Fractional-order Nonlinear Systems Research Articles

Observer-based adaptive control for fractional-order strict-feedback nonlinear systems with state quantisation and input quantisation

This paper investigates the adaptive control problems of fractional-order strict-feedback nonlinear systems with state quantisation and input quantisation. Both Caputo and Riemann–Liouville derivative definitions are considered. Based on these two definitions, two fractional-order state observers are constructed to smooth the discontinuous quantised states, respectively. Then, new error variables and the dynamic filters are introduced to facilitate the design of the adaptive quantised controller. It is shown that the designed controller is valid for fractional-order strict-feedback nonlinear systems under both Caputo derivative definition and Riemann–Liouville derivative definition. Finally, three examples are given to demonstrate the effectiveness of the proposed method.

Fractional-order input-to-state stability and its converse Lyapunov theorem

This paper establishes input-to-state stability (ISS) for fractional-order nonlinear systems, providing a stability constraint framework for fractional-order systems affected by external inputs. The general fractional comparison principle aids in constructing Lyapunov functions, offering criteria for input-to-state stability of fractional-order nonlinear systems. The concept of weak robustness and the converse Lyapunov theorem for fractional-order nonlinear systems further enhance the completeness of input-to-state stability in these systems. By designing systems to maintain fractional-order input-to-state stability (FOISS), the system can remain robust against the effects of bounded external inputs. Serving as a foundational ”toolbox,” this work provides insights into the stability analysis of fractional-order nonlinear systems, benefiting researchers and practitioners in the field of control theory.

Decentralized state feedback guaranteed cost control for conformable fractional-order nonlinear large-scale systems

Decentralized state feedback guaranteed cost control for conformable fractional-order nonlinear large-scale systems

Finite-Time Resource Allocation Algorithm for Networked Fractional Nonlinear Agents

This paper investigates finite-time resource allocation problems (RAPs) for uncertain nonlinear fractional-order multi-agent systems (FOMASs), considering global equality and local inequality constraints. Each agent is described by high-order dynamics with multiple-input multiple-output and only knows its local objective function. Due to the characteristics of dynamic systems, the outputs of agents are inconsistent with their inputs, making it challenging to satisfy the inequality constraints when solving RAPs. To address this complex optimization control problem, a novel hierarchical algorithm is proposed, consisting of a distributed estimator and a local controller. Specifically, the distributed estimator is established by adopting the ϵ-exact penalty function and the gradient descent method. This estimator enables the system states to reach the optimal solution of RAPs within a finite time. Furthermore, the local controller is presented based on the fractional-order tracking differentiator and adaptive neural control approach. Under this controller, the system states are slaved to track the optimal signals generated by the estimator within a finite time. In both the estimator and controller algorithms, the finite-time stability is uniformly guaranteed with the help of Lyapunov functions. Finally, the effectiveness of our algorithm is demonstrated through three simulation examples.

Stabilization for a Class of Partially Observable Uncertain Fractional-Order Nonlinear Systems With Time-Varying Delays and Disturbance

Stabilization for a Class of Partially Observable Uncertain Fractional-Order Nonlinear Systems With Time-Varying Delays and Disturbance

Command Filter‐Based Prescribed Performance Adaptive Control for Fractional Order Non‐Strict System With Unmodeled Dynamics and Input Delay

ABSTRACTThis article focuses on adaptive control of a class of non‐strict feedback nonlinear fractional order systems with input delay and unmodeled dynamics under prescribed performance constraints. Command filtering backstepping design method is employed to avoid explosion of computation. Additionally, an error compensation mechanism is established to mitigate any errors introduced by the command filter. Radial basis function neural network is utilized to approximate the nonlinear function. Auxiliary signal processing variables are introduced to handle unmodeled dynamics. To address the input delay problem, a Pade approximation technique is employed. The stability analysis of the controller is conducted using Lyapunov stability theory, ensuring that the tracking error converges within a narrow predefined performance range. Finally, simulation results are presented to demonstrate the effectiveness of the proposed controller.

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.

Global Stability of Fractional Nonlinear Differential Systems With State-Dependent Delayed Impulses.

The fractional-order (FO) nonlinear differential system with state-dependent (SD) delayed impulses (DI) is considered in this brief. The considered impulses are related to the delayed state of the system and the delays are SD. A novel lemma for the monotonicity of the solution of Caputo's FO derivative equation is given. By means of linear matrix inequality (LMI) and several comparative arguments, criteria of uniform stability, uniform asymptotical stability, and Mittag-Leffler stability are obtained. Compared with other works on integer-order (IO) impulsive delayed systems with SD delays or fixed delays, how to impose constraints on parameters and impulses is explored, without imposing the boundedness on the state delays. Two examples are implemented to examine the practicality and sharpness of our theoretical analysis.

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.