A matrix eigenvalue method is applied to analyse the thermodynamic stability of two-component interacting fermions. The non-relativistic and ultra-relativistic d = 1, 2, 3 dimensions have been discussed in detail, res...A matrix eigenvalue method is applied to analyse the thermodynamic stability of two-component interacting fermions. The non-relativistic and ultra-relativistic d = 1, 2, 3 dimensions have been discussed in detail, respectively. The corresponding stability region has been given according to the two-body interaction strength and the particle number density ratio.展开更多
The concept of edge polynomials with variable length is introduced. Stability of such polynomials is analyzed. Under the condition that one extreme of the edge is stable, the stability radius of edge polynomials with ...The concept of edge polynomials with variable length is introduced. Stability of such polynomials is analyzed. Under the condition that one extreme of the edge is stable, the stability radius of edge polynomials with variable length is characterized in terms of the real spectral radius of the matrix H -1 ( f 0) H (g) , where both H (f 0) and H (g) are Hurwitz like matrices. Based on this result, stability radius of control systems with interval type plants and first order controllers are determined.展开更多
With the rapid development of power-electronicsenabled power systems,the new converter-based generators are deteriorating the small-signal stability of the power system.Although the numerical differentiation method ha...With the rapid development of power-electronicsenabled power systems,the new converter-based generators are deteriorating the small-signal stability of the power system.Although the numerical differentiation method has been widely used for approximately calculating the eigenvalue sensitivities,its accuracy has not been carefully investigated.Besides,the element-based formulation for computing closed-form eigenvalue sensitivities has not been used in any commercial software due to the average efficiency,complicated formulation,and errorprone characteristics.Based on the matrix calculus,this paper proposes an easily manipulated formulation of the closed-form eigenvalue sensitivities with respect to the power generation.The distinguishing feature of the formulation is that all the formulas consist of vector and matrix operations,which can be performed by developed numerical algorithms to take full advantages of architectural features of the modern computer.The tests on WSCC 3-machine 9-bus system,New England 10-machine 39-bus system,and IEEE 54-machine 118-bus system show that the accuracy of the proposed formulation is superior to the numerical differentiation method and the efficiency is also greatly improved compared to the element-based closed-form formulation.The proposed formulation will be helpful to perform a more accurate and faster stability analysis of a power grid with converter-based devices.展开更多
An efficient and stable structure preserving algorithm, which is a variant of the QR like (SR) algorithm due to Bunse-Gerstner and Mehrmann, is presented for computing the eigenvalues and stable invariant subspaces of...An efficient and stable structure preserving algorithm, which is a variant of the QR like (SR) algorithm due to Bunse-Gerstner and Mehrmann, is presented for computing the eigenvalues and stable invariant subspaces of a Hamiltonian matrix. In the algorithm two strategies are employed, one of which is called dis-unstabilization technique and the other is preprocessing technique. Together with them, a so-called ratio-reduction equation and a backtrack technique are introduced to avoid the instability and breakdown in the original algorithm. It is shown that the new algorithm can overcome the instability and breakdown at low cost. Numerical results have demonstrated that the algorithm is stable and can compute the eigenvalues to very high accuracy.展开更多
A code developed recently by the authors, for counting and computing the eigenvalues of a complex tridiagonal matrix, as well as the roots of a complex polynomial, which lie in a given region of the complex plane, is ...A code developed recently by the authors, for counting and computing the eigenvalues of a complex tridiagonal matrix, as well as the roots of a complex polynomial, which lie in a given region of the complex plane, is modified to run in parallel on multi-core machines. A basic characteristic of this code (eventually pointing to its parallelization) is that it can proceed with: 1) partitioning the given region into an appropriate number of subregions;2) counting eigenvalues in each subregion;and 3) computing (already counted) eigenvalues in each subregion. Consequently, theoretically speaking, the whole code in itself parallelizes ideally. We carry out several numerical experiments with random complex tridiagonal matrices, and random complex polynomials as well, in order to study the behaviour of the parallel code, especially the degree of declination from theoretical expectations.展开更多
Some properties of characteristic polynomials, eigenvalues, and eigenvectors of the Wilkinson matrices W2n＋1 and W2n＋1 are presented. It is proved that the eigenvalues of W2n＋1 just are the eigenvalues of its leadi...Some properties of characteristic polynomials, eigenvalues, and eigenvectors of the Wilkinson matrices W2n＋1 and W2n＋1 are presented. It is proved that the eigenvalues of W2n＋1 just are the eigenvalues of its leading principal submatrix Vn and a bordered matrix of Vn. Recurrence formula are given for the characteristic polynomial of W2＋n＋1 . The eigenvectors of W2＋n＋1 are proved to be symmetric or skew symmetric. For W2n＋1 , it is found that its eigenvalues are zero and the square roots of the eigenvalues of a bordered matrix of Vn2. And the eigenvectors of W2n＋1 , which the corresponding eigenvahies are opposite in pairs, have close relationship.展开更多
We present in this paper a new method for solving polynomial eigenvalue problem. We give methods that decompose a skew-Hamiltonian matrix using Cholesky like-decomposition. We transform first the polynomial eigenvalue...We present in this paper a new method for solving polynomial eigenvalue problem. We give methods that decompose a skew-Hamiltonian matrix using Cholesky like-decomposition. We transform first the polynomial eigenvalue problem to an equivalent skew-Hamiltonian/Hamiltonian pencil. This process is known as linearization. Decomposition of the skew-Hamiltonian matrix is the fundamental step to convert a structured polynomial eigenvalue problem into a standard Hamiltonian eigenproblem. Numerical examples are given.展开更多
The signless Laplacian matrix of a graph is the sum of its diagonal matrix of vertex degrees and its adjacency matrix. Li and Feng gave some basic results on the largest eigenvalue and characteristic polynomial of adj...The signless Laplacian matrix of a graph is the sum of its diagonal matrix of vertex degrees and its adjacency matrix. Li and Feng gave some basic results on the largest eigenvalue and characteristic polynomial of adjacency matrix of a graph in 1979. In this paper, we translate these results into the signless Laplacian matrix of a graph and obtain the similar results.展开更多
DC microgrids(DCMGs)are made up of a network of sources and loads that are connected by a number of power electronic converters(PECs).The increase in the number of these PECs instigates major concerns in system stabil...DC microgrids(DCMGs)are made up of a network of sources and loads that are connected by a number of power electronic converters(PECs).The increase in the number of these PECs instigates major concerns in system stability.While interconnecting the microgrids to form a cluster,the system stability must be ensured.This paper proposes a novel stepby-step system matrix building(SMB)algorithm to update the system matrix of an existing DCMG cluster when a new microgrid is added to the cluster through a distribution line.The stability of the individual DCMGs and the DCMG cluster is analyzed using the eigenvalue method.Further,the particle swarm optimization(PSO)algorithm is used to retune the controller gains if the newly formed cluster is not stable.The simulation of the DCMG cluster is carried out in MATLAB/Simulink environment to test the proposed algorithm.The results are also validated using the OP4510 real-time simulator(RTS).展开更多
This paper concerns the numerical stability of an eikonal transformation based splitting method which is highly effective and efficient for the numerical solution of paraxial Helmholtz equation with a large wave numbe...This paper concerns the numerical stability of an eikonal transformation based splitting method which is highly effective and efficient for the numerical solution of paraxial Helmholtz equation with a large wave number.Rigorous matrix analysis is conducted in investigations and the oscillation-free computational procedure is proven to be stable in an asymptotic sense.Simulated examples are given to illustrate the conclusion.展开更多
The asymptotic stability of delay differential equation x′(t)=Ax(t)+Bx(t τ) is concerned with,where A,B∈C d×d are constant complex matrices, x(t τ)=(x 1(t-τ 1),x 2(t-τ 2),...,x d(t-τ d))T,τ k>...The asymptotic stability of delay differential equation x′(t)=Ax(t)+Bx(t τ) is concerned with,where A,B∈C d×d are constant complex matrices, x(t τ)=(x 1(t-τ 1),x 2(t-τ 2),...,x d(t-τ d))T,τ k>0(k=1,...,d) stand for constant delays. Two criteria through evaluation of a harmonic function on the boundary of a certain region are obtained. The similar results for neutral delay differential equation x′(t)=Lx(t)+Mx(t-τ)+Nx′(t-τ) are also obtained,where L,M and N∈C d×d are constant complex matrices and τ>0 stands for constant delay. Numerical examples are showed to check the results which are more general than those already reported.展开更多
To facilitate stability analysis of discrete-time bidirectional associative memory (BAM) neural networks, they were converted into novel neural network models, termed standard neural network models (SNNMs), which inte...To facilitate stability analysis of discrete-time bidirectional associative memory (BAM) neural networks, they were converted into novel neural network models, termed standard neural network models (SNNMs), which interconnect linear dynamic systems and bounded static nonlinear operators. By combining a number of different Lyapunov functionals with S-procedure, some useful criteria of global asymptotic stability and global exponential stability of the equilibrium points of SNNMs were derived. These stability conditions were formulated as linear matrix inequalities (LMIs). So global stability of the discrete-time BAM neural networks could be analyzed by using the stability results of the SNNMs. Compared to the existing stability analysis methods, the proposed approach is easy to implement, less conservative, and is applicable to other recurrent neural networks.展开更多
Two methods for calculating eigenvalues of linear systems are proposed. One is to use the root locus method to calculate the eigenvalues step by step from low order systems to high order systems. The other is to use t...Two methods for calculating eigenvalues of linear systems are proposed. One is to use the root locus method to calculate the eigenvalues step by step from low order systems to high order systems. The other is to use the property of Schwartz matrix to determine the factors of the characteristic equation by the searching method. A method for determining the number of eigenvalues in each half complex plane is also presented.展开更多
The problem of checking robust D-stability of multi-in and multi-out （MIMO） systems was studied. Three system models were introduced, i.e. multilinear polynomial matrix, polytopic polynomial matrix and feedback syst...The problem of checking robust D-stability of multi-in and multi-out （MIMO） systems was studied. Three system models were introduced, i.e. multilinear polynomial matrix, polytopic polynomial matrix and feedback system model. Furthermore, the convex property of each model with respect to the parametric uncertainties was estabilished respectively. Based on this, sufficient conditions for D-stability were expressed in terms of linear matrix inequalities （LMIs） involving only the convex vertices. Therefore, the robust D-stability was tested by solving an LMI optimal problem.展开更多
基金Project supported by the Scientific Starting Research Fund of Central China Normal University of Chinathe National Natural Science Foundation of China (Grant Nos 10675052 and 10875050)Ministry of Education of China (Grant No IRT0624)
文摘A matrix eigenvalue method is applied to analyse the thermodynamic stability of two-component interacting fermions. The non-relativistic and ultra-relativistic d = 1, 2, 3 dimensions have been discussed in detail, respectively. The corresponding stability region has been given according to the two-body interaction strength and the particle number density ratio.
文摘The concept of edge polynomials with variable length is introduced. Stability of such polynomials is analyzed. Under the condition that one extreme of the edge is stable, the stability radius of edge polynomials with variable length is characterized in terms of the real spectral radius of the matrix H -1 ( f 0) H (g) , where both H (f 0) and H (g) are Hurwitz like matrices. Based on this result, stability radius of control systems with interval type plants and first order controllers are determined.
基金supported by National Natural Science Foundation of China(No.51967001,No.51967002)Guangxi Provincial Natural Science Foundation of China(No.2018JJA160164)。
文摘With the rapid development of power-electronicsenabled power systems,the new converter-based generators are deteriorating the small-signal stability of the power system.Although the numerical differentiation method has been widely used for approximately calculating the eigenvalue sensitivities,its accuracy has not been carefully investigated.Besides,the element-based formulation for computing closed-form eigenvalue sensitivities has not been used in any commercial software due to the average efficiency,complicated formulation,and errorprone characteristics.Based on the matrix calculus,this paper proposes an easily manipulated formulation of the closed-form eigenvalue sensitivities with respect to the power generation.The distinguishing feature of the formulation is that all the formulas consist of vector and matrix operations,which can be performed by developed numerical algorithms to take full advantages of architectural features of the modern computer.The tests on WSCC 3-machine 9-bus system,New England 10-machine 39-bus system,and IEEE 54-machine 118-bus system show that the accuracy of the proposed formulation is superior to the numerical differentiation method and the efficiency is also greatly improved compared to the element-based closed-form formulation.The proposed formulation will be helpful to perform a more accurate and faster stability analysis of a power grid with converter-based devices.
文摘An efficient and stable structure preserving algorithm, which is a variant of the QR like (SR) algorithm due to Bunse-Gerstner and Mehrmann, is presented for computing the eigenvalues and stable invariant subspaces of a Hamiltonian matrix. In the algorithm two strategies are employed, one of which is called dis-unstabilization technique and the other is preprocessing technique. Together with them, a so-called ratio-reduction equation and a backtrack technique are introduced to avoid the instability and breakdown in the original algorithm. It is shown that the new algorithm can overcome the instability and breakdown at low cost. Numerical results have demonstrated that the algorithm is stable and can compute the eigenvalues to very high accuracy.
文摘A code developed recently by the authors, for counting and computing the eigenvalues of a complex tridiagonal matrix, as well as the roots of a complex polynomial, which lie in a given region of the complex plane, is modified to run in parallel on multi-core machines. A basic characteristic of this code (eventually pointing to its parallelization) is that it can proceed with: 1) partitioning the given region into an appropriate number of subregions;2) counting eigenvalues in each subregion;and 3) computing (already counted) eigenvalues in each subregion. Consequently, theoretically speaking, the whole code in itself parallelizes ideally. We carry out several numerical experiments with random complex tridiagonal matrices, and random complex polynomials as well, in order to study the behaviour of the parallel code, especially the degree of declination from theoretical expectations.
基金The Fundamental Research Funds for the Central Universities, China (No.10D10908)
文摘Some properties of characteristic polynomials, eigenvalues, and eigenvectors of the Wilkinson matrices W2n＋1 and W2n＋1 are presented. It is proved that the eigenvalues of W2n＋1 just are the eigenvalues of its leading principal submatrix Vn and a bordered matrix of Vn. Recurrence formula are given for the characteristic polynomial of W2＋n＋1 . The eigenvectors of W2＋n＋1 are proved to be symmetric or skew symmetric. For W2n＋1 , it is found that its eigenvalues are zero and the square roots of the eigenvalues of a bordered matrix of Vn2. And the eigenvectors of W2n＋1 , which the corresponding eigenvahies are opposite in pairs, have close relationship.
文摘We present in this paper a new method for solving polynomial eigenvalue problem. We give methods that decompose a skew-Hamiltonian matrix using Cholesky like-decomposition. We transform first the polynomial eigenvalue problem to an equivalent skew-Hamiltonian/Hamiltonian pencil. This process is known as linearization. Decomposition of the skew-Hamiltonian matrix is the fundamental step to convert a structured polynomial eigenvalue problem into a standard Hamiltonian eigenproblem. Numerical examples are given.
基金Foundation item： the National Natural Science Foundation of China （No. 10871204） Graduate Innovation Foundation of China University of Petroleum （No. S2008-26）.
文摘The signless Laplacian matrix of a graph is the sum of its diagonal matrix of vertex degrees and its adjacency matrix. Li and Feng gave some basic results on the largest eigenvalue and characteristic polynomial of adjacency matrix of a graph in 1979. In this paper, we translate these results into the signless Laplacian matrix of a graph and obtain the similar results.
文摘DC microgrids(DCMGs)are made up of a network of sources and loads that are connected by a number of power electronic converters(PECs).The increase in the number of these PECs instigates major concerns in system stability.While interconnecting the microgrids to form a cluster,the system stability must be ensured.This paper proposes a novel stepby-step system matrix building(SMB)algorithm to update the system matrix of an existing DCMG cluster when a new microgrid is added to the cluster through a distribution line.The stability of the individual DCMGs and the DCMG cluster is analyzed using the eigenvalue method.Further,the particle swarm optimization(PSO)algorithm is used to retune the controller gains if the newly formed cluster is not stable.The simulation of the DCMG cluster is carried out in MATLAB/Simulink environment to test the proposed algorithm.The results are also validated using the OP4510 real-time simulator(RTS).
文摘This paper concerns the numerical stability of an eikonal transformation based splitting method which is highly effective and efficient for the numerical solution of paraxial Helmholtz equation with a large wave number.Rigorous matrix analysis is conducted in investigations and the oscillation-free computational procedure is proven to be stable in an asymptotic sense.Simulated examples are given to illustrate the conclusion.
文摘The asymptotic stability of delay differential equation x′(t)=Ax(t)+Bx(t τ) is concerned with,where A,B∈C d×d are constant complex matrices, x(t τ)=(x 1(t-τ 1),x 2(t-τ 2),...,x d(t-τ d))T,τ k>0(k=1,...,d) stand for constant delays. Two criteria through evaluation of a harmonic function on the boundary of a certain region are obtained. The similar results for neutral delay differential equation x′(t)=Lx(t)+Mx(t-τ)+Nx′(t-τ) are also obtained,where L,M and N∈C d×d are constant complex matrices and τ>0 stands for constant delay. Numerical examples are showed to check the results which are more general than those already reported.
基金Project (No. 60074008) supported by the National Natural Science Foundation of China
文摘To facilitate stability analysis of discrete-time bidirectional associative memory (BAM) neural networks, they were converted into novel neural network models, termed standard neural network models (SNNMs), which interconnect linear dynamic systems and bounded static nonlinear operators. By combining a number of different Lyapunov functionals with S-procedure, some useful criteria of global asymptotic stability and global exponential stability of the equilibrium points of SNNMs were derived. These stability conditions were formulated as linear matrix inequalities (LMIs). So global stability of the discrete-time BAM neural networks could be analyzed by using the stability results of the SNNMs. Compared to the existing stability analysis methods, the proposed approach is easy to implement, less conservative, and is applicable to other recurrent neural networks.
基金Supported by the National Natural Science Foundation of China(2 0 0 0 CG0 1 0 3) the Fund of"The Developing Program for Outstanding Person"in NPUS & T Innovation Foundation for Young Teachers of Northwestern Polytechnical University.
文摘In this paper, the spectrum and characteristic polynomial for a special kind of symmetric block circulant matrices are given.
文摘Two methods for calculating eigenvalues of linear systems are proposed. One is to use the root locus method to calculate the eigenvalues step by step from low order systems to high order systems. The other is to use the property of Schwartz matrix to determine the factors of the characteristic equation by the searching method. A method for determining the number of eigenvalues in each half complex plane is also presented.
基金Project supported by Post-Doctoral Science Foundation of China
文摘The problem of checking robust D-stability of multi-in and multi-out （MIMO） systems was studied. Three system models were introduced, i.e. multilinear polynomial matrix, polytopic polynomial matrix and feedback system model. Furthermore, the convex property of each model with respect to the parametric uncertainties was estabilished respectively. Based on this, sufficient conditions for D-stability were expressed in terms of linear matrix inequalities （LMIs） involving only the convex vertices. Therefore, the robust D-stability was tested by solving an LMI optimal problem.
