Structure-preserving discretization and model order reduction of boundary-controlled 1D port-Hamiltonian systems.
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| Title: | Structure-preserving discretization and model order reduction of boundary-controlled 1D port-Hamiltonian systems. |
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| Authors: | Toledo-Zucco, Jesus-Pablo1 (AUTHOR), Matignon, Denis2 (AUTHOR), Poussot-Vassal, Charles3 (AUTHOR), Le Gorrec, Yann4 (AUTHOR) |
| Source: | Systems & Control Letters. Dec2024, Vol. 194, pN.PAG-N.PAG. 1p. |
| Subjects: | Partial differential equations, Finite element method, Wave equation, Discretization methods, Interpolation |
| Abstract: | This paper presents a systematic methodology for the discretization and reduction of a class of one-dimensional Partial Differential Equations (PDEs) with inputs and outputs collocated at the spatial boundaries. The class of system that we consider is known as Boundary-Controlled Port-Hamiltonian Systems (BC-PHSs) and covers a wide class of Hyperbolic PDEs with a large type of boundary inputs and outputs. This is, for instance, the case of waves and beams with Neumann, Dirichlet, or mixed boundary conditions. Based on a Partitioned Finite Element Method (PFEM), we develop a numerical scheme for the structure-preserving spatial discretization for the class of one-dimensional BC-PHSs. We show that if the initial PDE is passive (or Impedance Energy Preserving), the discretized model also is. In addition and since the discretized model or Full Order Model (FOM) can be of large dimension, we recall the standard Loewner framework for the Model Order Reduction (MOR) using frequency domain interpolation. We recall the main steps to produce a Reduced Order Model (ROM) that approaches the FOM in a given range of frequencies. We summarize the steps to follow in order to obtain a ROM that preserves the passive structure as well. Finally, we provide a constructive way to build a projector that allows to recover the physical meaning of the state variables from the ROM to the FOM. We use the one-dimensional wave equation and the Timoshenko beam as examples to show the versatility of the proposed approach. [ABSTRACT FROM AUTHOR] |
| Copyright of Systems & Control Letters is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
| FullText | Text: Availability: 0 |
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| Header | DbId: egs DbLabel: Engineering Source An: 181161485 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Structure-preserving discretization and model order reduction of boundary-controlled 1D port-Hamiltonian systems. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Toledo-Zucco%2C+Jesus-Pablo%22">Toledo-Zucco, Jesus-Pablo</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Matignon%2C+Denis%22">Matignon, Denis</searchLink><relatesTo>2</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Poussot-Vassal%2C+Charles%22">Poussot-Vassal, Charles</searchLink><relatesTo>3</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Le+Gorrec%2C+Yann%22">Le Gorrec, Yann</searchLink><relatesTo>4</relatesTo> (AUTHOR) – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Systems+%26+Control+Letters%22">Systems & Control Letters</searchLink>. Dec2024, Vol. 194, pN.PAG-N.PAG. 1p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Partial+differential+equations%22">Partial differential equations</searchLink><br /><searchLink fieldCode="DE" term="%22Finite+element+method%22">Finite element method</searchLink><br /><searchLink fieldCode="DE" term="%22Wave+equation%22">Wave equation</searchLink><br /><searchLink fieldCode="DE" term="%22Discretization+methods%22">Discretization methods</searchLink><br /><searchLink fieldCode="DE" term="%22Interpolation%22">Interpolation</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: This paper presents a systematic methodology for the discretization and reduction of a class of one-dimensional Partial Differential Equations (PDEs) with inputs and outputs collocated at the spatial boundaries. The class of system that we consider is known as Boundary-Controlled Port-Hamiltonian Systems (BC-PHSs) and covers a wide class of Hyperbolic PDEs with a large type of boundary inputs and outputs. This is, for instance, the case of waves and beams with Neumann, Dirichlet, or mixed boundary conditions. Based on a Partitioned Finite Element Method (PFEM), we develop a numerical scheme for the structure-preserving spatial discretization for the class of one-dimensional BC-PHSs. We show that if the initial PDE is passive (or Impedance Energy Preserving), the discretized model also is. In addition and since the discretized model or Full Order Model (FOM) can be of large dimension, we recall the standard Loewner framework for the Model Order Reduction (MOR) using frequency domain interpolation. We recall the main steps to produce a Reduced Order Model (ROM) that approaches the FOM in a given range of frequencies. We summarize the steps to follow in order to obtain a ROM that preserves the passive structure as well. Finally, we provide a constructive way to build a projector that allows to recover the physical meaning of the state variables from the ROM to the FOM. We use the one-dimensional wave equation and the Timoshenko beam as examples to show the versatility of the proposed approach. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Systems & Control Letters is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1016/j.sysconle.2024.105947 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 1 StartPage: N.PAG Subjects: – SubjectFull: Partial differential equations Type: general – SubjectFull: Finite element method Type: general – SubjectFull: Wave equation Type: general – SubjectFull: Discretization methods Type: general – SubjectFull: Interpolation Type: general Titles: – TitleFull: Structure-preserving discretization and model order reduction of boundary-controlled 1D port-Hamiltonian systems. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Toledo-Zucco, Jesus-Pablo – PersonEntity: Name: NameFull: Matignon, Denis – PersonEntity: Name: NameFull: Poussot-Vassal, Charles – PersonEntity: Name: NameFull: Le Gorrec, Yann IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 12 Text: Dec2024 Type: published Y: 2024 Identifiers: – Type: issn-print Value: 01676911 Numbering: – Type: volume Value: 194 Titles: – TitleFull: Systems & Control Letters Type: main |
| ResultId | 1 |
