Balanced Truncation of Descriptor Systems with a Quadratic Output

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.