$[10$ points$]$ Consider the following fluid system.

The state variable equations are:

$A_1{h}_1^{}=q_{mi}-\frac{g{h}_1-g{h}_2}{R_1}$

$A_2{h}_2^{}=\frac{g{h}_1-g{h}_2}{R_1}-\frac{g{h}_2}{R_2}$

Assume that $g=10\frac{m^2}{s},=1,R_1=1,R_2=3,A_1=10,$ and $A_2=4.$

a$.$ From the state variable equations, analytically determine the steady$-$state values for $h_1$

and $h_2,$ assuming the input $q_{mi}$ is the unit step function.

Apply MATLAB to answer the following questions:

b$.$ Plot $h_1$ and $h_2$ as a function of time when the input $q_{mi}$ is the unit step function.

Assume both tanks start in the empty state. Give all commands used to generate your

plot.

c$.$ Provide a plot of the flows across $R_1$ and $R_2$ as a function of time. Describe the system

dynamics qualitatively over time.

d$.$ From your graphs in $($b$)$ and $($c$),$ what are the water heights in each tank and the flows

across each pipe in steady state? $[$The first answer should match your answer in $($a$)]$

e$.$ Repeat part $($b$),$ but now assume that the initial heights of the water are $h_1=4$ and $h_2$

$=5.$