The equation for the transfer function is the input over the output
$$H(s)=\frac{Y(s)}{U(s)}$$
The transfer function of a state-space system can be accomplished using the following equations. The standard state-space representation is
$$\begin{matrix}
\dot{x(t)}=Ax(t)+Bu(t) \\
y(t) = Cx(t)+Du(t)
\end{matrix}$$
The Laplace transform of the state-space equations are
$$\begin{matrix}
sX(s) = AX(s)+BU(s) \\
Y(s)=CX(s)+DU(s)
\end{matrix}$$
We can solve the upper equation for $X(s)$
$$\begin{matrix}
sX(s)-AX(s)=BU(s) \\
(sI-A)X(s)=BU(s) \\
X(s)=(sI-A)^{-1}BU(s)
\end{matrix}$$
This can then be substituted into the output state-space equation
$$Y(s)=C(sI-A)^{-1}BU(s)+DU(s)$$
We can then obtain the transfer function $\frac{Y(S)}{U(S)}$
$$\frac{Y(s)}{U(s)}=\frac{C(sI-A)^{-1}BU(s)+DU(s)}{U(s)}$$
Which simplifies to
$$\frac{Y(s)}{U(s)}=C(sI-A)^{-1}B+D$$
Where \(s=\sigma+j\omega\) is a complex frequency.
- TransformationTransferFunction
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