Singular value analysis of linear systems

Abstract

This paper is a condensed version of a recent two part report on applied analysis of linear multivariable systems [1], [2]. The basic thrust of the work is this: Singular value analysis, of proven power in applied analysis of systems of linear equations, can be applied in a very direct way to systems of linear differential equations as well. The essential fact revealed and exploited is that norm characteristics (l2 norm) of relevant maps are reflected, with no distortion, by norm characteristics of associated grammian matrices. The basic tools developed in the beginning of the paper are used to develop a framework where "near" uncontrollability/unobservability is well defined and is directly related to near redundancy of state variables. In this setting there is continuity between Kalman's minimal realization theory and model reduction based on elimination of nearly redundant state variables. One can view this process as the application of the mechanics of Kalman decomposition using working values of the relevant subspaces.