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Constrained Predictive Control of an Irrigation Canal1
A.´
Alvarez1, M.A.Ridao2, D.R.Ramirez3
L.S´anchez4
2
ABSTRACT3
This paper presents the application of a Distributed Model Predictive Controller (DMPC)4
to the control of an accurate model of an actual irrigation canal in Spain. The canal is5
modelled using the Saint-Venant equations and implemented using the well known modelling6
software for irrigation canals SIC. The DMPC algorithm has been implemented in Matlab7
and interfaced to SIC. In the distributed control algorithms, the local controllers exchange8
information so that their control policies are optimal in the sense of getting the best values9
of a performance index. The results show that the proposed distributed control algorithm10
obtains better control performance than a more conventional decentralized control scheme11
without information exchange. This better performance translates directly into money and12
resource savings.13
Keywords: Model Predictive Control, irrigation canal, distributed control, control14
algorithms15
1 INTRODUCTION16
Water is a limited resource. In addition, nowadays there are some regions in Europe17
and all over the world with long seasons of drought. As a consequence, the development of18
innovative control techniques that optimize water management is a relevant issue.19
1Dpto. Ingenier´ıa de Sistemas y Autom´atica, Escuela T´ecnica Superior de Ingenieros, Universidad de
Sevilla, Camino Descubrimientos, s/n., 41092 Sevilla, Espa˜na. E-mail: antonio.alvarez@cartuja.us.es
2Dpto. Ingenier´ıa de Sistemas y Autom´atica, Escuela T´ecnica Superior de Ingenieros, Universidad de
Sevilla, Camino Descubrimientos, s/n., 41092 Sevilla, Espa˜na. E-mail: miguelridao@us.es
3Dpto. Ingenier´ıa de Sistemas y Autom´atica, Escuela T´ecnica Superior de Ingenieros, Universidad de
Sevilla, Camino Descubrimientos, s/n., 41092 Sevilla, Espa˜na. E-mail: danirr@cartuja.us.es
4AECOM INOCSA S.L.U. (SPAIN). E-mail: laura.sanchezmora@aecom.com
1
The main objective of irrigation canals is to supply water to farmers according to a20
specific schedule. An irrigation canal is composed by several reaches, connected by gates,21
and usually following a tree structure. In a typical irrigation canal the length can be hundred22
of kilometers, there are tens of gates and hundreds of off-take points, used by farmers to take23
water from the canal.24
Irrigation canals management involves operating gates, pumps and valves in order to25
satisfy user demands and minimize costs and water loses. In addition, a set of constraints26
imposed by the physical system and management policies has to be considered, for example,27
maximum and minimum water level and flow.28
Automatic control techniques are widely used in irrigation canals, most of them based on29
a local control of gates using classic approaches as PI (Proportional-Integral) controllers (See30
(Malaterre et al. 1998) for a detailed classification of these algorithms). These decentralized31
approaches provide reasonable behavior in many cases, but as the coupling effect among32
the different local controllers (agents) is not taken into account, sometimes they produce33
important loss in the control performance.34
Another approach based on PI is discussed in (Ooi and Weyer 2008), where the controller35
is a PI controller augmented with a first order low pass filter in order not to amplify waves36
present in the channel. The developed routine for controller design is based on frequency37
response design, and configurations with and without feedforward from downstream gate are38
considered.39
The use of a single global controller for the control of the whole system (centralized40
control) is an alternative to deal with this problem. Model Predictive Control (MPC) (See41
(Camacho and Bordons 2004)) approaches have been widely and successfully applied in water42
systems. However, MPC is a technique with strong computational requirements that hinder43
its application to large-scale systems such as water networks in a centralized way. Moreover,44
the communication difficulties in a system extended in a geographical area of hundreds of45
kilometers make not sensible the use of a centralized real-time control system based on long46
2
distance communications. Another problem to use centralized approaches is the fact that47
sometimes different sections of the canal can be managed by different control centers and48
even by different organizations.49
Distributed Model Predictive Control techniques to optimize the management of water50
in irrigation canals provide a reasonable trade-off between complexity and performance.51
Basically, the idea is to provide communication among local controllers, in such a way that52
agents can exchange information or even negotiate and reach agreements. In this paper, a53
Distributed MPC algorithm is presented where the communication requirements are adapted54
to the complexity of the control of the different subsystems, from a simple information55
exchange to a negotiation in problematic reaches.56
There are several works that address the canal control with Predictive control techniques57
with decentralized and centralized approaches.58
In (Rodellar et al. 1993), a model predictive algorithm is presented to control the59
downstream discharge of a canal reach. (G´omez et al. 2002) presents a decentralized60
predictive control for an irrigation canal composed by a series of pools. In order to decouple61
the system, the controller used an estimation of the future discharges and the hypothesis62
of being linearly approaching the reference, to finally reach it, at the end of the prediction63
horizon. Because the control law solution was given in terms of reach’s inflow discharge,64
they used a local controller to adjust the gate opening to the required discharge.65
In (Sawadogo et al. 1998), and later in (Sawadogo et al. 2000), a similar decentralized66
adaptive predictive control is presented , but that used the reach’s head gate opening as67
controllable variable and the reach’s tail gate opening and the irrigation off-take discharge68
as known disturbances.69
Several centralized MPC approaches also have been proposed. (Malaterre and Rodellar70
1997) performed a multivariable predictive control of a two reaches canal using a state space71
model. They observed that the increase of the prediction horizon produced a change in72
the controller behavior, varying the control perspective from a local to a global problem.73
3
(Wahlin 2004) tested a Multivariable Constrained Predictive controller using a state space74
model based on Schuurmans first-order Integrator Delay model (Schuurmans et al. 1999).75
They performed tests where the controller either knew or did not know the canal parameters76
and with and without the minimum gate movement restriction.77
In (Silva et al. 2007), a predictive controller, based on a linearization of the Saint-Venant78
equations, has been also implemented on an experimental water canal. (Begovich et al.79
2004; Begovich 2007) proposed a multivariable predictive controller with constraints which80
was implemented in real-time to regulate the downstream levels of a four-pool irrigation81
canal prototype. In (Lemos et al. 2009), several control structures are applied to a pilot82
canal, ranging from decentralized MPC, multivariate control using only neighbour reaches,83
to centralized multivariable control. Also, an adaptive MPC based on multiple models is84
evaluated. A complete state-of the art of MPC applications can be found in (van Overloop85
2006) and (Sepulveda 2007).86
Distributed control has been also a focus of research during the last few years. (Tricaud87
and Chen 2007; Li and Cantoni 2008; Li and De Schutter 2010) presented different distributed88
approaches based on control techniques different to MPC. (Negenborn et al. 2009) presented89
a distributed MPC based on Lagrange multipliers. At every sample interval the controllers90
perform several iterations of local optimization problem and communication with their91
neighbour based on a serial communication scheme. Finally, in (Zafra-Cabeza et al. 2011) a92
distributed MPC method based on game theory for multiple agents is applied to irrigation93
canals. The controller were tested by a simulation in which the canals were modelled using94
the integrator delay model. Also, the controlled variables were water flows at each gate,95
assuming an underlying low level control structure that managed to get the flows set by the96
distributed MPC controller. The distributed MPC algorithm used, presented in (Maestre97
et al. 2011), provides a reasonable trade-off between performance and low communication98
requirements needed to reach a cooperative solution.99
In this paper we present the modelling of a section of a real canal in the South-East of100
4
Spain and its control using predictive controllers based on the distributed MPC algorithm101
presented in (Maestre et al. 2011) and also on feedforward techniques. The controller uses an102
iterative game theory algorithm for the two most coupled subsystems and for the remaining103
a non iterative distributed scheme in which the information exchanged for each controller is104
used to compensate the interactions in a feedforward manner.105
The model of the canal used to test the control structure is a very realistic one developed106
using the well known SIC software (Simulation of Irrigation Canals), which is based on a107
mathematical model that can simulate the hydraulic behaviour of most of the irrigation108
canals or rivers, under steady and unsteady flow conditions. The SIC hydraulic model solves109
the complete Saint Venant equations using the classical implicit Preissmann scheme.110
Moreover, different control scenarios are illustrated in the paper, and in each of them111
different control structures are tested. The performance of each control structure is112
illustrated by means of a performance index and an estimation of the economical costs113
incurred by each controller. These merit figures shows that distributed decentralized114
predictive controllers obtain the best results compared with decentralized local controllers.115
The rest of the paper is organized as follows: Sections 2 and 3 present the irrigation canal116
benchmark and some issues regarding the control of canals. The proposed control strategy is117
presented in section 4 and experimental results for several simulations are shown in section118
5. Finally the conclusions are presented in section 6.119
2 CONTROL OF IRRIGATION CANALS120
The control of irrigation canals presents some specific details that should be considered121
before choosing any control strategy. First, it is important to start with the variables involved122
in the control scheme. In the case of irrigation canals, there are two types of variables that123
one could wish to control (i.e., controlled variables or system outputs): water levels and flows124
(measured at each or some of the gates). On the other hand, to achieve the control goals,125
two variables can be manipulated (i.e., control inputs or manipulated variables): the degree126
of gate aperture and also flows (usually only at the head of the canal). Note that in the case127
5
that the manipulated variable is chosen to be the flow at some specific gate (instead of the128
head of the canal), then a lower level controller has to be used to attain that flow using the129
gate aperture.130
An important issue in the control of the canal system is the location of controlled variable131
relative to the control structure (i.e, gates). Mainly, two alternatives are considered. In132
downstream control strategy, control structure adjustments are based upon information133
measured by a sensor located downstream. Downstream control transfers the downstream134
canal-side off-takes demands to the upstream water supply source (or canal head works).135
On the other hand, in upstream control, control structure adjustments are based upon136
information from upstream. Upstream control transfers the upstream water supply (or137
inflow) downstream to points of diversion or to the end of the canal. Upstream control138
has to be used when the flow at the head of the canal is fixed, normally by an external139
organization. In any other circumstances, downstream control has demonstrated to be more140
efficient.141
Measurable disturbances play an important role in the control of irrigation canals.142
This is because the coupled nature of irrigation channels, which extend for hundreds of143
kilometers and have multiple controllers that disturb their neighbours with each change in144
their manipulated variables. Specifically, downstream control actions mean disturbances145
that could be considered when computing a control action somewhere in the canal. When146
calculating the opening/closing of any gate at any sample period, the opening/closing of the147
following downstream gate could be considered as a measurable disturbance and its effect148
could be taken into account in the optimal control sequence calculation.149
Off-takes and in-takes comprise another kind of disturbance. An off-take is a point150
where water is taken for a particular purpose (for example, irrigation).The flows are151
usually scheduled, so their value and moment of apparition can be predicted in advance.152
Nevertheless, the off-take gates are manipulated directly by farmers, so an uncertainty must153
be considered in this prediction. Sometimes there is only partial information about off-take154
6
flows, for example, an aggregate value of the flows of the off-take in a determined area. Also155
in-takes can be considered, for example, rainfall. The operation of off-takes and in-takes is156
considered as a measurable disturbance in the same manner as the gate movements, with157
exactly the same treatment.158
Finally, the aforementioned coupled nature of irrigation canals together with the usual159
geographical dispersion found in the actual control hardware used leads to the consideration160
of distributed control schemes as a practical control solution. Thus, the overall performance161
of the canal control system will be greatly improved if distributed control strategies are used162
at least in those segments of the canal in which the coupling is so strong that a measurable163
disturbance management only is not enough.164
2.1 The irrigation canal of La Pedrera (Murcia, Spain)165
This work is focused on the control of a section of the ”postrasvase Tajo-Segura” in the166
South-East of Spain. The ”postrasvase Tajo-Segura” is a set of canals which distribute water167
coming from the Tajo River in the basin of the Segura River. This water is mainly used for168
irrigation (78%), although 22% of it is drinking water. The selected section is a Y-shape169
canal (see Figure 1), a main canal that splits into two canals with a gate placed at the input170
of each one of them:171
•”Canal de la Pedrera”, 6.680 kilometres long.172
•”Canal de Cartagena”, 17.444 kilometres long.173
It is a gravity-fed canal without pumps in the considered section. The total length of the174
canals is approximately of 24 kilometers with a trapezoidal section. There are five main175
sluice-type overshot gates (in red in Figure 1) and 5 gravity off-take gates in the section176
selected (green arrows in Figure 1).177
The objective of this paper is to control the distance downstream level at each one of178
the reaches in Canal de Cartagena (ref1, ref2, ref4 and ref5 in 1) and the flow at the head of179
Canal de la Pedrera (ref3). To reach this objective, the control system will manipulate the180
7
flow at the head of the main canal (f1) and the position of the main gates (g2 to g5).181
Notice that typically the source of water at the head of the canal (a reservoir or a main182
river, as in this benchmark) is managed by a different organization than the canal operator.183
That includes also the control or supervision of the gate at the head of the canal (typically184
a set of undershot sluice gates), fixing at any time the head flow of the canal or, at least,185
the constraints on the flow. In this benchmark, the proposed control system provide flow186
set-points to the head gate but considering flow limits imposed by the external organization187
The following constraints are also considered:188
•Minimum level to guarantee that off-take points are submerged.189
•Maximum level to prevent the canal from bursting its banks and causing floods.190
•The flow at the head of the canal is limited.191
•Maximum and minimum gates opening. Maximum gate opening is fixed by the water192
level, when the whole gate becomes above the water line and increasing its opening is193
pointless. Besides, gate opening has a physical limit which depends on the gate itself.194
A combination of local and distributed MPC approaches is proposed for the control of this195
section of the canal.196
3 CONSTRAINED PREDICTIVE CONTROL OF IRRIGATION CANALS197
This section presents control algorithms and techniques used to control the irrigation198
canal described in the previous section. These techniques are briefly reviewed, and only the199
main ideas are presented. Thus, the reader interested in more technical details is encouraged200
to consult the Model Predictive Control works cited in this section. Section 3.1 presents201
the main constrained predictive control used to compute the control signal applied to each202
gate. How to take into account the effect of measurable disturbances in the control signal203
computation is shown in section 3.2. Finally, the coordination of a pair of local controllers204
using a Distributed Model Predictive Control scheme is presented in section 3.3.205
8
3.1 Constrained Model Predictive Control206
Model Predictive Control (MPC) is one of the most popular techniques in the field of207
automatic control. It is in fact one of the few advanced control techniques that are nowadays208
available in commercial industrial control solutions. The reasons of that success are mainly209
the ability to consider constraints in the computation of the control signal, the possibility210
of taking into account measurable disturbances and process dead-time and, also, that the211
extension to the multivariable case is relatively straightforward. All these features yield a212
control performance that can be much better than that obtained with conventional control213
methods (i.e., PID controllers). Furthermore, MPC can be applied to a wide range of214
problems and the tuning of MPC controllers involve the choice of a reasonable number of215
design parameters.216
All MPC strategies are based on a process model that is used to predict the evolution217
of the system state or output1along an interval of time called the prediction horizon (see218
figure 2). The prediction of the system output are computed iteratively using the prediction219
model using present and past values of the system input and output as initial conditions.220
Predicted values for system output or input at time k+jusing the information available at221
time kare denoted by yk+j|kor uk+j|k. On the other hand, the prediction horizon comprises222
all sampling times between k+N1and k+N2. There is also a control horizon, comprising223
sampling times between kand k+Nu−1, after which the system input is considered to be224
constant (see figure 2).225
Usually, the process model (or prediction model) considered is a discrete time model that226
can be nonlinear or linear. The theory of MPC using linear models is much more developed227
than that of nonlinear models, thus almost all commercial implementations are based on a228
linear prediction model. From a practitioner point of view, the most natural and easy choice229
is an input-output model based on the transfer function of the process to be controlled.230
Thus, we propose the use of a CARIMA (Controlled Auto-Regressive Integrated Moving231
1In this work we use input-output models, thus we consider here predictions of the system output.
9
Average) model, which in time domain can be written as:232
yk=a1yk−1+···+anayk−na +b0∆uk−d−1+···+bnb∆uk−d−nb−1+ek(1)233
where dis the model dead time measured in sampling times and ykand ∆ukdenote the234
value of the process output and process input increment at sampling time k. Furthermore,235
this model considers a noisy disturbance denoted as ekwhich can be modelled as a white236
noise. Note also that, in practice, the ai,biparameters would be usually obtained through237
identification from the real process to be controlled.238
Model (1) can be used to obtain at time kpredictions of the future values of the process239
output along a prediction horizon defined by the sampling times k+j,j∈[N1, N2]. Those240
predicted values of the process output computed at time kwill be denoted as yk+j|k. Note241
that these predictions will depend on some information that is readily available at time242
k(namely the present and past values of the process output and the past values of the243
input increments) and also they will depend on the present and future values of the input244
increments, which have to be computed by the predictive controller. These present and245
future values of the input increments will be considered along a control horizon defined by246
the sampling times k+jwith j∈[0, Nu−1]2.247
As mentioned at the beginning of this section, one of the most remarkable features of248
MPC is that constraints can be taking into account in the computation of the control signal.249
Thus, here constraints on the values of the input signal, input increments and predicted250
outputs are considered:251
u≤uk+i|k≤u, i = 0,...,Nu−1
∆u≤∆uk+i|k≤∆u, i = 0,...,Nu−1
y≤yk+i|k≤y, i =N1,...,N2
(2)252
2Note that if the control horizon is smaller than the prediction horizon, i.e., Nu< N2, then, the input
increments after the control horizon are assumed to be zero, i.e, ∆uk= 0, k∈[Nu, N2].
10
Note that, being model (1) linear, all these constraints are linear on the input increments,253
so they can be rewritten as:254
Ru≤c(3)255
where Rand care a matrix and a vector of appropiate dimensions and uis the sequence256
of present and future input increments defined as u=∆uk|k,∆uk+1|k,...,∆uk+Nu−1|k(see257
(Camacho and Bordons 2004) for details on how to find Rand c). Only those sequences u258
that satisfy (3) will be considered as admissible by the controller.259
Once the admissible sequences uare characterized, the next step to the formulation of a260
MPC is to provide some means of getting a measure of how good is a sequence uin terms261
of control performance. This can be achieved by means of a quadratic cost function of the262
future set point tracking errors plus a term weighting the input increments is added:263
J(u) =
N2
X
j=N1
(y(k+j|k)−r(k+j))2+λ
Nu−1
X
j=0
∆u(k+j|k)2(4)264
where λ > 0 is the weighting factor for present and future input increments3. Note that265
this term is added to penalize the use of unnecessary arbitrarily large values of the input266
increments as these increments are usually related to economical costs. With this definition267
of J(u) the best control sequence will be that which obtains the smallest tracking errors with268
the smallest control input increments. This sequence will be the one that minimizes the cost269
function J(u).270
With all the previous elements, the optimal control sequence u∗produced by the MPC271
controller is defined as the solution of the following optimization problem:272
u∗= arg min
uJ(u)
s.t. Ru≤c
(5)273
3Note that more complex weighting schemes exist (like using time variable weight factors or weighting
both terms in (4)). We use here the scheme proposed in (Clarke et al. 1987) as practice shows that a similar
performance can be achieved with the added benefit of a simpler tuning procedure (as only one weighting
factor has to be tuned).
11
The solution to this optimization problem is applied using a receding horizon scheme, that274
means that every sampling time problem (5) is solved, and at each sampling time, only the275
first component of u∗is in fact applied to the system, whereas the remaining components276
are discarded. The reason to use such receding horizon scheme is to close the control loop,277
that otherwise would result in an open-loop control scheme. Note that, being the model278
and constraints linear and the cost function quadratic, the optimization problem (5) is a279
Quadratic Program that can be efficiently solved using the current computer hardware.280
There are some different ways to implement MPC algorithms. All of them are discussed281
in depth in (Camacho and Bordons 2004). We have chosen Generalized Predictive Control282
(GPC), which can easily be extended for the distributed case.283
3.2 Consideration of measurable disturbances in the computation of the control284
signal285
Measurable disturbances can be easily included in an MPC scheme like the one presented286
so far in this section. The only modification that has to be done is in the prediction model,287
that now has two deterministic inputs: the manipulated input u(which it is used to control288
the system) and the disturbance v(which has to be measurable). Thus, model (1) will be289
rewritten as:290
yk=a1yk−1+···+anayk−na +b0∆uk−du−1+···+bnb∆uk−du−nb−1+ (6)291
+d0∆vk−dv−1+···+dnd∆vk−dv−nd−1+ek
292
Note that the delay from each input to the output is not necessarily equal, and that the293
measurable disturbance behaves just like an extra input that it is not under our control.294
Besides this modification, the MPC controller formulation remains the same. This way of295
taken into account measurable disturbance is essentially the same as in the classic feedforward296
disturbance compensation techniques (Camacho and Bordons 2004).297
12
3.3 Cooperative Distributed MPC298
The MPC strategy discussed so far involves a number of controllers that operate299
independently without exchanging any information about the optimal sequences computed300
by each one. Thus, each controller operates independtly, having its own data, which are just301
a part of the whole information. However, it is possible to establish a communication link302
between two or more controllers in order to share information and to work in a collaborative303
manner. In this way, the controllers would have more information available, which would304
improve the overall control performance. This observation leads to the development of305
cooperative Distributed MPC (DMPC) strategies (see (Zafra-Cabeza et al. 2011)). The306
algorithm used here, which is discussed in detail in (Maestre et al. 2011), involves only a307
pair of controllers (although it can be extended to consider any number of controllers), and308
it is based on cooperative game theory. The goal is to control a pair of constrained coupled309
linear systems where a communication link is established between controllers. Each controller310
has only a part of the information related to the model and the state of the overall system,311
although they can exchange information about their optimal control sequences. Game theory312
is used to implement a coordination scheme in which both controllers have to cooperate to313
achieve their control goals, even in the case of conflicting goals. The coordination problem314
is reduced to a cooperative game where each agent have to make a choice among three315
possibilities. Only two communication cycles will be required for each choice.316
The proposed distributed MPC algorithm,for a pair of controllers, is the following:317
1. At sample time kEach controller i∈[1,2] reads its controlled variables. Denote the318
optimal sequence computed in the previous sample time as US
i(k).319
2. Each controller i∈[1,2] solves its local MPC problem minimizing its own cost320
function Jiand considering the effect of the control actions of the other controller as321
a measurable disturbance. It is assumed that the other controller will keep applying322
the optimal control sequence computed in the previous sample time (that is, US
j(k))4.323
4Given i∈[1,2] and j∈[1,2], when i=1 then j=2 and vice versa. So, in general terms, we use the
13
Denote the optimal control sequence as U∗
i(k).324
3. Each controller i∈[1,2], assuming that it applies the optimal sequence previously325
obtained in step 2, computes the control sequence for neighbour jthat gets the326
smallest value of its own cost function Ji. That is, each controller computes the327
neighbour input that it is more beneficial for its own performance. Denote this328
sequence as Uw
j(k)5. Note that each controller assumes that its neighbour behaves in329
an altruist way, thus it will ”agree” to use Uw
j(k) instead of U∗
j(k).330
4. Both controllers communicate the sequences computed in the previous steps.331
Controller 1 sends to controller 2 the sequences U∗
1(k) and Uw
2(k), whereas controller332
2 sends to controller 1 U∗
2(k) and Uw
1(k). Thus, at the end of this step both controllers333
know all the sequences that have been computed so far.334
5. Each controller ievaluates its own cost function for all the sequences it could choose.
That is, controller 1 computes the set:
J1=J1(US
1(k)), J1(U∗
1(k)), J1(Uw
1(k))
and controller 2 computes the set:335
J2=J2(US
2(k)), J2(U∗
2(k)), J2(Uw
2(k)).
6. Both controllers communicate the values obtained in the previous step. That is,336
controller 1 sends the set J1to controller 2, whereas controller 2 sends the set J2to337
controller 1.338
7. Both controllers consider the 9 possible pairs (J1, J2) of optimal costs in J=J1×J2
339
and pick the one that gives the minimum sum J=J1+J2. Note that this pair has340
a pair of associated optimal sequences, which will be denoted as Ud
1(k) and Ud
2(k)341
respectively.342
subindex iwhen referring to the controller we are dealing with and jwhen referring to the other one
5Note that controller 1 computes Uw
2(k) and controller 2 computes Uw
1(k).
14
8. Each controller iapply to the controlled system the first component of Ud
i(k) and the343
whole procedure is repeated again at the next sampling time.344
To summarize the procedure, the goal is to construct a 3x3 matrix. Each row contains a345
possible optimal sequence which can be chosen by controller 1, and each column contains a346
possible optimal sequence which can be chosen by controller 2. Cells contain the sum of cost347
functions for each of the possible optimal sequence combinations. Thus, there are 9 options,348
and the combination that minimize the cost function sum will be chosen.349
4 MODELING AND CONTROL STRUCTURE350
Models that involve water movement are generally obtained making use of simplifications351
of the Navier-Stokes equations, because of the complexity in dealing directly with them. For352
irrigation canals, one of the most accepted and used model in simulations is the system353
given by the Saint-Venant Equations, because of its capacity to represent the dynamic354
characteristics of real interest. However, this system is a nonlinear partial differential355
equation system, which has analytical solution only in very special cases, forcing the356
employment of numerical methods to solve it properly. Since the early 60s researchers357
have devoted important efforts to developing efficient solutions methods for those equations.358
Most numerical methods can be included in the finite difference or finite element categories.359
As a model for computational simulation it is very accurate, but as model for control, it360
is clearly not appropriate because of its complexity. Linearizations or simplifications of the361
Saint-Venant equations are used for control purposes.362
Making use of Saint Venant equations, a reach can be modelled by two partial differential363
equations representing a mass balance (continuity equation) and a momentum balance.364
∂q(t,z)
∂z +∂s(t,z)
∂t = 0
1
g
∂
∂t q(t,z)
s(t,z)+1
2g
∂
∂z q2(t,z)
s2(t,z)+∂h(t,z)
∂z +If(t, z)−I0(z) = 0
(7)365
The variables represent the following quantities:366
15
•zis the spatial variable which increases along the flow main direction;367
•q(t, z) is the river flow (or discharge) at time tand space coordinate z;368
•s(t, z) is the wetted surface;369
•h(t, z) is the water level w.r.t. the river bed;370
•gis the gravitational acceleration;371
•If(t, z) is the friction slope;372
•I0(z) is the river bed slope.373
Different approaches have been used to model the friction slope such as the Gauckler-374
Manning-Strickler equation:375
Sf(t, z) = q(t, z)2(p(t, z))4/3
k2
str (s(t, z))10/3(8)376
where p(z) is the wet section perimeter and kstr is the Gauckler-Manning-Strickler coefficient.377
The Gauckler-Manning-Strickler coefficient changes accordingly to the kind of river bed378
surface.379
In order to have a realistic simulation of the irrigation canal of La Pedrera, Saint-Venant380
equations, the well known SIC (Simulation of Irrigation Canals) software has been used.381
SIC provides a mathematical model which can simulate the hydraulic behaviour of most of382
the irrigation canals or rivers, under steady and unsteady flow conditions. Steady flow and383
unsteady flow computations can be performed on any type of hydraulic networks (linear,384
looped or branched). Any reach can be composed of a minor, a medium and a major385
bed. Storage pools can also be modelled. The SIC model is an efficient tool allowing386
canal managers, engineers and researchers to quickly simulate a large number of hydraulic387
conditions at the design or management level. Moreover, it can be interfaced (by means388
of its regulation module) to different mathematical software like Matlab and Scilab, a very389
convenient feature for research purposes.390
The SIC model of the irrigation canal of La Pedrera comprises a set of data which are391
16
obtained from a topographic source or planes. From the software processing point of view,392
hydraulic canals are usually described on the basis of a set of cross-sections. Each section393
has some associated information such as the section shape (circular, square, trapezoidal), the394
coordinates of the significant points of the section (usually vertices), the position measured395
from the origin of the canal, the Manning coefficient or water leakage losses. In addition,396
points where either water injection or extraction exists are indicated by using nodes. As397
indicated above, these points are called off-takes. A reach is a canal portion situated between398
a pair of nodes. SIC has a editing tool (see figure 3)which allows to characterize the canal399
by introducing the data related to each cross-section(see figure 4).400
The Saint-Venant model of the irrigation canal is a very realistic one and it will be used401
as a test bed for the control structure proposed in this paper. But for control purposes a402
less complex model is usually needed. Moreover, the control structure proposed here relies403
on model predictive controllers, which use a linear prediction model to compute the gate404
openings that are necessary to attain the target flows at each gate. Thus a Multi Input405
Multi Output (MIMO) model of five inputs and five outputs has been identified using the406
well known Least Squares method (this method and others used in identification processes407
are discussed in depth in (Cellier and Greifeneder 1991; Johansson 1993; Landau and Landau408
1990)) around the operating point shown in table 1. In the model, the five inputs are the409
flow at the head of the main canal (u1), and the position of the main gates g2 to g5 (u2 to u5410
in table 1). On the other hand, the five outputs that are to be controlled are water level at411
each one of the reaches in Canal de Cartagena (y1, y2, y4 and y5) and the flow at the head412
of Canal de la Pedrera (y3). The linear models for each input-output pair are first or second413
order models plus a transport delay (system modelling and concretely first and second order414
approaches are deeply discussed in (Ogata 2010)) caused by the distance between reaches.415
Note that being the model a MIMO one, there can be couplings between different pairs of416
input-outputs, thus a given output can be affected not only by its paired input but also by417
any other input in the model. These couplings or interactions can be weaker or more intense,418
17
and in this latter case they cannot be neglected when designing the control structure.419
Once the hydraulic canal has been modelled as a MIMO plant, the following step is420
to design an optimal control structure. Firstly an appropriate input-output pairing must421
be chosen. During this research several pairings were tested. The chosen input-output422
pairing is detailed in table 2. In this table, an input-output pair is detailed in every row and423
information about the involved magnitudes and the measurement points is shown. Two data424
are necessary to locate these points: the branch where they are situated and the kilometric425
distance to the end of the branch. Figure 5 gives an idea of the location of both inputs and426
outputs and distances between them. Some control structures will be explained below.427
Figure 6 shows a totally decentralized control structure based on predictive control.428
Five GPC controllers govern each input-output pairing aforementioned. GPC1 tracks a429
downstream water level reference by regulating the incoming water flow at the canal head430
gate. GPC3 monitors the downstream flow through its corresponding gate by manipulating431
its degree of aperture. Finally, GPC2, GPC4 and GPC5 track a water level reference using432
the degree of gate aperture as a manipulated variable.433
An hydraulic canal is such a coupled system that every control command sent to the434
plant in order to obtain a desirable behaviour at one of the outputs significantly affects435
the rest of them. This may be taken into account at every sample time when computing436
the following control action. Every single controller can consider control actions computed437
by its neighbours as a measurable disturbance. This disturbance is easily included in the438
control action calculation using a feedforward compensation as explained in section 3.2.439
Figure 7 shows how this theory is applied to this research. Each GPC will have two kinds of440
inputs: on the one hand the measurement of its output and the corresponding reference (red441
arrows), and on the other hand the measured disturbances (green arrows). Disturbances442
could be considered both upstream and downstream, but in this case only downstream443
disturbances were taken into account, in order to simplify the problem. Moreover, the444
implementation of this feedforward compensation will be done in a sequential manner. That445
18
is, the control actions to be applied at each gate are computed sequentially, starting from446
the most downstream gate and proceeding upwards to the first (upstream) gate. Then, when447
computing the optimal aperture of a given gate, the aperture of the nearest downstream gate448
(which was computed in the previous step of this sequence) is considered as a measurable449
disturbance. This feedforward scheme is later referred in the text as a sequential feedforward.450
In section 5 results will show a significant improvement of the canal control performance by451
considering downstream couplings as disturbances when computing control actions.452
Finally, two controllers can cooperate, as explained in section 3.3, to obtain an optimal453
control sequence, by using an algorithm based on game theory. To implement this algorithm,454
a communication channel between the controllers (or agents) is necessary. This is a455
distributed control schema. Starting with the structure presented in figure 7, the distributed456
control algorithm is implemented in controllers GPC1 and GPC2. A communication link457
is established between them and each controller takes into account the control actions458
performed by the other one for calculating its own control actions. The neighbour control459
actions will be considered as measurable disturbances.460
5 RESULTS461
Different scenarios and control approaches have been tested in simulation. The canal462
benchmark has been modeled in SIC. The operation point has been established with 12m3/s463
at the head of the canal and gates positions of 1mfor the first gate and 0.5mfor all the other464
gates. Table 1 shows lavels and flows at significant positions of the canal for the operation465
point. Three predictive approaches have been tested in the different scenarios:466
•Control Schema 1: Downstream local MPC in each one of the gates.467
•Control Schema 2: Local MPC with sequential feedforward.468
•Control Schema 3: Distributed MPC in the two first gates and local MPC in the469
others with sequential feedforward.470
Sampling time has been fixed to 6 minutes. The duration of the simulation tests is four471
19
days. A comparison among the three approaches had been performed using the following472
control performance and economic indexes:473
•Performance Index (PJ). The sum of the cost functions Jin equation 4 of each one474
of the controller has been used as control performance index.475
•Economic Index (EI ) considers lost water and unsatisfied water demand. In all the476
test cases, the demand of lateral off-takes has been satisfied properly, but also a477
flow demand at the end of each one of the canal branches has been considered. The478
flow demand has been considered constant along the simulation time, and different479
economic penalization for flows over the demand (lost water) and under demand480
(unsatisfied demand) are applied. Figure 8 shows the penalization that has been used481
in the following test cases.482
qLWi(t) =
qoi(t)−qd
oi if qoi(t)> qd
oi i= 1,2
0 ifqoi(t)≤qd
oi
qUDi(t) =
qd
oi −qoi(t) if qoi(t)< qd
oi i= 1,2
0 ifqoi(t)≥qd
oi
LWi=CLW (Rtf
0qLWidt)
UDi=CUD (Rtf
0qUDidt)
EI =LW1+LW2+U D1+U D2
(9)483
Where:484
•EI : Economic index485
•qoi: Flow at the tail of branch i(i= 1, La Pedrera branch and i= 2, Campo de486
20
Cartagena branch as shown in Figure 1).487
•qod
i: Flow demand at the tail of branch i.488
•LWi: Lost water (m3) in branch i.489
•UDi: Unsatisfied demand (m3) in branch i.490
•CLW : Cost of lost water (0.2 Euros/m3).491
•CLW : Cost of water unsatisfied demand (0.5 Euros/m3).492
These two indexes Jand EI are presented for each one of the control schemas and493
different tests in the following subsections.494
5.1 Test 1: Set point changes495
To compare the controllers under similar operation conditions a experiment is defined496
where a set of reference changes in the levels of reaches (ref1 and ref2) and in the flow (ref3).497
Reference 1 is increased 0.2mat the beginning of the second day, reference 2 is increased also498
0.2mat the beginning of the third day and finally the flow reference 3 is increased 0.5m3/s499
at the beginning of the third day. Figure 9 shows the controlled variables (level in ref1, ref2,500
ref4 and ref5 and flow in ref3) using the three test controllers (green dashed line for control501
schema 1, blue dashed-dotted for schema 2 and red for schema 3). The best performance is502
obtained using distributed control in reaches 1 and 2. Level zero in the figures corresponds503
to the operating point value (See Table 1). It can be seen that in set-point change in reach 1504
(Figure 9a), damping appearing in control schemes 1 and 2 is considerably reduced. Notice505
that the disturbance of the setpoint change of ref1 in the behavior of the level of reach 2 is506
dramatically diminished (Figure 9b).507
Table 3 shows the global control performance index considered as the sum of the local508
cost function for each of one of the controllers as defined in equation (4). The last column509
shows that the economic index using the control schema 3 is a third of the economic index510
of control schema 1.511
21
5.2 Test 2: Off-take flow changes512
This second test is devoted to analyze the behavior of the tested controllers when changes513
in off-take flow are produced. Off-takes are considered as perturbations since the farmers514
decided at any time the flow they need for their local irrigation (nevertheless, they usually515
follow a previous established irrigation plan). For this reason the off-take prediction is516
considered in the MPC control.517
In the presented test, a flow of 1m3/s is extracted from the canal in points off1 and off4518
(See Figure 1) from the beginning of the second day to the end of the simulation period.519
Figures 10 shows the controlled variables of the controllers and 11 the manipulated520
variables (gate position in g2, g3, ga4 and g5 and flow at the head of the canal). Notice that521
in gate 2 only the distributed controller is able to maintain the set point level, and with local522
controllers (schema 1), this gate reaches the maximum opening limit. The reason is the lack523
of communication between controllers 1 and 2. Controller 1 takes the decision independently524
of the needs of controller 2 in schema 1 (and even in schema 2) but in distributed controller525
both controller act in a coordinated way reaching a satisfactory performance.526
Table 4 presents the performance index for test 2. Again, best results are obtained with527
the distributed controller in reaches 1 and 2.528
5.3 Test 3: Off-take flow and references changes529
This last test is a more complex situation with several simultaneous level and flow530
references and off-take flow changes. This test will show the coupling of the different531
subsystems and the effect of upstream perturbations at the downstream part of the canal.532
The following reference changes and off-take flow modification have been considered:533
•Change of 0.4m. in the level reference of gate 1 (Reference 1) at the beginning of the534
second day535
•Increase 0.4m. in reference 2 at the beginning of the third day536
•Change of 5m3/s in reference 3 at the beginning of day 3537
22
•Change of 0.1m. in the level reference of gate 4 at the beginning of the forth day538
•A flow of 1m3/s is extracted from the canal in points off1 and off4 since the beginning539
of the second day540
Figures 12 show the controlled variables of the controllers. Again, the best behavior is541
obtained using schema 3 and the worst performance with schema 1.542
Figure 12d shows the evolution of the level at the end of reach 4. Notice the effect of543
perturbations during the second and third day. Most of them are due to changes produced544
upstream. The behavior is quite oscillatory, but the amplitude of the oscillations is quite545
small (around 2cm.).546
Table 5 presents economic and performance indicator of the three approaches. Notice547
that an important decrease of both indexes is obtained when control schema 3 is applied.548
6 CONCLUSIONS AND FUTURE WORKS549
In this paper a distributed predictive controller has been proposed to control irrigation550
canals. An accurate model of a real irrigation canal in Spain has been used as a551
test bed for the controller. The model has been developed using the well known SIC552
software. This software uses the Saint-Venant equations to model the dynamics of the553
canal with better accuracy than other methods. The SIC software has been interfaced to554
the predictive controller which has been developed using Matlab. The results show that555
the proposed distributed control algorithm achieves better control performance than a local556
based controller scheme without information exchange (which is by far the most usual control557
scheme in automated irrigation canals). The improvements in control performance will lead558
to a better and more efficient management of irrigation canals that ultimately results in559
money and resource savings.560
Future work will be focused on the development of more complex algorithms and in the561
validation of the controller in the actual irrigation canal. One interesting feature of the562
control of irrigation canals is that the dynamics are relatively slow so that complex control563
23
algorithms can be used even when the available control hardware has moderate computing564
capabilities. Thus, the use of nonlinear prediction models and the consideration of uncertain565
or non measurable disturbances are possibilities that can be explored. On the other hand,566
the validation of the control scheme in the actual irrigation canal imply the implementation567
of the control algorithms in preexistent hardware with the least possible addition of new568
control hardware (that for budget and reliability reasons will be based on microcontrollers).569
24
APPENDIX I. REFERENCES570
Begovich, O. (2007). “Predictive control with constraints of a multi-pool irrigation canal571
prototype.” Latin American Applied Research, 37.572
Begovich, O., Aldana, C., Ruiz, V., Georges, D., and Besan¸con, G. (2004). “Real-time573
predictive control with constraints of a multi-pool open irrigation canal.” Proceedings of574
the XI Congreso Latinoamericano de Control Automatico, CLCA2004, La Habana, Cuba.575
Camacho, E. F. and Bordons, C. (2004). Model Predictive Control. Springer, second edition.576
Cellier, F. E. and Greifeneder, J. (1991). Continuous System Modeling. Springer-Verlag577
(May).578
Clarke, D., Mohtadi, C., and Tuffs, P. (1987). “Generalized predictive control–Part I. The579
basic algorithm.” Automatica, 23(2), 137 – 148.580
G´omez, M., Rodellar, J., and Mantec´on, J. A. (2002). “Predictive control method for581
decentralized operation of irrigation canals.” Applied Mathematical Modelling, 26(11), 1039582
– 1056.583
Johansson, R. (1993). System Modelling and Identification. Prentice H.584
Landau, I. D. and Landau, Y. D. (1990). System Identification and Control Design. Prentice585
Hall.586
Lemos, J., Machado, F., Nogueira, N., Rato, L., and Rijo, M. (2009). “Adaptive and non-587
adaptive model predictive control of an irrigation channel.” Networks and Heterogeneous588
Media, 4(2), 303 – 324.589
Li, . and Cantoni, M. (2008). “Distributed controller design for open water channels.”590
Proceedings of the 17th IFAC World Congress, Seoul, Korea.591
Li, Y. and De Schutter, B. (2010). “Performance analysis of irrigation channels with592
distributed control.” Control Applications (CCA), 2010 IEEE International Conference593
on, 2148 –2153.594
Maestre, J., de la Pe˜na, D. M., and Camacho, E. F. (2011). “Distributed model predictive595
control based on a cooperative game.” Optimal Control Applications and Methods, 32,596
25
153–176.597
Malaterre, P. and Rodellar, J. (1997). “Multivariable predictive control of irrigation canals.598
design and evaluation on a 2-pool model.” Proceedings of the International workshop on599
regulation of irrigation canals, Marrakech, Morocco.600
Malaterre, P., Rogers, D., and Schuurmans, J. (1998). “Classification of canal control601
algorithms.” ASCE Journal of Irrigation and Drainage Engineering, 124.602
Negenborn, R., Overloop, P., Keviczky, T., and Shutter, B. D. (2009). “Distributed model603
predictive control of irrigation canals.” Networks and Heterogeneous Media, 4(2), 359 –604
380.605
Ogata, K. (2010). Modern Control Engineering. 5th edition.606
Ooi, S. and Weyer, E. (2008). “Control design for an irrigation channel from physical data.”607
Control Engineering Practice, 16, 1132–1150.608
Rodellar, J., G´omez, M., and Bonet, L. (1993). “Control method for on-demand operation609
of open-channel flow.” Journal of Irrigation and Drainage Engineering, 119(2), 225–241.610
Sawadogo, S., Faye, R., Benhammou, A., and Akouz, K. (2000). “Decentralized adaptive611
predictive control of multi-reach irrigation canal.” Systems, Man, and Cybernetics, 2000612
IEEE International Conference on, Vol. 5, 3437 –3442.613
Sawadogo, S., Faye, R., Malaterre, P., and Mora-Camino, F. (1998). “Decentralized614
predictive controller for delivery canals.” Systems, Man, and Cybernetics, 1998. 1998615
IEEE International Conference on, Vol. 4, 3880 –3884 (oct).616
Schuurmans, J., Clemmens, A., Dijkstra, S., Hof, A., and Brouwer, R. (1999). “Modeling of617
irrigation and drainage canals for controller design.” Journal of Irrigation and Drainage618
Engineering, 125(6), 338–344.619
Sepulveda, C. (2007). “Instrumentation, model identification and control of an experimental620
irrigation canal.” Ph.D. thesis, Universidad Polit´ecnica de Catalu˜na, Universidad621
Polit´ecnica de Catalu˜na.622
Silva, P., Boto, M. A., Figueiredo, J., and Rijo, M. (2007). “Model predictive control of an623
26
experimental water canal.” Proceedings of the European Control Conference, Kos, Greece,624
2977–2984.625
Tricaud, C. and Chen, Y. Q. (2007). “Cooperative control of water volumes of parallel ponds626
attached to an open channel based on information consensus with minimum diversion627
water loss.” Mechatronics and Automation, 2007. ICMA 2007. International Conference628
on, 3283 –3288.629
van Overloop, P. (2006). “Model predictive control on open water systems.” Ph.D. thesis,630
TU Delft, TU Delft.631
Wahlin, B. T. (2004). “Performance of model predictive control on asce test canal 1.” Journal632
of Irrigation and Drainage Engineering, 130(3), 227–238.633
Zafra-Cabeza, A., Maestre, J., Ridao, M. A., Camacho, E. F., and S´anchez, L. (2011). “A634
hierarchical distributed model predictive control approach to irrigation canals: A risk635
mitigation perspective.” Journal of Process Control, 21(5), 787 – 799.636
27
List of Tables637
1 Operating point used for prediction model identification. . . . . . . . . . . . 29638
2 Canal control: input-output pairing . . . . . . . . . . . . . . . . . . . . . . . 30639
3 Table of the performance indexes of each schema for Test 1 in a four-day640
simulation. Control performance in second column and economic performance641
inthirdcolumn .................................. 31642
4 Table of the performance indexes of each schema for Test 2 in a four-day643
simulation. Control performance in second column and economic performance644
inthirdcolumn .................................. 32645
5 Table of the performance indexes of each schema for Test 3. Control646
performance in second column and economic performance in third column . . 33647
28
u1 Flow (m3·s−1) 12 y1 Water level (m) 82.951
u2 Gate opening (m) 1 y2 Water level (m) 82.073
u3 Gate opening (m) 0.5 y3 Flow (m3·s−1) 5.41
u4 Gate opening (m) 0.5 y4 Water level (m) 81.269
u5 Gate opening (m) 0.5 y5 Water level (m) 80.643
TABLE 1: Operating point used for prediction model identification.
29
u1 Flow (m3·s−1) Head Gate y1 Water level (m) Branch 1/ RS 4.275
u2 Gate opening (m) Branch 1/ RS 4.27 y2 Water level (m) Branch 1/ RS 0
u3 Gate opening (m) Branch 1/ RS 6.672 y3 Flow (m3·s−1) Branch 1/ RS 6.669
u4 Gate opening (m) Branch 2/ RS 12.964 y4 Water level (m) Branch 2/ RS 6.972
u5 Gate opening (m) Branch 2/ RS 6.969 y5 Water level (m) Branch 2/ RS 3.021
TABLE 2: Canal control: input-output pairing
30
Performance PJEI (Euros)
Control Schema 1 28.45 905
Control Schema 2 18.79 662
Control Schema 3 5.30 402
TABLE 3: Table of the performance indexes of each schema for Test 1 in a four-day
simulation. Control performance in second column and economic performance in third
column
31
Performance PJEI (Euros)
Control Schema 1 125.23 1674
Control Schema 2 52.20 1101
Control Schema 3 16.03 816
TABLE 4: Table of the performance indexes of each schema for Test 2 in a four-day
simulation. Control performance in second column and economic performance in third
column
32
Performance PJEI (Euros)
Control Schema 1 157.49 12845
Control Schema 2 116.43 7767
Control Schema 3 68.92 4660
TABLE 5: Table of the performance indexes of each schema for Test 3. Control performance
in second column and economic performance in third column
33
List of Figures648
1 Section of the Postrasvase Tajo-Segura. . . . . . . . . . . . . . . . . . . . . . 35649
2 Prediction in Model Predictive Control . . . . . . . . . . . . . . . . . . . . . 36650
3 View of the SIC tool for editing the canal hydraulic model . . . . . . . . . . 37651
4 Introducing data related to a cross-section . . . . . . . . . . . . . . . . . . . 38652
5 Canal control: location for inputs and outputs . . . . . . . . . . . . . . . . . 39653
6 Canal control structure based on decentralized GPC predictive controllers . . 40654
7 Canal control structure based on decentralized GPC predictive controllers.655
Consideration of measurable disturbances . . . . . . . . . . . . . . . . . . . . 41656
8 Economic index computation . . . . . . . . . . . . . . . . . . . . . . . . . . 42657
9 Test 1: Level at ref1 to ref5 ((a) to (e) figures) position with control schema658
1 (dashed green), schema 2 (dotted-dashed blue) and schema 3 (solid red) . . 43659
10 Test 2: Controlled variables at ref1 to ref5 ((a) to (e) figures) position with660
control schema 1 (dashed green), schema 2 (dotted-dashed blue) and schema661
3(solidred).................................... 44662
11 Test 2: Manipulated variables. Flow at the head a) and gate positions at663
points ref2 to ref5 ((b) to (e) figures) with control schema 1 (dashed green),664
schema 2 (dotted-dashed blue) and schema 3 (solid red) . . . . . . . . . . . . 45665
12 Test 3: Controlled variables at ref1 to ref5 ((a) to (e) figures) position with666
control schema 1 (dashed green), schema 2 (dotted-dashed blue) and schema667
3(solidred).................................... 46668
34
FIG. 1: Section of the Postrasvase Tajo-Segura.
35
k-1 k k+1
….
k+N1k+j k+N2
Prediction Horizon
uk|k
uk+j|k
yk+j|k
yk
k+Nu-1
….
Control Horizon
uk-1
… …
FIG. 2: Prediction in Model Predictive Control
36
FIG. 3: View of the SIC tool for editing the canal hydraulic model
37
FIG. 4: Introducing data related to a cross-section
38
LJϮ
ƵϮ
LJϭ
LJϯ
Ƶ ϯ
Ƶϰ
LJϰ LJϱ
Ƶϱ
Z ^Z sK /Z
Ƶϭ
LJϭ ͗ůĞǀĞůƐĞŶƐŽƌ
LJϮ ͗ůĞǀĞůƐĞŶƐŽƌ
LJϯ͗ĐĂƵĚĂůŝŵĞ ƚĞƌ
LJϰ ͗ůĞǀĞůƐĞŶƐŽƌ
LJϱ ͗ůĞǀĞůƐĞŶƐŽƌ
FIG. 5: Canal control: location for inputs and outputs
39
' W Ϯ
' W ϭ
'W ϯ
' W ϰ ' W ϱ
ǁ Ϯ
LJϮ
ǁ ϭ
ƵϮ
LJϭ
ǁϯ
LJϯ
Ƶϯ
ǁ ϰ
Ƶϰ LJ ϰ
ǁ ϱ
LJϱ
Ƶϱ
Z ^ Zs K/ Z
Ƶϭ
FIG. 6: Canal control structure based on decentralized GPC predictive controllers
40
'W Ϯ
'W ϭ
'W ϯ
'W ϰ ' Wϱ
ǁϮ
LJϮ
Ƶϭ
ǁϭ
ƵϮ
LJϭ
ǁϯ
LJ ϯ
Ƶ ϯ
ǁϰ
Ƶϰ LJϰ
ǁϱ
LJϱ
Ƶϱ
Z ^ZsK/Z
ǀϰ
ǀϮ
ǀϮΖ
ǀϭ
FIG. 7: Canal control structure based on decentralized GPC predictive controllers.
Consideration of measurable disturbances
41
FIG. 8: Economic index computation
42
0 200 400 600 800 1000
−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
Level (m.)
Time (sampling units)
(a)
0 200 400 600 800 1000
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
Level (m.)
Time (sampling units)
(b)
0 200 400 600 800 1000
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
Time (sampling units)
Flow (m3/s.)
(c)
0 200 400 600 800 1000
−0.05
0
0.05
0.1
Time (sampling units)
Level (m.)
(d)
0 200 400 600 800 1000
−0.05
0
0.05
0.1
Time (sampling units)
Level (m.)
(e)
FIG. 9: Test 1: Level at ref1 to ref5 ((a) to (e) figures) position with control schema 1
(dashed green), schema 2 (dotted-dashed blue) and schema 3 (solid red)
43
0 200 400 600 800 1000
−0.1
−0.05
0
0.05
0.1
0.15
0.2
Time (sampling units)
Level (m.)
(a)
0 200 400 600 800 1000
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
Level (m.)
Time (sampling units)
(b)
0 200 400 600 800 1000
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
Time (sampling units)
Flow (m3/s)
(c)
0 200 400 600 800 1000
−0.1
−0.05
0
0.05
0.1
0.15
0.2
Time (sampling units)
Level (m.)
(d)
0 200 400 600 800 1000
−0.1
−0.05
0
0.05
0.1
0.15
0.2
Time (sampling time)
Level (m.)
(e)
FIG. 10: Test 2: Controlled variables at ref1 to ref5 ((a) to (e) figures) position with control
schema 1 (dashed green), schema 2 (dotted-dashed blue) and schema 3 (solid red)
44
0 200 400 600 800 1000
−0.5
0
0.5
1
1.5
2
2.5
3
3.5
Time (sampling units)
Flow (m3/s.)
(a)
0 200 400 600 800 1000
−0.5
0
0.5
1
1.5
2
Time (sampling units)
Gate opening (m.)
(b)
0 200 400 600 800 1000
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
Time (sampling units)
Gate opening (m.)
(c)
0 200 400 600 800 1000
−0.5
0
0.5
1
1.5
2
2.5
Time (sampling units)
Gate opening (m.)
(d)
0 200 400 600 800 1000
−0.1
−0.05
0
0.05
0.1
0.15
Time (sampling units)
Gate opening (m.)
(e)
FIG. 11: Test 2: Manipulated variables. Flow at the head a) and gate positions at points ref2
to ref5 ((b) to (e) figures) with control schema 1 (dashed green), schema 2 (dotted-dashed
blue) and schema 3 (solid red)
45
0 200 400 600 800 1000
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
Time (sampling units)
Level (m.)
(a)
0 200 400 600 800 1000
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
Time (sampling units)
Level (m.)
(b)
0 200 400 600 800 1000
0
1
2
3
4
5
6
7
Flow (m3/h)
(c)
0 200 400 600 800 1000
−0.05
0
0.05
0.1
Level (m.)
(d)
0 200 400 600 800 1000
−0.05
0
0.05
0.1
Time (sampling units)
Level (m.)
(e)
FIG. 12: Test 3: Controlled variables at ref1 to ref5 ((a) to (e) figures) position with control
schema 1 (dashed green), schema 2 (dotted-dashed blue) and schema 3 (solid red)
46
