Further Improvements on Non-Negative Edge Consensus of Networked Systems

• Published in: IEEE transactions on cybernetics Vol. 52; no. 9; pp. 9111 - 9119
• Main Authors: Liu, Jason J. R., Lam, James, Kwok, Ka-Wai
• Format: Journal Article
• Language: English
• Published: United States IEEE 01.09.2022; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Controllers; Criteria; Eigenvalues; Eigenvalues and eigenfunctions; Laplace equations; Linear programming; Linear systems; Matrices (mathematics); Networked systems; non-negative edge consensus; Parameterization; positive linear system; Programming; Protocols; Semidefinite programming; State feedback; Symmetric matrices; Topology; undirected graphs; Upper bounds
• ISSN: 2168-2267; 2168-2275
• DOI: 10.1109/TCYB.2021.3052833

In this article, the non-negative edge consensus problem is addressed for positive networked systems with undirected graphs using state-feedback protocols. In contrast to existing results, the major contributions of this work included: 1) significantly improved criteria of consequentiality and non-negativity, therefore leading to a linear programming approach and 2) necessary and sufficient criteria giving rise to a semidefinite programming approach. Specifically, an improved upper bound is given for the maximum eigenvalue of the Laplacian matrix and the (out-) in-degree of the degree matrix, and an improved consensuability and non-negativevity condition is obtained. The sufficient condition presented only requires the number of edges of a nodal network without the connection topology. Also, with the introduction of slack matrix variables, two equivalent conditions of consensuability and non-negativevity are obtained. In the conditions, the system matrices, controller gain, as well as Lyapunov matrices are separated, which is helpful for parameterization. Based on the results, a semidefinite programming algorithm for the controller is readily developed. Finally, a comprehensive analytical and numerical comparison of three illustrative examples is conducted to show that the proposed results are less conservative than the existing work.