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Model Predictive Control of Parabolic PDE Systems under Chance Constraints

Author

Listed:
  • Ruslan Voropai

    (Group of Process Optimization, Institute for Automation and Systems Engineering, Technische Universität Ilmenau, P.O. Box 100565, 98684 Ilmenau, Germany)

  • Abebe Geletu

    (German Research Chair, African Institute of Mathematical Sciences (AIMS), KN 3 Rd, Kigali, Rwanda)

  • Pu Li

    (Group of Process Optimization, Institute for Automation and Systems Engineering, Technische Universität Ilmenau, P.O. Box 100565, 98684 Ilmenau, Germany)

Abstract

Model predictive control (MPC) heavily relies on the accuracy of the system model. Nevertheless, process models naturally contain random parameters. To derive a reliable solution, it is necessary to design a stochastic MPC. This work studies the chance constrained MPC of systems described by parabolic partial differential equations (PDEs) with random parameters. Inequality constraints on time- and space-dependent state variables are defined in terms of chance constraints. Using a discretization scheme, the resulting high-dimensional chance constrained optimization problem is solved by our recently developed inner–outer approximation which renders the problem computationally amenable. The proposed MPC scheme automatically generates probability tubes significantly simplifying the derivation of feasible solutions. We demonstrate the viability and versatility of the approach through a case study of tumor hyperthermia cancer treatment control, where the randomness arises from the thermal conductivity coefficient characterizing heat flux in human tissue.

Suggested Citation

  • Ruslan Voropai & Abebe Geletu & Pu Li, 2023. "Model Predictive Control of Parabolic PDE Systems under Chance Constraints," Mathematics, MDPI, vol. 11(6), pages 1-23, March.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:6:p:1372-:d:1094852
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    References listed on IDEAS

    as
    1. Deng, Zhong-Shan & Liu, Jing, 2001. "Blood perfusion-based model for characterizing the temperature fluctuation in living tissues," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 300(3), pages 521-530.
    2. Ockendon, John & Howison, Sam & Lacey, Andrew & Movchan, Alexander, 2003. "Applied Partial Differential Equations," OUP Catalogue, Oxford University Press, number 9780198527718.
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