Nonlinear system modeling based on constrained Volterra series estimates
For practitioners needing nonlinear system identification with short data, this provides a theoretically grounded method with finite-sample guarantees.
The paper proposes a constrained Volterra series estimation algorithm for nonlinear system modeling that works with limited prior knowledge and short data records, achieving a model error of order √(N⁻¹ ln D) even when D ≥ N. Tests on the Wiener-Hammerstein benchmark show that q=1 yields sparser models with smaller input-output errors, while q>1 better characterizes system nature.
A simple nonlinear system modeling algorithm designed to work with limited \emph{a priori }knowledge and short data records, is examined. It creates an empirical Volterra series-based model of a system using an $l_{q}$-constrained least squares algorithm with $q\geq 1$. If the system $m\left( \cdot \right) $ is a continuous and bounded map with a finite memory no longer than some known $τ$, then (for a $D$ parameter model and for a number of measurements $N$) the difference between the resulting model of the system and the best possible theoretical one is guaranteed to be of order $\sqrt{N^{-1}\ln D}$, even for $D\geq N$. The performance of models obtained for $q=1,1.5$ and $2$ is tested on the Wiener-Hammerstein benchmark system. The results suggest that the models obtained for $q>1$ are better suited to characterize the nature of the system, while the sparse solutions obtained for $q=1$ yield smaller error values in terms of input-output behavior.