Connections Between Georgiou And Smith'S Robust Stability Type Theorems And The Nonlinear Small-Gain Theorems

Nonlinear robust stability theorems due to Georgiou and Smith use the concept of the nonlinear gap metric. While the link between gap metric uncertainty and the small-gain theorem is clearly drawn already in Georgiou and Smith's original seminal work, the contribution of this paper is to provide an extensive treatise on the relation between these robust stability type theorems and the nonlinear small-gain type theorems in multiple scenarios. Here a special case of the nonlinear small-gain theorem using gain functions is shown to be equivalent to a fundamental robust stability theorem of Georgiou and Smith for feedback systems. Since existence of solutions is a fundamental requirement of a feedback system and typically it is harder to establish than the uniqueness of solutions, we also show that both global and local forms of the nonlinear small-gain theorem which establish existence and boundedness properties simultaneously imply the corresponding types of Georgiou and Smith's robust stability theorem in gain function formulation. Moreover, if we only consider the boundedness for both theorems in the local setting, we show that the equivalence between them can be similarly established as in the global setting.