Question

Consider the system described by the following state variable equation
$\left[\dot{X}\right]=\left[\begin{array}{cc}0& 1\\ 3& -4\end{array}\right]X+\left[\begin{array}{c}0\\ 1\end{array}\right]u$

Y = [1 1]X
Which of the below given statements are correct regarding the above system?

• A. 1 and 2 only
• B. 2 and 3 only
• C. 1 and 3 only
• D. 1, 2 and 3

Answer 1

The correct option is C 1 and 3 only

For the given system,
Characteristic equation : |sI - A|

Where, $A=\left[\begin{array}{cc}0& 1\\ 3& -4\end{array}\right]$

So, $|sI-A|=\left[\begin{array}{cc}s-0& -1\\ -3& s+4\end{array}\right]=s(s+4)-3=s^2+4s-3$

$s=\frac{-4\pm \sqrt{16+4\times 3}}{2}=\frac{-4\pm 5.292}{2}=0.646,-4.646$

$\therefore$ One root of characteristic equation lies on LHS of s-plane.
Checking for controllability:

$Q_C=[B:AB]suchthat|Q_C|\ne 0$

$B=\left[\begin{array}{c}0\\ 1\end{array}\right],AB=\left[\begin{array}{cc}0& 1\\ 3& -4\end{array}\right]\left[\begin{array}{c}0\\ 1\end{array}\right]=\left[\begin{array}{c}1\\ -4\end{array}\right]$

So, $Q_C=\left[\begin{array}{cc}0& 1\\ 1& -4\end{array}\right]and|Q_C|=-1$

as $|Q_C|\ne 0\therefore$ given system is controllable

Checking for observability,

$Q_0=\left[\begin{array}{c}C\\ CA\end{array}\right]$ such that $|Q_0|\ne 0$

C = [1 1]
CA = [1 1] $\left[\begin{array}{cc}0& 1\\ 3& -4\end{array}\right]$ = [3 -3]

So, $Q_o=\left[\begin{array}{c}C\\ CA\end{array}\right]=\left[\begin{array}{cc}1& 1\\ 3& -3\end{array}\right]and|Q_0|=-6$

$\therefore$ the given system is observable as
$|Q_0|\ne 0$