Identification of non-linear characteristics of a class of block-oriented non-linear systems via Daubechies wavelet-based models

The paper deals with the identification of non-linear characteristics of a class of block-oriented dynamical systems. The systems are driven by random stationary white processes (i.i.d. random input sequences) and disturbed by a zero-mean stationary, white or coloured, random noise. The prior knowledge about non-linear characteristics is non-parametric excluding implementation of standard parametric identification methods. To recover non-linearities, a class of Daubechies wavelet-based models using only input-output measurement data is introduced and their accuracy is investigated in the global MISE error sense. It is shown that the proposed models converge with a growing collection of data to the true non-linear characteristics (or their versions), provided that the complexity of the models is appropriately fitted to the number of measurements. Suitable rules for optimum model size selection, maximizing the convergence speed, are given and the asymptotic rate of convergence of the MISE error for optimum models is established. It is shown that in some circumstances the rate is the best possible that can be achieved in non-parametric inference. We also show that the convergence conditions and the asymptotic rate of convergence are insensitive to the correlation of the noise and are the same for known and unknown input probability density function (assumed to exist). The theory is illustrated by simulation examples.