A New Nonlinear Control Methodology for Irrigation Canals Based on a Delayed Input Model Zohra Benayache, Gildas Besan¸ con and Didier Georges * * Control Systems Department (ex-LAG), GIPSA-lab, ENSIEG, BP 46, 38402 Saint-Martin d’H` eres, France.Email {Zohra.Benayache,Gildas.Besancon, Didier.Georges}@inpg.fr. Abstract: This paper is devoted to nonlinear feedback design for irrigation canals. Such systems are classically described by Saint-Venant nonlinear partial differential equations. Here instead, an ordinary differential equation model (still nonlinear) with a state-dependent input delay is used, on the basis of a model previously proposed in Litrico et al [2003]. The control design approach is based on a state prediction computation and the state predictor is constructed from a dynamic inversion in the same spirit as in Georges et al [2007]. The proposed methodology is analyzed and tested in simulation, first on the basis of the control model, and then using some ”more accurate” model. 1. INTRODUCTION Irrigation canals are used to conduct water from its up- stream source towards downstream users. Managing ir- rigation canals efficiently (i.e satisfying users needs) and at the same time reducing water waste is an increasingly important issue. For these reasons, control approaches have been more investigated in the last decade (see for example [Mareels et al, 2005, Litrico et al, 2005, Halleux et al, 2003, Georges et al, 2002, ...] and references therein). Canals belong to the class of transport systems, where the delay plays a major role in the dynamics. Those dynam- ics are usually mathematically described by Saint-Venant nonlinear hyperbolic partial differential equations. Saint- Venant equations don’t have a known analytical solution, unless for special cases with no friction and no slope (Malaterre et al [1998]). To obtain an approximate solution of Saint-Venant equations different techniques have been developed. The most used numerical scheme for hydro- dynamic is the implicit Preissmann scheme (Chaudhry [1987]). There are two classical politics to control irrigation canals: the local upstream control and the distant downstream control (for more details see Malaterre et al [1998]). In the present paper, we focus on the distant downstream politics which consist in controlling the downstream water level using the upstream control variable. Its main advantage lies in reducing water waste. Different methodologies have been used to design con- trollers which are classified from linear to nonlinear ones. Among the most cited linear controllers, we find the clas- sical linear PID approach (Malaterre et al [1998]). Despite of its simple implementation, it does not however take into account the time delay explicitly, and including a Smith predictor makes the control sensitive to modeling errors. In order to better take into account perturbations and modeling errors, robust approaches have been developed ( Malaterre et al [2000], Litrico et al [2006]). A predictive approach has also been investigated in Rodellar et al [1989] or in Bogovich et al [2007]. Those controllers on the other hand neglect the nonlin- ear dynamics of the canal, which might limit the per- formances obtained with linear controllers. Therefore, a nonlinear approach can be of interest. Dulhoste et al [2004] have developed a nonlinear control law based on an Input/Output linearization method. Interesting results have been obtained in simulation, as well as in real-time experiments performed by Besan¸ con et al [2004]. However, this approach is limited so far to upstream local control and the time delay is not explicitly taken into account in the model. In the approaches cited above, the control law is determined from a finite dimension model. In Coron et al [2007], by means of a Lyapunov approach a stabilizing boundary control laws is proposed from the Saint-Venant equations. The objective of the present work is to develop a nonlinear control law for irrigation canal based on a nonlinear model where the delay explicitly appears. To that end, the canal is described by a nonlinear state- dependent input delay model of the form: ˙ x(t)= A(x(t))x(t)+ B(x(t))u(t − τ (x(t))) (1) derived from the recent work of Litrico et al [2003]. The control law is based on a state predictor which is constructed by a dynamic inversion. In order to illustrate the control performances (transient and steady-state responses), we first test the method on a model (1), and then on a more realistic model given by a Preissmann scheme. The paper is organized as follows: section 2 focuses on the canal modeling. We will first recall classical Saint- Venant equations and then present the input-delay model that will be considered for control design. In section 3, the proposed control approach is presented on the basis of a state predictor. Section 4 then gives some corresponding Proceedings of the 17th World Congress The International Federation of Automatic Control Seoul, Korea, July 6-11, 2008 978-1-1234-7890-2/08/$20.00 © 2008 IFAC 2544 10.3182/20080706-5-KR-1001.3292