An indirect adaptive servocompensator for signals of unknown frequencies with application to nanopositioning

We propose an adaptive servocompensator utilizing frequency estimation and slow adaptation for systems subject to inputs of unknown frequencies. We show that the proposed controller can achieve zero tracking error for a class of periodic references and disturbances, including scenarios specifically relevant to piezo-actuated nanopositioning systems. In particular, for the case of a sinusoidal reference input, we establish the exponential stability of the closed-loop system in the presence of harmonic disturbances, under certain conditions on the amplitudes of the reference and disturbances. We also prove exponential stability in the case of sinusoidal reference and disturbance with two distinct frequencies. Additionally, we show that the proposed method, in conjunction with approximate hysteresis inversion, can attenuate the effect of hysteresis nonlinearity preceding linear dynamics and ensure the boundedness of the closed-loop system. Experiments conducted on a commercially available nanopositioner confirm our theoretical analysis and demonstrate the effectiveness of the proposed method as compared to Iterative Learning Control, a competitive technique in nanopositioning for tracking periodic references.