Some Combined Results from Eneström–Kakeya and Rouché Theorems on the Generalized Schur Stability of Polynomials and the Stability of Quasi-Polynomials-Application to Time-Delay Systems
This paper derives some generalized Schur-type stability results of polynomials based on several forms and generalizations of the Eneström–Kakeya theorem combined with the Rouché theorem. It is first investigated, under sufficiency-type conditions, the derivation of the eventually generalized Schur...
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2024-09-01
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| author | Manuel De la Sen |
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| author_sort | Manuel De la Sen |
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| description | This paper derives some generalized Schur-type stability results of polynomials based on several forms and generalizations of the Eneström–Kakeya theorem combined with the Rouché theorem. It is first investigated, under sufficiency-type conditions, the derivation of the eventually generalized Schur stability sufficient conditions which are not necessarily related to the zeros of the polynomial lying in the unit open circle. In a second step, further sufficient conditions were introduced to guarantee that the above generalized Schur stability property persists within either the same above complex nominal stability region or in some larger one. The classical weak and, respectively, strong Schur stability in the closed and, respectively, open complex unit circle centred at zero are particular cases of their corresponding generalized versions. Some of the obtained and proved results are further generalized “ad hoc” for the case of quasi-polynomials whose zeros might be interpreted, in some typical cases, as characteristic zeros of linear continuous-time delayed time-invariant dynamic systems with commensurate constant point delays. |
| format | Article |
| id | doaj-art-9de3770fe20a45d9900ee2b72a8ee000 |
| institution | OA Journals |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-09-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj-art-9de3770fe20a45d9900ee2b72a8ee0002025-08-20T02:16:50ZengMDPI AGMathematics2227-73902024-09-011219302310.3390/math12193023Some Combined Results from Eneström–Kakeya and Rouché Theorems on the Generalized Schur Stability of Polynomials and the Stability of Quasi-Polynomials-Application to Time-Delay SystemsManuel De la Sen0Institute of Research and Development of Processes, Department of Electricity and Electronics, Faculty of Science and Technology, University of the Basque Country (UPV/EHU), 48940 Leioa, Bizkaia, SpainThis paper derives some generalized Schur-type stability results of polynomials based on several forms and generalizations of the Eneström–Kakeya theorem combined with the Rouché theorem. It is first investigated, under sufficiency-type conditions, the derivation of the eventually generalized Schur stability sufficient conditions which are not necessarily related to the zeros of the polynomial lying in the unit open circle. In a second step, further sufficient conditions were introduced to guarantee that the above generalized Schur stability property persists within either the same above complex nominal stability region or in some larger one. The classical weak and, respectively, strong Schur stability in the closed and, respectively, open complex unit circle centred at zero are particular cases of their corresponding generalized versions. Some of the obtained and proved results are further generalized “ad hoc” for the case of quasi-polynomials whose zeros might be interpreted, in some typical cases, as characteristic zeros of linear continuous-time delayed time-invariant dynamic systems with commensurate constant point delays.https://www.mdpi.com/2227-7390/12/19/3023Eneström–Kakeya theoremRouché theoremHurwitz stabilitySchur stabilityquasi-polynomialtime-delay dynamic systems |
| spellingShingle | Manuel De la Sen Some Combined Results from Eneström–Kakeya and Rouché Theorems on the Generalized Schur Stability of Polynomials and the Stability of Quasi-Polynomials-Application to Time-Delay Systems Mathematics Eneström–Kakeya theorem Rouché theorem Hurwitz stability Schur stability quasi-polynomial time-delay dynamic systems |
| title | Some Combined Results from Eneström–Kakeya and Rouché Theorems on the Generalized Schur Stability of Polynomials and the Stability of Quasi-Polynomials-Application to Time-Delay Systems |
| title_full | Some Combined Results from Eneström–Kakeya and Rouché Theorems on the Generalized Schur Stability of Polynomials and the Stability of Quasi-Polynomials-Application to Time-Delay Systems |
| title_fullStr | Some Combined Results from Eneström–Kakeya and Rouché Theorems on the Generalized Schur Stability of Polynomials and the Stability of Quasi-Polynomials-Application to Time-Delay Systems |
| title_full_unstemmed | Some Combined Results from Eneström–Kakeya and Rouché Theorems on the Generalized Schur Stability of Polynomials and the Stability of Quasi-Polynomials-Application to Time-Delay Systems |
| title_short | Some Combined Results from Eneström–Kakeya and Rouché Theorems on the Generalized Schur Stability of Polynomials and the Stability of Quasi-Polynomials-Application to Time-Delay Systems |
| title_sort | some combined results from enestrom kakeya and rouche theorems on the generalized schur stability of polynomials and the stability of quasi polynomials application to time delay systems |
| topic | Eneström–Kakeya theorem Rouché theorem Hurwitz stability Schur stability quasi-polynomial time-delay dynamic systems |
| url | https://www.mdpi.com/2227-7390/12/19/3023 |
| work_keys_str_mv | AT manueldelasen somecombinedresultsfromenestromkakeyaandrouchetheoremsonthegeneralizedschurstabilityofpolynomialsandthestabilityofquasipolynomialsapplicationtotimedelaysystems |