Pendulum-Like System And Coupling Synchronization Based On Pendulum-Like Systme

The pendulum-like system is a special nonlinear system with multiple equilibria. Its dynamic properties such as dichotomy property, gradient-like property, Lagrange stability and the periodic solutions, have attracted atten-tions of many researchers.This paper mainly investigates the nonexistence of second periodic solu-tions for the pendulum-like systems and the synchronization of complex networks consisted of the pendulum-like systems.First, based on KYP lemma, the frequency-domain condition about the nonexistence of second periodic solutions for the pendulum-like systmes can be converted to the time-domain condition. According to this time-domain condition, the desigh methods of the state-feedback controller and the ob-server-based state-feedback controller are given.Then, the design method of decentralized controller to guarantee the syn-chronization of complex networks consisted of pendulum-like systems is developed. And the controller is derived by studying the asymptotic problem of the corresponding error systems.Finally, the synchronization of complex network composed of multi-input multi-output pendulum-like systems is studied. The novel delay-dependent synchronous criterion of the complex networks with time-invarying delays is obtained through considering the asymptotic problem of the corresponding error systems. The characteristic of this method is that no linearization tech-nique is involved but turning the high dimensional error system into the multiple low dimensional systems so that it can reduce the conservatism. And a simple synchronous condition in form of LMI is given. For synchro-nization of complex network with time-varying coupling delays, asymptotic criterion of the low dimensional error system is given based on appropriate Lyapunov function. It is not involved the linearization technique either. Numerical examples illustrate the effectiveness of the proposed criteria.