Introduction of fractional elements for improvising the performance of PID controller for heating furnace using AMIGO tuning technique

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Basu, Amlan and Mohanty, Sumit and Sharma, Rohit (2016) Introduction

of fractional elements for improvising the performance of PID controller

for heating furnace using AMIGO tuning technique. Perspectives in

Science, 8. pp. 323-326. ISSN 2213-0209 ,

http://dx.doi.org/10.1016/j.pisc.2016.04.065

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Introduction

of

fractional

elements

for

improvising

the

performance

of

PID

controller

for

heating

furnace

using

AMIGO

tuning

technique

夽

Amlan

Basu

∗

_,

_Sumit

_Mohanty

1

_,

_Rohit

_Sharma

2

ITMUniversity,Gwalior,MadhyaPradesh,India

Received5February2016;accepted10April2016

Availableonline 27April2016

KEYWORDS Nelder-Mead; Astrom-Hagglund; Grunwald-Letnikov; IOM; FOM

Summary Thepaperdemonstratesaboutmeliorationofintegerorderandfractionalorder modelofheatingfurnace.Bothmodelsarebeingplacedinclosedloopalongwiththe propor-tionalintegralderivative(PID)controllerandfractionalorderproportionalintegralderivative (FOPID)controllersothatthevarioustimedomainperformancecharacteristicsoftheheating furnacecanbemeliorated.Thetuningparameters(Kp,KiandKd)ofthecontrollershasbeen foundusingtheAstrom-Hagglundtuningtechniqueandthediffer-integrals(and)arefound usingtheNelder-Meadoptimisationtechnique.

Introduction

In mathematical modelling, an understanding is made of

those feelings into the vernacular of science (Lawson

夽 _This_article_belongs_to_the_special_issue_on_Engineering_and

Mate-rialSciences.

∗_{Corresponding}_author._Tel.:₊₉₁_9981135396.

E-mailaddresses:[email protected](A.Basu),

[email protected](S.Mohanty),

[email protected](R.Sharma).

1 _Tel.:₊₉₁_8959763079. 2 _Tel.:₊₉₁_8269991055.

and Marion, 2008). Heating furnace is a mechanical gad-get that is used to warm distinctive substances at the

required temperature. There are bunches of

parame-ters that are not up to the imprint in the warming

heater like overshoot, steady state error and settling time. The controllers are designed and tuned utilising diverse tuning methods and streamlining systems. Tuning methods are the techniques for achieving the different tuning parameters of the controllers. Optimising proce-duresare additionally the same yet they likewisedeliver the estimations of two additional parameters called the differ-integrals which assume an essential part in the designing of the controller of fractional order controller (Figs.1and2).

http://dx.doi.org/10.1016/j.pisc.2016.04.065

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324 A.Basuetal.

Figure1 StepresponseoftheheatingfurnaceEq.(11).

IOPID

and

FOPID

controller

IOPID controlleris mathematically defined as(Shahri and Balochian,2012), u(t)=Kpe(t)+Ki t 0 e() d+Kd de dt (1)

where, Kp is the gain of proportionality, Ki is the gain of

Integral,KdisthegainofDerivative,eistheError,tsignifies

theinstantaneoustimeandisthevariableofintegration. OnperformingtheLaplacetransformoftheEq.(1)whichis thePIDcontrollerequationis,

LI(s)=Kp+

Ki

s +Kds (2)

Numerically, the FOPID controller can be defined as

(RastogiandTiwari,2013), u(t)=KPe(t)+KiD

−

t e(t)+KdD

te(t) (3) OnperformingtheLaplace transformoftheEq.(3)we get(ShahriandBalochian,2012),

Lf(s)=Kp+

Ki

s+Kds

(4) Where,Kp is the gain of proportionality, Ki is the gain of

Integral,Kd isthe gainof Derivative and and arethe

differential-integral’sorderforFOPIDcontroller.

Astrom-Hagglund

or

AMIGO

tuning

technique

The other name for this tuning method is AMIGO which

stands for approximate M-constrained integral gain opti-misation method for tuning. The tuningprocedure of the AMIGOisasfollows(AstromandHagglund,1995),

Kp= 1 K 0.2+0.45T L (5) Ki= 0.4L+0.8T L+0.1T L (6) Kd= 0.5LT 0.3L+T (7)

Nelder-Mead

optimisation

technique

The different operations in Nelder-Mead optimisation

methodare(Wright,2012),Takingafunctionf(x),x ∈ Rn

which is to beminimised in which thecurrent pointsare

x1, x2...xn+1. (i). Order: On the basis of valuesat the

vertices,f(x1)≤f(x2)≤...≤f(xn+1).(ii).Calculate

thecentroidofallpoints(x0)exceptxn+1.(iii).Reflection:

Calculatexr=x0+˛(x0−xn+1).Ifthe reflectedpoint is not

betterthan thebestandisbetterthan thesecond worst, that is, f(x1)≤f(xr)<f(xn). After this by putting back the

worstpointxn+1withreflectedpointxrtogetanewsimplex

andgotothefirststep.(iv).Expansion:Ifwehavethebest reflected part then f(xr)<f(x1), then solve the expanded

point xe=x0+␥(x0-xn+1). Ifthe reflected point is not

bet-terthanexpanded point,thatis,[f(xe)<f(xr)]theneither

by replacingthe mostawful point xn+1 by expanded point

xe toget newsimplex andthen go tothefirst stepor by

replacingthemostawfulpointxn+1byreflectedpointxrto

acquireorgetanewsimplexandthengobacktothefirst step.Elseifthereflectedpointisnotwellagainthan sub-sequentworstthenmovetothefifthstep.(v).Contraction: Herewe know thatf(xr)≥f(xn), contractedpointis tobe

calculatedxc=x0+␳(x0-xn+1),iff(xc)<f(xn+1)thatisthe

con-tracted pointis betterthan themost awfulpoint then by replacingthe most awfulpoint xn+1 withcontracted point

xc toachieve a new simplex and then go to first stepor

proceedtosixthstep.(vi). Reduction:reinstatethe point withxi=x1+␴(xi-x1)foralli ∈ {2,...,n+1},thengoto

thefirststep.

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Table1 Outputparametervalues.

PID+IOM FOPID+IOM PID+FOM FOPID+FOM

Overshoot(%) 19.5 41 16.3 0

Settlingtime(sec.) 435 171.7 2345 133

Results

Foranyphysicalsystem,thetotalforceisequaltothe sum-mationof individualforcesexertedbymass(m),damping (b) and spring (k) element.Mathematically, we can state thesameas,

F=ma+bv+kx (8)

In the Eq.(8) acceleration is signifiedasa, velocityis signifiedasvanddisplacementissignifiedasx.Therefore, thedifferentialequationofEq.(8)is,

F=md 2 x dt2 +b dx dt+kx (9)

Note:Fordesigninganetwork-basedPID,theabove equa-tion or model is a rough process behaviour description. Therefore,thedifferentialequationoftheheatingfurnace using the aboveequation becomes (Zhao etal., 2005,Vic Dannon,2009), F=73043d 2 x dt2 +4893 dx dt +1.93x (10)

TheLaplacetransferfunctionofEq.(10)isgivenas(Maiti andKonar,2008),

GI(s)= 1

73043s2₊₄₈₉₃_s₊₁_.₉₃ (11)

InEq.(20)‘s’istheLaplaceoperator.TheFOPDTmodel ofEq.(12)is,

GIOM-FOPDT(s)=

0.518133 1+2520.04se

−15.2189 ₍₁₂₎

The FOM of the heating furnace is obtained using the Grunwald-Letnikovequationforfractionalcalculuswhichis giveninEq.(13), aD˛ tf(t)=lim 1 (˛)h˛ (t −a) h k=0 (˛+k) (k+1) f(t−kh) (13) Therefore,theFOMofheatingfurnacewhichcomesout tobe(Tepljakovetal.,2011),

GF(s)=

1

14494s1.31₊₆₀₀₉_.₅_s0.97₊₁_.₆₉ (14)

Therefore,theFOPDTmodelfortheEq.(14)is, GFOM-FOPDT(s)=

0.404257 1+3440.71se

−66.9314 ₍₁₅₎

The PID controllerequation deduced using the FOPDT

modeloftheIOMis, LI1(s)=144.198+

1.25211

s +1095.28s (16)

The PID controllerequation deduced using the FOPDT

modelofFOMis,

Lf₁(s)=57.7181+ 0.127522

s +1920.37s (17) The FOPID controllerequation deduced for the IOM of heatingfurnaceis,

LI₂(s)=144.198+1.25211

s0.4876 +1095.28s

0.01011 ₍₁₈₎

The FOPIDcontrollerequationdeduced forthe FOMof heatingfurnaceis,

Lf₂(s)=57.7181+ 0.127522

s0.3636 +1920.37s

0.12483 ₍₁₉₎

WhentheEqs.(16)and(17)are,respectively,putinthe closedloopsystemalongwithEq.(11)theoutputsobtained are H1(s)= 1095.3s2₊₁₄₄_.₂_s₊₁_.₂₅₂₁ 73043s3₊₅₉₈₈_.₃_s2₊₁₄₆_.₁₃_s₊₁_.₂₅₂₁ (20) H2(s)= 1920.4s2₊₅₇_.₇₁₈_s₊₀_.₁₂₇₅₂ 14994s2.31₊₁₉₂₀_.₄_s2₊₆₀₀₉_.₅_s1.97₊₅₉_.₄₀₈_s₊₀_.₁₂₇₅₂ (21)

When the Eqs.(18) and (19) are,respectively, put in the closedloopsystemalongwithEq.(14)theoutputsobtained are, H3(s)= 1095.3s0.49771 +144.2s0.4876 +1.2521 73043s2.4876₊₄₈₉₃_s1.4876₊₁₀₉₅_.₃_s0.49771₊₁₄₆_.₁₃_s0.4876₊₁_.₂₅₂₁ (22) H4(s)= 1920.4s0.48843 +57.718s0.3636 +0.12752 14994s1.6736₊₆₀₀₉_.₅_s1.3336₊₁₉₂₀_.₄_s0.48843₊₅₉_.₄₀₈_s0.3636₊₀_.₁₂₇₅₂ (23)

Discussion

It is clear that the IOM transfer function of heating fur-naceexhibitsverypoorresponsewithasteady-stateerrorof morethan50%.SoaPIDcontrollerisdesignedusingAMIGO tuning technique. But the overshoot of the system then became19.5%where asthesettling became435s. There-fore,Nelder-MeadOptimisationalgorithmwasusedtothis PID tofind the fractional elements & , so that FOPID

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326 A.Basuetal. can be designed for the System. But it also exhibited an

overshoot and settling time of 41% & 170s, respectively. Therefore,PID is designed based onthe Fractional Order Modelof Heating FurnaceTransfer function. When AMIGO methodwasappliedtoFOMforthetuningparameters,the finalsystembecamestablewithan exhibitedovershootof 16%,whereasthesettlingtimeincreaseddrasticallyupto 2400s. Therefore to improvise the response Nelder-Mead optimisationalgorithmswasusedtotunethealreadytuned tuning parametersusing AMIGO method and also to opti-mise the differ-integral parameters. It is clear from the stepresponsethatthesystemovershootdecreasedtoazero valuewhereasthesettlingtimewasaround130s(Table1).

Conclusion

Theplotsoftimeresponsecharacteristicsclearedthatthe PID & Fractional order PID designed for the IOM gave a verydisturbedresponse.FOMoffurnacegavecomparatively goodresponsewhileusedwithAMIGO tuningmethod,but itexhibitedahighovershootandalsoasluggishresponse. Astheovershootinfurnacegeneratessuddenhighpressure whichmayendangerthelifeofworkersandproperties,this methodwasavoided.Butwhenfractionalelements ofPID wereoptimisedusingNelder-Meadoptimisation,thesystem exhibitedalmost negligiblerange of overshootand alsoa comparativelylowsettlingtime.Therefore,itcanbe con-cluded that more the fractional elements are introduced moretheresultwillbesmoothandswift.

References

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Lawson,D.,Marion,G.,2008.AnIntroductiontoMathematical Mod-eling.BioinformaticsandstatisticsScotland.

Maiti, D., Konar, A., 2008. Approximation of a fractional order systembyanintegerordermodelusingparticleswarm optimiza-tiontechnique.In:IEEESponsoredConferenceonComputational Intelligence,ControlAndComputerVisionInRobotics& Automa-tion.

Rastogi, A.,Tiwari, P.,2013. Optimaltuningof fractional order PIDcontrollerforDCmotorspeedcontrolusingparticleswarm optimization.IJSCE3(May(2)).

Shahri,M.E.,Balochian,S.,2012. FractionalPID controllerfor a highperformancedrillingmachine.AMEA2(4),232—235.

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