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Control-Oriented Modeling of Fluid Networks: A Time-Delay Approach

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Recent Results on Nonlinear Delay Control Systems

Part of the book series: Advances in Delays and Dynamics ((ADVSDD,volume 4))

Abstract

Fluid networks are characterized by complex interconnected flows, involving high order nonlinear dynamics and transport phenomena. Classical lumped models typically capture the interconnections and nonlinear effects but ignore the transport phenomena, which may strongly affect the transient response. To control such flows with regulators of reduced complexity, we improve a classical lumped model (obtained by combining Kirchhoff’s laws and graph theory) by introducing the effect of advection as a time delay. The model is based on the isothermal Euler equations to describe the dynamics of the fluid through the pipe. The resulting hyperbolic system of partial differential equations (PDEs) is diagonalized using Riemann invariants to find a solution in terms of delayed equations, obtained analytically using the method of the characteristics. Conservation principles are applied at the nodes of the network to describe the dynamics as a set of (possibly non linear) delay differential equations. Both linearized and nonlinear Euler equations are considered.

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References

  1. Banda, M., Herty, M., Klar, A.: Coupling conditions for gas networks governed by the isothermal euler equations. Netw. Heterogenous Media 1(2), 295–314 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  2. Banda, M., Herty, M., Klar, A.: Gas flow in pipeline networks. Netw. Heterogenous Media 1(1), 41–56 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bradu, B., Gayet, P., Niculescu, S.-I., Witrant, E.: Modeling of the very low pressure helium flow in the lhc cryogenic distribution line after a quench. Cryogenics 50(2), 71–77 (2010)

    Article  Google Scholar 

  4. Bresch-Pietri, D., Leroy, T., Petit, N.: Control-oriented time-varying input-delayed temperature model for si engine exhaust catalyst. In: Proceedings of the American Control Conference, pp. 2189–2195 (2013)

    Google Scholar 

  5. Colombo, R., Guerra, G., Herty, M., Schleper, V.: Optimal control in networks of pipes and canals. SIAM J. Control Optim. 48(3), 2032–2050 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cooke, K., Krumme, D.: Differential-difference equations and non-linear partial-boundary value problems for linear hyperbolic partial differential equations. J. Math. Anal. Appl. 24(2), 362–387 (1968)

    Article  MathSciNet  Google Scholar 

  7. Corriga, G., Sanna, S., Usai, G.: Sub-optimal constant-volume control for open channel networks. Appl. Math. Model. 7(4), 262–267 (1983)

    Article  MATH  Google Scholar 

  8. Cross, H.: Analysis of flow in networks of conduits or conductors. University of Illinois Engineering Experiment Station Bulletin 286 (1936)

    Google Scholar 

  9. de Halleux, J., Prieur, C., Coron, J.-M., d’Andrea Novel, B., Bastin, G.: Boundary feedback control in networks of open channels. Automatica 39(8), 1365–1376 (2003)

    Google Scholar 

  10. de Saint-Venant, B.: Theorie du mouvement non-permanent des eaux avec applications aux crues des rivieres et a l’introduction des marees dans leur lit. Comptes-rendus de l’Academie des Sciences 73, 148–154 (1871)

    Google Scholar 

  11. Dick, M., Gugat, M., Herty, M., Steffensen, S.: On the relaxation approximation of boundary control of the isothermal Euler equations. Int. J. Control 8(11), 1766–1778 (2012)

    Article  MathSciNet  Google Scholar 

  12. Evans, L.: Partial Differential Equations. Series Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence (1998)

    Google Scholar 

  13. Gugat, M., Herty, M., Schleper, V.: Flow control in gas networks: exact controllability to a given demand. Math. Methods Appl. Sci. 34(7), 745–757 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  14. Gugat, M., Dick, M.: Time-delayed boundary feedback stabilization of the isothermal Euler equations with friction. Math. Control Relat. Fields 1(4), 469–491 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hartman, H., Mutmansky, J., Ramani, R., Wang, Y.: Mine Ventilation and Air Conditioning, 3rd edn. Wiley, New York (1997)

    Google Scholar 

  16. Hu, Y., Koroleva, O., Krstic, M.: Nonlinear control of mine ventilation networks. Syst. Control Lett. 49(4), 239–254 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  17. Karafyllis, I., Krstic, M.: On the relation of delay equations to first-order hyperbolic partial differential equations. ESAIM Control Optim. Calc. Var. 20(3), 894–923 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  18. Koroleva, O., Krstic, M.: Averaging analysis of periodically forced fluid networks. Automatica 41(1), 129–135 (2005)

    MATH  MathSciNet  Google Scholar 

  19. Litrico, X., Georges, D.: Robust continuous-time and discrete-time flow control of a dam-river system. Appl. Math. Model. 23(11), 809–827 (1999)

    Article  MATH  Google Scholar 

  20. Litrico, X., Fromion, V.: Analytical approximation of open-channel flow for controller design. Appl. Math. Model. 28(7), 677–695 (2004)

    Article  MATH  Google Scholar 

  21. Petrov, N., Shishkin, M., Dmitriev, V., Shadrin, V.: Modeling mine aerology problems. J. Min. Sci. 28(2), 185–191 (1992)

    Article  Google Scholar 

  22. Rasvan, V.: Functional differential equations and one-dimensional distortionless propagation. Tatra Mountains Math. Publ. 43(1), 215–228 (2009)

    MATH  MathSciNet  Google Scholar 

  23. Rasvan, V.: Delays. Propagation. Conservation Laws. In: Sipahi, R., Vyhlidal, T., Niculescu, S-I., Pepe, P. (eds.) Time delay systems: Methods, Applications and New Trends. Series Lecture Notes in Control and Information Sciences, vol. 423, pp. 147–159. Springer, New York (2012)

    Google Scholar 

  24. Rasvan, V.: Three lectures on neutral functional differential equations. J. Control Eng. Appl. Inf. 11(9), 49–55 (2009)

    Google Scholar 

  25. Toro, E.: Riemann Solvers and Numerical Methods for Fluid Dynamics, 3rd edn. Springer, New York (2009)

    Book  MATH  Google Scholar 

  26. Treiber, M., Kesting, A.: Traffic Flow Dynamics: Data. Models and Simulation. Springer, Berlin Heidelberg (2013)

    Book  Google Scholar 

  27. Witrant, E., Johansson, K.: Air flow modeling in deepwells: Application to mining ventilation. In: Proceedings of the IEEE International Conference on Automation Science and Engineering, pp. 845–850 (2008)

    Google Scholar 

  28. Witrant, E., Marchand, N.: Modeling and feedback control for air flow regulation in deep pits. In: Sivasundaram, S. (ed.) Mathematical Problems in Engineering. Aerospace and Sciences. Cambridge Scientific Publishers, Cambridge (2011)

    Google Scholar 

  29. Witrant, E., Niculescu, S.-I.: Modeling and control of large convective flows with time-delays. Math. Eng. Sci. Aerosp. 1(2), 191–205 (2010)

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. IPN, ESIME Culhuacan, Santa Ana 1000, C.P. 44300, Mexico City, Mexico

    David Fernando Novella Rodriguez

  2. Université de Grenoble, GIPSA-lab, Grenoble, France

    Emmanuel Witrant & Olivier Sename

Authors
  1. David Fernando Novella Rodriguez
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  2. Emmanuel Witrant
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  3. Olivier Sename
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Corresponding author

Correspondence to David Fernando Novella Rodriguez .

Editor information

Editors and Affiliations

  1. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Iasson Karafyllis

  2. Department of Mathematics, Louisiana State University (LSU), Baton Rouge, Louisiana, USA

    Michael Malisoff

  3. CNRS-CentraleSupélec-Université Paris Sud, EPI Inria “DISCO”, Laboratory of Signals and Systems (L2S, UMR CNRS 8506), Gif-sur-Yvette, France

    Frederic Mazenc

  4. Department of Information Engineering, Computer Science, and Mathematics, University of L’Aquila, L’Aquila, Italy

    Pierdomenico Pepe

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Novella Rodriguez, D.F., Witrant, E., Sename, O. (2016). Control-Oriented Modeling of Fluid Networks: A Time-Delay Approach. In: Karafyllis, I., Malisoff, M., Mazenc, F., Pepe, P. (eds) Recent Results on Nonlinear Delay Control Systems. Advances in Delays and Dynamics, vol 4. Springer, Cham. https://doi.org/10.1007/978-3-319-18072-4_14

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  • DOI: https://doi.org/10.1007/978-3-319-18072-4_14

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-18071-7

  • Online ISBN: 978-3-319-18072-4

  • eBook Packages: EngineeringEngineering (R0)

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