Question

A control system is represented by following state space representation
$\left[\begin{array}{c}{\dot{x}}_1\\ {\dot{x}}_2\end{array}\right]=\left[\begin{array}{cc}-5& 1\\ 0& -4\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{c}0\\ 1\end{array}\right]u\left(t\right)$

$y=\left[\begin{array}{cc}1& 4\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]$
If two such systems are connected in parallel then overall transfer function of the combined system is

• A. $\frac{4s+21}{(s+4)(s+5)}$
• B. $\frac{4s+36}{(s+4)(s+5)}$
• C. $\frac{2(s+9)}{(s+4)(s+5)}$
• D. $\frac{2(4s+21)}{(s+4)(s+5)}$

Answer 1

The correct option is D $\frac{2(4s+21)}{(s+4)(s+5)}$

The transfer function from steady state model can be written as

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

$[sI-A{]}^{-1}=\frac{adj[sI-A]}{|sI-A|}$

$[sI-A]=\left[\begin{array}{cc}s+5& -1\\ 0& s+4\end{array}\right]$

$[sI-A{]}^{-1}=\frac{1}{(s+5)(s+4)}\left[\begin{array}{cc}s+4& 1\\ 0& s+5\end{array}\right]$

$T\left(s\right)=\frac{1}{(s+5)(s+4)}\left[\begin{array}{cc}1& 4\end{array}\right]\left[\begin{array}{cc}s+4& 1\\ 0& s+5\end{array}\right]\left[\begin{array}{c}0\\ 1\end{array}\right]$

$T\left(s\right)=\frac{1}{(s+5)(s+4)}\left[\begin{array}{cc}1& 4\end{array}\right]\left[\begin{array}{c}1\\ s+5\end{array}\right]$

$T\left(s\right)=\frac{1+4(s+5)}{(s+4)(s+5)}=\frac{4s+21}{(s+4)(s+5)}$


If two systems are connected in parallel
Overall transfer function,

$T\left(s\right)=T\left(s\right)+T\left(s\right)=\frac{2(4s+21)}{(s+4)(s+5)}$