关于双随机循环矩阵中的素元(英文)

非负矩阵中的素元分类问题在控制和系统论中有重要的应用.本文将研究由G.Picci等所提出的关于双随机循环矩阵中素元的一个问题和一个猜想,得到了一个判别具有位数5的n阶双随机循环矩阵不是素元的充要条件,给出了猜想成立的一些充分条件.     

某些特殊循环矩阵的逆

贵刊1986年第10期,姚存峰给出了求循环矩阵的逆矩阵的一个方法。此法虽然解决了循环矩阵的求逆问题,但在实际应用中因有大量的三角函数运算等问题,因此此法使用起来不太方便.本文就某些特殊类型的循环矩阵的求逆问题进行探讨,给出一些简便方法. 设循环矩阵A为     

Pascal矩阵与Vandermonde矩阵的关系

EI-Mikkawy M证明了对称Pascal矩阵Q_n和Vlandermonde矩阵V_n之间满足矩阵方程Q_n=T_nV_n,这里T_n是一个随机矩阵。本文证明了随机矩阵T_n能够分解成第一类Stirling矩阵和对角矩阵的乘积,得到了矩阵T_n的元素之间的递推关系,从而回答了EI-Mikkawy M的一个公开问题。同时得到了一些与Stirling数相关的组合恒等式。     

g—轮换矩特征值的公式解

通过建立ｇ－轮换矩阵的分解定理和降价定理，本文解决了ｇ－轮换矩阵特征值的计算问题，给出了一个公式解法。     

轻矩阵及其推广

在R.E.Hartwig工作的基础上进一步研究了轻矩阵,得到了轻矩阵的一些特性,且把轻矩阵推广到了广义轻矩阵,得到了广义轻矩阵的一个充要条件;部分回答了文[1]中R.E.Harwtig提出的一个公开问题：当A与B都为n除非负矩阵时,刻划方程AX=XB的非负解。     

关于Z—矩阵的弱正则分裂

本文主要考虑一类Ｚ－矩阵弱正则分裂的迭代矩阵的谱半径和特征值１的指数问题，推广了Ｒｏｓｅ在１９８４年得到的一个定理。     

益智游戏“汉诺塔”中的矩阵运算

汉诺塔(Tower of Hanoi)问题源于印度一个古老传说,据此做成了益智游戏,蕴含大量的数学思想与方法.本文采用矩阵描述汉诺塔状态和圆盘移动过程,将圆盘从一个位置移动到另一个位置转化为矩阵的加法,进而构造由若干可能状态矩阵组成的图的邻接矩阵,计算其幂矩阵,由此很方便地求得完成汉诺塔游戏的所有可能的圆盘移动方案,求解过程简单,含义清晰,易于理解和实现.     

左（右）逆矩阵计算的误差估计

我们在实际中常常遇到满秩长方矩阵的求逆问题,并希望知道所求逆矩阵的精度。目前常用的估计式仅给出了一个粗糙的上界(见[1]),这个上界往往放大许多,因而极不准确,本文给出了实用和较准确的估计式。     

二阶矩阵快速乘法的一个新的算法集合

文献[1]—[4]从不同角度研究了二阶矩阵快速乘的各种问题,所有算法分属于以S算法与W算法为基础的两个算法集合.本文作者深入研究了算法的结构和性质,通过计算机检索,得到一个不属于上述两集合的算法和相应的包含有1048576个算法的封闭的算法集合.     

用试验数据修正振动系统的双对称阻尼矩阵与刚度矩阵

讨论用试验数据修正振动系统的双对称阻尼矩阵与刚度矩阵问题.依据特征方程、阻尼矩阵与刚度矩阵的双对称性,利用代数二次特征值反问题的理论和方法,研究了该问题解的存在性与唯一性,提出了修正阻尼矩阵与刚度矩阵的一个新方法.利用双对称矩阵的性质研究了方程的双对称解.给出了二次特征值反问题双对称解的一般表达式,讨论了对任意给定矩阵的最佳逼近问题,并给出了问题的最佳逼近解.用该方法修正的阻尼矩阵与刚度矩阵不仅满足二次特征方程,而且是唯一的双对称矩阵.     

一个状态控制问题的数学建模与求解

对互联网上的一个数学游戏的控制问题,进行了一般性推广,运用数学建模的方法给出了其基于有限域上的线性方程组的数学模型及求解.     

“关灯”游戏的数学建模与求解

对互联网上的一个数学游戏的控制问题,进行了一般性推广,运用数学建模的方法给出了其基于有限域上的线性方程组的数学模型及求解.     

Global optimization for the Biaffine Matrix Inequality problem

It has recently been shown that an extremely wide array of robust controller design problems may be reduced to the problem of finding a feasible point under a Biaffine Matrix Inequality (BMI) constraint. The BMI feasibility problem is the bilinear version of the Linear (Affine) Matrix Inequality (LMI) feasibility problem, and may also be viewed as a bilinear extension to the Semidefinite Programming (SDP) problem. The BMI problem may be approached as a biconvex global optimization problem of minimizing the maximum eigenvalue of a biaffine combination of symmetric matrices. This paper presents a branch and bound global optimization algorithm for the BMI. A simple numerical example is included. The robust control problem, i.e., the synthesis of a controller for a dynamic physical system which guarantees stability and performance in the face of significant modelling error and worst-case disturbance inputs, is frequently encountered in a variety of complex engineering applications including the design of aircraft, satellites, chemical plants, and other precision positioning and tracking systems.     

A structure‐preserving doubling algorithm for Lur'e equations

We introduce a numerical method for the numerical solution of the Lur'e equations, a system of matrix equations that arises, for instance, in linear‐quadratic infinite time horizon optimal control. We focus on small‐scale, dense problems. Via a Cayley transformation, the problem is transformed to the discrete‐time case, and the structural infinite eigenvalues of the associated matrix pencil are deflated. The deflated problem is associated with a symplectic pencil with several Jordan blocks of eigenvalue 1 and even size, which arise from the nontrivial Kronecker chains at infinity of the original problem. For the solution of this modified problem, we use the structure‐preserving doubling algorithm. Implementation issues such as the choice of the parameter γ in the Cayley transform are discussed. The most interesting feature of this method, with respect to the competing approaches, is the absence of arbitrary rank decisions, which may be ill‐posed and numerically troublesome. The numerical examples presented confirm the effectiveness of this method. Copyright © 2015 John Wiley & Sons, Ltd.     

Elastoplastic stress analysis of thick laminated metal-matrix composite plates by the finite-element method

The elastoplastic stress state of a laminated stainless-steel-fiber-reinforced aluminum-matrix plates, with or without a hole, subjected to a pressure on their top is examined by using the finite-element method. The analysis is carried out for three layouts: (0/90/0/90)_s, (45/-45/45/-45)_s, and (30/60/30/60)_s. The Newton-Raphson method is used to solve the nonlinear problem. The distributions of equivalent stresses and the plastic zones of the plates without a hole and with a hole of various diameters are determined. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 42, No. 4, pp. 531–544, July–August, 2006.     

一类非线性系统的时滞相关H∞控制

本文研究一类非线性控制系统的时滞相关日。控制问题．首先通过构造恰当的Lyapunov泛函,设计非线性H∞控制器,得到了该类非线性控制系统渐近稳定的一个充分条件．其次通过建立具有线性矩阵不等式（LMI）约束的凸优化问题,得到了该类非线性系统的时滞相关H∞性能的一个充分条件．MATLAB仿真验证了本文方法的可行性．     

Sampled-data control of Lur’e systems

The main purpose of this paper is to fill a gap in the literature concerning the problem of designing a sampled-data control law for continuous-time Lur’e systems. The goal is to design a state feedback sampled-data control for this class of nonlinear systems preserving global asymptotic stability and minimizing a guaranteed quadratic cost. The main challenge towards the solution of the proposed problem is to handle this class of nonlinear system in order to propose less conservative design conditions expressed through differential linear matrix inequalities - (DLMIs). Bellman’s Principle of Optimality applied together with the Popov–Lyapunov function that emerges from the celebrated Popov Stability Criterion is the key issue to obtain the reported results. Two examples are solved for illustration.     

网络化马尔可夫跳变系统的鲁棒无源控制

针对基于网络的凸多面体不确定离散时间马尔可夫跳变系统,研究其鲁棒无源控制问题.在网络诱导时滞是时变且有界的情况下,基于李雅普诺夫稳定性理论,通过构造参数依赖的随机李雅普诺夫泛函和运用广义系统变换,提出了不依赖模态的无源控制器存在的时滞依赖充分条件.所设计的鲁棒无源控制器保证了相应的闭环系统是鲁棒随机稳定且具有指定耗散率.将鲁棒无源控制器设计问题转化为一组线性矩阵不等式的可解性问题.仿真算例证明了本文方法的有效性.     

Fuzzy guaranteed cost controller for trajectory tracking in nonlinear systems

In this paper, we propose a fuzzy logic based guaranteed cost controller for trajectory tracking in nonlinear systems. Takagi–Sugeno (T–S) fuzzy model is used to represent the dynamics of a nonlinear system and the controller design is carried out using this fuzzy model. State feedback law is used for building the fuzzy controller whose performance is evaluated using a quadratic cost function. For designing the fuzzy logic based controller which satisfies guaranteed performance, linear matrix inequality (LMI) approach is used. Sufficient conditions are derived in terms of matrix inequalities for minimizing the performance function of the controller. The performance function minimization problem with polynomial matrix inequalities is then transformed into a problem of minimizing a convex performance function involving standard LMIs. This minimization problem can be solved easily and efficiently using the LMI optimization techniques. Our controller design method also ensures that the closed-loop system is asymptotically stable. Simulation study is carried out on a two-link robotic manipulator tracking a reference trajectory. From the results of the simulation study, it is observed that our proposed controller tracks the reference trajectory closely while maintaining a guaranteed minimum cost.     

Optimal control of lumped parameter systems via shifted Legendre polynomial approximation

Shifted Legendre polynomial functions are employed to solve the linear-quadratic optimal control problem for lumped parameter system. Using the characteristics of the shifted Legendre polynomials, the system equations and the adjoint equations of the optimal control problem are reduced to functional ordinary differential equations. The solution of the functional differential equations are obtained in a series of the shifted Legendre functions. The operational matrix for the integration of the shifted Legendre polynomial functions is also introduced in the simulation step in order to simplify the computational procedure. An illustrative example of an optimal control problem is given, and the computational results are compared with those of the exact solution. The proposed method is effective and accurate.