Calculation
u(t)=u₀+e₀·t/TI
An integral controller does not respond with a jump; it changes output at a constant rate for as long as control error persists.
Zacher/Reuter 2024, Abschnitt 4.3.2, Gleichungen 4.38 bis 4.41: I-Regeleinrichtung
I controller: output for a constant error
An integral controller does not respond with a jump; it changes output at a constant rate for as long as control error persists.
MINTSI
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Result
Select a target and calculate.
Understand the inputs
- Controller output u(t) — Integral-controller output after the stated error duration. Compare with actuator limits; exceeding them causes saturation in practice.
- Integral time TI — Time after which constant error e₀ changes output by exactly e₀. For equally scaled input and output, TI is the reciprocal of integral gain; take it from controller settings or design.
- Constant control error e₀ — Setpoint minus measured value, assumed constant for the complete duration. Its sign determines whether controller output rises or falls.
- Error duration t — Time elapsed since the constant control error began. Determine it from measurement timestamps or the simulation instant of interest.
- Initial integrator output u₀ — Stored controller output at the start of the interval. Take it from controller state; for a reset integrator use u₀=0.
Example
TI=5 s, e₀=3, t=4 s and u₀=5 give a change of 3·4/5=2.4 and hence u=7.4. A negative error produces a negative change.
Assumptions and limits
Ideal continuous integral controller, constant error over the interval and consistently scaled signals. Output saturation, anti-windup, leaky integration, sampling and actuator dynamics are excluded.
Understand I controller: output for a constant error
This calculator completes the existing P, PI, PD and PID pages with a pure integral controller. It shows how a persistent setpoint-minus-output difference accumulates and why controller output rises or falls continuously without an immediate proportional term.
What does this quantity describe?
An I controller acts on the time integral of control error e(t). A constant error therefore creates a straight output ramp. TI is integral time; for equally scaled signals, integral gain is KIR=1/TI.
Formula and variables
u(t)=u₀+e₀·t/TI
General: u(t)=u₀+(1/TI)·∫e(t)dtConstant error: u(t)=u₀+e₀·t/TIIntegral gain for equal scaling: KIR=1/TI
| Symbol / input | Meaning |
|---|---|
| Controller output u(t) | Integral-controller output after the stated error duration. Compare with actuator limits; exceeding them causes saturation in practice. |
| Integral time TI | Time after which constant error e₀ changes output by exactly e₀. For equally scaled input and output, TI is the reciprocal of integral gain; take it from controller settings or design. |
| Constant control error e₀ | Setpoint minus measured value, assumed constant for the complete duration. Its sign determines whether controller output rises or falls. |
| Error duration t | Time elapsed since the constant control error began. Determine it from measurement timestamps or the simulation instant of interest. |
| Initial integrator output u₀ | Stored controller output at the start of the interval. Take it from controller state; for a reset integrator use u₀=0. |
Choose the inputs correctly
Take TI from controller settings or design. e₀ is setpoint minus measured value and is assumed constant over the interval. t is elapsed time since this error began. u₀ is the stored integrator output at interval start, obtained from controller state or set to zero after reset.
How to use the calculator
First verify controller action sign. Read TI and current integrator state from controller configuration. Subtract measured value from setpoint on one scale and enter the actual duration over which error is approximately constant. Compare the result with actuator minimum and maximum.
Worked example
With TI=5 s, e₀=3, t=4 s and u₀=5, output increases by e₀·t/TI=2.4 to u=7.4. After exactly TI=5 s, the change would equal error value 3.
Understand the result and units
A positive constant error produces positive slope e₀/TI; a negative error reduces output. Smaller TI means faster integration. Once error becomes zero, an ideal I controller holds its most recent output.
TI and t must use the same time dimension and convert internally through seconds. e₀, u₀ and u(t) use a consistent normalised signal scale. Physical units are possible when controller scaling is defined accordingly.
Typical applications
Use it for controller fundamentals, checking an integrator trace, estimating time to an actuator limit and comparing pure integral control with P and PI structures.
Assumptions, limits and common mistakes
The calculator assumes an ideal continuous integrator and constant error. Real output limits, anti-windup, discrete sampling, leaky integration, deadband and actuator dynamics are excluded. Beyond an actuator limit, do not interpret the linear value as actual output.
Common mistake: Do not confuse u₀ with a proportional-controller bias: here it is stored integrator state. Do not enter integral gain as TI; larger TI makes integration slower. For changing error, do not multiply the final error by the complete time—integrate piecewise instead.
Frequently asked questions
What is “Output of a pure integral controller” used for?
Use it for controller fundamentals, checking an integrator trace, estimating time to an actuator limit and comparing pure integral control with P and PI structures.
Where do the input values come from?
Take TI from controller settings or design. e₀ is setpoint minus measured value and is assumed constant over the interval. t is elapsed time since this error began. u₀ is the stored integrator output at interval start, obtained from controller state or set to zero after reset.
What does the result not cover?
The calculator assumes an ideal continuous integrator and constant error. Real output limits, anti-windup, discrete sampling, leaky integration, deadband and actuator dynamics are excluded. Beyond an actuator limit, do not interpret the linear value as actual output.