$\mathcal{H}_\infty$ model order reduction for quadratic output systems

Abstract: Linear time-invariant quadratic output (LTIQO) systems generalize linear time-invariant systems to nonlinear regimes. Problems of this class occur in multiple applications naturally, such as port-Hamiltonian systems, optimal control, and stochastical problems. We introduce an $\mathcal{H}\infty$-norm for LTIQO systems with one or multiple outputs and propose an algorithm to optimize a reduced order model (ROM) to be close in the $\mathcal{H}\infty$-norm to a given full order model. We illustrate the applicability and the performance with an established numerical example and compare the resulting ROMs with results from balanced truncation and $\mathcal{H}_2$-focussed algorithms.