Abstract
This paper presents an algebraic method to design a linear feedback control for regulating the water flow in open channels. We deal with a hyperbolic system of partial differential equations describing the behavior of the water flow and the sediment transport. By using an a priori estimation techniques and the Faedo–Galerkin method, we build a stabilizing boundary control. This control law ensures a decrease of the energy and convergence of the controlled system.
Similar content being viewed by others
References
Balogun OS, Hubbard M, De-Vries JJ (1988) Automatic control of canal flow using linear quadratic regulator theory. J Hydraulic Eng 114(1):75–102
Callaghan DP, Saint-Cast F, Nielsen P, Ebaldock T (2006) Numerical solutions of the sediment conservation law: a review and improved formulation for coastal morphological modeling. Coastal Eng 53:557–571
Castro-Diaz MJ, Nieto EDF, Ferreiro AM (2008) Sediment transport model in shallow water equations and numerical approach by high order finite volume methods. Comput Fluids 37:299–316
Cen LH, Xi YG (2009) Stability of boundary feedback control based on weighted Lyapunov function in networks of open channels. Acta Autom Sin 35(1):97–102
Coron J-M, d’Andréa-Novel B, Bastin G (2007) A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws. IEEE Trans Autom Control 52(1):2–11
De-Halleux J, Prieur C, Coron J-M, d’Andréa-Novel B, Bastin G (2003) Boundary feedback control in networks of open-channels. Automatica 39:1365–1376
Diagne A, Bastin G, Coron J-M (2012) Lyapunov exponential stability of linear hyperbolic systems of balance laws. Automatica 48(1):109–114
Dos-Santos V, Bastin G, Coron J-M, d’Andréa-Novel B (2008) Boundary control with integral action for hyperbolic systems of conservation laws: stability and experiments. Automatica 44(5):1310–1318
Durdu OF (2006) Control of transient flow in irrigation canals using Lyapunov fuzzy filter-based Gaussian regulator. Int J Numer Methods Fluids 50(4):491–509
Feliu-Batlle V, Pérez RR, Rodriguez LS (2007) Fractional robust control of main irrigation canals with variable dynamic parameters. Control Eng Practice 15(6):673–686
Girault V, Raviart PA (1986) Finite element methods for Navier-Stokes equations. Theory and algorithms. Springer, Berlin
Gómez M, Rodellar J, Mantecòn JA (2002) Predictive control method for decentralized operation of irrigation canals. Appl Math Model 26(11):1039–1056
Goudiaby M, Séne A, Kreiss G (2011) Irrigation: types, sources and problems. Book 3, ISBN:979-953-307-706-1
Grass AJ (1981) Sediment transport by waves and currents. Technical report, SERC London, Cent. Mar. Technol, Report No. FL29
Hudson J, Sweby P (2003) Formulations for numerically approximating hyperbolic systems governing sediment transport. J Sci Comput 19:225–252
Kubatko EJ, Westerink JJ, Dawson C (2006) An unstructured grid morphodynamic model with a discontinuous Galerkin method for bed evolution. Ocean Model 15:71–89
Leugering G, Schmidt J-P (2002) On the modelling and stabilisation of flows in networks of open canals. SIAM J Control Optim 41(1):164–180
LeVeque RJ (2002) Finite volume methods for hyperbolic problems. Cambridge texts in applied mathematics
Litrico X, Fromion V (2006) \(H_{\infty }\) control of an irrigation canal pool with a mixed control politics. IEEE Trans Autom Control Syst Technol 14(1):99–111
Malaterre PO (1998) Pilote: linear quadratic optimal controller for irrigation canals. J Irrig Drain Eng 124:187–194
Malaterre PO, Rogers DC, Schuurmans J (1998) Classification of canal control algorithms. J Irrig Drain Eng ASCE 124:3–10
Meyer-Peter E, Muller R (1948) Formula for bed-load transport. In: 2nd Meeting of the International Association of Hydraulic Structures Research (Stockholm 1948), pp 39–64
Ndiaye N, Bastin G (2004) Commande frontiere adaptative d’un bief de canal avec prélévements inconnus. RS-JESA 38:347–372
Parés C, Castro M (2004) On the well-balance property of Roe’s method for nonconservative hyperbolic systems. Applications to shallow-water systems. Math Model Numer Anal 38(5):821–852
Pognant-Gros P, Fromion V, Baume JP (2001) Canal controller design: a multivariable approach using \(H_\infty \). In: Proceedings of the European control conference (September 2001, Porto, Portugal), pp 3398–3403
Prieur C, De-Halleux J (2004) Stabilization of a 1-D tank containing a fluid modeled by the shallow water equations. Syst Control Lett 52(3–4):167–178
Reddy JM (1995) Kalman filtering in the control of irrigation canals. Appl Math Model 19(4):201–209
Rijn V (1984) Sediment transport. Part 1: bed load transport. J Irrig Drain Eng ASCE 110:1431–1456
Séne A, Wane BA, Le-Roux DY (2008) Control of irrigation channels with variable bathymetry and time dependent stabilization rate. C R Acad Sci Paris Ser I 346:1119–1122
Toumbou B, Le-Roux DY, Séne A (2007) An existence theorem for a 2-D coupled sedimentation shallow-water model. C R Acad Sci Paris 344(7):443–446
Toumbou B, Le-Roux DY, Séne A (2008) A shallow-water sedimentation model with friction and Coriolis: an existence theorem. J Differ Equ 244:2020–2040
Tsien LT (2004) Exact controllability for quasilinear hyperbolic systems and its application to unsteady flows in a network of open canals. Math Methods Appl Sci 27:1089–1114
Wang SSY, Wu W (2005) Computational simulation of river sedimentation and morphology: a review of the state of the art. Int J Sediment Res 20(1):7–29
Weyer E (2003) LQ control of irrigation channels. Decis Control 1:750–755
Xu CZ, Sallet G (2002) Exponential stability and transfer functions of processes governed by symmetric hyperbolic systems. ESAIM Control Optim Calculus Var 7:421–442
Yang CT (1996) Sediment transport theory and practice. McGraw-Hill, New York
Zabsonré JD, Lucas C, Nieto EFD (2009) An energitically consistent viscous sedimentation model. Math Models Methods Appl Sci 19(3):477–499
Zic C, Vukovic S, Sopta L (2003) Extension of ENO and WENO schemes to one dimensional sediment transport equations. Comput Fluids 33:31–56
Author information
Authors and Affiliations
Corresponding author
Additional information
This work is supported by grants from International Science Programme Program-Sweden, Aires-Sud Project-France, UMMISCO-IRD Project-France and Fond National Suisse.
Rights and permissions
About this article
Cite this article
Diagne, A., Sène, A. Control of shallow water and sediment continuity coupled system. Math. Control Signals Syst. 25, 387–406 (2013). https://doi.org/10.1007/s00498-012-0101-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00498-012-0101-3