A remark on abstract multiplier conditions for robustness problems

This paper presents a generalization of recent results of Açıkmeşe and Corless [B. Açıkmeşe, M. Corless, Stability analysis with quadratic Lyapunov functions: Some necessary and sufficient multiplier conditions, Systems Control Letters 57 (2008) 78–94] concerning multiplier conditions of quadratic stability of uncertain/nonlinear systems. Abstract full block S-procedure results are formulated, extending the results of Scherer [C.W. Scherer, LPV control and full block multipliers, Automatica 37 (2001) 361–375] for such cases when the sets representing the uncertainties do not necessarily have subspace structure. The main contribution of the present work is the investigation of the conditions under which the results of Scherer and Açıkmeşe and Corless can be formulated in a unified framework.