We develop an adaptive-optics system based on a Mach-Zehnder radial shearing interferometer with liquid-crystal-device (LCD) phase-shift interferometry(PSI). Using accurate phase calibration and transient nematic driving of the LCD, the developed three-step PSI procedure can be achieved in a time of 5 ms. The proposed Mach-Zehnder radial shearing PSI method reconstructs the phase information using a digital signal processor (DSP). The DSP then computes appropriate control signals to drive a deformable mirror in such a way as to eliminate the wavefront distortion. The current adaptive-optics system is capable of suppressing low-frequency thermal disturbances with a signal-to-noise ratio improvement of more than 20 dB and a steady-state phase error of less than 0.02-π root mean square when the control loop is operated at a frequency of 30 Hz.