Observability of general linear pairs

In this work, we deal with the observability of a general linear pair (X, πK) on G which is a connected Lie group with Lie algebra g. By definition, the vector field X belongs to the normalizer of g related to the Lie algebra of all smooth vector fields on G. K is a closed Lie subgroup of G and πK is the canonical projection of G onto the homogeneous space G/K. We compute the Lie algebra of the equivalence class of the identity element, and characterize local and global observability of (X, πK). We extend the well-known observability rank condition of linear control systems on n and generalize the results appearing in [1].