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 and disadvantages

• The integral term in a PI controller causes the steady-state error to reduce to zero, which is not the case for proportional-only control in general.
• The lack of derivative action may make the system more steady in the steady state in the case of noisy data. This is because derivative action is more sensitive to higher-frequency terms in the inputs.
• Without derivative action, a PI-controlled system is less responsive to real (non-noise) and relatively fast alterations in state and so the system will be slower to reach setpoint and slower to respond to perturbations than a well-tuned PID system may be.

See also

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