Commutativity of Cascaded Connected Fractional Order Linear Time-Varying Systems
In this work, we present a comprehensive study of the commutativity of fractional-order linear time-varying systems (LTVSs). Commutativity is a fundamental property in the analysis and control of dynamic systems and is often used to simplify the design of controllers. Fractionalorder systems, which...
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| Main Authors: | , , , , |
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| Format: | Article |
| Language: | en |
| Published: |
World Scientific
2025
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| Subjects: | |
| Online Access: | http://eprints.uthm.edu.my/12727/1/J19696_8ff506a5cb4f6606cdbd7debde056937.pdf http://eprints.uthm.edu.my/12727/ https://doi.org/10.1142/S0218126625500951 |
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| Summary: | In this work, we present a comprehensive study of the commutativity of fractional-order linear time-varying systems (LTVSs). Commutativity is a fundamental property in the analysis and control of dynamic systems and is often used to simplify the design of controllers. Fractionalorder systems, which are characterized by a noninteger-order derivative, have been widely
studied in recent years due to their ability to model a wide range of phenomena. However, the commutativity of fractional-order LTVSs has not been widely explored. In this work, we present a comprehensive study of the commutativity of fractional-order LTVSs. We ¯rst provide a mathematical de¯nition of commutativity for these systems and demonstrate that it is equivalent to the commutativity of their transfer functions. We then propose a method for verifying the general condition for commutativity of fractional-order LTVSs under zero initial conditions (ICs) and prove it mathematically. Based on our ¯ndings, we realized that the commutative requirements, properties, theories, and conditions are general for fractional-order LTVSs, please observed that some fractional-order LTVSs are commutative, some are not commutative, while
some are commutative under certain conditions. Based on this fact, we can say that not all fractional-order LTVSs are commutative. We apply explicit commutative results to several examples of fractional-order LTVSs. Our
theoretical and simulation results show a good agreement and prove that our fractional-order LTVSs are commutative under certain conditions, moreover, the commutativity property holds for certain conditions and classes of fractional-order LTVSs, but not for others. Because of the application of fraction commutativity in various ¯elds of science and engineering, we ¯nd it necessary to come up with explicit results for the ¯rst time. |
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