PI command after an integral update calculator

An integral update adds the current error accumulated over the interval to its prior state. The output also includes proportional response and bias; this model excludes saturation and anti-windup.

Units

u = u_{\text{bias}} + K_{p} \times e + I_{\text{prev}} + K_{i} \times e \times \text{dt}

Calculator

Enter the values you know and leave one box empty. Calculate finds the empty one, the result or any input.

Worked example

Given

$u_{\text{bias}}$ = 20 %
$K_{p}$ = 3 %/Δ°F
$e$ = 4 Δ°F
$I_{\text{prev}}$ = 15 %
$K_{i}$ = 0.5 %/(Δ°F·min)
$\text{dt}$ = 10 min

u = 20 + 3 \times 4 + 15 + 0.5 \times 4 \times 10 = 67

Symbols and units

Symbol	Meaning	Unit
$u$	unsaturated output command	%
$u_{\text{bias}}$	output bias	%
$K_{p}$	signed proportional gain	%/Δ°F
$e$	signed current error held across the interval	Δ°F
$I_{\text{prev}}$	prior integral contribution	%
$K_{i}$	signed integral gain	%/(Δ°F·min)
$\text{dt}$	update duration	min

Symbol	Meaning	Unit
$u$	unsaturated output command	%
$u_{\text{bias}}$	output bias	%
$K_{p}$	signed proportional gain	%/K
$e$	signed current error held across the interval	K
$I_{\text{prev}}$	prior integral contribution	%
$K_{i}$	signed integral gain	%/(K·min)
$\text{dt}$	update duration	min

Related calculators

Results are estimates for study and screening. Check the inputs, units and assumptions before you use them for design, billing or contract decisions.