When there is an error (= setpoint - output), the error is multiplied by I term and added to the control output, and the term is accumulated (integral) in internal memory and will be applied in the next control output sampling, whereas the last error for P term will not be remembered. that is the fundamental difference between the P and I.
Assuming that there is an error E. A PI control apply some output to correct it. But in the next sample, it is found that the E remain unchanged. The P term will apply the same output since it is based only on the sampled error, but the I term will apply double the last control output, since the last E x I is "integrated" and added to the current "E x I". And if the next sample it finds that the E remain the same (i.e. control output did not manage to result any changes to the error), then additional output will be applied due to the cumulative effect of the I term until the output "saturated" or the "limit" is reached.
Hence, if the sample time of your control system is small, but the system inertial is large (meaning system changes slowly with respect to change of control output) the I term can accumulate quickly and drive the control output to saturation and the system can fluctuate wildly, so use it with caution. It is not always necessary to have very short cycle time for digital control, it really depends on the dynamics of the system. If the system is slow then the system does not change rapidly in response to change of control output then a longer sample period may be more beneficial since it allows time for the system to react to the output change.
