PI Control
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## 1 Introduction to Proportional-Integral Controller (PI) Design

A PI-Lead controller is a proportional gain in parallel with an integrator; both in series with a lead controller. The proportional gain provides fast error response. The integrator drives the system to a 0 steady-state error.

## 2 Mathematics of the Proportional-Integral Controller

Figure 2: PI Controller Example

The Proportional-Integral (PI) Controller is a proportional controller (simple gain $LaTeX: k_{p}$) and an integrator $LaTeX: \left(\frac{k_{i}}{s}\right)$. Examples of PI controllers with different zeros are on the right.

 $LaTeX: K\left(s\right)=k_{p} + \frac{k_{i}}{s}$

 $LaTeX: K\left(s\right)=\frac{k_{p}s}{s} + \frac{k_{i}}{s}$

 $LaTeX: K\left(s\right)=\frac{k_{p}s + k_{i}}{s}$

 $LaTeX: K\left(s\right)=k_{p}\frac{s + \frac{k_{i}}{k_{p}}}{s}$ Eqn. 1

The MATLAB command to create a PI controller is

 PIcomp = @(z) tf([1 z],[1 0]);


where

z is the zeros in a PI controller.

The MATLAB command presented ignores the proportional gain, $LaTeX: k_{p}$. The proportional gain is set so that the open loop system can achieve the desired open loop crossover frequency.