Adaptive rejection of disturbances having two sinusoidal components with close and unknown frequencies

The paper considers the problem of rejecting disturbances with two sinusoidal components in the case where the frequencies are unknown and closely spaced. A natural approach consists of cancelling the components using two separate adaptive algorithms combined in a single scheme. However, experiments in active noise control applications have shown that convergence using such an approach could be very slow. The alternative approach of this paper consists of representing the disturbance signal as a single sinusoid with time-varying magnitude and phase. The theoretical basis and the limitations of this representation are first discussed. Then, an adaptive disturbance rejection algorithm is proposed and the resulting nonlinear system is analyzed using some approximations. Active noise control experiments demonstrate that the proposed algorithm has better convergence properties than an algorithm designed to cancel the two frequency components separately. In some cases, however, the cost is a small residual error on the output signal.