The controllability of multiagent systems governed by node and edge dynamics on signed graphs is investigated under distinct partition rules and control algorithms. Motivated by a social network case study, dynamic models for nodes and edges are established. A novel concept “edge bidirectional mapping” is introduced, along with an associated matrix that rigorously characterizes the relationship between node and edge Laplacian matrices. Necessary and sufficient conditions for controllability equivalence between node and edge dynamics are derived across three topological regimes, determined by the cardinality relationships between the number of nodesnand edgesm, namelyn > m, n = m, andnm. Furthermore, more explicit and comprehensive criteria for controllability properties of node and edge dynamics are formulated under distinct control algorithms and partition rules. Specifically, the proposed algorithms are considered in both absolute and relative forms, while the partition rules, distance partition (DP), generalized almost equitable partition (GAEP), and generalized relaxed almost equitable partition (GRAEP), are distinguished based on the number of positive, negative, and total connections between vertices (nodes or edges) within a cell and their neighboring connections to other cells. Notably, this study systematically analyzes their hierarchical inclusion relationships for the first time. Upper and lower bounds on the dimensions of the node and edge controllable subspaces are presented. The lower bound reflects DP with edge weight considerations, while the upper bound is obtained via partition algorithms that yield the best LGRAEP. Additionally, the computational complexity and convergence properties of the proposed partition rules (DP, GEP, GREP), Algorithms 1-2, and the derived bounds for the controllable subspace dimensions are analyzed. Theoretical results are comprehensively validated through multi-level simulations, including verification of theoretical bounds, analysis of clustering effects, and comparative model evaluation.

The paper considers the event-triggered stabilizability of multi-agent systems (MAS). To reduce the frequency of control input update and information transmission, a novel distributed event-triggered control strategy with state estimation feedback is designed to achieve stabilizability. Event-trigger rules involving dynamic threshold are constructed, which ensure the convergence of systems states in finite time. By utilizing iSCC (independent strongly connected component) partition and Lyapunov stability theory, sufficient criteria for achieving finite time stabilizability have been obtained. Moreover, formation control under event-trigger scheme is addressed based on the obtained stabilizability results. Besides, it has been proven that the event-trigger interval has a positive lower bound, which can avoid Zeno behavior. Finally, the effectiveness of the theoretical results is verified by simulation.Note to Practitioners—In the fields of control and engineering, system stability has always been a research hotspot. Traditional stability analysis often focuses on the asymptotic stability. However, many practical scenarios require the system to reach a stable state within a finite time, i.e., finite time stabilizability, such as rapid response systems, emergency braking systems, and responding to emergencies. Hence, the research on finite time stabilizability has important practical value. In order to reduce the frequency of control updates, this paper proposes the finite time stabilizability under event-trigger scheme and designs a novel feedback controller based on state estimation, which has not been discussed before. At the same time, to explore the impact of network topology on stabilizability, the sufficient conditions for achieving finite time stabilizability of the system are obtained by iSCC graph partition. Dynamic threshold in event-triggering conditions to guarantee the implementation of finite time stabilizability is constructed. Finally, the theoretical results are applied to formation control. The research is of great significance for improving system performance and meeting real-time requirements.

For some natural networks, edge dynamics are a scientific representation, and physical quantities can be better characterized by edges than nodes, such as transportation and social networks. Topology is a paramount determinant for characterizing system performance. To bridge the gaps between the topology structure and stabilizability, we propose a technique to achieve the desired independent strongly connected component (iSCC) partition by adding edges to change the topology structure. Besides, based on iSCC partition as the central tool for grasping the stabilizability of node and edge dynamics, it has been proven that the stabilizability realization of first-order multiagent systems directly depends on the topology structure. Furthermore, the relationship between node stabilizability and edge stabilizability is explored from a graph theory perspective through the transformation mechanism from a node digraph to an edge digraph. In particular, the stabilizability results of second-order multiagent systems reveal that the stabilizability of node dynamics and edge dynamics depends not only on the topology, but also on the feedback coefficientsk1,k2. Ultimately, simulation experiments are provided to verify the correctness and effectiveness of the proposed control protocol.

Establishing the graph-based criterion for the selection of any location and any number of leaders is the main difficulty in the complete graphic characterization of multiagent controllability. This greatly increases the complexity of the study, compared with the results derived for only one or several classes of leaders. Through a detailed analysis of graphs of six nodes, this article presents a systematic design and identification process for the complete graphic characterization by taking advantage of controllability destructive nodes. The topologies obtained by the proposed method allow directly determining controllability at the network topology level. The results are not only applicable to any leader's selection but also reveal the difficulty and complexity in the study of complete controllability graphic characterizations. Moreover, by comparing graphs composed of five and six nodes, the results reveal the graph-theory-based controllability complexity caused by adding only one node. Finally, results are derived to show how to design topology structures to ensure the controllability under any selection of leaders.

In this article, by analyzing the eigenvalues and eigenvectors of Laplacian L , we investigate the controllability of multiagent systems under equitable partitions. Two classes of nontrivial cells are defined according to the different numbers of links between them, which are completely connected nontrivial cells (CCNCs) and incompletely connected nontrivial cells. For the system with CCNCs, a necessary condition for controllability is found to be choosing leaders from each nontrivial cell, the number of which should be one less than the cardinality of the cell. It is shown that the controllability is affected by three factors: 1) the number of the links between nontrivial cells; 2) the rank of the connection matrix; and 3) the odevity of the capacity of the nontrivial cells. In the case of nontrivial cells under the equitable partition, there are automorphisms of interconnection graph G , which induce the eigenvectors of L with zero entries. For the system with automorphisms, by taking advantage of the property of eigenvectors associated with L , we propose several graphical necessary conditions for controllability. In addition, by the PBH rank criterion, the controllable subspaces of the system with different classes of nontrivial cells are compared. Finally, a necessary and sufficient condition for controllability under minimum inputs is given.

Recently, graphical characterization of multiagent controllability has been studied extensively. A major effort in the study is to determine controllability directly from topology structures of communication graphs. In this paper, we proposed the concept of controllability destructive nodes, which indicates that the difficulty in graphical characterization turns out to be the identification of topology structures of controllability destructive nodes. It is shown that each kind of double and triple controllability destructive nodes happens to have a uniform topology structure which can be defined similarly. The definition, however, is verified not to be applicable to the topology structures of quadruple controllability destructive (QCD) nodes. Even so, a design method is proposed to uncover topology structures of QCD nodes for graphs with any size, and a complete graphical characterization is presented for the graphs consisting of five vertices. One advantage of the established complete graphical characterization is that the controllability of any graph with any selection of leaders can be determined directly from the identified/defined destructive topology structures. The results generate several necessary and sufficient graphical conditions for controllability. A key step of arriving at these results is the discovery of a relationship between the topology structure of the controllability destructive nodes and a corresponding eigenvector of the Laplacian matrix.

The consensus problem for second-order multiagent systems with absolute velocity damping under directed topologies is investigated. In contrast to the existing results, which rely on a sufficiently large common absolute velocity damping gain above a lower bound dependent on global information, this paper focuses on novel algorithms to overcome this limitation. A novel consensus algorithm, where different agents use different absolute velocity damping gains, is first proposed. In the absence of delays, based on a system transformation method, the consensus problem for second-order multiagent systems is converted into that for first-order multiagent systems with the agent number doubled. Necessary and sufficient conditions are then derived under directed topologies by relating the topologies associated with the doubled number of agents and the original team of agents. In the presence of multiple constant delays, based on a further system transformation method, the consensus problem for second-order multiagent systems is converted into the stability problem for corresponding systems. Necessary and sufficient conditions are presented to guarantee consensus under a directed fixed topology. For systems with a uniform constant delay, more concrete necessary and sufficient conditions on how large the delay can be to guarantee consensus is given. Numerical simulations are provided to illustrate the effectiveness of the obtained theoretical results.

We study the stabilization of discrete-time switched linear systems and discrete-time uncertain switched systems when an arbitrary switching rule is imposed on the observer-based static output feedback controllers. Several linear matrix inequality (LMI) based conditions are established to this problem with switched Lyapunov function method for both cases.