Question

The state space representation of a system is given by
$\left[\begin{array}{c}{\dot{x}}_1\left(t\right)\\ {\dot{x}}_2\left(t\right)\end{array}\right]=\left[\begin{array}{cc}-1& -1\\ 1& -3\end{array}\right]\left[\begin{array}{c}{x}_1\left(t\right)\\ x_2\left(t\right)\end{array}\right]+\left[\begin{array}{c}0\\ 1\end{array}\right]u\left(t\right)$

$y\left(t\right)=\left[\begin{array}{cc}1& 0\end{array}\right]\left[\begin{array}{c}{x}_1\left(t\right)\\ x_2\left(t\right)\end{array}\right]$
Resolvent matrix of the system is

• A. $\frac{1}{(s+2{)}^2}\left[\begin{array}{cc}(s+1)& 1\\ -1& (s+3)\end{array}\right]$
• B. $\frac{1}{(s+2{)}^2}\left[\begin{array}{cc}(s+3)& -1\\ 1& (s+1)\end{array}\right]$
• C. $\frac{1}{(s+2{)}^2}\left[\begin{array}{cc}(s+1)& -1\\ 1& (s+3)\end{array}\right]$
• D. $\frac{1}{(s+2{)}^2}\left[\begin{array}{cc}(s+3)& -1\\ -1& (s+1)\end{array}\right]$

Answer 1

The correct option is B $\frac{1}{(s+2{)}^2}\left[\begin{array}{cc}(s+3)& -1\\ 1& (s+1)\end{array}\right]$

State space model of system is given by,

$\dot{x}\left(t\right)=Ax\left(t\right)+Bu\left(t\right)$
$y\left(t\right)=Cx\left(t\right)$

Comparing the equations we get,

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

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

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

Resolvent matrix is given by

$\varphi \left(s\right)=[sI-A{]}^{-1}$

$[sI-A]=\left[\begin{array}{cc}s& 0\\ 0& s\end{array}\right]-\left[\begin{array}{cc}-1& -1\\ 1& -3\end{array}\right]$

$[sI-A]=\left[\begin{array}{cc}s+1& 1\\ -1& s+3\end{array}\right]$

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

$\therefore$ $\varphi \left(s\right)=\frac{1}{(s+2{)}^2}\left[\begin{array}{cc}(s+3)& -1\\ 1& (s+1)\end{array}\right]$