Find an analytical expression for y(t), given Y1 (s) = (3*s^3 + 5*s^2 + 10*s + 40) / (s^4 + 16*s^3 + 69*s^2 + 94*s + 40) Y 2(s) = 2*s^2 + 5*s + 7) / (s^3 + 3*s^2 + 7*s + 5)

Do the partial fraction expansion in Matlab.

Problem 2

Plot the unit step response curves for the two systems defined by the transfer functions

X(s)/ U(s) = (25)/(s^2 + 5*s + 25)

Y (s)/U(s) = (5*s + 25)/ (s^2 + 5*s + 25)

in one plot (You should specify which line belongs to which transfer function). Then plot each curve in a separate figure in two subplots with proper titles. Plot the responses for 0 t 3 with t = 0.05s. Do not forget to include labels for your plot axis.

Problem 3

Consider the following differential equation

y(t) + 3y(t) + 2y(t) = 5u(t) (5)

Calculate the transfer function corresponding to this system and plot its step response.

Assume u(t) = 1(t), y(0) = 1, and y(0) = 1. Find an expression for the Laplace transform of y(t), i.e. Y (s). Then,

plot the impulse response of Y (s) in an individual figure.

plot the step response of sY (s) in another individual figure.

Explain what signal each of these plots represent.