Control of systems with hysteresis via servocompensation and its application to nanopositioning

Partly motivated by the nanopositioning application in AFM and SPM systems, we consider the problem of tracking periodic signals for a class of systems consisting of linear dynamics preceded by a hysteresis operator, where uncertainties exist in both the dynamics and the hysteresis. A servocompensator is proposed, in combination with an approximate hysteresis inverse, to achieve high-precision tracking. The servocompensator accommodates the internal model of the reference signal and a finite number of harmonic terms. It is shown that, with a Prandtl-Ishlinskii (PI) hysteresis operator, the closed-loop system admits a unique, asymptotically stable, periodic solution, which justifies treating the inversion error as an exogenous periodic disturbance. Consequently, the asymptotic tracking error can be made arbitrarily small when the servocompensator accommodates a sufficient number of harmonic terms. Robustness with respect to uncertainties in the dynamics is also established. Simulation and experimental results are presented to validate the approach and evaluate the role of hysteresis inversion. In particular, experiments on a nanopositioner show that, with the proposed method, tracking can be achieved for a 200 Hz reference signal with 0.6% mean error and 2.3% peak error for a travel range of 20 μm.