page_title: PID controller
section_title: PI controller
hierarchical_section_title: PID controller / Modifications to the algorithm / PI controller
context_page_description: A proportional–integral–derivative controller is a control loop mechanism employing feedback that is widely used in industrial control systems and a variety of other applications requiring continuously modulated control. A PID controller continuously calculates an error value as the difference between a desired setpoint and a measured process variable and applies a correction based on proportional, integral, and derivative terms, hence the name.
  In practical terms it automatically applies accurate and responsive correction to a control function. An everyday example is the cruise control on a car, where ascending a hill would lower speed if only constant engine power were applied. The controller's PID algorithm restores the measured speed to the desired speed with minimal delay and overshoot by increasing the power output of the engine.
  The first theoretical analysis and practical application was in the field of automatic steering systems for ships, developed from the early 1920s onwards. It was then used for automatic process control in the manufacturing industry, where it was widely implemented in pneumatic, and then electronic, controllers.
context_section_description: A PI controller (proportional-integral controller) is a special case of the PID controller in which the derivative (D) of the error is not used.
  The controller output is given by
  where  is the error or deviation of actual measured value (PV) from the setpoint (SP).
  A PI controller can be modelled easily in software such as Simulink or Xcos using a "flow chart" box involving Laplace operators:
  where
   = proportional gain
   = integral gain
  Setting a value for  is often a trade off between decreasing overshoot and increasing settling time.
  The lack of derivative action may make the system more steady in the steady state in the case of noisy data. This is because derivative action is more sensitive to higher-frequency terms in the inputs.
  Without derivative action, a PI-controlled system is less responsive to real (non-noise) and relatively fast alterations in state and so the system will be slower to reach setpoint and slower to respond to perturbations than a well-tuned PID system may be.
caption: Basic block of a PI controller