A new kernel-based approach for overparameterized Hammerstein system identification
This work addresses system identification for Hammerstein models, which are used in control and signal processing, but it appears incremental as it builds on existing kernel-based methods with a tailored kernel.
The authors tackled the problem of identifying Hammerstein systems, which combine a static nonlinearity with a linear dynamic system, by proposing a new kernel-based approach that estimates an overparameterized vector to reconstruct both components; they demonstrated through numerical experiments that this method compares very favorably with two standard identification methods.
In this paper we propose a new identification scheme for Hammerstein systems, which are dynamic systems consisting of a static nonlinearity and a linear time-invariant dynamic system in cascade. We assume that the nonlinear function can be described as a linear combination of $p$ basis functions. We reconstruct the $p$ coefficients of the nonlinearity together with the first $n$ samples of the impulse response of the linear system by estimating an $np$-dimensional overparameterized vector, which contains all the combinations of the unknown variables. To avoid high variance in these estimates, we adopt a regularized kernel-based approach and, in particular, we introduce a new kernel tailored for Hammerstein system identification. We show that the resulting scheme provides an estimate of the overparameterized vector that can be uniquely decomposed as the combination of an impulse response and $p$ coefficients of the static nonlinearity. We also show, through several numerical experiments, that the proposed method compares very favorably with two standard methods for Hammerstein system identification.