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A collection of converse theorems for integral quadratic constraints (IQCs)is established for linear time-invariant systems. It is demonstrated that whena system interconnected in feedback with an arbitrary system satisfying an IQCis stable, then the given system must necessarily satisfy the complementaryIQC. These theorems are specialized to derive multiple versions of conversepassivity results. They cover standard notions of strict passivity as well aspassivity indices that characterize the trade-offs between passivity surplusand deficit. Converse frequency-weighted small-gain and passivity theorems arealso established.

Converse Theorems for Integral Quadratic Constraints