Hyperexponential Stabilization of Double Integrator with Unmatched Perturbations

In this letter, we propose linear time-varying state feedback controllers for a double integrator system subject to bounded disturbances. Under the proposed controls, the first state of the double integrator converges to zero at a hyperexponential rate (faster than any exponential decay) uniformly with respect to the disturbances. Meanwhile, the second state stays bounded or approaches the negative of an unmatched differentiable perturbation. These results are applied to design a novel differentiator.