Question

1. (20 pts) An LTCI system is defined by the equation *0) + 1) + 4y0) = 10) (a) find the characteristic polynomial, characteristic function equation, characteristic root and characteristic modes of this system. (b) Find yo(t), the zero-input component of the responses y(t) for t > 0 with initial conditions yo(0) = 2 and y(0) = -1. 2. (20 pts) Repeat the previous problem if Vt) + 9y(t) = 351) + 2FE) and yo(0) = 0, Y%(0) = 6.

Answer 1

D^2/dt^2 y(t) + 4 d/dt y(t) + 4 y(t) = d/dt f(t) a) characteristic polynomial: (D^2 + 4D + 4) = 0 characteristic roots: D^2 + 2 times 2D + (2)^2 = 0 (D + 2)^2 = 0 D = -2, -2 characteristic function (C F) = (C_1 + C_2 t) e^-2t b) Zero - input rightarrow F(t0 = 0 so: d^2/dt^2 y_0 (t) + 4 d/dt y_0 (t) + 4 y_0 (t) = 0 CF calculation (D^2 + $D + 4) = 0 (delta + 2)^2 = 0 delta = -2, -2 c F = (C_1 + C_2 t) e^-2t y_0 (t) = (C_1 + C_2 t) e^-2t rightarrow (1) Put y_o (0) = 2 in (1) y_0 (0) = (C_1 + C_2 times 0) e^0 2 = C_1 Put y'_0 (0) = -1 y'_0 (t) = -2 c_2 r e^-2t + e^-2t C_1 - 2 C_1 e^-2t y'_0 (0) = 0 + c_2 - 2c_1 Rightarrow c_2 = -1 + 2c_1 c_2 = 3 so, y_0 (t) = (2 + 3t) e^-2t a. d^2 y/dt^2 y(t) + 9 y(t) = 3 d/dt f(t) + 2 f(t) characteristic polynomial (D^2 +) = 0 (D^2 + (3)^2) = 0 characteristic roots rightarrow D = plusminus 3i