Article Contents
Article Contents

# Feedback stabilization for a coupled PDE-ODE production system

• * Corresponding author: Simone Göttlich
• 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:

• 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

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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Tables(2)