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Page 1: Stabilit As

STABILITAS

Page 2: Stabilit As

StabilitasGU

GpGvGc

Gs

+

Se point

disturbance

+

+𝑅

π‘ˆ

π‘ŒControlled variabel

π‘¦π‘œ+βˆ† 𝑦

Harga ygdiinginkan

toleransi

π‘Œ=𝐺𝑝 .𝐺𝑣 .𝐺𝑐

1+𝐺𝑝 .𝐺𝑣 .𝐺𝑐 .𝐺𝑠𝑅+

𝐺𝑒1+𝐺𝑝+𝐺𝑣+𝐺𝑐+𝐺𝑠

π‘ˆ

1) Operasi normal (regulated variabel)2) Operasi tak normal (servo operation)3) Regulated & servo operation

𝑅=0 (π‘ π‘’π‘‘π‘π‘œπ‘–π‘›π‘‘ π‘¦π‘”π‘‘π‘’π‘‘π‘Žπ‘)π‘ˆ=0

π‘ˆ β‰ 0 ,𝑅≠ 0

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Stabilitas tjd saat kapan dan dimana ??

Kriteria stabilitas :

Stabil ← Nilai controlled variabel (y(t) pd tβ†’ ~, tertentuTak stabil ←nilai controlled variabel (y(t) pd tβ†’~, Β± ~

π‘Œ=𝑁 (𝑠)𝐷(𝑠)

= π‘π‘œπ‘šπ‘–π‘›π‘Žπ‘‘π‘œπ‘Ÿπ‘‘π‘’π‘›π‘œπ‘šπ‘–π‘›π‘‘π‘œπ‘Ÿ

+...................+

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𝑦𝑖 (𝑑 )=π‘’πœ‡π‘– .𝑑 { πœ‡>0β†’ π‘™π‘–π‘šπ‘‘β†’βˆž 𝑦𝑖 (𝑑 )=0β†’π‘ π‘‘π‘Žπ‘π‘–π‘™ΒΏπœ‡=0β†’ π‘™π‘–π‘šπ‘‘β†’βˆž 𝑦 (𝑑 )=π›Όβ†’π‘ π‘‘π‘Žπ‘π‘–π‘™ π‘‘π‘’π‘›π‘”π‘Žπ‘›π‘œπ‘“π‘“ 𝑠𝑒𝑑 (π‘Žπ‘‘π‘Žπ‘‘π‘œπ‘™π‘’π‘Ÿπ‘Žπ‘›π‘ π‘–π‘›π‘¦π‘Ž)

ΒΏπœ‡<0β†’ π‘™π‘–π‘šπ‘‘β†’βˆžβ†’ π‘‘π‘‘π‘˜π‘ π‘‘π‘Žπ‘π‘–π‘™

Stabilitas harga dipengaruhi Denominator (D(s)) yang untuk ke-3 kasus (1, 2 dan 3)Mempunyai D(s) yg sama yaitu D(s)=1 + Gp .Gv. Gs. S Pers. karakteristik

1 + Gp.Gv.Gc. S = 0

Sifat kestabilan ini akan terkontrolPada akar-akar pers. Tsb. Pada Tabel berikut :

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Akar-akar pers. Karakteristik, si = -Β΅ - Ο‰i

riil imajiner

Penyebab kestabilan

πœ‡>0 ,π‘ π‘‘π‘Žπ‘π‘–π‘™ πœ‡<0 , π‘‘π‘‘π‘˜π‘ π‘‘π‘Žπ‘π‘–π‘™

Real

Imajiner

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Faktor polinomial karakateristik

Akar-akar karakteristik

Respon (I L T)

Stabilitas Letak Akar Gmbar

S=Β΅ -Β΅ k.

s2 +Ο‰2 S=Β±Ο‰i k sin Ο‰t Batas stabil

(s+Β΅)2+Ο‰2 S2 Β΅ Β±Ο‰i k e-Ο‰t sin (Ο‰t+Ø)

Im

Re-Β΅

Im

ReΒ΅

y(t)

ty(t)

tIm

Re

Ο‰

Ο‰

y(t)

t

Re

ImΒ΅

Β΅

Ø y(t)

t

Re

Imy(t)

t

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Faktor polinomial karakateristik

Akar-akar karakteristik

Respon (I L T)

Stabilitas Letak Akar Gmbar

S S=0 k

s2 S=0,0 Ko+k1.t Tdk pernah stabil

(s+Β΅)2 S= Β΅1-Β΅ k e-Β΅t (ko+k1t)

Stabil jk Β΅ > 0

lm

Re1 akar

t

y(t)

lm

Re2 akar

t

y(t)

lm

Re-Β΅

t

y(t)

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Fungsi Sinus : π‘₯ (𝑑 )=a sinπœ”π‘‘y(t)

t π‘₯ (𝑠)= π‘Žπœ”π‘ 2+πœ”2

y(t)

t π‘₯ (𝑠)= π‘Žπ‘ π‘ 2+πœ”2

Digeser 𝛼 (𝑑 )=π‘Ž cosπœ”π‘‘

y(t)

t ?

Digeser sebesar Ø (Ο‰t+Ø)

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= k. sin = k. cos sin + k sin cos t = k1 sin t + k2 cos t

βˆ…a=k1

b=k2

Sin Ø= b = k sin Ø

Cos Ø= a = k cos Ø

tg Ø= Ø = arctg

π‘˜=βˆšπ‘˜12+π‘˜2❑2

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Stabilitas s atau pers. Karakteristik

1 + Gp. Gv. Gc. Gs = 0a1.sn + a2 sn-1 + .......=0

n=1 akar mudah carin=2 rumus ABCn 3 ???

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G(s)𝑋 (𝑠) π‘Œ (𝑠)

b = 2 f c = 1

=

𝑠1=βˆ’ π‘“πœ

+ √ 𝑓 2βˆ’1𝜏

𝑠2=βˆ’ π‘“πœβˆ’ √ 𝑓 2βˆ’1

𝜏

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π‘ˆπ‘›π‘‘π‘’π‘˜ :√ 𝑓 2βˆ’1

1. Jika f > 1 2 akar riil yg bebeda

Over damped / non osilasi

2. Jika f=0 2 akar riil yg sama

Crtically Damped

3. Jika f < 1 Sepasang akar bil. komplek

Osilasi (under lumped)

f1

f2

f3

f4f5

ho

f1> f2 > f3 > f4 > f5

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π‘₯ (𝑠 )=𝑠𝑑𝑒𝑝 π‘“π‘’π‘›π‘π‘‘π‘–π‘œπ‘›=0𝑠 π‘₯ (𝑑 )=π‘Ž

𝑦 (𝑠)π‘’π‘›π‘‘π‘’π‘˜ 𝑓 <1 𝑦 (𝑑 )=1βˆ’ 1

√1βˆ’ 𝑓 2π‘’βˆ’ 𝑓 𝑑

𝜏 sin (√1βˆ’ 𝑓 2π‘‘πœ

+π‘‘π‘Žπ‘›βˆ’ 1 √1βˆ’ 𝑓2

𝑠 )𝑦 (𝑠)π‘’π‘›π‘‘π‘’π‘˜ 𝑓 =1 𝑦 (𝑑 )=1βˆ’(1+ 𝑑

𝜏 )π‘’βˆ’ π‘“πœ

𝑦 (𝑠)π‘’π‘›π‘‘π‘’π‘˜ 𝑓 >1 𝑦 (𝑑 )=1βˆ’π‘’βˆ’ 𝑓 𝑑

𝜏 ( hπ‘π‘œπ‘  √ 𝑓 2βˆ’1 π‘‘πœ

+ 𝑓

√ 𝑓 2h𝑠𝑖𝑛 √ 𝑓 2βˆ’1 𝑑

𝜏 ) =

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Respon time limit

ts B

A

T

c

1. Over shoot = simpangan terbesar = exp ( -ΒΆ f /

2. Decay Ratio Peredaman = c/A f > c/A > 3. Rise Time =tr wkt pertama kali y(t) berharga sebesar B tr > f >

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4. Response Time

wkt y(t) = yt ~ Β± βˆ†y

-5 % < βˆ†y < +5%

5. Periode T

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Stabilitas

akar-akar persamaan F2(s)=0

Bentuk umum :λj=-μ ±iω

j=1,2,.....ΞΌ =bagian realΞ©=bagian imajiner

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Bila Ο‰=0 β†’ bil. Real β†’

Bila Ο‰β‰ 01. ΞΌ=0 β†’bil. Imajiner pasangan akar

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k sin(Ο‰t+Ø) =K

k1=Kcos

k2=K sin

tan = 2+k2

2 =K2

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2. ΞΌβ‰ 0a. ΞΌ > 0 Ξ»j=- ΞΌ Β± ib. ΞΌ <0

Κ†{y(t)} =F(s)Κ†{y(t)e-at} =F(s+a)

𝐡1+𝐡2(π‘ βˆ’πœ‡)(π‘ βˆ’πœ‡)2+πœ”2

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Β΅<0 ; e +ΞΌt

Β΅>0 ; e -ΞΌt

βˆ’πœ™πœ”

K sin(Ο‰t+Ø)eΒ΅t ; ΞΌ<0 ; stabil

k sin(Ο‰t+Ø)

K sin(Ο‰t+Ø)eΒ΅t ; ΞΌ>0 ; tidak stabil

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Contoh.

Gc=Kc

A

Gv=Kv

X L/min

SolventU L/min

Gs=Ks

Y gmol/L

Konsentrasi A dikendalikan

t=0 ; U=Uot β‰₯0 ; U=Uo+a

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F2(s)=s { s3 +6s2 +12s +8 + 8Kc}

(s+Ξ»1)(s+ Ξ»2)(s+Ξ»3) (s+Ξ»4)

Ξ»1=0

Yang dipengaruhi adalah harga Kc yang digunakan.

Bila kc=1 ; λ2 = -4; λ3= -1± i√3 ; ¡>0 ; stabil

Bila Kc=8 ; λ2 = -6; λ3,4= i(2√3) ; ¡=0 ; keadaan batas

Bila Kc=27 ; λ2 = -8; λ3,4= 1 ± i(3√3) ; ¡<0 ; tidak stabil

Kesimpulan :Harga kc > 8 sistem tidak stabilHarga kc < 8 sistem stabil

Dari segi keamanan harga Kc yg diambil jauh lebih kecil dari 8 Hp jangan terlalu kecil krn merugikan

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Contoh.

Kc =1

Sehingga kita punya bentuk :

= = offset

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Dilihat dari offset : Kc<<< - offset besar - stabil Harus ada kompromi antara stabilitas dan offset.Offset bisa dihilangkan dgn jenis kontroller lain, tetapi kontroller tidak bisa menghilangkan stabilitas.Stabilitas diketahui dengan mencari akar F2(s)=0Dgn real λ =-μ ± iω akar-akar = - μ ; μ >0 stabil

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β€œFREQUENSI RESPONSE”G(I)

I (s) O (s)

O(s) = G(I) . I(s)

I(t) = a sin (Ο‰t)

Amplitudo

t=T

Asin (Ο‰t)

t

t=-Ø/w

K

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O(s) = G(s) . I(s)O(s) = G(s) a

+....

Ksin(

A1sin(t)+A2 cos(t)

t=besar

Bil. Stabil 0

O(t)=K sin (t+Ø

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Sistem yang stabilInput = a sin Ο‰t output pd t >> , O =K sin (t+Ø) sinusoida fungsi sinusoida

Merupakan ciri dg perb. Amplitudo K/a= |G(i )| f1()Transf. Func. System perb. Phase = Ø = G(i ) f2()

*

G(s) awS2+Ο‰2 = 0S=i Ο‰

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aω=

S=iw

G(iw)=

tanØ = A2/A1k2 = A1

2 +A22 im

ReA1/a

A2/a

Ø|G|