Predefined-time output stabilization with second order sliding mode generation

In this article, a novel second order sliding mode control strategy is proposed for relative-degree-2 nonlinear uncertain single- input-single-output (SISO) systems. To design the strategy, the phase portrait of the second order system in normal form associated with the formulated sliding mode control problem is studied. A sliding surface switching between arcs of parabolas is conceived to ensure the convergence in a predefined-time to the desired second order sliding mode. Lyapunov-based analysis and the reformulation of LaSalle-Yoshizawa results for nonsmooth systems are used to prove the uniform finite-time stability of the equilibrium consisting in the second order sliding mode enforcement. This in turn allows to prove the asymptotic convergence of the original plant state to the origin in spite of the uncertainties.