Robust adaptive compensation of periodic disturbances with unknown frequency

An indirect adaptive output feedback compensator is designed to attenuate the effects of periodic disturbances with unknown frequency on a known stable linear system with no zeros on the imaginary axis. The compensator contains an adaptive internal model which rejects a fixed number of harmonics in the Fourier series expansion of the periodic disturbance. It also contains an independent frequency estimator whose estimation error converges into an arbitrarily small neighborhood of the origin, provided that a design parameter is chosen to be sufficiently large. The adaptive compensator ensures the global exponential convergence of the closed loop error signals to a neighborhood of the origin, whose size depends on the higher order harmonics which are neglected by the adaptive internal model.

I. Introduction

The problem of rejecting unknown sinusoidal disturbances was posed and studied in [1], [2], [3], [4] for known continuous time linear systems using the adaptive notch filters which were introduced in [5] and in [6] for known discrete time linear systems using the Youla-Kucera parametrization. An adaptive observer approach was followed in [7] and shown to work for multiple sinusoidal disturbances as well in [8]. Adaptive notch filters for on-line estimation of the frequency of a periodic signal have been studied in [9], [10], [11]. The problem of frequency estimation from noisy measurements has been addressed in [12], while the adaptive cancellation of unknown sinusoidal disturbances by state derivative feedback has been studied in [13]. When the periodic disturbance is not purely sinusoidal and contains infinite harmonics in its Fourier series expansion, according to the internal model principle formulated in [14], a finite dimensional compensator can not achieve disturbance rejection even though the frequency is known: this paper addresses precisely this difficult control problem in the case of unknown frequency.