Calculation
V=Kp·Ks; Kcl=V/(1+V); Tcl=T/(1+V); x(t)=Δw·Kcl·[1−e^(−t/Tcl)]
Closing the loop speeds up first-order dynamics, but proportional-only control leaves a steady setpoint error.
Zacher/Reuter 2024, Abschnitte 2.2, 2.6.2 und 3.2: geschlossener Regelkreis mit P-Regler und PT1-Strecke
P controller with first-order process: closed-loop step response
Closing the loop speeds up first-order dynamics, but proportional-only control leaves a steady setpoint error.
MINTSI
02
Result
Select a target and calculate.
Calculated step response
Move the pointer or finger across the curve to read time and output. The chart updates directly with the inputs.
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Understand the inputs
- Measured-output change x(t) — Change in controlled process output from its operating point at the selected time. Add the pre-step measured value for an absolute reading; the chart also shows the final value.
- Controller gain Kp — Actuator-output change per unit of control error. Take it from the active controller setting; it must match the signal scaling used for process gain.
- Static process gain Ks — Final process-output change divided by a small actuator-output change. Determine it from two steady operating points or an identified first-order model.
- Open-loop time constant T — First-order time constant of the uncontrolled process. After an actuator step, 63.2% of final process change is reached after T seconds.
- Setpoint step Δw — New minus old setpoint on the same scale as measured output. Enter only the change, not the new absolute setpoint.
- Time since setpoint step t — Elapsed time since the setpoint change. The calculated point is marked on the interactive curve.
Example
Kp=4, Ks=2, T=5 s and Δw=10 give V=8, final value 8.889 and closed-loop time constant 0.556 s. At t=5 s, x(t)≈8.888 and is practically settled.
Assumptions and limits
Linear negative unity feedback, ideal proportional controller and stable first-order process without dead time, additional poles, saturation or disturbance. Controller and process gains are positive and consistently scaled.
Understand P controller with first-order process: closed-loop step response
This calculator shows the controller and process acting together in a closed loop. It calculates how quickly measured output follows a setpoint step and the steady residual error left by proportional-only control.
What does this quantity describe?
A first-order process has proportional static behaviour and one lag. Kp is proportional-controller gain and Ks is static process gain. Under negative unity feedback, the closed loop remains first order, with gain Kcl=Kp·Ks/(1+Kp·Ks) and shorter time constant Tcl=T/(1+Kp·Ks).
Formula and variables
V=Kp·Ks; Kcl=V/(1+V); Tcl=T/(1+V); x(t)=Δw·Kcl·[1−e^(−t/Tcl)]
Loop gain: V=Kp·KsClosed-loop gain: Kcl=V/(1+V)Closed-loop time constant: Tcl=T/(1+V)Response: x(t)=Δw·Kcl·[1−exp(−t/Tcl)]
| Symbol / input | Meaning |
|---|---|
| Measured-output change x(t) | Change in controlled process output from its operating point at the selected time. Add the pre-step measured value for an absolute reading; the chart also shows the final value. |
| Controller gain Kp | Actuator-output change per unit of control error. Take it from the active controller setting; it must match the signal scaling used for process gain. |
| Static process gain Ks | Final process-output change divided by a small actuator-output change. Determine it from two steady operating points or an identified first-order model. |
| Open-loop time constant T | First-order time constant of the uncontrolled process. After an actuator step, 63.2% of final process change is reached after T seconds. |
| Setpoint step Δw | New minus old setpoint on the same scale as measured output. Enter only the change, not the new absolute setpoint. |
| Time since setpoint step t | Elapsed time since the setpoint change. The calculated point is marked on the interactive curve. |
Choose the inputs correctly
Kp comes from the controller setting. Obtain Ks and T from a small actuator step on the uncontrolled process or an identified model. Δw is new minus old setpoint and t is elapsed time since that step. All gains must use the same signal scaling, for example percent actuator output and percent measurement range.
How to use the calculator
Identify the process around a safe operating point. Determine Ks from final output change divided by actuator step and T from the 63.2% time. Enter Kp and the planned setpoint step. Compare final value and curve with accuracy and speed requirements; check actuator range and real stability separately.
Worked example
Kp=4 and Ks=2 give loop gain V=8. With T=5 s, Tcl=5/9=0.556 s. A setpoint step of 10 reaches only 80/9=8.889 at steady state; after one closed-loop time constant, 63.2% of that value is reached.
Understand the result and units
Higher Kp shortens the ideal response and moves final value closer to setpoint. Residual error Δw/(1+Kp·Ks) nevertheless remains. In a real plant, additional lags and dead time limit safe controller gain.
T and t may use seconds, minutes or hours and convert internally through seconds. Kp·Ks must be dimensionless. Δw and x(t) use the same normalised scale.
Typical applications
Use it for teaching, preliminary proportional-loop design, comparing Kp settings, checking a recorded setpoint response, and estimating response time and steady accuracy.
Assumptions, limits and common mistakes
The result assumes linear negative unity feedback, an ideal proportional controller and exactly one process lag. Dead time, additional poles, sensor filters, actuator dynamics, saturation, disturbances and noise are excluded. Arbitrary speed-up at large Kp is therefore not a commissioning proof.
Common mistake: Do not confuse open-loop T with Tcl. Do not enter absolute setpoint instead of Δw. Do not identify Ks on percentage signals and then use Kp on a different physical scale. Do not infer steady error from a curve observed for too little time.
Frequently asked questions
What is “Closed-loop response of a P controller and first-order process” used for?
Use it for teaching, preliminary proportional-loop design, comparing Kp settings, checking a recorded setpoint response, and estimating response time and steady accuracy.
Where do the input values come from?
Kp comes from the controller setting. Obtain Ks and T from a small actuator step on the uncontrolled process or an identified model. Δw is new minus old setpoint and t is elapsed time since that step. All gains must use the same signal scaling, for example percent actuator output and percent measurement range.
What does the result not cover?
The result assumes linear negative unity feedback, an ideal proportional controller and exactly one process lag. Dead time, additional poles, sensor filters, actuator dynamics, saturation, disturbances and noise are excluded. Arbitrary speed-up at large Kp is therefore not a commissioning proof.