Anti-causal identification of Hammerstein models

Muscle response to Functional Electrical Stimulation (FES) is frequently modeled in Hammerstein form, which consists of a static nonlinearity followed by a linear transfer function. To identify these dynamics, mainly forward approaches are used. The advantage, provided that the nonlinearity and the dynamics are linear in the parameters, is that a simple least-squares solution can be found. For model-based control with input-output linearization, the inverse nonlinearity is needed. Depending on the parameterization, the identified forward nonlinearity is not necessarily invertible. Furthermore, muscle recruitment is generally of saturation characteristic, complicating a linear parameterization with a low number of parameters. In this paper, a reverse identification is performed, changing the structure to Wiener type. The number of parameters can be very low, exploiting the fact that an inverted saturation characteristic is approximated well by a simple third-order polynomial. The algorithm is tested to model FES response of human quadriceps and hamstrings, and it is compared to forward identification approaches with diverse basis functions, and to linear identification. When inverted again, estimation performance of the reversely identified model is comparable to that obtained by forward identification.