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This article addresses the distributed consensus problem for identical continuous-time positive linear systems with state-feedback control. Existing works of such a problem mainly focus on the case where the networked communication topologies are of either undirected and incomplete graphs or strongly connected directed graphs. On the other hand, in this work, the communication topologies of the networked system are described by directed graphs each containing a spanning tree, which is a more general and new scenario due to the interplay between the eigenvalues of the Laplacian matrix and the controller gains. Specifically, the problem involves complex eigenvalues, the Hurwitzness of complex matrices, and positivity constraints, which make analysis difficult in the Laplacian matrix. First, a necessary and sufficient condition for the consensus analysis of directed networked systems with positivity constraints is given, by using positive systems theory and graph theory. Unlike the general Riccati design methods that involve solving an algebraic Riccati equation (ARE), a condition represented by an algebraic Riccati inequality (ARI) is obtained for the existence of a solution. Subsequently, an equivalent condition, which corresponds to the consensus design condition, is derived, and a semidefinite programming algorithm is developed. It is shown that, when a protocol is solved by the algorithm for the networked system on a specific communication graph, there exists a set of graphs such that the positive consensus problem can be solved as well. [ABSTRACT FROM AUTHOR]

This technical note investigates the positive consensus problem of multiagent systems with directed communication topologies, where all the agents have identical continuous-time positive linear dynamics. Existing works of such a problem mainly focus on the case, where networked communication topologies are of either undirected and incomplete graphs, or strongly connected directed graphs. In contrast to them, we study this problem in which the communication topologies of the multiagent system are described by directed graphs each containing a spanning tree, which is a more general and new scenario due to the interplay between the eigenvalues of the Laplacian matrix and the controller gains. Based on the existing results in spectral graph theory and positive linear systems theory, several necessary, and sufficient conditions on positive consensus of the directed multiagent system are derived through using linear matrix inequality techniques. A primal-dual iterative algorithm is developed for the computation of solutions. Finally, several numerical simulations are provided to illustrate the effectiveness of the proposed theoretical results. [ABSTRACT FROM AUTHOR]