Consistent identification of Hammerstein systems using an ersatz nonlinearity

We develop a method for identifying SISO Hammerstein systems with an unknown static nonlinearity, linear dynamics, white input noise and colored output noise. We use least squares with a μ-Markov model to estimate the Markov parameters of the linear time-invariant dynamical system. Since the input to the linear system is not available, we use a substitute (ersatz) nonlinearity to transform the input for use in the regressor matrix. We prove that the Markov parameters of the system can be estimated consistently up to a constant scalar as the amount of data increases. This method is demonstrated with several numerical examples.