Lyapunov Approach for Analysis and Design of Second Order Sliding Mode Algorithms

Lyapunov functions are a basic tool for analysis and design in the modern control theory, and there are many different design methodologies based on Lyapunov theory. Second Order Sliding Modes, and in particular, the Super-Twisting Algorithm (STA), are a powerful tool for the design of controllers, observers and differentiators having very attractive dynamic features: they converge in finite time, even in presence of persistently acting bounded perturbations. This property, that we will call exactness, can be achieved because of the discontinuous nature of the STA. The design of control or observation algorithms based on Second Order Sliding Modes has been performed until now using either geometric or homogeneous approaches, but not Lyapunov methods. The reason for this situation is simple: only recently has been possible to find adequate Lyapunov functions for some of these algorithms. In this paper some recent advances in this direction will be presented and extended.