(560e) Equation-Free Multiparametric Model Predictive Control for Dissipative PDEs

Constrained model predictive control (MPC) is a powerful control strategy where on-line optimization and a receding horizon are employed to compute the optimum trajectory for the manipulated variables [1]. The use of MPC for distributed parameter systems, consisting of partial differential equations (PDEs) is computationally expensive [2]. The dissipative nature of such systems, expressed as a separation of scales in the eigenvalues of the linearized models [3], has been exploited for model order reduction (MOR) in control of (usually parabolic) PDEs [4], projecting the infinite dimensional states onto a small finite subspace. Nevertheless, the real-time control of PDEs requires the knowledge of the first principle equations that model the system’s dynamics. To tackle this problem equation-free methodologies have been proposed [5]–[7] that solves the non-linear dynamic optimization problem; although the computational burden can significantly be dropped with the use of the reduced order gradients, they usually require a considerable amount of cpu-time, which can make them unsuitable for real-time control applications. The computation of the on-line control action can be enhanced with the use of multiparametric (mp) optimization [8] that computes an off-line map of the control law as a function of the states. Therefore, the computational effort is redistributed off-line, overcoming the on-line computational burden of the classic MPC. Nevertheless, despite the fact that the majority of computations are performed off-line, the large number of variables of the physical system may produce an intractable computational approach making the use of model reduction necessary. Rivotti et al. [9] combine non-linear model order reduction, based on balancing of empirical gramians to reduce the size of the system with mp approximate non-linear MPC, which can still be prohibitive for complex large-scale systems, modelled by a large set of PDEs.

In this work, an equation-free multiparametric MPC is proposed where there is no use of equations and only an available black-box simulator (such as COMSOL [10]) is employed, performing input-output tasks. An equation-free non-linear dynamic optimization has been developed based on its static counterpart [11] utilizing an equation-free model reduction. The mp-optimization takes advantage of the reduced gradients produced by the non-linear dynamic optimizer. The multiparametric approach aims to compute an off-line map that approximates the reduced non-linear problem efficiently. First, an initial solution is computed for the non-linear optimization problem employing the matrix-free Arnoldi iterations [12] for constructing a model reduction orthonormal basis without the use of full gradients. As a result a reduced order piecewise affine (PWA) model is produced as a good approximation of the model. Then, the mp-optimization is solved computing an initial set of critical regions (CR). Afterward, the CRs are refined using the non-linear dynamic optimization in order to approximate the non-linear problem within an arbitrary tolerance. The results show that only a small number of CR regions are necessary in order to sufficiently approximate the problem. The CRs are constructed in a form of a search tree, due to the refinement algorithm, which accelerates the on-line search. In order to demonstrate the effectiveness of this algorithm, the aforementioned methodology has been applied to a chemical engineering application modelled by COMSOL.

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[10] “COMSOL Multiphysics® v. 5.2. www.comsol.com. COMSOL AB, Stockholm, Sweden.” .

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