A Dissipation Inequality Formulation for Stability Analysis with Integral Quadratic Constraints

Integral quadratic constraints (IQCs) provide a general framework for robustness analysis of feedback interconnections. The main IQC stability theorem by Megretski and Rantzer was formulated with frequency domain conditions that depend on the IQC multiplier. Their proof of this theorem uses a homotopy method and operator theory. An interesting aspect of this theory is that input/output stability (defined as uniformly bounded gain over all finite horizons) is established using integral constraints that only hold, in general, on infinite time horizons. The use of IQCs that only hold over infinite time horizons is related to the use of noncausal multipliers in absolute stability theory. This paper shows that if the conditions of the IQC stability theorem are satisfied by any rational IQC multiplier then a dissipation inequality is satisfied by a quadratic storage function. This provides a new interpretation for IQC analysis in terms of quadratic storage functions and a causal, finite-horizon dissipation inequality.