Calculation
u=u₀+Kp·[e₁+Tv·(e₁−e₀)/Δt]
The proportional part evaluates current error; the derivative part also responds to how quickly that error changes.
Zacher/Reuter 2024, Abschnitt 4.3.5: PD-Regler
PD controller: output from an error ramp
The proportional part evaluates current error; the derivative part also responds to how quickly that error changes.
MINTSI
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Result
Select a target and calculate.
Understand the inputs
- Controller output u — PD-controller output at the end of the sampling interval. Check it against actuator minimum and maximum before use.
- Proportional gain Kp — Gain applied to the complete bracketed expression. Obtain it from controller design or a controlled tuning test.
- Derivative time Tv — Weighting of error rate. A larger Tv strengthens anticipatory action but also amplifies measurement noise.
- Error at interval start e₀ — Setpoint minus measured value at the first timestamp. Take it from the control log or data acquisition.
- Current error e₁ — Setpoint minus measured value at the end of the interval; this value forms the proportional contribution.
- Sampling interval Δt — Time between e₀ and e₁. Both readings must come from equally scaled signals.
- Bias u₀ — Output for error-free steady operation; obtain it from plant operation or use 0.
Example
Kp=2, Tv=0.5 s, e₀=2, e₁=4, Δt=1 s and u₀=9 give u=19: proportional contribution 8, derivative contribution 2.
Assumptions and limits
Ideal continuous PD controller and linear error between two samples; no filter time constant, saturation or noise suppression. Real derivative action usually requires a low-pass filter.
Understand PD controller: output from an error ramp
Calculate a proportional-derivative controller output from two error readings and their sampling interval.
What does this quantity describe?
The proportional part evaluates current error; the derivative part also responds to how quickly that error changes. 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=u₀+Kp·[e₁+Tv·(e₁−e₀)/Δt]
| Symbol / input | Meaning |
|---|---|
| Controller output u | PD-controller output at the end of the sampling interval. Check it against actuator minimum and maximum before use. |
| Proportional gain Kp | Gain applied to the complete bracketed expression. Obtain it from controller design or a controlled tuning test. |
| Derivative time Tv | Weighting of error rate. A larger Tv strengthens anticipatory action but also amplifies measurement noise. |
| Error at interval start e₀ | Setpoint minus measured value at the first timestamp. Take it from the control log or data acquisition. |
| Current error e₁ | Setpoint minus measured value at the end of the interval; this value forms the proportional contribution. |
| Sampling interval Δt | Time between e₀ and e₁. Both readings must come from equally scaled signals. |
| Bias u₀ | Output for error-free steady operation; obtain it from plant operation or use 0. |
Choose the inputs correctly
Controller output u: PD-controller output at the end of the sampling interval. Check it against actuator minimum and maximum before use. Proportional gain Kp: Gain applied to the complete bracketed expression. Obtain it from controller design or a controlled tuning test. Derivative time Tv: Weighting of error rate. A larger Tv strengthens anticipatory action but also amplifies measurement noise. Error at interval start e₀: Setpoint minus measured value at the first timestamp. Take it from the control log or data acquisition. Current error e₁: Setpoint minus measured value at the end of the interval; this value forms the proportional contribution. Sampling interval Δt: Time between e₀ and e₁. Both readings must come from equally scaled signals. Bias u₀: Output for error-free steady operation; obtain it from plant operation or use 0.
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, Tv=0.5 s, e₀=2, e₁=4, Δt=1 s and u₀=9 give u=19: proportional contribution 8, derivative contribution 2.
Understand the result and units
The proportional part evaluates current error; the derivative part also responds to how quickly that error changes. 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
PD controller: output from an error ramp: Such basic calculations support plausibility checks, early component selection and preparation of a complete verification.
Assumptions, limits and common mistakes
Ideal continuous PD controller and linear error between two samples; no filter time constant, saturation or noise suppression. Real derivative action usually requires a low-pass filter.
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.