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

TL;DR

This work develops an $\mathcal{H}_2$-optimal reduction framework for linear dynamical systems with quadratic-output terms (LQO systems) by recasting the MOR problem as a multivariate rational interpolation of the linear and quadratic transfer functions $G_1$ and $G_2$. Central to the approach is a pole-residue, mixed-multipoint tangential interpolation theory, which yields first-order optimality conditions that couple $G_1$, $G_2$, and their sum, generalizing the Meier–Luenberger framework for $\mathcal{H}_2$-optimal MOR of LTI systems. The authors show how to enforce these conditions via Petrov–Galerkin projection and propose the linear quadratic-output IRKA ($\mathsf{LQO}$-IRKA), an iterative method that updates interpolation data from the ROM’s poles and residues and uses shifted linear solves to build real-valued reduced models. Numerical experiments on a 1D advection–diffusion benchmark demonstrate that $\mathsf{LQO}$-IRKA delivers high-fidelity reduced models with significantly lower $\mathcal{H}_2$ errors compared to non-optimal interpolatory or Gramian-based approaches, while remaining scalable to large-scale problems. Overall, the framework provides a principled, efficient path to accurate surrogates for LQO systems applicable to control, estimation, and design tasks.

Abstract

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 interpolation-based 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 interpolation data, we show how to enforce these mixed-multipoint tangential interpolation conditions explicitly by Petrov-Galerkin projection of the full-order model matrices. 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 a reduced linear quadratic-output system that satisfies 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.

Figures (3)

• Figure 1: Output magnitudes and pointwise relative errors \ref{['eq:pointwiseOutputError']} of the full-order and order $r=30$ reduced models driven by $u_1(t)=u_{\mathop{\mathrm{sinc}}\nolimits}(t)$ and $u_1(t)=u_{\exp}(t)$ in \ref{['eq:inputs']}.
• Figure 2: Relative $\mathcal{H}_2$ errors of the intermediate reduced models computed by $\mathsf{LQO}\hbox{-}\mathsf{IRKA_{eigs}}$ and $\mathsf{LQO}\hbox{-}\mathsf{IRKA_{imag}}$ for the first $50$ iterations.
• Figure 3: Relative $\mathcal{H}_2$ errors \ref{['eq:relH2error']} due to the hierarchy of reduced models for orders $r = 2, 4, \ldots, 30$.

Theorems & Definitions (13)

• Lemma 2.1
• proof
• Definition 2.1
• Theorem 2.1
• proof
• Theorem 3.1
• proof : Proof of \ref{['thm:lqoH2OptInterpolationCons']}
• Remark 3.1
• Theorem 3.2
• proof : Proof of \ref{['thm:enforceMixedInterp']}