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Fractal Fract
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Advances in Fractional-Order Control for Nonlinear Systems
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Advances in Fractional-Order Control for Nonlinear Systems

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

Deadline for manuscript submissions: 16 November 2026 | Viewed by 2409

Editors

Dr. Masoud Sotoodeh-Bahraini

E-Mail Website
Guest Editor
Department of Mechanical Engineering, School of Engineering, University of Birmingham, Birmingham B15 2FG, UK
Interests: autonomous robots; industrial robots; human-robot interaction; visual odometry; visual SLAM; fractional control
Dr. Pedro Ferreira

E-Mail Website
Guest Editor
Intelligent Automation Centre, Wolfson School of Mechanical, Electrical and Manufacturing Engineering, Loughborough University, Leicestershire LE11 3TU, UK
Interests: symbiotic assembly systems; agent-based industrial control; cyber physical systems; human–machine interaction; collaborative robotics; self-adapting systems; self-learning systems; semantic technology; modular assembly systems; mechatronic systems; I4.0 engineering education
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This Special Issue brings together recent advances and innovative approaches in the field of fractional-order control applied to nonlinear systems. Fractional calculus, with its ability to model memory and hereditary properties, has emerged as a powerful tool for analysing and controlling the complex dynamical behaviours frequently observed in nonlinear systems. The articles in this collection present cutting-edge research on theoretical developments, stability analysis, controller design, robust and adaptive strategies, optimization techniques, and real-world implementations of fractional-order controllers.

The contributions also address challenges in foundational theory, practical implementation, computational complexity, parameter tuning, and real-time deployment—offering insights into future research directions in this rapidly evolving field.

Key topics include the following:

  • Novel stability criteria for fractional-order nonlinear systems, generalized Lyapunov-based analysis methods, and convergence proofs for adaptive fractional controllers.
  • Innovative control architectures such as fractional-order PID variants, variable-order controllers, fractional backstepping designs, and hybrid fractional–integer schemes tailored to nonlinear dynamics.
  • Applications of fractional-order control in robotics, autonomous systems with complex dynamics, and human–robot interaction.
  • Applications in power electronics and motor drives with inherent nonlinearities.
  • Applications in biomedical systems exhibiting memory-dependent responses.
  • Applications in industrial automation.
  • Applications in process control within chemical engineering.
  • Applications in mechatronic systems with hysteresis and friction effects.

This Special Issue aims to serve as a valuable resource for researchers, practitioners, and graduate students seeking to understand and apply fractional-order control methods to enhance the performance, robustness, and adaptability of nonlinear systems.

Dr. Masoud Sotoodeh-Bahraini
Dr. Pedro Ferreira
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 250 words) can be sent to the Editorial Office for assessment.

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-anonymized 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 control
  • nonlinear systems
  • fractional calculus
  • variable-order controllers
  • adaptive control
  • robust control
  • stability analysis
  • Lyapunov methods
  • fractional PID controllers
  • backstepping control
  • memory-dependent systems
  • hybrid fractional–integer control
  • computational complexity in control
  • real-time control systems
  • human–robot interaction
  • power electronics control
  • biomedical systems control
  • mechatronic systems
  • industrial automation
  • process control

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Published Papers (4 papers)

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21 pages, 5869 KB  
Article
Adaptive Fractional-Order Sliding-Mode Control with Extended State Observer for Autonomous Underwater Vehicles Under Uncertain Disturbances
by Nanmu Hui, Changjin Dong, Baoju Wu, Binbin Tu, Yan Huo and Zehao Wang
Fractal Fract. 2026, 10(6), 398; https://doi.org/10.3390/fractalfract10060398 - 10 Jun 2026
Viewed by 154
Abstract
In this paper, a composite control framework integrating feedback linearization, an extended state observer, and an adaptive fractional-order sliding-mode controller is presented for autonomous underwater vehicles operating under uncertain hydrodynamics and external disturbances. The proposed algorithm, named adaptive fractional-order sliding-mode control with extended [...] Read more.
In this paper, a composite control framework integrating feedback linearization, an extended state observer, and an adaptive fractional-order sliding-mode controller is presented for autonomous underwater vehicles operating under uncertain hydrodynamics and external disturbances. The proposed algorithm, named adaptive fractional-order sliding-mode control with extended state observer, aims to enhance trajectory-tracking accuracy, disturbance rejection, and robustness against model uncertainties beyond what is offered by conventional active disturbance rejection control and integer-order sliding-mode control. First, a fractional-order sliding surface with an extended state observer is introduced to estimate and compensate lumped disturbances, where the fractional operator provides intrinsic filtering and memory effects to reduce chattering. Second, an adaptive exponential reaching law with smooth switching is formulated to overcome the trade-off between convergence speed and chattering, and a Levant differentiator is employed for sensorless velocity estimation. Finally, the uniform ultimate boundedness of the closed-loop system is proved via Lyapunov stability theory. Comparative simulation studies on step, sinusoidal, and circular trajectories under external disturbances, measurement noise, and 50% parametric uncertainties demonstrate that the proposed controller achieves zero overshoot, suppresses position fluctuations by 97%, and reduces root mean square tracking errors by 38–70% relative to conventional methods, confirming its superior performance. Full article
(This article belongs to the Special Issue Advances in Fractional-Order Control for Nonlinear Systems)
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29 pages, 1925 KB  
Article
Practical Exponential Stability of Tempered ϖ-Fractional Systems: Lyapunov Criteria and Applications to Perturbed and Controlled Systems
by Ayed R. A. Alanzi, Raouf Fakhfakh, Abdellatif Ben Makhlouf and Omar Naifar
Fractal Fract. 2026, 10(5), 344; https://doi.org/10.3390/fractalfract10050344 - 19 May 2026
Viewed by 340
Abstract
In this paper, we investigate the practical exponential stability of a class of nonlinear systems governed by the tempered ϖ-Caputo fractional derivative. A new Lyapunov-based criterion is established to derive sufficient conditions ensuring ϖ-practical exponential stability. The obtained result is formulated [...] Read more.
In this paper, we investigate the practical exponential stability of a class of nonlinear systems governed by the tempered ϖ-Caputo fractional derivative. A new Lyapunov-based criterion is established to derive sufficient conditions ensuring ϖ-practical exponential stability. The obtained result is formulated in a general framework involving suitable growth bounds on the Lyapunov function together with a tempered fractional derivative inequality and a boundedness condition on a weighted integral term. The proposed theorem provides an explicit practical exponential estimate for the system trajectories and extends existing stability results that are available for standard fractional and tempered fractional systems. To demonstrate the applicability of the developed theory, two applications are presented. First, the general criterion is applied to a class of perturbed tempered ϖ-fractional systems, for which verifiable sufficient conditions are derived in terms of quadratic Lyapunov functions and perturbation bounds. Second, a state-feedback stabilization result is established for a class of nonlinear tempered fractional control systems, showing that the proposed theorem can be used as an effective tool for closed-loop practical exponential stabilization. Finally, numerical examples are provided to validate the theoretical developments and to illustrate the effectiveness of the proposed approach. An additional test case with η3>0 is included to demonstrate the nontrivial range of Theorem 1. Furthermore, a socio-economic tempered fractional cobweb model is incorporated to show how the proposed criterion applies to price-adjustment dynamics with memory and persistent market perturbations. Full article
(This article belongs to the Special Issue Advances in Fractional-Order Control for Nonlinear Systems)
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20 pages, 683 KB  
Article
Exploring Fixed-Time Synchronization of Fractional-Order Fuzzy Cellular Neural Networks with Information Interactions and Time-Varying Delays via Adaptive Multi-Module Control
by Hongguang Fan, Kaibo Shi, Anran Zhou, Fei Meng and Liang Jiang
Fractal Fract. 2026, 10(4), 253; https://doi.org/10.3390/fractalfract10040253 - 13 Apr 2026
Cited by 3 | Viewed by 401
Abstract
This article focuses on the fixed-time synchronization problem for fractional-order fuzzy cellular neural networks (FOFCNNs) with information interactions and time-varying delays. To capture the complex dynamics of practical networks, nonlinear activation functions along with fuzzy AND and OR operators are incorporated into the [...] Read more.
This article focuses on the fixed-time synchronization problem for fractional-order fuzzy cellular neural networks (FOFCNNs) with information interactions and time-varying delays. To capture the complex dynamics of practical networks, nonlinear activation functions along with fuzzy AND and OR operators are incorporated into the master–slave systems. To achieve fixed-time synchronization despite these complexities, a novel adaptive multi-module controller is proposed. This controller integrates three functionally distinct components to accelerate the convergence rate, eliminate the effects of delays, and introduce negative feedback during communication, respectively. By employing fractional calculus tools, inequality techniques, and the proposed control law, sufficient criteria for the synchronization of the considered systems are rigorously established. Compared with existing synchronization works, this paper has significant advantages in model generality and controller design. Additionally, an explicit settling-time estimate is derived, which depends solely on control parameters and is independent of the initial conditions. Full article
(This article belongs to the Special Issue Advances in Fractional-Order Control for Nonlinear Systems)
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30 pages, 1488 KB  
Article
Beyond Quaternions: Adaptive Fixed-Time Synchronization of High-Dimensional Fractional-Order Neural Networks Under Lévy Noise Disturbances
by Essia Ben Alaia, Slim Dhahri and Omar Naifar
Fractal Fract. 2025, 9(12), 823; https://doi.org/10.3390/fractalfract9120823 - 16 Dec 2025
Cited by 1 | Viewed by 808
Abstract
This paper develops a unified synchronization framework for octonion-valued fractional-order neural networks (FOOVNNs) subject to mixed delays, Lévy disturbances, and topology switching. A fractional sliding surface is constructed by combining I1−μeg with integral terms in powers of [...] Read more.
This paper develops a unified synchronization framework for octonion-valued fractional-order neural networks (FOOVNNs) subject to mixed delays, Lévy disturbances, and topology switching. A fractional sliding surface is constructed by combining I1−μeg with integral terms in powers of |eg|. The controller includes a nonsingular term −ρ2gsgc2sign(sg), a disturbance-compensation term −θ^gsign(sg), and a delay-feedback term −λgeg(t−τ), while dimension-aware adaptive laws ,CDtμρg=k1gN∥sg∥c2 and ,CDtμθ^g=k2gN∥sg∥ ensure scalability with network size. Fixed-time convergence is established via a fractional stochastic Lyapunov method, and predefined-time convergence follows by a time-scaling of the control channel. Markovian switching is treated through a mode-dependent Lyapunov construction and linear matrix inequality (LMI) conditions; non-Gaussian perturbations are handled using fractional Itô tools. The architecture admits observer-based variants and is implementation-friendly. Numerical results corroborate the theory: (i) Two-Node Baseline: The fixed-time design drives ∥e(t)∥1 to O(10−4) by t≈0.94s, while the predefined-time variant meets a user-set Tp=0.5s with convergence at t≈0.42s. (ii) Eight-Node Scalability: Sliding surfaces settle in an O(1) band, and adaptive parameter means saturate well below their ceilings. (iii) Hyperspectral (Synthetic): Reconstruction under Lévy contamination achieves a competitive PSNR consistent with hypercomplex modeling and fractional learning. (iv) Switching Robustness: under four modes and twelve random switches, the error satisfies maxt∥e(t)∥1≤0.15. The results support octonion-valued, fractionally damped controllers as practical, scalable mechanisms for robust synchronization under non-Gaussian noise, delays, and time-varying topologies. Full article
(This article belongs to the Special Issue Advances in Fractional-Order Control for Nonlinear Systems)
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