Statistical analysis of a two-ellipsoid overlap test for real-time failure detection

Real-time failure detection for systems having linear stochastic dynamical truth models has been posed in terms of two confidence region sheaths in [1]-[3]. One confidence region sheath is about the expected nominal no-failure trajectory; the other is about the Kalman estimate of the state(s) being monitored for failures. The implementation of a necessary and sufficient test of whether these two confidence regions of elliptical cross section are disjoint at any time instant is shown to result in a scalar test statistic that is compared to a prespecified decision threshold at each check-time in making failure/no-failure decisions. The motivating theoretical basis of the test statistic is briefly discussed, the implementation equations and theoretical milestones previously encountered in guaranteeing algorithm convergence and establishing convergence rate are Summarized, then the details are presented for: 1) the derivation and analytic evaluation of the expressions for the probabilities of false alarm and correct detection that serve as a basis for subsequent tradeoffs in setting the threshold level; and 2) the derivation of an expression for the decision threshold and a technique for its calculation from the covariance of the Kalman filter. The probability of correct detection is shown to be a monotonely increasing function of the underlying fundamental signal-to-noise ratio response of the Kalman filter estimate of the failure mode state to a particular magnitude of failure. Real data results are provided to illustrate application of this technique for the two-dimensional case to detect failures in an inertial navigation system having two-degree-of-freedom gyros. This is the application for which the technique was developed.