design of a centralized controller for an irrigation...

Design of a Centralized Controller for an Irrigation Channel Using H∞

Loop-shaping

Yuping Li, Michael Cantoni and Erik Weyer

Abstract— This paper describes the design of a multivariableH∞ loop-shaping controller for water level regulation on the Haughton Main Chan-nel, Queensland Australia. A one-degree-of-freedom controller structureis applied to three successive pools. Analysis of robust performance is pre-sented in terms of the structured singular value. Closed-loop performanceis also evaluated through simulations with high fidelity models of thechannel. The controller shows better performance than decentralized PIcontrol, and attains similar performance to a multivariable LQ controller.The H∞ loop-shaping controller is, however, much easier to tune thanan LQ controller. This becomes important when the number of pools tobe controlled is increased. As such, it is an appealing design paradigmwhen high performance control of an irrigation channel is required.

I. INTRODUCTION

An irrigation channel must be able to deliver water to farmers ondemand, whilst minimizing water losses. Currently, water losses aretypically large, and since water is a scarce resource in many parts ofthe world, this is becoming an increasingly important issue. Continuedinvestment in the instrumentation and automation of irrigation net-works, will give rise to a great potential for satisfying water demandswhilst reducing water losses. Indeed, in recent years this area hasreceived significant attention, see e.g. [1 – 11].

In the literature, two approaches are commonly employed forcontrolling irrigation channels, namely, decentralized PI control ([8],[9], [10]), and multivariable (centralized) LQ or predictive control([3], [7], [11]). Naturally we attain better norminal performance witha multivariable LQ controller than with decentralized PI control, butan LQ controller is much more difficult to tune, particularly for largesystems. Decentralized PI control, on the other hand, is advantageousin terms of ease of design and implementation. The channel weconsider in this work is the Haughton Main Channel (HMC), situatedin Queensland Australia. Very accurate simulation models exist for thissystem (see [12] and [13]). Both decentralized PI controllers ([10],[14], [22]) and a centralized LQ controller ([11], [15]) have beendesigned for the HMC and tested in the field with promising results.In this paper we employH∞ loop-shaping to design a multivariablecontroller for the same part of the HMC. The controller obtained yieldsperformance comparable to the LQ controllers of [11,15]. However,it is much easier to design, which is a significant advantage from theperspective of designing controllers for similar irrigation systems ona larger scale.

The paper is organized as follows. In section II, a descriptionof the HMC and the models used for control design are presented.We then describe the design of theH∞ loop-shaping controller insection III. Section IV is devoted to a discussion of the robustness andperformance of the resulting control system in terms of the structuredsingular value. By simulating the controller in closed-loop with highfidelity models of the HMC, a comparison with decentralized PIcontrol and multivariable LQ control is given in Section V. Finallysome conclusions are given in section VI.

II. MODELS OF THE HMC

A top view of an irrigation channel is shown in Fig 1. The stretchof a channel between two gates is referred to as a pool. Along the

Research funded in part by the Commonwealth of Australia through theCo-operative Research Centre for Sensor, Signal and Information Processing(CSSIP).

Yuping Li and Erik Weyer are with CSSIP, Department of Electricaland Electronic Engineering, University of Melbourne, Parkville VIC 3010,Australia{yuping,e.weyer}@ee.mu.oz.au

Michael Cantoni is with the Department of Electrical and Electron-ics Engineering, University of Melbourne, Parkville VIC 3010, [email protected]

channel there are offtakes to farms and secondary channels feedingoff the main channel. In most cases we do not have measurements ofthe offtakes, and as such they are usually treated as disturbances.

Farm

Farm

Gate8 Gate9Pool8

Gate10

Gate

Gate

Gate

Branch Channel

Main Channel

Fig. 1. Topview of an irrigation channel

The water levels in the channel are controlled by over-shot gateslocated along the channel as sketched in Fig 2. The measurementsavailable for control are the water levels upstream of each gate and thegate positions. All water levels are given in mAHD (meters AustralianHeight Datum), relative to a reference level. The height of water abovethe gate is called the head over the gate.

y8

y9

h8

h9p8

p9Pool8

Fig. 2. Sideview of an irrigation channel

As the offtakes are gravity fed, the water level in each pool mustbe regulated in order to satisfy the demand for water from farmers.To the end of formulating models for the water levels, a basic volumebalance gives

dV (t)

dt= Qin(t) − Qout(t), (1)

where V is the volume of one pool, and Qin and Qout are the inflowsand outflows respectively. For an overshot gate, Qin and Qout areusually approximated (see [16]) by:

Q(t) = ch3/2(t), (2)

where Q is the flow, h the head over the gate and c an unknownparameter. Note that this approximation assumes that the gate is infree flow, which means that the top of the gate is above the immediatedownstream water level. At the HMC, there is a drop in bed elevationafter each gate, so all gates are in free flow.

Assuming that the volume in a pool is proportional to the waterlevel, we arrive at

y9(t) = c8,2h3/28 (t − τ8) − c9,1h

3/29 (t) − d8(t), (3)

where y9(t) is the water level of the pool to be controlled, h8(t) isthe head over the upstream gate of that pool, and h9(t) is the headover the downstream gate – see Fig 2. d8(t) represents the offtake ofwater and τ8 is the time delay in pool 8 – i.e. the time it takes forthe water to flow from the upstream gate to the downstream gate. Theunknown parameters c8,2 and c9,1 have been determined from systemidentification experiments – see [12] and [13].

Control 2004, University of Bath, UK, September 2004 ID-071

TABLE IPARAMETERS OFTHE PLANT MODEL

Pool Length Wave Freq Time Delay ci,2 ci+1,1i (meter) (rad/min) τ (min)8 1601 0.42 6 0.0138 0.01679 853 0.74 3 0.0454 0.041710 3129 0.20 16 0.0092 0.0100

Setting u8(t) = h3/28 (t) and u9(t) = h

3/29 (t), and taking the

Laplace transform, the plant model for pool 8 can be expressed as:

y9(s) =c8,2e

−τs

su8(s) −

c9,1

su9(s) −

1

sd8(s). (4)

Modelling pools 9 and 10 the same way, yields the following plantmodel:

y(s) = P (s)u(s) − Pd(s)d(s) (5)

wherey = (y9, y10, y11), u = (u8, u9, u10) = (h3/28 , h

3/29 , h

3/210 ) and

d = (d8, d9, d10, h3/211 ) (h11 is the head over the downstream gate of

pool 10).P (s) is a 3 × 3 transfer matrix:

P (s) =

c8,2e−τ8s

s

−c9,1

s0

0c9,2e−τ9s

s

−c10,1

s

0 0c10,2e−τ10s

s

(6)

andPd(s) is a 3 × 4 transfer matrix:

Pd(s) =

1s

0 0 00 1

s0 0

0 0 1s

c11,1

s

(7)

The parametersci,2, ci+1,1 for i = 8, 9, 10, the length, the frequencyof the dominant wave and the time delay for each pool are given inTable I.

III. DESIGNING THE LOOP-SHAPING CONTROLLER

In this section we first discuss the control objectives, beforespecifying a control system configuration. A multivariable controlleris then designed via theH∞ loop-shaping technique of McFarlaneand Glover [17].

A. Control Objectives

As the offtakes to the farms are gravity fed, i.e. there is no pumping,the water levels of each pool should be maintained at certain levels.Offtakes by farmers can be treated as load disturbances that mustbe rejected by the control system. When these disturbances occur,the water level for each pool should be controlled to remain at therespective setpoint value through the movement of gates. Since gatemovements can induce wave motions, gate movements around thewave frequencies should be avoided. Therefore, the main controlobjectives include:

• High loop gain at low frequencies for rejection of load distur-bances due to offtakes;

• Roll-off rate of approximately 20dB/dec at the desired bandwidth(limited by wave frequencies) to yield sufficient “phase margin”for the loop gain and hence stability of the closed-loop system;

• High roll-off rates at frequencies beyond the desired bandwidth,to give low loop gain at wave frequencies in order to suppresseffects of the waves and mitigate uncertainty in the plant modelat high frequency.

Water is usually lost when it passes the last gate without beingused. In irrigation channels where only the upper part is automatedwe should therefore ensure that the flow into the lower non-automatedpart is regulated according to the downstream demand. The flow overthe last gate is proportional to the head over the last gate to the powerof 3/2 (see (2)), and the head is given by:

h11 = y11 − p11 (8)

wherep11 is the position of gate 11. If gate 11 were the first non-automated gate,p11 would be adjusted manually about once a dayin order to meet the demand for water further downstream. Whencalculating the daily position for gate 11 one assumes that the waterlevel y11 is on setpoint, so water level control is also important fordelivering the right flow to the downstream plant of the system.

B. Control System Configuration

Since any water level setpoint change or load disturbance dueto offtakes is reflected in the error between the setpoints and themeasured values of the water levels in the pools, we set these errorsas the input signal to the controller. By choosing the head over theupstream gate of each pool as the output signal of the controller, weget a control system configuration as shown in Fig 3, wherer andyrepresent the setpoints and measured value of the water levels of pool8 to 10 respectively,d represents the offtakes from pool 8 to 10 andany change in the head over gate 11. In designing the controllerK

−

−

d

d

K(s) P (s)

Pd(s)

ru

y

Fig. 3. Closed loop Control System

we aim to minimize the effect of the disturbanced on the outputy.

C. Design of Centralized Controller UsingH∞ Loop-Shaping

Towards achieving the control objectives just identified, we candesign weights (which will ultimately appear as a part of the “con-troller”) to shape the (open) loop gain according to the specificationsdescribed above. A robust stabilization problem is then formulated,and solved, for the weighted plant. Indeed, this is precisely theH∞

loop-shaping procedure of McFarlane and Glover [17–19].Two steps are involved in theH∞ loop-shaping procedure. First,

the singular values of the nominal plant,P , are shaped, usingW1 andW2, to give a desired (open) loop gain shapeW2PW1. A stabilizingcontroller K∞ is then synthesised to achieve a bound on a measureof closed-loop robustness. The final feedback controller is constructedby combiningK∞ and the weights asW1K∞W2.

Step 1. Loop ShapingThe nominal plantP and the shaping functionsW1, W2 are

combined to form the shaped plantPs = W2PW1. The weightsW1

andW2 are designed to ensure that the shaped plant has the followingproperties:

• σ(Ps) � 1 in the low-frequency range, which corresponds tothe control objective of high loop gain at low frequencies forofftake load disturbance rejection;

• σ(Ps) � 1 in the high-frequency range, which corresponds tothe control objective of low loop gain at high frequencies tosuppress wave resonance and to mitigate under-modelled highfrequency dynamics.

For the plant model (6) of pool 8 to 10, we select the post-compensatorW2 = I and the pre-compensatorW1 as a diagonal3 × 3 matrix:

W1(s) = diag

[

ki(1 + Tcis)

s(1 + Tfis)

]

, i = 8, 9, 10 (9)

In order to ensure complete disturbance rejection (asymptotically), weinclude an integrator1

sin the weight functionW1(s). To suppress the

wave resonances, a low-pass filter11+Tf s

is included to ensure a lowloop gain at the wave frequencies. We also ensure that the roll-off rateof σ(Ps) be around 20dB/dec around the desired bandwidth, so that


in the subsequentH∞ optimization step, which yields a stabilizingcontroller for the weighted plant, an acceptable level of robust stabilityis achieved. This is achieved by including the term(1+Tcs) in W1(s),which essentially provides some phase lead around cross-over.

We see that since the plant modelP in (6) is nearly diagonal andeach term inP is rather simple, the weight functionW1 has a simplestructure. A plot of the singular values of the nominal plantP andthe shaped plantPs is shown in Fig 4. We see that compared withP , Ps has higher gain at low frequencies and lower gain at the wavefrequencies. In contrast to classical loop-shaping, the shaping of the

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10−1

100

101

10−6

10−4

10−2

100

102

104

106

rad/min

mag

nitu

de

Singular Value of The Shaped Plant

Singular Value of The Nominal Plant

Fig. 4. Singular values of the shaped plant

loop gain is carried out without explicit regard for the phase of thenominal plant model and hence, without explicit regard for closed-loopstability (although completely ignoring the phase typically results in apoor robust stability margin in the second step, which yields a robustlystabilising controller [18,19]).

Step 2. Controller SynthesisIn this step of the design procedure we consider the robust

stabilization of a left coprime factorizationM−1N = Ps, with[ M N ][ M N ]∼ = I (i.e. the factorisation is normalised).In particular,K∞ is synthesised to ensure that

∥

∥

∥

∥

[

K∞

I

]

(I + PsK∞)−1M−1

∥

∥

∥

∥

∞

≤ 1/ε (10)

for some

ε ≤ emax :=√

1 − ‖[ N M ]‖2H < 1,

where‖·‖H is theHankelnorm. This guarantess closed-loop stabilityfor any perturbationPs∆ = (M + ∆M )−1(N + ∆N ), of the shapedplat, that satisfies‖[ ∆N ∆M ]‖∞ < ε. Note that the numberεmax

gives an indication of the compatibility between nominal performanceobjectives and the closed-loop stability requirements. In particular,εmax should not be too small (typically≥ 0.25). If εmax � 1, theweightsW1 andW2 must be adjusted.

Based on the shaped plant we get by combiningW1(s) and thenominal plantP (s) for pools 8 to 10, theH∞ optimization problemdescriped in step 2, is solved using the matlab functionncfsyn (see[21]), to obtainK∞. In order to use this it is necessary to have finite-dimensional models of the weighted plant, and as such, the timedelays in (6) are replaced by first order Pade approximation (notethat the loop-gain at the corresponding non-minimum phase zeros islow). A slightly suboptimal controller achievingε = 0.3 is obtained,which guarantees stability in the face of up to30% uncertainty inthe coprime factors. As described above, the final feedback controllerK = W1K∞W2 is formed by combining the weights andK∞. Thesingular values ofPK (the final open loop-gain) are shown in Fig 5.Compared with the singular values of the shaped plantPs, we seethat becauseε = 0.3 is sufficiently large, the loop gain obtained withthe controllerK = W1K∞ and the original plantP does not changemuch with respect to the loop-shape designed in step 1 (as Theorems18.9 and 18.10 in [19] predict).

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100

101

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10−6

10−4

10−2

100

102

104

106

rad/min

mag

nitu

de

Singular Values of The Open Loop Gain

Singular Values of The Shaped Plant

Fig. 5. Singular values of the open loop gain

Note that the main design work is done in the first step, inwhich we select the weight functions to shape the nominal plant toreflect performance and robustness objectives. In the second step, thestabilizing controllerK∞ is synthesized and the stability marginε iscalculated. This is also used to check whether the weight functionschosen in step 1 are appropriate.

IV. ASSESSING ROBUST PERFORMANCE WITHµ

Consider, the closed-loop interconnection structure of a perturbedmodel of the nominal plant and loop-shaping controller shown inFig 6. The dashed box labelledPp represents the nominal plant

Wdel

∆G

yu

++

+ +P

K

Wp

Pp

w

z

e

dist

Fig. 6. The closed-loop interconnection structure

model with multiplicative uncertainties. Inside the boxWdel and∆G

parameterize the uncertainty.Wdel is designed to reflects the amountof uncertainty in the model, so that we may assume that‖∆G‖∞ < 1(i.e. σ(∆G(jω)) < 1 for all ω). Good performance is taken to meanthat the transfer function fromdist to e be small, in the‖ · ‖∞ sense.Indeed we want this to be true for all possible uncertainty∆G, i.e.

‖Wp(I + P (I + ∆GWdel)K)−1‖∞ < 1 (11)

for all stable∆G with ‖∆G‖∞ < 1. The weightWp allow us tonormalise the right hand side of the inequality so that when (11)is satisfied, the size of the perturbed sensitivity function(I + P (I +∆GWdel)K)−1 is guaranteed to be smaller than the size of the inverseof the weightWp, at any frequency. Finally, note the input signaldisthere can be interpreted as the reference signalr or disturbance signald (whenPd is absorbed intoWp) in Fig 3.

The problem of determining whether the closed-loop remains stableand whether (11) is satisfied for all stable∆G with ‖∆G‖∞ < 1can be formulated as aµ-analysis problem [19], [20]. Central tothis analysis is the LFT configuration shown in Fig 7. Here,G =(

G11 G12 G13

G21 G22 G23

G31 G32 G33

)

is a generalized plant constructed from a nominal

model and appropriate weights, so that:

1) Fu

((

G11 G13

G31 G33

)

, ∆G

)

:= G33 + G31∆G(I − G11∆G)−1G13

describes the uncertain plant set as∆G varies over somestructured set – i.e. within the context of the robust performance


problem just describedFu

((

G11 G13

G31 G33

)

, ∆G

)

must correspondto the transfer function fromu to y in Fig 6; and

2) For a given controllerK, FL

((

G22 G23

G32 G33

)

, K)

:= G22 +

G23K(I − G33K)−1G32 corresponds to all nominal closed-loop transfer functions by which performance is to be gaugedin terms of theH∞ norm – i.e. within the context of our robustperformance problem the transfer function fromdist to e.

∆G

∆p

G

K

w z

y u

diste

Fig. 7. Robust performance test loop

The total uncertainty block in Fig 6 has the following structure:

∆TOT = diag[∆G, ∆p],

where ∆G can be structured, and the unstructured∆p is used toconvert the corresponding the robust performance problem charac-terised by the inequality (11) into a structured robust stability analysisproblem. Indeed, including∆p in the uncertainty block, we have thefollowing sufficient and necessary condition for robust performanceof the system:

Theorem 1:The loop shown above is well-posed, internally sta-ble, and ‖Fu(Fl(G, K), ∆G)‖∞ ≤ 1 for all stable ∆G(s) with‖∆G‖∞ < 1, if and only if

supω∈R

µ∆T OT(Fl(G, K)(jω)) ≤ 1,

whereµ denotes the structured singular value.For proof of Theorem 1 and details aboutµ-analysis for robustperformance, see [19] and [20].

For the robust performance problem at hand (see (11), we use astructured∆G to represent plant uncertainty and select a diagonaluncertainty weightWdel to reflect that at low frequency, there ispotentially a20% modelling error, and at the wave frequencies forpool 8 to 10, the uncertainties in the model are up to100% at0.02rad/min,158% at 0.42rad/min,181% at 0.74rad/min, and get evenlarger at higher frequencies – see Fig 8 For the performance weight

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101

102

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Frequency(rad/min)

Fig. 8. Multiplicative Uncertainty Weight Function

Wp, we use a3 × 3 diagonal function

Wp(s) = 0.6s+0.0552s+0.055×(1e−4)

I3.

The singular values of the inverse of this are shown in Fig 9. Assuch, when theµ test corresponding to Thm 1 is satisfied, we cansay that all perturbed closed-loop sensitivity functions will be small

10−4

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10−1

100

101

102

10−3

10−2

10−1

100

101

Frequency(rad/min)

Fig. 9. The Inverse of the Performance Weight Function

at low frequency (i.e. good set-point tracking and load disturbancerejection).

Using the matlab functionmu (see [21]) we calculate the upper andlower bounds for the structured singular value to obtain theµ-curvesshown in Fig 10. Since the value ofµ is less than 1, we conclude,

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102

0

0.1

0.2

0.3

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0.5

0.6

0.7

0.8

0.9

1

Frequency(rad/min)

upper boundlower bound

Fig. 10. The structured singular value

according to Theorem 1, that the closed-loop system achieves robustperformance with respect to the uncertainty weight functionWdel

and performance weight functionWp. Also note from Fig 10, that thevalue ofµ is about0.2 at low frequencies and0.3 at high frequencies.Two possible interpretations can be given for this:

• In order to satisfy the same performance requirements (asreflected by the weightWp), the system could include moreuncertainty over these frequency ranges.

• In order to achieve robust performance for the uncertainty char-acterised byWdel, the system could achieve more demandingperformance requirements (as characterised byWp) over thisfrequency range.

To see this, consider for example, the robust performance problemcorresponding to the sameWp as used above, but for which the un-certainty is now characterised in terms of the weight shown in Fig 11.We see that the newµ-curves are still less than 1 at all frequencies andhence robust performance is achieved. Similarly, consider the robust

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1

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3

4

5

6

7

8

Frequency(rad/min)

new uncertainty weightold uncertainty weight

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102

0

0.1

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0.5

0.6

0.7

0.8

0.9

1

Frequency(rad/min)

new upper bound for munew lower bound for muold upper bound for muold lower bound for mu

Fig. 11. Change ofµ for more uncertainty

performance problem corresponding to the uncertainty weight shown


in Fig 9, but for which the performance weightWp is taken to bethe one shown in Fig 12 (which corresponds to a more demandingperformance requirement). Again we see thatµ is still less than 1over all frequency.

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101

102

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101

Frequency(rad/min)

inverse of performance weight (old)inverse of performance weight (new)

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101

102

0

0.1

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0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency(rad/min)

new upper bound for munew lower bound for muold upper bound for muold lower bound for mu

Fig. 12. Change ofµ for more demanding performance requirement

V. SIMULATION RESULTS

A. Comparisons with Decentralized PI Control And Centralized LQControl

We simulate the centralizedH∞ loop-shaping control schemeusing accurate higher order system identification models of the plant(see [12], [13] for details) and compare the results with those fordecentralized PI control and centralized LQ control.

The PI controllers operate in distant downstream mode, where agate is controlling the water level immediately upstream of the nextdownstream gate, and feedforward action from the downstream gateis also used. For gate 8 the head over gate is calculated as follows

u8(s) = C8(s)(y9,setpoint(s) − y9(s))

h3/28 (s) = u8(s) + KffF (s) 1

Kgh

3/29 (s)

and similarly for the other heads.Kff is the feedforward gain, andit is equal to0.75. F (s) is a second order Butterworth filter with cutoff frequency around half the wave frequency in the pool (see TableI). Kg is the ratio c8,2

c9,1for pool 8.C(s) is a PI controller augmented

with a first order lowpass filter, which has the same structure as thediagonal elements in the weight functionW1(s) for the H∞ loop-shaping controller. Details are given in [14] and [22]. The input-outputstructure of the LQ centralized controller is similar to that of theH∞

loop-shaping centralized controller. Further details are given in [15].The following scenario was simulated. At time 0 minutes all water

levels were in steady state at setpoints of 26.50, 23.85, 21.15mAHDfor pool 8, 9 and 10 respectively. An offtake in pool 10 started at time200 min and finished at 1000 min. At time 1600 min the position ofgate 11 was moved from 1200 to 1300mm, which caused a changeof the head over gate 11. At time 2100 min the setpoint for the waterlevel in pool 9 was reduced from 21.85 to 21.80 mAHD.

The simulation results for pool 8, 9 and 10 are shown in Fig 13. Wesee that all the three methods give acceptable responses. The waterlevels recover smoothly from disturbances without large deviationsfrom setpoint and without inducing excessive wave motions.

The LQ controller is slightly different in that it also controlsh11

through manipulation of gate 11. The step in gate 11 at time 1600 minwas substituted with a step in the setpoint forh11 in the LQ controlsuch that the steady state gate positions were 1200 and 1300mm beforeand after the step.

The simulation results show that in all the three pools, betterperformance is achieved by the centralized controllers compared to thedecentralized controller. In pool 8 and 10, when the offtake happensin pool 10, the LQ controller shows a better performance than theH∞ loop-shaping controller and the PI controller, especially with thewater level going back faster to setpoint. The centralizedH∞ loop-shaping controller achieves a smaller deviation from the water levelsetpoint, compared to the decentralized PI controller. In pool 9, the

0 500 1000 1500 2000 250026.46

26.47

26.48

26.49

26.5

26.51

26.52pool 8

(min)

Wat

er le

vel (

mHA

D)

setpointwith H−infinity controlwith PI controlwith LQ control

0 500 1000 1500 2000 250023.78

23.79

23.8

23.81

23.82

23.83

23.84

23.85

23.86

23.87

23.88pool 9

(min)

Wat

er le

vel (

mHA

D)


0 500 1000 1500 2000 2500

21.08

21.1

21.12

21.14

21.16

21.18

21.2

21.22

pool 10

(min)

Wat

er le

vel (

mHA

D)


Fig. 13. Water level changes of pool 8, 9 and 10

H∞ loop-shaping controller shows a better performance than both theLQ controller and the PI controllers.

On the other hand, the simulation shows that when there is a set-point change in pool 9 at 2100 min, there are water level fluctuations inpool 10 using the centralized controllers while there are no water levelfluctuations in pool 10 using the decentralized PI controller. It wouldappear that centralized control distributes the effect of a disturbanceover all the three pools, while decentralized control localizes the effectof disturbances to the upstream pools.

In practice due to dead zones on gate positions, a setpoint error ofless than a couple of centimeters is regarded as being “on setpoint”.With this in mind we observe that there is little difference in theperformance achieved by the LQ andH∞ loop-shaping controller. Itis noted however, that by virtue of the almost diagonal structure ofthe plant model for the irrigation channel the design of theH∞ loop-shaping controller, via weights that shape the loop gain, was simplerthan the design of the LQ controller. Indeed, significantly more timewas spent designing the LQ controller in order to achieve the samelevel of performance. Furthermore, the almost diagonal structure ofthe plant allows us to use adiagonal weighting functionW1, in the“design” step 1 of theH∞ loop-shaping procedure. Importantly, thisstructure for the weighting function can be maintained when additionalpools are added to the plant model. It is not as clear, on the other hand,how the design parameters in a LQ framework should be modified asthe number of pools is increased.

B. Robustness of TheH∞ Loop-shaping Controller

In the last simulation we applied a higher order system identificationmodel which is expected to captured the high frequency outputuncertainty. In the real irrigation control system, there are input


uncertainties in addition to the high frequency output uncertainty.There are two main sources for input uncertainties:

• Due to errors in the look up tables for the gates, the actualgate positions may be different from the calculated ones. Gatepositions errors of±10% are often left unattended.

• Due to the traffic on the radio network, there is time delaybetween the calculations of the gate positions and when thegates start moving. There is also time delay between the waterlevel measurements being taken and when they were used forcalculations. The uncertainty in time delay from one relay stationto the next one is expected to be less than 20 seconds.

In the simulation, we include an error of10% for each gate position,i.e. the actual gate positions are10% more open than the calculatedones. For time delay in signal transmission, we include a time delayof 1 minute for the signals from the field to the central computer and1 minute from the central computer to the field. The scenario is thesame as in the last simulation. The results are shown in Fig 14. Theresponses get a bit more oscillatory but they are still acceptable. Thetime delay could in fact have been 5 minutes return without muchchange in the response.

0 500 1000 1500 2000 250026.48

26.485

26.49

26.495

26.5

26.505

26.51

26.515

26.52pool 8

(min)

Wat

er le

vel (

mHA

D)

setpointwith uncertaintywithout uncertainty

0 500 1000 1500 2000 250023.76

23.77

23.78

23.79

23.8

23.81

23.82

23.83

23.84

23.85

pool 9

(min)

Wat

er le

vel (

mHA

D)


0 500 1000 1500 2000 2500

21.1

21.12

21.14

21.16

21.18

21.2

pool 10

(min)

Wat

er le

vel (

mHA

D)


Fig. 14. Water level changes of pool 8, 9 and 10

VI. CONCLUSIONS

In this paper we have usedH∞ loop-shaping to design a multivari-able (centralized) controller for an irrigation channel with overshotgates. From the simulation results, we see that the resulting centralizedH∞ loop-shaping control system achieves better performance thandecentralized PI control and comparable nominal performance tomultivariable LQ control. Bothµ analysis and the simulation results

show that theH∞ loop-shaping control also achieves a good level ofrobust performance.

The advantage of usingH∞ loop-shaping, compared to LQ basedcontrol, is clear in terms of ease of design/tuning. Indeed, the requiredweight functions are simple and easy to adjust, and to design aH∞

loop-shaping controller for more pools, we need simply augment theweights with similar diagonal terms adjusted appropriately to accountfor difference in the dynamics of any additional pools.

REFERENCES

[1] Journal of Irrigation and Drainage Engineering, Vol. 124, No. 1, 1998,pp. 1-62.

[2] Proceedings of IEEE Conference on System, Man and Cybernetics, SanDiego, 1998, pp. 3850-3920.

[3] Garcia A., M. Hubbard, and J.J. de Vries (1992), “Open ChannelTransient Flow Control by Discrete Time LQR Methods”,Automatica,Vol. 28, no. 2, pp. 255-264.

[4] de Halleux, J., C. Prieur, J.-M. Coro, B. d’Andrea-Novel and G. Bastin(2003), “Boundary feedback control in networks of open channels”,Automatica, Bol. 39, no. 8, pp. 1365-1376.

[5] Litrico, X. (2002), “Robust IMC Flow Control of SIMO Dam-RiverOpen-Channel Systems”,IEEE Trans. on Control Systems Technology,Vol. 10, no. 3, pp. 432-437.

[6] Litrico, X., V. Fromion, J.-P. Baume, and M. Rijo, “Modelling andPI control of an irrigation channel”,Proceedings of European ControlConference ECC03, Cambridge, UK, September 2003.

[7] Malaterre, P.O., “PILOTE: Linear Quadratic Optimal Controller forirrigation Canals”,Journal of Irrigation and Drainage Engineering, Vol.124, no. 4, pp. 187-194.

[8] Malaterre, P.O. and B.P. Baume, “Modeling and regulationof irrigationcanals: existing applications ans ongoing researches”,Proceedings ofIEEE Conference on System, Man and Cybernetics, San Diego, 1998,pp. 3850-3855.

[9] Schuurmanns J., A. Hof, S. Dijkstra, O.H. Bosgra and R. Brouwer,“Simple water level controller for irrigation and drainage canals”,Journalof Irrigation and Drainage Engineering, Vol. 125. no. 4, pp. 189-195.

[10] Weyer E., “Control of open water channels. Part I: Decentralised control”,Submitted for publication, 2003.

[11] Weyer E., “Control of open water channels. Part II: Centralised LQcontrol”, Submitted to for publication, 2003.

[12] Ooi S.K. and Weyer E., “Closed loop identification of an irrigationchannel”,Procedings of the 40th IEEE CDC, Orlando, USA, pp. 4338-4343, 2001.

[13] E. Weyer, “System identification of an open water channel”, ControlEngineering Practice, Vol. 9, pp. 1289-1299, 2002.

[14] Ooi S.K. and Weyer E., “Control design for an irrigationchannel fromphysical data ”,Procedings of European Control Conference ECC03,Cambridge, UK, 2003.

[15] E. Weyer, “LQ control of an irrigation channel”,Procedings of the 42ndIEEE CDC, Hawaii, USA, pp. 750-755, 2003.

[16] Bos, M.G(Ed.), “Discharge measurement structures”,International Insti-tute for Land Reclamation and Inprovement/ILRI, Waageningen, Nether-lands, 1978.

[17] D. McFarlane and K. Glover, “Robust Controller Design Using Normal-ized Coprime Factor Plant Descriptions, Vol. 138 ofLecture Notes inControl and Informantion Science, Springer-Verlag, Berlin; 1990.

[18] S. Skogestad and I. Postlethwaite,Multivariable Feedback Control, JohnWiley and Sons, Baffins Lane, Chichester; 1996.

[19] K. Zhou, J.C. Doyle and K. Glover,Robust And Optimal Control, PrenticeHall, Upper Saddle River, NJ; 1996.

[20] A. Packard and J. Doyle, “The complex structured singular value”,Automatica, 29(1), pp. 71-109, 1993

[21] G.J. Balas, J.C. Doyle, K. Glover, A. Packard and R. Smith, µ-Analysisand Synthesis Toolbox: User’s Guide, The MathWorks, Inc.; 1996, Foruse with MATLAB.

[22] Weyer E. ”Decentralised PI control of an open water channel”. InProceedings of IFAC 15th World Congress, Barcelona, Spain,July 2002.


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