Transcript Sistem LTI dan Persamaan Diferensial Ir. Risanuri Hidayat, M.Sc.
Sistem LTI dan Persamaan Diferensial
4/29/2020 Ir. Risanuri Hidayat, M.Sc.
LTI 1
Sistem LTI
• A Linear Time Invariant System is one that: 1. Is unaffacted by time. That is, if you perform an experiment on Monday to find the systems response to a sine wave, you will get the same result if you do the experiment again on Wednesday. 2. Is Linear. Given two input signals (ax, cx) and that they produce two output signals (by, dy), the system is linear if, and only if, the input signal ax + cx produces the output signal by + dy 4/29/2020 LTI 2
Sistem LTI
• More formally: In a Time Invariant system: – Jika •
x(t)
y(t)
– Maka •
x(t-t 0 )
y(t-t 0 )
• In a Linear System: – Jika • • •
ax
by cx
dy ax + cx
dan dan
by + dy
– Maka Sistem tersebut linear 4/29/2020 LTI 3
Sistem LTI
• Sistem yang mempunyai sifat linearitas dan time-invariant • Isyarat dapat dirumuskan dengan
x
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)
y
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( ) (
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h
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y
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h
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x
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d
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) *
h
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) LTI
x
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k
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].
[
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Impuls Response
• Tanggapan Impuls
h(t)
Fungsi keluaran ketika sistem diberi masukan impuls
h
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y
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Persamaan Differensial pada System LTI • Sistem LTI dapat dirumuskan secara matematis dengan
a n
persamaan differensial
y
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a n
1
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1 ) ..
a
0
b m x
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b m
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d n y
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dt
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n x
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d m x
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m
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Pers. Differensial
• Contoh: y´´ + 5 y´ + 6 y = 3 e 2t y(t) = ?
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Pers. Differensial
y´´ + 5 y´ + 6 y = x 6y = x - y´´ - 5 y´ x -5 -1 6 ’ y’ ’ y’’ y Penyelesaian ada 2: 1.
Homogen, y h 2.
Particular, y p 4/29/2020 LTI 8
Pers. Differensial
y h y´´ + 5 y´ + 6 y = 0 Misal: y h = A e st y h ´ = A s e st y h ´´ = A s 2 e st A e st A e st ( s 2 + 5 s + 6 ) = 0 (s + 2)(s + 3) = 0, s 1 = -2, s 2 = -3 y h = A 1 e -2t + A 2 e -3t 4/29/2020 LTI 9
Pers. Differensial
• Y p = B e 2t Fungsi y p berbeda mengikuti fungsi masukan dengan amplitudo y p ´ = B 2 e 2t y p ´´ = B 4 e 2t B 4 e 2t + B 10 e 2t + B 6 e 2t = 3 e 2t 20 B = 3, B = 0.15, Y P = 0.15 e 2t 4/29/2020 LTI 10
Pers. Differensial
• Penyelesaian keseluruhan y = y h y = A 1 + y p e -2t + A 2 e -3t + 0.15 e 2t A 1 dan A 2 dapat diketahui jika kondisi awal diketahui. Misalnya kondisi awal y(0)=0 dan y´(0)=0 y(0) = 0 = A 1 + A 2 y´(0)= 0 = -2A 1 + 0.15
–3A 2 + 0.3, A 1 = -0.75, A 2 = 0.6
Penyelesaian akhir,
y = -0.75 e -2t
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+ 0.6 e -3t + 0.15 e 2t
11
X
RANGKAIAN
2 1 Y 1.
2.
Tentukan Pers. Differensial sistem, Cari Y(t) jika diketahui X(t)=5 u(t) Tentukan impulse response sistem, Cari Y(t) dengan konvolusi jika X(t)=5 u(t). Bandingkan hasilnya dengan soal nomor 1.
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