Is set modeling of white noise a good tool for robust H2 analysis?

A new deterministic characterization of the H₂ norm has recently been proposed, using a new norm (||.||w_η), based on (approximate) set membership modeling of white noise. The main result of Paganini (1995, 1996) shows that under mild conditions, for a fixed system the gap between the H₂ and W_η norms can be made arbitrarily small. Motivated by these results it has been argued that the ||.||w_η norm provides a useful tool for analyzing robust H₂ controllers, specially since in this context LMI based necessary and sufficient conditions for robust performance are available. However, we show with an example involving a very simple plant that the worst case ||.||w_η^m norm can be conservative by at least a factor of √m (where m denotes the dimension of the exogenous signal) for the original robust H₂ problem. Thus, at this point the problem of finding non-conservative bounds on the worst H₂ norm under LTI or slowly-varying LTV perturbations still remains open