Eigenfactor solution of the matrix Riccati equation--A continuous square root algorithm

This paper introduces a new algorithm for solving the matrix Riccati equation. Differential equations for the eigenvalues and eigenvectors of the solution matrix are developed in which their derivatives are expressed in terms of the eigenvalues and eigenvectors themselves and not as functions of the solution matrix. The solution of these equations yields, then, the time behavior of the eigenvalues and eigenvectors of the solution matrix. A reconstruction of the matrix itself at any desired time is immediately obtained through a trivial similarity transformation. This algorithm serves two purposes. First, being a square root solution, it entails all the advantages of square root algorithms such as nonnegative definiteness and accuracy. Secondly, it furnishes the eigenvalues and eigenvectors of the solution matrix continuously without resorting to the complicated route of solving the equation directly and then decomposing the solution matrix into its eigenvalues and eigenvectors. The algorithm which handles cases of distinct as well as multiple eigenvalues is tested on several examples. Through these examples it is seen that the algorithm is indeed more accurate than the ordinary one. Moreover, it is seen that the algorithm works in cases where the ordinary algorithm fails and even in cases where the closed-form solution cannot be computed as a result of numerical difficulties.     

A square root of a matrix approach to obtain the solution to a steady-state matrix Riccati equation

By utilizing the square root of a matrix approach, a method of generating the solution to a steady-state matrix Riccati type equation (under certain restrictions) is presented. This approach not only yields a closed form expression for the Riccati solution, but also converts the original Riccati equation into other equations which may have numerical or computational advantages. An example is worked out for a second-order case.     

On the computation of a matrix inverse square root

The inverse square root of a matrix plays a role in the computation of an optimal symmetric orthogonalization of a set of vectors. We suggest two iterative techniques to compute an inverse square root of a given matrix. The two schemes are analyzed and their numerical stability properties are investigated.     

An extension on a new formulation of the algebraic Riccati equation problem

In a recent paper [3], by utilizing the square root of a matrix approach a new formulation of the algebraic Riccati equation has been presented and a closed form solution has been obtained under the hypothesis that certain commutativity conditions are fulfilled on a transformed space. In this note the imposed conditions are weakened and the formulation is extended to a wider class of Riccati problems.     

基于平方根求积卡尔曼滤波器的弹道目标跟踪

针对再入阶段的弹道目标跟踪问题,提出运用平方根求积卡尔曼滤波器(SRQKF)估计目标的状态.所提出的算法是求积卡尔曼滤波(QKF)算法的平方根实现.该算法传播了目标状态的均值和协方差的平方根,确保了协方差矩阵的对称性和半正定性,改进了数值精度和稳定性,但其计算复杂性稍有增加.仿真实验表明,所提出算法的估计精度优于QKF算法和扩展卡尔曼滤波(EKF)算法,是一种很有效的非线性滤波方法.     

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。     

广义分布参数系统的初值问题

本文定义了广义左逆、广义Ｆｏｕｒｉｅｒ变换，矩产方根等概念，论述了由带奇异系数矩耦合偏微分方程描述的广义分布参数系统，由广义Ｆｕｒｉｅｒ变换定理讨论了广义分布参数系统的初值问题，得到了该系统的解及其相容的初值条件。     

Order-recursive factorization of the pseudoinverse of a covariancematrix

A numerically reliable algorithm is developed to recursively update the square root of a pseudoinverse matrix using the square root of a lower dimension pseudoinverse matrix. The numerical computations are based on a generalized singular value decomposition which is used to do a canonical correlation analysis. An operation count is given for sequential and parallel implementation of a partitioned order-recursive algorithm. These methods are useful for covariance analysis to determine the contributions due to various modeling errors in the design of a Kalman filter     

Discrete square root filtering: A survey of current techniques

The conventional Kalman approach to discrete filtering involves propagation of a state estimate and an error covariance matrix from stage to stage. Alternate recursive relationships have been developed to propagate a state estimate and a square root error covariance instead. Although equivalent algebraically to the conventional approach, the square root filters exhibit improved numerical characteristics, particularly in ill-conditioned problems. In this paper, current techniques in square root filtering are surveyed and related by applying a duality association. Four efficient square root implementations are suggested, and compared with three common conventional implementations in terms of computational complexity and precision. The square root computational burden should not exceed the conventional by more than 50 percent in most practical problems. An examination of numerical conditioning predicts that the square root approach can yield twice the effective precision of the conventional filter in ill-conditioned problems. This prediction is verified in two examples. The excellent numerical characteristics and reasonable computation requirements of the square root approach make it a viable alternative to the conventional filter in many applications, particularly when computer word length is limited, or the estimation problem is badly conditioned.     

平方根体积积分卡尔曼滤波

体积积分是一种新的具有较高代数精度的积分方法。为了提高非线性滤波算法的精度和数值稳定性,将体积积分规则和平方根分解引入卡尔曼滤波框架中,提出了平方根体积积分卡尔曼滤波算法（SRCQKF）。新算法采用球半径体积规则和高斯-拉盖尔积分规则计算积分点,利用矩阵的QR分解得到协方差矩阵的平方根并传播平方根。两个典型的非线性系统的实验结果表明,与体积卡尔曼滤波相比,新算法提高了非线性状态的估计精度,具有较高的数值稳定性。