Fractional order sliding mode control based on single parameter adaptive law for nano‐positioning of piezoelectric actuators

A fractional order sliding mode control (FOSMC) based on single parameter adaptive law for nano-positioning of Piezoelectric Actuators (PEAs) is proposed. First, the Bouc–Wen (B–W) model is used to describe the hysteresis of the nano-position platform based on PEAs, which provides a mathematical model for the subsequent controller design. Then, theoretical support is provided to design the FOSMC based on adaptive law of different parameters, which are proposed for the displacement tracking problem of PEAs, and the position error convergence is also proved. Moreover, the core parameters of FOSMC based on single parameter adaptive law are identified by hybrid differential evolution (HDE) and adaptive differential evolution (ADE), which require considering the relationship between the scaling factor and the cross-probability factor. Finally, experiments have been conducted with the displacement signals mixed with multiple frequencies and multiple amplitudes and the results obtained from them show that the proposed control scheme can produce a faster response and smaller tracking errors in PEAs system as compared to traditional control algorithms.