Skip to main content
Log in

Delay-dependent Stability Criterion for Discrete-time Uncertain State-delayed Systems Employing Saturation Nonlinearities

  • Research Article - Systems Engineering
  • Published:
Arabian Journal for Science and Engineering Aims and scope Submit manuscript

Abstract

A new criterion for the global asymptotic stability of a class of uncertain discrete-time state-delayed systems employing saturation nonlinearities is presented. The proposed criterion takes the form of a linear matrix inequality which is delay-dependent. Numerical examples are given to illustrate the applicability of the presented criterion.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Ebert P.M., Mazo J.E., Taylor M.G.: Overflow oscillations in digital filters. Bell Syst. Tech. J. 48, 2999–3020 (1969)

    Article  Google Scholar 

  2. Mills W.L., Mullis C.T., Roberts R.A.: Digital filter realizations without overflow oscillations. IEEE Trans. Acoust. Speech Signal Process. 26, 334–338 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  3. Sandberg I.W.: The zero-input response of digital filters using saturation arithmetic. IEEE Trans. Circuits Syst. 26, 911–915 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  4. Erickson K.T., Michel A.N.: Stability analysis of fixed-point digital filters using computer generated Lyapunov functions—Part I: direct form and coupled form filters. IEEE Trans. Circuits Syst. 32, 113–132 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ritzerfeld J.H.F.: A condition for the overflow stability of second-order digital filters that is satisfied by all scaled state-space structures using saturation. IEEE Trans. Circuits Syst. 36, 1049–1057 (1989)

    Article  Google Scholar 

  6. Singh V.: A generalized approach for the absolute stability of discrete-time systems utilizing the saturation nonlinearity, based on passivity properties. IEEE Trans. Circuits Syst. 37, 444–447 (1990)

    Article  MATH  Google Scholar 

  7. Singh V.: Elimination of overflow oscillations in fixed-point state-space digital filters using saturation arithmetic. IEEE Trans. Circuits Syst. 37, 814–818 (1990)

    Article  Google Scholar 

  8. Bose T., Chen M.-Q.: Overflow oscillations in state-space digital filters. IEEE Trans. Circuits Syst. 38, 807–810 (1991)

    Article  Google Scholar 

  9. Liu D., Michel A.N.: Asymptotic stability of discrete-time systems with saturation nonlinearities with applications to digital filters. IEEE Trans. Circuits Syst. I 39, 789–807 (1992)

    Article  Google Scholar 

  10. Kar H., Singh V.: A new criterion for the overflow stability of second-order state-space digital filters using saturation arithmetic. IEEE Trans. Circuits Syst. I 45, 311–313 (1998)

    Article  Google Scholar 

  11. Kar H., Singh V.: Stability analysis of discrete-time systems in a state-space realisation with partial state saturation nonlinearities. IEE Proc. Control Theory Appl. 150, 205–208 (2003)

    Article  Google Scholar 

  12. Ooba T.: On companion systems with state saturation nonlinearity. IEEE Trans. Circuits Syst. I 50, 1580–1584 (2003)

    Article  MathSciNet  Google Scholar 

  13. Ooba T.: Stability of linear discrete dynamics employing state saturation arithmetic. IEEE Trans. Automat. Control 48, 626–630 (2003)

    Article  MathSciNet  Google Scholar 

  14. Kar H., Singh V.: Elimination of overflow oscillations in fixed-point state-space digital filters with saturation arithmetic: an LMI approach. IEEE Trans. Circuits Syst. II 51, 40–42 (2004)

    Article  Google Scholar 

  15. Kar H., Singh V.: Elimination of overflow oscillations in digital filters employing saturation arithmetic. Digital Signal Process. 15, 536–544 (2005)

    Article  Google Scholar 

  16. Singh V.: Elimination of overflow oscillations in fixed-point state-space digital filters using saturation arithmetic: an LMI approach. Digital signal process. 16, 45–51 (2006)

    Article  Google Scholar 

  17. Singh V.: Modified form of Liu-Michel’s criterion for global asymptotic stability of fixed-point state-space digital filters using saturation arithmetic. IEEE Trans. Circuits Syst. II 53, 1423–1425 (2006)

    Article  Google Scholar 

  18. Kar H.: An LMI based criterion for the nonexistence of overflow oscillations in fixed-point state-space digital filters using saturation arithmetic. Digital Signal Process. 17, 685–689 (2007)

    Article  Google Scholar 

  19. Chen S.-F.: Asymptotic stability of discrete-time systems with time-varying delay subject to saturation nonlinearities. Chaos Solit. Fract. 42, 1251–1257 (2009)

    Article  MATH  Google Scholar 

  20. Kandanvli V.K.R., Kar H.: Robust stability of discrete-time state-delayed systems employing generalized overflow nonlinearities. Nonlinear Anal. Theory Meth. Appl. 69, 2780–2787 (2008)

    Article  MATH  Google Scholar 

  21. Kandanvli V.K.R., Kar H.: Robust stability of discrete-time state-delayed systems with saturation nonlinearities: linear matrix inequality approach. Signal Process. 89, 161–173 (2009)

    Article  MATH  Google Scholar 

  22. Kandanvli V.K.R., Kar H.: An LMI condition for robust stability of discrete-time state-delayed systems using quantization/overflow nonlinearities. Signal Process. 89, 2092–2102 (2009)

    Article  MATH  Google Scholar 

  23. Ji X., Liu T., Sun Y., Su H.: Stability analysis and controller synthesis for discrete linear time-delay systems with state saturation nonlinearities. Int. J. Syst. Sci. 42, 397–406 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Xu S., Lam J., Yang C.: Quadratic stability and stabilization of uncertain linear discrete-time systems with state delay. Syst. Control Lett. 43, 77–84 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  25. Xu S.: Robust H filtering for a class of discrete-time uncertain nonlinear systems with state delay. IEEE Trans. Circuits Syst. I 49, 1853–1859 (2002)

    Article  Google Scholar 

  26. Xu S., Lam J., Chen T.: Robust H control for uncertain discrete stochastic time-delay systems. Syst. Control Lett. 51, 203–215 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lu G., Ho D.W.C.: Robust H observer for nonlinear discrete systems with time delay and parameter uncertainties. IEE Proc. Control Theory Appl. 151, 439–444 (2004)

    Article  Google Scholar 

  28. Palhares R.M., de Souza C.E., Peres P.L.D.: Robust H filtering for uncertain discrete-time state-delayed systems. IEEE Trans. Signal Process. 49, 1696–1703 (2001)

    Article  MathSciNet  Google Scholar 

  29. Chen W.-H., Guan Z.-H., Lu X.: Delay-dependent guaranteed cost control for uncertain discrete-time systems with both state and input delays. J. Franklin Inst. 341, 419–430 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  30. Guan X., Lin Z., Duan G.: Robust guaranteed cost control for discrete-time uncertain systems with delay. IEE Proc. Control Theory Appl. 146, 598–602 (1999)

    Article  Google Scholar 

  31. Bakule L., Rodellar J., Rossell J.M.: Robust overlapping guaranteed cost control of uncertain state-delay discrete-time systems. IEEE Trans. Automat. Control 51, 1943–1950 (2006)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  33. Kim J.H., Ahn S.J., Ahn S.: Guaranteed cost and H filtering for discrete-time polytopic uncertain systems with time delay. J. Franklin Inst. 342, 365–378 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  34. Mahmoud M.S.: Robust control and filtering for time-delay systems. Marcel-Dekker, New York (2000)

    MATH  Google Scholar 

  35. Mahmoud M.S., Boukas El-K., Ismail A.: Robust adaptive control of uncertain discrete-time state-delay systems. Comp. Math. Appl. 55, 2887–2902 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  36. Kau S.-W., Liu Y.-S., Hong L., Lee C.-H., Fang C.-H., Lee L.: A new LMI condition for robust stability of discrete-time uncertain systems. Systems Control Lett. 54, 1195–1203 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  37. Mahmoud M.S.: New results on robust control design of discrete-time uncertain systems. IEE Proc. Control Theory Appl. 152, 453–459 (2005)

    Article  Google Scholar 

  38. Xie L., Fu M., De Souza C.E.: H control and quadratic stabilization of systems with parameter uncertainty via output feedback. IEEE Trans. Automat. Control 37, 1253–1256 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  39. Xu S., Lam J., Lin Z., Galkowski K.: Positive real control for uncertain two-dimensional systems. IEEE Trans. Circuits Syst. I 49, 1659–1666 (2002)

    Article  MathSciNet  Google Scholar 

  40. Gao H., Lam J., Wang C., Wang Y.: Delay-dependent output-feedback stabilisation of discrete-time systems with time-varying state delay. IEE Proc. Control Theory Appl. 151, 691–698 (2004)

    Article  Google Scholar 

  41. Gao H., Chen T.: New results on stability of discrete-time systems with time-varying state delay. IEEE Trans. Automat. Control 52, 328–334 (2007)

    Article  MathSciNet  Google Scholar 

  42. Xu S., Chen T.: Robust H control for uncertain discrete-time systems with time-varying delays via exponential output feedback controllers. Syst. Control Lett. 51, 171–183 (2004)

    Article  MATH  Google Scholar 

  43. He Y., Liu G.-P., Rees D., Wu M.: H filtering for discrete-time systems with time-varying delay. Signal Process. 89, 275–282 (2009)

    Article  MATH  Google Scholar 

  44. Chen Y., Xue A., Zhou S., Lu R.: Delay-dependent robust control for uncertain stochastic time-delay systems. Circuits Syst. Signal Process. 27, 447–460 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  45. Guan X., Chen C., Shi P.: On robust stability for uncertain time-delay systems: a polyhedral Lyapunov-Krasovskii approach. Circuits Syst. Signal Process. 24, 1–18 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  46. Li H., Chen B., Zhou Q., Lin C.: A delay-dependent approach to robust H control for uncertain stochastic systems with state and input delays. Circuits Syst. Signal Process. 28, 169–183 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  47. 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)

    Article  MathSciNet  MATH  Google Scholar 

  48. Paszke W., Lam J., Galkowski K., Xu S., Lin Z.: Robust stability and stabilisation of 2D discrete state-delayed systems. Syst. Control Lett. 51, 277–291 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  49. He Y., Wu M., Liu G.-P., She J.-H.: Output feedback stabilization for a discrete-time system with a time-varying delay. IEEE Trans. Automat. Control 53, 2372–2377 (2008)

    Article  MathSciNet  Google Scholar 

  50. Zhang L., Shi P., Boukas E.-K.: 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 

  51. Xu S., Lam J.: A survey of linear matrix inequality techniques in stability analysis of delay systems. Int. J. Syst. Sci. 39, 1095–1113 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  52. Luo Y., Zhang H.: Approximate optimal control for a class of nonlinear discrete-time systems with saturating actuators. Progress Nat. Sci. 18, 1023–1029 (2008)

    Article  MathSciNet  Google Scholar 

  53. Liu W., Zhang H., Wang Z.: A novel truncated approximation based algorithm for state estimation of discrete-time Markov jump linear systems. Signal Process. 91, 702–712 (2011)

    Article  MATH  Google Scholar 

  54. Gahinet, P.; Nemirovski, A.; Laub, A.J.; Chilali, M.: LMI Control Toolbox-For use with Matlab. The MATH Works Inc., Natic (1995)

  55. Boyd, S.; El Ghaoui, L.; Feron, E.; Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia (1994)

  56. Kandanvli V.K.R., Kar H.: Delay-dependent LMI condition for global asymptotic stability of discrete-time uncertain state-delayed system using quantization/overflow nonlinearities. Int. J. Robust Nonlinear Control. 21, 1611–1622 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to V. Krishna Rao Kandanvli.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kandanvli, V.K.R., Kar, H. Delay-dependent Stability Criterion for Discrete-time Uncertain State-delayed Systems Employing Saturation Nonlinearities. Arab J Sci Eng 38, 2911–2920 (2013). https://doi.org/10.1007/s13369-013-0613-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13369-013-0613-2

Keywords

Navigation