Design and Simulation of PSO Based PIControllers for a Non-Linear Two-Tank

Interacting Level Process R.Soundariya and S.Arulselvi

Abstract— In this work, a non linear two-tank interacting level process is taken-up for study. The mathematical model ofthe two-tank interacting process isderived. A conventional PI controller isdesigned based on process reaction curvewith ZN techniques. To improve theperformance of PI controller, a ParticleSwarm Optimization (PSO) is proposed andimplemented. The servo and regulatoryresponses with PI and Particle SwarmOptimization are obtained and discussed.From the results and performancemeasures, it is observed that ParticleSwarm Optimization is producing betterresults than PI controller.

Keywords— Two-tank interacting process, PI controller and PSO.

I. Introduction The process industries require theliquids to be pumped, stored in tanksand then pumped to another tank. Manytimes the liquid will be processed bychemical or mixing treatment in thetanks, but always the level of the fluidin the tanks must be controlled. A levelthat is too high may upset the reactionequilibrium, cause damage to theequipment or result spillage of valuableor hazardous material [1]. If the levelis too low, it may have bad consequencesfor the sequential operations. Hence,control of liquid level is an importantand common in process industries.Conventional controllers are widely usedin industries since they are simplerobust and familiar to the field of

operator. The most basic and pervasivecontroller algorithm used in thefeedback control is proportionalintegral controller algorithm. PIcontroller is widely used controlstrategy to control most of industrialautomation process because of itsremarkable efficiency and simplicity.Hence, a mathematical model is derivedand a simulation is carried out for thegiven mathematical equation. Theconventional PI controller parametersare designed based on Ziegler-Nicholas.However, it will not give satisfactoryresponse for change in operating point.Hence, the Particle Swarm Optimizationis proposed and implemented]. In thiswork, a simple performancecriterion in time domain isproposed for evaluating theperformance of a PSO-PIDcontroller that was applied to ainteracting two-tank process

II. PROCESS DESCRIPTION

Fig.1 shows the photograph of thelaboratory level process station. Itconsists of three pumps, two motorizedcontrol valves, six process tanks, twooverhead tanks, two differentialpressure transmitters, five leveltransmitters androtameters .Instrumentation panelconsists of two PID controllers, main

R.Soundariya, PG scholar, Electronicsand Instrumentation Engineering,Annamalai University,Annamalai Nagar,Chidambaram-608002, Tamilnadu, India.Mail-soundariya2015@gmail.comDr. S.Arulselvi,Professor Electronicsand InstrumentationEngineering,Annamalai University,

power supply switch, pump switches,motorized control valve switches andauxiliary switches for individualcomponents.

A. Working principles

Fluid level in the tank is measuredby level transmitter (LT). Output of LTis given to the data acquisition setup.It consists of ADC and DAC. Thedifferential pressure level transmitter(DPLT) measures the flow by sensing thedifference in level between the tanks.The DPLT then transmits a current signal(4-20mA) to the I/V converter. Theoutput of the I/V converter is given tothe interfacing hardware associated withthe personal computer (PC). Controlalgorithms are implemented in Lab viewsoftware. It compares and takescorrective action on the motorizedcontrol valve. Based on the valveopening flow rate is manipulated.Rotameter can visualize the flow rate.

Fig.1. Experimental setup of a two-tankinteracting process.

The controller compares the controlledvariable against set point and generatesmanipulated variable as current signal

(4-20mA) [2]. Here the controlledvariable is the level (h2) and themanipulated variable is the flow rate(qin).Specification dimension aretabulated in Table I and TableII,respectively.

TABLE ISPECIFICATION OF TWO-TANK INTERACTING

PROCESSCOMPONENTS SPECIFICATIONS

MOTORISEDCONTROL VALVE

Flow rateCharacteri

sticsValveaction

50LPHEqual%

motorizedcontrol

ROTAMETER TypeRangefloat

material

variablearea

(0-100)LPH i.e (0-16663/min)

SS 316PUMP RPM

DischargeVoltage

45001000(LPH)

220/230 volts AC&DC

PROCESS TANK CapacityHeightDiameter

3 litres300 mm120 mm

LEVELTRANSMITTER

InputHeightType

24V DC0-400 mm WC

Fcapacitance

DIFFERENTIALPRESSURE

TRANSMITTER

InputSupplyOutput

0-400mm H2O24V DC

4-20mA at24V DC

TABLE IIDIMENSIONS AND VARIABLES FOR TANK1 AND

TANK 2.

Parameter DimensionArea(A1,A2) 113.0973 cm2

Height(max) 25 cm

Diameter 12 cmInflow

rate(MV),qin0-1666 cm3/min

Process Variable(PV),h2

0-25 cm

Flow rate(LV),qL1,qL2

0-500 cm3/min

g Gravity(9.81N/m2)

a1 and a2 Area of thepipe outlet

(a1=3.5735cm2 ,a2=3.9012cm2)

cd Discharge co-efficient( cd=0.08)

mv- manipulated variable pv- process variable Lv- Load variable

B. Non linear mathematical modelling of atwo-tank interacting level

process Fig.2 shows the processconsists of two interacting liquidtanks. The volumetric flow into thetank1 is , the volumetric flow ratefrom tank1 to tank2 is q1(cm3/min), andthe volumetric flow rate from tank2 isqo(cm3/min) [2]. The height of theliquid level is h1(cm) in tank1 andh2(cm) in tank2 (cm), respectively Bothtanks have the same cross sectional areaand A2(cm2), qL1 is the inflow of tank1as load disturbance (cm3/min) and qL2is the inflow of tank2 as loaddisturbance (cm3/min) [1and 2]..

Fig.2.Two-tank interacting processMass balance equation for tank 1

Assuming non -linear resistance to flow :

Mass balance equation for tank 2

Assuming non-linear resistance

to flow:

C. Open loop responses for two-tank interactingprocess

Fig.3 shows the simulated open loopresponse of interacting process. Thelevel (h2) changes from 0 to 8.5 cm, when

tank1 tank2

(1)

h1R1

h2

A1

R2

A2

q1

qin(m qL1 (Load) qL2(Load)

(2)

(

(3)

applying a step input inqin(50*16.66cm3/min) also the level (h1)changes from 0 to 18 cm due tointeraction. The simulated processreaction curve(PRC) of h2 for step changein qin for ±499.8cm3/min is shown inFig.4.

Fig.3. Simulated open loop response ofh1 and h2 of interacting Process.

1000 2000 3000 4000 50005

6

7

8

9

10

Time(sec)

h 2(cm)

Fig.4. Simulated PRC of h2 for step change in qin for

±499.8cm3/min From the fig.4 the transferfunctions are obtained andtabulated in Table III. From theaverage transfer function, thecontroller parameters are obtainedusing Z-N tuning rule [4]&[5]. Fortwo-tank interacting process the PIcontroller parameters are tabulated inTable IV.

TABLE IIITRANSFER FUNCTIONMODEL OF TWO-TANK

INTERACTING PROCESSStepInput

TransferFunction

AverageTransferFunction

PositiveStepInputNegativeStepInput

TABLE IVPI CONTROLLER SETTINGS FOR TWO-TANK

INTERACTING PROCESS.

ModePI 7.9351 56.61PSO 2.03791 0.1005

D. PARTICLE SWARM OPTIMIZATION (PSO)ALGORITHM

Particle swarm optimization (PSO)algorithm, which is tailored foroptimizing difficult numerical functionsand based on metaphor of human socialinteraction, is capable of mimicking theability of human societies to processknowledge. It has roots in two maincomponent methodologies: artificial life(such as bird flocking, fish schoolingand swarming) and evolutionarycomputation. Although the PSO algorithmis initially developed as a tool formodelling social behaviour, it has been

0 500 1000 15000

5

10

15

20

25

Time(sec)

Level (cm)

h1h2

applied in differentareas.

Fig.5 . PSO based PI controller

to control the level of two- tank interacting process.The block diagram of PSO based PI

controller is shown in Fig 5.1. Also theflow chart to optimize the PI controlleris shown in Fig 5.2. 5.2.1 The PSO learning algorithm isrepresented by following stepsStep1 Select the number of iterations

(n) and the PSO learning rate(c1, c2).

Step2 Randomly generate positionvector X and associated velocityV of all particles in thepopulation.

Step3 Calculate each individual’sfitness value, and then compareeach individual’s evaluationvalue with best global particlevalue gbest the personal bestvalue pbest . Finally select thenew best value (gbest and pbest)

Step4 For every particle, update itsown velocity and position value.

Step5 If iteration=n, then go to exit,otherwise go to step 3.

Step6 The best particle’s value willbe selected as the finialparameter set to form desiredPI controller.

IV. SIMULATION RESULTES

A. Servo responses of levels with PI,FLC and PSO The servo responses for two tanklevel interacting process for stepchange in h2 from 6 to 8 cm, from 8 to6cm is shown in fig.6 by implementingPI, FLC and PSO. The correspondingchange in h1 and inlet flowrate qin arealso shown in fig.7 and fig.8,respectively. The performance measuresfor step change in h2 from 6 to 8 cm istabulated in Table VI. From the resultsand Table VI, it is observed that theFLC gives better performances than PIcontroller. .

0 1000 2000 3000 4000 5000 6000 7000 80000

2

4

6

8

10

Time(sec)

h 2(cm)

PSOPIFLC

Fig.6. Servo response of h2 with PI ,fuzzy logic controller and PSO.

.

0 1000 2000 3000 4000 5000 6000 7000 80000

5

10

15

20

25

Time(sec)

h 1(cm)

PSOPIFLC

Fig.7.Servo response of h1 with PI ,FLCand PSO

Computer performan

ce Criterion for each agent

Process

C(s)PSO

E(s)

R(s)

U(s)

Y(s)+

-+

0 2000 4000 6000 80000

100

200

300

400

500

600

Time(sec)

q in(cm

3 /min)

PSOPIFLC

Fig.8. Response of PI, FLC and PSOoutput (qin) for servo

response

TABLE VCOMPARISON OF PERFORMANCE MEASURES

LEVEL WITH PI ,FLC AND PSO FOR SERVORESPONSE (6-8) CM

B.Regulatory response for Positive step change load variable qL1(+)

A sudden load disturbance of+10% is given in inlet flow rate of tank1at 4000 sample from qL1(+) . Due to thislevel in h1 increases for PI from 13.3 to14.5 cm , FLC it increases from 13.2 to13.3cm and PSO it increases from 13.3 to15.4 as shown in fig.10. The level h2 alsoincreases for PI from 6 to 6.5 cm , FLCit increases from 6 to 6.1 cm and PSO itincreases from 6 to 6.9cm as shown in fig10. The PI , FLC and PSO Controllertakes necessary action to reduce theflowrate qin from 420 to 350 in orderbring back the level h2 back to 6cm asshown in fig.9.

4000 4500 5000 55005.8

6

6.2

6.4

6.6

6.8

7

time(sec)

h 2(cm)

PSOPIFLC

Fig.9. Regulatory response of h2 with PI ,FLC and PSO due to load variationin +10% from qL1.

4000 4500 5000 550012

12.5

13

13.5

14

14.5

15

15.5

time(sec)

h 1(cm)

PSOPIFLC

Fig.10. Regulatory response of h1 with PI ,FLC and PSO due to Load variation in +10% from qL1.

Servo response for h2 (6-8)cmts(sec) %overshoo

tISE

FLC 707 - 262.6PI 1988 12.825 353.4PSO 978 33.66 354.2

3800 4000 4200 4400 4600 4800 5000 5200 5400 5600340

360

380

400

420

time(sec)

q in(cm

3 /min)

PSOPIFLC

Fig.11. Response of PI ,FLC and PSO output(qin) for load variation in +10% from qL1.

C. Regulatory response for negative step change load variable qL1(-)

A sudden load disturbance of -10% isgiven in inlet flow rate of tank1 at 4000sample from qL1(-) . Due to this level inh1 decreases for PI from 13.3 to 12.1 cm ,FLC it decreases from 13.2 to 13.3cmand PSO it decreases from 13.3 to 11.1 asshown in Fig 13. The level h2 alsodecrease for PI from 6 to 5.5 cm , FLCit decreases from 6 to 6.1 cm and PSO itdecreases from 6 to 5.1 cm as shown infig 12. The PI , FLC and PSO Controllertakes necessary action to increase theflowrate qin from 425 to 500 in orderbring back the level h2 back to 6cm asshown in fig 6.12.

4000 4500 5000 5500

5.2

5.4

5.6

5.8

6

6.2

time(sec)

h 2(cm)

PSOPIFLC

Fig.12. Regulatory response of h2 with PI ,FLC and PSO due to load variation in

4000 4500 5000 5500

11.5

12

12.5

13

13.5

14

time(sec)

h 1(cm)

PSOPIFLC

Fig.13. Regulatory response of h1 with PI ,FLC and PSO due to load variation in -10% from qL1.

3800 4000 4200 4400 4600 4800 5000 5200 5400 5600 5800

410

420430

440450

460

470480

490

500510

time(sec)

q in(cm3min)

PSOPIFLC

Fig.14. Response of PI ,FLC and PSO output(qin) for load variation in -10% from qL1.

TABLE VICOMPARISON OF PERFORMANCE MEASURE OF

LEVEL WITH PI ,FLC and PSO FORREGULATORY RESPONSE.

Regulatory Response h2 (6-8)cmController

+10% from qL1 -10% from qL1

ts(sec)

ISE ts(sec)

ISE

PSO 1032 2822 1046 2821PI 1460 9316 1275 9316

Fuzzy 460 1210 609 1209

-10% from

D.Regulatory response for positivestep change load variable

qL2(+) A sudden load disturbance of +10% isgiven in inlet flow rate of tank1 at 4000sample from qL2(+) . Due to this level inh1 increases for PI from 13.3 to 13.5 cm ,FLC it decreases from 13.2 to 11.2 cmand PSO it increases from 13.3 to 13.6cmas shown in fig 16. The level h2 alsoincreases for PI from 6 to 7 cm , FLC itincreases from 6 to 6.2 cm and PSO itincreases from 6 to 6.9cm as shown in fig15. The PI , FLC and PSO Controller takesnecessary action to reduce the flowrateqin from 420 to 200 in order bring backthe level h2 back to 6cm as shown in fig15.

3800 4000 4200 4400 4600 4800 5000 5200 54005.5

6

6.5

7

time(sec)

h 2(cm)

PSOPIFLC

Fig.15. Regulatory response of h2

with PI ,FLC and PSO due to load

variation in +10% from qL2

3800 4000 4200 4400 4600 4800 5000 5200 540011

11.5

12

12.5

13

13.5

time(sec)

h 1(cm)

PSOPIFLC

. .

Fig.16. Regulatory response of h1 withPI ,FLC and PSO due to load variation in +10% from qL2 .

4000 4200 4400 4600 4800 5000200

250

300

350

400

450

500

time(sec)

q in(cm3/min)

PSOPIFLC

Fig.17. Regulatory response of qin with PI and FLC due to load

variation in +10% from qL2.E. Regulatory response for Negative step change load variable qL2(-)A sudden load disturbance of -10% isgiven in inlet flow rate of tank1 at4000 sample from qL2(-) . Due to thislevel in h1 increases for PI from 13.3to 15.6 cm , FLC it increases from 13.2to 16 cm and PSO it increases from 13.3to 15 as shown in fig 19. The level h2

also decreases for PI from 6 to 5.6 cm ,FLC it decreases from 6 to 5.8 cm andPSO it decreases from 6 to 5.8cm asshown in fig 18. The PI , FLC and PSOController takes necessary action toincrease the flowrate qin from 420 to750 in order bring back the level h2

back to 6cm as shown in fsig 18.

4000 4500 5000 55005

5.5

6

time(sec)

h 2(cm)

PSOPIFLC

Fig.18. Regulatory response of h2 with PI ,FLC and PSO due to load variation in -10% from qL2.

4000 4500 5000 5500

1313.514

14.515

15.516

time(sec)

h 1(cm)

PSOPIFLC

Fig.19. Regulatory response of h1 with PI,FLC and PSO due to load variation in -10% from qL2.

3800 4000 4200 4400 4600 4800 5000 5200 540011

11.5

12

12.5

13

13.5

time(sec)

h 1(cm)

PSOPIFLC

Fig.20. Regulatory response of qin with PI,FLC and PSO due to load variation in -10% from qL2.

TABLE VIICOMPARISON OF PERFORMANCE MEASURE OF WITH PI AND FLC FOR REGULATORY

Regulatory Response h2 (6-8)cmController

+10% from qL2 -10% from qL2

ts(sec)

ISE ts(sec)

ISE

PSO 1036 2824 1045 2826PI 1460 9321 1277 9311

Fuzzy 461 1211 609 1209

V. CONCLUSION The non-linear mathematical model ofa two-tank interacting process wasderived. To improve the performance ofclosed loop control , a Fuzzy LogicController was designed and implementedfor a two-tank interacting process. Theservo and regulatory responses wereobtained with Fuzzy Logic Controller. Theperformances of Fuzzy Logic Controllerwere compared with that of PI and PSOcontroller in simulation. The performancemeasures were tabulated. It is observedthat the FLC gives is better performanceof less in terms of integral squareerror, faster settling time and withoutoscillation.

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