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Estimation of Nonlinear Systems via a Chebyshev Approximation Approach

Moussa Yahia, Pascal Acco, and Malek Benslama
International Journal of Control, Automation, and Systems, vol. 9, no. 6, pp.1021-1027, 2011

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 compu-tation complexity and allow real time applications. Simulations under the Additive White Gaussian Noise (AWGN) hypothesis, show relevant performances of this algorithm compared to classical nonli-near estimators.

Keyword : Chebychev polynomials, estimation, Kalman filter, polynomial maps.

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