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Vaclav Kosar
Software, Machine Learning, & Business

PID Controller: A Simple Control Loop Mechanism

Proportional–integral–derivative controller calculates feedback to reduce the error in the next step.

When to use PID controller?

  • measured value (process variable) is a time series
  • ideal value (setpoint) is known
  • can correct via a feedback in the next steps
  • tuning (training) data is available
  • cannot model of the process
  • in non-linear systems may not work
  • example: cruise control
    • process variable = speed
    • setpoint = ideal speed
    • error = ideal speed - actual speed
    • feedback = gas pedal

What is PID controller?

  • is sometimes called three-term controller
  • tool to stay close to the ideal (control loop mechanism)
  • uses distance from setpoint (error) to produce feedback
    • error = setpoint - process variable
  • defined as sum of 3 terms:
    • proportional term
      • current error
      • corrects for error in the previous step
    • integral term
      • sum of errors till now
      • corrects for error in the same direction in the past
    • derivative
      • current derivative of the error
      • can cause instability and not used often
      • or low pass filtering
      • corrects for sudden change in error
  • mathematical form:
    • error: \( e(t) \)
    • proportional coefficient: \( K_p \)
    • integral coefficient: \( K_i \)
    • derivative coefficient: \( K_d \)
    • time: \( t \)
    • feedback value (control function): \( u(t) \)
    • equation: \( u(t) = K_p e(t) + K_i \int_0^t e(t) dt + K_d \frac{de(t)}{dt} \)


  • grey dots represent setpoint
    • here: constantly zero
  • blue represents original process variable
    • uncontrolled process variable
  • red represents process variable after corrective feedback
    • here: process variable minus feedback
  • try changing the input function
  • Find the demo source below

  sine input function

Self-tuning PID using Kalman filter

  • Kalman filter uses linear relationship between measured values
  • to estimate true values and uncertainty
  • in the paper relationship between the PID parameters defines the Kalman filter tuning

Demo Source Code

Created on 21 May 2021.

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