Robust Control of Multiscale Process Systems Under Uncertainty: An Application to Epitaxial Thin Film Growth

Multiscale processes are characterized by inherently coupled chemical and physical phenomena occurring over different temporal and spatial scales [1]. In such systems, control of the events evolving at the microscopic scale is essential to product quality while efficient operations require manipulated variables at the macroscopic scale for real-time feedback control [2]. Accordingly, to describe the behavior of these processes, it is common to employ a multiscale model encompassing macro-scale continuum equations (e.g., Partial Differential Equations) embedded with micro-scale simulations (e.g., Kinetic Monte Carlo). Although such a model is capable to provide a fair representation of a system, the microscopic processes are subject to model parameter uncertainty that can significantly affect the control and optimization objectives. As such, to achieve the desired objectives, it is crucial to develop control strategies that are robust to these uncertainties.

This study explores a systematic framework to analyze model parameter uncertainty for the purpose of robust control and optimization in multiscale models. The difficulties in analysing uncertainty in such systems arise due to the computational intensity of the fine-scale simulations (typically KMC simulations). Moreover, the lack of a closed formulation between the process optimization objectives and the model parameters, and absence of sensors to measure microscopic features at fine scales make this analysis even more challenging. Motivated by these observations in this work, Power Series Expansion (PSE), which obtains a simpler representation of the actual system, is employed to analyze model uncertainty propagation [3].

The thin film growth process has been used in this study to effectively exemplify a multiscale process system. Strong dependence of the electrical and mechanical properties of thin films on their microstructure requires accurate control of the product’s quality (e.g., film’s roughness and thickness) at the fine scale. For uncertainty analysis of the thin film process, the probability distributions of microscopic events are determined using the PSE-based models that rely on the prior distribution of the uncertain parameters. Subsequently, the upper and lower bounds on desired outputs (e.g., film roughness) are determined at a given confidence level of the distributions obtained for the states of the system [4]. Moreover, multiple reduced-order lattices are employed in KMC simulations to reduce the computational costs. The potential application of the proposed method can be illustrated in open-loop optimization or closed-loop purposes. In open-loop optimization, the optimal temperature profile is obtained to optimize the end-point properties of the thin film. For closed-loop purposes, on the other hand, measurements are not available as frequent as required for the development of an effective feedback control strategy. The proposed algorithm can therefore be applied to develop a robust estimator that is able to provide the estimate of the controlled output using online measurements. Furthermore, the feedback control of the film microstructure can be achieved via offline identification of a low-order model from lower and upper bounds on the process outputs in the presence of uncertainty.

[1] Vlachos, D. G., (2005) A review of multiscale analysis: Examples from systems biology, materials engineering, and other fluid-surface interacting systems. Advances in Chemical Engineering, 30, 1-61.

[2] Christofides, P. D., Armaou, A., Lou, Y. & Varshney, A., (2008) Control and optimization of multiscale process systems. Boston: Birkhäuser.

[3] Nagy, Z. K. & Braatz, R. D., (2007) Distributional uncertainty analysis using power series and polynomial chaos expansions. Journal of Process Control, 17, 229-240

[4] Rasoulian, S., Ricardez-Sandoval, L.A. “Uncertainty analysis and robust optimization of multiscale process systems with application to epitaxial thin film growth” Chemical Engineering Science, In Press.