Basis Functions for a Transient Analysis of Linear Commensurate Fractional-Order Systems
In this paper, the possibilities of expressing the natural response of a linear commensurate fractional-order system (FOS) as a linear combination of basis functions are analyzed. For all possible types of <i>s<sup>α</sup></i>-domain poles, the corresponding basis functions a...
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MDPI AG
2023-07-01
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| Series: | Algorithms |
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| Online Access: | https://www.mdpi.com/1999-4893/16/7/335 |
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| author | Dalibor Biolek Viera Biolková Zdeněk Kolka Zdeněk Biolek |
| author_facet | Dalibor Biolek Viera Biolková Zdeněk Kolka Zdeněk Biolek |
| author_sort | Dalibor Biolek |
| collection | DOAJ |
| description | In this paper, the possibilities of expressing the natural response of a linear commensurate fractional-order system (FOS) as a linear combination of basis functions are analyzed. For all possible types of <i>s<sup>α</sup></i>-domain poles, the corresponding basis functions are found, the kernel of which is the two-parameter Mittag–Leffler function <i>E<sub>α</sub></i><sub>,<i>β</i></sub>, <i>β</i> = <i>α</i>. It is pointed out that there are mutually unambiguous correspondences between the basis functions of FOS and the known basis functions of the integer-order system (IOS) for <i>α</i> = 1. This correspondence can be used to algorithmically find analytical formulas for the impulse responses of FOS when the formulas for the characteristics of IOS are known. It is shown that all basis functions of FOS can be generated with Podlubny‘s function of type <i>ε<sub>k</sub> (t</i>, <i>c</i>; <i>α</i>, <i>α</i>), where <i>c</i> and <i>k</i> are the corresponding pole and its multiplicity, respectively. |
| first_indexed | 2024-03-11T01:22:32Z |
| format | Article |
| id | doaj.art-6bf2e64d189545448a82febdc201c0dc |
| institution | Directory Open Access Journal |
| issn | 1999-4893 |
| language | English |
| last_indexed | 2024-03-11T01:22:32Z |
| publishDate | 2023-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Algorithms |
| spelling | doaj.art-6bf2e64d189545448a82febdc201c0dc2023-11-18T17:59:12ZengMDPI AGAlgorithms1999-48932023-07-0116733510.3390/a16070335Basis Functions for a Transient Analysis of Linear Commensurate Fractional-Order SystemsDalibor Biolek0Viera Biolková1Zdeněk Kolka2Zdeněk Biolek3Department of Electrical Engineering, University of Defence Brno, 662 10 Brno, Czech RepublicDepartment of Radio Electronics, Brno University of Technology, 616 00 Brno, Czech RepublicDepartment of Radio Electronics, Brno University of Technology, 616 00 Brno, Czech RepublicDepartment of Electrical Engineering, University of Defence Brno, 662 10 Brno, Czech RepublicIn this paper, the possibilities of expressing the natural response of a linear commensurate fractional-order system (FOS) as a linear combination of basis functions are analyzed. For all possible types of <i>s<sup>α</sup></i>-domain poles, the corresponding basis functions are found, the kernel of which is the two-parameter Mittag–Leffler function <i>E<sub>α</sub></i><sub>,<i>β</i></sub>, <i>β</i> = <i>α</i>. It is pointed out that there are mutually unambiguous correspondences between the basis functions of FOS and the known basis functions of the integer-order system (IOS) for <i>α</i> = 1. This correspondence can be used to algorithmically find analytical formulas for the impulse responses of FOS when the formulas for the characteristics of IOS are known. It is shown that all basis functions of FOS can be generated with Podlubny‘s function of type <i>ε<sub>k</sub> (t</i>, <i>c</i>; <i>α</i>, <i>α</i>), where <i>c</i> and <i>k</i> are the corresponding pole and its multiplicity, respectively.https://www.mdpi.com/1999-4893/16/7/335Mittag–Leffler functioncommensurate fractional-order systembasis functionimpulse response |
| spellingShingle | Dalibor Biolek Viera Biolková Zdeněk Kolka Zdeněk Biolek Basis Functions for a Transient Analysis of Linear Commensurate Fractional-Order Systems Algorithms Mittag–Leffler function commensurate fractional-order system basis function impulse response |
| title | Basis Functions for a Transient Analysis of Linear Commensurate Fractional-Order Systems |
| title_full | Basis Functions for a Transient Analysis of Linear Commensurate Fractional-Order Systems |
| title_fullStr | Basis Functions for a Transient Analysis of Linear Commensurate Fractional-Order Systems |
| title_full_unstemmed | Basis Functions for a Transient Analysis of Linear Commensurate Fractional-Order Systems |
| title_short | Basis Functions for a Transient Analysis of Linear Commensurate Fractional-Order Systems |
| title_sort | basis functions for a transient analysis of linear commensurate fractional order systems |
| topic | Mittag–Leffler function commensurate fractional-order system basis function impulse response |
| url | https://www.mdpi.com/1999-4893/16/7/335 |
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