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

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

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University of Manchester
University of Manchester
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 systemâ€™s dynamics. To tackle this problem equation-free methodologies have been proposed [5]â€“[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.

[1] Camacho and Bordons, Model Predictive Control, vol. 7, no. 11. 2007.

[2] T. F. Edgar, W. J. Campbell, and C. Bode, â€œModel-based control in microelectronics manufacturing,â€ Proc. 38th IEEE Conf. Decis. Control (Cat. No.99CH36304), vol. 4, no. December, pp. 85â€“91, 1999.

[3] W. Xie, I. Bonis, and C. Theodoropoulos, â€œData-driven model reduction-based non-linear MPC for large-scale distributed parameter systems,â€ J. Process Control, vol. 35, pp. 50â€“58, Nov. 2015.

[4] A. Varshney, S. Pitchaiah, and A. Armaou, â€œFeedback control of dissipative PDE systems using adaptive model reduction,â€ AIChE J., vol. 55, no. 4, pp. 906â€“918, 2009.

[5] C. Theodoropoulos, Y.-H. Qian, and I. G. Kevrekidis, â€œâ€˜Coarseâ€™ stability and bifurcation analysis using time-steppers: A reaction-diffusion example,â€ Proc. Natl. Acad. Sci., vol. 97, no. 18, pp. 9840â€“9843, Aug. 2000.

[6] C. Theodoropoulos and E. Luna-ortiz, â€œA Reduced Input / Output Dynamic Optimisation Method,â€ in Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena, 2006, pp. 535â€“560.

[7] A. Armaou, I. G. Kevrekidis, and C. Theodoropoulos, â€œEquation-free gaptooth-based controller design for distributed complex/multiscale processes,â€ Comput. Chem. Eng., vol. 29, no. 4, pp. 731â€“740, Mar. 2005.

[8] E. N. Pistikopoulos, N. A. Diangelakis, R. Oberdieck, M. M. Papathanasiou, I. Nascu, and M. Sun, â€œPAROCâ€”An integrated framework and software platform for the optimisation and advanced model-based control of process systems,â€ Chem. Eng. Sci., vol. 136, pp. 115â€“138, Nov. 2015.

[9] P. Rivotti, R. S. C. Lambert, and E. N. Pistikopoulos, â€œCombined model approximation techniques and multiparametric programming for explicit non-linear model predictive control,â€ Comput. Chem. Eng., vol. 42, pp. 277â€“287, Jul. 2012.

[10] â€œCOMSOL MultiphysicsÂ® v. 5.2. www.comsol.com. COMSOL AB, Stockholm, Sweden.â€ .

[11] I. Bonis and C. Theodoropoulos, â€œModel reduction-based optimization using large-scale steady-state simulators,â€ Chem. Eng. Sci., vol. 69, no. 1, pp. 69â€“80, 2012.

[12] Y. Saad, â€œNumerical methods for large eigenvalue problems,â€ Algorithms Archit. Adv. Sci. Comput., p. 346 p., 1992.