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New LMI Criteria to the Global Asymptotic Stability of Uncertain Discrete-Time Systems with Time Delay and Generalized Overflow Nonlinearities

  • Pushpendra Kumar GuptaEmail author
  • V. Krishna Rao Kandanvli
Conference paper
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 587)

Abstract

This paper investigates the problem of stability analysis of discrete-time systems under the effect of generalized overflow nonlinearities, parameter uncertainties, and time delay. The systems under assumption involve norm-bounded parameter uncertainties. Two stability criteria based on Linear Matrix Inequality (LMI) approach are presented. The usefulness of the presented criteria is numerically proved.

Keywords

Generalized overflow nonlinearity Global asymptotic stability Time delay Parameter uncertainty Lyapunov method Linear matrix inequality 

Notes

Acknowledgements

The corresponding author wishes to thank the TEQIP-III grant, MNNIT Allahabad for providing scholarship to pursue his research work. The authors of the paper wish to thank the reviewers for their comments and suggestions.

References

  1. 1.
    Li, X., Zhang, X., Wang, X.: Stability analysis for discrete-time markovian jump systems with time-varying delay: a homogeneous polynomial approach. IEEE Access 5, 27573–27581 (2017).  https://doi.org/10.1109/ACCESS.2017.2775606CrossRefGoogle Scholar
  2. 2.
    Wu, L., Lam, J., Yao, X., Xiong, J.: Robust guaranteed cost control of discrete‐time networked control systems. Optim. Control. Appl. Methods 32(1), 95–112 (2011).  https://doi.org/10.1002/oca.932MathSciNetCrossRefGoogle Scholar
  3. 3.
    Zhang, C.K., He, Y., Jiang, L., Wang, Q.G., Wu, M.: Stability analysis of discrete-time neural networks with time-varying delay via an extended reciprocally convex matrix inequality. IEEE Trans. Cybern. 47(10), 3040–3049 (2017).  https://doi.org/10.1109/TCYB.2017.2665683CrossRefGoogle Scholar
  4. 4.
    Zhang, D., Shi, P., Zhang, W.A., Yu, L.: Energy-efficient distributed filtering in sensor networks: a unified switched system approach. IEEE Trans. Cybern. 47(7), 1618–1629 (2017).  https://doi.org/10.1109/TCYB.2016.2553043CrossRefGoogle Scholar
  5. 5.
    Kandanvli, V.K.R., Kar, H.: Robust stability of discrete-time state-delayed systems employing generalized overflow nonlinearities. Nonlinear Anal.: Theory, Methods Appl. 69(9), 2780–2787 (2008).  https://doi.org/10.1016/j.na.2007.08.050MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kandanvli, V.K.R., Kar, H.: Robust stability of discrete-time state-delayed systems with saturation nonlinearities: Linear matrix inequality approach. Signal Process 89(2), 161–173 (2009).  https://doi.org/10.1016/j.sigpro.2008.07.020CrossRefzbMATHGoogle Scholar
  7. 7.
    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(11), 2092–2102 (2009).  https://doi.org/10.1016/j.sigpro.2009.04.024CrossRefzbMATHGoogle Scholar
  8. 8.
    Guan, X., Lin, Z., Duan, G.: Robust guaranteed cost control for discrete-time uncertain systems with delay. IEE Proc. Control Theory Appl. 146(6), 598–602 (1999).  https://doi.org/10.1049/ip-cta:19990714CrossRefGoogle Scholar
  9. 9.
    Bakule, L., Rodellar, J., Rossell, J.M.: Robust overlapping guaranteed cost control of uncertain state-delay discrete-time systems. IEEE Trans. Automat. Control 51(12), 1943–1950 (2006).  https://doi.org/10.1109/TAC.2006.886536MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    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(4), 412–416 (2003).  https://doi.org/10.1049/ip-cta:20030572CrossRefGoogle Scholar
  11. 11.
    Mahmoud, M.S.: Robust Control and Filtering for Time-Delay Systems. CRC Press, Marcel-Dekker, New York (2000)CrossRefGoogle Scholar
  12. 12.
    Mahmoud, M.S., Boukas, E.K., Ismail, A.: Robust adaptive control of uncertain discrete-time state-delay systems. Comput. Math. Appl. 55(12), 2887–2902 (2008).  https://doi.org/10.1016/j.camwa.2007.11.021MathSciNetCrossRefGoogle Scholar
  13. 13.
    Xu, S.: Robust H filtering for a class of discrete-time uncertain nonlinear systems with state delay. IEEE Trans. Circuits Syst. I 49(12), 1853–1859 (2002).  https://doi.org/10.1109/tcsi.2002.805736CrossRefGoogle Scholar
  14. 14.
    Boyd, S., Ghaoui, L.E., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia, PA (1994)Google Scholar
  15. 15.
    Xu, S., Lam, J., Lin, Z., Galkowski, K.: Positive real control for uncertain two-dimensional systems. IEEE Trans. Circuits Syst. I: Fundam. Theory Appl. 49(11), 1659–1666 (2002).  https://doi.org/10.1109/TCSI.2002.804531MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Bakule, L., Rodellar, J., Rossell, J.M.: Robust overlapping guaranteed cost control of uncertain state-delay discrete-time systems. IEEE Trans. Autom. Control 51(12), 1943–1950 (2006).  https://doi.org/10.1109/TAC.2006.886536MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kandanvli, V.K.R., Kar, H.: A delay-dependent approach to stability of uncertain discrete-time state-delayed systems with generalized overflow nonlinearities. ISRN Comput. Math. (2012).  https://doi.org/10.5402/2012/171606CrossRefGoogle Scholar
  18. 18.
    Liu, D., Michel, A.N.: Asymptotic stability of discrete-time systems with saturation nonlinearities with applications to digital filters. IEEE Trans. Circuits Syst. I: Fundam. Theory Appl. 39(10), 798–807 (1992).  https://doi.org/10.1109/81.199861CrossRefzbMATHGoogle Scholar
  19. 19.
    Rani, P., Kokil, P., Kar, H.: l2l suppression of limit cycles in interfered digital filters with generalized overflow nonlinearities. Circuits, Syst. Signal Process. 36(7), 2727–2741 (2017).  https://doi.org/10.1007/s00034-016-0433-1CrossRefzbMATHGoogle Scholar
  20. 20.
    Dey, A., Kar, H.: LMI-based criterion for the robust stability of 2D discrete state-delayed systems using generalized overflow nonlinearities. J. Control Sci. Eng. 23 (2011).  https://doi.org/10.1155/2011/271515MathSciNetCrossRefGoogle Scholar
  21. 21.
    Kar, H.: An LMI based criterion for the nonexistence of overflow oscillations in fixed-point state-space digital filters using saturation arithmetic. Digit. Signal Proc. 17(3), 685–689 (2007).  https://doi.org/10.1016/j.dsp.2006.11.003CrossRefGoogle Scholar
  22. 22.
    Kokil, P., Kandanvli, V.K.R., Kar, H.: A note on the criterion for the elimination of overflow oscillations in fixed-point digital filters with saturation arithmetic and external disturbance. AEU-Int. J. Electron. Commun. 66(9), 780–783 (2012).  https://doi.org/10.1016/j.aeue.2012.01.004CrossRefGoogle Scholar
  23. 23.
    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: Fundam. Theory Appl. 45(3), 311–313 (1998).  https://doi.org/10.1109/81.662720CrossRefGoogle Scholar
  24. 24.
    Nam, P.T., Pathirana, P.N., Trinh, H.: Discrete wirtinger-based inequality and its application. J. Franklin Inst. 352(5), 1893–1905 (2015).  https://doi.org/10.1016/j.jfranklin.2015.02.004MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Tadepalli, S.K., Kandanvli, V.K.R., Vishwakarma, A.: Criteria for stability of uncertain discrete-time systems with time-varying delays and finite wordlength nonlinearities. Trans. Inst. Meas. Control 40(9), 2868–2880 (2017).  https://doi.org/10.1177/0142331217709067CrossRefGoogle Scholar
  26. 26.
    Kokil, P., Parthipan, C.G., Jogi, S., Kar, H.: Criterion for realizing state-delayed digital filters subjected to external interference employing saturation arithmetic. Cluster Comput. 1–8 (2018)Google Scholar
  27. 27.
    Kokil, P., Arockiaraj, S.X., Kar, H.: Criterion for limit cycle-free state-space digital filters with external disturbances and generalized overflow nonlinearities. Trans. Inst. Meas. Control 40(4), 1158–1166 (2018).  https://doi.org/10.1177/0142331216680287CrossRefGoogle Scholar
  28. 28.
    Lofberg, J.: YALMIP: a toolbox for modeling and optimization in MATLAB. In: 2004 IEEE International Symposium on Computer Aided Control Systems Design, pp. 284–289. IEEE, New Orleans, LA, USA (2001).  https://doi.org/10.1109/cacsd.2004.1393890
  29. 29.
    Mary, T.J., Rangarajan, P.: Delay-dependent stability analysis of microgrid with constant and time-varying communication delays. Electr. Power Compon. Syst. 44(13), 1441–1452 (2016).  https://doi.org/10.1080/15325008.2016.1170078CrossRefGoogle Scholar
  30. 30.
    Razeghi-Jahromi, M., Seyedi, A.: Stabilization of distributed networked control systems with constant feedback delay. In: IEEE 52nd Annual Conference on Decision and Control (CDC), pp. 4619–4624. IEEE, Florence, Italy (2013).  https://doi.org/10.1109/cdc.2013.6760612

Copyright information

© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  • Pushpendra Kumar Gupta
    • 1
    Email author
  • V. Krishna Rao Kandanvli
    • 1
  1. 1.Department of Electronics and Communication EngineeringMotilal Nehru National Institute of Technology AllahabadPrayagrajIndia

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