Skip to main content
SpringerLink
Log in

ℋ ∞ Filter for Uncertain Markovian Jump Nonlinear Systems: An LMI Approach

  • Published:
Circuits, Systems & Signal Processing Aims and scope Submit manuscript

Abstract

The paper addresses the problem of designing a robust filter for a class of uncertain Markovian jump nonlinear systems described by a Takagi–Sugeno fuzzy model with Markovian jumps. LMI-based sufficient conditions for the existence of a robust filter that guarantees the ℒ2-gain from an exogenous input to an estimation error to be less than a prescribed value are derived. A tunnel diode circuit is used to illustrate the effectiveness of the proposed design techniques.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aliyu, M.D.S., Boukas, E.K.: H ∞ control for Markovian jump nonlinear systems. In: 37th IEEE Conference on Decision and Control, pp. 766–771, Tampa, FL, Dec. 1998

  2. Arrifano, N.S.D., Oliveira, V.A.: Robust ℋ∞ fuzzy control design for Markovian jump systems. In: Proc. 44th IEEE Conf. on Decision and Control, and European Control Conference, CDC-ECC ’05, pp. 3243–3248, Seville, Spain, Dec. 2005

  3. Assawinchaichote, W., Nguang, S.K.: Fuzzy observer-based controller design for singularly perturbed nonlinear systems: an LMI approach. In: Proc. 41st IEEE Conference on Decision and Control, pp. 2165–2170, Las Vegas, NV, Dec. 2002

  4. Assawinchaichote, W., Nguang, S.K.: ℋ∞ filtering for nonlinear singularly perturbed systems with pole placement constraints: an LMI approach. IEEE Trans. Signal Process. 52(6), 1659–1667 (2004)

    Article  MathSciNet  Google Scholar 

  5. Assawinchaichote, W., Nguang, S.K.: Robust ℋ∞ state-feedback control for uncertain fuzzy Markovian jump systems: LMI-based design. Int. J. Comput. Intell. 3, 187–194 (2006)

    Google Scholar 

  6. Boukas, E.K., Haurie, A.: Manufacturing flow control and preventive maintenance: a stochastic control approach. IEEE Trans. Autom. Control 35, 1024–1031 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  7. Boukas, E.K., Shi, P.: Stochastic stability and guaranteed cost control of discrete-time uncertain systems with Markovian jumping parameters. Int. J. Robust Nonlinear Control 8, 1155–1167 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  8. Boukas, E.K., Zhang, Q., Yin, G.: Robust production and maintenance planning in stochastic manufacturing systems. IEEE Trans. Autom. Control 40, 1098–1102 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  9. Boukas, E.K., Shi, P., Nguang, S.K., Agarwal, R.K.: Robust H ∞ control of a class of nonlinear systems with Markovian jumping parameters. In: In Proc. IEEE American Control Conference, pp. 970–976, San Diego, CA, June 1999

  10. Boukas, E.K., Shi, P., Nguang, S.K.: Robust control for linear Markovian jump systems with unknown nonlinearities. J. Math. Anal. Appl. 282(1), 241–255 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  11. Cao, S.G., Ree, N.W., Feng, G.: Quadratic stability analysis and design of continuous-time fuzzy control systems. Int. J. Syst. Sci. 27, 193–203 (1996)

    Article  MATH  Google Scholar 

  12. Chen, B.S., Tseng, C.S., Uang, H.J.: Mixed H 2/H ∞ fuzzy output feedback control design for nonlinear dynamic systems: an LMI approach. IEEE Trans. Fuzzy Syst. 8, 249–265 (2000)

    Article  Google Scholar 

  13. Chen, C.L., Chen, P.C., Chen, C.K.: Analysis and design of fuzzy control system. Fuzzy Sets Syst. 57, 125–140 (1993)

    Article  MATH  Google Scholar 

  14. de Souza, C.E., Fragoso, M.D.: H ∞ control for linear systems with Markovian jumping parameters. Control Theory Adv. Technol. 9, 457–466 (1993)

    Google Scholar 

  15. Dynkin, E.B.: Markov Process. Springer, Berlin (1965)

    Google Scholar 

  16. Feng, X.K., Loparo, K.A., Ji, Y., Chizeck, H.J.: Stochastic stability properties of jump linear system. IEEE Trans. Autom. Control 37, 38–53 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  17. Ito, K., McKean, H.P.: Diffusion Processes and Their Sample Paths. Springer, Berlin (1965)

    MATH  Google Scholar 

  18. Ji, Y., Chizeck, H.J.: Controllability, stabilizability, and continuous-time Markovian jump linear quadratic control. IEEE Trans. Autom. Control 35, 777–788 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  19. Krasovskii, N.N., Lidskii, E.A.: Analysis design of controller in systems with random attributes—Part 1. Autom. Remote Control 22, 1021–1025 (1961)

    MathSciNet  Google Scholar 

  20. Ma, X.J., Sun, Z.Q., He, Y.Y.: Analysis and design of fuzzy controller and fuzzy observer. IEEE Trans. Fuzzy Syst. 6, 41–51 (1998)

    Article  Google Scholar 

  21. Nguang, S.K.: Robust nonlinear ℋ∞ output feedback control. IEEE Trans Autom. Control 41, 1003–1008 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  22. Nguang, S.K., Assawinchaichote, W.: ℋ∞ filtering for fuzzy dynamical systems with pole placement constraints. IEEE Trans. Circuits Syst. Part 1 5(11), 1503–1508 (2003)

    Article  MathSciNet  Google Scholar 

  23. Nguang, S.K., Shi, P.: Stabilisation of a class of nonlinear time-delay systems using fuzzy models. In: Proc. 39th IEEE Conf. on Decision and Control, pp. 4415–4419, Sydney, Australia, 2000

  24. Nguang, S.K., Shi, P.: ℋ∞ fuzzy output feedback control design for nonlinear systems: an LMI approach. IEEE Trans. Fuzzy Syst. 11, 331–340 (2003)

    Article  Google Scholar 

  25. Pan, G., Bar-Shalom, Y.: Stabilization of jump linear Gaussian systems without mode observations. Int. J. Control 6, 631–661 (1996)

    Article  MathSciNet  Google Scholar 

  26. Pan, G., Bar-Shalom, Y.: A stabilizing controller for jump linear Gaussian systems with noisy state observations. Eur. J. Control 2, 227–238 (1996)

    MATH  Google Scholar 

  27. Pan, Z., Basar, T.: Robustness of minimax controllers to nonlinear perturbations. J. Optim. Theory Appl. 87, 631–678 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  28. Sethi, S.P., Zhang, Q.: Hierarchical Decision Making in Stochastic Manufacturing Systems. Birkhäuser, Boston (1994)

    MATH  Google Scholar 

  29. Shi, P., Boukas, E.K.: H ∞ control for Markovian jumping linear systems with parametric uncertainty. J. Optim. Theory Appl. 95, 75–99 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  30. Shi, P., Boukas, E.K., Agarwal, R.K.: Control of Markovian jump discrete-time systems with norm bounded uncertainty and unknown delays. IEEE Trans. Autom. Control 44, 2139–214 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  31. Shi, P., Boukas, E.K., Agarwal, R.K.: Robust control for Markovian jumping discrete-time systems. Int. J. Syst. Sci. 30, 787–797 (1999)

    Article  MATH  Google Scholar 

  32. Takagi, T., Sugeno, M.: Fuzzy identification of systems and its applications to modelling and control. IEEE Trans. Syst. Man. Cybern. 15, 116–132 (1985)

    MATH  Google Scholar 

  33. Tanaka, K.: Stability and stabilizability of fuzzy neural linear control systems. IEEE Trans. Fuzzy Syst. 3, 438–447 (1995)

    Article  Google Scholar 

  34. Tanaka, K., Sugeno, M.: Stability analysis and design of fuzzy control systems. Fuzzy Sets Syst. 45, 135–156 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  35. Tanaka, K., Ikeda, T., Wang, H.O.: Robust stabilization of a class of uncertain nonlinear systems via fuzzy control: quadratic stabilizability, H ∞ control theory, and linear matrix inequality. IEEE Trans. Fuzzy Syst. 4, 1–13 (1996)

    Article  Google Scholar 

  36. Teixeira, M., Zak, S.H.: Stabilizing controller design for uncertain nonlinear systems using fuzzy models. IEEE Trans. Fuzzy Syst. 7, 133–142 (1999)

    Article  Google Scholar 

  37. Wang, H.O., Tanaka, K., Griffin, M.F.: An approach to fuzzy control of nonlinear systems: stability and design issues. IEEE Trans. Fuzzy Syst. 4, 14–23 (1996)

    Article  Google Scholar 

  38. Wang, L.X.: A Course in Fuzzy Systems and Control. Prentice-Hall, Englewood Cliffs (1997)

    MATH  Google Scholar 

  39. Wonham, W.M.: Random differential equations in control theory. Probab. Methods Appl. Math. 2, 131–212 (1970)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical and Computer Engineering, University of Auckland, Private Bag 92019, Auckland, New Zealand

    Sing Kiong Nguang

  2. Department of Electronic and Telecommunication Engineering, King Mongkut’s University of Technology Thonburi, 91 Suksawads 48 Rd, Bangkok, 10140, Thailand

    Wudhichai Assawinchaichote

  3. Division of Mathematics and Statistics, School of Technology, University of Glamorgan, Pontypridd, Wales, CF37 1DL, UK

    Peng Shi

Authors
  1. Sing Kiong Nguang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wudhichai Assawinchaichote
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Peng Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sing Kiong Nguang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nguang, S.K., Assawinchaichote, W. & Shi, P. ℋ ∞ Filter for Uncertain Markovian Jump Nonlinear Systems: An LMI Approach. Circuits Syst Signal Process 26, 853–874 (2007). https://doi.org/10.1007/s00034-007-9002-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-007-9002-y

Keywords

Search

Navigation