On the identification of Hammerstein – Wiener systems

Special classes of nonlinear systems applied in engineering are nonlinear systems with both block-oriented Hammerstein and Wiener structures, respectively [1, 3, 4, 7, 8, 14]. There are a lot of papers devoted to the different aspects of the parametric identification of Hammerstein and Wiener systems and much less on that of the Hammerstein–Wiener (H-W) systems with so-called hard nonlinearities [2, 6, 12, 13]. On the other hand, the abovementioned systems are common in nonlinear control applications where hard nonlinearities such as the saturation, preload, dead-zone, etc., are present [5]. Especially frequently saturation nonlinearities as an input or an output nonlinearity are observed here, too. In such a case, respective observations of a nonlinear system to be identified could be partitioned into distinct data sets according to different descriptions. However the boundaries of sets of observations depend on the value of unknown thresholds – observations are divided into regimes dependent on whether some observed threshold variable is smaller or larger than the threshold. Therefore, the problem of identification of unknown parameters of linear blocks of the H-W systems could be solved, if a simple way of partitioning the available data sets were found in the case of unknown thresholds of both saturations. Afterwards the estimates of parameters of regression functions could be calculated by processing particles of non-clipped observations to be determined. Comparing with [9, 10, 11] we extend here our research on the parametric identification of linear parts of block-oriented H-W systems with saturation nonlinearities by processing input-output observations.