Robust Fixed-Time Stability: Application to Sliding-Mode Control

Abstract

This article deals with robust fixed-time stability and stabilization. First, new global robust fixed-time stability results are proposed for scalar systems by using constant and variable exponent coefficients. Then, they are applied to global robust fixed-time stabilization of a class of uncertain nonlinear secondorder systems by using sliding mode control. All the results are illustrated in simulation.

Fixed-time sliding mode control with mismatched disturbances

“…where δ 3 " δ 2 `2βλ 1 p1 `µ1 qpδ 1 `k2 q. By using Theorem 2 in [8], one deduces that system (6) starting at sp0q " s 0 reaches the sliding surface ts " 0u in a fixedtime satisfying T ps 0 q ď…”

“…By using one more time Theorem 2 in [8], it is deduced that x 1 ptq starting at x 1 p0q " x 10 reaches the origin in a fixed-time satisfying T px 10 q ď…”

“…However, the proposed controllers were rather complex to implement due to a problem of singularity avoidance coming from the fixed-time stability strategy. In [8], simple robust controllers are proposed by using state-dependent variable exponent coefficient in the sliding surfaces and the controllers.…”

“…In the previous works, the control problems are solved by using observers to estimate mismatched disturbances. In our article, we use a fixed-time SMC strategy adapted from [8] which is known to be robust in order to reject the mismatched disturbance of a secondorder system without using disturbance observers. This renders this new strategy easier to tune.…”

“…To the best of the authors' knowledge, this is the first time that a mismatched disturbance is rejected by using a fixed-time sliding surface known for its robustness properties. Thus, we adapt the sliding-mode controller developed in [8] in order to be used with a mismatched disturbance. The results are tested in simulation on the double integrator in order to show the efficiency of the proposed method.…”