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Feedback stabilization for a coupled PDE-ODE production system

  • * Corresponding author: Simone Göttlich

    * Corresponding author: Simone Göttlich 
Abstract Full Text(HTML) Figure(7) / Table(2) Related Papers Cited by
  • We consider an interlinked production model consisting of conservation laws (PDE) coupled to ordinary differential equations (ODE). Our focus is the analysis of control laws for the coupled system and corresponding stabilization questions of equilibrium dynamics in the presence of disturbances. These investigations are carried out using an appropriate Lyapunov function on the theoretical and numerical level. The discrete $ L^2- $stabilization technique allows to derive a mixed feedback law that is able to ensure exponential stability also in bottleneck situations. All results are accompanied by computational examples.

    Mathematics Subject Classification: Primary: 65Mxx; Secondary: 93D05, 90B30.

    Citation:

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  • Figure 1.  Serial production system with two processors and two queues

    Figure 2.  Feedback loop with control input $ u_1(t) $

    Figure 3.  Discrete Lyapunov function with parameters: $ q_e(0) = 0 $, $ {v}_e = 1 $, $ b_e-a_e = 1 $, $ \mu_1 = 10,\; \mu_2 = 9,\; \mu_3 = 8 $, $ f_1(0,x) = 10 $, $ f_2(0,x) = 9 $, $ f_3(0,x) = 8 $, $ \eta_e = \tilde{\eta}_e = 0.1 $ and $ p_e = c_e = 1 $

    Figure 4.  Discrete Lyapunov functions for linear and mixed feedback with parameters: $ q_1(0) = 0 $, $ q_2(0) = 1 $, $ {v}_e = 1 $, $ b_e-a_e = \frac{1}{2} $, $ \mu_1 = 6,\; \mu_2 = 4 $, $ f_1(0,x) = 4 $, $ f_2(0,x) = 4 $, $ \eta_e = \tilde{\eta}_e = \frac{1}{2} $ and $ p_e = c_e = 1 $

    Figure 5.  Discrete Lyapunov functions for linear and mixed feedback: log-plot

    Figure 6.  Production system with one increasing queue with parameters: $ q_1(0) = 0 $, $ q_2(0) = 0 $, $ {v}_e = 1 $, $ b_e-a_e = \frac{1}{2} $, $ \mu_1 = 6,\; \mu_2 = 4 $, $ f_1(0,x) = 6 $, $ f_2(0,x) = 4 $, $ \eta_e = \tilde{\eta}_e = 0.2 $ and $ p_e = c_e = 1 $

    Figure 7.  Lyapunov functions and feedback laws

    Table 1.  Convergence of the decay rate $ \nu $ for $ \eta = \tilde{\eta} = 0.575 $ and first-order convergence of the discretization for a CFL constant equal to 1, $ N = \frac{1}{2h} $ and different velocities

    $ {v}_e=1 $
    $ N $ $ \lVert \cdot\rVert_{\infty} $ Conv. Rate $ \lVert \cdot\rVert_{L^2} $ Conv. Rate $ \nu $
    $ 10 $ $ 0.0754 $ - $ 0.1326 $ - $ 0.5668 $
    $ 50 $ $ 0.0151 $ $ 0.99 $ $ 0.0265 $ $ 1.00 $ $ 0.5734 $
    $ 100 $ $ 0.0075 $ $ 1.01 $ $ 0.0132 $ $ 1.00 $ $ 0.5742 $
    $ 200 $ $ 0.0038 $ $ 0.99 $ $ 0.0066 $ $ 1.00 $ $ 0.5746 $
    $ 400 $ $ 0.0019 $ $ 1.00 $ $ 0.0033 $ $ 1.00 $ $ 0.5748 $
    $ 800 $ $ 0.0009 $ $ 1.05 $ $ 0.0017 $ $ 0.97 $ $ 0.5749 $
    $ {v}_e=5 $
    N $ \lVert \cdot\rVert_{\infty} $ Conv. Rate $ \lVert \cdot\rVert_{L^2} $ Conv. Rate $ \nu $
    10 0.0834 - 0.2007 - 0.2834
    50 0.0153 1.05 0.0380 1.03 0.2867
    100 0.0076 1.01 0.0189 1.01 0.2871
    200 0.0038 1.00 0.0094 1.01 0.2873
    400 0.0019 1.00 0.0047 1.00 0.2874
    800 0.0009 1.08 0.0023 1.03 0.2874
     | Show Table
    DownLoad: CSV

    Table 2.  Dependence of the Lyapunov function on $ \kappa $

    $ \kappa $ $ \frac{V^T}{V^0} $ $ \eta $ $ \tilde{\eta} $ $ \nu $
    $ 0.1 $ $ 3.75e^{-60} $ $ 4.6052 $ $ 0.5752 $ $ 0.5750 $
    $ 0.25 $ $ 1.40e^{-36} $ $ 2.7726 $ $ 0.5752 $ $ 0.5750 $
    $ 0.5 $ $ 1.05e^{-18} $ $ 1.3863 $ $ 0.5752 $ $ 0.5750 $
    $ 0.75 $ $ 3.20e^{-8} $ $ 0.5752 $ $ 0.5752 $ $ 0.5750 $
     | Show Table
    DownLoad: CSV
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