Accurate control of magnetic fields is crucial for cold-atom experiments, often necessitating custom-designed control systems due to limitations in commercially available power supplies. Here, we demonstrate precise and flexible control of a static magnetic field by employing a field-programmable gate array and a feedback loop. This setup enables us to maintain exceptionally stable current with a fractional stability of 1 ppm within 30 s. The error signal of the feedback loop exhibited a noise level of 10^-5 A·Hz^-1/2 for control bandwidths below 10 kHz. Utilizing this precise magnetic field control system, we investigate the second-order Zeeman shift in the context of cold-atom coherent population-trapping (CPT) clocks. Our analysis reveals the second-order Zeeman coefficient to be 574.21 Hz/G², with an uncertainty of 1.36 Hz/G². Consequently, the magnetic field stabilization system we developed allows us to achieve a second-order Zeeman shift below 10^-14, surpassing the long-term stability of current cold-atom CPT clocks.