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Delays and Interconnections: Methodology, Algorithms and Applications pp 3–17Cite as

Singular Perturbation Approach for Linear Coupled ODE-PDE Systems

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Part of the Advances in Delays and Dynamics book series (ADVSDD,volume 10)

Abstract

This paper focuses on a class of linear coupled ODE-PDE systems whose dynamics evolve in two time scales. The fast time scale modeled by a small positive perturbation parameter is introduced to the dynamics either of the ODE or of the PDE. By setting the perturbation parameter to zero, two subsystems, namely the reduced and the boundary-layer subsystems, are formally computed. Firstly, we propose a sufficient stability condition for the full coupled system. This stability condition implies the stability of both subsystems. Then, we state an approximation of the full coupled ODE-PDE systems by the subsystems based on the singular perturbation method. The error between the solution of the full system and that of the subsystems is the order of the perturbation parameter. Finally, numerical simulations on academic examples illustrate the theoretical results.

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  • DOI: 10.1007/978-3-030-11554-8_1
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References

  1. Bastin, G., Coron, J.-M.: Stability and Boundary Stabilization of 1-D Hyperbolic Systems. PNLDE Subseries in Control, Birkhäuser (2016)

    Google Scholar 

  2. Bastin, G., Coron, J.-M., Tamasoiu, S.: Stability of linear density-flow hyperbolic systems under PI boundary control. Automatica 53, 37–42 (2015)

    MathSciNet  CrossRef  Google Scholar 

  3. Coron, J.-M., Bastin, G., d’Andréa-Novel, B.: Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems. SIAM J. Control Optim. 47(3), 1460–1498 (2008)

    MathSciNet  CrossRef  Google Scholar 

  4. Daafouz, J., Tucsnak, M., Valein, J.: Nonlinear control of a coupled PDE/ODE system modeling a switched power converter with a transmission line. Syst. Control Lett. 70, 92–99 (2014)

    MathSciNet  CrossRef  Google Scholar 

  5. Dos Santos, V., Bastin, G., Coron, J.-M., d’Andréa-Novel, B.: Boundary control with integral action for hyperbolic systems of conservation laws: stability and expriments. Automatica 44, 1013–1318 (2008)

    MATH  Google Scholar 

  6. Khalil, H.K.: Nonlinear Systems. Prentice-Hall (1996)

    Google Scholar 

  7. Kokotović, P., Haddad, A.: Singular perturbations of a class of time optimal controls. IEEE Trans. Autom. Control 20, 163–164 (1975)

    CrossRef  Google Scholar 

  8. Kokotović, P., Khalil, H., O’Reilly, J.: Singular Perturbation Methods in Control: Analysis and Design. Academic Press (1986)

    Google Scholar 

  9. Kokotović, P., Sannuti, P.: Singular perturbation method for reducing the model order in optimal control design. IEEE Trans. Autom. Control 13, 377–384 (1968)

    CrossRef  Google Scholar 

  10. Kokotović, P., Yackel, R.: Singular perturbation of linear regulators: basic theorems. IEEE Trans. Autom. Control 17, 29–37 (1972)

    MathSciNet  CrossRef  Google Scholar 

  11. Krstic, M., Smyshlyaev, A.: Backtepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays. Syst. Control Lett. 57, 750–758 (2008)

    CrossRef  Google Scholar 

  12. Littman, W., Markus, L.: Exact boundary controllability of a hybrid system of elasticity. Arch. Ration. Mech. Anal. 103(3), 193–236 (1988)

    MathSciNet  CrossRef  Google Scholar 

  13. Tang, Y., Mazanti, G.: Stability analysis of coupled linear ODE-hyperbolic PDE systems with two time scales. Automatica (accepted)

    Google Scholar 

  14. Tang, Y., Prieur, C., Girard, A.: Tikhonov theorem for linear hyperbolic systems. Automatica 57, 1–10 (2015)

    MathSciNet  CrossRef  Google Scholar 

  15. Tang, Y., Prieur, C., Girard, A.: Singular perturbation approximation by means of a \({H}^2\) Lyapunov function for linear hyperbolic systems. Syst. Control Lett. 88, 24–31 (2016)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Université de Lorraine, CRAN UMR 7039 and CNRS, 2 avenue de la Forêt de Haye, 54516, Vandoeuvre-lès-Nancy, France

    Ying Tang

  2. GIPSA-lab, Grenoble Campus, 11 rue des Mathématiques, BP 46, 38402, Saint Martin d’Hères Cedex, France

    Christophe Prieur

  3. Laboratoire des signaux et systèmes (L2S), CNRS, CentraleSupélec, Université Paris-Sud, Université Paris-Saclay, 3 rue Joliot-Curie, 91192, Gif-sur-Yvette Cedex, France

    Antoine Girard

Authors
  1. Ying Tang
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  2. Christophe Prieur
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  3. Antoine Girard
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Corresponding author

Correspondence to Antoine Girard .

Editor information

Editors and Affiliations

  1. Laboratoire des Signaux et Systèmes, CNRS-CentraleSupèlec-Université Paris-Saclay, Gif-sur-Yvette, France

    Prof. Giorgio Valmorbida

  2. Laboratoire d’Analyse et d’Architecture de Systèmes-CNRS, Toulouse, France

    Dr. Alexandre Seuret

  3. Laboratoire des Signaux et Systèmes, CNRS-CentraleSupèlec-Université Paris-Saclay, Gif-sur-Yvette, France

    Islam Boussaada

  4. Mechanical and Industrial Engineering, Northeastern University, Boston, MA, USA

    Prof. Rifat Sipahi

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Tang, Y., Prieur, C., Girard, A. (2019). Singular Perturbation Approach for Linear Coupled ODE-PDE Systems. In: Valmorbida, G., Seuret, A., Boussaada, I., Sipahi, R. (eds) Delays and Interconnections: Methodology, Algorithms and Applications. Advances in Delays and Dynamics, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-030-11554-8_1

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