An Advanced Data-Driven Fault Diagnosis Framework based on Symmetric Positive-Definite Matrices for Industrial Processes

This dissertation is devoted to the systematic investigation of symmetric positive-definite matrices from the perspective of Riemannian geometry for the purpose of fault detection and diagnosis. Meeting the essential requirements in industry, an advanced unified data-driven fault diagnosis framework based on symmetric positive-definite matrices in the Riemannian manifold is developed as an alternative to state-of-the-art schemes established in the Euclidean space. Initially, the data-driven fault diagnosis framework utilising symmetric positive-definite matrices is introduced. Leveraging the mathematical tools from the field of Riemannian geometry and the geometric properties of symmetric positive-definite matrices, an alternative fundamental framework for numerous data-driven fault diagnosis schemes is provided. Encompassing the realisation of fault diagnosis schemes with the core ideas of multivariate analysis-based, machine learning-based and model-based approaches, the novel framework is contextualised with control-theoretic interpretations. An intuitive model based on the Riemannian mean constitutes a basic fault detection scheme and serves as a key component in the subsequent development of advanced fault diagnosis. It is demonstrated and analysed on a simulated numeric example and experiment data obtained from the Three-Tank-System benchmark. Based on the prior introduction of the framework and the intuitive model, machine learning-based realisations of the framework are presented against the background of multimode process monitoring. In particular, a clustering and a classification method are studied using symmetric positive-definite matrices and Riemannian geometry. To enable a flexible and scalable process monitoring, a novel piecewise multi-geodesic model incorporating just-in-time learning to be applied to unknown operating points is developed subsequently. Finally, the modelling of a linear time-invariant system based on symmetric positive-definite matrices is proposed. The solution of the matrix differential equation provides the time-dependent state matrix representing the system eigendynamics. With the help of parallel transport, the state matrices at different time instances are made comparable and evaluated for detection purpose. The options for data-driven implementation are provided and discussed as well.