Robust Tracking Error Feedback Control for a One-Dimensional Schrödinger Equation

Abstract

In this article, we consider robust output tracking for a Schrödinger equation with external disturbances in all possible channels. The challenge of the problem comes from the fact that the observation operator is unbounded and the regulated output and the control are noncollocated. An observer-based approach is adopted in investigation. We first select specially some coefficients of the disturbances to obtain a nominal system, which is a coupled PDE+ODE system. For this nominal system, we design a feedforward control by solving related regulator equation. An observer is then designed for the nominal system in terms of the tracking error only. As a result, an error feedback control is, thus, designed by replacing the state and disturbances in the feedforward control with their estimates obtained from the observer. We show that this observer based error feedback control is robust to disturbances in all possible channels and system uncertainty. The stability of the closed loop and convergence are established by the Riesz basis approach. Some numerical simulations are presented to validate the results.