Guaranteed Lower and Upper Bounds on the Finite-Frequency H2 Norm of Uncertain Linear Systems

The $H_{2}$ norm plays a key role in control applications when the input of the system is a random noise. It is therefore important to evaluate how this metric is affected by uncertainties in the model. Different approaches have been proposed in the literature to compute an upper bound on the worst-case $H_{2}$ norm of an uncertain system. No method, however, is available to check if the $H_{2}$ norm remains larger than a given threshold on an entire set of uncertainties. In this context, the present paper introduces a sufficient condition to compute a guaranteed lower bound on the $H_{2}$ norm of an uncertain system based on Linear Matrix Inequalities (LMIs). G-scaling matrices are also introduced to improve the accuracy of the computed bounds in the presence of real parametric uncertainties. The theoretical results are eventually implemented and applied to a test example.