Expectation maximization estimation for a class of input nonlinear state space systems by using the Kalman smoother

The parameter estimation for a class of single-input single-output (SISO) Hammerstein state space systems is considered in this paper. The nonlinear block in the discussed system is represented by a polynomial in the input signal with unknown coefficients. By applying the over-parameterization method, the SISO Hammerstein state space model is transformed to a multiple-input single-output linear state space model. The unknown system states and parameters are estimated interactively. The Kalman smoother is used to calculate the state estimates. Under the principle of the expectation maximization, an identification algorithm is derived to realize the joint estimation for the unknown model parameters and states. Although the over-parameterization method increases the number of redundant parameters, it simplifies the identification problem of the input nonlinear state space model in this paper. A numerical simulation example and an experiment carried out on the multitank system are provided to demonstrate that the derived identification method is effective.