Question

The state variable formulation of a system is given as
$\left[\begin{array}{c}{\dot{x}}_1\\ {\dot{x}}_2\end{array}\right]=\left[\begin{array}{cc}-2& 0\\ 0& -1\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{c}1\\ 1\end{array}\right]$u, $x_1\left(0\right)=0,x_2\left(0\right)=0andy=\left[\begin{array}{cc}1& 0\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]$

The system is

• A. Controllable but not observable
• B. Not controllable but observable
• C. Both controllable and observable
• D. Both not controllable and not observable

Answer 1

The correct option is A Controllable but not observable

$A=\left[\begin{array}{cc}-2& 0\\ 0& -1\end{array}\right],B=\left[\begin{array}{c}1\\ 1\end{array}\right]$

$AB=\left[\begin{array}{cc}-2& 0\\ 0& -1\end{array}\right]\left[\begin{array}{c}1\\ 1\end{array}\right]=\left[\begin{array}{c}-2\\ -1\end{array}\right]$

For controllability,
| B : AB | $\ne$ 0

$or\left|\begin{array}{cc}1& -2\\ 1& -1\end{array}\right|=-1-(-2)=1\ne 0$

The system is controllable.

$C^T=\left[\begin{array}{c}1\\ 0\end{array}\right]$

$A^T{C}^T=\left[\begin{array}{cc}-2& 0\\ 0& -1\end{array}\right]\left[\begin{array}{c}1\\ 0\end{array}\right]=\left[\begin{array}{c}-2\\ 0\end{array}\right]$

For observability,

$|C^T:A^T{C}^T|\ne$ 0

$or\left|\begin{array}{cc}1& -2\\ 0& 0\end{array}\right|$ = 0

The system is not observable.