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This article analyzes the implementation process of a fractional-order control system using available toolboxes, software, and hardware. The main objective is to showcase the current state-of-the-art hardware implementation of fractional-order control, comparing its potential to classical counterparts, and emphasizing the benefits of its utilization in industry. The article covers theoretical aspects of fractionalorder calculus and provides an example implementation of a Fractional-order PID controller with a NI MyRIO-1900 measurement board with FPGA module, comparing it with simulations for a given control plant.

This article abandons the traditional Laplace transform and proposes a new method for studying fractional-order circuits, which is the Loop-By-Loop Progressive Iterative Method(LPIM). Firstly, in order to demonstrate the correctness of LPIM, the fractance circuit, which is a relatively mature and simple form in fractional-order circuits, was chosen as the research object. The output signals of fractance circuit were studied for the first time using Laplace transform and LPIM, respectively. The results showed that the conclusions obtained by LPIM were completely consistent with those obtained by Laplace transform method and existing theories, thus verifying the correctness of LPIM. Then, a brand new Fractional-Order Memory Systems (FMS) model is constructed, and based on this model, LPIM is used for the first time to simulate the output signal of Flux-Controlled Fractional-Order Memory Systems (FFMS) that has not been studied so far. The results show that when a sine signal is used as the excitation signal, the output signal of the FFMS intersects at two points, and the output signal is modulated by the frequency of the excitation signal. Finally, combining existing theories, predict the output commonalities of FMS.

A fractor is a simple fractional-order system. Its transfer function is 1 / F s α ; the coefficient, F , is called the fractance, and α is called the exponent of the fractor. This paper presents how a fractor can be realized, using RC ladder circuit, meeting the predefined specifications on both F and α . Besides, commonly reported fractors have α between 0 and 1. So, their constant phase angles (CPA) are always restricted between 0 ∘ and - 90 ∘ . This work has employed GIC topology to realize fractors from any of the four quadrants, which means fractors with α between - 2 and +2. Hence, one can achieve any desired CPA between + 180 ∘ and - 180 ∘ . The paper also exhibits how these GIC parameters can be used to tune the fractance of emulated fractors in real time, thus realizing dynamic fractors. In this work, a number of fractors are developed as per proposed technique, their impedance characteristics are studied, and fractance values are tuned experimentally.

This paper proposes a novel fractance device (FD) based on simple R-C pairs for realizing fractional-order analog filters. Unlike conventional R-C ladder network-based FDs, which are complex and bulky, the proposed design employs only two passive components. This simplification results in lower cost, compact hardware, and improved power efficiency. The proposed FD is implemented in a VDTA-based fractional-order universal filter to evaluate its performance. The accuracy and validity of the proposed design are verified through SPICE and MATLAB simulations. Further, Monte Carlo analysis confirms the robustness and stability of the proposed circuit under parameter variations. The novelty of this work lies in its minimal-component realization of fractance behavior without compromising performance. The proposed approach supports SDG 7 (Affordable and Clean Energy) and SDG 9 (Industry, Innovation, and Infrastructure), advancing energy-efficient and innovative solutions for modern communication and signal processing systems.

Having an approximate realization of the fractance device is an essential part of fractional-order filter design and implementation. This encouraged researchers to introduce many approximation techniques of fractional-order elements. In this paper, the fractional-order KHN low-pass and high-pass filters are investigated based on four different approximation techniques: Continued Fraction Expansion, Matsuda, Oustaloup, and Valsa. Fractional-order filter fundamentals are reviewed then a comparison is made between the ideal and actual characteristic of the filter realized with each approximation. Moreover, stability analysis and pole movement of the filter with respect to the transfer function parameters using the exact and approximated realizations are also investigated. Different MATLAB numerical simulations, as well as SPICE circuit results, have been introduced to validate the theoretical discussions. Also, to discuss the sensitivity of the responses to component tolerances, Monte Carlo simulations are carried out and the worst cases are summarized which show good immunity to component deviations. Finally, the KHN filter is tested experimentally.

In this paper, three novel fractional-order CFOA-based inverse filters are introduced. The inverse low-pass, high-pass and band-pass responses are investigated using different approximation techniques. The studied approximations for the fractional-order Laplacian operator are the continued fraction expansion and Matsuda approximations. A comparison is held between the ideal filter characteristic and the realized ones from each approximation. A comparative study is summarized between the proposed circuits with some of the released inverse filters introduced in the literature. Foster-I realization is employed to transform the obtained fractional-order capacitor (FOC) from the investigated approximations into an RC parallel–series circuit topology. Additionally, to discuss the sensitivity of the FOC to component tolerances, Monte Carlo simulations are carried out which shows immunity to the component tolerances. Numerical examples, as well as SPICE circuit simulations, have been introduced to validate the theoretical discussions. Finally, the three CFOA inverse filters are tested experimentally.

The scaling fractal-chuan fractance approximation circuit (SFCFAC), which can realize the rational approximation of arbitrary-order fractances and has an excellent approximation performance, is presented in this paper. For an original SFCFAC, the progression ratios of resistance and capacitance are limited to the range 0–1. However, it is possible for the values of both progression ratios to be greater than one. The impedance function of an SFCFAC can be represented by an irregular scaling equation. By solving the scaling equation approximately, the operational order of an SFCFAC can be obtained using both progression ratios as \\[\\mu = -\\lg \\alpha /\\lg (\\alpha \\beta )\\]. Therefore, the SFCFAC has fractional operational characteristics, which is explained in theory. Oscillation phenomena are inherent to the SFCFAC. It is necessary to learn about these oscillation characteristics. The approximation performance can be improved by adding a series resistor and a series capacitor to the SFCFAC. The corresponding resistance and capacitance are determined by both progression ratios. The optimization has the advantages of simple operation, evident influence, and good practicality, which will make the SFCFAC competitive in future studies. Moreover, the values of the operational order of SFCFACs are extended from \\[-1<\\mu <0\\] to \\[0<|\\mu |<2\\] without using inductors, which is feasible for practical applications.

In this research, many novel expressions for time domain responses of fractance device to various often cited inputs have been proposed. Unlike the previous ones, our expressions have been derived based on the Caputo fractional derivative by also concerning the dimensional consistency with the integer order device based responses and the different between two types of fractance device i.e. fractional order inductor and fractional order capacitor. These previous expressions have been derived based on the Riemann-Liouvielle fractional derivative which has certain features that leads to contradictions and additional modeling difficulties unlike the Caputo fractional derivative. Our new expressions are applicable to both fractional order inductor and capacitor with arbitrary order. They are also applicable to any subject which its electrical characteristic can be modeled based on the fractance device. With our expressions and numerical simulations, the time domain behavioral analysis of both fractance device and such subject can be directly performed without requiring any time to frequency domain conversion and its inverse as already presented in this work. Therefore our work has been found to be beneficial to various fractance device related disciplines e.g. biomedical engineering, control system, electronic circuit and electrical engineering etc.

The objectives of this work are (i) to verify experimentally the theoretical claim that neither Riemann–Liouville nor Caputo fractional derivative can be used to predict the time response of fractional order systems without proper correction for the system’s past history in terms of an initialization function and (ii) to study quantitatively how the error incurred due to ignoring initialization depends on the nature of the past history and the system parameters. The entire analysis is restricted to a special class of single input single output linear time invariant fractional order system which can be realized by a simple electrical circuit consisting of a resistance and a fractance in series. This work involves two different realizations of fractances or constant phase elements whose characteristic parameters are first determined based on their respective impedance frequency responses and then used to simulate the time responses of the circuit with the input same as the one used for experimentation using a numerical method for two cases: (i) taking past history into account and (ii) without taking past history into account. Thereafter, an integral square error criterion is presented and variation of the same is studied with respect to system parameters and nature of the history function to have a relative idea of how much partial past history in the absence of a complete one should suffice in practical applications.

The growing interest shows the importance of the control of chaos in fractional-order systems in recent years. This paper investigates in the hybrid projective synchronization of two chaotic systems with fractional-order, which were derived from Euler equations of rigid body motion. Theoretical analyses of the proposed methods are validated by numerical simulation in the time domain. Moreover, the synchronization system is realized using electronic circuits with fractance in the frequency domain.