System Identification of Khaira DistributorySaad Abdul Aleem, Asad Khalid, Hassan Zaheer, Sarmad Jabbar, Abubakr Muhammad

Department of Electrical Engineering, Lahore University of Management Sciences, Lahore, Pakistan 54792Email: abubakr@lums.edu.pk

Abstract—Water has become a scarce resource for agro-basedeconomy of Pakistan. In this work, we derive channel modelsfor regulation in irrigation canals using system identificationtechniques. System identification models are computationallyinexpensive and accurate for simulating water profiles. We takeinspiration from recent developments in control of large-scaleirrigation networks and perform experimental identification ofdistributory irrigation channels in Pakistan. The channel modelsare estimated and validated on operational data from KhairaDistributory off BRB Link Canal, Lahore. Our results showthat system identification techniques capture the efficacy ofcanal models appropriately and will play a vital role in buildingaccurate models for controlled irrigation canals in Pakistan1.

Keywords: System Identification; Open Channel Flows;Controlled Irrigation Canals; Parameter Estimation

I. INTRODUCTION

Irrigation networks play a fundamental part in the agri-culture system of various countries. According to [1], manydeveloping countries, like Pakistan, are suffering from waterscarcity due to mismanagement and inefficiencies of theirexisting irrigation networks. This is why, control of irrigationcanals continues to attract great interest from the researchcommunity [2], [3], [4], [5]. The necessity of efficient controlhas transformed the canal irrigation methodology from asupply-driven one to one which is demand-driven. This is whyan efficient and well managed irrigation system is of immenseimportance for Pakistan, where current regulation follows apre-defined schedule and infrastructure is operated manually.

For control design, channel models are required whichcapture physical dynamics of the irrigation system. Thesemodels can be derived either from physical modelling ofwater flow in open channels or from system identificationexperiments. In literature, St. Venant’s equations have beentraditionally used to describe open channel flows. Theseequations develop a mathematical model based on physicalprinciples, involving mass and energy balances [6]. Simu-lations from St. Venant’s equations agree considerably withreal data from open channels [7], allowing them to serve asa starting point for simulating flow and water level profiles(called hydrographs) in irrigation canals [3], [8], [9]. However,St. Venant’s equations suffer from modelling complexity and

1This work has been carried out at the Laboratory of Cyber PhysicalNetworks & Systems (CYPHYNETS) at LUMS. It has been co-sponsored by apartnership with International Water Management Institute (IWMI), a NationalGrassroots Research Initiative student grant by ICT R&D Fund, Pakistan;and via the logistical and technical support of PMIU, Punjab IrrigationDepartment, Pakistan.

lack parsimony; thus cannot be directly used for efficientcontrol design. Additionally, in the context of control and pre-diction, St. Venant’s equations are computationally intensiveand, sometimes, infeasible to implement [10]. For efficientcontrol design, we require relatively simple models, withoutlosing much of the system dynamics.

These constraints have led to the development of SystemIdentification for irrigation systems. These models are obtaineddirectly from the observed data and have been widely usedfor control of irrigation networks because of their low com-putational overhead (see e.g. [11]). In the presence of repre-sentative data, system identification processes can effectivelycapture the relevant dynamics of irrigation channels [10], [12].Results of system identification for irrigation canals in [13]and [14] confirm that these models are parsimonious, havelow computational overhead and are fairly accurate for controldesign.

This paper attempts to address system identification aspectswhich play a crucial role in developing a controlled irrigationnetwork. We take inspiration from existing literature on iden-tifying system parameters and perform system identificationof irrigation channels in Pakistan. We performed extensiveexperimentation on Khaira Distributory, which stems fromBRB Link Canal, Lahore, to generate a representative data ofhydrographs. In this paper, we present our complete systemidentification procedure, from experimental design to modelvalidation, and we hope that our findings will help in buildingaccurate models for controlled irrigation canals in Pakistan.The work reported in this paper runs in parallel to the effortsat CYPHYNETS Lab in Lahore University of ManagementSciences, to build a scalable hydrometry infrastructure for theIndus basin canal network [9], [15], [16]. In particular, the datarecordings have been obtained using smart sensors developedby the authors’ group.

The remainder of the paper is organized as follows. InSection II, we present theoretical background of St. Venant’sequations and System Identification procedure. In Section III,we describe canal gates to model flow through irrigationchannels and discuss Step Response of a first-order systemas an illustrative example. In Section IV, we present ourexperimental results in identifying lumped parametric modelsof irrigation canals. Additionally, we outline the method inwhich a relevant model structure is developed, taking intoaccount prior physical information about the system dynamicsand gate models. The performance of our estimated model iscompared to the actual response of the physical system, in

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Section V, where we use independent set of input signals forModel Validation. We conclude in Section VI.

II. THEORETICAL BACKGROUND

A. Physical Modelling: St. Venant’s Equations

In constructing physical models of an open channel, St.Venant’s equations are used as a starting point for simulatingflow and level profiles [3], [8]. These equations developmathematical models based on physical principles, involvingmass and energy balances [6], and are given by:

∂𝐴

∂𝑡+

∂𝑄

∂𝑥= 0,

∂𝑄

∂𝑡+(𝑔𝐴𝐵

− 𝑄2

𝐴2

)∂𝐴∂𝑥

+2𝑄

𝐴

∂𝑄

∂𝑥+ 𝑔𝐴𝑆𝑓 − 𝑔𝐴𝑆𝑜 = 0,

where 𝑥 is the distance coordinate; 𝑡 is time; 𝐴 is the cross-sectional area of the channel; 𝐵 is the width of water surface;𝑄 is the flow (discharge); 𝑔 is the gravitational constant (takenas 9.81𝑚/𝑠2); 𝑆𝑓 and 𝑆𝑜 are frictional and bed slope of thechannel respectively. The frictional slope is modeled with theclassical Manning formula i.e. 𝑆𝑓 = 𝑄2𝜂2

𝐴2𝑅43

where, 𝜂 is theManning coefficient, 𝑅 is the hydraulic radius, defined by 𝑅 =𝐴𝑃 with 𝑃 as the wetted perimeter [6].

St. Venant’s equations model the channel flows accurately,if channel geometry and operating conditions are known, andhave been used extensively in hydraulic and civil engineering[4]. However, St. Venant’s equations suffer from modellingcomplexity. Additionally, these equations are computationallyintensive and are infeasible to implement in the context ofcontrol design [10].

B. System Identification

System Identification is a data based modelling technique,used to generate a viable transfer function between informativeinput and output data. Methods in system identification providevalidity for only certain input signals and around certain oper-ating points [17]. In order to use these techniques effectively,we must have informative and representative input/output data.As mentioned in the previous section, St. Venant equationsare hyperbolic partial differential equations, and hence, theyare difficult to use directly for control design. On the otherhand, models developed through system identification areeasier to use for control design [3] because of their lowcomputational demands [13]. Similarly, it has been shown,in [10] and [12], that models developed through the systemidentification process can capture relevant dynamics of theirrigation channel as they build on raw measurements from theirrigation canal and give accurate description of the underlyingphysical process. For our purpose, we have used grey boxmodelling technique in identifying open channel irrigationsystems. We will show, while deriving the model structurein later sections, that a simplified mass balance relationshipcan be used as sufficient prior information for the estimationproblem. Some important steps involved in the process ofsystem identification are shown as a flowchart in Fig. 1 [18].

Experimentation:

Collection of

Representative Data

Model Hypothesis:

Structure, Order, Delay,

etc.

Estimation of Model

Parameters

Model Validation,Decision:

Is ModelGood Enough?

Prior

Knowledge

about

Physical

System

System

IdentifiedYESNO

Fig. 1. System identification procedure for grey box modelling.

As discussed, model structure and parameter estimation playan important part in system identification. Subsequently, wewill discuss the role of prior information and appropriate gatemodels in Model Selection.

III. MATHEMATICAL OVERVIEW

In this section we present mathematical overview of modelselection for system identification technique which will beused in deriving channel representation.

A. Model Selection

In using grey box technique for system identification, weneed to use some prior information about the physical processfor modelling a mathematical relationship between representa-tive input/output data. In our model, downstream water level isthe controlled variable, and the gate position is the manipulated(input) variable.

In this procedure, we used some fundamental physicalinsight about the channel and tried to develop a representativeparametric model of the system. The methodology in derivingthe model structure for open channel pools has been explainedin [9], [13], [14]. We begin with a mass (or volume) balanceequation

𝑑𝑉 (𝑡)

𝑑𝑡= 𝑄𝑖𝑛(𝑡)−𝑄𝑜𝑢𝑡(𝑡), (1)

where 𝑉 is the volume in a pool, 𝑡 is the time, 𝑄𝑖𝑛 isthe inflow at the upstream gate and 𝑄𝑜𝑢𝑡 is the outflowfrom the downstream gate. Our objective is to reduce into adifferential equation involving water level (more importantlydownstream level) in the channel. For that we need to studyflow relationships, governed by canal gates.

Canal gates are essential hydraulic structures which playan important role in regulating the water flow in irrigationchannels [14]. Two such hydraulic structures, widely used inregulating the flow of irrigation canals, are undershot gatesand overshot gates (as shown in Fig. 2).

The stretch of a channel between two gates is referred to asa pool [13]. In the literature related with hydraulics and flows(for example in [19]), the flow equations for both overshot andundershot gates can be found (see Fig. 2 for the flow relationsof overshot and undershot gates). It is important to keep in

20

yu

h

pyd

Q

yu

ydp

Q

Fig. 2. Schematic side view of Overshot gate (left) in free flow andUndershot gate (right) in submerged flow. For an overshot gate, under freeflow conditions, the flow over the gate is expressed as 𝑄 ≈ 𝜃ℎ

32 and for

undershot gates, under submerged flow conditions, the flow is approximatedby 𝑄 ≈ 𝜃𝑝

√𝑦𝑢 − 𝑦𝑑; where 𝑄 - water flow, 𝑦𝑢 - upstream water level, 𝑦𝑑

- downstream water level, 𝑝 - gate position and ℎ - head over the gate.

mind that these flow expressions are, at best, approximationswhich model the flow well under free flow conditions, forovershot gates, and under submerged flow conditions, forundershot gates.

B. Parameter Identification from Step Response

We now discuss the step response of a linear first-ordersystem with delay as an illustrative example of estimatingparameters of the system. Estimating these parameters reflectthe output of a linear (first order) system and is considereda good method to get quick and easy insights into timedelays, static gains and time constants. In system identificationof our irrigation channels, these ‘step tests’ provide usefulinformation for choice of sampling interval and experimentdesign. In our experimental design, we have tried to modelstep inputs at upstream to obtain fairly approximate heuristicsabout how fast or slow the channel response is.

Consider the first order differential equation

��(𝑡) = 𝑎𝑥(𝑡) + 𝑏𝑢(𝑡− 𝜏) (2)

Assuming initial conditions at time 𝑡0 as 𝑥(𝑡0), if we applya step input to this system, i.e. 𝑢(𝑡) = 𝑢0 for 𝑡 > 𝑡0, then theresponse of the system is given by

𝑥(𝑡) = 𝑒−𝑡−𝑡0−𝜏

𝑇 𝑥(𝑡0 + 𝜏) +𝐾𝑢0

(1− 𝑒−

𝑡−𝑡0−𝜏𝑇

)(3)

where 𝑇 = − 1𝑎 is called time constant, 𝐾 = − 𝑏

𝑎 is thestatic gain and 𝜏 is known as the time delay of the system.Essentially, these three parameters give us insights about theresponse of a first order system to the applied step input. 𝑇determines the speed of response of the system, 𝜏 gives us thetime before input has any effect on the output and 𝐾 givesthe ratio of change in steady state output to the height of inputstep (see Fig. 3).

Note that we can completely characterize the step responseof a linear system by estimating 𝜏 , 𝑇 and 𝐾. A quickinspection of Fig. 3 allows us to estimate these parameters,where we note that static gain 𝐾 can be estimated by dividingchange in steady state with the applied step height, the timetaken by the system to 63% of the steady state value (from

0 2 4 6 8 10 12

0

0.2

0.4

0.6

0.8

1

Step ResponseStep Input

Height of Input Step (u0)

Changein

SteadyState

Output(K)

63% ofFinal Value

Final Value

5 Time Constants(5T)

Time Delay

τ 1 TimeConstant (T)

Fig. 3. Step Response of First Order System

Fig. 4. Google Earth image of Project Site

rest) corresponds to one time constant 𝑇 and the time delay,𝜏 , can be visually estimated from the graph.

IV. EXPERIMENTAL DESIGN

In this section we apply the estimation techniques to identifythe lumped parametric models of irrigation canals, using datafrom actual canal experimentation at Khaira Distributory offBRB Link Canal, Lahore.

A. Khaira Distributory

We carried out extensive experiments and collected datapoints from Khaira distributory (see Fig. 4). It stems fromBRB Link Canal and is approximately 18 km long, withchannel width of 10 ft and height 4 ft. Additionally, theregular discharge from head (upstream) gate is 87 cusecs.This distributory has 3 minors branching from it. It is aperennial canal and was originally designed to irrigate mostof the eastern-suburban villages. We performed most of ourexperimentation at the upstream of distributory.

In these experiments, we only had control over the upstreamgate (which was undershot) and there was no physical (hy-draulic) structure present at the downstream ends of the canal,which happened to be points of measurement. We modelled itas an always open, hypothetical, overshot gate (see Fig. 5).

B. Channel Model

To derive the channel model, we start with Eq.(1) andsubstitute appropriate flow approximations at the upstream anddownstream (see Fig. 5 for illustration). If we assume thatthe volume in the pool is proportional to the water level at

21

y1u

p1

y1dy2

p2

h2

Upstream

Downstream

Fig. 5. Side-view sketch of Khaira Distributory Pool with undershotgate at upstream and ‘hypothetical’ overshot gate downstream (point ofmeasurement). The value of 𝑝2 is always taken to b 0 to model an alwaysopen overshot gate.

the downstream (i.e. assuming uniform channel cross-section)then we can write Eq.(1) as a non-linear differential equationin terms of downstream water level 𝑦2. Additionally, we knowthat there is a time delay for the water at the upstream to reachdownstream.

Assuming free flow conditions for the system, we can writeEq.(1) as

𝑑𝑦2(𝑡)

𝑑𝑡= 𝑐1𝑝1(𝑡− 𝜏)

√𝑦𝑢 − 𝑦𝑑(𝑡− 𝜏) + 𝑐2

(𝑦2(𝑡)− 𝑝2(𝑡)

) 32

,

(4)where 𝜏 is the time delay for the water at upstream gate toreach downstream water, 𝑝1(𝑡) is the opening of upstreamgate and ℎ2(𝑡) is substituted as 𝑦2(𝑡) − 𝑝2(𝑡). It is to benoted that in case of multiple hydraulic structures at theupstream/downstream, we will have to introduce indexing andsummation to represent cumulative 𝑄𝑖𝑛 and 𝑄𝑜𝑢𝑡.

For implementation point of view, we are interested in dis-cretization of the continuous first order, non-linear, differentialequation into a first order difference equation. This will giveus a discretized model for the open channel system. Using theEuler’s approximations for the derivative term, with samplingtime 𝑇 (as explained in [13] and [14]), we obtain the followingmodel structure

𝑦2[𝑘 + 1] = 𝑦2[𝑘] + 𝜃1𝑝1[𝑘 − 𝜏𝑘][√

𝑦𝑢 − 𝑦𝑑[𝑘 − 𝜏𝑘]]

+𝜃2

(𝑦2[𝑘]− 𝑝2[𝑘]

) 32

(5)

where 𝜃1 (equivalent to 𝑇𝑐1)and 𝜃2 (equivalent to 𝑇𝑐2) arethe unknown system parameters and 𝜏 is the discretized timedelay (which is obtained by dividing the actual time delay 𝜏by the sampling time 𝑇 ). Our resulting discretized model ofdyanmical system is in linear regression form [20], i.e. it islinear in unknown parameters (𝜃1, 𝜃2). The system parameters[𝜃1, 𝜃2]

𝑇 are estimated using the methods of least squares.It is important to mention that the discretized time delay,𝜏 (in minutes), was approximated to the nearest integer tofacilitate computations in least square algorithm for parameterestimation.

C. Experimental Procedure

In our field work, we used Ultra-Sonic range sensors [15].These sensors were deployed (usually at bridges) at differentlengths along the downstream channel. In addition, we also

Fig. 6. Experimental procedure: sensor deployment.

Fig. 7. Upstream gate at Khaira Distributory.

monitored the water level close to the opening of the upstreamgate (𝑦𝑑). A linear potentiometric encoder was used to measurethe current opening of the upstream gate. Our objective was toexcite the upstream gate in the form of step input and recordchannel response for different pool lengths.

The sampling time of our experiment was kept at 10seconds. For system identification to work, we need uniformlysampled data. As our readings were transmitted through GSMmodule, it introduced some delays between successive mes-sages. Our measurements were ‘batched’, i.e. we received agroup of readings with the ‘inter-batch’ interval more thanthe ‘inter-sample’ interval (10 s) within each batch. To caterfor this discrepancy, we used data interpolation to obtain auniformly sampled data. The measurements relayed by theultra-sonic sensors were time-stamped and we adjusted foruniform sample time of 10 seconds by linear interpolation(as shown in Fig. 8). This pre-processing was necessary togenerate a uniform data for least squares to work.

D. Parameter Estimation

We tried to model a step input at upstream. In our ex-periment we closed the upstream gate and then re-openedit. The water level was measured at 50m, 300m and 550mdownstream lengths, after every 10s (see Fig. 9). For channel

0 2000 4000 6000 8000 1000020

40

60

80

100

Time/sec

Wat

er L

evel

/cm

Data Processing: Interpolation

Raw DataInterpolated Data

Fig. 8. Data interpolation of measurements.

22

050

100150 0

200400

6000

20

40

60

80

100

120

Sensor Location / m

Hydrographs for System Identification − 3D Overlay

Time (min)

Wat

er L

evel

/ cm

Gate HeightWater Level at 50mWater Level at 350mWater Level at 550m

Fig. 9. Experimentation: 3D overlay of hydrographs.

0 50 100 1500

0.2

0.4

0.6

0.8

1

1.2

1.4

Time/ (min)

Wat

er L

evel

/ (m

)

Estimated Water LevelActual Water LevelUpstream Gate Height

Fig. 10. Actual and estimated response for Khaira distributory.

modelling, we used the data from 300m and 550m sensors.The corresponding data was processed and interpolated. Forestimation process, the linear regression model Eq.(5) wasattempted to fit the observed response. In our estimation, 𝑦𝑢was assumed to be constant and 𝑝2[𝑘] was taken to be zeroto model an always opened’, hypothetical, overshot gate. Inaddition, 𝑦𝑑 was measured near the opening of the upstreamgate. The response delay, 𝜏𝑘, was inspected from the raw data.In order to obtain a better estimate of the time constant forthe pool, the upstream gate was opened and closed suddenly(as if to model a sudden impulse). This accounts for theapparent ‘jump’ in the gate input and the corresponding blipin hydrographs. The results of our experiments, along withthe estimated parameters, have been tabulated in Table I. InFig. 10, the response of 550m sensor along with respectiveupstream gate input and estimated water level is shown.

Our estimated parameters make intuitive sense from aphysical point of view. 𝜃1 is positive as it is related with theinflow of the water (from upstream), while 𝜃2 is negative asit is associated with the outflow of water.

Pool 1 2Length (m) 300 550𝜏 (sec) 200 350𝜃1 0.0161 x 10−3 0.0152 x 10−3

𝜃2 -0.3271 x 10−3 -0.2737 x 10−3

Avg. Sq. Error (𝑚2) 0.0164 0.0155

TABLE IEXPERIMENTAL & ESTIMATION RESULTS FOR KHAIRA DISTRIBUTORY

Pool 1 Pool 2Length (m) 300 550

Prediction Error(m2) 0.0077 0.0014

TABLE IIPREDICTION ERROR IN CANAL EXPERIMENTATION

V. MODEL VALIDATION

In this section, we complete our analysis by validatingthe proximity of our model to that of actual response. It isimportant to compare the performance of our estimated modelwith the actual response of the physical system to establishsufficient confidence in our hypothesized model. In our modelverification process, we generated a predicted response, for anindependent set of inputs, from our model and compared itwith the actual water level.

In order to verify our predicted model, we carried outanother set of experiments at the canal (with different set ofinput data). For the new side information, we compared theresponse of the estimated water level to that of the actual waterlevel. This is shown in Fig. 11. It can be seen that when were-open the gate, there is some poor tracking exhibited by theestimated model. However, the predicted response begins totrack the actual level after some time.

0 5 10 15 20 25 30 35 40 45 500

0.2

0.4

0.6

0.8

1

1.2

1.4

Time/ (min)

Wat

er L

evel

/ (m

)

Model Verification for Canal Experimentation

Upstream Gate HeightActual Water LevelPredicted Water Level

Fig. 11. Model verification for canal experimentations.

We also used average squared prediction error, given by1𝑁

𝑁∑𝑖=1

𝜖2(𝑖, 𝜃) where 𝜖(𝑖, 𝜃) = 𝑦(𝑖)− 𝑦(𝑖, 𝜃), as a performance

metric for model validity. 𝜃 is the vector of estimated parame-ters and 𝑦(𝑖, 𝜃) is the predicted value given 𝜃, obtained from anARX type predictor [21] defined from the model structure inequation (5). The results of our model verification have beentabulated in Table II

VI. CONCLUSION

In this paper, further steps have been taken towards realizingcontrolled irrigation systems in Pakistan. We have presentedcomplete system identification procedure, from experimentaldesign to model validation, for actual irrigation canals. It wasfound that a first-order non-linear model, derived from priorknowledge about the physical system, gave accurate resultsin predicting downstream water level of an actual irrigationcanal.

23

In this study we have verified that:

∙ Design of an accurate canal model can be obtainedthrough elaborate experimentation, which captures dy-namics of the underlying physical system.

∙ The proposed methodology of system identification issuitable for introducing controlled irrigation systems indeveloping countries.

Moreover, the estimated models are simple which makesthem suitable for control design. These results illustrate thatsystem identification, for control in irrigation canal, can playan important part in managing water resources for developingcountries.

ACKNOWLEDGEMENT

The authors would like to express our sincere appreciationfor Dr. Erik Weyer’s feedback on our experimentation results.We thank Mr. Zahoor Ahmed for development of sensors usedin our study. Finally, we are thankful to Mr. Habibullah Bodhlafrom Punjab irrigation Department for granting us access tothe Khaira distributory.

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