NONPARAMETRIC IDENTIFICATION OF WIENER SYSTEMS - A MEDIAN BASED APPROACH GRZEGORZ MZYK Institute of Engineering Cybernetics, Wroclaw University of Technology, 50-372 Wroclaw, Poland, tel. (071) 320-25-49, grmz@ict.pwr.wroc.pl Abstract. A new identification algorithm for estimation of nonlinear characteristic in Wiener system with FIR linear element under existence of random noise is pro- posed. The assumptions imposed on unknown characteristic are weak. The algorithm is based on the idea of local median computing. The convergence in probability of the estimate is proved for each continuity point of the unknown characteristic. The question of the convergence rate is discussed. Sample illustrative experimental results are included. Key Words. Nonparametric identification, block-oriented models, Wiener systems, random processes. 1. INTRODUCTION A nonlinear characteristic µ() of the Wiener sys- tem (Fig. 1) with the FIR filter {λ j } S j=0 and the random output noise {z k } is estimated. The prob- lem has fundamental meaning in practice ([2], [3], [12], [14]), particularly in such domains as biocy- bernetics, artificial neural networks, modeling of distillation and fermentation processes. The state of art in Wiener and Wiener-Hammerstein sys- tem identification is still not satisfying. Since the linear dynamics precedes the nonlinear element, the identification of Wiener system is in general more difficult in comparison to identification of Hammerstein system. The linear dynamics pro- duces the ”system-noise” {δ k } (Fig. 2) which is transfered through the nonlinear block. In conse- quence we obtain error-in problem (see [17], [18]) and standard approaches leads to biased estimates (for discussion see [16]). Correlation methods pro- posed in 80’s ([4]) assumed polynomial form of nonlinear characteristic µ() and white Gaussian noise {z k }. Nonparametric methods based on the kernel regression do not involve parametric knowl- edge of µ() ([7]-[9]), but requires µ() to be glob- ally invertible. Main advantages of nonparamet- ric approach to nonlinear systems identification have been widely discussed in the literature ([10], [11]). In this paper we propose a new, alterna- tive method which imposes extremely weak a pri- ori restrictions on the nonlinear subsystem and the random disturbance. The algorithm is simple and consists principally in data sorting. Thanks to good statistical properties the empirical median is commonly used in a lot of real issues (e.g. product quality testing, jurors’ grading scale processing). The convergence proof of the proposed estimate is intuitive and strict. To illustrate practical aspects of the method, sample experimental simulations are enclosed. 2. STATEMENT OF THE PROBLEM 2.1. Wiener system Consider a system shown in Fig. 1, where u k and y k denotes system input and output at time k re- spectively, z k is a random noise, µ() is the non- linear characteristic of the static subsystem and {λ j } S j=0 — the impulse response of the linear FIR filter. The intrernal signal x k is not available.