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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.

This paper describes a state-space representation of the parallel-cascade model of ankle joint stiffness whose parameters are directly related to the underlying dynamics of the system. It then proposes a two step subspace method to identify this model. In the first step, the intrinsic stiffness is estimated using proper orthogonal projections. In the second step, the reflexive pathway is estimated by iterating between estimating its nonlinear and linear components. The identified models can be easily converted to continuous-time for physiological interpretation. Monte-Carlo studies using simulated data which replicate closely the experimental conditions, were used to compare the performance of the new method with the previous parallel-cascade, and subspace methods. The new method is more robust to noise and is guaranteed to converge.

This paper presents an algorithm for the identification of Hammerstein cascades with hard nonlinearities. The nonlinearity of the cascade is described using a B-spline basis with fixed knot locations; the linear dynamics are described using a state-space model. The algorithm automatically estimates both the order of the linear system and the number and locations of the knots used to characterize the nonlinearity. Therefore, it significantly reduces the a priori knowledge about the underlying system required for identification. A simulation study on a model of reflex stiffness shows that the new method estimates the nonlinearity accurately in the presence of output noise.

Reflex stiffness is often modeled as a Hammerstein system comprising a cascade of a static nonlinear element and a linear dynamic element. The nonlinearity is frequently modeled as a half wave rectifier so that changes in the reflex response can only be modeled by changes in the parameters of the linear element. This is an oversimplification since there are physiological mechanisms that could change both the threshold of the nonlinearity and the linear dynamics. This study explores the ability of a new subspace identification algorithm to distinguish changes in parameters of the nonlinear element from those of the linear element. Simulation studies demonstrate that the method does so very effectively even in the presence of substantial output noise. Pilot experiments in which the method was applied to stretch reflex EMG data revealed that both the threshold of the nonlinearity and the gain of the linear element change with muscle activation.