Identification of Hammerstein Systems with Random Fourier Features and Kernel Risk Sensitive Loss

Identification of Hammerstein systems with polynomial features and mean square error loss has received a lot of attention due to their simplicity in calculation and solid theoretical foundation. However, when the prior information of nonlinear subblock of a Hammerstein system is unknown or some outliers are involved, the performance of related methods may degenerate seriously. The main reason is that the used polynomial just has finite approximation capability to an unknown nonlinear function, and mean square error loss is sensitive to outliers. In this paper, a new identification method based on random Fourier features and kernel risk sensitive loss is therefore proposed. Since the linear combination of random Fourier features can well approximate any continuous nonlinear function, it is expected to be more powerful to characterize the nonlinear behavior of Hammerstein systems. Moreover, since the kernel risk sensitive loss is a similarity measure that is insensitive to outliers, it is expected to have excellent robustness. Based on the mean square convergence analysis, a sufficient condition to ensure the convergence and some theoretical values regarding the steady-state excess mean square error of the proposed method are also provided. Simulation results on the tasks of Hammerstein system identification and electroencephalogram noise removal show that the new method can outperform other popular and competitive methods in terms of accuracy and robustness.