Subspace identification of SISO Hammerstein systems: application to stretch reflex identification

This paper describes a new subspace-based algorithm for the identification of Hammerstein systems. It extends a previous approach which described the Hammerstein cascade by a state-space model and identified it with subspace methods that are fast and require little a priori knowledge. The resulting state-space models predict the system response well but have many redundant parameters and provide limited insight into the system since they depend on both the nonlinear and linear elements. This paper addresses these issues by reformulating the problem so that there are many fewer parameters and each parameter is related directly to either the linear dynamics or the static nonlinearity. Consequently, it is straightforward to construct the continuous-time Hammerstein models corresponding to the estimated state-space model. Simulation studies demonstrated that the new method performs better than other well-known methods in the nonideal conditions that prevail during practical experiments. Moreover, it accurately distinguished changes in the linear component from those in the static nonlinearity. The practical application of the new algorithm was demonstrated by applying it to experimental data from a study of the stretch reflex at the human ankle. Hammerstein models were estimated between the velocity of ankle perturbations and the EMG activity of triceps surae for voluntary contractions in the plantarflexing and dorsiflexion directions. The resulting models described the behavior well, displayed the expected unidirectional rate sensitivity, and revealed that both the gain of the linear element and the threshold of the nonlinear changed with contraction direction.