Eigenvector Sensitivity: A Kharitonov Result

Abstract

This study is motivated by a need to effectively determine the difference between a system fault and normal system operation under parametric uncertainty using eigenstructure analysis. This involves computational robustness of eigenvectors in linear state space systems dependent upon uncertain parameters. The work involves the development of practical algorithms which provide for computable robustness measures on the achievable set of eigenvectors associated with certain state space matrix constructions. To make connections to a class of systems for which eigenvalue and characteristic root robustness are well understood, the work begins by focusing on companion form matrices associated with a polynomial whose coefficients lie in specified intervals. The work uses an extension of the well known theories of Kharitonov that provides computational efficient tests for containment of the roots of the polynomial (and eigenvalues of the companion matrices) in "desirable" regions, such as the left half of the complex plane.Keywords: Eigenvectors; Kharitonov; Robustness BackgroundThis body of work extends the concept of fault detection to a condition on the robustness of eigenvectors to modeled parametric uncertainty. This can be thought of as a problem of finding the space of eigenvectors relative to the space of system parameters. We consider a set of linear functions formulated as linear matrix equations. Many systems can be described as having a nominal or expected parametric structure. However, no physical system can be realistically described with only certain parameters. In fact, one may realistically describe a system as having parameters that fall within some interval. For any real or complex interval, this becomes a daunting task to generate mappings from the system parameters to eigenvectors. Much work has been accomplished in the area of complexity analysis to analyze this very structure to categorize it as an NP-hard problem. See where  is a diagonal matrix of eigenvalues. Eigenvector robustness is a relatively recent area of exploration with the bulk of the existing related literature in the area of eigenstructure assignment and stability under parametric uncertainty; see [6] and [7]. Kato [8] considers perturbations in eigenspace and provides a functional relationship between eigenvalue perturbations (sufficiently small) and corresponding eigenvectors but this functional relationship requires the computation of a matrix inverse and definite integral. The following sections detail the study of achievable sets of eigenvectors which are associated with sets of uncertain matrices, examining special features in various matrix constructions. We focus on matrix constructions based on interval polynomials. We seek a new result which provides for a robustness measure on eigenvector variation over sets of matrices, but does not require matrix integration, extensive Monte Carlo and other sampling methods determine the set of