分数阶系统状态空间描述的数值算法

利用Grünwald-Letnicov分数微积分定义计算分数微积分的数值解,计算精度仅为1阶,不能满足快速收敛性要求.给出并证明了分数阶微积分的高阶近似所应满足的条件,并在此基础上推导出分数阶线性定常系统状态空间描述的数值计算公式.本法不但公式简单易编程,而且具有计算精度高、运算速度快等优点.给出一个粘弹性动态系统的仿真实例,验证了其有效性.     

分数阶线性定常系统的稳定性及其判据

介绍了分数阶微分方程和分数阶系统 ,给出分数阶线性定常系统的传递函数描述和状态空间描述 .给出了分数阶线性定常系统的稳定性条件 ,并结合分数阶状态方程给出定理的证明 .直接从复分析中的辐角原理出发 ,推导出分数阶线性定常系统 2个有效的稳定性判据 :分数阶系统奈奎斯特判据和分数阶系统对数频率判据 .通过实例验证了其有效性     

分数阶线性系统的稳定性证明

提出一种分数阶线性系统稳定性证明方法。首先分析了双参数Mittag-Leffler函数估值定理中的限制条件, 通过证明得出了该定理收敛域有误, 因此改进了双参数Mittag-Leffler函数估值定理, 并将它的参数由实数推广到矩阵。然后提出了可适用于分数阶线性系统的稳定性理论, 并利用改进的双参数Mittag-Leffler函数估值定理进行了证明。仿真结果验证了理论的正确性。     

分数阶线性系统的静态输出反馈鲁棒稳定性控制

主要对一类范数有界参数的分数阶线性控制系统,研究了使闭环系统全局渐进稳定的静态输出反馈控制器的设计问题。以线性矩阵不等式（LMI）的形式,证明了上述不确定系统的全局渐进稳定问题,等价于求一个线性矩阵不等式的可行解问题,并且利用该线性矩阵不等式的可行解,构造出了上述系统输出反馈控制器。     

基于分数阶积分器的分数阶混沌系统状态观测器同步研究

提出了一种基于分数阶积分器的分数阶混沌系统状态观测器同步算法。通过引入一个新的变量,该变量是将驱动系统的输出信号与传输信道中干扰的和进行分数阶积分处理,然后再作为输入信号加到观测系统中,以便实现分数阶混沌系统的状态观测系统同步。然后利用Lyapunov稳定性理论和线性矩阵不等式证明了该方法的正确性。将该同步方法应用于分数阶Chen混沌系统,得出了同步误差曲线,仿真结果表明了该同步方法的有效性,最终实现了分数阶混沌系统的状态观测器同步。     

Caputo分数阶切换线性系统的指数稳定性

众所周知,与整数阶切换系统不同, Caputo分数阶切换系统的积分下界不能随子系统的切换而被更新,意味着在下界非一致的任意区间内不能直接取分数阶导数的分数阶积分.对此,本文给出了一个不等式(文中引理6)克服这一问题,并用1个数值例子进行了验证.通过这一不等式,然后分别利用多Lyapunov函数方法和模型依赖平均驻留时间(MDADT)方法,给出了Caputo分数阶切换线性系统指数稳定的条件,并利用2个数值例子进行验证.     

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。     

分数阶系统的分数阶PID控制器设计

对于一些复杂的实际系统,用分数阶微积分方程建模要比整数阶模型更简洁准确.分数阶微积分也为描述动态过程提供了一个很好的工具.对于分数阶模型需要提出相应的分数阶控制器来提高控制效果.本文针对分数阶受控对象,提出了一种分数阶PID控制器的设计方法.并用具体实例演示了对于分数阶系统模型,采用分数阶控制器比采用古典的PID控制器取得更好的效果.     

分数阶线性定常系统的状态反馈镇定

研究了在分数阶受控系统中稳定性的问题。假定分数阶系统是在线性定常的情况下,利用状态反馈的方法,构造状态反馈矩阵以实现对系统的稳定性控制;给出了分数阶系统由状态反馈镇定的条件及其证明,并给出了状态反馈镇定的综合算法。仿真实例证明了采用状态反馈实现系统镇定的可行性和有效性。     

含有分数阶有色关联噪声的分数阶系统的卡尔曼滤波器设计

提出基于Tustin生成函数的分数阶卡尔曼滤波器设计方法,以解决含有相互关联的分数阶有色过程噪声和分数阶有色测量噪声的连续时间线性分数阶系统的状态估计问题.通过Tustin生成函数方法,对连续时间线性分数阶系统进行离散化,将分数阶系统的微分方程转化为差分方程.利用增广向量法,将分数阶状态方程和分数阶有色噪声作为新的增广...     

Finite‐time stability of linear fractional‐order time‐delay systems

In this paper, a finite‐time stability results of linear delay fractional‐order systems is investigated based on the generalized Gronwall inequality and the Caputo fractional derivative. Sufficient conditions are proposed to the finite‐time stability of the system with the fractional order. Numerical results are given and compared with other published data in the literature to demonstrate the validity of the proposed theoretical results.     

Routh table test for stability of commensurate fractional degree polynomials and their commensurate fractional order systems

A Routh table test for stability of commensurate fractional degree polynomials and their commensurate fractional order systems is presented via an auxiliary integer degree polynomial. The presented Routh test is a classical Routh table test on the auxiliary integer degree polynomial derived from and for the commensurate fractional degree polynomial. The theoretical proof of this proposed approach is provided by utilizing Argument principle and Cauchy index. Illustrative examples show efficiency of the presented approach for stability test of commensurate fractional degree polynomials and commensurate fractional order systems. So far, only one Routh-type test approach [1] is available for the commensurate fractional degree polynomials in the literature. Thus, this classical Routh-type test approach and the one in [1] both can be applied to stability analysis and design for the fractional order systems, while the one presented in this paper is easy for peoples, who are familiar with the classical Routh table test, to use.     

A note on the stability of fractional order systems

In this paper, a new approach is suggested to investigate stability in a family of fractional order linear time invariant systems with order between 1 and 2. The proposed method relies on finding a linear ordinary system that possesses the same stability property as the fractional order system. In this way, instead of performing the stability analysis on the fractional order systems, the analysis is converted into the domain of ordinary systems which is well established and well understood. As a useful consequence, we have extended two general tests for robust stability check of ordinary systems to fractional order systems.     

Finite‐time stability of fractional order impulsive switched systems

This paper considers the finite‐time stability of fractional order impulsive switched systems. First, by using the fractional order Lyapunov function, Mittag–Leffler function, and Gronwall–Bellman lemma, two sufficient conditions are given to verify the finite‐time stability of fractional order nonlinear systems. Then, the concept of finite‐time stability is extended to fractional order impulsive switched systems. A sufficient condition is given to verify the finite‐time stability of fractional order impulsive switched systems by combining the method of average dwell time with fractional order Lyapunov function. Finally, two numerical examples are provided to illustrate the theoretical results. Copyright © 2014 John Wiley & Sons, Ltd.     

Arnoldi-based model reduction for fractional order linear systems

In this paper, the Arnoldi-based model reduction methods are employed to fractional order linear time-invariant systems. The resulting model has a smaller dimension, while its fractional order is the same as that of the original system. The error and stability of the reduced model are discussed. And to overcome the local convergence of Padé approximation, the multi-point Arnoldi algorithm, which can recursively generate a reduced-order orthonormal basis from the corresponding Krylov subspace, is used. Numerical examples are given to illustrate the accuracy and efficiency of the proposed methods.     

Local observability and controllability of nonlinear discrete-time fractional order systems based on their linearisation

The concepts of local controllability and observability of nonlinear discrete-time systems with the Caputo-, Riemann–Liouville- and Grünwald–Letnikov-type h-difference fractional order operators are studied. The Implicit Function Theorem is used in order to show that the nonlinear systems are locally observable or controllable in a finite number of steps if their linear approximations are observable or controllable, respectively, in the same number of steps.     

On the componentwise stability of linear systems

The componentwise asymptotic stability (CWAS) and componentwise exponential asymptotic stability (CWEAS) represent stronger types of asymptotic stability, which were first defined for symmetrical bounds constraining the flow of the state‐space trajectories, and then, were generalized for arbitrary bounds, not necessarily symmetrical. Our paper explores the links between the symmetrical and the general case, proving that the former contains all the information requested by the characterization of the CWAS/CWEAS as qualitative properties. Complementary to the previous approaches to CWAS/CWEAS that were based on the construction of special operators, we incorporate the flow‐invariance condition into the classical framework of stability analysis. Consequently, we show that the componentwise stability can be investigated by using the operator defining the system dynamics, as well as the standard ε?δ formalism. Although this paper explicitly refers only to continuous‐time linear systems, the key elements of our work also apply, mutatis mutandis, to discrete‐time linear systems. Copyright © 2004 John Wiley & Sons, Ltd.