Observer‐based output feedback control for a boundary controlled fractional reaction diffusion system with spatially‐varying diffusivity

Abstract

This study is concerned with observer design and observer-based output feedback control for a fractional reaction diffusion (FRD) system with a spatially-varying (non-constant) diffusion coefficient by the backstepping method. The considered FRD system is endowed with only boundary measurable and actuation available. The contribution of this study is divided into three parts: first is the backstepping-based observer design for the FRD system with non-constant diffusivity, second is the output feedback controller generated by the integration of a separately backstepping-based feedback controller and the proposed observer to stabilise the FRD system with non-constant diffusivity, and the last is the Mittag–Leffler stability analysis of the observer error and the closed-loop FRD systems. Specifically, anti-collocated location of actuator and sensor is considered in the stabilisation problem of this system with Robin boundary condition at x = 0 and the boundary feedback controller for Dirichlet actuation at x = 1 . By designing an invertible coordinate transformation to convert the observer error system into a Mittag–Leffler stable target system, the observer gains are obtained. They are used to design the output feedback control law for stabilising the closed-loop system. Finally, a numerical example is shown to validate the effectiveness of the authors' proposed method.