$\mathcal{H}_2$-optimal model reduction of linear quadratic-output systems by multivariate rational interpolation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909602120990720 |
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| author | Reiter, Sean Gosea, Ion Victor Duff, Igor Pontes Gugercin, Serkan |
| author_facet | Reiter, Sean Gosea, Ion Victor Duff, Igor Pontes Gugercin, Serkan |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_03057 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $\mathcal{H}_2$-optimal model reduction of linear quadratic-output systems by multivariate rational interpolation Reiter, Sean Gosea, Ion Victor Duff, Igor Pontes Gugercin, Serkan Numerical Analysis Systems and Control Dynamical Systems Optimization and Control 34C20, 41A05, 49K15, 65J05, 65F99, 93A15, 93C10, 93C80 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. |
| title | $\mathcal{H}_2$-optimal model reduction of linear quadratic-output systems by multivariate rational interpolation |
| topic | Numerical Analysis Systems and Control Dynamical Systems Optimization and Control 34C20, 41A05, 49K15, 65J05, 65F99, 93A15, 93C10, 93C80 |
| url | https://arxiv.org/abs/2505.03057 |