Abstract

A sufficient condition for the open loop system to be regular, causal and stochastically stable is proposed for a class of discrete-time descriptor Markov jump systems with partly unknown transition probabilities. The proposed criteria are in the form of a set of strict linear matrix inequalities and convenient for numerical realization. The presented condition used the information of unknown transition probabilities in an effective way and is less conservative. Furthermore, the stabilization control of the researching systems is realized by designing the state feedback controller to make the close-looped systems be regular, causal and stochastically stable. At last, a numerical example is given to demonstrate the validity of the proposed results.

Keywords:

Descriptor Markov jump systems, Linear Matrix Inequality (LMI), partially unknown, stability analysis, stabilization control, Transition Probabilities (TPs),

References

1. Boukas, E.K., 2005. Stochastic Switching Systems: Analysis and Design. Birkhauser, Berlin.
   
2. Boukas, E.K., 2008. Control of Singular Systems with Random Abrupt Changes' in 'Series: Communications and Control Engineering. Springer, Berlin.
   
3. Boukas, E.K. and Y. Xia, 2008. Descriptor discrete-time systems with random abrupt changes: Stability and stabilisation. Int. J. Control, 81(8): 1311-1318.
   CrossRef    
4. Chaibi, N. and E.H. Tissir, 2012. Delay dependent robust stability of singular systems with time-varying delay. Int. J. Control Automat. Syst., 10(3): 632-638.
   CrossRef    
5. Chang, H., Y. Fang and S. Lou, 2012. Stabilization of discrete-time singular Markov jump systems. J. Nanjing Univ. Aeronaut. Astronaut., 44(1): 65-69.
   
6. Che, W. and J. Wang, 2010. Static output feedback H8 control for discrete-time Markov jump linear systems. Proceeding of the 8th IEEE International Conference on Control and Automation, 2010: 2278-2283.
   
7. Costa, O.L.V., M.D. Fragoso and R.P. Marques, 2005. Discrete Time Markov Jump Linear Systems. Springer-Verlag, London.
   CrossRef    
8. Dai, L., 1989. 'Singular control systems' in 'Volume 118 of Lecture Notes in Control and Information Sciences'. Springer-Verlag, Berlin, Germany.
   
9. Feng, Z. and J. Lam, 2012. Robust reliable dissipative filtering for discrete delay singular systems. Signal Process., 92(12): 3010-3025.
   CrossRef    
10. Huang, L. and X. Mao, 2011. Stability of singular stochastic systems with markovian switching. IEEE T. Automat. Control, 56(2): 424-429.
    CrossRef    
11. Liberzon, D., 2003. Switching in Systems and Control. Birkhauser, Berlin.
    
12. Ma, S., C. Zhang and S. Zhu, 2011. Robust stability for discrete-time uncertain singular Markov jump systems with actuator saturation. IET Control Theory A., 5(2): 255-262.
    CrossRef    
13. Xia, Y., E.K. Boukas and P. Shi, 2009. Stability and stabilization of continuous-time singular hybrid systems. Automatica, 45(6): 1504-1509.
    CrossRef    
14. Xu, S. and J. Lam, 2006. Control and ?ltering of Singular Systems. Springer, Berlin.
    
15. Zhang, L. and E.K. Boukas, 2009a. Stability and stabilization of Markovian jump linear systems with partly unknown transition probabilities. Automatica, 45(2): 463-468.
    CrossRef    
16. Zhang, L. and E.K. Boukas, 2009b. Mode-dependent H8 ?ltering for discrete-time Markovian jump linear systems with partly unknown transition probabilities. Automatica, 45(6): 1462-1467.
    CrossRef    

Competing interests

The authors have no competing interests.

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