Total and partial observation–detection in linear dynamical systems with characterized sources: finite-dimensional cases

In this work, we address the partial observation–detection problem for finite-dimensional dynamical linear systems that may not be fully observable or detectable.  We introduce the concepts of `observation–detection' and `partial observation–detection,' which involve reconstructing either the entirety or a portion of the system state and the source reacting on the system, even when the system is not fully observable or detectable.  We provide characterizations of `observable–detectable systems' and `observable–detectable spaces.'  The reconstruction of the state and source on the observable–detectable subspace is achieved through orthogonal projection, leveraging the algebraic structure of the given finite-dimensional system.  Additionally, we present examples to illustrate our approach.