On kernel design for regularized Volterra series identification of Wiener–Hammerstein systems

There have been increasing interests in the Volterra series identification with the kernel-based regularization method. The major difficulties are on the kernel design and efficiency of the corresponding implementation. In this paper, we first assume that the underlying system to be identified is the Wiener–Hammerstein (WH) system with polynomial nonlinearity. We then show how to design kernels with nonzero off-diagonal blocks for Volterra maps by taking into account the prior knowledge of the linear blocks and the structure of WH systems. Moreover, exploring the structure of the designed kernels leads to the same computational complexity as the state-of-the-art result, i.e., $O\left(N^3\right)$, where $N$ is the sample size, but with the significant difference that the proposed kernels are designed in a direct and flexible way. In addition, for a special case of the kernel and a class of widely used input signals, further exploring the separable structure of the output kernel matrix can lower the computational complexity from $O\left(N^3\right)$ to $O\left(N{\gamma }^2\right)$, where $\gamma$ is the separability rank of the output kernel matrix and can be much smaller than $N$. We finally run Monte Carlo simulations to demonstrate the proposed kernels and the obtained theoretical results.