The standard form of a PID controller is

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  1. Kp + Kds + K1s2
  2. Kp . s+ Kds2 + K1s

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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:

Applying Laplace transform, we get:

Transfer function will be:

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.

  

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:

It is equivalent to a High-pass filter.

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