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Jinwei Wang - One of the best experts on this subject based on the ideXlab platform.
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robust and minimum norm partial quadratic eigenvalue assignment in vibrating systems a new optimization approach
Mechanical Systems and Signal Processing, 2010Co-Authors: Biswa Nath Datta, Jinwei WangAbstract:Abstract The partial quadratic eigenvalue assignment problem (PQEVAP) concerns reassigning a few undesired eigenvalues of a quadratic Matrix pencil to suitably chosen locations and keeping the other large number of eigenvalues and Eigenvectors unchanged (no spill-over). The problem naturally arises in controlling dangerous vibrations in structures by means of active feedback control design. For practical viability, the design must be robust, which requires that the norms of the feedback matrices and the condition number of the closed-loop Eigenvectors are as small as possible. The problem of computing feedback matrices that satisfy the above two practical requirements is known as the Robust Partial Quadratic Eigenvalue Assignment Problem (RPQEVAP). In this paper, we formulate the RPQEVAP as an unconstrained minimization problem with the cost function involving the condition number of the closed-loop Eigenvector Matrix and two feedback norms. Since only a small number of eigenvalues of the open-loop quadratic pencil are computable using the state-of-the-art Matrix computational techniques and/or measurable in a vibration laboratory, it is imperative that the problem is solved using these small number of eigenvalues and the corresponding Eigenvectors. To this end, a class of the feedback matrices are obtained in parametric form, parameterized by a single parametric Matrix, and the cost function and the required gradient formulas for the optimization problem are developed in terms of the small number of eigenvalues that are reassigned and their corresponding Eigenvectors. The problem is solved directly in quadratic setting without transforming it to a standard first-order control problem and most importantly, the significant “no spill-over property” of the closed-loop eigenvalues and Eigenvectors is established by means of a mathematical result. These features make the proposed method practically applicable even for very large structures. Results on numerical experiments show that the proposed method considerably reduces both feedback norms and the sensitivity of the closed-loop eigenvalues. A study on robustness of the system responses of the method under small perturbations show that the responses of the perturbed closed-loop system are compatible with perturbations.
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robust and minimum norm partial quadratic eigenvalue assignment in vibrating systems a new optimization approach
Mechanical Systems and Signal Processing, 2010Co-Authors: Zhengjian Bai, Biswa Nath Datta, Jinwei WangAbstract:Abstract The partial quadratic eigenvalue assignment problem (PQEVAP) concerns reassigning a few undesired eigenvalues of a quadratic Matrix pencil to suitably chosen locations and keeping the other large number of eigenvalues and Eigenvectors unchanged (no spill-over). The problem naturally arises in controlling dangerous vibrations in structures by means of active feedback control design. For practical viability, the design must be robust, which requires that the norms of the feedback matrices and the condition number of the closed-loop Eigenvectors are as small as possible. The problem of computing feedback matrices that satisfy the above two practical requirements is known as the Robust Partial Quadratic Eigenvalue Assignment Problem (RPQEVAP). In this paper, we formulate the RPQEVAP as an unconstrained minimization problem with the cost function involving the condition number of the closed-loop Eigenvector Matrix and two feedback norms. Since only a small number of eigenvalues of the open-loop quadratic pencil are computable using the state-of-the-art Matrix computational techniques and/or measurable in a vibration laboratory, it is imperative that the problem is solved using these small number of eigenvalues and the corresponding Eigenvectors. To this end, a class of the feedback matrices are obtained in parametric form, parameterized by a single parametric Matrix, and the cost function and the required gradient formulas for the optimization problem are developed in terms of the small number of eigenvalues that are reassigned and their corresponding Eigenvectors. The problem is solved directly in quadratic setting without transforming it to a standard first-order control problem and most importantly, the significant “no spill-over property” of the closed-loop eigenvalues and Eigenvectors is established by means of a mathematical result. These features make the proposed method practically applicable even for very large structures. Results on numerical experiments show that the proposed method considerably reduces both feedback norms and the sensitivity of the closed-loop eigenvalues. A study on robustness of the system responses of the method under small perturbations show that the responses of the perturbed closed-loop system are compatible with perturbations.
Biswa Nath Datta - One of the best experts on this subject based on the ideXlab platform.
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robust and minimum norm partial quadratic eigenvalue assignment in vibrating systems a new optimization approach
Mechanical Systems and Signal Processing, 2010Co-Authors: Biswa Nath Datta, Jinwei WangAbstract:Abstract The partial quadratic eigenvalue assignment problem (PQEVAP) concerns reassigning a few undesired eigenvalues of a quadratic Matrix pencil to suitably chosen locations and keeping the other large number of eigenvalues and Eigenvectors unchanged (no spill-over). The problem naturally arises in controlling dangerous vibrations in structures by means of active feedback control design. For practical viability, the design must be robust, which requires that the norms of the feedback matrices and the condition number of the closed-loop Eigenvectors are as small as possible. The problem of computing feedback matrices that satisfy the above two practical requirements is known as the Robust Partial Quadratic Eigenvalue Assignment Problem (RPQEVAP). In this paper, we formulate the RPQEVAP as an unconstrained minimization problem with the cost function involving the condition number of the closed-loop Eigenvector Matrix and two feedback norms. Since only a small number of eigenvalues of the open-loop quadratic pencil are computable using the state-of-the-art Matrix computational techniques and/or measurable in a vibration laboratory, it is imperative that the problem is solved using these small number of eigenvalues and the corresponding Eigenvectors. To this end, a class of the feedback matrices are obtained in parametric form, parameterized by a single parametric Matrix, and the cost function and the required gradient formulas for the optimization problem are developed in terms of the small number of eigenvalues that are reassigned and their corresponding Eigenvectors. The problem is solved directly in quadratic setting without transforming it to a standard first-order control problem and most importantly, the significant “no spill-over property” of the closed-loop eigenvalues and Eigenvectors is established by means of a mathematical result. These features make the proposed method practically applicable even for very large structures. Results on numerical experiments show that the proposed method considerably reduces both feedback norms and the sensitivity of the closed-loop eigenvalues. A study on robustness of the system responses of the method under small perturbations show that the responses of the perturbed closed-loop system are compatible with perturbations.
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robust and minimum norm partial quadratic eigenvalue assignment in vibrating systems a new optimization approach
Mechanical Systems and Signal Processing, 2010Co-Authors: Zhengjian Bai, Biswa Nath Datta, Jinwei WangAbstract:Abstract The partial quadratic eigenvalue assignment problem (PQEVAP) concerns reassigning a few undesired eigenvalues of a quadratic Matrix pencil to suitably chosen locations and keeping the other large number of eigenvalues and Eigenvectors unchanged (no spill-over). The problem naturally arises in controlling dangerous vibrations in structures by means of active feedback control design. For practical viability, the design must be robust, which requires that the norms of the feedback matrices and the condition number of the closed-loop Eigenvectors are as small as possible. The problem of computing feedback matrices that satisfy the above two practical requirements is known as the Robust Partial Quadratic Eigenvalue Assignment Problem (RPQEVAP). In this paper, we formulate the RPQEVAP as an unconstrained minimization problem with the cost function involving the condition number of the closed-loop Eigenvector Matrix and two feedback norms. Since only a small number of eigenvalues of the open-loop quadratic pencil are computable using the state-of-the-art Matrix computational techniques and/or measurable in a vibration laboratory, it is imperative that the problem is solved using these small number of eigenvalues and the corresponding Eigenvectors. To this end, a class of the feedback matrices are obtained in parametric form, parameterized by a single parametric Matrix, and the cost function and the required gradient formulas for the optimization problem are developed in terms of the small number of eigenvalues that are reassigned and their corresponding Eigenvectors. The problem is solved directly in quadratic setting without transforming it to a standard first-order control problem and most importantly, the significant “no spill-over property” of the closed-loop eigenvalues and Eigenvectors is established by means of a mathematical result. These features make the proposed method practically applicable even for very large structures. Results on numerical experiments show that the proposed method considerably reduces both feedback norms and the sensitivity of the closed-loop eigenvalues. A study on robustness of the system responses of the method under small perturbations show that the responses of the perturbed closed-loop system are compatible with perturbations.
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an optimization approach for minimum norm and robust partial quadratic eigenvalue assignment problems for vibrating structures
Journal of Sound and Vibration, 2009Co-Authors: Sanjoy Brahma, Biswa Nath DattaAbstract:The partial quadratic eigenvalue assignment problem (PQEVAP) concerns the reassignment of a small number of undesirable eigenvalues of a quadratic Matrix pencil, while leaving the remaining large number of eigenvalues and the corresponding Eigenvectors unchanged. The problem arises in controlling undesirable resonance in vibrating structures and in stabilizing control systems. The solution of this problem requires computations of a pair of feedback matrices. For practical effectiveness, these feedback matrices must be computed in such a way that their norms and the condition number of the closed-loop Eigenvector Matrix are as small as possible. These considerations give rise to the minimum norm partial quadratic eigenvalue assignment problem (MNPQEVAP) and the robust partial quadratic eigenvalue assignment problem (RPQEVAP), respectively. In this paper we propose new optimization based algorithms for solving these problems. The problems are solved directly in a second-order setting without resorting to a standard first-order formulation so as to avoid the inversion of a possibly ill-conditioned Matrix and the loss of exploitable structures of the original model. The algorithms require the knowledge of only the open-loop eigenvalues to be replaced and their corresponding Eigenvectors. The remaining open-loop eigenvalues and their corresponding Eigenvectors are kept unchanged. The invariance of the large number of eigenvalues and Eigenvectors under feedback is guaranteed by a proven mathematical result. Furthermore, the gradient formulas needed to solve the problems by using the quasi-Newton optimization technique employed are computed in terms of the known quantities only. Above all, the proposed methods do not require the reduction of the model order or the order of the controller, even when the underlying finite element model has a very large degree of freedom. These attractive features, coupled with minimal computational requirements, such as solutions of small diagonal Sylvester equations make the proposed algorithms ideally suited for application to large real-life structures. Numerical results show significant improvement in feedback norms and in the condition number of the closed-loop system. Also, the closed-loop eigenvalues have acceptable accuracy.
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an optimization approach for minimum norm and robust partial quadratic eigenvalue assignment problems for vibrating structures
Journal of Sound and Vibration, 2009Co-Authors: Sanjoy Brahma, Biswa Nath DattaAbstract:The partial quadratic eigenvalue assignment problem (PQEVAP) concerns the reassignment of a small number of undesirable eigenvalues of a quadratic Matrix pencil, while leaving the remaining large number of eigenvalues and the corresponding Eigenvectors unchanged. The problem arises in controlling undesirable resonance in vibrating structures and in stabilizing control systems. The solution of this problem requires computations of a pair of feedback matrices. For practical effectiveness, these feedback matrices must be computed in such a way that their norms and the condition number of the closed-loop Eigenvector Matrix are as small as possible. These considerations give rise to the minimum norm partial quadratic eigenvalue assignment problem (MNPQEVAP) and the robust partial quadratic eigenvalue assignment problem (RPQEVAP), respectively. In this paper we propose new optimization based algorithms for solving these problems. The problems are solved directly in a second-order setting without resorting to a standard first-order formulation so as to avoid the inversion of a possibly ill-conditioned Matrix and the loss of exploitable structures of the original model. The algorithms require the knowledge of only the open-loop eigenvalues to be replaced and their corresponding Eigenvectors. The remaining open-loop eigenvalues and their corresponding Eigenvectors are kept unchanged. The invariance of the large number of eigenvalues and Eigenvectors under feedback is guaranteed by a proven mathematical result. Furthermore, the gradient formulas needed to solve the problems by using the quasi-Newton optimization technique employed are computed in terms of the known quantities only. Above all, the proposed methods do not require the reduction of the model order or the order of the controller, even when the underlying finite element model has a very large degree of freedom. These attractive features, coupled with minimal computational requirements, such as solutions of small diagonal Sylvester equations make the proposed algorithms ideally suited for application to large real-life structures. Numerical results show significant improvement in feedback norms and in the condition number of the closed-loop system. Also, the closed-loop eigenvalues have acceptable accuracy.
Zhengjian Bai - One of the best experts on this subject based on the ideXlab platform.
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robust and minimum norm partial quadratic eigenvalue assignment in vibrating systems a new optimization approach
Mechanical Systems and Signal Processing, 2010Co-Authors: Zhengjian Bai, Biswa Nath Datta, Jinwei WangAbstract:Abstract The partial quadratic eigenvalue assignment problem (PQEVAP) concerns reassigning a few undesired eigenvalues of a quadratic Matrix pencil to suitably chosen locations and keeping the other large number of eigenvalues and Eigenvectors unchanged (no spill-over). The problem naturally arises in controlling dangerous vibrations in structures by means of active feedback control design. For practical viability, the design must be robust, which requires that the norms of the feedback matrices and the condition number of the closed-loop Eigenvectors are as small as possible. The problem of computing feedback matrices that satisfy the above two practical requirements is known as the Robust Partial Quadratic Eigenvalue Assignment Problem (RPQEVAP). In this paper, we formulate the RPQEVAP as an unconstrained minimization problem with the cost function involving the condition number of the closed-loop Eigenvector Matrix and two feedback norms. Since only a small number of eigenvalues of the open-loop quadratic pencil are computable using the state-of-the-art Matrix computational techniques and/or measurable in a vibration laboratory, it is imperative that the problem is solved using these small number of eigenvalues and the corresponding Eigenvectors. To this end, a class of the feedback matrices are obtained in parametric form, parameterized by a single parametric Matrix, and the cost function and the required gradient formulas for the optimization problem are developed in terms of the small number of eigenvalues that are reassigned and their corresponding Eigenvectors. The problem is solved directly in quadratic setting without transforming it to a standard first-order control problem and most importantly, the significant “no spill-over property” of the closed-loop eigenvalues and Eigenvectors is established by means of a mathematical result. These features make the proposed method practically applicable even for very large structures. Results on numerical experiments show that the proposed method considerably reduces both feedback norms and the sensitivity of the closed-loop eigenvalues. A study on robustness of the system responses of the method under small perturbations show that the responses of the perturbed closed-loop system are compatible with perturbations.
Guangming Pan - One of the best experts on this subject based on the ideXlab platform.
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on asymptotics of Eigenvectors of large sample covariance Matrix
arXiv: Probability, 2007Co-Authors: Zhidong Bai, Baiqi Miao, Guangming PanAbstract:Let \{$X_{ij}$\}, $i,j=...,$ be a double array of i.i.d. complex random variables with $EX_{11}=0,E|X_{11}|^2=1$ and $E|X_{11}|^4<\infty$, and let $A_n=\frac{1}{N}T_n^{{1}/{2}}X_nX_n^*T_n^{{1}/{2}}$, where $T_n^{{1}/{2}}$ is the square root of a nonnegative definite Matrix $T_n$ and $X_n$ is the $n\times N$ Matrix of the upper-left corner of the double array. The Matrix $A_n$ can be considered as a sample covariance Matrix of an i.i.d. sample from a population with mean zero and covariance Matrix $T_n$, or as a multivariate $F$ Matrix if $T_n$ is the inverse of another sample covariance Matrix. To investigate the limiting behavior of the Eigenvectors of $A_n$, a new form of empirical spectral distribution is defined with weights defined by Eigenvectors and it is then shown that this has the same limiting spectral distribution as the empirical spectral distribution defined by equal weights. Moreover, if \{$X_{ij}$\} and $T_n$ are either real or complex and some additional moment assumptions are made then linear spectral statistics defined by the Eigenvectors of $A_n$ are proved to have Gaussian limits, which suggests that the Eigenvector Matrix of $A_n$ is nearly Haar distributed when $T_n$ is a multiple of the identity Matrix, an easy consequence for a Wishart Matrix.
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on asymptotics of Eigenvectors of large sample covariance Matrix
Annals of Probability, 2007Co-Authors: Zhidong Bai, Baiqi Miao, Guangming PanAbstract:Let {X ij }, i, j = ..., be a double array of i.i.d. complex random variables with EX 11 = 0, E|X 11 | 2 = 1 and E|X 11 | 4 <∞, and let An = (1 N T 1/2 n X n X* n (T 1/2 n , where T 1/2 n is the square root of a nonnegative definite Matrix T n and X n is the n x N Matrix of the upper-left comer of the double array. The Matrix An can be considered as a sample covariance Matrix of an i.i.d. sample from a population with mean zero and covariance Matrix T n , or as a multivariate F Matrix if T n is the inverse of another sample covariance Matrix. To investigate the limiting behavior of the Eigenvectors of An, a new form of empirical spectral distribution is defined with weights defined by Eigenvectors and it is then shown that this has the same limiting spectral distribution as the empirical spectral distribution defined by equal weights. Moreover, if { X ij } and T n are either real or complex and some additional moment assumptions are made then linear spectral statistics defined by the Eigenvectors of An are proved to have Gaussian limits, which suggests that the Eigenvector Matrix of An is nearly Haar distributed when T n is a multiple of the identity Matrix, an easy consequence for a Wishart Matrix.
Guoping Liu - One of the best experts on this subject based on the ideXlab platform.
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technical communique complete parametric approach for eigenstructure assignment in a class of second order linear systems
Automatica, 2002Co-Authors: Guangren Duan, Guoping LiuAbstract:This note deals with eigenstructure assignment in the second-order linear system [email protected][email protected]?-Cq=Bu using the proportional-plus-derivative feedback controller u=K"0q+K"[email protected]?. Under the controllability condition of the Matrix pair [AB], very simple, general, and complete parametric expressions in direct closed forms for both the closed-loop Eigenvector Matrix and the feedback gains are established in terms of the closed-loop eigenvalues and a group of parameter vectors. The main computations are two sets of elementary Matrix transformations, which can be replaced by a series of singular value decompositions when the closed-loop eigenvalues are chosen a priori. The approach utilizes directly the original system data A, B and C, and involves manipulations on only n-dimensional matrices. An example illustrates the effect of the proposed approach.
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complete parametric approach for eigenstructure assignment in a class of second order linear systems
IFAC Proceedings Volumes, 1999Co-Authors: G R Duan, D Howe, Guoping LiuAbstract:Abstract This paper deals with eigenstructure assignment in the second-order linear systems using proportional plus derivative feedback controllers. Under the controllability condition of the Matrix pair [A B], veiy simple, general, and complete parametric expressions in direct closed forms for both the closed-loop Eigenvector Matrix and the proportional and derivative feedback gains are proposed. The approach utilises directly the original system data A, B and C, and involves manipulations on only n-dimensional matrices. An example illustrates the effect of the proposed approach.