A Wronskian Approach to Observability and the Unobservable Subspace through the Null-Space Analysis

This paper discusses a connection between the Wronski matrix and observability of linear time-invariant systems as well as their unobservable subspaces. For a linear time-invariant single-output system, its observability is closely related to linear independence of a set of functions associated with the output. Hence, there is a connection between observability and the Wronski matrix, which is often used for deciding linear independence of a set of functions. In discussing observability and the unobservable subspace in this context, it matters whether the Wronski matrix can describe the necessary and sufficient condition for an arbitrary set of functions to be linearly independent. This paper first reviews the known facts about the relationship between linear independence of a set of functions and the Wronski matrix, as well as an additional condition under which linear independence can be decided in a necessary and sufficient fashion through the Wronski matrix. Then, well-known conditions on observability and characterization of the unobservable subspace of linear time-invariant systems are revisited from the viewpoint of the null space of the Wronski matrix.