Stability analysis of fractional-order Cohen–Grossberg neural networks with time delay

In this paper, the existence and finite-time stability of equilibrium point for a class fractional-order Cohen–Grossberg neural networks with time delay are investigated. Moreover, some sufficient conditions for the finite time stability are obtained by using Bellman–Gronwall inequality and differential mean value theorem,and contraction mapping. Finally, an example is given to illustrate the effectiveness of the results.     

Finite-time stability analysis for fractional-order Cohen–Grossberg BAM neural networks with time delays

In this paper, the problem of finite-time stability for a class of fractional-order Cohen–Grossberg BAM neural networks with time delays is investigated. Using some inequality techniques, differential mean value theorem and contraction mapping principle, sufficient conditions are presented to ensure the finite-time stability of such fractional-order neural models. Finally, a numerical example and simulations are provided to demonstrate the effectiveness of the derived theoretical results.

     

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.     

Finite-time stability criteria for a class of fractional-order neural networks with delay

Finite-time stabilities of a class of fractional-order neural networks delayed systems with order \(\alpha {:}\) \(0<\alpha \le 0.5\) and \(0.5<\alpha <1\) are addressed in this paper, respectively. By using inequality technique, two new delay-dependent sufficient conditions ensuring stability of such fractional-order neural networks over a finite-time interval are obtained. Obtained conditions are less conservative than that given in the earlier references. Two numerical examples are given to show the effectiveness of our proposed method.     

Robust finite time stability of fractional-order linear delayed systems with nonlinear perturbations

This letter is concerned with robust stability of fractional-order linear delayed system with nonlinear perturbations over finite time interval. By using inequality technique, two new sufficient conditions for the finite time stability for such systems with order α: 0 α ≤ 0.5 and 0.5 < α < 1 are presented, respectively. A numerical example is presented to demonstrate the validity and feasibility of the obtained results.     

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.     

高速列车非线性系统的分数阶有限时间控制器设计

针对具有输入非线性, 不确定的气动阻力, 未知的车间力, 外部扰动以及未知的执行器故障等特征的高速列车非线性系统, 结合分数阶稳定性原理以及有限时间控制理论, 本文设计了一种分数阶有限时间控制器以实现高速列车更快速且更高精度的跟踪控制. 该控制器能够直接补偿高速列车的不确定性和非线性以及执行器故障而不需任何“试错”过程, 且稳定时间可由控制参数的不同选择来调整. 仿真研究验证了所设计控制器的有效性和优越性.     

Necessary and sufficient stability condition of fractional-order interval linear systems

This paper establishes a necessary and sufficient stability condition of fractional-order interval linear systems. It is supposed that the system matrix A is an interval uncertain matrix and fractional commensurate order belongs to 1≤α<2. Using the existence condition of Hermitian P=P^∗ for a complex Lyapunov inequality, we prove that the fractional-order interval linear system is robust stable if and only if there exists Hermitian matrix P=P^∗ such that a certain type of complex Lyapunov inequality is satisfied for all vertex matrices. The results are directly extended to the robust stability condition of fractional-order interval polynomial systems.     

具有通信时延的二阶多智能体系统有限时间一致性跟踪控制

针对具有通信时延的二阶多智能体系统的有限时间一致性控制问题,分别研究了具有固定拓扑和切换拓扑网络结构情形下的二阶多智能体系统的有限时间一致性。为使多智能体系统能在有限时间内可以达到一致,引入一致性控制增益矩阵并设计了相应的基于相对位置和相对速度的时延状态误差有限时间一致性控制算法,利用系统模型转换,泛函微分方程稳定性理论和有限时间Lyapunov稳定性定理得到了使系统在有限时间内达到一致跟踪的最大时延上界值。最后,仿真实验结果验证了所得理论的正确性和有效性。     

Synchronisation of a fractional-order chaotic system using finite-time input-to-state stability

The issue of synchronisation for a fractional-order chaotic system with uncertainties and disturbance is studied in this paper. The finite-time input-to-state stable theory of fractional-order dynamical system is presented for the first time. A linear feedback controller is proposed to achieve synchronisation of this fractional-order system and guarantee the bounded state error for any bounded interference in finite time. Since the chaotic system displays special dynamical behaviours as invariable Lyapunov exponent spectrums and controllable signal amplitude, one can achieve complete synchronisation and projective synchronisation by only adjusting the system parameter. Numerical simulations are shown to verify the feasibility of the presented synchronisation scheme.     

Further results on finite-time stabilisation for stochastic Markovian jump systems with time-varying delay

In this paper, we study the issue of finite-time stabilisation for stochastic Markovian jump systems with time-varying delay by considering a new criterion on finite-time stability. By constructing more appropriate Lyapunov–Krasovskii functional, some new conditions for verifying the finite-time stability of the plant as well as controller synthesis are established in standard linear matrix inequalities. The practical example about a single-link robot arm model demonstrates the validity of the main results.     

Wright分数阶时滞微分方程的离散化过程

引进了一种离散化方法对分数阶时滞微分方程进行离散化求解。首先考察Wright分数阶时滞微分方程;其次分析相应具有分段常数变元的Wright分数阶时滞微分方程,并应用离散化过程对模型进行数值求解;然后根据不动点理论讨论该合成动力系统不动点的稳定性;最后借助MATLAB对模型进行数值仿真,并结合Lyapunov指数、相图、时间序列图、分岔图探讨模型更多复杂的动力学现象。结果显示,提出方法成功对Wright分数阶时滞微分方程进行离散。     

A dissipative dynamical systems approach to stability analysis of time delay systems

In this paper the concepts of dissipativity and the exponential dissipativity are used to provide sufficient conditions for guaranteeing asymptotic stability of a time delay dynamical system. Specifically, representing a time delay dynamical system as a negative feedback interconnection of a finite‐dimensional linear dynamical system and an infinite‐dimensional time delay operator, we show that the time delay operator is dissipative with respect to a quadratic supply rate and with a storage functional involving an integral term identical to the integral term appearing in standard Lyapunov–Krasovskii functionals. Finally, using stability of feedback interconnection results for dissipative systems, we develop sufficient conditions for asymptotic stability of time delay dynamical systems. The overall approach provides a dissipativity theoretic interpretation of Lyapunov–Krasovskii functionals for asymptotically stable dynamical systems with arbitrary time delay. Copyright © 2004 John Wiley & Sons, Ltd.     

Robust Stability Analysis of Smith Predictor Based Interval Fractional-Order Control Systems: A Case Study in Level Control Process

The robust stability study of the classic Smith predictor-based control system for uncertain fractional-order plants with interval time delays and interval coefficients is the emphasis of this work.Interval uncertainties are a type of parametric uncertainties that cannot be avoided when modeling real-world plants.Also,in the considered Smith predictor control structure it is supposed that the controller is a fractional-order proportional integral derivative(FOPID) controller.To the best of the a...     

Finite-time stability and stabilisation for a class of nonlinear systems with time-varying delay

This paper is concerned with the problems of finite-time stability (FTS) and finite-time stabilisation for a class of nonlinear systems with time-varying delay, which can be represented by Takagi–Sugeno fuzzy system. Some new delay-dependent FTS conditions are provided and applied to the design problem of finite-time fuzzy controllers. First, based on an integral inequality and a fuzzy Lyapunov–Krasovskii functional, a delay-dependent FTS criterion is proposed for open-loop fuzzy system by introducing some free fuzzy weighting matrices, which are less conservative than other existing ones. Then, the parallel distributed compensation controller is designed to ensure FTS of the time-delay fuzzy system. Finally, an example is given to illustrate the effectiveness of the proposed design approach.     

Stability analysis for a fractional differential model of HIV infection of CD4 T-cells with time delay

In this paper, we introduce fractional-order derivatives into a model of HIV infection of CD4⁺ T-cells with time delay. We deal with the stability of both the viral free equilibrium and the infected equilibrium. Criteria are given to ensure that both the equilibria are asymptotically stable for all delay under some conditions. Numerical simulations are carried out to illustrate the results.     

Differential games,finite-time partial-state stabilisation of nonlinear dynamical systems,and optimal robust control

Two-player zero-sum differential games are addressed within the framework of state-feedback finite-time partial-state stabilisation of nonlinear dynamical systems. Specifically, finite-time partial-state stability of the closed-loop system is guaranteed by means of a Lyapunov function, which we prove to be the value of the game. This Lyapunov function verifies a partial differential equation that corresponds to a steady-state form of the Hamilton–Jacobi–Isaacs equation, and hence guarantees both finite-time stability with respect to part of the system state and the existence of a saddle point for the system's performance measure. Connections to optimal regulation for nonlinear dynamical systems with nonlinear-nonquadratic cost functionals in the presence of exogenous disturbances and parameter uncertainties are also provided. Furthermore, we develop feedback controllers for affine nonlinear systems extending an inverse optimality framework tailored to the finite-time partial-state stabilisation problem. Finally, two illustrative numerical examples show the applicability of the results proven.     

Parameter Estimation and Topology Identification of Uncertain General Fractional-order Complex Dynamical Networks with Time Delay

Complex networks have attracted much attention from various fields of sciences and engineering in recent years. However, many complex networks have various uncertain information, such as unknown or uncertain system parameters and topological structure, which greatly affects the system dynamics. Thus, the parameter estimation and structure identification problem has theoretical and practical importance for uncertain complex dynamical networks. This paper investigates identification of unknown system parameters and network topologies in uncertain fractional-order complex network with time delays (including coupling delay and node delay). Based on the stability theorem of fractional-order differential system and the adaptive control technique, a novel and general method is proposed to address this challenge. Finally two representative examples are given to verify the effectiveness of the proposed approach.      

Finite-time control for discrete-time switched systems with time delay

The problem of finite-time state feedback control is considered in this paper for a class of discrete-time switched systems with time delay. Firstly, the concepts of finite-time stability and finitetime boundedness are extended to discrete-time switched systems with time delay, respectively. Then, by resorting to the average dwell time approach and Lyapunov-Krasovskii functional technology, some new delay-dependent criteria guaranteeing finite-time boundedness and stability are developed. The state feedback controller is also obtained by virtue of a cone complement linearization (CCL) method. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed results.     

线性电机伺服系统的自适应鲁棒控制

通过对一类音圈电机直驱式精密伺服系统运动控制问题的研究,在只考虑系统标称模型的情况下,采用分数阶形式的趋近律,得到了一种有限时间滑模控制算法;与此同时,当系统存在参数不确定和外界干扰的情况下,将有限时间滑模控制方法和自适应控制理论相结合,构造了一种自适应滑模控制律.另一方面,本文还考虑了系统具有输入饱和约束的情况,通过引入一个辅助系统,进而设计了一种自适应抗饱和控制律.而且,文中对闭环系统的稳定性进行了证明.最后,仿真和实验对所提出控制策略的有效性和伺服精度进行了对比、分析和验证.