Consistent parameter bounding identification for linearly parametrized model sets

Abstract

In standard parameter bounding identification a time-domain bound on the noise signal is used to construct parameter bounds. One of the properties of this standard approach is that there is no consistency if a conservative noise bound has been chosen. It is shown that alternative bounds on the noise signal give rise to consistent parameter bounding identification methods; i.e., asymptotically in the number of data samples, the feasible parameter set converges to the true parameter vector. The first alternative noise bound that is introduced is a bound on the cross-covariance between the noise and some instrumental (input) signal. The noise bound is represented by a small number of linear inequalities, which can be used in parameter bounding by linear programming. It is shown that there is consistency under fairly general conditions, even when a conservative bound has been chosen. Additionally, a procedure is presented to estimate the cross-covariance bound from data. Similar consistency results are shown for two other types of noise bounds: a bound on the discrete Fourier transform of the noise in combination with sinusoidal excitation, and a time-domain bound on the noise after measurement averaging with periodic excitation.