An LPV rational order control model of an irrigation canal is derived from system identification experiments. This model is experimentally obtained by using the described LPV fractional identification procedure. This procedure consists of the identification of a rational order model in each operation point in an experimental test canal. Global LPV model is obtained from polynomial interpolation of local model parameters. Validation results demonstrate that rational order models are more accurate than integer order models. Therefore rational order control models have an important role to play in management and efficient use of water resources.
Water is becoming a precious and very scarce resource in many countries due to the increase of industrial and agricultural demands, as well as population growth. Irrigation is the main water consuming activity in the world, as it represents about 80
Unfortunately, for control design purposes control techniques and their implementation are directly proportional to the complexity of proposed control models. Then, it is essentially a noncomplex and simple control model that represents in a precise way water behavior of openflow canals. However, this type of systems corresponds to long distributed systems with complex dynamics. Furthermore, these systems involve mass energy transport phenomena which behave as intrinsically distributed parameter systems, and their characteristics are very complex such as the variation of parameters with operation points, large delays that vary with operation point, and numerous interactions between different consecutive subsystems and strong nonlinearity. Their complete dynamics is represented by nonlinear partial differential hyperbolic equations (PDEs) that depend on the time as well as the spatial coordinates: SaintVenant's equations. This equation system has unknown analytical solution in real geometry and it has to be solved numerically (characteristic method, Preissman implicit scheme, etc.) [
Resulting time consuming simulation models are therefore suitable for scientific purposes but they are too complex for online applications and control needs. Moreover, linearizations or simplifications of SaintVenant's equations are currently studied by irrigation control research community [
Normally, classical identification methods [
One of the main motivations for using LPV gain scheduling control versus classical gain scheduling control is that the former, as opposed to the latter, rigorously guarantees system stability [
Alternatively, a second approach that can be carried out in “one shot” , by assuming a linear dependence of parameters with operating points. Here, according to [
The last two decades have witnessed considerable development in the use of fractional differentiation in various fields. Fractional control is now mature enough and is widely used to design control for representing systems that present diffusive phenomena, electromechanical diffusion, and transport phenomena. This last phenomenon corresponds to the case of irrigation pools. In this section, LPV identification methodology used for the experimental modelling of a pilot canal plant is described.
An experimental canal prototype (this experimental test canal is a part of a more complex laboratory research canal available at Automatic Control Dept, UPC, Barcelona) is used in the research presented in this paper (Figure
Frontal view of the experimental prototype canal.
Full structure of the plant.
Top side view of the tank, converted into a pool.
The mathematical definition of fractional derivatives has been the subject of several different approaches [
The fractional order models are clasificated in commensurable and noncommensurable order models. In this work, commensurable models are used.
So, the differential equation (
From the previous definition and based on the property of “q”,
Consider the fractional discrete linear system, described by the statespace equations
LPV identification method used in this article is a twostep procedure where (
As the LPV model is interpolated between local rational models, varying parameters of LPV model can be locally interpreted as parameters of the interpolated rational model. Varying parameters in each operation point are interpolated in a polynomical way. This polynomial depends on a scheduling parameter vector
In this paper, this system identification procedure is used to obtain a reliable dynamic model of a main irrigation canal when the design of a modelbased control system is requested.
For identification of the pilot canal system different experiments have been carried out. These canal pools are operated by means of a downstream water level regulation method. Available measurements are downstream water levels (
The appearance of integrator pole, or in other words, the fact that a reach has similarities with a swimming pool or a tank, is not a real surprise and is, in some case, expected. As mentioned before, this pole appears clearly in the uniform case regime and has been successfully included in several simplified models proposed in other works (Integrator Delay (ID) model [
To obtain data containing the maximum information about the canal pools dynamic behaviour, pools must be excited with a persistent input signal that contains the largest number of frequencies representative of the system dynamics [
The sampling time
Operation points for pool
Pool 
Operation range [cm]; 











Operation points for pool
Pool 
Operation range [cm]; 

















The model structure selection constitutes one of the most important and difficult decisions in system identification procedure because model complexity influences the accuracy of the description of the real process and the control schemes. SaintVenant equations [
The secondorder system behaviour can be clearly observed in Figures
Downstream level for pool
Downstream level for pool
In order to identify the canal system, the continuous model is discretized by using zeroorder hold method. Furthermore we assume that control model is LPV (as it is explained in Section
Observing and analyzing the PRBS responses obtained at each operation point (see Figures
For
Parameters of models (
The estimation method used in this work is the previously mentioned in Section
In the case of integer model, there exists a delay which is estimated using correlation analysis [
Parameters of models obtained in both pools,
Model parameters obtained by identification in each operating point
Parameters 







17  10  8  6  5 







0.9537  0.9309  0.9051  0.9011  0.8747 

0.0010  0.0022  0.0043  0.0051  0.0088 

0.0203 


0.0063 








0.0370  0.0333  0.0261  0.0226  0.0212 

0.0019  0.0052  0.0104  0.0139  0.0198 

0.0995  0.1224  0.1793  0.1969  0.3952 

633.48  606.00  553.53  544.37  229.55 







55.0820  74.408  82.686  95.255  62.955 







2.0729  1.8516  2.1183  2.3379  2.0016 







0.4176  0.7331  0.4440  0.5309  0.4104 







0.0133  0.0560  0.0151  0.0185  0.0162 







0.0464  0.6397  0.0990  0.1061  0.2949 



0.9473  0.9293  1.8894 



0.8899  0.8545  1.1611 



0.9887  0.9411  0.6570 

0.3424  0.5942  1.4846  1.4310  0.9577 

2.2736  2.1238  2.0835  2.0224  1.8793 

2.1646  1.7134  1.2674  1.1950  1.5309 






Model parameters obtained by identification in each operating point
Parameters 







11  10  9  8  7 







0.9434  0.9270  0.9153  0.9005  0.8904 

0.0015  0.0026  0.0038  0.0055  0.0074 



0.0075  0.0147  0.0205 







0.0377  0.0484  0.0356  0.0178  0.0100 

0.0032  0.0017  0.0069  0.0154  0.0199 

0.1347  0.3226  0.2270  0.1694  0.1214 

603.9000  117.29  307.20  726.79  989.96 







51.8040  15.713  48.386  122.65  207.87 







1.9327  1.6503  2.1372  2.4464  2.5364 







0.3661  0.4019  0.4440  0.5688  0.6308 







0.0111  0.0283  0.0151  0.0197  0.0241 







0.0434  0.4807  0.1050  0.1112  0.1667 

2.1585 



1.1259 

1.4147 



0.0930 

0.8165 





1.0115  0.6744  1.1870  1.1744  0.5429 

1.7673  2.4956  2.8789  2.7318  2.4285 

0.9012  1.3293  1.2377  1.1205  1.6084 






Model parameters obtained by identification in each operating Point
Parameters 





5  4  3 





0.8907  0.8891  0.8791 

0.0081  0.0088  0.0094 

0.0309  0.0447  0.0720 

0.1722  0.2216  0.0919 





0.0434  0.0509  0.0437 

0.3951  0.4421  0.2546 

722.21  566.36  757.35 





133.11  136.39  203.67 





2.2776  2.6077  2.8311 





0.5203  0.6903  0.8029 





0.0282  0.0369  0.0390 





0.7180  0.7576  0.4943 

5.6247  3.8395  13.4920 

2.3606  2.4912 








7.8987 

0.1248 



5.8407  6.5926  8.5167 




Parameters are estimated experimentally by applying the set of input PRBSs, explained in Section
For instance, for noninteger model
Values of
Coefficients 




















Values of
Coefficients 




















Polynomial aproximations of
Polynomial aproximations of
Model validation is the core of the identification problem because it makes possible to evaluate the model quality, that is, if the method fits the measured experimental data with accuracy enough, if it is valid for its purpose, and if the model describes correctly the real process [
Model output in pool
Model output in operation point
Model output in pool
Model output in operation point
In order to assess how suitable models respect validation data set, mean absolute error (MAE) is quantified as
Mean absolute error (MAE) in every operation point: pool
Operation points 





0.0294  0.0152  0.0148 

0.0228  0.0145  0.0067 

0.0164  0.0120  0.0163 

0.0191  0.0168  0.0180 

0.0187  0.0172  0.0179 
Mean absolute error (MAE) in every operation point: pool
Operation points 





0.0160  0.0093  0.0074 

0.0133  0.0095  0.0073 

0.0121  0.0098  0.0188 

0.0133  0.0123  0.0106 

0.0117  0.0114  0.0108 

0.0129  0.0124  0.0093 

0.0128  0.0125  0.0099 

0.0101  0.0097  0.0077 
However, the lower the value of
In this article, an LPV rational order modelbased controloriented system identification procedure for irrigation canals has been developed. This identification procedure has been applied in an experimental prototype canal. In this case, rational local models for an irrigation pool in different operation points have been obtained and interpolated to reach the complete model: the LPV rational model. Resulting LPV rational order control model normally describes the plant with a lower error than the corresponding LPV integer order control model. The lower the
Authors would like to thank InterMinisterial Commission of Spanish Government (DPI20045414, MCYT), Catalan Autonomous Government (VIS, Consolidated Research Group), European Commission (FP62005IST6, URUS045062), and Cooperation Spanish Agency for International Development (AECID) for funding this research. Authors also want explicitly to thank reviewers' comments that have largely improved this article.