Recursive least squares identification for piecewise affine Hammerstein models

Purpose The purpose of this paper is to probe the recursive identification of piecewise affine Hammerstein models directly by using input-output data. To explain the identification process of a parametric piecewise affine nonlinear function, the authors prove that the inverse function corresponding to the given piecewise affine nonlinear function is also an equivalent piecewise affine form. Based on this equivalent property, during the detailed identification process with respect to piecewise affine function and linear dynamical system, three recursive least squares methods are proposed to identify those unknown parameters under the probabilistic description or bounded property of noise. Design/methodology/approach First, the basic recursive least squares method is used to identify those unknown parameters under the probabilistic description of noise. Second, multi-innovation recursive least squares method is proposed to improve the efficiency lacked in basic recursive least squares method. Third, to relax the strict probabilistic description on noise, the authors provide a projection algorithm with a dead zone in the presence of bounded noise and analyze its two properties. Findings Based on complex mathematical derivation, the inverse function of a given piecewise affine nonlinear function is also an equivalent piecewise affine form. As the least squares method is suited under one condition that the considered noise may be a zero mean random signal, a projection algorithm with a dead zone in the presence of bounded noise can enhance the robustness in the parameter update equation. Originality/value To the best knowledge of the authors, this is the first attempt at identifying piecewise affine Hammerstein models, which combine a piecewise affine function and a linear dynamical system. In the presence of bounded noise, the modified recursive least squares methods are efficient in identifying two kinds of unknown parameters, so that the common set membership method can be replaced by the proposed methods.