Structured Joint Sparse Principal Component Analysis for Fault Detection and Isolation

Abstract

In order to improve the performance of fault isolation and diagnosis of principal component analysis (PCA) based methods, this article proposes a novel fault detection and isolation approach using the structured joint sparse PCA (SJSPCA). The objective function involves two regularization terms: the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$l_{2,1}$</tex-math></inline-formula> norm and the graph Laplacian. By imposing the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$l_{2,1}$</tex-math></inline-formula> norm, SJSPCA is able to achieve row-wise sparsity, and introducing the graph Laplacian term can incorporate structured variable correlation information. The row-sparsity property of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$l_{2,1}$</tex-math></inline-formula> norm ensures that the score indices associated with normal variables approaching zero and the graph Laplacian constraint helps the isolation of correlated faulty variables. Once a fault is detected, a two-stage fault-isolation strategy is considered and a score index is calculated for each variable. It is proved that the proposed two-stage strategy is capable of isolating faulty variables. The improved fault-isolation performance of SJSPCA is illustrated by a simulation example and a gas flow fault observed in an industrial blast furnace iron-making process.