"How do you test a yaw damper system? "
Are you asking about testing for the initial certification of an aircraft type ...or simply for maintenance purposes "cwatters"?
Modern aircraft systems have automatic self testing/fault reporting as well as pushbutton testing. Yaw Damper computers monitor themselves and associated hardware for correct operation from power up.... and then store a list of faults internally and/or send the list to a Central Maintenance Computer. Also, the YD computer(s) can be made to run through a test routine with the push of a button (or two or three). For example, on a Boeing 767, there is a test switch in the cockpit for the L & R Yaw Damper Systems. You power up the Inertial Reference and hydraulic systems and simply toggle the test switch and watch the rudder cycle back and forth (there is a CRT (TV screen) in the cockpit which shows, symbolically, the movement of the rudder).
Cheers.
Q.