Eigenvalue Perturbation Research Articles

Sharp spectral bounds for complex perturbations of the indefinite Laplacian

We derive quantitative bounds for eigenvalues of complex perturbations of the indefinite Laplacian on the real line. Our results substantially improve existing results even for real potentials. For L1-potentials, we obtain optimal spectral enclosures which accommodate also embedded eigenvalues, while our result for Lp-potentials yield sharp spectral bounds on the imaginary parts of eigenvalues of the perturbed operator for all p∈[1,∞). The sharpness of the results are demonstrated by means of explicit examples.

Analysis of the Heterogeneous Vectorial Network Model of Collective Motion

We analyze the vectorial network model, a stochastic protocol that describes collective motion of groups of agents, randomly mixing in a planar space. Motivated by biological and technical applications, we focus on a heterogeneous form of the model, where agents have different propensity to interact with others. By linearizing the dynamics about a synchronous state and leveraging an eigenvalue perturbation argument, we establish a closed-form expression for the mean-square convergence rate to the synchronous state in the absence of additive noise. These closed-form findings are extended to study the effect of added noise on the agents’ coordination, captured by the polarization of the group. Our results reveal that heterogeneity has a detrimental effect on both the convergence rate and the polarization, which is nonlinearly moderated by the average number of connections in the group. Numerical simulations are provided to support our theoretical findings.

Stability Analysis of Vehicle Platooning With Limited Communication Range and Random Packet Losses

Control performance of vehicle platooning relies on the information flow topology and quality of wireless communications. In this article, we investigate the constant-time-headway-spacing-policy-based vehicle platooning problem, where multiple predecessors' information is used by the following vehicles and communication impairments, i.e., limited communication range and random packet losses, are considered. In this article, first, when the leading vehicle moves at a constant speed, we obtain the sufficient and necessary conditions on sampling time, control gains, and internal lag, to ensure the stability of the vehicle platoon based on matrix polynomials' stability for ideal communications. Second, for time-independent homogeneous random packet losses, we provide the upper bound for the loss rate to maintain convergence in expectation by matrix eigenvalue perturbation theory when no input is set for lossy information. We also provide sufficient conditions to guarantee mean-square convergence for heterogeneous time-independent random packet losses and show the convergence time for any given accuracy and probability. Third, when historically latest information is used for input, the sufficient and necessary conditions are provided to ensure the internal stability and string stability by Markov jump linear system theory. Furthermore, we discuss the controller design when no feasible solution exists to guarantee the string stability. Extensive numerical results validate our analysis.

Non-Fragile H ∞ Order Reduction of LTI Controllers

The eigenvalue perturbation theorem is used to propose a convex fragility criterion with application to control system design. The criterion can be considered as a non-normality measure of the controller state-space matrix. Non-normality of a matrix is defined as its distance to the nearest real normal matrix within a convex normal subspace. Based on the criterion, an $\mathcal {H}_{\infty }$ method for the order reduction of linear time-invariant (LTI) controllers is developed which leads to non-fragile reduced order controllers.

Dynamic Heterogeneous Information Network Embedding With Meta-Path Based Proximity

Heterogeneous information network (HIN) embedding aims at learning the low-dimensional representation of nodes while preserving structure and semantics in a HIN. Existing methods mainly focus on static networks, while a real HIN usually evolves over time with the addition (deletion) of multiple types of nodes and edges. Because even a tiny change can influence the whole structure and semantics, the conventional HIN embedding methods need to be retrained to get the updated embeddings, which is time-consuming and unrealistic. In this paper, we investigate the problem of dynamic HIN embedding and propose a novel Dynamic HIN Embedding model (DyHNE) with meta-path based proximity. Specifically, we introduce the meta-path based first- and second-order proximities to preserve structure and semantics in HINs. As the HIN evolves over time, we naturally capture changes with the perturbation of meta-path augmented adjacency matrices. Thereafter, we learn the node embeddings by solving generalized eigenvalue problem effectively and employ eigenvalue perturbation to derive the updated embeddings efficiently without retraining. Experiments show that DyHNE outperforms the state-of-the-arts in terms of effectiveness and efficiency.

Beyond the limitations of perturbation methods for real random eigenvalue problems using Exceptional Points and analytic continuation

A numerical method is proposed to approximate the solution of parametric eigenvalue problem when the variability of the parameters exceed the radius of convergence of low order perturbation methods. The radius of convergence of eigenvalue perturbation methods, based on Taylor series, is known to decrease when eigenvalues are getting closer to each other. This phenomenon, known as veering in structural dynamics, is a direct consequence of the existence of branch point singularity in the complex plane of the varying parameters where some eigenvalues are defective. When this degeneracy, referred to as Exceptional Point (EP), is close to the real axis, the veering becomes stronger.The main idea of the proposed approach is to combined a pair of eigenvalues to remove this singularity. To do so, two analytic auxiliary functions are introduced and are computed through high order derivatives of the eigenvalue pair with respect to the parameter. This yields a new robust eigenvalue reconstruction scheme which is compared to Taylor and Puiseux series. In all cases, theoretical bounds are established and all approximations are compared numerically on a three degrees of freedom toy model. This system illustrates the ability of the method to handle the vibrations of a structure with a randomly varying parameter. Computationally efficient, the proposed algorithm could also be relevant for actual numerical models of large size, arising from other applications involving parametric eigenvalue problems, e.g., waveguides, rotating machinery or instability problems such as squeal or flutter.

Integrating genetics, biophysical, and demographic insights identifies critical sites for seagrass conservation.

The eelgrass Zostera marina is an important foundation species of coastal areas in the Northern Hemisphere, but is continuing to decline, despite management actions. The development of new management tools is therefore urgent in order to prioritize limited resources for protecting meadows most vulnerable to local extinctions and identifying most valuable present and historic meadows to protect and restore, respectively. We assessed 377 eelgrass meadows along the complex coastlines of two fjord regions on the Swedish west coast-one is currently healthy and the other is substantially degraded. Shoot dispersal for all meadows was assessed with Lagrangian biophysical modeling (scale: 100-1,000m) and used for barrier analysis and clustering; a subset (n=22) was also assessed with population genetic methods (20 microsatellites) including diversity, structure, and network connectivity. Both approaches were in very good agreement, resulting in seven subpopulation groupings or management units (MUs). The MUs correspond to a spatial scale appropriate for coastal management of "waterbodies" used in the European Water Framework Directive. Adding demographic modeling based on the genetic and biophysical data as a third approach, we are able to assess past, present, and future metapopulation dynamics to identify especially vulnerable and valuable meadows. In a further application, we show how the biophysical approach, using eigenvalue perturbation theory (EPT) and distribution records from the 1980s, can be used to identify lost meadows where restoration would best benefit the present metapopulation. The combination of methods, presented here as a toolbox, allows the assessment of different temporal and spatial scales at the same time, as well as ranking of specific meadows according to key genetic, demographic and ecological metrics. It could be applied to any species or region, and we exemplify its versatility as a management guide for eelgrass along the Swedish west coast.

Galloping Stability Criterion for 3-DOF Coupled Motion of an Ice-Accreted Conductor

AbstractThis study presents an analytical galloping stability criterion for a three-degree-of-freedom (3-DOF) coupled motion of an ice-accreted conductor by using the eigenvalue perturbation method...

Distributed algorithms for resource allocation in cyber-physical energy systems with uniform/nonuniform communication delays

As the communication delays in cyber-physical energy systems are considered, the distributed optimal resource allocation problem has not been well studied. In this paper, distributed algorithms are proposed to solve this problem and the effects of communication delays are analyzed. Firstly, as the uniform and nonuniform communication delays in cyber-physical energy system are taken into account, two distributed resource allocation (RA) algorithms are proposed respectively. By using the eigenvalue perturbation theory and the Generalized Nyquist Criterion, the impacts of time delays on the distributed algorithms are strictly analyzed and the maximum allowable delay bounds are derived. Then, as the generation capacity constraints are further considered, the distributed iterative method based on the unconstrained RA solutions is presented. Finally, several simulation studies are carried out to validate the theoretical results.

Accuracy of approximate projection to the semidefinite cone

When a projection of a symmetric or Hermitian matrix to the positive semidefinite cone is computed approximately (or to working precision on a computer), a natural question is to quantify its accuracy. A straightforward bound invoking standard eigenvalue perturbation theory (e.g. Davis-Kahan and Weyl bounds) suggests that the accuracy would be inversely proportional to the spectral gap, implying it can be poor in the presence of small eigenvalues. This work shows that a small gap is not a concern for projection onto the semidefinite cone, by deriving error bounds that are gap-independent.

A Decentralized Optimization Strategy for Distributed Generators Power Allocation in Microgrids Based on Load Demand–Power Generation Equivalent Forecasting

In order to guarantee the economic and reliable operation of renewable Distributed Generators (DGs) in microgrids, a decentralized optimization strategy for DGs power allocation is proposed in this paper. According to the method, all processes and parameters are designed in a fully distributed way. To achieve decentralization and to maintain the balance between power supply and load demand, a load demand–power generation equivalent forecasting method is proposed to improve the strategy through replacing information of load demand by predicted power output, which removes the load prediction center and load sensor devices. The data of historical power generation, which is used for prediction, has already satisfied the balance constraint between power supply and load demand. Therefore, when the balance between the real power output and the predicted power output is gained, the balance constraint of power supply and load demand is achieved. Meanwhile, the uncertainty and forecasting errors of renewable generation are taken into account in the cost functions to optimize the expense of DG operation comprehensively. Then, the proposed algorithm is expounded in detail and the convergence is proved by eigenvalue perturbation theory. Finally, various cases are simulated to verify the accuracy and effectiveness of the proposed method. In summary, the proposed method are effective tools for DGs economic power allocation and the decentralization of microgrid system.

Ecological coherence of Marine Protected Areas: New tools applied to the Baltic Sea network

AbstractSpatial connectivity is an essential process to consider in the design and assessment of Marine Protected Areas (MPAs). To help maintain and restore marine populations and communities MPAs should form ecologically coherent networks. How to estimate and implement connectivity in MPA design remains a challenge.Here a new theoretical framework is presented based on biophysical modelling of organism dispersal, combined with a suite of tools to assess different aspects of connectivity that can be integrated in MPA design. As a demonstration, these tools are applied to an MPA network in the Baltic Sea (HELCOM MPA).The tools are based on the connectivity matrix, which summarizes dispersal probabilities, averaged over many years, between all considered areas in the geographic target area. The biophysical model used to estimate connectivity included important biological traits that affect dispersal patterns where different trait combinations and habitat preferences will produce specific connectivity matrices representing different species.Modelled connectivity matrices were used to assess local retention within individual MPAs, which offers indications about the adequacy of size when MPAs are considered in isolation. The connectivity matrix also provides information about source areas to individual MPAs, e.g. sources of larvae or pressures such as contaminants. How well several MPAs act as a network was assessed within a framework of eigenvalue perturbation theory (EPT). With EPT, the optimal MPA network with respect to connectivity can be identified. In addition, EPT can suggest optimal extensions of existing MPA networks to enhance connectivity. Finally, dispersal barriers can be identified based on the connectivity matrix, which may suggest boundaries for management units.The assessment of connectivity for the HELCOM MPA are discussed in terms of possible improvements, but the tools presented here could be applied to any region.

Leader-follower consensus on activity-driven networks.

Social groups such as schools of fish or flocks of birds display collective dynamics that can be modulated by group leaders, which facilitate decision-making toward a consensus state beneficial to the entire group. For instance, leaders could alert the group about attacking predators or the presence of food sources. Motivated by biological insight on social groups, we examine a stochastic leader-follower consensus problem where information sharing among agents is affected by perceptual constraints and each individual has a different tendency to form social connections. Leveraging tools from stochastic stability and eigenvalue perturbation theories, we study the consensus protocol in a mean-square sense, offering necessary-and-sufficient conditions for asymptotic stability and closed-form estimates of the convergence rate. Surprisingly, the prediction of our minimalistic model share similarities with observed traits of animal and human groups. Our analysis anticipates the counterintuitive result that heterogeneity can be beneficial to group decision-making by improving the convergence rate of the consensus protocol. This observation finds support in theoretical and empirical studies on social insects such as spider or honeybee colonies, as well as human teams, where inter-individual variability enhances the group performance.

Reductional Method in Perturbation Theory of Generalized Spectral E. Schmidt Problem

In this a paper perturbations of multiple eigenvalues of E. Schmidt spectral problems is considered. At the usage of the reductional method suggested in the articles [10, 11] the investigation of the multiple E. Schmidt perturbation eigenvalues is reduced to the investigation of perturbation of simple ones. At the end, as application of the obtained results the problem about the boundary perturbation for the system of two Sturm-Liouville problems with E. Schmidt spectral parameter is considered.

Perturbation theory for Hermitian quadratic eigenvalue problem – damped and simultaneously diagonalizable systems

The main contribution of this paper is a novel approach to the perturbation theory of a structured Hermitian quadratic eigenvalue problems (λ2M+λD+K)x=0. We propose a new concept without linearization, considering two structures: general quadratic eigenvalue problems (QEP) and simultaneously diagonalizable quadratic eigenvalue problems (SDQEP). Our first two results are upper bounds for the difference |∥X2*MX˜1∥F2−∥X2*MX1∥F2|, and for ∥X2*MX˜1−X2*MX1∥F, where the columns of X1=[x1,…,xk] and X2=[xk+1,…,xn] are linearly independent right eigenvectors and M is positive definite Hermitian matrix. As an application of these results we present an eigenvalue perturbation bound for SDQEP. The third result is a lower and an upper bound for ∥sinΘ(X1,X˜1)∥F, where Θ is a matrix of canonical angles between the eigensubspaces X1 and X˜1,X1 is spanned by the set of linearly independent right eigenvectors of SDQEP and X˜1 is spanned by the corresponding perturbed eigenvectors. The quality of the mentioned results have been illustrated by numerical examples.

Passivity Enforcement of FDNEs via Perturbation of Singularity Test Matrix

This paper presents a fast and efficient alternative method to enforce passivity of frequency-dependent network equivalents (FDNE). Based on the eigenvalue perturbation theory, the proposed technique relocates the eigenvalues of the half-size singularity matrix by iteratively perturbing the residues of the FDNE model until the rational function becomes passive. It is demonstrated that the proposed technique achieves significant computational savings compared to traditional techniques. Three illustrative case studies involving different conditions of passivity violations are presented to validate the proposed method. Its effectiveness is demonstrated via comparison with two commonly used techniques, i.e., Fast Residue Perturbation (FRP) and Hamiltonian Matrix Perturbation (HMP).

Consensus Over Activity-Driven Networks

The problem of self-coordination of a network of dynamical systems toward a common state is often referred to as the consensus problem. In view of its wide range of applications, the consensus problem has been extensively studied in the past few decades. However, most of the available results focus on static networks, challenging our mathematical understanding of coordination in temporal networks. In this article, we study discrete-time stochastic consensus over temporal networks, modeled as activity-driven networks. In this paradigm, each node has a specific tendency to create links in the network, measured through an activity potential. Differences in the activity potential of nodes favor the evolution of heterogeneous networks, in which some nodes are more involved in the process of information sharing than others. Through stochastic stability theory, we characterize the expected consensus state, which is found to be dominated by low-activity nodes. By further leveraging eigenvalue perturbation techniques, we derive a closed-form expression for the convergence rate in a mean-square sense, which points at a detrimental effect of moderate levels of heterogeneity for large networks. Simulations are conducted to support and illustrate our analytical findings.

Updating boundary conditions for bridge structures using modal parameters

A method for updating the additional rotational and horizontal constraints of bridge boundary conditions using modal parameters is proposed. The updating method is expressed as an optimization problem whose main goal is to iteratively minimize modal parameter residuals between the analytical model and the actual bridge structure. The weighted least-square method is utilized for solving the problem, and first-order perturbations of eigenvalues and eigenvectors are introduced in order to improve efficiency. Corresponding programs for optimization have been written in the MATLAB software package. To validate the proposed method, a maglev railway girder finite element (FE) model was taken as a numerical example. The result shows that the additional boundary constraints of the girder have been successfully updated within just four to seven iterations through only the first few modal parameters. Meanwhile, the sensitivity study on noise shows fairly good robustness of the method. According to prestressed concrete (PC) beams of the straddle-type monorail transit system in Chongqing City, based on on-site natural vibration test results and FE model calculations, the additional torsional constraints stiffnesses were successfully updated. The updated FE model with additional torsional constraints can be a benchmark model for use in damage identification based on modal parameters.

Dynamically encircling exceptional points in a three-mode waveguide system

Dynamically encircling exceptional points (EPs) in non-Hermitian systems has attracted considerable attention recently, but all previous studies focused on two-state systems, and the dynamics in more complex multi-state systems is yet to be investigated. Here we consider a three-mode non-Hermitian waveguide system possessing two EPs, and study the dynamical encircling of each single EP and both EPs, the latter of which is equivalent to the dynamical encircling of a third-order EP that has a cube-root behavior of eigenvalue perturbations. We find that the dynamics depends on the location of the starting point of the loop, instead of the order of the EP encircled. Compared with two-state systems, the dynamical processes in multi-state systems exhibit more non-adiabatic transitions owing to the more complex topological structures of energy surfaces. Our findings enrich the understanding of the physics of multi-state non-Hermitian systems and may lead to the design of new wave manipulation schemes.

Coherency Identification and Aggregation in Grid-Forming Droop-Controlled Inverter Networks

There is an increasing need to apply rigorous model-order-reduction techniques in the analysis of large-scale networks of inverter-based distributed generation resources due to the limitations of existing simulation tools. Various coherency-based aggregation techniques have long been used to construct reduced-order dynamic models of large-scale synchronous machine (SM) networks. Such techniques have the advantage of preserving the nonlinear nature of the dynamic model throughout the order-reduction process, enabling the efficient and accurate analysis of large-scale network dynamics during large disturbances such as fault events. This paper proposes the application of a rigorous coherency-based aggregation technique to the analysis of large-scale networks of grid-forming droop-controlled inverters. A rapid and powerful generalized eigenvalue perturbation technique for coherency identification, previously only applied to SM networks, is adapted to grid-forming droop-controlled inverter networks. The resulting reduced-order models are physically insightful and are capable of accurately reproducing the system response in the aftermath of large disturbances. For some networks, a rigorously-derived condition of coherency can be difficult to achieve, given the expected range of L–C–L filter impedances. To remedy this limitation, the potential for high-bandwidth inverter control to enforce the conditions that allow for coherency of droop-controlled inverters has been investigated and confirmed using a controller hardware-in-the loop testbed. Using this approach, the use of simple nonlinear aggregate inverter models to accurately model large sections of the inverter network can be more rigorously justified.