Adaptive gains of dual level to super-twisting algorithm for sliding mode design

A gain-adaption mechanism of a dual level to the super-twisting algorithm (STA) for adaptive sliding mode design is studied. The proposed dual level method first tunes a third-order sliding mode observer to exactly estimate the magnitude level of external disturbances, and then adjusts the two gains (a, beta) of STA online simultaneously such that a second-order sliding mode can take place with small rectifying gains. The gains of the third-order sliding mode observer are adjusted by exploring the homogeneous property such that only one auxiliary parameter L is needed to be tuned. The magnitude of this parameter L increases until the error between the observer output and actual disturbance disappears. While driving the sliding variable to the sliding mode surface of STA, one gain beta of the STA automatically converges to an adjacent area of the perturbation magnitude in finite time. The other gain a is adjusted by the gain beta to guarantee the robustness of the STA. This method requires no intervention during adaptation. The usefulness is illustrated by an example of designing an equivalent control-based sliding mode control with the proposed adaptive STA for a perturbed linear time-invariant system.