Question

The state equations of a system are given below:
$\dot{x_1}=x_1+x_2+u$
$\dot{x_2}=-x_2$
$y=x_1$
The system is said to be

• A. Controllable as we as observable
• B. Controllable but not observable
• C. Observable but not controllable
• D. Neither controllable nor observable

Answer 1

The correct option is C Observable but not controllable

$\left[\begin{array}{c}\dot{x_1}\\ \dot{x_2}\end{array}\right]=\left[\begin{array}{cc}1& 1\\ 0& -1\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{c}1\\ 0\end{array}\right]u$
$y=\left[10\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]$
$\text{Here,}A=\left[\begin{array}{cc}0& 1\\ 0& -1\end{array}\right]$
$B=\left[\begin{array}{c}1\\ 0\end{array}\right]$
$\text{and}C=\left[10\right]$
check for controllability,
$Q_C=[B:AB]$
$AB=\left[\begin{array}{cc}1& 1\\ 0& -1\end{array}\right]\left[\begin{array}{c}1\\ 0\end{array}\right]=\left[\begin{array}{c}1\\ 0\end{array}\right]$
$Q_C=\left[\begin{array}{cc}1& 1\\ 0& 0\end{array}\right]$
$|Q_C|=0\Rightarrow \text{uncontrollable}$
check for observability,
$Q_0=[C^T;A^T{C}^T]$
$C^T=\left[\begin{array}{c}1\\ 0\end{array}\right],A^T=\left[\begin{array}{cc}1& 0\\ 1& -1\end{array}\right]$
$A^T{C}^T=\left[\begin{array}{cc}1& 0\\ 1& -1\end{array}\right]\left[\begin{array}{c}1\\ 0\end{array}\right]=\left[\begin{array}{c}1\\ 1\end{array}\right]$
$Q_0=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$
$|Q_0|\ne 0\Rightarrow \text{observable}$