An interactive maximum likelihood estimation method for multivariable Hammerstein systems

For a multivariable Hammerstein controlled autoregressive moving average system (CARMA) system, the identification difficulty is hard to parameterize the system into an quasi auto-regression form to which the standard least square method can apply. By using an interactive maximum likelihood (IML) estimation method, this paper interactively maximizes the logarithmic likelihood function over multiple parameter vectors in a more general model, respectively. The details include: (1) reframe the system into a sum of some bilinear functions about the parameter vectors of the nonlinear part and the linear part; (2) interactively maximize the logarithmic likelihood function over each parameter vector to get their estimates; (3) when updating one parameter vector, substitute other parameter vectors or unknown information vectors by their estimates. The advantage of the IML algorithm is that it overcomes the limit on an autoregressive model form with one parameter vector. The IML method is simple to understand and easy to implement. Numerical simulations indicate that the explored IML algorithm is capable of generating accurate parameter estimates, and easy to implement on-line. (C) 2020 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.