6.3 Integral Control
The contribution from the integral term is proportional to both the magnitude of the error
and the duration of the error. Summing the instantaneous error over time (integrating the
error) gives the accumulated offset that should have been corrected previously. The
accumulated error is then multiplied by the integral gain and added to the controller
output. The magnitude of the contribution of the integral term to the overall control
action is determined by the integral gain, Ki.
The integral term is given by:
Where
•
D
out
:
Derivative output

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•
K
d
:
Derivative Gain
, a tuning parameter
•
e
:
Error
=
SP
−
PV
•
t
:
Time
or instantaneous time (the present)
6.3.1 Limitations
Windup
A basic knowledge of the concept of windup is useful before describing a specific type.
Windup is defined as the situation when the feedback controller surpasses the saturation
(i.e. maximum) limits of the system actuator and is not capable of instantly responding to
the changes in the control error. The concept of the control variable reaching the
actuator’s operation limits is reasonable considering the wide variety of operating
conditions that are possible. When windup occurs the actuator constantly runs at its
saturation limit despite any output the system might have. This means that the system
now runs with an open loop instead of a constant feedback loop.
Integrator Windup
The most common type of windup that occurs is integrator windup. This occurs when the
input into the system receives a sudden positive step command and causes a positive error
when the system first responds to the actuator. If the rate of integration is larger than the
actual speed of the system the integrator’s output may exceed the saturation limit of the
actuator. The actuator will then operate at its limit no matter what the process outputs.
The error will also continue to be integrated and the integrator will grow in size or “wind
up”. When the system output finally reaches the desired value, the sign of the error
reverses (e.g.
) and causes the integrator to “wind down” until things return
back to normal. Through the wind down the control signal is still maximum for a long
period of time and the response becomes delayed. The integrator takes a long period of
time to fully recover to the operating range of the actuator. Integrator windup may occur
from large set point changes, significant disturbances, or equipment malfunctions.

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The illustration is a flowchart showing the specific steps that take place through the
integrator controller. It shows the input (
) and output of the system (
) , along with
the integrator, the actuator, the system, and sensor involved in the process. The sigma
used in each flowchart is used to represent the summation of all variables inputed to it.