False alarm and correct detection probabilities over a time interval for restricted classes of failure detection algorithms

The statistical analysis of failure detection decisions in terms of the instantaneous probabilities of false alarm and correct detection for a specified failure magnitude at each check-time have previously been performed for several different failure detection techniques that utilize a Kalman filter. By performing a discrete-time specialization of a result of Gallager and Helstrom on a tightened upper bound for continuous-time level-crossing probabilities, upper bounds on the probabilities of false alarm and correct detection over a time interval have been obtained for the specific technique of CR2 tailnre detection (to allow an accounting for the effect of time correlations of the filter estimates). When these upper bounds are optimized to be as tight as possible to the desired probabilities, the resulting optimization problem for discrete-time is a collection of quadratic programming (QP) problems, which may easily be solved exactly without recourse to approximate solutions as were resorted to in the continuous-time formulation. This technique for evaluating tightened upper bounds on the false alarm and correct detection probabilities may be of general interest, since it can be applied to any failure detection technique or signal detection technique that can relate an exceeding of the deterministic decision threshold by the test statistic directly to a deterministic level being exceeded by a scalar Gaussian random process.