Sistem LTI dan Persamaan Diferensial Ir Risanuri Hidayat

Sistem LTI dan Persamaan Diferensial Ir. Risanuri Hidayat, M. Sc. 11/27/2020 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 11/27/2020 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 dan • cx dy dan • ax + cx by + dy – Maka Sistem tersebut linear 11/27/2020 LTI 3

Sistem LTI • Sistem yang mempunyai sifat linearitas dan time-invariant • Isyarat dapat dirumuskan dengan 11/27/2020 LTI 4

Impuls Response • Tanggapan Impuls h(t) Fungsi keluaran ketika sistem diberi masukan impuls 11/27/2020 LTI 5

Persamaan Differensial pada System LTI • Sistem LTI dapat dirumuskan secara matematis dengan persamaan differensial 11/27/2020 LTI 6

Pers. Differensial • Contoh: y´´ + 5 y´ + 6 y = 3 e 2 t y(t) = ? 11/27/2020 LTI 7

Pers. Differensial y´´ + 5 y´ + 6 y = x 6 y = x - y´´ - 5 y´ -5 y ’ y’ ’ -1 y’’ Penyelesaian ada 2: 1. Homogen, yh 2. Particular, yp 11/27/2020 LTI 8

Pers. Differensial yh y´´ + 5 y´ + 6 y = 0 Misal: yh = A est yh´ = A s est yh´´ = A s 2 est A est ( s 2 + 5 s + 6 ) = 0 A est (s + 2)(s + 3) = 0, s 1 = -2, s 2 = -3 yh = A 1 e-2 t + A 2 e-3 t 11/27/2020 LTI 9

Pers. Differensial • Yp = B e 2 t Fungsi yp mengikuti fungsi masukan dengan amplitudo berbeda yp´ = B 2 e 2 t yp´´ = B 4 e 2 t + B 10 e 2 t + B 6 e 2 t = 3 e 2 t 20 B = 3, B = 0. 15, YP = 0. 15 e 2 t 11/27/2020 LTI 10

Pers. Differensial • Penyelesaian keseluruhan y = yh + yp y = A 1 e-2 t + A 2 e-3 t + 0. 15 e 2 t 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 + 0. 15 y´(0)= 0 = -2 A 1 – 3 A 2 + 0. 3, A 1 = -0. 75, A 2 = 0. 6 Penyelesaian akhir, y = -0. 75 e-2 t + 0. 6 e-3 t + 0. 15 e 2 t 11/27/2020 LTI 11

RANGKAIAN X 2 1 Y 1. Tentukan Pers. Differensial sistem, Cari Y(t) jika diketahui X(t)=5 u(t) 2. Tentukan impulse response sistem, Cari Y(t) dengan konvolusi jika X(t)=5 u(t). Bandingkan hasilnya dengan soal nomor 1. 11/27/2020 LTI 12