Make a unit test to investigate integral oscillations

Are we able to re-create integral oscillations with a linear pid model and feeback to a linear unit model?

If most of the integral oscillations are interpreted as an oscillating disturbance, then re-tuning the pid-controller cannot completely get rid of the integral oscillations?

There is something unclear about how a linear system can have a standing oscillation with one tuning and then no oscillations with another tuning?

Do you need a nonlinear model to be able to explain integral oscillations? Do we need to make certain assumptions about the disturbance?