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Three-Tank-SystemTS200 Practicalnstructions

Practicalnstructions

\ '-, Date: 5.05.1996

Practical nstructions


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Three Tank SystemDTS200 lntroduction

1 IntroductionThe control of nonlinear systems, particularlymulti-variable systems,plays a more andmore mportantpart n the scopeof theadvancingautomationof technicalprocesses.Due to the ever increasingrequirementsofprocesscontrol (e.g. response ime, precision, ransferbehavior) nonlinear controller designs are neccessary.One design method with practical applications wasdevelopedamong othersby Prof. Dr.-Ing. E. Freundandsuccessfullyusedfor trajectorycontrol of robots. n thiscontext it is about the so-called principle of nonlinearcontrol and decoupling. The decoupling refers to theinput/output behavior, i.e.: after successfuldecouplingevery nput affects only the colrespondingoutput.Thus tis possible, o subdivide he multi-variablesystem ntosubsystems which are mutually decoupled. Thesesubsystems re inear, .e. simpleto analyze.Bymeansofdetermining freely choosable parameters, he transferbehavior of the subsystemscan be designedsubject tosome estrictions.

During this laboratoryexperiment, he application of thisprinciple to a three-tank-system with two inputs andthreemeasurable tatevariables s to be examined.To thatend, he step esponse nd he disturbancebehaviorof thecontrol system ollowing thecontrollerdesign s analyzedand compared with those of a standard Pl-controlledsystem.Because he theroretical oundationsare formulated n ageneral manner, you should be able to apply the"Nonlinear System Decoupling and Control" even tomore complicated systems after carrying out theexperiment.

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Practical nstructions 1 - 1


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Three Tank SystemDTS200 Theoretical oundations

2 TheoreticalFoundationsThe nonlinear control and decoupling is applicable to

nonlinear, time-variant multi-variable systems. Thesystemdescriptionmust have he following form:

A fundamental equirements that he numbersof systeminputsand outputsmustbe dentical.Now a state eedback n the ollowine form is ntroduced:u(0=F(x,t)+G(x,t) (t), 8q.2.3where:F(x,t) : column ectorof dimensionm

G(x,t) : in (x,t) non-singular m x m)-matrixw(0 : new m-dimensional nput vector (setpoint

vector)Figure2.1 showsgraphically heclosed oop systemwithD=0:Now onehas o determineF andG such hat he -th inputw, (i=1,...,m)only affects he -th outputy'. Moreover, tis possible o adjust the dynamics of the decoupeledsubsystems y placing he poles.The difference orderd referring to the i-th output signalyi is of great importance for the decoupling. Thedifference order indicateswhich total time derivative ofthe output y, is directly affected by the input u,. It is ameasure of the number of poles that can be placed

dx(t)/dt=A(x,t)+B(4,t) u(t)y(t)=C(x,t)+D(x,t) u(t)

(0u(t)v(0A(x,t)B(x,t)C(x,t)D(x,0

statevectorwith dimension (xr(t),xr1t;,....,xn(t))input vectorof dimensionmoutput vectorof dimensionm(n x l)-matrix(n x m)-matrix(m x l)-matrix(m x m)-matrix

8q.2.1E.q.2.2

with the nitial conditions: (tp=fo and he definitions:

Figure2.1 : Block diagramwith systemdecoupling

Plant(x,t)F(x,t)

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Theoretical oundations Three Tank SystemDTS200

arbitrarily yi'(t) =/t C,(X,t)+ {/x C,(x,t)J]A(1t)+B(x,t)(t)}In order to define the difference order, in the followingthe -th componentof the outputvector s used nsteadof =/t C;(X,t)all the m components: + {/x [C'(x,t)l] A8,t)+ {/xCi(x,t)} (x,t) (t) Eq.2.8y1(t)=C1(x,t)+Q(x,t)(t) 8q.2.4

For simplification, now the following nonlinear,where: time-variantoperato.Nou is defined:C,(x,t) : i-th component f the vectore(x,t) Nok C,(x,t)= /t (t o* .t C,(x,t))+ {/x No*-tCig,t))}A(x,0 8q.2.9Q(x,t) : i-th row vector of the matrix D(x,t) where:In caseQ(x,t) * 0 , the difference rderd, s now defined k=|,2,...as ollows:

and the initial value:di=o

Noo C,{x,t)= Ci(x,t)In this case,aswill be shown ater,decoupling s possibleusing directcompensation. Using this operator n 8q.2.8, heresult s:In caseD,(x.t)=O.he result s:| ' _ . , _ .dir' Yi'(t) ; h ,,l,?c,E,t)J)(x,t)(t) Eq.2.r0In order o determine heexactvalue n this case. he -th In case:output equation: ta/axNoo i(r,t))lB(x,t) o,y1=Ci(x,t) 8q.2.5

the differenceorder di=1 is defined. f in other case hisis repeatedlydifferentiatedwith respect o time, which is term is equal to the null vector, the seconddiffentiationshown exemplaryly in the following. The first must be carriedout:diffentiationyields: __ ) _Y;"(t) ; ilh ,".t'cig,t))r(x,t)(t; Eq.2.rly1'(t) =dldt C,(x,OJ=/tC'(x,t)+ /x CiE,t)]x'(t) 8q.2.6 Proceedingn this manner, he final resultof the -thdifferentiations:where: y,0)= Noj c,{x,ty+ tE/Ex No"t c,(x,t))J (x,t)u(t)/x Cift,t)]= (/x.,C;(x,t)],....,/x,C,(x,t)])8q.2.7 where ventually:Substitutingqq.2.lnto8q.2.7 esultsn: 1/xNoFt ci8,t))l B(x,t) 0

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Three Tank SystemDTS200 Theoretical Foundations

for the irst time.This means, he nputvectoru(t) directlyaffectsthe j-th differentiation of the output signal y,(t).With this, the difference order of the i-th subsystemresults n:di = jIn summary the difference order can be defined asfollows:a) f Q(x,t) * 0 : d.=0 8q.2.12U; f Q(x,t) = 0 : d,=mln1i:[/x oj-t Ct(x,t))]B(x,t)*Q) F,q.2.13For simplification t is assumedn the ollowing , that d,(i=1,..,m) s constantwith respect o all x(t) and .Using the above mentioned method for all m outputs ofthe nonlinear, ime-variantsystem, he result s:y*(t)=g* x,r)+D*(x,t) (t) Eq.2.14

where:y*(o=(yI(ot)(r),....,y,,rtdm)1t;)T

C*(x,t) : m-dimensionalcolumn vector\ D*(x,t) : (m x m)-matrix;

In thiscase he -th component f C*(x,t) canbe statedasfollows:c*,(x,t)=Nodi c,{x,t) Eq.2.15Moreover the i-th row vector of the matrix B*1x,t; isdefined bv:^* , lDi (l't) fr dt=oD-,(x't)=lrurtf*X'-rc, {x,t)l (x,t) rd,*0Eq.2.r6On the assumption hat the rank of the matrix D*(x,t) istl-", constant, ll its row vectorsareunequal o thenull vector.

8q.2.14 is the initial equation o derive the decouplingmatricesand he introductionof assignable ynamicsofthe decoupled ubsystems.If the relation:u(t)=F, ^,t)=-D*-lx,t)C (x,t)is substitutedn 8q.2.14,he esults:r.(t)=0

F,q.Z.17

Eq.2.18

Eq.2.2l

This means,each of the m outputs of the multi-variablesystem s decoupled. xtending he above elation n thefollowing manner:u(t)=pt1*,t)+GG,t)w(t)where:G(x,t)=B*-l(x,t) L

=d iag{11} , ( i=1,2 , . . . ,m)yields:y'(t)=L w(t)

Eq.2.l9

8q.2.20

8q.2.22

8q.2.238q.2.24

This form of feedback additionally allows a freeadjustment of the input amplification of the i-thsubsystem.n order o make t possible o influence hedynamic of the decoupled ubsystem he above elation schangedas ollows:u(t)=F1^,1)+G(x,t) (t )where:F(x,t)=F, x,t)+Fr(x,t)F"(x,t)=-P*-l (*,t) M*1x,t;andM*1x,t; : m-dimensional ector

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Theoretical Foundations Three Tank SystemDTS200{

Substitutinghis relation nEq.2.l4 yields:y*(t)=-M*(x,t)+L w(t) Eq.Z.25The vector M*(x,t) has to be determinedsuch that thedynamicof the m subsystemsan be changed singpoleshifting andnew couplings are avoidedas well.A suitable elation of the i-th componentof the vector

rfiM (x,t) canbe statedasfo frd,=o l "M ,(x,t)=jr^ 11k , . . : : t : t1g,.o6,..o , (x,t) frdr*O Eq.Z.26The constant actorsaki are freely assignablewith:i=| ,2, . . ,m ndk=0,1, . . , (di- lIn can be shown (substitutingEq.2.26 nEq.2.25) thateachcomponentof 8q.2.25 with di*O, using the abovechoice for M*,(x,t), can be described n the followingform:

y1(di){t)*u(0,-,),y,(ot-l)1t;*...*u61y1(t)=1,,(t) F,q.2.27Accordingly each subsystem results in a lineardifferentialequationof d,-th order.Thus, he nonlinear, ime-variant, multi-variable systemis decoupled.The transfer behavior of the subsystemswith d,*0 is describedby8q.2.27.In summaryF and G aredeterminedas ollows:F(x,t)=-p*-t (o,t) { c*(x,t)+M*(*,t) }G(x,t)=D*-l(x,t)LMoreover t has o be considered hat hedifferenceorderd, is invariant with respect to a transformationof thesystem into different coordinate systems. Therequirements f the determinationof d, howeverdependstronglyon the choice of the basis system.This is alsotrue with respect o the further stepsof determining healgorithmsof decouplingand control.

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Three Tank SystemDTS200 The System Three-Tank-System"

3 TheSystem"Three-Tank-System3,1 The PlantThe following figure 3.1 shows he principal structureoftheplant.The usedabbreviationsare described n the following.

The plant consitsof threeplexiglascylindersTl, T3 andT2 with the crosssectionA. Theseare connectedseriallywith each otherby cylindrical pipeswith the crosssectionSn.Located atT2 is the singleso callednominal outflowvalve. It also has a circular cross section Sn. Theoutflowing liquid (usuallydistilled water) is collected na reservoir, which supplies he pumps I and 2. Here thecircle s closed.H-u* denotes hehighestpossible iquid level. n case heliquid level of Tl or T2 exceeds this value thecorrespondingpump will be switched off automatically.Q1 and Q2 are he flow ratesof the pumps I and2.

Technicaldata:- A=0.01 4 mz- Sn=5*10-5m2- H**=62cm (+/- 1cm)- Q **=Q2**= I 00mltr/sec=6'0ltr/minThe remarkable feature of the used "Three-section-diaphragm-pumps"with fixed pistonstroke Pump I and2) is the well defined flow per rotation.They are drivenby DC motors.For the purposeof simulating clogs or operatingerrors,the connecting pipes and the nominal outflow areequippedwith manually adjustableball valves, whichallow to close he corresponding ipe.For thepurposeof simulating eakseach ankadditionallyhas a circular opening with the cross section St and amanually adjustableball valve in series.The followingpipeends n the reservoir.The pump flow ratesQ1 and Q2 denote he input signals,the iquid levelsof Tl andT2 denote he outputsignals,which have to be decoupled,of the controlledplant. The

Pump PumpA

Hr"*

,S,It - -r)Sn

Connection i

\ l\Leakage pening

Figure 3.1 : Structureof theplant



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The System Three-Tank-System Three Tank SystemDTS200 (

necessary level measurementsare carried out bypiezo-resistivedifferencepressure ensors.At eachof ita pressure ine measureshe pressure f the atmosphere.3.1.1 TheControllerHere a digital controller realized on a microcumputersystem s used.The connection o the controlledplant scarriedout by l2-bit converters.TheA/D-converter is required o read he iquid levelsofthe tanksT1, T3 andT2.It is initialized or the voltagerange: -10V...10V. This results in a resolution of4.88mV. Two- D/A-converters are used to control thepumps. The following figure 3.2 shows the principalstructureof the completecontrol loop.The control software s designed uch hat the decoupledsubsystems can be controlled additionally byPl-controllers. Furthermorea Pl-control of the liquidlevelsof tank 1 and2 spossible venwithoutdecoupling.The respective ontroller configurationcan beseenat thescreen.

Each controlleradjustment s well as he setpointscanbechanged n-line.The controlsignalsand he statevectorcomponents i.e.the pump flow ratesand the liquid levels) can be storedin the RAM of the microcomputerand can be plottedafterwardson aplotter.The controllerstaysactiveduringthe graphical output, but inputs cannot be carried outduring this time.Operating he programwill be described ater in detail.3.1.2 TheSignalAdaptionUnitThe purposeof the signal adaption unit is to adapt thevoltage evelsof theplantand he converter o eachother.Here the output voltagesof the sensors re adapted o themaximum resolutionof the A/D- convertersand on theother hand the output voltage range of theD/A-converters is adapted the servo amplifier of theconesponding ump.

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L-

PC Actuator PlantDatascreen

A/D.D/AConverter

Plotter

3-Tank ystem

Figure3.2: Control oop structure



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Three Tank SystemDTS200 TheSystem Three-Tank-System

3.1.3 TheDisturbance odulUsing switches, ensor ailuresof the evel measurementcan be simulated. The corresponding sensor outputsupplies a voltage which colresponds o the liquid level0.Alternatively it is possible to scale the measured iquidlevel between the real height (L00Vo)and the height 0()Eo)using a potentiometer.Two otherpotentiometers llow to simulatedefectsof thepumps.To do this the control signal can be reducedup to}Voof its original value,which is equivalent o decreasingthepump flow rate.3.2 Mathematical odel

,,' The following figure 3.3 once again shows he structureof the plant to define the variablesand theparameters.

: outflow coefficients [ ]: liquid levels m]

flow rates m3/sec]supplying low rates m3/sec]section f cylinder m2]

sectionof leak opening m2]sectionof connectionpipe [m2]

qijQrQzAslsn

A dh,/dt=Qr-QrsA dhr/dt=Q,r-Q:zA dhy'dt=Q2+Q3;Q2s

where -7,2,3 and i )=[ (1,3);(3,2)(2,0)]If the balanceequation:A dh/dt=Sumof all occuring low rates Eq.3.1is used or all of the three anks, he result is:

Eq.3.2Eq.3.3F,q.3.4zj

hi

Figure 3.3 : Principle structure o define thevariablesandparameters

Pump Pump2

A

h 5T3

lt'h 2odzt

Q''t Qsz dzz

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TheSystem"Three-Tank-System" Three Tank SystemDTS200 (

The still unknown quantitiesQrg, Q:z and Qro can bedeterminedusing the generalizedTorricelli-ruIe. It canbe statedasdh/dt=a6)+BQ : state quationy=(hl,h2)T : output equation

Eq.3.r4Eq.3.15

sgn(z) : sign of the argumentzAh : liquid level differencebetween wo tanks

connectedo eachotheraz : outflow coefficient (correcting factor,

dimensionless,eal values angingfrom 0to 1)q : resulting low rate n the connectingpipeSo the result or the unknown quantities s:

q=azsngn(ah)e lln l)1/2where:g : earthacceleration

Q13=az,n gn(h,-h) (2e n,-nr ;t/2Q32=azrSngn(hr-h,) (2e lnr-nzlftzQ2s=azrSn(2g hr)t''With the vectordefinitions:h=(h1,h2,\)TQ=(Ql,Q2)rA(D=(-Qrs,Qgz-Qzo,Qs-Q:z)r tlAy=(h1,h2)rand hematrixdefinition:

With this the plant is described ompletely. Because hephysical values h and Q are not accessible directly,convertersand sensorsa"re sed. Theseare parts of theplant and are described using their characteristics(characteristics f pu-ps and sensors). urthermore thevaluesof the outflow coefficients are required to get aquantitativedescriptionof 8q.3.14 andto carry out thenonlinear decoupling in practice. Therefore, theneccessarydata are determined nteractively using thecontrolprogrambefore he actualcontrol starts. (The model described y Eq.3.1418q.3.15s now usedasthe initial basis for the nonlinear decoupling of theThree-Tanks-System. The corresponding controllerdesign is carried out within the framework of thepreparations.The solutions of these preparations aretherefore the fundamental basis to cuurv out theexperiment.

t

Eq.3.5

Eq.3.6F,q.3.7Eq.3.8

8q.3.9Eq .3 . l 0Eq .3 .11Eq.3.l2

E=*[il Eq.3.13the abovesystemof equationscanbe transformed o:3,-4 Practical nstructions


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Three - Tank - System DTS200 Preparationsor the LaboratoryExperiihent

4 Preparationsor theLaboratoryExperiment4.1 GontrollerDesign

a) Determine he differenceordersdt,dz of the model asdescribed y Eq.3.14 8q.3.15.Hint: Either use the general definition or transformEq.3.I4tEq.3.15 irectly o a form similar o Eq.2.14withD.,(x,t) unequal .What is the behaviour of each subsystem after thedecoupling?b) Determine the vector C*1x,t; and the matrix D*1x,t;.What s he ank of D*(x,tX Can hesystem edecoupled?c) Determine he vector Fr(x,t) and he matrix G(x,t).d) Determine the vector M*1x,t) with freely assignableparametersa* using a suitableformulation. Determinethe vectorF^(x.t).e) Determine the overall transfer behaviour of thedecoupled ubsystems. oughlydrawthe step esponses.What is theconditioneachof the eal controllerparameterl, and a* has o meetso that the transferbehaviour of thesubsystemss stablewith an overallamplification of one?

i-thdecoupledsubsystemi=1 2

Fisure4.1 : Pl-control of the subsvstems

4.2 Disturbance ehaviourncaseof a leak n tank2a) How do themodelequations q.3.1418q.3.15hangein case of a leak with the section St and the outflowcoefficientazrintank 2 ?The eak s assumedo be ocatedat the evel of the connection ipes.b) What is the behaviour of the subsystemsn case hecontrollerdesigndeterminedn 4.1 is used?To do this,formulatethe correspondingdifferential equationsof thesystems. re the subsystemstill decoupled?c) Compute the stationary output value of the Zndsubsystem outputsignalhr) in caseof this disturbanceand a constant input signal. Is there a steady statedifferencecomparedwith the disturbance-freease?f so,what s its value?4.3 Pl-Controlof the decoupledsubsystemsThe decoupled subsystems can be Pl-controlledadditionally in accordance with the following blockdiagram:a) Compute the overall transfer function:Gg(s)=Hi(s)AVEt(s)

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Preparationsor the LaboratoryExperiment Three Tank SystemDTS200 (-

b) What is the condition the real parameters ,,k, and P,have to fulfill so that each Pl-controlled loop is stablewith an overall amplificationof one?c) What is the overall transferbehaviour n casePi=O?

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Three Tank SystemDTS200 Carrying-Outhe Experiment

5 Carrying-OutheExperimentStart he controller program (furtherdetailswill be found

in the chapter nstructions or operating he computer).5.1 Determination fCharacteristics nd OutflowGoefficientsBefore you start with the experiments he system has obe calibrated. To do this you select the menu itemParameter from the upper menu bar.A pulldown menuwill appear offering further items for the systemcalibration.a) Use the menu item Characteristic Liquid LevelSensor to determine the characteristic: iquid level ADC input voltage by experiment.Control the pumpsmanually n this process switches o "Manual").b) Change o the determinationof the outflow coefficientsby meansof the menu item Outflow Coefficient and useexperimentalmeasuring.Approximately 60 sec after theinteractive start, the determination s performed by theprogramwhich computes he following equations:azr- Ah l'l(-Sn (2g (h,-h, ))r/2)azr- A(h,' +hr' +h3' /(-Sn 1Zg tr)r/z)azr- A(h,' +hr' )/(-Sn Qg 6r-hr))r/?)Theseequations anbederived rom themodellequationsEq.3.l4 with qr=qz=O and hr>=hr>=hr.Record he outflow coefficients.c) Use the menu item Characteristic Pump FIow Rateto determine he characteristic:DAC output voltagepump supply using experimentalmeasuring(Do not usethe defaultvalues ).

Here, according o an amountof 8 DAC output voltages,the difference of levels which is measured n a timeintervallof 20 sec s used o determinehepumpflow rates(supplies).Please ollow exactlv the instructions of the controllerprogram.After the determinationof theplant parameters he menuitem File Save Parameter may be used to store thesevalues o the hard disk.The window called "DTS200Monitor" is located n thelower part of the screen.ts first line displays he currentsystemstate respectively he selectedcontrol structure.The following lines containcurrent data of the plant andof the controller like liquid levels and flowrates. Thestatusof theplot buffer used o store he measurementssdisplayed n addition.A graphical evaluation of any experiment result ispossibleonly if previously the storageof measurementswas switchedon.Now open he connectionvalvesand henominal outflowvalve Start the controller by meansof the menu itemRUN.5.2 Behaviorof Reference ndDisturbance ariablewithout Pl-controlThe following tasksrequire the control structurewith adecoupling network (select the menu itemDecoupling-Controller from the menuRUN andpromptwith 'OK'). The accompanying arameters adjustableby selection of the menu item Decoupling-Controllerfrom thepulldown menu Parameter. As mentionedwiththe preparation asks he parameters re set according o'Decoup'=li=aoi meaning that the amplification of thedecoupled ubsystemss equal o 1.



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Carrying-Outhe Experiment Three Tank SystemDTS200{

a) StepResponsesAt first adjust the decoupling parameter to'Decoup'=0.03and choose he setpointswr-32cm (tankl) and w2=20cm (tank 2) (Use the menu item AdjustSetpoint from the menu RUN, enter the correspondingsetpointvalues with a constantsignal shapeandpromptwith 'OK'). Wait until the steady tate s reached.Nowswitch on the storing of measured ata with a measuringtime=240 sec (Use the menu item Measuring from themenu RUN, enter appropriatevalues and prompt with'oK').Now generatea step of the setpointw, (tank 1) to 37cm(Use the menu item Adjust Setpoint from the menuRUN, enter the correspondingsetpoint value with aconstantsignal shapeand prompt with 'OK'). After thesteadystate s reached approximatelyafter 5OVo f themeasuring time) change the decoupling parameterto'Decoup'=0.04and generate step of the setpoint w,(tank 2) to 25cm (Use the menu item Adjust Setpointfrom the menu RUN, enter the correspondingsetpointvaluewith a constant ignalshapeandpromptwith 'OK')The measuring s finished when he status ine of the plotbuffer displays the message Buffer filled". Using themenuView and ts menuitem Plot Measured Data themeasured ataaredisplayable n a graphic epresentationon the screen.The graphic may be sent to a plotter orprinter n addition.Furthermore he datamay be saved na file using the menu tem Files - Save Plot Data.Notes or the user:The seriesof events,storing data and plot output willoccurduring all of the following experiments.Thereforeit will not be mentionedexplicitly in the following.

b) Disturbance behaviour in caseof leaksAdjust hedecouplingparametertoDecoup'=0.2and hesetpoints o 40cm (tank 1) and 15cm tank2) .

Storemeasurements150 sec):Create a leak in tank 3 by opening the correspondingvalve. Close the valve after 30sec. Now use'Decoup'=0.lsecand createa leak n tank 2. Wait untilthe steadystates f thewater evels are eached.Close hevalve after this.

c) Disturbance behaviour in case of closedconnection valveAdjust the setpoints o 25cm (tank 1) und 2}cm (tank 2)with 'Decoup'=Q.1.Wait until the steadystatesof thewater evels are eached.Storemeasurements21Osec):Close he connection alve between ank 3 and tank 2.Wait until the water levels of tank 3 and tank 1 are thesame.Now create stepof the setpointw, to 50cm.At the end of the measuring do not forget to open theconnectionvalve between ank 3 and ank 2 again.

d) Disturbance behaviour in case of sensorfailureIf the disturbancemodul is not present his experimentcannotbe performed.

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Choose the parameter Decoup'=S.l and adjust thesetpoints o 25cm (tank 1) and 20cm (tank2).Store measurements 00sec):Create a sensor failure in tank I until 50Vo of themeasuring ime is reached Knob down, switch "Tank1"of the disturbancemodule). Then switch on the sensoragain.

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Three Tank SystemDTS200 Carrying-Outhe Experiment

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5.3 Behaviour fVariablewith the ReferencePl-ControlTo obtain an overall amplification of I for the closedcontrol loop the controller parameters , are set equal tovi .

a) Reference behaviour of the decoupled,Pl-controlled subsvstemsChoose the parameter Decoup'-0.05 and adjust theparameters f the Pl-controller o Pi = 0, Ki = O.lsec-l(Use the menu item Pl-Controller from the menuParameter, enter the correspondingvalues and promptwith'OK'). Now adjustthesetpointso 30cm tank 1)and20cm (tank 2). Then activate the Pl-controller withdecouplingmode (Select he menu item Pl-Controllerfrom the pulldown menu RUN, activatedecoupled andprompt with 'OK'). Wait until the steadystatesof thewater evels are reached.

Storemesurements120sec):Createa stepof the setpointw, to 34cm.

b) Reference behaviour of the coupled,Pl-controlled subsvstemsActivate the decouplingcontroller by using hemenu temDecoupling Controller. Adjust the setpoints to 30cm(tank 1) and 15cm (tank 2). Close the connectionvalvebetween ank 3 and tank 2. Wait until the water evelsh,and h, are settled o steadystate.Adjust the proportionalportionof the Pl-controller o 0 and select i=0.5sec-1.In the ollowing, only the referencebehaviourof thewaterlevel h, will be considered.Storemeasurements450sec):Now activate he Pl-controller without decoupling.Thewater levels of tank I and tank 2 are now not any longerdecoupled; hey are controlledbe thePl-controller.

Adjust the parameterP.=l5sec after approximately 50Voof the measuring ime.

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Three Tank SystemDTS200 Evaluation f the Experiments

6 Evaluation f theExperimentsref.5.2a):

Explainthe characteristic ehaviourandexamine he imeconstants.ref. 5.2b):A leak in tank 3 does not change he steadystatesof theouptput variables(water levels of tank I and tank 2). Aleak in tank 2 results in difference of the steadystates.Why? Use this difference o calculate he opening of theleak valve. Use a value of 0.7 for the outflow coefficientof the eak valve.Will a Pl-controller settle he eak disturbancen tank 2?Use the differential equationsof the system o show thison the assumption hat he control oop is stable.ref 5.2c):Explain the characteristicbehaviour.Does the nonlinearcontroller designdecouple he subsystemsurthermore ncaseof such a failure?ref.5.2d):Explain the characteristicbehaviour.Does the nonlinearcontroller design furthermore decouple the outputvariablehr? What would be the result of a sensor ailurein tank 3?

ref .5.3a):Will such parameters ead to stable or unstablecontrolloops?Explain he systembehaviour.ref .5.3b):Is the control of the output variable h, stable f you usePz=O r Pr=lQsec?Show his by linearization f theplantmodel of tank 2 around the reference variable and bycalculating he transfer unction of the control loop.Which is the order of the linearizedplant in the nominalstate?

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