## Abstract and Applied Analysis

### Finite-Time ${H}_{\infty }$ Control for Discrete-Time Markov Jump Systems with Actuator Saturation

#### Abstract

This paper investigates the finite-time control problem for discrete-time Markov jump systems subject to saturating actuators. A finite-state Markovian process is given to govern the transition of the jumping parameters. The finite-time ${H}_{\infty }$ controller via state feedback is designed to guarantee that the resulting system is mean-square locally asymptotically finite-time stabilizable. Based on stochastic finite-time stability analysis, sufficient conditions that ensure stochastic control performance of discrete-time Markov jump systems are derived in the form of linear matrix inequalities. Finally, a numerical example is provided to illustrate the effectiveness of the proposed approach.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 182613, 7 pages.

Dates
First available in Project Euclid: 6 October 2014

https://projecteuclid.org/euclid.aaa/1412605755

Digital Object Identifier
doi:10.1155/2014/182613

Mathematical Reviews number (MathSciNet)
MR3198157

Zentralblatt MATH identifier
07021885

#### Citation

Li, Bo; Zhao, Junjie. Finite-Time ${H}_{\infty }$ Control for Discrete-Time Markov Jump Systems with Actuator Saturation. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 182613, 7 pages. doi:10.1155/2014/182613. https://projecteuclid.org/euclid.aaa/1412605755

#### References

• Y. Hong, J. Wang, and D. Cheng, “Adaptive finite-time control of nonlinear systems with parametric uncertainty,” IEEE Transactions on Automatic Control, vol. 51, no. 5, pp. 858–862, 2006.
• S. He and F. Liu, “Finite-time ${H}_{\infty }$ filtering of time-delay stochastic jump systems with unbiased estimation,” Journal of Systems and Control Engineering, vol. 224, no. 8, pp. 947–959, 2010.
• S. He and F. Liu, “Observer-based finite-time control of time-delayed jump systems,” Applied Mathematics and Computation, vol. 217, no. 6, pp. 2327–2338, 2010.
• S. Li, H. Liu, and S. Ding, “A speed control for a PMSM using finite-time feedback control and disturbance compensation,” Transactions of the Institute of Measurement and Control, vol. 32, no. 2, pp. 170–187, 2010.
• H. Song, L. Yu, D. Zhang, and W.-A. Zhang, “Finite-time ${H}_{\infty }$ control for a class of discrete-time switched time-delay systems with quantized feedback,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 12, pp. 4802–4814, 2012.
• Q. Lan, S. Li, J. Yang, and L. Guo, “Finite-time soft landing on asteroids using nonsingular terminal sliding mode control,” Transactions of the Institute of Measurement and Control, 2013.
• Y. Hong, Y. Xu, and J. Huang, “Finite-time control for robot manipulators,” Systems & Control Letters, vol. 46, no. 4, pp. 243–253, 2002.
• S. He and F. Liu, “Finite-time ${H}_{\infty }$ fuzzy control of nonlinear jump system with time delays via dynamic observer-based state feedback,” IEEE Transactions on Fuzzy Systems, vol. 20, no. 4, pp. 605–614, 2012.
• Y.-Y. Cao and Z. Lin, “Stability analysis of discrete-time systems with actuator saturation by a saturation-dependent Lyapunov function,” Automatica, vol. 39, no. 7, pp. 1235–1241, 2003.
• M. L. Corradini and G. Orlando, “Linear unstable plants with saturating actuators: robust stabilization by a time varying sliding surface,” Automatica, vol. 43, no. 1, pp. 88–94, 2007.
• 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.
• X. Song, J. Lu, S. Xu, H. Shen, and J. Lu, “Robust stabilization of state delayed T-S fuzzy systems with input saturation via dynamic anti-windup fuzzy design,” International Journal of Innovative Computing, Information and Control, vol. 7, no. 12, pp. 6665–6676, 2011.
• B. M. Chen, T. H. Lee, K. Peng, and V. Venkataramanan, “Composite nonlinear feedback control for linear systems with input saturation: theory and an application,” IEEE Transactions on Automatic Control, vol. 48, no. 3, pp. 427–439, 2003.
• W. Lan and J. Huang, “Semiglobal stabilization and output regulation of singular linear systems with input saturation,” IEEE Transactions on Automatic Control, vol. 48, no. 7, pp. 1274–1280, 2003.
• T. Hu, A. R. Teel, and L. Zaccarian, “Anti-windup synthesis for linear control systems with input saturation: achieving regional, nonlinear performance,” Automatica, vol. 44, no. 2, pp. 512–519, 2008.
• J. M. Araujo, P. R. Barros, and C. E. T. Dórea, “Dynamic output feedback control of constrained descriptor systems,” Transactions of the Institute of Measurement and Control, vol. 35, no. 8, pp. 1129–1138, 2013.
• H. Shen, X. Song, and Z. Wang, “Robust fault-tolerant control of uncertain fractional-order systems against actuator faults,” IET Control Theory & Applications, vol. 7, no. 9, pp. 1233–1241, 2013.
• H. Gao and T. Chen, “New results on stability of discrete-time systems with time-varying state delay,” IEEE Transactions on Automatic Control, vol. 52, no. 2, pp. 328–334, 2007.
• P. Park and J. Wan Ko, “Stability and robust stability for systems with a time-varying delay,” Automatica, vol. 43, no. 10, pp. 1855–1858, 2007.
• 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.
• M. Wu, Y. He, J.-H. She, and G.-P. Liu, “Delay-dependent criteria for robust stability of time-varying delay systems,” Automatica, vol. 40, no. 8, pp. 1435–1439, 2004.
• Z. Wu, P. Shi, H. Su, and J. Chu, “Passivity analysis for discrete-time stochastic Markovian jump neural networks with mixed time delays,” IEEE Transactions on Neural Networks, vol. 22, no. 10, pp. 1566–1575, 2011.
• Z. Wu, P. Shi, H. Su, and J. Chu, “Stability analysis for discrete-time Markovian jump neural networks with mixed time-delays,” Expert Systems with Applications, vol. 39, no. 6, pp. 6174–6181, 2012.
• 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, no. 6, pp. 1796–1806, 2013.
• H. Liu, F. Sun, and E. K. Boukas, “Robust control of uncertain discrete-time Markovian jump systems with actuator saturation,” International Journal of Control, vol. 79, no. 7, pp. 805–812, 2006.
• J. Raouf and E. K. Boukas, “Stabilisation of singular Markovian jump systems with discontinuities and saturating inputs,” IET Control Theory & Applications, vol. 3, no. 7, pp. 971–982, 2009.
• Z. 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.
• H. Shen, J. H. Park, L. Zhang, and Z.-G. Wu, “Robust extended dissipative control for sampled-data Markov jump systems,” International Journal of Control, 2014.
• H. Shen, Z. Wang, X. Huang, and J. Wang, “Fuzzy dissipative control for nonlinear Markovian jump systems via retarded feedback,” Journal of the Franklin Institute, 2013.
• H. Li, B. Chen, Q. Zhou, and W. Qian, “Robust stability for uncertain delayed fuzzy hopfield neural networks with Markovian jumping parameters,” IEEE Transactions on Systems, Man, and Cybernetics B: Cybernetics, vol. 39, no. 1, pp. 94–102, 2009.
• S. Ma and E. K. Boukas, “Robust ${H}_{\infty }$ filtering for uncertain discrete Markov jump singular systems with mode-dependent time delay,” IET Control Theory & Applications, vol. 3, no. 3, pp. 351–361, 2009.
• H. Shen, S. Xu, X. Song, and J. Luo, “Delay-dependent robust stabilization for uncertain stochastic switching systems with distributed delays,” Asian Journal of Control, vol. 11, no. 5, pp. 527–535, 2009.
• H. Shen, S. Xu, X. Song, and G. Shi, “Passivity-based control for Markovian jump systems via retarded output feedback,” Circuits, Systems, and Signal Processing, vol. 31, no. 1, pp. 189–202, 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. \endinput