一类求解非线性奇异方程组的牛顿改进算法

受一个求解非线性奇异方程组迭代格式的启示,将两种牛顿改进算法推广成一般形式,并将其发展为一类求解具有奇异雅可比矩阵的非线性方程组的牛顿改进算法.首先,描述这类新算法的迭代格式,并导出其收敛阶,该新格式每步迭代仅需计算一次函数值和一次导函数值;然后,对测试函数进行检验,并与牛顿算法及其他奇异牛顿算法进行比较,从而验证该算法的快速收敛性;最后,通过两个实际问题验证所提出算法的有效性.     

求多变量线性矩阵方程组自反解的迭代算法

利用矩阵分解的方法求多变量线性矩阵方程组的自反解是很困难的.本文建立了一种迭代方法来解决这个问题,利用此迭代方法可以判断多变量线性矩阵方程组的可解性,且当矩阵方程组相容时,可以在有限步迭代后得到其自反解.选取特殊的初始矩阵时,能够求得矩阵方程组的极小范数自反解.进一步,通过求新的线性矩阵方程组的极小范数自反解,能够求得给定矩阵的最佳逼近矩阵.数值算例表明,迭代算法是有效的.     

一类离散时间代数Riccati矩阵方程对称解的双迭代算法

利用逆矩阵的Neumann级数形式,将在线性二次优化问题中遇到的含未知矩阵之逆的离散时间代数Riccati矩阵方程(DTARME)转化为高次多项式矩阵方程,然后采用牛顿算法求高次多项式矩阵方程的对称解,并采用修正共轭梯度法求由牛顿算法每一步迭代计算导出的线性矩阵方程的对称解或者对称最小二乘解,建立求DTARME的对称解的双迭代算法。双迭代算法仅要求DTARME有对称解,不要求它的对称解唯一,也不对它的系数矩阵做附加限定。数值算例表明双迭代算法是有效的。     

特定消谐PWM开关角数值计算方法研究

本文建立了特有谐PWM的数学模型，根据开关角非线性方程组的特点，提出了等效转化牛顿迭代解法，该方法剔除了直接牛顿迭代过程中的三角函数计算，减少迭代过程中的截断误差，综合了计算时间，使算法的效率得到了较在的提高。     

基于GaBP算法的快速潮流计算方法

研究在潮流迭代求解过程中雅可比矩阵方程组的迭代求解方法及其收敛性。首先利用PQ分解法进行潮流迭代求解,并针对求解过程中雅可比矩阵对称且对角占优的特性,对雅可比矩阵方程组采用高斯置信传播算法(GaBP)进行求解,再结合Steffensen加速迭代法以提高GaBP算法的收敛性。对IEEE118、IEEE300节点标准系统和两个波兰互联大规模电力系统进行仿真计算后结果表明：随着系统规模的增长,使用Steffensen加速迭代法进行加速的GaBP算法相对于基于不完全LU的预处理广义极小残余方法(GMRES)具有更好的收敛性,为大规模电力系统潮流计算的快速求解提供了一种新思路。     

一类Lyapunov型矩阵方程组的中心对称解及其最佳逼近

建立了求矩阵方程组AiXBi＋GiXDi=Fi（i=1,2）的中心对称解的迭代算法.使用该方法不仅可以判断矩阵方程组是否有中心对称解,而且在有中心对称解时,还能够在有限步迭代计算之后得到矩阵方程组的极小范数中心对称解.同时,也能够在矩阵方程组的中心对称解集合中求得给定矩阵的最佳逼近.     

拟牛顿法在航空发动机特性仿真中的应用

航空发动机特性仿真中常用牛顿迭代法求解非线性方程组,牛顿法每一步迭代计算都需要计算Jacobi矩阵,这需要多次发动机气动热力过程计算.因此避免大量重复计算Jacobi矩阵可以减少发动机计算整机的气动热力计算次数,从而提高发动机特性计算的速度.文中采用基于Broyden原理的拟牛顿法求解发动机非线性方程组,这种方法可以直接求出第一步迭代后的Jacobi矩阵,从而大幅度提高计算速度.将拟牛顿法应用于某型涡喷和涡扇发动机特性计算,通过分析计算结果,证明了采用拟牛顿法可以提高发动机特性模拟的计算速度.     

加权范数下矩阵方程组的对称解及其最佳逼近

应用复合最速下降法,给出了求解矩阵方程组[(AXB=E,CXD=F)]加权范数下对称解及最佳逼近问题的迭代解法。对任意给定的初始矩阵,该迭代算法能够在有限步迭代计算之后得到矩阵方程组的对称解,并且在上述解集合中也可给出指定矩阵的最佳逼近矩阵。     

线性矩阵方程异类约束最小二乘解的迭代算法

多矩阵变量线性矩阵方程(LME)约束解的计算问题在参数识别、结构设计、振动理论、自动控制理论等领域都有广泛应用。本文借鉴求线性矩阵方程(LME)同类约束最小二乘解的迭代算法,通过构造等价的线性矩阵方程组,建立了求多矩阵变量LME的一种异类约束最小二乘解的迭代算法,并证明了该算法的收敛性。在不考虑舍入误差的情况下,利用该算法不仅可在有限步计算后得到LME的一组异类约束最小二乘解,而且选取特殊初始矩阵时,可求得LME的极小范数异类约束最小二乘解。另外,还可求得指定矩阵在该LME的异类约束最小二乘解集合中的最佳逼近解。算例表明,该算法是有效的。     

非线性微分方程组两点边值问题的参数微分——牛顿迭代解法

引言 在利用变分法或最大值原理求解最优控制问题时,常常最后归结为求解一个微分方程组的两点边值问题,对于线性常微分方程组的两点边值问题,可采用共轭函数、补足函数等法通过一次积分来求解,而对于非线性常微分方程组,则要运用上法或与牛顿法联合使用通过迭代来求解,而其中以牛顿片比较有效。在牛顿迭代程序中,偏导数矩阵     

漂浮基空间机械臂分解运动控制的增广自适应算法

本文讨论了载体位置与姿态均不受控制的漂浮基两杆空间机械臂系统的逆运动学问题 ,推导了系统的运动学、动力学方程．分析表明,结合系统动量守恒及动量矩守恒关系得到 的系统广义Jacobi关系为系统惯性参数的非线性函数．本文证明了,借助于增广变量法可以 将增广广义Jacobi矩阵表示为一组适当选择的惯性参数的线性函数．并在此基础上,给出了 系统参数未知时由空间机械臂末端惯性空间期望轨迹产生机械臂关节铰期望角速度、角加速 度的增广自适应控制算法．仿真运算,证实了方法的有效性．     

空间机械臂本体与末端抓手协调运动的自适应控制方法

讨论了载体位置不受控制的漂浮基空间机械臂本体与末端抓手协调运动的自适应控制问题. 对系统的运动学、动力学分析表明, 结合系统动量守恒关系得到的系统动力学方程及协调运动的增广广义Jacobi矩阵可以表示为适当选择的组合惯性参数的线性函数. 以此为基础, 对于系统存在未知参数的情况, 设计了本体姿态与机械臂末端抓手惯性空间轨迹协调运动的自适应控制方案. 上述控制方案的显著优点在于: 不需要测量、反馈飞行器本体的位置、移动速度及移动加速度. 仿真运算, 证实了上述控制方案的有效性.     

Comparison of Extended Jacobian and Lagrange Multiplier Based Methods for Resolving Kinematic Redundancy

Several methods have been proposed in the past for resolving the control of kinematically redundant manipulators by optimizing a secondary criterion. The extended Jacobian method constrains the gradient of this criterion to be in the null space of the Jacobian matrix, while the Lagrange multiplier method represents the gradient as being in the row space. In this paper, a numerically efficient form of the Lagrange multiplier method is presented and is compared analytically, computationally, and operationally to the extended Jacobian method. This paper also presents an improved method for tracking algorithmic singularities over previous work.     

带时延补偿的图像雅可比矩阵在线估计方法

传统的在线估计图像雅可比矩阵的方法没有考虑时延因素,因此具有较大的估计误差。为了补偿时间延迟,提出一种新的带时延补偿的图像雅可比矩阵估计方法。该方法利用卡尔曼滤波估计特征点在图像空间中当前时刻的位置和速度,进而计算当前时刻较为准确的图像雅可比矩阵估计值。仿真和实验结果表明,该方法显著地提高了系统的性能,从而验证了提出的带时延补偿的图像雅可比矩阵在线估计方法的可行性和优越性。     

Incomplete Jacobian Newton method for nonlinear equations

This paper presents a new modified Newton method for nonlinear equations. This method uses a part of elements of the Jacobian matrix to obtain the next iteration point and is refereed to as the incomplete Jacobian Newton (IJN) method. The IJN method may be fit for solving large scale nonlinear equations with dense Jacobian. The conditions of linear, superlinear and quadratic convergence of the IJN method are given and the local convergence results are analyzed and proved. Some special IJN algorithms are designed and numerical experiments are given. The results show that the IJN method is promising.     

Calibration-free robotic eye-hand coordination based on an auto disturbance-rejection controller

This paper addresses the calibration-free robotic eye-hand coordination in a way other than the conventional image Jacobian matrix approach that has been studied extensively in literature. A nonlinear mapping rather than the linear mapping used in the image Jacobian matrix between the image space and the robotic control space is proposed. This mapping is regarded as the system's unmodeled dynamics expressed in system state equations. An extended state observer is designed first to estimate the unmodeled dynamics as well as the external disturbance of the system. With the estimation results as the compensation, a system controller is designed based on the nonlinear state-error feedback control strategy. Convergence of the extended state observer as well as the overall controller for a typical eye-hand coordination system is proved. Compared with the conventional calibration-free robotic eye-hand coordination with a Jacobian matrix, the proposed controller is independent of specific tasks and system configurations. Thus, a general design procedure is proposed for the calibration-free robotic eye-hand coordination. Simulation and experiment results demonstrate the satisfactory performance and effectiveness of the proposed approach.     

Mapping the Properties of Geological Objects with Allowance for Anisotropy Based on the Simulation of the Deformation Transformation

The statement for the problem on mapping the properties of geological objects is considered. Examples of its solution with allowance for anisotropy based on modeling the deformation transformation of an initially isotropic space within the variational method for geological mapping are given. The problem is reduced to modifying the type of model conditions represented by partial differential equations using the Jacobian matrix corresponding to the coordinate transformation for the simulated deformation. The proposed approach is applicable to the solution of two-dimensional and three-dimensional problems of geological mapping taking into account both the homogeneous and locally varying space anisotropy of the properties of geological objects     

Digital control of space robot manipulators with velocity type joint controller using transpose of generalized Jacobian matrix

For free floating space robots having manipulators, we have proposed a discrete-time tracking control method using the transpose of Generalized Jacobian Matrix (GJM). Control inputs of the control method are joint torques of the manipulator. In this paper, the control method is augmented for angular velocity inputs of the joints. Computer simulations have shown the effectiveness of the augmented method. This work was presented in part and awarded as Best Paper Award at the 13th International Symposium on Artificial Life and Robotics, Oita, Japan, January 31–February 2, 2008     

Singularity-free simulation of closed loop multibody systems by using null space of Jacobian matrix

Numerical simulation of closed loop multibody systems is associated with numerical solution of equations of motion which are, in general, in the form of DAE’s index-3 systems. For assuring continuous simulation, one should overcome some difficulties such as stabilization of the constraint equations, singular configuration of the system. In this paper, the system equations of motion with the Lagrange multipliers is rewritten by introducing generalized reaction forces. The combination with the condition of ideality of constraints leads to the system of equations which can be solved by numerical techniques smoothly, even over singular positions. Based on the new criterion of ideality of constraints, which relates generalized reaction forces and the null space matrix of Jacobian matrix, it is possible also to remove reaction forces and use only the reduced system of equations with null space matrix for passing singular positions. In order to prevent the constraint equations from the accumulated errors of integral time, the method of position and velocity projection has been exploited. Some numerical experiments are carried out to verify the proposed approach.