  This paper studies the super-twisting algorithm (STA) for adaptive sliding
mode design. The proposed method tunes the two gains of STA on line
simultaneously such that a second order sliding mode can take place with small
rectifying gains. The perturbation magnitude is obtained exactly by employing a
third-order sliding mode observer in opposition to the conventional
approximations by using a first order low pass filter. While driving the
sliding variable to the sliding mode surface, one gain of the STA automatically
converges to an adjacent area of the perturbation magnitude in finite time. The
other gain is adjusted by the above gain to guarantee the robustness of the
STA. This method requires only one parameter to be adjusted. The adjustment is
straightforward because it just keeps increasing until it fulfills the
convergence constraints. For large values of the parameter, chattering in the
update law of the two gains is avoided by employing a geometry based backward
Euler integration method. The usefulness is illustrated by an example of
designing an equivalent control based sliding mode control (ECBC-SMC) with the
proposed adaptive STA for a perturbed LTI system.
