Question

The second order dynamic system
$\frac{dX}{dt}=PX+Qu$

y = RX has the matrices, P, Q and R as follows:

$P=\left[\begin{array}{cc}-1& 1\\ 0& -3\end{array}\right]Q=\left[\begin{array}{c}0\\ 1\end{array}\right]$ $R=\left[01\right]$

The system has the following controllability and observability properties:

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

Answer 1

The correct option is C Controllable but not observable

Given:
$\dot{x}\left(t\right)=Px+Qu$

y = Rx

$P=\left[\begin{array}{cc}-1& 1\\ 0& -3\end{array}\right],Q=\left[\begin{array}{c}0\\ 1\end{array}\right],$

$R=\left[01\right]$

For controllability,

$Q_c=\left[QPQ\right]$

$\left[PQ\right]=\left[\begin{array}{cc}-1& 1\\ 0& -3\end{array}\right]\left[\begin{array}{c}0\\ 1\end{array}\right]=\left[\begin{array}{c}1\\ -3\end{array}\right]$

$\therefore$ $Q_c=\left[\begin{array}{cc}0& 1\\ 1& -3\end{array}\right]$

$\therefore$ $\left|Q_c\right|=-1\ne 0....controllable$

Also, for observability,

$Q_0=\left[\begin{array}{cc}{R}^T& P^T{R}^T\end{array}\right]$

$\left[\begin{array}{cc}{P}^T& R^T\end{array}\right]=\left[\begin{array}{cc}-1& 0\\ 1& -3\end{array}\right]\left[\begin{array}{c}0\\ 1\end{array}\right]=\left[\begin{array}{c}0\\ -3\end{array}\right]$

$\therefore$ $Q_0=\left[\begin{array}{cc}0& 0\\ 1& -3\end{array}\right]$

$\therefore \left|Q_0\right|=(0\times -3-0\times 1)=0...Non-observable$

$Since,Q_c\ne 0and{Q}_0=0$

$\therefore$ System is controllable but not observable.