Real-time failure detection: A nonlinear optimization problem that yields a two-ellipsoid overlap test

Abstract

Real-time failure detection for systems having linear stochastic dynamical truth models 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 pointx* whose presence in both ellipsoids is necessary and sufficient for overlap. Thus, the overlap test is contorted into a search forx*, shown to be the solution of a nonlinear optimization problem that is easily solved once an associated scalar Lagrange multiplier is known. A successive approximations iteration equation for λ is obtained and is shown to converge as a contraction mapping. The method was developed to detect failures in an inertial navigation system that appear as uncompensated gyroscopic drift rate. For simulated gyroscopic failures, the iterations converged very quickly, easily allowing real-time failure detection.