Synchronization on Stiefel manifolds: a theory on almost global consensus

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https://hh.diva-portal.org/smash/project.jsf?pid=project:2698

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Grant for employment or scholarship

Title [sv]

Synkronisering på Stiefelmångfalder: en teori om nästan global konsensus

Title [en]

Synchronization on Stiefel manifolds: a theory on almost global consensus

Abstract [en]

Synchronization is studied in engineering, physics, and biology in contexts such as multi-robot coordination, coupled oscillators, and neural signaling. Such systems contain large numbers of agents whose interactions are described by a graph and whose states evolve with time. The problem is to control, or determine conditions, such that states synchronize. This project addresses a key such synchronization problem, the consensus problem, where all the states shall converge to the same point. The states are often confined to be in non-trivial sets that impose constraints on the convergence analysis. We study the consensus problem for sets that are Stiefel manifolds, which comprise matrices with orthonormal columns. This includes, rotation matrices and permutation matrices. A challenge is to show convergence. The project aims to extend previous convergence results, by developing a complete theory for Almost Global Consensus (AGC) on Stiefel manifolds. The project is conducted by the applicant over four years. It comprises five work packages with partial overlap. The first four cover theory development, whereas the last covers applications and outreach. The theory will leverage results from dynamical systems theory, control systems, optimization, and linear algebra. It aims to unify convergence results for synchronization of oscillators, direction alignment of animal flocks, and formation control for multi-robot systems—all described as consensus problems on Stiefel manifolds.