  This work discusses model reduction for differential-algebraic systems with
quadratic output equations. Under mild conditions, these systems can be
transformed into a Weierstra{\ss} canonical form and, thus, be decoupled into
differential equations and algebraic equations. The corresponding decoupled
states are referred to as proper and improper states. Due to the quadratic
function of the state as an output, the proper and improper states are coupled
in the output equation, which imposes a challenge from a model reduction
viewpoint. Keeping the coupling in mind, our goal in this work is to find
important subspaces of the proper and improper states and to reduce the system
accordingly. To that end, we first propose the system's matrices, the so-called
Gramians, to characterize the system's dominant subspaces. We pay particular
attention to the computation of the observability Gramians that take into
account the nonlinear coupling between the proper and the improper states. We
furthermore show that the proposed Gramians are related to certain kernel
functions, which are used to identify important subspaces. This allows us to
propose a reduction algorithm to obtain reduced-order systems by removing the
subspaces that are difficult to reach, as well as, difficult to observe.
Moreover, we quantify the error between the full-order and reduced-order models
and demonstrate the proposed methodology using three numerical experiments.
