An auto-focusing system based on a step motor (without position detection sensor) and image quality feedback for an astronomical telescope was introduced, which consists of a step motor position open-loop control subsystem and an image definition evaluation subsystem. Based on the characteristics of astronomical images, the Ful Width at Half Maximum( FWHM)of point source images of stars was used as image definition evaluation parameters. In order to achieve high efficiency of focusing process, the error of the position control system was analyzed. Then, the focusing equivalent, focal depth, system backlash and the step pulse equivalence were measured to correct the system position errors effectively. By analyzing and measuring the correlation between temperature change and image quality change, the qualitative relationship between focusing direction and temperature change was obtained. Moreover, the focal depth edge positioning policy of the focal plane was designed and a fast process focusing was achieved. The system has fault-tolerant function and self learning ability, so it shows higher levels on automation and intelligence. The design above mentioned was applied to a Ground-based Wide-Angle Camera(GWAC) array of SVOM astronomy satellite. Measurement results and astronomical observations show that positioning accuracy of the auto-focusing system is ±3.0 μm and the focusing accuracy is 0.05 pixel. As it can obtain the best image quality only by one focusing process, it has higher focusing efficiency while containing stable running.