Decentralized Consensus Protocols on <i>SO</i>(4)<sup><i>N</i></sup> and <i>TSO</i>(4)<sup><i>N</i></sup> with Reshaping

Decentralized Consensus Protocols on <i>SO</i>(4)<sup><i>N</i></sup> and <i>TSO</i>(4)<sup><i>N</i></sup> with Reshaping

Consensus protocols for a multi-agent networked system consist of strategies that align the states of all agents that share information according to a given network topology, despite challenges such as communication limitations, time-varying networks, and communication delays. The special orthogonal...

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Bibliographic Details
Main Authors: Eric A. Butcher, Vianella Spaeth
Format: Article
Language:English
Published: MDPI AG 2025-07-01
Series:Entropy
Subjects:
special orthogonal groups
consensus protocols
Morse–Lyapunov function
almost global asymptotic stability
Online Access:https://www.mdpi.com/1099-4300/27/7/743
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Summary:Consensus protocols for a multi-agent networked system consist of strategies that align the states of all agents that share information according to a given network topology, despite challenges such as communication limitations, time-varying networks, and communication delays. The special orthogonal group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>O</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula> plays a key role in applications from rigid body attitude synchronization to machine learning on Lie groups, particularly in fields like physics-informed learning and geometric deep learning. In this paper, N-agent consensus protocols are proposed on the Lie group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>O</mi><mo>(</mo><mn>4</mn><mo>)</mo></mrow></semantics></math></inline-formula> and the corresponding tangent bundle <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mi>S</mi><mi>O</mi><mo>(</mo><mn>4</mn><mo>)</mo></mrow></semantics></math></inline-formula>, in which the state spaces are <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>O</mi><msup><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow><mi>N</mi></msup></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mi>S</mi><mi>O</mi><msup><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow><mi>N</mi></msup></mrow></semantics></math></inline-formula>, respectively. In particular, when using communication topologies such as a ring graph for which the local stability of non-consensus equilibria is retained in the closed loop, a consensus protocol that leverages a reshaping strategy is proposed to destabilize non-consensus equilibria and produce consensus with almost global stability on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>O</mi><msup><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow><mi>N</mi></msup></mrow></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>T</mi><mi>S</mi><mi>O</mi><msup><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow><mi>N</mi></msup></mrow></semantics></math></inline-formula>. Lyapunov-based stability guarantees are obtained, and simulations are conducted to illustrate the advantages of these proposed consensus protocols.
ISSN:1099-4300

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