The robust performance of traditional partial least squares (PLS) method is poor because the objective function of PLS is to maximize the square of L ₂ norm between the input and output data spaces, and this method is sensitive to outliers but insensitive to local features and causes nuisance alarms. Conversely, the L ₁ norm can perfectly maintain the signal relative size in global statistical features extraction, and its feature extraction results are robust to outliers. Therefore, in this article a novel dual robustness projection to latent structure method, which includes the robustness of feature extraction and regression coefficients, based on the L ₁ norm (L ₁ -PLS) is proposed. Then, more attention is paid to the robustness of both the input and output spaces and the robustness relationship between those spaces in the dual robustness L ₁ PLS method. The corresponding quality-relevant monitoring strategy is also established. Finally, the robustness of the L ₁ -PLS method is illustrated with the Tennessee Eastman process simulation platform. The results show that the proposed method is insensitive to outliers and maintains the signal relative size, and the fault monitoring results are consistent with the actual situation.