Time-varying feedback for regulation of normal-form nonlinear systems in prescribed finite time

While non-smooth approaches (including sliding mode control) provide explicit feedback laws that ensure finite-time stabilization but in terminal time that depends on the initial condition, fixed-time optimal control with a terminal constraint ensures regulation in prescribed time but lacks the explicit character in the presence of nonlinearities and uncertainties. In this paper we present an alternative to these approaches, which, while lacking optimality, provides explicit time-varying feedback laws that achieve regulation in prescribed finite time, even in the presence of non-vanishing (though matched) uncertain nonlinearities. Our approach employs a scaling of the state by a function of time that grows unbounded towards the terminal time and is followed by a design of a controller that stabilizes the system in the scaled state representation, yielding regulation in prescribed finite time for the original state. The achieved robustness to right-hand-side disturbances is not accompanied by robustness to measurement noise, which is also absent from all controllers that are nonsmooth or discontinuous at the origin.