Global asymptotic stability of pendulum-like system with state delay

This paper considers robust global asymptotic stability for the delayed nonlinear pendulum-like systems with ploytopic uncertainties. The delay-dependent criteria, guaranteeing the global asymptotic stability for the pendulum-like systems with state delay, for the first time, are established in terms of linear matrix inequalities (LMIs) which can be checked by resorting to recently developed algorithms solving LMIs. Furthermore, based on the derived delay-dependent global asymptotic stability results, LMI characterizations are developed to ensure the robust global asymptotic stability for delayed pendulum-like systems under convex polytopic uncertainties. The new extended LMIs involve no product of the Lyapunov matrix and the system matrices. It enables one to check the global asymptotic stability by using parameter-dependent Lyapunov methods. Finally, a concrete application to phase-locked loop (PLL) illustrates the validity of the proposed approach.