Problem $2$

Consider the linear dynamic model for our $1$D mobile robot from class:

$\left[\begin{array}{c}{x}_1^{}\\ x_2^{}\end{array}\right]=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{c}0\\ \frac{1}{m}\end{array}\right]u$

where $n{R}^2$ is the state of the robot $($position $x_1$ and speed $x_2)$ and uinR is the force input.

Choose a full$-$state feedback controller of the form

$u=-Kx=-\left[\begin{array}{cc}{k}_1& k_2\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]$

to stabilize the system to the equilibrium point $x=0.$

Find the values of $k_1$ and $k_2$ that put the poles of the closed$-$loop system at ${}_1=-1$ and

${}_2=-2($both of which are in the left$-$half complex plane$).$

Draw a block diagram of the closed$-$loop system, labelling all the blocks and signals.