Boundary feedback control in networks of open channels☆
Introduction
The so-called Saint-Venant equations are the partial differential equations (PDE) that are commonly used in hydraulics to describe the flow of water in open channels (see e.g. the textbooks in Chow, 1954 or Graf, 1998). These equations are a standard tool for solving engineering problems regarding the dynamics of canals and rivers. In this paper we will focus our attention on canals made up of a cascade of reaches delimited by underflow gates. Such systems typically occur in canalized water-ways and irrigation networks. But we shall see that the results of the paper are directly applicable to more complicated networks of canals and other kinds of control gates.
We address the problem of regulating the water level and the water velocity in a channel by using the gate openings as control actions.
This problem has been considered for a long time in the literature as reported in the survey paper Malaterre, Rogers and Schuurmans (1998) which involves a comprehensive bibliography. Starting from rudimentary and heuristic feedback control approaches, various advanced control methods where progressively investigated. Among other relevant references, we may mention for instance:
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LQ control methods which have been especially developed and studied in Balogun, Hubbard, and De Vries (1988), Garcia, Hubbard, and De Vries (1992) and Malaterre (1998). On the basis of finite-dimensional discrete linear approximations of the Saint-Venant equations.
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Robust H∞ control design techniques which are developed in Litrico and Georges (2001) and Litrico (2001) on the basis of a model approximation by a simple linear diffusive wave equation.
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Boundary PI regulation which is analyzed in Xu and Sallet (1999) on the basis of a linear PDE model around a steady state.
In Section 2, we start our analysis with the flow modelling in the special case of a single reach. Two different forms of the model, respectively, in terms of flow velocity and Riemann invariants are successively established. Sufficient conditions for the system stability are then stated in Theorem 1. This theorem is due to Greenberg and Li (1984).
Section 3 deals with the boundary control design in a single horizontal reach without friction. A control law is proposed on the basis of the Riemann invariants whose stabilizability is analyzed as an application of Theorem 1. Some illustrative simulation experiments of the control law are given in Section 4.
The main result of the paper is presented in Section 5. The aim is to generalize the previous result to open channels made up of several interconnected reaches in cascade. For the sake of clarity, we treat the special case of two reaches in cascade. Our stability result is given in Theorem 4 which, as we will see in Appendix, is a consequence of a theorem due to Li (1994).
The theorem provides a sufficient stability condition which can be applied to the stability analysis of canal networks having more general topologies (like for instance the star configurations considered in Leugering & Schmidt, 2002).
Some conclusions are given in Section 6.
Section snippets
Saint-Venant equations
Let us consider a one-dimensional portion of a canal delimited by two underflow gates as depicted in Fig. 1 under the following modelling assumptions:
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the canal is horizontal,
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the canal is prismatic with a constant rectangular cross section and a unit width,
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the friction effects due to walls are neglected.
The dynamics of the system are then described by the Saint-Venant equations Saint-Venant (1871) (also called shallow water equations):where A(H,V) is the characteristic matrix
Statement of the control problem
The control objective is to regulate system (1) at the set point . The control actions are the two gate openings u1 and u2. The water levels H(0,t) and H(L,t) are supposed to be measured online at each time instant t. The external constant water levels Hup and Hdo are known.
Control design based on Riemann invariants
From (7), it is obvious that the set point (H̄, V̄) expressed in the (α, β) coordinates isThe control objective can thus be reformulated as the problem of finding boundary controls able to regulate α(x,t)
Comparison with a unit-step open-loop control law
The control design method is illustrated with some realistic simulation experiments. In this section, we consider a small channel which is typical in local irrigation networks. Simulations for larger waterways will be given in the next section. The simulation parameters are , width , , , , , , .
The Saint-Venant equations are integrated numerically using a standard Preissman scheme (see e.g. Graf, 1998, Chapter 5) with a spatial
Control of multireach canals
The aim of this section is to generalize the previous sufficient stability condition to open channels made up of several interconnected reaches and to show how this condition can be used for control law design.
However, for the sake of clarity, we shall treat explicitly the special case of two reaches in cascade separated by three underflow gates as depicted in Fig. 7.
Conclusions
In this paper, a general sufficient stability condition for water velocities and water levels in open channels has been described and analyzed. A control law design based on this stability condition has been proposed and applied to reaches in cascade.
The main theoretical result of the paper is an application of a previous result of Li Ta-tsien given in Theorem 6. For the sake of simplicity, the theorem has been applied to a prototype canal made up of two horizontal reaches in cascade with a
Acknowledgements
This paper present research results of the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian State Prime Minister's Office for Science, Technology and Culture. The scientific responsibility rests with its authors.
Jonathan de Halleux received the Applied Mathematics engineering degree in 2000 from Université Catholique de Louvain, Louvain-la-Neuve, Belgium. He is presently doing a Ph.D. in the same institution on the stabilization of nonlinear partial differential equations.
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Cited by (0)
Jonathan de Halleux received the Applied Mathematics engineering degree in 2000 from Université Catholique de Louvain, Louvain-la-Neuve, Belgium. He is presently doing a Ph.D. in the same institution on the stabilization of nonlinear partial differential equations.
Christophe Prieur was born in Essey-les-Nancy, France, in 1974. He graduated in Mathematics from the Ecole Normale Supérieure, France in 2000. He received the Ph.D. degree in 2001 in Applied Mathematics from the Universite Paris-Sud, France. Since 2002 he is research associate CNRS at the Laboratoire SATIE, Ecole Normale Supérieure de Cachan, France. His current research interest include nonlinear control theory, robust control and control of nonlinear partial differential equations.
Jean-Michel Coron obtained the diploma of engineer from Ecole polytechnique in 1978 and from the Corps des Mines in 1981. He got his Thèse d'Etat in 1982. He was researcher at Ecole Nationale Supérieure des Mines de Paris, associate professor at Ecole polytechnique. He is currently Professor at Université Paris-Sud. His research interests include nonlinear partial differential equations, calculus of variations and nonlinear control theory.
Brigitte d'Andréa-Novel graduated from Ecole Supérieure d'Informatique Electronique Automatique in 1984. She received the Doctorate degree from Ecole Nationale Supérieure des Mines de Paris in 1987 and her Habilitation degree from Université Paris-Sud in 1995. She is currently Professor of Systems Control Theory and responsible for a research group on “Advanced Controlled Systems” at the Centre de Robotique—Ecole des Mines de Paris. Her current research interests include nonlinear control theory and applications to underactuated mechanical systems, control of wheeled vehicles with application to automated highways and boundary control of flexible mechanical systems coupling ODEs and PDEs.
Georges Bastin received the electrical engineering degree and the Ph.D. degree, both from Université Catholique de Louvain, Louvain-la-Neuve, Belgium. He is presently Professor in the Center for Systems Engineering and Applied Mechanics (CESAME) at the Université Catholique de Louvain and Associate Professor at the Ecole des Mines de Paris. His main research interests are in system identification and nonlinear control theory with applications to mechanical systems and robotics, biological and chemical processes, environmental problems and communication networks.
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This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Henk Nijmeijer under the direction of Editor Hassan Khalil.