Non-existence of finite-time stable equilibria in fractional-order nonlinear systems

We note that in the literature it is often taken for granted that for fractional-order system without delays, whenever the system trajectory reaches the equilibrium, it will stay there. In fact, this is the well-known phenomenon of finite-time stability. However, in this paper, we will prove that for fractional-order nonlinear system described by Caputo’s or Riemann–Liouville’s definition, any equilibrium cannot be finite-time stable as long as the continuous solution corresponding to the initial value problem globally exists. In addition, some examples of stability analysis are revisited and linear Lyapunov function is used to prove the asymptotic stability of positive fractional-order nonlinear systems.     

Stability analysis for fractional-order neural networks with time-varying delay

This paper focuses on the stability analysis for fractional-order neural networks with time-varying delay. A novel Lyapunov's asymptotic stability determination theorem is proved, which can be used for fractional-order systems directly. Different from the classical Lyapunov stability theorem, constraint condition on the derivative of Lyapunov function is revised as an uniformly continuous class-K function in the fractional-order case. Based on this novel Lyapunov stability theorem and free weight matrix method, a new sufficient condition on Lyapunov asymptotic stability of fractional-order Hopfield neural networks is derived by constructing a suitable Lyapunov function. Moreover, two numerical examples are provided to illustrate the effectiveness of these criteria.     

On a method for studying stability of nonlinear dynamic systems

A method for studying the stability of the equilibrium points of linearized, nonlinear dynamic systems of arbitrary order is considered. The method is based on the fact that, due to the nature of the mutual arrangement of the trajectories of the corresponding linearized system, and the boundaries of some simply-connected, bounded neighborhood of its equilibrium point, one can judge the asymptotic stability and instability of both this point and the equilibrium point of the nonlinear system. Necessary and sufficient conditions of asymptotic stability and sufficient conditions of instability of equilibrium points of linear systems are given. Together with the theorems of the first Lyapunov method, these conditions determine the sufficient conditions of asymptotic stability and instability of equilibrium points of nonlinear systems. In some cases, the proposed conditions may turn out to be preferable to the known ones.     

Fractional neural observer design for a class of nonlinear fractional chaotic systems

In this paper, a novel observer structure for nonlinear fractional-order systems is presented to estimate the states of fractional-order nonlinear chaotic system with unknown dynamical model. A new fractional error back-propagation learning algorithm is derived to adapt weights of the artificial neural network, by taking advantage of the Lyapunov stability strategy of fractional-order systems which is called Miattag–Leffler stability. The main contribution is the extension of neural observer for fractional dynamics in a way that satisfies Miattag–Leffler conditions. Observer design procedure guarantees the convergence of observer error to the neighborhood of zero. Furthermore, the robustness of the proposed estimator against uncertainties and external disturbances are the main benefits of the proposed method. Simulation results show the effectiveness and capabilities of the proposed observer.

     

Non-fragile observer design for fractional-order one-sided Lipschitz nonlinear systems

This paper is concerned with the problem of the full-order observer design for a class of fractional-order Lipschitz nonlinear systems. By introducing a continuous frequency distributed equivalent model and using an indirect Lyapunov approach, the sufficient condition for asymptotic stability of the full-order observer error dynamic system is presented. The stability condition is obtained in terms of LMI, which is less conservative than the existing one. A numerical example demonstrates the validity of this approach.     

Lyapunov Functional Approach to Stability Analysis of Riemann‐Liouville Fractional Neural Networks with Time‐Varying Delays

This paper is concerned with the globally asymptotic stability of the Riemann‐Liouville fractional‐order neural networks with time‐varying delays. The Lyapunov functional approach to stability analysis for nonlinear fractional‐order functional differential equations is discussed. By constructing an appropriate Lyapunov functional associated with the Riemann‐Liouville fractional integral and derivative, the asymptotic stability criteria of fractional‐order neural networks with time‐varying delays and constant delays are derived. The advantage of our proposed method is that one may directly calculate the first‐order derivative of the Lyapunov functional. Two numerical examples are also presented to illustrate the validity and feasibility of the theoretical results. With the increasing of the order of fractional derivatives, the state trajectories of neural networks show that the speeds of converging toward zero solution are faster and faster.     

Design of a chatter-free terminal sliding mode controller for nonlinear fractional-order dynamical systems

This paper investigates the problem of robust control of nonlinear fractional-order dynamical systems in the presence of uncertainties. First, a novel switching surface is proposed and its finite-time stability to the origin is proved. Subsequently, using the sliding mode theory, a robust fractional control law is proposed to ensure the existence of the sliding motion in finite time. We use a fractional Lyapunov stability theory to prove the stability of the system in a given finite time. In order to avoid the chattering, which is inherent in conventional sliding mode controllers, we transfer the sign function of the control input into the fractional derivative of the control signal. The proposed chattering-free sliding mode technique is then applied for stabilisation of a broad class of three-dimensional fractional-order chaotic systems via a single variable driving control input. Simulation results reveal that the proposed fractional sliding mode controller works well for chaos control of fractional-order hyperchaotic Chen, chaotic Lorenz and chaotic Arneodo systems with no-chatter control inputs.     

分数阶非线性多智能体系统的一致性研究

研究固定拓扑结构下的分数阶非线性多智能体系统协调控制的动力学模型问题。由于实际多智能体系统中,系统的状态变量难以全部测量,为了克服这一困难,利用状态观测器对系统状态进行重构并基于重构状态进行状态反馈。利用分数阶Lyapunov稳定性理论,证明了当反馈增益矩阵满足一定的线性矩阵不等式（LMI）条件时,系统中的智能体最终趋于所给定的目标状态。最后利用分数阶微积分的预估-校正算法进行数值仿真验证了理论分析的有效性和可行性。     

一类不确定分数阶混沌系统同步的自适应滑模控制方法

针对一类系统不确定及受外界干扰的分数阶混沌系统,本文首先将分数阶微积分应用到滑模控制中,构造了一个具有分数阶积分项的滑模面.针对系统不确定及外界干扰项,基于分数阶Lyapunov稳定性理论与自适应控制方法,设计了一种滑模控制器以及分数阶次的参数自适应律,实现了两不确定分数阶混沌系统的同步控制,并辨识出相应误差系统中不确定项及外界干扰项的边界.在分数阶系统稳定性分析中使用的分数阶Lyapunov稳定性理论及相关函数都可以很好地运用到其它分数阶系统同步控制方法中.最后数值仿真验证了所提控制方法的可行性与有效性.     

BIBO stability of fractional-order controlled nonlinear systems

The BIBO stability of fractional-order controlled nonlinear systems is investigated in this paper. By introducing the properties of some Mittag-Leffler functions and using an inequality satisfied by the Caputo derivative of a Lyapunov function, sufficient conditions to guarantee the BIBO stability are firstly presented. Then, the boundedness of solutions of the fractional financial system and the fractional low-order atmospheric circulation system are proved by the established stability results. Besides, two illustrative examples verify the theoretical results.     

Dynamic analysis of a 5D fractional-order hyperchaotic system

In this paper, the fractional-order 5D hyperchaotic system is proposed based on the hyperchaotic Lorenz system. Fractional-order chaotic systems are often three- or four-dimensional. There are few results about high dimension fractional-order systems. For this 5D hyperchaotic system, the stability of equilibrium points is analyzed by means of the stability theory of fractional systems. Then the fractional bifurcation is investigated. It is found that the system admits bifurcations with varying fractional-order and parameters, respectively. Under different bifurcation parameters, some conditions ensuring the bifurcations are presented. Finally, numerical simulations are presented to confirm the given analytical results.     

Full‐Order and Reduced‐order Observer Design for a Class of Fractional‐order Nonlinear Systems

The paper is concerned with problem of the full‐order and reduced‐order observer design for a class of fractional‐order one‐sided Lipschitz nonlinear systems. By introducing a continuous frequency distributed equivalent model and using indirect Lyapunov approach, the sufficient condition for asymptotic stability of the full‐order observer error dynamic system is presented. Furthermore, the proposed design method was extended to reduced‐order observer design for fractional‐order nonlinear systems. All the stability conditions are obtained in terms of LMI, which are less conservative than some existing ones. Finally, a numerical example demonstrates the validity of this approach.     

Stabilization for nonlinear fractional-order time-varying switched systems

This paper deals with the global asymptotic stabilization problem for discontinuous nonlinear time-varying switched systems in the sense of fractional-order. Via the designed periodic switching law, some criteria are firstly developed. It is shown that the derivatives of the constructed Lyapunov functions are allowed to be indefinite. Several corollaries are further derived, and some existing results can be regarded as special cases of our obtained results. Finally, our findings are verified by three numerical examples and one experiment.     

Dissipativity and synchronization of fractional-order output-coupled neural networks with multiple adaptive coupling weights

Different from the majority of existing results that focus on the passivity analysis of nonlinear systems, this paper attempts to analyze the more general QSR-dissipativity of fractional-order neural networks that own output coupling and multiple coupled weights, where outer coupling weights can be adjusted online by developing an adaptation law. Using the linear matrix inequality technique, several sufficient criteria that not only guarantee the QSR-dissipativity of the network system but also achieve global synchronization even under zero input are established. Of particular significance is the proposal of a fractional Lyapunov-like lemma, which plays a crucial role in verifying the asymptotic stability of synchronization errors. Finally, a simulation example is presented to verify the plausibility of the theoretical results.     

Average dwell time based stability analysis for nonautonomous continuous‐time switched systems

Inspired by the idea of multiple Lyapunov functions and the average dwell time, we address the stability analysis of nonautonomous continuous‐time switched systems. First, we investigate nonautonomous continuous‐time switched nonlinear systems and successively propose sufficient conditions for their (uniform) stability, global (uniform) asymptotic stability, and global (uniform) exponential stability, in which an indefinite scalar function is utilized to release the nonincreasing requirements of the classical multiple Lyapunov functions. Afterwards, by using multiple Lyapunov functions of quadratic form, we obtain the corresponding sufficient conditions for (uniform) stability, global (uniform) asymptotic stability, and global exponential stability of nonautonomous switched linear systems. Finally, we consider the computation issue of our current results for a special class of nonautonomous switched systems (ie, rational nonautonomous switched systems), associated with two illustrative examples.     

Robust Finite-time Synchronization of Non-identical Fractional-order Hyperchaotic Systems and Its Application in Secure Communication

This paper proposes a novel adaptive sliding mode control (SMC) method for synchronization of non-identical fractional-order (FO) chaotic and hyper-chaotic systems. Under the existence of system uncertainties and external disturbances, finite-time synchronization between two FO chaotic and hyperchaotic systems is achieved by introducing a novel adaptive sliding mode controller (ASMC). Here in this paper, a fractional sliding surface is proposed. A stability criterion for FO nonlinear dynamic systems is introduced. Sufficient conditions to guarantee stable synchronization are given in the sense of the Lyapunov stability theorem. To tackle the uncertainties and external disturbances, appropriate adaptation laws are introduced. Particle swarm optimization (PSO) is used for estimating the controller parameters. Finally, finite-time synchronization of the FO chaotic and hyper-chaotic systems is applied to secure communication.      

Uniform bounded input bounded output stability of fractional‐order delay nonlinear systems with input

The bounded input bounded output (BIBO) stability for a nonlinear Caputo fractional system with time‐varying bounded delay and nonlinear output is studied. Utilizing the Razumikhin method, Lyapunov functions and appropriate fractional derivatives of Lyapunov functions some new bounded input bounded output stability criteria are derived. Also, explicit and independent on the initial time bounds of the output are provided. Uniform BIBO stability and uniform BIBO stability with input threshold are studied. A numerical simulation is carried out to show the system's dynamic response, and demonstrate the effectiveness of our theoretical results.     

Asymptotic Stabilisation of Multiple Input Nonlinear Systems via Sliding Modes

A dynamic feedback controller design method is proposed for multiple input systems. The method uses a novel choice of sliding surface to effect asymptotic linearisation of nonlinear differential input output systems and a class of state space systems. The stability of the overall system, that is a canonical state space form with a dynamic feedback, is analysed with a generalised Lyapunov approach plus an asymptotic analysis in a neighbourhood of the origin. The nonlinear system does not have to be expressed in regular form as is the case in many other sliding mode control approaches. A type of zero dynamics, which are the dynamics of the control, are involved. The resulting dynamic feedback is shown to provide chatter free control if the system is minimum phase with respect to the zero dynamics. The theoretical results are applied to Gas Jet systems with two controls.     

Method of Radial Drift for Qualitative Study of the Properties of the Nonlinear Dynamic Systems. II. Study of Asymptotic System Stability

The method of monotonic radial drift proposed in the first part of this paper was used to study the conditions for asymptotic stability (in large) of the equilibrium states of dynamic nonautonomous nonlinear systems of an arbitrary order. It enabled the author to obtain new conditions for asymptotic stability of these states which allow one to carry out studies mostly on the basis of direct information about the right-hand side of the system. The method enables one to study systems to which neither the first nor second Lyapunov method is applicable.     

Projective synchronization of two different fractional-order chaotic systems via adaptive fuzzy control

In this paper, the projective synchronization problem of two fractional-order different chaotic (or hyperchaotic) systems with both uncertain dynamics and external disturbances is considered. More particularly, a fuzzy adaptive control system is investigated for achieving an appropriate projective synchronization of unknown fractional-order chaotic systems. The adaptive fuzzy logic systems are used to approximate some uncertain nonlinear functions appearing in the system model. These latter are augmented by a robust control term to compensate for the unavoidable fuzzy approximation errors and external disturbances as well as residual error due to the use of the so-called e-modification in the adaptive laws. A Lyapunov approach is adopted for the design of the parameter adaptation laws and the proof of the corresponding stability as well as the asymptotic convergence of the underlying synchronization errors towards zero. The effectiveness of the proposed synchronization system is illustrated through numerical experiment results.