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Fractional Order Controllers: Design and Applications

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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: closed (20 March 2023) | Viewed by 21027

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Special Issue Editors

Dr. Isabela Roxana Birs

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1. Research Group of Dynamical Systems and Control, Faculty of Engineering and Architecture, Ghent University, Technologiepark 125, B-9052 Ghent, Belgium
2. Automation Department, Technical University of Cluj-Napoca, Memorandumului 28, 400114 Cluj-Napoca, Romania
Interests: fractional calculus; fractional order control; biomedical systems; vibration suppression; non-Newtonian models
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Dr. Cristina I. Muresan

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Faculty of Automation and Computer Science, Department of Automation, Technical University of Cluj-Napoca, Memorandumului 28, 400014 Cluj-Napoca, Romania
Interests: fractional calculus; predictive control; biomedical engineering; dead-time compensation
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Prof. Dr. Clara Ionescu

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Guest Editor

Department of Electrical Energy, Systems and Automation, Ghent University, 9000 Ghent, Belgium
Interests: fractional calculus; biomedical engineering; anaesthesia control; viscoelastic phenomena; control engineering
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Special Issue Information

Dear Colleagues,

Fractional calculus is an emerging field in system identification and control engineering. It is well known that a fractional order model is more suitable for accurately describing complex physical phenomena such as viscoelasticity, diffusion in porous media and various biomedical processes. From the control perspective, fractional calculus extends the widely popular proportional integral derivative (PID) controller to a more versatile fractional order proportional integral derivative controller (FOPID) by adding two additional parameters to the controller’s transfer function consisting of arbitrary, non-integer orders of integration and differentiation. The additional parameters enable the FOPID controller to satisfy a more restrictive set of specifications than the PID controller, resulting in more degrees of freedom, increased stability and improved performance of the closed loop system. Furthermore, another major advantage consists of being able to directly impose robustness specification (usually through the isodamping property) in the tuning procedure. Apart from the FOPID controller, there are also other control strategies that have been extended using fractional calculus with promising results.

This Special Issue focuses on design strategies of fractional order controllers and its various applications. The aim is to present the latest advances in theory, design, implementation and validation of any kind of fractional order control strategy for both integer and fractional order processes. Topics that are invited for submission include (but are not limited to):

Theoretical aspects of fractional order control (e.g., stability analysis)
Fractional order controller design strategies
Digital and analog approximations of fractional order elements
Implementation of fractional order controllers (discrete-time implementation strategies, control effort assessment, etc.)
Comparisons between fractional and integer order controllers
Experimental implementation and validation
Fractional order control in Industry 4.0
Applications of fractional order control strategies

Dr. Isabela Roxana Birs
Dr. Cristina I. Muresan
Prof. Dr. Clara Ionescu
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.

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Published Papers (9 papers)

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Research

15 pages, 3071 KiB

Open AccessArticle

Trajectory Tracking Control of Euler–Lagrange Systems Using a Fractional Fixed-Time Method

by Saim Ahmed, Ahmad Taher Azar, Mohamed Tounsi and Zeeshan Anjum

Fractal Fract. 2023, 7(5), 355; https://doi.org/10.3390/fractalfract7050355 - 27 Apr 2023

Cited by 15 | Viewed by 1440

Abstract

The results of this research provide fixed-time fractional-order control for Euler–Lagrange systems that are subject to external disturbances. The first step in the process of developing a new system involves the introduction of a method known as fractional-order fixed-time non-singular terminal sliding mode control (FoFtNTSM). The advantages of fractional-order calculus and NTSM are brought together in this system, which result in rapid convergence, fixed-time stability, and smooth control inputs. Lyapunov analysis reveals whether the closed-loop system is stable over the duration of the time period specified. The performance of the suggested method when applied to the dynamics of the Euler–Lagrange system is evaluated and demonstrated with the help of computer simulations. Full article

(This article belongs to the Special Issue Fractional Order Controllers: Design and Applications)

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13 pages, 778 KiB

Open AccessArticle

Approximate Analytical Methods for a Fractional-Order Nonlinear System of Jaulent–Miodek Equation with Energy-Dependent Schrödinger Potential

by Saleh Alshammari, M. Mossa Al-Sawalha and Rasool Shah

Fractal Fract. 2023, 7(2), 140; https://doi.org/10.3390/fractalfract7020140 - 2 Feb 2023

Cited by 40 | Viewed by 1835

Abstract

In this paper, we study the numerical solution of fractional Jaulent–Miodek equations with the help of two modified methods: coupled fractional variational iteration transformation technique and the Adomian decomposition transformation technique. The Jaulent–Miodek equation has applications in several related fields of physics, including control theory of dynamical systems, anomalous transport, image and signal processing, financial modelings, nanotechnology, viscoelasticity, nanoprecipitate growth in solid solutions, random walk, modeling for shape memory polymers, condensed matter physics, fluid mechanics, optics and plasma physics. The results are presented as a series of quickly converging solutions. Analytical solutions have been performed in absolute error to confirm the proposed methodologies are trustworthy and accurate. The generated solutions are visually illustrated to guarantee the validity and applicability of the taken into consideration algorithm. The study’s findings show that, compared to alternative analytical approaches for analyzing fractional non-linear coupled Jaulent–Miodek equations, the Adomian decomposition transform method and the variational iteration transform method are computationally very efficient and accurate. Full article

(This article belongs to the Special Issue Fractional Order Controllers: Design and Applications)

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15 pages, 5558 KiB

Open AccessArticle

Fractional-Order PID Controller Based on Immune Feedback Mechanism for Time-Delay Systems

by Adel Makhbouche, Badreddine Boudjehem, Isabela Birs and Cristina I. Muresan

Fractal Fract. 2023, 7(1), 53; https://doi.org/10.3390/fractalfract7010053 - 1 Jan 2023

Cited by 8 | Viewed by 2652

Abstract

The control of processes with time delays is crucial in process industries such as petrochemical, hydraulic, and manufacturing. It is a challenging task for automation engineers, as it may affect both phase and gain margins. In this case, a robust control system is preferred. This article presents a novel controller structure combining computational intelligence (CI) and fractional-order control. A fractional-order PID (FOPID) controller based on a bio-inspired immune feedback mechanism (IFM) is developed for controlling processes described as first-order plus time-delay systems (FOPTD). A genetic algorithm (GA) is used to optimize the controller parameters. Fractional-order control has been used to give extra flexibilities and an immune feedback mechanism for its self-adaptability. Numerical simulations are presented to validate the proposed control strategy in terms of reference tracking and disturbance rejection. Comparative simulation results with an immune integer-order PID controller are also included to demonstrate the efficiency of the proposed fractional-order method. Full article

(This article belongs to the Special Issue Fractional Order Controllers: Design and Applications)

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20 pages, 4485 KiB

Open AccessArticle

Fractional Transformation-Based Intelligent H-Infinity Controller of a Direct Current Servo Motor

by Muhammad Zia Ur Rahman, Víctor Leiva, Carlos Martin-Barreiro, Imran Mahmood, Muhammad Usman and Mohsin Rizwan

Fractal Fract. 2023, 7(1), 29; https://doi.org/10.3390/fractalfract7010029 - 28 Dec 2022

Cited by 17 | Viewed by 2409

Abstract

Direct current (DC) servo motors are central to many complex systems, such as electrical, electro-mechanical, and electro-hydraulic frameworks. In practice, these systems can have nonlinear characteristics and parameter variations. Accurate model representation and position tracking of DC motors are the main issues in many real systems, such as twin rotors, aircraft, airships, and robot manipulators. The precise position tracking of these systems has already been achieved using conventional H-infinity (H

_{\infty}

) controllers. However, the order and structure become more intricate when employing complex weights to shape the closed-loop system, which limits the current proposals. To overcome the above-mentioned limitations, in this article, we provide a precise angular position tracking of a DC servo motor utilizing an intelligent, robust linear controller based on a fixed-structure linear fractional transformation. The conventional H

_{\infty}

controllers are based on the minimization of an unstructured linear fractional transformation objective function that leads to a complex design of these controllers. The main advantage of the proposed intelligent H

_{\infty}

synthesis is the fixed and simple structure that increases its practical implementation. The methodology is formulated in the MATLAB software for the robust design of the proposed synthesis based on an intelligent fixed-structure H

_{\infty}

optimization. Simulation results are compared with conventional H

_{\infty}

and proportional-integral-derivative controllers. The results are also validated experimentally. Full article

(This article belongs to the Special Issue Fractional Order Controllers: Design and Applications)

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18 pages, 5447 KiB

Open AccessArticle

Fractional Order Model Identification of a Person with Parkinson’s Disease for Wheelchair Control

by Mircea Ivanescu, Ioan Dumitrache, Nirvana Popescu and Decebal Popescu

Fractal Fract. 2023, 7(1), 23; https://doi.org/10.3390/fractalfract7010023 - 26 Dec 2022

Cited by 1 | Viewed by 1330

Abstract

The paper focuses on the design of an intelligent interface that compensates for the incapacity of a person with Parkinson’s disease to drive a wheelchair. The fractional order model that defines a person with Parkinson’s disease is investigated. An identification technique based on the analysis of the frequency behavior of the movement of a wheelchair driven by a with Parkinson’s disease person on the test trajectory is proposed and a delay time crossover model with fractional order exponent

β = 1.5

is inferred. The fractional dynamic model of the “disabled man-wheelchair” system is discussed and a control system is proposed to compensate for the inability of the wheelchair driver. The conditions that ensure the stability of the closed loop control system are inferred. An experimental technique for analyzing movement performance is developed and a quality index is proposed to evaluate these experiments. The values of this index on the tests performed on Parkinson’s patients are analyzed and discussed. Full article

(This article belongs to the Special Issue Fractional Order Controllers: Design and Applications)

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21 pages, 4377 KiB

Open AccessArticle

A Robust Controller of a Reactor Electromicrobial System Based on a Structured Fractional Transformation for Renewable Energy

by Muhammad Zia Ur Rahman, Rabia Liaquat, Mohsin Rizwan, Carlos Martin-Barreiro and Víctor Leiva

Fractal Fract. 2022, 6(12), 736; https://doi.org/10.3390/fractalfract6120736 - 12 Dec 2022

Cited by 7 | Viewed by 1594

Abstract

The focus on renewable energy is increasing globally to lessen reliance on conventional sources and fossil fuels. For renewable energy systems to work at their best and produce the desired results, precise feedback control is required. Microbial electrochemical cells (MEC) are a relatively new technology for renewable energy. In this study, we design and implement a model-based robust controller for a continuous MEC reactor. We compare its performance with those of traditional methods involving a proportional integral derivative (PID), H-infinity (H

_{\infty}

) controller and PID controller tuned by intelligent genetic algorithms. Recently, a dynamic model of a MEC continuous reactor was proposed, which describes the complex dynamics of MEC through a set of nonlinear differential equations. Until now, no model-based control approaches for MEC have been proposed. For optimal and robust output control of a continuous-reactor MEC system, we linearize the model to state a linear time-invariant (LTI) state-space representation at the nominal operating point. The LTI model is used to design four different types of controllers. The designed controllers and systems are simulated, and their performances are evaluated and compared for various operating conditions. Our findings show that a structured linear fractional transformation (LFT)-based H

_{\infty}

control approach is much better than the other approaches against various performance parameters. The study provides numerous possibilities for control applications of continuous MEC reactor processes. Full article

(This article belongs to the Special Issue Fractional Order Controllers: Design and Applications)

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14 pages, 5725 KiB

Open AccessArticle

Design and Experimental Results of an Adaptive Fractional-Order Controller for a Quadrotor

by Daniel D. Timis, Cristina I. Muresan and Eva-H. Dulf

Fractal Fract. 2022, 6(4), 204; https://doi.org/10.3390/fractalfract6040204 - 6 Apr 2022

Cited by 17 | Viewed by 2745

Abstract

The use of multi-copter systems started to grow over the last years in various applications. The designed solutions require high stability and maneuverability. To fulfill these specifications, a robust control strategy must be designed and integrated. Focusing on this challenge, this research proposes an adaptive control design applied to a physical model of a quadrotor prototype. The proposed adaptive structure guarantees robustness, control flexibility, and stability to the whole process. The prototype components, structure, and laboratory testing equipment that are used to run the experiments are presented in this paper. The study is focused on the performance comparison of a classical PID controller and a fractional-order controller, which are both integrated into the adaptive scheme. Fractional-order controllers are preferred due to their recognized ability to increase the robustness of the overall closed-loop system. Furthermore, this work covers the design and the tuning method of this control approach. The research concludes with the actual results obtained for this comparative study that highlights the advantages of the fractional-order controller. Full article

(This article belongs to the Special Issue Fractional Order Controllers: Design and Applications)

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33 pages, 3636 KiB

Open AccessArticle

The Multi-Switching Sliding Mode Combination Synchronization of Fractional Order Non-Identical Chaotic System with Stochastic Disturbances and Unknown Parameters

by Weiqiu Pan, Tianzeng Li and Yu Wang

Fractal Fract. 2022, 6(2), 102; https://doi.org/10.3390/fractalfract6020102 - 11 Feb 2022

Cited by 8 | Viewed by 1649

Abstract

This paper deals with the issue of the multi-switching sliding mode combination synchronization (MSSMCS) of fractional order (FO) chaotic systems with different structures and unknown parameters under double stochastic disturbances (SD) utilizing the multi-switching synchronization method. The stochastic disturbances are considered as nonlinear uncertainties and external disturbances. Our theoretical part considers that the drive-response systems have the same or different dimensions. Firstly, a FO sliding surface is established in terms of the fractional calculus. Secondly, depending on the FO Lyapunov stability theory and the sliding mode control technique, the multi-switching adaptive controllers (MSAC) and some suitable multi-switching adaptive updating laws (MSAUL) are designed. They can ensure that the state variables of the drive systems are synchronized with the different state variables of the response systems. Simultaneously, the unknown parameters are assessed, and the upper bound values of stochastic disturbances are examined. Selecting the suitable scale matrices, the multi-switching projection synchronization, multi-switching complete synchronization, and multi-switching anti-synchronization will become special cases of MSSMCS. Finally, examples are displayed to certify the usefulness and validity of the scheme via MATLAB. Full article

(This article belongs to the Special Issue Fractional Order Controllers: Design and Applications)

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17 pages, 918 KiB

Open AccessArticle

A Look-Up Table Based Fractional Order Composite Controller Synthesis Method for the PMSM Speed Servo System

by Weijia Zheng, Runquan Huang, Ying Luo, YangQuan Chen, Xiaohong Wang and Yong Chen

Fractal Fract. 2022, 6(1), 47; https://doi.org/10.3390/fractalfract6010047 - 15 Jan 2022

Cited by 10 | Viewed by 2726

Abstract

Considering the performance requirements in actual applications, a look-up table based fractional order composite control scheme for the permanent magnet synchronous motor speed servo system is proposed. Firstly, an extended state observer based compensation scheme was adopted to suppress the motor parametric uncertainties [...] Read more.

P D^{μ}

) controller was adopted as the speed controller to provide the optimal step response performance for the servo system. A universal look-up table was established to estimate the derivative order of the

P D^{μ}

controller, according to the optimal samples collected by an improved differential evolution algorithm. With the look-up table, the optimal

P D^{μ}

controller can be tuned analytically. Simulation and experimental results show that the servo system using the composite control scheme can achieve optimal tracking performance and has robustness to the motor parametric uncertainties and disturbance torques. Full article

(This article belongs to the Special Issue Fractional Order Controllers: Design and Applications)

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