Our PIDs act on angles and angular rates. You can think of a simplified toy example of a controller that acts to stop a rolling ball at a certain distance, only knowing the velocity and position of the system.
Taking an example case of the diagram above, the ball is moving to the right at 5m/s. If the ball is supposed to be at rest at position at position 0, then it has to stop (v=0) after 8m. Assuming we can only provide a constant force, this can be solved purely analytically through the equations of motion:
$$ v^2=u^2+2as $$
Where:
v=final velocity (0m/s)
u=initial velocity (5m/s)
s=displacement (8m)
All units are SI, so no conversions are needed, and we can solve for acceleration
$$ 0^2=5^2+2*8a \newline -25=16a \newline a=-\frac{25}{16} $$
Since F=ma, we can evaluate that the force needed to stop the ball at 0 is the ball’s mass in kg multiplied by the desired acceleration. If the ball has a mass of 16kg, the force needed would be 25N to the left (or -25N)
This system works to stop the ball precisely where needed, but one key limitation here is that it assumes there are no external forces acting on the system. If this whole system was on an incline, for example, gravity would cause an additional leftward acceleration, that would move our net acceleration to a lower number. For example, if our system was at a 5 degree incline, the net acceleration would be
$$ -\frac{25}{16}+9.81*sin(5)=-0.7075m/s^2
$$
Plugging this acceleration back into our equation gives us an updated stopping distance
$$ 0^2=5^2+2*-0.7075s \newline s=11.01 $$