That's the "power stage", in integrated H bridge.
The control is typically done in a microcontroller that has a suitable PWM module & analog / digital I/O for motor voltage / current feedback, the position feedback device and control system.
Other than that, you are looking at a ready-made servo controller, but decent dual-loop ones for CNC or robotics tend to be relatively expensive.
Those hall transducers are pretty good equivalents to a industrial CNC grade optical encoder - eg. if you are using 10mm pitch leadscrews with a 3:1 reduction from the motor, which is pretty typical, one micron of movement is around 1/3000 of a motor turn. A 2000 line optical encoder (so 8000 counts) gives better than that & the hall encoders are four times that resolution.
With 5mm pitch screws, you double the mechanical resolution for the same counts.
Or you could use something like this, an economy grade optical encoder; connect it to the driven end of the leadscrew with either a bellows or oldham coupling. With a 5mm pitch screw, that would give (5000 / 2400) roughly 2 micron increments, or 4 micron with 10mm pitch screw.
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If connected to the motor before the belt reduction, you would get better than 1um counts with 5mm pitch, but with a slight loss of absolute position due to give in the belt.
For really high resolution, you are looking at something more like these - but don't get carried away, most cheaper ballscrew, ballnut and bearing combinations cannot get all that near 1um accuracy to start with.
Note that "brushless DC" motors and drives have near enough completely replaced conventional DC drives in machine tools and robotics now.
A new, good, servo drive to suit those is relatively cheap - and the motors inherently have internal position sensors, which are often made available on the drives to give position feedback to the CNC unit.
Example:
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