BCR method for solving generalized coupled Sylvester equations over centrosymmetric or anti-centrosymmetric matrix

This paper introduces another version of biconjugate residual method (BCR) for solving the generalized coupled Sylvester matrix equations over centrosymmetric or anti-centrosymmetric matrix. We prove this version of BCR algorithm can find the centrosymmetric solution group of the generalized coupled matrix equations for any initial matrix group within finite steps in the absence of round-off errors. Furthermore, a method is provided for choosing the initial matrices to obtain the least norm solution of the problem. At last, some numerical examples are provided to illustrate the efficiency and validity of methods we have proposed.     

Finite iterative algorithms for the generalized Sylvester-conjugate matrix equation {AX+BY=E\overline{X}F+S}

This paper investigates the generalized Sylvester-conjugate matrix equation, which includes the normal Sylvester-conjugate, Kalman–Yakubovich-conjugate and generalized Sylvester matrix equations as its special cases. An iterative algorithm is presented for solving such a kind of matrix equations. This iterative method can give an exact solution within finite iteration steps for any initial values in the absence of round-off errors. Another feature of the proposed algorithm is that it is implemented by original coefficient matrices. By specifying the proposed algorithm, iterative algorithms for some special matrix equations are also developed. Two numerical examples are given to illustrate the effectiveness of the proposed methods.     

Efficient iterative solutions to general coupled matrix equations

Linear matrix equations are encountered in many systems and control applications.In this paper,we consider the general coupled matrix equations（including the generalized coupled Sylvester matrix equations as a special case）l t=1EstYtFst = Gs,s = 1,2,···,l over the generalized reflexive matrix group（Y1,Y2,···,Yl）.We derive an efcient gradient-iterative（GI） algorithm for fnding the generalized reflexive solution group of the general coupled matrix equations.Convergence analysis indicates that the algorithm always converges to the generalized reflexive solution group for any initial generalized reflexive matrix group（Y1（1）,Y2（1）,···,Yl（1））.Finally,numerical results are presented to test and illustrate the performance of the algorithm in terms of convergence,accuracy as well as the efciency.     

A note on combined generalized Sylvester matrix equations

The solution of two combined generalized Sylvester matrix equations is studied. It is first shown that the two combined generalized Sylvester matrix equations can be converted into a normal Sylvester matrix equation through extension, and then with the help of a result for solution to normal Sylvester matrix equations, the complete solution to the two combined generalized Sylvester matrix equations is derived. A demonstrative example shows the effect of the proposed approach.     

关于一般耦合矩阵方程的迭代对称解

本文构造了一个有效的迭代方法(CGL)去求解一般耦合矩阵方程的对称解.若一般耦合矩阵方程关于对称解相容,则对于任意给定的初始对称矩阵组,利用所构造的迭代算法,都能在有限步迭代出所求问题的一组对称解,若选用一些特殊的初值,则可获得矩阵方程的极小范数对称解.最后的数值例子表明了所给算法的有效性.     

离散时间Itˆo 型跳变系统Lyapunov 方程的有限次迭代求解算法

针对离散时间Itˆo 型马尔科夫跳变系统Lyapunov 方程的求解给出一种迭代算法. 经证明, 在误差允许的范围内, 该算法可以在确定的有限次数内收敛到系统的精确解, 收敛速度较快, 具有良好的数值稳定性, 并且该算法为显式迭代, 可避免迭代过程中求解其他矩阵方程对结果精度产生的影响. 最后通过一个数值算例对该算法的有效性进行了验证.

     

Generalized Lyapunov equation and factorization of matrix polynomials

This paper presents an algorithm for the construction of a solution of the generalized Lyapunov equation. It is proved that the polynomial matrix factorization relative to the imaginary axis may be reduced to the successive solution of Lyapunov equations, i.e. the factorization is reduced to the solution of a sequence of generalized Lyapunov equations, not to the solution of generalized Riccati equation.     

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。     

Generalized Chandrasekhar algorithms: Time-varying models

The Riccati equation (RE) plays a fundamental role in optimal control theory, linear estimation, radiative transfer, neutron transport theory, etc. Its effective, numerical solution constitutes the integral prerequisite to the solution of important problems in the above and related fields. A computationally advantageous approach to the solution of matrix Re's is the so-calledx-yor Chandrasekhar algorithm through which the matrix RE is replaced by two coupled differential equations of lesser dimensionality. These previous Chandrasekhar algorithms were, however, restricted to the case of time-invariant models. In this short paper, generalizedx-yor Chandrasekhar algorithms are presented that are applicable to time-varying models as well as time-invariant ones. Backward and forward time differentiations are introduced that readily yield the generalized Chandrasekhar algorithms as well as provide several interesting interpretations of these results. Furthermore, the possible computational advantages, as well as the theoretical significance of the generalized Chandrasekhar algorithms are explored.     

Coupled Cross-correlation Neural Network Algorithm for Principal Singular Triplet Extraction of a Cross-covariance Matrix

This paper proposes a novel coupled neural network learning algorithm to extract the principal singular triplet (PST) of a cross-correlation matrix between two high-dimensional data streams. We firstly introduce a novel information criterion (NIC), in which the stationary points are singular triplet of the crosscorrelation matrix. Then, based on Newton's method, we obtain a coupled system of ordinary differential equations (ODEs) from the NIC. The ODEs have the same equilibria as the gradient of NIC, however, only the first PST of the system is stable (which is also the desired solution), and all others are (unstable) saddle points. Based on the system, we finally obtain a fast and stable algorithm for PST extraction. The proposed algorithm can solve the speed-stability problem that plagues most noncoupled learning rules. Moreover, the proposed algorithm can also be used to extract multiple PSTs effectively by using sequential method.      

矩阵方程AXB=C的D反对称最佳逼近解的算法

本文利用广义共轭梯度法设计求矩阵方程AXB=C的D反对称最佳逼近解的算法,证明了算法的有限步收敛性,数值实例说明算法是有效的。     

An explicit solution to right factorization with application in eigenstructure assignment

Based on the well-known Leverrier algorithm, a simple explicit solution to right factorization of a linear system is established.This solution is expressed by the controllability matrix of the given system and a symmetric operator matrix.Applications of this solution to a type of generalized Sylvester matrix equations and the problem of parametric eigenstructure assignment by state feedback are investigated,and general complete parametric solutions to these two problems are deduced. These new solutions are simple,and possess desirable structural properties which render the solutions readily implementable.An example demonstrates the effect of the proposed results.     

Discrete generalized algebraic Riccati equations and polynomial matrix factorization

This paper presents an algorithm for solving discrete generalized algebraic Riccati equations with the help of an orthogonal projector. A generalization of the procedure of forming and correcting the orthogonal projector is considered and also that of correcting the proper solution by the Newton-Raphson scheme. The possibility to use the discrete generalized Riccati equations for polynomial matrix factorization with respect to the unit circle is demonstrated. A numerical example is given.     

Tracing post-limit-point paths with reduced basis technique

A reduced basis technique and a problem-adaptive computational algorithm are presented for predicting the post-limit-point paths of structures. In the proposed approach the structure is discretized by using displacement finite element models. The nodal displacement vector is expressed as a linear combination of a small number of vectors and a Rayleigh-Ritz technique is used to approximate the finite element equations by a small system of nonlinear algebraic equations.To circumvent the difficulties associated with the singularity of the stiffness matrix at limit points, a constraint equation, defining a generalized arc-length in the solution space, is added to the system of nonlinear algebraic equations and the Rayleigh-Ritz approximation functions (or basis vectors) are chosen to consist of a nonlinear solution of the discretized structure and its various order derivatives with respect to the generalized arc-length. The potential of the proposed approach and its advantages over the reduced basis-load control technique are outlined. The effectiveness of the proposed approach is demonstrated by means of numerical examples of structural problems with snap-through and snap-back phenomena.     

Convergence of gradient-based iterative solution of coupled Markovian jump Lyapunov equations

The solution of coupled discrete-time Markovian jump Lyapunov matrix equations (CDMJLMEs) is important in stability analysis and controller design for Markovian jump linear systems. This paper presents a simple and effective iterative method to produce numerical solutions to this class of matrix equations. The gradient-based algorithm is developed from an optimization point of view. A necessary and sufficient condition guaranteeing the convergence of the algorithm is established. This condition shows that the algorithm always converges provided the CDMJLMEs have unique solutions which is evidently different from the existing results that converge conditionally. A simple sufficient condition which is easy to test is also provided. The optimal step size in the algorithm such that the convergence rate of the algorithm is maximized is given explicitly. It turns out that an upper bound of the convergence rate is bounded by a function of the condition number of the augmented coefficient matrix of the CDMJLMEs. Some parameters are introduced to the algorithm that will potentially reduce the condition number and thus increase the convergence rate of the algorithm. A numerical example is used to illustrate the efficiency of the proposed approach.     

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。     

On a general optimal algorithm for multirate output feedbackcontrollers for linear stochastic periodic systems

A modified optimal algorithm for multirate output feedback controllers of linear stochastic periodic systems is developed. By combining the discrete-time linear quadratic regulation (LQR) control problem and the discrete-time stochastic linear quadratic regulation (SLQR) control problem to obtain an extended linear quadratic regulation (ELQR) control problem, one derives a general optimal algorithm to balance the advantages of the optimal transient response of the LQR control problem and the optimal steady-state regulation of the SLQR control problem. In general, the solution of this algorithm is obtained by solving a set of coupled matrix equations. Special cases for which the coupled matrix equations can be reduced to a discrete-time algebraic Riccati equation are discussed. A reducable case is the optimal algorithm derived by H.M. Al-Rahmani and G.F. Franklin (1990), where the system has complete state information and the discrete-time quadratic performance index is transformed from a continuous-time one     

Solution to Generalized Sylvester Matrix Equations

An explicit solution to the generalized Sylvester matrix equation AX+BY= EXF is established. This solution is expressed in terms of the R-controllability matrix of (E, A, B), a generalized symmetric operator matrix and an observability matrix. Moreover, based on this solution, solutions to some other matrix equations are also derived. The results may provide great convenience for the analysis and synthesis problems related to these equations.     

Efficient numerical solution of constrained multibody dynamics systems

A numerical algorithm for conducting coupled system dynamical simulation is presented. The interconnected system, comprising numerous modules, is treated as a constrained multibody dynamics system. Of particular focus is the efficient solution of coupled system simulation without sacrificing the independence of the separate dynamical modules. The proposed algorithm, Maggi’s equations with perturbed iteration (MEPI) emanates from numerical methods for differential-algebraic equations. Separate treatment of the constraint equations from the resolution of subsystem dynamical responses marks MEPI’s main characteristic.