Adaptive robust output-feedback boundary control of an unstable parabolic PDE subjected to unknown input disturbance

In this paper, an adaptive robust boundary controller is designed for an unstable heat equation with external disturbance flowing into the control end. The disturbance and uncertainty effect is cancelled out in the closed-loop system by using on-line approximation of disturbance upper-bound provided as an output of adaptive robust update law. The asymptotic stability of the controlled system in the presence of unknown disturbance and/or parametric uncertainties is proved utilising the Lyapunov theorem. Unlike previous researches, the control implementation does not require prior knowledge of unknown disturbance. Moreover, the controller does not require measuring the system states or estimating the system states by an observer. The adaptive robust controller can be regarded as a combination of the best qualities of the adaptive controller and the robust controller with no prior knowledge of system uncertainty, and asymptotic tracking error performance. Numerical simulations and comparisons are provided to illustrate the effectiveness of the proposed method.