Generalized Criteria for Admissibility of Singular Fractional Order Systems
Unified frameworks for fractional order systems with fractional order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><<...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2023-04-01
|
| Series: | Fractal and Fractional |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2504-3110/7/5/363 |
| _version_ | 1797600016304439296 |
|---|---|
| author | Longxin Zhang Jin-Xi Zhang Xuefeng Zhang |
| author_facet | Longxin Zhang Jin-Xi Zhang Xuefeng Zhang |
| author_sort | Longxin Zhang |
| collection | DOAJ |
| description | Unified frameworks for fractional order systems with fractional order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>2</mn></mrow></semantics></math></inline-formula> are worth investigating. The aim of this paper is to provide a unified framework for stability and admissibility for fractional order systems and singular fractional order systems with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>2</mn></mrow></semantics></math></inline-formula>, respectively. By virtue of the LMI region and GLMI region, five stability theorems are presented. Two admissibility theorems for singular fractional order systems are extended from Theorem 5, and, in particular, a strict LMI stability criterion involving the least real decision variables without equality constraint by isomorphic mapping and congruent transform. The equivalence between the admissibility Theorems 6 and 7 is derived. The proposed framework contains some other existing results in the case of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>α</mi><mo><</mo><mn>2</mn></mrow></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>. Compared with published unified frameworks, the proposed framework is truly unified and does not require additional conditional assignment. Finally, without loss of generality, a unified control law is designed to make the singular feedback system admissible based on the criterion in a strict LMI framework and demonstrated by two numerical examples. |
| first_indexed | 2024-03-11T03:43:29Z |
| format | Article |
| id | doaj.art-4706d1730c804c55bb6eb6b532417f4e |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-11T03:43:29Z |
| publishDate | 2023-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-4706d1730c804c55bb6eb6b532417f4e2023-11-18T01:26:03ZengMDPI AGFractal and Fractional2504-31102023-04-017536310.3390/fractalfract7050363Generalized Criteria for Admissibility of Singular Fractional Order SystemsLongxin Zhang0Jin-Xi Zhang1Xuefeng Zhang2College of Sciences, Northeastern University, Shenyang 110819, ChinaState Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110819, ChinaCollege of Sciences, Northeastern University, Shenyang 110819, ChinaUnified frameworks for fractional order systems with fractional order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>2</mn></mrow></semantics></math></inline-formula> are worth investigating. The aim of this paper is to provide a unified framework for stability and admissibility for fractional order systems and singular fractional order systems with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>2</mn></mrow></semantics></math></inline-formula>, respectively. By virtue of the LMI region and GLMI region, five stability theorems are presented. Two admissibility theorems for singular fractional order systems are extended from Theorem 5, and, in particular, a strict LMI stability criterion involving the least real decision variables without equality constraint by isomorphic mapping and congruent transform. The equivalence between the admissibility Theorems 6 and 7 is derived. The proposed framework contains some other existing results in the case of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>α</mi><mo><</mo><mn>2</mn></mrow></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>. Compared with published unified frameworks, the proposed framework is truly unified and does not require additional conditional assignment. Finally, without loss of generality, a unified control law is designed to make the singular feedback system admissible based on the criterion in a strict LMI framework and demonstrated by two numerical examples.https://www.mdpi.com/2504-3110/7/5/363admissibilitygeneralized criteriastabilitysingular fractional order systems |
| spellingShingle | Longxin Zhang Jin-Xi Zhang Xuefeng Zhang Generalized Criteria for Admissibility of Singular Fractional Order Systems Fractal and Fractional admissibility generalized criteria stability singular fractional order systems |
| title | Generalized Criteria for Admissibility of Singular Fractional Order Systems |
| title_full | Generalized Criteria for Admissibility of Singular Fractional Order Systems |
| title_fullStr | Generalized Criteria for Admissibility of Singular Fractional Order Systems |
| title_full_unstemmed | Generalized Criteria for Admissibility of Singular Fractional Order Systems |
| title_short | Generalized Criteria for Admissibility of Singular Fractional Order Systems |
| title_sort | generalized criteria for admissibility of singular fractional order systems |
| topic | admissibility generalized criteria stability singular fractional order systems |
| url | https://www.mdpi.com/2504-3110/7/5/363 |
| work_keys_str_mv | AT longxinzhang generalizedcriteriaforadmissibilityofsingularfractionalordersystems AT jinxizhang generalizedcriteriaforadmissibilityofsingularfractionalordersystems AT xuefengzhang generalizedcriteriaforadmissibilityofsingularfractionalordersystems |