian jump systems subject to inter-mittent observations and partially known transition proba-bilities,” Systems and Control Letters, vol. 62, no. 12, pp. 1114–1124, 2013. [click]
E. G. Tian, D. Yue, and G. L. Wei, “Robust control for Markovian jump systems with partially known transition probabilities and nonlinearities,” Journal of the Franklin Institute, vol. 350, no. 8, pp. 2069–2083, 2013.
Y. J. Wang, Z. Q. Zuo and Y. L. Cui, “Stochastic stabi-lization of Markovian jump systems with partial unknown transition probabilities and actuator saturation,” Circuits, Systems and Signal Processing, vol. 31, no. 1, pp. 371–383, 2012. [click]
Y. Q. Liu and F. Liu, “Disturbance rejection for Markov jump systems with partly unknown transition probabilities and saturation,” Circuits, Systems and Signal Processing, vol. 32, no. 6, pp. 2783–2797, 2013. [click]
J. J. Zhao, J. Wang, and H. Shen, “Dynamic anti-windup control design for Markovian jump delayed systems with input saturation,” Circuits Systems and Signal Processing vol. 32, no. 5, pp. 2213–2229, 2013. [click]
J. J. Zhao, J. H. Wang, J. H. Park, and H. Shen, “Memory feedback controller design for stochastic Markov jump dis-tributed delay systems with input saturation and partially known transition rates,” Nonlinear Analysis: Hybrid Sys-tems, vol. 15, pp. 52–62, 2015. [click]
D. Yang and J. Zhao, “Robust finite-time output feedback H ∞ control for stochastic jump systems with incomplete transition rates,” Circuits Systems and Signal Processing vol. 34, no. 6, pp. 1799–1824, 2015. [click]
J. P. Richard, “Time-delay systems: An overview of some recent advances and open problems,” Automatica, vol. 39, no. 10, pp. 1667–1694, 2003.
H. Y. Shao, “Delay-dependent approaches to globally ex-ponential stability for recurrent neural networks,” IEEE Transactions on Circuits Systems II, vol. 55 no. 6, pp. 591–595, 2008.
H. Y. Shao, “Delay-dependent stability for recurrent neu-ral networks with time-varying delays,” IEEE Transactions on Neural Networks, vol. 19, no. 9, pp. 1647–1651, 2008. [click]
Z. H. Hu, H. Y. Zhu, and J. M. Zhao, “Further results on H ∞ filtering for a class of discrete-time singular systems with interval time-varying delay,” Circuits, Systems and Signal Processing, vol. 32, no. 3, pp. 1081–1095, 2013.
H. Y. Zhu, Z. Zhang, and S. C. Cui, “Further results on H ∞ control for discrete-time uncertain singular systems with interval time-varying delays in state and input,” Optimal Control Applications and Methods, vol. 34, no. 3, pp. 328–347, 2013. [click]
Q. X. Zhu, “Stabilization of stochastically singular nonlinear jump systems with unknown parameters and continuously distributed delays,” International Journal of Control, Automation and Systems, vol. 11, no. 4, pp. 683–691, 2013.
F. S. Yang and H. G. Zhang, “Delay dependent stability conditions of static recurrent neural networks: a non-linear convex combination method,” IET Control Theory and Applications, vol. 8, no. 14, pp. 1396–1404, 2014. [click]
S. Tarbouriech, G. Garcia, J. M. G. d. S. Jr, I. Queinnec, Stability and Stabilization of Linear Systems with Saturating Actuators, Springer-Verlag London Limited, 2011.
H. K. Khalil, Nonlinear Systems, MacMillan, London, 1992.
G. Grimm, J. Hatfield, I. Postlethwaite, A. R. Teel, M. C. Turner, and L. Zaccarian, “Anti-windup for stable linear systems with input saturation: an LMI based synthesis,” IEEE Transactions on Automatic Control, vol. 48, no. 9, pp. 1509–1525, 2003. [click]
F. Wu, K. M. Grigoriadis, and A. Packard, “Anti-windup controller design using linear parameter-varying control methods,” Internatioanl Journal of Control, vol. 73, no. 12, pp. 1104–1114, 2000. [click]
H. P. Liu, E. K. Boukas, F. C. Sun, and D. W. C. Ho, “Controller design for Markov jump systems subject to actuator saturation,” Automatica, vol. 42, no. 3, pp. 459–465, 2006. [click]
S. P. Huang, Z. R. Xiang, and H. R. Karimi, “Robust L2-gain control for 2D nonlinear stochastic systems with time–varying delays and actuator saturation,” Journal of the Franklin Institute, vol. 350, no. 7, pp. 1865–1885, 2013. [click]
M. S. Zhang, “Robust stabilization for uncertain stochastic multiple time-delay systems with actuator saturation: An LMI Approach,” Procedia Engineering, vol. 29, pp. 935–939, 2012.
L. Xie, X. He, W. D. Zhang, and X. P. Xu, “Robust H ∞ control for uncertain stochastic saturating systems with time delays,” Journal of Systems Engineering and Electronic, vol. 15, no. 4, pp. 563–567, 2004.
G. F. Song, F. Chen, S. Y. Xu, and Y. Zhou, “Disturbance tolerance and rejection of discrete-time stochastic systems with saturating actuators,” Journal of the Franklin Institute, vol. 350, no. 6, pp. 1488–1499, 2013.
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Recommended by Associate Editor Jiuxiang Dong under the direction of Editor PooGyeon Park. This work is supported by Key Program of National Natural Science Foundation of China (61433004).
Xianwen Gao received his B.S. degree from Shenyang University of Chemical Technology in 1978 and M.S. degree from Northeastern University in 1993. In 1998, he received his Ph.D. degree in control theory and control engineering from Northeastern University. He is currently a professor in Northeastern University. His main research interests are modeling of complex industry process and intelligent control, stochastic jump systems, etc.
Lian Lian was born in Dandong, Liaoning Province, P. R. China, in 1981. She received her B.S. degree in telecommunication from Jilin University in 2004 and M.S. degree from Northeastern University in 2009. Now, she is a Ph.D. candidate in Northeastern University, Shen yang, P.R. China. Her research work focus on Markovian systems, stochastic time delay systems, etc.
Wenhai Qi was born in Taian, Shandong Province, P. R. China, in 1986. He received his B.S. degree in automation from Qufu Normal University in 2008 and his M.S. degree from Qufu Normal University in 2013. Now, he is a Ph.D. candidate in Northeastern University,Shenyang, P. R. China. His research work focus on Markovian systems, networked control systems, etc.
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Gao, X., Lian, L. & Qi, W. H ∞ control for sochastic time-delayed Markovian switching systems with partly known transition rates and input saturation. Int. J. Control Autom. Syst. 14, 637–646 (2016). https://doi.org/10.1007/s12555-015-0032-0
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DOI: https://doi.org/10.1007/s12555-015-0032-0