$\mathcal{H}_2$ optimal model reduction of linear systems with multiple quadratic outputs
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| Format: | Preprint |
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2024
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| _version_ | 1866914789501960192 |
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| author | Reiter, Sean Duff, Igor Pontes Gosea, Ion Victor Gugercin, Serkan |
| author_facet | Reiter, Sean Duff, Igor Pontes Gosea, Ion Victor Gugercin, Serkan |
| contents | In this work, we consider the $\mathcal{H}_2$ optimal model reduction of dynamical systems that are linear in the state equation and up to quadratic nonlinearity in the output equation. As our primary theoretical contributions, we derive gradients of the squared $\mathcal{H}_2$ system error with respect to the reduced model quantities and, from the stationary points of these gradients, introduce Gramian-based first-order necessary conditions for the $\mathcal{H}_2$ optimal approximation of a linear quadratic output (LQO) system. The resulting $\mathcal{H}_2$ optimality framework neatly generalizes the analogous Gramian-based optimality framework for purely linear systems. Computationally, we show how to enforce the necessary optimality conditions using Petrov-Galerkin projection; the corresponding projection matrices are obtained from a pair of Sylvester equations. Based on this result, we propose an iteratively corrected algorithm for the $\mathcal{H}_2$ model reduction of LQO systems, which we refer to as LQO-TSIA (linear quadratic output two-sided iteration algorithm). Numerical examples are included to illustrate the effectiveness of the proposed computational method against other existing approaches. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_05951 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $\mathcal{H}_2$ optimal model reduction of linear systems with multiple quadratic outputs Reiter, Sean Duff, Igor Pontes Gosea, Ion Victor Gugercin, Serkan Numerical Analysis Systems and Control Dynamical Systems Optimization and Control 15A24, 46N10, 49K15, 93A15, 93C10, 93C80 In this work, we consider the $\mathcal{H}_2$ optimal model reduction of dynamical systems that are linear in the state equation and up to quadratic nonlinearity in the output equation. As our primary theoretical contributions, we derive gradients of the squared $\mathcal{H}_2$ system error with respect to the reduced model quantities and, from the stationary points of these gradients, introduce Gramian-based first-order necessary conditions for the $\mathcal{H}_2$ optimal approximation of a linear quadratic output (LQO) system. The resulting $\mathcal{H}_2$ optimality framework neatly generalizes the analogous Gramian-based optimality framework for purely linear systems. Computationally, we show how to enforce the necessary optimality conditions using Petrov-Galerkin projection; the corresponding projection matrices are obtained from a pair of Sylvester equations. Based on this result, we propose an iteratively corrected algorithm for the $\mathcal{H}_2$ model reduction of LQO systems, which we refer to as LQO-TSIA (linear quadratic output two-sided iteration algorithm). Numerical examples are included to illustrate the effectiveness of the proposed computational method against other existing approaches. |
| title | $\mathcal{H}_2$ optimal model reduction of linear systems with multiple quadratic outputs |
| topic | Numerical Analysis Systems and Control Dynamical Systems Optimization and Control 15A24, 46N10, 49K15, 93A15, 93C10, 93C80 |
| url | https://arxiv.org/abs/2405.05951 |