The standard form of a PID controller is

Proportion + Integral + Derivative:

The PID controller produces on output, which is the combination of outputs of proportional, integral, and derivative controllers.

This is defined in terms of differential equations as:

\(u\left( t \right) = {K_p}\;e\left( t \right) + {K_I}\smallint e\left( t \right) \cdot dt + {K_D} \cdot \frac{{de\left( t \right)}}{{dt}}\)

Applying Laplace transform, we get:

\(U\left( s \right) = \left( {{K_p} + \frac{{{K_I}}}{s} + {K_D}s} \right)E\left( s \right)\)

Transfer function will be:

\(\frac{{U\left( s \right)}}{{E\left( s \right)}} = {K_p} + \frac{{{K_I}}}{s} + {K_D} \cdot s\)

Integral control:

It is the control mode where the controller Output is proportional to the integral of the error with respect to time.

Integral controller output = k × integral of error with time, i.e.

\({G_C}\left( s \right) = \frac{{U\left( s \right)}}{{E\left( s \right)}} = \frac{{{K_I}}}{s}\)  

Proportional + Derivate:

The additive combination of proportional & Derivative control is known as P-D control.

Overall transfer function for a PD controller is given by:

\({G_C}\left( s \right) = \frac{{U\left( s \right)}}{{E\left( s \right)}} = {K_P} + s{K_D}\)

It is equivalent to a High-pass filter.