Question

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

• A. controllable
• B. uncontrollable
• C. observable
• D. unstable

Answer 1

The correct option is B uncontrollable

Let,
$\left[\begin{array}{c}\dot{x_1}\\ \dot{x_2}\end{array}\right]=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{c}{x}_1\left(t\right)\\ x_2\left(t\right)\end{array}\right]+\left[\begin{array}{c}e\\ f\end{array}\right]u\left(t\right)$
$\left[B\right]=\left[\begin{array}{c}e\\ f\end{array}\right],\left[AB\right]=\left[\begin{array}{cc}ae& bf\\ ce& df\end{array}\right]$

$Since,x_1\left(t\right)=x_2\left(t\right)$ and $\dot{x_1}\left(t\right)=\dot{x_2}\left(t\right)$

$\Rightarrow e=fanda=candb=d$

$\therefore |Q_C|=|BAB|=\left|\begin{array}{cc}e& ae+bf\\ f& ce+df\end{array}\right|=0$
$\therefore$Uncontrollable.