Analysis of closed-loop systems involving hysteresis is important to both the understanding of these systems and the synthesis of control schemes. However, such analysis is challenging due to the nonsmooth nature of hysteresis nonlinearities. In this paper, singular perturbation techniques are employed to derive an analytical approximation to the tracking error for a system consisting of fast linear dynamics preceded by a piecewise linear hysteresis nonlinearity, which is motivated by applications such as piezo-actuated nanopositioning. The control architecture considered combines hysteresis inversion and proportional-integral feedback, with and without a constant feedforward control. The analysis incorporates the effect of uncertainty in the hysteresis model, and offers insight into how the tracking performance depends on the system parameters and the references, thereby offering guidance in the controller design. Simulation and experimental results on a piezo-actuated nanopositioning system are presented to support the analysis. In particular, the control scheme incorporating the feedforward element consistently outperforms the classical PI controller in tracking a variety of references.
Complementarity and passivity for piecewise linear feedback systems