Necessary and Sufficient Conditions for Incremental Stability of Feedback Systems Using Integral Quadratic Constraints

This paper addresses the robust incremental stability of feedback interconnections between a linear timeinvariant (LTI) plant and a nonlinear dynamic uncertainty. Traditional integral quadratic constraint (IQC) methods rely on homotopy arguments and require stability over a continuous path of uncertainties, leading to conservatism. We propose a novel criterion that avoids this limitation by leveraging a robust version of Finsler's lemma and a global inverse function theorem. The result is a set of necessary and sufficient conditions for robust incremental stability based solely on local linearizations, enabling certification for disconnected or non-star-shaped uncertainty sets. For specific uncertainty classes, such as LTI operators and memoryless nonlinearities with symmetric Jacobians, the conditions simplify and reduce computational burden. We demonstrate the effectiveness of the approach on time-delay systems with uncertain bounded delays. Numerical results show tight, non-conservative bounds where standard IQC methods fail.