APPLICATION OF INSTRUMENTAL VARIABLE METHOD TO THE IDENTIFICATION OF HAMMERSTEIN-WIENER SYSTEMS GRZEGORZ MZYK† †Wrocław University of Technology, Institute of Engineering Cybernetics, ul. Janiszewskiego 11/17, 50-372 Wrocław, Poland, grmz@diuna.ict.pwr.wroc.pl Abstract. Application of least squares and instrumental variables to recovering parameters of nonlinear complex dynamic block-oriented systems is examined. For a system with the Hammerstein-Wiener structure the instrumental variable algorithm is designed and compared with the least squares algorithm for estimating system parameters. The advantages of the proposed instrumental variable estimator are discussed and in particular its weak consistency, even in the presence of correlated noise, is shown. The problem of generating optimal values of instrumental variables is analysed, rate of convergence of the proposed estimator is evaluated and simulation examples are included. The paper provides an extension of the results introduced in [1]. Key Words. System identification, complex systems, parameter estimation, nonlinear models. 1. INTRODUCTION Instrumental variable method is one of the most popular and universal methods used for determining parameters of single-element linear dynamic objects. The main aim of this paper is applying of the instrumental variable method to identify parameters of nonlinear dynamic systems with composite structure, and comparing it with the traditional least squares method. Applicability analysis is made on the example of the system with Hammerstein-Wiener structure described in detail in section 2. The Hammerstein-Wiener system can be viewed as a serial connection of two typical cascade structures: the Hammerstein system (Fig. 1) and the Wiener system (Fig. 2). In spite of occuring in many fields ([6],[7],[9],[11],[13]), the Hammerstein-Wiener structure has received little attention in the literature. In section 3 the least squares algorithm proposed recently in [1] for the white noise case is presented and its asymptotic bias is shown in the presence of correlated noise. To reduce the bias, the new estimator based on the instrumental variable method is proposed in section 4. The procedure is based on the ideas presented in [10] and [12] for single- element linear systems. The proposed method is next compared with the least squares algorithm given in [1]. Asymptotic behaviour is studied, and the advantages of the proposed instrumental variable estimator are discussed. In particular its weak consistency, even in the presence of correlated noise, is shown. Also the problem of generating optimal values of instrumental variables is analysed, and rate of convergence of the proposed estimator is evaluated. Comparative simulation study results are presented. N ( • ) B(z) A(z) u y z w Fig. 1. Hammerstein system. B(z) A(z) N ( • ) u y z w Fig. 2. Wiener system.