What sort of control mechanisms (i.e., PID controllers) do modern FBW aircraft use?
As others have commented, you've underestimated how fast an Intel 80186 found on an early A320 is. Its clock rate is 6 MHz to 25 MHz. While it's very slow compared to what's available nowadays, that's still 6-25 million cycles a second.

The A320's first flight was in 1987, here's what the Air Force Flight Dynamics Laboratory had to say about the subject (Theory of Closed Loop Flight Control Systems) 20 years earlier:


  The electronics field has miniaturized even more so. Microelectronic circuits enable mixing, blending, voting, and, in general, response shaping to be done on a relatively small number of easily replaced cards. Metal oxide semiconductor (MOS) techniques decrease the size of these components by several orders of magnitude. Using these techniques it is now possible to redundacize electronics at a functional module-level with a resulting decrease in size, weight, and cost and a net increase in system reliability.


For modern aircraft, say an A380, we're looking at 40-98 MHz computers:


(Cranfield University, 2016)