Regulation under disturbances with multiple harmonics of unknown frequency

In this paper, we address the regulation problem for a linear plant driven by a neutrally stable exosystem. It is assumed that the plant is subjected to an exogenous disturbance, and that this signal is formed of known harmonics of a sinusoidal reference signal whose frequency is unknown. Such an exosystem is motivated by the case where the system possesses an input nonlinearity like hysteresis. We propose an adaptive servocompensator requiring estimation of only the primary frequency. Using two-time-scale analysis, we establish the semi-global convergence of the parameter error to an arbitrarily small neighborhood of zero, when the adaptation gain is small enough and when the disturbance is sufficiently small comparing to the reference signal. The proposed control scheme involves far fewer adaptation variables than existing methods, and its performance is verified in both simulation and experiments conducted on a nanopositioning stage.