Advertisement

Robust \(H_{\infty }\) Filters for Uncertain Systems with Finite Frequency Specifications

  • Abderrahim El-amrani
  • Bensalem BoukiliEmail author
  • Abdelaziz Hmamed
  • El Mostafa El Adel
Article

Abstract

This paper deals with \(H_{\infty }\) filtering problem of linear discrete-time uncertain systems with finite frequency input signals. The uncertain parameters are supposed to reside in a polytope. By applying the generalized Kalman–Yakubovich–Popov lemma, polynomially parameter-dependent Lyapunov function and some key matrices to eliminate the product terms between the filter parameters and the Lyapunov matrices, an improved condition is obtained for analyzing the \(H_{\infty }\) performance of the filtering error system. Then sufficient condition in terms of linear matrix inequality is established for designing filters with a guaranteed \(H_{\infty }\) filtering performance level. Finally, a numerical examples are used to demonstrate the effectiveness of the proposed method.

Keywords

Discrete time system Finite frequency GKYP lemma \(H_{\infty }\) Filtering Uncertain systems Linear matrix inequalities (LMIs) 

References

  1. Benzaouia, A., Hmamed, A., & Tadeo, F. (2016). Two-dimensional systems. Studies in Systems Decision and Control, 28. doi: 10.1007/978-3-319-20116-0-3.
  2. Boukili, B., Hmamed, A., Benzaouia, A., & El Hajjaji, A. (2013). \(H_{\infty }\) filtering of two-dimensional T–S fuzzy systems. Circuits, Systems and Signal Processing, 33(6), 1737–1761.CrossRefMathSciNetGoogle Scholar
  3. Boukili, B., Hmamed, A., & Tadeo, F. (2016). Robust \(H_{\infty }\) filtering for 2-D discrete roesser systems. Journal of Control, Automation and Electrical Systems, 27(5), 497–505. doi: 10.1007/s40313-016-0251-5.CrossRefGoogle Scholar
  4. Boukili, B., Hmamed, A., & Tadeo, F. (2016). Reduced-order \(H_{\infty }\) filtering with intermittent measurements for a class of 2D systems. Journal of Control, Automation and Electrical Systems, 27(6), 597–607.CrossRefGoogle Scholar
  5. Chang, X. H., Park, J. H., & Tang, Z. (2015). New approach to \(H_{\infty }\) filtering for discrete-time systems with polytopic uncertainties. Signal Processing, 113, 147–158.CrossRefGoogle Scholar
  6. Chen, Y., Zhang, W., & Gao, H. (2010). Finite frequency \(H_{\infty }\) control for building under earthquake excitation. Mechatronics, 20(1), 128–142.CrossRefGoogle Scholar
  7. Duan, Z., Zhang, J., Zhang, C., & Mosca, E. (2006). Robust \(H_{2}\) and \(H_{\infty }\) filtering for uncertain linear systems. Automatica, 42(11), 1919–1926.CrossRefzbMATHMathSciNetGoogle Scholar
  8. Dong, J., & Yang, G. H. (2013). Robust static output feedback control synthesis for linear continuous systems with polytopic uncertainties. Automatica, 49, 1821–1829.CrossRefzbMATHMathSciNetGoogle Scholar
  9. De Oliveira, M. C., Bernussou, J., & Geromel, J. C. (1999). A new discrete-time robust stability condition. Systems & Control Letters, 37(4), 261–265.CrossRefzbMATHMathSciNetGoogle Scholar
  10. De Souza, C., Xie, L., & Coutinho, D. (2010). Robust filtering for 2D discrete-time linear systems with convex-bounded parameter uncertainty. Automatica, 46(4), 673–681.CrossRefzbMATHMathSciNetGoogle Scholar
  11. Ding, D., & Yang, G. (2009). Finite frequency \(H_{\infty }\) filtering for uncertain discrete-time switched linear systems. Progress in Natural Science, 19(11), 1625–1633.CrossRefMathSciNetGoogle Scholar
  12. Li, X., Lam, J., Gao, H., & Xiong, J. (2016). \(H_{\infty }\) and \(H_{2}\) filtering for linear systems with uncertain Markov transitions. Automatica, 67, 252–266.CrossRefzbMATHMathSciNetGoogle Scholar
  13. El-Kasri, C., Hmamed, A., & Tadeo, F. (2013). Reduced-order \(H_{\infty }\) filters for uncertain 2D continuous systems, via LMIs and polynomial matrices. Journal of Circuits Systems and Signal Processing, 33(4), 1189–1214.CrossRefGoogle Scholar
  14. El-amrani, A., Hmamed, A., Boukili, B. & El hajaji, A. (2016). \(H_{\infty }\) Filtering of T–S fuzzy systems in finite frequency domain. In 5th International conference on systems and control (ICSC). doi: 10.1109/ICoSC.2016.7507038.
  15. Elsayed, A., & Grimble, M. J. (1989). A new approach to \(H_{\infty }\) design of optimal digital linear filters. IMA Journal of Mathematical Control and Information, 6(2), 233–251.CrossRefzbMATHMathSciNetGoogle Scholar
  16. Gao, H., & Li, X. (2014). Robust filtering for uncertain systems. A parameter-dependent approach. New York: Springer - Communications and Control Engineering Series.CrossRefzbMATHGoogle Scholar
  17. Gao, C. Y., Duan, G. R., & Meng, X. Y. (2008). Robust \(H_{\infty }\) filter design for 2D discrete systems in Roesser model. International Journal of Automation and Computing, 5(4), 413–418.CrossRefGoogle Scholar
  18. Gao, H., & Li, X. (2011). \(H_{\infty }\) filtering for discrete-time state-delayed systems with finite frequency specifications. IEEE Transactions on Automatic Control, 56(12), 2935–2941.CrossRefzbMATHMathSciNetGoogle Scholar
  19. Geromel, J. C., & Levin, G. (2006). Suboptimal reduced-order filtering through an LMI-based method. IEEE Transactions on Signal Processing, 54(7), 2588–2595.CrossRefzbMATHGoogle Scholar
  20. Grimble, M., & El Sayed, A. (1990). Solution of the \(H_{\infty }\) optimal linear filtering problem for discrete-time systems. IEEE Transactions on Acoustics, Speech, and Signal Processing, 38(7), 1092–1104.CrossRefzbMATHMathSciNetGoogle Scholar
  21. Iwasaki, T., & Hara, S. (2005). Generalized KYP lemma: Unified frequency domain inequalities with design applications. IEEE Transactions on Automatic Control, 50(1), 41–59.CrossRefzbMATHMathSciNetGoogle Scholar
  22. Iwasaki, T., Hara, S., & Fradkov, A. L. (2005). Time domain interpretations of frequency domain inequalities on (semi) finite ranges. Systems Control Letters, 54(7), 681–691.CrossRefzbMATHMathSciNetGoogle Scholar
  23. Iwasaki, T., Hara, S., & Fradkov, A. L. (2011). Restricted frequency inequalities equivalent to restricted dissipativity. In The 43rd IEEE conference on decision and control, pp. 14–17.Google Scholar
  24. Lee, D. H. (2013). An improved finite frequency approach to robust \(H_{\infty }\) filter design for LTI systems with polytopic uncertainties. International Journal of Adaptive Control and Signal Processing, 27(11), 944–956.CrossRefzbMATHMathSciNetGoogle Scholar
  25. Li, H., & Fu, M. (1977). A linear matrix inequality approach to robust \(H_{\infty }\) filtering. IEEE Transactions on Signal Processing, 45(9), 2338–2350.Google Scholar
  26. Li, X. J., & Yang, G. H. (2014). Fault detection in finite frequency domain for TakagiSugeno fuzzy systems with sensor faults. IEEE Transactions on Cybernetics, 44(8), 1446–1458.CrossRefGoogle Scholar
  27. Li, X., & Gao, H. (2012). Robust finite frequency \(H_{\infty }\) filtering for uncertain 2-D Roesser systems. Automatica, 48(6), 1163–1170.CrossRefzbMATHMathSciNetGoogle Scholar
  28. Li, X., & Gao, H. (2013). Robust finite frequency \(H_{\infty }\) filtering for uncertain 2-D systems: The FM model case. Automatica, 49, 2446–2452.CrossRefzbMATHMathSciNetGoogle Scholar
  29. Lacerda, M. J., Oliveira, R. C. L. F., & Peres, P. L. D. (2011). Robust \(H_{2}\) and \(H_{\infty }\) filter design for uncertain linear systems via LMIs and polynomial matrices. Signal Processing, 91(5), 1115–1122.CrossRefzbMATHGoogle Scholar
  30. Lofberg, J. (2004). YALMIP: A toolbox for modeling and optimization in MATLAB. In Proceedings of the IEEE computer-aided control system design conference, Taipei, pp. 284-289.Google Scholar
  31. Nagpal, K. M., & Khargonekar, P. P. (1991). Filtering and smoothing in an \(H_{\infty }\) setting. IEEE Transactions on Automatic Control, 36(2), 152–166.CrossRefzbMATHMathSciNetGoogle Scholar
  32. Palhares, R. M., & Peres, P. L. D. (2001). LMI approach to the mixed \(H_{2}/H_{\infty }\) filtering design for discrete-time uncertain systems. IEEE Transactions on Aerospace and Electronic Systems, 37(1), 292–296.CrossRefGoogle Scholar
  33. Qiu, J., Feng, G., & Yang, J. (2008). Robust mixed \(H_{2}/H_{\infty }\) filtering design for discrete-time switched polytopic linear systems. The Institution of Engineering and Technology, 2(5), 420–430.MathSciNetGoogle Scholar
  34. Romao, L. B. R. R., De Oliveira, M. C., Peres, P. L. D. & Oliveira, R. C. L. F. (2016). State-feedback and filtering problems using the generalized KYP lemma. In IEEE conference on computer aided control system design (CACSD), pp. 1054–1059.Google Scholar
  35. Rotstein, H., Sznaier, M. & Idan, M. (1996). \(H_{2}/H_{\infty }\) filtering theory and an aerospace application. International Journal of Robust and Nonlinear Control, 6(4), 347–366.Google Scholar
  36. Sturm, J. F. (1999). Using SeDuMi 1.02. A MATLAB toolbox for optimization over symmetric cones. Optimization Methods and Software, 11(1–4), 625–653.Google Scholar
  37. Takaba, K., & Katayama, T. (1996). Discrete-time \(H_{\infty }\) algebraic Riccati equation and parametrization of all H filters. International Journal of Control, 64(6), 1129–1149.CrossRefzbMATHMathSciNetGoogle Scholar
  38. Wang, H., Peng, L. Y., Ju, H. H., & Wang, Y. L. (2013). \(H_{\infty }\) state feedback controller design for continuous-time T–S fuzzy systems in finite frequency domain. Information Sciences, 223, 221–235.Google Scholar

Copyright information

© Brazilian Society for Automatics--SBA 2017

Authors and Affiliations

  • Abderrahim El-amrani
    • 1
  • Bensalem Boukili
    • 1
    Email author
  • Abdelaziz Hmamed
    • 1
  • El Mostafa El Adel
    • 2
  1. 1.Department of Physics, Faculty of Sciences Dhar El MehrazUniversity of Sidi Mohamed Ben AbdellahFez-AtlasMorocco
  2. 2.LSIS-UMR 6168University of Paul CzanneAix-MarseilleFrance

Personalised recommendations