Wouldn't it be better to model imperfections separately rather than rely on PID oscillations? I can see how this can easily fail and cause some dramatic and unnatural camera movements.
It was, like, 15 minutes of work to find a range of acceptable PID values, then generate a random variable from a normal distribution with a mean at the expected value (we just hard-trimmed the random value at 1 std deviation, because we didn't want a 99th percentile result to cause craziness).
Other than the hilarity that ensued while I was playing around with PID values to find the acceptable ranges, I can't recall ever seeing a bug that was problematic from this, so I didn't really give it much thought. Honestly, there were much bigger things to work on.
The nice thing about this system though, is it created "personality" for the camera. If a virtual camera operator had a lower D-value, they'd overshoot just a little bit, semi-consistently. Another one might have a slightly lower I-value, and end up just a little bit further back in the tracking.
It was a super quick way to give a little bit of organic variance to the camera operation, so it didn't feel "samey" every time.
Generally though, It would be pretty trivial to add noise for this stuff. You would probably want some noise on the orientation if you wanted to model more of a camera shake. Perlin noise is my go to if I want to introduce randomness to positions really quickly.