Improvements on stability analysis and stabilisation of constant- and variable-gain second-order sliding mode algorithms via exact convex representations

Abstract

A novel Lyapunov-based methodology for both stability analysis and stabilisation of second-order sliding mode algorithms in the presence of exogenous disturbances is presented. Nowadays, second-order sliding modes such as the well-known super-twisting algorithm are widely recognised for their usefulness in the design of controllers, observers, and exact differentiators. However, a theoretical proof for an optimal and systematical selection of parameters for the algorithm to be stable is of particular interest. The solution proposed in this paper is based on combining exact convex representations of the nonlinear algorithms with a strict Lyapunov function, which leads to stability or design conditions in terms of linear matrix inequalities that can be efficiently solved via convex optimisation techniques, allowing us to provide an enlarged and optimal parameter setting in contrast with existing results; moreover, the systematic nature of the proposal is able to deal with different scenarios avoiding ad-hoc solutions of previous works. Effectiveness of the results is verified via simulation examples.