A small-gain approach for consensus of heterogeneous linear multi-agent systems

This paper proposes a small-gain approach for consensus of heterogeneous linear multi-agent systems with a directed graph topology. From the perspective of Laplacian, the decomposition of the entire system results in two blocks with null dynamics and error dynamics. Since one block with null dynamics is uncontrollable, this block is assumed not to diverge. The other block with error dynamics includes the average of errors between states of each agent and the other agents. The two blocks are connected with each other in the loop, whereas for homogeneous systems, the two blocks are connected but not in the loop. By merging these two blocks into one block, this paper constructs a stabilizing condition, where the state of all agents achieves consensus with the state of null dynamics while the errors converge to zero as time progresses. Based on constructed conditions, an iterative algorithm is suggested to find a Laplacian-associated gain matrix with the loop gain index, which is less than one for consensus. Two illustrative examples and one practical example demonstrate the effectiveness of the proposed approaches.