PI controller

Sampled PI control with output saturation and a bounded integral state.

signal
Gradara equations
3 ports
4 parameters

BlockPI controllerKindpiLibraryControl

Description

A sampled PI controller acting on the error between a reference and a measurement. At each sample it updates a clamped integral and computes a saturated output, which then holds until the next sample.

Use it for loops such as motor speed or output-voltage regulation where a fixed controller rate matters.

Example

PI and PID loops The same first-order plant under a sampled PI, a continuous PID, and a sampled PID controller.

Set point (PI)PI (sampled)Plant (PI)y (PI)Set point (PID)ErrorPIDPlant (PID)y (PID)Set point (sampled PID)PID (sampled)Plant (sampled PID)y (sampled PID)

In Gradara, select a PI controller block and press F1, then choose Open example. It opens as a new model in My models, ready to run.

Also in this example:Subtract Step Gain Low-pass filter PID Discrete PID

Ports

Inputs 2

refreferencesignal

Setpoint.
measmeasuredsignal

Measured value, in the same units as the reference.

Outputs 1

outysignal

Controller output, held between samples and limited to ±limit.

Parameters

Proportional gainkp0.6

Proportional gain.
Integral gainki2

Integral gain, per second.
Voltage limitlimit24 V

≥ 0.1

Output limit, at least 0.1. Clamps both the output and the integral to ±limit. Labeled in volts, but it applies in whatever unit the output uses.
Sample periodsamplePeriod0.001 s

≥ 0.0001

Sample period Ts, in seconds, at least 0.1 ms. The first sample is at t = 0.

Equations

Modelica

error = reference − measured
At t = k · samplePeriod:
  integral = max(−limit, min(limit, integral⁻ + samplePeriod · ki · error))
  y = max(−limit, min(limit, kp · error + integral))
integral⁻ is the value from the previous sample; integral starts at 0.

Implementation

The block’s Modelica model

discrete Real integral(start=0, fixed=true);
Real error;
error = reference - measured;
when sample(0, samplePeriod) then
  integral = max(-limit, min(limit, pre(integral) + samplePeriod*ki*error));
  y = max(-limit, min(limit, kp*error + integral));
end when;

Assumptions and limitations

The integral is updated with the current error (backward Euler), so a step in error affects y at the same sample, with no computation delay.
Anti-windup is a clamp on the integral at ±limit, not back-calculation. The integral can still sit at the limit while the proportional term alone saturates the output.
No derivative term. For one, use Discrete PID.
The error is read only at sample instants; nothing between samples is seen.

Tips

Choose samplePeriod well below the loop’s time constants; each sample is a simulation event.

Used in

These larger examples use it too. Open them from Examples in the app.

Data center cooling control
Motor speed control
480 VAC → 24 VDC flyback