Complete Characterization of Minimum-Order Functional Observers from Darouach to Luenberger

Abstract

This paper gives an order-by-order characterization of linear functional observers between the minimum dimension permitted by the algebraic observer condition and the upper endpoint determined by functional observability. Functional observability indices determine the minimum algebraically admissible order $$ q_0=\sum_{j=1}^rη_j, $$ where $r$ is the number of independent functionals to be estimated and $η_j$ is the functional observability index associated with the $j$th functional. For an $n$-dimensional system with $p$ independent measured outputs, let $n_0$ denote the rank of the observability matrix. Under functional observability, the upper endpoint of the functional-observer order range is $n_0-p$. The algebraic and spectral observer conditions are characterized at each prescribed order between these two endpoints. A nullspace representation in the original state space yields rank and matrix-pencil characterizations of the algebraic and spectral conditions, respectively. The minimum observer order is obtained as the smallest spectrally feasible order in the resulting order spectrum. The two endpoints include the classical Darouach and Luenberger orders as special cases: the lower endpoint reduces to the Darouach order when $q_0=r$, while, under complete observability, the upper endpoint reduces to the Luenberger order $n-p$.