Neumann 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 induced 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 finally the developed controls are applied on the stochastic system, which boundary is disturbed with an unmeasured white noise. The regulation error that emerges in stochastic control is analysed and proved to be a zero-mean, bounded-variance Gaussian variable that is correlated temporarily. As an example of such a system, any electrochemical reduction or oxidation process where diffusion mass transfer is important can be considered.

KEYWORDS:

• Distributed parameter system
• stochastic process
• boundary control

Acknowledgment

The author wish to thank Alexander Mendelson for ideas of proof of Theorems 3.1 and 3.2.

Disclosure statement

No potential conflict of interest was reported by the author.