Recursive prediction error algorithms for joint time delay and parameter estimation of certain classes of nonlinear systems

The problem of identifying certain classes of nonlinear systems which incorporate an unknown, possibly time varying, delay is addressed. In particular, the Wiener system which consists of a dynamic linear part followed by a static nonlinearity and the Hammerstein system in which the order of these two blocks is reversed are studied. The linear part of respective system is assumed to consist of an IIR filter in series with a pure time delay and the nonlinearity is represented by a polynomial. A recursive prediction error method, RPEM is derived for simultaneous identification of the parameters of the IIR filter, the delay and the coefficients of the polynomial. The proposed algorithm is combined with a method for online adjustment of the forgetting factor in order to enable tracking in case the parameters are time varying. In addition, Lagrange interpolation of the input signal is used in order to enable estimation of noninteger time delays. The usefulness of the proposed scheme is illustrated by means of numerical examples.