Consensus problems for agent networks

Abstract

This thesis focuses on consensus problems for agent networks, including linear networks, multi-agent networks, and complex dynamical networks.

Regarding consensus problems, the research work can be roughly classified into five aspects: a) For networks over finite fields instead of real numbers, the consensus problem is first investigated with time-delays, aiming to compare the results over finite fields with counterparts over real numbers, which really indicates that several conclusions are intrinsically different from what are already known over real numbers. For example, consensus of a network over finite fields can always be achieved in finite time as long as this network indeed can reach consensus, while it is not the case over real numbers. Furthermore, along this line the consensus problem is further studied with switching topology and time-delays over finite fields, in which a slice of necessary and sufficient conditions are provided in detail; b) Inspired by previous research study, finite-time consensus tracking problem is taken into consideration for secondorder multi-agent networks under directed interaction topology by a structural approach, where one continuous structural control strategy with only position measurements is presented for tackling the finite-time consensus tracking problem in a distributed way; c) For more practical scenarios, the joint effect of quantization, sampled data, and general Markovian interaction links on consensus of general linear networks is addressed with a leader under directed graphs, in which a sufficient condition is derived for ensuring convergence of all encoded states and, subsequently, a necessary and sufficient condition is obtained for achieving consensus tracking in mean-square sense. Finally, a sufficient condition on coupling gain is provided by proposing an optimal LQR-based gain matrix to ensure mean-square consensus tracking; d) Given that a large number of controllers depend on some eigenvalues of the Laplacian matrix or the adjacency matrix, some distributed bounds on the algebraic connectivity and spectral radius of graphs are theoretically established. In doing so, one first considers a directed graph with a leader node, deriving some bounds on the spectral radius and the smallest real part of all eigenvalues of M associated with this graph. Then distributed bounds on the algebraic connectivity and spectral radius of an undirected connected graph are provided in the sense of only knowing the information of edge weights’ bounds and the number of nodes in a graph, without using any information of inherent structures of the graph. Thus, these bounds can be, in some sense, applied to agent networks for reducing the conservatism where control gains in control protocols depend on some eigenvalues of matrices M or L, which is global information; e) Besides linear agent networks with identical nodes, general nonidentical networks are investigated for global bounded consensus problem with nonlinear dynamics and distributed time-delays, in which the distributed time-delays are distinct among each other. To ensure global bounded consensus, sufficient delay-dependent conditions are derived with the aid of constructing a Lyapunov-Krasovskii functional and utilizing the technique of integral partitioning.