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

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

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  • 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) $

    Figure 2.  Explanation of how the choice of $ P $ influences the quantity $ E_{N,P} $

    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 $ \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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