Sums Of Hermitian Matrices Research Articles

Analysis of a bilinear model of an electric power system using spectral decompositions of Lyapunov functions

In this paper, the spectral decompositions of Lyapunov functions are applied for the first time to the analysis of the behavior of a bilinear model of a two-area electric power system. In contrast to the technique of normal forms and modal series methods, we consider the spectral decomposition not for the dynamics of state variables, but for Lyapunov functions, which characterize the L2-norms of variables or signals in the time domain. The solution of the generalized Lyapunov equation for a bilinear system is represented as the sum of Hermitian matrices corresponding to individual eigenvalues of the system or their pairwise combinations. An iterative algorithm for calculating spectral terms is developed for stable bilinear systems. In a test experiment for the purpose of transient stability analysis, we evaluate the value of individual eigenmodes and their pairwise combinations depending on the magnitude of bilinear terms. The obtained results are consistent with an intuitive interpretation derived from the model equations and eigenvalue analysis. The spectral decompositions of Lyapunov functions allowed us to indicate the range of applicability of linear model and to reveal dominant eigenmodes in the analysis of transient stability of electric power system.

Comparison of Sub-Gramian Analysis with Eigenvalue Analysis for Stability Estimation of Large Dynamical System

In earlier works, solutions of Lyapunov equations were represented as sums of Hermitian matrices corresponding to individual eigenvalues of the system or their pairwise combinations. Each eigen-term in these expansions are called a sub-Gramian. In this paper, we derive spectral decompositions of the solutions of algebraic Lyapunov equations in a more general formulation using the residues of the resolvent of the dynamics matrix. The qualitative differences and advantages of the sub-Gramian approach are described in comparison with the traditional analysis of eigenvalues when estimating the proximity of a dynamical system to its stability boundary. These differences are illustrated by the example of a system with a multiple root and a system of two resonating oscillators. The proposed approach can be efficiently used to evaluate resonant interactions in large dynamical systems.

Fast calculation of modal interaction in large power systems using spectral decompositions of Gramians

Abstract The solutions of Lyapunov and Sylvester equations can be represented as a sum of Hermitian matrices or sub-Gramians corresponding either to individual eigenvalues of the system or their pairwise combinations. The sub-Gramians corresponding to a given part of the spectrum of dynamical system can be used to estimate the modal interaction in this part of the spectrum. In this paper we propose an algorithm for the fast calculation of sub-Gramians for large electric power systems. Our numerical tests demonstrate that the sub-Gramians corresponding to the selected part of the spectrum can be calculated much faster than all sub-Gramians of the system. Therefore, the proposed algorithm can serve as a basis for the fast calculation of modal interactions in large-scale power systems in real time.

Participation factors and sub-Gramians in the selective modal analysis of electric power systems

The solution of Lyapunov equations can be represented as a sum of Hermitian matrices corresponding either to particular eigenvalues of the system matrix, or to pairwise combinations of these eigenvalues. These eigen-parts or sub-Gramians proved to be useful for the stability analysis of electric power systems. In this paper we compare and contrast the sub-Gramians and participation factors as applied to the power system state estimation. Using the sub-Gramian approach we introduce the energy participation factor as a new indicator for selective modal analysis. For a stable system it characterizes the participation of i-th mode and initial k-th state in the integrated energy produced in k-th state. We explain the conceptual meaning and practical usefulness of energy participation factors and contrast them with the conventional participation factors in a selective modal analysis of the IEEE 57-bus test model.

Determinants of sums of two real matrices and their extensions

We study the determinants of the sum of two real matrices under the action of SO(n) ⊗ SO(n). The extremal determinants are determined. The result is a refinement of the results of Li and Mathias [C.K. Li and R. Mathias, The determinant of the sum of two matrices, Bull. Aust. Math. Soc. 52 (1995), pp. 425–429]. We also study the problem in the context of real classical simple Lie algebras. The results of Fiedler [M. Fiedler, Bounds for the determinant of the sum of Hermitian matrices, Proc. Amer. Math. Soc. 30 (1971), pp. 27–31], Li and Mathias 4 and Tam and Thompson [T.Y. Tam and M.C. Thompson, Determinant and Pfaffian of sum of skew symmetric matrices, Linear Algebra Appl. 433 (2010), pp. 412–423] are some special cases in this context. Complete solutions are obtained.

Eigenvalues of sums of pseudo-Hermitian matrices

We study analogues of classical inequalities for the eigenvalues of sums of Hermitian matrices for the cone of admissible elements in the pseudo-Hermitian case. In particular, we obtain analogues of the Lidskii-Wielandt inequalities.

Off-diagonal submatrices of a Hermitian matrix

A necessary and sufficient condition is given to a p × q p\times q complex matrix X X to be an off-diagonal block of an n × n n\times n Hermitian matrix C C with prescribed eigenvalues (in terms of the eigenvalues of C C and singular values of X X ). The proof depends on some recent breakthroughs in the study of spectral inequalities on the sum of Hermitian matrices by Klyachko and Fulton. Some interesting geometrical properties of the set S S of all such matrices are derived from the main result. These results improve earlier ones that only give partial information for the set S S .

Local Systems on $\mathbb P$1 − S for S a Finite Set

I give the necessary and sufficient conditions for the existence of Unitary local systems with prescribed local monodromies on P1 − S where S is a finite set. This is used to give an algorithm to decide if a rigid local system on P1 − S has finite global monodromy, thereby answering a question of N. Katz. The methods of this article (use of Harder–Narasimhan filtrations) are used to strengthen Klyachko's theorem on sums of Hermitian matrices. In the Appendix, I give a reformulation of Mehta–Seshadri theorem in the SU(n) setting.

Direction finding using noise covariance modeling

We consider the problem of direction finding in the presence of colored noise whose covariance matrix is unknown. We show that the ambient noise covariance matrix can be modeled by a sum of Hermitian matrices known up to a multiplicative scalar. Using this model, we estimate jointly the directions of arrival of the signals and the noise model parameters. We show that under certain conditions, it is possible to obtain unbiased and efficient estimates of the signal direction. The Cramer-Rao bound is used as the principal analysis tool. Computer simulations using the maximum likelihood estimator provide a validation of the analytical results. >