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Asymptotic stabilisation of coupled delayed time fractional reaction diffusion systems with boundary input disturbances via backstepping sliding-mode control

In this paper we study the asymptotic stabilisation for coupled time fractional reaction diffusion (FRD) systems with time varying delays and input disturbances by backstepping-based boundary sliding-mode control. Here, the spatially varying diffusion coefficients can be same or distinct. To stabilise the system, we first use an invertible backstepping transformation to convert an original dynamics into a target dynamics with new manipulable inputs and perturbations. Then, we employ the sliding-mode algorithm to design this discontinuous controller to suppress disturbances. In this case, we obtain the combined backstepping/sliding-mode controller of the original dynamics. By using the fractional Halanay's inequality, the delay-independent stability condition is obtained for the asymptotic stabilisation of the delayed target system and therefore of the delayed original system under the combined controller. Fractional numerical examples are given to verify the efficiency of the proposed synthesis.