tar-34-1-6-0903-47

45

Research Article

Turk J Agric For34 (2010) 45-58 TBTAKdoi:10.3906/tar-0903-47

Linear-like discrete-time fuzzy control in the regulation ofirrigation canals

mer Faruk DURDU*Adnan Menderes University, Water Resources Research Center (SUARGE), 09100 Aydn - TURKEY

Received: 25.03.2009

Abstract: A linear-like discrete-time fuzzy controller was designed to control and stabilize a single-pool irrigation canal.Saint Venant equations for open-channel flow were linearized using the Taylor series and a finite-difference approximationof the original nonlinear partial differential equations. Using the linear optimal control theory, a traditional linearquadratic regulator (LQR) was first developed for an irrigation canal with a single-pool, and the results were observed.Then a linear-like global system representation of a discrete-time fuzzy system was proposed by viewing a discrete-timefuzzy system in a global concept and unifying the individual matrices into synthetic matrices. This linear-likerepresentation aided development of a design scheme for a global optimal fuzzy controller in the way of the generallinear quadratic approach. Based on this kind of system representation, a discrete-time optimal fuzzy control law that canachieve global minimum effect was developed. An example problem with a single-pool was considered for evaluating theperformance of the discrete-time optimal fuzzy controller in the control of irrigation canals. The results obtained withthe optimal fuzzy controller were compared to the results obtained with a traditional linear quadratic regulator. Thediscrete-time fuzzy controller was the best for the operation of the canal system, reaching the optimal performance indexunder unknown demands.

Key words: Optimal fuzzy controller, linear quadratic regulator, canal automation

Dorusal ayrk zamanl bulank kontrol metodu ile sulama kanallarnn reglasyonu

zet: Bu almada tek havuzlu bir sulama kanalnda akn kontrolu ve reglasyonu iin dorusal ayrk zamanl bulankkontrol yntemine gre bir kontrol algoritmas gelitirilmitir. Sulama kanallarnda ak ifade eden Saint Venanteitlikleri, Taylor serileri ve dorusal olmayan ksmi diferansiyel eitliklerin sonlu-farklar yaklam kullanlarak dorusalhale getirilmitir. Optimal kontrol teorisi kullanlarak tek havuzlu sulama kanal iin geleneksel dorusal karesel regulator(LQR) gelitirilmi ve simulasyon sonular gzlemlenmitir. Sonraki aamada, sistem matrislerinin sentetik matriseklinde btnletirilmesi ile mevcut problemin ayrk zamanl bulank model hali tanmlanmtr. Bu model ile globaloptimal bulank kontrol tekniinin dorusal karesel reglator yaklam eklinde ifade edilmesi salanmtr. Bu yaklammodelinden hareket ile, dardan gelen etkilerden mimimum dzeyde etkilenen ayrk zamanl optimal bulank kontrolmodeli tasarlanmtr. Gelitirilen bu modelin sulama kanallarnn kontrolnde gsterecei performans deerlendirmekiin tek havuzlu bir sulama kanal rnek problem olarak seilmitir. Simulasyon sonras hem optimal bulank kontrolcuhemde gelenksel dorusal karesel reglatorden elde edilen sonular karlatrlmtr. Sulama kanallar iletimindekanaldan bilinmeyen taleplerin olmas durumunda ayrk zamanl bulank kontrolcunun, dorusal karesel reglatregre daha iyi performans gsterdii ve ak dzeni salad tespit edilmitir.

Anahtar szckler: Optimal bulank kontrolcu, dorusal karesel regulator, kanal otomasyon

* E-mail: [email protected]

IntroductionUncertainty is always bothersome in controlling a

real system, as a physical system is usually only partlyknown and difficult to describe, and has fewmeasurements available. Irrigation canals are operatedusing a variety of delivery schedules. Providing theright quantity of water at the right time increasesagricultural production. Supply-oriented systemshave not been able to provide the needed flexibility,in terms of the quantity of water and timing, toimprove crop yields and water-use efficiency. Thiscalls for a more flexible delivery schedule, calleddemand delivery.

The demand delivery schedule provides moreflexibility to water users than other delivery schedulesthat are in use today. With demand delivery operationof irrigation canals, variations in water withdrawalrates into lateral canals (disturbances) are not knownin advance; hence, these variations in flow rates areclassified as random disturbance actions on the supplycanal. In other words, the level of uncertainty in thedemand delivery schedule is high. In the absence ofinformation on the disturbances being obtained inadvance, meeting the random demands whenoperating canals becomes a difficult task.

In the past, the concepts of optimal control theoryhave been applied to derive closed-loop controlalgorithms for real-time control of irrigation canals(Balogun 1985; Reddy et al. 1992; Malaterre 1994;Begovich et al. 2007a, 2007b). Balogun et al. (1988),Hubbard et al. (1987), Reddy (1999), and Durdu(2003) applied the linear quadratic regulator (LQR)technique to open-channel flow control using alinearized, spatially discretized version of the SaintVenant equations; however, most of these studies dealtwith the traditional linear quadratic regulatortechnique. None of the above studies used an optimaldiscrete-time fuzzy controller to control irrigationcanals. Begovich et al. (2003) developed a controllerfor the real-time implementation and evaluation of afuzzy gain scheduling control regulating thedownstream levels at the end of the pools of a 4-poolopen irrigation canal prototype. Durdu (2005)designed an optimal fuzzy filter that employed theLyapunov function to formulate the fuzzy interferencerules to solve the state estimation problem ofcontrolled irrigation canals. Most fuzzy control

studies are based on the Takagi-Sugeno (T-S)-typefuzzy model, combined with the parallel distributioncompensation (PDC) concept and application ofLyapunovs method for stability analysis (Wu and Lin2002).

Wu and Lin (2002) developed a fuzzy systemrepresentation proposed to maturate the formulationand simplification of the quadratic optimal fuzzycontrol problem. This linear-like representationmotivates one to develop the design scheme for afuzzy controller using the general linear quadratic(LQ) approach. Fuzzy modeling can mimic a realsystem well and fuzzy control can support morerobust control than linear control is capable of (Wuand Lin 2002). Irrigation canals are regulated usingspatially distributed control structures (gates). In aglobal fuzzy control algorithm, the variation in theopening of the gate in the system is computed basedupon the information on water levels in the pools. Thefuzzy controller of an irrigation canal needs to knowthe current water level and must be able to set the gate.The controllers input will be the water level error(desired water level minus actual water level) and itsoutput will be the rate at which the gate is opened orclosed. The goal of the present study was to determinethe effectiveness of a global optimal fuzzy controllerfor the operation of irrigation canals in the presenceof arbitrary external disturbances (unknowndemands), and to evaluate the performance of thecontroller algorithm, as compared to a traditionallinear quadratic regulator.

Materials and methodsMathematical modeling of open-channel flowIn the operation of irrigation canals, decisions

regarding changes in gate opening in response toarbitrary (random) changes in water withdrawal ratesinto lateral or branch canals are required to maintainthe flow rate into the lateral canals close to the desiredvalue. This is accomplished by maintaining the depthof flow or the volume of water in a given pool at atarget value. This problem is similar to the processcontrol problem in which the state of the system ismaintained close to the desired value using real-timefeedback control. Linear control theory is welldeveloped and is easier to apply than nonlinear

Linear-like discrete-time fuzzy control in the regulation of irrigation canals

46

control theory (Reddy 1991). In the present study,therefore, a linear-like system representation of afuzzy system was employed.

Water conveyance equationsThe Saint Venant equations, presented below, were

used to model flow in a canal

(1)

(2)

in which Q = flow in the canal (m3 s-1), A = wettedarea (m2, ql = lateral flow (m

2 s-1), y = water depth (m),t = time (s), x = longitudinal direction of the channel(m), g = gravitational acceleration (m2 s-1), S0 = canalbottom slope (m/m), R = hydraulic radius, A/P (m), P= wetted perimeter (m), n = roughness coefficient(s/m1/3), and Sf = the friction slope (m/m), which isdefined as:

Sf = Q|Q|/K2 (3)

in which K = hydraulic conveyance of the canal(AR2/3/n) and R = hydraulic radius (m). In derivingEq. (2), the effect of the net acceleration termsstemming from removal of a fraction of the surfacestream was assumed to be negligible.

Lateral flow ratesLateral canals in the main canal are usually

scattered throughout the length of the supply canal.Manually controlled discharge regulators are used atthe head of lateral canals. The mathematicalrepresentation of flow through these structures isgiven as follows:

ql = Cdblwl(2g(Z-Zl))1/2 for submerged flow (4)

ql = Cd blwl(2g(Z-Es))1/2 for free flow (5)

in which ql = lateral discharge rate (m3 s-1), Cd = outlet

discharge coefficient, bl = width of the outlet structure(m), wl = height of gate opening of the outlet structure(m), Z = water surface elevation in the supply canal(m), Zl = water surface elevation in the lateral canal(m), and Es = sill elevation of the head regulator (m).Obviously, the flow rate through a head regulatordepends upon the water surface elevation in thesupply canal. The water surface elevation in the lateralcanal is a function of the discharge rate through thehead regulator. As such, this equation is an implicitequation. In the case of free flow, the discharge ratethrough the head regulator is independent of thewater surface elevation in the lateral canal; therefore,once the required discharge into a lateral is specified,the gate opening is adjusted to obtain the requiredflow rate through the head regulator, assuming thatthe water surface elevation in the supply canal ismaintained constant at the target level. When amanually controlled head regulator is used, forsimulation purposes the gate opening or the variationin the gate opening is specified as a function of time.Conversely, when an automated discharge rateregulator is used, for simulation purposes the lateraldischarge rate, as a function of time, is specified as aknown input, i.e. ql = fq(t).

Control structures (gates)In the regulation of the main canal, decisions

regarding the opening of gates in response to randomchanges in water withdrawal rates into lateral canalsare required in order to maintain the flow rate intolaterals close to the desired value. This isaccomplished by either maintaining the depth of flowin the immediate vicinity of the turnout structures inthe supply canal constant or by maintaining thevolume of water in the canal pools at the target value.When the latter option is used, the outlets are oftenfitted with discharge rate regulators. The water levelsor the volume of water stored in the canal pools areregulated using a series of spatially distributed gates(control elements); hence, irrigation canals aremodeled as distributed control systems. As such, inthe solutions of Eqs. (1) and (2), additional boundaryconditions are specified at the control structures, interms of the flow continuity and the gate dischargeequations, which are given by:

. F. DURDU

47

tA

xQ ql2

222+ =

( / ) t xQ A gA x

y S SQ 00 f2

22

22

22+ + + =c m

Qi-1,N = Qgi = Qi,1 (continuity) (6)

Qgi = Cdibiui(2g(Zi-1,N Zi,1))1/2 (gate discharge) (7)

in which Qi-1,N = flow rate through the downstreamgate (or node N) of pool i-1 (m3 s-1), Qgi = flow ratethrough the upstream gate of pool I (m3 s-1), Qi,1 = flowrate through the upstream gate (or node 1) of pool i(m3 s-1), Cdi = discharge coefficient of gate I, bi = widthof gate i (m), ui = opening of gate i (m), Zi-1,N = watersurface elevation at node N of pool i-1 (m), Zi,1 = watersurface elevation at node 1 of pool i (m), and i = poolindex (i = 0 refers to the upstream constant level ofthe reservoir).

Linearization and discretization of systemequations

The Saint Venant open-channel equations arelinearized about equations and the average operatingcondition of the canal is used to apply the linearcontrol theory concepts to the problem (Malaterre1997). After applying a finite-differenceapproximation and the Taylor series expansions toEqs. (1) and (2), a set of linear ordinary differentialequations is obtained for the canal with control gatesand turnouts (Durdu 2003):

A11Q+

j + A12z+

j + A13Q+

j+1 + A14z+

j+1 =A 11Qj + A12 zj + A 13Qj+1 + A 14 zj+1 + C1

(8)

A21Q+j + A22z

+j + A23Q

+j+1 + A24z

+j+1 =

A 21Qj + A22 zj + A 23Qj+1 + A24 zj+1 + C2(9)

where Q+j and z+

j = discharge and water-levelincrements from time level t + 1 at node j, Qj and zj= discharge and water-level increments from timelevel t at node j, and A11, A21,. A12, A22 are thecoefficients of the continuity and momentumequations, respectively, computed with known valuesat time level (t). Similar equations are derived forchannel segments that contain a gate structure, a weir,or some other type of hydraulic structure. The matrixform of the about equations for the canal can bedefined as follows (Malaterre 1994):

(10)

where f is any dependent variable on Q or z. From thematrix form of the above equations, the state of thesystem equation at any sampling interval (k) can bewritten, in a compact form as follows:

ALx(k+1) = AR x(k) + Bu(k) + Cq(k) (11)

where AL = n n system feedback matrix [left-handside coefficients of Eqs. (8) and (9)], AR = n n systemfeedback matrix [right-hand side coefficients of theEqs. (8) and (9)], B = n m control distributionmatrix, C = p n disturbance matrix, x(k = n 1state vector, u(k) = m 1 control vector, q =variation in demands (or disturbances) at the turnouts(m2 s-1), n = number of dependent (state) variables inthe system, m = number of controls (gates) in thecanal, p = number of outlets in the canal, and k = timeincrement (s). The elements of matrices A, B, and Cdepend upon the initial condition. The dimensions ofcontrol distribution matrix B depend on the number


48

( ) 1( )

A A A AA A A A

zf

zf

QzQzQzA

11 12 13 14

21 22 23 24

22

2

je

je

j

j

j l

j l

j

jL

22

22

2

2

2

2

2

2

=+

+

+

++

++

++

++

R

T

SSSSS

R

T

SSSSSSSSS

V

X

WWWWW

V

X

WWWWWWWWW1 2 344444 44444

( ) ( )

A A A AA A A A

zf

zf

A

QzQzQz

1j

ej

e

j

j

j l

j l

j

jR

11 1 1 1

21 22 23 24

22

2

2 3 4

22

22

2

2

2

2

2

2

++

+

+

+

+

l l l l

l l l l

R

T

SSSSS

R

T

SSSSSSSSS

V

X

WWWWW

V

X

WWWWWWWWW1 2 344444 44444

( )u

f A AA A

QQ

B C

00

je

p

p

11 11

21 2122 Od d

d

++

l

l

R

T

SSSSS = >

V

X

WWWWW G H

1 2 344 44 \

of state variables and the number of gates in the canal.The dimensions of disturbance matrix C depend onthe number of disturbances acting on the canal systemand the number of dependent state variables. Eq. (11)can be written in a state-variable form, along with theoutput equations as follows (Reddy 1991):

x(k + 1) = x(k) + u(k) + q(k) (12)

y(k) = H x(k) (13)

where = (AL)-1 *AR, = (AL)

-1*B, = (AL)-1*C, y(k)

= r 1 vector of output (measured variables), H = r n output matrix, and r = number of outputs. Theelements of matrices , , and depend upon thecanal parameters, the sampling interval, and theassumed average operating condition of the canal. InEq. (12), the vector of state variables is defined asfollows:

x = (Qi,1, Zi,2, Qi,2,Zi,N-1, Qi,N-1, Qi,N)

(14)

Linear optimal quadratic controlIn the irrigation control literature much attention

has been devoted to linear quadratic regulator designproblems, largely as a result of their elegant problemformulation, solution tractability, and robustproperties with respect to fairly large variations insystem parameters. The problem of designing a linearfeedback control system to minimize the quadraticperformance index can be reduced to the problem ofobtaining a positive definite solution of a matrixRiccati equation. An important characteristic oftransient performance of an open canal is its stability.Once a canal is disturbed from its original equilibriumcondition responses to the disturbance will result in astable, neutral, or unstable condition.

The stability requirement of any system is definedin terms of eigenvalues, which are the roots of thecharacteristic equation of matrix and must havevalues less than unity. The oscillatory behavior of acanals water surface is associated with the presenceof complex roots in the solution of the characteristic

equation of the system. The response amplitude growscontinuously if the absolute value of the complex rootsis greater than unity, decays to zero if the absolutevalue is less than unity, and oscillates at a constantamplitude if the real part of the roots is zero.Additionally, because of inertia, it is almost impossibleto derive the deviation in water surface elevation(error) instantaneously to zero. Thus, the output ofthe system lags the desired input and results inovershoot or oscillation of the water level about itsequilibrium position.

The objective of control theory is to find a controllaw that will bring an initially disturbed water surfaceto the desired target water level in the presence ofexternal disturbances acting on the canal. This can beaccomplished by applying a large proportional controlin which change in gate opening is proportional tochanges in flow depths and flow rates, as follows(Reddy 1991):

u(k) = -K(k) x(k) (15)

where K(k) = controller gain matrix. Controllabilityensures the stability of the system and maintains thewater level at any desired value by suppressing theinfluence of external disturbances. A canal is said tobe controllable if it is possible to derive it from anyinitial water level to any specified water level (state)within a finite number of steps. Eq. (15), which wasused throughout the study, is called the discrete stateequation and control law. This equation describes thecondition or evolution of the basic internal variablesof the system. The variables in the equation (i.e. x)are called the state variables. In optimal control theorythe elements of gain matrix K can be obtained byformulating the control problem as an optimizationproblem in which the cost function to be minimizedis given as follows (Reddy 1999):

(16)

subject to the constraint that:

x(k+1) + x(k) + u(k) = 0k = 0,,K (17)

. F. DURDU

49

( ) ( ) ( ) ( )J x k Qx x k u k R u ki l

KT

nxnT

mxmd d d d= +=

3

6 @/

where K = number of sampling intervals consideredto derive the steady state controller, Qxnxn = state costweighting matrix, and Rmxm = control cost weightingmatrix. The matrices Qx and R are symmetric, and tosatisfy the non-negative definite condition they areusually selected to be diagonal, with all diagonalelements positive or zero.

The first term in Eq. (16) represents the penalty onthe deviation of the state variables from the averageoperating (or target) condition, whereas the secondterm represents the cost of control. This term isincluded in an attempt to limit the magnitude of thecontrol signal u(k). Unless a cost is imposed for theuse of control, the design that emerges is liable togenerate control signals that cannot be achieved bythe actuator. In this case saturation of the controlsignal will occur, resulting in system behavior thatdiffers from the closed loop system behavior that waspredicted assuming that saturation will not occur.Therefore, the control signal weighting matrixelements are selected to be large enough to avoidsaturation of the control signal under normaloperating conditions. Eqs. (16) and (17) constitutea constrained-minimization problem that can besolved using the method of Lagrange multipliers. Thisproduces a set of coupled difference equations thatmust be solved recursively backwards in time;however, because irrigation canals run for a long timeand the dynamics of the canals are usually very slow,a steady state controller is more desirable. For thesteady state case, the solution for u(k) is the sameform as Eq. (15), except that K is given by:

K = [R + T P]-1 T P (18)

P is a solution of the discrete algebraic Riccatiequation (DARE):

T P T P S-1 T P + Qx = P (19)

where S = R + TS , R = RT > 0, and Qx = QxT = HTH 0. The solution of the discrete algebraic Riccatiequation is fundamental to the implementation ofoptimal control. The control law defined by Eq. (15)brings an initially disturbed system to an equilibrium

condition in the absence of any external disturbancesacting on the system (Reddy 1991). In the presence ofthese external disturbances, the system cannot bereturned to the equilibrium condition using Eq. (15).An integral control, in which the cumulative (orintegrated) deviation of a selected output variable isused in the feedback control loop, is required to returnthe system to the equilibrium condition in thepresence of external disturbances (Kwakernaak andSivan 1972; Kailath 1980). Integral control is achievedby appending additional variables of the followingform to the system dynamic equation (Reddy et al.1992):

xl(k+1) = Dx(k) + xl (k) (20)

in which xl = integral state variables and D = theintegral feedback matrix. This produces a new controllaw to the form:

u(k) = Kx(k) - Kl xl (k) (21)

The first term in Eq. (21) accounts for initialdisturbances, whereas the second term accounts forexternal disturbances. Eq. (21) predicts the desiredgate openings as a function of the measureddeviations in the values of the state variables (Reddyet al. 1992). In hydraulic engineering problems, thedepth of flow, flow rate, and velocity as a function ofdistance can be considered as the state or internalvariables. Sometimes, the volume of water in a givenreach of a canal can also be considered as a statevariable. In the present study the water surfaceelevation and flow rate were considered the statevariables. Given initial conditions [x(0)], u, and q,Eq. (16) can be solved for variations in flow depth andflow rate as a function of time (Reddy 1990). If thesystem is really at equilibrium [i.e. x (0) = 0 at time t= 0] and there is no change in the lateral withdrawalrates (disturbances), the system would continue to beat equilibrium forever; then there is no need for anycontrol action (Reddy 1990). Conversely, in thepresence of disturbances (known or random) thesystem would deviate from the equilibrium condition.The actual condition of the system may be either


50

above or below the equilibrium condition, dependingupon the sign and magnitude of the disturbances. Ifthe irrigation canal system departs from the stabilitycondition, the discharge rates into the turnouts mightbe different (either more or less) than the targetedrates. However, in canal conveyance systems, thepurpose is to keep these departures to a minimum sothat a nearly constant rate of discharge is maintainedthrough the turnouts.

Optimal quadratic fuzzy controllerIn the present study a linear-like quadratic fuzzy

control problem, developed by Wu and Lin (2002),was used to formulate the single-pool irrigation canalsystem representation (Figure 1). This systemrepresentation maturates the formulation of thequadratic optimal fuzzy control problem and a soundunification of the individual matrices into syntheticmatrices to generate a linear-like global systemrepresentation of a fuzzy system, which aided in thederivation of a theoretical design scheme for thequadratic optimal fuzzy controller (Wu and Lin 2002).The considered T-S type fuzzy model for the single-pool irrigation canal is as follows:

Ri : If xn is T1i,......, xn is Tni, thenx(k+1) = i(k)x(k) + i(k)u(k) (22)y(k) = H(k)x(k), i = 1,......,r

where Ri = ith rule of the fuzzy model, x1, .. xn =system states, T1i,..Tni = input fuzzy terms in the i

th

rule, x(k) = [x1,xn]T = state vector, y(k) = the

output vector (measured variables), and u(k) = =system input (i.e. control output or changes in gateopenings). i(k), i(k), and H(k) are, respectively, n n, n m, and n n matrices, whose elements are real-value functions defined on non-negative realnumbers, N. Throughout in this report, it is assumedthat i(k) is nonsingular for all k to ensure nodeadbeat response; in that case, i(k + 1) and u(k)cannot define i(k) uniquely, and the poles of theresultant closed-loop system are all located at zeropoints (Wu and Lin 2002). If the desired controller isa rule-based non-linear fuzzy controller, then theform of the equation is

Ri : If y1 is S1i,......, xn is Sni, then u(k) = ri(k),

i = 1,......, (23)

where y1,...yn = elements of output vector y(k),S1i,.Sni = input fuzzy terms in the i

th control rule, andu(k) or ri(k) = the control output (changes in gateopening) vector (Wu and Lin 2000). To describe aquadratic optimal fuzzy control problem for the givenT-S type rule-based fuzzy system in Eq. (22) withx(k0) = x0 and a rule-based non-linear fuzzycontroller in Eq. (23), k [k0, k1 1], find a control input(changes in gate opening) u(.) that can minimize thequadratic cost function

over all possible u(.) of class piecewise continuous,where Qx and R belong to symmetric positive semi-definite n n matrices. Since each penalty term in theperformance index [Eq.( 24)] is in reference to theentire fuzzy system and the controller, it is possible toformulate the distributed fuzzy subsystems and rule-based fuzzy controller into one equation from theglobal concept. As such, the well-known T-S-typefuzzy model was used to obtain the system stateequations as follows (Wu and Lin 2002):

and the control law is

(26)

hi(x(k)) and wj(y(k)) denote, respectively, thenormalized firing strength of the ith rule of the

. F. DURDU

51

( (.)) ( ) ( ) ( ) ( )

( )( ) ( )

J u x k Qx x k u k u k

x k R x k24

k k

kT

nxnT

Tmxm

1

1 1

0

1

d d d d d

d d

= + +=

6 @/

( ) ( ( )) ( ) ( )

( ( )) ( ( )) ( ) ( ) ( )

( ) ( ) ( )

x k h x k k x k

h x k w y k k r k

y k H k x k

1

25i

r

i i

i

r

j li j i j

1

1

d d d

d d

d d

U

C

+ = +

=

m

=

= =

/

/ /

( ) ( ( )) ( )u k w y k r kj l

j jd d=m

=/

( ( )) ( ( )) 1,with h x k and w y k where1i jj li l

r

d d= =l

==//

discrete-time fuzzy model and that of the ith fuzzycontrol rule (Wu and Lin 2002). Given the entiresystem state equations in Eq. (25), with the fuzzycontroller law u(k) in Eq. (26) and x(k0) = x0 , k [k0,k1 1], find a control input law (changes in gateopening), ri, i = 1,,, to minimize the quadraticperformance index

This kind of quadratic optimal control problem isobviously still too tough to deal with; therefore, thefollowing synthetic matrices, H((x)), W(y(k)), (k),(k), and U(k), can be used (Wu and Lin 2002):

where In and Im denote the identity matrices ofdimension n and m, respectively. Based on thesesynthetic matrices, Eqs. (25) and (28) can be rewrittenas follows:

x(k + 1) = H(x(k))(k) x(k) +H(x(k))(k)W(y(k))U(k), y(k) = H(k)x(k) (29)

With x(k0) = x0 , find the optimal control law(changes in gate opening), U(.), to minimize thequadratic performance index

Eqs. (29) and (30) represent the entire T-S-typefuzzy system that materializes the design of the globaloptimal fuzzy controller in the way of the generallinear quadratic (LQ) approach (Figure 2). It isnecessary for the process of integrating all distributedfuzzy subsystems into one equation to describe theentire fuzzy system in order to determine the globaloptimal controller. Eq. (25) provides a practical way towork out the global optimal solution; however, eventhough each fuzzy subsystem in the T-S model islinear, Eq. (25) is complicated and non-linear.Therefore, the synthetic form of system state equation[Eq. (29)] is lower down the order and adds to thedifficulty of the problem (Wu and Lin 2002). Now, adiscrete-time optimal fuzzy controller for a single-pool irrigation canal can be described using Eqs. (29)and (30). For each segmental dynamic fuzzy system

x(k + 1) = Hi(k) x(k) + HiWi(k)U(k),y(k)=Hx(k) (31)

where Hi = ith stage of H, Wi = i

th stage of W, andx(k0

i) is known. Then there is a unique n nsymmetric positive semi-definite solution (P) of thediscrete time algebraic Riccati-like equation

K = -WiT[WiWi

T]-1THiTP[In +

HiTHi

TP]-1Hi (32)

then the optimal control law will be u(k) =K x(k),more clearly

u(k) = WiT[WiWi

T]-1THiTP[In +

HiTHi

TP]-1Hi x(k) k [k0, ] (33)

Eq. (33) will minimize the quadratic performanceindex


52

( (.)) [ ( ) ( )

( ( )) ( ) ( ) ( )] (27)

( ) ( )

J u x k Qx x k

w y k w y k r k r k

x k R x k

k k

kT

lxl

i l lj j i

Tj

Tmxm l

j

1

1

0

1

d d d

d d d d

d d

= +

+

m l=

= =

/

/ /

( ( )) [ ( ( )) ..... ( ( )) ]( ( )) [ ( ( )) ..... ( ( )) ]

( )

( )( )

.( )

( )( )

.( )

( )( )

.( )

H x k h x k I h x k IW y k w x k I w x k I

kk

kk

k

kk

k

k

28

n r n

m m

r r r

1

1

1 1 1

d d d

d d d

d

d

U

U

U

C

C

C

U

==

= = =

l

R

T

SSSS

R

T

SSSS

R

T

SSSS

V

X

WWWW

V

X

WWWW

V

X

WWWW

( (.)) [ ( ) ( )

( ) ( ( )) ( ( )) ( )] ( )( ) ( )

J U x k Qx x k

U k W y k W y k U kx k R x k

30

k k

kT

nxn

T T

Tmxm

1

1 1

l

0

d d

d d

d d

= +

+

=/ ( (.)) [ ( ) ( )

( )( ) ( )]

J U x k Qx x k

U k W W U k34

k k

Tlxl

TiT

i

i0

d d= +3

=

/

The first term in Eq. (34) represents the penaltyon the deviation of the state variables (flow depthand flow rates) from the average operationcondition, whereas the second term represents thecost of control. Eqs. (34) and (33) constitute adiscrete-time optimal fuzzy control problem (aconstrained-minimization problem) that can besolved using the calculus of variations methodscombined with the Lagrange multiplier method toobtain the necessary and sufficient condition forglobal optimum (Wu and Lin 2002). This producesa set of coupled difference equations that must besolved recursively, backward in time. This procedureyields a time-varying controller gain matrix (K),which is defined by Eq. (32). A commonly employednumerical approach to finding the solution to Eq.(32) is by defining a Hamiltonian matrix (H), asfollows (Tewari 2002):

Of course, the definition of the Hamiltonianmatrix by Eq. (35) requires that be non-singular.Finding the eigenvectors of the inverse of theHamiltonian matrix helps obtain the steady stateelements of the controller gain matrix (K) (Reddy1999). Once the equations of the discrete-fuzzy

optimal controller are obtained and measured valuesfor one or more state variables in a given pool areavailable, the dynamics of the linear system can besimulated for any arbitrarily selected values ofexternal disturbance. In the present study, a singlereach of a canal was considered. This model predictsthe flow rate, Q(x,t), and the depth of flow, y(x,t),given the initial boundary conditions. The fuzzyoptimal controller equations were added assubroutines to this program. The controllers inputwill be the water level error (desired water level minusactual water level) and its output will be the rate atwhich the gate is opened or closed. A first pass atwriting a fuzzy controller for this system is as follows:

1. If (water level at downstream is okay) then(upstream gate is unchanged)

2. If (water level at downstream is lower than targetdepth) then (upstream gate is opened rapidly)

3. If (water level at downstream is higher thantarget depth) then (upstream gate is closedrapidly)

These 3 rules are not sufficient, as the water leveltends to oscillate around the desired level; therefore,another input, the water levels rate of change, shouldbe added to slow down the gate movement when thewater level gets close to the right level.

4. If (water level at downstream is higher thantarget depth) and (variation in water level rateis positive), then (upstream gate is closedslowly)

5. If (water level at downstream is lower than targetdepth) and (variation in water level rate isnegative), then (upstream gate is openedslowly)

. F. DURDU

53

( )( )

[ [ ][ ] [ ]

( )( )

( )

x kP k

H H H Q H H HH Q H

Xx k

P k

11

35

i iT T

iT

iT

iT T

iT

TiT T

iT

1

1 1

d

d

U CC U CC U

U U

++

=

-

>

>

>

H

H

H

1 2 344444444444 44444444444

upstream pool

downstream pool

y(i+1,N) u(i+1)

lateral withdrawal

gate 2 gate 1

y(i-1,N)

u(i)

Q

1 2 3 4 5

Nodes

y

Figure 1. Schematic of an irrigation canal pool.

q(z) disturbances

LQR Controller Canal

G(z) H(z)

U(z) Input Output

Figure 2. A feedback control system scheme.

These if-then rule statements are used to formulatethe conditional statements that comprise fuzzy logic(Wu and Lin 2002). In other words, to minimize thequadratic performance index [Eq. (34)] the bestoptimal control law [Eq. (33)] should be obtained byfinding the controller gain matrix [Eq. (32)]. In thoseif-then rules, water level is higher/lower than targetdepth and variations in water level rate is positiveare called the antecedent or premise, while upstreamgate is opened slowly is called the consequence orconclusion. Note that target depth and positive arerepresented as a number between 0 and 1, and so theantecedents are interpretations that return a singlenumber between 0 and 1. On the other hand, openedslowly is represented as a fuzzy set, and so theconsequence is an assignment that assigns the entirefuzzy set opened slowly to the output variable gateopening. In general, the input to an if-then rule is thecurrent value for the input variable (in this case,water level at downstream and variation in waterlevel) and the output is an entire fuzzy set (in thiscase, upstream gate opening fast/slow). Given theinitial flow rate and the target depth at thedownstream end of the pool, the model computed thebackwater surface elevation. Later, the downstreamflow requirement and the withdrawal rate into thelateral were provided as a boundary condition.Known state variables (flow depths and flow rates)were used in the controller subroutine to compute thechange in the upstream gate opening in order to bringthe depth at the downstream end of the pool close tothe target depth. Based upon this gate opening, thenew flow rate into the pool at the upstream end wascalculated and used as the boundary condition at theupstream end of the pool. This process was repeatedduring the entire simulation period (7000 s).

ResultsTo demonstrate and compare the feasibility of the

linear-like fuzzy controller, an optimal regulationproblem for a discrete-time single pool irrigationcanal was simulated. Using 5 nodes in the pool, astate-variable model with 8 variables was formulated.The state variable equation was supplemented with anoutput equation, in which the output variables werethe flow depths at the nodes of the pool. The variationin the depth of flow at the downstream end of the pool

was the controlled variable, and the control objectivewas to maintain the flow depth at the downstream endof the pool at a constant in the presence of randomdisturbance actions on the system. An exampleproblem obtained from Reddy (1990) was used, thedata of which was as follows: length of canal reach =5000 m, number of nodes = 5, number of sub-reachesused = 4, x = 1250 m, channel slope = 0.0003, sideslope = 1.0, bottom width = 1.7 m, turnout demand =2.5 m3 s-1, discharge required at the end of the canal =0.52 m3 s-1, upstream reservoir elevation = 103.2 m,downstream reservoir elevation = 101.14 m, targetdepth at the downstream end = 1.2 m, gate width =1.7 m, and gate discharge coefficient = 0.75. Thesedata were used to calculate the steady state values,which in turn were used to compute the initial gateopenings and the elements of the , , and H matrices[Eq. (12)], using a sampling interval of 30 s.

The initial gate opening was 0.59 m and 0.37 m,respectively, for the gates upstream and downstreamof the given pool. Figure 3 illustrates variations in flowdepth in the pool for both the linear-like fuzzyquadratic controller and a standard linear quadraticregulator (LQR). The system response was simulatedfor an unknown disturbance of +0.50 m3 s-1. Thepositive sign indicates an increase in the withdrawalrate from the turnouts. At the beginning of thesimulation, there were oscillations in the depth of flowat node 1. Later, with the introduction of waterrelease, the flow depth gradually increased and thevariations in flow depth approached a maximumdeviation of 0.0340 m for the standard LQR and0.0285 m for the linear-like fuzzy controller at 5000 s.After 5000 s, variations in flow depth at node 1reached a constant level for both controllers.

It is obvious that at node 1 the variations in flowdepth for the fuzzy optimal controller were less thanthose for the standard LQR controller. At node 2 thedepth of flow had some oscillatory behavior at first,because of downstream demand. After theintroduction of water release the flow depth increasedgradually, and the deviations in flow depth reached aconstant value of 0.0150 m for the linear-like fuzzycontroller and 0.0175 m for the standard LQR at theend of the simulation. At node 3 the variations in flowdepth had some oscillations at the beginning of thesimulation, and reached a value of 0.02 m for the


54

linear-like fuzzy controller and 0.021 m for thestandard LQR. With the release of water fromupstream of the pool, the depth of flow graduallyincreased, and the deviations approached a constantvalue of 0.019 m for the fuzzy controller and 0.021 mfor the standard LQR controller. At node 4, followinginitial oscillatory behavior the variations in flow depthgradually decreased, and reached 0.015 m for thelinear-like fuzzy controller and 0.018 m for thestandard LQR controller at the end of the simulation.The variations in depth of flow at node 5, in other

words at the downstream end of the pool, graduallyincreased at the beginning of the disturbance period,and finally approached a negative deviation of 0.041m for the standard LQR and 0.027 m for the linear-like fuzzy controller at the end of the simulation. Thenegative sign indicates that there was a decrease inflow depth at node 5 because of the demand at thedownstream end.

Figure 4 shows that the incremental gate openingwas lower for the fuzzy optimal controller than for thestandard LQR controller; in other words, gatemovement with the fuzzy optimal controller was morestable. At the beginning of the simulation thedeviations in gate opening reached a small negativevalue of 9.5 103 m for the standard LQR, versus 3 103 for the linear-like fuzzy controller. After 4500s the incremental gate opening reached anequilibrium position for both controllers. Figure 5indicates that to bring the initially disturbed system

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55

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0 1 2 3 4 5 6 7

0 1 2 3 4 5 6 7

0 1 2 3 4 5 6 7

Duration of simulation, sec (thousands)

Standard LQRLinear-like fuzzynode 1

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05


node 3

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02


node 5

Var

iatio

ns in

flow

dep

th, m

V

aria

tions

in fl

ow d

epth

, m

Var

iatio

ns in

flow

dep

th, m

Figure 3. Flow depth variations for standard LQR and optimalfuzzy controllers.

-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0 1 2 3 4 5 6 7Duration of simulation, sec (thousands)

Incr

emen

tal g

ate o

peni

ng, m

x 10-3

Linear-like fuzzyStandard LQR

-0.04-0.03-0.02-0.01

00.010.020.030.040.050.06


Cum

ulat

ive g

ate o

peni

ng, m

Standard LQRLinear-like fuzzy

Figure 4. Incremental gate opening for standard LQR and fuzzyoptimal controllers.

Figure 5. Cumulative gate opening for standard LQR and fuzzyoptimal controllers.

to an equilibrium position, the cumulative gateopening reached 0.048 m for the standard LQR and0.040 m for the fuzzy optimal controller at the end ofthe simulation. Lastly, the final gate openings for boththe linear-like fuzzy controller and the standard LQRcontroller reached a final value of 0.63 m and 0.638m, respectively (Figure 6).

DiscussionThe present study used 1 control variable (the

upstream gate) per pool and only 1 variable (eitherthe volume of water in the pool or the depth of flow atany one point in the pool) was maintained at thetarget value (Kwakernaak and Sivan 1972). Therefore,the upstream end gate alone was used for constantlevel control by freezing the opening of thedownstream end gate at its initial position. Aftercomputing steady state values, the control algorithm,written with MATLAB (1992), formulated a standardLQR controller and the results were obtained. Later,the algorithm designed a linear-like fuzzy controllerand we compared its results to those obtained withthe LQR controller.

The analysis began by evaluating system stability.All the eigenvalues of the feedback matrix werepositive and had values less than one. The system wasalso both controllable and observable. In thederivation of the control matrix elements () it wasassumed that both the upstream and downstreamgates of each reach could be manipulated to control

the system dynamics. The downstream end gateposition was frozen at the original steady state value,and only the upstream end gate of the given reach wascontrolled to maintain the system at the equilibriumcondition. The effect of variations in the opening ofthe downstream gate must be taken into accountthrough real-time feedback of the actual depthsimmediately upstream and downstream of thedownstream gate (node N). In the derivation of thefeedback gain matrix (K), R was set equal to 1000,whereas Qx was set equal to an identity matrix ofdimension 8 (the dimensions of the system). In theabsence of a well-defined procedure for selecting theelements of these matrices, these values were selectedbased upon trial and error.

The downstream end of the pool (node 5)illustrated best that the linear-like fuzzy controllerresulted in less deviation in flow depth than did thestandard LQR controller. As additional water that wasreleased into the pool reached the downstream end,the depth of flow gradually returned to the steadylevel, i.e. the depth at the downstream end of the poolwas maintained constant. In all the nodes consideredmaximum deviation in depth of flow occurred at thefirst and last nodes of the reach. Because the turnoutswere located at the downstream end of the reach, asthe flow rate into the lateral increased, the depth offlow at the downstream end decreased rapidly.Conversely, maximum increase in depth of flowoccurred at the upstream end of the reach. This wasdue to the increased opening of the upstream gate andcompensated for the disturbances at the downstreamend or turnout. At all the nodes variations in flowdepth obtained with the optimal fuzzy controller wereless than those obtained with the standard LQRcontroller. Thus, the fuzzy optimal controllerprovided better stability for the controlled single-poolirrigation canal.

Considering the position of the upstream gate,which was close to the equilibrium value at the end ofthe simulation period, it is evident that the systemwould eventually return to the equilibrium condition.During the simulation use of the fuzzy optimalcontroller resulted in less up and down movement of


56

0.550.560.570.580.590.6

0.610.620.630.64


Fina

l gat

e ope

ning

, m

Standard LQRLinear-like fuzzy

Figure 6. Final gate opening for standard LQR and fuzzy optimalcontrollers

the gate, and provided enough water for thedownstream end of the canal reach. In the absence ofany other disturbances (changes in withdrawal rates)the gate will return to its equilibrium position atsteady state. Once again, the performance of bothcontrollers was evaluated, assuming that values for allthe state variables (8 in this case) were available.Variation in upstream gate opening and deviation inthe depth of flow at the downstream end of the poolwere satisfactory (Figures 3 and 5). Obviously, therewere some differences between the linear-like fuzzyand standard LQR controllers in terms of gateopening and flow depth. Both models maintained thedownstream depth at the target value; however,application of the linear-like fuzzy model algorithmmaintained the downstream depth at the target valuewith less deviation in gate opening and flow depth.The overall results of this study show that theproposed linear-like fuzzy controller provides betterstability and offers an efficient alternative to atraditional LQR controller when dealing withuncertainty (increase in withdrawal rate from theturnout). Based on the results of the simulationmodel, it is clear that the optimal fuzzy controlleralgorithm is more suitable than the traditional LQRmethod for the regulation of irrigation canals.

ConclusionsA quadratic optimal fuzzy control problem was

formulated for constant-level control of irrigationcanals. A T-S-type fuzzy system that materialized thedesign of a fuzzy controller based on the way ofgeneral linear quadratic (LQ) approach wasconsidered. Individual matrices were unified intosynthetic matrices to generate a linear-like globaldiscrete-time fuzzy system. For minimizing thequadratic performance index, the discrete-time fuzzycontrol law was shown to be the best for the single-pool system. The performance of the linear-like fuzzycontroller was compared with that of a standard LQRcontroller, in terms of variations in the depth of flowand the upstream gate opening. The linear-like fuzzycontroller provided both good stability andperformance under unknown withdrawals from theirrigation canal. Overall, the performance of thelinear-like fuzzy control technique for constant-levelcontrol was better than that of the full-state feedbackLQR controller (assuming all the state variables in thesystem were measured). In the present study it wasassumed that all the state variables were measured inthe canal. As it is very expensive to measure all flowdepths and flow rates for each node, an optimalestimator should be used in the design of a regulatorand fuzzy controller in any subsequent research.

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Balogun OS (1985) Design of Real-time Feedback Control for CanalSystems Using Linear Quadratic Regulator Theory. PhD thesis,University of California at Davis, Department of MechanicalEngineering, p. 230.

Balogun OS, Hubbard M, DeVries JJ (1988) Automatic control ofcanal flow using linear quadratic regulator theory. Journal ofIrrigation and Drainage Engineering 114: 75-101.

Begovich O, Salinas JA, Ruiz VM (2003) Real-Time implementationof a fuzzy gain scheduling control in a multi-pool openirrigation canal prototype. In: Proceedings of the InternationalSymposium on Intelligent Control, Huston, Texas, pp. 304-309.

Begovich O, Ruiz VM, Besancon G (2007a) Predictive control withconstraints of a multi-pool irrigation canal prototype. LatinAmerican Applied Research 37: 177-185.

Begovich O, Felipe JC, Ruiz VM (2007b) Real-time implementation ofa decentralized control for an open irrigation canal prototype.Asian Journal of Control 9: 170-179.

Durdu OF (2003) Robust Control of Irrigation Canals. PhDDissertation, Colorado State University, Department of CivilEngineering, Fort Collins CO, p.280.

Durdu OF (2005) Control of transient flow in irrigation canals usingLyapunov fuzzy filter-based Gaussian regulator. InternationalJournal of Numerical Methods in Fluids 50: 491-509.

Hubbard M, DeVries JJ, Balogun OS (1987) Feedback control of open-channel flow with guaranteed stability. In: Proceedings of the22nd Congress of IAHR, Lausanne, pp. 408-413.

Kailath T (1980) Linear Systems. Prentice-Hall, Englewood Cliffs, N.J.

Kwakernaak H, Sivan R (1972) Linear Optimal Control Systems, JohnWiley and Sons, New York, N.Y.

Malaterre PO (1997) Multivariable predictive control of irrigationcanals. design and evaluation on an a 2-pool model. In:Proceedings of International Workshop on Regulation ofIrrigation Canals. Morocco, pp. 230-238.

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Matlab and Simulink (1992) A program for Simulating DynamicSystems, MathWorks Inc., New York.

Malaterre PO (1994) Modelisation, Analysis and LQR OptimalControl of an Irrigation

Canal. PhD dissertation. LAAS-CNRS-ENGREF-Cemagref, France.

Reddy MJ (1990) Local optimal control of irrigation canals. Journal ofIrrigation and Drainage Engineering 116: 616-631.

Reddy MJ (1991) Design of a combined observer-controller forirrigation canals. Water Resources Management 5: 217-231.

Reddy JM, Dia A, Oussou A (1992) Design of control algorithm foroperation of irrigation canals. Journal of Irrigation andDrainage Engineering 118: 852-867.

Reddy JM (1999) Simulation of feedback controlled irrigation canals.In: Proceedings of the USCID Workshop, Modernization ofIrrigation Water Delivery Systems: pp. 605-617.

Tewari A (2002) Modern Control Design with Matlab and Simulink,Wiley, New York.

Wu JS, Lin CT (2000) Optimal fuzzy controller design: local conceptapproach. IEEE Transactions on Fuzzy Systems 8: 171-185.

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