On the conservatism of upper bound tests for structured singular value analysis

Because of the well-known difficulties of exact real/mixed p computation [19, 16, 31, efficiently computable upper bound tests are of great importance for both p 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 p analysis. It shown that any upper bound test, p, satisfying the condition p ( M ) _< p ( M ) 5 C dim(M)l-€ p ( M ) , must itself be NP-hard to compute. In other words, unless LLP # NP” is false, for any efficiently computable upper bound test, p, the worst case gap between the upper bound and the exact p is not bounded by O(dim(M)’-‘). Therefore, unless “P # NP” is false, no matter which efficiently computable upper bound test we choose, there will be examples with arbitrarily large j i /p ratios, i.e. with arbitrarily large conservatism.