Let's try to give a comprehensive answer and stay on topic!
As you said in your question there is one PID per axis. This means 3 PIDs in total: roll, pitch, and yaw.
Also the motors input is a PWM high-level time, typically in the 1000-2000us range. There are 4 motors input: front, back, left, and right. One subtlety of the motors is that there is a value under which the motors stop completely, for instance 1100ms.
There is another variable not mentioned in the question: the throttle.
The first step in my quadcopter project was to tune the throttle. My throttle values were between -100 and 100, so I applied the following formula to each motor: motor = 300 * throttle / 100 + 1550. The constant values are arbitrary and give us a motor value in the 1250-1850 range, other values of the same magnitude would work.
Only then do we apply the PIDs using the motor mix formula. For instance for my +4 quadcopter the formula is:
- Front = Throttle + PitchPID - YawPID
- Back = Throttle - PitchPID - YawPID
- Left = Throttle + RollPID + YawPID
- Right = Throttle - RollPID + YawPID
The PID output has no unit. Choosing the right P, I, and D constants shall give us values which can stabilise the quadcopter. One caveat is that depending on the PID values the motor input can exceed the bounds: it could go over 2000us or under 1100us (which is our example motor cut value). In order to avoid such a situation we can analyse the higher and the lower of our motors inputs and add a constant to get all the motors within bounds.
For instance if the motor mix formula gave us:
- Front = 1900
- Back = 1800
- Left = 2100
- Right = 1700
We would remove 100 from each motor and get:
- Front = 1800
- Back = 1700
- Left = 2000
- Right = 1600
This would give us motors input values within bounds and maintain the difference between motors.
As a side note you said in a comment that the thrust vs PWM input is not linear. This is true but PIDs can drive non-linear systems with more or less success. For quadcopters the consensus is that it works reasonably well :)
If anything was unclear please let me know!