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This paper deals with the identification of Hammerstein–Wiener models. The novelty lies in the fact that the front nonlinear block is allowed to be the memory of hysteresis type. The latter is any rate-independent hysteresis operator, that is characterised by closed limit cycles when excited by periodic input signals called ‘loading-unloading’ (or ‘increasing-decreasing’). Designing appropriate ‘loading-unloading’ periodic inputs will prove to be crucial in the proposed identification method. By using such signals, the intermediary linear subsystem is made transparent and the hysteresis front block is equivalent to two static nonlinearities (i.e. the lateral borders of the limit cycle). Then, the input and output signals turn out to be related by two composed functions. A decomposition algorithm is then resorted to extract the back nonlinearity and both lateral borders of the front hysteresis operator. Finally, the impulse response of the intermediary linear subsystem is determined using a least-squares like estimator. The quality of the estimated model is checked using a theoretical analysis and simulation results.

Identification of Hammerstein–Wiener models with hysteresis front nonlinearities