For this system, the rod is considered ideal with a length of $a+b($massless

and frictionless at the pivot$),$ and $$ is an angle that remains small so that all

forces and displacements behave in a translational manner along the horizontal

axis. The displacements $d_1$ and $d_2,$ as well as $,$ are measured from the equilib$-$

rium position that occurs when the input $f_1=0N($rod perpendicular to the

horizontal axis$).$ Additionally, $v_1$ and $v_2$ represent the velocities of the centers

of mass of $M_1$ and $M_2.$

$($a$)(30\%)$ If $x=\left[\begin{array}{c}{d}_1\\ v_1\\ d_2\\ v_2\end{array}\right],$ determine a linear state$-$space model $($MVE$)$ for this

system if the output is $y=\left[\begin{array}{c}{d}_1\\ d_2\end{array}\right].$

$($b$)$ Explain whether it is appropriate to consider $$ as a state variable of the

system in addition to the four previously mentioned state variables.

$($c$)$ Simulate the system's response to a unit step change at $t=1s$ using

ode$45$ in MATLAB, considering zero initial conditions for the four state

variables. Also, assume $k_1=2\frac{N}{m},k_2=1\frac{N}{m},M_1=2kg,M_2=0\mathrm{.}75$

$kg,a=0\mathrm{.}8m,b=0\mathrm{.}4m,B_1=0\mathrm{.}5N\frac{s}{m},$ and $B_2=0\mathrm{.}25N\frac{s}{m}.$ Observe

the obtained results and indicate whether they are consistent with the

expected behavior. You must show both the input and output signals of

the system in a single plot and also include the MATLAB code.