Przemysław Śliwiński

Nonlinear system identification by Haar wavelets

Introduction.- Hammerstein systems.- Identification goal.- Haar orthogonal bases.- Identification algorithms.- Computational algorithms. - Final remarks. - Technical derivations.

On-line wavelet estimation of Hammerstein system nonlinearity

On-line wavelet estimation of Hammerstein system nonlinearity A new algorithm for nonparametric wavelet estimation of Hammerstein system nonlinearity is proposed. The algorithm works in the on-line regime (viz., past measurements are not available) and offers a convenient uniform routine for nonlinearity estimation at an arbitrary point and at any moment of the identification process. The pointwise convergence of the estimate to locally bounded nonlinearities and the rate of this convergence are both established.

A simple scheme for semi-recursive identification of Hammerstein system nonlinearity by Haar wavelets

Abstract A simple semi-recursive routine for nonlinearity recovery in Hammerstein systems is proposed. The identification scheme is based on the Haar wavelet kernel and possesses a simple and compact form. The convergence of the algorithm is established and the asymptotic rate of convergence (independent of the input density smoothness) is shown for piecewise- Lipschitz nonlinearities. The numerical stability of the algorithm is verified. Simulation experiments for a small and moderate number of input-output data are presented and discussed to illustrate the applicability of the routine.

Model Selection of Hammerstein System Nonlinearity under Heavy Noise

Abstract The paper deals with the problem of automatic model selection of the nonlinear characteristic in a block-oriented dynamic system. We look for the parametric model of Hammerstein system nonlinearity. From the finite set of candidate classes of parametric models we select the best one on the basis of the input-output measurement data, using the concept of nearest neighbour borrowed from pattern recognitions techniques. The algorithm uses the pattern of the true characteristic generated by its nonparametric estimates on the grid of fixed (e.g. equidistant) points. Each class generates parametrized learning sequence through the values on the same grid of points. Next, for each class, the optimal parameters are computed by the least squares method. Finally, the nearest neighbour approach is applied for the selection of the best model in the mean square sense. The idea is presented on the exemplary competition between polynomial, exponential and piece-wise linear models of the same complexity (i.e. number of parameters needed to be stored in memory). For all classes, the upper bounds of the integrated approximation errors of the true characteristic are computed and compared.

Mixed parametric-nonparametric identification of Hammerstein and Wiener systems - a survey1

Abstract The paper surveys the ideas of cooperation between parametric and nonparametric (kernel-based) algorithms of nonlinear block-oriented system identification. Various strategies are proposed, dependently on the system structure, number of data and the prior knowledge. The estimates are consistent and their rates of convergence are presented. The aim of the paper is to show some recent results in the field in a systematic, ordered way.

Quasi-parametric recovery of Hammerstein system nonlinearity by smart model selection

In the paper we recover a Hammerstein system nonlinearity. Hammerstein systems, incorporating nonlinearity and dynamics, play an important role in various applications, and effective algorithms determining their characteristics are not only of theoretical but also of practical interest. The proposed algorithm is quasi-parametric, that is, there are several parametric model candidates and we assume that the target non-linearity belongs to the one of the classes represented by the models. The algorithm has two stages. In the first, the neural network is used to recursively filter (estimate) the nonlinearity from the noisy measurements. The network serves as a teacher/trainer for the model candidates, and the appropriate model is selected in a simple tournament-like routine. The main advantage of the algorithm over a traditional one stage approach (in which models are determined directly from measurements), is its small computational overhead (as computational complexity and memory occupation are both greatly reduced).

Non-linear system modelling based on constrained Volterra series estimates

A simple non-linear system modelling algorithm designed to work with limited a priori knowledge and short data records, is examined. It creates an empirical Volterra series-based model of a system using an l q -constrained least squares algorithm with q ≥ 1 . If the system m ⋅ is a continuous and bounded map with a finite memory no longer than some known τ , then (for a D parameter model and for a number of measurements N) the difference between the resulting model of the system and the best possible theoretical one is guaranteed to be of order N − 1 ln ⁡ D , even for D ≥ N . The performance of models obtained for q = 1 , 1.5 and 2 is tested on the Wiener–Hammerstein benchmark system. The results suggest that the models obtained for q > 1 are better suited to characterise the nature of the system, while the sparse solutions obtained for q = 1 yield smaller error values in terms of input-output behaviour.

On Hammerstein System Nonlinearity Identification Algorithms Based on Order Statistics and Compactly Supported Functions

Nonparametric algorithms recovering the nonlinearity in Hammerstein systems are examined. The algorithms are based on ordered measurements and on compactly supported functions. The contribution of the note consists in that the probability density function of the input signal does not need to be strictly bounded from zero but can vanish in a finite number of points. In this setting, the convergence is established for nonlinearities being piecewise-Lipschitz functions. It is also verified that for p times locally differentiable nonlinearities, the algorithms attain the convergence rate O(n −2p/(2p+1)), the best possible nonparametric one. Noteworthy, the rate is not worsened by irregularities of the input probability density function.

Haar Orthogonal Bases

Two Haarwavelet bases, classic and the recently introduced unbalanced one, are presented together with the corresponding fast wavelet transforms (in the classic and lifting versions). Both linear and nonlinear (derived from the EZW algorithm) Haar approximation schemes are examined. The effectiveness of these schemes for Lipschitz and piecewise-Lipschitz functions is compared.

Empirical recovery of input nonlinearity in distributed element models

Abstract Two algorithms recovering an input nonlinearity in a nonlinear distributed element modeled as a Hammerstein system are proposed. The first is based on the empirical distribution function while the other on the empirical Haar orthogonal series. Both algorithms self-adjust their accuracy to a local density of the input measurements.