树形交互图上的特殊正交群SO(3)分布式连续时间优化
Distributed Continuous-Time Optimization on the Special Orthogonal Group SO(3) over Tree Interaction Graphs
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中文总结 AI 辅助
针对树形图上SO(3)姿态分布式优化，提出非光滑协议结合黎曼梯度下降与符号一致性反馈，证明有限时间一致性与指数收敛。
中文摘要 AI 辅助
我们研究了在无向树图上三维旋转群上的连续时间分布式优化问题，其中具有异构局部成本的智能体寻求一个共同姿态，该姿态在测地线凸运行球上最小化其成本的测地线强凸和。我们提出了一种具有固定增益的非光滑协议，该协议将局部黎曼梯度下降与基于相邻智能体之间相对旋转的李群对数的符号一致性反馈相结合。由于测地线强凸性本身并不能保证不变性，正如一个经过验证的反例所证明的那样，我们施加了一个内向边界条件，并证明了所有智能体在球内初始化的每个Filippov解都保持在球内。我们构造了一个非光滑的内在分歧李雅普诺夫函数，并通过利用树结构的簇收缩论证推导出一个均匀耗散界。在局部梯度有界的情况下，该界给出了一个显式的增益比条件，在该条件下，我们为每个这样的解建立了精确的有限时间一致性，并获得了一个不需要除智能体数量之外的结构图常数的稳定时间估计。在一致性之后，我们将滑动动力学识别为求和目标的缩放黎曼梯度流。当最小化器位于球内部时，动力学在测地线距离上指数收敛到它。仿真验证了我们的理论结果。
英文摘要
We study continuous-time distributed optimization on the three-dimensional rotation group over undirected tree graphs, where agents with heterogeneous local costs seek a common attitude minimizing the geodesically strongly convex sum of their costs on a geodesically convex operating ball. We propose a nonsmooth protocol with fixed gains that combines local Riemannian gradient descent with signum-based consensus feedback computed from the Lie-group logarithms of relative rotations between neighboring agents. Since geodesic strong convexity alone does not guarantee invariance, as demonstrated by a certified counterexample, we impose an inward-pointing boundary condition and prove that every Filippov solution with all agents initialized in the ball remains there. We construct a nonsmooth intrinsic disagreement Lyapunov function and derive a uniform dissipation bound through a cluster-contraction argument exploiting the tree structure. With bounded local gradients, this bound gives an explicit gain-ratio condition under which we establish exact finite-time consensus for every such solution and obtain a settling-time estimate requiring no structural graph constant beyond the number of agents. After consensus, we identify the sliding dynamics as a scaled Riemannian gradient flow of the summed objective. When the minimizer lies in the interior of the ball, the dynamics converge to it exponentially in geodesic distance. Simulations illustrate our theoretical results.
发表机构
- University of California, Riverside（加州大学河滨分校）
机构由 AI 辅助整理，请以论文原文为准。
