The volumetric singular value and robustness of feedback control systems

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 μ_υ. In contrast to the theory underlying the structured singular value μ, the volumetric theory includes no implicit assumption that all components Δ_i of the uncertainty Δ are expanded by the same factor r⩾0. In making the transition from stability to instability, we allow for separate bounds r_i for each component Δ _i of Δ. Within this new framework, it becomes possible to systematically study the tradeoffs associated with various uncertainty components. To this end, given a complex n×n matrix M and a positive diagonal matrix R of uncertainty bounds, we provide a definition of μ_υ(M) which involves maximization of the natural volumetric measure vol(R)≐(det R)¹n/ subject to the usual stability preservation constraint det(I+MΔ≠0). From a computational point of view, it turns out that μ_υ(M) enjoys many of the nice properties enjoyed by μ(M)