Optimal Fault Detection for Stochastic Time-Varying Systems via Hotelling’s $T^{2}$ Statistic

In this article, the problem of optimal fault detection (FD) is studied for stochastic linear time-varying (LTV) systems based on Hotelling’s$\boldsymbol{T}^{\boldsymbol{2}}$statistic. The considered system involves noise with unknown covariance matrices, which limits the application of traditional$\boldsymbol{\chi}^{\boldsymbol{2}}$-based detection methods. A unified FD framework is proposed to describe model-based detection methods, including the processes of residual generation and evaluation. An optimization objective is formulated to minimize the missed detection rate (MDR) under a specified false alarm rate. To achieve the optimal objective, the fault detector is designed from two perspectives: the dimension of the residual and the residual parameter matrix. First, for a given dimension of the residual, the residual parameter matrix is constructed by maximizing the noncentrality parameter of the Hotelling’s$\boldsymbol{T}^{\boldsymbol{2}}$statistic. Then, the relationship between the MDR and the dimension of the residual is analyzed, revealing that the achievable MDR increases with the dimension. Finally, comparative simulations and experiments are conducted to validate the effectiveness of the proposed methods on the rotary steerable drilling tool system.