Adaptive band-limited disturbance rejection in linear discrete-time systems

The problem of adaptively rejecting disturbances consisting of a linear combination of sinusoids with unknown and/or time varying frequencies for SISO LTI discrete-time systems is considered. The rejection of the disturbance input is achieved by constructing the set of stabilizing controllers using the Youla parametrization and adjusting the Youla parameter to achieve asymptotic disturbance rejection. First, a numerical off-line controller design algorithm, which assumes knowledge of only the disturbance input values, is presented and the structure of the resulting controller analyzed. It is shown, under some mild assumptions, that if the off-line designed controller achieves regulation, then it must include a model of the disturbance input. Two on-line (adaptive) controller design algorithms are then introduced. Conditions under which the on-line algorithms yield an asymptotic controller that achieves regulation are presented. Conditions for both the case where the disturbance input properties are constant but unknown and for the case where they are unknown and time varying are given. The on-line controller construction amounts to an adaptive implementation of the internal model principle. The performance of the two adaptive algorithms is illustrated with a simulation example