A Crame´r-Rao-type estimation lower bound for systems with measurement faults

A Crame´r-Rao-type lower bound is presented for systems with measurements prone to discretely-distributed faults, which are a class of hybrid systems. Lower bounds for both the state and the Markovian interruption variables (fault indicators) of the system are derived, using the recently presented sequential version of the Crame´r-Rao lower bound (CRLB) for general nonlinear systems. Because of the hybrid nature of the systems addressed, the CRLB cannot be directly applied due to violation of its associated regularity conditions. To facilitate the calculation of the lower bound, the hybrid system is first approximated by a system in which the discrete distribution of the fault indicators is replaced by an approximating continuous one. The lower bound is then obtained via a limiting process applied to the approximating system. The results presented herein facilitate a relatively simple calculation of a nontrivial lower bound for the state vector of systems with fault-prone measurements. The CRLB-type lower bound for the interruption process variables turns out to be trivially zero, however, a nontrivial, non-CRLB-type bound for these variables has been recently presented elsewhere by the authors. The utility and applicability of the proposed lower bound are demonstrated via a numerical example involving a simple global positioning system (GPS)-aided navigation system, where the GPS measurements are fault-prone due to their sensitivity to multipath errors.