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

Because of the well-known difficulties of exact real/mixed μ computation, efficiently computable upper bound tests are of great importance for both μ 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 μ analysis. It shown that any upper bound test, μ¯, satisfying the condition μ(M)⩽μ¯(M)⩽ C dim(M)^1-ε μ(M), must itself be 𝒩𝒫-hard to compute. In other words, unless “𝒫≠𝒩𝒫” is false, for any efficiently computable upper bound test, μ¯, the worst case gap between the upper bound and the exact μ is not bounded by 𝒪(dim(M)^1-ε). Therefore, unless “𝒫≠𝒩𝒫 is false, no matter which efficiently computable upper bound test we choose, there will be examples with arbitrarily large μ¯/μ ratios, i.e. with arbitrarily large conservatism