$\mathcal{H}_2$-optimal model reduction of linear quadratic-output systems by multivariate rational interpolation

This paper addresses the $\mathcal{H}_2$-optimal approximation of linear dynamical systems with quadratic-output functions, also known as linear quadratic-output systems. Our major contributions are threefold. First, we derive interpolatory first-order optimality conditions for the linear quadratic-output $\mathcal{H}_2$ minimization problem. These conditions correspond to the mixed-multipoint tangential interpolation of the full-order linear- and quadratic-output transfer functions, and generalize the Meier-Luenberger optimality framework for the $\mathcal{H}_2$-optimal model reduction of linear time-invariant systems. Second, given the optimal interpolation data, we show how to enforce the interpolatory optimality conditions explicitly by Petrov-Galerkin projection of the full-order model. Third, to find the optimal interpolation data, we build on this projection framework and propose a generalization of the iterative rational Krylov algorithm for the $\mathcal{H}_2$-optimal model reduction of linear quadratic-output systems, called LQO-IRKA. Upon convergence, LQO-IRKA produces reduced linear quadratic-output systems that satisfy the interpolatory optimality conditions. The method only requires solving shifted linear systems and matrix-vector products, thus making it suitable for large-scale problems. Numerical examples are included to illustrate the effectiveness of the proposed method.