Consensus of Positive Networked Systems on Directed Graphs

Area of Science:

• Control Systems Engineering
• Networked Systems Theory
• Positive Systems Theory

Background:

• Distributed consensus is crucial for networked systems.
• Existing research often assumes undirected or strongly connected directed graphs.
• Positive linear systems with state-feedback control present unique analytical challenges.

Purpose of the Study:

• To address the distributed consensus problem for identical continuous-time positive linear systems.
• To investigate novel communication topologies described by directed graphs with spanning trees.
• To develop a new condition and algorithm for consensus analysis and controller design.

Main Methods:

• Utilizing positive systems theory and graph theory for consensus analysis.
• Deriving a necessary and sufficient condition for consensus in directed networked systems with positivity constraints.
• Employing an algebraic Riccati inequality (ARI) instead of traditional algebraic Riccati equations (ARE).
• Developing a semidefinite programming algorithm for consensus controller design.

Main Results:

• A necessary and sufficient condition for consensus analysis in directed networked systems with positivity constraints was established.
• An algebraic Riccati inequality (ARI) condition was derived for the existence of a consensus solution.
• An equivalent condition for consensus design was obtained, leading to a semidefinite programming algorithm.
• The algorithm's effectiveness was demonstrated for solving the positive consensus problem.

Conclusions:

• The proposed methods offer a generalized approach to distributed consensus for positive linear systems.
• The use of ARI and semidefinite programming provides an efficient design pathway.
• The findings extend the applicability of consensus control to more general network topologies.