A Lyapunov approach to output feedback control using second-order sliding modes

The objective of this paper is to provide for the second-order sliding modes (SOSM) output feedback (OF) a Lyapunov-based approach, in the same spirit as it is done for the observer-based output feedback control in conventional non-linear control. To reach this objective, we consider first the Lyapunov design of a state feedback (SF) control based on the twisting algorithm. We propose a (new) Lipschitz continuous strict Lyapunov function for the twisting controller, which assures its finite-time convergence and its robustness under bounded perturbations. In a second step, a Lyapunov design of a finite-time convergent and robust observer estimating the state of the plant is considered. It generalizes the super-twisting differentiator. In a final step, the properties of the interconnection of the observer and the controller in the OF controller are established by Lyapunov arguments, much in the same form as in the conventional non-linear control literature. It is shown that the SOSM-based OF controller (almost) satisfies a separation principle, a result similar to the linear case. Here, however, in order to guarantee the robustness of the OF, the observer has to be designed taking into account the SF controller.