Location Of Eigenvalues Research Articles

On the location of eigenvalues of matrix polynomials

A number $\lambda \in \mathbb C $ is called an {\it eigenvalue} of the matrix polynomial $P(z)$ if there exists a nonzero vector $x \in \mathbb C^n$ such that $P(\lambda)x = 0$. Note that each finite eigenvalue of $P(z)$ is a zero of the characteristic polynomial $\det(P(z))$. In this paper we establish some (upper and lower) bounds for eigenvalues of matrix polynomials based on the norm of their coefficient matrices and compare these bounds to those given by N.J. Higham and F. Tisseur, J. Maroulas and P. Psarrakos.

Modeling and Analysis of Vehicle Shimmy with Consideration of the Coupling Effects of Vehicle Body

Based on the Lagrange equation, a 9‐degrees‐of‐freedom shimmy model with consideration of the coupling effects between the motions of vehicle body and the shimmy of front wheels and a 5‐degrees‐of‐freedom shimmy model ignoring these coupling effects for a vehicle with double‐wishbone independent front suspensions are presented here to study the problem of vehicle shimmy. According to the eigenvalue loci of system’s Jacobian matrix plotted on the complex plane, the Hopf bifurcation characteristics of nonlinear shimmy are studied and the conditions for the generation of limit cycle are analyzed. Numerical calculation and simulation are used to study the dynamic behavior of vehicle shimmy. By comparing the dynamic responses of two different shimmy models, the coupling effects of vehicle body on vehicle shimmy are studied. Finally, the relationship between the amplitude of each DoF and vehicle velocity and the influences of vehicle parameters such as the mass of vehicle body, the longitudinal position of the center of gravity of vehicle body, and the inclination angle of front suspension on shimmy are studied.

Stability analysis on flow parameters in coolant temperature control system of lead-cooled fast reactor

Lead-cooled Fast Reactor (LFR), one of the promising nuclear energy technologies for future advanced system, is well foreseen in meeting the goals of Generation IV fission reactor. In LFR, Coolant Temperature Control System (CTCS) is one of the critical parts, and the influence of coolant flow parameters on the stability analysis is worth investigating and studying. In this paper, we use the CLEAR-S, an integrated multifunctional non-nuclear test facility, as reference and the stability analysis of CTCS, concerned to coolant flow rate has been completed based on simplified coolant thermal circulation model. Location of eigenvalues and calculation results of real stability radius, both qualitatively and quantitatively reflect the influence of coolant flow rate on stability respectively. The analysis results show that the feed-water flow rate in steady state is a critical parameter for CLEAR-S, because larger feed-water flow rate can improve stability apparently. However, LBE coolant flow rate has little impact on system stability. The results of this work will provide an effective reference for design and operation of LFR.

Systems with Fractional Variable-Order Difference Operator of Convolution Type and Its Stability

In the paper the Grunwald–Letnikov-type linear fractional variable order discrete-time systems are studied. The conditions for stability and instability are formulated. The regions of the systems stability are determined accordingly to locus of eigenvalues of a matrix associated to the considered system.DOI: http://dx.doi.org/10.5755/j01.eie.24.5.21846

A note on eigenvalue bounds for Schrödinger operators

We obtain a new bound on the location of eigenvalues for a non-self-adjoint Schrödinger operator with complex-valued potentials by obtaining a weighted L2 estimate for the resolvent of the Laplacian.

Eventually DSDD Matrices and Eigenvalue Localization

Firstly, the relationships among strictly diagonally dominant ( S D D ) matrices, doubly strictly diagonally dominant ( D S D D ) matrices, eventually S D D matrices and eventually D S D D matrices are considered. Secondly, by excluding some proper subsets of an existing eigenvalue inclusion set for matrices, which do not contain any eigenvalues of matrices, a tighter eigenvalue inclusion set of matrices is derived. As its application, a sufficient condition of determining non-singularity of matrices is obtained. Finally, the infinity norm estimation of the inverse of eventually D S D D matrices is derived.

Eigenwaves in Sommerfeld–Goubau Line: Spectrum

The problem on normal waves in an inhomogeneous metal-dielectric waveguide structure is reduced to a boundary value problem for the longitudinal components of the electromagnetic field in Sobolev spaces. Nonhomogeneous filling and entering of spectral parameter in transmission conditions leads to necessity of special definition of solution of the problem. We formulate the definition of solution using variational relation. The variational problem is reduced to the study of an operator function. We investigate properties of the operators of the operator function needed for the analysis of its spectral properties. We prove theorem of discrete spectrum and theorem of localization of eigenvalues of the operator-function on complex plane.

Eigenvalue location in graphs of small clique-width

Finding a diagonal matrix congruent to A−cI for constants c, where A is the adjacency matrix of a graph G allows us to quickly tell the number of eigenvalues in a given interval. If G has clique-width k and a corresponding k-expression is known, then diagonalization can be done in time O(poly(k)n) where n is the order of G.

Equivalence classes of e-matrices and associated eigenvalue localization regions

ABSTRACT This paper is a continuation of a series of papers on related topics by the authors. It is divided into two main parts. In the first part, the set of e-matrices (real constant row-sum matrices) is partitioned into equivalence classes which are called e-similarity classes. The relationships between the spectra of different matrices belonging to the same equivalence class, as well as their left and generalized left eigenspaces, are discussed. In the second part, the results obtained in the first part are applied to improve the location of eigenvalues of e-matrices. Associated localization Gershgorin regions of the so-called second type are obtained. A number of examples are provided.

Exclusion sets in eigenvalue localization sets for tensors

ABSTRACTBy excluding some proper subsets, which do not include any eigenvalues of tensors, from an existing eigenvalue localization set provided by Zhao and Sang, a new eigenvalue localization set for tensors is given. As an application, a sufficient condition such that the determinant of a tensor is not zero is provided.

On the Convergence of Discrete-Time Linear Systems: A Linear Time-Varying Mann Iteration Converges IFF Its Operator Is Strictly Pseudocontractive

We adopt an operator-theoretic perspective to study convergence of linear fixed-point iterations and discrete-time linear systems. We mainly focus on the so-called Krasnoselskij–Mann iteration, ${x}$ ( $k + 1$ ) = ( $1-\alpha _{k}$ ) ${x}$ ( ${k}$ ) + $\alpha _{k} A~{x}$ ( ${k}$ ), which is relevant for distributed computation in optimization and game theory, when $A$ is not available in a centralized way. We show that strict pseudocontractiveness of the linear operator $x \mapsto Ax$ is not only sufficient (as known) but also necessary for the convergence to a vector in the kernel of $I-A$ . We also characterize some relevant operator-theoretic properties of linear operators via eigenvalue location and linear matrix inequalities. We apply the convergence conditions to multi-agent linear systems with vanishing step sizes, in particular, to linear consensus dynamics and equilibrium seeking in monotone linear-quadratic games.

On decentralized integral controllability of stable negative-imaginary systems and some related extensions

This paper exploits the decentralized integral controllability (DIC) of stable negative-imaginary (NI) systems with non-singular DC-gain matrix to develop a constant input tracking framework with failure tolerance property in the event of sensor and/or actuator faults. By virtue of the DIC property of NI systems, it is shown that a positive feedback interconnection of two NI systems, of which one is a decentralized integrator system with non-negative gains and the other one is a stable NI system having negative definite DC-gain matrix, is internally stable. This is in contrast to the existing results where, for internal stability, at least one of the systems needs to be strictly negative-imaginary (SNI). Further it is shown that, in a positive feedback interconnection, stable NI systems with negative definite DC-gain matrix exhibit integral controllability (IC) property for any positive value of the integral gain k, which is a distinct property of stable NI systems having negative definite DC-gain matrix since the existing notion of IC for LTI systems offers only a finite range of k. Using the DIC property and the eigenvalue loci technique, some new results and research directions for the NI systems have also been presented in this paper.

Dashnic–Zusmanovich type matrices: A new subclass of nonsingular H-matrices

A new class of matrices, called Dashnic–Zusmanovich type (DZ-type) matrices, which is similar to, but different from Dashnic–Zusmanovich (DZ) matrices, is introduced and proved to be a subclass of nonsingular H-matrices. The relations between strictly diagonally dominant matrices, doubly strictly diagonally dominant matrices, DZ matrices and DZ-type matrices are discussed. Finally, by the non-singularity of DZ-type matrices, a new eigenvalue localization set for matrices is given.

Mode decoupling robust eigenstructure assignment applied to the lateral-directional dynamics of the F-16 aircraft

In this paper, a robust controller is designed which is based on the eigenstructure assignment technique. With this technique, the physical modes of the system are decoupled by the appropriate shaping of the eigenvectors. While keeping the desired eigenvector structure as much as possible, Nelder–Mead optimization algorithm is implemented for the search of the optimal robust solution. Different optimization cost functions are investigated in terms of eigenvalue sensitivity and stability robustness. Design solutions are compared with each other and with a fundamental solution in terms of robustness measures. Furthermore, a small amount of flexibility is included in the selection of the eigenvalue locations and the improvement in the robustness is evaluated by singular value analysis. The effectiveness of the control system is demonstrated with nonlinear simulations on the F-16 aircraft mathematical model.

Dynamic modeling, control design and stability analysis of railway active power quality conditioner

Two-phase three-wire converter has been proposed to compensate the power quality issues in AC electrified railway systems. In this paper, a step-by-step modeling procedure is presented and applied to the compensator to obtain a linear model. The model is then used in a linear control system design and stability analysis of the compensator. The first step is to obtain the bi-linear form of the exact (switched) model of the compensator representing the high and low frequency behavior of the converter. Then, the generalized state space averaging is applied to the bi-linear model to obtain the continuous time model of the compensator. The linear small signal model is derived by differentiating the averaged model around its equilibrium point. Linear control system design is established based on the linear small-signal model to improve DC-link voltage performance. The simulated model of compensator is verified by quantitatively comparing to the previously reported experimental results. The verified simulation model is used to validate the obtained models. Finally, the open loop stability analysis is performed by studying the locus of system eigenvalues with respect to the circuit parameters variations.

Current-Mode Pulse Train Controlled Quadratic Boost Converter Operating in Discontinuous Conduction Mode

Unlike the conventional pulse width modulation control, a unique current-mode pulse train (CMPT) control technique presents a different nonlinear dynamics associated with the power electronics system. The quadratic boost converter in discontinuous conduction mode (DCM) is examined under CMPT control. The control performance of the system is studied by analyzing the combination of high-power control pulse PH and low-power control pulse PL of a control pulse repetition cycle in regulating the output voltage. The computer simulation is accomplished to capture the gist of the system dynamics revealing the periodic transition manifested due to border collision bifurcation under the influence of the variation in input voltage and load resistance. The stability of the system is assessed with the evidence of loci of eigenvalues from two-dimensional discrete-time model and maximal Lyapunov exponents obtained using QR decomposition algorithm. An experimental setup is also developed to verify the simulated and analytical dynamics witnessed.

Improving the dynamic response of frequency and power in a wind integrated power system by optimal design of compensated superconducting magnetic energy storage

ABSTRACTThe frequency deviation and power fluctuation need to be controlled in a wind-integrated power system (WIPS) for keeping the balance between system power generation and demand, which support the quality and stability of overall power system. The present paper addresses this problem while concerning the integration of intermittent wind power and load disturbance into the WIPS. With this intent, it proposes the compensated superconducting magnetic energy storage (CSMES) system with proportional integral derivative (PID) controller for improving the frequency and power deviation profile. A novel swarm intelligence-based artificial bee colony (ABC) algorithm is used for optimal design of PID-CSMES system. Robustness of the proposed ABC-based PID-CSMES control strategy is tested in WIPS under various disturbance patterns of load and wind power. To demonstrate the improved dynamic response, their simulation results are compared with particle swarm optimization-based PID-CSMES, PID with SMES, and only PID controller technique. The performance indices and transient response characteristics of frequency and power deviation are used to evaluate and compare the accuracy and efficiency of each controller. Stability of various system configurations is analyzed using eigenvalue location. Comparing the results of different controller in WIPS indicates a substantial improvement in the dynamic response of system frequency and power deviations by utilizing the proposed control strategy.

Parabolic transmission eigenvalue-free regions in the degenerate isotropic case

We study the location of the transmission eigenvalues in the isotropic case when the restrictions of the refraction indices on the boundary coincide. Under some natural conditions we show that there exist parabolic transmission eigenvalue-free regions.

On bounding the eigenvalues of matrices with constant row-sums

ABSTRACTFor every real matrix A having as an eigenvector e, the all-ones vector, an upper bound is obtained for the absolute value of its eigenvalues except, in some cases, the one associated with e. Comparisons are made between this bound and some existing bounds for stochastic matrices as well as other types of matrices such as Laplacian matrices and Randić matrices. Other examples and results are also provided. An interesting recent article ‘An eigenvalue localization theorem for stochastic matrices and its application to Randić matrices’, Linear Algebra Appl. 2016;505:85–96, by A. Banerjee and R. Mehatari, is useful for this present paper.

High-frequency approximation of the interior Dirichlet-to-Neumann map and applications to the transmission eigenvalues

We study the high-frequency behavior of the Dirichlet-to-Neumann map for an arbitrary compact Riemannian manifold with a non-empty smooth boundary. We show that far from the real axis it can be approximated by a simpler operator. We use this fact to get new results concerning the location of the transmission eigenvalues on the complex plane. In some cases we obtain optimal transmission eigenvalue-free regions.