## Abstract and Applied Analysis

### Finite-Time Control for Markovian Jump Systems with Polytopic Uncertain Transition Description and Actuator Saturation

Zhongyi Tang

#### Abstract

The problem of finite-time ${L}_{2}\text{-}{L}_{\infty }$ control for Markovian jump systems (MJS) is investigated. The systems considered time-varying delays, actuator saturation, and polytopic uncertain transition description. The purpose of this paper is to design a state feedback controller such that the system is finite-time bounded (FTB) and a prescribed ${L}_{2}\text{-}{L}_{\infty }$ disturbance attenuation level during a specified time interval is guaranteed. Based on the Lyapunov method, a linear matrix inequality (LMI) optimization problem is formulated to design the delayed feedback controller which satisfies the given attenuation level. Finally, illustrative examples show that the proposed conditions are effective for the design of robust state feedback controller.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 314812, 7 pages.

Dates
First available in Project Euclid: 6 October 2014

https://projecteuclid.org/euclid.aaa/1412605741

Digital Object Identifier
doi:10.1155/2014/314812

Mathematical Reviews number (MathSciNet)
MR3214418

#### Citation

Tang, Zhongyi. Finite-Time Control for Markovian Jump Systems with Polytopic Uncertain Transition Description and Actuator Saturation. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 314812, 7 pages. doi:10.1155/2014/314812. https://projecteuclid.org/euclid.aaa/1412605741

#### References

• H. Shen, S. Xu, J. Lu, and J. Zhou, “Passivity-based control for uncertain stochastic jumping systems with mode-dependent round-trip time delays,” Journal of the Franklin Institute, vol. 349, no. 5, pp. 1665–1680, 2012.
• H. Shen, S. Xu, J. Zhou, and J. Lu, “Fuzzy ${H}_{\infty }$ filtering for nonlinear Markovian jump neutral systems,” International Journal of Systems Science, vol. 42, no. 5, pp. 767–780, 2011.
• Z.-G. Wu, P. Shi, H. Su, and J. Chu, “Asynchronous ${l}_{2}$-${l}_{\infty }$ filtering for discrete-time stochastic Markov jump systems with randomly occurred sensor nonlinearities,” Automatica, vol. 50, no. 1, pp. 180–186, 2014.
• S. He and F. Liu, “Robust peak-to-peak filtering for Markov jump systems,” Signal Processing, vol. 90, no. 2, pp. 513–522, 2010.
• J. Dong and G.-H. Yang, “Robust ${H}_{2}$ control of continuous-time Markov jump linear systems,” Automatica, vol. 44, no. 5, pp. 1431–1436, 2008.
• 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.
• S. Ma and C. Zhang, “${H}_{\infty }$ control for discrete-time singular Markov jump systems subject to actuator saturation,” Journal of the Franklin Institute, vol. 349, no. 3, pp. 1011–1029, 2012.
• H. Liu, E.-K. Boukas, F. Sun, and D. W. C. Ho, “Controller design for Markov jumping systems subject to actuator saturation,” Automatica, vol. 42, no. 3, pp. 459–465, 2006.
• G. V. Kamenkov, “On stability of motion over a finite interval of time,” Journal of Applied Mathematics and Mechanics, vol. 17, pp. 529–540, 1953.
• F. Amato, M. Ariola, and C. Cosentino, “Finite-time stability of linear time-varying systems: analysis and controller design,” IEEE Transactions on Automatic Control, vol. 55, no. 4, pp. 1003–1008, 2010.
• F. Amato, R. Ambrosino, M. Ariola, and C. Cosentino, “Finite-time stability of linear time-varying systems with jumps,” Automatica, vol. 45, no. 5, pp. 1354–1358, 2009.
• Y. Zhang, C. Liu, and H. Sun, “Robust finite-time ${H}_{\infty }$ control for uncertain discrete jump systems with time delay,” Applied Mathematics and Computation, vol. 219, no. 5, pp. 2465–2477, 2012.
• S. He and F. Liu, “Finite-time ${H}_{\infty }$ fuzzy control of nonlinear jump systems with time delays via dynamic observer-based state feedback,” IEEE Transactions on Fuzzy Systems, vol. 20, pp. 605–614, 2012.
• Y. Zhang and C. Liu, “Observer-based finite-time ${H}_{\infty }$ control of discrete-time Markovian jump systems,” Applied Mathematical Modelling, vol. 37, no. 6, pp. 3748–3760, 2013.
• Y. Yang, J. Li, and G. Chen, “Finite-time stability and stabilization of Markovian switching stochastic systems with impulsive effects,” Journal of Systems Engineering and Electronics, vol. 21, no. 2, pp. 254–260, 2010. \endinput