Lyapunov function design for finite-time convergence analysis of “twisting” and “super-twisting” second order sliding mode controllers

A new method of a Lyapunov functions design is presented. It is based on the characteristic method application and related to resolving of the first-order partial differential equation of a special type. A successful resolution of this equation guaranties of finite-time convergence for the corresponding dynamics given by an ordinary differential equation with a discontinuous right-hand side and allows to estimate convergence time. The suggested method is illustrated by its application to the ldquotwistingrdquo and ldquosuper-twistingrdquo controllers stability analysis. The constructed Lyapunov function as well its level lines sections is graphically illustrated.