The picosecond-laser system at Shanghai Astronomical Observatory is analyzed and its performances, such as output powers and output waveforms, under different repetition rates are investigated. A stable laser output with repetition rate of 4 kHz, power of 3 W, and wavelength of 532 nm is realized. The 4 kHz repetition rate ranging system is established and the 4 kHz repetition rate satellite laser ranging is achieved. Compared with those for 1 kHz repetition rate, the amount of observation data and the precision of normal point data are increased by approximate 2.62 times and 1.62 times, respectively, which indicating the performance of satellite observation is improved. Moreover, the average spin rate of Ajisai satellite obtained by the Lomb algorithm is 0.4234 Hz with an accuracy of 0.0054 Hz, higher than those from the 1 kHz laser observation, and the measurement accuracy of satellite spin is also improved.