A two ellipsoid overlap test for real time failure detection and isolation by confidence regions

Real time failure detection for systems having a linear stochastic dynamical truth model is posed in terms of two confidence region sheaths. One confidence region sheath is about the expected no failure trajectory, the other is about the Kalman estimate. If these two confidence regions of ellipsoidal cross-section are disjoint at any time instant, a failure is declared. A test for two ellipsoid overlap is developed which involves finding a single point x* whose presence in both ellipsoids is necessary and sufficient for overlap. Thus, the overlap test is contorted into a search for x*, shown to be the solution of a nonlinear static optimization problem that is easily solved once an associated scalar Lagrange multiplier ?? is known. A successive approximations iteration equation for ?? was obtained and shown to converge as a contraction mapping. In an application of detecting simulated gyro failures in an Inertial Navigation System (INS), the iterations converged very quickly, easily allowing real time failure detection.