Elektrik - Elektronik Fakültesi

. Elektrik - Elektronik Fakültesi KON314 Kontrol Sistem Tasar m Ödev #1 Birol Çapa-4645 Doç. Dr. Mehmet Turan Söylemez 23.3.29 1

1.a.Amaç Transfer fonksiyonu ( n 1 ve n üzerine konulan bir kontrolör ile kontrol edilmek istenmektedir. bir kontrolörü F( s) = KP n 1 =1 ve n =2 seçmelidir. 2. a. AppendTo[$Path,"P:\\Muhendis\Mathematica\macsybox"]; <<Control` Gs= 3.2+ n s 2 + H3.5+.3n 1 L s+ 4+.2n n= olmak üzere 3.2 4 + 5. s + s 2 transfer fonksiyonu F= K p Kontrolörün ifadesi olmak üzere ileri yol ifadesi L=F Gs 2

3.2 K p 4 + 5. s + s 2 o T= TogetherAExpandA L 1+ L EE 3.2 K p 4. + 5. s + s 2 + 3.2 K p pcs=denominator[t] 4.+ 5. s + s 2 + 3.2 K p Asim =.3; Log@AsimD ζdesired = è!!!!!!!!!!!!!!!!!!!!!!!!!!!!! π 2 + Log@AsimD 2 b ζdesired=.74484 Buna uygun bir karakteristik polinom elde etmek için pds= s 2 + 2ζ w n s+ w n 2 ê. ζ ζdesired s 2 + 1.48961 s w n + w n 2 polinomdur. 3

sol=solve[coefficientlist[pcs,s] CoefficientList[pds,s]] 88K p 2.2784, w n 3.35659<< ç Kontrolör K p K p = 2.2784 w n = 3.35659 settime= 4 ζdesired w n ê.sol@@1dd AchievedTs=T/.sol[[1]] 7.2667 11.2667 + 5. s + s 2 TimeDomainCharacteristics[AchievedTs,ShowMessages Settling Time (Ts) : 1.72129 sec Overshoot Time (Tp): 1.427 sec Overshoot :.299999 Delay Time (Td) :.438147 sec 4

Rise Time (Tr) :.675998 sec SteadyStateError[AchievedTs] -.35529 dir y ss = AchievedTs ê.s.644971 e ss = Simplify@1 y ss D.35529, 3. a.simülasyon Step[AchievedTs]; YHtL 1.8.6.4.2 Time Response.5 1 1.5 2 2.5 3 Time 5

2.2784 Gain 3.2 s 2+5s+4 Transfer Fcn Step Scope 4. a.sonuç Sonuç olarak, %. 6

1. b.i.ii.iii.amaç Sonuç1.a sonuçta elde bir faz ilerlemeli kontrolör tasarlanmak isteniyor: Matlab veya Mathematica kullanara hal 2.b..i.ii.iii. AppendTo[$Path,"P:\\Muhendis\Mathematica\macsybox"]; <<Control` n 1 = 5 n = Sistem Gs= 3.2+ n s 2 + H3.5+.3n 1 L s+ 4+.2n 3.2` 4+ 5.`s+s 2 pzm=polezeromap[gs] 7

1 Pole Zero Map.5 I m HsL -.5-1 PolesZeros[Gs] {{-4.,-1.},{}} Sisteme RootLoci[Gs] -4-3.5-3 -2.5-2 -1.5-1 ReHsL 3 2 1 Root Locus Plot I m HsL -1-2 -3-4 -3.5-3 -2.5-2 -1.5-1 ReHsL RootLoci[Gs,KRange {-2,2.5,.1}] 8

3 2 1 Root Locus Plot I m HsL -1-2 -3-4 -3.5-3 -2.5-2 -1.5-1 ReHsL rl=rootloci[gs,joingraphics {pzm}] 3 2 1 Root Locus Plot I m HsL -1-2 -3-4 -3.5-3 -2.5-2 -1.5-1 ReHsL Show[{rl,pzm}] 3 2 1 Root Locus Plot I m HsL -1-2 -3-4 -3.5-3 -2.5-2 -1.5-1 ReHsL 9

3 2 1 Root Locus Plot I m HsL -1-2 -3 Kontrolör ifadesi Asim=.3; Buradan ζ: -4-3.5-3 -2.5-2 -1.5-1 ReHsL Fs= K s+ z 1 s+ p 1 K Hs+ z 1 L s+ p 1 ζdesired = Log@AsimD è!!!!!!!!!!!!!!!!!!!!!!!!!!!!! π 2 + Log@AsimD 2.74484 SettlingTime=1.721291223883845` 1.72129 targetsettlingtime=settlingtime/2.86645611941923` Buradanw n 1

wdesired = 4 ζdesired targetsettlingtime 6.2413777929637` Bu iki yeni veriye göre tasarlanan karakteristik polinom pds= s 2 + 2ζ w n s+ w n 2 ê. 8ζ ζdesired, wn wdesired< 38.93931951272934`+ 9.29535344921142`s+ s 2 L=Gs Fs 3.2`K Hs+ z 1 L H4+ 5.`s+s 2 L Hs+ p 1 L T fazil = TogetherAExpandA faz ilerlemeli kontrolöre sahip transfer fonksiyonunun karakteristik denklemi L 1+ L EE 3.2`K s+ 3.2`K z 1 4.`s+ 3.2`K s+5.`s 2 + s 3 + 4.` p 1 + 5.`s p 1 + s 2 p 1 + 3.2`K z 1 p fazil = Denominator@T fazil D 4.`s+ 3.2`K s+ 5.`s 2 + s 3 + 4.` p 1 + 5.`s p 1 + s 2 p 1 + 3.2`K z 1 residue polinomu gerekli: p istenen = Expand@pds Hs+ ald 38.93931951272934` a+ 38.93931951272934` s + 9.29535344921142`as+ 9.29535344921142`s 2 + as 2 +s 3 karakteristik denklemler çözülürse 11

clst1= CoefficientList@p fazil, sd 84.` p 1 + 3.2`K z 1, 4.`+ 3.2`K + 5.` p 1, 5.`+ p 1, 1< clst2 = CoefficientList@p istenen, sd {38.9393 a,38.9393+9.29535 a,9.29535+a,1} ve bu çözüm eklenen a kutbunu serbest parametre olarak kabul edilerek verilirse sol2= Solve@clst1 clst2, 8K, p 1, z 1 <D 2.9248788731324`*^23 + 5.947896937521918`*^23a 99z 1 2.291798346191674`*^23 + 7.31219718282553`*^22a, p 1 2.36273273599356`*^-8 H1.81795992`*^8 + 4.2323873`*^7 al, K 3.67139426215336`*^-24 H1.145899173545838`*^24+ 3.656985914127515`*^23 al== olsun ve O halde a z 1 1 1 1 1 in a - r ifade K c (s)g(s) ifadesindeki G c üsttekine göre çok - 12

sol3=sol2/.a 4 88z 1 4., p 1 8.29535, K 9.57624<< 88z 1 3.999999999999999`, p 1 8.29535344921142`, K 9.5762394519936`<< edilebilir Bu çözüme göre Kontr yeniden düzenlenirse Fs K Hs+ z 1 L s+ p 1 Fbulunan=Fs/.sol3[[1]] 9.5762394519936` H3.999999999999999` + sl 8.29535344921142`+ s Lbulunan=L/.sol3[[1]] 3.64396696351797` H3.999999999999999` + sl PolesSISO[Lbulunan] {-8.29535,-4.,-1.} ZerosSISO[Lbulunan] {-4.} H8.29535344921142`+sL H4+ 5.`s+ s 2 L Tbulunan= T fazil ê.sol3@@1dd 122.5758643854716`+3.64396696351797`s 155.75727859173`+76.127331295751`s+13.29535344921142`s 2 +s 3 PolesZeros[Tbulunan] {{-4.64768-4.16394,-4.64768+4.16394,-4.},{-4.}} 13

pzm2=polezeromap[lbulunan] 1 Pole Zero Map.5 I m HsL -.5-1 -8-7 -6-5 -4-3 -2-1 ReHsL pzm3=polezeromap[tbulunan] 4 Pole Zero Map 2 I m HsL -2-4 -4.6-4.5-4.4-4.3-4.2-4.1-4 ReHsL rloci=rootloci[lbulunan,joingraphics {pzm2}] 14

1 Root Locus Plot 5 I m HsL -5-1 -8-7 -6-5 -4-3 -2-1 ReHsL Tbulunan 122.5758643854716` +3.64396696351797` s 155.75727859173`+ 76.127331295751`s+13.29535344921142`s 2 + s 3 Step[Tbulunan] YHtL 1.8.6.4.2 Time Response.5 1 1.5 2 Time Zaman bölgesi analizi: TimeDomainCharacteristics[Tbulunan,ShowMessages True] Reducing tmax to 1.1986 Settling Time (Ts) :.92757 sec Overshoot Time (Tp):.7547 sec 15

Overshoot :.3 Delay Time (Td) :.234569 sec Rise Time (Tr) :.36492 sec {.3,.7547,.92757,.234569,.36492} SteadyStateError[Tbulunan] -.21333 3. b..i.ii.iii.simülasyon Step[Tbulunan] YHtL 1.8.6.4.2 Time Response.5 1 1.5 2 Time Step 9.57624 Gain s+4 s+8.29534 Transfer Fcn 1 3.2 s 2+5s+4 Transfer Fcn Scope Display.787 16

4. b..i.ii.iii.sonuç.92757 1.72129 1 53.8583. 17

4 Pole Zero Map 2 I m HsL -2-4 -4.6-4.5-4.4-4.3-4.2-4.1-4 ReHsL (Kutup- Burada dikkat çekilecek husus: S - -4.64768-4.16394,-4.64768+4.16394, -4) bu 18

1.b.iv.Amaç 4. b..i.ii.iii.sonucuna dayanarak b ilerlemeli-gerilemeli AppendTo[$Path,"P:\\Muhendis\Mathematica\macsybox"]; <<Control`. z 1 = 3.999999999999999` p 1 = 8.29535344921142` K = 9.5762394519936` 4. 8.29535 9.57624 F= K Hs+ z 1L Hs+ z 2 L Hs+ p 1 L Hs+ p 2 L 9.57624 H4. + sl Hs + z 2 L H8.29535+ sl Hs + p 2 L Bir e ss1 =.21332829314812`.21333 e ss = e ss1 ê1.21333 Kp= H1êe ss L 1 19

45.9411. n 1 = 5 n = Gs= 3.2+ n s 2 + H3.5+.3n 1 L s+ 4+.2n 3.2 4 + 5. s + s 2 : L=F Gs K p = lim s Gc ( s) G p ( s) : Kpl=L/.s 3.644 H4. + sl Hs+ z 2 L H8.29535 + sl H4 + 5. s + s 2 L Hs+ p 2 L 3.69411 z 2 p 2 tasarlanan sistemin sol=solve[kpl-kp ] 88z 2 12.4363 p 2 << Normalde Sistemin transfer fonksiyonunun -1 ve - Faz ilerlemeli kontrolör ile - ve - Bu kontrolör ile sistemin -4 teki kutbunun etkisini - Geriye - - ku 2

- eklenmelidir. O yüzen p 2 kutbu da z 2 - Bu fikirden hareketle.8121 1.679 -. bulunabilir: F Gs L p 2 =.8121 z 2 = 12.397344418426517` p 2 9.57624 H1.679 + sl H4. + sl H.8121+ sl H8.29535 + sl 3.2 4 + 5. s + s 2 3.644 H1.679 + sl H4.+ sl H.8121 + sl H8.29535 + sl H4 + 5. s + s 2 L PolesSISO[L] ZerosSISO[L] {-8.29535,-4.,-1.,-.8121} {-4.,-1.679} 21

T fazilgeri = TogetherAExpandA l. {-4.18393-3.7786,-4.18393+3.7786,-4.,-1.871} {-4.,-1.679} pzml=polezeromap[l] L 1+ L EE 123.48 + 153.428 s + 3.644 s 2 126.13 + 19.32 s + 77.24 s 2 + 13.3766 s 3 + s 4 pch= Denominator@T fazilgeri D 126.13+ 19.32 s + 77.24 s 2 + 13.3766 s 3 + s 4 PolesSISO@T fazilgeri D ZerosSISO@T fazilgeri D 1 Pole Zero Map.5 I m HsL -.5-1 -8-6 -4-2 ReHsL Graphics rloci=rootloci[l,joingraphics {pzml}] 22

1 Root Locus Plot 5 I m HsL -5-1 -8-6 -4-2 ReHsL Graphics pzmt= PoleZeroMap@T fazilgeri D Pole Zero Map I m HsL 3 2 1-1 -2-3 -4-3.5-3 -2.5-2 -1.5-1 ReHsL Graphics TimeDomainCharacteristics@T fazilgeri, ShowMessages TrueD Reducing tmax to 1.14331 Settling Time (Ts) : 1.4155 sec Overshoot Time (Tp):.84647 sec Overshoot :.373 Delay Time (Td) :.261817 sec Rise Time (Tr) :.4717 sec 23

{.373,.84647,1.4155,.261817,.4717} SteadyStateError@T fazilgeri D -.213688 1.b.iv.Simülasyon Step@T fazilgeri D 1 Time Response.8.6 YHtL.4.2 2 4 6 Time Sistem Matlab Step 9.57624 Gain s+4 s+8.29534 Transfer Fcn 1 s+1.679 s+.8121 Transfer Fcn 2 3.2 s 2+5s+4 Transfer Fcn Scope.9786 Display 24

1.b.iv.Sonuç Sonuç olarak, n.373 % 3, bir önceki sistemde.92757 idi. 1.4155 olarak - -1.679 de bulunan kut.213688 k 1/ görülebilir. 25

1.c.Amaç AppendTo[$Path,"P:\\Muhendis\Mathematica\macsybox"]; <<Control` n 1 = 5 n = Gs= 3.2+ n s 2 + H3.5+.3n 1 L s+ 4+.2n 3.2 4 + 5. s + s 2 Fs= K p + K d s s K d + K p Ls=Fs Gs 3.2 Hs K d + K p L 4+ 5. s + s 2 T PD = TogetherAExpandA 3.2 s K d + 3.2 K p Ls 1+ Ls EE 4. + 5. s + s 2 + 3.2 s K d + 3.2 K p p PD = Denominator@T PD D 4.+ 5. s + s 2 + 3.2 s K d + 3.2 K p Asim=.3; 26

ζdesired = Log@AsimD è!!!!!!!!!!!!!!!!!!!!!!!!!!!!! π 2 + Log@AsimD 2.74484 SettlingTime= 1.721291223883845` 1.72129 targetsettlingtime=settlingtime/2.86645 wdesired = 4 ζdesired targetsettlingtime 6.2414 pds= s 2 2 + 2ζ w n s+ w n ê. 8ζ ζdesired, wn wdesired< 38.9393+ 9.29535 s + s 2 clst1 = CoefficientList@p PD, sd 84.+ 3.2 K p, 5. + 3.2 K d, 1< clst2=coefficientlist[pds,s] {38.9393,9.29535,1} sol= Solve@clst1 clst2, 8K p, K d <D 88K d 1.3423, K p 1.9185<< Fsfix=Fs/.sol[[1]] 1.9185 +1.3423 s Solve[Fsfix ] {{s -8.13421}} AchievedL=Ls/.sol[[1]] 3.2 H1.9185 + 1.3423 sl 4+ 5. s + s 2 27

PolesSISO[AchievedL] {-4.,-1.} ZerosSISO[AchievedL] {-8.13421} AchievedTs2 = T PD ê.sol@@1dd 34.9393 + 4.29535 s 38.9393 + 9.29535 s + s 2 poles=polessiso[achievedts2] {-4.64768-4.16394,-4.64768+4.16394 } ZerosSISO[AchievedTs2] {-8.13421} TimeDomainCharacteristics[AchievedTs2,ShowMessages True] Reducing tmax to.91173 Settling Time (Ts) :.828166 sec Overshoot Time (Tp):.544822 sec Overshoot :.531135 Delay Time (Td) :.11844 sec Rise Time (Tr) :.257834 sec {.531135,.544822,.828166,.11844,.257834} SteadyStateError[AchievedTs2] -.12724 28

1.c.Simülasyon Step[AchievedTs2] YHtL 1.8.6.4.2 Time Response.25.5.75 1 1.25 1.5 Time GraphicsArray 1.c.Sonuç Sonuç olarak, b.828166 saniye %96 %5.3 bulundu. Bu sonuç beklenenden hayli fazla. Bunun sebebi ise -.12724.35529 ev etkeninin 29

1.d.Amaç {- - Bu takdirde PID Kontrolörü yüzünden eklenecek ikinci yine onun kutbun et F( s) = KP + KDs kontrolörünün kökü -8 idi. Ki F( s) = KP + KDs + s KP + KDs Ki F( 8.13421) = KP + KDs + 8.13421 terim K i eksi yönde bir ötelemeye neden olur. O halde bu negatif ötelemenin bir pozitif 8.13421 öteleme ile giderilmesi gerekir ki F( s) K P + K Ds - - K P terimi K Ds Ki teriminden büyük olacak ve bu pozitif fark, s AppendTo[$Path,"P:\\Muhendis\Mathematica\macsybox"]; <<Control`, Fs= TogetherAExpandAK p + K d s+ K i s EE 3

s 2 K d + K i + s K p s Fsn=Numerator[Fs] s 2 K d + K i + s K p Solve[Fsn,{s}] - -8 den biraz küçük) olsun daha ::s K p "###################### 4 K d K i + K2 p 2 K d >, :s K p + "###################### 4 K d K i + K2 p 2 K d >> f= K p "###################### 4K d K i + K2 p 2K d H 7.619L 7.619+ K p "###################### 4 K d K i + K2 p 2 K d sol=solve[f ] 88K i 1. 1 6 H 5.8492 1 7 K d + 7.619 1 6 K p L<< düzenlenirse Fsyeni=Fs/.sol[[1]] s 2 K d + s K p + 1. 1 6 H 5.8492 1 7 K d + 7.619 1 6 K p L s n 1 = 5 n = Gs= 3.2+ n s 2 + H3.5+.3n 1 L s+ 4+.2n 31

3.2 4 + 5. s + s 2 Ls=Gs Fsyeni H3.2 Hs 2 K d + s K p + 1. 1 6 H 5.8492 1 7 K d + 7.619 1 6 K p LLL Hs H4+ 5. s + s 2 LL = TogetherA Ls 1+ Ls E H 185.757 K d + 3.2 s 2 K d + 24.388 K p + 3.2 s K p L H4. s+ 5. s 2 + s 3 185.757 K d + 3.2 s 2 K d + 24.388 K p + 3.2 s K p L 4. s+ 5. s 2 + s 3 185.757 K d + 3.2 s 2 K d + 24.388 K p + 3.2 s K p bulunur. Asim=.3; ζdesired = Log@AsimD è!!!!!!!!!!!!!!!!!!!!!!!!!!!!! π 2 + Log@AsimD 2.74484 SettlingTime= 1.721291223883845` 1.72129 targetsettlingtime=settlingtime/2.86645 wdesired = 4 ζdesired targetsettlingtime 6.2414 32

pds= s 2 + 2ζ w n s+ w n 2 ê. 8ζ ζdesired, wn wdesired< 38.9393+ 9.29535 s + s 2 residue i olsun: pes=s+a a+s pds2=expand[pds pes] 38.9393 a+ 38.9393 s + 9.29535 a s + 9.29535 s 2 + a s 2 + s 3 8 185.757 K d + 24.388 K p, 4. + 3.2 K p, 5. + 3.2 K d, 1< clst2=coefficientlist[pds2,s] {38.9393 a,38.9393 +9.29535 a,9.29535 +a,1} sol2=solve[clst1 clst2] 88a.644357, K d 1.54366, K p 12.793<< Bu noktadan hareketle K p,k d,k i ifadeleri yerlerine konursa: Fs s 2 K d + K i + s K p s K i =.1`H 66969.`K d + 813.`K p L ê.sol2@@1dd 1.95374 Fsyeni/.sol2[[1]] 33

7.8489 + 12.793 s + 1.54366 s 2 s LT=Ls/.sol2[[1]] 3.2 H7.8489 + 12.793 s + 1.54366 s 2 L s H4+ 5. s + s 2 L RootLoci[LT] Root Locus Plot 4 2 I m HsL -2-4 -15-125 -1-75 -5-25 ReHsL Graphics 25.98 + 4.9288 s + 4.93971 s 2 25.98 + 44.9288 s + 9.93971 s 2 + s 3 PolesZeros[AT] {{-4.64768-4.16394,-4.64768+4.16394,-.644357},{-7.619, -.666677}} -7.619 da olan Yine -7.619 daki kutup sayesinde etkisini giderebilecek bir kutbun o bölgeye geldi. 34

(-.666677) kutbu sayesinde(-.644357) etkisiz hale geliyor. beklenir. Zaman Bölges TimeDomainCharacteristics[AT,ShowMessages True] Reducing tmax to.721259 Settling Time (Ts) :.659231 sec Overshoot Time (Tp):.534453 sec Overshoot :.35389 Delay Time (Td) :.116844 sec Rise Time (Tr) :.26472 sec {.35389,.534453,.659231,.116844,.26472} SteadyStateError[AT]. 1.d.Simülasyon Step[AT] 1.8 Time Response YHtL.6.4.2 2 4 6 8 1 12 Time GraphicsArray 35

1.d.Sonuç Sonuç olarak, n.35389 % 3,5.86645.659231 olarak bir sonraki. 36

2.1.Amaç Sistem transfer fonksiyonunun Gs= 3.2+ n Hs 2 + H3.5+.3n 1 L s + 4+.2n L H.1s + 1L v -Nichols (osilasyon) yöntemiyle belirleyiniz. Mathematica nda Sonuç olarak verilen sistemi kontrol 2.2. AppendTo[$Path,"P:\\Muhendis\Mathematica\macsybox"]; <<Control` n 1 = 5 n = Sistemin transfer fonksiyonu: Gs= 3.2+ n Hs 2 + H3.5+.3n 1 L s + 4+.2n L H.1s + 1L Ziegler- F=K K L=Gs K 3.2 H1 +.1 sl H4 + 5. s + s 2 L 3.2 K H1 +.1 sl H4 + 5. s + s 2 L T= TogetherAExpandA L 1+ L EE 37

3.2 K 4. + 3.2 K + 5.4 s + 1.5 s 2 +.1 s 3 polinom pcs=denominator[t] 4.+ 3.2 K + 5.4 s + 1.5 s 2 +.1 s 3 rtblnichols=routhtabulation[pcs] :.1, 1.5, 5.19.34762 K,.975238 H 164.125+ KL H1.25 + KL 5.19.34762 K Map[(#>)&,rtblnichols] > :True, True, 5.19.34762 K >,.975238 H 164.125+ KL H1.25 + KL 5.19.34762 K > > cond=apply[and,map[(#>)&,rtblnichols]] 5.19.34762 K > &&.975238 H 164.125+ KL H1.25 + KL 5.19.34762 K > Reduce[cond] -1.25<K<164.125-1.25<K< pcsnichols=pcs/.k 164.125 529.2+ 5.4 s + 1.5 s 2 +.1 s 3 s1=solve[pcsnichols ] 88s 15.<, 8s 4.14939 1 17 22.4499 <, 8s 4.14939 1 17 + 22.4499 << 38

K c = 164.125 164.125 : w c = 22.449 22.449 Buradan Ziegler- P c = 2 π w c.279887 K p =.6K c 98.475 T r =.5P c.139944 T d = P c 8.349859 i FNichols= ExpandAK p j1+ 1 k T r s + T d s y ze { 98.475+ 73.677 s Ziegler- Gs + 3.44524 s 39

LNichols=Gs FNichols ç 3.2 H1 +.1 sl H4 + 5. s + s 2 L 3.2 I98.475 + 73.677 + 3.44524 sm s H1 +.1 sl H4 + 5. s + s 2 L TNichols= TogetherAExpandA Step[TNichols] LNichols 1+ LNichols EE 2251.77 + 315.12 s + 11.248 s 2 2251.77 + 319.12 s + 16.648 s 2 + 1.5 s 3 +.1 s 4 Time Response YHtL 1.5 1.25 1.75.5.25.5 1 1.5 2 2.5 3 Time TimeDomainCharacteristics[TNichols,ShowMessages True] Settling Time (Ts) : 1.51467 sec Overshoot Time (Tp):.159916 sec Overshoot :.642757 Delay Time (Td) :.482765 sec Rise Time (Tr) :.54311 sec {.642757,.159916,1.51467,.482765,.54311} 4

Bir F PID = 1 s I7.8488521683325`+ 12.79264945564488`s + 1.5436595785453735`s 2 M 7.8489 + 12.793 s + 1.54366 s 2 s L PID = GsF PID 3.2 H7.8489 + 12.793 s + 1.54366 s 2 L H1 +.1 sl s H4 + 5. s+ s 2 L T PID = TogetherAExpandA L PID 1+ L PID EE 25.98 + 4.9288 s + 4.93971 s 2 25.98 + 44.9288 s + 9.97971 s 2 + 1.5 s 3 +.1 s 4 Buna göre elde edilen transfer fonksiyonunun Step@T PID D Time Response 1.8 YHtL.6.4.2 2 4 6 8 1 12 Time Yine TimeDomainCharacteristics@T PID, ShowMessages TrueD 41

Reducing tmax to.734446 Settling Time (Ts) :.676425 sec Overshoot Time (Tp):.513378 sec Overshoot :.39457 Delay Time (Td) :.121918 sec Rise Time (Tr) :.24758 sec {.39457,.513378,.676425,.121918,.24758} 42

2.3.Simülasyon Step[TNichols] YHtL 1.5 1.25 1.75.5.25 TimeDomainCharacteristics[TNi chols,showmessages True] Settling Time (Ts) : 1.51467 sec Overshoot Time (Tp):.159916 sec Overshoot :.642757 Time Response.5 1 1.5 2 2.5 3 Time Delay Time (Td) :.482765 sec Rise Time (Tr) :.54311 sec Step[T PID ] YHtL 1.8.6.4.2 2 4 6 8 1 12 Time TimeDomainCharacteristics [T PID,ShowMessages True] Settling Time (Ts) :.676425 sec Overshoot Time (Tp):.513378 sec Overshoot :.39457 Time Response Delay Time (Td) :.121918 sec Rise Time (Tr) :.24758 sec {.642757,.159916,1.51467,.482765,. 54311} {.39457,.513378,.676425,.121918,.247 58} 43

2.4.Sonuç - PID çok daha olumsuz sonuçlar Ziegler- Nichols c göre; Ziegler- Nichols, önceki 3 kat ve önceki Buna göre önceki PID Verilen sistemi kontrol etmek için 44