On the conservatism of upper bound tests for structured singular value analysis
Onur Toker · 2002
Because of the well-known difficulties of exact real/mixed /spl mu/ computation, efficiently computable upper bound tests are of great importance for both /spl mu/ analysis and synthesis. However, another important issue is the introduced conservatism, and in this paper, we consider the worst case conservatism of these efficiently computable upper bound tests for real/mixed /spl mu/ analysis. It shown that any upper bound test, /spl mu/~, satisfying the condition /spl mu/(M)/spl les//spl mu/~(M)/spl les/ C dim(M)/sup 1-/spl epsiv// /spl mu/(M), must itself be /spl Nscr//spl Pscr/-hard to compute. In other words, unless "/spl Pscr//spl ne//spl Nscr//spl Pscr/" is false, for any efficiently computable upper bound test, /spl mu/~, the worst case gap between the upper bound and the exact /spl mu/ is not bounded by /spl Oscr/(dim(M)/sup 1-/spl epsiv//). Therefore, unless "/spl Pscr//spl ne//spl Nscr//spl Pscr/ is false, no matter which efficiently computable upper bound test we choose, there will be examples with arbitrarily large /spl mu/~//spl mu/ ratios, i.e. with arbitrarily large conservatism.