Explicit Boundary Controls for Finite Diffusion Process

Abstract

Three types of boundary controls are considered: proportional control, feed-forward control and integrated-by-past-values control. They stabilise the diffusion process exponentially at the desired reference on the controlled boundary. The process is actuated through the Neumann boundary while on other boundaries it is given with the Dirichlet data and, in higher dimensions, also with the homogeneous Neumann boundary condition. Two systems, deterministic and stochastic are compared as an ideal and real description of a physical system. Disturbance in the stochastic system is originated by a white noise on the controlled boundary. Physically this noise represents different types of uncertainties like a side-reaction mass-flux and other uncertainties that are not part of a deterministic model. At first the proportional control is analysed on the deterministic system and then it is represented in the feedback integrated past controls form, and lastly the developed controls are modified as a version of stochastic control that applied on the stochastic system, which boundary is disturbed with an unmeasured noise. The regulation error that emerges in the stochastic system control is analysed and proved to be a zero-mean, bounded-variance Gaussian variable that is correlated spatially and temporarily. The process control is demonstrated on an electrodeposition process example.