How Dynamics Improve Fault Detection: A Gaussian LTI Case

Fault detection has witnessed a rapid development and many approaches have been proposed in recent decades. However, many of them are based on heuristic solutions that do not check the applicability and optimality of the resulting fault-detection systems. This paper starts from a hypothesis test that subsumes all fault-detection problems, based on which a unified optimisation problem is formulated. The resulting optimal solution defines the deemed-normal region of the system. It is proven that dynamic information shrinks the deemed-normal region and improves detection performance in Gaussian LTI systems. The theoretical results are verified on a simulated three-tank system. Compared with the static method, the fault-detection rate (FDR) of the dynamic method based on a Kalman filter increases from 96% and 56% to 99% and 100% for two faults, respectively, while the false-alarm rate (FAR) decreases from 7.38% and 0.88% to 0.75% and 0.63%. This paper provides a theoretical foundation for understanding fault detection for Gaussian LTI systems, and avoids using any heuristic proposals and solutions for the problem of fault detection. The rigorous justification of the well-known fact that incorporating dynamic information improves fault-detection performance implies a roadmap towards advanced methods for more complex cases. In addition, the analysis including the idea of a deemed-normal region has the potential to be extended to a physically meaningful framework for performance assessment of fault diagnosis and fault-tolerant control systems.