Fractional-order Nonlinear Systems Research Articles

Securing Consensus in Fractional-Order Multi-Agent Systems: Algebraic Approaches Against Byzantine Attacks

This paper investigates the behavior of fractional-order nonlinear multi-agent systems subjected to Byzantine assaults, specifically focusing on the manipulations of both sensors and actuators. We employ weighted graphs, both directed and undirected, to illustrate the system's topology. Our methodology combines algebraic graph theory with fractional-order Lyapunov techniques to develop algebraic requirements for leader-following consensus, providing a robust framework for analyzing consensus dynamics in these complex systems. We present quantitative results demonstrating the effectiveness of our approach, including two numerical examples that validate the proposed requirements for consensus evaluation. Notably, our work highlights the novelty of using fractional-order systems to enhance resilience against adversarial conditions, contributing significantly to the field of multi-agent systems. By clarifying key terms and streamlining our language, we ensure accessibility for a broader audience while emphasizing the implications of our findings for real-world applications.

Event-Triggered Backstepping Control of Fractional-Order Chaotic Systems with Dead Zone Via Disturbance Observer

This paper comes up with an adaptive event-triggered tracking control scheme for a kind of fractional-order strict feedback nonlinear systems with actuator dead-zone inputs and uncertain external disturbances. The presented strategy can overcome the defect in standard backstepping control scheme which leads to the “explosion of complexity” problem of virtual signals, and strengthen the transmission efficiency by an event-triggered scheme. An error compensation mechanism is presented for promoting the tracking accuracy and reducing the effect filter error. The designed controller guarantees that the tracking error converges into a small neighborhood near the origin, and all closed-loop signals are bounded. Moreover, Zeno behavior will be avoided. The numerical simulation is given to verify the accuracy and effectiveness of the designed technique.

Event-Triggered Fuzzy Adaptive Predefined-Time Control for Fractional-Order Nonlinear Systems with Time-Varying Deferred Constraints and Its Application

This paper focuses on the fuzzy adaptive predefined-time control for fractional-order nonlinear systems with time-varying deferred constraints. First, a modified dynamic surface control technique is introduced to address the problem of computational complexity exposed in the backstepping framework, and the interval type-2 fuzzy logic systems are applied to model the unknown nonlinearities of the systems. Next, a shifting function and the barrier Lyapunov function with variational barrier bounds are formulated to deal with the constraints issue. Particularly, the constraint conditions can be satisfied within a predetermined time, even if they are transgressed initially. Furthermore, a switching threshold event-triggered controller is devised to balance the control energy and communication resources. With the help of the predefined-time stability criterion, it is proven that the presented predefined-time event-triggered controller can ensure that all the signals involved in the closed-loop system are bounded and the tracking error fluctuates to a small neighborhood of the origin in a predefined-time interval. Finally, two simulation examples are provided to confirm the effectiveness of the put-forward control algorithm.

Modified Hermite Wavelet Discrete Matrix Approach for the Brusselator Chemical Model

AbstractThe primary goal of this study is to create a wavelet collocation technique that can be used to solve nonlinear fractional order systems of ordinary differential equations, which are equations that arise in modeling problems related to auto‐catalytic chemical reactions. Using the Hermite wavelet collocation method (HWCM), the system of nonlinear ordinary differential equations of integer and fractional order is numerically solved. The nonlinear Brusselator system is transformed into an algebraic equation system using the collocation method and the fractional derivative operational matrices. The Newton‐Raphson method is used to solve these algebraic equations, and the approximate values of the derived unknown coefficients are substituted. Through the numerical examples, the method's computational effectiveness and correctness are illustrated with various model constraints. A numerical comparison is made between the current approach ND solver, RK method, and Haar wavelet method (HWM). The efficiency and reliability of the developed strategy's performance are shown in graphs and tables. The created Hermite wavelet collocation method is resilient and has good accuracy compared to current methods found in the literature. Numerical computations are performed through Mathematica, a mathematical software.

Adaptive neural self-triggered bipartite consensus control for nonlinear fractional-order multi-agent systems with actuator fault

Adaptive neural self-triggered bipartite consensus control for nonlinear fractional-order multi-agent systems with actuator fault

Distributed Adaptive Formation Control for Fractional-Order Multi-Agent Systems with Actuator Failures and Switching Topologies

In this paper, a class of distributed adaptive formation control problems are investigated for second-order nonlinear fractional-order multi-agent systems with actuator failures and switching topologies. To address these challenges, two adaptive coupling gains based on agents’ position and velocity are incorporated into the control protocol. Using the Lyapunov method along with graph theory and matrix analysis, sufficient conditions for system stability are derived in the presence of actuator failures and switching topologies. The effectiveness of the proposed control protocol is demonstrated through numerical simulations, which show its capability to maintain stable formation control under these challenging conditions.

Containment control for non-linear fractional-order multi-agent systems via refined sample data controller

Abstract This manuscript concentrates on the problem of designing a sampled data controller (SDC) for the consensus of a fractional-order multi-agent system (FOMAS) with Lipschitz non-linearity via an algebraic approach. The solution of the FOMAS is represented by using the Laplace transform approach. An upper bound of the sampling period is determined through various integral inequality techniques. Distinguished from the existing works, the estimate for an upper bound is more accurate which involves the Lipschitz constant of the non-linear function. Finally, numerical examples are given to validate the correctness of results. Furthermore, the comparison results are presented to show the proposed method determines a better upper bound of the sampling period.

Finite-time contractive stability for fractional-order nonlinear systems with delayed impulses: Applications to neural networks

This paper deals with finite-time stability (FTS) and finite-time contractive stability (FTCS) criteria of fractional-order nonlinear systems (FONSs), where the state of the systems consists of a finite number of delayed impulsive instants. To achieve these results, we have extended the theories on FTS and FTCS criteria from integer-order NSs to fractional-order systems. Employing the fractional-order Lyapunov stability theory, the sufficient conditions of the above-mentioned stability criteria are derived for a class of FONSs with state-dependent delayed impulses. Then, to ensure the applicability of the proposed results, we have analyzed the desired performances for several neural network (NN) models, such as non-autonomous NNs (NANNs), Cohen–Grossberg NNs (CGNNs), switched NNs (SNNs) and bidirectional associative memory NNs (BAMNNs). Finally, four representative numerical examples are illustrated to demonstrate the assurance of the obtained stability conditions of the respective state-dependent impulsive fractional-order NNs (FONNs).

Fuzzy Adaptive Control for Consensus Tracking in Multiagent Systems with Incommensurate Fractional-Order Dynamics: Application to Power Systems

This paper presents a novel approach to address the consensus tracking problem within a class of incommensurate fractional-order nonlinear non-affine systems. Our method employs an adaptive fuzzy technique that integrates newly developed fractional adaptive algorithms based on the Lyapunov method into the controller's design process. This method develops stability based on the global representation of the follower and leader systems, reducing assumptions on the system dynamics to address non-affinity. Additionally, it introduces a simplified approach to designing controllers for incommensurate fractional-order multiagent systems. The proposed controller effectively discerns uncertainties and external disturbances, compelling follower agents to seamlessly follow the desired trajectories set by the leader. Notably, compared to the existing literature, our method exhibits key advantages, including reduced assumptions regarding the system's non-affinity and a simpler design for controlling incommensurate systems. We demonstrate the efficacy of the proposed incommensurate fractional controller through simulations conducted using MATLAB on a fractional-order multiagent power system.

Stability analysis of random fractional-order nonlinear systems and its application

The research on stability analysis and control design for random nonlinear systems have been greatly popularized in recent ten years, but almost no literature focuses on the fractional-order case. This paper explores the stability problem for a class of random Caputo fractional-order nonlinear systems. As a prerequisite, under the globally and the locally Lipschitz conditions, it is shown that such systems have a global unique solution with the aid of the generalized Gronwall inequality and a Picard iterative technique. By resorting to Laplace transformation and Lyapunov stability theory, some feasible conditions are established such that the considered fractional-order nonlinear systems are respectively Mittag-Leffler noise-to-state stable, Mittag-Leffler globally asymptotically stable. Then, a tracking control strategy is established for a class of random Caputo fractional-order strict-feedback systems. The feasibility analysis is addressed according to the established stability criteria. Finally, a power system and a mass–spring-damper system modeled by the random fractional-order method are employed to demonstrate the efficiency of the established analysis approach. More critically, the deficiency in the existing literatures is covered up by the current work and a set of new theories and methods in studying random Caputo fractional-order nonlinear systems is built up.

Fault-tolerant controller design with connectivity maintenance in fractional-order multi-agent systems using a super-twisting sliding mode algorithm

This article explores fault analysis in distributed fractional multi-agent systems, specifically addressing fault detection in nonlinear fractional-order multi-agent systems by introducing a sliding surface. To achieve fault detection, we employ a super-twisting sliding mode observer designed for accurate fault estimation within individual agents. Building on this, we present a new fractional-order sliding surface accompanied by a potential function, strategically designed to ensure connectivity maintenance. Drawing on the estimated fault and the fractional-order sliding surface, we propose a fault-tolerant controller. This controller operates within a finite time frame, simultaneously guaranteeing fault tolerance and connectivity maintenance. To demonstrate the effectiveness of our approach, we conduct simulations on three numerical examples. These examples represent heterogeneous systems with both directed and undirected graph topologies, showcasing the versatility and applicability of our proposed methodology.

Finite-time adaptive NN dynamic surface control for nonstrict nonlinear FOSs subject to input dead-zone and full-states constraints

This article focuses on a kind of nonstrict nonlinear fractional-order systems (FOSs) suffering from state constraints and dead-zone input. Meanwhile, a finite-time adaptive dynamic surface control (DSC) approach based on backstepping technology and approximation principle of radial basis function neural network (RBFNN) is developed. To overcome the problem of inherent computational complexity, a fractional-order filter is applied to approach the virtual controller and its fractional-order derivative in each step of the backstepping procedure. The barrier Lyapunov function (BLF) is employed to handle the state constraints, and finite-time stability criteria on the basis of fractional-order Lyapunov method are introduced to prove the finite-time convergence of the tracking error into a small region around the origin. It is shown that all the solutions of the closed-loop system are bounded, while the state constraints are satisfied within a predetermined finite time. Finally, two examples are provided to demonstrate the effectiveness of the presented control scheme.

Securing Bipartite Nonlinear Fractional-Order Multi-Agent Systems against False Data Injection Attacks (FDIAs) Considering Hostile Environment

This study investigated the stability of bipartite nonlinear fractional-order multi-agent systems (FOMASs) in the presence of false data injection attacks (FDIAs) in a hostile environment. To tackle this problem we used signed graph theory, the Razumikhin methodology, and the Lyapunov function method. The main focus of our proposed work is to provide a method of stability for FOMASs against FDIAs. The technique of Razumikhin improves the Lyapunov-based stability analysis by supporting the handling of the intricacies of fractional-order dynamics. Moreover, utilizing signed graph theory, we analyzed both hostile and cooperative interactions between agents within the MASs. We determined the system stability requirements to ensure robustness against erroneous data injections through comprehensive theoretical investigation. We present numerical examples to illustrate the robustness and efficiency of our proposed technique.

Deviation analysis in transient response of fractional-order systems: An elementary function-based lower bound

In this paper, transient response of commensurate fractional-order systems with non-zero initial conditions is investigated in the viewpoint of the existence of deviation from initial condition. In particular, firstly peak effect for a class of linear fractional order systems is inspected by presenting a lower bound for their responses. Such a lower bound is described by an elementary function, where the commensurate order of the system is considered as a rational number. Furthermore, it has been demonstrated that under particular circumstances the derived lower bound can be extended to apply for deviation analysis in response of a class of fractional-order nonlinear systems. Moreover, included are various illustrative examples intended to assess the applicability of the obtained lower bound in prediction of the deviation value and the time instance at which the peak effect occurs.

Sliding mode tracking control of a class of fractional-order nonstrict-feedback nonlinear systems

Since the Leibniz rule for integer-order derivatives of the product of functions, which includes a finite number of terms, is not true for fractional-order (FO) derivatives of that, all sliding mode control (SMC) methods introduced in the literature involved a very limited class of FO nonlinear systems. This article presents a solution for the unsolved problem of SMC of a class of FO nonstrict-feedback nonlinear systems with uncertainties. Using the Leibniz rule for the FO derivative of the product of two functions, which includes an infinite number of terms, it is shown that only one of these terms is needed to design a SMC law. Using this point, an algorithm is given to design the controller for reference tracking, that significantly reduces the number of design parameters, compared to the literature. Then, it is proved that the algorithm has a closed-form solution which presents a straightforward tool to the designer to obtain the controller. The solution is applicable to the systems with a mixture of integer-order and FO dynamics. Stability and finite-time convergence of the offered control law are also demonstrated. In the end, the availability of the suggested SMC is illustrated through a numerical example arising from a real system.

Robust Consensus Analysis in Fractional-Order Nonlinear Leader-Following Systems with Delays: Incorporating Practical Controller Design and Nonlinear Dynamics

This article investigates the resilient-based consensus analysis of fractional-order nonlinear leader-following systems with distributed and input lags. To enhance the practicality of the controller design, an incorporation of a disturbance term is proposed. Our modeling framework provides a more precise and flexible approach that considers the memory and heredity aspects of agent dynamics through the utilization of fractional calculus. Furthermore, the leader and follower equations of the system incorporate nonlinear functions to explore the resulting changes. The leader-following system is expressed by a weighted graph, which can be either undirected or directed. Analyzed using algebraic graph theory and the fractional-order Razumikhin technique, the case of leader-following consensus is presented algebraically. To increase robustness in multi-agent systems, input and distributive delays are used to accommodate communication delays and replicate real-time varying environments. This study lays the groundwork for developing control methods that are more robust and flexible in complex networked systems. It does so by advancing our understanding and practical application of fractional-order multi-agent systems. Additionally, experiments were conducted to show the effectiveness of the design in achieving consensus within the system.

Reduced-Order Observer-Based Adaptive Fuzzy Self-Triggered Control for Fractional Order Nonlinear Systems Without Feasibility Conditions

Reduced-Order Observer-Based Adaptive Fuzzy Self-Triggered Control for Fractional Order Nonlinear Systems Without Feasibility Conditions

Optimal fuzzy event-triggered fault-tolerant control of fractional-order nonlinear stochastic systems

This paper considers the optimal fuzzy fault-tolerant control issue for fractional-order nonlinear stochastic systems (FONSS) using event-triggered mechanisms with fractional-order derivative α∈(0,1). First, in order to weaken the impacts of actuator and sensor faults, a fuzzy observer and augmented system are developed. Since stochastic disturbance can adversely affect stability, based on fractional Barbalat lemmas and Lyapunov theory, an adaptive fault-tolerant controller and the corresponding adaptive update law are designed to ensure the stability of FONSS. Then, the architecture of identifier-critic-actor NN, which is more concise than the traditional one, is described. This NN architecture recognizes unknown dynamics and completes control actions after evaluating the control performance, driving control toward optimization. Furthermore, the event-triggered mechanism is devised to save control resources and avoid Zeno behavior. Finally, simulation results demonstrate that under equivalent conditions, the proposed optimal event-triggered control reduces the number of information transmissions compared to traditional time-triggered control, saving communication resources and enhancing the overall control performance.

Numerical simulation of fractional-order Duffing system with extended Mittag-Leffler derivatives

In this paper, we studied the dynamics of a nonlinear fractional-order Duffing system combined with Mittag-Leffler derivatives in order to provide dynamic behaviors different from existing ones. The Mittag-Leffler derivative is a generalized version of the exponential kernel derivative. To achieve this goal, we introduced a modified extension to higher-order Mittag-Leffler derivatives to overcome the initialization problem. Moreover, we discussed some properties and relationships of the studied derivatives. Then we presented numerical schemes to handle fractional extensions of the considered oscillatory system including the Mittag-Leffler and the Caputo derivatives. Numerical simulations are carried out and the resulting simulation dynamics of the studied fractional oscillatory system are compared.

Adaptive Fuzzy Control for Fractional-Order Networked Control Systems with Input Time Delay and Data Loss

This paper considers the adaptive fuzzy control of fractional-order nonlinear networked control systems subjected to network-induced input delay and data loss. To approximate unknown functions, fuzzy logic systems are employed. Furthermore, the Pade approximation method and an intermediate variable are introduced to eliminate the impact of input delay, and an adaptive fuzzy controller is designed using backstepping technology. Based on fractional-order Lyapunov stability theory, the proposed method can ensure that all signals are uniformly ultimately bounded, and the tracking error can converge to a small region of the origin. Two simulation examples are provided to verify the viability of the control method.