Matakuliah
Tahun
Versi
: H0332/Simulasi dan Permodelan
: 2005
: 1/1
Pertemuan #12
Runge Kutta (OFC)
1
Learning Outcomes
Pada akhir pertemuan ini, diharapkan mahasiswa
akan mampu :
• Mahasiswa dapat menerapkan metode
Runge Kutta pada perhitungan sistem
dinamik (C3)
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Outline Materi
• Runge Kutta
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1. Mathematic Representations
• Classical Differential Equations
d mx
d 2x
dx
d nu
d 2u
du
am m  ...  a2 2  a1  a0 x  bn n  ...  b2 2  b1
 b0u
dt
dt
dt
dt
dt
dt
• Transfer Function
bn s n  ...  b2 s 2  b1s  b0
X

U am s m  ...  a2 s n  a1s  a0
• State-Space Equation
d
x A yBu
dt
y C yDu
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2. Time Response and Digital Simulation
• Runge-Kutta Method
System Equation
dx
 x  f ( x , u , t )
dt
x1  f1 ( xm ,..., x2 , x1, un ,..., u2 , u1, t )
x2  f 2 ( xm ,..., x2 , x1, un ,..., u2 , u1, t )
...
xm  f m ( xm ,..., x2 , x1, un ,..., u2 , u1, t )
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• Runge-Kutta Method (cont.)
Solution
h
x j 1  x j  ( A  2 B  3C  D )
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where
A  f (x j ,u j ,t j )
h
h
B  f ( x j  A, u j , t j  )
2
2
h
h
C  f (x j  B ,u j , t j  )
2
2
D  f ( x j  hC , u j , t j  h)
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3. Input Signals
• Step Input
u (t )  0
for
u (t )  u0 for
• Ramp Input
t 0
t 0
• Impulse Input
u (t )  0
for
for
t 0
u (t )  u0t for t  0
• Pulse Input
t 0
0
 u (t )dt  u0
0
u (t )  0
for
t  0 and t  T1
u (t )  u0
for
0  t  T1
• Ramped Step Input
• Harmonic Input
u (t )  0
u (t )  0
for
t 0
u(t )  u0 sin t for t  0
u (t )  0
u (t )  u0
u (t )  u0
for
t 0
for
t  T1
t
for 0  t  T1
T1
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