Calculation
u(t)=u₀+Kp·e₀·(1+t/Tn)
The proportional part responds immediately; the integral part keeps growing while setpoint and measured value differ.
Zacher/Reuter 2024, Abschnitt 4.3.3: PI-Regler
PI controller: output for a constant error
The proportional part responds immediately; the integral part keeps growing while setpoint and measured value differ.
MINTSI
02
Result
Select a target and calculate.
Understand the inputs
- Controller output u(t) — Controller output at time t. It is sent to the actuator; compare it with the actuator's permissible range before use.
- Proportional gain Kp — Immediate output change per unit of control error. Obtain it from controller design, simulation or a controlled tuning test.
- Constant control error e₀ — Setpoint minus measured value, assumed constant over the considered time. Both values must use the same scale.
- Reset time Tn — Time after which the integral contribution equals the proportional contribution for constant error. Tn is a controller setting; a smaller value means faster integration.
- Error duration t — Time for which control error e₀ has already persisted unchanged.
- Bias u₀ — Output that holds the operating point at e=0. Take it from steady operation or use 0 when no bias is modelled.
Example
Kp=2, e₀=3, Tn=5 s, t=4 s and u₀=10 give u=20.8: 6 units proportional plus 4.8 units integral contribution.
Assumptions and limits
Ideal continuous PI controller, constant error from t=0, zero initial integral state and no output saturation or anti-windup function.
Understand PI controller: output for a constant error
Calculate the time-dependent output of a proportional-integral controller for a constant control error.
What does this quantity describe?
The proportional part responds immediately; the integral part keeps growing while setpoint and measured value differ. This calculator represents a clearly bounded technical relationship between the displayed quantities. The definition helps put inputs into the same reference state before interpreting the result.
Formula and variables
u(t)=u₀+Kp·e₀·(1+t/Tn)
| Symbol / input | Meaning |
|---|---|
| Controller output u(t) | Controller output at time t. It is sent to the actuator; compare it with the actuator's permissible range before use. |
| Proportional gain Kp | Immediate output change per unit of control error. Obtain it from controller design, simulation or a controlled tuning test. |
| Constant control error e₀ | Setpoint minus measured value, assumed constant over the considered time. Both values must use the same scale. |
| Reset time Tn | Time after which the integral contribution equals the proportional contribution for constant error. Tn is a controller setting; a smaller value means faster integration. |
| Error duration t | Time for which control error e₀ has already persisted unchanged. |
| Bias u₀ | Output that holds the operating point at e=0. Take it from steady operation or use 0 when no bias is modelled. |
Choose the inputs correctly
Controller output u(t): Controller output at time t. It is sent to the actuator; compare it with the actuator's permissible range before use. Proportional gain Kp: Immediate output change per unit of control error. Obtain it from controller design, simulation or a controlled tuning test. Constant control error e₀: Setpoint minus measured value, assumed constant over the considered time. Both values must use the same scale. Reset time Tn: Time after which the integral contribution equals the proportional contribution for constant error. Tn is a controller setting; a smaller value means faster integration. Error duration t: Time for which control error e₀ has already persisted unchanged. Bias u₀: Output that holds the operating point at e=0. Take it from steady operation or use 0 when no bias is modelled.
How to use the calculator
Select the target quantity, enter the other known values with units, then check the result against the worked example and model limits.
Worked example
Kp=2, e₀=3, Tn=5 s, t=4 s and u₀=10 give u=20.8: 6 units proportional plus 4.8 units integral contribution.
Understand the result and units
The proportional part responds immediately; the integral part keeps growing while setpoint and measured value differ. Read the result as a model value for the selected operating point and check units, sign, order of magnitude and application boundary conditions.
Use the displayed units and convert afterwards. Prefixes such as k-, m- and µ- are common sources of mistakes.
Typical applications
PI controller: output for a constant error: Such basic calculations support plausibility checks, early component selection and preparation of a complete verification.
Assumptions, limits and common mistakes
Ideal continuous PI controller, constant error from t=0, zero initial integral state and no output saturation or anti-windup function.
Common mistake: A formally correct result can still be unsuitable when load case, reference state or units do not match the application.
Frequently asked questions
How do I check the result?
Compare unit and order of magnitude with a second calculation and vary inputs one at a time.
Are assumptions automatically satisfied?
No. The calculator exposes a model; real boundary conditions require separate review.
Can I mix arbitrary units?
Only when the calculator converts within the same physical quantity type.