Delay and especially delay in the transmission of agents’ information, is one of the most important causes of disruption to achieving consensus in a multi-agent system. This paper deals with achieving consensus in delayed fractional-order multi-agent systems (FOMAS). The aim in the present note is to find the exact maximum allowable delay in a FOMAS with non-uniform delay, i.e., the case in which the interactions between agents are subject to non-identical communication time-delays. By proving a stability theorem, the results available for non-delayed networked fractional-order systems are extended for the case in which interaction links have nonequal communication time-delays. In this extension by considering a time-delay coordination algorithm, necessary and sufficient conditions on the time delays and interaction graph are presented to guarantee the coordination. In addition, the delay-dependent stability region is also obtained. Finally, the dependency of the maximum allowable delay on two parameters, the agent fractional-order and the largest eigenvalue of the graph Laplacian matrix, is exactly determined. Numerical simulation results are given to confirm the proposed methodologies.

This study outlines the necessary and sufficient criteria for swarm stability asymptotically, meaning consensus in a class of fractional-order multi-agent systems (FOMAS) with interval uncertainties for both fractional orders 0 < α < 1 and 1 < α < 2. The constraints are determined by the graph topology, agent dynamics, and neighbor interactions. It is demonstrated that the fractional-order interval multi-agent system achieves consensus if and only if there are some Hermitian matrices that satisfy a particular kind of complex Lyapunov inequality for all of the system vertex matrices. This is done by using the existence condition of the Hermitian matrices in a Lyapunov inequality. To do this, at first it is shown under which conditions a multi-agent system with unstable agents can still achieve consensus. Then, using a lemma and a theory, the Lyapunov inequality regarding the negativity of the maximum eigenvalue of an augmented matrix of a FOMAS is used to find some Hermitian matrices by checking only a limited number of system vertex matrices. As a result, the necessary and sufficient conditions to reach consensus in a FOMAS in the presence of internal uncertainties are obtained according to the Lyapunov inequalities. Using the main theory of the current paper, instead of countless matrices, only a limited number of vertex matrices need to be used in Lyapunov inequalities to find some Hermitian matrices. As a confirmation of the notion, some instances from numerical simulation are also provided at the end of the paper.

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