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Closed loop control of gas flow in a pipe: stability for a transient model

Regelung des Gasflusses in einer Pipeline: Stabilität eines transienten Modells
  • Martin Gugat

    Martin Gugat obtained his doctoral degree (summa cum laude) in applied mathematics in 1994 and the Habilitation in Mathematics in 1999 from the University of Trier. From 2000 to 2003 he joined the Department of Mathematics at the Technical University of Darmstadt. Since 2003 he has been a member of the Department of Mathematics at the FAU Erlangen-Nürnberg where he works as a professor with research in optimal control, exact control and stabilization for systems governed by partial differential equations. He was a visiting scholar at IPAM at UCLA, LJLL in Paris and Fudan University in Shanghai.

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    , Falk M. Hante

    Falk M. Hante received his Doctorate degree in mathematics in 2010 from FAU Erlangen-Nürnberg, Germany. He held postdoctoral positions at INRIA Grand Est in Nancy, France, and Universität Heidelberg, Germany, before joining the Department of Mathematics at FAU Erlangen-Nürnberg in 2012 as a Research Associate. Since 2020, he has been Professor at Humboldt-Universität zu Berlin, Germany, in applied mathematics with a specialization in optimization of complex dynamics. His research interests include control, including optimal control; switched and hybrid dynamical systems; network dynamics and optimization.

    and Li Jin

    Li Jin is an Assistant Professor at the New York University Tandon School of Engineering. He received B.Eng. degree in Mechanical Engineering from Shanghai Jiao Tong University in 2011, M.S. degree in Mechanical Engineering from Purdue University in 2012, and Ph.D. degree in Transportation from the Massachusetts Institute of Technology in 2018. He was also a visiting scholar at the University of Erlangen-Nuremberg in 2016. His background is stochastic processes, dynamic control, and optimization. His research focuses on developing resilient control algorithms for cyber-physical systems with guarantees of efficiency in nominal settings, robustness against random perturbations, and survivability under strategic disruptions.

Abstract

This contribution focuses on the analysis and control of friction-dominated flow of gas in pipes. The pressure in the gas flow is governed by a partial differential equation that is a doubly nonlinear parabolic equation of p-Laplace type, where p=32. Such equations exhibit positive solutions, finite speed of propagation and satisfy a maximum principle. The pressure is fixed on one end (upstream), and the flow is specified on the other end (downstream). These boundary conditions determine a unique steady equilibrium flow.

We present a boundary feedback flow control scheme, that ensures local exponential stability of the equilibrium in an L2-sense. The analysis is done both for the PDE system and an ODE system that is obtained by a suitable spatial semi-discretization. The proofs are based upon suitably chosen Lyapunov functions.

Zusammenfassung

Unser Beitrag befasst sich mit einem reibungsdominiertem Modell für den Gasfluss in Pipelines. Das Modell für den Druck ist dabei eine doppelt nichtlineare parabolische p-Laplace Gleichung mit p=32. Solche Gleichungen haben positive Lösungen mit endlicher Ausbreitungsgeschwindigkeit und erfüllen ein Maximumprinzip. Am einen Rohrende wird der Druck und am anderen Rohrende die Flussrate vorgegeben. Durch diese Randbedingungen wird eindeutig ein stationärer Fluss festgelegt.

Wir beschreiben eine Regelung über die Randbedingungen, die Flüsse generiert, welche lokal exponentiell schnell im L2-Sinn gegen den stationären Fluss konvergieren. Die Analyse wird sowohl für das System mit der partiellen Differentialgleichung als auch für eine gewöhnliche Differentialgleichung durchgeführt, die man durch eine Semidiskretisierung im Ort erhält. Die Beweise basieren auf passend gewählten Lyapunovfunktionen. Numerische Beispiele werden präsentiert.

Award Identifier / Grant number: Transregio 154

Award Identifier / Grant number: A03

Award Identifier / Grant number: C03

Award Identifier / Grant number: C05

Award Identifier / Grant number: Z01

Funding statement: This work is supported by DFG Collaborative Research Centre CRC/Transregio 154, Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks, projects A03, C03, C05 and Z01; moreover it is supported by NYU Tandon School of Engineering.

About the authors

Martin Gugat

Martin Gugat obtained his doctoral degree (summa cum laude) in applied mathematics in 1994 and the Habilitation in Mathematics in 1999 from the University of Trier. From 2000 to 2003 he joined the Department of Mathematics at the Technical University of Darmstadt. Since 2003 he has been a member of the Department of Mathematics at the FAU Erlangen-Nürnberg where he works as a professor with research in optimal control, exact control and stabilization for systems governed by partial differential equations. He was a visiting scholar at IPAM at UCLA, LJLL in Paris and Fudan University in Shanghai.

Falk M. Hante

Falk M. Hante received his Doctorate degree in mathematics in 2010 from FAU Erlangen-Nürnberg, Germany. He held postdoctoral positions at INRIA Grand Est in Nancy, France, and Universität Heidelberg, Germany, before joining the Department of Mathematics at FAU Erlangen-Nürnberg in 2012 as a Research Associate. Since 2020, he has been Professor at Humboldt-Universität zu Berlin, Germany, in applied mathematics with a specialization in optimization of complex dynamics. His research interests include control, including optimal control; switched and hybrid dynamical systems; network dynamics and optimization.

Li Jin

Li Jin is an Assistant Professor at the New York University Tandon School of Engineering. He received B.Eng. degree in Mechanical Engineering from Shanghai Jiao Tong University in 2011, M.S. degree in Mechanical Engineering from Purdue University in 2012, and Ph.D. degree in Transportation from the Massachusetts Institute of Technology in 2018. He was also a visiting scholar at the University of Erlangen-Nuremberg in 2016. His background is stochastic processes, dynamic control, and optimization. His research focuses on developing resilient control algorithms for cyber-physical systems with guarantees of efficiency in nominal settings, robustness against random perturbations, and survivability under strategic disruptions.

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Received: 2020-05-01
Accepted: 2020-10-13
Published Online: 2020-11-27
Published in Print: 2020-11-18

© 2020 Walter de Gruyter GmbH, Berlin/Boston

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