Adaptive dual layer second-order sliding mode control and observation

This paper considers the problem of developing a 2-SM super-twisting-like structure with adaptive gains. The objective is to allow the gains to adapt (to both increase and decrease as appropriate) in such a way that they are as small as possible, and yet large enough to sustain a sliding motion. The concept of `equivalent control´ is used as a mechanism for assessing the extent to which the gains over-bound the uncertainty, and this information is used in the adaptation process. The resulting adaptation law has a dual-layer structure which is formally analyzed using Lyapunov techniques. In the paper, a new additional time varying gain is introduced into the traditional super-twisting structure, which has the effect of simplifying the stability analysis, whilst still ensuring the 2-SM properties of the traditional super-twisting scheme. Furthermore in the case when adaptation stops and the gains become constant, the traditional super-twisting structure is recovered.