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Electronics
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Fractional-Order Circuits & Systems Design and Applications
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Fractional-Order Circuits & Systems Design and Applications

A special issue of Electronics (ISSN 2079-9292). This special issue belongs to the section "Circuit and Signal Processing".

Deadline for manuscript submissions: closed (31 December 2022) | Viewed by 28453

Special Issue Editors

Dr. Dražen Jurišić

E-Mail Website
Guest Editor
Faculty of Electrical Engineering and Computing, University of Zagreb, HR-10000 Zagreb, Croatia
Interests: fractional calculus; analysis methods for fractional order circuits; design methods for fractional order circuits; circuit theory; circuits and systems for signal processing; analog filters; analog integrated circuits; fault analysis in analog filters
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Dominik Sierociuk

E-Mail Website
Guest Editor
Faculty of Electrical Engineering; Institute of Control & Industrial Electronics Warsaw University of Technology, 00-662 Warsaw, Poland
Interests: fractional calculus; variable fractional order operators; analog modellig of fractional constant and variable order systems

Special Issue Information

Dear Colleagues,

Fractional calculus is a generalization of traditional differential calculus into the case whereas derivatives and integrals can be not only of integer orders but also of non-integer–fractional orders. This calculus was found to be a very efficient and valuable tool in many areas of science. One excellent example of using fractional order calculus in electrical engineering is modelling ultracapacitors. Based on an accurate fractional order model, it was possible to explain some interesting phenomena like loss of capacity equivalence with respect to growing frequency and that the resonance frequency of an ultracapacitor-coil circuit is not equal to the frequency for maximum current.

These and other results of modelling electrical elements and devices based on fractional calculus provided strong motivation to explore fractional circuits theory and applications.

The purpose of this Special Issue is to gather original research articles reflecting the latest developments in both theory and applications of fractional order circuits. The areas of interest include, but are not limited to, modelling of electrical and electronic devices like the new generation of ultracapacitors, batteries, fuel cells, analog modelling of constant and variable order operators and systems, fractional order filters, analysis, synthesis, and design methods for circuits with fractional order elements.

Prof. Dr. Dražen Jurišić
Prof. Dr. Dominik Sierociuk
Guest Editor

Manuscript Submission Information

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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. Electronics is an international peer-reviewed open access semimonthly 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 2400 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 caluclus
  • Modelling electronic and electrical devices
  • Analog modelling of constant and variable order operators and systems
  • Analysis methods for fractional order circuits
  • Design methods for fractional order circuits
  • Circuits for chaotic fractional order systems

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Published Papers (11 papers)

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Research

22 pages, 20068 KiB  
Article
Comparative Performance of UPQC Control System Based on PI-GWO, Fractional Order Controllers, and Reinforcement Learning Agent
by Marcel Nicola, Claudiu-Ionel Nicola, Dumitru Sacerdoțianu and Adrian Vintilă
Electronics 2023, 12(3), 494; https://doi.org/10.3390/electronics12030494 - 17 Jan 2023
Cited by 16 | Viewed by 1987
Abstract
In this paper, based on a benchmark on the performance of a Unified Power Quality Conditioner (UPQC), the improvement of this performance is presented comparatively by using Proportional Integrator (PI)-type controllers optimized by a Grey Wolf Optimization (GWO) computational intelligence method, fractional order [...] Read more.
In this paper, based on a benchmark on the performance of a Unified Power Quality Conditioner (UPQC), the improvement of this performance is presented comparatively by using Proportional Integrator (PI)-type controllers optimized by a Grey Wolf Optimization (GWO) computational intelligence method, fractional order (FO)-type controllers based on differential and integral fractional calculus, and a PI-type controller in tandem with a Reinforcement Learning—Twin-Delayed Deep Deterministic Policy Gradient (RL-TD3) agent. The main components of the UPQC are a series active filter and an Active Parallel Filter (APF) coupled to a common DC intermediate circuit. The active series filter provides the voltage reference for the APF, which in turn corrects both the harmonic content introduced by the load and the VDC voltage in the DC intermediate circuit. The UPQC performance is improved by using the types of controllers listed above in the APF structure. The main performance indicators of the UPQC-APF control system for the controllers listed above are: stationary error, voltage ripple, and fractal dimension (DF) of the VDC voltage in the DC intermediate circuit. Results are also presented on the improvement of both current and voltage Total harmonic distortion (THD) in the case of, respectively, a linear and nonlinear load highly polluting in terms of harmonic content. Numerical simulations performed in a MATLAB/Simulink environment demonstrate superior performance of UPQC-APF control system when using PI with RL-TD3 agent and FO-type controller compared to classical PI controllers. Full article
(This article belongs to the Special Issue Fractional-Order Circuits & Systems Design and Applications)
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20 pages, 3082 KiB  
Article
Operator-Based Fractional-Order Nonlinear Robust Control for the Spiral Heat Exchanger Identified by Particle Swarm Optimization
by Guanqiang Dong and Mingcong Deng
Electronics 2022, 11(17), 2800; https://doi.org/10.3390/electronics11172800 - 5 Sep 2022
Cited by 1 | Viewed by 1621
Abstract
Fractional-order calculus and derivative is extended from integral-order calculus and derivative. This paper investigates a nonlinear robust control problem using fractional order and operator theory. In order to improve the tracking performance and antidisturbance ability, operator- and fractional-order-based nonlinear robust control for the [...] Read more.
Fractional-order calculus and derivative is extended from integral-order calculus and derivative. This paper investigates a nonlinear robust control problem using fractional order and operator theory. In order to improve the tracking performance and antidisturbance ability, operator- and fractional-order-based nonlinear robust control for the spiral counter-flow heat exchanger described by the parallel fractional-order model (PFOM) is proposed. The parallel fractional-order model for the spiral counter-flow heat exchanger was identified by particle swarm optimization (PSO) and the parameters of a fractional-order PID (FOPID) controller were optimized by the PSO. First, the parallel fractional-order mathematical model for a spiral counter-flow heat exchanger plant was identified by PSO. Second, a fractional-order PID controller and operator controller for the spiral heat exchanger were designed under the identified parallel fractional-order mathematical model. Third, the parameters of the operator and fractional-order PID were optimized by PSO. Then, tracking and antidisturbance performance of the control system were analyzed. Finally, comparisons of two control schemes were performed, and the effectiveness illustrated. Full article
(This article belongs to the Special Issue Fractional-Order Circuits & Systems Design and Applications)
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17 pages, 887 KiB  
Article
Mittag–Leffler Synchronization of Caputo-Delayed Quaternion BAM Neural Networks via Adaptive and Linear Feedback Control Designs
by Renyu Ye, Jingshun Cheng, Axiu Shu and Hai Zhang
Electronics 2022, 11(11), 1746; https://doi.org/10.3390/electronics11111746 - 31 May 2022
Viewed by 1434
Abstract
The Mittag–Leffler synchronization (MLS) issue for Caputo-delayed quaternion bidirectional associative memory neural networks (BAM-NNs) is studied in this paper. Firstly, a novel lemma is proved by the Laplace transform and inverse transform. Then, without decomposing a quaternion system into subsystems, the adaptive controller [...] Read more.
The Mittag–Leffler synchronization (MLS) issue for Caputo-delayed quaternion bidirectional associative memory neural networks (BAM-NNs) is studied in this paper. Firstly, a novel lemma is proved by the Laplace transform and inverse transform. Then, without decomposing a quaternion system into subsystems, the adaptive controller and the linear controller are designed to realize MLS. According to the proposed lemma, constructing two different Lyapunov functionals and applying the fractional Razumikhin theorem and inequality techniques, the sufficient criteria of MLS on fractional delayed quaternion BAM-NNs are derived. Finally, two numerical examples are given to illustrate the validity and practicability. Full article
(This article belongs to the Special Issue Fractional-Order Circuits & Systems Design and Applications)
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19 pages, 856 KiB  
Article
Practical Applications of Diffusive Realization of Fractional Integrator with SoftFrac
by Jerzy Baranowski, Waldemar Bauer and Rafał Mularczyk
Electronics 2021, 10(15), 1767; https://doi.org/10.3390/electronics10151767 - 24 Jul 2021
Cited by 1 | Viewed by 1999
Abstract
Fractional calculus has found multiple applications around the world. It is especially prevalent in the domains of control and electronics. One of the key elements of fractional applications is the fractional integral (or integrator) which is a backbone of famous PIλD [...] Read more.
Fractional calculus has found multiple applications around the world. It is especially prevalent in the domains of control and electronics. One of the key elements of fractional applications is the fractional integral (or integrator) which is a backbone of famous PIλD controller. It gives advantages of traditional PID with a limited phase lag. The are, however, issues with implementation, which will allow good low-frequency behavior. In this paper, we consider a diffusive realization of a fractional integrator with the use of quadratures. We implemented this method in numerical package SoftFrac, and we illustrate how different quadratures work for this purpose. We show superiority of bounded domain integration with logarithmic transformation and explain issues with behavior for extremely low frequencies. Full article
(This article belongs to the Special Issue Fractional-Order Circuits & Systems Design and Applications)
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10 pages, 4726 KiB  
Article
Power-Law Compensator Design for Plants with Uncertainties: Experimental Verification
by Stavroula Kapoulea, Costas Psychalinos, Ahmed S. Elwakil and Mohammad Saleh Tavazoei
Electronics 2021, 10(11), 1305; https://doi.org/10.3390/electronics10111305 - 30 May 2021
Cited by 7 | Viewed by 2121
Abstract
A power-law compensator scheme for achieving robust frequency compensation in control systems including plants with an uncertain pole, is introduced in this work. This is achieved through an appropriate selection of the compensator parameters, which guarantee that the Nyquist diagram of the open-loop [...] Read more.
A power-law compensator scheme for achieving robust frequency compensation in control systems including plants with an uncertain pole, is introduced in this work. This is achieved through an appropriate selection of the compensator parameters, which guarantee that the Nyquist diagram of the open-loop system compensator-plant crosses a fixed point independent of the plant pole variations. The implementation of the fractional-order compensator is performed through the utilization of a curve-fitting-based technique and the derived rational integer-order transfer function is realized on a Field-Programmable Analog Array device. The experimental results confirm that the the phase margin is well preserved, even for ±40% variation in the pole location of the plant. Full article
(This article belongs to the Special Issue Fractional-Order Circuits & Systems Design and Applications)
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16 pages, 696 KiB  
Article
The Frequency and Real-Time Properties of the Microcontroller Implementation of Fractional-Order PID Controller
by Krzysztof Oprzędkiewicz, Maciej Rosół and Jakub Żegleń-Włodarczyk
Electronics 2021, 10(5), 524; https://doi.org/10.3390/electronics10050524 - 24 Feb 2021
Cited by 10 | Viewed by 2447
Abstract
The paper presents time, frequency, and real-time properties of a fractional-order PID controller (FOPID) implemented at a STM 32 platform. The implementation uses CFE approximation and discrete version of a Grünwald–Letnikov operator (FOBD). For these implementations, experimental step responses and Bode frequency responses [...] Read more.
The paper presents time, frequency, and real-time properties of a fractional-order PID controller (FOPID) implemented at a STM 32 platform. The implementation uses CFE approximation and discrete version of a Grünwald–Letnikov operator (FOBD). For these implementations, experimental step responses and Bode frequency responses were measured. Real-time properties of the approximations are also examined and analyzed. Results of tests show that the use of CFE approximation allows to better keep the soft real-time requirements with an accuracy level a bit worse than when using the FOBD. The presented results can be employed in construction-embedded fractional control systems implemented at platforms with limited resources. Full article
(This article belongs to the Special Issue Fractional-Order Circuits & Systems Design and Applications)
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13 pages, 545 KiB  
Article
Time-Domain Analysis of Fractional Electrical Circuit Containing Two Ladder Elements
by Ewa Piotrowska and Krzysztof Rogowski
Electronics 2021, 10(4), 475; https://doi.org/10.3390/electronics10040475 - 17 Feb 2021
Cited by 7 | Viewed by 2557
Abstract
The paper is devoted to the theoretical and experimental analysis of an electric circuit consisting of two elements that are described by fractional derivatives of different orders. These elements are designed and performed as RC ladders with properly selected values of resistances and [...] Read more.
The paper is devoted to the theoretical and experimental analysis of an electric circuit consisting of two elements that are described by fractional derivatives of different orders. These elements are designed and performed as RC ladders with properly selected values of resistances and capacitances. Different orders of differentiation lead to the state-space system model, in which each state variable has a different order of fractional derivative. Solutions for such models are presented for three cases of derivative operators: Classical (first-order differentiation), Caputo definition, and Conformable Fractional Derivative (CFD). Using theoretical models, the step responses of the fractional electrical circuit were computed and compared with the measurements of a real electrical system. Full article
(This article belongs to the Special Issue Fractional-Order Circuits & Systems Design and Applications)
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15 pages, 3147 KiB  
Article
Fractional PIλD Controller Design for a Magnetic Levitation System
by Waldemar Bauer and Jerzy Baranowski
Electronics 2020, 9(12), 2135; https://doi.org/10.3390/electronics9122135 - 13 Dec 2020
Cited by 23 | Viewed by 2806
Abstract
Currently, there are no formalized methods for tuning non-integer order controllers. This is due to the fact that implementing these systems requires using an approximation of the non-integer order terms. The Oustaloup approximation method of the sα fractional derivative is intuitive and [...] Read more.
Currently, there are no formalized methods for tuning non-integer order controllers. This is due to the fact that implementing these systems requires using an approximation of the non-integer order terms. The Oustaloup approximation method of the sα fractional derivative is intuitive and widely adopted in the design of fractional-order PIλD controllers. It requires special considerations for real-time implementations as it is prone to numerical instability. In this paper, for design and tuning of fractional regulators, we propose two methods.The first method relies on Nyquist stability criterion and stability margins. We base the second on parametric optimization via Simulated Annealing of multiple performance indicators. We illustrate our methods with a case study of the PIλD controller for the Magnetic Levitation System. We illustrate our methods’ efficiency with both simulations and experimental verification in both nominal and disturbed operation. Full article
(This article belongs to the Special Issue Fractional-Order Circuits & Systems Design and Applications)
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23 pages, 7662 KiB  
Article
Random Number Generator with Long-Range Dependence and Multifractal Behavior Based on Memristor
by María Téllez, Johan Mejía, Hans López and Cesar Hernández
Electronics 2020, 9(10), 1607; https://doi.org/10.3390/electronics9101607 - 1 Oct 2020
Cited by 7 | Viewed by 3550
Abstract
Random number generators are used in areas such as encryption and system modeling, where some of these exhibit fractal behaviors. For this reason, it is interesting to make use of the memristor characteristics for the random number generation. Accordingly, the objective of this [...] Read more.
Random number generators are used in areas such as encryption and system modeling, where some of these exhibit fractal behaviors. For this reason, it is interesting to make use of the memristor characteristics for the random number generation. Accordingly, the objective of this article is to evaluate the performance of a chaotic memristive system as a random number generator with fractal behavior and long-range dependence. To achieve the above, modeling memristor and its corresponding chaotic systems is performed, from which a random number generator is constructed. Subsequently, the Hurst parameter for the detection of long-range dependence is estimated and a fractal analysis of the synthesized data is performed. Finally, a comparison between the model proposed in the research and the β-MWM algorithm is made. The results obtained show that the data synthesized from the proposed generator have a variable Hurst parameter and both monofractal and multifractal behavior. The main contribution of this research is the proposal of a new model for the synthesis of traces with long-range dependence and fractal behavior based on the non-linearity of the memristor. Full article
(This article belongs to the Special Issue Fractional-Order Circuits & Systems Design and Applications)
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15 pages, 4651 KiB  
Article
Modeling and Analysis of the Fractional-Order Flyback Converter in Continuous Conduction Mode by Caputo Fractional Calculus
by Chen Yang, Fan Xie, Yanfeng Chen, Wenxun Xiao and Bo Zhang
Electronics 2020, 9(9), 1544; https://doi.org/10.3390/electronics9091544 - 21 Sep 2020
Cited by 22 | Viewed by 3067
Abstract
In order to obtain more realistic characteristics of the converter, a fractional-order inductor and capacitor are used in the modeling of power electronic converters. However, few researches focus on power electronic converters with a fractional-order mutual inductance. This paper introduces a fractional-order flyback [...] Read more.
In order to obtain more realistic characteristics of the converter, a fractional-order inductor and capacitor are used in the modeling of power electronic converters. However, few researches focus on power electronic converters with a fractional-order mutual inductance. This paper introduces a fractional-order flyback converter with a fractional-order mutual inductance and a fractional-order capacitor. The equivalent circuit model of the fractional-order mutual inductance is derived. Then, the state-space average model of the fractional-order flyback converter in continuous conduction mode (CCM) are established. Moreover, direct current (DC) analysis and alternating current (AC) analysis are performed under the Caputo fractional definition. Theoretical analysis shows that the orders have an important influence on the ripple, the CCM operating condition and transfer functions. Finally, the results of circuit simulation and numerical calculation are compared to verify the correctness of the theoretical analysis and the validity of the model. The simulation results show that the fractional-order flyback converter exhibits smaller overshoot, shorter setting time and higher design freedom compared with the integer-order flyback converter. Full article
(This article belongs to the Special Issue Fractional-Order Circuits & Systems Design and Applications)
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13 pages, 400 KiB  
Article
Analog Realization of a Fractional Recursive Variable-Type and Order Operator for a Particular Switching Strategy
by Dominik Sierociuk, Michal Macias, Wiktor Malesza and Michał Sławomir Wiraszka
Electronics 2020, 9(5), 855; https://doi.org/10.3390/electronics9050855 - 21 May 2020
Cited by 3 | Viewed by 2474
Abstract
In this paper, we propose a method of practical realization and an actual, physical hardware implementation of a fractional variable-type and order difference operator that switches between two (i.e., B - and D -type) variable-order definitions. After the theoretical model of such a [...] Read more.
In this paper, we propose a method of practical realization and an actual, physical hardware implementation of a fractional variable-type and order difference operator that switches between two (i.e., B - and D -type) variable-order definitions. After the theoretical model of such a switch, we report the experimental validation on an analog model to prove its adequacy. The tests prove with great certainty that the proposed model and the realization behave correctly. They also let the authors assume that the proposed method is the only one suitable for this case, based on the counterexamples presented. Full article
(This article belongs to the Special Issue Fractional-Order Circuits & Systems Design and Applications)
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