Figure 1.
Illustration of the quantities used for the computation of the performance indices $ \varepsilon_{N,P}^2(K) $ and $ \varepsilon_{N,P}^3(K) $
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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 |
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.
Mathematics Subject Classification:
Primary: 93-08, 93B52.
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 and 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 |
| [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. |
| [3] |
L. Grüne,
Economic receding horizon control without terminal constraints, Automatica, 49 (2013), 725-734.
doi: 10.1016/j.automatica.2012.12.003. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [13] |
K. Kunisch and S. Volkwein,
Galerkin proper orthogonal decomposition methods for parabolic problems, Numerische Mathematik, 90 (2001), 117-148.
doi: 10.1007/s002110100282. |
| [14] |
B. Lincoln and A. Rantzer,
Relaxing dynamic programming, IEEE Transactions on Automatic Control, 51 (2006), 1249-1260.
doi: 10.1109/TAC.2006.878720. |
| [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 |
| [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. |
| [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. |
| [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. |
| [19] |
M. Ohlberger and S. Rave., Reduced basis methods: success, limitations and future challenges., Proceedings of the Conference Algoritmy, 1–12, 2016. |
| [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. |
| [21] |
F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications, American Math. Society, Providence, 2010. |
| [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. |
show all references
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 |
| [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. |
| [3] |
L. Grüne,
Economic receding horizon control without terminal constraints, Automatica, 49 (2013), 725-734.
doi: 10.1016/j.automatica.2012.12.003. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [13] |
K. Kunisch and S. Volkwein,
Galerkin proper orthogonal decomposition methods for parabolic problems, Numerische Mathematik, 90 (2001), 117-148.
doi: 10.1007/s002110100282. |
| [14] |
B. Lincoln and A. Rantzer,
Relaxing dynamic programming, IEEE Transactions on Automatic Control, 51 (2006), 1249-1260.
doi: 10.1109/TAC.2006.878720. |
| [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 |
| [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. |
| [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. |
| [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. |
| [19] |
M. Ohlberger and S. Rave., Reduced basis methods: success, limitations and future challenges., Proceedings of the Conference Algoritmy, 1–12, 2016. |
| [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. |
| [21] |
F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications, American Math. Society, Providence, 2010. |
| [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. |
Figure 3.
Boundary $ \Gamma = \Gamma_{\mathsf{out}}\cup \Gamma_{\mathsf{c}} $ and velocity field $ \boldsymbol v(x) $ (green = inflow, blue = outflow)
Figure 4.
MPC closed loop cost and relative performance index
Figure 5.
Plots of the individual components of the absolute performance index
Figure 6.
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} $
Figure 7.
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
Figure 8.
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 $
Figure 9.
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 $
Table 1.
Test results for $ N = 200 $ , $ P = 30 $ and $ K = 120 $ for full (FE) and reduced (POD) order models
| Method | rel_err_ |
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 | rel_err_ |
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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