Computing and Estimating the Reaching Time of the Super-Twisting Algorithm

Abstract

The super-twisting algorithm is a second-order sliding-mode algorithm that may be used either for control or for observation purposes. An important performance characteristic of this algorithm is the so-called reaching or convergence time, the time it takes for the controller to reach the sliding surface or for the estimates to converge. In this chapter, three techniques are discussed to estimate, i.e., upper bound, and in some cases even compute this reaching time in the presence of additive perturbations, which are Hölder continuous in the state or Lipschitz continuous in the time. The first is obtained from an analytic computation of the unperturbed reaching time; the second is based on a family of quadratic Lyapunov functions; and the third is derived from a necessary and sufficient stability criterion. For each approach, the range of permissible perturbations, its asymptotic properties with respect to parameters and perturbation bounds, and, when applicable, the selection of parameters are discussed. Numerical comparisons illustrate the results obtained with each approach.