# Is the integral in a PID controller definite or indefinite?

Would the integral calculated by the PID controller be considered definite or indefinite?

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According to the definition given in this WP article, the relevant integral is of the form

$$\int_{0}^{t}{e(\tau)}\,{d\tau}.$$

If this is regarded as a function of $t$ (say in a theoretical treatment), it would be an indefinite integral; however, your question concerns the integral calculated by the PID controller, and this is a definite integral for each specified value of $t$.

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On the other hand if it is a discrete time controller, you first filter it out so $e(t)$ gets convoluted with a low-pass filter and then due to saturation problems you might wish to make the $I$ parameter gain dependent. If the error is too large you decrease the $I$ parameter of the controller etc.