Take the following with a pinch of salt as it's based on a ttrs with mag ride and the sports button. An RS3 valve control will be similar...... (no mag ride)
It may not be exactly correct, but it's an example of how it may be connected/ work.
N321 on the left is the solenoid that controls the exhaust flap via vacuum.
It's got a switched ignition feed permanently, and the Ecu controls the ground side.
So the Ecu controls the flap by giving N321 a complete circuit ( by ground) or not.
The switch just changes the mapping for the flap so it's a bit more vocal in certain revs/speeds/gears. ( it's not necessarily 100% in all gears , open in S)
The switch E541 goes to the Ecu, and the latching circuit/ led in the switch L156 is taken care of by the Ecu.
K91 and all the stuff on the right relates to mag ride. ( on a TTRS)
Legend
E461 Front Center Console Control Head
E541 Sport Program Button
G76 Left Rear Level Control System Sensor
G77 Right Rear Level Control System Sensor
G78 Left Front Level Control System Sensor
G79 Accelerator Pedal Position Sensor
G185 Accelerator Pedal Position Sensor 2
G289 Right Front Level Control System Sensor
J250 Electronic Damping Control Module
J623
Electronic Control Module
K91 Automatic transmission Sport Mode Indicator Lamp
L156 Switch Illumination Bulb
N321 Exhaust Door Valve 1
N336 Left Front Damping Adjustment Valve
N337 Right Front Damping Adjustment Valve
N338 Left Rear Damping Adjustment Valve
N339 Right Rear Damping Adjustment Valve
S Fuses