An integro-differential formula on the Wiener kernels and its application to sandwich system identification

It is well known that a nonlinear system with a white Gaussian noise input can be characterized in terms of kernels using the celebrated Wiener theory. In a practical use of the method, however, one may encounter difficulty in obtaining higher order kernels except for the first few because of, for instance, the excessive computational requirement. In this paper, we give an integro-differential formula on the kernels and as its application, an algorithm to identify a cascade system of a linear, a memoryless nonlinear, and linear subsystems, which we call a sandwich system as a whole. According to the formula, kernels up to the second order for different power levels of the input noise are required to identify the subsystems. Impulse response functions of the two linear subsystems can be obtained under appropriate normalization conditions, while the nonlinear subsystem is estimated in the form of a truncated Hermite polynomial expansion. As illustrated examples, two such systems are identified using the algorithm.