Bipartite Consensus on Matrix-Valued Weighted Networks

Abstract

This paper examines the consensus problem on time-varying matrix-weighed undirected networks. First, we introduce the matrixweighted integral network for the analysis of such networks. Under mild assumptions on the switching pattern of the time-varying network, necessary and/or sufficient conditions for which average consensus can be achieved are then provided in terms of the null space of matrix-valued Laplacian of the corresponding integral network. In particular, for periodic matrix-weighted time-varying networks, necessary and sufficient conditions for reaching average consensus is obtained from an algebraic perspective. Moreover, we show that if the integral network with period T > 0 has a positive spanning tree over the time span [0, T ), average consensus for the node states is achieved. Simulation results are provided to demonstrate the theoretical analysis.of weight matrices are further introduced in [14], [15]. In the meantime, the notion of network connectivity can be further extended for matrix-valued networks. For instance, one can identify edges with positive/negative definite matrices as "strong" connections;whereas an edge weighted by positive/negative semi-definite matrices can be considered as a "weak" connection [16].

On the Controllability of Matrix-Weighted Networks

“…Matrix-weighted networks are a natural extension of scalar-valued networks; they have been examined in scenarios such as graph effective resistance (motivated by distributed estimation and control) [20], [21], logical inter-dependency of multiple topics in opinion evolution [22], [23], bearing-based formation control [24], as well as the array of coupled LC oscillators [25]. More recently, consensus and synchronization problems on matrix-weighted networks have been examined in [26], [27], [28], [29].…”

Event-triggered Consensus of Matrix-weighted Networks Subject to Actuator Saturation

“…In signed networks, the concept of structural balance (can be tracked back to the seminal work [34]) turns out to be an important graph-theoretic object playing a critical role in bipartite consensus problems [35]. This concept has been extended to the matrix-weighted networks in [3].…”

“…C ONSENSUS problem on matrix-weighted networks is becoming a recent concern since, as an immediate generalization of scalar-weighted networks, they naturally captures interdependencies among higher-dimensional states of neighboring agents in a multi-agent network [1], [2], [3], [4], [5], [6]. Actually, matrix-weighted networks arise in scenarios such as graph effective resistance based distributed control and estimation [7], [8], logical inter-dependency of multiple topics in opinion evolution [9], bearing-based formation control [10], array of coupled LC oscillators [11] as well as consensus and synchronization on matrix-weighted networks [3], [4].…”

“…In contrast to scalar-weighted networks, connectivity alone does not translate to achieving consensus on matrix-weighted networks, properties of weight matrices now play an important role in the characterization of consensus as well as the design of interaction protocols subject to physical constraints. In literatures, positive/negative definiteness/semi-definiteness weight matrices have been employed to provide consensus conditions [3], [12], [2], [4]. In the meantime, it is worth noting that the matrix-weighted network is a more general category of multiagent networks, recent trends in this line of research seems to involve the constraints of physical systems encountered in realworld applications.…”

Fast Convergence of Multi-quadrotor Cooperation Using Weighted-Neighbor-based Control

“…In ref. [19], the bipartite consensus problem is considered, where the edge weights are matrix-valued. The objective is different from that of our paper.…”