Controllability

Q. Consider a second order system whose state space representation is of the form $\dot{X}=AX+BU$. If $x_1\left(t\right)=x_2\left(t\right)$, then system is

1. controllable
2. uncontrollable
3. observable
4. unstable

Q. The state variable description of an LTI system given by
$\left[\begin{array}{c}\dot{x_1}\\ \dot{x_2}\\ \dot{x_3}\end{array}\right]=\left[\begin{array}{ccc}0& a_1& 0\\ 0& 0& a_2\\ a_3& 0& 0\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]u$

$y=\left[\begin{array}{ccc}1& 0& 0\end{array}\right]$

Where y is the ouput and u is the input. The system controllable for

1. $a_1\ne 0.a_2=0,a_3=0$
2. $a_1=0.a_2\ne 0,a_3\ne 0$
3. $a_1=0.a_2\ne 0,a_{3}0$
4. $a_1\ne 0.a_2\ne 0,a_3=0$

Q. $Forthesystem\dot{X}=\left[\begin{array}{cc}2& 3\\ 0& 5\end{array}\right]X+\left[\begin{array}{c}1\\ 0\end{array}\right]u,$ which of the following statement is true?

1. The system is controllable but unstable
2. The system is uncontrollable and unstable
3. The system is controllable and stable
4. Theable system is uncontrollable and stable.

Q. The system $\dot{X}$ = AX + BU and $A=\left[\begin{array}{cc}-1& 2\\ 0& 2\end{array}\right],B=\left[\begin{array}{c}0\\ 1\end{array}\right]$ is

1. Stable and controllable
2. Stable but uncontrollable
3. Unstable but controllable
4. Unstable and uncontrollable