PI controller

Basic block of a 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.

PI Controller Model

The controller output is given by

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

where $\Delta$ is the error or deviation of actual measured value (PV) from the set-point (SP).

$\Delta$ = SP - PV.

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

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

where

$G=K_{P}$ = proportional gain

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

Setting a value for $G$ is often a trade off between decreasing overshoot and increasing settling time.

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 the steady-state errors quicker.

Advantages of a Proportional-Integral Controller

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

See also

• Control theory
• Feedback
• Instability
• Oscillation
• PID controller