Allowing Nonlinear Stability Margins in Interconnection of iISS Dissipation Inequalities

Abstract

Abstract: This paper tackles the problem of constructing Lyapunov functions of interconnected systems when gain margins are not linear. Nonlinear small-gain criteria such as (i)ISS small-gain theorems allow systems to have gains which are not necessarily linear. However, allowing nonlinear gains does not mean coping with nonlinear gain margins. Gain margins quantify how small the map of the (nonlinear) loop gain is with respect to the identity map. In the integral input-to-state stability (iISS) framework which do not require input-to-state stability (ISS), in contrast to trajectory-based arguments, construction of Lyapunov functions in the literature has been often limited to the linear gain margin case. For constructing Lyapunov functions of interconnected systems whose components are not necessarily ISS and whose linear gain margins allowed to be zero, this paper develops a symmetric formula which has several clear advantages.