ADRC Dynamic Stabilization of an Unstable Heat Equation

In this article, we consider the exponential compensation of an unstable heat equation through a dynamic boundary active disturbance rejection control (ADRC) compensator, which is described by a first-order system of two ordinary differential equations. The one internal point temperature of the heat equation is measured and fluxed into the compensator while the output of the compensator is forced into the boundary of heat equation. The resulting closed-loop system has been shown to be well-posed by the semigroup approach. The eigenvalues of the closed-loop system are tested to be located at the open left-half plane by the Nyquist criterion for distributed parameter systems. Then the Riesz basis method is adopted to show that the closed-loop system is exponentially stable. Numerical simulations are presented to show the effectiveness of the ADRC compensator to stabilize the unstable heat equation with one unstable pole.