Question

If the transfer function of a control system is given below:
$\frac{Y\left(s\right)}{U\left(s\right)}=\frac{1}{s^3+4s^2+5s+3}$

The matrix [A] and [B] in state variable representation of the system will be

• A. $\left[A\right]=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -4& -5& -3\end{array}\right];\left[B\right]=\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]$
• B. $\left[A\right]=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -3& -5& -4\end{array}\right];\left[B\right]=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]$
• C. $\left[A\right]=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -8& -10& -6\end{array}\right];\left[B\right]=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]$
• D. $\left[A\right]=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -6& -10& -18\end{array}\right];\left[B\right]=\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]$

Answer 1

The correct option is B $\left[A\right]=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -3& -5& -4\end{array}\right];\left[B\right]=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]$

For the given system we can write,
$Y\left(s\right)[s^3+4s^2+5s+3]=U\left(s\right)\left[1\right]$ ....(i)

Let, $x_1=y$
So, $x_2={\dot{x}}_1=\dot{y}$ ...(ii)
$x_3={\dot{x}}_2=\ddot{y}$ ....(iii)

The equation (i) can be written as

${\dot{x}}_3+4x_3+5x_2+3x_1=u$

i.e. ${\dot{x}}_3=-3x_1-5x_2-4x_3+u$ ...(iv)

Using equation (ii), (iii) and (iv) we can write in state variable form as below,

$\left[\begin{array}{c}{\dot{x}}_1\\ {\dot{x}}_2\\ {\dot{x}}_3\end{array}\right]=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -3& -5& -4\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]u$

So $\left[A\right]=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -3& -5& -4\end{array}\right]$

and $\left[B\right]=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]$

Hence option (b) is correct.