Question

The state space equation of a system is described by $\dot{x}=Ax+Bu$, $y=Cx$ where $x$ is state vector, $u$ is input, $y$ is output and $A=\left[\begin{array}{cc}0& 1\\ 0& -2\end{array}\right]$, $B=\left[\begin{array}{c}0\\ 1\end{array}\right],C=\left[10\right]$

The transfer function G(s) if this system will be

• A. $\frac{s}{(s+2)}$
• B. $\frac{s+1}{s(s-2)}$
• C. $\frac{s}{(s-2)}$
• D. $\frac{1}{s(s+2)}$

Answer 1

The correct option is D $\frac{1}{s(s+2)}$

$\dot{x}=Ax+Bu$

$andy=Cx$

Where,

$A=\left[\begin{array}{cc}0& 1\\ 0& -2\end{array}\right],B=\left[\begin{array}{c}0\\ 1\end{array}\right]$;

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

$[sI-A]=\left[\begin{array}{cc}s& 0\\ 0& s\end{array}\right]-\left[\begin{array}{cc}0& 1\\ 0& -2\end{array}\right]=\left[\begin{array}{cc}s& -1\\ 0& s+2\end{array}\right]$

$[sI-A{]}^{-1}=\frac{1}{s(s+2)}\left[\begin{array}{cc}s+2& 1\\ 0& s\end{array}\right]$

Transfer function

= $C[sI-A{]}^{-1}B+D$

= $\left[\begin{array}{cc}1& 0\end{array}\right]$ $\frac{\left[\begin{array}{cc}s+2& 1\\ 0& s\end{array}\right]\left[\begin{array}{c}0\\ 1\end{array}\right]}{s(s+2)}$

= $\left[\begin{array}{cc}1& 0\end{array}\right]$ $\frac{\left[\begin{array}{c}1\\ s\end{array}\right]}{s(s+2)}=\frac{1}{s(s+2)}$