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CALCULATORS / CONTROL SYSTEMS AND DYNAMIC RESPONSE

PID output from error, integral and derivative calculator

One scalar PID output sample with supplied error integral and derivative; not a closed-loop solve.

controls-pid.napkin / editable
✓ checks pass

SCENARIO COMPARISON

Compare the numbers.

This source describes an analytical model. Its quantities and checks are the view.

Scenariooutput
0.55
0.47

All calculated quantities are listed with the inputs. Switch to Source to inspect the equations.

SCENARIOS / CALCULATED VALUESSAME SOURCE

Change an input to see the consequences.

output saturation screen0.55 ✓ passes

BASELINE INPUTS

CALCULATED

output0.55

Edits change the baseline source.

Calculated from your source · not saved
Baseline · Baseline
output saturation screen0.55 ✓ passes
alternative · alternative
output saturation screen0.47 ✓ passes

Illustrative first-order example. Checks are editable requirements, not design standards. The broader topic may require additional equations, data, and validation.

Formula and units

This example uses the following equations. Input names and units match the editable source above.

output
output = kp*error+ki*integratedError+kd*errorRate

Original worked example

These are the baseline values in the downloadable source. Edits above let you test your own values.

Given

kp
2
ki
1 s^-1
kd
0.1 s
error
0.2
integrated error
0.1 s
error rate
0.5 s^-1

Calculated

output
0.55

Assumptions and limits

One scalar PID output sample with supplied error integral and derivative; not a closed-loop solve.

The requirement checks use the limits written in this example. Passing a check does not establish that the model covers every load, material behavior, or operating condition in a real design.

Download this .napkin example ↓

Take it further with your agent

Napkin keeps the inputs, equations, checks and assumptions together in one study. Your agent sets it up; you change the numbers.

Try with Claude ↗

codex 'Use Napkin (https://napkin.engineering/mcp, or the local Napkin plugin) to build a study: PID response exploration. Inputs: plant model; controller gains; sample time; limits. Outputs: closed-loop response; saturation behavior. Model scope: Declared linear or low-order nonlinear plant models. Read napkin_capabilities and the current snapshot. Ask me for the values that matter, label assumptions with units, author .napkin source, validate it and apply it with napkin_source using the current revision, then open the study. Add checks for the limits I supply. Keep unsupported analyses explicit.'

Claude needs the Napkin connector; Codex needs the Napkin plugin. Set up your agent ↗

The broader study

The topic brief below describes the broader scope. The worked example above states the particular calculation it implements.

  • Inputs: plant model, controller gains, sample time, limits
  • Outputs: closed-loop response, saturation behavior
  • Modes: calculate, compare, sweep
  • Drawing: Response and stability plots

EXECUTABLE WORKED EXAMPLE · FIRST-ORDER MODEL · CHECK THE SCOPE & ASSUMPTIONS

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