Fractional-order Systems Research Articles

A Study on Fractional Power-Law Applications and Approximations

The frequency response of the fractional-order power-law filter can be approximated by different techniques, which eventually affect the expected performance. Fractional-order control systems introduce many benefits for applications like compensators to achieve robust frequency and additional degrees of freedom in the tuning process. This paper is a comparative study of five of these approximation techniques. The comparison focuses on their magnitude error, phase error, and implementation complexity. The techniques under study are the Carlson, continued fraction expansion (CFE), Padé, Charef, and MATLAB curve-fitting tool approximations. Based on this comparison, the recommended approximation techniques are the curve-fitting MATLAB tool and the continued fraction expansion (CFE). As an application, a low-pass power-law filter is realized on a field-programmable analog array (FPAA) using two techniques, namely the curve-fitting tool and the CFE. The experiment aligns with and validates the numerical results.

Fractional Trigonometric Function Approximations in a Regular System

Introduction: Linear fractional differential equations of the commensurable order, specifically those related to fractional trigonometry, pose challenges in terms of analytical solutions. background: This study presents an approach to the solution of fractional differential equations with commensurate order, specifically those associated with fractional trigonometry. Methods: The purpose of this article is to present a novel approximate analytical technique to solve linear fractional differential equations of the commensurable order, which are related to fractional trigonometry. This method is used to obtain approximate solutions by forming linear combinations of appropriate fractional basic functions, for which Laplace transforms are irrational functions. This approximation technique enables us to represent these functions using timeinvariant linear system models. Also, the implementation of analog circuits of these basic fractional order systems can be obtained using approximation of rational functions. Results: The research demonstrates the efficacy and precision of this method through illustrative examples, showcasing its effectiveness in solving linear fractional systems. Conclusion: The newly introduced approximate analytical method presents a promising approach to solving linear fractional differential equations of the commensurable order, providing accurate solutions and possibly offering applications in various fields.

Forecasting and dynamical modeling of reversible enzymatic reactions with a hybrid proportional fractional derivative

Chemical kinetics is a branch of chemistry that investigates the rates of chemical reactions and has applications in cosmology, geology, and physiology. In this study, we develop a mathematical model for chemical reactions based on enzyme dynamics and kinetics, which is a two-step substrate–enzyme reversible reaction, applying chemical kinetics-based modeling of enzyme functions. The non-linear differential equations are transformed into fractional-order systems utilizing the constant proportional Caputo–Fabrizio (CPCF) and constant proportional Atangana–Baleanu–Caputo (CPABC) operators. The system of fractional differential equations is simulated using the Laplace–Adomian decomposition method at different fractional orders through simulations and numerical results. Both qualitative and quantitative analyses such as boundedness, positivity, unique solution, and feasible concentration for the proposed model with different hybrid operators are provided. The stability analysis of the proposed scheme is also verified using Picard’s stable condition through the fixed point theorem.

Modeling and Analysis of Caputo–Fabrizio Definition-Based Fractional-Order Boost Converter with Inductive Loads

This paper proposes a modeling and analysis method for a Caputo–Fabrizio (C-F) definition-based fractional-order Boost converter with fractional-order inductive loads. The proposed method analyzes the system characteristics of a fractional-order circuit with three state variables. Firstly, this paper constructs a large signal model of a fractional-order Boost converter by taking advantage of the state space averaging method, providing accurate analytical solutions for the quiescent operating point and the ripple parameters of the circuit with three state variables. Secondly, this paper constructs a small signal model of the C-F definition-based fractional-order Boost converter by small signal linearization, providing the transfer function of the fractional-order system with three state variables. Finally, this paper conducts circuit-oriented simulation experiments where the steady-state parameters and the transfer function of the circuit are obtained, and then the effect of the order of capacitor, induced inductor, and load inductor on the quiescent operating point and ripple parameters is analyzed. The experimental results show that the simulation results are consistent with those obtained by the proposed mathematical model and that the three fractional orders in the fractional model with three state variables have a significant impact on the DC component and steady-state characteristics of the fractional-order Boost converter. In conclusion, the proposed mathematical model can more comprehensively analyze the system characteristics of the C-F definition-based fractional-order Boost converter with fractional-order inductive loads, benefiting the circuit design of Boost converters.

Supplemental Stability Criteria with New Formulation for Linear Time-Invariant Fractional-Order Systems

In this paper, new stability criteria for linear time-invariant fractional-order systems (LTIFOSs) based on linear matrix inequalities (LMIs) are derived. The solved variable of the existing LMI formulations is generalized to a complex one. In addition, based on the congruent transformation, a new LMI formulation is obtained, which is different from those in the existing literature. To deal with the above LMIs more conveniently with simulation software, the complex LMIs are converted to equivalent real LMIs. Finally, numerical examples are presented to validate the effectiveness of our theoretical results.

Group consensus of fractional-order heterogeneous multi-agent systems with random packet losses and communication delays

The group consensus problem of heterogeneous fractional-order multi-agent systems with data packet losses and communication delays is investigated in this paper, and data packet losses are described by the Bernoulli-distribution. Inspired by genetic and the infinite memory property of the Caputo fractional derivative, a novel group consensus control protocol based on sampled data is designed. Sufficient conditions for mean-square group consensus of heterogeneous fractional-order multi-agent systems are derived by using matrix theory, Gerschgorin disc theorem and graph theory. Finally, numerical simulation examples are given to verify the correctness of the theoretical results.

A composite fractional order active disturbance rejection control method for a class of uncertain systems with multiple disturbances

SummaryThe existing active disturbance rejection control (ADRC) method may not provide sufficient disturbance rejection to multiple mismatched disturbances for the fractional order systems. In this paper, a composite disturbance rejection approach is developed for a class of fractional order uncertain systems, by synthesizing the fractional order ADRC (FOADRC) approach and a disturbance observer (DO)‐based compensation scheme. Taking advantage of more disturbance information and a filter structure, an improved DO is developed to achieve precise estimation of disturbances in the presence of sensor noises. In addition, a state transformation is developed to convert the system into a simple integral chain model with only matched disturbances. Then a composite control law is designed to compensate the disturbances and provide satisfying dynamic performance. The efficiency of the proposed method is demonstrated by a numerical simulation and an actual servo control simulation, as well as the comparison with two kinds of the existing ADRC methods and the commonly used integral sliding mode control (I‐SMC) method.

Observer-Based Adaptive Fuzzy Quantized Control for Fractional-Order Nonlinear Time-Delay Systems with Unknown Control Gains

This paper investigates the observer-based adaptive fuzzy quantized control problem for a class of fractional-order nonlinear time-delay systems with unknown control gains based on a modified fractional-order dynamic surface control (FODSC) technique and an indirect Lyapunov method. First, a fractional-order, high-gain state observer is constructed to estimate unavailable state information. Furthermore, the Nussbaum gain technique and a fractional-order filter are adopted to cope with the problem of unknown control gains and to reduce the computational complexity of the conventional recursive procedure, respectively. Moreover, through integration with the compensation mechanism and estimation model, the adaptive fuzzy quantized controllers and adaptive laws are designed to ensure that all the signals of the closed-loop system are bounded. In the end, the proposed controller is applied to a numerical example and a single-machine-infinite bus (SMIB) power system; the simulation results show the validity, superiority, and application potential of the developed control strategy.

Designing hardware for a robust high-speed cryptographic key generator based on multiple chaotic systems and its FPGA implementation for real-time video encryption

Recent advancements in communication technologies have highlighted the pivotal role of information security for all individuals and entities. In response, researchers are increasingly focusing on cryptographic solutions to ensure the reliability of confidential information. Recognizing the superiority of chaotic systems preference as entropy source of cryptographic systems, this paper proposes a novel true random number generator (TRNG) design by combining four different chaotic systems outputs, tailored for real-time video encryption application. These chaotic systems are continuous-time Lorenz and fractional-order Chen-Lee systems, as well as discrete-time Logistic and Tent maps. This study generates true random bit (TRB) sequences at a high bit rate (25 Mbps) through the hardware implementations of four distinct chaotic systems to have the best statistical randomness in the resulting output. Then, the cryptographic true random key bits (8-bit at 25 MHz frequency) are employed in the post-processing with real-time video data by using the XOR operation, a fundamental post-processing algorithm. The real-time video encryption application is executed on an experimental assembly, composed of a Field Programmable Gate Array (FPGA) development kit, an OV7670 camera module, a VGA monitor, and prototype circuit boards for the chaotic systems. To evaluate the effectiveness of the proposed encryption system, several security assessments are conducted. These include NIST SP 800 − 22 statistical tests, FIPS 140-1 standards, chi-square tests, histogram and correlation analysis, and NPCR and UACI differential attack resilience tests. Consequently, the findings suggest that the presented real-time embedded cryptosystem is robust and suitable for secure communications, particularly in the realm of video transmission.

Mathematical modelling with computational fractional order for the unfolding dynamics of the communicable diseases

Mathematical models based on computational fractional orders, employed for accurate modelling of complex dynamic systems, can ensure the implementation of various analytical, numerical and computing methods encompassing their applications to emerging and ever-varying real-world problems. Tracking, managing and controlling communicable diseases, one being monkeypox with different features, virological and taxonomic attributes, are oriented towards high-risk groups concerning global public health. This study, accordingly, is devoted to the presentation of the piecewise global derivative model of the monkeypox virus by applying the Caputo and Atangana Baleanu fractional-order derivatives in the partitioned two sub-intervals. The model includes eight compartments with two categories of human and rodent populations. The cases which take part in some sense for the said infection are investigated along with connection in this format. The existence and uniqueness of the solution in the framework of the piecewise global derivative are analyzed for both sub-intervals using fixed point theory. The detailed investigation of the dynamics of fractional-order systems and among many other dynamic features, stability is addressed. The stability of the solution is, thus, examined using the idea of Ulam Hyers concept. For the best fitting values of the parameters, the results are simulated using the monkeypox data. Using the method of Newton polynomial, different piecewise dynamics of each compartment are simulated on different fractional orders and time durations. This kind of a proposed approach is thought to lay a foundation where the transmission takes place to control epidemic events and other infectious medical conditions through vaccines or taking preventive measures to maintain and advance global public health while fully optimizing the clinical care of the diseases to manage complications, alleviate symptoms as well as prevent the long-term sequelae. This analysis also deals with sudden variation in monkeypox dynamics and also for crossover dynamics along with removal of discontinuity through modification of piecewise global analysis.

Chaotic hidden attractor in a fractional order system modeling the interaction between dark matter and dark energy

In this paper the dynamics of a fractional order system modeling the interaction between dark matter and dark energy is analytically and numerically studied. It is shown for the first time that systems modeling the interaction between dark matter and dark energy, chaotic hidden attractors can be present. The chaotic attractor coexists with two asymptotically stable equilibria. Equilibria of the linearized system exhibit a center-like behavior. The numerical integration is done by means of the Adams–Bashforth–Moulton scheme and the finite Lyapunov exponents are numerically determined with a dedicated Matlab code. The 3D representation of the chaotic hidden attractor reveals the fact it is not connected with the equilibria, being “hidden” somewhere in the considered phase space.

Event-triggered hybrid impulsive control for synchronization of fractional-order multilayer signed networks under cyber attacks

In this paper, we consider the exponential bipartite synchronization (EBS) problem of fractional-order multilayer signed networks with time-varying delays (FO-MSNT) under random cyber attacks. In contrast to the existing literature, the proposed hybrid event-triggered controller combines the advantages of feedback controller and impulsive controller, and the event-triggered condition is constructed by applying the network topology and the Lyapunov function of the subsystem, rather than the state function of the subsystem. Based on the Lyapunov–Razumikhin method and the graph theory, some sufficient conditions for achieving EBS of FO-MSNT under cyber attacks which are related to the topology of networks, the event-triggered parameters, the order of fractional derivative and the signal sent by the enemy are obtained. Furthermore, fractional-order coupled Chua’s circuits model and fractional-order power systems built on MSNT are established and the EBS issues under cyber attacks are analyzed. Numerical examples and simulations are provided to show the validity of our theories.

Practical Mittag-Leffler stability of quasi-one-sided Lipschitz fractional order systems

Practical Mittag-Leffler stability of quasi-one-sided Lipschitz fractional order systems

Privacy Preservation of Nabla Discrete Fractional-Order Dynamic Systems

This article investigates the differential privacy of the initial state for nabla discrete fractional-order dynamic systems. A novel differentially private Gaussian mechanism is developed which enhances the system’s security by injecting random noise into the output state. Since the existence of random noise gives rise to the difficulty of analyzing the nabla discrete fractional-order systems, to cope with this challenge, the observability of nabla discrete fractional-order systems is introduced, establishing a connection between observability and differential privacy of initial values. Based on it, the noise magnitude required for ensuring differential privacy is determined by utilizing the observability Gramian matrix of systems. Furthermore, an optimal Gaussian noise distribution that maximizes algorithmic performance while simultaneously ensuring differential privacy is formulated. Finally, a numerical simulation is provided to validate the effectiveness of the theoretical analysis.

FRACTIONAL ORDER ANALYSIS OF THE 4-DIMENSIONAL HYPERCHAOTIC PANG SYSTEM AND ITS ADAPTIVE SYNCHRONIZATION

Fractional calculus is an effective method used to analyze the dynamics of nonlinear systems and provide more precise results. In this study, firstly, the 4-dimensional Pang system is introduced and its dynamic analyses demonstrating the hyperchaotic structure are given. Then, fractional-order calculations of the system are presented and the dynamics of the system for different fraction orders are investigated. At this point, according to the results obtained from Lyapunov exponents and phase-space representation, the Pang system exhibits periodic, chaotic, and hyperchaotic behaviors in different fractional orders. The results obtained at the end of this study present that the system is hyperchaotic for the fractional order of 3.52 and it is also confirmed that more accurate results are obtained than the integer-order analysis. In the next part of the study, adaptive synchronization of the fractional-order system is performed. Three different cases are examined and it is demonstrated that synchronization is achieved in all cases.

Effects of high-order interactions on synchronization of a fractional-order neural system

In this study, effects of high-order interactions on synchronization of the fractional-order Hindmarsh–Rose neuron models have been examined deeply. Three different network situations in which first-order coupling, high-order couplings and first-plus second-order couplings included in the neuron models, have been considered, respectively. In order to find the optimal values of the first- and high-order coupling parameters by minimizing the cost function resulted from pairwise and triple interactions, the particle swarm optimization algorithm is employed. It has been deduced from the numerical simulation results that the first-plus second-order couplings induce the synchronization with both reduced first-order coupling strength and total cost compared to the first-order coupled case solely. When the only first-order coupled case is compared with the only second-order coupled case, it is determined that the neural network with only second-order couplings involved could achieve synchronization with lower coupling strength and, as a natural result, lower cost. On the other hand, solely second- and first-plus second-order coupled networks give very similar results each other. Therefore, high-order interactions have a positive effect on the synchronization. Additionally, increasing the network size decreases the values of the both first- and high-order coupling strengths to reach synchronization. However, in this case, total cost should be kept in the mind. Decreasing the fractional order parameter causes slower synchronization due to the decreased frequency of the neural response. On the other hand, more synchronous network is possible with increasing the fractional order parameter. Thus, the neural network with higher fractional order as well as high-order coupled is a good candidate in terms of the neural synchronization.

Investigation of the hyperchaos and control in the fractional order financial system with profit margin

This research study proposes a novel hyperchaotic finance system with profit margin and then utilizes the Adomian Decomposition Method (ADM) to tackle the solution of the associated Caputo-derivative based fractional-order hyperchaotic finance system with profit margin. Numerical simulations and analyses, such as the evaluation of Lyapunov exponents (LE), the creation of bifurcation diagrams (BD), the complexity analysis (CA), and the 0-1 test, are used to get a full picture of the system. The results of our study demonstrate the occurrence of periodic, chaotic, and hyperchaotic dynamics in the system. Furthermore, the overall criteria for the existence and the uniqueness of the exact solutions for Caputo fractional-order models are presented. In addition, we present a control methodology for the fractional-order hyperchaotic financial system utilizing the Laplace transform and linear feedback control. Significantly, our research showcases a noteworthy correlation between the analytical results and numerical simulations, emphasizing the soundness and effectiveness of our methodology.

Mittag–Leffler stability and stabilization of delayed fractional-order memristive neural networks based on a new Razumikhin-type theorem

The Mittag–Leffler stability and stabilization of delayed fractional-order memristive neural networks(DFMNNs) are investigated in this paper. First, two new fractional Halanay inequalities are established by solving two fractional-order non-autonomous differential inequalities. Next, by using the proposed fractional Halanay inequalities, a novel Razumikhin-type theorem for Mittag–Leffler stability of delayed fractional-order systems is presented, which is an extension of the so-called Razumikhin theorem for integer-order delayed differential systems. Applying the Razumikhin-type theorem to the DFMNNs, several Mittag–Leffler stability and stabilization criteria are obtained. Finally, the validity of the proposed results is shown by two numerical examples.

A novel adaptive synchronization algorithm for a general class of fractional-order complex-valued systems with unknown parameters, and applications to circuit realization and color image encryption

This article proposes an adaptive synchronization (AS) algorithm to synchronize a general class of fractional-order complex-valued systems with completely unknown parameters, which may appear in physical and engineering problems. The analytical and theoretical concepts of the algorithm rely on the mathematical framework of the Mittag-Leffler global stability of fractional-order systems. A specific control system is established analytically based on the fractional-order adaptive laws of parameters, and the corresponding numerical results are executed to verify the accuracy of the AS algorithm. The proposed synchronization method is evaluated using the fractional-order complex Rabinovich system as an attractive example. The electronic circuits of the new system with different fractional orders are designed. By utilizing the Multisim electronic workbench software, various chaotic/hyperchaotic behaviors have been observed, and a good agreement is found between the numerical results and experimental simulation. In addition, the approximation of the transfer function for different fractional-order are presented. And the corresponding resistor and capacitor values in the chain ship model (CSM) are estimated, which can be utilized in designing electronic circuits for other fractional-order systems. Furthermore, two strategies for encrypting color images are proposed using the AS algorithm and fractional-order adaptive laws of parameters. In the first strategy, the color image is treated as a single package and divided into two vectors. The first vector is embedded into transmitter parameters, while the second vector is injected into the transmitter state signals. In the second strategy, the primary RGB channel components of the original color image are extracted and separated into two vectors, and the same process is followed as in the first strategy. These strategies complicate the decryption task for intruders. Different scales of white Gaussian noise are added to color images to examine the robustness of the proposed color images encryption strategies.

Adaptive iterative learning control for a class of parameterized fractional-order systems subjected to backlash-like hysteresis

In this article, a fractional-order adaptive iterative learning control scheme is proposed for a class of parameterized fractional-order systems with unknown control gain and backlash-like hysteresis nonlinearity under disturbance. Based on the sufficient condition for the stability of linear fractional-order systems, a sliding mode surface of tracking errors is constructed to facilitate the controller design and stability analysis. A new boundary layer function is designed by using Mittag-Leffler function to relax the restriction of the identical initial condition of iterative learning control design. The fractional-order differential-type adaptive laws and difference-type learning law are designed to estimate unknown constant parameters and time-varying parameter, respectively. To deal with the influence of backlash-like hysteresis nonlinearity reminder term and unknown bounded external disturbance, a robust control term is designed by employing the hyperbolic tangent function with a convergent series sequence which can guarantee the learning convergence along iteration axis. By constructing Lyapunov-like composite energy function, the stability analysis is presented to prove the convergence of the system output to a small neighborhood of the desired trajectory and the boundedness of all the closed-loop signals. Finally, a simulation example of second-order nonlinear fractional-order system is presented, which demonstrates the effectiveness of the proposed fractional-order adaptive iterative learning control scheme.