Observability and observer design of time-delay linear systems with unknown inputs

In this chapter, we present a study of the observability problem of a general class of linear time-delay systems affected by unknown inputs. We give two different notions of observability. The unknown input observability (UIO) allows us to reconstruct the trajectories of the system by using past, present, and future values of the system output, or in other words, the reconstruction comes with time delay. Thus we introduce the backward UIO, in which the state reconstruction is allowed using only past and present values of the system output. Once the structural property of BUIO observability is analyzed, we see the usefulness of it by designing a Luenberger-like observer. Some basic concepts of modern algebra are used for the study carried. To cover this point, we briefly review the required mathematical tools used here, like matrices over polynomial rings and the concept of a Noetherian module.