Robust analysis, sectors, and quadratic functionals

Working from a topological separation framework it is shown that integral quadratic constraints useful for robust analysis must have the form ∫_-∞^∞z*S*diag{I,-I}Szdω, for some invertible (possibly frequency dependent) matrix S. It is further shown that many of the integral quadratic constraints used in robustness analysis may be put into a positivity form with a fixed or known generalized sector transform and a “free” multiplier. The work presented here opens the way to a positivity/linear matrix inequality framework for robust analysis based on quadratic functional constraints