PI controller

In control engineering, a PI Controller (proportional-integral controller) is a feedback controller which drives the plant to be controlled with a weighted sum of the error (difference between the output and desired set-point) and the integral of that value. It is a special case of the common PID controller in which the derivative (D) of the error is not used.

The controller output is given by

$K_{P}\Delta +K_{I}\int \Delta \,dt$

where $\Delta$ is the set-point error.

Advantages of a Proportional Plus Integral Controller

The integral term in a PI controller causes the steady-state error to be zero.

PI Controller Model

A PI controller can be modelled easily in software such as Simulink using a "flow chart" box involving Laplace operators:

$C={\frac {G(1+s\tau )}{s\tau }}$

where

$G=K_{P}$ = proportional gain

$G/\tau =K_{I}$ = integral gain

Finding a value for G

Setting a value for $G$ is a trade off between decreasing rise time and increasing settling time. One rule of thumb for $G$ is between 4 and 8.

Finding a value for $\tau$

Finding a proper value for $\tau$ is an iterative process.

1) Set a value for $G$ from the optimal range.

2) View the Nichols Plot for the open-loop response of the system. Observe where the response curve crosses the 0dB line. This frequency is known as the cross-over frequency ($f_{\mathrm {c} }$).

3) The value of $\tau$ can be calculated as:

$\tau =1/f_{\mathrm {c} }$

4) Decreasing $\tau$ decreases the phase margin, however it eliminates a greater proportion of the steady-state errors.

Disadvantages of a Proportional Plus Integral Controller

The problem with using a PI controller is that it introduces a phase-lag. This means that on a Nichols Plot, the stability margin (the phase margin) decreases. So careful design considerations with respect to the gain must be considered.