Mixed zonotopes: a set representation suitable for unbounded systems and its application to set-based state estimation and active fault diagnosis of descriptor systems

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.