Table of Contents
ISRN Applied Mathematics
Volume 2014 (2014), Article ID 861759, 10 pages
http://dx.doi.org/10.1155/2014/861759
Research Article

Stability Criteria for Uncertain Discrete-Time Systems under the Influence of Saturation Nonlinearities and Time-Varying Delay

Department of Electronics and Communication Engineering, Motilal Nehru National Institute of Technology Allahabad, Allahabad 211004, India

Received 23 January 2014; Accepted 12 February 2014; Published 1 April 2014

Academic Editors: F. Ding, C.-Y. Lu, and Q. Song

Copyright © 2014 Siva Kumar Tadepalli et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. T. A. C. M. Claasen, W. F. G. Mecklenbräuker, and J. B. H. Peek, “Effects of quantization and overflow in recursive digital filters,” Institute of Electrical and Electronics Engineers. Transactions on Audio and Electroacoustics, vol. ASSP-24, no. 6, pp. 517–529, 1976. View at Google Scholar · View at MathSciNet
  2. H. J. Butterweck, J. H. F. Ritzerfeld, and M. J. Werter, “Finite wordlength in digital filters—a review,” EUT Report 88-E-205, Eindhoven University of Technology, Eindhoven, The Netherlands, 1988. View at Google Scholar
  3. T. Ooba, “Stability of linear discrete dynamics employing state saturation arithmetic,” Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, vol. 48, no. 4, pp. 626–630, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  4. P. M. Ebert, J. E. Mazo, and M. G. Taylor, “Overflow oscillations in digital filters,” Bell System Technical Journal, vol. 48, no. 9, pp. 2999–3020, 1969. View at Google Scholar · View at Scopus
  5. W. L. Mills, C. T. Mullis, and R. A. Roberts, “Digital filter realizations without overflow oscillations,” Institute of Electrical and Electronics Engineers. Transactions on Audio and Electroacoustics, vol. 26, no. 4, pp. 334–338, 1978. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. I. W. Sandberg, “The zero-input response of digital filters using saturation arithmetic,” Institute of Electrical and Electronics Engineers. Transactions on Circuits and Systems, vol. 26, no. 11, pp. 911–915, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. J. H. F. Ritzerfeld, “A Condition for the overflow stability of second-order digital filters that is satisfied by all scaled state-space structures using saturation,” IEEE Transactions on Circuits and Systems, vol. 36, no. 8, pp. 1049–1057, 1989. View at Publisher · View at Google Scholar · View at Scopus
  8. T. Bose and M.-Q. Chen, “Overflow oscillations in state-space digital filters,” IEEE Transactions on Circuits and Systems, vol. 38, no. 7, pp. 807–810, 1991. View at Publisher · View at Google Scholar · View at Scopus
  9. D. Liu and A. N. Michel, “Asymptotic stability of discrete-time systems with saturation nonlinearities with applications to digital filters,” IEEE Transactions on Circuits and Systems I, vol. 39, no. 10, pp. 798–807, 1992. View at Publisher · View at Google Scholar · View at Scopus
  10. H. Kar and V. Singh, “A new criterion for the overflow stability of second-order state-space digital filters using saturation arithmetic,” IEEE Transactions on Circuits and Systems I, vol. 45, no. 3, pp. 311–313, 1998. View at Publisher · View at Google Scholar · View at Scopus
  11. H. Kar and V. Singh, “Elimination of overflow oscillations in fixed-point state-space digital filters with saturation arithmetic: an LMI approach,” IEEE Transactions on Circuits and Systems II, vol. 51, no. 1, pp. 40–42, 2004. View at Publisher · View at Google Scholar · View at Scopus
  12. V. Singh, “Elimination of overflow oscillations in fixed-point state-space digital filters using saturation arithmetic,” IEEE Transactions on Circuits and Systems, vol. 37, no. 6, pp. 814–818, 1990. View at Publisher · View at Google Scholar · View at Scopus
  13. V. Singh, “Elimination of overflow oscillations in fixed-point state-space digital filters using saturation arithmetic: an LMI approach,” Digital Signal Processing, vol. 16, no. 1, pp. 45–51, 2006. View at Publisher · View at Google Scholar · View at Scopus
  14. H. Kar, “An LMI based criterion for the nonexistence of overflow oscillations in fixed-point state-space digital filters using saturation arithmetic,” Digital Signal Processing, vol. 17, no. 3, pp. 685–689, 2007. View at Publisher · View at Google Scholar · View at Scopus
  15. S.-F. Chen, “Asymptotic stability of discrete-time systems with time-varying delay subject to saturation nonlinearities,” Chaos, Solitons and Fractals, vol. 42, no. 2, pp. 1251–1257, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. V. K. R. Kandanvli and H. Kar, “Robust stability of discrete-time state-delayed systems with saturation nonlinearities: linear matrix inequality approach,” Signal Processing, vol. 89, no. 2, pp. 161–173, 2009. View at Publisher · View at Google Scholar · View at Scopus
  17. V. K. R. Kandanvli and H. Kar, “Delay-dependent LMI condition for global asymptotic stability of discrete-time uncertain state-delayed systems using quantization/overflow nonlinearities,” International Journal of Robust and Nonlinear Control, vol. 21, no. 14, pp. 1611–1622, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. V. K. R. Kandanvli and H. Kar, “Delay-dependent stability criterion for discrete-time uncertain state-delayed systems employing saturation nonlinearities,” Arabian Journal for Science and Engineering, vol. 38, no. 10, pp. 2911–2920, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. X. Ji, T. Liu, Y. Sun, and H. Su, “Stability analysis and controller synthesis for discrete linear time-delay systems with state saturation nonlinearities,” International Journal of Systems Science, vol. 42, no. 3, pp. 397–406, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. M. S. Mahmoud, Robust Control and Filtering for Time-Delay Systems, vol. 5 of Control Engineering, Marcel Dekker, New York, NY, USA, 2000. View at MathSciNet
  21. S. Xu and J. Lam, “A survey of linear matrix inequality techniques in stability analysis of delay systems,” International Journal of Systems Science, vol. 39, no. 12, pp. 1095–1113, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. S. Xu, J. Lam, and Y. Zou, “Improved conditions for delay-dependent robust stability and stabilization of uncertain discrete time-delay systems,” Asian Journal of Control, vol. 7, no. 3, pp. 344–348, 2005. View at Google Scholar · View at Scopus
  23. H. Huang and G. Feng, “Improved approach to delay-dependent stability analysis of discrete-time systems with time-varying delay,” IET Control Theory & Applications, vol. 4, no. 10, pp. 2152–2159, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  24. H. Shao and Q.-L. Han, “New stability criteria for linear discrete-time systems with interval-like time-varying delays,” Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, vol. 56, no. 3, pp. 619–625, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  25. W.-H. Chen, Z.-H. Guan, and X. Lu, “Delay-dependent guaranteed cost control for uncertain discrete-time systems with delay,” IEE Proceedings Control Theory and Applications, vol. 150, no. 4, pp. 412–416, 2003. View at Publisher · View at Google Scholar · View at Scopus
  26. Q.-L. Han, “Improved stability criteria and controller design for linear neutral systems,” Automatica, vol. 45, no. 8, pp. 1948–1952, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. Y. Wu, Y. Wu, and Y. Chen, “Mean square exponential stability of uncertain stochastic neural networks with time-varying delay,” Neurocomputing, vol. 72, no. 10–12, pp. 2379–2384, 2009. View at Publisher · View at Google Scholar · View at Scopus
  28. X. Jiang, Q.-L. Han, and X. Yu, “Stability criteria for linear discrete-time systems with interval-like time-varying delay,” in Proceedings of the American Control Conference (ACC '05), pp. 2817–2822, Portland, Ore, USA, June 2005. View at Scopus
  29. J. Liu and J. Zhang, “Note on stability of discrete-time time-varying delay systems,” IET Control Theory & Applications, vol. 6, no. 2, pp. 335–339, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  30. X. Meng, J. Lam, B. Du, and H. Gao, “A delay-partitioning approach to the stability analysis of discrete-time systems,” Automatica, vol. 46, no. 3, pp. 610–614, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. P. Kokil, H. Kar, and V. K. R. Kandanvli, “Stability analysis of linear discrete-time systems with interval delay: a delay-partitioning approach,” ISRN Applied Mathematics, vol. 2011, Article ID 624127, 10 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. V. K. R. Kandanvli and H. Kar, “A delay-dependent approach to stability of uncertain discrete-time statedelayed systems with generalized over flow nonlinearities,” ISRN Computational Mathematics, vol. 2012, Article ID 171606, 8 pages, 2012. View at Publisher · View at Google Scholar
  33. Y. He, M. Wu, G.-P. Liu, and J.-H. She, “Output feedback stabilization for a discrete-time system with a time-varying delay,” Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, vol. 53, no. 10, pp. 2372–2377, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  34. B. Zhang, S. Xu, and Y. Zou, “Improved stability criterion and its applications in delayed controller design for discrete-time systems,” Automatica, vol. 44, no. 11, pp. 2963–2967, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. H. Gao and T. Chen, “New results on stability of discrete-time systems with time-varying state delay,” Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, vol. 52, no. 2, pp. 328–334, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  36. H. Gao, J. Lam, C. Wang, and Y. Wang, “Delay-dependent output-feedback stabilisation of discrete-time systems with time-varying state delay,” IEE Proceedings Control Theory and Applications, vol. 151, no. 6, pp. 691–698, 2004. View at Google Scholar
  37. O. M. Kwon, M. J. Park, J. H. Park, S. M. Lee, and E. J. Cha, “Improved delay-dependent stability criteria for discrete-time systems with time-varying delays,” Circuits, Systems, and Signal Processing, vol. 32, no. 4, pp. 1949–1962, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  38. L. Bakule, J. Rodellar, and J. M. Rossell, “Robust overlapping guaranteed cost control of uncertain state-delay discrete-time systems,” Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, vol. 51, no. 12, pp. 1943–1950, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  39. P. Park, J. W. Ko, and C. Jeong, “Reciprocally convex approach to stability of systems with time-varying delays,” Automatica, vol. 47, no. 1, pp. 235–238, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  40. L. Xie, M. Fu, and C. E. de Souza, “H-control and quadratic stabilization of systems with parameter uncertainty via output feedback,” Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, vol. 37, no. 8, pp. 1253–1256, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  41. S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, vol. 15 of SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  42. J. Löfberg, “YALMIP: a toolbox for modeling and optimization in MATLAB,” in Proceedings of the IEEE International Symposium on Computer Aided Control System Design, pp. 284–289, Taipei, Taiwan, September 2004. View at Scopus
  43. J. F. Sturm, “Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones,” Optimization Methods and Software, vol. 11/12, no. 1–4, pp. 625–653, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet