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Networks & Heterogeneous Media

June 2009 , Volume 4 , Issue 2

Special Issue on Irrigation Channels and Related Problems

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Preface
Georges Bastin, Alexandre M. Bayen, Ciro D'Apice, Xavier Litrico and Benedetto Piccoli
2009, 4(2): i-v doi: 10.3934/nhm.2009.4.2i +[Abstract](429) +[PDF](87.6KB)
Abstract:
1. Introduction: Management of canal networks at the age of information technology. With the miniaturization of sensors and their decreasing costs, the paradigm of instrumentation of the built infrastructure and the environment has now been underway for several years, leading to numerous successful and sometimes spectacular realizations such as the instrumentation of the Golden Gate with wire- less sensors a few years ago. The convergence of communication, control and sensing on numerous platforms including multi-media platforms has enabled engineers to augment physical infrastructure systems with an information layer, capable of real- time monitoring, with particular success in the health monitoring community. This paradigm has reached a level of maturity, revealed by the emergence of numerous technologies usable to monitor the built infrastructure. Supervisory Control And Data Acquisition (SCADA) systems are a perfect example of such infrastructure, which integrate sensing, communication and control. In the context of management of irrigation networks, the impact of this technology on the control of such systems has the potential of significantly improving efficiency of operations.

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On Lyapunov stability of linearised Saint-Venant equations for a sloping channel
Georges Bastin, Jean-Michel Coron and Brigitte d'Andréa-Novel
2009, 4(2): 177-187 doi: 10.3934/nhm.2009.4.177 +[Abstract](776) +[PDF](1316.6KB)
Abstract:
We address the issue of the exponential stability (in $L^2$-norm) of the classical solutions of the linearised Saint-Venant equations for a sloping channel. We give an explicit sufficient dissipative condition which guarantees the exponential stability under subcritical flow conditions without additional assumptions on the size of the bottom and friction slopes. The stability analysis relies on the same strict Lyapunov function as in our previous paper [5]. The special case of a single pool is first treated. Then, the analysis is extended to the case of the boundary feedback control of a general channel with a cascade of $n$ pools.
Methods for the localization of a leak in open water channels
Nadia Bedjaoui, Erik Weyer and Georges Bastin
2009, 4(2): 189-210 doi: 10.3934/nhm.2009.4.189 +[Abstract](566) +[PDF](327.1KB)
Abstract:
In this paper, we present two methods for determining the position of a leak in an open water channel. The available measurements are the water level and the gate position at the upstream and downstream end of a channel reach. We assume that the size of the leak and the time it started are already estimated by a leak-detection method. Both of the proposed methods make use of a nonlinear Saint-Venant equation model of the channel where the leak is modelled as a lateral outflow. The first method makes use of a bank of $N$ models corresponding to $N$ possible positions of the leak along the channel. The estimated position of the leak is determined by the model which minimizes a quadratic cost function. The second method is based on the same principle except that it uses observers instead of pure models. The methods are tested on both real and simulated data from the Coleambally Channel 6 in Australia. It is further shown that the determination of the position of a leak is an inherently difficult problem.
Towards nonlinear delay-based control for convection-like distributed systems: The example of water flow control in open channel systems
Gildas Besançon, Didier Georges and Zohra Benayache
2009, 4(2): 211-221 doi: 10.3934/nhm.2009.4.211 +[Abstract](595) +[PDF](418.2KB)
Abstract:
In this paper, the driving idea is to use a possible approximation of partial differential equations with boundary control by ordinary differential equations with time-varying delayed input, for a control purpose. This results in the development of a specific nonlinear control methodology for such delayed-input systems. The case of water flow control in open channel systems is used as a motivating and illustrative example, with corresponding simulation results.
Sensor systems on networked vehicles
João Borges de Sousa, Bernardo Maciel and Fernando Lobo Pereira
2009, 4(2): 223-247 doi: 10.3934/nhm.2009.4.223 +[Abstract](600) +[PDF](1399.2KB)
Abstract:
The future role of networked unmanned vehicles in advanced field studies is discussed in light of the recent technological advances and trends. Visions for systems which could have not been designed before are contrasted to the legal, technological and societal challenges facing the deployment of these systems. The discussion is illustrated with examples of developments from the Underwater Systems and Technologies Laboratory (LSTS) from Porto University.
A Hamiltonian perspective to the stabilization of systems of two conservation laws
Valérie Dos Santos, Bernhard Maschke and Yann Le Gorrec
2009, 4(2): 249-266 doi: 10.3934/nhm.2009.4.249 +[Abstract](728) +[PDF](334.6KB)
Abstract:
This paper aims at providing some synthesis between two alternative representations of systems of two conservation laws and interpret different conditions on stabilizing boundary control laws. The first one, based on the invariance of its coordinates, is the representation in Riemann coordinates which has been applied successfully for the stabilization of linear and non-linear hyperbolic systems of conservation laws. The second representation is based on physical modelling and leads to port Hamiltonian systems which are extensions of infinite-dimensional Hamiltonian systems defined on Dirac structure encompassing pairs of conjugated boundary variables. In a first instance the port Hamiltonian formulation is recalled with respect to a canonical Stokes-Dirac structure and then derived in Riemann coordinates. In a second instance the conditions on the boundary feedback relations derived with respect to the Riemann invariants are expressed in terms of the port boundary variable of the Hamiltonian formulation and interpreted in terms of the dissipation inequality of the Hamiltonian functional. The p-system and the Saint-Venant equations arising in models of irrigation channels are the illustrating examples developed through the paper.
Infinite-dimensional nonlinear predictive control design for open-channel hydraulic systems
Didier Georges
2009, 4(2): 267-285 doi: 10.3934/nhm.2009.4.267 +[Abstract](702) +[PDF](565.3KB)
Abstract:
A nonlinear predictive control design based on Saint Venant equations is presented in this paper in order to regulate both water depth and water flow rate in a single pool of an open-channel hydraulic system. Thanks to variational calculus, some necessary optimality conditions are given. The adjoint partial differential equations of Saint Venant partial differential equations are also derived. The resulting two-point boundary value problem is solved numerically by using both time and space discretization and operator approximations based on nonlinear time-implicit finite differences. The practical effectiveness of the control design is demonstrated by a simulation example. A extension of the predictive control scheme to a multi-pool system is proposed by using a decomposition-coordination approach based on two-level algorithm and the use of an augmented Lagrangian, which can take advantage of communication networks used for distributed control. This approach may be easily applied to other problems governed by hyperbolic PDEs, such as road traffic systems.
Traffic flow models with phase transitions on road networks
Paola Goatin
2009, 4(2): 287-301 doi: 10.3934/nhm.2009.4.287 +[Abstract](700) +[PDF](346.1KB)
Abstract:
The paper presents a review of the main analytical results available on the traffic flow model with phase transitions described in [10]. We also introduce a forthcoming existence result on road networks [14].
Adaptive and non-adaptive model predictive control of an irrigation channel
João M. Lemos, Fernando Machado, Nuno Nogueira, Luís Rato and Manuel Rijo
2009, 4(2): 303-324 doi: 10.3934/nhm.2009.4.303 +[Abstract](750) +[PDF](698.5KB)
Abstract:
The performance achieved with both adaptive and non-adaptive Model Predictive Control (MPC) when applied to a pilot irrigation channel is evaluated. Several control structures are considered, corresponding to various degrees of centralization of sensor information, ranging from local upstream control of the different channel pools to multivariable control using only proximal pools, and centralized multivariable control relying on a global channel model. In addition to the non-adaptive version, an adaptive MPC algorithm based on redundantly estimated multiple models is considered and tested with and without feedforward of adjacent pool levels, both for upstream and downstream control. In order to establish a baseline, the results of upstream and local PID controllers are included for comparison. A systematic simulation study of the performances of these controllers, both for disturbance rejection and reference tracking is shown.
Modal decomposition of linearized open channel flow
Xavier Litrico and Vincent Fromion
2009, 4(2): 325-357 doi: 10.3934/nhm.2009.4.325 +[Abstract](706) +[PDF](599.0KB)
Abstract:
Open channel flow is traditionally modeled as an hyperbolic system of conservation laws, which is an infinite dimensional system with complex dynamics. We consider in this paper an open channel represented by the Saint-Venant equations linearized around a non uniform steady flow regime. We use a frequency domain approach to fully characterize the open channel flow dynamics. The use of the Laplace transform enables us to derive the distributed transfer matrix, linking the boundary inputs to the state of the system. The poles of the system are then computed analytically, and each transfer function is decomposed in a series of eigenfunctions, where the influence of space and time variables can be decoupled. As a result, we can express the time-domain response of the whole canal pool to boundary inputs in terms of discharges. This study is first done in the uniform case, and finally extended to the non uniform case. The solution is studied and illustrated on two different canal pools.
Distributed model predictive control of irrigation canals
Rudy R. Negenborn, Peter-Jules van Overloop, Tamás Keviczky and Bart De Schutter
2009, 4(2): 359-380 doi: 10.3934/nhm.2009.4.359 +[Abstract](1355) +[PDF](281.3KB)
Abstract:
Irrigation canals are large-scale systems, consisting of many interacting components, and spanning vast geographical areas. For safe and efficient operation of these canals, maintaining the levels of the water flows close to pre-specified reference values is crucial, both under normal operating conditions as well as in extreme situations.
   Irrigation canals are equipped with local controllers, to control the flow of water by adjusting the settings of control structures such as gates and pumps. Traditionally, the local controllers operate in a decentralized way in the sense that they use local information only, that they are not explicitly aware of the presence of other controllers or subsystems, and that no communication among them takes place. Hence, an obvious drawback of such a decentralized control scheme is that adequate performance at a system-wide level may be jeopardized, due to the unexpected and unanticipated interactions among the subsystems and the actions of the local controllers.
   In this paper we survey the state-of-the-art literature on distributed control of water systems in general, and irrigation canals in particular. We focus on the model predictive control (MPC) strategy, which is a model-based control strategy in which prediction models are used in an optimization to determine optimal control inputs over a given horizon. We discuss how communication among local MPC controllers can be included to improve the performance of the overall system. We present a distributed control scheme in which each controller employs MPC to determine those actions that maintain water levels after disturbances close to pre-specified reference values. Using the presented scheme the local controllers cooperatively strive for obtaining the best system-wide performance. A simulation study on an irrigation canal with seven reaches illustrates the potential of the approach.
A salinity sensor system for estuary studies
Thanh-Tung Pham, Thomas Green, Jonathan Chen, Phuong Truong, Aditya Vaidya and Linda Bushnell
2009, 4(2): 381-392 doi: 10.3934/nhm.2009.4.381 +[Abstract](614) +[PDF](1031.6KB)
Abstract:
In this paper, we present the design, development and testing of a salinity sensor system for estuary studies. The salinity sensor was designed keeping size, cost and functionality in mind. The target market for this sensor is in hydrology where many salinity sensors are needed at low cost. Our sensor can be submersed in water for up to two weeks (all electronics are completely sealed) while salinity is recorded on-board at user-defined intervals. The data is then downloaded to a computer in the laboratory, after which the sensor is recharged, cleaned for biofouling and ready to be used again. The system uses a software program to download, display and analyze the sensor data. Our initial laboratory testing shows the salinity sensor system is functional. The novelty of this work is in the use of toroidal (inductive) conductivity sensors, the resulting low cost and simple design.
Control of systems of conservation laws with boundary errors
Christophe Prieur
2009, 4(2): 393-407 doi: 10.3934/nhm.2009.4.393 +[Abstract](723) +[PDF](295.9KB)
Abstract:
The general problem under consideration in this paper is the stability analysis of hyperbolic systems. Some sufficient criteria on the boundary conditions exist for the stability of a system of conservation laws. We investigate the problem of the stability of such a system in presence of boundary errors that have a small $\mathcal{C}^1$-norm. Two types of perturbations are considered in this work: the errors proportional to the solutions and those proportional to the integral of the solutions. We exhibit a sufficient criterion on the boundary conditions such that the system is locally exponentially stable with a robustness issue with respect to small boundary errors. We apply this general condition to control the dynamic behavior of a pipe filled with water. The control is defined as the position of a valve at one end of the pipe. The potential application is the study of hydropower installations to generate electricity. For this king of application it is important to avoid the waterhammer effect and thus to control the $\mathcal{C}^1$-norm of the solutions. Our damping condition allows us to design a controller so that the system in closed loop is locally exponential stable with a robustness issue with respect to small boundary errors. Since the boundary errors allow us to define the stabilizing controller, small errors in the actuator may be considered. Also a small integral action to avoid possible offset may also be added.
Comparison of two data assimilation algorithms for shallow water flows
Issam S. Strub, Julie Percelay, Olli-Pekka Tossavainen and Alexandre M. Bayen
2009, 4(2): 409-430 doi: 10.3934/nhm.2009.4.409 +[Abstract](618) +[PDF](953.1KB)
Abstract:
This article presents the comparison of two algorithms for data assimilation of two dimensional shallow water flows. The first algorithm is based on a linearization of the model equations and a quadratic programming (QP) formulation of the problem. The second algorithm uses Ensemble Kalman Filtering (EnKF) applied to the non-linear two dimensional shallow water equations. The two methods are implemented on a scenario in which boundary conditions and Lagrangian measurements are available. The performance of the methods is evaluated using twin experiments with experimentally measured bathymetry data and boundary conditions from a river located in the Sacramento Delta. The sensitivity of the algorithms to the number of drifters, low or high discharge and time sampling frequency is studied.

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