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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]

In this paper, we investigate the positive edge consensus of nodal networks with Luenberger observer-based dynamic output-feedback controllers. The line graph of a nodal network, which is used to model the interaction among edges, is directed and assumed to contain a spanning tree. Based on the results in graph theory and positive systems theory, positive edge consensus problem is analyzed and the protocol design condition is given. Moreover, a semi-definite programming algorithm is developed to solve the positive edge consensus problem. Compared with the recent work [24] where the nodal graph is assumed to be strongly connected and balanced, we study the positive edge consensus problem in a more general case, where the line graph is assumed to contain a spanning tree. Finally, the numerical simulations are given to verify the effectiveness of the proposed results. [ABSTRACT FROM AUTHOR]