Observer design for tracking consensus in second-order multi-agent systems: Fractional order less than two

This technical note studies the leader-following tracking consensus problem of a class of multi-agent systems where the dynamics of the leader is described by second-order systems. In order to track the states of the leader, observers for the followers are designed by fractional-order multi-agent systems where the relative velocity information is unavailable. It is interestingly found that the followers can track the leader with second-order dynamics even if the fractional order is less than two by only using the position information of the local neighbors, which is different from the existing results. A novel fractional-order observer is first proposed, whose order is surprisingly less than the original leader system. It is also shown that leader-following consensus can be ensured if some carefully selected followers are informed and the relative position-based protocols are appropriately designed with fractional order being between one and two. A necessary and sufficient condition for the leader-following consensus in multi-agent systems without control-input delay is proposed. The results are then extended to the case with constant control-input delay. It is found that, in both cases, the real and imaginary parts of the eigenvalues of the augmented Laplacian matrix of the topology play an important role in achieving consensus.