On the Relation Between Observability of a Class of Linear Systems With Norm Outputs and Navigation With Range Measurements

This paper formulates sufficient conditions for observability of a class of linear time invariant systems with a norm output. The observability of these systems depends on the input and will be studied for any input that satisfies a linear autonomous difference equation. It is shown that the observability conditions are related to the navigation problem of finding a position fix from several range measurements by intersecting spheres. This offers a new geometric interpretation of observability. Moreover, it will be shown that the initial condition of the system can still be estimated in the least squares sense if noise is added to the output. An example shows the application of the methodology. Additionally the example shows that the initial condition of the noiseless system is obtained by inverting a lower dimensional upper triangular matrix. Thus, the proposed method yields a matrix inversion with less computations when compared to the augmented state linear time-varying system approaches in the literature. A second example shows how the proposed methodology can be extended to perform state estimation.