Solve this Systems Analysis question from a worksheet

Figure $8$ illustrates a two degree$-$of$-$freedom mass$-$spring$-$damper system. The equations of motion for this

system are given in scalar form as

$q_1^{}+5000(q_1-q_2)=(F_t+F_c)$

$q_2^{}+20q_2^{}+10,000q_2+5000(q_2-q_1)=(-F_c)$

Part $($a$).$ Consider that this system has two inputs and two outputs $u=\left[\begin{array}{c}{F}_e\\ F_c\end{array}\right],y=\left[\begin{array}{c}{q}_1\\ 5000(q_1-q_2)\end{array}\right].$

Represent the system in Natural Second$-$Order Form $($NSOF$)$ using the equations supplied on the formula

sheet $-$ showing clearly what the vectors, $q$ and $f$ contain as well as the matrices, $S_L,S_R$ and $M,D,K.$

Part $($b$).$ Show how this system can be represented in state$-$space form $($see formula sheet$).$ State the

contents of the vector $x$ as well as the matrices, $A,B,C,D$

Part $($c$).$ Show how the state$-$space form would be modified if a control law was put in place such that

$F_t=(q_1-q_2)$ for some real constant, $$

Part $($d$).$ Show how the state$-$space form would be modified further if the control law put in place was

$F_c=(q_1-q_2)+(q_1^{}-q_2^{})+{\displaystyle \underset{0}{\overset{1}{}}}(q_1-q_2)d$

Part $($e$).$ Do EITHER ONE of the following, not BOTH.

e$1.$ Sketch a SIMULINK diagram which could represent the system of part $($d$).$

OR

e$2.$ Indicate how you could determine whether any given state$-$space system is

stable, observable and controllable. No calculations are expected here.