On estimation of errors caused by non-linear undermodelling in system identification

The problem of quantifying errors due to non-linear undermodelling is addressed. In particular, the Wiener system which consists of a linear dynamic block followed by a static non-linearity and the Hammerstein system in which the order of these two blocks is reversed are studied. In either case it is assumed that the objective is to identify the linear part of the system using a purely linear model. These scenarios can be interpreted as linear identification applied to systems where non-linearity errors may occur and hence need to be quantified. The stochastic embedding approach is therefore applied to capture the on-average properties of the undermodelling due to inadequate representation of the non-linear effects. As compared to previous methods, the priors on the covariance matrix of the embedding parameters are reduced. As a result an expression for the amplitude error bound of the estimated transfer function, that does not require knowledge of the true system parameters, is obtained. The quality of this measure depends on the degree of accuracy with which the unknown non-linearity can be represented using a set of known polynomials. The proposed method simultaneously delivers error bounds on the estimated transfer function and an indirect estimate of the size of the non-linearity, both quantities being central in controller design.