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<p>This paper deals with the distributed solving problem of a specific class of linear algebraic equations (LAEs) with block Toeplitz structures. To reduce the communication burden and achieve computation efficiency, a distributed iterative algorithm from the communication-efficient perspective is proposed by incorporating the specific structure of the coefficient matrix tied to any given LAE over a multi-agent network. Each agent possesses a state vector of size smaller than the dimensions of unknown variables related to the LAE and receives information from its neighbors. It is shown that the presented distributed iterative algorithm can solve the specific class of LAEs without requiring any initialization conditions, irrespective of whether it admits a unique solution or multiple solutions. Moreover, an equivalent relation is established between the problem of solving LAEs and the tracking problem of iterative learning control (ILC) systems. The proposed distributed iterative algorithm is leveraged to obtain the distributed control law for ILC systems to realize the tracking objective. Theoretical guarantees are provided for our developed solution results of LAEs, and the effectiveness of them is also verified through simulation examples.</p>

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Consensus-Based Distributed Solution Algorithms for Linear Equations with Block Toeplitz Structures

  • 期刊论文

  • 发表日期：

    2025年7月3日

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  • Shufen Ding,
  • Deyuan Meng,
  • Kaiquan Cai,
  • Juntao Li,
  • Qiang Song

摘要

This paper deals with the distributed solving problem of a specific class of linear algebraic equations (LAEs) with block Toeplitz structures. To reduce the communication burden and achieve computation efficiency, a distributed iterative algorithm from the communication-efficient perspective is proposed by incorporating the specific structure of the coefficient matrix tied to any given LAE over a multi-agent network. Each agent possesses a state vector of size smaller than the dimensions of unknown variables related to the LAE and receives information from its neighbors. It is shown that the presented distributed iterative algorithm can solve the specific class of LAEs without requiring any initialization conditions, irrespective of whether it admits a unique solution or multiple solutions. Moreover, an equivalent relation is established between the problem of solving LAEs and the tracking problem of iterative learning control (ILC) systems. The proposed distributed iterative algorithm is leveraged to obtain the distributed control law for ILC systems to realize the tracking objective. Theoretical guarantees are provided for our developed solution results of LAEs, and the effectiveness of them is also verified through simulation examples.