Active disturbance rejection control to stabilization of coupled delayed time fractional-order reaction–advection–diffusion systems with boundary disturbances and spatially varying coefficients

In this paper we consider the boundary asymptotic stabilization of a coupled time fractional-order reaction–advection–diffusion (FRAD) system with boundary input disturbances, spatially varying coefficients and time varying delays. The complex system composition makes it difficult to analyze system property directly. For this issue, we here address it by introducing two transformations to convert the original system into the one where theories of fractional evolution equations and operator semigroup are easily applicable. The active disturbance rejection control (ADRC) and backstepping control are also utilized in investigation. Using ADRC, the disturbance is first estimated by constructing two auxiliary systems and then canceled through an approximation in the feedback-loop. For such auxiliary systems, one is used to take disturbances from the original system and to put them into a stable system. Another is used to estimate disturbances. In the second part, we use backstepping to develop a boundary state feedback control law with disturbance approximation to ’eliminate’ the disturbances and to achieve the closed-loop stability. The well-posedness and disturbance estimation are established by theories of fractional evolution equations. With fractional Halanay’s inequality, sufficient conditions are obtained to make the controlled system asymptotically stable and the auxiliary system bounded. Fractional simulation scheme is constructed to test the proposed synthesis generated by ADRC when the explicit solution of kernel equations does not exist.