Exact and Inexact Lifting Transformations of Nonlinear Dynamical Systems: Transfer Functions, Equivalence, and Complexity Reduction
In this work, we deal with the problem of approximating and equivalently formulating generic nonlinear systems by means of specific classes thereof. Bilinear and quadratic-bilinear systems accomplish precisely this goal. Hence, by means of exact and inexact lifting transformations, we are able to re...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-02-01
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| Series: | Applied Sciences |
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| Online Access: | https://www.mdpi.com/2076-3417/12/5/2333 |
| _version_ | 1797475786619355136 |
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| author | Ion Victor Gosea |
| author_facet | Ion Victor Gosea |
| author_sort | Ion Victor Gosea |
| collection | DOAJ |
| description | In this work, we deal with the problem of approximating and equivalently formulating generic nonlinear systems by means of specific classes thereof. Bilinear and quadratic-bilinear systems accomplish precisely this goal. Hence, by means of exact and inexact lifting transformations, we are able to reformulate the original nonlinear dynamics into a different, more simplified format. Additionally, we study the problem of complexity/model reduction of large-scale lifted models of nonlinear systems from data. The method under consideration is the Loewner framework, an established data-driven approach that requires samples of input–output mappings. The latter are known as generalized transfer functions, which are appropriately defined for both bilinear and quadratic-bilinear systems. We show connections between these mappings as well as between the matrices of reduced-order models. Finally, we illustrate the theoretical discussion with two numerical examples. |
| first_indexed | 2024-03-09T20:48:47Z |
| format | Article |
| id | doaj.art-9d6452942b9b470c9edf30271a46ecfc |
| institution | Directory Open Access Journal |
| issn | 2076-3417 |
| language | English |
| last_indexed | 2024-03-09T20:48:47Z |
| publishDate | 2022-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Applied Sciences |
| spelling | doaj.art-9d6452942b9b470c9edf30271a46ecfc2023-11-23T22:39:08ZengMDPI AGApplied Sciences2076-34172022-02-01125233310.3390/app12052333Exact and Inexact Lifting Transformations of Nonlinear Dynamical Systems: Transfer Functions, Equivalence, and Complexity ReductionIon Victor Gosea0Data-Driven System Reduction and Identification Group, Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstrasse 1, 39114 Magdeburg, GermanyIn this work, we deal with the problem of approximating and equivalently formulating generic nonlinear systems by means of specific classes thereof. Bilinear and quadratic-bilinear systems accomplish precisely this goal. Hence, by means of exact and inexact lifting transformations, we are able to reformulate the original nonlinear dynamics into a different, more simplified format. Additionally, we study the problem of complexity/model reduction of large-scale lifted models of nonlinear systems from data. The method under consideration is the Loewner framework, an established data-driven approach that requires samples of input–output mappings. The latter are known as generalized transfer functions, which are appropriately defined for both bilinear and quadratic-bilinear systems. We show connections between these mappings as well as between the matrices of reduced-order models. Finally, we illustrate the theoretical discussion with two numerical examples.https://www.mdpi.com/2076-3417/12/5/2333nonlinear systemscomplexity model reductiondata-driven methodsbilinear quadratic structuretransfer functionslifting transformations |
| spellingShingle | Ion Victor Gosea Exact and Inexact Lifting Transformations of Nonlinear Dynamical Systems: Transfer Functions, Equivalence, and Complexity Reduction Applied Sciences nonlinear systems complexity model reduction data-driven methods bilinear quadratic structure transfer functions lifting transformations |
| title | Exact and Inexact Lifting Transformations of Nonlinear Dynamical Systems: Transfer Functions, Equivalence, and Complexity Reduction |
| title_full | Exact and Inexact Lifting Transformations of Nonlinear Dynamical Systems: Transfer Functions, Equivalence, and Complexity Reduction |
| title_fullStr | Exact and Inexact Lifting Transformations of Nonlinear Dynamical Systems: Transfer Functions, Equivalence, and Complexity Reduction |
| title_full_unstemmed | Exact and Inexact Lifting Transformations of Nonlinear Dynamical Systems: Transfer Functions, Equivalence, and Complexity Reduction |
| title_short | Exact and Inexact Lifting Transformations of Nonlinear Dynamical Systems: Transfer Functions, Equivalence, and Complexity Reduction |
| title_sort | exact and inexact lifting transformations of nonlinear dynamical systems transfer functions equivalence and complexity reduction |
| topic | nonlinear systems complexity model reduction data-driven methods bilinear quadratic structure transfer functions lifting transformations |
| url | https://www.mdpi.com/2076-3417/12/5/2333 |
| work_keys_str_mv | AT ionvictorgosea exactandinexactliftingtransformationsofnonlineardynamicalsystemstransferfunctionsequivalenceandcomplexityreduction |