A delayed feedback control for network of open canals
Tóm tắt
This paper proposes an algebraic method to design time-delay boundary feedback controllers to regulate the water flow and level in a network of open canals. The network is modeled as
$$n$$
canals with one junction where all canals come together. The Saint-Venant equations are linearized around a prescribed steady state. We consider steady subcritical flow condition and an energy estimate, to build feedback boundary conditions. These boundary conditions depend on data at earlier times and ensure the exponential decrease in
$$L^2$$
-norm of the solution of the linearized model. A single canal is first treated and afterward the analysis is extended to the network. Finally, the controllers are applied numerically to the nonlinear Saint-Venant equations.
Tài liệu tham khảo
de SAINT-VENANT B (1871) Théorie du mouvement non-permanent des eaux avec application aux crues des riviéres et á l’introduction des marées dans leur lit. C R Acad Sci 73(148–154):237–240
Russel DL (1978) Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. SIAM Rev 20(4):639–739
Malaterre PO, Roge DC (1998) Classification of canal control algorithms. J Irrig Drain Eng 124(1):3–10
Bastin G, Bayen AM, D’apice C, Litrico X, Piccoli B (2009) Open problems and research perspectives for irrigation channels. Netw Heterog Media 4(2):1–4
Balogun O, Hubbard M, De Vries JJ (1988) Automatic control of canal flow using linear quadratic regulator theory. J Hydraul Eng 114(1):75–102
Malaterre PO (1998) PILOTE linear quadratic optimal controller for irrigation canals. J Irrig Drain Eng 124(4):187–193
Weyer E (2003) LQ control of irrigation channel. Decis Control 1:750–755
Chen M, Georges D, Lefevre L (2002) Infinite dimensional LQ control of an open-channel hydraulic system. ASCC, Singapore
Xu CZ, Sallet G (1999) Proportional and integral regulation of irrigation canal systems governed by the Saint-Venant equation. In: Proceedings of the 14th world congress IFAC. Elsevier, Beijing, pp 147–152
Litrico X, Vincent F (2006) \(H_{\infty }\) control of an irrigation canal pool with a mixed control politics. IEEE Trans Autom Control Syst Technol 14(1):99–111
Gugat M, Dick M (2011) Time-delayed boundary feedback stabilization of the isothermal Euler equations with friction. Math Control Relat Fields 1(4):469–491
Leugering G, Georg Schmidt EJP (2002) On the modelling and stabilization of flows in networks of open canal. SIAM J Control Optim 41(1):164–180
Gugat M, Leugering G, Georg Schmidt EJP (2004) Global controllability between steady supercritical flows in channel networks. Math Methods Appl Sci 27:781–802
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(8):1365–1376
Ndiaye M, Bastin G (2004) Commande fontiere adptative d’un bief de canal avec prélévements inconnus. RS-JESA 38:374–371
Cen L.-H., Xi Y.-G. (2009) Stability of boundary feedback control based on weighted Lyapunov function in networks of open channels. Acta Autom Sin 35(1):97–102
Bastin G, Coron JM, D’Andréa-Novel B (2009) On Lyapunov stability of linearised Saint-Venant equations for a sloping channel. Netw Heterog Media 4(2):177–187
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 I 346:1119–1122
Misra R, Mohan Kumar MS, Sridharan K (1991) Analysis of transients in a canal network. Sādhanā 16(1):85–89
X. Litrico (2001) Robust flow control of single input multiple outputs regulated rivers. J Irrig Drain Eng 127(5):281–286
Li T (2005) Exact boundary controlability of unsteady flows in a network of open canals. Math Nachr 278(3):278–289
Qilong G, Li T (2008) Exact boundary observability of unsteady flows in a tree-like network of open canals. Math Methods Appl Sci. doi:10.1002/mma.1043
Qilong G (2008) Exact boundary controllability of unsteady supercritical flows in a tree-like network of open canals. Math Methods Appl Sci 31:1497–1508
Gugat M, Dick M, Leugering G (2013) Stabilization of the gas flow in star-shaped networks by feedback controls with varying delay. Syst Model Optim IFIP Adv Inf Commun Technol 391(4):255–265
Leveque RJ (2002) Finite volume methods for hyperbolic problems. Cambridge texts in applied mathematics. Cambridge University Press, Cambridge
Toro EF (1999) Riemann solvers and numerical methods for fluid dynamics: a practical introduction. Springer, Berlin
Dos Santos V, Bastin G, Coron JM, D’Andréa-Novel B (2003) Boundary control with integral action for hyperbolic systems of conservation laws: stability and experiments. Automatica 44:1310–1318
Girault V, Raviart PA (1986) Finite elements methods for Navier–Stokes equations. Theory and algorithms. Springer, Berlin