ANZIAM J. 49 (2007), no. 1, pp. 111–129.
Delay-dependent stability and stabilization of uncertain discrete-time Markovian jump singular systems with time delay
Shuping Ma Xinzhi Liu† Chenghui Zhang
School of Mathematics and System Science
Shandong University
Jinan, 250100
China
mashup@sdu.edu.cn. 		Department of Applied Mathematics
University of Waterloo
Waterloo, Ontario
Canada N2L 3G1
xzliu@uwaterloo.ca.		 School of Control and Engineering
Shandong University
Jinan, 250061
China.
Received 9 December, 2006; revised 5 June, 2007

Abstract

This paper discusses robust stochastic stability and stabilization of time-delay discrete Markovian jump singular systems with parameter uncertainties. Based on the restricted system equivalent (RES) transformation, a delay-dependent linear matrix inequalities condition for time-delay discrete-time Markovian jump singular systems to be regular, causal and stochastically stable is established. With this condition, problems of robust stochastic stability and stabilization are solved, and delay-dependent linear matrix inequalities are obtained. A numerical example is also given to illustrate the effectiveness of this method.

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2000 Mathematics Subject Classification: primary 39A12; secondary 93C55
(Metadata: XML, RSS, BibTeX) MathSciNet: MR2378153 Z'blatt-MATH: 1132.93349
†indicates author for correspondence

References

  1. J. D. Aplevich, Implicit Linear Systems (Springer-Verlag, Berlin, 1991). MR1100921
  2. E. K. Boukas, “Static output feedback control for stochastic hybrid systems: LMI approach”, Automatica J. IPAC 42 (2006) 183–188. MR2183081
  3. E. K. Boukas and N. F. Al-Muthairi, “Delay-dependent stabilization of singular linear systems with delays”, Int. J. Innovative Comput. Information Contr. 2 (2006) 283–291.
  4. E. K. Boukas and Z. K. Liu, “Robust stability and H_{∞} control of discrete-time jump linear systems with time-delays: an LMI approach”, in Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, Australia, (1999), 1527–1532.
  5. E. K. Boukas and Z. K. Liu, “Robust H_{∞} control of discrete-time Markovian jump linear systems with mode-dependent time-delays”, IEEE Trans. Automat. Contr. 46 (2001) 1918–1924. MR1878213
  6. Y. Y. Cao and J. Lams, “Stochastic stabilizability and H_{∞} control for discrete-time jump linear systems with time delay”, J. Franklin Inst. 336 (1999) 1263–1281. MR1748735
  7. B. Chen, J. Lam and S. Xu, “Memory state feedback guaranteed cost control for neutral delay systems”, Int. J. Innovative Comput. Information Contr. 2 (2006) 293–303.
  8. W. H. Chen, Z. H. Guan and P. Yu, “Delay-dependent stability and H_{∞} control of uncertain discrete-time Markovian jump systems with mode-dependent time delays”, Systems Control Lett. 52 (2004) 361–376. MR2074379
  9. L. Dai, Singular Control Systems. Lecture Notes in Control and Information Sciences (Springer-Verlag, New York, 1989). MR986970
  10. E. Fridman and U. Shaked, “A descriptor system approach to H_{∞} control of linear time-delay systems”, IEEE Trans. Automat. Contr. 47 (2002) 253–270. MR1881892
  11. E. Fridman and U. Shaked, “An improved stabilization method for linear time-delay systems”, IEEE Trans. Automat. Contr. 47 (2002) 1931–1937. MR1937712
  12. Y. He, M. Wu, J. H. She and G. P. Liu, “Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays”, Systems Control Lett. 51 (2004) 57–65. MR2026262
  13. Y. He, M. Wu, J. H. She and G. P. Liu, “Parameter-dependent Lyapunov functional for stability of time-delay systems with polytopic type uncertainties”, IEEE Trans. Automat. Contr. 49 (2004) 828–832. MR2057826
  14. S. P. Ma and Z. L. Cheng, “An LMI approach to robust stabilization for uncertain discrete-time singular systems”, in Proceedings of the 41st IEEE CDC, Las Vegas, Nevada, USA, (2002), 1090–1095.
  15. S. P. Ma and Z. L. Cheng, “Delay-dependent robust stabilization for uncertain discrete-time singular systems with time-delay”, in Proceedings of the Sixth World Congress on Intelligent Control and Automation, Dalian, China, (2006), 2081–2085.
  16. I. R. Petersen, “A stabilization algorithm for a class of uncertain linear systems”, Systems Control Lett. 8 (1987) 351–357. MR884885
  17. P. Shi and E. K. Boukas, “On H_{∞} control design for singular continuous-time delay systems with parametric uncertainties”, Nonlinear Dyn. Syst. Theory 4 (2004) 59–71. MR2055111
  18. P. Shi, E. K. Boukas and K. Agarwal, “Control of Markovian jump discrete-time systems with norm bounded uncertainty and unknown delay”, IEEE Trans. Automat. Contr. 44 (1999) 2139–2144. MR1735731
  19. M. Wu, Y. He and J. H. She, “New delay-dependent stability criteria and stabilizing method for neutral systems”, IEEE Trans. Automat. Contr. 49 (2004) 2266–2271. MR2106758
  20. M. Wu, Y. He, J. H. She and G. P. Liu, “Delay-dependent criteria for robust stability of time-varying delay systems”, Automatica J. IPAC 40 (2004) 1435–1439. MR2153058
  21. S. Xu, P. V. Dooren, R. Stefan and J. Lam, “Robust stability and stabilization for singular systems with state delay and parameter uncertainty”, IEEE Trans. Automat. Contr. 47 (2002) 1122–1128. MR1911484
  22. S. Xu and J. Lam, “Robust stability and stabilization of discrete singular systems: An equivalent characterization”, IEEE Trans. Automat. Contr. 49 (2004) 568–574. MR2049814
  23. S. Xu, J. Lam and C. Yang, “Robust H_{∞} control for discrete singular systems with state delay and parameter uncertainty”, Dyn. Contin. Discrete Impuls. Syst Ser. B Appl. Algorithms 9 (2002) 539–554. MR1935624
  24. D. Yue, J. Lam and D. W. C. Ho, “Reliable H_{∞} control of uncertain descriptor systems with multiple delays”, IEE Proceedings - Control Theory and Applications 150 (2003) 557–564.