Figure $1\mathrm{.}27$ shows a hydraulic system where liquid is stored in an open tank.

The cross$-$sectional area of the tank, $A(h),$ is a function of $h,$ the height of the liquid

level above the bottom of the tank. The liquid volume $v$ is given by $v={\displaystyle \underset{0}{\overset{h}{}}}A()d.$

For a liquid of density $,$ the absolute pressure $p$ is given by $p=gh+p_a,$ where

$p_a$ is the atmospheric pressure $($assumed constant$)$ and $g$ is the acceleration due

to gravity. The tank receives liquid at a flow rate $w_i$ and loses liquid through a

valve that obeys the flow$-$pressure relationship $w_o=k\sqrt[\phantom{2}]{p}.$ In the current case,

$p=p-p_a.$ Take $u=w_i$ to be the control input and $y=h$ to be the output.

$($a$)$ Using $h$ as the state variable, determine the state model.

$($b$)$ Using $p-p_a$ as the state variable, determine the state model.

$($c$)$ Find $u_{ss}$ that is needed to maintain the output at a constant value $r.$

l$)$ Propose linearized equation of motion in the vicinity of $u_{ss}$ and $r.$