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Article
Publication date: 11 May 2010

Tadeusz Sobczyk

The purpose of this paper is to reduce issues arising when computing steady‐state solutions for AC machine models using the harmonic balance method.

Abstract

Purpose

The purpose of this paper is to reduce issues arising when computing steady‐state solutions for AC machine models using the harmonic balance method.

Design/methodology/approach

Generally, currents at steady‐states of AC machines are described by periodic or quasi‐periodic time functions, which Fourier spectra are determined by an infinite set of algebraic equations obtained from a harmonic balance method. To solve them, after reducing to finite dimensions, an iterative algorithm is developed in this paper. It bases on the LU decomposition of an infinite matrix representing the inductance matrix of an AC machine. Since that decomposition is done separately, due to a band type form of this matrix, the equation set determining the Fourier spectra of currents is solved recurrently.

Findings

An algorithm for the LU decomposition of an infinite matrix representing the inductance matrix of an AC machine and an iterative algorithm for determining AC machine steady‐state currents in a recursive manner.

Research limitations/implications

The approach is limited to solving of so‐called “circuital” models of AC voltage supplied machines. The approach breaks the large dimension barrier when solving steady‐state equations for AC machines.

Practical implications

Reducing computer requirements in terms of computer memory, workload and computing time to determine a steady‐state solution for AC machines.

Originality/value

A separation of the LU decomposition of an infinite matrix representing the inductance matrix in AC machine steady‐state model from the solution method.

Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, vol. 29 no. 3
Type: Research Article
ISSN: 0332-1649

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Article
Publication date: 15 November 2011

Mehdi Dehghan and Masoud Hajarian

The purpose of this paper is to find the efficient iterative methods for solving the general matrix equation A1X+ XA2+A3XH+XHA4=B (including Lyapunov and Sylvester matrix

Abstract

Purpose

The purpose of this paper is to find the efficient iterative methods for solving the general matrix equation A1X+ XA2+A3XH+XHA4=B (including Lyapunov and Sylvester matrix equations as special cases) with the unknown complex (reflexive) matrix X.

Design/methodology/approach

By applying the principle of hierarchical identification and the Hermitian/skew‐Hermitian splitting of the coefficient matrix quadruplet A1; A2; A3; A4 the authors propose a shift‐splitting hierarchical identification (SSHI) method to solve the general linear matrix equation A1X+XA2+A3XH+XHA4=B. Also, the proposed algorithm is extended for finding the reflexive solution to this matrix equation.

Findings

The authors propose two iterative methods for finding the solution and reflexive solution of the general linear matrix equation, respectively. The proposed algorithms have a simple, neat and elegant structure. The convergence analysis of the methods is also discussed. Some numerical results are given which illustrate the power and effectiveness of the proposed algorithms.

Originality/value

So far, several methods have been presented and used for solving the matrix equations by using vec operator and Kronecker product, generalized inverse, generalized singular value decomposition (GSVD) and canonical correlation decomposition (CCD) of matrices. In several cases, it is difficult to find the solutions by using matrix decomposition and generalized inverse. Also vec operator and Kronecker product enlarge the size of the matrix greatly therefore the computations are very expensive in the process of finding solutions. To overcome these complications and drawbacks, by using the hierarchical identification principle and the Hermitian=skew‐Hermitian splitting of the coefficient matrix quadruplet (A1; A2; A3; A4), the authors propose SSHI methods for solving the general matrix equation.

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Article
Publication date: 1 December 2005

Dragan B. Kandić and Branimir D. Reljin

To investigate the general necessary condition for synthesis of square, real rational matrices of complex frequency as admittance matrices of active multiports with…

Abstract

Purpose

To investigate the general necessary condition for synthesis of square, real rational matrices of complex frequency as admittance matrices of active multiports with resistors, inductors, capacitors and possibly multiport transformers and to prove that this condition is also sufficient for synthesis of stable, square, real rational matrices of complex frequency as admittance matrices of balanced active multiports having only resistors, capacitors and voltage‐amplifiers with sufficiently large amplifications. The main aim of the paper is to provide a new and general method for stable admittance matrices synthesis and to develop strict realization algorithm by active balanced transformerless multiport networks.

Design/methodology/approach

The objectives of the paper are achieved by using factorization of regular polynomial matrices in complex frequency with certain degree as products of other regular polynomial matrices with specified degrees. A set of sufficient conditions for such a factorization is presented and derived a pertinent algorithm as the starting point for investigation and solving network synthesis problem and generation of class of equivalent realizations.

Findings

Theorem 1 states that sufficient condition for factorization of Pth order, generally regular polynomial matrix P(s) in complex frequency s with degree L, whose determinant has K distinct zeros, in form P(s)=P1(sP2(s), where 1≤p2=P20L−1 is degree of polynomial matrix P2(s), reads: K>(P−1)·L+p2−1. The coefficient‐matrices of s, s2,… in P1(s) and P2(s) are real or complex depending on whether distinct zeros of det P(s) are real or complex, respectively. Theorem 2 states that: (a) for realization of Pth order matrix of real rational functions in complex frequency s (i.e. RRF matrix) as admittance matrix of active balanced RLC P‐port network with multiport transformers, or without them, P generalized controlled‐sources and P controlling‐ports are necessary, in general; and (b) P balanced voltage‐controlled voltage‐sources (VCVSs) with real and by module greater than unity controlling coefficients (“voltage amplifications”) are sufficient for realization of stable admittance RRF matrix by active, balanced, transformerless, RC P‐port network.

Originality/value

This is a research paper with the following two main contributions (original results). First, a theorem on sufficient conditions for factorization of regular polynomial matrices in complex frequency; and second, a theorem relating to sufficient conditions for synthesis of matrices of real rational functions in complex frequency by active, balanced, transformerless networks. The results may be interesting for network theorists and researchers in the field of electric circuits and systems.

Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, vol. 24 no. 4
Type: Research Article
ISSN: 0332-1649

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Article
Publication date: 20 April 2015

Yixiong Wei, Qifu Wang, Yunbao Huang, Yingjun Wang and Zhaohui Xia

The purpose of this paper is to present a novel strategy used for acceleration of free-vibration analysis, in which the hierarchical matrices structure and Compute Unified…

Abstract

Purpose

The purpose of this paper is to present a novel strategy used for acceleration of free-vibration analysis, in which the hierarchical matrices structure and Compute Unified Device Architecture (CUDA) platform is applied to improve the performance of the traditional dual reciprocity boundary element method (DRBEM).

Design/methodology/approach

The DRBEM is applied in forming integral equation to reduce complexity. In the procedure of optimization computation, ℋ-Matrices are introduced by applying adaptive cross-approximation method. At the same time, this paper proposes a high-efficiency parallel algorithm using CUDA and the counterpart of the serial effective algorithm in ℋ-Matrices for inverse arithmetic operation.

Findings

The analysis for free-vibration could achieve impressive time and space efficiency by introducing hierarchical matrices technique. Although the serial algorithm based on ℋ-Matrices could obtain fair performance for complex inversion operation, the CUDA parallel algorithm would further double the efficiency. Without much loss in accuracy according to the examination of the numerical example, the relative error appeared in approximation process can be fixed by increasing degrees of freedoms or introducing certain amount of internal points.

Originality/value

The paper proposes a novel effective strategy to improve computational efficiency and decrease memory consumption of free-vibration problems. ℋ-Matrices structure and parallel operation based on CUDA are introduced in traditional DRBEM.

Details

Engineering Computations, vol. 32 no. 2
Type: Research Article
ISSN: 0264-4401

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Article
Publication date: 13 July 2012

Mehdi Dehghan and Masoud Hajarian

The purpose of this paper is to find two iterative methods to solve the general coupled matrix equations over the generalized centro‐symmetric and central antisymmetric matrices.

Abstract

Purpose

The purpose of this paper is to find two iterative methods to solve the general coupled matrix equations over the generalized centro‐symmetric and central antisymmetric matrices.

Design/methodology/approach

By extending the idea of conjugate gradient (CG) method, the authors present two iterative methods to solve the general coupled matrix equations over the generalized centro‐symmetric and central antisymmetric matrices.

Findings

When the general coupled matrix equations are consistent over the generalized centro‐symmetric and central anti‐symmetric matrices, the generalized centro‐symmetric and central anti‐symmetric solutions can be obtained within nite iterative steps. Also the least Frobenius norm generalized centrosymmetric and central anti‐symmetric solutions can be derived by choosing a special kind of initial matrices. Furthermore, the optimal approximation generalized centrosymmetric and central anti‐symmetric solutions to given generalized centro‐symmetric and central anti‐symmetric matrices can be obtained by finding the least Frobenius norm generalized centro‐symmetric and central anti‐symmetric solutions of new matrix equations. The authors employ some numerical examples to support the theoretical results of this paper. Finally, the application of the presented methods is highlighted for solving the projected generalized continuous‐time algebraic Lyapunov equations (GCALE).

Originality/value

By the algorithms, the solvability of the general coupled matrix equations over generalized centro‐symmetric and central anti‐symmetric matrices can be determined automatically. The convergence results of the iterative algorithms are also proposed. Several examples and an application are given to show the efficiency of the presented methods.

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Article
Publication date: 1 January 2013

A. Kaveh, H. Rahami, S.R. Mirghaderi and M. Ardalan Asl

In the analysis of some near‐regular structures one can solve the regular part independently and then superimpose the effect of the additional part. For such models, the…

Abstract

Purpose

In the analysis of some near‐regular structures one can solve the regular part independently and then superimpose the effect of the additional part. For such models, the matrices corresponding to regular part have canonical forms and their eigensolution or inversion can easily be performed. The effect of member changing the regular to a near‐regular structure can then be added. The purpose of this paper is to analyze near‐regular structures using the force method.

Design/methodology/approach

The paper uses the force method, and instead of selecting a statically determinate basic structure (standard method), the paper employs the regular part of the structure as the basic structure.

Findings

A new algebraic method is introduced for the force method of analysis for efficient analysis of large near‐regular structures.

Originality/value

In this paper, the force method is used, however, instead of selecting a statically determinate basic structure, the regular part of the structure is employed as the basic structure. Those additional elements are considered as redundant elements. This method is applied to truss and frame structures. In the present approach it is possible to have missing elements instead of additional elements.

Details

Engineering Computations, vol. 30 no. 1
Type: Research Article
ISSN: 0264-4401

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Article
Publication date: 7 August 2009

Pavel Pakshin and Sergey Soloviev

The purpose of this paper is to provide a parametric description (parametrization) of all static output feedback stabilizing controllers for linear stochastic…

Abstract

Purpose

The purpose of this paper is to provide a parametric description (parametrization) of all static output feedback stabilizing controllers for linear stochastic discrete‐time systems with Markovian switching, applications of this result to simultaneous and robust stabilization problems and obtaining of algorithms for computing stabilizing gains.

Design/methodology/approach

The proposed approach presents parameterization in terms of coupled linear matrix equations and quadratic matrix inequalities which depend on parameter matrices similar to weight matrices in linear quadratic regulator (LQR) theory. To avoid implementation problems, a convex approximation technique is used and linear matrix inequalities (LMI)‐based algorithms are obtained for computing of stabilizing gain.

Findings

The algorithms obtained in this paper are non‐iterative and used computationally efficient LMI technique. Moreover, it is possible to use well‐known LQR methodology in the process of controller design.

Originality/value

As a result of this paper, a new unified approach to design of static output feedback stabilizing control is developed. This approach leads to efficient stabilizing gain computation algorithms for both stochastic systems with Markovian switching and deterministic systems with polytopic uncertainty.

Details

Kybernetes, vol. 38 no. 7/8
Type: Research Article
ISSN: 0368-492X

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Article
Publication date: 7 November 2016

Yunfei Liu, Jun Lv and Xiaowei Gao

The purpose of this paper is to introduce a new method called simultaneous elimination and back-substitution method (SEBSM) to solve a system of linear equations as a new…

Abstract

Purpose

The purpose of this paper is to introduce a new method called simultaneous elimination and back-substitution method (SEBSM) to solve a system of linear equations as a new finite element method (FEM) solver.

Design/methodology/approach

In this paper, a new technique assembling the global stiffness matrix will be proposed and meanwhile the direct method SEBSM will be applied to solve the equations formed in FEM.

Findings

The SEBSM solver for FEM with the present assembling technique has distinct advantages in both computational time and memory space occupation over the conventional methods, such as the Gauss elimination and LU decomposition methods.

Originality/value

The developed solver requires less memory space no matter the coefficient matrix is a typical sparse matrix or not, and it is applicable to both symmetric and unsymmetrical linear systems of equations. The processes of assembling matrix and dealing with constraints are straightforward, so it is convenient for coding. Compared to the previous solvers, the proposed solver has favorable universality and good performances.

Details

Engineering Computations, vol. 33 no. 8
Type: Research Article
ISSN: 0264-4401

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Article
Publication date: 28 January 2014

Qiao-Xing Li, Sifeng Liu and Nai-Ang Wang

This paper attempts to establish the general formula for computing the inverse of grey matrix, and the results are applied to solve grey linear programming. The inverse of…

Abstract

Purpose

This paper attempts to establish the general formula for computing the inverse of grey matrix, and the results are applied to solve grey linear programming. The inverse of a grey matrix and grey linear programming plays an important role in establishing a grey computational system.

Design/methodology/approach

Starting from the fact that missing information often appears in complex systems, and therefore that true values of elements are uncertain when the authors construct a matrix, as well as calculate its inverse. However, the authors can get their ranges, which are called the number-covered sets, by using grey computational rules. How to get the matrix-covered set of inverse grey matrix became a typical approach. In this paper, grey linear programming was explained in detail, for the point of grey meaning and the methodology to calculate the inverse grey matrix can successfully solve grey linear programming.

Findings

The results show that the ranges of grey value of inverse grey matrix and grey linear programming can be obtained by using the computational rules.

Practical implications

Because the matrix and the linear programming have been widely used in many fields such as system controlling, economic analysis and social management, and the missing information is a general phenomenon for complex systems, grey matrix and grey linear programming may have great potential application in real world. The methodology realizes the feasibility to control the complex system under uncertain situations.

Originality/value

The paper successfully obtained the ranges of uncertain inverse matrix and linear programming by using grey system theory, when the elements of matrix and the coefficients of linear programming are intervals and the results enrich the contents of grey mathematics.

Details

Grey Systems: Theory and Application, vol. 4 no. 1
Type: Research Article
ISSN: 2043-9377

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Article
Publication date: 9 October 2009

A. Kaveh and B. Alinejad

The purpose of this paper is to introduce a general equation for eigensolution. Eigenvalues and eigenvectors of graphs have many applications in combinatorial optimization…

Abstract

Purpose

The purpose of this paper is to introduce a general equation for eigensolution. Eigenvalues and eigenvectors of graphs have many applications in combinatorial optimization and structural mechanics. Some important applications of graph products consist of nodal ordering and graph partitioning for structuring the structural matrices and finite element subdomaining, respectively.

Design/methodology/approach

In the existing methods for the eigensolution of Laplacian matrices, members have been added to the model of a graph product such that for its Laplacian matrix an algebraic relation between blocks become possible. These methods are categorized as topological approaches. Here, using concepts of linear algebra a general algebraic method is developed.

Findings

A new algebraic method is introduced for calculating the eigenvalues of Laplacian matrices in graph products.

Originality/value

The present method provides a simple tool for calculating the eigenvalues of the Laplacian matrices without using the configurational model and merely by using the Laplacian matrices. The developed formula for calculating the eigenvalues contains approximate terms which can be managed by the analyst.

Details

Engineering Computations, vol. 26 no. 7
Type: Research Article
ISSN: 0264-4401

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