Q$2.$Discretize the above system for a sampling period of $100ms.$ Which of the following represent the state$-$

space of the discrete double integrator system?

a$.A=\left[\begin{array}{cc}1\mathrm{.}005& 0\mathrm{.}1\\ 0\mathrm{.}1& 1\mathrm{.}005\end{array}\right],B=\left[\begin{array}{c}0\mathrm{.}1052\\ 0\mathrm{.}1002\end{array}\right],C=\left[\begin{array}{cc}1& 0\end{array}\right],D=[0]$

b$.A=\left[\begin{array}{cc}1\mathrm{.}105& 0\mathrm{.}0525\\ 0& 1\mathrm{.}158\end{array}\right],B=\left[\begin{array}{c}0\mathrm{.}1052\\ 0\end{array}\right],C=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right],D=[0]$

c$.A=\left[\begin{array}{cc}1& 0\mathrm{.}1\\ 0& 1\end{array}\right],B=\left[\begin{array}{c}0\mathrm{.}005\\ 0\mathrm{.}1\end{array}\right],C=\left[\begin{array}{cc}1& 0\end{array}\right],D=[0]$

d$.A=\left[\begin{array}{cc}1\mathrm{.}105& 0\mathrm{.}1\\ 0\mathrm{.}1& 1\mathrm{.}158\end{array}\right],B=\left[\begin{array}{c}0\\ 0\mathrm{.}005\end{array}\right],C=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right],D=[0]$

Q$3.$Given a final state of the discrete double integrator system after $t$ units of time, is it possible to conclude a unique initial state for a sequence of input signals?

a$.$ Yes

b$.$ No

Q$4.$Find out the observability matrix for the discrete double integrator system in Question $2,$ where the second state is considered to be the output of the system.