Time-delay control scheme with adaptive fixed-time convergent super-twisting fractional-order nonsingular terminal sliding mode for piezoelectric displacement amplifier

Piezoelectric displacement amplifiers (PDAs) have been widely used in precision positioning fields. However, the inherent hysteresis and creep nonlinear effect of piezoelectric actuators (PEAs) and time-varying lumped disturbances bring extreme challenges to the precise motion control of PDAs. Although various control schemes based on PEAs have been developed and have shown significant results. However, due to the high sensitivity of precision positioning to environmental variations, the development and identification of accurate models and the control timeliness often become obstacles in engineering. To realize precise motion control of PDAs under complex lumped disturbances, a new time-delay control scheme (AFSTA-FONTSM) using an adaptive fixed-time convergent super-twisting algorithm (AFSTA) and a fractional-order nonsingular terminal sliding mode (FONTSM) is proposed. Specifically, the time-delay information obtained by time-delay estimation technology is used to estimate the lumped dynamic characteristic of the system, thus establishing a simple control framework without a system dynamic model. FONSTM is constructed as a sliding mode manifold, and satisfactory error dynamic characteristic is obtained. A new AFSTA is designed as the reaching law in the sliding mode phase. AFSTA has fixed-time convergence when the upper bound of lumped disturbances exists, which ensures the control timeliness. Benefiting from the newly designed adaptive algorithm, the upper bound value of lumped disturbances is no longer needed to determine the control gains, which effectively prevents overestimation of the control gains. Correspondingly, the convergence time of AFSTA is estimated, and the stability of the closed-loop system is analyzed by the Lyapunov theory. Three existing time-delay control schemes, namely MSTA-FONTSM, AMSTA-FONTSM, and ASTA-FONTSM are selected, and four scenes are designed for comparative experiments. The experimental results show that MSTA-FONTSM has the worst control performance among the four control schemes. For the step, and continuous cosine trajectories with periods of T = 1 s and T = 2 s, the root-mean-square error of the proposed AFSTA-FONTSM is reduced by 56.86%, 54.03%, and 50.24% compared with MSTA-FONTSM. For disturbance experiments under different loads, the control performance of the proposed AFSTA-FONTSM is still superior to the other three control schemes without load.