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.