Advertisement

Numerical Algorithms

, Volume 50, Issue 4, pp 439–467 | Cite as

Numerical differentiation with annihilators in noisy environment

  • Mamadou MboupEmail author
  • Cédric Join
  • Michel Fliess
Original Paper

Abstract

Numerical differentiation in noisy environment is revised through an algebraic approach. For each given order, an explicit formula yielding a pointwise derivative estimation is derived, using elementary differential algebraic operations. These expressions are composed of iterated integrals of the noisy observation signal. We show in particular that the introduction of delayed estimates affords significant improvement. An implementation in terms of a classical finite impulse response (FIR) digital filter is given. Several simulation results are presented.

Keywords

Numerical differentiation Operational calculus Orthogonal polynomials Linear filtering 

Mathematics Subject Classifications (2000)

65D25 44A40 44A10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions. Dover Publications (1965)Google Scholar
  2. 2.
    Al-Alaoui, M.A.: A class of second-order integrators and low-pass differentiators. IEEE Trans. Circuits Syst. I 42(4), 220–223 (1995)CrossRefGoogle Scholar
  3. 3.
    Alpay, D.: Algorithme de Schur, espaces à noyau reproduisant et théorie des systèmes, Panoramas et Synthèses, vol. 6. Société mathématique de France (1998)Google Scholar
  4. 4.
    Aronszajn, N.: Theory of reproducing kernels. Trans. AMS 68(3), 337–404 (1950)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Chen, C.K., Lee, J.H.: Design of high-order digital differentiators using L 1 error criteria. IEEE Trans. Circuits Syst. II 42(4), 287–291 (1995)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Chitour, Y.: Time-varying high-gain observers for numerical differentiation. IEEE Trans. Automat. Contr. 47, 1565–1569 (2002)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Dabroom, A.M., Khalil, H.K.: Discrete-time implementation of high-gain observers for numerical differentiation. Int. J. Control 72, 1523–1537 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Diop, S., Grizzle, J.W., Chaplais, F.: On numerical differentiation algorithms for nonlinear estimation. In: Proc. CDC. Sydney (2000)Google Scholar
  9. 9.
    Duncan, T.E., Mandl, P., Pasik-Duncan, B.: Numerical differentiation and parameter estimation in higher-order linear stochastic systems. IEEE Trans. Automat. Contr. 41, 522–532 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Fliess, M.J.C., Sira Ramírez, H.: Closed-loop fault-tolerant control for uncertain nonlinear systems. In: Meurer, T., Graiche, K., Gilles, E. (eds.) Control and Observer Design for Nonlinear Finite and Infinite Dimensional Systems. Lect. Notes Control Informat. Sci., vol. 322, pp. 217–233. Springer (2005)Google Scholar
  11. 11.
    Fliess, M., Join, C., Mboup, M., M., H.S.R.: Compression différentielle de transitoires bruités. CRAS, Série 1, Mathématiques 339, 821–826 (2004)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Fliess, M., Join, C., Mboup, M., Sira Ramírez, H.: Analyse et représentation de signaux transitoires : application à la compression, au débruitage et à la détection de ruptures. In: Actes 20e Coll. GRETSI. Louvain-la-Neuve (2005). http://hal.inria.fr/inria-00001115
  13. 13.
    Fliess, M., Join, C., Sedoglavic, A.: Estimation des dérivées d’un signal multidimensionnel avec applications aux images et aux vidéos. In: Actes 20e Coll. GRETSI. Louvain-la-Neuve (2005). http://hal.inria.fr/inria-00001116
  14. 14.
    Fliess, M., Mboup, M., Mounier, H., Sira-Ramírez, H.: Questioning some paradigms of signal processing via concrete examples. In: Sira-Ramírez, H., Silva-Navarro, G. (eds.) Algebraic Methods in Flatness, Signal Processing and State Estimation. Innovación ed. Lagares, México (2003)Google Scholar
  15. 15.
    Fliess, M., Sira-Ramírez, H.: An algebraic framework for linear identification. In: ESAIM: COCV, vol. 9, pp. 151–168. SMAI (2003). http://www.esaim-cocv.org/
  16. 16.
    Fliess, M., Sira Ramírez, H.: Control via state estimations of some nonlinear systems. In: Proc. Symp. Nonlinear Control Systems (NOLCOS’04). Stuttgart (2004). http://hal.inria.fr/inria-00001096
  17. 17.
    Haykin, S., Van Veen, B.: Signals and Systems, 2nd edn. John Wiley & Sons (2002)Google Scholar
  18. 18.
    Ibrir, S.: Online exact differentiation and notion of asymptotic algebraic observers. IEEE Trans. Automat. Contr. 48, 2055–2060 (2003)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Ibrir, S.: Linear time-derivatives trackers. Automatica 40, 397–405 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Ibrir, S., Diop, S.: A numerical procedure for filtering and efficient high-order signal differentiation. Int. J. Appl. Math. Compt. Sci. 14, 201–208 (2004)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Ismail, M.E.H., Li, X.: Bound on the extreme zeros of orthogonal polynomials. Proceedings of the AMS 115(1), 131–140 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Levant, A.: Higher-order sliding modes, differentiation and output-feedback control. Int. J. Control 76, 924–941 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Lorentz, G.G.: Bernstein Polynomials, 2nd edn. AMS Chelsea Publishing (1986)Google Scholar
  24. 24.
    Massart, P.: Concentration inequalities and model selection. Lecture Notes in Mathematics, vol. 1896. Springer, Berlin (2007)Google Scholar
  25. 25.
    Mboup, M.: Parameter estimation via differential algebra and operational calculus. Tech. rep., Submitted to Signal Processing (2007)Google Scholar
  26. 26.
    Mboup, M., Join, C., Fliess, M.: A revised look at numerical differentiation with an application to nonlinear feedback control. In: 15th Mediterranean conference on Control and automation (MED’07). Athenes, Greece (2007)Google Scholar
  27. 27.
    Mikusiǹski, J.: Operational Calculus, vol. 1. PWN Varsovie & Oxford University Press, Oxford (1983)Google Scholar
  28. 28.
    Mikusiǹski, J., Boehme, T.K.: Operational Calculus, vol. 2. PWN Varsovie & Oxford University Press, Oxford (1987)Google Scholar
  29. 29.
    Rader, C.M., Jackson, L.B.: Approximating noncausal IIR digital filters having arbitrary poles, including new Hilbert transformer designs, via forward/backward block recursion. IEEE Trans. Circuits Syst. I 53(12), 2779–2787 (2006)CrossRefMathSciNetGoogle Scholar
  30. 30.
    Richard, J.: Time delay systems: an overview of some recent advances and open problems. Automatica 10, 1667–1694 (2003)CrossRefMathSciNetGoogle Scholar
  31. 31.
    Roberts, R.A., Mullis, C.T.: Digital Signal Processing. Addison-Wesley (1987)zbMATHGoogle Scholar
  32. 32.
    Saitoh, S.: Theory of Reproducing Kernels and its Applications. Pitman Research Notes in Mathematics. Longman Scientic & Technical, UK (1988)Google Scholar
  33. 33.
    Seuret, A., Dambrine, M., Richard, J.: Robust exponential stabilization for systems with time-varying delays. In: 5th IFAC Workshop on Time Delay Systems. Leuven, Belgium (2004)Google Scholar
  34. 34.
    Su, Y.X., Zheng, C.H., Mueller, P.C., Duan, B.Y.: A simple improved velocity estimation for low-speed regions based on position measurements only. IEEE Trans. Control Syst. Technology 14, 937–942 (2006)CrossRefGoogle Scholar
  35. 35.
    Szegö, G.: Orthogonal Polynomials, 3rd edn. AMS, Providence, RI (1967)Google Scholar
  36. 36.
    Yosida, K.: Operational Calculus - A Theory of Hyperfunctions. Springer, New York (1984)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2008

Authors and Affiliations

  1. 1.UFR Mathématiques et Informatique, CRIP5Université Paris DescartesParis cedex 06France
  2. 2.CRAN (CNRS-UMR 7039)Université Henri Poincaré (Nancy I)Vandœuvre-lès- NancyFrance
  3. 3.Équipe MAX, LIX (CNRS-UMR 7161)École polytechniquePalaiseauFrance
  4. 4.EPI ALIEN INRIAParisFrance

Personalised recommendations