Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems
•New Lemmas relating the Caputo Fractional Derivatives of general quadratic forms and the trace of the product of matrices.•Possibility to use general quadratics Lyapunov functions in the fractional-order extension of Lyapunov direct method.•Uniform stability for fractional systems using Lyapunov fu...
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| Veröffentlicht in: | Communications in nonlinear science & numerical simulation 2015-05, Vol.22 (1-3), p.650-659 |
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| creator | Duarte-Mermoud, Manuel A. Aguila-Camacho, Norelys Gallegos, Javier A. Castro-Linares, Rafael |
| description | •New Lemmas relating the Caputo Fractional Derivatives of general quadratic forms and the trace of the product of matrices.•Possibility to use general quadratics Lyapunov functions in the fractional-order extension of Lyapunov direct method.•Uniform stability for fractional systems using Lyapunov functions.•Stability analysis of Fractional Order Model Reference Adaptive Control (FOMRAC) schemes.
This paper presents two new lemmas related to the Caputo fractional derivatives, when α∈0,1, for the case of general quadratic forms and for the case where the trace of the product of a rectangular matrix and its transpose appear. Those two lemmas allow using general quadratic Lyapunov functions and the trace of a matrix inside a Lyapunov function respectively, in order to apply the fractional-order extension of Lyapunov direct method, to analyze the stability of fractional order systems (FOS). Besides, the paper presents a theorem for proving uniform stability in the sense of Lyapunov for fractional order systems. The theorem can be seen as a complement of other methods already available in the literature. The two lemmas and the theorem are applied to the stability analysis of two Fractional Order Model Reference Adaptive Control (FOMRAC) schemes, in order to prove the usefulness of the results. |
| doi_str_mv | 10.1016/j.cnsns.2014.10.008 |
| format | Article |
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This paper presents two new lemmas related to the Caputo fractional derivatives, when α∈0,1, for the case of general quadratic forms and for the case where the trace of the product of a rectangular matrix and its transpose appear. Those two lemmas allow using general quadratic Lyapunov functions and the trace of a matrix inside a Lyapunov function respectively, in order to apply the fractional-order extension of Lyapunov direct method, to analyze the stability of fractional order systems (FOS). Besides, the paper presents a theorem for proving uniform stability in the sense of Lyapunov for fractional order systems. The theorem can be seen as a complement of other methods already available in the literature. The two lemmas and the theorem are applied to the stability analysis of two Fractional Order Model Reference Adaptive Control (FOMRAC) schemes, in order to prove the usefulness of the results.</description><identifier>ISSN: 1007-5704</identifier><identifier>EISSN: 1878-7274</identifier><identifier>DOI: 10.1016/j.cnsns.2014.10.008</identifier><language>eng</language><publisher>Elsevier B.V</publisher><subject>Complement ; Computer simulation ; Derivatives ; Fractional adaptive systems ; Fractional calculus ; Fractional extension of Lyapunov direct method ; General quadratic Lyapunov functions ; Lyapunov functions ; Model reference adaptive control ; Nonlinearity ; Quadratic forms ; Stability ; Theorems ; Uniform stability of fractional order systems</subject><ispartof>Communications in nonlinear science & numerical simulation, 2015-05, Vol.22 (1-3), p.650-659</ispartof><rights>2014 Elsevier B.V.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c406t-b7cbf2eed1f31d3f6d4d49b44a859afa86306e04eab428373d23f256b11b0b653</citedby><cites>FETCH-LOGICAL-c406t-b7cbf2eed1f31d3f6d4d49b44a859afa86306e04eab428373d23f256b11b0b653</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://www.sciencedirect.com/science/article/pii/S100757041400481X$$EHTML$$P50$$Gelsevier$$H</linktohtml><link.rule.ids>314,776,780,3537,27901,27902,65306</link.rule.ids></links><search><creatorcontrib>Duarte-Mermoud, Manuel A.</creatorcontrib><creatorcontrib>Aguila-Camacho, Norelys</creatorcontrib><creatorcontrib>Gallegos, Javier A.</creatorcontrib><creatorcontrib>Castro-Linares, Rafael</creatorcontrib><title>Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems</title><title>Communications in nonlinear science & numerical simulation</title><description>•New Lemmas relating the Caputo Fractional Derivatives of general quadratic forms and the trace of the product of matrices.•Possibility to use general quadratics Lyapunov functions in the fractional-order extension of Lyapunov direct method.•Uniform stability for fractional systems using Lyapunov functions.•Stability analysis of Fractional Order Model Reference Adaptive Control (FOMRAC) schemes.
This paper presents two new lemmas related to the Caputo fractional derivatives, when α∈0,1, for the case of general quadratic forms and for the case where the trace of the product of a rectangular matrix and its transpose appear. Those two lemmas allow using general quadratic Lyapunov functions and the trace of a matrix inside a Lyapunov function respectively, in order to apply the fractional-order extension of Lyapunov direct method, to analyze the stability of fractional order systems (FOS). Besides, the paper presents a theorem for proving uniform stability in the sense of Lyapunov for fractional order systems. The theorem can be seen as a complement of other methods already available in the literature. The two lemmas and the theorem are applied to the stability analysis of two Fractional Order Model Reference Adaptive Control (FOMRAC) schemes, in order to prove the usefulness of the results.</description><subject>Complement</subject><subject>Computer simulation</subject><subject>Derivatives</subject><subject>Fractional adaptive systems</subject><subject>Fractional calculus</subject><subject>Fractional extension of Lyapunov direct method</subject><subject>General quadratic Lyapunov functions</subject><subject>Lyapunov functions</subject><subject>Model reference adaptive control</subject><subject>Nonlinearity</subject><subject>Quadratic forms</subject><subject>Stability</subject><subject>Theorems</subject><subject>Uniform stability of fractional order systems</subject><issn>1007-5704</issn><issn>1878-7274</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2015</creationdate><recordtype>article</recordtype><recordid>eNp9kD1PwzAQhiMEElD4BSweWVLsxImTgQEhvqRKLHS2HPuMXFK7-JxK_fe4LRIb030-r-7eorhhdM4oa-9Wc-3R47yijOfOnNLupLhgnehKUQl-mnNKRdkIys-LS8QVzVTf8Ivia4nOf5JP8BDVSL4nZaJKTpPFTm0mH7bETl4nFzySFMgmhi38zSbvbIhrgkkNbnRpR3JJbFQHIuuFaCAS3GGCNV4VZ1aNCNe_cVYsn58-Hl_LxfvL2-PDotSctqkchB5sBWCYrZmpbWu44f3AueqaXlnVtTVtgXJQA6-6WtSmqm3VtANjAx3app4Vt0fdfO33BJjk2qGGcVQewoSStUL0vO86kVfr46qOATGClZvo1iruJKNyb61cyYO1cm_tvpmtzdT9kYL8xdZBlKgdeA3GRdBJmuD-5X8A4PGGfg</recordid><startdate>20150501</startdate><enddate>20150501</enddate><creator>Duarte-Mermoud, Manuel A.</creator><creator>Aguila-Camacho, Norelys</creator><creator>Gallegos, Javier A.</creator><creator>Castro-Linares, Rafael</creator><general>Elsevier B.V</general><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>7TB</scope><scope>8FD</scope><scope>FR3</scope><scope>JQ2</scope><scope>KR7</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope></search><sort><creationdate>20150501</creationdate><title>Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems</title><author>Duarte-Mermoud, Manuel A. ; Aguila-Camacho, Norelys ; Gallegos, Javier A. ; Castro-Linares, Rafael</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c406t-b7cbf2eed1f31d3f6d4d49b44a859afa86306e04eab428373d23f256b11b0b653</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2015</creationdate><topic>Complement</topic><topic>Computer simulation</topic><topic>Derivatives</topic><topic>Fractional adaptive systems</topic><topic>Fractional calculus</topic><topic>Fractional extension of Lyapunov direct method</topic><topic>General quadratic Lyapunov functions</topic><topic>Lyapunov functions</topic><topic>Model reference adaptive control</topic><topic>Nonlinearity</topic><topic>Quadratic forms</topic><topic>Stability</topic><topic>Theorems</topic><topic>Uniform stability of fractional order systems</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Duarte-Mermoud, Manuel A.</creatorcontrib><creatorcontrib>Aguila-Camacho, Norelys</creatorcontrib><creatorcontrib>Gallegos, Javier A.</creatorcontrib><creatorcontrib>Castro-Linares, Rafael</creatorcontrib><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>Technology Research Database</collection><collection>Engineering Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Civil Engineering Abstracts</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Communications in nonlinear science & numerical simulation</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Duarte-Mermoud, Manuel A.</au><au>Aguila-Camacho, Norelys</au><au>Gallegos, Javier A.</au><au>Castro-Linares, Rafael</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems</atitle><jtitle>Communications in nonlinear science & numerical simulation</jtitle><date>2015-05-01</date><risdate>2015</risdate><volume>22</volume><issue>1-3</issue><spage>650</spage><epage>659</epage><pages>650-659</pages><issn>1007-5704</issn><eissn>1878-7274</eissn><abstract>•New Lemmas relating the Caputo Fractional Derivatives of general quadratic forms and the trace of the product of matrices.•Possibility to use general quadratics Lyapunov functions in the fractional-order extension of Lyapunov direct method.•Uniform stability for fractional systems using Lyapunov functions.•Stability analysis of Fractional Order Model Reference Adaptive Control (FOMRAC) schemes.
This paper presents two new lemmas related to the Caputo fractional derivatives, when α∈0,1, for the case of general quadratic forms and for the case where the trace of the product of a rectangular matrix and its transpose appear. Those two lemmas allow using general quadratic Lyapunov functions and the trace of a matrix inside a Lyapunov function respectively, in order to apply the fractional-order extension of Lyapunov direct method, to analyze the stability of fractional order systems (FOS). Besides, the paper presents a theorem for proving uniform stability in the sense of Lyapunov for fractional order systems. The theorem can be seen as a complement of other methods already available in the literature. The two lemmas and the theorem are applied to the stability analysis of two Fractional Order Model Reference Adaptive Control (FOMRAC) schemes, in order to prove the usefulness of the results.</abstract><pub>Elsevier B.V</pub><doi>10.1016/j.cnsns.2014.10.008</doi><tpages>10</tpages></addata></record> |
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| source | Elsevier ScienceDirect Journals |
| subjects | Complement Computer simulation Derivatives Fractional adaptive systems Fractional calculus Fractional extension of Lyapunov direct method General quadratic Lyapunov functions Lyapunov functions Model reference adaptive control Nonlinearity Quadratic forms Stability Theorems Uniform stability of fractional order systems |
| title | Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems |
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