Question

Consider the system given below, The matrix A of the system in state model when controllable canonical form is used is $\frac{Y\left(s\right)}{U\left(s\right)}=\frac{5}{s^3+6s+7}$

• A. $\left[\begin{array}{ccc}0& 1& 1\\ 0& 0& 1\\ 0& -7& -6\end{array}\right]$
• B. $\left[\begin{array}{ccc}1& 0& 1\\ -6& 0& 0\\ -7& 0& 1\end{array}\right]$
• C. $\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -7& -6& 0\end{array}\right]$
• D. $\left[\begin{array}{ccc}1& 0& 0\\ 1& 0& 1\\ 1& 6& 7\end{array}\right]$

Answer 1

The correct option is C $\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -7& -6& 0\end{array}\right]$

$Y\left(s\right)[s^3+6s+7]=5u\left(s\right)$
$\overline{y}+6y+7y=5u$
$x_1=y$
$x_2=y=x_1$
$x_2=y=x_1$
$x_3=\overline{y}=x_2$
$\overline{y}=-6y-7y+5u$
$x_3=-6x_2-7x_1+5u$
$x_1=x_2$
$x_3=-7x_1-6x_2+5u$
$\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -7& -6& 0\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ 5\end{array}\right]uA=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -7& -6& 0\end{array}\right]$
Method - II:
In controllable canonical form:
$\frac{Y\left(s\right)}{u\left(s\right)}=\frac{b_0{s}^n+b_1{s}^{n-1}+....+b_{n-1}s+b_n}{s^n+a_1{s}^{n-1}+...+a_{n-1}S+a_n}$
$A=\left[\begin{array}{ccc}0& 1& 0.....0\\ 0& 0& 1.....0\\ 0& 0& 0.....1\\ -a_n& -a_{n-1}& ....a_1\end{array}\right]=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -7& -6& 0\end{array}\right]$