Time space deformation approach to prescribed-time stabilization: Synergy of time-varying and non-Lipschitz feedback designs

Abstract

A novel approach is proposed to the prescribed-time stabilizing design of feedback linearizable controllable systems regardless of their initial conditions. The basic design idea consists in scaling an appropriate finite-time stabilizing non-Lipschitz feedback by means of a certain time deformation which squeezes the infinite time interval [0,∞) to the prescribed one [0,T), while also changing the state variables. The resulting nonsmooth feedback design relies on time-varying gains, uniformly bounded over the infinite horizon, thereby yielding an attractive implementation opportunity which is opposed to the original design (Song et al., 2017) with time-varying gains, escaping to infinity as time goes to the prescribed time instant. To facilitate the exposition the approach is first developed for a perturbed double integrator, driven by a twisting-wise controller, which is fed by a supertwisting-wise observer. Both the controller and observer, chosen for their robustness properties, are properly scaled according to the time space deformation to constitute an innovative prescribed-time stabilizing output feedback design of the double integrator, operating in the presence of uniformly bounded matched disturbances. To test the versatility of the approach it is then extended to the prescribed-time state feedback design of multi-input multi-output systems representable in the normal form. Theoretical results are additionally supported by simulations.