A New Stability Theory for Grünwald–Letnikov Inverse Model Control in the Multivariable LTI Fractional-Order Framework
The new general theory dedicated to the stability for LTI MIMO, in particular nonsquare, fractional-order systems described by the Grünwald−Letnikov discrete-time state−space domain is presented in this paper. Such systems under inverse model control, principally MV/perf...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2019-10-01
|
| Series: | Symmetry |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2073-8994/11/10/1322 |
| _version_ | 1798042363841478656 |
|---|---|
| author | Wojciech Przemysław Hunek Łukasz Wach |
| author_facet | Wojciech Przemysław Hunek Łukasz Wach |
| author_sort | Wojciech Przemysław Hunek |
| collection | DOAJ |
| description | The new general theory dedicated to the stability for LTI MIMO, in particular nonsquare, fractional-order systems described by the Grünwald−Letnikov discrete-time state−space domain is presented in this paper. Such systems under inverse model control, principally MV/perfect control, represent a real research challenge due to an infinite number of solutions to the underlying inverse problem for nonsquare matrices. Therefore, the paper presents a new algorithm for fractional-order perfect control with corresponding stability formula involving recently given <i>H</i>- and <inline-formula> <math display="inline"> <semantics> <mi>σ</mi> </semantics> </math> </inline-formula>-inverse of nonsquare matrices, up to now applied solely to the integer-order plants. On such foundation a new set of stability-related tools is introduced, among them the key role played by so-called control zeros. Control zeros constitute an extension of transmission zeros for nonsquare fractional-order LTI MIMO systems under inverse model control. Based on the sets of stable control zeros a minimum-phase behavior is specified because of the stability of newly defined perfect control law described in the non-integer-order framework. The whole theory is complemented by pole-free fractional-order perfect control paradigm, a special case of fractional-order perfect control strategy. A significant number of simulation examples confirm the correctness and research potential proposed in the paper methodology. |
| first_indexed | 2024-04-11T22:34:30Z |
| format | Article |
| id | doaj.art-1047c917bad64d679cf6b94730b930d6 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-04-11T22:34:30Z |
| publishDate | 2019-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-1047c917bad64d679cf6b94730b930d62022-12-22T03:59:15ZengMDPI AGSymmetry2073-89942019-10-011110132210.3390/sym11101322sym11101322A New Stability Theory for Grünwald–Letnikov Inverse Model Control in the Multivariable LTI Fractional-Order FrameworkWojciech Przemysław Hunek0Łukasz Wach1Department of Electrical, Control and Computer Engineering, Opole University of Technology, Prószkowska 76 Street, 45-758 Opole, PolandDepartment of Electrical, Control and Computer Engineering, Opole University of Technology, Prószkowska 76 Street, 45-758 Opole, PolandThe new general theory dedicated to the stability for LTI MIMO, in particular nonsquare, fractional-order systems described by the Grünwald−Letnikov discrete-time state−space domain is presented in this paper. Such systems under inverse model control, principally MV/perfect control, represent a real research challenge due to an infinite number of solutions to the underlying inverse problem for nonsquare matrices. Therefore, the paper presents a new algorithm for fractional-order perfect control with corresponding stability formula involving recently given <i>H</i>- and <inline-formula> <math display="inline"> <semantics> <mi>σ</mi> </semantics> </math> </inline-formula>-inverse of nonsquare matrices, up to now applied solely to the integer-order plants. On such foundation a new set of stability-related tools is introduced, among them the key role played by so-called control zeros. Control zeros constitute an extension of transmission zeros for nonsquare fractional-order LTI MIMO systems under inverse model control. Based on the sets of stable control zeros a minimum-phase behavior is specified because of the stability of newly defined perfect control law described in the non-integer-order framework. The whole theory is complemented by pole-free fractional-order perfect control paradigm, a special case of fractional-order perfect control strategy. A significant number of simulation examples confirm the correctness and research potential proposed in the paper methodology.https://www.mdpi.com/2073-8994/11/10/1322stability criteriafeedback control methodszero setspole zero assignmentminimum-phase systemsrobust controlmatrix inversionstate–space modelsmimo |
| spellingShingle | Wojciech Przemysław Hunek Łukasz Wach A New Stability Theory for Grünwald–Letnikov Inverse Model Control in the Multivariable LTI Fractional-Order Framework Symmetry stability criteria feedback control methods zero sets pole zero assignment minimum-phase systems robust control matrix inversion state–space models mimo |
| title | A New Stability Theory for Grünwald–Letnikov Inverse Model Control in the Multivariable LTI Fractional-Order Framework |
| title_full | A New Stability Theory for Grünwald–Letnikov Inverse Model Control in the Multivariable LTI Fractional-Order Framework |
| title_fullStr | A New Stability Theory for Grünwald–Letnikov Inverse Model Control in the Multivariable LTI Fractional-Order Framework |
| title_full_unstemmed | A New Stability Theory for Grünwald–Letnikov Inverse Model Control in the Multivariable LTI Fractional-Order Framework |
| title_short | A New Stability Theory for Grünwald–Letnikov Inverse Model Control in the Multivariable LTI Fractional-Order Framework |
| title_sort | new stability theory for grunwald letnikov inverse model control in the multivariable lti fractional order framework |
| topic | stability criteria feedback control methods zero sets pole zero assignment minimum-phase systems robust control matrix inversion state–space models mimo |
| url | https://www.mdpi.com/2073-8994/11/10/1322 |
| work_keys_str_mv | AT wojciechprzemysławhunek anewstabilitytheoryforgrunwaldletnikovinversemodelcontrolinthemultivariableltifractionalorderframework AT łukaszwach anewstabilitytheoryforgrunwaldletnikovinversemodelcontrolinthemultivariableltifractionalorderframework AT wojciechprzemysławhunek newstabilitytheoryforgrunwaldletnikovinversemodelcontrolinthemultivariableltifractionalorderframework AT łukaszwach newstabilitytheoryforgrunwaldletnikovinversemodelcontrolinthemultivariableltifractionalorderframework |