Robust Feedback Instability Analysis in the $\mathscr {L}_{2}$-Gap

The robustness of instability is investigated for feedback interconnections of rational transfer functions, motivated by the relevance of such in the study of oscillatory dynamics. An analysis is presented in terms of normalized coprime factorizations and the
$\mathscr {L}_{2}$
-gap metric. The main result is a necessary and sufficient condition for feedback instability to persist across a particular uncertainty set for one open-loop component of the interconnection, with the other component fixed. This uncertainty set is a subset of an
$\mathscr {L}_{2}$
-gap ball. It includes the associated
$\nu$
-gap metric ball, and other winding-number based partitions that enable necessity to be established for sufficiently large ball radius.