A procedure is presented for computing the derivatives of repeated eigenvalues and the corresponding eigenvectors of damped systems. The derivatives are calculated in terms of the eigenvalues and eigenvectors of the s...A procedure is presented for computing the derivatives of repeated eigenvalues and the corresponding eigenvectors of damped systems. The derivatives are calculated in terms of the eigenvalues and eigenvectors of the second-order system, and the use of rather undesirable state space representation is avoided. Hence the cost of computation is greatly reduced. The efficiency of the proposed procedure is illustrated by considering a 5-DOF non-proportionally damped system.展开更多
The sensitivity calculating formulas in structural dynamics was developed by utilizing the mathematical theorem and new definitions of sensitivities. So the singularity problem of sensitivity with repeated eigenvalues...The sensitivity calculating formulas in structural dynamics was developed by utilizing the mathematical theorem and new definitions of sensitivities. So the singularity problem of sensitivity with repeated eigenvalues is solved completely. To improve the computational efficiency, the reduction system is obtained based on Lanczos vectors. After incorporating the mathematical theory with the Lanczos algorithm, the approximate sensitivity solution can be obtained. A numerical example is presented to illustrate the performance of the method.展开更多
This paper proposes a new method based on principal component analysis to find the direction of an object in any pose.Experiments show that this method is fast,can be applied to objects with any pixel distribution and...This paper proposes a new method based on principal component analysis to find the direction of an object in any pose.Experiments show that this method is fast,can be applied to objects with any pixel distribution and keep the original properties of objects invariant.It is a new application of PCA in image analysis.展开更多
With the rapid integration of renewable energy,wide-band oscillations caused by interactions between power electronic equipment and grids have emerged as one of the most critical stability issues.Existing methods are ...With the rapid integration of renewable energy,wide-band oscillations caused by interactions between power electronic equipment and grids have emerged as one of the most critical stability issues.Existing methods are usually studied for local power systems with around one hundred nodes.However,for a large-scale power system with tens of thousands of nodes,the dimension of transfer function matrix or the order of characteristic equation is much higher.In this case,the existing methods such as eigenvalue analysis method and impedance-based method have difficulty in computation and are thus hard to utilize in practice.To fill this gap,this paper proposes a novel method named the smallest eigenvalues based logarithmic derivative(SELD)method.It obtains the dominant oscillation modes by the logarithmic derivative of the k-smallest eigenvalue curves of the sparse extended nodal admittance matrix(NAM).An oscillatory stability analysis tool is further developed based on this method.The effectiveness of the method and the tool is validated through a local power system as well as a large-scale power system.展开更多
Current common challenges such as high-dimensional data processing and steady-state analysis of complex systems have become increasingly prominent.Eigenvalues and eigenvectors,leveraging their unique mathematical prop...Current common challenges such as high-dimensional data processing and steady-state analysis of complex systems have become increasingly prominent.Eigenvalues and eigenvectors,leveraging their unique mathematical properties,play an irreplaceable role in fields such as data mining and system modeling,serving as a crucial bridge connecting theoretical mathematics with practical applications.Through literature review,this study investigates the application of matrix eigenvalues and eigenvectors in Principal Component Analysis(PCA)and Markov chain steady-state analysis.The results demonstrate that matrix eigenvalues and eigenvectors exhibit significant universality and effectiveness in cross-domain applications.Validated in scenarios including PCA and Markov chain steady-state analysis,they help address key issues including high-dimensional data dimensionality reduction,system steady-state prediction,and information prioritization,thereby providing mathematical support for technological optimization.Simultaneously,they can reveal intrinsic system patterns,reflecting a deep analytical capability for system structures.Future research may focus on optimizing algorithms for solving sparse matrix eigenvalues and exploring integration with deep learning and graph neural networks to expand their application boundaries in large-scale complex systems.展开更多
The principal component analysis(PCA)is one of the most celebrated methods in analysing multivariate data.An effort of extending PCA is projection pursuit(PP),a more general class of dimension-reduction techniques.How...The principal component analysis(PCA)is one of the most celebrated methods in analysing multivariate data.An effort of extending PCA is projection pursuit(PP),a more general class of dimension-reduction techniques.However,the application of this extended procedure is often hampered by its complexity in computation and by lack of some appropriate theory.In this paper,by use of the empirical processes we established a large sample theory for the robust PP estimators of the principal components and dispersion matrix.展开更多
On one hand, when the bridge stays in a windy environment, the aerodynamic power would reduce it to act as a non-classic system. Consequently, the transposition of the system’s right eigenmatrix will not equal its le...On one hand, when the bridge stays in a windy environment, the aerodynamic power would reduce it to act as a non-classic system. Consequently, the transposition of the system’s right eigenmatrix will not equal its left eigenmatrix any longer. On the other hand, eigenmatrix plays an important role in model identification, which is the basis of the identification of aerodynamic derivatives. In this study, we follow Scanlan’s simple bridge model and utilize the information provided by the left and right eigenmatrixes to structure a self-contained eigenvector algorithm in the frequency domain. For the purpose of fitting more accurate transfer function, the study adopts the combined sine-wave stimulation method in the numerical simulation. And from the simulation results, we can conclude that the derivatives identified by the self-contained eigenvector algorithm are more dependable.展开更多
基金Project supported by the Mathematical Tianyuan Foundation of China (No. 10626019)
文摘A procedure is presented for computing the derivatives of repeated eigenvalues and the corresponding eigenvectors of damped systems. The derivatives are calculated in terms of the eigenvalues and eigenvectors of the second-order system, and the use of rather undesirable state space representation is avoided. Hence the cost of computation is greatly reduced. The efficiency of the proposed procedure is illustrated by considering a 5-DOF non-proportionally damped system.
文摘The sensitivity calculating formulas in structural dynamics was developed by utilizing the mathematical theorem and new definitions of sensitivities. So the singularity problem of sensitivity with repeated eigenvalues is solved completely. To improve the computational efficiency, the reduction system is obtained based on Lanczos vectors. After incorporating the mathematical theory with the Lanczos algorithm, the approximate sensitivity solution can be obtained. A numerical example is presented to illustrate the performance of the method.
文摘This paper proposes a new method based on principal component analysis to find the direction of an object in any pose.Experiments show that this method is fast,can be applied to objects with any pixel distribution and keep the original properties of objects invariant.It is a new application of PCA in image analysis.
基金supported by the National Natural Science Foundation of China(No.52321004)the Delta Power Electronics Science and Education Development Program of Delta Group.
文摘With the rapid integration of renewable energy,wide-band oscillations caused by interactions between power electronic equipment and grids have emerged as one of the most critical stability issues.Existing methods are usually studied for local power systems with around one hundred nodes.However,for a large-scale power system with tens of thousands of nodes,the dimension of transfer function matrix or the order of characteristic equation is much higher.In this case,the existing methods such as eigenvalue analysis method and impedance-based method have difficulty in computation and are thus hard to utilize in practice.To fill this gap,this paper proposes a novel method named the smallest eigenvalues based logarithmic derivative(SELD)method.It obtains the dominant oscillation modes by the logarithmic derivative of the k-smallest eigenvalue curves of the sparse extended nodal admittance matrix(NAM).An oscillatory stability analysis tool is further developed based on this method.The effectiveness of the method and the tool is validated through a local power system as well as a large-scale power system.
文摘Current common challenges such as high-dimensional data processing and steady-state analysis of complex systems have become increasingly prominent.Eigenvalues and eigenvectors,leveraging their unique mathematical properties,play an irreplaceable role in fields such as data mining and system modeling,serving as a crucial bridge connecting theoretical mathematics with practical applications.Through literature review,this study investigates the application of matrix eigenvalues and eigenvectors in Principal Component Analysis(PCA)and Markov chain steady-state analysis.The results demonstrate that matrix eigenvalues and eigenvectors exhibit significant universality and effectiveness in cross-domain applications.Validated in scenarios including PCA and Markov chain steady-state analysis,they help address key issues including high-dimensional data dimensionality reduction,system steady-state prediction,and information prioritization,thereby providing mathematical support for technological optimization.Simultaneously,they can reveal intrinsic system patterns,reflecting a deep analytical capability for system structures.Future research may focus on optimizing algorithms for solving sparse matrix eigenvalues and exploring integration with deep learning and graph neural networks to expand their application boundaries in large-scale complex systems.
基金The researcb was partially supported by the National Natural Science Foundation of China under Grant No.19631040.
文摘The principal component analysis(PCA)is one of the most celebrated methods in analysing multivariate data.An effort of extending PCA is projection pursuit(PP),a more general class of dimension-reduction techniques.However,the application of this extended procedure is often hampered by its complexity in computation and by lack of some appropriate theory.In this paper,by use of the empirical processes we established a large sample theory for the robust PP estimators of the principal components and dispersion matrix.
基金supported by the State Key Program of National Natural Science Foundation of China (Grant No. 11032009)the National Natural Science Foundation of China (Grant No. 10772048)
文摘On one hand, when the bridge stays in a windy environment, the aerodynamic power would reduce it to act as a non-classic system. Consequently, the transposition of the system’s right eigenmatrix will not equal its left eigenmatrix any longer. On the other hand, eigenmatrix plays an important role in model identification, which is the basis of the identification of aerodynamic derivatives. In this study, we follow Scanlan’s simple bridge model and utilize the information provided by the left and right eigenmatrixes to structure a self-contained eigenvector algorithm in the frequency domain. For the purpose of fitting more accurate transfer function, the study adopts the combined sine-wave stimulation method in the numerical simulation. And from the simulation results, we can conclude that the derivatives identified by the self-contained eigenvector algorithm are more dependable.
