H∞ control design for systems approximated by piecewise linearity

In this paper, a linear robust controller is proposed for systems comprising a hysteresis operator preceding a linear stable system. The hysteresis operator is modeled with piecewise linear characteristics with uncertainties, and a nominal inverse operator is included to mitigate the hysteresis effect. An H_∞ control design is proposed to handle the remaining uncertainties in a two-degree-of freedom (2DOF) framework. Usually, the H_∞ control is designed for a nominal plant while hysteresis is treated as uncertainty. The hysteresis nonlinearity is not modeled or inverted in those methods. In our work we reduce the effect of hysteresis by inversion and only the remaining inversion error determines the size of the uncertainty. We compared the proposed H_∞ method with a PI controller when 2DOF is applied to both system. The PI controller usually has good performance at low frequencies, however, it does not take advantage of knowing the dynamics of the system as in H_∞ design, where we can improve the performance at high frequency. Simulation results on a model of piezoelectric actuator-based nanopositioner are presented to illustrate the design and analysis, where the hysteresis nonlinearity is represented by a Prandtal-Ishlinskii operator.