Approximate balanced truncation for linear structured systems based on greedy numerical integration

Abstract This paper studies balanced truncation for linear structured system (LSS). We focus on the efficient computation of controllability and observability Gramians. Unlike standard linear time-invariant (LTI) systems, there is no general algebraic Lyapunov equation that encodes the Gramians for LSS. Instead, natural approaches approximate the Gramians using quadrature rules, often requiring many nodes. We identify these approximations with the solutions to generalized Sylvester equations, allowing us to employ and adapt an existing active sampling strategy and compute low-rank factors of the Gramians as solutions to the matrix equations. Numerical examples show comparable accuracy to standard methods with similar computational time for medium-sized systems, and speed-up and reduced memory requirements for large dynamical systems.