What sort of control mechanisms (i.e., PID controllers) do modern FBW aircraft use? It's a well-established fact that aircraft like the Boeing 777, and all Airbus passenger aircraft that were designed after the A300, feature a digital fly-by-wire control system. Considering the Boeing 777 as an example, the pilot flying applies force onto their yoke and one of the flight computers (I'm not aware of the architecture(s) of Boeing FBW systems) interprets the input and changes the control surfaces of the plane accordingly.

I am wondering what control paradigms are used by Boeing and Airbus to translate pilot settings (i.e. trim), accelerometer and gyroscope data from the IMU, and feedback from the aircraft's control surfaces to re-adjust said control surfaces.

Would a simple PID controller suffice? If so, how would a single PID system be able to manage feedback data from different sources (i.e. how could a PID system handle feedback from a gyroscope and Airbus side stick at the same time)?

I understand that the accuracy of PID systems are fully dependent on the feedback they receive, so I would assume that software methods of error correction like Kalman filter would be employed. However, how can computer hardware from the 80s (such as the Intel i386) be able to sample data quickly and compute Kalman filter within acceptable response times?

I appreciate all answers, and I apologize if my questions require analyzing proprietary technical documents.
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)