Stabilisation of an anti-stable joint string with boundary disturbance

In this paper, we consider the boundary stabilisation for a one-dimensional joint anti-stable string equation with input disturbance entering the control end. A backstepping transformation is used to design a suitable boundary controller so that the system is equivalent to a target system which is exponentially stable when the disturbance is absent. The active disturbance rejection control approach (ADRC) is adopted to estimate and eliminate the external disturbance. The well-posedness and stability of the target system are presented. Some numerical stimulations indicate the effectiveness of the control law.