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Fractal Fract
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Fractional-Order System: Control Theory and Applications
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Fractional-Order System: Control Theory and Applications

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

Deadline for manuscript submissions: closed (31 July 2022) | Viewed by 24164

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editors

Dr. Thach Ngoc Dinh

E-Mail Website
Guest Editor
Conservatoire National des Arts et Métiers (CNAM), Cedric-Laetitia, 292 Rue St-Martin, 75141 Paris CEDEX 03, France
Interests: state estimation; interval observer; robust control; output feedback; positive fractional-order systems
Special Issues, Collections and Topics in MDPI journals
Dr. Shyam Kamal

E-Mail Website
Guest Editor
Department of Electrical Engineering, Indian Institute of Technology (Banaras Hindu University), Varanasi 221005, India
Interests: stability and stabilization of fractional order systems; sliding mode control; nonlinear observers; contraction analysis
Special Issues, Collections and Topics in MDPI journals
Dr. Rajesh Kumar Pandey

E-Mail Website
Guest Editor
Department of Mathematical Sciences, Indian Institute of Technology (BHU) Varanasi, Varanasi 221005, Uttar Pradesh, India
Interests: fractional differential equations; fractional variational problems; applications of fractional calculus in image processing; computational methods
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

In the last two decades, fractional differential equations have been used more frequently in physics, signal processing, fluid mechanics, viscoelasticity, mathematical biology, electrochemistry, and many other fields, opening a new and more realistic way to capture memory-dependent phenomena and irregularities inside systems using more sophisticated mathematical analysis.

As a result of the growing applications, the study of stability of fractional differential equations has attracted much attention. Furthermore, in recent years, an increasing amount of attention has been given to fractional-order controllers. Some of these applications include optimal control, CRONE controllers, fractional PID controllers, lead-lag compensators, and sliding mode control.

The focus of this Special Issue is to continue to advance research on topics relating to fractional-order control theory and its applications to practical systems modeled using fractional-order differential equations such as design, implementation, and application of fractional-order control to electrical circuits and systems, mechanical systems, chemical systems, biological systems, finance systems, etc.

Topics that are invited for submission include (but are not limited to):

  • Fractional-order control theory for fractional-order systems;
  • Fractional-order control theory for integer-order systems
  • Lyapunov-based stability and stabilization of fractional-order systems;
  • Feedback linearization-based controller and observer design for fractional-order systems;
  • Digital implementation of fractional-order control;
  • Sliding mode control of fractional-order systems;
  • Finite, fixed, and predefined-time stability and stabilization of fractional-order systems;
  • Interval observer and set-membership design for fractional-order systems;
  • High-gain based observers and differentiator design for fractional-order systems;
  • Event-based control of fractional-order systems;
  • Incremental stability of fractional-order systems;
  • Control of non-minimum phase systems using fractional-order theory;
  • New physical interpretation of fractional-order operators and their relationship to control design;
  • Design and development of efficient battery management and state of heath estimation using fractional-order calculus;
  • Applications of fractional-order control to electrical, mechanical, chemical, finance, and biological systems;
  • Verification and reachability analysis of fractional-order differential equations.

Dr. Thach Ngoc Dinh
Dr. Shyam Kamal
Dr. Rajesh Kumar Pandey
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 control theory
  • Fractional-order controllers, observers, and differentiator design
  • Event-based control of fractional-order systems
  • Lyapunov analysis of fractional-order differential equations
  • Fractional differential equations
  • Fractional variational problems and fractional control problems
  • Analytical and computational methods for fractional-order systems

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Related Special Issue

Published Papers (11 papers)

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Editorial

Jump to: Research, Review

4 pages, 183 KiB  
Editorial
Fractional-Order System: Control Theory and Applications
by Thach Ngoc Dinh, Shyam Kamal and Rajesh Kumar Pandey
Fractal Fract. 2023, 7(1), 48; https://doi.org/10.3390/fractalfract7010048 - 31 Dec 2022
Cited by 5 | Viewed by 1695
Abstract
(Fractional) differential equations have seen increasing use in physics, signal processing, fluid mechanics, viscoelasticity, mathematical biology, electrochemistry, and many other fields over the last two decades, providing a new and more realistic way to capture memory-dependent phenomena and irregularities inside systems using more [...] Read more.
(Fractional) differential equations have seen increasing use in physics, signal processing, fluid mechanics, viscoelasticity, mathematical biology, electrochemistry, and many other fields over the last two decades, providing a new and more realistic way to capture memory-dependent phenomena and irregularities inside systems using more sophisticated mathematical analysis (see, for example, [...] Full article
(This article belongs to the Special Issue Fractional-Order System: Control Theory and Applications)

Research

Jump to: Editorial, Review

20 pages, 349 KiB  
Article
Synchronization of Fractional-Order Uncertain Delayed Neural Networks with an Event-Triggered Communication Scheme
by M. Hymavathi, M. Syed Ali, Tarek F. Ibrahim, B. A. Younis, Khalid I. Osman and Kanit Mukdasai
Fractal Fract. 2022, 6(11), 641; https://doi.org/10.3390/fractalfract6110641 - 2 Nov 2022
Cited by 5 | Viewed by 1524
Abstract
In this paper, the synchronization of fractional-order uncertain delayed neural networks with an event-triggered communication scheme is investigated. By establishing a suitable Lyapunov–Krasovskii functional (LKF) and inequality techniques, sufficient conditions are obtained under which the delayed neural networks are stable. The criteria are [...] Read more.
In this paper, the synchronization of fractional-order uncertain delayed neural networks with an event-triggered communication scheme is investigated. By establishing a suitable Lyapunov–Krasovskii functional (LKF) and inequality techniques, sufficient conditions are obtained under which the delayed neural networks are stable. The criteria are given in terms of linear matrix inequalities (LMIs). Based on the drive–response concept, the LMI approach, and the Lyapunov stability theorem, a controller is derived to achieve the synchronization. Finally, numerical examples are presented to confirm the effectiveness of the main results. Full article
(This article belongs to the Special Issue Fractional-Order System: Control Theory and Applications)
17 pages, 2893 KiB  
Article
Exponential Enclosures for the Verified Simulation of Fractional-Order Differential Equations
by Andreas Rauh
Fractal Fract. 2022, 6(10), 567; https://doi.org/10.3390/fractalfract6100567 - 5 Oct 2022
Cited by 4 | Viewed by 1256
Abstract
Fractional-order differential equations are powerful tools for the representation of dynamic systems that exhibit long-term memory effects. The verified simulation of such system models with the help of interval tools allows for the computation of guaranteed enclosures of the domains of reachable pseudo [...] Read more.
Fractional-order differential equations are powerful tools for the representation of dynamic systems that exhibit long-term memory effects. The verified simulation of such system models with the help of interval tools allows for the computation of guaranteed enclosures of the domains of reachable pseudo states over a certain finite time horizon. In the previous work of the author, an iteration scheme—derived on the basis of the Picard iteration—was published that makes use of Mittag-Leffler functions to determine guaranteed pseudo-state enclosures. In this paper, the corresponding iteration is generalized toward the use of exponential functions during the evaluation of the iteration scheme. Such exponential functions are well-known from a verified solution of integer-order sets of differential equations. The aim of this work is to demonstrate that the use of exponential functions instead of pure box-type interval enclosures for Mittag-Leffler functions does not only improve the tightness of the computed pseudo-state enclosures but also reduces the required computational effort. These statements are demonstrated with the help of a close-to-life simulation model for the charging/discharging dynamics of Lithium-ion batteries. Full article
(This article belongs to the Special Issue Fractional-Order System: Control Theory and Applications)
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23 pages, 1232 KiB  
Article
A Plea for the Integration of Fractional Differential Systems: The Initial Value Problem
by Nezha Maamri and Jean-Claude Trigeassou
Fractal Fract. 2022, 6(10), 550; https://doi.org/10.3390/fractalfract6100550 - 28 Sep 2022
Cited by 10 | Viewed by 1349
Abstract
The usual approach to the integration of fractional order initial value problems is based on the Caputo derivative, whose initial conditions are used to formulate the classical integral equation. Thanks to an elementary counter example, we demonstrate that this technique leads to wrong [...] Read more.
The usual approach to the integration of fractional order initial value problems is based on the Caputo derivative, whose initial conditions are used to formulate the classical integral equation. Thanks to an elementary counter example, we demonstrate that this technique leads to wrong free-response transients. The solution of this fundamental problem is to use the frequency-distributed model of the fractional integrator and its distributed initial conditions. Using this model, we solve the previous counter example and propose a methodology which is the generalization of the integer order approach. Finally, this technique is applied to the modeling of Fractional Differential Systems (FDS) and the formulation of their transients in the linear case. Two expressions are derived, one using the Mittag–Leffler function and a new one based on the definition of a distributed exponential function. Full article
(This article belongs to the Special Issue Fractional-Order System: Control Theory and Applications)
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16 pages, 7277 KiB  
Article
Spiral Dive Control of Underactuated AUV Based on a Single-Input Fractional-Order Fuzzy Logic Controller
by Zhiyu Cui, Lu Liu, Boyu Zhu, Lichuan Zhang, Yang Yu, Zhexuan Zhao, Shiyuan Li and Mingwei Liu
Fractal Fract. 2022, 6(9), 519; https://doi.org/10.3390/fractalfract6090519 - 14 Sep 2022
Cited by 4 | Viewed by 1627
Abstract
Autonomous underwater vehicles (AUVs) have broad applications owing to their ability to undertake long voyages, strong concealment, high level of intelligence and ability to replace humans in dangerous operations. AUV motion control systems can ensure stable operation in the complex ocean environment and [...] Read more.
Autonomous underwater vehicles (AUVs) have broad applications owing to their ability to undertake long voyages, strong concealment, high level of intelligence and ability to replace humans in dangerous operations. AUV motion control systems can ensure stable operation in the complex ocean environment and have attracted significant research attention. In this paper, we propose a single-input fractional-order fuzzy logic controller (SIFOFLC) as an AUV motion control system. First, a single-input fuzzy logic controller (SIFLC) was proposed based on the signed distance method, whose control input is the linear combination of the error signal and its derivative. The SIFLC offers a significant reduction in the controller design and calculation process. Then, a SIFOFLC was obtained with the derivative of the error signal extending to a fractional order and offering greater flexibility and adaptability. Finally, to verify the superiority of the proposed control algorithm, comparative numerical simulations in terms of spiral dive motion control were conducted. Meanwhile, the parameters of different controllers were optimized according to the hybrid particle swarm optimization (HPSO) algorithm. The simulation results illustrate the superior stability and transient performance of the proposed control algorithm. Full article
(This article belongs to the Special Issue Fractional-Order System: Control Theory and Applications)
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16 pages, 433 KiB  
Article
Finite Difference–Collocation Method for the Generalized Fractional Diffusion Equation
by Sandeep Kumar, Rajesh K. Pandey, Kamlesh Kumar, Shyam Kamal and Thach Ngoc Dinh
Fractal Fract. 2022, 6(7), 387; https://doi.org/10.3390/fractalfract6070387 - 11 Jul 2022
Cited by 10 | Viewed by 1736
Abstract
In this paper, an approximate method combining the finite difference and collocation methods is studied to solve the generalized fractional diffusion equation (GFDE). The convergence and stability analysis of the presented method are also established in detail. To ensure the effectiveness and the [...] Read more.
In this paper, an approximate method combining the finite difference and collocation methods is studied to solve the generalized fractional diffusion equation (GFDE). The convergence and stability analysis of the presented method are also established in detail. To ensure the effectiveness and the accuracy of the proposed method, test examples with different scale and weight functions are considered, and the obtained numerical results are compared with the existing methods in the literature. It is observed that the proposed approach works very well with the generalized fractional derivatives (GFDs), as the presence of scale and weight functions in a generalized fractional derivative (GFD) cause difficulty for its discretization and further analysis. Full article
(This article belongs to the Special Issue Fractional-Order System: Control Theory and Applications)
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15 pages, 651 KiB  
Article
Fractional-Order Interval Observer for Multiagent Nonlinear Systems
by Haoran Zhang, Jun Huang and Siyuan He
Fractal Fract. 2022, 6(7), 355; https://doi.org/10.3390/fractalfract6070355 - 25 Jun 2022
Cited by 14 | Viewed by 1638
Abstract
A framework of distributed interval observers is introduced for fractional-order multiagent systems in the presence of nonlinearity. First, a frame was designed to construct the upper and lower bounds of the system state. By using monotone system theory, the positivity of the error [...] Read more.
A framework of distributed interval observers is introduced for fractional-order multiagent systems in the presence of nonlinearity. First, a frame was designed to construct the upper and lower bounds of the system state. By using monotone system theory, the positivity of the error dynamics could be ensured, which implies that the bounds could trap the original state. Second, a sufficient condition was applied to guarantee the boundedness of distributed interval observers. Then, an extension of Lyapunov function in the fractional calculus field was the basis of the sufficient condition. An algorithm associated with the procedure of the observer design is also provided. Lastly, a numerical simulation is used to demonstrate the effectiveness of the distributed interval observer. Full article
(This article belongs to the Special Issue Fractional-Order System: Control Theory and Applications)
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13 pages, 295 KiB  
Article
Finite Time Stability of Fractional Order Systems of Neutral Type
by Abdellatif Ben Makhlouf and Dumitru Baleanu
Fractal Fract. 2022, 6(6), 289; https://doi.org/10.3390/fractalfract6060289 - 26 May 2022
Cited by 14 | Viewed by 2128
Abstract
This work deals with a new finite time stability (FTS) of neutral fractional order systems with time delay (NFOTSs). In light of this, FTSs of NFOTSs are demonstrated in the literature using the Gronwall inequality. The innovative aspect of our proposed study is [...] Read more.
This work deals with a new finite time stability (FTS) of neutral fractional order systems with time delay (NFOTSs). In light of this, FTSs of NFOTSs are demonstrated in the literature using the Gronwall inequality. The innovative aspect of our proposed study is the application of fixed point theory to show the FTS of NFOTSs. Finally, using two examples, the theoretical contributions are confirmed and substantiated. Full article
(This article belongs to the Special Issue Fractional-Order System: Control Theory and Applications)
18 pages, 778 KiB  
Article
Design and Application of an Interval Estimator for Nonlinear Discrete-Time SEIR Epidemic Models
by Awais Khan, Xiaoshan Bai, Muhammad Ilyas, Arshad Rauf, Wei Xie, Peiguang Yan and Bo Zhang
Fractal Fract. 2022, 6(4), 213; https://doi.org/10.3390/fractalfract6040213 - 9 Apr 2022
Cited by 6 | Viewed by 2734
Abstract
This paper designs an interval estimator for a fourth-order nonlinear susceptible-exposed-infected-recovered (SEIR) model with disturbances using noisy counts of susceptible people provided by Public Health Services (PHS). Infectious diseases are considered the main cause of deaths among the top ten worldwide, as per [...] Read more.
This paper designs an interval estimator for a fourth-order nonlinear susceptible-exposed-infected-recovered (SEIR) model with disturbances using noisy counts of susceptible people provided by Public Health Services (PHS). Infectious diseases are considered the main cause of deaths among the top ten worldwide, as per the World Health Organization (WHO). Therefore, tracking and estimating the evolution of these diseases are important to make intervention strategies. We study a real case in which some uncertain variables such as model disturbances, uncertain input and output measurement noise are not exactly available but belong to an interval. Moreover, the uncertain transmission bound rate from the susceptible towards the exposed stage is not available for measurement. We designed an interval estimator using an observability matrix that generates a tight interval vector for the actual states of the SEIR model in a guaranteed way without computing the observer gain. As the developed approach is not dependent on observer gain, our method provides more freedom. The convergence of the width to a known value in finite time is investigated for the estimated state vector to prove the stability of the estimation error, significantly improving the accuracy for the proposed approach. Finally, simulation results demonstrate the satisfying performance of the proposed algorithm. Full article
(This article belongs to the Special Issue Fractional-Order System: Control Theory and Applications)
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20 pages, 1738 KiB  
Article
Multiweighted-Type Fractional Fourier Transform: Unitarity
by Tieyu Zhao and Yingying Chi
Fractal Fract. 2021, 5(4), 205; https://doi.org/10.3390/fractalfract5040205 - 8 Nov 2021
Cited by 4 | Viewed by 1586
Abstract
The definition of the discrete fractional Fourier transform (DFRFT) varies, and the multiweighted-type fractional Fourier transform (M-WFRFT) is its extended definition. It is not easy to prove its unitarity. We use the weighted-type fractional Fourier transform, fractional-order matrix and eigendecomposition-type fractional Fourier transform [...] Read more.
The definition of the discrete fractional Fourier transform (DFRFT) varies, and the multiweighted-type fractional Fourier transform (M-WFRFT) is its extended definition. It is not easy to prove its unitarity. We use the weighted-type fractional Fourier transform, fractional-order matrix and eigendecomposition-type fractional Fourier transform as basic functions to prove and discuss the unitarity. Thanks to the growing body of research, we found that the effective weighting term of the M-WFRFT is only four terms, none of which are extended to M terms, as described in the definition. Furthermore, the program code is analyzed, and the result shows that the previous work (Digit Signal Process 2020: 104: 18) based on MATLAB for unitary verification is inaccurate. Full article
(This article belongs to the Special Issue Fractional-Order System: Control Theory and Applications)
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Review

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25 pages, 6256 KiB  
Review
A Review of Recent Developments in Autotuning Methods for Fractional-Order Controllers
by Cristina I. Muresan, Isabela Birs, Clara Ionescu, Eva H. Dulf and Robin De Keyser
Fractal Fract. 2022, 6(1), 37; https://doi.org/10.3390/fractalfract6010037 - 11 Jan 2022
Cited by 45 | Viewed by 4289
Abstract
The scientific community has recently seen a fast-growing number of publications tackling the topic of fractional-order controllers in general, with a focus on the fractional order PID. Several versions of this controller have been proposed, including different tuning methods and implementation possibilities. Quite [...] Read more.
The scientific community has recently seen a fast-growing number of publications tackling the topic of fractional-order controllers in general, with a focus on the fractional order PID. Several versions of this controller have been proposed, including different tuning methods and implementation possibilities. Quite a few recent papers discuss the practical use of such controllers. However, the industrial acceptance of these controllers is still far from being reached. Autotuning methods for such fractional order PIDs could possibly make them more appealing to industrial applications, as well. In this paper, the current autotuning methods for fractional order PIDs are reviewed. The focus is on the most recent findings. A comparison between several autotuning approaches is considered for various types of processes. Numerical examples are given to highlight the practicality of the methods that could be extended to simple industrial processes. Full article
(This article belongs to the Special Issue Fractional-Order System: Control Theory and Applications)
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