  This paper studies the problem of state estimation for linear time-invariant
descriptor systems in their most general form. The estimator is a system of
ordinary differential equations (ODEs). We introduce the notion of partial
causal detectability and characterize this concept by means of a simple rank
criterion involving the system coefficient matrices. Also, several equivalent
characterizations for partial causal detectability are established. In
addition, we prove that partial causal detectability is equivalent to the
existence of functional ODE estimators. A numerical example is given to
validate the theoretical results.
