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Fractal Fract
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Fractional Calculus in the Design, Control and Implementation of Complex...
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Fractional Calculus in the Design, Control and Implementation of Complex Systems

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Complexity".

Deadline for manuscript submissions: closed (26 September 2024) | Viewed by 10109

Special Issue Editors

Dr. Miguel A. Platas-Garza

E-Mail Website1 Website2
Guest Editor
Facultad de Ingeniería Mecánica y Eléctrica, Universidad Autónoma de Nuevo León, San Nicolás de los Garza 66455, Mexico
Interests: analysis and control of systems; signal processing
Special Issues, Collections and Topics in MDPI journals
Dr. Cornelio Posadas-Castillo
*
E-Mail Website
Guest Editor
Department of Electrical and Mechanical Engineering, University of Nuevo Leon, San Nicolás de los Garza 66455, Mexico
Interests: complex systems; synchronization; control of non-linear behavior; fractional calculus; private communication
* We dedicate the memory of the editor, Dr. Cornelio Posadas Castillo, who passed away during this special issue period.
Dr. Ernesto Zambrano-Serrano

E-Mail Website1 Website2
Guest Editor
Facultad de Ingeniería Mecánica y Eléctrica, Universidad Autónoma de Nuevo León, San Nicolás de los Garza 66455, Mexico
Interests: non-linear dynamics and chaos; fractional calculus; biological mathematics; engineering applications
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Fractional calculus studies the generalization of the differentiation operator in the case where the order is permitted to be any real or complex number. Particularly, fractional derivatives are considered non-local operators because they provide the memory effect in temporary applications. The ability to describe the hereditary characteristics of a system and its memory is the most fundamental advantage of fractional calculus over integer calculus. If the fractional differential operator is introduced into a system, the system can produce new complex dynamic behaviors.

Complex systems are of great significance in practical applications such as encryption, secure communication, random sequence, key design, signal processing, and signal detection. As such, it would be significant and necessary to control and analyze the complexity of these systems. At present, many new analytical techniques have been proposed to analyze and increase the complexity of these systems. However, there are still various theoretical and technical issues that should be addressed.

This Special Issue aims to introduce and discuss new results and new methods related to fractional calculus applications, and new results for control and analysis of nonlinear complex systems.

We welcome original research and review articles relating to the themes of this Special Issue. The invited topics include, but are not limited to:

  • Analysis, control and implementation of fractional order complex systems.
  • Analysis, control and implementation of variable order complex systems.
  • Fractional order chaotic systems and their applications.
  • Analysis, control and synchronization of fractional order complex networks.
  • Identification and modeling of fractional order complex systems.
  • Digital implementation of fractional order systems.
  • Signal processing through fractional order models.
  • Fractional order chaos-based cryptography.
  • Fractional order neural networks.
  • Fractional order systems for engineering and medical applications.
  • Complex data analysis.
  • Complexity in social dynamics.
  • Chaotic or hyperchaotic fractional order systems with large Lyapunov exponent.
  • Chaotic or hyperchaotic fractional order systems with a wide frequency spectrum.
  • Fractional order chaotic systems with high complexity, multistability or hidden dynamics.
  • Chaos, chimeras and spiral waves in fractional order biological systems.
  • Melnikov analysis and chaos in fractional order mechanical systems.
  • Complexity analysis and complexity measurement of chaotic time series.
  • Fractional order chaos-based sound steganography, image encryption and sound encryption.
  • Fractional order chaos-based secure communications.

This Special Issue is dedicated to Dr. Cornelio Posadas Castillo on the occasion of his unexpected passing.

Dr. Miguel A. Platas-Garza
Dr. Cornelio Posadas-Castillo
Dr. Ernesto Zambrano-Serrano
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • fractional calculus
  • chaotic systems
  • complex networks
  • variable order calculus
  • mathematical modeling
  • control theory
  • complexity
  • circuit implementation
  • chaos based cryptography
  • neural networks
  • identification and modeling
  • signal processing
  • complexity measures
  • chaos theory
  • applied mathematics

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Further information on MDPI's Special Issue polices can be found here.

Related Special Issue

Published Papers (7 papers)

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Research

19 pages, 6656 KiB  
Article
Dynamic Analysis and FPGA Implementation of Fractional-Order Hopfield Networks with Memristive Synapse
by Andrés Anzo-Hernández, Ernesto Zambrano-Serrano, Miguel Angel Platas-Garza and Christos Volos
Fractal Fract. 2024, 8(11), 628; https://doi.org/10.3390/fractalfract8110628 - 24 Oct 2024
Viewed by 669
Abstract
Memristors have become important components in artificial synapses due to their ability to emulate the information transmission and memory functions of biological synapses. Unlike their biological counterparts, which adjust synaptic weights, memristor-based artificial synapses operate by altering conductance or resistance, making them useful [...] Read more.
Memristors have become important components in artificial synapses due to their ability to emulate the information transmission and memory functions of biological synapses. Unlike their biological counterparts, which adjust synaptic weights, memristor-based artificial synapses operate by altering conductance or resistance, making them useful for enhancing the processing capacity and storage capabilities of neural networks. When integrated into systems like Hopfield neural networks, memristors enable the study of complex dynamic behaviors, such as chaos and multistability. Moreover, fractional calculus is significant for their ability to model memory effects, enabling more accurate simulations of complex systems. Fractional-order Hopfield networks, in particular, exhibit chaotic and multistable behaviors not found in integer-order models. By combining memristors with fractional-order Hopfield neural networks, these systems offer the possibility of investigating different dynamic phenomena in artificial neural networks. This study investigates the dynamical behavior of a fractional-order Hopfield neural network (HNN) incorporating a memristor with a piecewise segment function in one of its synapses, highlighting the impact of fractional-order derivatives and memristive synapses on the stability, robustness, and dynamic complexity of the system. Using a network of four neurons as a case study, it is demonstrated that the memristive fractional-order HNN exhibits multistability, coexisting chaotic attractors, and coexisting limit cycles. Through spectral entropy analysis, the regions in the initial condition space that display varying degrees of complexity are mapped, highlighting those areas where the chaotic series approach a pseudo-random sequence of numbers. Finally, the proposed fractional-order memristive HNN is implemented on a Field-Programmable Gate Array (FPGA), demonstrating the feasibility of real-time hardware realization. Full article
(This article belongs to the Special Issue Fractional Calculus in the Design, Control and Implementation of Complex Systems)
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17 pages, 400 KiB  
Article
Asymptotic Synchronization for Caputo Fractional-Order Time-Delayed Cellar Neural Networks with Multiple Fuzzy Operators and Partial Uncertainties via Mixed Impulsive Feedback Control
by Hongguang Fan, Chengbo Yi, Kaibo Shi and Xijie Chen
Fractal Fract. 2024, 8(10), 564; https://doi.org/10.3390/fractalfract8100564 - 28 Sep 2024
Cited by 1 | Viewed by 442
Abstract
To construct Caputo fractional-order time-delayed cellar neural networks (FOTDCNNs) that characterize real environments, this article introduces partial uncertainties, fuzzy operators, and nonlinear activation functions into the network models. Specifically, both the fuzzy AND operator and the fuzzy OR operator are contemplated in the [...] Read more.
To construct Caputo fractional-order time-delayed cellar neural networks (FOTDCNNs) that characterize real environments, this article introduces partial uncertainties, fuzzy operators, and nonlinear activation functions into the network models. Specifically, both the fuzzy AND operator and the fuzzy OR operator are contemplated in the master–slave systems. In response to the properties of the considered cellar neural networks (NNs), this article designs a new class of mixed control protocols that utilize both the error feedback information of systems and the sampling information of impulse moments to achieve network synchronization tasks. This approach overcomes the interference of time delays and uncertainties on network stability. By integrating the fractional-order comparison principle, fractional-order stability theory, and hybrid control schemes, readily verifiable asymptotic synchronization conditions for the studied fuzzy cellar NNs are established, and the range of system parameters is determined. Unlike previous results, the impulse gain spectrum considered in this study is no longer confined to a local interval (−2, 0) and can be extended to almost the entire real number domain. This spectrum extension relaxes the synchronization conditions, ensuring a broader applicability of the proposed control schemes. Full article
(This article belongs to the Special Issue Fractional Calculus in the Design, Control and Implementation of Complex Systems)
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21 pages, 1350 KiB  
Article
Convergence Analysis of Iterative Learning Control for Initialized Fractional Order Systems
by Xiaofeng Xu, Jiangang Lu and Jinshui Chen
Fractal Fract. 2024, 8(3), 168; https://doi.org/10.3390/fractalfract8030168 - 14 Mar 2024
Viewed by 1172
Abstract
Iterative learning control is widely applied to address the tracking problem of dynamic systems. Although this strategy can be applied to fractional order systems, most existing studies neglected the impact of the system initialization on operation repeatability, which is a critical issue since [...] Read more.
Iterative learning control is widely applied to address the tracking problem of dynamic systems. Although this strategy can be applied to fractional order systems, most existing studies neglected the impact of the system initialization on operation repeatability, which is a critical issue since memory effect is inherent for fractional operators. In response to the above deficiencies, this paper derives robust convergence conditions for iterative learning control under non-repetitive initialization functions, where the bound of the final tracking error depends on the shift degree of the initialization function. Model nonlinearity, initial error, and channel noises are also discussed in the derivation. On this basis, a novel initialization learning strategy is proposed to obtain perfect tracking performance and desired initialization trajectory simultaneously, providing a new approach for fractional order system design. Finally, two numerical examples are presented to illustrate the theoretical results and their potential applications. Full article
(This article belongs to the Special Issue Fractional Calculus in the Design, Control and Implementation of Complex Systems)
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13 pages, 3975 KiB  
Article
System Identification and Fractional-Order Proportional–Integral–Derivative Control of a Distributed Piping System
by Xiaomeng Zhang, Shuo Zhang, Furui Xiong, Lu Liu, Lichuan Zhang, Xuan Han, Heng Wang, Yanzhu Zhang and Ranzhen Ren
Fractal Fract. 2024, 8(2), 122; https://doi.org/10.3390/fractalfract8020122 - 19 Feb 2024
Cited by 2 | Viewed by 1414
Abstract
The vibration of piping systems is one of the most important causes of accelerated equipment wear and reduced work efficiency and safety. In this study, an active vibration control method based on a fractional-order proportional–integral–derivative (PID) controller was proposed to suppress pipeline vibration [...] Read more.
The vibration of piping systems is one of the most important causes of accelerated equipment wear and reduced work efficiency and safety. In this study, an active vibration control method based on a fractional-order proportional–integral–derivative (PID) controller was proposed to suppress pipeline vibration and reduce pipeline damage. First, a mathematical model of the distributed piping system was established using the finite element analysis method, and the characteristics of the distributed piping system were studied effectively. Further, the time-frequency domain parameter identification method was used to realise the system identification of the cross-point vibration transfer function between the brake and sensor, and the particle swarm optimisation algorithm was utilised to further optimise the transfer function parameters to improve the system identification accuracy. Therefore, a fractional-order PID controller was designed using the D-decomposition method, and the optimal controller parameters were obtained. The experimental and numerical simulation results show that the improved system identification algorithm can significantly improve modelling accuracy. In addition, the designed fractional-order PID controller can effectively reduce the system’s overshoot, oscillation time, and adjustment time, thereby reducing the vibration response of piping systems. Full article
(This article belongs to the Special Issue Fractional Calculus in the Design, Control and Implementation of Complex Systems)
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17 pages, 406 KiB  
Article
Diffusion of an Active Particle Bound to a Generalized Elastic Model: Fractional Langevin Equation
by Alessandro Taloni
Fractal Fract. 2024, 8(2), 76; https://doi.org/10.3390/fractalfract8020076 - 24 Jan 2024
Cited by 2 | Viewed by 1663
Abstract
We investigate the influence of a self-propelling, out-of-equilibrium active particle on generalized elastic systems, including flexible and semi-flexible polymers, fluid membranes, and fluctuating interfaces, while accounting for long-ranged hydrodynamic effects. We derive the fractional Langevin equation governing the dynamics of the active particle, [...] Read more.
We investigate the influence of a self-propelling, out-of-equilibrium active particle on generalized elastic systems, including flexible and semi-flexible polymers, fluid membranes, and fluctuating interfaces, while accounting for long-ranged hydrodynamic effects. We derive the fractional Langevin equation governing the dynamics of the active particle, as well as that of any other passive particle (or probe) bound to the elastic system. This equation analytically demonstrates how the active particle dynamics is influenced by the interplay of both the non-equilibrium force and of the viscoelastic environment. Our study explores the diffusional behavior emerging for both the active particle and a distant probe. The active particle undergoes three different surprising and counter-intuitive regimes identified by the distinct dynamical time-scales: a pseudo-ballistic initial phase, a drastic decrease in the mobility, and an asymptotic subdiffusive regime. Full article
(This article belongs to the Special Issue Fractional Calculus in the Design, Control and Implementation of Complex Systems)
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23 pages, 5579 KiB  
Article
Fractional Order Weighted Mixed Sensitivity-Based Robust Controller Design and Application for a Nonlinear System
by Erdem Ilten
Fractal Fract. 2023, 7(10), 769; https://doi.org/10.3390/fractalfract7100769 - 19 Oct 2023
Cited by 2 | Viewed by 1308
Abstract
This paper focuses on fractional-order modeling and the design of a robust speed controller for a nonlinear system. An induction motor (IM), widely used in Electrical Vehicles (EVs), is preferred in this study as a well-known nonlinear system. The major challenge in designing [...] Read more.
This paper focuses on fractional-order modeling and the design of a robust speed controller for a nonlinear system. An induction motor (IM), widely used in Electrical Vehicles (EVs), is preferred in this study as a well-known nonlinear system. The major challenge in designing a robust speed controller for IM is the insufficiency of the machine model due to inherent machine dynamics. Fractional calculus is employed to model the IM using the small-signal method, accounting for model uncertainties. In this context, experimental data is approximated using a fractional-order small-signal transfer function. Consequently, a mixed sensitivity problem is formulated with fractional-order weighting functions. The primary advantage of these weighting functions is their greater flexibility in solving the mixed sensitivity problem by involving more coefficients. Hereby, three robust speed controllers are designed using the PID toolkit of the Matlab program and solving the H∞ mixed sensitivity problem, respectively. The novelty and contribution of the proposed method lie in maintaining the closed-loop response within a secure margin determined by fractional weighting functions while addressing the controller design. After evaluating the robust speed controllers with Bode diagrams, it is proven that all the designed controllers meet the desired nominal performance and robustness criteria. Subsequently, real-time implementations of the designed controllers are performed using the dsPIC microcontroller unit. Experimental results confirm that the designed H∞-based fractional-order proportional-integral-derivative (FOPID) controller performs well in terms of tracking dynamics, exhibits robustness against load disturbances, and effectively suppresses sensor noise compared to the robust PID and fixed-structured H∞ controller. Full article
(This article belongs to the Special Issue Fractional Calculus in the Design, Control and Implementation of Complex Systems)
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38 pages, 3959 KiB  
Article
An Improved Marine Predators Algorithm-Tuned Fractional-Order PID Controller for Automatic Voltage Regulator System
by Mohd Zaidi Mohd Tumari, Mohd Ashraf Ahmad, Mohd Helmi Suid and Mok Ren Hao
Fractal Fract. 2023, 7(7), 561; https://doi.org/10.3390/fractalfract7070561 - 20 Jul 2023
Cited by 30 | Viewed by 2039
Abstract
One of the most popular controllers for the automatic voltage regulator (AVR) in maintaining the voltage level of a synchronous generator is the fractional-order proportional–integral-derivative (FOPID) controller. Unfortunately, tuning the FOPID controller is challenging since there are five gains compared to the three [...] Read more.
One of the most popular controllers for the automatic voltage regulator (AVR) in maintaining the voltage level of a synchronous generator is the fractional-order proportional–integral-derivative (FOPID) controller. Unfortunately, tuning the FOPID controller is challenging since there are five gains compared to the three gains of a conventional proportional–integral–derivative (PID) controller. Therefore, this research work presents a variant of the marine predators algorithm (MPA) for tuning the FOPID controller of the AVR system. Here, two modifications are applied to the existing MPA: the hybridization between MPA and the safe experimentation dynamics algorithm (SEDA) in the updating mechanism to solve the local optima issue, and the introduction of a tunable step size adaptive coefficient (CF) to improve the searching capability. The effectiveness of the proposed method in tuning the FOPID controller of the AVR system was assessed in terms of the convergence curve of the objective function, the statistical analysis of the objective function, Wilcoxon’s rank test, the step response analysis, stability analyses, and robustness analyses where the AVR system was subjected to noise, disturbance, and parameter uncertainties. We have shown that our proposed controller has improved the AVR system’s transient response and also produced about two times better results for objective function compared with other recent metaheuristic optimization-tuned FOPID controllers. Full article
(This article belongs to the Special Issue Fractional Calculus in the Design, Control and Implementation of Complex Systems)
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