{"title":"Unified Predefined-Time Stability Theorem and Sliding Mode Control for Fractional-Order Nonlinear Systems","authors":"Jingang Liu, Ruiqi Li","doi":"10.1002/mma.10634","DOIUrl":null,"url":null,"abstract":"<div>\n \n <p>This paper proposes a class of predefined-time stability (PDTS) theorem and a fractional-order (FO) sliding mode control algorithm for the synchronization of FO nonlinear systems. Based on the simple tetration function, a new PDTS theorem and an FO sliding mode controller are established. Then, a new unified PDTS theorem is defined, which extends the existing Lyapunov function and provides a detailed mathematical proof. The Lyapunov function is used to construct a unified FO sliding mode control algorithm, and the mathematical proof that it satisfies the PDTS is given. Finally, the proposed framework is applied to the synchronization of two different chaotic systems, and numerical simulations prove that the unified approach can extend the existing PDTS theorems and is universal and effective.</p>\n </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"48 5","pages":"5755-5767"},"PeriodicalIF":2.1000,"publicationDate":"2025-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Methods in the Applied Sciences","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/mma.10634","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
This paper proposes a class of predefined-time stability (PDTS) theorem and a fractional-order (FO) sliding mode control algorithm for the synchronization of FO nonlinear systems. Based on the simple tetration function, a new PDTS theorem and an FO sliding mode controller are established. Then, a new unified PDTS theorem is defined, which extends the existing Lyapunov function and provides a detailed mathematical proof. The Lyapunov function is used to construct a unified FO sliding mode control algorithm, and the mathematical proof that it satisfies the PDTS is given. Finally, the proposed framework is applied to the synchronization of two different chaotic systems, and numerical simulations prove that the unified approach can extend the existing PDTS theorems and is universal and effective.
期刊介绍：
Mathematical Methods in the Applied Sciences publishes papers dealing with new mathematical methods for the consideration of linear and non-linear, direct and inverse problems for physical relevant processes over time- and space- varying media under certain initial, boundary, transition conditions etc. Papers dealing with biomathematical content, population dynamics and network problems are most welcome.
Mathematical Methods in the Applied Sciences is an interdisciplinary journal: therefore, all manuscripts must be written to be accessible to a broad scientific but mathematically advanced audience. All papers must contain carefully written introduction and conclusion sections, which should include a clear exposition of the underlying scientific problem, a summary of the mathematical results and the tools used in deriving the results. Furthermore, the scientific importance of the manuscript and its conclusions should be made clear. Papers dealing with numerical processes or which contain only the application of well established methods will not be accepted.
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