Robustness and Performance of Adaptive Suppression of Unknown Periodic Disturbances

In recent years, a class of adaptive schemes has been developed for suppressing periodic disturbance signals with unknown frequencies, phases, and amplitudes. The stability and robustness of these schemes with respect to inevitable unmodeled dynamics and noise disturbances in the absence of persistently exciting signals has not been established despite successful simulation results and implementations. The purpose of this technical note is to propose a robust adaptive scheme for rejection of unknown periodic components of the disturbance and analyze its stability and performance properties. First, we consider the ideal case (non-adaptive) when complete information about the characteristics of the disturbance is available. We show that the rejection of periodic terms may lead to amplification of output noise and in some cases lead to a worse output performance. The way to avoid such undesirable noise amplification is to increase the size of the feedback control filter in order to have the flexibility to achieve rejection of the periodic disturbance terms while minimizing the effect of the noise on the output. The increased filter order leads to an over-parameterized scheme where persistence of excitation is no longer possible, and this shortcoming makes the use of robust adaptation essential. With this important insight in mind, the coefficients of the feedback filter whose size is over parameterized are adapted using a robust adaptive law. We show analytically that the proposed robust adaptive control scheme guarantees stability, performance and robustness with respect to unmodeled dynamics and bounded broadband noise disturbances. We use numerical simulations to demonstrate the results.