Skip to main content
SpringerLink
Log in

l ∞ Fuzzy filter design for nonlinear systems with missing measurements: Fuzzy basis-dependent Lyapunov function approach

  • Regular Papers
  • Control Theory and Applications
  • Published:
International Journal of Control, Automation and Systems Aims and scope Submit manuscript

Abstract

In this paper, l ∞ fuzzy filtering problem is dealt for nonlinear systems with both persistent bounded disturbances and missing probabilistic sensor information. The Takagi–Sugeno (T–S) fuzzy model is adopted to represent a nonlinear dynamic system. The measurement output is assumed to contain randomly missing data, which is modeled by a Bernoulli distributed with a known conditional probability. To design the l ∞ fuzzy filter and guarantee tracking performance, the effect of the perturbation against persistent bounded disturbances is reduced by using the minimum l ∞ performance. By using the fuzzy basis-dependent Lyapunov function approach, a sufficient condition is established that ensure the mean square exponential stability of the filtering error. The proposed sufficient condition is represented as some linear matrix inequalities (LMIs), and the filter gain is obtained by the solution to a set of LMIs. Finally, the effectiveness of the proposed design method is shown via an example.

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. U. Shaked and N. Berman, “H ∞ nonlinear filtering of discrete-time process,” IEEE Trans. Signal Proces., vol. 43, no. 9, pp. 2205–2209, 1995.

    Article  Google Scholar 

  2. M. Vidyasagar, “Optimal rejection of persistent bounded disturbances,” IEEE Trans. Autom. Control, vol. 32, no. 6, pp. 527–534, 1986.

    Article  MathSciNet  MATH  Google Scholar 

  3. M. Vidyasagar, “Further results on the optimal rejection of persistent bounded disturbances,” IEEE Trans. Autom. Control, vol. 36, no. 6, pp. 642–652, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  4. D. O. Brian Anderson, and J. B. Moore, Optimal Filtering, Dover Publications, Inc., 1979.

    MATH  Google Scholar 

  5. L. Chisci, J. A. Rossiter, and G. Zappa, “Systems with persistent disturbances: Predictive control with restricted constraints,” Automatica, vol. 37, no. 7, pp. 1019–1028, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  6. J. S. Shamma, “Nonlinear state-feedback for l ∞ optimal control,” Syst. Control Lett., vol. 21, pp. 264–270, 1993.

    Article  MathSciNet  Google Scholar 

  7. J. S. Shamma and M. A. Dahleh, “Time-varying versus time-invariant compensation for rejection of persistent bounded disturbances and robust stabilization,” IEEE Trans. Autom. Control, vol. 36, no. 7, pp. 838–847, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  8. C. T. Lin and C. F. Juang, “An adaptive neural fuzzy filter and its applications,” IEEE Trans. Syst. Man, Cybern. Part B, vol. 27, no. 4, pp. 635–656, 1997.

    Article  Google Scholar 

  9. C. S. Tseng, “Robust fuzzy filter design for nonlinear systems with persistent bounded disturbances,” IEEE Trans. Syst. Man, Cybern. Part B, vol. 36, no. 4, pp. 940–945, 2006.

    Article  Google Scholar 

  10. D. H. Lee and Y. H. Joo, “Extended robust H 2 and H ∞ filter design for discrete-time invariant linear systems,” Circuits Syst. Signal Proces., vol. 33, no. 2, pp. 393–419, 2014.

    Article  Google Scholar 

  11. D. H. Lee, Y. H. Joo, and M. H. Tak, “Periodically timevarying H ∞ memory filter design for discrete-time LTI systems with polytopic uncertainty,” IEEE Trans. Autom. Control, vol. 59, no. 5, pp. 1380–1385, 2014.

    Article  MathSciNet  Google Scholar 

  12. D. C. W. Ramos and P. L. D. Peres, “An LMI condition for the robust stability of uncertain continuous-time linear systems,” IEEE Trans. Autom. Control, vol. 47, no. 4, pp. 675–678, 2002.

    Article  MathSciNet  Google Scholar 

  13. Z. Wang, F. Yang, D.W. C. Ho, and X. Liu, “Robust H ∞ filtering for stochastic time-delay systems with missing measurements,” IEEE Trans. Signal Proces., vol. 54, no. 7, pp. 594–599, 2006.

    Google Scholar 

  14. H. Gao, Y. Zhao, J. Lam, and K. Chen, “H ∞ fuzzy filtering of nonlinear systems with intermittent measurements,” IEEE Trans. Fuzzy Syst., vol. 17, no. 2, pp. 291–300, 2009.

    Article  Google Scholar 

  15. D. H. Lee, Y. H. Joo, and M. H. Tak, “Linear matrix inequality approach to local stability analysis of discretetime Takagi–Sugeno fuzzy systems,” IET Control Theory Appl., vol. 7, no. 9, pp. 1309–1318, 2013.

    Article  MathSciNet  Google Scholar 

  16. D. H. Lee, Y. H. Joo, and M. H. Tak, “Local stability analysis of continuous-time Takagi–Sugeno fuzzy systems: A fuzzy Lyapunov function approach,” Inf. Sci., vol. 257, no. 1, pp. 163–175, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  17. D. H. Lee and Y. H. Joo, “On the generalized local stability and local stabilization conditions for discrete-time Takagi–Sugeno fuzzy systems,” IEEE Trans. Fuzzy Syst., vol. 22, no. 6, pp. 1654–1668, 2014.

    Article  Google Scholar 

  18. D. H. Lee, J. B. Park, and Y. H. Joo, “Approaches to extended non-quadratic stability and stabilization conditions for discrete-time Takagi–Sugeno fuzzy systems,” Automatica, vol. 47, no. 3, pp. 534–538, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  19. G. B. Koo, J. B. Park, and Y. H. Joo, “Intelligent digital redesign for nonlinear systems using a guaranteed cost control method,” Int. J. Control, Autom. Syst., vol. 11, no. 6, pp. 1075–1083, 2013.

    Article  Google Scholar 

  20. H. C. Sung, J. B. Park, and Y. H. Joo, “Observer-based sampled-data control for nonlinear systems: Robust intelligent digital redesign approach,” Int. J. Control, Autom. Syst., vol. 12, no. 4, pp. 486–496, 2014.

    Article  Google Scholar 

  21. M. K. Song, J. B. Park, and Y. H. Joo, “Stability and stabilization for discrete-time Markovian jump fuzzy systems with time-varying delays: Partially known transition probabilities case,” Int. J. Control, Autom. Syst., vol. 11, no. 1, pp. 136–146, 2013.

    Article  Google Scholar 

  22. J. B. Burl, “H ∞ estimation for nonlinear systems,” IEEE Trans. Signal Proces. Lett., vol. 5, no. 8, pp. 199–202, 1998.

    Article  Google Scholar 

  23. B. S. Chen, C. L. Tsai, and Y. F. Chen, “Mixed H 2/H ∞ fil-tering design in multirate trans-multiplexer systems: LMI approach,” IEEE Trans. Signal Proces., vol. 49, no. 11, pp. 2693–2701, 2001.

    Article  Google Scholar 

  24. W. Zhang, B. S. Chen, and C. S. Tseng, “H ∞ robust filter-ing for nonlinear stochastic systems,” IEEE Trans. Signal Proces., vol. 53, no. 2, pp. 589–598, 2005.

    Article  MathSciNet  Google Scholar 

  25. C. S. Tseng, “Robust fuzzy filter design for a class of non-linear stochastic systems,” IEEE Trans. Fuzzy Syst., vol. 15, no. 2, pp. 261–274, 2007.

    Article  Google Scholar 

  26. C. Weber and J. Allebach, “Convolution and Hankel opera-tor norms for linear systems,” IEEE Trans. Autom. Control, vol. 34, no. 1, pp. 94–98, 1989.

    Article  Google Scholar 

  27. H. J. Lee, J. B. Park, and G. Chen, “Robust fuzzy control of nonlinear systems with parametric uncertainties,” IEEE Trans. Fuzzy Syst., vol. 9, no. 2, pp. 369–379, 2001.

    Article  Google Scholar 

  28. Z. Wang, D. W. C. Ho, and X. Liu, “Variance-constrained filtering for uncertain stochastic systems with missing mea-surements,” IEEE Trans. Autom. Control, vol. 48, no. 7, pp. 1254–1258, 2003.

    Article  MathSciNet  Google Scholar 

  29. Z. Wang, D. W. C. Ho, and X. Liu, “Variance-constrained control for uncertain stochastic systems with missing mea-surements,” IEEE Trans. Syst. Man, Cybern. Part A, vol. 35, no. 5, pp. 746–753, 2005.

    Article  Google Scholar 

  30. B. Sinopoli, L. Schenato, and M. Franceschetti, “Kalman filtering with intermittent observations,” IEEE Trans. Au-tom. Control, vol. 49, no. 9, pp. 1453–1464, 2004.

    Article  MathSciNet  Google Scholar 

  31. H. Katayama and A. Ichikawa, “H ∞ control for sampled-data fuzzy systems,” Proc. American Control Conf., 2003, pp. 4237–4242.

    Google Scholar 

  32. J. V. D. Oliveira, J. Bernussou, and J. C. Geromel, “A new discrete-time robust stability condition,” Syst. Control Lett., vol. 37, pp. 261–265, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  33. J. Daafouz and J. Bernussou, “Parameter dependent Lya-punov functions for discrete time systems with time-varying parametric uncertainties,” Syst. Control Lett., vol. 43, pp. 355–359, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  34. B.-S. Chen, “Optimal tracking design for stochastic fuzzy systems,” IEEE Trans. Fuzzy Syst., vol. 11, no. 6, pp. 796–813, 2003.

    Article  Google Scholar 

  35. D. W. Kim, H. J. Lee, and J. B. Park, “Fuzzy stabilization of nonlinear systems under sampled-data feedback: An exact discrete-time model approach,” IEEE Trans. Fuzzy Syst., vol. 18, no. 2, pp. 251–260, 2010.

    MathSciNet  Google Scholar 

  36. A. N. Shiryayev, Probability, Graduate Texts in Mathematics, Springer-Verlag, New York, 1984.

    Book  Google Scholar 

  37. S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia PA, 1994.

    Book  MATH  Google Scholar 

  38. E. Kim and H. Lee, “New approaches to relaxed quadratic stability condition of fuzzy control systems,” IEEE Trans. Fuzzy Syst., vol. 8, no. 5, pp. 523–534, 2000.

    Article  Google Scholar 

  39. R. Skelton, T. Iwasaki, and K. Grigoriadis, A Unified Alge-braic Approach to Linear Control Design, Taylor and Francis, London, 1997.

    Google Scholar 

  40. S. Son, J. B. Park, and Y. H. Joo, “Fuzzy c-means clustering-based smart tracking model for three-dimensional manoeuvring target including un-known acceleration input,” IET Radar, Sonar & Navigation, vol. 7, no. 6, pp. 623–634, 2013.

    Article  Google Scholar 

  41. H. Lee, M. H. Tak, and Y. H. Joo, “A Lyapunov functional approach to robust stability analysis of continuous-time uncertain linear systems in polytopic domains,” Int. J. Control, Autom. Syst., vol. 11, no. 3, pp. 460–469, 2013.

    Article  Google Scholar 

  42. D. H. Lee, M. H. Tak, and Y. H. Joo, “A Lyapunov functional approach to robust stability analysis of continuoustime uncertain linear systems in polytopic domains,” Int. J. Control, Autom. Syst., vol. 11, no. 3, pp. 460–469, 2013.

    Article  Google Scholar 

  43. D. H. Lee, Y. H. Joo, M. H. Tak, “Periodically time-varying memory static output feedback control design for discretetime LTI systems,” Automatica, vol. 52, no. 1, pp. 47–54, 2015.

    Article  MathSciNet  MATH  Google Scholar 

  44. H. S. Son, J. B. Park, and Y. H. Joo, “Segmentalized FCM-based tracking algorithm for zigzag maneuvering target,” Int. J. Control, Autom. Syst., vol. 13, no. 1, pp. 231–237, 2015.

    Article  Google Scholar 

  45. S. Y. Noh, J. B. Park, and Y. H. Joo, “L ∞ fuzzy filter for non-linear systems with intermittent measurement and persistent bounded disturbances,” IET Radar, Sonar & Navigation, vol. 7, no. 5, pp. 489–496, 2013.

    Article  Google Scholar 

  46. G. B. Koo, J. B. Park, and Y. H. Joo, “LMI condition for sampled-data fuzzy control of nonlinear systems,” Electronics Letters, vol. 51, no. 1, pp. 29–31, 2015.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Nuclear Fusion Technology Division, Korea Atomic Energy Research Institute, Daejeon, 305-353, Korea

    Sun Young Noh

  2. Department of Electrical and Electronic Engineering, Yonsei University, Seoul, 120-749, Korea

    Geun Bum Koo & Jin Bae Park

  3. Department of Control and Robotics Engineering, Kunsan National University, Kunsan, Chonbuk, 573-701, Korea

    Young Hoon Joo

Authors
  1. Sun Young Noh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Geun Bum Koo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jin Bae Park
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Young Hoon Joo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jin Bae Park.

Additional information

Recommended by Associate Editor Do Wan Kim under the direction of Editor Euntai Kim. This work was supported by the national Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (NRF-2015R1A2A2A05001610).

Sun Young Noh received the B.S. degree in Electrical and Electronic Engineering from Myongji University, Gyeonggi-do, Korea, in 1999, the M.S. degree in Electrical Engineering and Ph.D. degree in Electrical and Electronic Engineering from Yonsei University, Seoul, Korea, in 2005 and 2013 respectively. She is currently a researcher in the Nuclear Technology Fusion Department, Korea Atomic Energy Research Institute, Daejeon. Her research interests include optimal state estimation, energy-efficient actuator design and control.

Geun Bum Koo received the B.S. and Ph.D. degrees in Electrical and Electronic Engineering from Yonsei University, Seoul, Korea, in 2007 and 2015, respectively. His current research interests include largescale systems, decentralized control, sampled-data control, intelligent digital redesign, nonlinear control, and fuzzy systems.

Jin Bae Park received the B.S. degree in electrical engineering from Yonsei University, Seoul, Korea, and the M.S. and Ph.D. degrees in electrical engineering from Kansas State University, Manhattan, KS, USA, in 1977, 1985, and 1990, respectively. Since 1992, he has been with the Department of Electrical and Electronic Engineering, Yonsei University, where he is currently a Professor. His major research interests include robust control and filtering, nonlinear control, intelligent mobile robot, fuzzy logic control, neural networks, adaptive dynamic programming, chaos theory, and genetic algorithms. Dr. Park served as the Editor-in-Chief for the International Journal of Control, Automation, and Systems (IJCAS) (2006–2010) and the President for the Institute of Control, Robot, and Systems Engineers (ICROS) (2013). He was served as the Senior Vice-Present for Yonsei University(2014–2015).

Young Hoon Joo received the B.S., M.S., and Ph.D. degrees in Electrical Engineering from Yonsei University, Seoul, Korea, in 1982, 1984, and 1995, respectively. He worked with Samsung Electronics Company, Seoul, Korea, from 1986 to 1995, as a project manager. He was with the University of Houston, Houston, TX, from 1998 to 1999, as a visiting professor in the Department of Electrical and Computer Engineering. He is currently a professor in the Department of Control and Robotics Engineering, Kunsan National University, Korea. His major interest is mainly in the field of intelligent robot, intelligent control, human-robot interaction, intelligent surveillance systems, and wind farm control. He severed as the President for Korea Institute of Intelligent Systems (KIIS) (2008–2009) and he severed as President for Korea Institute of Intelligent Systems (KIIS) (2008–2009) and is serving as the Vice-President for the Korean Institute of Electrical Engineers (KIEE) (2016-present)) and Institute of Control, Robotics and Systems (ICROS)(2016-present), and the Editor-in-Chief for the International Journal of Control, Automation, and Systems (IJCAS) (2014-present).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Noh, S.Y., Koo, G.B., Park, J.B. et al. l ∞ Fuzzy filter design for nonlinear systems with missing measurements: Fuzzy basis-dependent Lyapunov function approach. Int. J. Control Autom. Syst. 14, 425–434 (2016). https://doi.org/10.1007/s12555-014-0535-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12555-014-0535-0

Keywords

Search

Navigation