This paper studies the average consensus tracking problem of first-order multi-agent systems under directed topologies. A control protocol is constructed based on the neighbors’ agent sampled information. With the help of the matrix theory and frequency domain analysis, the necessary and sufficient condition can be induced for the average consensus tracking of multi-agent systems, although the weight-balanced condition is not satisfied. In addition, a numerical simulation example is given to illustrate the effectiveness of the proposed results.

The intention of this paper is to explore the distributed control issues for directed signed networks in the face of external disturbances under strongly connected topologies. A new class of nonsingular transformations is provided by introducing an output variable, with which the consensus can be equivalently transformed into the output stability regardless of whether the associated signed digraphs are structurally balanced or not. By taking advantage of the standard robust $H_{\infty }$ control theory, the bipartite consensus and state stability results can be built for signed networks under structurally balanced and unbalanced conditions, respectively, in which the desired disturbance rejection performances can also be satisfied. Furthermore, the mathematical expression can be given for the terminal states of signed networks under the influence of external disturbances. In addition, two simulations are presented to verify the correctness of our developed results.