IV.2.6 Insight into the cause of offset

The existence of offset in a proportionally controlled process indicates that the input manipulation has reached some sort of saturation before the controlled output could even reach the setpoint. As a result, manipulated input no longer changes even if the error between setpoint and controlled output is non-zero. The above fact can be demonstrated mathematically as follows.

The equations of P and PI controllers are:

	(IV.35)
	(IV.36)

The gradient of the dynamics of above control actions can be derived by simple differentiation,

	(IV.37)
	(IV.38)

The above equations show that in case of P controller gradient of control action becomes zero if there is no change in the value of error. In other words, when the error dynamics reaches a saturation point, the control action reaches saturation too. Mere existence of non-zero error does not yield any change in control action. On the other hand, gradient of control action in a PI controller not only depends upon the gradient of error dynamics but also upon the absolute value of the error at that moment of time. Hence, until the error and its gradient both are zero, the gradient of control action will not be zero. This fact ensures offset-free control action by PI controller.