Nonlinear Dynamics

, Volume 57, Issue 3, pp 401–410 | Cite as

Stability and implementable ℋ filters for singular systems with nonlinear perturbations

Original Paper


In this paper, we investigate the problem of designing ℋ filter for a class of continuous-time uncertain singular systems with nonlinear perturbations, which can be realized in practice. The perturbation is a time-varying function of the system state and satisfies a Lipschitz constraint. The design objective is to guarantee that a prescribed upper bound on an ℋ performance of the robust filter is attained for all possible energy-bounded input disturbances and all admissible uncertainties and which can be implemented on-line to get a good replica of the state. We first establish sufficient condition for the existence and uniqueness of solution to the singular system connected with the normal filter. Using a linear matrix inequality (LMI) format, we then provide a sufficient condition for the asymptotic stability of the realizable ℋ filter. Then by means of a convex analysis procedure the filter gain matrices are derived and an important special case is readily deduced. Finally, a numerical example is presented to illustrate the theoretical developments.


filter Singular systems Robustness LMIs 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Anderson, B.D.O., Moore, J.B.: Optimal Filtering. Prentice Hall, New York (1979) MATHGoogle Scholar
  2. 2.
    Aplevich, J.D.: Implicit Linear Systems. Springer, Berlin (1991) MATHCrossRefGoogle Scholar
  3. 3.
    Bernstein, D.S., Haddad, W.M.: Steady-state Kalman filtering with an ℋ error bound. Syst. Control Lett. 16, 309–317 (1991) CrossRefMathSciNetGoogle Scholar
  4. 4.
    Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia (1994) MATHGoogle Scholar
  5. 5.
    Campbell, S.L.: Singular System of Differential Equations. Pitman, San Francisco (1982) Google Scholar
  6. 6.
    Chen, S.J., Lin, J.L.: Robust stability of discrete time-delay uncertain singular systems. In: IEE Proceedings Control Theory and Applications, vol. 151, pp. 45–51 (2004) Google Scholar
  7. 7.
    Dai, L.: Singular Control Systems. Springer, Berlin (1989) MATHCrossRefGoogle Scholar
  8. 8.
    El Ghaoui, L., Oustry, F., Ait Rami, M.: A cone complementarity linearization algorithm for static output feedback control and related problems. IEEE Trans. Automat. Control. 42, 1171–1176 (1997) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Fu, M., de Souza, C.E., Xie, L.: H -estimation for uncertain systems. Int. J. Robust Nonlinear Control 2, 87–105 (1992) MATHCrossRefGoogle Scholar
  10. 10.
    Gahinet, P., Apkarian, P.: A linear matrix inequality approach to ℋ control. Int. J. Robust Nonlinear Control 4, 421–448 (1994) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Geromel, J.C.: Optimal linear filtering with parameter uncertainty. IEEE Trans. Signal Process. 47, 168–175 (1999) MATHCrossRefGoogle Scholar
  12. 12.
    Ho, D.W.C., Lu, G.: Robust stabilization of a class of discrete-time nonlinear systems via output feedback: the unified LMI approach. Int. J. Control 76, 105–115 (2003) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Khalil, H.K.: Nonlinear Systems, 3rd edn. Prentice Hall, New York (2002) MATHGoogle Scholar
  14. 14.
    Lewis, F.L.: A survey of linear singular systems. Circuits Syst. Signal Process. 5, 3–36 (1986) MATHCrossRefGoogle Scholar
  15. 15.
    Lin, C., Wang, J.L., Soh, C.B.: Robustness of uncertain descriptor systems. Syst. Control Lett. 31, 129–138 (1997) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Lu, G., Ho, D.W.C.: Generalized quadratic stability for continuous-time singular systems with nonlinear perturbation. IEEE Trans. Automat. Control 51, 818–823 (2006) CrossRefMathSciNetGoogle Scholar
  17. 17.
    Mahmoud, M.S., Al-Muthairi, N.F., Bingulac, S.: Robust Kalman filtering for continuous time-lag systems. Syst. Control Letters 38(4), 309–319 (1999) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Mahmoud, M.S., Boujarwah, A.A.: Robust ℋ filtering for a class of linear parameter-varying systems. IEEE Trans. Circuits Syst. I 48(9), 1131–1138 (2001) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Mahmoud, M.S., Shi, P.: Robust Kalman filtering for continuous time-lag systems with jump parameters. IEEE Trans. Circuits Syst. I 50(1), 98–105 (2003) CrossRefMathSciNetGoogle Scholar
  20. 20.
    Mahmoud, M.S.: Resilient linear filtering of uncertain systems. Automatica 40(10), 1797–1802 (2004) MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Mahmoud, M.S.: Resilient Control of Uncertain Dynamical Systems. Springer, Berlin (2004) MATHGoogle Scholar
  22. 22.
    Mahmoud, M.S.: Resilient ℒ2/ℒ filtering of polytopic systems with state-delays. In: Proceedings of IET Control Theory and Applications, vol. 1(1), pp. 141–150 (2007) Google Scholar
  23. 23.
    Nagpal, K.M., Khargonekar, P.P.: Filtering and smoothing in ℋ setting. IEEE Trans. Automat. Control 36, 152–166 (1991) MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Rehm, A., Allgower, F.: Self-scheduled ℋ output feedback control of descriptor systems. Comput. Chem. Eng. 24, 279–284 (2000) CrossRefGoogle Scholar
  25. 25.
    Xu, S., Dooren, P.V., Stefan, R., Lam, J.: Robust stability and stabilization for singular systems with state-delay and parameter uncertainty. IEEE Trans. Automat. Control. 47, 1122–1228 (2002) CrossRefMathSciNetGoogle Scholar
  26. 26.
    Xu, S., Lam, J., Zou, Y.: ℋ filtering for singular systems. IEEE Trans. Automat. Control. 48, 2217–2222 (2003) CrossRefMathSciNetGoogle Scholar
  27. 27.
    Xu, S., Lam, J.: Robust stability and stabilization of discrete singular systems: an equivalent characterization. IEEE Trans. Automat. Control. 49, 568–574 (2004) CrossRefMathSciNetGoogle Scholar
  28. 28.
    Yue, D., Han, Q.L.: Robust ℋ filter design of uncertain descriptor systems with discrete and distributed delays. IEEE Trans. Signal Process. 52, 3200–3212 (2004) CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Systems Engineering DepartmentKing Fahd University of Petroleum and MineralsDhahranSaudi Arabia
  2. 2.Electrical Engineering DepartmentKuwait UniversitySafatKuwait

Personalised recommendations