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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2025-07-01
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| author | Eric A. Butcher Vianella Spaeth |
| author_facet | Eric A. Butcher Vianella Spaeth |
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| description | 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. |
| format | Article |
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| institution | DOAJ |
| issn | 1099-4300 |
| language | English |
| publishDate | 2025-07-01 |
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| series | Entropy |
| spelling | doaj-art-ef30442f61f241c59c4e0dbd745be0052025-08-20T02:45:45ZengMDPI AGEntropy1099-43002025-07-0127774310.3390/e27070743Decentralized Consensus Protocols on <i>SO</i>(4)<sup><i>N</i></sup> and <i>TSO</i>(4)<sup><i>N</i></sup> with ReshapingEric A. Butcher0Vianella Spaeth1Department of Aerospace and Mechanical Engineering, University of Arizona, 1130 N Mountain Avenue, Tucson, AZ 85721, USADepartment of Mathematics, University of Arizona, 1130 N Mountain Avenue, Tucson, AZ 85721, USAConsensus 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.https://www.mdpi.com/1099-4300/27/7/743special orthogonal groupsconsensus protocolsMorse–Lyapunov functionalmost global asymptotic stability |
| spellingShingle | Eric A. Butcher Vianella Spaeth 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 Entropy special orthogonal groups consensus protocols Morse–Lyapunov function almost global asymptotic stability |
| title | 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 |
| title_full | 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 |
| title_fullStr | 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 |
| title_full_unstemmed | 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 |
| title_short | 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 |
| title_sort | 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 |
| topic | special orthogonal groups consensus protocols Morse–Lyapunov function almost global asymptotic stability |
| url | https://www.mdpi.com/1099-4300/27/7/743 |
| work_keys_str_mv | AT ericabutcher decentralizedconsensusprotocolsonisoi4supinisupanditsoi4supinisupwithreshaping AT vianellaspaeth decentralizedconsensusprotocolsonisoi4supinisupanditsoi4supinisupwithreshaping |