Parameter-dependent Lyapunov function method for a class of uncertain nonlinear systems with multiple equilibria

First, a new linear matrix inequality (LMI) characterization of extended strict positive realness is presented for linear continuous-time systems. Then a class of nonlinear systems with multiple equilibria subject to polytopic uncertainty is addressed by the parameter-dependent Lyapunov function method. New sufficient conditions for global convergence are presented. This allows the Lyapunov function to be parameter dependent. Furthermore, an LMI-based controller design method is also given, and reduced-order controllers can be designed by performing a structural constraint on the introduced slack variables. Several numerical examples are included to demonstrate the applicability of the proposed method.