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Modeling, Optimization, and Control of Fractional-Order Neural Networks and Nonlinear Systems

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

Deadline for manuscript submissions: 1 October 2025 | Viewed by 14058

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Special Issue Editors

Prof. Dr. Kaibo Shi

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Guest Editor

School of Information Science and Engineering, Chengdu University, Chengdu 610106, China
Interests: fractional-order neural networks; nonlinear systems; networked control systems; control theory and application of neural network
Special Issues, Collections and Topics in MDPI journals

Dr. Zhinan Peng

E-Mail Website
Guest Editor

School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China
Interests: multiagent systems; reinforcement learning; robot control
Special Issues, Collections and Topics in MDPI journals

Prof. Dr. Xin Wang

E-Mail Website
Guest Editor

School of Cyber Science and Engineering, Sichuan University, Chengdu 610065, China
Interests: information physical systems; DoS attacks; artificial intelligence
Special Issues, Collections and Topics in MDPI journals

Dr. Xiao Cai

E-Mail Website
Guest Editor

1. School of Information and Software Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China
2. Cyberspace Institute of Advanced Technology, Guangzhou University, Guangzhou 510006, China
Interests: cyber-physical system; networked control system; cyber security control and optimization

Special Issue Information

Dear Colleagues,

With the increasing application of fractional-order theory in the fields of neural networks and nonlinear systems, the Special Issue "Modeling, Optimization, and Control of Fractional-Order Neural Networks and Nonlinear Systems" aims to provide a platform for researchers to showcase their latest research findings, innovative methods and application cases in this field.

The purpose of launching this Special Issue is to bring together significant developments concerning the modeling, optimization and control of fractional-order neural networks and nonlinear systems, and facilitate research collaboration and the exchange of ideas surrounding this topic. Potential topics include, but are not limited to, the following:

Modeling and analysis of fractional-order neural networks;

Reinforcement learning control and optimization of fractional-order neural networks;

Intelligent learning and adaptive control of fractional-order neural networks;

Robust distributed control methods of nonlinear systems;

Sampled-data and event-triggered intelligent control;

Distributed intelligent control and optimization applications;

Security control of networked control systems.

Prof. Dr. Kaibo Shi
Dr. Zhinan Peng
Prof. Dr. Xin Wang
Dr. Xiao Cai
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 order neural networks
nonlinear systems
networked control systems
modelling and analysis
control and optimization
intelligent learning
multiagent systems

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Published Papers (11 papers)

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Research

24 pages, 1057 KiB

Open AccessArticle

Synchronization for Delayed Fractional-Order Memristive Neural Networks Based on Intermittent-Hold Control with Application in Secure Communication

by Xueqi Yao, Jingxi Shi, Shouming Zhong and Yuanhua Du

Fractal Fract. 2024, 8(9), 519; https://doi.org/10.3390/fractalfract8090519 - 30 Aug 2024

Viewed by 614

Abstract

This article investigates the dynamic behaviors of delayed fractional-order memristive fuzzy cellular neural networks via the Lyapunov method. To address the delay terms of fractional-order systems, a novel lemma is provided to make the solutions of the systems exponentially stable. Furthermore, two new intermittent-hold controllers are designed to improve the robustness of the system and reduce the cost of the controller. One intermittent-hold controller is based on the feedback control strategy, while the other one integrates an adaptive control strategy. Moreover, two crucial theorems are derived from the proposed lemma and controllers, guaranteeing the exponential synchronization between drive and response systems. Finally, the superior performance of the controllers in achieving exponential synchronization is demonstrated through simulations. Full article

(This article belongs to the Special Issue Modeling, Optimization, and Control of Fractional-Order Neural Networks and Nonlinear Systems)

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21 pages, 358 KiB

Open AccessArticle

Analysis of Caputo Sequential Fractional Differential Equations with Generalized Riemann–Liouville Boundary Conditions

by Nallappan Gunasekaran, Murugesan Manigandan, Seralan Vinoth and Rajarathinam Vadivel

Fractal Fract. 2024, 8(8), 457; https://doi.org/10.3390/fractalfract8080457 - 5 Aug 2024

Viewed by 787

Abstract

This paper delves into a novel category of nonlocal boundary value problems concerning nonlinear sequential fractional differential equations, coupled with a unique form of generalized Riemann–Liouville fractional differential integral boundary conditions. For single-valued maps, we employ a transformation technique to convert the provided system into an equivalent fixed-point problem, which we then address using standard fixed-point theorems. Following this, we evaluate the stability of these solutions utilizing the Ulam–Hyres stability method. To elucidate the derived findings, we present constructed examples. Full article

(This article belongs to the Special Issue Modeling, Optimization, and Control of Fractional-Order Neural Networks and Nonlinear Systems)

30 pages, 3489 KiB

Open AccessArticle

Design of the Novel Fractional Order Hybrid Whale Optimizer for Thermal Wind Power Generation Systems with Integration of Chaos Infused Wind Power

by Abdul Wadood, Babar Sattar Khan, Hani Albalawi and Aadel Mohammed Alatwi

Fractal Fract. 2024, 8(7), 379; https://doi.org/10.3390/fractalfract8070379 - 27 Jun 2024

Cited by 2 | Viewed by 772

Abstract

This article introduces a novel optimization approach known as fractional order whale optimization algorithm (FWOA). The proposed optimizer incorporates the idea of fractional calculus (FC) into the mathematical structure of the conventional whale optimization algorithm (WOA). To validate the efficiency of the proposed FWOA, it is applied to address the challenges associated with the economic load dispatch (ELD) problem, which is a nonconvex, nonlinear, and non-smooth optimization problem. The objectives associated with ELD such as fuel cost and wind power generation cost minimization are achieved by taking into consideration different practical constraints like valve point loading effect (VPLE), transmission line losses, generator constraints, and stochastically variation of renewable energy sources (RES) integration. RES, particularly wind energy, has garnered more attention in recent times due to a range of environmental and economic factors. Stochastic wind (SW) power is also included in the ELD problem formulation. The incomplete gamma function (IGF) quantifies the influence of wind power. To assess its efficacy, the suggested approach is applied to a range of power systems including 3 generating units, 13 generating units and 40 generating units, consisting of 37 thermal units and 3 wind power units. To further strengthen the performance of the optimizer, the FWOA is hybridized with the interior point algorithm (IPA) to further refine the outcomes of the FWOA. The FWOA and IPA are used to address the problem of ELD while including the unpredictable nature of wind power. The simulation results of the suggested technique are compared with the most advanced heuristic optimization methods available, and it has been observed that the proposed optimizer obtained a superior and refined solution when compared to other state of the art optimization techniques. Furthermore, the efficacy of the suggested strategy in enhancing the solution of the ELD issue is validated through statistical analysis in terms of minimum fitness value. Full article

(This article belongs to the Special Issue Modeling, Optimization, and Control of Fractional-Order Neural Networks and Nonlinear Systems)

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30 pages, 6730 KiB

Open AccessArticle

An Enhanced Multiple Unmanned Aerial Vehicle Swarm Formation Control Using a Novel Fractional Swarming Strategy Approach

by Abdul Wadood, Al-Fahad Yousaf and Aadel Mohammed Alatwi

Fractal Fract. 2024, 8(6), 334; https://doi.org/10.3390/fractalfract8060334 - 3 Jun 2024

Cited by 1 | Viewed by 887

Abstract

This paper addresses the enhancement of multiple Unmanned Aerial Vehicle (UAV) swarm formation control in challenging terrains through the novel fractional memetic computing approach known as fractional-order velocity-pausing particle swarm optimization (FO-VPPSO). Existing particle swarm optimization (PSO) algorithms often suffer from premature convergence and an imbalanced exploration–exploitation trade-off, which limits their effectiveness in complex optimization problems such as UAV swarm control in rugged terrains. To overcome these limitations, FO-VPPSO introduces an adaptive fractional order β and a velocity pausing mechanism, which collectively enhance the algorithm’s adaptability and robustness. This study leverages the advantages of a meta-heuristic computing approach; specifically, fractional-order velocity-pausing particle swarm optimization is utilized to optimize the flying path length, mitigate the mountain terrain costs, and prevent collisions within the UAV swarm. Leveraging fractional-order dynamics, the proposed hybrid algorithm exhibits accelerated convergence rates and improved solution optimality compared to traditional PSO methods. The methodology involves integrating terrain considerations and diverse UAV control parameters. Simulations under varying conditions, including complex terrains and dynamic threats, substantiate the effectiveness of the approach, resulting in superior fitness functions for multi-UAV swarms. To validate the performance and efficiency of the proposed optimizer, it was also applied to 13 benchmark functions, including uni- and multimodal functions in terms of the mean average fitness value over 100 independent trials, and furthermore, an improvement at percentages of 29.05% and 2.26% is also obtained against PSO and VPPSO in the case of the minimum flight length, as well as 16.46% and 1.60% in mountain terrain costs and 55.88% and 31.63% in collision avoidance. This study contributes valuable insights to the optimization challenges in UAV swarm-formation control, particularly in demanding terrains. The FO-VPPSO algorithm showcases potential advancements in swarm intelligence for real-world applications. Full article

(This article belongs to the Special Issue Modeling, Optimization, and Control of Fractional-Order Neural Networks and Nonlinear Systems)

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24 pages, 469 KiB

Open AccessArticle

Time-Varying Function Matrix Projection Synchronization of Caputo Fractional-Order Uncertain Memristive Neural Networks with Multiple Delays via Mixed Open Loop Feedback Control and Impulsive Control

by Hongguang Fan, Yue Rao, Kaibo Shi and Hui Wen

Fractal Fract. 2024, 8(5), 301; https://doi.org/10.3390/fractalfract8050301 - 20 May 2024

Cited by 11 | Viewed by 953

Abstract

This paper shows solicitude for the generalized projective synchronization of Caputo fractional-order uncertain memristive neural networks (FOUMNNs) with multiple delays. By extending the constant scale factor to the time-varying function matrix, we establish an extraordinary synchronization mode called time-varying function matrix projection synchronization (TFMPS), which is a generalized version of traditional matrix projection synchronization, modified projection synchronization, complete synchronization, and anti-synchronization. To achieve the goal of TFMPS, we design a novel mixed controller including the open loop feedback control and impulsive control, which employs the state information in the time-delayed interval and the sampling information at the impulse instants. It has a prominent advantage that impulse intervals are not restricted by time delays. To establish the connection between the error system and the auxiliary system, a generalized fractional-order comparison theorem with time-varying coefficients and impulses is established. Applying the stability theory, the comparison theorem, and the Laplace transform, new synchronization criteria of FOUMNNs are acquired under the mixed impulsive control schemes, and the derived synchronization theorem and corollary can effectively expand the correlative synchronization achievements of fractional-order systems. Full article

(This article belongs to the Special Issue Modeling, Optimization, and Control of Fractional-Order Neural Networks and Nonlinear Systems)

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22 pages, 835 KiB

Open AccessArticle

Exponential H_∞ Output Control for Switching Fuzzy Systems via Event-Triggered Mechanism and Logarithmic Quantization

by Jiaojiao Ren, Can Zhao, Jianying Xiao, Renfu Luo and Nanrong He

Fractal Fract. 2024, 8(5), 290; https://doi.org/10.3390/fractalfract8050290 - 15 May 2024

Viewed by 835

Abstract

This paper investigates the problem of exponential

H_{\infty}

output control for switching fuzzy systems, considering both impulse and non-impulse scenarios. Unlike previous research, where the average dwell time (ADT:

τ_{a}

) and the upper bound of inter-event intervals (IEIs: T) [...] Read more.

This paper investigates the problem of exponential

H_{\infty}

output control for switching fuzzy systems, considering both impulse and non-impulse scenarios. Unlike previous research, where the average dwell time (ADT:

τ_{a}

) and the upper bound of inter-event intervals (IEIs: T) satisfy the condition

τ_{a} \geq \frac{ln μ + (α + β) T}{α} = \frac{ln μ + β T}{α} + T

, implying that frequent switching is difficult to achieve, this paper demonstrates that by adopting the mode-dependent event-triggered mechanism (ETM) and a switching law, frequent switching is indeed achieved. Moreover, the question of deriving the normal

L_{2}

norm constraint is solved through the ADT method, although only a weighted

L_{2}

norm constraint was obtained previously. Additionally, by constructing a controller-mode-dependent Lyapunov function and adopting logarithmic quantizers, the sufficient criteria of exponential

H_{\infty}

output control problem are presented. The validity of established results is demonstrated by a given numerical simulation. Full article

(This article belongs to the Special Issue Modeling, Optimization, and Control of Fractional-Order Neural Networks and Nonlinear Systems)

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21 pages, 6171 KiB

Open AccessArticle

Magnetically Suspended Control Sensitive Gyroscope Rotor High-Precision Deflection Decoupling Method using Quantum Neural Network and Fractional-Order Terminal Sliding Mode Control

by Yuan Ren, Lei Li, Weijie Wang, Lifen Wang and Weikun Pang

Fractal Fract. 2024, 8(2), 120; https://doi.org/10.3390/fractalfract8020120 - 17 Feb 2024

Cited by 2 | Viewed by 1507

Abstract

To achieve high-precision deflection control of a Magnetically Suspended Control and Sensitive Gyroscope rotor under high dynamic conditions, a deflection decoupling method using Quantum Radial Basis Function Neural Network and fractional-order terminal sliding mode control is proposed. The convergence speed and time complexity of the neural network controller limit the control accuracy and stability of rotor deflection under high-bandwidth conditions. To solve the problem, a quantum-computing-based structure optimization method for the Radial Basis Function Neural Network is proposed for the first time, where the input and the center of hidden layer basis function of the neural network are quantum-coded, and quantum rotation gates are designed to replace the Gaussian function. The parallel characteristic of quantum computing is utilized to reduce the time complexity and improve the convergence speed of the neural network. On top of that, in order to further address the issue of input jitter, a fractional-order terminal sliding mode controller based on the Quantum Radial Basis Function Neural Network is designed, the fractional-order differential sliding mode surface and the fractional-order convergence law are proposed to reduce the input jitter and achieve finite-time convergence of the controller, and the Quantum Radial Basis Function Neural Network is used to approximate the residual coupling and external disturbances of the system, resulting in improving the rotor deflection control accuracy. The semi-physical simulation experiments demonstrate the effectiveness and superiority of the proposed method. Full article

(This article belongs to the Special Issue Modeling, Optimization, and Control of Fractional-Order Neural Networks and Nonlinear Systems)

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13 pages, 506 KiB

Open AccessArticle

Privacy Preservation of Nabla Discrete Fractional-Order Dynamic Systems

by Jiayue Ma, Jiangping Hu and Zhinan Peng

Fractal Fract. 2024, 8(1), 46; https://doi.org/10.3390/fractalfract8010046 - 11 Jan 2024

Viewed by 1244

Abstract

This article investigates the differential privacy of the initial state for nabla discrete fractional-order dynamic systems. A novel differentially private Gaussian mechanism is developed which enhances the system’s security by injecting random noise into the output state. Since the existence of random noise gives rise to the difficulty of analyzing the nabla discrete fractional-order systems, to cope with this challenge, the observability of nabla discrete fractional-order systems is introduced, establishing a connection between observability and differential privacy of initial values. Based on it, the noise magnitude required for ensuring differential privacy is determined by utilizing the observability Gramian matrix of systems. Furthermore, an optimal Gaussian noise distribution that maximizes algorithmic performance while simultaneously ensuring differential privacy is formulated. Finally, a numerical simulation is provided to validate the effectiveness of the theoretical analysis. Full article

(This article belongs to the Special Issue Modeling, Optimization, and Control of Fractional-Order Neural Networks and Nonlinear Systems)

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21 pages, 701 KiB

Open AccessArticle

Improved Results on Delay-Dependent and Order-Dependent Criteria of Fractional-Order Neural Networks with Time Delay Based on Sampled-Data Control

by Junzhou Dai, Lianglin Xiong, Haiyang Zhang and Weiguo Rui

Fractal Fract. 2023, 7(12), 876; https://doi.org/10.3390/fractalfract7120876 - 11 Dec 2023

Cited by 1 | Viewed by 1284

Abstract

This paper studies the asymptotic stability of fractional-order neural networks (FONNs) with time delay utilizing a sampled-data controller. Firstly, a novel class of Lyapunov–Krasovskii functions (LKFs) is established, in which time delay and fractional-order information are fully taken into account. Secondly, by combining with the fractional-order Leibniz–Newton formula, LKFs, and other analysis techniques, some less conservative stability criteria that depend on time delay and fractional-order information are given in terms of linear matrix inequalities (LMIs). In the meantime, the sampled-data controller gain is developed under a larger sampling interval. Last, the proposed criteria are shown to be valid and less conservative than the existing ones using three numerical examples. Full article

(This article belongs to the Special Issue Modeling, Optimization, and Control of Fractional-Order Neural Networks and Nonlinear Systems)

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19 pages, 4068 KiB

Open AccessArticle

Multi-Machine Power System Transient Stability Enhancement Utilizing a Fractional Order-Based Nonlinear Stabilizer

by Arman Fathollahi and Björn Andresen

Fractal Fract. 2023, 7(11), 808; https://doi.org/10.3390/fractalfract7110808 - 7 Nov 2023

Cited by 8 | Viewed by 2266

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. Full article

(This article belongs to the Special Issue Modeling, Optimization, and Control of Fractional-Order Neural Networks and Nonlinear Systems)

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20 pages, 11275 KiB

Open AccessArticle

Development of an Efficient Variable Step-Size Gradient Method Utilizing Variable Fractional Derivatives

by Luotang Ye, Yanmao Chen and Qixian Liu

Fractal Fract. 2023, 7(11), 789; https://doi.org/10.3390/fractalfract7110789 - 30 Oct 2023

Viewed by 1382

Abstract

The fractional gradient method has garnered significant attention from researchers. The common view regarding fractional-order gradient methods is that they have a faster convergence rate compared to classical gradient methods. However, through conducting theoretical convergence analysis, we have revealed that the maximum convergence rate of the fractional-order gradient method is the same as that of the classical gradient method. This discovery implies that the superiority of fractional gradients may not reside in achieving fast convergence rates compared to the classical gradient method. Building upon this discovery, a novel variable fractional-type gradient method is proposed with an emphasis on automatically adjusting the step size. Theoretical analysis confirms the convergence of the proposed method. Numerical experiments demonstrate that the proposed method can converge to the extremum point both rapidly and accurately. Additionally, the Armijo criterion is introduced to ensure that the proposed gradient methods, along with various existing gradient methods, can select the optimal step size at each iteration. The results indicate that, despite the proposed method and existing gradient methods having the same theoretical maximum convergence speed, the introduced variable step size mechanism in the proposed method consistently demonstrates superior convergence stability and performance when applied to practical problems. Full article

(This article belongs to the Special Issue Modeling, Optimization, and Control of Fractional-Order Neural Networks and Nonlinear Systems)

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