Classification of the Poles and Zeros of the Best Linear Approximations of Wiener-Hammerstein Systems*

The parameters of a Wiener-Hammerstein model, a nonlinear block structure comprising two linear filters separated by a memoryless nonlinearity, may be identified using an iterative nonlinear least squares optimization, however avoiding suboptimal local minima in the error surface requires a good initial estimate of the parameter vector. The Best Linear Approximation (BLA) of a Wiener-Hammerstein model will contain all the poles and zeros of both linear elements, but does not provide any information regarding which poles and/or zeros should be assigned to either of the linear elements. This information is contained in the nonlinear terms in the system response. One such nonlinear term is the BLA fitted between a suitably chosen nonlinear transformation of the input, and the output residuals remaining after all linear terms have been removed. The poles and zeros present in this nonlinear transfer function are used to classify the poles and zeros in the initial linear fit as belonging to either the first or second linear element in the Wiener-Hammerstein model. The procedure is illustrated by applying it to experimental data from a Wiener-Hammerstein benchmark system.