Panahi Shirin, Lodi Matteo, Storace Marco, Sorrentino Francesco
Department of Mechanical Engineering, University of New Mexico, Albuquerque, New Mexico 80131, USA.
DITEN, University of Genoa, Via Opera Pia 11A, 16154 Genova, Italy.
Chaos. 2022 Nov;32(11):113111. doi: 10.1063/5.0090095.
In this paper, we study the network pinning control problem in the presence of two different types of coupling: (i) node-to-node coupling among the network nodes and (ii) input-to-node coupling from the source node to the "pinned nodes." Previous work has mainly focused on the case that (i) and (ii) are of the same type. We decouple the stability analysis of the target synchronous solution into subproblems of the lowest dimension by using the techniques of simultaneous block diagonalization of matrices. Interestingly, we obtain two different types of blocks, driven and undriven. The overall dimension of the driven blocks is equal to the dimension of an appropriately defined controllable subspace, while all the remaining undriven blocks are scalar. Our main result is a decomposition of the stability problem into four independent sets of equations, which we call quotient controllable, quotient uncontrollable, redundant controllable, and redundant uncontrollable. Our analysis shows that the number and location of the pinned nodes affect the number and the dimension of each set of equations. We also observe that in a large variety of complex networks, the stability of the target synchronous solution is de facto only determined by a single quotient controllable block.
在本文中，我们研究了存在两种不同类型耦合时的网络牵制控制问题：（i）网络节点之间的节点到节点耦合，以及（ii）从源节点到“被牵制节点”的输入到节点耦合。先前的工作主要集中在（i）和（ii）为同一类型的情况。我们通过使用矩阵同时块对角化技术，将目标同步解的稳定性分析解耦为最低维的子问题。有趣的是，我们得到了两种不同类型的块，驱动块和非驱动块。驱动块的总维数等于适当定义的可控子空间的维数，而所有其余的非驱动块都是标量。我们的主要结果是将稳定性问题分解为四组独立的方程，我们称之为商可控、商不可控、冗余可控和冗余不可控。我们的分析表明，被牵制节点的数量和位置会影响每组方程的数量和维数。我们还观察到，在各种各样的复杂网络中，目标同步解的稳定性实际上仅由单个商可控块决定。
