Design of Continuous Twisting Algorithm

For the double integrator with matched Lipschitz disturbances we propose a continuous homogeneous controller providing finite-time stability of the origin. The disturbance is compensated exactly in finite time using a discontinuous function through an integral action. Since the controller is dynamic, the closed loop is a third order system that achieves a third order sliding mode in the steady state. The stability and robustness properties of the controller are proven using a smooth and homogeneous strict Lyapunov function (LF). In a first stage, the gains of the controller and the LF are designed using a method based on Pólya's Theorem. In a second stage the controller's gains are adjusted through a sum of squares representation of the LF.