分数阶微积分在滑模控制中的应用特性

针对分数阶微积分算子的信息记忆与遗传特性,从分数阶滑模趋近律与分数阶滑模控制律两方面,对分数阶微积分算子在滑模控制理论中的应用特性进行了研究。首先,从传统滑模控制理论的几种趋近律入手,引出分数阶滑模趋近律并分析其收敛特性。其次,针对航天器姿态控制系统,设计了一种分数阶滑模控制器。最后,对比数值仿真验证了所设计控制器的良好性能,与传统滑模趋近律和传统滑模控制律相比,分数阶滑模趋近律具有较好的平滑特性,分数阶滑模控制律具有更好的抗干扰性与强鲁棒性。     

基于混合反馈控制的分数阶混沌系统多延时完全同步

为提高保密通信的安全性,本文首次将完全同步和延时投影同步的思想相结合提出了多延时完全同步。针对Lorenz、金融分数阶两个不同的混沌系统,基于分数阶稳定性理论以及混合反馈控制方法,通过设计能够自动调节反馈增益系数的混合反馈控制器,实现了分数阶多延时完全同步。根据前人提出的修正的预估校正算法对混沌系统进行数值仿真,结果表明由驱动系统与响应系统构建的多延时完全同步误差系统将最终稳定于零点,从而验证了混合反馈控制器的有效性和可行性。     

分数阶Chen混沌系统的完全同步与反相同步

以分数阶 Chen 系统为例,分析了该系统的混沌特性,并对其分别进行了完全同步与反相同步控制.首先,基于分数阶系统稳定性理论和非线性动力学理论,构造出相应的非线性控制器,由此实现了分数阶 Chen 混沌系统的完全同步与反相同步控制;其次,利用分数阶稳定性理论,对上述同步给出了严格的数学证明;最后,借助于 Adams-Bashforth-Moultom 算法,利用数值模拟验证了所提方法的有效性.     

结构整数阶模型的分数阶控制方法研究

为了把分数阶控制方法引入到控制策略中,假设原整数阶模型为分数阶模型,将其表示为状态方程形式;通过系统输出相同的方式建立分数阶和整数阶系统之间的联系;为保证分数阶系统控制器能够用于整数阶系统,采用了基于系统输出的控制策略。算例结果表明:在相同控制能耗条件下,与整数阶控制策略相比,采用本文方法能使位移降低30%。     

Caputo导数下分数阶Birkhoff系统的准对称性与分数阶Noether定理

应用分数阶模型可以更准确地描述和研究复杂系统的动力学行为和物理过程,同时Birkhoff力学是Hamilton力学的推广,因此研究分数阶Birkhoff系统动力学具有重要意义.分数阶Noether定理揭示了Noether对称变换与分数阶守恒量之间的内在联系,但是当变换拓展为Noether准对称变换时,该定理的推广遇到了很大的困难.本文基于时间重新参数化方法提出并研究Caputo导数下分数阶Birkhoff系统的Noether准对称性与守恒量.首先,将时间重新参数化方法应用于经典Birkhoff系统的Noether准对称性与守恒量研究,建立了相应的Noether定理;其次,基于分数阶Pfaff作用量分别在时间不变的和一般单参数无限小变换群下的不变性给出分数阶Birkhoff系统的Noether准对称变换的定义和判据,基于Frederico和Torres提出的分数阶守恒量定义,利用时间重新参数化方法建立了分数阶Birkhoff系统的Noether定理,从而揭示了分数阶Birkhoff系统的Noether准对称性与分数阶守恒量之间的内在联系.分数阶Birkhoff系统的Noether对称性定理和经典Birkhoff系统的Noether定理是其特例.最后以分数阶Hojman-Urrutia问题为例说明结果的应用.     

岩石流变分数阶模型研究

以Kelvin流变模型为研究对象,提出了一种分数阶Kelvin流变模型。首先,把Kelvin模型中的整数阶导数改为分数阶导数,考虑到岩石材料的频率通常不超过1000 Hz,在分数阶拟合时,拟合频段选取为[0 1000],进而利用Oustalop滤波算法把分数阶表示为整数阶模式;其次,利用试验数据对分数阶模型进行参数识别,考虑到分数阶Kelvin模型具有强非线性的特点,引入了Levenberg-Marquardt优化算法来确定未知参数;最后,对于频域表示的流变方程,利用Laplace逆变换获得流变精确表达式。仿真实例表明本文方法可以很好地反映岩石流变特性。     

分数阶导数粘弹性模型的有限元法

在分析分数阶导数三元件模型理论的基础上,把分数阶导数三元件模型引入有限元模型中,推导出具有分数阶导数三元件本构关系的粘弹性结构动力学有限元格式。同时,应用分数阶导数型粘弹性结构动力学方程的数值算法求解了该有限元格式的数值解。并以二维沥青路面结构为例进行了路面动态粘弹性响应分析。算例分析表明,该方法能够正确有效地进行路面动态粘弹性分析。     

三种分形和分数阶导数阻尼振动模型的比较研究

标准的整数阶导数方程不能准确描述粘弹性材料的记忆性参考文献[1]和阻尼的分数次幂频率依赖[2],因此分形导数、分数阶导数及正定分数阶导数被用于描述粘弹性介质中的阻尼振动.该文通过分析模型和数值模拟,比较了三种模型描述的振动过程.结果显示,当p小于约O.75或大于约1.9时(p为非整数阶导数的阶数),分形导数模型衰减最快;当P大于约0.75且小于约1.9时,正定分数阶导数模型衰减最快,衰减最慢的分别为分数阶导数模型(p1).且正定分数阶导数模型衰减快于分数阶导数模型,当p接近2时,两种模型较为相近.     

短脉冲激光加热分数阶导热及其热应力研究

短脉冲激光加热引起材料内部复杂的传热过程及热变形,现有的以Fourier定律或Cattaneo-Vernotte松弛方程结合弹性理论为框架建立起来热应力理论在刻画其热物理过程存在严重缺陷.本文基于分数阶微积分理论,以半空间为研究对象,建立了分数阶Cattaneo热传导方程和相应的热应力方程,给出了问题的初始条件和边界条件,采用拉普拉斯变换方法,给出了非高斯时间分布激光热源辐射下温度场和热应力场的解析解,研究了短脉冲激光加热的温度场及热应力场的热物理行为.数值计算中,首先对理论解进行数值验证,然后取分数阶变量p=0.5研究温度场和热应力场的变化特点及激光参数对温度和热应力的影响,最后数值计算分数阶参数对温度和热应力场的影响.计算结果表明,分数阶Cattaneo传热方程和热应力方程描述的温度和热应力任然具有波动特性,与经典的Fourier传热模型和标准的Cattaneo传热模型相比,分数阶阶次越大,热波波速越小,热波波动性越明显;反之,则热波波速越大,热扩散性越强.激光加热和冷却的速度越快,温度上升和下降的速度越快,压应力和拉应力交替变化越快,温度变化幅值越小,热应力幅值影响不明显.     

Dynamic behavior of fractional order Duffing chaotic system and its synchronization via singly active control

With the increasingly deep studies in physics and technology,the dynamics of fractional order nonlinear systems and the synchronization of fractional order chaotic systems have become the focus in scientific research.In this paper,the dynamic behavior including the chaotic properties of fractional order Duffing systems is extensively investigated.With the stability criterion of linear fractional systems,the synchronization of a fractional non-autonomous system is obtained.Specifically,an effective singly active control is proposed and used to synchronize a fractional order Duffing system.The numerical results demonstrate the effectiveness of the proposed methods.     

A theorem on potential of isotropic symmetric second-order tensor function

In this paper we propose and prove the following theorem: If the second-order tensorH is an isotropic function of a symmetric second-order tensorT, and there exists a potential function forH, then there will certainly exist a potential function forT, too.     

Characteristic fractional step finite difference method for nonlinear section coupled system

For the section coupled system of multilayer dynamics of fluids in porous media, a parallel scheme modified by the characteristic finite difference fractional steps is proposed for a complete point set consisting of coarse and fine partitions. Some tech- niques, such as calculus of variations, energy method, twofold-quadratic interpolation of product type, multiplicative commutation law of difference operators, decomposition of high order difference operators, and prior estimates, are used in theoretical analysis. Optimal order estimates in 12 norm are derived to show accuracy of the second order approximation solutions. These methods have been used to simulate the problems of migration-accumulation of oil resources.     

Analytical solution of time fractional Cattaneo heat equation for finite slab under pulse heat flux

A fractional Cattaneo model is derived for studying the heat transfer in a finite slab irradiated by a short pulse laser. The analytical solutions for the fractional Cattaneo model, the classical Cattaneo-Vernotte model, and the Fourier model are obtained with finite Fourier and Laplace transforms. The effects of the fractional order parameter and the relaxation time on the temperature fields in the finite slab are investigated. The results show that the larger the fractional order parameter, the slower the thermal wave. Moreover, the higher the relaxation time, the slower the heat flux propagates. By comparing the fractional order Cattaneo model with the classical Cattaneo-Vernotte and Fourier models, it can be found that the heat flux predicted using the fractional Cattaneo model always transports from the high temperature to the low one, which is in accord with the second law of thermodynamics. However, the classical Cattaneo-Vernotte model shows that the unphysical heat flux sometimes transports from the low temperature to the high one.     

Modified characteristic finite difference fractional step method for moving boundary value problem of nonlinear percolation system

A fractional step scheme with modified characteristic finite differences running in a parallel arithmetic is presented to simulate a nonlinear percolation system of multilayer dynamics of fluids in a porous medium with moving boundary values. With the help of theoretical techniques including the change of regions, piecewise threefold quadratic interpolation, calculus of variations, multiplicative commutation rule of difference operators, multiplicative commutation rule of difference operators, decomposition of high order difference operators, induction hypothesis, and prior estimates, an optimal order in l ² norm is displayed to complete the convergence analysis of the numerical algorithm. Some numerical results arising in the actual simulation of migration-accumulation of oil resources by this method are listed in the last section.     

Stability analysis of linear fractional differential system with multiple time delays

In this paper, we study the stability of n-dimensional linear fractional differential equation with time delays, where the delay matrix is defined in (R ⁺)^n×n. By using the Laplace transform, we introduce a characteristic equation for the above system with multiple time delays. We discover that if all roots of the characteristic equation have negative parts, then the equilibrium of the above linear system with fractional order is Lyapunov globally asymptotical stable if the equilibrium exist that is almost the same as that of classical differential equations. As its an application, we apply our theorem to the delayed system in one spatial dimension studied by Chen and Moore [Nonlinear Dynamics 29, 2002, 191] and determine the asymptotically stable region of the system. We also deal with synchronization between the coupled Duffing oscillators with time delays by the linear feedback control method and the aid of our theorem, where the domain of the control-synchronization parameters is determined.     

Influence of the time delay of signal transmission on synchronization conditions in drive-response systems

Conditions for complete and lag synchronizations in drive-response systems are considered under the unified framework of generalized synchronization. The question is addressed that whether the synchronization conditions achieving complete synchronization is still valid for lag synchronization when the time delay of signal transmission between the drive and response systems increases from 0. Theoretical and numerical results show that whether the synchronization conditions is stable for the influence of the time delay of signal transmission depends on a particular form of equilibria of the drive and response systems. Furthermore, it seems that the less the number of the equilibria of the drive system, the more likely the synchronization conditions are stable for the time delay of signal transmission.     

高黏性流动有限元模拟的迭代稳定分步算法

分步算法已被广泛应用于数值求解不可压缩N-S方程. Guermond等认为时间步长必须大于 某个临界值方能使算法稳定. 然而在高黏性流动模拟中,已有的显式和半隐式分步算法由于 其显式本质,必须采用小时间步长计算,不但降低了计算效率,同时也常与为使分步算法稳 分步算法已被广泛应用于数值求解不可压缩N-S方程. Guermond等认为时间步长必须大于 某个临界值方能使算法稳定. 然而在高黏性流动模拟中,已有的显式和半隐式分步算法由于 其显式本质,必须采用小时间步长计算,不但降低了计算效率,同时也常与为使分步算法稳 定必须满足的最小时间步长要求冲突. 本文目的是构造一种含迭代格式的分步算法,它能在 保证精度的前提下大幅度地增大时间步长. 方腔流和平面Poisseuille流数值计算结果证实 了此特点,该方法被有效应用于充填流动过程的数值模拟.     

基于有限差分-谱方法的分数阶Burgers流体的非稳态驻点流动

研究了分数阶Burgers流体通过拉伸平板的非稳态驻点流动问题。将分数阶导数引入Burgers流体模型可以更好地模拟流动过程,但也增加了模型的复杂性和求解难度。首次运用有限差分-谱方法求解分数阶Burgers流体模型,离散格式构造简单有效。采用谱方法对控制方程中的空间项进行离散,利用有限差分方法分别结合L-1和L-2算法离散控制方程中的时间项,给出了两种离散格式,并且通过构造数值算例证明了离散格式的收敛性。结果表明,在靠近平板处,速度随着分数阶导数的增加而减小,而无穷远处的流体速度呈现出相反的趋势,体现了分数阶导数的记忆特性。此外,雷诺数越小,流体的粘度越大,导致流体速度越大。由于松弛时间参数的松弛特性,靠近平板处松弛时间参数对速度分布有抑制作用,远离平板处松弛时间促进流体流动。     

时间同步误差对船体角形变测量的影响

针对激光陀螺船体角形变测量,分析评估了两组激光陀螺组合体时间同步误差的影响,并提出了一种时间同步误差的在线估计算法.严格推导了考虑了时间同步误差的惯性姿态匹配方程,从方程可见,船体在波浪摇摆条件下时间同步误差将导致额外的Kalman滤波观测量波动误差,直接影响船体角形变测量精度.另一方面,基于新推导的惯性姿态匹配方程,在滤波状态中增加时间延迟变量,通过Kalman滤波能够在线估计时间延迟大小.基于实测远望船体姿态和角变形数据进行了仿真,仿真测试表明大的时间延迟将导致大的船体角形变测量误差,同时验证了时间延迟在线估计方法的有效性.