Calculation
e∞ = Δw / (1 + Kp·Ks)
A static proportional-control loop needs a remaining error to produce a sustained actuator command; higher loop gain reduces that error.
Zacher/Reuter 2024, Abschnitt 2.6.2: statischer Regelfaktor
P-control loop: steady-state error
A static proportional-control loop needs a remaining error to produce a sustained actuator command; higher loop gain reduces that error.
MINTSI
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Result
Select a target and calculate.
Understand the inputs
- Steady-state error e∞ — Setpoint minus measured value after all transients have decayed. Compare it with the permitted process error; if it is too large, proportional-only control may be insufficient.
- Controller gain Kp — Controller-output change per unit of control error. Take it from controller settings or documentation. The value applies to the chosen signal scaling.
- Static process gain Ks — Final controlled-variable change divided by a small actuator-output change. Determine it from two steady operating points; sign and scaling must match negative feedback.
- Setpoint step Δw — New minus old setpoint on the same scale as the controlled variable. For the final error after an operating-point change, enter only the setpoint change.
Example
Kp=4, Ks=2 and Δw=10 give loop gain 8 and e∞=10/9≈1.111. The measured output therefore achieves 8.889 units of the requested change.
Assumptions and limits
Linear negative unity feedback, proportional controller, process with finite static gain and a stable closed loop. Disturbances, feedforward, bias, integral action, saturation and measurement scaling factors are excluded.
Understand P-control loop: steady-state error
This calculator shows the persistent setpoint-minus-output difference after a setpoint step when a process is controlled only by a proportional controller. It supports an early decision on whether proportional-only control can meet the steady accuracy requirement.
What does this quantity describe?
A proportional controller changes its output by Kp times control error. Ks is static process gain. In negative unity feedback their product forms static loop gain. Because proportional control requires an error to sustain actuator output, a finite gain leaves steady-state error e∞.
Formula and variables
e∞ = Δw / (1 + Kp·Ks)
Static loop gain: V₀=Kp·KsStatic regulation factor: RF=1/(1+V₀)Steady-state error: e∞=RF·Δw
| Symbol / input | Meaning |
|---|---|
| Steady-state error e∞ | Setpoint minus measured value after all transients have decayed. Compare it with the permitted process error; if it is too large, proportional-only control may be insufficient. |
| Controller gain Kp | Controller-output change per unit of control error. Take it from controller settings or documentation. The value applies to the chosen signal scaling. |
| Static process gain Ks | Final controlled-variable change divided by a small actuator-output change. Determine it from two steady operating points; sign and scaling must match negative feedback. |
| Setpoint step Δw | New minus old setpoint on the same scale as the controlled variable. For the final error after an operating-point change, enter only the setpoint change. |
Choose the inputs correctly
Kp comes from the active controller setting. Determine Ks from two nearby steady operating points as output change divided by actuator-output change. Δw is only the setpoint change, new minus old, on the same scale as the controlled measurement. The equation uses positive gain magnitudes for correctly closed negative feedback.
How to use the calculator
First scale controller and process signals consistently. Determine Ks from a small safe actuator step in manual operation and take Kp from the controller. Enter the planned setpoint step and compare e∞ with permitted steady error. Assess dynamic stability separately.
Worked example
For Kp=4, Ks=2 and Δw=10, Kp·Ks=8. Thus e∞=10/(1+8)=1.111 units remain and controlled output changes by only 8.889 rather than 10.
Understand the result and units
Kp=0 means no control action and the whole setpoint step remains as error. More loop gain reduces e∞ but cannot eliminate it at finite values. If the error is unacceptable, consider integral action, suitable feedforward or process redesign, each with its own stability assessment.
The equation assumes consistently scaled signals. Physically Kp has actuator-output units per error and Ks has controlled-output units per actuator output, making their product dimensionless. Δw and e∞ share a unit but are entered as normalised values in this general model.
Typical applications
Use it for initial controller-structure selection, teaching experiments, estimating steady setpoint accuracy and comparing proportional gains.
Assumptions, limits and common mistakes
The calculation assumes a stable linear loop, negative unity feedback and a process with finite static gain. It says nothing about overshoot, oscillation or actuator saturation. A controller or process integrator, feedforward, disturbance or non-unity measurement feedback changes the steady equation.
Common mistake: Do not enter absolute setpoint instead of its change. Do not mix Kp and Ks from differently scaled percentage and physical signals. A small calculated error is not proof of stability; excessive Kp can destabilise the real loop.
Frequently asked questions
What is “Steady-state error of a proportional-control loop” used for?
Use it for initial controller-structure selection, teaching experiments, estimating steady setpoint accuracy and comparing proportional gains.
Where do the input values come from?
Kp comes from the active controller setting. Determine Ks from two nearby steady operating points as output change divided by actuator-output change. Δw is only the setpoint change, new minus old, on the same scale as the controlled measurement. The equation uses positive gain magnitudes for correctly closed negative feedback.
What does the result not cover?
The calculation assumes a stable linear loop, negative unity feedback and a process with finite static gain. It says nothing about overshoot, oscillation or actuator saturation. A controller or process integrator, feedforward, disturbance or non-unity measurement feedback changes the steady equation.