Numerical differentiation with annihilators in noisy environment

Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions. Dover Publications (1965)

Al-Alaoui, M.A.: A class of second-order integrators and low-pass differentiators. IEEE Trans. Circuits Syst. I 42(4), 220–223 (1995)

Alpay, D.: Algorithme de Schur, espaces à noyau reproduisant et théorie des systèmes, Panoramas et Synthèses, vol. 6. Société mathématique de France (1998)

Aronszajn, N.: Theory of reproducing kernels. Trans. AMS 68(3), 337–404 (1950)

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)

Chitour, Y.: Time-varying high-gain observers for numerical differentiation. IEEE Trans. Automat. Contr. 47, 1565–1569 (2002)

Dabroom, A.M., Khalil, H.K.: Discrete-time implementation of high-gain observers for numerical differentiation. Int. J. Control 72, 1523–1537 (1999)

Diop, S., Grizzle, J.W., Chaplais, F.: On numerical differentiation algorithms for nonlinear estimation. In: Proc. CDC. Sydney (2000)

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)

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)

Fliess, M., Join, C., Mboup, M., M., H.S.R.: Compression différentielle de transitoires bruités. CRAS, Série 1, Mathématiques 339, 821–826 (2004)

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 20 e Coll. GRETSI. Louvain-la-Neuve (2005). http://hal.inria.fr/inria-00001115

Fliess, M., Join, C., Sedoglavic, A.: Estimation des dérivées d’un signal multidimensionnel avec applications aux images et aux vidéos. In: Actes 20 e Coll. GRETSI. Louvain-la-Neuve (2005). http://hal.inria.fr/inria-00001116

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)

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/

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

Haykin, S., Van Veen, B.: Signals and Systems, 2nd edn. John Wiley & Sons (2002)

Ibrir, S.: Online exact differentiation and notion of asymptotic algebraic observers. IEEE Trans. Automat. Contr. 48, 2055–2060 (2003)

Ibrir, S.: Linear time-derivatives trackers. Automatica 40, 397–405 (2004)

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)

Ismail, M.E.H., Li, X.: Bound on the extreme zeros of orthogonal polynomials. Proceedings of the AMS 115(1), 131–140 (1992)

Levant, A.: Higher-order sliding modes, differentiation and output-feedback control. Int. J. Control 76, 924–941 (2003)

Lorentz, G.G.: Bernstein Polynomials, 2nd edn. AMS Chelsea Publishing (1986)

Massart, P.: Concentration inequalities and model selection. Lecture Notes in Mathematics, vol. 1896. Springer, Berlin (2007)

Mboup, M.: Parameter estimation via differential algebra and operational calculus. Tech. rep., Submitted to Signal Processing (2007)

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)

Mikusiǹski, J.: Operational Calculus, vol. 1. PWN Varsovie & Oxford University Press, Oxford (1983)

Mikusiǹski, J., Boehme, T.K.: Operational Calculus, vol. 2. PWN Varsovie & Oxford University Press, Oxford (1987)

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)

Richard, J.: Time delay systems: an overview of some recent advances and open problems. Automatica 10, 1667–1694 (2003)

Roberts, R.A., Mullis, C.T.: Digital Signal Processing. Addison-Wesley (1987)

Saitoh, S.: Theory of Reproducing Kernels and its Applications. Pitman Research Notes in Mathematics. Longman Scientic & Technical, UK (1988)

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)

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)

Szegö, G.: Orthogonal Polynomials, 3rd edn. AMS, Providence, RI (1967)

Yosida, K.: Operational Calculus - A Theory of Hyperfunctions. Springer, New York (1984)