Application of Backstepping Stability Control in the Diffusion Process

Abstract

In this paper, 1-D mathematical model of the coagulation process of the polyacrylonitrile (PAN) carbon fiber is established using Fick diffusion law. Boundary stabilization for a linear parabolic diffusion-reaction partial differential equation (PDE) is considered. We use the method of backstepping to implement the boundary control of the concentration diffusion in the forming process of carbon fiber. By using the coordinate transformation, we transform the original system to a standard static system. The transformation depends on a so called gain kernel function, and we can design the boundary feedback controller using the kernel function. For the model in this paper, the kernel function itself is a hyperbolic PDE, and there is no explicit formation. Therefore, we use numerical methods to obtain the kernel function, and give the simulation results for the closed-loop control response. The simulation results show that the open-loop unstable system is stabilized by a boundary feedback.