Cluster consensus in discrete-time networks of multi-agents with inter-cluster nonidentical inputs

The multi-agent systems have broad applications. The consensus problems of multi-agent systems have attracted increasing interests from many fields, such as physics, control engineering, and biology. In network of agents, consensus means that all agents will converge to some common state. A consensus algorithm is an interaction rule how agents update their states. The consensus algorithm has also been used in social learning models. Social learning focuses on the opinion dynamics in the society, which has attached a growing interests. In social learning models, individuals engage in communication with their neighbors in order to learn from their experiences. A large amount of papers concerning consensus algorithms have been published most of which focused on the average principle,i.e., the current state of each agent is an average of the previous states of its own and its neighbors, which is implemented by communication between agents and can be described by the following difference equations for the discrete-time cases: To realize consensus, the stability of the underlying dynamical system is curial. Since the network can be regarded as a graph, the issues can be depicted by the graph theory. In the most existing literature, the concept of spanning tree is widely use to describe the communicability between agents in networks that can guarantee the consensus There were a lot of literature, in which the stability analysis of are investigated. Most of their results can be derived from the theories of infinite nonnegative matrix product and ergodicity of inhomogeneous Markov chain. Among them, the followings should be highlighted. the compression of the differences among rows in a stochastic matrix when multiplied by another stochastic matrix that is scrambling. In , it was proved that a scrambling stochastic matrix could be obtained if a certain number of stochastic matrices that have spanning trees for their corresponding graphs were multiplied. The sufficient conditions were expressed in terms of spanning trees in the union graph across time intervals of a given length. The references therein. Besides, communication delays were also widely investigated and nonlinear consensus algorithms were proposed.