Partial detectability and generalized functional observer design for linear descriptor systems

In this study, we present two primary contributions to the theory of detectability for linear time-invariant descriptor systems. First, we establish a necessary and sufficient condition for the partial detectability by proving that it holds if and only if a specific rank condition involving the system’s coefficient matrices is satisfied. We define partial detectability through the system’s behavior approach and also provide a geometric characterization of this concept in terms of Wong sequences. Additionally, we discuss particular cases of the rank characterization in detail and establish a novel algebraic characterization of partial observability for state-space systems. The second major contribution is demonstrating that partial detectability is equivalent to the existence of a generalized functional estimator. However, we show that while partial detectability is necessary for the existence of generalized functional observers, it is not a sufficient condition. We derive an additional requirement that, when combined with partial detectability, ensures the construction of a generalized functional observer. The results are illustrated through numerical examples.