Fractional-order Systems Research Articles

A Novel Approach to Modeling Incommensurate Fractional Order Systems Using Fractional Neural Networks

This research explores the application of the Riemann–Liouville fractional sigmoid, briefly RLFσ, activation function in modeling the chaotic dynamics of Chua’s circuit through Multilayer Perceptron (MLP) architecture. Grounded in the context of chaotic systems, the study aims to address the limitations of conventional activation functions in capturing complex relationships within datasets. Employing a structured approach, the methods involve training MLP models with various activation functions, including RLFσ, sigmoid, swish, and proportional Caputo derivative PCσ, and subjecting them to rigorous comparative analyses. The main findings reveal that the proposed RLFσ consistently outperforms traditional counterparts, exhibiting superior accuracy, reduced Mean Squared Error, and faster convergence. Notably, the study extends its investigation to scenarios with reduced dataset sizes and network parameter reductions, demonstrating the robustness and adaptability of RLFσ. The results, supported by convergence curves and CPU training times, underscore the efficiency and practical applicability of the proposed activation function. This research contributes a new perspective on enhancing neural network architectures for system modeling, showcasing the potential of RLFσ in real-world applications.

Loewner integer-order approximation of MIMO fractional-order systems

A state–space integer–order approximation of commensurate–order systems is obtained using a data–driven interpolation approach based on Loewner matrices. Precisely, given the values of the original fractional–order transfer function at a number of generalised frequencies, a descriptor–form state–space model matching these frequency response values is constructed from a suitable Loewner matrix pencil, as already suggested for the reduction of high–dimensional integer–order systems. Even if the stability of the resulting integer–order system cannot be guaranteed, such an approach is particularly suitable for approximating (infinite–dimensional) fractional–order systems because: (i) the order of the approximation is bounded by half the number of interpolation points, (ii) the procedure is more robust and simple than alternative approximation methods, and (iii) the procedure is fairly flexible and often leads to satisfactory results, as shown by some examples discussed at the end of the article. Clearly, the approximation depends on the location, spacing and number of the generalised interpolation frequencies but there is no particular reason to choose the interpolation frequencies on the imaginary axis, which is a natural choice in integer–order model reduction, since this axis does not correspond to the stability boundary of the original fractional–order system.

Cyber secure consensus of fractional order multi-agent systems with distributed delays: Defense strategy against denial-of-service attacks

Leader-follower cyber secure consensus problem in cybersecurity is crucial as it directly addresses the need for secure and stable communication in a networked system, which is paramount in safeguarding against cyber threats like denial-of-service attacks. In this article, by utilizing control strategy with distributed delays in denial-of-service attack's presence for discrete-time multi-agent system of fractional-order, the leader-following cyber secure consensus problem is examined. Nonlinear functions have been assumed in the leader and follower equation of the system to study variations due to this function in the system. As multi-agent system is networked environment, in response to numerous threats, their security control becomes critically desirable, for example, denial of service. The consensus performance may be destabilized by resulting topologies which are caused by Denial-of-Service attacks. The connectivity between agents is destroyed, especially under connectivity-broken attacks. In order to overcome these problems, the strategy of novel defense which consists of consensus (with distributed delays) control is introduced. By using Caputo fractional difference operator sufficient criteria which includes the condition in terms LMI is derived and by using Lyapunov function approach, average dwell time and algebraic graph theory for security of addressed system's cyber secure consensus this is done for determining the obtained system of error's stability. Finally, by showing some numerical examples on the introduced systems, the effectiveness of the obtained results is determined.

Predefined Time Fuzzy Adaptive Control of Switched Fractional-Order Nonlinear Systems with Input Saturation

Article Predefined Time Fuzzy Adaptive Control of Switched Fractional-Order Nonlinear Systems with Input Saturation Lusong Ding, and Weiwei Sun * 1 Institute of Automation, Qufu Normal University, Qufu 273165, China * Correspondence: wwsun@hotmail.com Received: 31 August 2023 Accepted: 31 October 2023 Published: 21 December 2023 Abstract: This article investigates the predefined-time fuzzy adaptive tracking control problem for a class of nonlinear switched fractional-order systems with input saturation and external disturbances under a nonstrict feedback structure. By combining the backstepping technique and the common Lyapunov function method, a predefined-time switching control method is constructed based on a novel fractional-order auxiliary function. The fuzzy logic system and the adaptive method are introduced to identify unknown compounded continuous functions. Moreover, the issue of calculating explosion and the problem of singularity are tackled through the newly proposed predefined-time and filter-based dynamic surface control. Especially, the construction of a continuous term in the controller eliminates possible chattering. The developed control strategy achieves that the closed-loop system is practically predefined-time stable under arbitrary switchings, where the upper bound of the settling-time can be defined by users in advance. Finally, two simulation examples are illustrated to prove the feasibility and effectiveness of the presented scheme.

Stability Analysis of Finite Time for a Class of Nonlinear Time-Delay Fractional-Order Systems

In this study, we delve into the examination of Finite Time Stability (FTS) within a specific class of Fractional-Order Systems (FOS) with time delays. By applying a fixed-point theorem, we establish novel sufficient conditions to ensure FTS for time-delayed FOS within 1<σ<2. Moreover, we investigate the existence and uniqueness of global solutions for this particular system. To demonstrate the credibility of our results, we substantiate our findings through the presentation of two illustrative examples.

Anti-windup scheme-based control for fractional-order systems subject to actuator and sensor saturation

This paper addresses the problem of dissipative-based stabilization for fractional-order nonlinear control systems with uncertainties, time-delay, external disturbances, actuator faults, actuator and sensor saturations. Specifically, a non-fragile reliable dynamic output feedback controller is designed for the purpose of stabilization of the addressed system. Due to the effect of saturation phenomena occurring in both the actuator and sensor, the anti-windup loops framed with the help of the observer design are incorporated into the controller design. Some sufficient conditions are formulated in terms of linear matrix inequalities by choosing a suitable Lyapunov function thereby assuring the dissipative-based stabilization. The derived results are thus validated by two numerical examples including the practical model of two-stage chemical reactor.

Parametric fractional-order analysis of Arneodo chaotic system and microcontroller-based secure communication implementation

Chaotic systems are often used in cryptology applications thanks to their complex dynamic structures. By increasing the dynamic diversity of chaotic systems through fractional-order analysis, more complex systems for data security can be obtained. In this study, fractional calculus analyses of the Arneodo chaotic jerk system with cubic nonlinearity and its microcontroller-based chaotic masking application are presented. Numerical analyses such as bifurcation diagrams, spectral entropy complexity, and Lyapunov spectra are performed to examine the dynamic characteristics of the fractional-order system. The parametric analysis indicates the presence of chaotic states in the system for nearly all fractional-order values ranging from 0.55 to 1. In this manner, the study focuses on investigating the minimum applicable fractional-order value for use in secure communication applications, constituting the main contribution of this research. Consequently, the dynamic diversity and unpredictability of the cubic system are greatly increased. Moreover, two fractional-order Arneodo chaotic (FOAC) systems with distinct initial conditions are synchronized using a single-state fractional-order sliding mode controller (FOSMC). Finally, an innovative secure communication system based on a microcontroller is designed and implemented by employing the chaotic masking method. As anticipated, the voltage outputs of the microcontroller-based secure communication application demonstrated good agreement with the numerical simulations.

Exploiting Newly Designed Fractional-Order 3D Lorenz Chaotic System and 2D Discrete Polynomial Hyper-Chaotic Map for High-Performance Multi-Image Encryption

Chaos-based image encryption has become a prominent area of research in recent years. In comparison to ordinary chaotic systems, fractional-order chaotic systems tend to have a greater number of control parameters and more complex dynamical characteristics. Thus, an increasing number of researchers are introducing fractional-order chaotic systems to enhance the security of chaos-based image encryption. However, their suggested algorithms still suffer from some security, practicality, and efficiency problems. To address these problems, we first constructed a new fractional-order 3D Lorenz chaotic system and a 2D sinusoidally constrained polynomial hyper-chaotic map (2D-SCPM). Then, we elaborately developed a multi-image encryption algorithm based on the new fractional-order 3D Lorenz chaotic system and 2D-SCPM (MIEA-FCSM). The introduction of the fractional-order 3D Lorenz chaotic system with the fourth parameter not only enables MIEA-FCSM to have a significantly large key space but also enhances its overall security. Compared with recent alternatives, the structure of 2D-SCPM is simpler and more conducive to application implementation. In our proposed MIEA-FCSM, multi-channel fusion initially reduces the number of pixels to one-sixth of the original. Next, after two rounds of plaintext-related chaotic random substitution, dynamic diffusion, and fast scrambling, the fused 2D pixel matrix is eventually encrypted into the ciphertext one. According to numerous experiments and analyses, MIEA-FCSM obtained excellent scores for key space (2541), correlation coefficients (<0.004), information entropy (7.9994), NPCR (99.6098%), and UACI (33.4659%). Significantly, MIEA-FCSM also attained an average encryption rate as high as 168.5608 Mbps. Due to the superiority of the new fractional-order chaotic system, 2D-SCPM, and targeted designs, MIEA-FCSM outperforms many recently reported leading image encryption algorithms.

On the dynamical behaviors in fractional-order complex PMSM system and Hamilton energy control

On the dynamical behaviors in fractional-order complex PMSM system and Hamilton energy control

Dynamical Analysis and Misalignment Projection Synchronization of a Novel RLCM Fractional-Order Memristor Circuit System

In this paper, a simple and novel fractional-order memristor circuit is established, which contains only resistance, inductance, capacitance and memristor. By using fractional calculus theory and the Adomian numerical algorithm, special bifurcations, chaotic degradation, C0 and Spectral Entropy (SE) complexity under one-dimensional and two-dimensional parameter variations with different orders, parameters and initial memristor values of the system were studied. Meanwhile, in order to better utilize the applications of fractional-order memristor systems in communication and security, a misalignment projection synchronization scheme for fractional-order systems is proposed, which overcomes the shortcomings of constructing Lyapunov functions for fractional-order systems to prove stability and designing controllers for the Laplace transform matrix.

On the Relationship Between the Pontryagin Maximum Principle and the Hamilton–Jacobi–Bellman Equation in Optimal Control Problems for Fractional-Order Systems

We consider the optimal control problem of minimizing the terminal cost functional for a dynamical system whose motion is described by a differential equation with Caputo fractional derivative. The relationship between the necessary optimality condition in the form of Pontryagin’s maximum principle and the Hamilton–Jacobi–Bellman equation with so-called fractional coinvariant derivatives is studied. It is proved that the costate variable in the Pontryagin maximum principle coincides, up to sign, with the fractional coinvariant gradient of the optimal result functional calculated along the optimal motion.

Synchronization of fractional-order fuzzy complex networks with time-varying couplings and proportional delay

The topic of synchronization of fractional-order fuzzy complex networks with time-varying couplings and proportional delay is addressed in this paper. By combining graph theory, the Lyapunov method, and the Razumikhin method, the synchronization criteria are obtained by designing a new time-varying graph-theoretical Lyapunov function. Different from the typical quadratic Lyapunov function, the time-varying graph-theoretical Lyapunov function contains the product of three-term functions and poses the difficulty of fractional derivative due to the unavailability of the general Leibniz formula and the chain rule. A new inequality about the fractional derivative rule of the time-varying graph-theoretical Lyapunov function is established based on the convex function creatively. In addition, compared with relevant literatures, the synchronization criteria are related to the proportionality coefficient and the order of fractional derivative. Then, the synchronization of fractional-order power system established on fuzzy complex networks with time-varying couplings and proportional delay is studied for the first time by applying the feedback control. Lastly, some simulations are exhibited to demonstrate the effectiveness of the theoretical results.

A new 3D fractional-order chaotic system with complex dynamics

Compared to integer-order chaotic systems, fractional-order chaotic systems have more complex dynamical features due to the introduction of order. The application of fractional-order chaotic systems to chaotic cryptosystems makes the cryptosystems with higher security properties. In this paper, we developed a new 3D fractional-order chaotic system from a 3D integer-order chaotic system, and investigate the dynamical behaviors of this fractional-order system with different parameters and orders. Moreover, self-excited attractors appeared at lower orders through circuit simulations. Furthermore, the synchronization of the new fractional-order chaotic system in the presence of systematic uncertainties and perturbations was achieved using the sliding mode control technique, which sets the stage for the implementation of communication. Finally, offset boosting control was used to investigate the utility of the new chaotic system in engineering applications.

An effective QLM-based Legendre matrix algorithm to solve the coupled system of fractional-order Lane-Emden equations

The purpose of this study is to propose a computationally effective algorithm for the numerical evaluation of a fractional-order system of singular Lane-Emden type equations arising in physical problems. The fractional operator considered is in the sense of the Liouville-Caputo derivative. The presented matrix collocation method is based upon a combination of the quasilinearization method (QLM) and the shifted Legendre functions (SLFs) and is called QLM-SLFs method. By applying first the QLM to the nonlinear underlying system, we get a family of linear equations. Hence, a spectral matrix collocation scheme relied on the SLFs is designed to solve the resulting sequence of linear system of equations at very few iterations. The uniform convergence of the shifted Legendre expansion series solution is established. To illustrate the effectiveness of the proposed QLM-SLFs technique in the present paper, three test examples are carried out. The applicability and validity of the proposed method are testified through comparisons with the outcomes of other existing procedures in the literature. The proposed QLM-SLFs method is efficient and easy to implement. The approximation obtained by the method also converges quickly to the solutions of the underlying model problem. In comparison with available existing computational procedures, the QLM-SLFs approach shows that the use of Legendre functions together with QLM provides solutions with high accuracy and exponential convergence rate.

Effects of coupling on extremely multistable fractional-order systems

Multistability is a phenomenon in which a dynamical system possesses multiple attractors within the same set of parameters. This study explores the effects of coupling on the dynamics of a recently introduced 2D single-neuron neural network model. The system is in its original form bistable, but its extension to include fractional-order derivatives yields an unprecedented level of complexity that is characterized by extreme multistability. We provide a comprehensive analysis of the behavior of this system when coupled. We find a diverse range of attractors for two coupled systems emerging from different initial conditions, and we elucidate the transition from periodic to chaotic behavior as the coupling strength increases. We also show that the synchronization dynamics in this case ranges from anti-phase synchronization under weak coupling to complete synchronization under stronger coupling. Finally, we study the collective behavior of several coupled extremely multistable fractional-order systems, which compared to bistable systems exhibit more complexity and uncertainty as they oscillate between different attractors, or even between emerging attractors.

The adaptive sliding synchronization of uncertain Duffing-Holmes fractional-order chaotic systems with dead-zone

This article investigates the problem of adaptive sliding synchronization for Duffing-Holmes fractional-order chaotic systems in the presence of dead-zone, disturbance, and uncertainty. It starts by estimating dead-zone parameters, and then adaptive laws are used to compensate the dead-zone parameters. In the second step, a sliding mode controller is designed so that the slave system can follow the master system. The fractional type of Lyapunov is implemented to prove stability. The proposed adaptive sliding mode controller guarantees the asymptotic stability of system despite the presence of the dead-zone and uncertainty. Simulation results show the validity and effectiveness of the proposed controller for synchronization of Duffing-Holmes fractional-order chaotic systems perturbed by the dead-zone, disturbance, and uncertainty.

The analysis of a new fractional model to the Zika virus infection with mutant

We present a new mathematical model to analyze the dynamics of the Zika virus (ZV) disease with the mutant under the real confirmed cases in Colombia. We give the formulation of the model initially in integer order derivative and then extend it to a fractional order system in the sense of the Mittag-Leffler kernel. We study the properties of the model in the Mittag-Leffler kernel and establish the result. The basic reproduction of the fractional system is computed. The equilibrium points of the Zika virus model are obtained and found that the endemic equilibria exist when the threshold is greater than unity. Further, we show that the model does not possess the backward bifurcation phenomenon. The numerical procedure to solve the problem using the Atangana-Baleanu derivative is shown using the newly established numerical scheme. We consider the real cases of the Zika virus in Colombia outbreak are considered and simulate the model using the nonlinear least square curve fit and computed the basic reproduction number R0=0.4942, whereas in previous work (Alzahrani et al., 2021) [1], the authors computed the basic reproduction number R0=0.5447. This is due to the fact that our work in the present paper provides better fitting to the data when using the fractional order model, and indeed the result regarding the data fitting using the fractional model is better than integer order model. We give a sensitivity analysis of the parameters involved in the basic reproduction number and show them graphically. The results obtained through the present numerical method converge to its equilibrium for the fractional order, indicating the proposed scheme's reliability.

A robust scheme for reduction of higher fractional-order systems

A robust scheme for reduction of higher fractional-order systems

Fractional Order Modeling and Control of an Articulated Robotic Arm

This paper presents a fractional order system modeling of a robotic arm and the development of a Fractional Order PID (FOPID) controller applied to the system. The controller technique originated from non-integer calculus, which improves the robotic arm's overall stability and positioning. The robotic arm system is modeled using the non-integer order technique in order to improve system accuracy. Thus, a non-integer order Proportional Integral Derivative (PID) control method is implemented to stabilize the plant positioning. Using MATLAB/Simulink the FOPID controller simulations were confirmed and compared to the Integer Order PID (IOPID) controller for tracking the robotic arm positioning. Simulation outcomes imply that the proposed non-integer controller increases the system stability and position with/without external disturbances being present in the environment.

Solutions For Fractional Linear Systems With Hattaf Derivative

This paper deals with the analytical solutions of the fractional-order linear systems characterized by DX(t) = AX(t)fractional differential equations. Here D, is the fractional derivative in the Hattaf sense and A is the (2x2)-typenon-singular real coefficient matrix. We present some results related to the existence of analytical solutions of the linearequation systems of the Hattaf fractional derivative. These solutions are provided for three different fractional-orderlinear systems.