Question

Let the state- space representation of an LTI system be $\dot{X}\left(t\right)=AX\left(t\right)+Bu\left(t\right),y\left(t\right)=CX\left(t\right)+Du\left(t\right)$ where $A,B,C$ are matrices, $D$ is a scalar, $u\left(t\right)$ is the input to the system, and $y\left(t\right)$ is its output. Let $B=[001{]}^T$ and $d=0$. Which one of the following option for $A$ and $C$ will ensure that the transfer function ofthis LTI system is
$H\left(s\right)=\frac{1}{s^3+3s^2+2s+1}$

• A. $A=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -3& -2& -1\end{array}\right]\text{and}C=\left[001\right]$
• B. $A=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -1& -2& -3\end{array}\right]\text{and}C=\left[001\right]$
• C. $A=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -1& -2& -3\end{array}\right]\text{and}C=\left[100\right]$
• D. $A=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -3& -2& -1\end{array}\right]\text{and}C=\left[100\right]$

Answer 1

The correct option is C $A=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -1& -2& -3\end{array}\right]\text{and}C=\left[100\right]$

$X\left(t\right)=Ax\left(t\right)+Bu\left(t\right)$
$y\left(t\right)=Cx\left(t\right)$
$B=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]$
$\frac{Y\left(s\right)}{U\left(s\right)}=\frac{Y\left(s\right)}{X_1\left(s\right)}\times \frac{X_1\left(s\right)}{U\left(s\right)}=1\times \frac{1}{s^3+3s^2+2s+1}$

$X_1\left(s\right)[s^3+3s^2+2s+1]=U\left(s\right)$

$x_2\left(t\right)={\dot{x}}_1\left(t\right)$

$\Rightarrow X_2\left(s\right)=s{X}_1\left(s\right)$

$x_3={\dot{x}}_2\left(t\right)$

$\Rightarrow X_3\left(s\right)=s{X}_2\left(s\right)=s^2{X}_1\left(s\right)$

So, $s{X}_3\left(s\right)=-X_1\left(s\right)-2X_2\left(s\right)-3X_3\left(s\right)+U\left(s\right)$

${\dot{x}}_3\left(t\right)=-x_1\left(t\right)-2x_2\left(t\right)-3x_3\left(t\right)+u\left(t\right)$

$y\left(t\right)=x_1\left(t\right)$

$\dot{x}\left(t\right)=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -1& -2& -3\end{array}\right]x\left(t\right)+\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]u\left(t\right)$

$y\left(t\right)=\left[100\right]x\left(t\right)$