Tuning rules for second order sliding mode based output feedback synthesis

Tuning rules to a-priori guarantee an arbitrarily specified settling time for the output feedback synthesis are studied for planar systems. The estimate of one of the states is given by a second order sliding mode (super-twisting) observer. This is utilised in conjunction with the measured state to form the so-called `twisting´ controller. Existing results on finite settling time convergence based on strict and non-strict Lyapunov functions are used to construct a new combined Lyapunov function for the output feedback system. A finite upper bound on the closed-loop system trajectories is computed explicitly under output feedback synthesis, thereby proving that the trajectories cannot escape to infinity. Then, the twisting controller essentially coincides with the state feedback case. Furthermore, tuning rules for the observer are developed to ensure that the observer error converges to zero before the closed-loop trajectories of the output feedback system escapes to an arbitrarily specified region. Existing tuning rules for the twisting controller are utilised to ensure finite time convergence from the arbitrarily specified region. The results prove a separation principle for the second order sliding mode output feedback problem.