In this article, we deduce a procedure to apply balanced truncation to parameterdependent differential-algebraic systems. For that, we solve projected Lyapunov equations to compute the Gramians that are required for the truncation procedure. This process would lead to high computational costs if we perform it for a large number of parameters. Hence, we combine this approach with the reduced basis method that determines a reduced representation of the Lyapunov equation solutions for the parameters of interest. Residualbased error estimators are then used to evaluate the quality of the approximations. To apply the error estimators, a uniformly strictly dissipative state-space realization of the system is needed. We demonstrate how this property can be enforced by suitable statespace transformations. We illustrate the effectiveness of our approach on several models from fluid dynamics and mechanics. We further consider an application of the method in the context of damping optimization.

In this paper, we consider the problem of finding surrogate models for large-scale second-order linear time-invariant systems with inhomogeneous initial conditions. For this class of systems, the superposition principle allows us to decompose the system behavior into three independent components. The first behavior corresponds to the transfer between the input and output having zero initial conditions. In contrast, the other two correspond to the transfer between the initial position and the initial velocity conditions having zero input, respectively. Based on this superposition of systems, our goal is to propose model reduction schemes allowing to preserve the second-order structure in the surrogate models. To this aim, we introduce tailored secondorder Gramians for each system component and compute them numerically, solving Lyapunov equations. As a consequence, two methodologies are proposed. The first one consists in reducing each of the components independently using a suitable balanced truncation procedure. The sum of these reduced systems provides an approximation of the original system. This methodology allows flexibility on the order of the reduced-order model. The second proposed methodology consists in extracting the dominant subspaces from the sum of Gramians to build the projection matrices leading to a surrogate model. Additionally, we discuss error bounds for the overall output approximation. Finally, the proposed methods are illustrated by means of benchmark problems.

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