  In this paper, cluster consensus of multi-agent systems is studied via
inter-cluster nonidentical inputs. Here, we consider general graph topologies,
which might be time-varying. The cluster consensus is defined by two aspects:
the intra-cluster synchronization, that the state differences between each pair
of agents in the same cluster converge to zero, and inter-cluster separation,
that the states of the agents in different clusters are separated. For
intra-cluster synchronization, the concepts and theories of consensus including
the spanning trees, scramblingness, infinite stochastic matrix product and
Hajnal inequality, are extended. With them, it is proved that if the graph has
cluster spanning trees and all vertices self-linked, then static linear system
can realize intra-cluster synchronization. For the time-varying coupling cases,
it is proved that if there exists T>0 such that the union graph across any
T-length time interval has cluster spanning trees and all graphs has all
vertices self-linked, then the time-varying linear system can also realize
intra-cluster synchronization. Under the assumption of common inter-cluster
influence, a sort of inter-cluster nonidentical inputs are utilized to realize
inter-cluster separation, that each agent in the same cluster receives the same
inputs and agents in different clusters have different inputs. In addition, the
boundedness of the infinite sum of the inputs can guarantee the boundedness of
the trajectory. As an application, we employ a modified non-Bayesian social
learning model to illustrate the effectiveness of our results.
