To address the shortcomings of existing frequency-stabilized lasers with low frequency stability and short stabilization periods, a laser frequency stabilization system is designed based on the first-harmonic saturable absorption scheme. The saturation absorption method was used to obtain the saturation absorption spectra of ⁸⁷Rb D₂ line F = 2 and ⁸⁵Rb F = 3 in the rubidium gas chamber. To obtain an error signal reflecting the changes in laser frequency, modulation and demodulation techniques were applied. Additionally, automatic locking of the laser frequency was achieved by keeping the laser temperature unchanged and controlling current. In addition to the frequency stabilization scheme, the study designed a differential current/voltage conversion circuit, spectral conditioning circuit, and low-noise negative-feedback composite amplifier constant-current source circuit based on the Howland structure. These components facilitate the spectral signal conversion, demodulation, and drive and feedback control of the LD. Furthermore, a peak detection procedure based on minimum convolutional root-mean-square is designed and developed using QT uplink and data acquisition card, incremental PID, and other procedures. The experiment used the saturated absorption spectrum of the ⁸⁷Rb D₂ line F=2 in the cross peak $\mathrm{C}{\mathrm{O}}_{\mathrm{3}}^{\mathrm{2}-}$ to stabilize the frequency of a 780 nm semiconductor laser. The results demonstrate successful power-on auto-locking with frequency stability analysis yielding a second-level stability index of 1.64×10^-10.