Cluster synchronisation of non‐linearly coupled Lur'e networks with identical and non‐identical nodes and an asymmetrical coupling matrix

Abstract

In this study, the cluster synchronisation problem of Lur'e networks is focused with non-linear coupling. There are both identical and non-identical nodes in the dynamical system and the coupling matrix is asymmetrical. By using the linear and non-linear negative feedback control schemes, the Lyapunov stability theorem and linear matrix inequality, sufficient conditions are obtained that guarantee the realisation of the cluster synchronisation pattern for all initial values. Numerical simulation results are also given to support the validity of the main results.

Synchronization of nonlinear master–slave systems under input delay and slope‐restricted input nonlinearity

“…The details of the inequality (6) in Assumption 1 can be seen from [24,25]. Assumption 2 can facilitate the design of controller for the synchronization of nonlinear systems under input nonlinearity.…”

“…Theorem 1 provides a synchronization controller design criterion, by selection of an appropriate value of K , that depends on both the upper and the lower bounds of the input delay by using the LK functional and the LPV theories (see, for instance, [13][14][15] and [23][24][25][26][27] and references therein). Theorem 2 is more pragmatic because it allows computation of the controller gain K for delay-rangedependent synchronization of the master-slave systems under slope-restricted unknown input nonlinearity.…”

“…Linear parameter varying (LPV) method is a new and less conservative approach for design, analysis, and synthesis of feedback controllers for nonlinear systems. In the past few years, LPV approach has received more attention as it has been successfully applied to the gain scheduling, H 1 controller design [23,24], system identification, and state feedback control [25,26].…”

“…To design a controller applicable to both small and large input delays, the delay-rangedependent control approach, rather than the conventional delay-dependent and delay-independent methodologies, is adopted by considering time-varying nature of the input delay. Based on an intricate Lyapunov-Krasovskii (LK) functional [15], rigorous matrix algebra, LPV approach [24][25][26], properties of nonlinear dynamics, and Jensen's inequality [27], a state-feedback control design schema is derived. The feedback controller gain for the synchronization controller can be determined by solving matrix inequalities for feasibility problem to achieve asymptotic synchronization of the master-slave systems.…”

Adaptive synchronization of Cohen–Grossberg neural network with mixed time-varying delays and stochastic perturbation

“…[23][24][25]31,32,36,37,40], Different from LMI method, our synchronization criterion does not solve any linear matrix inequality and is easily verified. The adaptive feedback controller is easy to be designed.…”