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Selection of a Closed-Form Expression Polynomial Orthogonal Basis for Robust Nonlinear System Identification

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Abstract

Polynomial nonlinear system identification suffers from numerical instability related to the ill-conditioning of the involved matrices. Orthogonal methods consist in conditioning the input signal in order to reduce the eigenvalues spread of the correlation matrix. The selection of an appropriate orthogonalization procedure, for robustness improvement, resort to signal statistics. Consequently, several orthogonal polynomial bases are proposed in the literature. Most of them use iterative processes and imply a considerable computational cost. Actually, with the growth of real-time applications, it is important to generate non-iterative orthogonal procedures, allowing optimization of algorithm-architecture adequacy. Our paper’s motivation is based on the complexity reduction aspect related to the orthogonalization process. Therefore we propose to focus on closed-form expressions of commonly used orthogonal polynomials bases namely the Shifted-Legendre orthogonal polynomials and the Hermite polynomials. A robustness study in terms of numerical stability enhancement of the two bases is carried on. Through comparative simulations results, the basis allowing the best matrix conditioning and an ease generalization for real-time applications with less restrictive hypothesis is selected in order to study the robustness of a polynomial nonlinear system identification scheme. Computer simulations are carried out to emphasize the advantages of the proposed scheme using different performance criteria for both the optimal case and the adaptive case, in terms of numerical stability and convergence rate. We propose to experiment the identification of the power amplifiers, for radio mobile applications and the loudspeakers for audio applications.

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Authors and Affiliations

  1. Signals and Systems research Unit, National Engineering School of Tunis, BP 37, Le Belvédère, 1002, Tunisia

    Mariem Kallel Smaoui & Mériem Jaidane

  2. Department of Informatics and Applied Mathematics, National Engineering School of Sfax, Sfax, 3038, Tunisia

    Yousra Ben Jemâa

Authors
  1. Mariem Kallel Smaoui
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  2. Yousra Ben Jemâa
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  3. Mériem Jaidane
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Correspondence to Yousra Ben Jemâa.

Appendix

Appendix

For a matrix A, the condition number is defined as:

$$ K(\mathbf{A})=|\frac{{\uplambda}_{\textit{max}}}{{\uplambda}_{\textit{min}}}| $$
(33)

where λ m a x and λ m i n are respectively the maximum and the minimum eigenvalue of the matrix.

A matrix is said to be ill-conditioned if the condition number is much larger than 1, which induces difficulties to inverse correctly the matrix and hence to estimate accurately the NL system coefficients set with relation A x = b.

In fact, if we introduce perturbation on system:

$$ \mathbf{(A+{\Delta} A)(x+{\Delta} x)=b+{\Delta} b} $$
(34)

Δ x is the error caused by perturbations Δ A and Δ b.

In [30], it is demonstrated that the relative error estimation of x o p t noted \(\delta (\mathbf {x})\;=\;\frac {\|\mathbf {\Delta x}\|}{\|\mathbf {x}\|}\) is quantified as:

$$ \delta(\mathbf{x})\leq K(\mathbf{A})\left(\frac{\|{\Delta} \mathbf{A}\|}{\|\mathbf{A}\|}+ \frac{\|{\Delta} \mathbf{b}\|}{\|\mathbf{b}\|}\right) $$
(35)

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Smaoui, M.K., Jemâa, Y.B. & Jaidane, M. Selection of a Closed-Form Expression Polynomial Orthogonal Basis for Robust Nonlinear System Identification. J Sign Process Syst 81, 285–303 (2015). https://doi.org/10.1007/s11265-014-0948-2

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  • DOI: https://doi.org/10.1007/s11265-014-0948-2

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