Documentation sections

PID Controller Tuning (Ziegler-Nichols)

POSThttps://rftools.io/api/py/v1/calculate

Slug pid-tuning. Calculate PID controller gains using the Ziegler-Nichols open-loop (reaction curve) method from process gain, dead time, and time constant.

Inputs

Send every one of these in inputs, in the units shown. Most are required: a call without one is refused (400) and costs nothing; one the calculator can do without takes the default shown. An input outside its range is still computed and flagged in the provenance.

Inputs of pid-tuning
Name	Unit	Range	Default	Description
processGainnumber	none	at least 0.001	1.5	Output change / Input step size from step test
deadTimenumber	s	at least 0.001	0.1	Time before response begins
timeConstantnumber	s	at least 0.001	1	Time for 63.2% of final step response
overshootBudgetnumber	%	0.1 … 200	25	The overshoot the loop may have on a setpoint step; Ziegler–Nichols tuning aims at about 25 % (Ogata, Modern Control Engineering, 5th ed., ch. 8)

Outputs

Returned in values, by key. A value that is unbounded or undefined is null.

Outputs of pid-tuning
Field	Unit	Description
kpPIDheadline	none	Kp (PID)
kiPID	1/s	Ki (PID)
kdPID	s	Kd (PID)
kpPI	none	Kp (PI only)
tiPI	s	Ti (PI only)
tiPID	s	Ti (PID)
tdPID	s	Td (PID)
overshootPct	%	How far the process overshoots a setpoint step under the PID gains, simulated with the derivative on the measurement
settlingTime	s	When the simulated step response last leaves ±2 % of the setpoint

Calling it

Calculate: one call, the request and response in full, and its errors.
Batch: this calculator with others, or a sweep, in one request.
Solve for a target: the value of one input that brings an output to a target.

curl -X POST https://rftools.io/api/py/v1/calculate \
  -H "X-API-Key: $RFTOOLS_API_KEY" \
  -H "Content-Type: application/json" \
  -d '{
    "slug": "pid-tuning",
    "inputs": {
      "processGain": 1.5,
      "deadTime": 0.1,
      "timeConstant": 1,
      "overshootBudget": 25
    }
  }'

import rftools

client = rftools.Client()  # reads RFTOOLS_API_KEY, or the key `rftools login` saved

result = client.calculate("pid-tuning", {
    "processGain": 1.5,
    "deadTime": 0.1,
    "timeConstant": 1,
    "overshootBudget": 25,
})
print(result.values)
print(result.warnings, result.provenance.formula_ref)

const response = await fetch("https://rftools.io/api/py/v1/calculate", {
  method: "POST",
  headers: {
    "X-API-Key": process.env.RFTOOLS_API_KEY,
    "Content-Type": "application/json",
  },
  body: JSON.stringify({
    "slug": "pid-tuning",
    "inputs": {
      "processGain": 1.5,
      "deadTime": 0.1,
      "timeConstant": 1,
      "overshootBudget": 25
    }
  }),
});
console.log(response.status, await response.json());

package main

import (
	"bytes"
	"fmt"
	"io"
	"net/http"
	"os"
)

func main() {
	body := []byte(`{
	  "slug": "pid-tuning",
	  "inputs": {
	    "processGain": 1.5,
	    "deadTime": 0.1,
	    "timeConstant": 1,
	    "overshootBudget": 25
	  }
	}`)
	req, err := http.NewRequest("POST", "https://rftools.io/api/py/v1/calculate", bytes.NewReader(body))
	if err != nil {
		panic(err)
	}
	req.Header.Set("X-API-Key", os.Getenv("RFTOOLS_API_KEY"))
	req.Header.Set("Content-Type", "application/json")
	res, err := http.DefaultClient.Do(req)
	if err != nil {
		panic(err)
	}
	defer res.Body.Close()
	out, _ := io.ReadAll(res.Body)
	fmt.Println(res.Status, string(out))
}

application/json

{
  "slug": "pid-tuning",
  "values": {
    "kpPID": 7.999999999999998,
    "kiPID": 39.999999999999986,
    "kdPID": 0.3999999999999999,
    "tiPID": 0.2,
    "tdPID": 0.05,
    "kpPI": 5.999999999999999,
    "tiPI": 0.333,
    "overshootPct": 73.24945355372432,
    "settlingTime": 0.9511583255229012
  },
  "warnings": [],
  "errors": [],
  "provenance": {
    "method": "calculator:pid-tuning",
    "version": "api@3f2c1a9b04de",
    "formulaRef": "Ziegler & Nichols, 1942",
    "assumptions": [],
    "validRange": {
      "status": "inside",
      "bounds": {
        "processGain": {
          "min": 0.001,
          "max": null,
          "unit": ""
        },
        "deadTime": {
          "min": 0.001,
          "max": null,
          "unit": "s"
        },
        "timeConstant": {
          "min": 0.001,
          "max": null,
          "unit": "s"
        },
        "overshootBudget": {
          "min": 0.1,
          "max": 200,
          "unit": "%"
        }
      },
      "outside": [],
      "model": null
    },
    "computedAt": "2026-09-24T14:05:11.003Z",
    "inputs": {
      "processGain": 1.5,
      "deadTime": 0.1,
      "timeConstant": 1,
      "overshootBudget": 25
    },
    "seed": null,
    "elapsedSeconds": 0.003
  }
}

{
  "detail": [
    {
      "type": "float_parsing",
      "loc": [
        "body",
        "inputs",
        "processGain"
      ],
      "msg": "Input should be a valid number, unable to parse string as a number",
      "input": "abc"
    }
  ]
}

An input sent as text. Nothing was counted.

{
  "detail": "More than 120 requests in 60 seconds from this address.",
  "errorKind": "rate_limited",
  "limit": 120,
  "windowSeconds": 60,
  "upgradeUrl": "https://rftools.io/pricing/"
}

Too many requests in a minute, with Retry-After.

Same inputs on the calculator pagePID Controller Tuning (Ziegler-Nichols), at these values