Optimal and Robust Operations of Complex Processes withNon-Gaussian Distributed Uncertain Variables under ChanceConstraints - Extension to Model Predictive Control of Parabolic Partial Differential Equation Systems
Final Report Abstract
The chance constrained optimization problem and model predictive control of partial differential equations systems (CCPDE and CCMPCE of PDE respectively) pose an enormous theoretical and computational challenge. In general, numerical solutions for such problems are obtained by an approximation method. In the first part of the research in this project, our smoothing inner-outer approximation approach was extended and aplied by analyzing its properties on elliptic CCPDE problems with space dependent random fields in the boundary value condition of PDE systems. For solving this problem, we derived and proved the existence and uniqueness of the solution as well as main structural properties. Further more, we analyzed generalized derivatives of our approximation method and derived the convexity for both inner and outer approximation problems, thus, insuring sufficiency of the first-order optimality condition to numerically address such problems. Our second step was to conduct the investigation for more general problems of parabolic CCPDE with uncertainty in coefficients, in forcing term, and in boundary value condition. In this study we have already proved the existence and uniqueness of the solution and some structural properties, but this work is still in progress. The second part of our work focused on the development of CCMPC of parabolic PDE system with random parameters. We proposed a new CCMPC scheme that gains the following advantages: feasibility is guarantied irrespective of the distribution of input uncertainties (bounded or unbounded uncertainties), through construction of deterministic tubes the feasibility is ensured on the whole control horizon yielding robust constraint satisfaction, and the deterministic tubes are constructed automatically, which neither incurs additional computation expenses nor does it require additional simplifying assumptions on the problem. Validity and generality of the proposed scheme were demonstrated on the case study of hyperthermia treatment of cancerous tumor with random conductivity coefficient. To implement the underlying CCMPC scheme, we applied a series of transformations to the original problem to convert it into a couple of high-dimensional deterministic state-constrained NLP problems, solution of which is an approximation to the solution of the initial problem. Numerical results demonstrate the viability of the proposed scheme, verify the stated properties, and also indicate some directions of improvement. Specifically, the developed approach proved to be quite computationally demanding, indicating the need for the reduction of the prediction horizon. The latter can be accomplished by analyzing the stability properties and obtaining a minimum stabilising prediction horizon of CCMPC. These and other questions will be the main focus of our future research.
Publications
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Analytic approximation and differentiability of joint chance constraints. Optimization, 68, 10, 2019
Geletu A., Hoffmann A., Li P.
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A Computation approach to chance Constrained optimization of boundary-Value parabolic partial differential equation systems. 21st IFAC World Congress, Berlin, Germany, July 12-17, 2020
Nida, K., Geletu, A., Li, P.
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Chance constrained optimization of elliptic PDE systems with a smoothing convex approximation. ESAIM: Control, Optimisation and Calculus of Variations, 26, 70, 2020
Geletu A., Hoffmann A., Schmidt P., Li P.
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Chance constrained model predictive control of parabolic PDE systems with random parameters. 31st European Conference on Operational Research, Athens, Greece, July 11-14, 2021
Voropai, R., Geletu, A., Li, P.
