Fractional-order Nonlinear Systems Research Articles

Event-triggered impulsive control for synchronization in finite time of fractional-order reaction–diffusion complex networks

This paper is concerned with the finite-time synchronization (FTS) issue for fractional-order reaction–diffusion complex networks (RDCNs). A finite-time stability principle, which plays a key role in the synchronization analysis later, is developed for fractional-order nonlinear impulsive system. A novel hybrid controller, which consists of a event-triggered controller and a impulsive controller, is designed to realize the global FTS objective for the considered network. By applying the Lyapunov stability theory and the fractional calculus, the global FTS conditions are addressed in the form of algebraic inequalities. In addition, the exclusion of Zeno behavior are proved for the designed event-triggered strategy. Finally, a numerical example is provided to illustrate the feasibility of the proposed control approach and the correctness of the theoretical results.

Embedded adaptive fractional-order sliding mode control based on TSK fuzzy system for nonlinear fractional-order systems

An adaptive fractional-order sliding mode control (AFOSMC) is proposed to control a nonlinear fractional-order system. This scheme combines the features of sliding mode control and fractional control for improving the response of nonlinear systems. The structure of AFOSMC includes two units: fractional-order sliding mode control (FOSMC) and the tuning unit that employs a certain Takagi–Sugeno–Kang fuzzy logic system for online adjusting the parameters of FOSMC. Tuning the parameters of the FOSMC improves its performance with various control problems. Moreover, stability analysis of the proposed controller is studied using Lyapunov theorem. Finally, the developed control scheme is introduced for controlling a fractional-order gyroscope system. The proposed AFOSMC is implemented practically using a microcontroller where the test is carried out using the hardware-in-the-loop simulation. The practical results indicate the improvements and enhancements introduced by the developed controller under external disturbance, uncertainties and random noise effects.

Cluster consensus of general fractional-order nonlinear multi agent systems via adaptive sliding mode controller

Cluster consensus of general fractional-order nonlinear multi agent systems via adaptive sliding mode controller

Saturated delayed impulsive effects for fractional order nonlinear system with piecewise Caputo derivative and its application

Fractional order nonlinear systems with saturated delayed impulses and short-term memory is investigated in this paper. Models in this paper are described by piecewise Caputo derivative. First, sufficient conditions for local exponential stability are derived based on the fractional order Lyapunov function method and saturated control theory. Then, the optimization model to estimate the domain of attraction is built, which is constrained by some LMIs. Third, the application of synchronization for fractional order complex dynamical networks via distributed saturated delayed impulsive control is studied. Finally, two numerical examples are given to illustrate the effectiveness of the theoretical results.

Adaptive observer-based control of arbitrarily switched fractional-order uncertain nonlinear system subject to input nonlinearities and asymmetric output constraints

This paper addresses an adaptive observer-based control for a switched fractional-order uncertain nonlinear system subject to input nonlinearities and asymmetric output constraints. The switching signal is arbitrary, and the designed controller needs no information regarding this signal. By employing a universal framework, the designed controller can cope with different input nonlinearities in different modes without having any information about their characteristics. It is assumed that only system output is measured, and a designed observer estimates the system states for control implementation. The uncertainties are approximated by intelligent approximators like fuzzy logic systems (FLS) or neural networks (NN). The time-varying asymmetric output constraints have been satisfied using a common barrier Lyapunov function (BLF). Using the common Lyapunov function (CLF) method and the backstepping control technique, the virtual controllers, adaptive law and control input have been designed, and the boundedness of closed-loop signals has been guaranteed. The effectiveness of the designed control scheme has been demonstrated through a numerical and two practical simulation examples.

Lyapunov theorem for stability analysis of nonlinear nabla fractional order systems

This paper addresses the issue of stability analysis of nabla fractional order systems, which is crucial when applying fractional calculus to control problems in engineering applications. In contrast to the conventional Lyapunov theorem, the newly developed theorems presented here are practical and versatile. Firstly, two enhanced stability criteria are established for Mittag-Leffler stability and asymptotic stability, respectively. Subsequently, Lyapunov theorems based on class K functions are derived to assess boundedness and attractiveness. Finally, illustrative examples are also provided to substantiate the theoretical results.

Event-triggered adaptive neural networks tracking control for incommensurate fractional-order nonlinear systems with external disturbance

The event-triggered adaptive neural networks (NNs) tracking control issue is considered in this paper for a type of incommensurate fractional-order systems (IFOSs) with disturbance. Unlike the existing works for IFOSs, the hypothesis of disturbance-like function and the frequency distributed model are avoided in this study by utilizing the continuity of fractional-order derivative (FOD). Then, based on the radial basis function (RBF) NNs and backstepping method, the adaptive control scheme is proposed to ensure that the reference signal can be tracked by the system output, where the derivative order of adaptive laws does not depend on the one of IFOS. Additionally, to further reduce the burden of communication network, an exponentially convergent term is introduced into traditional dynamic event-triggered mechanism (ETM) and the Zeno behavior is avoided. Finally, two illustrative examples are exhibited to verify the performance of designed control scheme.

Distributed output-based bipartite consensus for nonlinear fractional-order multi-agent systems with continuous and intermittent communications

This paper studies the distributed output-based bipartite consensus problem in a nonlinear fractional-order multi-agent system (FOMAS) subjected to heterogeneous nonlinear functions and disturbances over a signed directed graph. Two types of distributed bipartite consensus algorithms are proposed to solve the bipartite consensus problem by utilizing continuous and intermittent output measurements, respectively. A novel discontinuous algorithm with a nominal controller is first developed to derive that all agents reach the sliding-mode surface in finite time, and meanwhile, each agent’s dynamics can be described by a linear FOMAS with a nominal controller. When each agent can access its neighbors’ output measurements continuously all the time, a continuous output-based nominal controller is designed for the linear FOMAS to realize bipartite consensus asymptotically. When each agent can only access its neighbors’ output measurements intermittently in some disconnected time intervals, an intermittent output-based nominal controller is further designed, where the communication rate is required to be greater than a derived threshold value. Numerical simulations are performed to validate the effectiveness of the main results.

SOS Approach for Practical Stabilization of Tempered Fractional-Order Power System

Fractional systems have been widely utilized in various fields, such as mathematics, physics and finance, providing a versatile framework for precise measurements and calculations involving partial quantities. This paper aims to develop a novel polynomial controller for a power system (PS) with fractional-order (FO) dynamics. It begins by studying the practical stability of a general class of tempered fractional-order (TFO) nonlinear systems, with broad applicability and potential for expanding its applications. Afterward, a polynomial controller is designed to guarantee the practical stability of the PS, encompassing the standard constant controller as a specific instance. The design conditions for this controller are resolved using the sum of squares (SOS) approach, a powerful technique for guaranteeing stability and control design. To showcase the practical value of the analytical findings, simulations of the PS are conducted utilizing SOSTOOLS.

Event‐triggered fuzzy adaptive control of incommensurate fractional order nonlinear systems

AbstractThis paper focuses on the event‐triggered control of nonstrict‐feedback incommensurate fractional order systems with external disturbances. To release more communication resources and reduce the triggered time instant, we adopt the relative‐threshold‐based event‐triggered strategy and prove that the Zeno behavior does not exist. We utilize the fuzzy systems to approximate the nonlinear systems and design an adaptive update law to estimate the approximation errors and external disturbances. To conquer the dimension explosion phenomenon in the backstepping process, we design a fractional order filter and apply the dynamic surface control technique. The closed‐loop system is semiglobally stable and the tracking error converges to a small neighborhood of equilibrium point under the proposed control approach. Finally, the validity of the developed controller is verified by the simulation examples.

Quantized iterative learning control for singular nonlinear fractional-order time-delay multi-agent systems with iteration-varying reference trajectories and switching topologies

This paper addresses the consensus tracking of the leader-following singular nonlinear fractional-order system with state time-delay in face of iteration-varying communication topologies and reference trajectories via an iterative learning control scheme. Firstly, by means of the neighbor information, the closed-loop Dα-type quantized iterative learning protocol is constructed based on output tracking errors. Secondly, sufficient conditions of consensus errors between each follower agent and leader agent are given in a fixed time interval and convergence analysis is also presented under the fixed topology and reference trajectory. Thirdly, the extension to iteration-varying reference trajectory tracking and switching topologies cases are investigated. It is shown that the developed protocol can also effectively work along the iteration axis on a finite time interval, although the communication topologies and reference trajectories vary dynamically with respect to iteration. Finally, numerical examples are shown to verify the effectiveness of the obtained results.

Event-Triggered Distributed Sliding Mode Control of Fractional-Order Nonlinear Multi-Agent Systems

In this study, the state consensus problem is investigated for a class of nonlinear fractional-order multi-agent systems (FOMASs) by using a dynamics event-triggered sliding mode control approach. The main objective is to steer all agents to some bounded position based on their own information and the information of neighbor agent. Different from the existing results, both asymptotic consensus problem and Zeno-free behavior are ensured simultaneously. To reach this objective, a novel event-triggered sliding mode control approach is proposed, composed of distributed dynamic event-triggered schemes, event-triggered sliding mode controllers, and auxiliary switching functions. Moreover, to implement the distributed control scheme, the fractional-order adaptive law is also developed to tuning the coupling weight, which is addressed in distributed protocol. With the improved distributed control scheme, all signals in the fractional-order closed-loop systems are guaranteed to be consensus and bounded, and a novel approach is developed to avoid the Zeno behavior. Finally, the availability and the effectiveness of the above-mentioned approach are demonstrated by means of a numerical example.

A mathematical study unfolding the transmission and control of deadly Nipah virus infection under optimized preventive measures: New insights using fractional calculus

This paper presents a new mathematical model governed by a nonlinear fractional-order system of differential equations to investigate the dynamics and optimal control interventions of the Nipah virus using the Caputo derivative. The model describes the effects of inappropriate contact with an infectious corpse as a potential route for virus transmission. We considered two transmission modes while formulating the proposed model: Food-borne transmission and human-to-human transmission. Initially, the model is evaluated with constant controls, and the fundamental analysis is carried out. The model analysis identified three equilibrium states: Nipah virus-free equilibrium, infected flying foxes free equilibrium, and Nipah virus endemic equilibrium state. In addition, a thorough theoretical analysis including stability of the Nipah virus-free equilibrium and Nipah virus endemic equilibrium points of the fractional model is performed. Furthermore, sensitivity analysis of the model parameters is performed to obtain effective time-dependent controls. Based on the sensitivity indices, a fractional optimal control model is presented and the most effective strategy for disease eradication is determined through numerical simulations. The model is optimized using optimal control theory, and Pontryagin’s maximum principle is used to obtain primary optimality conditions. Simulation results are performed to verify the theoretical results.

Practical stability of fractional-order nonlinear fuzzy systems

In this paper, we present a practical Mittag–Leffler stability for a new class of fractional-order nonlinear Takagi–Sugeno fuzzy uncertain systems. Based on fractional-order Lyapunov stability theory and the parallel distributed compensation (PDC) controller techniques, we show the convergence of the solutions of the closed-loop considered system toward a neighborhood of the origin. Furthermore, a numerical example is given to show the applicability of the main result.

Fuzzy Adaptive Command-Filter Control of Incommensurate Fractional-Order Nonlinear Systems.

This paper focuses on the command-filter control of nonstrict-feedback incommensurate fractional-order systems. We utilized fuzzy systems to approximate nonlinear systems, and designed an adaptive update law to estimate the approximation errors. To overcome the dimension explosion phenomenon in the backstepping process, we designed a fractional-order filter and applied the command filter control technique. The closed-loop system was semiglobally stable, and the tracking error converged to a small neighbourhood of equilibrium points under the proposed control approach. Lastly, the validity of the developed controller is verified with simulation examples.

Event-Triggered Adaptive Fuzzy PI Control of Uncertain Fractional-Order Nonlinear Systems With Full-State Constraints

This article is devoted to the adaptive fuzzy proportional-integral (PI) control of uncertain fractional-order (FO) strict-feedback nonlinear systems subject to full-state constraints (FSCs) and external disturbances via an event-triggered (ET) control technique. First, fuzzy logic systems are introduced to approximate unknown nonlinear functions. Second, a novel adaptive FOPI controller with a simple structure and low calculation cost is developed, and it is able to determine its PI gains adaptively and automatically without involving manual operation or trial and error procedures, which is an improvement compared with the traditional PI control scheme. Third, a relative-threshold-based ET controller is constructed so that the communication load from the controller to the actuator can be significantly decreased, and there is no Zeno behavior. Then, a barrier Lyapunov function is introduced into the backstepping recursive process to avoid FSC violations. Moreover, all closed-loop signals are bounded, and the output is driven to follow the desired output through the FO Lyapunov stability analysis. In particular, the proposed approach does not require a priori knowledge of the FO derivative of the reference signal. Finally, the feasibility of the proposed scheme is verified by numerical simulations.

State Feedback Controller Design for a Class of Generalized Proportional Fractional Order Nonlinear Systems

The state feedback controller design for a class of Generalized Proportional Fractional Order (GPFO) Nonlinear Systems is presented in this paper. The design is based on the combination of the One-Sided Lipschitz (OSL) system class with GPFO modeling. The main contribution of this study is that, to the best of the authors’ knowledge, this work presents the first state feedback control design for GPFO systems. The suggested state feedback controller is intended to ensure the system’s generalized Mittag Leffler (GML) stability and to deliver optimal performance. The findings of this paper show that the proposed strategy is effective in stabilizing Generalized Proportional Fractional Order Nonlinear Systems. A numerical example is presented to demonstrate the usefulness of the stated theoretical conclusions.

Cluster consensus and cluster formation for nonlinear fractional-order multi-agent systems

Cluster consensus and cluster formation for nonlinear fractional-order multi-agent systems

Forecasting education expenditure with a generalized conformable fractional-order nonlinear grey system model

As an important human capital investment, education is an effective means to improve the comprehensive quality of people. Education expenditure is an important material guarantee for the development of educational undertakings. Education expenditure data is highly susceptible to numerous economic and social factors that complicate its nonlinear structure. In order to model the complex nonlinear problems of the system, this paper proposes a generalized conformable fractional-order nonlinear grey prediction model for the first time by analyzing the traditional time series-based modeling method in a nonlinear grey domain. The proposed model expands on the classical grey Bernoulli model by introducing the generalized conformable fractional accumulation as a new accumulation generator and utilizes error minimization principles in the modeling process. By altering the optimal order of the model and the cumulative generation operator, this model can adapt to various time series and reduce errors. Finally, the model is applied to education expenditure forecasting, and it is proved that the proposed model achieved good results and has higher accuracy than other models.

Asymptotic behavior of fractional-order nonlinear systems with two different derivatives

This paper addresses the asymptotic behavior of systems described by nonlinear differential equations with two fractional derivatives. Using the Mittag–Leffler function, the Laplace transform, and the generalized Gronwall inequality, a sufficient asymptotic stability condition is derived for such systems. Numerical examples illustrate the theoretical results.