The volumetric singular value and robustness of feedback control systems

Abstract

This paper provides an overview of results which are fully described in the authors's recent paper (Techn. Report ECE-93-9, University of Wisconsin-Madison (1993)). The focal point is a new concept-the volumetric singular valve /spl mu//sub /spl upsi//. In contrast to the theory underlying the structured singular value /spl mu/, the volumetric theory includes no implicit assumption that all components /spl Delta//sub i/ of the uncertainty /spl Delta/ are expanded by the same factor r/spl ges/0. In making the transition from stability to instability, we allow for separate bounds r/sub i/ for each component /spl Delta//sub i/ of /spl Delta/. Within this new framework, it becomes possible to systematically study the tradeoffs associated with various uncertainty components. To this end, given a complex n/spl times/n matrix M and a positive diagonal matrix R of uncertainty bounds, we provide a definition of /spl mu//sub /spl upsi//(M) which involves maximization of the natural volumetric measure vol(R)/spl esdot/(det R)/sup 1/n/ subject to the usual stability preservation constraint det(I+M/spl Delta//spl ne/0). From a computational point of view, it turns out that /spl mu//sub /spl upsi//(M) enjoys many of the nice properties enjoyed by /spl mu/(M). >