Feedback stabilization for a coupled PDE-ODE production system

Baumgärtner Vanessa; Göttlich Simone; Knapp Stephan

doi:10.3934/mcrf.2020003

Article Contents

2020, Volume 10, Issue 2: 405-424. Doi: 10.3934/mcrf.2020003

This issue Previous Article Necessary condition for optimal control of doubly stochastic systems Next Article Singular control of SPDEs with space-mean dynamics

Feedback stabilization for a coupled PDE-ODE production system

University of Mannheim, Department of Mathematics, 68131 Mannheim, Germany

^* Corresponding author: Simone Göttlich
^* Corresponding author: Simone Göttlich

Received: February 28, 2019

Revised: June 30, 2019

Early access: November 2019

Published: April 2020

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Abstract

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.

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

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

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

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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 $

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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 $

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

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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 $

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Figure 7. Lyapunov functions and feedback laws

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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

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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 $

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