Editing Proportional–integral–derivative controller (section)

===Series/interacting form===
Another representation of the PID controller is the series, or ''interacting'' form
:<math>G(s) = K_c (\frac{1}{\tau_i{s}}+1) (\tau_d{s}+1)</math>
where the parameters are related to the parameters of the standard form through
:<math>K_p = K_c \cdot \alpha</math>, <math>T_i = \tau_i \cdot \alpha</math>, and
:<math>T_d = \frac{\tau_d}{\alpha}</math>
with
:<math>\alpha = 1 + \frac{\tau_d}{\tau_i}</math>.

This form essentially consists of a PD and PI controller in series. As the integral is required to calculate the controller's bias this form provides the ability to track an external bias value which is required to be used for proper implementation of multi-controller advanced control schemes.