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

The paper considers the problem of rejecting disturbances composed of two sinusoidal signals with frequencies that are close to each other. A natural approach consists in treating the two components separately. However, when the frequencies are close and unknown, the convergence of a double frequency estimator is typically slow, due to the difficulty in distinguishing the components. The alternative approach taken in this paper consists in representing the disturbance signal as a single sinusoid with time-varying magnitude and phase. The theoretical basis and the limitations of such a representation are first discussed. Then, an adaptive disturbance rejection algorithm is proposed and the resulting nonlinear system is analyzed using various approximations. Simulation results show 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.