Electrical – Doubt with the conversion from output PID signal to PWM on inverted pendulum control problem

Your magnet approach is the only one that is likely to work, but there are a number of gotchas. Be aware that stabilizing an attractive magnetic setup (rather than a repulsive one) is not for the faint of heart, and especially not for beginners. The problem is that the attractive force can get very high very quickly if the two magnets get close together, so the magnets tend to "snap" together faster than your system can respond.

First, there is no need to make the magnet currents independent. I'd advise making the two linearly dependent. If the lower coil current varies from 0 to 1 amp, make the top coil vary from 1 to 0 amp, with the sum constant. This is hardly optimal in terms of performance, since the two repulsive forces will not be linearly related with respect to displacement from the center position, but it will be far easier to control and stabilize. Think ahead - upgrading to independent loops will be a no-brainer for further credit.

Second, your encoder scheme, while OK in principle, is unlikely to work well. The problem is that, unless you have access to a really high-resolution encoder your angle measurements will be too coarse to give the sort of accurate position information you need. Not only that, but measuring angular velocity (the D in PID) will be very coarse in granularity, and for low rates will have a very slow update rate. Get hold of your encoder specs and pendulum arm length, and calculate just what your vertical quantization is. You cannot do better than this.

Third, you'll be able to do a simple open-loop calibration of your system simply by putting differing currents into your coils and seeing where the moveable magnet gets to, although you'll have to be careful about damping out residual motion. This is probably a better bet than trying to calculate position from first principles. See https://en.wikipedia.org/wiki/Force_between_magnets for a discussion of inter-magnet forces. Be aware that these results are very sensitive to small changes in geometry, including the size of your magnet cores.

Do you connect the PID such that the controller sets the current to a certain level, or do you connect the PID such that the controller adds/subtracts current to the instantaneous current to reach the desired level?

There are different PID algorithms that do either of those two things. The latter type is called a velocity algorithm (because it controls the velocity of a hypothetical mechanical valve motion rather than the position of the valve). Since the valve performs the integration there were once some advantages to this method- for one thing it automatically stops integrating when the actuator hits the limit, which helps with overshoot due to integrator windup. However, if you are using derivative control it must be double differentiated- which tends to make that contribution very noisy.

By the way, controlling the current through the resistor is usually not a good idea. What you want to do (typically) is to control the power. The resulting nonlinearity from voltage control will usually make the controller hard to tune well over a range of setpoints (it will either be too sluggish or not stable enough).