In process industries, multivariate statistical process monitoring (MSPM) has become an important technique for enhancing product quality and operation safety. Among the family of MSPM methods, principal component analysis (PCA) may be the most commonly used one, due to its capabilities of dimensionality reduction and highlighting variation in a data set. However, the performance of a PCA model often degrades significantly when there are gross sparse errors, i.e. outliers, contained in the training data, since PCA assumes that the training data matrix only contains an underlying low-rank structure corrupted by dense noise. In this paper, a robust matrix recovery method named stable principal component pursuit (SPCP) is adopted to solve this problem, based on which a process modeling and online monitoring procedure is developed. Such a method inherits the benefits of PCA while being robust to outliers. The effectiveness of the SPCP-based monitoring method is illustrated using the benchmark Tennessee Eastman process.