Estimation of nonlinear systems via a Chebyshev approximation approach

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.

Use of Chebyshev Polynomial Kalman Filter for pseudo-blind demodulation of CD3S signals

“…In this paper, we used the Chebyshev Polynomial Kalman Filter method proposed in [5] and [6] in the dual Kalman filtering scheme to achieve pseudo blind demodulation of the signal generated through a CD3S communication system. The performance of this new method in terms of BER has been compared to traditional and popular Unscented Kalman Filter for different values of Eb/N0.…”

“…In this section we recap shortly the algebraic procedure required to implement the Chebyshev Polynomial Kalman Filter (we refer to [5] for more details). For a nonlinear non-polynomial transformation of a random variable x, say f (x) := Ω → Ω, where Ω = [−1, 1] (it is worth to point out that the same hold true for a nonlinear nonpolynomial transformation f (x) := [a, b] → [a, b] with a, b ∈ R, see for instance [6]), it is not possible to apply the Exact Polynomial Kalman Filter introduced in [4] as it works only for polynomial systems.…”

“…For a nonlinear non-polynomial transformation of a random variable x, say f (x) := Ω → Ω, where Ω = [−1, 1] (it is worth to point out that the same hold true for a nonlinear nonpolynomial transformation f (x) := [a, b] → [a, b] with a, b ∈ R, see for instance [6]), it is not possible to apply the Exact Polynomial Kalman Filter introduced in [4] as it works only for polynomial systems. To overcome the problem, in [5] the authors proposed first to exploit the orthogonality properties of Chebyshev polynomials to fit a polynomial g (x) to the nonlinear and non-polynomial function f (x), and then to use the ExPKF on the polynomial g (x) to estimate the original signal. The technique is summarized in the following.…”

“…The main drawbacks are the computational cost and the impossibility to work with nonlinear non polynomial functions, such as piecewise linear functions. In order to overcome both problems, recently [5] and [6] reformulated the Ex-PKF algorithm. The reformulation is based on the Chebyshev series approximation of the original model which allows to exploit the Chebyshev polynomial properties to derive and express in vector-matrix notation the closedform solutions for the moment propagation ensuring a computationally efficient implementation.…”

Differential equation description and Chebyshev approximation of linear time‐invariant circuits

“…The Chebyshev approximation theory 13 uses Chebyshev polynomials (CPs) 14 to approximate arbitrary continuous functions accurately, 15 especially the smooth function, 16 and is extensively applied in numeric analysis and computer simulation technology. 17 The redundant CSs 18 or Chebyshev expansions 19 of the desired degree are cut off to find the Nth order polynomial approximating with the largest feasible leading coefficients.…”