Problem $3$:

Consider the following three transfer functions.

$H_1(s)=\frac{4}{s+10},H_2(s)=\frac{4s-6}{4s^2-9},H_3(s)=\frac{4}{s^2+13s+30}.$

For each part below, answer the following questions for $H_i,$ where $i=1,2,3$ :

a$)$ What are the poles and zeros?

b$)$ What is the general form of the free response?

c$)$ What is the general form of the forced response when the input follows a unit step?

d$)$ Assuming all initial conditions are zero, determine the total output response $y_{\text{t}\text{o}\text{t}\text{a}\text{l}}(t).$ Also, assess if the system is stable and sketch the total output response $y_{\text{t}\text{o}\text{t}\text{a}\text{l}}(t).$

e$)$ Draw the block diagram assuming the outputs of $H_1(s),H_2(s),$ and $H_3(s)$ are added together to represent one output.

f$)$ From part e$),$ draw the signal flow graph.