The Consensus Problem Of Nonlinear Coupling Systems

The nonlinear coupling system is defined as two or more than two nonlinearsubsystems coupling together or the coupling way among these nonlinearity. How tomake sure these systems reach consensus is the topic of this paper. In this field, westudied modeling shape memory effect by dynamical neural networks, chaossynchronization between non-autonomous master system and autonomous slave system,neutral coupling complex networks.The main contents of this dissertation are as follows:①Dynamical neural network (DNN) couples unknown function. In the project ofmodeling and controlling the smart materials and structure, regarding the shape memoryalloy’s hysteresis behavior as unknown function, coupling the dynamic neural networkand the unknown function, make sure that the two systems could reach consensus andthe dynamical neural network could accurately model the hysteresis behavior. That is,the consensus between the DNN and the unknown function could be reached withreasonable accuracy. When predicting voltage, the DNN acts as the inverse function ofthe shape memory effect. This kind of DNN is called IDNN. Based on the IDNN, weproposed an adaptive inverse control strategy which driven the shape memory alloyaccurately tracking the desired command.②Time delay dynamic neural network (TDDNN) couples unknown function.Considering the past states’ effect on the hysteresis, a new TDDNN which introduces atime delay between the input and output response is proposed for modeling thehysteresis online. Experimental results demonstrated the time delay’s importance andthe effectiveness of the TDDNN. That is, the consensus between the TDDNN and theunknown function could be reached with reasonable accuracy.③Autonomous slave system couples non-autonomous master system. In thefield of robust synchronization of chaotic systems with unknown phase term in thetriangular function of master chaos system, through using the properties of thetriangular function, a novel slave system whose dimension is larger than the mastersystem is proposed. Most literatures investigated the synchronization effects with theassumed phase difference and the assumed amplitude of the sinusoidal forcing term.With unknown phase difference and unknown amplitude of the sinusoidal forcing termin the master system, numerical simulations show that the effectiveness of the proposed and novel slave system. That is, the consensus between the master system and the novelslave system could be reached with reasonable accuracy.④The neutral coupling among nodes in complex networks. The coupling waysnot only include the past states of one node’s effect on each other, but also include thederivative past states of one node’s effect on each other. In other words, the complexnetworks not only consider the delay coupling but also the neutral delay coupling.Using the transformation, we can transform the synchronization problem intostabilization problem. Based on these new complex models, we derive asymptotical andexponential criteria via delay fraction approach. Numerical examples are given toillustrate the effectiveness of our scheme, which shows that our results are better thanthat of the recent proposals.