  This paper proposes new methods for set-based state estimation and active
fault diagnosis (AFD) of linear descriptor systems. In contrast to simple set
representations, such as intervals, ellipsoids, and zonotopes, linear static
constraints on the state variables, typical of descriptor systems, can be
directly incorporated in the mathematical description of constrained zonotopes.
Thanks to this feature, set-based methods using constrained zonotopes, as
proposed in previous works, could provide less conservative enclosures.
However, an enclosure on the states was assumed to be known for all $k \geq 0$.
Such assumption is violated in the case of unstable descriptor systems. In this
context, this paper proposes a new representation for unbounded sets, which
allows to develop methods for state estimation and tube-based AFD of stable and
unstable linear descriptor systems. The new set representation inherits most of
the properties of constrained zonotopes, including efficient complexity
reduction methods, while allowing to describe different classes of sets, such
as strips, hyperplanes, and the entire $n$-dimensional Euclidean space. The
advantages of the proposed approaches with respect to constrained zonotope
methods are highlighted in numerical examples.
