In this article, the consensus problem of linear systems is revisited from a novel geometric perspective. The interaction network of these systems is assumed to be piecewise fixed. Moreover, it is allowed to be disconnected at any time but holds a quite mild joint connectivity property. The system matrix is marginally stable and the input matrix is not of full-row rank. By directly examining the subspace determined by the network, we first establish convergence by resorting to an observability condition. Then, according to joint connectivity, we are able to extend this convergence uniformly to the entire orthogonal complement of the consensus manifold. In this way, we work out the necessary and sufficient condition for exponential consensus. It turns out that, with a suitably designed feedback matrix, exponential consensus can be realized globally and uniformly if and only if a jointly (δ,T)(\delta,T) -connected condition and an observability condition relying only on the system and input matrices are satisfied. We also characterize the lower bound of the convergence rate. Simple yet effective examples are presented to illustrate the findings.

中文翻译：

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交换网络上线性系统的指数一致性：建立必要性和充分性的子空间方法


在本文中，从新颖的几何角度重新审视线性系统的共识问题。假设这些系统的交互网络是分段固定的。而且，它允许随时断开，但具有相当温和的联合连接属性。系统矩阵是边际稳定的，并且输入矩阵不是全行秩的。通过直接检查网络确定的子空间，我们首先利用可观测性条件建立收敛。然后，根据联合连通性，我们能够将这种收敛性统一扩展到共识流形的整个正交补集。这样，我们就得出了指数级共识的充要条件。事实证明，通过适当设计的反馈矩阵，当且仅当联合 (δ,T)(\delta,T) 连接条件和仅依赖于系统的可观察性条件时，才可以全局一致地实现指数共识满足输入矩阵。我们还描述了收敛速度的下限。提出了简单而有效的例子来说明研究结果。