Leader-Following Consensus of Multi-order Fractional Multi-agent Systems

The current study investigates the leader-following consensus problem for fractional-order multi-agent systems with different fractional orders under a fixed undirected graph. A virtual leader with the desired path is assumed, while the agents are chosen as fractional-order integrators with various orders. It is proved that the leader-following consensus problem for this multi-agent system is equivalent to the stability analysis of a multi-order fractional system. At first, the Laplace transform is employed to verify the asymptotic stability of a particular case of multi-order fractional systems. It is shown that if the state matrix is negative definite and a certain inequality between the fractional orders is met, the mentioned system is asymptotically stable. This inequality can be easily checked without any need for complex calculations. Accordingly, it is demonstrated that if a certain inequality is met among the fractional orders of a multi-order multi-agent system, the leader-following consensus of the mentioned heterogeneous multi-agent system can be realized. Numerical examples demonstrate the accuracy of the established leader-following consensus protocol.