PI

PI controller with limits and anti-windup (Modelica LimPID: back-calculation with Ni), its integral relaxed to the reset value while disabled

FastSmoothBuildings.Controls.PI, model (source)

Purpose

A PI controller with limits and anti-windup, and a reset without events: while it is disabled, its integral relaxes to the value that gives the reset output, instead of being reset by a trigger. The limits are smoothLimit, which reaches them exactly.

Equations

With the control error e and the integral I,

y_unl = k (e + I),   y = smoothLimit(y_unl, y_Min, y_Max, dY),

dI/dt = enable (e/T_i + (y − y_unl) / (k N_i T_i)) + (1 − enable) (y_Reset/k − e − I) / τ_Reset,   I(0) = y_Reset/k.

• enable = 1: the integral of e/T_i, with back-calculation: when the output is limited, y − y_unl draws the integral back with the anti-windup gain Ni.
• enable = 0: the integral relaxes with the time constant tauReset so that the unlimited output tends to yReset; the controller restarts from there when it is enabled again.

enable is a Real and the two parts are blended by it, so it can be the output of a Hysteresis. The output is not multiplied by enable: a model that needs no output while disabled multiplies it, as HeatPumpControl does.

Parameters

Name	Default	Meaning
k, Ti		gain and integral time
yMin, yMax	0, 1	limits of the output
Ni	0.9	anti-windup gain
yReset	yMin	output after a reset
tauReset	30 s	time constant of the reset while disabled
dY	10−3 (yMax − yMin)	half-width of the smooth limits

dY and tauReset are the smoothing of the controller: a calibration or an optimal control problem can widen them.

Relation to the Modelica Standard Library

The limits and the anti-windup of LimPID (back-calculation with Ni); the reset of a trigger is replaced by the relaxation above.

Use

In EmissionPI, HeatPumpControl and HVAC.IdealHeatingCoolingPI. See Validation.PIControl.

Parameters

Parameter	Type	Unit	Default	Description
k	Real			Gain
Ti	Real	s		Integral time
yMin	Real		0
yMax	Real		1
Ni	Real		0.9	Anti-windup gain
yReset	Real		yMin	Output after a reset
tauReset	Real	s	30	Time constant of the reset while disabled
dY	Real		1e-3*(yMax - yMin)	Width of the smooth limits

Revisions

• October 2026: moved from trano's FastModel.Controls.