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September  2021, 11(3): 579-599. doi: 10.3934/mcrf.2021013

## Performance estimates for economic model predictive control and their application in proper orthogonal decomposition-based implementations

 *. University of Bayreuth, Universitätsstraße 30, 95447 Bayreuth, Germany **. University of Konstanz, Universitätsstraße 10, 78464 Konstanz, Germany

Received  May 2019 Revised  January 2020 Published  March 2021

Fund Project: The authors are supported by DFG grants GR 1569/16-1 and VO 1658/4-1

In this paper performance indices for economic model predictive controllers (MPC) are considered. Since existing relative performance measures, designed for stabilizing controllers, fail in the economic setting, we propose alternative absolute quantities. We show that these can be applied to assess the performance of the closed loop trajectories on-line while the controller is running. The advantages of our approach are demonstrated by simulations involving a convection-diffusion-system. The method is also combined with proper orthogonal decomposition, thus demonstrating the possibility for both efficient and performant MPC for systems governed by partial differential equations.

Citation: Lars Grüne, Luca Mechelli, Simon Pirkelmann, Stefan Volkwein. Performance estimates for economic model predictive control and their application in proper orthogonal decomposition-based implementations. Mathematical Control & Related Fields, 2021, 11 (3) : 579-599. doi: 10.3934/mcrf.2021013
##### References:
 [1] J. Andrej, Modeling and Optimal Control of Multiphysics Problems Using the Finite Element Method, Ph.D. Thesis, 2019. Available at http://nbn-resolving.de/urn:nbn:de:gbv:8-diss-251049 Google Scholar [2] D. Angeli, R. Amrit and J. B. Rawlings, On average performance and stability of economic model predictive control, IEEE Transactions on Automatic Control, 57 (2012), 1615-1626.  doi: 10.1109/TAC.2011.2179349.  Google Scholar [3] L. Grüne, Economic receding horizon control without terminal constraints, Automatica, 49 (2013), 725-734.  doi: 10.1016/j.automatica.2012.12.003.  Google Scholar [4] L. Grüne, Approximation properties of receding horizon optimal control, Jahresbericht der Deutschen Mathematiker-Vereinigung, 118 (2016), 3-37.  doi: 10.1365/s13291-016-0134-5.  Google Scholar [5] L. Grüne and J. Pannek, Practical NMPC suboptimality estimates along trajectories, Systems & Control Letters, 58 (2009), 161-168.  doi: 10.1016/j.sysconle.2008.10.012.  Google Scholar [6] L. Grüne and J. Pannek, Nonlinear Model Predictive Control. Theory and Algorithms, 2$^{nd}$ edition, Springer, 2017. doi: 10.1007/978-3-319-46024-6.  Google Scholar [7] L. Grüne and S. Pirkelmann, Closed-loop performance analysis for economic model predictive control of time-varying systems, in Proceedings of the 56th IEEE Conference on Decision and Control (CDC 2017), (eds. R. Middleton and D. Nesic), 2017, 5563–5569. Google Scholar [8] L. Grüne and S. Pirkelmann, Economic model predictive control for time-varying system: Performance and stability results, Optimal Control Applications and Methods, 41 (2020), 42-64.  doi: 10.1002/oca.2492.  Google Scholar [9] L. Grüne, and S. Pirkelmann, Numerical verification of turnpike and continuity properties for time-varying PDEs, IFAC-PapersOnLine, 52 (2019), 7–12. doi: 10.1016/j.ifacol.2019.08.002.  Google Scholar [10] M. Gubisch and S. Volkwein, Proper orthogonal decomposition for linear-quadratic optimal control, in Model Reduction and Approximation: Theory and Algorithms, (eds. M. Ohlberger, P. Benner, A. Cohen and K. Willcox), SIAM (2017), Philadelphia, PA, 3–63. doi: 10.1137/1.9781611974829.ch1.  Google Scholar [11] S. Gugercin and A. C. Anthoulas., A survey of model reduction by balanced truncation and some new results, International Journal of Control, 77 (2004), 748-766.  doi: 10.1080/00207170410001713448.  Google Scholar [12] P. Holmes, J. L. Lumley, G. Berkooz and C. W. Rowley, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press, Cambridge, 2012.  doi: 10.1017/CBO9780511919701.  Google Scholar [13] K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for parabolic problems, Numerische Mathematik, 90 (2001), 117-148.  doi: 10.1007/s002110100282.  Google Scholar [14] B. Lincoln and A. Rantzer, Relaxing dynamic programming, IEEE Transactions on Automatic Control, 51 (2006), 1249-1260.  doi: 10.1109/TAC.2006.878720.  Google Scholar [15] L. Mechelli, POD-based State-Constrained Economic Model Predictive Control of Convection-Diffusion Phenomena, Ph.D. Thesis, 2019. Available at http://nbn-resolving.de/urn:nbn:de:bsz:352-2-2zoi8n9sxknm1 Google Scholar [16] L. Mechelli and S. Volkwein, POD-based economic model predictive control for heat-convection phenomena, Lecture Notes in Computational Science and Engineering, 126, Springer, Cham, 2019, 663–671. doi: 10.1007/978-3-319-96415-7_61.  Google Scholar [17] L. Mechelli and S. Volkwein, POD-based economic optimal control of heat-convection phenomena, in Numerical Methods for Optimal Control Problems, (M. Falcone, R. Ferretti, L. Grüne and W. M. McEneaney), Springer 2018, 63–87.  Google Scholar [18] M. A. Müller and L. Grüne, Economic model predictive control without terminal constraints for optimal periodic behavior, Automatica, 70 (2016), 128-139.  doi: 10.1016/j.automatica.2016.03.024.  Google Scholar [19] M. Ohlberger and S. Rave., Reduced basis methods: success, limitations and future challenges., Proceedings of the Conference Algoritmy, 1–12, 2016. Google Scholar [20] F. Tröltzsch and S. Volkwein, POD a-posteriori error estimates for linear-quadratic optimal control problems, Computational Optimization and Applications, 44 (2009), 83-115.  doi: 10.1007/s10589-008-9224-3.  Google Scholar [21] F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications, American Math. Society, Providence, 2010. Google Scholar [22] M. Zanon, L. Grüne and M. Diehl, Periodic optimal control, dissipativity and MPC, IEEE Transactions on Automatic Control, 62 (2017), 2943-2949.  doi: 10.1109/TAC.2016.2601881.  Google Scholar

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##### References:
 [1] J. Andrej, Modeling and Optimal Control of Multiphysics Problems Using the Finite Element Method, Ph.D. Thesis, 2019. Available at http://nbn-resolving.de/urn:nbn:de:gbv:8-diss-251049 Google Scholar [2] D. Angeli, R. Amrit and J. B. Rawlings, On average performance and stability of economic model predictive control, IEEE Transactions on Automatic Control, 57 (2012), 1615-1626.  doi: 10.1109/TAC.2011.2179349.  Google Scholar [3] L. Grüne, Economic receding horizon control without terminal constraints, Automatica, 49 (2013), 725-734.  doi: 10.1016/j.automatica.2012.12.003.  Google Scholar [4] L. Grüne, Approximation properties of receding horizon optimal control, Jahresbericht der Deutschen Mathematiker-Vereinigung, 118 (2016), 3-37.  doi: 10.1365/s13291-016-0134-5.  Google Scholar [5] L. Grüne and J. Pannek, Practical NMPC suboptimality estimates along trajectories, Systems & Control Letters, 58 (2009), 161-168.  doi: 10.1016/j.sysconle.2008.10.012.  Google Scholar [6] L. Grüne and J. Pannek, Nonlinear Model Predictive Control. Theory and Algorithms, 2$^{nd}$ edition, Springer, 2017. doi: 10.1007/978-3-319-46024-6.  Google Scholar [7] L. Grüne and S. Pirkelmann, Closed-loop performance analysis for economic model predictive control of time-varying systems, in Proceedings of the 56th IEEE Conference on Decision and Control (CDC 2017), (eds. R. Middleton and D. Nesic), 2017, 5563–5569. Google Scholar [8] L. Grüne and S. Pirkelmann, Economic model predictive control for time-varying system: Performance and stability results, Optimal Control Applications and Methods, 41 (2020), 42-64.  doi: 10.1002/oca.2492.  Google Scholar [9] L. Grüne, and S. Pirkelmann, Numerical verification of turnpike and continuity properties for time-varying PDEs, IFAC-PapersOnLine, 52 (2019), 7–12. doi: 10.1016/j.ifacol.2019.08.002.  Google Scholar [10] M. Gubisch and S. Volkwein, Proper orthogonal decomposition for linear-quadratic optimal control, in Model Reduction and Approximation: Theory and Algorithms, (eds. M. Ohlberger, P. Benner, A. Cohen and K. Willcox), SIAM (2017), Philadelphia, PA, 3–63. doi: 10.1137/1.9781611974829.ch1.  Google Scholar [11] S. Gugercin and A. C. Anthoulas., A survey of model reduction by balanced truncation and some new results, International Journal of Control, 77 (2004), 748-766.  doi: 10.1080/00207170410001713448.  Google Scholar [12] P. Holmes, J. L. Lumley, G. Berkooz and C. W. Rowley, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press, Cambridge, 2012.  doi: 10.1017/CBO9780511919701.  Google Scholar [13] K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for parabolic problems, Numerische Mathematik, 90 (2001), 117-148.  doi: 10.1007/s002110100282.  Google Scholar [14] B. Lincoln and A. Rantzer, Relaxing dynamic programming, IEEE Transactions on Automatic Control, 51 (2006), 1249-1260.  doi: 10.1109/TAC.2006.878720.  Google Scholar [15] L. Mechelli, POD-based State-Constrained Economic Model Predictive Control of Convection-Diffusion Phenomena, Ph.D. Thesis, 2019. Available at http://nbn-resolving.de/urn:nbn:de:bsz:352-2-2zoi8n9sxknm1 Google Scholar [16] L. Mechelli and S. Volkwein, POD-based economic model predictive control for heat-convection phenomena, Lecture Notes in Computational Science and Engineering, 126, Springer, Cham, 2019, 663–671. doi: 10.1007/978-3-319-96415-7_61.  Google Scholar [17] L. Mechelli and S. Volkwein, POD-based economic optimal control of heat-convection phenomena, in Numerical Methods for Optimal Control Problems, (M. Falcone, R. Ferretti, L. Grüne and W. M. McEneaney), Springer 2018, 63–87.  Google Scholar [18] M. A. Müller and L. Grüne, Economic model predictive control without terminal constraints for optimal periodic behavior, Automatica, 70 (2016), 128-139.  doi: 10.1016/j.automatica.2016.03.024.  Google Scholar [19] M. Ohlberger and S. Rave., Reduced basis methods: success, limitations and future challenges., Proceedings of the Conference Algoritmy, 1–12, 2016. Google Scholar [20] F. Tröltzsch and S. Volkwein, POD a-posteriori error estimates for linear-quadratic optimal control problems, Computational Optimization and Applications, 44 (2009), 83-115.  doi: 10.1007/s10589-008-9224-3.  Google Scholar [21] F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications, American Math. Society, Providence, 2010. Google Scholar [22] M. Zanon, L. Grüne and M. Diehl, Periodic optimal control, dissipativity and MPC, IEEE Transactions on Automatic Control, 62 (2017), 2943-2949.  doi: 10.1109/TAC.2016.2601881.  Google Scholar
Illustration of the quantities used for the computation of the performance indices $\varepsilon_{N,P}^2(K)$ and $\varepsilon_{N,P}^3(K)$
Explanation of how the choice of $P$ influences the quantity $E_{N,P}$
Boundary $\Gamma = \Gamma_{\mathsf{out}}\cup \Gamma_{\mathsf{c}}$ and velocity field $\boldsymbol v(x)$ (green = inflow, blue = outflow)
MPC closed loop cost and relative performance index
Plots of the individual components of the absolute performance index
Plots of performance index $\varepsilon^3_{N,P}$ for longer simulation times and of the influence of $P$ on the value of $E_{N,P}$
Comparison between POD-FE and POD-POD relative errors for $\sum_{k = 0}^{K-1} \varepsilon^1_N(k)$ with $\ell = 25$ POD basis functions
Relative error between the FE and POD cumulative performance index $\varepsilon_N^1$ over time $k$ for different $N$ and number of POD basis functions $\ell = 15,20,25$
Relative error between the FE and POD $E_{N,P}$ over time $k$ for $P = 30$, different $N$ and number of POD basis functions $\ell = 15,20,25$
Test results for $N = 200$, $P = 30$ and $K = 120$ for full (FE) and reduced (POD) order models
 Method $\ell$ rel_err_$u$ $J_{120}^{cl}(y_{\circ},\mu_{200})$ $E_{200,30}(30)$ Alg. Time Speed-up FE – – 7791.565 -0.0014 1182 s – POD-FE 20 0.00055 7791.591 0.0031 346 s 3.4 POD-FE 25 0.00008 7791.569 -0.0041 396 s 3.0 POD-POD 20 0.00070 7791.614 0.0042 312 s 3.8 POD-POD 25 0.00012 7791.577 -0.0038 302 s 3.9
 Method $\ell$ rel_err_$u$ $J_{120}^{cl}(y_{\circ},\mu_{200})$ $E_{200,30}(30)$ Alg. Time Speed-up FE – – 7791.565 -0.0014 1182 s – POD-FE 20 0.00055 7791.591 0.0031 346 s 3.4 POD-FE 25 0.00008 7791.569 -0.0041 396 s 3.0 POD-POD 20 0.00070 7791.614 0.0042 312 s 3.8 POD-POD 25 0.00012 7791.577 -0.0038 302 s 3.9
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