June  2020, 10(2): 405-424. doi: 10.3934/mcrf.2020003

Feedback stabilization for a coupled PDE-ODE production system

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

* Corresponding author: Simone Göttlich

Received  March 2019 Revised  July 2019 Published  June 2020 Early access  November 2019

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.

Citation: Vanessa Baumgärtner, Simone Göttlich, Stephan Knapp. Feedback stabilization for a coupled PDE-ODE production system. Mathematical Control and Related Fields, 2020, 10 (2) : 405-424. doi: 10.3934/mcrf.2020003
References:
[1]

D. ArmbrusterC. De BeerM. FreitagT. Jagalski and C. Ringhofer, Autonomous control of production networks using a pheromone approach, Phys. A, 363 (2006), 104-114.  doi: 10.1016/j.physa.2006.01.052.

[2]

D. ArmbrusterP. Degond and C. Ringhofer, A model for the dynamics of large queuing networks and supply chains, SIAM J. Appl. Math., 66 (2006), 896-920.  doi: 10.1137/040604625.

[3]

D. Armbruster and C. Ringhofer, Thermalized kinetic and fluid models for reentrant supply chains, Multiscale Model. Simul., 3 (2005), 782-800.  doi: 10.1137/030601636.

[4]

M. K. Banda and M. Herty, Numerical discretization of stabilization problems with boundary controls for systems of hyperbolic conservation laws, Math. Control Relat. Fields, 3 (2013), 121-142.  doi: 10.3934/mcrf.2013.3.121.

[5]

M. BarreauA. SeuretF. Gouaisbaut and L. Baudouin, Lyapunov stability analysis of a string equation coupled with an ordinary differential system, IEEE Trans. Automat. Control, 63 (2018), 3850-3857.  doi: 10.1109/TAC.2018.2802495.

[6]

G. Bastin and J.-M. Coron, Stability and Boundary Stabilization of 1-D Hyperbolic Systems, PNLDE Subseries in Control, Progress in Nonlinear Differential Equations and Their Applications, 88, Birkhäuser/Springer, 2016. doi: 10.1007/978-3-319-32062-5.

[7]

G. Bastin, J.-M. Coron and B. D'Andrea-Novel, Using hyperbolic systems of balance laws for modeling, control and stability analysis of physical networks, Lecture notes for the Pre-Congress Workshop on Complex Embedded and Networked Control Systems, 17th IFAC World Congress, Seoul, Korea, 2008.

[8]

G. BastinB. HautJ.-M. Coron and B. D'andréa-Novel, Lyapunov stability analysis of networks of scalar conservation laws, Netw. Heterog. Media, 2 (2007), 751-759.  doi: 10.3934/nhm.2007.2.751.

[9]

G. BrettiC. D'ApiceR. Manzo and B. Piccoli, A continuum-discrete model for supply chains dynamics, Netw. Heterog. Media, 2 (2007), 661-694.  doi: 10.3934/nhm.2007.2.661.

[10]

F. Castillo, L. Dugard, C. Prieur and E. Witrant, Dynamic boundary stabilization of linear and quasi-linear hyperbolic systems, 51st IEEE Conference on Decision and Control, Maui, HI, 2012, 2952–2957. doi: 10.1109/CDC.2012.6426802.

[11]

J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, 136, American Mathematical Society, Providence, RI, 2007.

[12]

J.-M. Coron and G. Bastin, Dissipative boundary conditions for one-dimensional quasilinear hyperbolic systems: Lyapunov stability for the $C^1$-norm, SIAM J. Control Optim., 53 (2015), 1464-1483.  doi: 10.1137/14097080X.

[13]

J.-M. CoronG. Bastin and B. D'Andrea-Novel, A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws, IEEE Trans. Automat. Control, 52 (2007), 2-11.  doi: 10.1109/TAC.2006.887903.

[14]

J.-M. CoronM. Kawski and Z. Wang, Analysis of a conservation law modeling a highly re-entrant manufacturing system, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1337-1359.  doi: 10.3934/dcdsb.2010.14.1337.

[15]

J.-M. Coron and Z. Wang, Controllability for a scalar conservation law with nonlocal velocity, J. Differential Equations, 252 (2012), 181-201.  doi: 10.1016/j.jde.2011.08.042.

[16]

C. D'Apice, S. Göttlich, M. Herty and B. Piccoli, Modeling, Simulation, and Optimization of Supply Chains. A Continuous Approach, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2010. doi: 10.1137/1.9780898717600.

[17]

C. D'Apice and R. Manzo, A fluid dynamic model for supply chains, Netw. Heterog. Media, 1 (2006), 379-398.  doi: 10.3934/nhm.2006.1.379.

[18]

A. DiagneG. Bastin and J.-M. Coron, Lyapunov exponential stability of 1-D linear hyperbolic systems of balance laws, Automatica J. IFAC, 48 (2012), 109-114.  doi: 10.1016/j.automatica.2011.09.030.

[19]

S. GöttlichM. Herty and A. Klar, Modelling and optimization of supply chains on complex networks, Commun. Math. Sci., 4 (2006), 315-330.  doi: 10.4310/CMS.2006.v4.n2.a3.

[20]

S. GöttlichM. Herty and P. Schillen, Electric transmission lines: Control and numerical discretization, Optimal Control Appl. Methods, 37 (2016), 980-995.  doi: 10.1002/oca.2219.

[21]

S. Göttlich and P. Schillen, Numerical discretization of boundary control problems for systems of balance laws: Feedback stabilization, Eur. J. Control, 35 (2017), 11-18.  doi: 10.1016/j.ejcon.2017.02.002.

[22]

M. Krstic and A. Smyshlyaev, Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays, Systems Control Lett., 57 (2008), 750-758.  doi: 10.1016/j.sysconle.2008.02.005.

[23]

T. Li, Controllability and Observability for Quasilinear Hyperbolic Systems, AIMS Series on Applied Mathematics, 3, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2010.

[24]

S. Tang and C. Xie, State and output feedback boundary control for a coupled PDE-ODE system, Systems Control Lett., 60 (2011), 540-545.  doi: 10.1016/j.sysconle.2011.04.011.

[25]

H.-N. Wu and J.-W. Wang, Observer design and output feedback stabilization for nonlinear multivariable systems with diffusion PDE-governed sensor dynamics, Nonlinear Dynam., 72 (2013), 615-628.  doi: 10.1007/s11071-012-0740-4.

[26]

H.-N. Wu and J.-W. Wang, Static output feedback control via PDE boundary and ODE measurements in linear cascaded ODE-beam systems, Automatica J. IFAC, 50 (2014), 2787-2798.  doi: 10.1016/j.automatica.2014.09.006.

show all references

References:
[1]

D. ArmbrusterC. De BeerM. FreitagT. Jagalski and C. Ringhofer, Autonomous control of production networks using a pheromone approach, Phys. A, 363 (2006), 104-114.  doi: 10.1016/j.physa.2006.01.052.

[2]

D. ArmbrusterP. Degond and C. Ringhofer, A model for the dynamics of large queuing networks and supply chains, SIAM J. Appl. Math., 66 (2006), 896-920.  doi: 10.1137/040604625.

[3]

D. Armbruster and C. Ringhofer, Thermalized kinetic and fluid models for reentrant supply chains, Multiscale Model. Simul., 3 (2005), 782-800.  doi: 10.1137/030601636.

[4]

M. K. Banda and M. Herty, Numerical discretization of stabilization problems with boundary controls for systems of hyperbolic conservation laws, Math. Control Relat. Fields, 3 (2013), 121-142.  doi: 10.3934/mcrf.2013.3.121.

[5]

M. BarreauA. SeuretF. Gouaisbaut and L. Baudouin, Lyapunov stability analysis of a string equation coupled with an ordinary differential system, IEEE Trans. Automat. Control, 63 (2018), 3850-3857.  doi: 10.1109/TAC.2018.2802495.

[6]

G. Bastin and J.-M. Coron, Stability and Boundary Stabilization of 1-D Hyperbolic Systems, PNLDE Subseries in Control, Progress in Nonlinear Differential Equations and Their Applications, 88, Birkhäuser/Springer, 2016. doi: 10.1007/978-3-319-32062-5.

[7]

G. Bastin, J.-M. Coron and B. D'Andrea-Novel, Using hyperbolic systems of balance laws for modeling, control and stability analysis of physical networks, Lecture notes for the Pre-Congress Workshop on Complex Embedded and Networked Control Systems, 17th IFAC World Congress, Seoul, Korea, 2008.

[8]

G. BastinB. HautJ.-M. Coron and B. D'andréa-Novel, Lyapunov stability analysis of networks of scalar conservation laws, Netw. Heterog. Media, 2 (2007), 751-759.  doi: 10.3934/nhm.2007.2.751.

[9]

G. BrettiC. D'ApiceR. Manzo and B. Piccoli, A continuum-discrete model for supply chains dynamics, Netw. Heterog. Media, 2 (2007), 661-694.  doi: 10.3934/nhm.2007.2.661.

[10]

F. Castillo, L. Dugard, C. Prieur and E. Witrant, Dynamic boundary stabilization of linear and quasi-linear hyperbolic systems, 51st IEEE Conference on Decision and Control, Maui, HI, 2012, 2952–2957. doi: 10.1109/CDC.2012.6426802.

[11]

J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, 136, American Mathematical Society, Providence, RI, 2007.

[12]

J.-M. Coron and G. Bastin, Dissipative boundary conditions for one-dimensional quasilinear hyperbolic systems: Lyapunov stability for the $C^1$-norm, SIAM J. Control Optim., 53 (2015), 1464-1483.  doi: 10.1137/14097080X.

[13]

J.-M. CoronG. Bastin and B. D'Andrea-Novel, A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws, IEEE Trans. Automat. Control, 52 (2007), 2-11.  doi: 10.1109/TAC.2006.887903.

[14]

J.-M. CoronM. Kawski and Z. Wang, Analysis of a conservation law modeling a highly re-entrant manufacturing system, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1337-1359.  doi: 10.3934/dcdsb.2010.14.1337.

[15]

J.-M. Coron and Z. Wang, Controllability for a scalar conservation law with nonlocal velocity, J. Differential Equations, 252 (2012), 181-201.  doi: 10.1016/j.jde.2011.08.042.

[16]

C. D'Apice, S. Göttlich, M. Herty and B. Piccoli, Modeling, Simulation, and Optimization of Supply Chains. A Continuous Approach, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2010. doi: 10.1137/1.9780898717600.

[17]

C. D'Apice and R. Manzo, A fluid dynamic model for supply chains, Netw. Heterog. Media, 1 (2006), 379-398.  doi: 10.3934/nhm.2006.1.379.

[18]

A. DiagneG. Bastin and J.-M. Coron, Lyapunov exponential stability of 1-D linear hyperbolic systems of balance laws, Automatica J. IFAC, 48 (2012), 109-114.  doi: 10.1016/j.automatica.2011.09.030.

[19]

S. GöttlichM. Herty and A. Klar, Modelling and optimization of supply chains on complex networks, Commun. Math. Sci., 4 (2006), 315-330.  doi: 10.4310/CMS.2006.v4.n2.a3.

[20]

S. GöttlichM. Herty and P. Schillen, Electric transmission lines: Control and numerical discretization, Optimal Control Appl. Methods, 37 (2016), 980-995.  doi: 10.1002/oca.2219.

[21]

S. Göttlich and P. Schillen, Numerical discretization of boundary control problems for systems of balance laws: Feedback stabilization, Eur. J. Control, 35 (2017), 11-18.  doi: 10.1016/j.ejcon.2017.02.002.

[22]

M. Krstic and A. Smyshlyaev, Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays, Systems Control Lett., 57 (2008), 750-758.  doi: 10.1016/j.sysconle.2008.02.005.

[23]

T. Li, Controllability and Observability for Quasilinear Hyperbolic Systems, AIMS Series on Applied Mathematics, 3, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2010.

[24]

S. Tang and C. Xie, State and output feedback boundary control for a coupled PDE-ODE system, Systems Control Lett., 60 (2011), 540-545.  doi: 10.1016/j.sysconle.2011.04.011.

[25]

H.-N. Wu and J.-W. Wang, Observer design and output feedback stabilization for nonlinear multivariable systems with diffusion PDE-governed sensor dynamics, Nonlinear Dynam., 72 (2013), 615-628.  doi: 10.1007/s11071-012-0740-4.

[26]

H.-N. Wu and J.-W. Wang, Static output feedback control via PDE boundary and ODE measurements in linear cascaded ODE-beam systems, Automatica J. IFAC, 50 (2014), 2787-2798.  doi: 10.1016/j.automatica.2014.09.006.

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
$ {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 $
$ \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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