Fractional-order Systems Research Articles

Neural network control of fractional-order chaotic systems with unknown control direction

In this paper, neural network control of fractional order chaotic systems (FOCSs) with input saturation and unknown sign of the controller gain is addressed by employing the Nussbaum function, where neural networks are utilized to model system uncertainties. To get rid of the limitation that reaching phase should be active before sliding motion in the traditional sliding mode control, a stable sliding surface is constructed. Then, by using the integer order Nussbaum gain control method, a novel controller with neural network sliding mode variable structure is designed. Finally, the practicality of the designed method is confirmed by a simulation experiment.

Sparse Identification of Fractional Chaotic Systems based on the time-domain data

The complexity inherent in defining and computing fractional derivatives renders the development of a data-driven modeling framework for fractional-order systems more formidable than for their integer-order counterparts. This study introduces a methodology for sparsely identifying fractional chaotic systems by leveraging time-domain data to ascertain the system’s governing equations. Initially, we establish a sparse identification framework specifically tailored for fractional order systems and introduce a joint iterative thresholding method designed to identify these systems’ fractional and integer order components concurrently. Moreover, this study conducts a comparative analysis of two error criteria for model selection. It introduces a strategy predicated on minimizing the distance to balance sparsity and model fitting error optimally. To validate the efficacy and precision of the proposed methodology, numerical simulations were conducted on fractional order simplified Lorenz and Chua’s circuits, affirming the robustness and accuracy of our approach.

An adaptive unscented particle filter for a nonlinear fractional-order system with unknown fractional-order and unknown parameters

An unscented particle filter (UPF) is proposed for a nonlinear fractional-order system (NFOS) with an unknown order (UO) and unknown parameters. The Grünwald–Letnikov difference is used to discretize the continuous-time NFOS and the corresponding difference equation is acquired. For each sampled particle, a unscented transformation is applied, and the particles are afterwards optimized using a resampling algorithm. Furthermore, the augmented equations of the states, UO, and unknown parameters are established by an augmented vector method. The proposed fractional-order UPF is more accurate in estimating states than the fractional-order unscented Kalman filter and the fractional-order particle filter. Besides, the adaptive fractional-order UPF effectively estimate the UO and unknown parameters. Finally, two numerical examples and a practical example are used to verify the effectiveness of the proposed algorithm.

Fractional order system of integro-differential equations: theoretical and numerical simulations using Appell-type Changhee polynomials

We examine the behavior of the solutions for a fractional-order integro-differential equation (FIDE) system using an efficient simulation presented in this paper. The FIDEs are treated by a presumably novel approximation method based on Changhee polynomials of Appell type (ACPs). In this work, we use the method of spectral collocation (SCM), which is based on the advantages of ACPs. Using this technique on the given model, it generates an algebraic equation system. Through the evaluation of the residual error function (REF), we verify the efficiency of the approach that has been offered. To verify the effectiveness and originality of the suggested algorithm, the results are compared with the precise solutions. Our results show that the method employed provides a straightforward and efficient tool to simulate the solution for such models. The suggested method's primary benefits are that it only requires a few easy steps and doesn't generate secular terms.

State estimation-based parameter identification for a class of nonlinear fractional-order systems

Parametric identification is an important part of system theory since knowledge of the parameters allows the analysis and control of the system. The aim of this paper is to propose a novel robust (against measurement noise) parameter identification method for a class of nonlinear fractional-order systems. In order to solve the parametric identification we carry out this problem to a state estimation problem, we introduce a Fractional Algebraic Identifiability (FAI) property which allows to represent the system parameters as a function of the inputs and outputs of the system, this parameter identification method provides an on-line identification process (while the system is operating), we also propose a fractional-order differentiator which allows to reduce the effect of measurement noise as well as to provide the estimation of a fractional-order derivative of the system output. Moreover, we use the Mittag–Leffler boundedness to demonstrate the convergence of this method, a different approach for this stability analysis method is given in this paper. Finally, we illustrate the accuracy and robustness of our proposed method by means of the parametric identification of two nonlinear fractional-order systems: a time-varying nonlinear fractional-order system and a nonlinear fractional-order mathematical model of a simple pendulum.

Mittag–Leffler kernel operator on prey-predator model interfusing intra-specific competition and prey fear factor

In the present research, a prey-predator system that classifies predator species into premature and veteran species was proposed. Mathematical models of the system of species that include the impact of panic and a variety of functional reactions to the species have been recently reviewed. The consequence of fear is a reduction factor in the fundamental reproduction of the prey species. Relying on the prior research, we propose a mathematical predator-prey model employing the concept of fractional order system that contains two crucial components: intra-specific competition under mature predator density and fear, which affects the progeny of the prey community. Incorporating self-similarity characteristics is the significance of adopting a fractional order system. Following that, using memory kernel, we have interpret this model into the perspective of Caputo, that is, Atangana_Baleanu_Caputo (A_B_C) kind of fractional order derivatives. Singularity and non-locality are two of the main properties of fractional operators that make it simpler to figure out the natural behaviour of complicated systems. Details regarding the unique solution to the established model's derivation are provided. The dichotomy operator investigates the stability of the system under Hyper-Ulam`s stability method. The numerical interpolation to the suggested model generates the best precise approximate result. Through computer simulations, a thorough investigation on the predator species exhibiting intra-specific factor and prey species exhibiting fear factor is investigated. Using Adam's semi-analytical technique, we were able to derive the numerical estimates and determine the solution. The purpose to employ the Adams-Moulton approach is that it will provide more implicitly accurate results. Extensive numerical experiments are carried out using MATLAB software in order to portray the dynamical behaviour of the system. These observations lead us to the conclusion that total population expansion in the environment would be managed by the successful adoption of the fear factor in prey and the intra-species rivalry in predators.

Adaptive fuzzy event-triggered cooperative control for fractional-order delayed multi-agent systems with unknown control direction

In this paper, the cooperative control of fractional-order multi-agent systems with time delay and unknown control direction is studied by combining frequency distributed model and event-triggered mechanism. The Nussbaum function is employed to address the unmeasured control direction. Then, through the event-triggered mechanism, a corresponding controller is designed, which can guarantee that all signals are bounded while saving resources. The proposed theme is convenient for analyzing the stability of fractional-order systems based on a transformation of frequency distributed model. In particular, in order to enhance the universality of the proposed scheme in the fractional-order system stability analysis, a generalized fractional-order generalized lemma with respect to the Nussbaum function is given. Finally, the availability of the proposed method is confirmed by a simulation example.

Global exponential convergence and synchronization for exponential numerical competitive neural networks with different time scales and fuzzy logic

For nonlocal fuzzy delayed competitive neural networks, a novel difference model is first developed in this article using the Mittag-Leffler Euler difference method. Secondly, the existence of a sole globally bounded solution and globally exponential stability to the discrete-time networks are addressed based on Banach contractive mapping principle and the constant variation formula for sequences. Besides, exponential synchronization of the discrete-time networks is investigated and a feedback controller is designed from the viewpoint of the theory of controls. It is the first time to consider nonlocal fuzzy delayed competitive neural networks by using Mittag-Leffler Euler differences. In contrast to nonlocal discrete approaches, such difference competitive neural networks could preferably figure the features of continuous networks. The difference method in this paper extends and improves the traditional difference approaches, for example, Adams-Bashforth Euler differences. Moreover, this article explores some avenue for the research of discrete-time fractional-order systems and develops a set of theories and methodologies for the future investigations.

Separation of solutions and the attractivity of fractional-order positive linear delay systems with variable coefficients

This article has two purposes. First, we study the separation of solutions to mixed-order positive linear delay systems with variable coefficients and the lower estimates of the separation of solutions to those systems. Second, we consider the global attractivity of fractional-order positive linear delay systems and describe precisely the rate of convergence of solutions to their equilibrium in some specific cases. To do this, the unified approach we use here is that the comparison principles have been modified to accommodate fractional-order delay systems. In addition, numerical simulations are introduced to illustrate the validity of the theoretical results.

Response analysis of the vibro-impact system under fractional-order joint random excitation

As a kind of good damping material, viscoelastic material is widely used in machinery, civil engineering, and other fields. In this paper, the viscoelasticity of the system is described by fractional differentiation. The dynamic response of a unilateral vibro-impact system with a viscoelastic oscillator under joint random excitation is studied, in which joint random excitation is composed of additive and multiplicative white noise. The fractional-order derivative was calculated based on Caputo’s definition, and the fractional derivative was equivalent to the corresponding linear damping force and linear restoring force. As a result, a new random system without fractional-order terms was obtained. A non-smooth transformation was introduced, which was equivalent to the original system to a new system without a velocity jump. The steady-state probability density functions of fractional-order vibro-impact systems under joint random excitation are solved by using the random average method and non-smooth transformation. In addition, the effects of parameters on the steady-state response of the system are analyzed.

Combination-combination synchronization of chaotic fractional order systems

The quest for simple and practically implementable synchronization control functions is the fundamental motive for this research work because most of the control functions for synchronization are bulky thereby leading to complexity in implementation and high cost. This research paper examines the combination-combination synchronization of chaotic fractional order systems of identical order evolving from different initial conditions via the backstepping nonlinear control technique with the aim of reducing the control function’s complexity and cost. So, based on the stability theory of fractional order systems, stable synchronization are designed via a backstepping technique using chaotic fractional order Duffing and Aneodo as the paradigm. Numerical simulations are presented to confirm the feasibility of the analytical technique. The number of control functions has been sufficiently reduced compared to previous work hence reduction in control functions complexity and reduced cost of implementation. The outcome of this work may be useful to give a better understanding of the underlying dynamical behaviour among several interacting particles particularly in particle physics.

Stability Analysis of SARS-CoV-2 with Heart Attack Effected Patients and Bifurcation.

The aim of this study is to analyze and investigate the SARS-CoV-2 (SC-2) transmission with effect of heart attack in United Kingdom with advanced mathematical tools. Mathematical model is converted into fractional order with the help of fractal fractional operator (FFO). The proposed fractional order system is investigated qualitatively as well as quantitatively to identify its stable position. Local stability of the SC-2 system is verified and test the proposed system with flip bifurcation. Also system is investigated for global stability using Lyponove first and second derivative functions. The existence, boundedness, and positivity of the SC-2 is checked which are the key properties for such of type of epidemic problem to identify reliable findings. Effect of global derivative is demonstrated to verify its rate of effects of heart attack in united kingdom. Solutions for fractional order system are derived with the help of advanced tool FFO for different fractional values to verify the combine effect of COVID-19 and heart patients. Simulation are carried out to see symptomatic as well as a symptomatic effects of SC-2 in the United Kingdom as well as its global effects, also show the actual behavior of SC-2 which will be helpful to understand the outbreak of SC-2 for heart attack patients and to see its real behavior globally as well as helpful for future prediction and controlstrategies.

Order-Dependent Sampling Control of Uncertain FO Linear System

This paper proposes a sampled-data control problem for the fractional-order system with parameter uncertainties. First, according to the input delay approach, the dynamics of the considered fractional-order system is modeled by a delay system. The main purpose of the problem addressed is to design a sampled-data controller, such that the closed-loop fractional-order system guarantees the asymptotic stability. Then, the Lyapunov-Krasovskii functional is constructed to derive the stable criterion. The new delay-dependent and order-dependent stability conditions are presented in the form of linear matrix inequalities. Eventually, a new sampled-data controller is designed to ensure the stability and stabilization of fractional-order system. Finally, a numerical example is given to illustrate the effectiveness and superiority of the derived criterion.

Finite-time stability of fractional-order nonlinear systems.

The paper studies the finite-time stability (FTS) of equilibrium points (EPs) in fractional-order nonlinear systems (FONSs). Classifying the EPs into initial EP and finite-time EP innovatively, equations of the EPs in FONSs are investigated comprehensively, and a unified definition depicting the EPs in the FONSs is proposed. Furthermore, sufficient conditions of the FTS of EPs in FONSs are given. The proposed results are verified with an illustrative example. Thus, different from existing works that declare non-existence of finite-time stable equilibria in FONSs, the existence of finite-time stable EPs in FONSs is confirmed in this paper.

Event-Triggered Adaptive Finite-Time Containment Control for Fractional-Order Nonlinear Multiagent Systems.

This article investigates the finite-time containment control problem for fractional-order nonlinear multiagent systems (FOMASs) with event-triggered inputs. First, a new Lyapunov stability lemma is developed, which provides a basic approach for the completely unknown nonlinear fractional-order system to realize finite-time convergence. Considering the containment control for FOMASs, the restricted assumption that the derivatives of leaders' trajectories are known to the followers is removed, and only the boundedness of derivatives is required in this article, whose upper bound need not be known. To reduce the burden of communication, an event-triggered condition consisting of the control input and a decreasing function related to the containment errors is devised, which can provide more design freedom to balance the system performance and communication resources. According to the proposed fractional-order finite-time convergence lemma, a distributed adaptive containment control scheme is developed, such that each follower can be steered to the convex hull spanned by the leaders in finite time. Simulation examples further demonstrate the effectiveness of our proposed method.

Adaptive Fuzzy Echo State Network Control of Fractional-Order Large-Scale Nonlinear Systems With Time-Varying Deferred Constraints

Adaptive Fuzzy Echo State Network Control of Fractional-Order Large-Scale Nonlinear Systems With Time-Varying Deferred Constraints

Fault-Tolerant Fuzzy-Resilient Control for Fractional-Order Stochastic Underactuated System With Unmodeled Dynamics and Actuator Saturation.

This article is considered on underactuated fractional-order stochastic systems (FOSSs) with actuator saturation and incrementally conic nonlinear terms, whose fractional-order α ∈ (0,1) . First, to bring FO dynamic signals, solving the unmodeled dynamics, in the meantime, the saturated nonlinear term of the control input is taken into account. At the time, to cope with the stability issue of FOSS under such situation, the fault tolerant resilient controller based on underactuated condition is designed. Then, according to the method of the Lyapunov and It∧ o differential formulation to design proper multiple Lyapunov-Krasovskii (L-K) functions, such that, a novel sufficient condition of the robustly asymptotically stability of fuzzy FOSS under underactuated conditions is rigorously proved in terms of linear matrix inequality (LMI). Furthermore, in order to research the mean square stability of the above-mentioned system, so the solution of FOSS is obtained to achieve this purpose. By applying the above method, which is proposed in this work that the controlled system can be obtained with faster response and higher control accuracy. At last, to display the superiority of the above-mentioned scheme is effective, tethered satellite system and numerical results are presented.

Discrete Impulsive Signal Observer for Fractional-Order Control Systems and Its Consumer Electronic Circuit Application

In the process of actual electronic equipment signal transmission, interference is inevitable. How to accurately estimate the device status to ensure valid signal travelling is very meaningful. This paper mainly investigates a new transmission signal estimation strategy for fractional order control systems with unknown inputs. Compared with integer order modeling, fractional order modeling can better describe and report the essential process of system signal transmission. By using the fractional order output derivative-based method and convex optimization technique, an effective discrete-time impulsive observer design algorithm is proposed to stabilize the error dynamic systems. Especially, the resulting criterion reflects well the relationship between the fractional order and the pulse signal, and then is applied on a fractional order electronic circuit. It implies that the given conclusions are of significance in signal estimation.

Proposing a Caputo-Land System for active tension. Capturing variable viscoelasticity

Accurate cell-level active tension modeling for cardiomyocytes is critical to understanding cardiac functionality on a subject-specific basis. However, cell-level models in the literature fail to account for viscoelasticity and inter-subject variations in active tension, which are relevant to disease diagnostics and drug screening, e.g., for cardiotoxicity. Thus, we propose a fractional order system to model cell-level active tension by extending Land's state-of-the-art model of cardiac contraction. Our approach features the (left) Caputo derivative of six state variables that identify the mechanistic origins of viscoelasticity in a myocardial cell in terms of the thin filament, thick filament, and length-dependent interactions. This proposed CLS is the first of its kind for active tension modeling in cells and demonstrates notable subject-specificity, with smaller mean square errors than the reference model relative to cell-level experiments across subjects, promising greater clinical relevance than its counterparts in the literature by highlighting the contribution of different cellular mechanisms to apparent viscoelastic cell behavior, and how it could vary with disease.

Study on Stability for Interconnected Uncertain Fractional-Order Systems Based on Vector-Bounded Technique

Study on Stability for Interconnected Uncertain Fractional-Order Systems Based on Vector-Bounded Technique