一类分数阶非线性混沌系统的同步控制

在分数阶非线性系统同步控制的研究中,针对一类分数阶非线性混沌系统,研究了基于分数阶控制器的同步方法.利用状态反馈方法和分数阶微积分定义,设计了分数阶混沌系统同步控制器.进一步,根据分数阶非线性系统稳定性理论、Mittag-Leffler函数、Laplace变换以及Gronwall不等式,证明了同步控制器的有效性.最后,通过数值仿真,实现了初始值不同的两个分数阶非线性混沌系统同步.误差响应曲线表明研究的分数阶非线性系统同步响应速度快,控制精度高,验证了本文所设计的混沌同步控制方案的可行性.     

State Feedback Control for a Class of Fractional Order Nonlinear Systems

Using the Lyapunov function method, this paper investigates the design of state feedback stabilization controllers for fractional order nonlinear systems in triangular form, and presents a number of new results. First, some new properties of Caputo fractional derivative are presented, and a sufficient condition of asymptotical stability for fractional order nonlinear systems is obtained based on the new properties. Then, by introducing appropriate transformations of coordinates, the problem of controller design is converted into the problem of finding some parameters, which can be certainly obtained by solving the Lyapunov equation and relevant matrix inequalities. Finally, based on the Lyapunov function method, state feedback stabilization controllers making the closed-loop system asymptotically stable are explicitly constructed. A simulation example is given to demonstrate the effectiveness of the proposed design procedure.      

Iterative Learning Control for a Class of Fractional Order Distributed Parameter Systems

This paper concerns a second‐order P‐type iterative learning control (ILC) scheme for a class of fractional order linear distributed parameter systems. First, by analyzing of the control and learning processes, a discrete system for P‐type ILC is established and the ILC design problem is then converted to a stability problem for such a discrete system. Next, a sufficient condition for the convergence of the control input and the tracking errors is obtained by using generalized Gronwall inequality, which is less conservative than the existing one. By incorporating the convergent condition obtained into the original system, the ILC scheme is derived. Finally, the validity of the proposed method is verified by a numerical example.     

基于反步法的耦合分数阶反应扩散系统边界输出反馈控制

针对具有空间依赖耦合系数的分数阶反应扩散系统, 利用反步法设计了基于观测器的边界输出反馈控制器, 证明了观测增益和控制增益核函数矩阵方程的适定性. 针对误差系统和输出反馈的闭环系统, 利用分数阶Lyapunov方法分析了系统的Mittag-Leffler稳定性, 且利用Wirtinger不等式改进了耦合系统稳定的条件. 当系统具有空间依赖的耦合系数时, 难以求得控制增益和观测增益核函数的解析解, 为此, 给出了核函数偏微分方程的数值解方法. 数值仿真验证了理论结果.     

一类不确定分数阶混沌系统的同步控制

针对一类参数未知,状态不能全部测量的分数阶混沌系统的同步控制问题,结合状态观测器和自适应方法,提出了一种更符合工程实际的新的控制方案,利用分数阶微积分稳定性理论,给出了基于状态观测器的控制律和自适应律。该同步方法理论严格,没有强加在系统上的限制条件,适用范围比较宽,便于实现,并且保留了非线性项,达到同步的时间短。以分数阶R~ssler系统为研究对象,实现了参数未知,状态不能全部测量的分数阶混沌系统同步。理论分析与计算机仿真结果证实了该方法的有效性。     

Bounded Real Lemmas for Fractional Order Systems

This paper derives the bounded real lemmas corresponding to L∞norm and H∞norm(L-BR and H-BR) of fractional order systems. The lemmas reduce the original computations of norms into linear matrix inequality(LMI) problems, which can be performed in a computationally efficient fashion. This convex relaxation is enlightened from the generalized Kalman-YakubovichPopov(KYP) lemma and brings no conservatism to the L-BR. Meanwhile, an H-BR is developed similarly but with some conservatism.However, it can test the system stability automatically in addition to the norm computation, which is of fundamental importance for system analysis. From this advantage, we further address the synthesis problem of H∞control for fractional order systems in the form of LMI. Three illustrative examples are given to show the effectiveness of our methods.     

Robust Output Feedback Control for Fractional Order Nonlinear Systems with Time-varying Delays

Robust controller design problem is investigated for a class of fractional order nonlinear systems with time varying delays. Firstly, a reduced-order observer is designed. Then, an output feedback controller is designed. Both the designed observer and controller are independent of time delays. By choosing appropriate Lyapunov functions, we prove the designed controller can render the fractional order system asymptotically stable. A simulation example is given to verify the effectiveness of the proposed approach.      

基于径向基神经网络的分数阶混沌系统控制

针对分数阶混沌系统的控制问题,提出了一种基于径向基函数（RBF）神经网络的控制方法．利用RBF 神经网络对混沌系统的非线性进行补偿,并且神经网络的权值可以通过调整律在线调整．在有参数干扰和外部扰动 的情况下,所设计的控制器仍能使得控制误差渐近收敛到零．以分数阶Liu 混沌系统为例施加控制,仿真结果验证 了该方法的有效性和鲁棒性．     

Robust D-Stability Test of LTI General Fractional Order Control Systems

This work deals with the robust D-stability test of linear time-invariant(LTI) general fractional order control systems in a closed loop where the system and/or the controller may be of fractional order. The concept of general implies that the characteristic equation of the LTI closed loop control system may be of both commensurate and non-commensurate orders, both the coefficients and the orders of the characteristic equation may be nonlinear functions of uncertain parameters, and the coefficients may be complex numbers. Some new specific areas for the roots of the characteristic equation are found so that they reduce the computational burden of testing the robust D-stability. Based on the value set of the characteristic equation, a necessary and sufficient condition for testing the robust D-stability of these systems is derived. Moreover, in the case that the coefficients are linear functions of the uncertain parameters and the orders do not have any uncertainties, the condition is adjusted for further computational burden reduction. Various numerical examples are given to illustrate the merits of the achieved theorems.     

Containment Control of Fractional Order Multi-Agent Systems With Time Delays

In complex environments, many distributed multiagent systems are described with the fractional-order dynamics. In this paper, containment control of fractional-order multiagent systems with multiple leader agents are studied. Firstly, the collaborative control of fractional-order multi-agent systems (FOMAS) with multiple leaders is analyzed in a directed network without delays. Then, by using Laplace transform and frequency domain theorem, containment consensus of networked FOMAS with time delays is investigated in an undirected network, and a critical value of delays is obtained to ensure the containment consensus of FOMAS. Finally, numerical simulations are shown to verify the results.      

分数阶控制系统

随着基于分数阶次的数学理论日益完善,分数阶控制系统也越来越广泛地被研究和讨论。为了完善分数阶控制系统的理论体系,给出了分数阶系统的总体综述。介绍了分数微积分的定义,给出了分数阶线性定常系统的传递函数和状态空间描述,简要介绍了分数阶控制系统的复频域分析和分数阶控制器及控制器的设计方法,并分析了几种设计方法的优缺点。分数系统的研究必然能找到合适的切入点广泛地进入现代控制领域,为真正实现工业控制自动化提供强有力的理论依据。     

Adaptive Synchronization for a Class of Fractional Order Time-delay Uncertain Chaotic Systems via Fuzzy Fractional Order Neural Network

International Journal of Control, Automation and Systems - Uncertainty and delay are common phenomena in chaotic systems, but their existence will increase the difficulty of synchronization. For...     

DPG Method with Optimal Test Functions for a Fractional Advection Diffusion Equation

We develop an ultra-weak variational formulation of a fractional advection diffusion problem in one space dimension and prove its well-posedness. Based on this formulation, we define a DPG approximation with optimal test functions and show its quasi-optimal convergence. Numerical experiments confirm expected convergence properties, for uniform and adaptively refined meshes.     

Optimal Control Based on Neuro Estimator for Fractional Order Uncertain Non-linear Continuous-Time Systems

Neural Processing Letters - In this paper, a novel method is presented for optimal control of fractional order systems in the presence of an unknown term in system dynamic where fractional order...     

Iterative Learning Consensus Control for Multi-agent Systems with Fractional Order Distributed Parameter Models

International Journal of Control, Automation and Systems - This paper concerns about the iterative learning consensus control scheme for a class of multi-agent systems (MAS) with distributed...     

分数阶区间系统的鲁棒稳定性分析

由于实际的物理系统常会存在各种不确定结构,这些不确定结构给系统的稳定性分析带来了困难和一定的影响,因此对这类带有不确定结构的分数阶系统控制问题的研究,具有广泛的实际应用意义。本文主要针对带有区间不确定性的分数阶控制系统,分析了该系统的鲁棒稳定性问题,首先将区间矩阵用分式线性变换(LFT)表示,然后基于分析得出了此类系统鲁棒稳定的充分必要条件。并给出两个数值算例阐释该方法,仿真结果表明了该方法的有效性。     

求解分数阶微分系统的一种数值算法

由于分数阶微分系统具有记忆功能,在其求解过程中计算量较大.文中的目的是针对分数阶Grunwald-Letnikov(6L)定义,研究并寻求一种求解分数阶微分方程的有效数值算法.首先由分数阶GL定义得出分数阶的数值计算公式,进而从理论上分析了算法中分数阶项计算系数的特点,结合计算机数值仿真的结果,得出了远离当前时间的无穷小项一般不可忽略的结论,并设计了一种合理有效的计算方法.计算机数值仿真的结果表明,所设计的求解分数阶微分方程的算法精度高,通用性好,且易于编程实现.     

High Performance Regulator for Fractional Order Systems: A Soft Variable Structure Control Approach

In this paper, soft variable structure control of fractional order systems is proposed for achieving high regulation rates and short settling times. Using the appropriate fractional order Lyapunov approach, a controller based on continuous fractional switch are designed to achieve high control performance. Due to the alleviation of chattering phenomenon soft VSC that is proposed for fractional order systems are quite useful in future for the practical systems.     

Consensus Control of Fractional‐Order Systems Based on Delayed State Fractional Order Derivative

In this paper, the delayed state fractional order derivative (DSFOD) is introduced into the existing traditional consensus protocol aiming to improve the robustness of fractional‐order multi‐agent systems against communication time delay. Both of communication channels with time‐delay and without time‐delay cases are considered. Based on the frequency‐domain analysis and algebraic graph theory, it is shown that properly choosing the intensity of DSFOD can improve the robustness of fractional‐order multi‐agent systems against communication delay. Finally, a simulated example with simulations is presented to confirm the correctness and effectiveness of the theoretical results.     

Fuzzy Adaptive Control of a Fractional Order Chaotic System With Unknown Control Gain Sign Using a Fractional Order Nussbaum Gain

In this paper we propose an improved fuzzy adaptive control strategy, for a class of nonlinear chaotic fractional order (SISO) systems with unknown control gain sign. The online control algorithm uses fuzzy logic sets for the identification of the fractional order chaotic system, whereas the lack of a priori knowledge on the control directions is solved by introducing a fractional order Nussbaum gain. Based on Lyapunov stability theorem, stability analysis is performed for the proposed control method for an acceptable synchronization error level. In this work, the Grünwald-Letnikov method is used for numerical approximation of the fractional order systems. A simulation example is given to illustrate the effectiveness of the proposed control scheme.