PI Control

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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.

3 See Also