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Estimation of nonlinear systems via a Chebyshev approximation approach

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  • Control Theory
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Abstract

This paper proposes to decompose the nonlinear dynamic of a chaotic system with Chebyshev polynomials to improve performances of its estimator. More widely than synchronization of chaotic systems, this algorithm is compared to other nonlinear stochastic estimator such as Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF). Chebyshev polynomials orthogonality properties is used to fit a polynomial to a nonlinear function. This polynomial is then used in an Exact Polynomial Kalman Filter (ExPKF) to run real time state estimation. The ExPKF offers mean square error optimality because it can estimate exact statistics of transformed variables through the polynomial function. Analytical expressions of those statistics are derived so as to lower ExPKF algorithm computation complexity and allow real time applications. Simulations under the Additive White Gaussian Noise (AWGN) hypothesis, show relevant performances of this algorithm compared to classical nonlinear estimators.

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Authors and Affiliations

  1. Department of Electronics, University of Jijel, Jijel, DZ 18000, Algeria

    Moussa Yahia

  2. “LASS-CNRS ” laboratory, University of Toulouse, Toulouse, 31000, France

    Pascal Acco

  3. Department of Electronics, University Mantouri of Constantine, Dz 25000, Constantine, Algeria

    Malek Benslama

Authors
  1. Moussa Yahia
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  2. Pascal Acco
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  3. Malek Benslama
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Corresponding author

Correspondence to Moussa Yahia.

Additional information

Recommended by Editorial Board member Duk-Sun Shim under the direction of Editor Jae Weon Choi.

Moussa Yahia was born in Fej-M’zala, Jijel, Algeria. He received his Diploma in Electrical Engineering and the Magister degree from the Department of Electronics, University of Constantine, Algeria in 1992 and 1997, respectively. He is finishing a Ph.D. degree in le Laboratoire de l’elétromagnétisme et Télécommunications (LET). Since 2001 he has been an adjunct lecturer in the department of Electronics, University of Jijel. His research activities concern nonlinear signal processing, secure communications relying on chaotic dynamics.

Pascal Acco received his Ph.D. degree in Automatic control from the Institut National des sciences Appliquée, Toulouse, France. Currently, he is an Assistant professor in Automatic Control and Electronics at the Institut National des sciences Appliquée and University of Toulouse. His research interests include the study of variable structure control applied to uncertain systems and chaos in telecommunication.

Malek Benslama was born in Constantine, Algeria, in 1953. He received the Diploma Engineer from Polytechnic School Algiers in 1978. He received DEA EEA, Doctor Enginner and State doctorat from ENSEEIHT Toulouse France in 1979, 1982, and 1992, respectively. Actually, He has a professor at Constantine University and he manages the Doctoral Space Telecommunications School. He has also member of scientific and technical Council of Algerian Space Agency. His research interests include communications systems, satellite systems and microwave engineering.

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Yahia, M., Acco, P. & Benslama, M. Estimation of nonlinear systems via a Chebyshev approximation approach. Int. J. Control Autom. Syst. 9, 1021–1027 (2011). https://doi.org/10.1007/s12555-011-0601-9

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  • DOI: https://doi.org/10.1007/s12555-011-0601-9

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