We consider a clustered network where connections inside the cluster are dense and between clusters are sparse. This leads us to a classical decoupling into fast (intra-cluster) and slow (inter-cluster) dynamics. Our objective is to provide a computationally efficient method to design control strategies that guarantee a certain bound on the cost for each cluster. Basically, we design a composite synchronizing controller with two terms: one responsible for the intra-cluster synchronization and the other achieving the synchronization between clusters. The first one does not require much computational effort since an analytic expression describes it. The second term is designed through a satisfaction equilibrium approach. In other words, the internal (fast) and external (slow) controllers are independently designed, and they ensure a guaranteed satisfactory cost for each cluster. Moreover, we show that the internal control affects the cluster cost only for a short time period. Finally, numerical simulations illustrate the theoretical results.

中文翻译：

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同构集群网络的计算高效保证成本控制设计

我们考虑一个集群网络，其中集群内部的连接密集而集群之间的连接稀疏。这导致我们将经典解耦为快速（簇内）和慢速（簇间）动态。我们的目标是提供一种计算有效的方法来设计控制策略，保证每个集群的成本有一定的限度。基本上，我们设计了一个具有两个术语的复合同步控制器：一个负责集群内同步，另一个实现集群之间的同步。第一个不需要太多的计算工作，因为解析表达式描述了它。第二项是通过满意度均衡方法设计的。换句话说，内部（快速）和外部（慢速）控制器是独立设计的，它们确保每个集群都有令人满意的成本。此外，我们表明内部控制仅在短时间内影响集群成本。最后，数值模拟说明了理论结果。