PI controller

In control theory, a proportional integral controller (PI) is a system controller that takes the form:

$$C(s)=k\left(1+\frac{1}{Ts}\right)=k\left(\frac{Ts+1}{Ts}\right)$$

where $k$ and $T$ are parameters to be designed. It takes the name because the output in the time domain includes an integral term.

We primarily use a PI controller to satisfy the internal model principle, especially when the system reference signal is a step input. Under the terms of the IMP, for a reference with a pole at the origin, we need a controller with a pole at the origin.

Design

The steps to determine the parameters are simple.

• Choose some value $k$ to meet asymptotic tracking and crossover frequency (${\omega }_c$) specifications.
• Find the crossover frequency ${\omega }_c$. This usually requires referring to the Bode plot.
• Find $T$ by setting an inequality such that:

$$\frac{1}{T}\le 0.1{\omega }_c$$