Observability analysis of piece-wise constant systems with application to inertial navigation

A method for analyzing the observability of time-varying linear systems which can be modeled as piecewise constant systems is presented. An observability matrix for such systems is developed for continuous and discrete time representations. A stripped observability matrix is introduced which simplifies the analysis in cases where the use of this matrix is legitimate. This approach circumvents the difficulty associated with the investigation of the observability Gramian of time-varying linear systems. It is shown that instead of investigating a Gramian, only a constant observability matrix needs to be investigated. Moreover, it is shown that if certain conditions on the null space of the dynamics matrix of the system are met, the observability matrix can be greatly simplified. A step-by-step observability analysis procedure is presented for this case. The method is applied to the analysis of in-flight alignment of inertial navigation systems whose estimability is known to be enhanced by maneuvers