Eigenvalue Perturbation Research Articles

The mathematics of eigenvalue optimization

Optimization problems involving the eigenvalues of symmetric and nonsymmetric matrices present a fascinating mathematical challenge. Such problems arise often in theory and practice, particularly in engineering design, and are amenable to a rich blend of classical mathematical techniques and contemporary optimization theory. This essay presents a personal choice of some central mathematical ideas, outlined for the broad optimization community. I discuss the convex analysis of spectral functions and invariant matrix norms, touching briefly on semidefinite representability, and then outlining two broader algebraic viewpoints based on hyperbolic polynomials and Lie algebra. Analogous nonconvex notions lead into eigenvalue perturbation theory. The last third of the article concerns stability, for polynomials, matrices, and associated dynamical systems, ending with a section on robustness. The powerful and elegant language of nonsmooth analysis appears throughout, as a unifying narrative thread.

Wielandt and Ky-Fan theorem for matrix pairs

The generalization of Wielandt and Ky-Fan theorem is given for Hermitian matrix pairs, and some new eigenvalue perturbation estimates are obtained. An application is made on a class of quadratic matrix pencils.

Optimization and Pseudospectra, with Applications to Robust Stability

The $\epsilon$-pseudospectrum of a matrix A is the subset of the complex plane consisting of all eigenvalues of all complex matrices within a distance $\epsilon$ of A. We are interested in two aspects of and pseudospectra. The first concerns maximizing the function real part over an $\epsilon$-pseudospectrum of a fixed matrix: this defines a function known as the $\epsilon$-pseudospectral abscissa of a matrix. We present a bisection algorithm to compute this function. Our second interest is in minimizing the $\epsilon$-pseudospectral abscissa over a set of feasible matrices. A prerequisite for local optimization of this function is an understanding of its variational properties, the study of which is the main focus of the paper. We show that, in a neighborhood of any nonderogatory matrix, the $\epsilon$-pseudospectral abscissa is a nonsmooth but locally Lipschitz and subdifferentially regular function for sufficiently small $\epsilon$; in fact, it can be expressed locally as the maximum of a finite number of smooth functions. Along the way we obtain an eigenvalue perturbation result: near a nonderogatory matrix, the eigenvalues satisfy a Holder continuity property on matrix space---a property that is well known when only a single perturbation parameter is considered. The pseudospectral abscissa is a powerful modeling tool: not only is it a robust measure of stability, but it also reveals the transient (as opposed to asymptotic) behavior of associated dynamical systems.

Habitat destruction, habitat restoration and eigenvector–eigenvalue relations

According to metapopulation theory, the capacity of a habitat patch network to support the persistence of a species is measured by the metapopulation capacity of the patch network. Mathematically, metapopulation capacity is given by the leading eigenvalue λ M of an appropriately constructed non-negative n× n matrix M, where n is the number of habitat patches. Both habitat destruction (in the sense of destruction of entire patches) and habitat deterioration (in the sense of partial destruction of patches) lower the metapopulation capacity of the patch network. The effect of gradual habitat deterioration is given by the derivative of λ M with respect to patch attributes and may be straightforwardly evaluated by sensitivity analysis. In contrast, destruction of entire patches leads to a rank modification of matrix M, the effect of which on λ M may be derived from eigenvector–eigenvalue relations. Eigenvector–eigenvalue relations have previously been analyzed only for symmetric matrices, which restricts their use in biological applications. In this paper I generalize some of the previous results by deriving eigenvector–eigenvalue relations for general non-symmetric matrices. In addition to the exact eigenvector–eigenvalue relations, I also derive eigenvalue perturbation formulae for rank-one modifications. These results lead to simple and intuitive approximation formulae, which may be used e.g. to assess the contribution of particular habitat patches to the metapopulation capacity of the landscape. The mathematical results presented are not restricted to the metapopulation context, but they should find a number of useful applications in biology, engineering and other applied sciences, where the removal (or addition) of matrix rows and columns often corresponds in a natural manner to decreasing (or increasing) the degrees of freedom of the focal system.

Examination of the effect of increasing the number of radiation beams on a radiation treatment plan

Within the confines of least-squares operations, it is possible to quantify the effect of the addition of treatment fields or beamlets to a treatment plan. Using linear algebra and eigenvalue perturbation theory, the effect of the increase in number of treatments is shown to be equivalent to adding a perturbation operator. The effect of adding additional fields will be negligible if the perturbation operator is small. The correspondence of this approach to an earlier work in beam-orientation optimization is also demonstrated. Results are presented for prostate, spinal and head and neck cases, and the connection to beam-orientation optimization is examined.

On perturbations of matrix pencils with real spectra, a revisit

This paper continues earlier studies by Bhatia and Li on eigenvalue perturbation theory for diagonalizable matrix pencils having real spectra. A unifying framework for creating crucial perturbation equations is developed. With the help of a recent result on generalized commutators involving unitary matrices, new and much sharper bounds are obtained.

Perturbation of eigenvalues embedded at a threshold

Results are obtained on perturbation of eigenvalues and half-bound states (zero-resonances) embedded at a threshold. The results are obtained in a two-channel framework for small off-diagonal perturbations. The results are based on given asymptotic expansions of the component Hamiltonians.

On perturbation of eigenvalues embedded at thresholds in a two channel model

We present some results on the perturbation of eigenvalues embedded at thresholds in a two channel model Hamiltonian with a small off-diagonal perturbation. Examples are given of the various types of behavior of the eigenvalue under perturbation.

OPTIMAL FINITE-PRECISION CONTROLLER AND FILTER IMPLEMENTATIONS USING FLOATING-POINT ARITHMETIC

In this paper, eigenvalue sensitivity measures are proposed that are suitable for assessing the fragility of digital controllers and filters implemented using floating-point arithmetic. Floating-point arithmetic parameter uncertainty is shown to be multiplicative. Based on first-order eigenvalue sensitivity analysis, an upper bound on the eigenvalue perturbations is derived. Consequently, open-loop and closed-loop eigenvalue sensitivity measures are proposed. These measures are dependent upon the filter/controller realization. Problems of obtaining the optimal realization with respect to both the open-loop and the closed-loop eigenvalue sensitivity measures are posed. The problem for the open-loop case is completely solved. The problem for the closed-loop case is solved using nonlinear programming. The problems are illustrated with a numerical example.

Relative perturbation theory for a class of diagonalizable Hermitian matrix pairs

Veselić and Slapničar gave a general perturbation result for the eigenvalues of the Hermitian matrix pair ( H, K), where K is positive definite. In this paper their result is generalized to a wider class of Hermitian matrix pairs. Especially, estimates for the relative perturbation of eigenvalues of definite pairs are also obtained.

Resonance Theory for Schrödinger Operators

Resonances which result from perturbation of embedded eigenvalues are studied by time dependent methods. A general theory is developed, with new and weaker conditions, allowing for perturbations of threshold eigenvalues and relaxed Fermi Golden rule. The exponential decay rate of resonances is addressed; its uniqueness in the time dependent picture is shown is certain cases. The relation to the existence of meromorphic continuation of the properly weighted Green's function to time dependent resonance is further elucidated, by giving an equivalent time dependent asymptotic expansion of the solutions of the Schr\"odinger equation. \keywords{Resonances; Time-dependent Schr\"odinger equation}

Robust Output Feedback Using Sliding Mode Control

A general sliding mode output feedback algorithm is developed that takes into account parametric uncertainty and external disturbances provided certain bounds are known. It has been assumed that the number of states is greater than the number of sensors and that there can be a mismatched disturbance. The details of an eigenvalue perturbation analysis for an uncertain matrix are presented. A boundary layer has been introduced around each sliding surface to eliminate the chattering phenomenon. The results from numerical simulations are presented to corroborate the validity of the proposed controller.

Note on the Norm Involved in the Definition of the ε-spectrum

There exist many definitions of the ε-spectrum. Unlike the spectrum, the ε-spectrum definitions depend on the choice of the norm. We propose to study the conditions on the norm that make all these definitions equivalent. The spectral radius of a related matrix is a useful tool in the eigenvalues perturbation analysis. The study of its connection with the ε-spectrum provides a new definition of the ε-spectrum. In most articles dealing with this subject, the chosen norm is the spectral one. In this paper, we intend to justify this choice.

Robust active control of a vibrating plate

This paper addresses several aspects of designing robust controllers for active control of vibration in a plate with piezo actuators and acceleration sensors. The discussion ranges over identification of the plant, design and implementation of controllers, and evaluation of their robustness and performance. All the concepts are tested on an experimental setup. Robustness issues are prominent, since in practice the plant differs from models obtained through identification, and moreover changes in ambient conditions induce variation over time in the plant’s characteristics such as modal frequencies and damping ratios. Consequently, a controller designed for a nominal plant model may yield a poor performance or even result in an unstable system if robustness issues are not considered properly. Several H∞ controllers are compared here. Besides using the usual multiplicative and additive uncertainty models, controllers are also designed using the eigenvalue perturbation approach. It is found that controllers designed using this approach are more robust than those obtained using conventional methods. Furthermore, the performance does not degrade severely (which usually occurs due to the robustness performance tradeoff). Hence, it is concluded that the approach is suitable for active vibration control of a plate or similar plants. a)Previously with Fac. Mathematical Sciences, Univ. Twente, The Netherlands.

Complex resonances in acoustic waveguides

We consider a two-dimensional infinitely long acoustic waveguide formed by two parallel lines containing an arbitrarily shaped obstacle. The existence of trapped modes that are the eigenfunctions of the Laplace operator in the corresponding domain subject to Neumann boundary conditions was proved by Evans, Levitin and Vassiliev (J. Fluid Mech. 261 1994) for obstacles symmetric about the centreline of the waveguide. In our paper we deal with the situation when the obstacle is shifted with respect to the centreline and study the resulting complex resonances, We are particularly interested in those resonances which are perturbations of (real) eigenvalues. We study how an eigenvalue becomes a complex resonance moving from the real axis into the upper half-plane as the obstacle is shifted from its original position. The shift of the eigenvalue along the imaginary axis is predicted theoretically and the result is compared with numerical computations. The total number of resonances lying inside a sequence of expanding circles is also calculated numerically.

Perturbation theory for the eigenvalues of factorised symmetric matrices

We obtain eigenvalue perturbation results for a factorised Hermitian matrix H=GJG ∗ where J 2=I and G has full row rank and is perturbed into G+δG, where δG is small with respect to G. This complements the earlier results on the easier case of G with full column rank. Applied to square factors G our results help to identify the so-called quasidefinite matrices as a natural class on which the relative perturbation theory for the eigensolution can be formulated in a way completely analogous to the one already known for positive definite matrices.

Mode Localization in Flexible Spacecraft: A Control Challenge

Mode localization is often an unexpected phenomenon in e exible space structures because it arises from very small manufacturing errors. It will occur, however, in structures comprising lightly damped, weakly coupled repeating members that have tightly clustered modes of vibration. If not properly taken into account, mode localization can potentially pose problems for high-performance e exible spacecraft controllers relying on the assumption of perfect symmetry. In cases where actuation and sensing are limited, local decentralized control design for each structural member relies on the propagation of energy from one member to the next through channels of weak coupling. Mode localization can severely inhibit this propagation leading to control performance degradation. A solution to this problem is to design controllers based on models consisting of multiple members of the structure or to centralize control design. Because mode shape errors can be large when modes localize, however, any uncertainty model that attempts to capture this information will lead to reduced control authority, overconservatism, and performance losses. It is shown that eigenvalue perturbation models are an effective means by which to design controllers fore exible space structures subject to mode localization. The successful control of a structurewith modelocalization isdemonstrated through experimentsconducted ona e exiblespacecraftemulator that exhibits the phenomenon.

EXACT SOLUTIONS FOR THE FREE AND FORCED VIBRATION OF A ROTATING DISK-SPINDLE SYSTEM

Spinning disk–spindle systems consisting of an elastic disk mounted on an elastic spindle by means of a three dimensional, rigid clamp extend the rich literature on spinning discs and shafts that are decoupled from each other. This work presents an exact, closed-form solution for the eigensolutions of such systems. The complex eigenfunctions have the classical properties of a gyroscopic system when the individual disk, spindle and clamp deflections for a given eigenfunction are collected in terms of an extended eigenfunction. Eigenvalue perturbations are calculated to determine the sensitivity of the zero speed eigenvalues and the critical speeds to system parameters. Additionally, critical speeds analogous to those of a rigidly supported (classical) spinning disk are examined for the coupled system. Whereas the rigidly supported disk does not experience critical speed instability in the one-nodal diameter eigenfunctions, the coupled system does. The exact solution admits a closed-form modal analysis for the forced response to disk, spindle and clamp excitation. Response is calculated for two examples that demonstrate the strong disk–spindle modal coupling that can exist and the potentially damaging transmission of excitation energy between the disk and spindle.

A Time-Dependent Theory of Quantum Resonances

In this paper we further develop a general theory of metastable states resulting from perturbation of unstable eigenvalues. We apply this theory to many-body Schrouml;dinger operators and to the problem of quasiclassical tunneling.

Asymptotic formula for the flexible bar

The postbuckling of an elastica bar is studied through an eigenvalue perturbation about the bifurcation point. The 4-term series is found to be highly accurate, valid up to the complete collapse of the bar. The result is particularly useful for analyzing systems containing flexible pinned–pinned elements. Two examples using such elastica links are illustrated. The present method is much more simpler than using elliptic functions or direct numerical integration.