Sistem LTI dan Persamaan Diferensial Ir. Risanuri Hidayat, M.Sc. 11/29/2018 LTI

Sistem LTI A Linear Time Invariant System is one that: 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. 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 11/29/2018 LTI

Sistem LTI More formally: In a Time Invariant system: Jika x(t)  y(t) Maka x(t-t0)  y(t-t0) In a Linear System: ax  by dan cx  dy dan ax + cx  by + dy Maka Sistem tersebut linear 11/29/2018 LTI

Sistem LTI Sistem yang mempunyai sifat linearitas dan time-invariant Isyarat dapat dirumuskan dengan 11/29/2018 LTI

Impuls Response Tanggapan Impuls h(t) Fungsi keluaran ketika sistem diberi masukan impuls 11/29/2018 LTI

Persamaan Differensial pada System LTI Sistem LTI dapat dirumuskan secara matematis dengan persamaan differensial 11/29/2018 LTI

Pers. Differensial Contoh: y´´ + 5 y´ + 6 y = 3 e2t y(t) = ? 11/29/2018 LTI

Pers. Differensial y 6 x y´´ + 5 y´ + 6 y = x 6y = x - y´´ - 5 y´ -5 ’ x y y’ y’’ -1 -5 6 y´´ + 5 y´ + 6 y = x 6y = x - y´´ - 5 y´ Penyelesaian ada 2: Homogen, yh Particular, yp 11/29/2018 LTI

Pers. Differensial yh  y´´ + 5 y´ + 6 y = 0 Misal: yh = A est yh´ = A s est yh´´ = A s2 est A est ( s2 + 5 s + 6 ) = 0 A est (s + 2)(s + 3) = 0,  s1 = -2, s2 = -3 yh = A1 e-2t + A2 e-3t 11/29/2018 LTI

Pers. Differensial Yp = B e2t Fungsi yp mengikuti fungsi masukan dengan amplitudo berbeda yp´ = B 2 e2t yp´´ = B 4 e2t B 4 e2t + B 10 e2t + B 6 e2t = 3 e2t 20 B = 3,  B = 0.15,  YP = 0.15 e2t 11/29/2018 LTI

Pers. Differensial Penyelesaian keseluruhan y = yh + yp y = A1 e-2t + A2 e-3t + 0.15 e2t A1 dan A2 dapat diketahui jika kondisi awal diketahui. Misalnya kondisi awal y(0)=0 dan y´(0)=0 y(0) = 0 = A1 + A2 + 0.15 y´(0)= 0 = -2A1 –3A2 + 0.3,  A1 = -0.75, A2 = 0.6 Penyelesaian akhir, y = -0.75 e-2t + 0.6 e-3t + 0.15 e2t 11/29/2018 LTI

RANGKAIAN 2 1 Y X 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. 11/29/2018 LTI