Question

The state variable description of an LTI system given by
$\left[\begin{array}{c}\dot{x_1}\\ \dot{x_2}\\ \dot{x_3}\end{array}\right]=\left[\begin{array}{ccc}0& a_1& 0\\ 0& 0& a_2\\ a_3& 0& 0\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]u$

$y=\left[\begin{array}{ccc}1& 0& 0\end{array}\right]$

Where y is the ouput and u is the input. The system controllable for

• A. $a_1\ne 0.a_2=0,a_3=0$
• B. $a_1=0.a_2\ne 0,a_3\ne 0$
• C. $a_1=0.a_2\ne 0,a_{3}0$
• D. $a_1\ne 0.a_2\ne 0,a_3=0$

Answer 1

The correct option is D $a_1\ne 0.a_2\ne 0,a_3=0$

$A=\left[\begin{array}{ccc}0& a_1& 0\\ 0& 0& a_2\\ a_3& 0& 0\end{array}\right]$
$B=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]$
$C={\left[\begin{array}{ccc}1& 0& 0\end{array}\right]}^T$
For system to be controllable, the matrix $Q_c$ must be non singular.

$Q_c=\left[\begin{array}{ccc}B& AB& A^{2}B\end{array}\right]$

$AB={\left[\begin{array}{ccc}0& a_2& 0\end{array}\right]}^T$
$A^2=\left[\begin{array}{ccc}0& 0& a_1{a}_2\\ a_2{a}_3& 0& 0\\ 0& a_1{a}_3& 0\end{array}\right]$

$A^{2}B={\left[\begin{array}{ccc}{a}_1{a}_2& 0& 0\end{array}\right]}^T$
$Q_c=\left[\begin{array}{ccc}0& 0& a_1{a}_2\\ 0& a_2& 0\\ 1& 0& 0\end{array}\right]$
$|Q_c|=-a_a{a}_2^2$
For $Q_c$ to be nonsingular
$|Q_c|\ne 0$
$\therefore a_1\ne 0and{a}_2\ne 0and{a}_3ϵR$