Revised barrier function-based adaptive finite- and fixed-time convergence super-twisting control

This paper presents an adaptive gain, finite- and fixed-time convergence super-twisting-like algorithm based on a revised barrier function, which is robust to perturbations with unknown bounds. It is shown that this algorithm can ensure a finite- and fixed-time convergence of the sliding variable to the equilibrium, no matter what the initial conditions of the system states are, and maintain it there in a predefined vicinity of the origin without violation. Also, the proposed method avoids the problem of overestimation of the control gain that exists in the current fixed-time adaptive control. Moreover, it shows that the revised barrier function can effectively reduce the computation load by obviating the need of increasing the magnitude of sampling step compared with the conventional barrier function. This feature will be beneficial when the algorithm is implemented in practice. After that, the estimation of the fixed convergence time of the proposed method is derived and the impractical requirement of the preceding fixed-time adaptive control that the adaptive gains must be large enough to engender the sliding mode at time $t=0$ is discarded. Finally, the outperformance of the proposed method over the existing counterpart method is demonstrated with a numerical simulation.