# Transfer Function

### From bio-physics-wiki

In control theory the **transfer function** describes the **relationship between the input and the output** of a linear time-invariant (autonomous) system. If a system represented by the equation
\begin{align}
a_0 \, y^{(n)} + a_1 \, y^{(n-1)} + \dots + a_{(n-1)} \, y + a_n = b_0 \, x^{(m)} + b_1 \, x^{(m-1)} + \dots + b_{m-1} \, x + b_m
\end{align}
has the input also called driving function $\sum_i b_i \, x^{(m-i)}$ (where powers of $x$ denote derivatives when written in brackets), then the output is given by the solution $y$ to the ODE. In compact notation a general systems can be describes by
\begin{align}
\sum_{i=0}^n a_i \, y^{(n-i)} = \sum_{i=0}^m b_i \, x^{(m-i)}
\end{align}

The **transfer function** $G(s)$ of a linear time-invariant system of differential equations is defined as the *ratio between the laplace transform of the ouput* (**response function**) *and the laplace transform of the input* (**driving function**), when all initial conditions are zero.
\begin{align}
G(s)=\left[ \frac{\mathcal{L}[output]}{\mathcal{L}[input]} \right]_{\text{zero i.c.}}=\frac{Y(s)}{X(s)}=\frac{\sum_i b_i \, y^{(m-i)}}{\sum_i a_i \, x^{(n-i)}}
\end{align}
By the use of Laplace transform it is possible to represent the system dynamics by *algebraic equations*.

## Transfer function for an Impulse input

For an impulse input the transfer function takes a very simple form. Acording to the property $\delta *x =x$ of the convolution we simply get
\begin{align}
Y(s)=\int_0^{\infty} \left\{ \delta*g \right\} \, e^{-st} dt=\int_0^{\infty} \left\{g \right\} \, e^{-st} dt=G(s) \\
\end{align}
\begin{align}
Y(s)=G(s) \\
\end{align}
The inverse Laplace transform of $g(t)=\mathcal{F}^{-1}G(s)$ is called the **impulse-response function** or **weighting function** of the system. Notice, that we can determine the weighting function by measuring the response of a system to a unit impulse. In this case the weighting function is the inverse Laplace transform of the system response
\begin{align}
g(t)=\mathcal{F}^{-1}Y(s)
\end{align}

Further Reading:

- Katsuhiko Ogata - Modern Control Engineering