# How is Transfer function G(s) derived?

The transfer function is commonly used in the analysis of single-input single-output electronic filters, for instance. It is mainly used in signal processing, communication theory, and control theory.

The term is often used exclusively to refer to linear, time-invariant systems (LTI). Most real systems have non-linear input/output characteristics, but many systems, when operated within nominal parameters (not ‘over-driven’) have behaviour that is close enough to linear that LTI system theory is an acceptable representation of the input/output behaviour.

## Transfer function of a closed-loop system

Diagram of a closed-loop system. Mason’s rule can be applied to determine the closed-loop transfer function.

We can deduce for the output of the system:

Y (s) = G(s)U(s) = G(s)Hc(s)E(s)

```
= G(s)Hc(s)[R(s) − M(s)Y (s)
= L(s)R(s) − L(s)M(s)Y (s)
```

With L(s) the transfer function of the open-loop system (controller and plant)

(1 + L(s)M(s))Y (s) = L(s)R(s)

Y (s) = (I + L(s)M(s))−1L(s)R(s) = T(S)R(s),

Therefore T(S)=Y(S)/R(S)

where T(s) is called the reference transfer function.