Problem $2$: Consider a basic cruise control system for a car. A simple model is illustrated in the block diagram below, where $y$ is the velocity of the car, $r$ is the desired velocity, $u$ is the throttle leve, $d$ is an external disturbance in the form of air resistance etc., and $m$ is the torque of the engine. $F(s)$ denotes here a PI$-$controller,

$F(s)=K_p+\frac{K_i}{s}.$

$-$ Set $K_p=3.$ Determine which values of $K_i$ yields a stable closed$-$loop system from $r$ to $y?[2$p$]$

Now we consider an alternative parameterization of the PI$-$controller,

$F(s)=K+\frac{K}{T_{\text{I}}s}=K\frac{T_{\text{I}}s+1}{T_{\text{I}}s},$ where $,K_p=K,K_i=\frac{K}{T_{\text{I}}}.$

The root$-$locus for the closed$-$loop system poles are shown below with respect to $K,$ while setting $T_{\text{I}}=3.$

$-$ Using the root$-$locus, predict and motivate the expected closed$-$loop system properties: i$)$ stability, ii$)$ quickness and iii$)$ oscillations, when $K$ is small versus large. $[2$p$]$

$-$Set $T_{\text{I}}=0\mathrm{.}8.$ Determine the open$-$loop system and plot the Nyquist curve of the system for some $K,$ e$.$g$.K=1.$ Then use the diagram to determinesome suitable $K$ that yields a stable closed$-$loop system. Plot the curve also for this choice of $K.[2$p$]$