With engine control, is the engine speed governed by power settings or pitch settings? I have always assumed that governor settings would be controlled by fuel input. In effect, if you go from cruise to climb, engine senses rev change and automatically inputs more fuel.
As with any other piston engine, engine speed and propeller RPM are tied together. But I think you're asking more than that!
The FADEC controlled manifold pressure (via turbo wastegate), fuel flow (via injector timing) and propeller pitch (via oil valve). From memory there were about sixty inputs into the FADEC to tell it what was going on in & around the engine.
The power lever angle is resolved into an electrical signal, so the FADEC knows what power is being requested. Each power setting is associated with a particular RPM, so the propeller control valve would move to change blade angle to change RPM to match the target value. Entering a climb, reducing speed, would cause the FADEC to command a smaller blade angle to maintain RPM. That doesn't set power however. The FADEC could set power independently of RPM, by adjusting MAP and injector timing.
Fuel flow would be set by the FADEC according to a specific schedule - obviously depending on inlet manifold temperature & pressure, fuel type (diesel, JetA, JetA1 etc), and many other variables. The FADEC would adjust fuel flow and manifold pressure to maintain the commanded power setting at any given TAS, altitude and RPM.
It worked very much more smoothly than traditional engines with three levers.
I once had a propeller control valve malfunction in flight. RPM was cycling between about 1200 and 2300 RPM, with a period of 20 seconds or so. At first I thought it was a sputtering engine, but power output was rock steady and there was no yaw. Strange feeling! But it illustrates the relative independence of RPM and power output in these engines.
Obviously I would be more impressed by an engine that didn't fail at all. But, given that was early days of the Thielert engine, things have improved since. It is IMO the way of the future.