Feedback Stability Analysis via Frequency Dependent Constraints

We establish robust stability of feedback interconnections of linear time-invariant systems with potentially marginally stable open-loop poles via frequency-varying quadratic constraints. These constraints may be used to characterise numerous system properties, such as small-gain, passivity, negative imaginariness, and their weighted variants, that are manifested on different band-limited frequencies. Such systems arise in practical mechanical settings where force actuators and displacement/velocity sensors are not colocated. We demonstrate that frequency-limited system properties may be characterised using the powerful Iwasaki–Hara lemma in terms of convex feasibility problems involving linear matrix inequalities, thereby complementing the utility of the Iwasaki–Hara lemma for robust feedback stability analysis involving mixed system properties on the frequency axis.