Parameter Identi fi cation of Nonlinearities in Hammerstein Systems with the Help of Nonparametric Regression Methods Zygmunt Hasiewicz and Grzegorz Mzyk Abstract– A combined, parametric-nonparametric routine for the identification of a static part of Hammerstein system is presented. Parameters of the input nonlinear characteristic of Hammerstein system are estimated for a wide range of ran- dom excitations and random noise, and without any knowl- edge of the parametric model of the output linear dynamics. The needed unmeasurable interaction inputs are estimated in a preliminary step by the nonparametric regression function estimation method. Next, they are used in the nonlinear optimization procedure for evaluating parameters of the static subsystem. Broad class of nonlinear characteristics including functions which are not linear in the parameters, as well as the infinite length impulse response of the linear dynamics are admitted. It is shown that the resulting parameter estimates are consistent for both white and colored noise. The analytical findings are validated using numerical simulation results. Index Terms– Hammerstein system, nonparametric regression, kernel estimation, nonlinear least squares, Levenberg-Marquardt method. I. INTRODUCTION The decentralized approach to the identification of block-oriented complex systems seems to be most nat- ural and desirable, as such an approach corresponds directly to the own nature of systems, composed of individual elements distinguished in the block structure. The Hammerstein system, built of a static non-linearity and a linear dynamics connected in a cascade (Fig. 1), is the simplest structure in the class and hence for the most part considered in the system identification literature (see e.g. [3] for the bibliography). Unfortu- nately, the popular, parametric, methods elaborated for Hammerstein system identification do not enable full decentralization of the system identification task, i.e. independent identification of a static nonlinearity and a linear dynamics in a completely decomposed manner — first of all, because of inaccessibility for measurements of the inner interconnection signal. They assume that the description of system components, i.e. of a static nonlinearity and a linear dynamics is known up to the parameters (a polynomial model along with the FIR dynamics representation is usually used) and these parameters are ”glued” when using the measured input- output data of the overall system (e.g. [1], [2]). On the other hand, in a nonparametric setting (the second class of the existing identification methods, see, e.g. [5], [12]) no preliminary assumptions concerning the structure of The authors are with The Institute of Computer Engineer- ing, Control and Robotics, Wrocław University of Technology, Janiszewskiego 11/17, 50-372 Wrocław, Poland, phone: (+48 71) 320-32-77, zygmunt.hasiewicz(grzegorz.mzyk)@pwr.wroc.pl subsystems are used and only the data decide about characteristics of the system components, but then any possible a priori knowledge about the description of subsystems is inevitably lost. We propose a method where the two approaches are combined. Namely, our idea is to join the results obtained in the nonparametric identification of nonlinear characteristics in Hammerstein systems (see the papers cited above), e.g. by using kernel regression or orthogonal series methods ([5], [9]), with parametric knowledge of subsystems and stan- dard results concerning nonlinear optimization methods (see, for instance, [10]), taking advantages of both. The paper is an extension of [6], where the combined parametric-nonparametric algorithm was proposed for the identification of parameters appearing linearly in the static nonlinear element. We generalize the approach introduced in [6] in the sense that we admit the static characteristic µ() not linear in the parameters. Similarly as in [6], the presented identification algorithm is two stage. Stage 1 of the algorithm (nonparametric), consists in nonparametric estimation of interaction inputs w k = µ(u k ) (Fig. 1) to cope which their inaccessibility for direct measurements. In stage 2 (parametric), using the obtained estimates e w k of w k , we identify parameters of the static characteristic, and no a priori knowledge of the linear dynamics is required. The latter may in particular be extremely inaccurate. In our consideration we assume that the system input is a random process, and that the random output noise can possess an arbitrary correlation structure. Due to the general form of subsystems and correlation of the output noise, the linear least squares approach presented in [6] fails, and the nonlinear least squares technique is applied instead of. We show that such an approach is computationally convenient and effective, i.e. leads to consistent models. Sufficient conditions for convergence in probability of the resulting parameter estimates are established. Per- formance of the method for moderate number of data illustrate simulation examples. We focus in the paper on the identification of a static nonlinear characteristic of a Hammerstein system because of the possible diversity of nonlinearities and that the nonlinearity is in fact the main distinguishing feature of each system. In turn, dynamical parts are each time linear.