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## Über dieses Buch

Open-channel hydraulics are described by hyperbolic equations, derived from laws of conservation of mass and momentum, called Saint-Venant equations. In conjunction with hydraulic structure equations these are used to represent the dynamic behavior of water flowing in rivers, irrigation canals, and sewers.

Building on a detailed analysis of open-channel flow modeling, this monograph constructs control design methodologies based on a frequency domain approach. In practice, many open-channel systems are controlled with classical input–output controllers that are usually poorly tuned. The approach of this book, fashioning pragmatic engineering solutions for the control of open channels is given rigorous mathematical justification. Once the control objectives are clarified, a generic control design method is proposed, first for a canal pool, and then for a whole canal. The methods developed in the book have been validated on several canals of various dimensions up to a large scale irrigation canal.

## Inhaltsverzeichnis

### Chapter 1. Introduction

Abstract
This chapter gives an overview of the issue of water management. We describe open channel systems, why they are difficult to manage, and why control engineering may provide useful tools to better manage such systems. We also emphasize the difficulties involved in the design of efficient controllers for open channels: they are distributed systems, with large delays between inputs and outputs, few measurements along the system, subject to large and unmeasured perturbations, with a nonlinear behavior. Our approach is based on an engineering background, where pragmatic but efficient solutions can be found to solve practical problems. We limit our study to the linearized Saint-Venant equations, which describe open channel flow around a given equilibrium regime. This is a classical approach in the control engineering community, linked to the gain-schedulingmethod for designing a controller for a nonlinear system, based on its linearizations.

### Chapter 2. Modeling of Open Channel Flow

Abstract
In this chapter, we present the classical model used to describe open channel hydraulics: the Saint-Venant equations. For completeness, the equations are rigorously derived in Appendix A. We first study some of their mathematical properties, such as the characteristic form. We briefly describe some numerical methods of resolution, and then consider the linearized equations that are valid for small variations around equilibrium regimes. These equations form the basis of all the methods developed in this book.

### Chapter 3. Frequency Domain Analysis of Open Channel Flow

Abstract
In this chapter, we analyze the linearized Saint-Venant equations. First the horizontal frictionless case, then the uniform flow case, and finally the nonuniform flow case are treated in depth, with a complete characterization of the Saint-Venant transfer matrix in terms of poles, delay, and series expansion. For the two former cases, the analysis of the system properties is possible due to the existence of closedform expressions for the transfer functions. By contrast, for the latter case, in order to bypass the absence of closed-form expressions, we provide a complete and new approach based on the use of efficient and convenient numerical schemes.

### Chapter 4. Finite Dimensional Models of Open Channel Flow

Abstract
This chapter examines various ways to obtain finite dimensional models for linearized open channel flow equations. We first study rational models obtained based on the modal decomposition derived in Chap. 3. Then we put the problem as a convex optimization one. Finally, we analyze the properties of a finite dimensional discrete linear model obtained based on the classical finite difference Preissmann scheme. These models are useful for simulation purposes and for controller design.

### Chapter 5. A Simplified Model of Open Channel Flow

Abstract
This chapter develops a simple model for open channel flow, called the IDZ model, for integrator delay zero. After the models developed in the previous chapters, we show that such a simple model can capture the main physical properties of the open channel dynamics. The main interest of this model is that it can be computed analytically from the physical parameters of the channel such as its geometry, slope and roughness coefficient, and the flow characteristics such as the average flow and the downstream water level. It is a simple yet accurate low frequency model of open channel flow, which will be used in Chap. 7 to design robust tuning rules for PI controllers.

### Chapter 6. Control of a Canal Pool with Hydraulic Structures

Abstract
The equations of open channel flow do not apply at sections where the flow varies very rapidly. This is the case at cross-sections equipped with hydraulic structures, such as gates or weirs. These structures are generally used to control the water level, or to deliver a discharge. In this chapter, we mainly deal with static hydraulic structures and study the behavior of the interaction between the flow and the structure, which is viewed as a local boundary controller imposing a feedback between the flow and the water level.

### Chapter 7. Classical Control Policies for a Canal Pool

Abstract
In this chapter, we examine the classical control policies distant downstream control and local upstream control. We develop tuning methods for feedback control of a canal pool using PI controllers. We also cast the problem into the H optimization method, which naturally incorporates robustness constraints in the design. Finally, we compare the PI controller and the H controller in the distant downstream case.

### Chapter 8. Mixed Control of a Canal Pool

Abstract
This chapter develops a mixed control policy for a canal pool. The classical control policies have advantages and drawbacks. The local upstream control policy ensures good performance, but bad water management. The distant downstream control ensures good water management, but relatively bad performance. We propose a mixed local upstream/distant downstream control that ensures good performance and at the same time efficient water management. The controller structure imposes a substitution between the two actuators. Two controller tuning methods are developed, a PI controller and an H controller.

### Chapter 9. Open-loop Control of a Canal Pool

Abstract
This chapter considers the design of feedforward boundary control laws for a canal pool. We first consider the problem of delivering water downstream according to a prespecified schedule by using the upstream discharge. The controller design is based on frequency domain methods. We then turn towards the similar problem of canceling known disturbances using boundary control. Perfect rejection of measured perturbations at one boundary is obtained by controlling the other boundary. Frequency domain comparisons and time domain simulations illustrate the good performance of the feedforward boundary controller.

### Chapter 10. Decentralized Control of a Multiple-pool Canal

Abstract
In this chapter, we study the extension of the control policies developed in Chaps. 7 and 8. We first study the case of a two-pool canal, controlled with distant downstream, local upstream, and mixed control policies. We study the stability and performance of decentralized controllers designed separately for each pool.We then extend the results to the case of a multiple-pool canal.

### Chapter 11. Experimental Results on a Small-scale Canal

Abstract
In this chapter, we present experimental results obtained on a small-scale canal located in Portugal. Most of the results presented in this book have been tested on this canal, from the modeling part to the control schemes, for one and multiple pools. The results show that the Saint-Venant equations for open channels can very accurately represent the system’s dynamics, and that the proposed control strategies are effective in practice.

### Chapter 12. Modeling and Control of Regulated Rivers

Abstract
This chapter presents another application of modeling and control of a hydrosystem. We consider regulated rivers where one dam or multiple dams located upstream enable control of the upstream flow into the river. We first study the case of a river with one dam and multiple discharge measurement points. We then study the case of a regulated river controlled with multiple dams, each having a different capacity. The controlled output can be regulated with one dam acting rapidly, but with a low capacity, or with a large dam located further upstream, or both, if a suitable control strategy is applied.

### Backmatter

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