基于伪谱法的翼伞系统归航轨迹容错设计

针对翼伞系统在归航过程中,控制电机工作异常致使控制性能发生变化,无法按原有规划轨迹到达目标点的问题,提出一种基于Gauss伪谱法的归航轨迹容错设计方法.首先根据翼伞系统控制特性的不同,分别建立了正常和单电机异常工作状态下的质点模型,并根据伞形参数确定了两种工作状态下的约束条件和目标函数;其次,利用Gauss伪谱法分别对两种工作状态下轨迹规划的最优控制问题求解,获得翼伞系统不同状态下的最优飞行轨迹.仿真结果表明,在约束情况下,翼伞系统无论在正常和单电机异常工作时都可以顺利到达目标点,获得高精度的飞行轨迹.     

退化高阶抛物型分布参数系统的迭代学习控制

研究一类高阶分布参数系统的迭代学习控制问题,该类系统由退化高阶抛物型偏微分方程构成.根据系统所满足的性质,基于P型学习算法构建得到迭代学习控制器.利用压缩映射原理,证明该算法能使得系统的输出跟踪误差于L~2空间内沿迭代轴方向收敛于零.最后,仿真算例验证了算法的有效性.     

解抛物型方程的Galerkin多层修正迭代算法

小波方法在微分方程数值解法中日益得到广泛应用.由于小波的紧支性、正交性使得离散后的代数方程组的系数矩阵具有稀疏性、层次性,在此基础上可以构造各种快速算法.基于多尺度空间,采用一组正交小波基来离散原方程,导出方程组的系数矩阵具有稀疏性和层次性,从而提出求抛物型微分方程的Galerkin多层修正迭代算法,并讨论了迭代修正算法的收敛性.提出的方案能容易地实现时间和空间方向的局部加密自适应修正过程.提供的数值算例说明了方法的有效性.     

具连续分布时滞的抛物型系统的变结构控制

基于比较原理,利用推广的向量Hanalay微分不等式,Dini导数,结合Green公式及不等式分析技术,研究一类具分布时滞的抛物型控制系统的变结构控制问题．首先对所导出的滑动模运动方程,在仅要求系数矩阵是个M-矩阵的条件下,获得了滑动模运动方程全局指数稳定性的充分条件,建立了滑动模运动方程全局指数稳定性定理．其次,设计了仅由状态函数描述的变结构控制器,给出了运动轨线到达滑动模态区的时间的估计．     

一类具有连续分布时滞的抛物型系统的无记忆滑模控制器设计

研究了一类具有连续分布时滞的抛物型系统的滑模控制问题.首先,通过构造辅助函数与使用矩阵范数不等式设计了无记忆功能的滑模控制器;其次,给出了滑模运动方程指数渐近稳定的充分条件;最后给出了从任意初始位置出发的轨线到达滑动模态区的时间估计.仿真结果说明了本文方法的有效性.     

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.     

空间可展开机构设计分析系统的研究

空间可展开机构设计过程中存在建模分析周期长、设计数据和设计流程不易管理等问题,导致设计过程效率低、设计结果不理想。基于对多种CAD/CAE软件的二次开发集成和数据库的应用,提出了空间可展开机构的自动化设计分析方法,基于VC++开发集成了空间可展开机构设计分析系统软件,实现了机构参数化建模、自动化力学分析,优化设计以及设计数据的管理,有效提高了产品设计效率和质量。     

基于函数展开与超混沌系统的图像加密

为有效保护数字图像的安全,提出一种基于小波展开函数与超混沌系统的数字图像加密算法。利用小波展开函数对图像进行置乱,通过超混沌系统扰乱原图像与加密图像之间的关系。在求解超混沌系统混沌序列的四阶Runge-Kutta公式中,插入多个参数以扩大参数空间。模拟实验结果表明,加密后图像灰度值分布伪随机性较好。     

一维抛物型方程的样条子域精细积分(SSPI)隐格式

对一维抛物型方程初边值问题的求解,以往已经有一些数值解法,它们或者无条件稳定但精度不高,或者精度高但仅为条件稳定,且稳定性条件严格.另外,以往的差分格式在处理第二、第三类边界条件问题时,对带导数边界条件都是进行简单的差分逼近,影响了数值解的精度.因此构造一个无条件稳定且对各类边值问题都具有良好精度的数值方法具有重要意义.为此,基于子域精细积分思想,结合三次样条函数,提出了求解一维抛物型方程初边值问题含参数的样条子域精细积分格式.该格式为绝对稳定且精度很高.由于三次样条函数的采用,避免了通常有限差分法中处理带导数边界条件时产生的逼近误差,大大提高了求解第二、三类边界条件问题时的精度.     

基于多项式展开的三角平动点垂直周期轨道解析构建方法研究

平动点是圆型限制性三体问题中的五个平衡解,其附近存在着大量的周期轨道,研究这些周期轨道的构建方法在深空探测中具有重要的理论及工程意义.本文从模态运动的角度出发,分析三角平动点附近周期轨道,通过多项式展开法构建出主坐标下周期轨道三个运动方向之间的渐近关系,从新的角度分析了系统的动力学特性和三维周期运动三个方向内在关联以及物理规律.同时可以为设计真实力学模型下的飞行器轨道提供借鉴.文中提出的方法可以被拓展至椭圆型限制性三体问题的三维周期轨道构建或共线平动点附近的轨道构建中.     

Identifying a control function in parabolic partial differential equations from overspecified boundary data

Determination of an unknown time-dependent function in parabolic partial differential equations, plays a very important role in many branches of science and engineering. In the current investigation, the Adomian decomposition method is used for finding a control parameter p(t) in the quasilinear parabolic equation u_t=u_xx+p(t)u+, in [0,1]×(0,T] with known initial and boundary conditions and subject to an additional condition in the form of which is called the boundary integral overspecification. The main approach is to change this inverse problem to a direct problem and then solve the resulting equation using the well known Adomian decomposition method. The decomposition procedure of Adomian provides the solution in a rapidly convergent series where the series may lead to the solution in a closed form. Furthermore due to the rapid convergence of Adomian’s method, a truncation of the series solution with sufficiently large number of implemented components can be considered as an accurate approximation of the exact solution. This method provides a reliable algorithm that requires less work if compared with the traditional techniques. Some illustrative examples are presented to show the efficiency of the presented method.     

Moving mesh methods for solving parabolic partial differential equations

A new adaptive method is described for solving nonlinear parabolic partial differential equations with moving boundaries, using a moving mesh with continuous finite elements. The evolution of the mesh within the interior of the spatial domain is based upon conserving the distribution of a chosen monitor function across the domain throughout time, where the initial distribution is selected based upon the given initial data. The mesh movement at the boundary is governed by a second monitor function, which may or may not be the same as that used to drive the interior mesh movement. The method is described in detail and a selection of computational examples are presented using different monitor functions applied to the porous medium equation (PME) in one and two space dimensions.     

Higher-order time accurate numerical methods for singularly perturbed parabolic partial differential equations

This article presents two numerical methods for singularly perturbed time-dependent reaction-diffusion initial–boundary-value problems. The spatial derivative is replaced by a hybrid scheme, which is a combination of the cubic spline and the classical central difference scheme in both the methods. In the first method, the time derivative is replaced by the Crank–Nicolson scheme, whereas in the second method the time derivative is replaced by the extended-trapezoidal scheme. These schemes are applied on the layer resolving piecewise-uniform Shishkin mesh. Some numerical examples are carried out to show the accuracy and efficiency of these methods.     

Sliding-mode boundary control for perturbed time fractional parabolic systems with spatially varying coefficients using backstepping

In this paper, we consider the boundary stabilization of an uncertain time fractional parabolic systems governed by time fractional parabolic partial differential equations (PDEs) with a boundary input disturbance and spatially varying coefficients (nonconstant coefficients) using a fractional-order sliding-mode controller. For this, the backstepping approach is used to transform an original system into a target system with a new manipulable input and perturbation. Then, the fractional-order sliding-mode algorithm is employed to design this new discontinuous boundary input to achieve the asymptotical stabilization of the target system (and, therefore, of the original system as well) by the fractional Lyapunov method. Apart from this, the well-posedness of the fractional parabolic system is analyzed theoretically. Fractional-order numerical simulations are provided to validate the developed technique.     

An efficient numerical algorithm for multi-dimensional time dependent partial differential equations

An efficient and robust numerical scheme based on Haar wavelets and finite differences is suggested for the solution of two-dimensional time dependent linear and nonlinear partial differential equations (PDEs). Excellent feature of the scheme is the conversion of linear and non-linear PDEs to algebraic equations which are comparatively easy to handle. Convergence of the scheme, which guarantees small error norm as the resolution level increases, is also an important part of this work. Different error norms are computed to check efficiency of the technique. Computations verify accuracy, flexibility and low computational cost of the method.     

An optimal control model for a system of degenerate parabolic integro-differential equations

An initial-boundary-value problem for a system of degenerate parabolic integro-differential equations is considered. The sufficient conditions for the existence and uniqueness of its generalized solution and for the existence of at least one optimal control for a given performance functional are obtained. A stable numerical solution to the initial-boundary-value problem is derived for a locally one-dimensional case and conditions are formulated for constructing a stable numerical algorithm of the optimal control problem on a class of piecewise-smooth control functions. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 90–102, November–December 2007.     

Fast methods for the iterative solution of linear elliptic and parabolic partial differential equations involving 2 space dimensions

Within the last decade, attention has been devoted to the introduction of several fast computational methods for solving the linear difference equations which are derived from the finite difference discretisation of many standard partial differential equations of Mathematical Physics.

In this paper, the authors develop and extend an exact factorisation technique previously applied to parabolic equations in one space dimension to the implicit difference equations which are derived from the application of alternating direction implicit methods when applied to elliptic and parabolic partial differential equations in 2 space dimensions under a variety of boundary conditions.     

A symbolic algorithm for computing recursion operators of nonlinear partial differential equations

A recursion operator is an integro-differential operator which maps a generalized symmetry of a nonlinear partial differential equation (PDE) to a new symmetry. Therefore, the existence of a recursion operator guarantees that the PDE has infinitely many higher-order symmetries, which is a key feature of complete integrability. Completely integrable nonlinear PDEs have a bi-Hamiltonian structure and a Lax pair; they can also be solved with the inverse scattering transform and admit soliton solutions of any order.

A straightforward method for the symbolic computation of polynomial recursion operators of nonlinear PDEs in (1+1) dimensions is presented. Based on conserved densities and generalized symmetries, a candidate recursion operator is built from a linear combination of scaling invariant terms with undetermined coefficients. The candidate recursion operator is substituted into its defining equation and the resulting linear system for the undetermined coefficients is solved.

The method is algorithmic and is implemented in Mathematica. The resulting symbolic package PDERecursionOperator.m can be used to test the complete integrability of polynomial PDEs that can be written as nonlinear evolution equations. With PDERecursionOperator.m, recursion operators were obtained for several well-known nonlinear PDEs from mathematical physics and soliton theory.