Problem$-1(30$ pts$)$

A process is modeled by the following ODE

$2\frac{dF}{dt}+F=2P(t)+Q(t)$

$\frac{dJ}{dt}-2F+J=2Q(t)$

Where $P(t)$ and $Q(t)$ are input variables. If $F,J,P$ and $Q$ are all written in deviation variables and the solution is given by:

$F(s)=G_1(s)P(s)+G_2(s)Q(s)$

$J(s)=G_3(s)P(s)+G_4(s)Q(s)$

Determine the transfer functions $G_1(s),G_2(s),G_3(s)$ and $G_4(s)$

Draw a block diagram of the system.

What is the time constant and steady state gain with respect to $P$ for $F?$

If the value of $P$ is suddenly changed from $4$ to $6$ what is the new steady state value of both $F$ and $J.$ Given: and $(\frac{\text{:}}{b}ar(J{)}_{\text{o}\text{l}\text{d}}=3$