Question

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

• A. The system is controllable but unstable
• B. The system is uncontrollable and unstable
• C. The system is controllable and stable
• D. Theable system is uncontrollable and stable.

Answer 1

The correct option is B The system is uncontrollable and unstable

$|Q_C|=\left[\begin{array}{cc}1& 2\\ 0& 0\end{array}\right]=0$
$\Rightarrow$ Uncontrollable.
Characteristic equation:
$|sI-A|=0$

$\left|\begin{array}{cc}s-2& -3\\ 0& s-5\end{array}\right|=0$

$=(s-2)(s-5)=0$

$s=2,5$

ie., roots lies on right side of s-plane.
$\Rightarrow$Unstable.