Problem $4.$ A truck is driven to follow another vehicle

using adaptive cruise control $($ACC$)$; see the figure.

The speed of the lead vehicle is v$\_($L$),$ the speed of the

truck is v$,$ the distance between the vehicles $($also

called as headway$)$ is h$,$ and the acceleration of the

truck is u$.$ The dynamics are governed by:

h$^()=$v$\_($L$)-$v

v$^()=$u$.$

The motion of the truck is regulated by the ACC law:

u$=\backslash$alpha $($V$($h$)-$v$)+\backslash$beta $($v$\_($L$)-$v$).$

with gains $\backslash$alpha and $\backslash$beta $.$ The second term makes the truck

match its speed v with the lead vehicle's speed v$\_($L$).$

The first term makes the truck approach a desired

velocity V that is a function of the headway h$.$ Let

V$($h$)=\backslash$kappa $($h$-$h$\_($st$)),$ i$.$e$.,$ the truck shall stop if the

headway is h$\_($st$)=5$m and it may increase its speed

proportionally to the headway with $\backslash$kappa $=0\mathrm{.}6$s$^(-1).$

av$\_($L$)^(*)=24$

$($m$)/($s$),$ calculate the equilibrium speed v$^(*)$ of the truck

and the equilibrium headway h$^(*).$

btilde$($v$)\_($L$)$ as input, tilde$($h$)$ and tilde$($v$)$

as states, and tilde$($v$)$ as output. Identify the A$,$B$,$C$,$D

matrices.

c$\backslash$alpha and $\backslash$beta gains that yield asymptotic stability.

d$\backslash$alpha and $\backslash$beta gains so that the characteristic

roots become s$\_(1)=-0\mathrm{.}4$ and s$\_(2)=-0\mathrm{.}6.$

e$|$T$($j$\backslash$omega $)|$

as a function of $\backslash$omega for $\backslash$omega in$[0,\backslash$pi $].$ Submit the figure

and your codes. If $|$T$($j$\backslash$omega $)|<mn>1</mn>$ for all $\backslash$omega $>0,$ then

the truck drives with smaller velocity fluctuations

than the lead vehicle. This is called string stability.

This helps the truck drive smoothly, which saves fuel

and may also help make the traffic behind the truck

smoother. Is your ACC design string stable?