Table of Contents
International Scholarly Research Notices
Volume 2014, Article ID 761959, 6 pages
http://dx.doi.org/10.1155/2014/761959
Research Article

An LMI Based Criterion for Global Asymptotic Stability of Discrete-Time State-Delayed Systems with Saturation Nonlinearities

Indian Institute of Information Technology, Design and Manufacturing, Kancheepuram, Chennai 600 127, India

Received 3 June 2014; Revised 3 August 2014; Accepted 8 August 2014; Published 29 October 2014

Academic Editor: Fernando Tadeo

Copyright © 2014 Priyanka Kokil. 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. J. H. F. Ritzerfeld, “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
  2. V. Singh, “A new realizability condition for limit cycle-free state-space digital filters employing saturation arithmetic,” IEEE Transactions on Circuits and Systems, vol. 32, no. 10, pp. 1070–1071, 1985. View at Publisher · View at Google Scholar · View at Scopus
  3. 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
  4. T. Bose and M. 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
  5. 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: Fundamental Theory and Applications, vol. 39, no. 10, pp. 798–807, 1992. View at Publisher · View at Google Scholar · View at Scopus
  6. 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
  7. H. Kar and V. Singh, “Stability analysis of discrete-time systems in a state-space realisation with partial state saturation nonlinearities,” IEE Proceedings: Control Theory and Applications, vol. 150, no. 3, pp. 205–208, 2003. View at Publisher · View at Google Scholar · View at Scopus
  8. T. Ooba, “Stability of linear discrete dynamics employing state saturation arithmetic,” IEEE Transactions on Automatic Control, vol. 48, no. 4, pp. 626–630, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. 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: Express Briefs, vol. 51, no. 1, pp. 40–42, 2004. View at Publisher · View at Google Scholar · View at Scopus
  10. V. Singh, “Stability analysis of discrete-time systems in a state-space realisation with state saturation nonlinearities: Linear matrix inequality approach,” IEE Proceedings Control Theory and Applications, vol. 152, no. 1, pp. 9–12, 2005. View at Publisher · View at Google Scholar
  11. H. Kar and V. Singh, “Elimination of overflow oscillations in digital filters employing saturation arithmetic,” Digital Signal Processing, vol. 15, no. 6, pp. 536–544, 2005. View at Publisher · View at Google Scholar · View at Scopus
  12. 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
  13. V. Singh, “Modified form of Liu-Michel's criterion for global asymptotic stability of fixed-point state-space digital filters using saturation arithmetic,” IEEE Transactions on Circuits and Systems II, vol. 53, no. 12, pp. 1423–1425, 2006. View at Publisher · View at Google Scholar · View at Scopus
  14. H. Kar, “An improved version of modified Liu-Michel’s criterion for global asymptotic stability of fixed-point state-space digital filters using saturation arithmetic,” Digital Signal Processing, vol. 20, no. 4, pp. 977–981, 2010. View at Publisher · View at Google Scholar · View at Scopus
  15. V. Singh, “Novel criterion for stability of discrete-time systems in a state-space realization utilizing saturation nonlinearities,” Applied Mathematics and Computation, vol. 218, no. 8, pp. 4305–4311, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  16. V. Singh, “New criterion for stability of discrete-time systems joined with a saturation operator on the state-space,” AEU-International Journal of Electronics and Communications, vol. 66, no. 6, pp. 509–511, 2012. View at Publisher · View at Google Scholar · View at Scopus
  17. V. Singh, “Stability of discrete-time systems joined with a saturation operator on the state-space: yet another version of Liu-Michel's criterion,” AEÜ International Journal of Electronics and Communications, vol. 66, no. 1, pp. 28–31, 2012. View at Publisher · View at Google Scholar · View at Scopus
  18. P. Kokil, V. K. R. Kandanvli, and H. Kar, “A note on the criterion for the elimination of overflow oscillations in fixed-point digital filters with saturation arithmetic and external disturbance,” AEÜ—International Journal of Electronics and Communications, vol. 66, no. 9, pp. 780–783, 2012. View at Publisher · View at Google Scholar · View at Scopus
  19. V. Krishna Rao Kandanvli and H. Kar, “Robust stability of discrete-time state-delayed systems employing generalized overflow nonlinearities,” Nonlinear Analysis, Theory, Methods and Applications, vol. 69, no. 9, pp. 2780–2787, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  20. V. Krishna Rao 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
  21. 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
  22. M. Malek-Zavarei and M. Jamshidi, Time-Delay Systems: Analysis, Optimization and Applications, North-Holland, Amsterdam, The Netherlands, 1987. View at MathSciNet
  23. J. K. Hale, Functional Differential Equations, Springer, New York, NY, USA, 1971.
  24. J. Richard, “Time-delay systems: an overview of some recent advances and open problems,” Automatica, vol. 39, no. 10, pp. 1667–1694, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. H. Shao, “Delay-dependent approaches to globally exponential stability for recurrent neural networks,” IEEE Transactions on Circuits and Systems II, vol. 55, no. 6, pp. 591–595, 2008. View at Publisher · View at Google Scholar · View at Scopus
  26. H. Huang and G. Feng, “Improved approach to delay-dependent stability analysis of discrete-time systems with time-varying delay,” IET Control Theory and Applications, vol. 4, no. 10, pp. 2152–2159, 2010. View at Publisher · View at Google Scholar · View at Scopus
  27. V. K. R. Kandanvli and H. Kar, “An LMI condition for robust stability of discrete-time state-delayed systems using quantization/overflow nonlinearities,” Signal Processing, vol. 89, no. 11, pp. 2092–2102, 2009. View at Publisher · View at Google Scholar · View at Scopus
  28. S. Xu, “Robust H filtering for a class of discrete-time uncertain nonlinear systems with state delay,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 49, no. 12, pp. 1853–1859, 2002. View at Publisher · View at Google Scholar · View at Scopus
  29. S. Xu, J. Lam, and T. Chen, “Robust H control for uncertain discrete stochastic time-delay systems,” Systems and Control Letters, vol. 51, no. 3-4, pp. 203–215, 2004. View at Publisher · View at Google Scholar · View at Scopus
  30. G. Lu and D. W. C. Ho, “Robust observer for nonlinear discrete systems with time delay and parameter uncertainties,” IEE Proceedings—Control Theory and Applications, vol. 151, no. 4, pp. 439–444, 2004. View at Google Scholar
  31. R. M. Palhares, C. E. de Souza, and P. L. D. Peres, “Robust H filtering for uncertain discrete-time state-delayed systems,” IEEE Transactions on Signal Processing, vol. 49, no. 8, pp. 1696–1703, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  32. W. Chen, Z. Guan, and X. Lu, “Delay-dependent guaranteed cost control for uncertain discrete-time systems with both state and input delays,” Journal of the Franklin Institute, vol. 341, no. 5, pp. 419–430, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  33. X. Guan, Z. Lin, and G. Duan, “Robust guaranteed cost control for discrete-time uncertain systems with delay,” IEE Proceedings-Control Theory and Applications, vol. 146, no. 6, pp. 598–602, 1999. View at Google Scholar
  34. L. Bakule, J. Rodellar, and J. M. Rossell, “Robust overlapping guaranteed cost control of uncertain state-delay discrete-time systems,” IEEE Transactions on Automatic Control, vol. 51, no. 12, pp. 1943–1950, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  35. 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
  36. J. H. Kim and S. J. Ahn, “Guaranteed cost and H filtering for discrete-time polytopic uncertain systems with time delay,” Journal of the Franklin Institute, vol. 342, no. 4, pp. 365–378, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  37. M. S. Mahmoud, E. Boukas, and A. Ismail, “Robust adaptive control of uncertain discrete-time state-delay systems,” Computers and Mathematics with Applications, vol. 55, no. 12, pp. 2887–2902, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  38. A. N. Michel, J. Si, and G. Yen, “Analysis and synthesis of a class of discrete-time neural networks described on hypercubes,” IEEE Transactions on Neural Networks, vol. 2, no. 1, pp. 32–46, 1991. View at Publisher · View at Google Scholar · View at Scopus
  39. P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, LMI Control Toolbox for use with Matlab, The MathWorks, Natick, Mass, USA, 1995.
  40. S. Boyd, L. El Ghaoui, and E. Feron, Linear Matrix Inequalities in System and Control Theory, vol. 15, SIAM, Philadelphia, Pa, USA, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  41. P. Kokil, H. Kar, and V. K. R. Kandanvli, “Stability analysis of discrete-time state-delayed systems with saturation nonlinearities,” in Proceedings of the 2011 International Conference on Computational Intelligence and Computing Research, Kanyakumari, India, 2011.
  42. 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 MathSciNet
  43. P. Kokil, V. K. R. Kandanvli, and H. Kar, “Delay-partitioning approach to stability of linear discrete-time systems with interval-like time-varying delay,” International Journal of Engineering Mathematics, vol. 2013, Article ID 291976, 7 pages, 2013. View at Publisher · View at Google Scholar
  44. P. Kokil and H. Kar, “An improved criterion for the global asymptotic stability of fixed-point state-space digital filters with saturation arithmetic,” Digital Signal Processing, vol. 22, no. 6, pp. 1063–1067, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  45. A. Benzaouia, F. Tadeo, and F. Mesquine, “The regulator problem for linear systems with saturations on the control and its increments or rate: an LMI approach,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 53, no. 12, pp. 2681–2691, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  46. M. Benhayoun, A. Benzaouia, F. Mesquine, and F. Tadeo, “Stabilization of 2D continuous systems with multi-delays and saturated control,” in Proceeding of the 18th Mediterranean Conference on Control and Automation (MED ’10), pp. 993–999, Marrakech, Morocco, June 2010. View at Publisher · View at Google Scholar · View at Scopus
  47. T. Hu, Z. Lin, and B. M. Chen, “Analysis and design for discrete-time linear systems subject to actuator saturation,” Systems and Control Letters, vol. 45, no. 2, pp. 97–112, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  48. T. Hu and Z. Lin, “The equivalence of several set invariance conditions under saturation,” in Proceeding of the 41st IEEE Conference on Decision and Control, pp. 4146–4147, Las Vegas, Nev, USA, December 2002. View at Scopus