Fractional robust control of main irrigation canals with variable dynamic parameters

https://doi.org/10.1016/j.conengprac.2006.11.018Get rights and content

Abstract

A new method is proposed for controlling main irrigation canals with variable dynamical parameters based on robust fractional order controllers. A methodology for designing PID controllers robust to changes in the time delay and the gain is presented first. Then this method is generalized to design fractional controllers that exhibit the same robustness as the previous PID to time delay and gain changes, but are noticeably more robust to variations in the dominant time constant of the process. This method is applied to control main irrigation canals. Extensive numerical simulations using the dynamic model of a real canal were carried out. Then experimental results were obtained in a prototype canal that proved the effectiveness of the proposed control method in terms of performance and robustness.

Introduction

Nowadays water is becoming a precious, rare and scarce resource all over the world. Thus there is a growing interest in the application of advanced management methods to prevent waste of this vital resource. Irrigation is the main water consuming activity around the world, since it represents about 80% of the available fresh water consumption. The most important objective of irrigation systems is to provide the demanded quantity of water to the different users at specified instants, and to guarantee the safety of the infrastructure (Malaterre, 1998).

Many irrigation systems are still being managed manually, leading to low efficiency in terms of delivered water versus water taken from the resource (Litrico & Georges, 1999). Automatic control leads to more efficient water management in irrigation systems which are based on open main canals subject to high losses (Kovalenko, 1983; Malaterre, 1995). The main objectives of these automatic control systems are: (1) to improve water efficiency and distribution, (2) to reduce water loss, and (3) to supply water consumers in due time.

Several control methods have been developed for water distribution canal networks: upstream control, downstream control, bival control, adjustable flow-rate control, etc. (Buyalski, Ehler, Falvey, Rogers, & Serfozo, 1991; Malaterre, 1995). Design of controllers for water distribution in main irrigation canals is a difficult task because these systems exhibit nonlinear dynamics distributed over long distances with significant time delays, and their dynamics change depending on the operating conditions (Litrico, Fromion, Baume, Arranja, & Rijo, 2005). Linear regulators are usually designed without considering robustness requirements—which are essential in time varying dynamics like these—leading to inefficient control.

The dynamics of water flowing in irrigation open canals is modeled by using the so-called Saint-Venant equations, which are nonlinear hyperbolic partial differential equations and are given by (Chow, 1988)At+Qx=q;Qt+Q2/Ax+gAzx=-gASf+kqV,where A(x,t)—canal cross section area; Q(x,t)—discharge through section A; q(x,t)—lateral discharge; V(x,t)—the mean velocity in section A; z(x,t)—absolute water surface elevation; x—distance along the canal; g—gravity acceleration; t—time variable; k—weighting coefficient, k=0 if q>0 and k=1 if q<0; Sf(x,t)—friction slope.

Nowadays different methods exist for the solution of the Saint-Venant equations, but all of them exhibit considerable mathematical complexities (Litrico & Fromion, 2004). Moreover, these equations are very difficult to use directly for controller design (Malaterre, 1995). Often, the Saint-Venant equations are linearized around a set point, and equivalent first-order systems plus a delay are used to model the canal dynamic behavior (Weyer, 2001). These models have the strong drawback that their parameters may experience considerable changes when the canal operation regime varies (e.g. Rivas Perez, 1990). Thus any controller to be designed for an irrigation canal has to be robust to variations in the parameters of the linearized model.

Over the past few years, fractional operators have been applied with satisfactory results to model and control processes with complex dynamics, most of them being distributed parameter processes (Machado, 1997; Odai & Hori, 2000; Podlubny, 1999; Vinagre, Podlubny, Hernandez, & Feliu, 2000). Recently, several works designed fractional PID controllers in the frequency domain with enhanced robustness properties (e.g. Barbosa, Tenreiro Machado, & Ferreira; 2004; Monje, Calderon, Vinagre, Chen & Feliu, 2004; Valerio & Costa, 2005). Robustness features in the frequency domain are also explored in this paper, and are applied to design robust controllers for effective water distribution control in main irrigation canals whose dynamic parameters vary over a wide range.

It is also worth mentioning that gain scheduling PI controllers (Bolea, Puig, Blesa, Gómez, & Rodellar, 2005) have been proposed for irrigation canals. These adaptive controllers reduce the effect of process parameter variations by changing its coefficients in function of some auxiliary variables, under real time operation conditions. This can lead to instabilities in the canal pool control system. These problems of adaptive controllers have been extensively reported in the literature. Guaranteeing their stability is a difficult task, which usually implies some Lyapunov analysis.

This paper is organized as follows. A mathematical model of a main irrigation canal is obtained in Section 2. An introduction to fractional order operators and controllers is presented in Section 3. A method to design robust PID controllers for main irrigation canals is proposed in Section 4. Section 5 generalizes the previous method to design robust fractional controllers. Section 6 describes the application of this new method to a particular main irrigation canal and its simulated results are compared with the results obtained from the standard PID controller of Section 4. Section 7 reports some experimental results in a canal prototype, and finally some conclusions are drawn in Section 8.

Section snippets

Main irrigation canal model for control

A typical main irrigation canal consists of several pools separated by gates that are used for regulating the water distribution from one pool to the next. Fig. 1 shows a scheme of a main irrigation canal with gates.

In automatically regulated canals, the controlled variables are the water levels (yi(t)), the manipulated variables are the gate positions (ui(t)), and the fundamental perturbation variables are the unknown offtake discharges (qi(t)). If the water levels are measured near the end of

Fractional order operators and fractional order control systems

Fractional calculus is a generalization of integration and differentiation to non-integer (fractional) order fundamental operators represented as aDtα where a and t are the limits and α(α∈ℜ) the order of the operation. Several definitions of this operator have been proposed (see, e.g. Podlubny, 1999). All of them generalize the standard differential/integral operator in two main senses: (a) they become the standard differential/integral operator of any order when a is an integer, (b) the

PID controller robust to parameter variations

There are three parameters to be tuned in the standard PID controller (8): Kp, Ti, Td. They can be chosen to verify the typical frequency specifications: phase margin (φm), crossover frequency (ωc), and gain margin (Mg). Generally speaking, the phase margin defines the damping of the system, and the crossover frequency fixes the speed of response. If a controller robust to parameter variations is desired for the canal described by model (4), it is clear that the phase margin defines the range

New fractional controller

In this section, a new fractional controller which exhibits enhanced robustness properties over the PID controller developed in the previous section is proposed.

The fractional PID controller of expression (9) has five parameters to be tuned: Kp, Ti, Td, λ, μ. In general, only three parameters are needed in order to fulfill specifications (a)–(c) of the previous section. Then the remaining two parameters can be used to obtain additional closed loop dynamics specifications. But this is not the

Comparison of controllers

The fractional control system of an irrigation canal pool whose block diagram is shown in Fig. 5 is considered. In this system it is assumed that parameters K, T1 and τ, which model the pool dynamics, can vary around nominal values K0=1.25, T10=300 s, and τ0=600 s, in the ranges defined by τmax=3τ0, Kmax=2.5K0, T1min=6 s, and T1max=6000 s. It is also assumed that the gate dynamics is invariant, exhibiting a time constant T2=60 s. The dynamics of the water level sensor plus the signal transmission

Experimental results

The previous results were obtained by using model (2) with parameters identified from a real canal in Cuba. In order to demonstrate the feasibility and practical characteristics of the proposed controller, experiments have been carried out in an experimental prototype canal in the Fluids Mechanics Laboratory of the Castilla-La Mancha University (Spain). This canal is 5 m long, 8 cm wide, and the height of the walls is 25 cm. It has three pools of different length separated by two submerged flow

Conclusions

A methodology to design PID controllers for main irrigation canals robust to changes in the time delay and the gain, which can be easily extended to the design of fractional PID controllers, has been proposed. Based on the previous methodology, a new fractional regulator to control a main irrigation canal has been designed which is robust to changes in the time delay, the gain, and the dominant time constant. We point out that the interest of such controllers is justified by the fact that the

Acknowledgments

The authors would like to acknowledge the support provided by the International Cooperation Program of the Universidad de Castilla-La Mancha (Spain) and to thank the anonymous reviewers for their constructive comments which helped to improve the quality of this manuscript.

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