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.
Highlights
This body of work extends the concept of fault detection to a condition on the robustness of eigenvectors to modeled parametric uncertainty
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
Property : An interval matrix, A, with property is one having all eigenvalues of A distinct and non-zero over the interval parameter domain of the interval matrix. This implies that a specific continuous eigenvalue path generated through an interval path of the interval polynomial corresponds to a single unique eigenvector path
Summary
This 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. Other works related to the design or analysis of matrix eigenstructures including [3] and [4] consider both eigenvalues and eigenvectors but have approaches based on the structure of the eigenvalue space. We consider direct robustness metrics on the eigenvector space We approach this problem through the use basic linear systems theory starting with the well-known equation relating eigenvalues and eigenvectors, AV V (1). We will use conj to denote complex conjugation and xT to denote the transpose of x where x is a matrix or vector quantity
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