Data-driven Fault Detection with Orthogonal Projection

This work proposes a data-driven fault detection approach for unknown linear systems with an orthogonal projection. The behavioral trajectory subspace is defined as the column space of a Hankel matrix constructed from nominal system data. Any nominal input-output trajectory lies within this subspace, and the rank of the Hankel matrix determines the dimension of the nominal system behavior. By performing a singular value decomposition (SVD) on the Hankel matrix, the orthonormal bases for the trajectory subspace and its orthogonal residual subspace are extracted, enabling complete decoupling of normal and abnormal behavior. Based on this SVD, a self-adjoint and idempotent projection operator is designed to generate a performance-based residual signal. Then, a residual evaluation function is developed to quantify the performance degradation by measuring the distance between the sampling input-output data and the nominal trajectory subspace. A system multiplicative fault is detected online when this residual evaluation signal exceeds a tolerable threshold defined by the gap metric between the nominal and uncertainty subspaces. The effectiveness of the projection-based detection results is validated through experimental applications on a three-tank platform.