Robust stability analysis using LMIs: Beyond small gain and passivity

A class of linear, time-invariant (LTI) systems, referred to as β-bounded LTI systems, which constrain both the gain and the phase of an LTI system, is introduced in this paper. This characterization corresponds to gain bounded systems, or bounded real systems, for β = 1, and it corresponds to phase bounded, positive real systems for β = 0. State space characterization of β-bounded LTI systems is presented in terms of linear matrix inequalities (LMIs), similar to the LMIs for bounded realness and positive realness. Robust stability results are developed in this paper for β-bounded systems, including stability of the feedback interconnection of β-bounded LTI systems and stability of uncertain systems with β-bounded uncertainties. Small gain and passivity conditions for robust stability are seen as special cases of these results for β-bounded systems. Finally, a convex optimization procedure is presented to select parameters for a tight characterization of uncertain LTI systems in terms of β-boundedness, and the technique is illustrated with a spring-mass-damper example.