Equivalent Liénard-type models for a fluid transmission line

Lizeth Torres; Jorge Alejandro Delgado Aguiñaga; Gildas Besançon; Cristina Verde; Ofelia Begovich

doi:10.1016/j.crme.2016.04.004

Equivalent Liénard-type models for a fluid transmission line

Lizeth Torres ^1,² ; Jorge Alejandro Delgado Aguiñaga ³; Gildas Besançon ^4,⁵; Cristina Verde ¹; Ofelia Begovich ³

¹ Instituto de Ingeniería, Universidad Nacional Autónoma de México, 04510, Coyoacán, Mexico City, Mexico
² Cátedras CONACYT, Mexico
³ Centro de Investigación y de Estudios Avanzados CINVESTAV Unidad Guadalajara, 45019 Zapopan, Jalisco, Mexico
⁴ Université Grenoble Alpes, GIPSA-Lab, 38000 Grenoble, France
⁵ CNRS, GIPSA-Lab, 38000 Grenoble, France

Comptes Rendus. Mécanique, Volume 344 (2016) no. 8, pp. 582-595.

Abstract
Résumé

The main contribution of this paper is the derivation of spatiotemporal Liénard-type models for expressing the dynamical behavior of a fluid transmission line. The derivation is carried out from a quasilinear hyperbolic system made of a momentum equation and a continuity one. An advantage of these types of models is that they are suitable for formulating estimation algorithms. This claim is confirmed in the present paper for the case of fluid dynamics, since the article presents the conception and evaluation of a Liénard model-based observer that estimates the parameters of a pipeline such as the friction factor, the equivalent length and the wave speed. To show the potentiality of the approach, results based on some simulation and experimental tests are presented.

Supplementary Materials:
Supplementary materials for this article are supplied as separate files:

Cet article envisage principalement la dérivation de modèles de type Liénard pour exprimer le comportement dynamique d'une ligne de transmission de fluides. Le calcul est effectué à partir d'un système hyperbolique quasi linéaire constitué d'une équation de quantité de mouvement et d'une équation de continuité. Dans des études précédentes, il a été montré que transformer les systèmes régis par une équation différentielle de type $\ddot{x} (t) + F_{0} (x (t)) \dot{x} (t) + G_{0} (x (t)) = 0$ dans la forme de Liénard peut s'avérer utile à la conception d'algorithmes d'estimation. Cette affirmation est confirmée dans cet article pour le cas des fluides, car la conception et l'évaluation d'un observateur basé sur un modèle de type Liénard, estimant les paramètres d'une canalisation, y sont présentées. L'approche est illustrée par des résultats de tests en simulation et expérimentaux.

Compléments :
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Received: 2016-03-02
Accepted: 2016-04-13
Published online: 2016-05-17

DOI: 10.1016/j.crme.2016.04.004

Keywords: Pipelines, Fluid dynamics, Liénard equation, State observers, Parameter identification
Mots-clés : Canalisations, Dynamique des fluides, Équation de Liénard, Observateurs d'états, Identification de paramètres

Author's affiliations:

Lizeth Torres ^{1, 2}; Jorge Alejandro Delgado Aguiñaga ³; Gildas Besançon ^{4, 5}; Cristina Verde ¹; Ofelia Begovich ³

¹ Instituto de Ingeniería, Universidad Nacional Autónoma de México, 04510, Coyoacán, Mexico City, Mexico
² Cátedras CONACYT, Mexico
³ Centro de Investigación y de Estudios Avanzados CINVESTAV Unidad Guadalajara, 45019 Zapopan, Jalisco, Mexico
⁴ Université Grenoble Alpes, GIPSA-Lab, 38000 Grenoble, France
⁵ CNRS, GIPSA-Lab, 38000 Grenoble, France

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     author = {Lizeth Torres and Jorge Alejandro Delgado Agui\~naga and Gildas Besan\c{c}on and Cristina Verde and Ofelia Begovich},
     title = {Equivalent {Li\'enard-type} models for a fluid transmission line},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {582--595},
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Lizeth Torres; Jorge Alejandro Delgado Aguiñaga; Gildas Besançon; Cristina Verde; Ofelia Begovich. Equivalent Liénard-type models for a fluid transmission line. Comptes Rendus. Mécanique, Volume 344 (2016) no. 8, pp. 582-595. doi : 10.1016/j.crme.2016.04.004. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2016.04.004/

References
Cited by

[1] L. Torres; G. Besançon; C. Verde Liénard type model of fluid flow in pipelines: application to estimation, Mexico City, Mexico, IEEE (2015), pp. 148-153

[2] S. Beck; N. Williamson; N. Sims; R. Stanway Pipeline system identification through cross-correlation analysis, Proc. Inst. Mech. Eng., E J. Process Mech. Eng., Volume 216 (2002) no. 3, pp. 133-142

[3] J. Yang; Y. Wen; P. Li Leak location using blind system identification in water distribution pipelines, J. Sound Vib., Volume 310 (2008) no. 1, pp. 134-148

[4] A.C. Zecchin; L.B. White; M.F. Lambert; A.R. Simpson Parameter identification of fluid line networks by frequency-domain maximum likelihood estimation, Mech. Syst. Signal Process., Volume 37 (2013) no. 1, pp. 370-387

[5] V. Ibarra-Junquera; H. Rosu PI-controlled bioreactor as a generalized Liénard system, Comput. Chem. Eng., Volume 31 (2007) no. 3, pp. 136-141

[6] B.V. der Pol; J.V. der Mark The heartbeat considered as a relaxation oscillation, and an electrical model of the heart, Philos. Mag. Ser. 7, Volume 6 (1928) no. 38, pp. 763-775

[7] H.N. Moreira Liénard-type equations and the epidemiology of malaria, Ecol. Model., Volume 60 (1992) no. 2, pp. 139-150

[8] J. Nagumo; S. Arimoto; S. Yoshizawa An active pulse transmission line simulating nerve axon, Proc. IRE, Volume 50 (1962) no. 10, pp. 2061-2070

[9] R. FitzHugh Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., Volume 1 (1961) no. 6, pp. 445-466

[10] A.L. Hodgkin; A.F. Huxley A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol., Volume 117 (1952) no. 4, pp. 500-544

[11] X.C. Huang Limit cycles in a continuous fermentation model, J. Math. Chem., Volume 5 (1990) no. 3, pp. 287-296

[12] P. Holmes; D. Rand Phase portraits and bifurcations of the nonlinear oscillator: $\ddot{x} + α \dot{x} + γ x^{2} \dot{x} + β x + δ x^{3} = 0$ , Int. J. Non-Linear Mech., Volume 15 (1980) no. 6, pp. 449-458

[13] P. Omari; G. Villari; F. Zanolin A survey of recent applications of fixed point theory to periodic solutions of the Liénard equation, Fixed Point Theory and Its Applications, vol. 72, American Mathematical Society, Providence, RI, USA, 1988, pp. 171-178

[14] P. Yu; M. Han Limit cycles in generalized Liénard systems, Chaos Solitons Fractals, Volume 30 (2006) no. 5, pp. 1048-1068

[15] S. Lynch Liénard systems and the second part of Hilbert's Sixteenth problem, Nonlinear Anal., Volume 30 (1997) no. 3, pp. 1395-1403

[16] T. Slight; B. Romeira; L. Wang; J.M.L. Figueiredo; E. Wasige; C.N. Ironside A Liénard oscillator resonant tunnelling diode–laser diode hybrid integrated circuit: model and experiment, IEEE J. Quantum Electron., Volume 44 (2008) no. 12, pp. 1158-1163

[17] E. Abd-Elrady; T. Soderstrom; T. Wigren Periodic signal modeling based on Liénard's equation, IEEE Trans. Autom. Control, Volume 49 (2004) no. 10, pp. 1773-1781

[18] A. Liénard Étude des oscillations entretenues, Rev. Gén. Électr., Volume 23 (1928), pp. 901-954

[19] G. Besançon; A. Voda Observer-based parameter estimation in Liénard systems, ALCOSP 2010, Anatolia, Turkey (2010)

[20] G. Besançon; A. Voda; G. Jouffroy A note on state and parameter estimation in a Van der Pol oscillator, Automatica, Volume 46 (2010), pp. 1735-1738

[21] M.H. Chaudhry Applied Hydraulic Transients, Van Nostrand Reinhold, New York, 1979

[22] G. Besançon Nonlinear Observers and Applications, Springer, 2007

[23] G. Besançon; G. Bornard; H. Hammouri Observer synthesis for a class of nonlinear control systems, Eur. J. Control, Volume 2 (1996), pp. 176-192

[24] J.A. Roberson; J.J. Cassidy; M.H. Chaudhry Hydraulic Engineering, Wiley, New York, 1998

[25] A. Navarro; O. Begovich; G. Besançon Calibration of fitting loss coefficients for modelling purpose of a plastic pipeline, Proc. 16th Conference on Emerging Technologies & Factory Automation (ETFA), IEEE, 2011, pp. 1-6

[26] J. Delgado-Aguiñaga; G. Besançon; O. Begovich; J. Carvajal Multi-leak diagnosis in pipelines based on extended Kalman filter, Control Eng. Pract., Volume 49 (2016), pp. 139-148

[27] E. Cruz-Zavala; J.A. Moreno; L.M. Fridman Uniform robust exact differentiator, IEEE Trans. Autom. Control, Volume 56 (2011) no. 11, pp. 2727-2733

[28] P.K. Swamee; A.K. Jain Explicit equations for pipe-flow problems, J. Hydraul. Div., Volume 102 (1976) no. 5, pp. 657-664

[29] C. Mataix Mecánica de Fluidos Y Máquinas Hidráulicas, El Castillo, 1986

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