Computing Steady$-$State Gains

Consider the model of the cruise controlled vehicle we have been working with

$m{y}^{}+ay=Eu-Fd$

where $y$ is the speed of the vehicle, $u$ the throttle input, and $d$ the grade of the roadway. The

model depends on several parameters which were previously identified to be $m=1000[kg]($the

mass of the vehicle$),a=60[k\frac{g}{sec}],E=720[\frac{m}{\text{v}\text{o}\text{l}\text{t}*sec}],$ and $.$

We choose to use a proportional control strategy

$u=K_{1}r-K_2{y}_m$

where $y_m=y+n$ is the measured velocity subject to additive noise $n,$ and $r$ is the reference

velocity set by the driver.

$($a$)$ Find the closed$-$loop steady$-$state gains from

$r$ to $y$

$d$ to $y$

$n$ to $y$

$($b$)$ Show that the chosen proportional control design cannot reject step disturbances and step

noises simultaneously at steady$-$state.