Multi-point PID calibration

nefelim4ag · April 1, 2026, 12:09am

@minicx, sorry to bother
Can you give it a try?

github.com/Klipper3d/klipper

RFC: Pid calibration use First Order Plus Dead Time model for PID estimation (#7243)

master ← nefelim4ag:pid-calibration-imc

opened 06:41PM - 28 Mar 26 UTC

nefelim4ag

+145 -23

TLDR: This is an attempt to make one PID calibrator to rule them all. The ba…sic idea is that one can make a model of the hotend. We have a gain over the initial heat-up, we have dead time from the relay test, and we can fit the time constant over the initial heat curve to estimate where the heater settles at a specific power. This data can be used within the internal model control method within the First-Order Plus Dead Time model. From this data, we can estimate the specific PID values for a specific heater. So, what it does: - Records the initial heating curve - Fit over this curve (it makes calculations more stable over different MAX_POWER values). - Calculate the gain, where I later used the estimated gain from the fit for the same purposes. It seems to produce okayish values for the PD, where I is always too small and, as a result, the output PID is sluggish. As we have relatively short dead time and low inertia on the hotend, there is a "hack" to increase I. Because normally the output is pretty conservative, because of how the model works (it tries not to overshoot and keep the control smooth), ~~lambda is zeroed to prefer fast responses~~. One can calibrate with reduced MAX_POWER as in the #7209, to get the initial heating up closer to the asymptote. It seems to work fine without that. There are 2 extreme scenarios for the tuning. One with 100% power (default), and with the least required power. Least required power should generally provide a better fit, because this is what the model tries to estimate. One can run the tuned heater at the target temperature, record the average PWM, for example 30%. That PWM is the least required power to do the test. It can be long, or it can trigger verify_heater errors, it is recommended to use some margin +5% for example. <img width="1348" height="528" alt="image" src="https://github.com/user-attachments/assets/5744d1c3-762b-435a-8727-ee124bb972d8" /> Just steady state graphs, extruder: <img width="1358" height="444" alt="image" src="https://github.com/user-attachments/assets/086af560-db9c-4492-863b-9fc1fbb33855" /> ``` Autotune: raw=7.618107/0.450000 | Tu=11.100615/DeadT=1.350078 Gain=1043.620833 tau=105.350765 | Kp=19.188908 Ki=3.553297 Kd=12.870788 Autotune: raw=-7.978697/0.450000 | Tu=11.400670/DeadT=1.350071 Gain=773.799418 tau=70.234177 | Kp=17.308440 Ki=3.205099 Kd=11.572582 Autotune: raw=7.834107/0.450000 | Tu=11.400545/DeadT=1.350055 Gain=597.414908 tau=46.823118 | Kp=15.017210 Ki=2.780851 Kd=9.992967 Autotune: raw=-7.293205/0.450000 | Tu=11.100456/DeadT=1.350052 Gain=483.753974 tau=62.430490 | Kp=24.639525 Ki=4.562698 Kd=16.454413 Autotune: raw=7.834107/0.450000 | Tu=11.400545/DeadT=1.400057 Gain=506.543530 tau=51.938975 | Kp=18.927158 Ki=3.379712 Kd=13.073347 Autotune: final: Kp=18.927158 Ki=3.379712 Kd=13.073347 ``` Where one can interpret the Gain and Tau (time constant), like this: Gain is where the heater will settle with the requested power percentage. Tau is 63.2%, which is a time where gain will reach ~63.2% of the set temperature. <img width="2560" height="1472" alt="image" src="https://github.com/user-attachments/assets/c13eb8ef-b13c-485d-9cd3-5b64d194530e" /> Bed: <img width="1358" height="444" alt="image" src="https://github.com/user-attachments/assets/a8ee92f4-84d3-4043-8013-6632a1c0d983" /> ``` Autotune: raw=-6.273359/1.000000 | Tu=161.700000/DeadT=3.150000 Gain=62.092383 tau=70.234177 | Kp=93.620563 Ki=7.430203 Kd=144.218294 Autotune: raw=6.180337/1.000000 | Tu=161.700000/DeadT=3.300000 Gain=46.271708 tau=93.645236 | Kp=159.140931 Ki=12.056131 Kd=258.036022 Autotune: raw=-6.173799/1.000000 | Tu=173.700000/DeadT=3.300000 Gain=56.845383 tau=78.037863 | Kp=108.323635 Ki=8.206336 Kd=175.033169 Autotune: raw=6.260284/1.000000 | Tu=134.100000/DeadT=2.300000 Gain=51.715770 tau=84.380185 | Kp=183.361755 Ki=19.930626 Kd=208.030810 Autotune: final: Kp=183.361755 Ki=19.930626 Kd=208.030810 ``` It is possible that it simply overfitted and produced critically dumped results only for me. So, I will be glad to see the data from others. Thanks, -Timofey --- _P.S. I would expect that MAX_POWER reduction suggested in #7209, which produces the symmetric relay test, should work fine. But it didn't work for the large/slow heaters, and didn't work for some with fast response times, so... I got myself drawn into the rabbit hole. Where I hope that now, I have some understanding of WHY it works in a way that it works - I think, it is hard to define magic ZN numbers for every possible case, and the "proper" relay method is more complicated, so here we are._ --- ~~Deadtime detection is suboptimal and may be susceptible to the noise of the sensor.~~ I hope the time between the control signal and the peak is a good enough estimation for our short dead time. --- Links, because I will lose them: - https://www.ijireeice.com/upload/2016/may-16/IJIREEICE%2047.pdf - IMC-PID - https://www.researchgate.net/publication/288897834_The_SIMC_method_for_smooth_PID_controller_tuning - Simple IMC PID - Sigurd Skogestad

PID_CALIBRATE .. MAX_POWER=1 - it should be sane
Check how it is holding temperature, for example, 250 °C
Remember the average power from 2
PID_CALIBRATE .. MAX_POWER=<average power +5% or so> - ideal case for calibration, where we have many samples, it is asymptote & etc
Same test, how does it go

Thanks,
-Timofey

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Multi-point PID calibration

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