On convergence analysis of an identification algorithm for Hammerstein-Wiener systems with unknown time-delay

Abstract

A Hammerstein-Wiener model with time delay is a specific class of nonlinear time delay systems where the time delay which involves the system input and the parameters are unknown and need to be estimated using input-output data. The main difficulty that has been encountered in this identification problem is the additional nonlinearity due to the presence of the time delay in the criterion to be minimized. As a solution of this problem, an alternative approach is applied, which consists in estimating separately the unknown parameters and time delay. However, the conventional optimization techniques are not directly applicable. Hence, we formulate the problem of estimating the unknown time delay as a continuous relaxation problem and we then apply the gradient approach to estimate all unknown variables. Furthermore, by using the martingale convergence theorem, the convergence analysis of the proposed algorithm is treated. A numerical example is offered to demonstrate the effectiveness of the proposed method.