Bounds on transient instability for complex ecosystems

Stability is a desirable property of complex ecosystems. If a community of interacting species is at a stable equilibrium point then it is able to withstand small perturbations to component species' abundances without suffering adverse effects. In ecology, the Jacobian matrix evaluated at an equilibrium point is known as the community matrix, which describes the population dynamics of interacting species. A system's asymptotic short- and long-term behaviour can be determined from eigenvalues derived from the community matrix. Here we use results from the theory of pseudospectra to describe intermediate, transient dynamics. We first recover the established result that the transition from stable to unstable dynamics includes a region of `transient instability', where the effect of a small perturbation to species' abundances---to the population vector---is amplified before ultimately decaying. Then we show that the shift from stability to transient instability can be affected by uncertainty in, or small changes to, entries in the community matrix, and determine lower and upper bounds to the maximum amplitude of perturbations to the population vector. Of five different types of community matrix, we find that amplification is least severe when predator-prey interactions dominate. This analysis is relevant to other systems whose dynamics can be expressed in terms of the Jacobian matrix. Our results will lead to improved understanding of how multiple perturbations to a complex system may irrecoverably break stability.Comment: 7 pages, two columns, 3 figures; text improved - Accepted for publication on PLoS On

Approximated structured pseudospectra

Pseudospectra and structured pseudospectra are important tools for the analysis of matrices. Their computation, however, can be very demanding for all but small-matrices. A new approach to compute approximations of pseudospectra and structured pseudospectra, based on determining the spectra of many suitably chosen rank-one or projected rank-one perturbations of the given matrix is proposed. The choice of rank-one or projected rank-one perturbations is inspired by Wilkinson's analysis of eigenvalue sensitivity. Numerical examples illustrate that the proposed approach gives much better insight into the pseudospectra and structured pseudospectra than random or structured random rank-one perturbations with lower computational burden. The latter approach is presently commonly used for the determination of structured pseudospectra

A Support Based Algorithm for Optimization with Eigenvalue Constraints

Optimization of convex functions subject to eigenvalue constraints is intriguing because of peculiar analytical properties of eigenvalues, and is of practical interest because of wide range of applications in fields such as structural design and control theory. Here we focus on the optimization of a linear objective subject to a constraint on the smallest eigenvalue of an analytical and Hermitian matrix-valued function. We offer a quadratic support function based numerical solution. The quadratic support functions are derived utilizing the variational properties of an eigenvalue over a set of Hermitian matrices. Then we establish the local convergence of the algorithm under mild assumptions, and deduce a precise rate of convergence result by viewing the algorithm as a fixed point iteration. We illustrate its applicability in practice on the pseudospectral functions.Comment: 18 pages, 2 figure

Dynamics-aware optimal power flow

Abstract — The development of open electricity markets has led to a decoupling between the market clearing procedure that defines the power dispatch and the security analysis that enforces predefined stability margins. This gap results in market inefficiencies introduced by corrections to the market solution to accommodate stability requirements. In this paper we present an optimal power flow formulation that aims to close this gap. First, we show that the pseudospectral abscissa can be used as a unifying stability measure to characterize both poorly damped oscillations and voltage stability margins. This leads to two novel optimization problems that can find operation points which minimize oscillations or maximize voltage stability margins, and make apparent the implicit tradeoff between these two stability requirements. Finally, we combine these optimization problems to generate a dynamics-aware optimal power flow formulation that provides voltage as well as small signal stability guarantees. I

Approximating the Real Structured Stability Radius with Frobenius Norm Bounded Perturbations

We propose a fast method to approximate the real stability radius of a linear dynamical system with output feedback, where the perturbations are restricted to be real valued and bounded with respect to the Frobenius norm. Our work builds on a number of scalable algorithms that have been proposed in recent years, ranging from methods that approximate the complex or real pseudospectral abscissa and radius of large sparse matrices (and generalizations of these methods for pseudospectra to spectral value sets) to algorithms for approximating the complex stability radius (the reciprocal of the $H_\infty$ norm). Although our algorithm is guaranteed to find only upper bounds to the real stability radius, it seems quite effective in practice. As far as we know, this is the first algorithm that addresses the Frobenius-norm version of this problem. Because the cost mainly consists of computing the eigenvalue with maximal real part for continuous-time systems (or modulus for discrete-time systems) of a sequence of matrices, our algorithm remains very efficient for large-scale systems provided that the system matrices are sparse