Moving-target detection using frequency-modulated continuous-wave (FMCW) radar systems has recently attracted attention. Detection tasks are more challenging with noise resulting from signals reflected from strong static objects or small moving objects(clutter) within radar range. Robust Principal Component Analysis (RPCA) approach for FMCW radar to detect moving objects in noisy environments is employed in this paper. In detail, compensation and calibration are first applied to raw input signals. Then, RPCA via Gradient Descents (RPCA-GD) is adopted to model the low-rank noisy background. A novel update algorithm for RPCA is proposed to reduce the computation cost. Finally, moving-targets are localized using an Automatic Multiscale-based Peak Detection (AMPD) method. All processing steps are based on a sliding window approach. The proposed scheme shows impressive results in both processing time and accuracy in comparison to other RPCA-based approaches on various experimental scenarios.