Optimal Fault Detection for Stochastic Linear Time-Varying Systems by $\chi ^{2}$ Test

This article is concerned with the problem of fault detection (FD) for stochastic linear time-varying systems. The unified framework of the residual generator is constructed by using all currently available inputs and outputs, which can describe the existing observer-based methods and the parity space method. Then, the
$\chi ^{2}$
test is introduced to evaluate the multivariate residual. In this article, the fault detector is said to be optimal if the missed detection rate (MDR) is minimal under a given false alarm rate. In theory, the relationship between the MDR and the residual parameter matrix is analyzed, which reveals how the dimension of a residual affects the MDR for the first time. When the fault amplitude is known, the optimal fault detector is derived, which is infeasible in practical applications. Next, a new recursive and feasible FD method is presented, where the residual parameter matrix gradually tends to the theoretically optimal one. Finally, an illustrative example is provided to demonstrate the feasibility and superiority of the obtained algorithm.