Question

The system $\dot{X}$ = AX + BU and $A=\left[\begin{array}{cc}-1& 2\\ 0& 2\end{array}\right],B=\left[\begin{array}{c}0\\ 1\end{array}\right]$ is

• A. Stable and controllable
• B. Stable but uncontrollable
• C. Unstable but controllable
• D. Unstable and uncontrollable

Answer 1

The correct option is C Unstable but controllable

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

$[sI-A]=\left[\begin{array}{cc}s& 0\\ 0& s\end{array}\right]-\left[\begin{array}{cc}-1& 2\\ 0& 2\end{array}\right]$

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

$[sI-A{]}^{-1}$ = $\frac{\left[\begin{array}{cc}s-2& 2\\ 0& s+1\end{array}\right]}{(s+1)(s-2)}$ ....(1)

$Transferfunction=C[sI-A{]}^{-1}B$

So denominator of equation (i) gives poles of the system.

(s+1)(s-2) = 0

s = -1 and 2

One pole lies in RHS of s-plane. Hence, the so, system is unstable.

For controllability, $Q_c$ is defined as

$Q_c$ = [B : AB]

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

$Q_c$ = [B : AB]

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

$\left|Q_c\right|\ne 0$

Hence the system is controllable.