Fractal and Fractional最新文献

{"title":"Fixed-Time Distributed Time-Varying Optimization for Nonlinear Fractional-Order Multiagent Systems with Unbalanced Digraphs","authors":"Kun Wang, Ping Gong, Zhiyao Ma","doi":"10.3390/fractalfract7110813","DOIUrl":"https://doi.org/10.3390/fractalfract7110813","url":null,"abstract":"This paper investigates the problem of fixed-time distributed time-varying optimization of a nonlinear fractional-order multiagent system (FOMAS) over a weight-unbalanced directed graph (digraph), where the heterogeneous unknown nonlinear functions and disturbances are involved. The aim is to cooperatively minimize a convex time-varying global cost function produced by a sum of time-varying local cost functions within a fixed time, where each time-varying local cost function does not have to be convex. Using a three-step design procedure, a fully distributed fixed-time optimization algorithm is constructed to achieve the objective. The first step is to design a fully distributed fixed-time estimator to estimate some centralized optimization terms within a fixed time T0. The second step is to develop a novel discontinuous fixed-time sliding mode algorithm with nominal controller to derive all the agents to the sliding-mode surface within a fixed time T1, and meanwhile the dynamics of each agent is described by a single-integrator MAS with nominal controller. In the third step, a novel estimator-based fully distributed fixed-time nominal controller for the single-integrator MAS is presented to guarantee all agents reach consensus within a fixed time T2, and afterwards minimize the convex time-varying global cost function within a fixed time T3. The upper bound of each fixed time Tm(m=0,1,2,3) is given explicitly, which is independent of the initial states. Finally, a numerical example is provided to validate the results.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":" 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135241307","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Research on Information Identification of Chaotic Map with Multi-Stability","authors":"You Li, Yuexi Peng","doi":"10.3390/fractalfract7110811","DOIUrl":"https://doi.org/10.3390/fractalfract7110811","url":null,"abstract":"Influenced by the rapid development of artificial intelligence, the identification of chaotic systems with intelligent optimization algorithms has received widespread attention in recent years. This paper focuses on the intelligent information identification of chaotic maps with multi-stability properties, and an improved sparrow search algorithm is proposed as the identification algorithm. Numerical simulations show that different initial values can lead to the same dynamic behavior, making it impossible to stably and accurately identify the initial values of multi-stability chaotic maps. An identification scheme without considering the initial values is proposed for solving this problem, and simulations demonstrate that the proposed method has the highest identification precision among seven existing intelligent algorithms and a certain degree of noise resistance. In addition, the above research reveals that chaotic systems with multi-stability may have more potential applications in fields such as secure communication.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":" 36","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135241750","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Disturbance Observer-Based Event-Triggered Adaptive Command Filtered Backstepping Control for Fractional-Order Nonlinear Systems and Its Application","authors":"Shuai Song, Xiaona Song, Inés Tejado","doi":"10.3390/fractalfract7110810","DOIUrl":"https://doi.org/10.3390/fractalfract7110810","url":null,"abstract":"This paper considers the disturbance observer-based event-triggered adaptive fuzzy tracking control issue for a class of fractional-order nonlinear systems (FONSs) with quantized signals and unknown disturbances. To improve the disturbance rejection ability, a fractional-order nonlinear disturbance observer (FONDO) is designed to estimate the unknown composite disturbances. Furthermore, by combining an improved fractional-order command-filtered backstepping control technique and an event-triggered control mechanism, an event-triggered adaptive fuzzy quantized control scheme is established, which guarantees the desired tracking performance can be achieved even in the presence of network constraint. Finally, the validity and superiority of the theoretic results are verified by a fractional-order horizontal platform system.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":" 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135243057","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Anisotropic Fractional Cosmology: K-Essence Theory","authors":"José Socorro, J. Juan Rosales, Leonel Toledo-Sesma","doi":"10.3390/fractalfract7110814","DOIUrl":"https://doi.org/10.3390/fractalfract7110814","url":null,"abstract":"In the particular configuration of the scalar field k-essence in the Wheeler–DeWitt quantum equation, for some age in the Bianchi type I anisotropic cosmological model, a fractional differential equation for the scalar field arises naturally. The order of the fractional differential equation is β=2α2α−1. This fractional equation belongs to different intervals depending on the value of the barotropic parameter; when ωX∈[0,1], the order belongs to the interval 1≤β≤2, and when ωX∈[−1,0), the order belongs to the interval 0<β≤1. In the quantum scheme, we introduce the factor ordering problem in the variables (Ω,ϕ) and its corresponding momenta (ΠΩ,Πϕ), obtaining a linear fractional differential equation with variable coefficients in the scalar field equation, then the solution is found using a fractional power series expansion. The corresponding quantum solutions are also given. We found the classical solution in the usual gauge N obtained in the Hamiltonian formalism and without a gauge. In the last case, the general solution is presented in a transformed time T(τ); however, in the dust era we found a closed solution in the gauge time τ.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":" 15","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135241607","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Computational Analysis of Fractional-Order KdV Systems in the Sense of the Caputo Operator via a Novel Transform","authors":"Mashael M. AlBaidani, Abdul Hamid Ganie, Adnan Khan","doi":"10.3390/fractalfract7110812","DOIUrl":"https://doi.org/10.3390/fractalfract7110812","url":null,"abstract":"The main features of scientific efforts in physics and engineering are the development of models for various physical issues and the development of solutions. In order to solve the time-fractional coupled Korteweg–De Vries (KdV) equation, we combine the novel Yang transform, the homotopy perturbation approach, and the Adomian decomposition method in the present investigation. KdV models are crucial because they can accurately represent a variety of physical problems, including thin-film flows and waves on shallow water surfaces. The fractional derivative is regarded in the Caputo meaning. These approaches apply straightforward steps through symbolic computation to provide a convergent series solution. Different nonlinear time-fractional KdV systems are used to test the effectiveness of the suggested techniques. The symmetry pattern is a fundamental feature of the KdV equations and the symmetrical aspect of the solution can be seen from the graphical representations. The numerical outcomes demonstrate that only a small number of terms are required to arrive at an approximation that is exact, efficient, and trustworthy. Additionally, the system’s approximative solution is illustrated graphically. The results show that these techniques are extremely effective, practically applicable for usage in such issues, and adaptable to other nonlinear issues.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":" 45","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135241311","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Fractional Soliton Wave Propagation of Non-Linear Volatility and Option Pricing Systems with a Sensitive Demonstration","authors":"Muhammad Bilal Riaz, Ali Raza Ansari, Adil Jhangeer, Muddassar Imran, Choon Kit Chan","doi":"10.3390/fractalfract7110809","DOIUrl":"https://doi.org/10.3390/fractalfract7110809","url":null,"abstract":"In this study, we explore a fractional non-linear coupled option pricing and volatility system. The model under consideration can be viewed as a fractional non-linear coupled wave alternative to the Black–Scholes option pricing governing system, introducing a leveraging effect where stock volatility corresponds to stock returns. Employing the inverse scattering transformation, we find that the Cauchy problem for this model is insolvable. Consequently, we utilize the Φ6-expansion algorithm to generate generalized novel solitonic analytical wave structures within the system. We present graphical representations in contour, 3D, and 2D formats to illustrate how the system’s behavior responds to the propagation of pulses, enabling us to predict suitable parameter values that align with the data. Finally, a conclusion is given.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":" 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135286115","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Multi-Machine Power System Transient Stability Enhancement Utilizing a Fractional Order-Based Nonlinear Stabilizer","authors":"Arman Fathollahi, Björn Andresen","doi":"10.3390/fractalfract7110808","DOIUrl":"https://doi.org/10.3390/fractalfract7110808","url":null,"abstract":"Given the intricate nature of contemporary energy systems, addressing the control and stability analysis of these systems necessitates the consideration of highly large-scale models. Transient stability analysis stands as a crucial challenge in enhancing energy system efficiency. Power System Stabilizers (PSSs), integrated within excitation control for synchronous generators, offer a cost-effective means to bolster power systems’ stability and reliability. In this study, we propose an enhanced nonlinear control strategy based on synergetic control theory for PSSs. This strategy aims to mitigate electromechanical oscillations and rectify the limitations associated with linear approximations within large-scale energy systems that incorporate thyristor-controlled series capacitors (TCSCs). To dynamically adjust the coefficients of the nonlinear controller, we employ the Fractional Order Fish Migration Optimization (FOFMO) algorithm, rooted in fractional calculus (FC) theory. The FOFMO algorithm adapts by updating position and velocity within fractional-order structures. To assess the effectiveness of the improved controller, comprehensive numerical simulations are conducted. Initially, we examine its performance in a single machine connected to the infinite bus (SMIB) power system under various fault conditions. Subsequently, we extend the application of the proposed nonlinear stabilizer to a two-area, four-machine power system. Our numerical results reveal highly promising advancements in both control accuracy and the dynamic characteristics of controlled power systems.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"5 8","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135432322","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"First-of-Its-Kind Frequency Enhancement Methodology Based on an Optimized Combination of FLC and TFOIDFF Controllers Evaluated on EVs, SMES, and UPFC-Integrated Smart Grid","authors":"Sultan Alghamdi, Mohammed Alqarni, Muhammad R. Hammad, Kareem M. AboRas","doi":"10.3390/fractalfract7110807","DOIUrl":"https://doi.org/10.3390/fractalfract7110807","url":null,"abstract":"The most recent advancements in renewable energy resources, as well as their broad acceptance in power sectors, have created substantial operational, security, and management concerns. As a result of the continual decrease in power system inertia, it is critical to maintain the normal operating frequency and reduce tie-line power changes. The preceding issues sparked this research, which proposes the Fuzzy Tilted Fractional Order Integral Derivative with Fractional Filter (FTFOIDFF), a unique load frequency controller. The FTFOIDFF controller described here combines the benefits of tilt, fuzzy logic, FOPID, and fractional filter controllers. Furthermore, the prairie dog optimizer (PDO), a newly developed metaheuristic optimization approach, is shown to efficiently tune the suggested controller settings as well as the forms of the fuzzy logic membership functions in the two-area hybrid power grid investigated in this paper. When the PDO results are compared to those of the Seagull Optimization Algorithm, the Runge Kutta optimizer, and the Chaos Game Optimizer for the same hybrid power system, PDO prevails. The system model incorporates physical constraints such as communication time delays and generation rate constraints. In addition, a unified power flow controller (UPFC) is put in the tie-line, and SMES units have been planned in both regions. Furthermore, the contribution of electric vehicles (EVs) is considered in both sections. The proposed PDO-based FTFOIDFF controller outperformed many PDO-based traditional (such as proportional integral derivative (PID), proportional integral derivative acceleration (PIDA), and TFOIDFF) and intelligent (such as Fuzzy PID and Fuzzy PIDA) controllers from the literature. The suggested PDO-based FTFOIDFF controller has excellent performance due to the usage of various load patterns such as step load perturbation, multi-step load perturbation, random load perturbation, random sinusoidal load perturbation, and pulse load perturbation. Furthermore, a variety of scenarios have been implemented to demonstrate the advantageous effects that SMES, UPFC, and EV units have on the overall performance of the system. The sensitivity of a system is ascertained by modifying its parameters from their standard configurations. According to the simulation results, the suggested PDO-based FTFOIDFF controller can improve system stability despite the multiple difficult conditions indicated previously. According to the MATLAB/Simulink data, the proposed method decreased the total fitness function to 0.0875, representing a 97.35% improvement over PID, 95.84% improvement over PIDA, 92.45% improvement over TFOIDFF, 83.43% improvement over Fuzzy PID, and 37.9% improvement over Fuzzy PIDA.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135590158","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Design and Performance Evaluation of a Second-Order Iterated Circular Minkowski Fractal Antenna for Ultra-Wideband Applications","authors":"Elijah Joseph, Pradeep Kumar, Thomas Afullo","doi":"10.3390/fractalfract7110806","DOIUrl":"https://doi.org/10.3390/fractalfract7110806","url":null,"abstract":"In this article, a second-order iterated Circular Minkowski fractal antenna (CMFA) tailored for ultra-wideband (UWB) applications is designed and developed. Leveraging the power of Minkowski fractal geometry, this antenna design achieves a high gain across the UWB frequency spectrum. The design utilizes a circular groove on the ground plane and an arc slot on the radiating element for improving the antenna performance. The proposed antenna is fabricated using cost-effective material, an FR-4 substrate. The antenna is simulated and optimized. The fabricated optimized antenna undergoes real-world testing. Measured results reveal an impressive 120.6% impedance bandwidth spanning from 3.37 GHz to 13.6 GHz, with resonant frequencies at 4.43 GHz, 6.07 GHz, and 9.3 GHz. Meanwhile, the simulated results indicate an impedance bandwidth of 118% ranging from 3.17 GHz to 12.44 GHz. Real-world measurements validate the anticipated UWB traits, closely aligning with the simulation data, and confirming efficient impedance matching with a VSWR of less than 2 across the 3.37 GHz to 13.6 GHz frequency range. The radiation pattern analysis demonstrates a robust bidirectional E-plane pattern and a nearly omnidirectional H-plane pattern. This research introduces a highly promising circular Minkowski fractal antenna for UWB applications, offering exceptional bandwidth and resonance characteristics. This antenna design holds excellent potential for multi-functional wireless systems and opens avenues for enhanced UWB communication and sensing capabilities in diverse applications.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"23 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135589745","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Mathematical Modeling of Breast Cancer Based on the Caputo–Fabrizio Fractal-Fractional Derivative","authors":"Muhammad Idrees, Abeer S. Alnahdi, Mdi Begum Jeelani","doi":"10.3390/fractalfract7110805","DOIUrl":"https://doi.org/10.3390/fractalfract7110805","url":null,"abstract":"Breast cancer ranks among the most prevalent malignancies affecting the female population and is a prominent contributor to cancer-related mortality. Mathematical modeling is a significant tool that can be employed to comprehend the dynamics of breast cancer progression and dissemination and to formulate novel therapeutic approaches. This paper introduces a mathematical model of breast cancer that utilizes the Caputo–Fabrizio fractal-fractional derivative. The aim is to elucidate and comprehend the intricate dynamics governing breast cancer cells and cytotoxic T lymphocytes in the context of the fractional derivative. The derivative presented herein offers a broader perspective than the conventional derivative, as it incorporates the intricate fractal characteristics inherent in the process of tumor proliferation. The significance of this study lies in its contribution to a novel mathematical model for breast cancer, which incorporates the fractal characteristics of tumor development. The present model possesses the capability to investigate the impacts of diverse treatment strategies on the proliferation of breast cancer, as well as to formulate novel treatment strategies that exhibit enhanced efficacy.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135635458","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}