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Switched Discrete-Time Delay Systems: Delay-Dependent Analysis and Synthesis

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Abstract

A delay-dependent analysis and synthesis approach is established for a class of linear discrete-time switched delay systems with convex bounded parameter uncertainties in all system matrices. New results are established for both constant and time-varying delays using switched Lyapunov–Krasovskii functionals. A delay-dependent analysis of the uncertain switched delay system is developed to guarantee that it is asymptotically stable with an ℒ2 gain smaller than a prescribed constant level. Delay-dependent switched control feedback is then designed, based on state and output measurements, to render the corresponding switched closed-loop system delay-dependent asymptotically stable with a prescribed ℒ2 gain measure. The developed results are cast as linear matrix inequalities (LMIs) and tested on representative examples.

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References

  1. P. Apkarian, H.D. Tuan, Continuous-time analysis, eigenstructure assignment, and ℋ2 synthesis with enhanced linear matrix inequalities (LMI) characterizations. IEEE Trans. Autom. Control 46, 1941–1946 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. A. Benzaouia, E. De Santis, P. Caravani, N. Daraoui, Constrained control of switching systems: a positive invariant approach. Int. J. Control 80, 1379–1387 (2007)

    Article  MATH  Google Scholar 

  3. E.K. Boukas, Z.K. Liu, Deterministic and Stochastic Time-delay Systems (Birkhauser, Boston, 2002)

    MATH  Google Scholar 

  4. E.K. Boukas, M.S. Mahmoud, A practical approach to control of nonlinear discrete-time state-delay systems. Opt. Control Appl. Methods 10, 1–21 (2007)

    MathSciNet  Google Scholar 

  5. M. Branicky, Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans. Autom. Control 43, 475–582 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. W.H. Chen, Z.H. Guan, X. Lu, Delay-dependent guaranteed cost control for uncertain discrete-time systems with delay. IEE Proc.-Control Theory Appl. 150, 412–416 (2003)

    Article  Google Scholar 

  7. F.A. Cuzzola, M. Morari, A generalized approach for analysis and control of discrete-time piecewise affine and hybrid systems, in Hybrid Systems: Computation and Control, ed. by M.D. Di Benedetto, A.L. San Giovanni-Vincentelli (Springer, Berlin, 2001)

    Google Scholar 

  8. J. Daafouz, P. Riedinger, C. Lung, Stability analysis and control synthesis for switched systems: a switched Lyapunov function approach. IEEE Trans. Autom. Control 47, 1883–1887 (2002)

    Article  Google Scholar 

  9. R. DeCarlo, M. Branicky, S. Pettersson, B. Lennartson, Perspectives and results on the stability and stabilizability of hybrid systems. Proc. IEEE 88, 1069–1082 (2000)

    Article  Google Scholar 

  10. D. Du, B. Jiang, S. Zhou, Delay-dependent robust stabilization of uncertain discrete-time switched linear systems with time-varying state delays. Int. J. Syst. Sci. 39, 305–313 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. E. Fridman, U. Shaked, Delay-dependent ℋ∞ control of uncertain discrete delay systems. Eur. J. Control 11, 29–37 (2005)

    Article  MathSciNet  Google Scholar 

  12. P. Gahinet, A.  Nemirovski, A.L. Laub, M. Chilali, LMI Control Toolbox (The MathWorks, Natick, 1995)

    Google Scholar 

  13. V. Kapila, W.M. Haddad, Memoryless ℋ∞ controllers for discrete-time systems with time-delay. Automatica 34, 1141–1144 (1998)

    Article  MATH  Google Scholar 

  14. X.D. Koutsoukos, P.J. Antsaklis, Design of stabilizing switching control laws for discrete- and continuous-time linear systems using piecewise-linear Lyapunov functions. Int. J. Control 75, 932–945 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. D. Liberzon, A. Morse, Basic problems in stability and design of switched systems. IEEE Control Syst. Mag. 19, 59–70 (1999)

    Article  Google Scholar 

  16. M.S. Mahmoud, Robust performance results for discrete-time systems. J. Math. Probl. Eng. Syst. 4, 17–38 (1997)

    Google Scholar 

  17. M.S. Mahmoud, Robust ℋ∞ control of discrete systems with uncertain parameters and unknown delays. Automatica 36, 627–635 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. M.S. Mahmoud, Robust Control and Filtering for Time-Delay Systems (Dekker, New York, 2000)

    MATH  Google Scholar 

  19. M.S. Mahmoud, Robust stability and ℋ∞ estimation for uncertain discrete systems with state-delay. J. Math. Probl. Eng. 7, 393–412 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. M.S. Mahmoud, Resilient Control of Uncertain Dynamical Systems (Springer, Berlin, 2004)

    MATH  Google Scholar 

  21. M.S. Mahmoud, P. Shi, Optimal guaranteed cost filtering for Markovian jump discrete-time systems. Math. Probl. Eng. 12, 33–48 (2004)

    Article  MathSciNet  Google Scholar 

  22. M.S. Mahmoud, L. Xie, Guaranteed cost control of uncertain discrete systems with delays. Int. J. Control 73, 105–114 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. M.S. Mahmoud, M. Zribi, ℋ∞ controllers for time-delay systems using linear matrix inequalities. J. Optim. Theory Appl. 100, 89–123 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  24. M.S. Mahmoud, M.N. Nounou, H.N. Nounou, Analysis and synthesis of uncertain switched discrete-time systems. IMA J. Math. Control Inf. 24, 245–257 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. M.S. Mahmoud, E.K. Boukas, P. Shi, Resilient feedback stabilization of discrete-time systems with delays. IMA J. Math. Control Inf. 24, 1–16 (2007)

    Google Scholar 

  26. A. Michel, Recent trends in the stability analysis of hybrid dynamical systems. IEEE Trans. Circuits Syst.-Part I 46, 120–134 (1999)

    Article  MATH  Google Scholar 

  27. S. Pettersson, B. Lennartson, Hybrid system stability and robustness verification using linear matrix inequalities. Int. J. Control 75, 1335–1355 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  28. P. Shi, E.K. Boukas, R.K. Agarwal, Optimal guaranteed cost control of uncertain discrete time-delay systems. J. Comput. Appl. Math. 157, 435–451 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  29. S.H. Song, J.K. Kim, ℋ∞ control of discrete-time linear systems with norm-bounded uncertainties and time-delay in state. Automatica 34, 137–139 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  30. Y.G. Sun, L. Wang, G. Xie, Delay-dependent robust stability and ℋ∞ control for uncertain discrete-time switched systems with mode-dependent time-delays. Appl. Math. Comput. 180, 428–435 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. X.M. Sun, J. Zhao, W. Wang, Two design schemes for robust adaptive control of a class of linear uncertain delay systems. Int. J. Innov. Comput., Inf. Control 3, 385–396 (2007)

    Google Scholar 

  32. X.M. Sun, W. Wang, G.P. Liu, J. Zhao, Stability analysis for linear switched systems with time-varying delays. IEEE Trans. Syst., Man Cybern.-Part B 38, 528–533 (2008)

    Google Scholar 

  33. R. Wang, J. Zhao, Exponential stability analysis for discrete-time switched linear systems with time-delay. Int. J. Innov. Comput., Inf. Control 3, 1557–1564 (2007)

    Google Scholar 

  34. S.M. Williams, R.G. Hoft, Adaptive frequency domain control of PPM switched power line conditioner. IEEE Trans. Power Electron. 6, 665–670 (1991)

    Article  Google Scholar 

  35. L. Yuan, L.E.K. Axhenie, W. Jiang, Robust ℋ∞ control of linear discrete-time systems with norm-bounded time-delay uncertainty. Syst. Control Lett. 27, 199–208 (1996)

    Article  MATH  Google Scholar 

  36. G. Zhai, H. Lin, P.J. Antsaklis, Quadratic stabilizability of switched linear systems with polytopic uncertainties. Int. J. Control 76, 747–753 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  37. L. Zhang, P. Shi, C. Wang, H. Gao, Robust ℋ∞ filtering for switched linear discrete-time systems with polytopic uncertainties. Int. J. Adapt. Control Signal Process. 20, 291–304 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  38. L. Zhang, P. Shi, E.K. Boukas, ℋ∞ output-feedback control for switched linear discrete-time systems with time-varying delays. Int. J. Control 80, 1354–1365 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  39. X. Zhu, C. Hua, S. Wang, State feedback controller design of networked control systems with time delay in the plant. Int. J. Innov. Comput., Inf. Control 4, 283–290 (2008)

    Google Scholar 

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Authors and Affiliations

  1. Systems Engineering Department, King Fahd University of Petroleum and Minerals, P.O. Box 985, Dhahran, 31261, Saudi Arabia

    Magdi S. Mahmoud

  2. Department of Systems Engineering, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia

    Fouad M. Al-Sunni

  3. School of Engineering, Kyushu Tokai University, 9-9-1, Toroku, Kumamoto, 862-8652, Japan

    Yan Shi

Authors
  1. Magdi S. Mahmoud
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  2. Fouad M. Al-Sunni
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  3. Yan Shi
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Correspondence to Magdi S. Mahmoud.

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Mahmoud, M.S., Al-Sunni, F.M. & Shi, Y. Switched Discrete-Time Delay Systems: Delay-Dependent Analysis and Synthesis. Circuits Syst Signal Process 28, 735–761 (2009). https://doi.org/10.1007/s00034-009-9107-6

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  • DOI: https://doi.org/10.1007/s00034-009-9107-6

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