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Global stabilization of a class of delay discrete-time nonlinear systems via state and output feedback

  • Control Theory
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Abstract

This paper deals with stabilization of a class of delay discrete-time nonlinear systems through state and output feedback. We provide an explicit bounded state feedback law as an extension of the Jurdjevic-Quinn method, from nonlinear theory, to this class of systems. Next, we present a useful and systematic approach to design an observer for the same class of systems. Then, we show how the global stabilization problem via dynamic output feedback can be solved by using the two previous results. Finally, numerical examples are given to illustrate the effectiveness of the proposed design method.

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References

  1. K. Gu, “A further refinement of discretized Lyapunov functional method for the stability of timedelay systems,” Int. J. Contr., vol. 74, no. 10, pp. 967–976, January 2001.

    Article  MATH  Google Scholar 

  2. R. Wang and J. Zhao, “Exponential stability analysis for discrete-time switched linear systems with time-delay,” Int. J. Innov. Comp. Inf. Control, vol. 3, no. 6, pp. 1557–1564, December 2007.

    Google Scholar 

  3. Y. Xia, G. P. Liu, P. Shi, D. Rees, and E. J. C. Thomas, “New stability and stabilization conditions systems with time-delay,” Int. J. Syst. Sci., vol. 38, no. 1, pp. 17–24, January 2007.

    Article  MathSciNet  MATH  Google Scholar 

  4. P. Shi, E. K. Boukas, and R. K. Agarwal, “Control of Markovian jump discrete-time systems with norm bounded uncertainty and unknown delay,” IEEE Trans. on Automatic Control, vol. 44, no. 11, pp. 2139–2144, November 1999.

    Article  MathSciNet  MATH  Google Scholar 

  5. F. Wu, “Robust quadratic performance for time delayed uncertain linear systems,” Int. J. Robust & Nonlinear Contr., vol. 13, no. 2, pp. 153–172, February 2003.

    Article  MATH  Google Scholar 

  6. E. Fridman and U. Shaked, “An improved stabilization method for linear systems with time-delay,” IEEE Trans. on Automatic Control, vol. 42, no.11, pp. 1931–1937, November 2002.

    Article  MathSciNet  Google Scholar 

  7. J. P. Richard, “Time-delay systems: an overview of some recent advances and open problems,” Automatica, vol. 39, no. 10, pp. 1667–1694, October 2003.

    Article  MathSciNet  MATH  Google Scholar 

  8. K. E. Bouazza, M. Boutayeb, and M. Darouach, “State feedback stabilization of a class of nonlinear discrete-time delay systems,” Proc. Triennal IFAC World Congress, Prague, Czech Republic, 2005.

    Google Scholar 

  9. M. Mahmoud, E. Boukas, and P. Shi, “Resilient feedback stabilization of discrete-time systems with delays,” IMA Journal of Mathematical Control and Information, vol. 25, no. 2, pp. 141–156, June 2008.

    Article  MathSciNet  MATH  Google Scholar 

  10. P. Shi, “Robust filter design for sampled-data systems with interconnections,” Signal Processing, vol. 58, no. 2, pp. 131–151, April 1997.

    Article  MATH  Google Scholar 

  11. Z. Wang, B. Huang, and H. Unbehauen, “Robust H ∞ observer design of linear state delayed systems with parametric uncertainty: the discrete-time case,” Automatica, vol. 35, no. 6, pp. 1161–1167, June 1999.

    Article  MathSciNet  MATH  Google Scholar 

  12. X. Li, Z. Li, and H. Gao, “Further results on filtering for discrete time systems with state delay,” Int. J. Robust & Nonlinear Contr., vol. 21, no. 3, pp. 248–270, February 2011.

    Article  MATH  Google Scholar 

  13. S.-I. Niculescu, “Delay effects on stability: a robust approach control,” Lecture Notes in Control and Information Sciences, vol. 269, 2001.

  14. A. Benzaouia and F. Tadeo, “Stabilization of positive switching linear discrete-time systems,” Int. J. Innov. Comp. Inf. Control, vol. 6, no. 6, pp. 2427–2438, June 2010.

    Google Scholar 

  15. X. Mao, “Exponential stability of nonlinear differential delay equations,” Systems and Control Letters, vol. 28, no. 3, pp. 159–165, July 1996.

    Article  MathSciNet  MATH  Google Scholar 

  16. M. Guay and X. Li, “Stabilization of state-delayed nonlinear systems,” IFAC World Congress on Automatic Control, Barcelona, Spain, 2002.

    Google Scholar 

  17. M. Jankovic, “Control Lyapunov-Razumikhin functions and robust stabilization of time delay systems,” IEEE Trans. on Automatic Control, vol. 46, no. 7, pp. 1048–1060, July 2001.

    Article  MathSciNet  MATH  Google Scholar 

  18. Y. Fu, Z. Tian, and S. Shi, “State feedback stabilization for a class of stochastic time-delay nonlinear systems,” IEEE Trans. on Automatic Control, vol. 48, no.2, pp. 282–286, February 2003.

    Article  MathSciNet  Google Scholar 

  19. E. Fridman, “Output regulation of nonlinear systems with delay,” Systems and Control Letters, vol. 50, no. 2, pp. 81–93, October 2003.

    Article  MathSciNet  MATH  Google Scholar 

  20. M. S. Mahmoud, Robust Control and Filtering for Time-delay Systems, Marcel-Dekker, New York, 1999.

    Google Scholar 

  21. L. A. Marquez-Martinez, C. Moog, and M. Velasco- Villa, “Observability and observers for nonlinear systems with time-delays,” Proc. of the 2nd International Federation of Automatic Control Workshop on Time Delay Systems, Ancona, Italia, p. 52, 2000.

    Google Scholar 

  22. A. Germani, C. Manes, and P. Pepe, “A new approach to state observation of nonlinear systems with delayed output,” IEEE Trans. on Automatic Control, vol. 47, no. 1, p. 96, January 2002.

    Article  MathSciNet  Google Scholar 

  23. E. C. T. Ahmed-Ali and M. MSaad, “Cascade high gain observers for nonlinear systems with delayed output,” Proc. IEEE Conference on Decision and Control, (Shanghai, China), 2009.

    Google Scholar 

  24. S. Ibrir, “Observer-based control of a class of timedelay nonlinear systems having triangular structure,” Automatica, vol. 47, no. 2, pp. 388–394, February 2011.

    Article  MathSciNet  MATH  Google Scholar 

  25. A. Zemouche and M. Boutayeb, “Observer synthesis method for Lipschitz nonlinear discrete-time systems with time-delay: an LMI approach,” Applied Mathematics and Computation, vol. 218, no. 2, pp. 419–429, September 2011.

    Article  MathSciNet  MATH  Google Scholar 

  26. V. Jurdjevic and J. P. Quinn, “Controllability and stability,” J. Diff. Eq., vol. 28, no. 3, pp. 381–389, June 1978.

    Article  MathSciNet  MATH  Google Scholar 

  27. J. Gauthier and G. Bornard, Stabilisation des Systèmes Non Linéaires. Outils et méthodes math. pour l’automat. I.D. Landau, 1981.

    Google Scholar 

  28. N. Kalouptsidis and J. Tsinias, “Stability improvement of nonlinear systems by feedback,” IEEE Trans. on Automatic Control, vol. 29, no. 4, pp. 731–736, April 1984.

    Article  MathSciNet  Google Scholar 

  29. K. Lee and A. Araposthathis, “Remarks on smooth feedback stabilization of nonlinear systems,” Systems and Control Letters, vol. 10, no. 1, pp. 41–44, January 1988.

    Article  MathSciNet  MATH  Google Scholar 

  30. R. Outbib and G. Sallet, “Stabilizability of the angular velocity of a rigid body revisited,” Systems and Control Letters, vol. 18, no. 2, pp. 93–98, February 1992.

    Article  MathSciNet  MATH  Google Scholar 

  31. W. Lin, “Input saturation and global stabilization of nonlinear systems via state and output feedback,” IEEE Trans. on Automatic Control, vol. 40, no. 4, pp. 776–782, April 1995.

    Article  MATH  Google Scholar 

  32. W. Lin and C. I. Byrnes, “KYP lemma, state feedback and dynamic output feedback in discrete time bilinear systems,” Systems and Control Letters, vol. 23, no. 2, pp. 127–136, August 1994.

    Article  MathSciNet  MATH  Google Scholar 

  33. L. Hetel, J. Daafouz, and C. Iung, “Equivalence between the Lyapunov-Krasovskii functionals approach for discrete delay systems and that of the stability conditions for switched systems,” Nonlinear Analysis: Hybrid Systems, vol. 2, no. 3, pp. 697–705, August 2008.

    Article  MathSciNet  MATH  Google Scholar 

  34. E. Boukas, “Discrete-time systems with timevarying time delay: stability and stabilizability,” Mathematical Problems in Engineering, vol. 2006, Article ID 42489, pp. 1–10, 2006.

    Article  MathSciNet  Google Scholar 

  35. S. Xu and J. Lam, “On equivalence and efficiency of certain stability criteria for time-delay systems,” IEEE Trans. on Automatic Control, vol. 52, no. 1, pp. 95–101, January 2007.

    Article  MathSciNet  Google Scholar 

  36. M. Mahmoud, “Switched delay-dependent control policy for water-quality systems,” IEE Control Theory Appl., vol. 3, no. 12, pp. 1599–1610, December 2009.

    Article  Google Scholar 

  37. B. Wang and D. Zhao, “The optimal control of discrete-time delay nonlinear system with dual heuristic dynamic programming,” Lecture Notes in Computer Sciences, vol. 7663, pp. 664–672, 2012.

    Article  Google Scholar 

  38. V. K. Deolia, S. Purwar, and T. N. Sharma, “An LMI approach for the stabilization of discrete-time delay nonlinear systems using backstepping,” International Conference on Communication Systems and Network Technologies, 2012.

    Google Scholar 

  39. E. Boukas, “State feedback stabilization of nonlinear discrete-time systems with time-varying time delay,” Nonlinear Analysis: Theory, Methods and Applications, vol. 66, no. 6, pp. 1341–1350, March 2007.

    Article  MathSciNet  MATH  Google Scholar 

  40. H. Gao, J. Lam, C. Wang, and Y. Wang, “Delaydependent output-feedback stabilisation of discrete time-systems with time-varying state delay,” IEE Proc.-Control Theory Appl., vol. 151, no. 6, pp. 691–698, November 2004.

    Article  MathSciNet  Google Scholar 

  41. C. Byrnes, W. Lin, and B. Ghosh, “Stabilization of discrete-time nonlinear systems by smooth state feedback,” Systems and Control Letters, vol. 21, no. 3, pp. 255–263, September 1993.

    Article  MathSciNet  MATH  Google Scholar 

  42. J. Carr, Application of Centre Manifold Theory. Springer-Verlag, New York, 1981.

    Book  Google Scholar 

  43. P. Marti, Analysis and Design of Real-Time Control Systems with Time-varying Control Timing Constraints. Ph.D. Thesis, Automatic Control Department, Technical University of Catalonia, 2002.

    Google Scholar 

  44. C.-D. Zheng, H. Zhang, and Z. Wang, “An augmented LKF approach involving derivative information of both state and delay,” IEEE Trans. on Neural Networks, vol. 21, no. 7, pp. 1100–1109, July 2010.

    Article  Google Scholar 

  45. H. Shao, “Improved delay-dependent globally asymptotic stability criteria for neural networks with a constant delay,” IEEE Trans. Circ. Syst. II: Express Briefs, vol. 55, no. 10, pp. 1071–1075, October 2008.

    Article  Google Scholar 

  46. S. M. Karbassi, “Time-optimal control of disturbance-rejection tracking systems for discrete time time-delayed systems by state feedback,” Computers and Mathematics with Applications, vol. 50, no. 8–9, pp. 1415–1424, October-November 2005.

    Article  MathSciNet  MATH  Google Scholar 

  47. W. Zhiwen, G. Ge, and L. Dongsong, “Augmented state pole-placement for networked control system with time-delay,” Proc. of the 2nd International Conference on Intelligent Computation Technology and Automation, 2009.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, University College in Al Jamoom, Umm Al-Qura University, Al Jamoom, Saudi Arabia

    Kheir Eddine Bouazza

  2. Department of Computer Science, College of Sciences, Oran University, Oran, Algeria

    Mohammed Ouali

  3. Laboratoire d’Informatique et des Technologies de l’Information d’Oran (LITIO), University of Oran, Oran, Algeria

    Kheir Eddine Bouazza & Mohammed Ouali

Authors
  1. Kheir Eddine Bouazza
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  2. Mohammed Ouali
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Correspondence to Kheir Eddine Bouazza.

Additional information

Recommended by Editorial Board member Nam H. Jo under the direction of Editor Hyungbo Shim.

Kheir Eddine Bouazza received his B.Eng. degree in Computer Science from University of Oran, Algeria, in 1998, an M.Sc. in Automatic Control and Digital Signal Processing in 2000 and a Ph.D. degree in Automatic Control in 2004 from University of Nancy, France. From 2004 to 2005, he was a research associate (ATER) in the department of Computer Engineering in IUT de Longwy, University of Nancy. He is currently an associate Professor in the University of Umm Al-Qura, KSA. His main research interests are in areas of nonlinear observer-based control, time delay systems, networked control systems and wireless sensor networks.

Mohammed Ouali received the French Doctorate in real-time, robotics and control as well as the Ph.D. in Mathematics and Computer Science from Ecole des Mines de Paris (France) and the University of Sherbrooke (Canada) respectively. He has significant experience in developing embedded systems in aeronautics and automotive industry as well as CAD systems for VLSI reverse engineering. Dr. Ouali’s main research interests are in systems analysis, signal and image processing, and pattern recognition.

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Bouazza, K.E., Ouali, M. Global stabilization of a class of delay discrete-time nonlinear systems via state and output feedback. Int. J. Control Autom. Syst. 11, 1084–1094 (2013). https://doi.org/10.1007/s12555-012-0265-0

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  • DOI: https://doi.org/10.1007/s12555-012-0265-0

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