Positively invariant sets for continuous-time systems with the cone-preserving property

The main objective of this paper is to determine positively invariant and asymptotically stable polyhedral sets for a linear continuous-time system [xdot](t) = Ax(t) for which matrix e ^1A is a cone-preserving matrix, that is, e ^1A K ? K, for some proper cone K. Necessary and sufficient conditions guaranteeing that some bounded sets are positively invariant and contractive are given. These sets are obtained by means of the intersection of shifted cones. First, some results presented under a geometrical form and also in algebraic form allow characterization of systems having the cone-preserving property. Finally, as an application, the proposed results are used to determine a stability domain for a state feedback regulator with constraints on either or both states and controls.     

Algebraic approach to the interval linear static identification,tolerance, and control problems,or one more application of kaucher arithmetic

In this paper, theidentification problem, thetolerance problem, and thecontrol problem are treated for the interval linear equation Ax=b. These problems require computing an inner approximation of theunited solution set Σ_??(A, b)={x ∈ ? ⁿ | (?A ∈ A)(Ax ∈ b)}, of thetolerable solution set Σ_??(A, b)={x ∈ ? ⁿ | (?A ∈ A)(Ax ∈ b)}, and of thecontrollable solution set Σ_??(A, b)={x ∈ ? ⁿ | (?b ∈ b)(Ax ∈b)} respectively. Analgebraic approach to their solution is developed in which the initial problem is replaced by that of finding analgebraic solution of some auxiliary interval linear system in Kaucher extended interval arithmetic. The algebraic approach is proved almost always to give inclusion-maximal inner interval estimates of the solutionsets considered. We investigate basic properties of the algebraic solutions to the interval linear systems and propose a number of numerical methods to compute them. In particular, we present the simple and fastsubdifferential Newton method, prove its convergence and discuss numerical experiments.     

General optimal attenuation of harmonic disturbance with unknown frequencies

We present algorithms for optimal harmonic disturbance attenuation in standard discrete-time control structure, based on a parametrisation of (marginally) stabilising controllers. The Frobenius norm and the spectral norm of the closed-loop transfer matrix at the disturbance frequencies are minimised. If there is only one frequency of the disturbance, the controller has an observer–based form, which we obtain by solving a static output feedback (SOF) stabilisation control problem. Although the SOF stabilisation problem is hard, the generical case of nonsquare matrix G ₂₂ is solved by linear algebra methods. Numerical simulation results are presented. As a corollary, we transform the control problem with unit circle invariant zeros into a ?_∞ control problem without such zeros. The elimination of the unit circle invariant zeros is based on the fact that matrix Y(zI???A?+?BF)^?1 is stable, where (Y,?F) with Y?≥?0 is a solution of a discrete-time algebraic Riccati system.     

Extended version with the analysis of dynamic system for iterative refinement of solution

The extended version with the analysis of dynamic system for Wilkinson's iteration improvement of solution is presented in this paper. It turns out that the iteration improvement can be viewed as applying explicit Euler method with step size h=1 to a dynamic system which has a unique globally asymptotically stable equilibrium point, that is, the solution x*=A ^?1 b of linear system Ax=b with non-singular matrix A. As a result, an extended iterative improvement process for solving ill-conditioned linear system of algebraic equations with non-singular coefficients matrix is proposed by following the solution curve of a linear system of ordinary differential equations. We prove the unconditional convergence and derive the roundoff results for the extended iterative refinement process. Several numerical experiments are given to show the effectiveness and competition of the extended iteration refinement in comparison with Wilkinson's.     

On solving the Lyapunov and Stein equations for a companion matrix

When the matrix A is in companion form, the essential step in solving the Lyapunov equation PA + A^TP = −Q involves a linear n × n system for the first column of the solution matrix P. The complex dependence on the data matrices A and Q renders this system unsuitable for actual computation. In this paper we derive an equivalent system which exhibits simpler dependence on A and Q as well as improved complexity and robustness characteristics. A similar results is obtained also for the Stein equation P − A^TPA = Q.     

Stability radii of differential algebraic equations with structured perturbations

This paper deals with a formula for computing stability radii of a differential algebraic equation of the form AX^′(t)−BX(t)=0, where A,B are constant matrices. A computable formula for the complex stability radius is given and a key difference between the ordinary differential equation (ODEs for short) and the differential algebraic equation (DAEs for short) is pointed out. A special case where the real stability radius and the complex one are equal is considered.     

Further results on controllability and the linear equation Ax = b

The linear equation Ax = b, with A an n × n matrix and b an n × l matrix over a unique factorization domain R, is related to the controllability submodule U of the pair (A, b). It is shown that the above equation has a solution lying in V if, and only if, A is unimodular as an operator on U. An example is given of a matrix which is unimodular as an operator on the controllability submodule, but not as an operator on Rⁿ and sparseness of this occurrence is discussed.     

A representation of all solutions of the control algebraic Riccati equation for infinite-dimensional systems

We obtain a representation of all self-adjoint solutions of the control algebraic Riccati equation associated to the infinite-dimensional state linear system Σ(A,B,C) under the following assumptions: A generates a C ₀-group, the system is output stabilizable, strongly detectable and the dual Riccati equation has an invertible self-adjoint non-negative solution.     

Algorithms of constructing a stabilizing solution of the algebraic Riccati equation based on its linear reduction in the problems of analytical construction of controllers

In the problem of the stabilizing solution of the algebraic Riccati equation, the resolvent Θ(s) = (s I _2n ? H)^?1 of the Hamilton 2n × 2n-matrix H of the algebraic Riccati equation allows us to reduce the problem to a linear matrix equation. In [1], the constructions necessary for this and the theorem of existence and representation of the stabilized solutions to an algebraic Riccati equation was proposed. In this paper, the methods of constructing the resolvent and the linear reduction matrix defined by it necessary for the application of the theorem, and in addition, the algorithms of constructing stabilizing solution of the algebraic Riccati equation are proposed.     

On the numerical solution of the Burgers's equation

In this paper, we consider the linear heat equation arisen from the Burgers's equation using the Hopf–Cole transformation. Discretization of this equation with respect to the space variable results in a linear system of ordinary differential equations. The solution of this system involves in computing exp(α A)y for some vector y, where A is a large special tridiagonal matrix and α is a positive real number. We give an explicit expression for computing exp(α A)y. Finally, some numerical experiments are given to show the efficiency of the method.     

Correction to ‘A note on the discrete Lyapunov and Riccati matrix equations’

In Tran and Sawan (1984), we derived a lower bound for the determinant of the discrete algebraic Riccati equation where A, B, P, Q are n x n matrices, Q = Q ^T > 0 and Rank (B) = n. We assumed that BB ^T>Q and the matrix A is stable. The purpose of this note is to include an additional assumption about the above equation in order for the result of Theorem 1 given in Tran and Sawan (1984) to be valid. Without the additional assumption, this theorem would be invalid as has been pointed out by Kwon and Youn (1985). The additional assumption is given below with the same notation as in Tran and Sawan (1984).     

Insensitivity of optimal linear discrete-time regulators †

Insensitivity of the optimal linear discrete-time regulator x_k+1 = A_xk+ B_uk with quadratic performance index is investigated. Expressions are derived for finite changes which can occur in A or B without affecting the solution P of the algebraic matrix Riccati equation, and for simultaneous changes in A and B which leave both P and the optimal feedback law uk=— Kxt unaltered. If there is a predetermined variation in P, changes in A and B which leave K fixed are also given. The work complements an earlier treatment of the continuous-time case, and comparisons are noted.     

On the solution of the fuzzy Sylvester matrix equation

In this paper, we consider the fuzzy Sylvester matrix equation AX+XB=C,AX+XB=C, where A ? \mathbbR^{n ×n}A\in {\mathbb{R}}^{n \times n} and B ? \mathbbR^{m ×m}B\in {\mathbb{R}}^{m \times m} are crisp M-matrices, C is an n×mn\times m fuzzy matrix and X is unknown. We first transform this system to an (mn)×(mn)(mn)\times (mn) fuzzy system of linear equations. Then, we investigate the existence and uniqueness of a fuzzy solution to this system. We use the accelerated over-relaxation method to compute an approximate solution to this system. Some numerical experiments are given to illustrate the theoretical results.     

求解大规模谱聚类的近似加权核k-means算法

谱聚类将聚类问题转化成图划分问题,是一种基于代数图论的聚类方法.在求解图划分目标函数时,一般利用Rayleigh熵的性质,通过计算Laplacian矩阵的特征向量将原始数据点映射到一个低维的特征空间中,再进行聚类.然而在谱聚类过程中,存储相似矩阵的空间复杂度是O(n²),对Laplacian矩阵特征分解的时间复杂度一般为O(n³),这样的复杂度在处理大规模数据时是无法接受的.理论证明,Normalized Cut图聚类与加权核k-means都等价于矩阵迹的最大化问题.因此,可以用加权核k-means算法来优化Normalized Cut的目标函数,这就避免了对Laplacian矩阵特征分解.不过,加权核k-means算法需要计算核矩阵,其空间复杂度依然是O(n²).为了应对这一挑战,提出近似加权核k-means算法,仅使用核矩阵的一部分来求解大数据的谱聚类问题.理论分析和实验对比表明,近似加权核k-means的聚类表现与加权核k-means算法是相似的,但是极大地减小了时间和空间复杂性.     

Some consequences on the planar three-index transportation problem

In this study, some algebraic characterizations of the coefficient matrix A of the planar three-index transportation problem are derived and the equivalent formulation of this problem is obtained using the Kronecker product. It is shown that eigenvectors of the matrix G ⁺ G are characterized in terms of eigenvectors of the matrix A ⁺ A , where G ⁺ is the Moore–Penrose inverse of the coefficient matrix G of the equivalent problem.     

AN ALGORITHM FOR FINDING ALL FUNCTIONS EMBEDDED IN A RELATION

The problem is posed: find an algorithm which for any given n-dimensional relation R ? A₁ × A₂ × ? × A_n, defined on a set family A = { A₁, A₂, ?, A_nrcub;, n = 1,2, ?, determines all functional dependences between disjoint subsets of A which are embedded in R. A solution algorithm is presented, a theorem is proved that allows a simplification in the algorithm, and an efficient computer implementation (available through the General Systems Depository) is demonstrated.     

An accelerated randomized Kaczmarz method via low-rank approximation

The Kaczmarz method for finding the solution to an overdetermined consistent system of linear equation Ax=b(A∈R^m×n) is an iterative algorithm that has found many applications ranging from computer tomography to digital signal processing. Recently, Strohmer and Vershynin proposed randomized Kaczmarz, and proved its exponential convergence. In this paper, motivated by idea of precondition, we present a modified version of the randomized Kaczmarz method where an orthogonal matrix was multiplied to both sides of the equation Ax=b, and the orthogonal matrix is obtained by low-rank approximation. Our approach fits the problem when m is huge and m?n. Theoretically, we improve the convergence rate of the randomized Kaczmarz method. The numerical results show that our approach is faster than the standard randomized Kaczmarz.     

A note on Wilkinson's iterative refinement of solution with automatic step-size control

In a more recent paper [X. Wu and Y.L. Fang, Wilkinson's iterative refinement of solution with automatic step-size control for linear system of equations, Appl. Math. Comput. 193 (2007), pp. 506–513], the authors proposed an iterative improvement of solution with automatic step-size control for a linear system of algebraic equations. The convergence analysis of the iterative procedure is shown for both stationary and non-stationary iterative formulas, based on the accurate coefficient matrix A and its inverse. However, in each iteration, A is approximated by the LU decomposition in floating-point arithmetic, and then its inverse is also approximated by a matrix B, although the role of B is realized by solving something like Az=r from the LU factorization of A. This point should be addressed that usually BA is not equal to an identity matrix I, in particular, when A is ill-conditioned. Therefore, in this paper, a supplementary convergence analysis is presented based on the approximation.     

Some investigation on Hermitian positive-definite solutions of a nonlinear matrix equation

In this paper, the Hermitian positive-definite solutions of the matrix equation X^s+A*X^?tA=Q are considered. New necessary and sufficient conditions for the equation to have a Hermitian positive-definite solution are derived. In particular, when A is singular, a new estimate of Hermitian positive-definite solutions is obtained. In the end, based on the fixed point theorem, an iterative algorithm for obtaining the positive-definite solutions of the equation with Q=I is discussed. The error estimations are found.     

Optimization and pole placement for a single input controllable system

Given the system [xdot]=Ax+bu and the cost function J=dt, relations are to be determined among the open-loop characteristic polynomial, the closed-loop characteristic polynomial and the matrices A and Q. Those relations take a simple form if the system is in the standard controllable form. In this case the optimal control law can be found easily without solving the matrix Riccati equation while the minimum value of the cost function, if it is required, can be determined by solving a matrix equation of the form C ^T. X+XC= ?D