Recent Advance in Synchronization of Lur’e Dynamical Networks via Pinning Control

This book studies the synchronisation of Lur’e complex networks with different problems. In order to synchronise the Lur’e complex networks, various forms of pinning control techniques are used throughout the entire work. Sufficient requirements for different types of synchronisation on the Lur’e dynamical networks are obtained based on the Lyapunov method. And to show the effectiveness of the key outcomes, numerical simulations are presented. In conclusion, the following four categories are primarily found in this book.

Firstly, it considers the question of cluster synchronisation under pinning control for delay coupled Lur’e dynamic networks. Only those Lur’e systems that have direct connections to the other clusters are artificially managed in view of cluster topologies in the dynamical network. The convex combination theorem, S-procedure and the concept of delay rate are derived, according to delay interval dividing methods, sufficient conditions for local and global cluster synchronisation of the Lur’e dynamic networks are derived, respectively, by applying the pinning controller.

Secondly, it concerns the question of finite-time cluster synchronisation for complex networks consisting of discontinuous Lur’e structures. The discontinuous nonlinear functions are transformed into set-valued functions by incorporating the idea of Filippov differential inclusions. Assumptions are made for measurable functions corresponding to discontinuous nonlinear functions chosen in accordance with the theorem of measurable selection. Some conditions for achieving cluster synchronisation of the identical and non-identical Lur’e networks are obtained in finite time through the design of feedback finite time controllers. In addition, based on the finite time stability principle, the settling time for cluster synchronisation is assessed.

Thirdly, cluster synchronisation problems are explored with time-varying delay for nonlinearly coupled Lur’e dynamic networks. For the first time, an asymmetrical edge-based distributed adaptive pinning control protocol is proposed to obtain sufficient coupling weights using the local information of the nodes in view of the nonlinearly coupled Lur’e networks. The couplings belonging to a spanning tree are absolutely pinned in each cluster. Based on S-procedure, Kronecker product and Lyapunov stability theory, appropriate cluster synchronisation conditions for the identical and non-identical Lur’e network are extracted, respectively.

Finally, from the point of view of leader-following, the issue of the exponential synchronisation of coupled stochastic Lur’e dynamical networks with multiple time-varying delays is investigated. A sort of distributed impulsive controls with a constant impulsive effect are designed to synchronise the stochastic Lur’e dynamical network. By applying the comparison principle, the average impulsive period and the extended formula for the variance of parameters, appropriate conditions are extracted for the effective synchronisation of the Lur’e dynamical networks. Furthermore, according toThe definition of the impulsive solution equation is obtained by the exponential convergence rate. And the lower limit of time-varying delays is shown to lead to a faster synchronisation speed.

Author (s) Details

Ze Tang
Associate Professor, School of Internet of Things Engineering, Jiangnan University, Wuxi 214122, PR China.

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