IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 52, NO. 2, FEBRUARY 2005 427 Nonparametric Identification of Nonlinearities in Block-Oriented Systems by Orthogonal Wavelets With Compact Support Zygmunt Hasiewicz, Miroslaw Pawlak, Member, IEEE, and Przemyslaw S ´ liwin ´ski Abstract—The paper addresses the problem of identification of nonlinear characteristics in a certain class of discrete-time block-oriented systems. The systems are driven by random sta- tionary white processes (independent and identically distributed input sequences) and disturbed by stationary, white, or colored random noise. The prior information about nonlinear charac- teristics is nonparametric. In order to construct identification algorithms, the orthogonal wavelets of compact support are applied, and a class of wavelet-based models is introduced and examined. It is shown that under moderate assumptions, the proposed models converge almost everywhere (in probability) to the identified nonlinear characteristics, irrespective of the noise model. The rule for optimum model-size selection is given and the asymptotic rate of convergence of the model error is established. It is demonstrated that, in some circumstances, the wavelet models are, in particular, superior to classical trigonometric and Hermite orthogonal series models worked out earlier. Index Terms—Block-oriented systems, nonlinearity recovering, nonparametric approach, wavelet-based models. I. INTRODUCTION W E CONSIDER the problem of recovering nonlinear characteristics of a class of discrete-time block-oriented dynamical systems, i.e., structured objects where nonlinear static elements are separated from the rest of the system and embedded in a composite structure containing discrete-time linear dynamic blocks and other “nuisance” nonlinearities [1]. It is assumed that prior information about subsystems is small, and, in particular, that the nonlinearity of interest is merely a bounded function in the identification region. Since the nonlinear characteristics to be identified are not given in a parametric form, and no finite-dimensional parametric representation of a possible characteristic can be reasonably motivated, our problem is nonparametric and standard para- metric identification methods are not applicable. The problem of recovering static characteristics in intercon- nected composite systems has been extensively studied in the literature. For steady-state systems, it has been investigated in Manuscript received December 18, 2003; revised January 14, 2004, and April 16, 2004. This paper was recommended by Associate Editor W. X. Zheng. Z. Hasiewicz and P. S ´ liwin ´ski are with the Institute of Engineering Cy- bernetics, Wroclaw University of Technology, 50-372 Wroclaw, Poland (e-mail:zhas@ict.pwr.wroc.pl; slk@ict.pwr.wroc.pl). M. Pawlak is with the Department of Electrical and Computer Engi- neering, University of Manitoba, Winnipeg, MB R3T 2N2, Canada (e-mail: pawlak@ee.umanitoba.ca). Digital Object Identifier 10.1109/TCSI.2004.840288 a number of papers using parametric and nonparametric ap- proach (e.g., [2], [3]). For block-oriented dynamical systems, the problem has been initially examined under the assumption that the nonlinear characteristics are known up to the parame- ters and that are typically polynomials of known degree. Var- ious parametric approaches have been developed, with partic- ular attention applied to cascade and parallel systems. A thor- ough overview of the derived parametric identification methods can be found in [4]. A less demanding nonparametric approach, discarding the re- strictive assumption of parametric prior knowledge of the char- acteristics, has been initiated in [5] for Hammerstein systems. This approach, which originated from nonparametric estimation of a regression function, was next developed bringing about a collection of papers where two types of nonparametric identifi- cation algorithms were studied: kernel algorithms (e.g., [5], [6]) and orthogonal series algorithms applying conventional orthog- onal series expansions of characteristics (e.g., [7], [8]). The al- gorithms were elaborated for Hammerstein and Wiener systems, i.e., cascade connections of static nonlinearities and linear dy- namic blocks in a suitable order. In this paper, we propose and examine a class of nonpara- metric identification algorithms based on wavelet approxima- tions of functions. The algorithms exploit only input–output measurement data collected in the experiment and apply to built of the wavelet models the orthogonal wavelets with compact supports. There are four main reasons for using orthogonal com- pactly supported wavelets for the construction of the identifica- tion algorithms. 1) Ease of obtaining the orthogonal wavelet bases by only scaling and translating of a father and mother wavelet. 2) Extraordinary approximation capability and adaptivity as- sociated with the ability of shrinking of the wavelets do- main to any small interval. 3) Ease of obtaining models of increasing precision and sen- sitivity to the characteristic details. 4) Existence of fast algorithms for wavelet computations and ready-for-use program packages. Orthogonality of wavelets enables simple and convenient im- provement of the wavelet models by adding standard building blocks and compactness of the supports results in parsimonious representations of nonlinear characteristics, with good localiza- tion properties. For reconstruction of nonlinearities the models use only the most informative “local” data lying in a close neigh- borhood of a point at which the estimation is carried out, and 1057-7122/$20.00 © 2005 IEEE