Can flight control computers detect and help to avoid pilot-induced oscillations?
PIOs occur because the pilot senses a rapid movement and feels the urge to counter it. Key is the frequency of a few Hertz, so the pilot's reaction kicks in with a phase shift such that it promotes the movement, and low damping such that the oscillation amplitude grows over time. With a working FCS this kind of rapid movement should not happen.

However, once the FCS is not tuned to the situation, it can actually aggravate the situation and promote PIOs. The best known case is probably the 1992 crash landing of Tom Morgenfeld with the YF-22 prototype.


                
            

Here the gains in the FCS were set too high with afterburner on and gear up at slow speed.

Another case happened in the early phases of flight testing the V-22 Osprey. Here two of the three gyros in the control system were incorrectly wired and the pilot made a wrong control input.


                
            

So it is less the ability of the FCS to detect PIOs than its improved damping which never lets a rapid oscillation happen in the first place.