Energy-Based Approximation of Linear Systems with Polynomial Outputs
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916605392322560 |
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| author | Balicki, Linus Gugercin, Serkan |
| author_facet | Balicki, Linus Gugercin, Serkan |
| contents | Controllability and observability energy functions play a fundamental role in model order reduction and are inherently connected to optimal control problems. For linear dynamical systems the energy functions are known to be quadratic polynomials and various low-rank approximation techniques allow for computing them in a large-scale setting. For nonlinear problems computing the energy functions is significantly more challenging. In this paper, we investigate a special class of nonlinear systems that have a linear state and a polynomial output equation. We show that the energy functions of these systems are again polynomials and investigate under which conditions they can effectively be approximated using low-rank tensors. Further, we introduce a new perspective on the well-established balanced truncation method for linear systems which then readily generalizes to the nonlinear systems under consideration. This new perspective yields a novel energy-based model order reduction procedure that accurately captures the input-output behavior of linear systems with polynomial outputs via a low-dimensional reduced order model. We demonstrate the effectiveness of our approach via two numerical experiments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_19730 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Energy-Based Approximation of Linear Systems with Polynomial Outputs Balicki, Linus Gugercin, Serkan Dynamical Systems Controllability and observability energy functions play a fundamental role in model order reduction and are inherently connected to optimal control problems. For linear dynamical systems the energy functions are known to be quadratic polynomials and various low-rank approximation techniques allow for computing them in a large-scale setting. For nonlinear problems computing the energy functions is significantly more challenging. In this paper, we investigate a special class of nonlinear systems that have a linear state and a polynomial output equation. We show that the energy functions of these systems are again polynomials and investigate under which conditions they can effectively be approximated using low-rank tensors. Further, we introduce a new perspective on the well-established balanced truncation method for linear systems which then readily generalizes to the nonlinear systems under consideration. This new perspective yields a novel energy-based model order reduction procedure that accurately captures the input-output behavior of linear systems with polynomial outputs via a low-dimensional reduced order model. We demonstrate the effectiveness of our approach via two numerical experiments. |
| title | Energy-Based Approximation of Linear Systems with Polynomial Outputs |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2409.19730 |