Evolution of regulation errors in the diffusion layer in electrodeposition process control

• We analyse effect of uncertainties on the resulting control errors.
• We model the uncertainties in general settings with a white noise and colour noise.
• In the case of colour noise the resulting control errors can be suppressed effectively.
• In electrodeposition the model can be improved essentially in the crucial part of reaction at the cathode surface.

A random field induced by infinitely many noises emerges in numerous processes, such as a mixed boundary electrochemical deposition process, which is determined by the main reaction and several side reactions. Since the process is controlled on one boundary without knowledge of the side reactions, a model error emerges and the regulation error trails. The model considered in this paper allows for general settings and is defined for white noise and colour noise. The white noise is responsible for unpredictable model errors and the colour noise for systematic errors. A relatively simple boundary control is applied that, when lifted into the domain, acts as a smooth function. The paper demonstrates that the regulation error can be suppressed effectively even if the correlation of random noises is relatively weak. The regulation error is suppressed by the double effects of the Laplacian and noise correlation. The traces of covariance operators are found for several categories of the domain noises and for the boundary noises. A similar trace function in general settings satisfies a Kolmogorov equation in infinitely many state variables. The solution for the Kolmogorov equation is simplified to a form that makes numerical treatment possible. The regulation errors induced by white noise are compared with the errors induced by coloured noise in the numerical analysis.