This article studies the containment control problem for a group of linear systems, consisting of more than one leader, over switching topologies. The input matrices of these linear systems are not required to have full-row rank and the switching can be arbitrary, making the problem quite general and challenging. We propose a novel analysis framework from the viewpoint of a state transition matrix. Specifically, according to the inherent linearity, we successfully establish a connection between state transition matrices of the above multileader system and a virtual leader-following system obtained by combining those leaders. This enlightening result relates the containment problem to a consensus one. Then, by analyzing the property of the state transition matrix, we uncover that each component of any follower’s state converges to the convex hull spanned by the corresponding components of the leaders’, provided some mild conditions are satisfied. These conditions are derived in terms of the concept of a positive linear system. A special case of the second-order linear system is further discussed to illustrate these conditions. Moreover, two different design methods of the feedback gain matrix are provided, which additionally require that the network topology contains a united spanning tree all the time.

中文翻译：

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具有开关拓扑的线性系统的包容性：一种新的状态转移矩阵视角

本文研究了一组线性系统的包含控制问题，该系统由多个领导者组成，具有切换拓扑。这些线性系统的输入矩阵不需要具有全行秩，并且切换可以是任意的，这使得问题非常普遍且具有挑战性。我们从状态转移矩阵的角度提出了一种新颖的分析框架。具体来说，根据内在的线性，我们成功地建立了上述多领导者系统的状态转移矩阵和通过组合这些领导者获得的虚拟领导者跟随系统之间的连接。这一启发性的结果将遏制问题与共识问题联系起来。然后，通过分析状态转移矩阵的性质，我们发现，只要满足一些温和的条件，任何跟随者状态的每个组件都会收敛到由领导者的相应组件跨越的凸包。这些条件是根据正线性系统的概念得出的。进一步讨论二阶线性系统的一个特例来说明这些条件。此外，还提供了两种不同的反馈增益矩阵设计方法，另外还要求网络拓扑始终包含一棵联合生成树。