Tracking control for systems with slope‐restricted hysteresis nonlinearities

Abstract

Summary This paper addresses the reference tracking problem for plants that can primarily be modeled as a linear time‐invariant (LTI) system preceded by a slope‐restricted hysteresis nonlinearity. This structure is particularly interesting due to its wide use in modeling smart actuator dynamics such as shape‐memory alloy and piezoelectric. The hysteresis nonlinearity is considered to be slope restricted; however, no specific modeling framework is exploited. Based on the slope bounds of the nonlinearity, conditions for asymptotic tracking are proposed in terms of linear matrix inequalities (LMIs) when constant reference is considered. By solving the LMIs, the controller parameters can be obtained, which eventually ensures asymptotic convergence of tracking error to zero. The proposed model‐free design technique is further reformulated considering parametric uncertainties in the LTI part. The resulting conditions in LMI framework facilitate solving the robust tracking problem for hysteretic systems. The results are applicable for multiple‐input–multiple‐output systems as well, with decoupled hysteresis nonlinearity. Additionally, with only integral control, the proposed results are specialized for the case of stable LTI part having positive definite symmetric part of the DC gain matrix. It is shown that there always exists such a tracking controller for the hysteretic system. Numerical examples are presented to illustrate the effectiveness of the proposed results.