Conservatism of the standard upper bound test: Is sup(${{\bar \mu } \mathord{\left/ {\vphantom {{\bar \mu } \mu }} \right. \kern-\nulldelimiterspace} \mu }$) finite? Is it bounded by 2?

Abstract

This short note is about the conservatism of the standard upper bound, $\bar \mu$relative to the complex structured singular value, μ.This problem is first formulated by John Doyle. To follow the present note, only mathematical definitions of μ, and $\bar \mu$ are necessary. For a tutorial introduction about complex μ and its importance in system analysis and design, see [10]. If M∈ ℂ^{n ×n}, then

$$\mu \left( M \right): = \sup \left\{ {p\left( {M\Delta } \right):\,\Delta = diag\left( {{\delta _1}, \ldots ,{\delta _n}} \right),\,{\delta _1}, \ldots ,{\delta _n} \in D} \right\}\,,$$

$$\bar \mu \left( M \right): = \inf \left\{ {\bar \sigma \left( {{D^{ - 1}}MD} \right):D = diag\left( {{d_1}, \ldots ,{d_n}} \right),{d_1}, \ldots ,{d_n} > 0} \right\}.$$

.