Studies On Synchronization Of Complex Dynamical Networks Based On Impulsive Control Strategies

Complex dynamical networks (CDNs) describe a class of large-scale dynamical systems formed by multiple coupling nodes according to certain topological structure. Synchronization is one of the major collective behav-iors of CDNs. For the case where the CDNs cannot achieve synchroniza-tion by itself, the designing of appropriate collective controls such that the controlled CDNs reach state synchronization has become an important top-ic in the control of complex systems. In recent years, as the digital control technology gradually substitutes the analog control technology, the impul-sive control technology based on sampled-data at discrete instants has caught the attention of many scholars. Since the existing time-invariant Lyapunov functions based analysis methods cannot fully exploit the hybrid structure characteristics of the impulsively controlled CDNs, the resulting synchro-nization criteria have certain conservatism and cannot be used to solve the problems of impulsive controllers synthesis and impulsive H∞ control. Aim-ing at these problems, in this thesis, the weighted impulse-time-dependent Lyapunov functions/functional are introduced. Based on impulsive control strategies, the synchronization control problems of CDNs are revisited, and several novel impulsive synchronization criteria are derived. The main results derived in this thesis are listed as follows:(1) Based on the decentralized impulsive control strategy, the synchro-nization control problem of a class of CDNs with respect to the given goal signals is studied. Firstly, the problem of decentralized impulsive synchro-nization is transformed into the stabilization problem of the synchroniza-tion error system via decentralized impulsive control. Then, with respect to the fast-varying coupling delays and the slowly-varying coupling delays, the weighted impulse-time-dependent Lyapunov function/functional based anal-ysis techniques are introduced. Two criteria of decentralized impulsive syn-chronization are established in terms of linear matrix inequalities (LMIs). These new criteria remove the restriction that the spectral radius of impul-sive gain matrices must be strictly less than 1 imposed by the existing results, and can be used to solve the synthesis problem of decentralized impulsive controllers.(2) Based on the distributed impulsive control strategy, the impulsive H∞ synchronization problem for a class of CDNs with coupling delays and external disturbance is investigated. Firstly, the distributed impulsive H∞ syn-chronization problem is transformed into the impulsive H∞ control problem of the synchronization error system with lower dimension. In the case with fast-varying coupling delays, a Razumikhin-type analysis technique is devel-oped to estimate the quadratic performance index; in the case with slowly-varying coupling delays, a impulse-time-dependent Lyapunov functional is introduced to estimate the two performance index. Two criteria for finite L2-gain of synchronization error system are established. According to the criteria, for given L2-gain level, two algorithms for designing the distributed impulsive synchronization controllers with the minimum coupling strength are proposed.(3) The robustness of the distributed impulsive synchronization scheme with respect to small impulse input delays is studied. It is assumed that there may be input delays in the impulsive controllers. An impulse-time-dependent Lyapunov function is introduced. Through integrating the Razumikhin-type technique, Newton-Leibniz formula, and convex combination technique, a robust distributed impulsive synchronization criterion with respect to small impulse-input delay is derived.(4) The distributed sampled-data synchronization problem of a class of CDNs with coupling delays is considered. By employing a sampled-data-time-dependent Lyapunov functional based method combined with the used of the Jensen inequality for tackling the integral term, two criteria for dis-tributed sampled-data synchronization are obtained. Furthermore, an opti-mization algorithm for the designing of the distributed sampled-data con-trollers with the minimum coupling strength is presented.