Bab 5 Root Locus
July 5, 2019 | Author: Muhammad Irfan | Category: N/A
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Bab 5: Root Locus
EL303: Sistem Kendali
ROOT LOCUS
Ì Pendahuluan Ì Dasar Root Locus Ì Plot Root Locus Ì Aturan-Aturan Penggambaran Root Locus Ì Root Locus Melalui MATLAB Ì Kasus Khusus Ì Analisis Sistem Kendali Melalui Root Locus Ì Root
Locus
untuk
Sistem
dengan
Transport Lag
__________________________________________________________________________ Teknik Elektro ITB [EYS-1998] hal 5-1
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Bab 5: Root Locus
EL303: Sistem Kendali
Ì PENDAHULUAN n
Karakteristik tanggapan transient sistem loop tertutup dapat ditentukan dari lokasi pole-pole (loop tertutupnya).
n
Bila K berubah, maka letak pole-pole nya juga berubah. Perlu pemahaman pola perpindahan letak pole-pole dalam bidang s. Desain sistem kendali melalui gain adjusment: pilih K sehingga pole-pole terletak ditempat yang diinginkan. Desain sistem kendali melalui kompensasi: memindahkan letak pole yang tak diinginkan melalui pole-zero cancellation. Mencari akar-akar persamaan karakteristik untuk orde tinggi sulit, terlebih dengan K sebagai variabel. (Alternatif: gunakan MATLAB ?!) W.R. Evan mengembangkan metoda untuk mencari akar-akar persamaan orde tinggi : metoda Root Locus. Root Locus: tempat kedudukan akar-akar persamaan karakterstik dengan K = 0 sampai K = tak hingga.
n n n n n n
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Bab 5: Root Locus
EL303: Sistem Kendali
Ì DASAR ROOT LOCUS
Persamaan Karakteristik: s 2 + 2s + K =0
Akar-akar Persamaan Karakteristik : s=
K 0 1 2 10
− 2 ± 4 − 4 K 2
s1 0 -1 -1+j1 -1+j3
= −1 ± 1 − K
s2 -2 -1 -1+j1 -1+j3
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Bab 5: Root Locus
EL303: Sistem Kendali
n
Root Locus mempunyai sifat simetri terhadap sumbu nyata.
n
Root Locus bermula dari pole-pole G(s)H(s) (untuk K=0) dan berakhir di zero-zero G(s)H(s) (untuk K→∞) termasuk zero-zero pada titik takhingga.
n
Root Locus cukup bermanfaat dalam desain sistem kendali linear karena Root Locus dapat menunjukkan pole-pole dan zero-zero loop terbuka mana yang harus diubah sehingga spesifikasi unjuk kerja sistem dapat dipenuhi.
n
Pendekatan desain melalui Root Locus sangat cocok diterapkan untuk memperoleh hasil secara cepat.
n
Sistem kendali yang membutuhkan lebih dari 1 parameter untuk diatur masih dapat menggunakan pendekatan Root Locus dengan mengubah hanya 1 parameter pada satu saat.
n
Root Locus sangat memudahkan pengamatan pengaruh variasi suatu parameter (K) terhadap letak pole-pole.
n
Sketsa Root Locus secara manual tetap dibutuhkan untuk dapat memahaminya dan untuk memperoleh idea dasar secara cepat, meskipun MATLAB dapat melakukannya secara cepat dan akurat.
n
Spesifikasi transient (koefisien redaman) dapat ditentukan dengan mengatur nilai K melalui Root
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Bab 5: Root Locus
EL303: Sistem Kendali
Ì PLOT ROOT LOCUS
Persamaan Karakteristik: 1 + G(s)H(s) G(s)H(s ) = 0 Atau: G(s)H(s) = -1, Sehingga: ⊃G(s)H(s) = ! 1800(2k+1); (syarat sudut) k = 0, 1, 2, …. | G(s)H(s)| = 1
(syarat magnitude)
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Bab 5: Root Locus
EL303: Sistem Kendali
Ì PROSEDUR PENGGAMBARAN ROOT LOCUS 1. Letakkan pole-pole pole-pole dan zero-zero loop terbuka pada bidang s.
2. Tentukan Root Locus pada sumbu nyata.
• Syarat Sudut: ⊃G(s)H(s) = ! 1800(2k+1); •
k = 0, 1, 2, …. Ambil titik test : bila jumlah total pole dan zero dikanan titik ini ganjil, maka titik tsb terletak di Root Locus.
3. Tentukan asimtot Root Locus:
• Banyaknya asimtot = n – m n = banyaknya pole loop terbuka m= banyaknya zero loop terbuka
± 1800 (2k + 1) • Sudut-sudut asimtot = n−m k=0, 1, 2, …
• Titik Potong asimtot-asimtot pada sumbu nyata: σa
(letak pole berhingga )− ∑ (letak zero berhingga ) ∑ =
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Bab 5: Root Locus
EL303: Sistem Kendali
4. Tentukan titik-titik break-away dan titik-titik titik-titik break-in: Untuk Persamaan Karakteristik: B(s) + KA(s) = 0, Maka titik-titik tsb harus berada di Root Locus dan memenuhi persamaan:
dK ds
= −
B ' ( s) A( s ) − B ( s ) A' ( s ) 2
A ( s )
=0
5. Tentukan sudut-sudut datang / sudut-sudut berangkat untuk pole-pole / zero-zero kompleks sekawan. • Sudut datang (dari suatu pole kompleks) = 180 0 – (jumlah sudut vektor-vektor dari pole-pole lain ke pole kompleks tsb) + ( jumlah sudut vektor-vektor vektor-vektor dari dari zerozero ke pole kompleks tsb). • Sudut pergi (ke suatu zero kompleks) = 180 0 – (jumlah sudut vektor-vektor dari zero-zero lain ke zero kompleks tsb) + ( jumlah sudut vektor-vektor vektor-vektor dari polepole ke zero kompleks tsb).
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Bab 5: Root Locus
EL303: Sistem Kendali
6. Tentukan batas kestabilan mutlak sistem (K):
• Melalui Kriteria Routh Hurwitz. • Secara analitis: memotong sumbu imajiner: s = j 7. Sketsa Root Locus secara lebih teliti pada daerahdaerah selain sumbu nyata dan asimtot. 8. Tentukan letak pole-pole pole-pole melalui nilai K yang memenuhi syarat magnitude. Sebalikya, bila letak polepole ditentukan (pada Root Locus), maka nilai K yang memenuhi dapat dihitung secara grafis atau secara analitis: Secara grafis: K=
perkalian panjang garis - garis dari titik s ke pole - pole perkalian panjang garis - garis dari titik s ke zero - zero
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Bab 5: Root Locus
EL303: Sistem Kendali
CONTOH : Gambarkan Root Locus sistem balikan satuan dengan G(s ) =
K s ( s + 1)(s + 2)
Tentukan juga nilai K agar koefisien redaman pole-pole kompleks sekawan loop tertutup dominannya bernilai 0,5. Solusi :
1. Tentukan Root Locus pada sumbu nyata. jω
Titik uji 2 -2
-1
Titik uji 1
•
0
•
σ
Untuk titik uji 1 : Syarat sudut : − ∠s − ∠(s + 1) − ∠( s + 2) = 0 0 + 0 0 + 0 0 = 0 0 (tak terpenuhi). Untuk titik uji 2 : 0 0 0 0 Syarat sudut : − ∠s − ∠(s + 1) − ∠( s + 2) = −180 − 0 − 0 = −180 (terpenuhi).
2. Penentuan asimtot Root Locus Banyaknya asimtot = banyaknya pole (n) – banyaknya zero (m) = 3 - 0 = 3
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Bab 5: Root Locus
K s (s + 1)(s + 2) dK ds
EL303: Sistem Kendali
+1 = 0
3 2 atau K = −(s + 3s + 2 s ) , sehingga:
= −(3s 2 + 6 s + 2) = 0
Diperoleh s1 = −0,4226 (memenuhi) dan s 2 = −1,5774 (tak memenuhi) 4. Penentuan batas kestabilan sistem menggunakan kriteria Routh Hurwitz. s3 1 s2 3 6−K s1 3 s0 K
2 K
Syarat stabil tercapai bila 0 < K < 6. Bila dihitung, perpotongan Root Locus dengan sumbu khayal ini terjadi pada : s = ± j 2 . Cara lain untuk mengetahui titik potong ini adalah secara analisis: s = j ω (pada sumbu khayal). 5. Tentukan beberapa titik uji dekat titik pencar yang memenuhi syarat sudut Root Locus agar diperoleh plot Root Locus secara akurat.
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Bab 5: Root Locus
EL303: Sistem Kendali
6. Gambar Root Locus nya:
7. Penentuan letak pole-pole kompleks sekawan dominan yang memiliki koefisien 2 redaman 0,5. Anggap pole kompleks sekawan s = −ζω n ± jω n 1 − ζ . Dengan
memperhatikan gambar dibawah ini, maka terlihat bahwa ζ = cos β . Untuk
ζ = 0,5, maka β = 60 0 . Dengan menggunakan cara analitis akan diperoleh polepole dominan tersebut adalah : s = -0,3337 + j0,5780, dengan nilai K adalah: K = s ( s +)(s + 2) s = −0 ,3337+ j 0,5780 = 1,0383
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Bab 5: Root Locus
EL303: Sistem Kendali
Ì BEBERAPA CATATAN • Konfigurasi pole-zero yang sedikit bergeser dapat mengubah total bentuk Root Locus.
• Orde sistem dapat berkurang akibat pole-pole G(s) di
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Bab 5: Root Locus
EL303: Sistem Kendali
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Bab 5: Root Locus
EL303: Sistem Kendali
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Bab 5: Root Locus
EL303: Sistem Kendali
Ì ROOT LOCUS MELALUI MATLAB Root Locus = persamaan karakteristiknya, dalam MATLAB: 1+ K
num den
=0
num = (s + z1 )(s + z 2 ) L (s + z m )
= s m + (z 1 + z 2 + L + z m )s m −1 + L z 1 z 2 L z m den = (s + p1 )(s + p 2 ) L (s + p n )
= s n + (p1 + p 2 + L + p n )s n −1 + L + p1 p 2 L p n Perintah MATLAB untuk menggambar Root Locus (Konsep Fungsi Alih):
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Bab 5: Root Locus
Cara
lain
EL303: Sistem Kendali
penggambaran
Root
Locus
adalah
dengan
menggunakan arguman berikut ini :
[r,K] = rlocus(num,den) rlocus(num,den) [r,K] = rlocus(num,den,K) rlocus(num,den,K) [r,K] = rlocus(A,B,C,D) rlocus(A,B,C,D) [r,K] = rlocus(A,B,C,D,K) rlocus(A,B,C,D,K) Pada layar akan tampil matriks r dan vektor penguatan K. Perintah :
r=rlocus(num,den) plot(r,'o')
atau,
plot(r,'x')
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Bab 5: Root Locus
EL303: Sistem Kendali
Contoh : Plot Root Locus menggunakan MATLAB suatu sistem kendali balikan satuan: G (s ) =
K (s 2 + 2s + 4) s (s + 4)(s + 6)(s 2 + 1,4s + 1)
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Bab 5: Root Locus
EL303: Sistem Kendali
Program MATLAB nya:
%------Root-Locus %------Root-Locus -------- ----num = [0
0
0
den = [1
11.4
1
2
39
rlocus(num,den) Warning:Divide by zero
4];
43.6
24
0];
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Bab 5: Root Locus
EL303: Sistem Kendali
Ì KASUS KHUSUS ] Parameter K bukan penguatan loop terbuka. ] Umpanbalik positif.
] Parameter K bukan Penguatan Loop Terbuka.
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Bab 5: Root Locus
EL303: Sistem Kendali
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Bab 5: Root Locus
] Umpanbalik Positif.
EL303: Sistem Kendali
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Bab 5: Root Locus
Contoh:
EL303: Sistem Kendali
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Bab 5: Root Locus
EL303: Sistem Kendali
Sistem tidak stabil untuk K > 3 (Gunakan metoda Root Hurwitz untuk menghitungnya!). Sistem harus distabilkan dengan umpanbalik negatif diluarnya.
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Bab 5: Root Locus
EL303: Sistem Kendali
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Bab 5: Root Locus
EL303: Sistem Kendali
Ì ANALISIS SISTEM KENDALI • Ortogonalitas dan locus dengan penguatan konstan
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Bab 5: Root Locus
• Sistem Stabil Kondisional
EL303: Sistem Kendali
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