Interesting that you mention P-factor. It has been so long but I do recall having a trim or 'offset' for the rudder. Otherwise placing the rudder amidships (rudder angle = 0 degrees) did not result in moving straight ahead. However I don't recall at what "bells" (speeds) it applied.
Maybe a bit more about controls on submarines. Most of this is for the LA and Trident boats (before the fancy Seawolf / Virginia).
Our small cable/pulley based GA planes have no indicators on the panel for the commanded and actual position of the control surfaces. Really no need actually. Its mechanically direct. You can also see them. More importantly, things happen relative fast and you can immediately feel them since the plane is incredibly light (think 1-2 tons).
On a submarine its much different as
@bflynn said. First off you can't see outside. Next, you aren't really going that fast and often purposely very slow. The Trident was displacing around 18,000 tons!!!
So the guys at the controls sit and watch the course, depth, ships bubble (our pitch angle). The is another inclinometer for roll (forget the name) but that (for us, back then) was only to really measure rolls due to wx.
Our version of AP had 3 modes. Off (all manual). Auto (ships computer ran that surface). And a very simple assist that indicated what the computer would do but did not control the surface. Our "AP" was decent and straight and level. But the start of a turn, end of turn, start of depth change, and end of depth change has some oscillation. In a sense the PID controller probably need a bit more Ki and Kd (integrator and derivative). I am sure its all worked out now
But back then it lacked that "experience" element that knew to lead and check things for efficiency.
Example of a course change:
They turn the rudder via a yoke not a lot different than in our old 182. Note: NO PEDALS!!! There is no force feedback as your turn the yoke. A good helmsman will use enough rudder to get the big old cow swinging in the right direction - but not cavitate the rudder! You have to keep the rudder in that position to keep turning. That same good helmsman will have done that turn and speed combination before (or something like it) and will know to "check" with opposite rudder at the right time and amount to not overshoot course. Just zeroing the rudder in advance will take forever to swing it onto the final course. Plus the turn rate will not be consistent. Who decides the course change? Usually the Officer Of The Deck (OOD) will command the new course and rudder angle to use for the turn (exception: see Conning Officer below). They know the turn rates of the boat at different bells (speeds) so kind of like knowing standard turns. Its been a lot of years so some verbiage is probably wrong but if often goes like this. They are about 5-15ft from each other so no headset is needed:
OOD: "Helm right 20 degrees rudder, make your course 112"
Helm: "Right 20 degrees rudder, make my course 112 helm aye"
...a bit later...
Helm: "Rudder is right 20 degrees".
DOOW: Watches over the helm to make sure he maintains depth and the proper turn.
....a bit later...
Helm: "Passing 0 degrees to the right"
....a bit later...
Helm: "Steady on course 112".
The jargon is a bit different than aviation but very heavy with exact repeat backs.
Depth changes and bubble are somewhat similar but the DOOW (Diving Officer of the watch) gets very involved as its his responsibility.
OOD: "Dive 10 degrees up angle, make your depth 200ft"
DOOW: "10 degrees up angle, make depth 200ft, Dive aye"
The DOOW then privately "instructs" the helm (who also controls the front planes) and the stern planes station (pitch) to make the changes need.
Both stations lead with enough control changes to get it moving but in this case the stern planes can zero out once the angle is established (like a plane).
eventually the angle is reduced and fine depth control is established an you will hear something like:
DOOW: "Officer of the deck 200ft" {or something like that}
...that callout is very important when transitioning to periscope depth where fine adjustments will be needed.
The Conning Officer:
Underway the the Officer Of The Deck is actually standing two roles. The Officer Of The Deck and the Conning Officer. The Conning Officer is who orders turns, depth changes, etc. In movies (and in real life) the Captain will typically take the 'Con' during battle stations or drills or if he is generally displeased with what it going on regarding safety, etc. Then you might hear this interesting exchange:
Captain: "I have the con"
OOD/CON: "Captain has the con, I have the deck"
Helm: "Captain has the con helm aye"
An even more interesting exchange (a very unhappy Captain) can do this (I was there it was not pretty to hear)
Captain: "Lt ABC I relieve you, I have the deck and the con"
OOD/CON: "Captain has the deck and the con, I stand relieved"
Helm: "Captain has the con, helm aye"