filehost 6544062 daniel goleman inteligenta emotinala

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commonly used defuzzit1cation methods arc discussed and system responses withdifferent defuzzlfication methods are compared. Finally disturbance rejectioncapabilities of the designed cOTIlrollers are investigated,

2, DC motor modelIn annature conn'!.)l of separately excited DC motors, the vol::age applied to (he a:matureof the motor is adjusted wlthonl changing the voltage applied to the field, Figure 1shows a separately excited DC mo1nr equivalent mode-I.

Ra La I(,_Hili i 0V Eb" eE2s-:::l' t '0'

Figure L DC motor mode!

( , R '() ," '"CI) ( )va t);;;:;:; a' ta t + L". - ; ; t ' + eb t (1 )ehCt) = Kh , wet) (2)

' ";"(0 = Kr , i"CI) \j)'P "->-, J dw(r) ( )Im l t . ) = m'--;;:- + Bm w,t (4)where Va = amw.ture voltage (V) Ra ::::;;; 3rrnature reslstam:,e (D) La ;;; arrnature inciu.;.."tance (H) fa - armature current (A) Eo = hack emf (V) w = angular speed (radjs) ' i ~ ;;;;;;. mo1onorqnc (Nm) e == angular position of rotor shaft (rad) 1m :;;; rotor inertia (kgm?) Em ;;;:;: viscous friction coefficient (Nms/rad) KT = torque constant (Nm/A) Kb ;;;;: back emfconsmnt (Vsjrad) L';"4: \15 combine the uppet equations togetjec

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. di,(t)v,(t) = Ra.l,(t) + L a . ~ + K,.w(t) (5). dw(t)KTla(t) = m 8m.wet) (6)

Laplace transfonns o f (5) and (6) are:Va(S ) = Ra. laCs ) + La.1a,Cs) .s + Kt . H' (s ) (7)KT l ,(s) = 1m.W(s).s + 8m,W(s) (8)If current is obtained from (8) and substiruted in (7) we have

1vats) = W(s). KT . [La1m s ' + (R,1m + La 8m). S + (Ra8m+ K,. KT) l (9)Then Ihe relation between rotor shaf: s ; : ~ and applied armature voltage is representedby transfer funct ion:W(s) KT (10)Va(s) :::: La .1m 52 + (Ra.fm + La 8m ), s + (Ra 8m + KbKr)The relarion between position and speed is:

18(s) = -W (s) ( 11),Then the transfer [untrien between shan pos it ion and annarure voltage at no-load is:e(s) KT (12)Va(s) = La1m53 + eRa.1m + La Bm}, 52 + (Kr Kb + Ra 8m), SFigure 2 shows tbe DC motor model built in Simulink. MOlor model was converted to a2-m 2-out subsystem, lnput ports are a rmature voltage (Va) and load torque (Tlaad) andthe outpm ports are angu lar speed in (w) and position (teta).

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wT 1.

la.s. Ra Jm.s..Bm I 'itVa ' -S ntegratoromenl coefficient Transfer FooclionansferFunctionK b l + ~ - - - - _ . . . . . J

Back emf coetfdenlFigure 2. Simulink model

A 3.70 kW, 240V, 1750 rpm DC motor with the below parameters wa s used:Ra=1l .2f lLa = 0.1215 H1m = 0.02215 kgm '8m = 0.002953 Nms /radK, = 1.28 Nm/AKb = 1.28 Vs/rad

3. ProportionaljDtegral.derivati....e (PIO) controllerPID contro llers are wid ely used in industria l control applications due to th eir simplestructures, comprehensible control alg ori thms and low cos ts . Figure 3 shows theschematic model ora control system wi[h a PID controller.

+ u{c). ( L . ( t) K, PlANTl.KJd

+

K:) d i._--- - ---- --

Figure 3. PID control system Contro l s igna l u (t ) is a linear combinat ion of error e(t), its integral and derivati ve.

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de(t}uCt) = Kre(t) + K, e(t)dt + K D ~ (13)J " 1 ' deet))uCt) = Kp lett) +-J e(t)dt +Tn .-- (14)T, dt

where Kp ",.- proportional gaulK, ooc- integral gainKo:::: dcrjvative gainTJ integral timeT{) 0" derivative timerf the controller is digital, then the derivative lenn may be rcpjaced with a baclnvarddifference and the integral teml may be replaced with a sum. For a small constantsumpltng rime T:;p (14) can be a p p r o x i n u H ~ d as:

n" 1 ,u,n) = Kp ,B('l) + cUlT, + To e(nl - ; ~ ( n - 1)) (15)(3.1 Tuning PIn parameters

PID controllers are usuaHy tuned using hand-tuning: or Ziegler-Kicools methods(Jantzen, 2007).Hand-tuning 1s generally used by experienced control engineers based un the rulesshmVll in Table L Bm these mi.e;,; are nor ahvays yalid, For example if an integratorexists in the plant, then increasing Kp resuhs in a more stable controL

Table 1. Hand-tuning rules! i i i i i ~ t i o ~ ~ T ~ ~ ~ ~ ~ ~ ! t s i ~ _ b i i i t ; ; - - ] , i Fas:er I Increase,; Decreases'i - TD __ _ _ _ __ ~ ' S J ; ; : ~ ~ _ J ~ o e . ~ ! _ ~ ~ - c ~ J i ~ c - r ~ ~ : : i J '-----_yt.::L1 __ ~ ~ ~ _ s t e ! _ L I W : f ! ' ~ S e $ , _ ! ? e c r e ; ; _ : _ ~ j

A simple hand.tuntng procedure is as foUows:L Remove deriv0tive and integral actions by setting Tn = a anc :r ;;:;: 02. Tunc Kp such that it gives t'le desiTed response except the final offset valuc from

the set p o ~ n t 3. Increase Kp s.lightJy find adjusl TI) to dampen the overshoot4. Tune l/T/ such that final offset is removed5. Repeat steps from '3 until Kp is as large as possible

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The disadvantage of this method is that it should take a long tim.:: to find the optimalvalues. Another method to tune PID para:;neters is Z i e g l e r ~ N i c h o : s frequency responsemet..11od, The procedure is 33 follows:

1 Increase Xp until system response oscillates v!lth a constant amplitude andrecord that gdifl value as Ku (ulrimate gain)2. Calculate the oscillation period and record it as 'l'u3. Tune parameters using Table 2

Table 2, Z i e g l e r ~ N i c h o t s rulesc o n t r o l ! e , , ' + ~ K " : - f ~ ~ . ' . l ~ _ t ' Q P 0.5,'(


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Figure 6 shows output and control signals of PD control system with adjustedparameters .

4. Fuzzy logic controllerA fuzzy logic controller has four main components as shown in Figure 7: fuzzificationinterface , in ference mechanism, rule base and defuzzification interface.FLCs are complex, nonlinear controllers. Therefore it' s difficult to predict how the risetime, se ttling time or steady sta te error is affected when controller parameters or controLrul es are changed. On the contrary, PID controllers are simple, linear controllers whichconsist of linear combinations of th ree signals.

WN "' ' ' ' '",..,

e< ,00 '- ' .. , ..TImals) , 3.5 ., ,

""""""'"- - ~ ~ - - ~ ~ ~ ~ ~ - __ - - ~ ~ " \."

t - - - - - ~. . : : : : : : : : : : ; s ; : : : : : ~ ' : : : : ~ ' ~ 5 : : : : : : ' : = = = = = ~ ' ~ 5 : : : : : : : : : : : : . : : : : = l , Tlme{$j

F igure 6. Output and control signals for crisp PD contro l system

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r-

c.9Input ,-(Crisp) ."0.c-

'-

Inferencemechanismf _

Rule-base

r - 0:;, Output

(Crisp)"uQ' -

Figure 7. Fuzzy Logic contro ller

Implementation of an FLC requires the choice of four key factors (Mamdan i, 1977):number o f fuzzy sets that constitute linguistic var iab les, mapping of the measurementsonto the support sets, control protocol (hat determines the co ntro ller behaviour andshape of membership functions. Thus, FLCs can be tuned not just by adjustingcontroller parameters but also by changing control rules, membership functions etc.Rule base, inference mechanism and defuzzification methods are the sources ofnonlineari ties in FLCs. But it's possible to construct a rule base with linear input-outputcharacteristics. For an FLC to become a linear contro ller with a control signalU = E + CE where E is "error" and CE is "change of error", some conditions must besatisfied (Jantzen, 2007):I . Support sets of input linguistic variables must be large enough so that input

va lues stay in limits.2. Linguistic values must consist of syrnm eTric triangular fuzzy sets that interceptwith neighbouring sets at a membersh ip va lu e of J.l = 0.5 so that for any timein stant, membership va lues add to t .

3. Rule base must consist of A-combinations of a ll fuzzy sets.4. Outpullinguistic variables must cons ist ofs in glelon fuzzy sets (Si. l ) pos itioned

at the s um of the peak pos itions o f input fuzzy se ts.5. A should be multiplica tion and defuzzification method mus[ be "centre of

gravity" (COGS).

4.1 FPD controiler designFigure 8 shows an FPD controller that acts on the same signals with a PD controller butthe control strategy is constructed as fuzzy rules (Jantzen, 2007) .

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. e

u U- GULCde/dt CE~ G C E > __ IFigure 8. FPO controller

Control sjgnaJ U(n) is a nonlinear function of "error" and "change of error". Thus,U(n) = f(GE x e(n). GeE x e(n)) x GU (20)where f represents the control algorithm. A linear approximation should be obtainedwith a suitable choice:I(GE x e(n).GeE x e(n)) '" GE x e(n) + GeE x e(n) (21)ThenU(n) = (GE x e(n) + GeE x e(n))x GU (22)

GeE )U(n) = GE x GU x e(n) + GE x e(n) (23)(When we compare this equation w ith the control signal of a crisp PD controller . therelationship between ga ins of a PO conrro lJer and of an fPD controller is:GE x GU ::; Kp (24)GeE (25)GE = TDConsequeml y. parameter va lues of a linear FPO controller may be detemlined from atuned PD controller.Figure 9 shows the co ntro l sys tem wi th an FPD controller.

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Table 4. Fuzzy rules

P ~ l PBM I NS'H Z i PS,J\ NB NB I NB NO I NM NS ZB NM NS Z PSM I NB NB I NB N$ , Z PS PMS NB NB NM Z ! PS PM PB NO NM NSPS , PMS Z PB PBS NM

ni l NS PS PM I PB PB PBPB ' PBS PM PB PBB Z

Figure 11 shows the fuzzy PD control system designed in Simulink.

"'......I.....

OpIinl2:lti::ln

r I '"Kd

l -JoIVa 1116'sr:J 11 VI) I~ na ; , . ,l oc moJora l l ~ 1 T V I Loo'

""""Figure 11. Fuzzy PO contro l systemDifferent defuzzification met hods we re used to obta in the co ntrol signal. Table 5 show sthe tuned va lues of the co ntrollerparameters for different defuzzificat ion methods .

Table 5. Co ntroller parameters for different defuzzificacion methodsKpethod KD

2.2484 0.01isector4.1236 0.01OM 0.1901OM 4.55384.7623 0 .1649OM

Figure 12(a)-(d) show s the system responses and conrml signals for the fuzzy contro lsystem s with different defuzzification method s.Table 6 shows the va lues of the perfounance criteria for different defuzzifica tionmethod s with the tuned controller parameters.

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