Design and Analysis of Sliding Mode Controller Under Approximate Hysteresis Compensation

A sliding mode controller (SMC) is proposed for a class of systems comprising a hysteresis operator preceding a linear system with an all-pole transfer function. The hysteresis operator is modeled with uncertain piecewise linear characteristics, and a nominal inverse operator is included to mitigate the hysteresis effect. A classical SMC design typically uses a constant coefficient in the switching component, which is tuned via trial-and-error. In this paper, a state- and time-dependent coefficient is proposed based on the derived inversion error, which eliminates the need for parameter tuning and ensures the convergence of the sliding surface to the boundary layer without compactness assumptions. In addition, singular perturbation is used to analyze the system behavior within the sliding-surface boundary layer for the case of a constant coefficient in the classical SMC design. In particular, analytical insight is gained on the frequency-scaling behavior of the tracking error under a periodic reference. Simulation and experimental results based on a piezoelectric actuator-based nanopositioner are presented to illustrate the design and analysis, where the hysteresis nonlinearity is represented by a Prandtal-Ishlinskii operator.