Model Reduction of Parametric Differential-Algebraic Systems by Balanced Truncation

In this article, we deduce a procedure to apply balanced truncation to parameter-dependent 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. Residual-based 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 state-space 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.