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Consensus of Positive Networked Systems on Directed Graphs.

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    This study tackles distributed consensus for positive linear systems over general directed graphs with spanning trees. A novel algebraic Riccati inequality condition and semidefinite programming algorithm are proposed for controller design.

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    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.