Calculation
u=Kp·[e₁+(e₀+e₁)·Δt/(2Tn)+Tv·(e₁−e₀)/Δt]
A PID controller combines immediate response, accumulated error and anticipatory response to error change.
Zacher/Reuter 2024, Abschnitt 4.3.6: PID-Regler
PID controller: P, I and D contributions for an error ramp
A PID controller combines immediate response, accumulated error and anticipatory response to error change.
MINTSI
02
Result
Select a target and calculate.
Understand the inputs
- Controller output u at interval end — Sum of amplified P, I and D action. Compare with actuator range; saturation and bias must be added externally.
- Proportional gain Kp — Common gain of all three controller contributions, obtained from design, simulation or controlled tuning.
- Reset time Tn — Sets integration speed: a smaller Tn strengthens accumulation of past error.
- Derivative time Tv — Weights error change. Larger values respond earlier and more strongly but increase sensitivity to measurement noise.
- Error at interval start e₀ — Setpoint minus measured value at the beginning of the interval.
- Current error e₁ — Setpoint minus measured value at interval end; directly forms the proportional contribution.
- Sampling interval Δt — Time between e₀ and e₁. The calculator assumes linear error between them.
Example
Kp=2, Tn=5 s, Tv=0.5 s, e₀=2, e₁=4 and Δt=1 s give P=8, I=1.2 and D=2, total u=11.2.
Assumptions and limits
Ideal parallel PID controller, linear error in the first interval and zero integral state at interval start; bias, derivative filter, output saturation and anti-windup are excluded.
Understand PID controller: P, I and D contributions for an error ramp
Calculate a proportional-integral-derivative controller output from initial error, current error and sampling interval.
What does this quantity describe?
A PID controller combines immediate response, accumulated error and anticipatory response to error change. 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=Kp·[e₁+(e₀+e₁)·Δt/(2Tn)+Tv·(e₁−e₀)/Δt]
| Symbol / input | Meaning |
|---|---|
| Controller output u at interval end | Sum of amplified P, I and D action. Compare with actuator range; saturation and bias must be added externally. |
| Proportional gain Kp | Common gain of all three controller contributions, obtained from design, simulation or controlled tuning. |
| Reset time Tn | Sets integration speed: a smaller Tn strengthens accumulation of past error. |
| Derivative time Tv | Weights error change. Larger values respond earlier and more strongly but increase sensitivity to measurement noise. |
| Error at interval start e₀ | Setpoint minus measured value at the beginning of the interval. |
| Current error e₁ | Setpoint minus measured value at interval end; directly forms the proportional contribution. |
| Sampling interval Δt | Time between e₀ and e₁. The calculator assumes linear error between them. |
Choose the inputs correctly
Controller output u at interval end: Sum of amplified P, I and D action. Compare with actuator range; saturation and bias must be added externally. Proportional gain Kp: Common gain of all three controller contributions, obtained from design, simulation or controlled tuning. Reset time Tn: Sets integration speed: a smaller Tn strengthens accumulation of past error. Derivative time Tv: Weights error change. Larger values respond earlier and more strongly but increase sensitivity to measurement noise. Error at interval start e₀: Setpoint minus measured value at the beginning of the interval. Current error e₁: Setpoint minus measured value at interval end; directly forms the proportional contribution. Sampling interval Δt: Time between e₀ and e₁. The calculator assumes linear error between them.
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, Tn=5 s, Tv=0.5 s, e₀=2, e₁=4 and Δt=1 s give P=8, I=1.2 and D=2, total u=11.2.
Understand the result and units
A PID controller combines immediate response, accumulated error and anticipatory response to error change. 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
PID controller: P, I and D contributions for an error ramp: Such basic calculations support plausibility checks, early component selection and preparation of a complete verification.
Assumptions, limits and common mistakes
Ideal parallel PID controller, linear error in the first interval and zero integral state at interval start; bias, derivative filter, output saturation and anti-windup are excluded.
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.