Short PID 101:

PID = Proportional Integral Derivative

Proportional control means that the correction is proportional to the
error.

Correction = -1 * P * Perror = -1 * P * (Setpoint - CurrentValue)

Integral control means that the correction depends on the present and past
error. More exactly, the Integral of the error, which can be represented
as the sum of the errors so far. If Ierror0 is the error at the previous
measurement time and error is the Ierror at the present time then:

Ierror = Ierror + (Setpoint - CurrentValue)
Correction = -1 * I * Ierror

Derivative control means that the correction depends on the change rate of
the error. Therefore if Derror was the previous error:

Derror = Derror - (Setpoint - CurrentValue)
Correction = -1 * D * Derror
Derror = (Setpoint - CurrentValue)

By putting it all together and everything in the middle of a loop that
repeats at fixed time intervals:

Ierror = 0 // initial error integral is zero
do
  // Read input value
  CurrentValue = Read(Measure Feedback Value)

  // PID process
  Error = (Setpoint - CurrentValue) // the error
  // Integral term
  if(Error != 0)
    Ierror += Error // integral
  else
    Ierror = 0; // reset integrator if the we have reached the setpoint
  // Limit integral value
  if(Ierror > MAXIERROR) // limit integrator
    Ierror = MAXIERROR;
  if(Ierror < -1 * MAXIERROR)
    Ierror = -1 * MAXIERROR;
  // Derivative error term
  Derror -= Error // derivative (differential)
  // Limit it
  if(Derror > MAXDERROR)
    Derror = MAXDERROR;
  if(Derror < -1 * MAXDERROR)
    Derror = -1 * MAXDERROR
  // Compute PID correction
  Correction = -1 * ( P * Error + I * Ierror + D * Derror )
  Derror = Error  // store the 'previous' derivative for the next pass

  // Output correction
  Write(Correction)

while forever

Note: All the variables are signed.

Note2: The integral term will cause a lot of trouble. It is customary to
zero it whenever Error is 0 and to limit its growth both in the high
direction and in the low direction, to reduce trouble. The Derror is also
limited sometimes in the high and low direction to avoid giant kicks when
the input Setpoint changes by a large value.

Note3: PID tuning is a course by itself. It requires that the user
understand the mechanical equivalents of inertial mass and friction
losses.

Note4: If PID is applied to a mechanism that has backlash then the loop
will oscillate. Typical example: model servos.

The integral model used here is not very accurate (squares = Riemann sum).
Precision PID would use a better integration method.

Peter

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