Estimation of algebraic solution by limiting the solution set of an interval linear system

In this paper, we introduce a new method for estimating the algebraic solution of an interval linear system (ILS) whose coefficient matrix is real-valued and right-hand side vector is interval-valued. In the proposed method, we first apply the interval Gaussian elimination procedure to obtain the solution set of an interval linear system and then by limiting the solution set of related ILS by the limiting factors, we get an algebraic solution of ILS. In addition, we prove that the obtained solution by our method satisfies the related interval linear system. Finally, based on our method, an algorithm is proposed and numerically demonstrated.     

A new approach to obtain algebraic solution of interval linear systems

In this paper, an algebraic solution of interval linear system involving a real square matrix and an interval right-hand side vector is obtained. A new approach to solve such systems based on the new concept “inclusion linear system” is proposed. Moreover, new necessary and sufficient conditions are derived for obtaining the unique algebraic solution. Furthermore, based on our method, an algorithm is proposed and numerically demonstrated. Finally, we compare the result obtained by our method with that obtained by interval Gauss elimination procedure.     

Criterion for the convergence of the solution of the Riccati differential equation

The optimal control problem for a linear system with a quadratic cost function leads to the matrix Riccati differential equation. The convergence of the solution of this equation for increasing time interval is investigated as a function of the final state penalty matrix. A necessary and sufficient condition for convergence is derived for stabilizable systems, even if the output in the cost function is not detectable. An algorithm is developed to determine the limiting value of the solution, which is one of the symmetric positive semidefinite solutions of the algebraic Riccati equation. Examples for convergence and nonconvergence are given. A discussion is also included of the convergence properties of the solution of the Riccati differential equation to any real symmetric (not necessarily positive semidefinite) solution of the algebraic Riccati equation.     

区间矩阵系统低保守性鲁棒控制器的设计

对于系统矩阵和输入矩阵均为区间矩阵的不确定系统 ,提出了实对称矩阵集合的最小上界和最大下界的计算方法 ,并应用该方法设计区间矩阵系统鲁棒控制器 ,把确定多个矩阵不等式解的复杂问题简化为求解单代数Riccati矩阵方程 ,该设计方法具有较小保守性 .     

New results for the bounds of the solution for the continuousRiccati and Lyapunov equations

This paper presents upper and lower matrix bounds for the solution of the continuous algebraic matrix Riccati equation. Furthermore, a new lower matrix bound for the solution of the continuous algebraic Lyapunov equation is also developed. These are new results     

区间离散Lurie系统的绝对稳定性

用Lyapunov函数研究了具有单调扇形限制的多非线性项的区间离散Lurie系统的鲁棒绝对稳定性, 给出了此类区间离散Lurie系统的鲁棒绝对稳定性的矩阵不等式形式的代数判据, 并与区间对称矩阵稳定性建立了联系.     

Scaling of general quadratic matrix equations

A scaling framework for general quadratic algebraic matrix equations is presented. All algebraic quadratic equations can be considered as special cases of a single generalized algebraic quadratic matrix equation (GQME). Hence, the paper is focused on the analysis and solution of the scaling problem of that GQME. The presented scaling method is based on the assignment of predetermined values of the coefficients and the unknown matrices of the GQME. The proposed framework is independent of any numerical method and therefore its use is general. Implementations are presented for the special case of matrix algebraic Riccati equations (AREs). Some new results of matrix algebraic identities considering Kronecker and Hadamard products are also reported.     

Representation of the characteristic polynomial of a two-channel system by a norm of an algebraic function

A new representation of the characteristic polynomial of a two-channel system by a norm of an algebraic function is proposed. The norm is obtained from the solution of the second-order algebraic equation for the transfer matrix. The discriminant of the algebraic equation determines the type of solution: real, complex, irreducible, and with a radical. For each solution type, an applied example of a two-channel system in aviation and power engineering is presented.     

An exact solution of the time-invariant discrete Kalman filter

An exact solution is presented of the matrix Riccati difference equation associated with a time-invariant discrete Kalman filter. The time-varying solution is expressed by means of the corresponding steady-state algebraic solution. An exact solution of the closed-loop transition matrix is also presented.     

Algebraic Approach in the "Outer Problem" for Interval Linear Equations

The subject of our work is the classical "outer" problem for the interval linear algebraic System Ax = b with the square interval matrix A: find "outer" coordinate-wise estimates of the united solution set formed by all solutions to the point systems Ax = b with A A and b b. The purpose of this work is to advance a new algebraic approach to the formulated problem, in which it reduces to solving one noninterval (point) equation in the Euclidean space of double dimension. We construct a specialized algorithm (subdifferential Newton method) that implements the new approach, then present results of the numerical tests with it. These results demonstrate that the proposed algebraic approach combines unique computational efficiency with high quality enclosures of the solution set.     

Multiplication Distributivity of Proper and Improper Intervals

The arithmetic on an extended set of proper and improper intervals presents algebraic completion of the conventional interval arithmetic allowing thus efficient solution of some interval algebraic problems. In this paper we summarize and present all distributive relations, known by now, on multiplication and addition of generalized (proper and improper) intervals.     

Upper matrix bound of the solution for the discrete Riccatiequation

A new upper matrix bound of the solution for the discrete algebraic matrix Riccati equation is developed. This matrix bound is then used to derive bounds on the eigenvalues, trace, and determinant of the same solution. It is shown that these eigenvalue bounds are less restrictive than previous results     

A method for solving systems of linear interval equations applied to the Leontief input–output model of economics

A new approach to solving systems of linear interval equations based on the generalized procedure of interval extension is proposed. This procedure is based on the treatment of interval zero as an interval centered around zero, and for this reason it is called the “interval extended zero” method. Since the “interval extended zero” method provides a fuzzy solution to interval equations, its interval representations are proposed. It is shown that they may be naturally treated as modified operations of interval division. These operations are used for the modified interval extensions of known numerical methods for solving systems of linear equations and finally for solving systems of linear interval equations. Using a well known example, it is shown that the solution obtained by the proposed method can be treated as an inner interval approximation of the united solution and an outer interval approximation of the tolerable solution, and lies within the range of possible AE-solutions between the extreme tolerable and united solutions. Generally, we can say that the proposed method provides the results which can be treated as approximate formal solutions sometimes referred to as algebraic solutions. Seven known examples are used to illustrate the method’s efficacy and advantages in comparison with known methods providing formal (algebraic) solutions to systems of linear interval equations. It is shown that a new method provides results which are close to the so-called maximal inner solutions (the corresponding method was developed by Kupriyanova, Zyuzin and Markov) and the algebraic solutions obtained by the subdifferential Newton method proposed by Shary. It is important that the proposed method makes it possible to avoid inverted interval solutions. The influence of the system’s size and number of zero entries on the results is analyzed by applying the proposed method to the Leontief input–output model of economics.     

Robust controllability of linear interval systems with multiple control delays

The robust controllability problem for linear interval systems with multiple control delays is studied in this article. The rank preservation problem is converted to the nonsingularity analysis problem of the minors of the matrix in discussion. Based on some essential properties of matrix measures, a new sufficient algebraically elegant criterion for the robust controllability of linear interval systems with multiple control delays is established. A numerical example is given to illustrate the application of the proposed sufficient algebraic criterion.     

基于Delta算子的统一代数Lyapunov方程解的上下界

基于Delta算子描述,统一研究了连续代数Lyapunov方程(CALE)和离散代数Lyapunov方程(DALE)的定界估计问题.采用矩阵不等式方法,给出了统一的代数Lyapunov方程(UALE)解矩阵的上下界估计,在极限情形下可分别得到CALE和DALE的估计结果.计算实例表明了本文方法的有效性.     

On the methods for calculation of grammians and their use in analysis of linear dynamic systems

Consideration was given to the methods for solution of the differential and algebraic Lyapunov and Sylvester equations in the time and frequency domains. Their solutions are represented as various finite and infinite grammians. The proposed approach to calculation of the grammians lies in expanding them as the sums of the matrix bilinear or quadratic forms generated with the use of the Faddeev matrices and representing each the solution of the linear matrix algebraic equation corresponding to an individual matrix eigenvalue. A lemma was proved representing explicitly the finite and infinite grammians as the matrix exponents depending on the combined spectrum of the original matrices. This result is generalized to the cases where the spectrum of one matrix contains an eigenvalue of the multiplicity two. Examples illustrating calculation of the finite and infinite grammians were discussed.     

Solution of Hallen's integral equation using multiwavelets

An effective method based upon Alpert multiwavelets is proposed for the solution of Hallen's integral equation. The properties of Alpert multiwavelets are first given. These wavelets are utilized to reduce the solution of Hallen's integral equation to the solution of sparse algebraic equations. In order to save memory requirement and computation time, a threshold procedure is applied to obtain algebraic equations. Through numerical examples, performance of the present method is investigated concerning the convergence and the sparseness of resulted matrix equation.     

A Bessel polynomial approach for solving general linear Fredholm integro-differential–difference equations

In this paper, to find an approximate solution of general linear Fredholm integro-differential–difference equations (FIDDEs) under the initial-boundary conditions in terms of the Bessel polynomials, a practical matrix method is presented. The idea behind the method is that it converts FIDDEs to a matrix equation which corresponds to a system of linear algebraic equations and is based on the matrix forms of the Bessel polynomials and their derivatives by means of collocation points. The solutions are obtained as the truncated Bessel series in terms of the Bessel polynomials J _n (x) of the first kind defined in the interval [0, ∞). The error analysis and the numerical examples are included to demonstrate the validity and applicability of the technique.     

Robust absolute stability of Lurie interval control systems

This paper considers robust absolute stability of Lurie control systems. Particular attention is given to the systems with parameters having uncertain, but bounded values. Such so‐called Lurie interval control systems have wide applications in practice. In this paper, a number of sufficient and necessary conditions are derived by using the theories of Hurwitz matrix, M matrix and partial variable absolute stability. Moreover, several algebraic sufficient and necessary conditions are provided for the robust absolute stability of Lurie interval control systems. These algebraic conditions are easy to be verified and convenient to be used in applications. Three mathematical examples and a practical engineering problem are presented to show the applicability of theoretical results. Numerical simulation results are also given to verify the analytical predictions. Copyright © 2007 John Wiley & Sons, Ltd.     

Interval Householder Method for Complex Linear Systems

We consider interval Householder method for outer estimation of solution sets for interval linear algebraic systems with complex interval parameters. A numerical example is presented showing that interval Householder method may work better than interval Gaussian algorithm. This paper was presented at IMCP'04 workshop (see Reliable Computing 11 (5) (2005)).