### Working

As we know that, the proportional controller tells us how far to move, to achieve zero error, whereas integral controller tells us how fast to move to achieve zero error. The proportional controller cannot guarantee zero error in case of a transient change in load dynamics of the system, whereas integral control accumulates positive and negative errors and tries to eliminate steady state error.

Integral action is provided by summing the error over time, multiplying that sum by a gain, and adding the result to the present controller output. You can see that if the error makes random excursions above and below zero, the net sum will be zero, so the integral action will not contribute. But if the error becomes positive or negative for an extended period of time, the integral action will begin to accumulate and make changes to the controller output.

Mathematically it can be represented as,

Where,

- P(t) = Controller’s output
- KI= Integral gain usually motioned in terms of integral time (1/Ti)
- ep (t)= Desired Value of controlled variable – Measured Value
- dτ = Tiny slices of time
- P(0) = Initial controller’s output

If we differentiate above equation then,

Above equation shows that when an error occurs, the controller begins to increase or decrease its output at a certain rate that depends on the size of the error and the integral time constant. If the error is zero, then controller output will not change. If there is a positive error, the controller output begins to ramp up at a rate determined.

### Applications

There are no specific applications where integral control used individually since integral controller alone will cause transient overshoot and which may result in actuator saturation (means actuator cannot be operated beyond this limit).

### Advantages

- Offset error can be eliminated.
- Overall system stability increases.

### Disadvantages

**Reset Windup**: Due to a sudden change in desired value, there is a huge accumulation of error which will result in high controller output, but as we know that actuator can not react beyond a certain limit, therefore there is no significant change on system’s output. Such a scenario where the action of the controller not able to wind up error due to frequent resetting of actuator known as “Reset wind up” condition.

- Phase lag added, which will affect on overall settling time of the system.