Adaptive sliding mode control - convergence and gain boundedness revisited

This paper reviews the main adaptive sliding mode controller (ASMC) designs for nonlinear systems with finite uncertainties of unknown bounds. Different statements of convergence referring to uniformly ultimate boundedness (UUB), asymptotic convergence (AC) and finite-time convergence (FTC) for ASMC shown in recent papers are analysed. Weaknesses and incomplete proofs apropos FTC are pointed out. Thereafter, a new approach is proposed to successfully demonstrate FTC of the so-called sliding variable. We identify a compensating phase and a reaching phase during the ASMC process. A new explicit form for estimating the upper-bound reaching time is provided for any bounded perturbation. An amended form of the real ASMC is recalled showing improved accuracy and chattering reduction. Finally, numerical and experimental applications are performed to convey the discussed results.