On the LTI properties of adaptive feedforward systems with tap delay-line regressors (Q2723881)

A fundamental result of J. R. Glover jun. (1977) showed that an adaptive system with a sinusoidal tap-delay-line regressor becomes linear time-invariant (LTI) in the limit as the number \(n\) of taps is increased to infinity. This result represents the basis of designing many present-day adaptive algorithms devoted to solving the fundamental problem of cancelling a sinusoidal disturbance having unknown frequency content.NEWLINENEWLINENEWLINEThe purpose of this paper is to extend Glover's result, showing that the adaptive system may be represented as a parallel connection of an LTI and LTV (linear time-variant) blocks, and putting the problem into a modern robust control setting. Treating the LTV part as a perturbation, the main result of the paper is to compute an explicit norm-bound (induced 2-norm) on the LTV block. For \(n\) finite, the norm bound is expressed as a function of the number \(n\) of taps, the adaptation gain, the number of tones, and the tone spacing. The norm bound goes to zero asymptotically as \(1/n\), when \(n\) goes to infinity, thus recovering Glover's result. The availability of this norm-bound in the more realistic case when the number \(n\) of taps is finite opens up new opportunities for analyzing adaptive systems within the framework of modern robust control theory.