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A cluster consensus algorithm for nonlinear multi-agent systems under directed graph topology is proposed in this paper. Cluster consensus is the convergence of states/outputs of agents in the same cluster to consistent values which are different from those of other clusters. Cluster consensus has been obtained based on Lyapunov stability and matrix theory in terms of some sufficient conditions. A feedback control law is provided using Linear Matrix Inequality (LMI) to achieve cluster consensus for multi-agent systems. Moreover, cluster consensus for nonlinear multi-agent systems in the presence of time delay has been studied in this paper. Finally, simulation results are presented for different number of clusters to validate theoretical analysis. Examples are provided for first-order and second-order and also general high-order systems. Furthermore, first-order system with time delay is simulated for a single-link flexible joint manipulator.

A novel synthesis of 2-phenylH-imidazio[1,2-α] pyridines is described from a one-pot, three-component reaction between pyridine, guanidine (urea or thiourea) and α-bromoketones under microwave irradiation and solvent-free conditions in excellent yields.

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