Controllability for multi-agent systems with matrix-weight-based signed network
Introduction
For MASs, controllability is a basic concept [1], [2], which is closely related to system structure, mainly studied by using various traditionally classic criteria based on system topology structure, such as the Rank test and the PBH test [3], [4], [5], [6].
Considerable studies on controllability of MASs have brought into sharp focus on the weights of interactions among agents in the network being characterized by non-negative real numbers, that is, the cooperative system. However, networks with both positive and negative edge weights, called as signed networks first proposed by Heider [7] to investigate the interactions between negative and positive relations in the triangular relationship of human being as cognitive subject based on the theory of social psychology, are widespread in nature or engineering [8], in which the positive and negative edge weights can describe the relationships such as friends and enemies, cooperation and competition, approval and opposition, as well as attraction and repulsion. Altafini first modelled a first-order continuous-time MAS with antagonistic interactions described by a signed graph, proposed the concept of bipartite consensus and established the necessary and sufficient conditions for bipartite consensus under structural balance [9]. Topological graphs are called as structurally balanced, i.e., all agents are divided into two groups, where the edge weights between two agents within the same group are nonnegative, while the edge weights between two ones within different groups are negative. Most of researches on structurally balanced graphs mainly turned signed graphs into nonnegative weighted graphs by Gauge Transformation, therefore, many methods for nonnegative weights can be applied to analyze coordination control problems of MASs. Literature [10] further extended the results in Altafini [9] to the case of time-varying signed networks and gave the looser topological conditions for bipartite consensus. Literature [11] discussed finite-time consensus of first-order continuous-time MASs with cooperative and antagonistic interactions and proposed several kinds of protocols to prove that finite-time consensus can be achieved whether the topology is structural balanced or structural imbalanced. Ji et al. discussed the fast consensus, decentralized stabilizability and formation control, and bipartite consensus of MASs with antagonistic interactions in Qu et al. [12], Liu et al. [13], Wang et al. [14], respectively. Recently, the controllability of MASs with antagonistic interactions was investigated in Sun et al. [15]. Liu et al. [16] studied the controllability of directed MASs with signed networks by graph partitions. Guan et al. discussed the controllability of MASs over signed networks with fixed topology [17] and switching topology [18], respectively. Leaders’ selection to ensure controllability on signed multi-agent networks was studied in She et al. [19].
It is noteworthy that, in most existing works, on coordination control problems are obtained for MASs with scalar weights among agents [20], namely assuming the weights of information interaction edges between all agents are constants (numerical weights/ scalar network). However, in reality, matrix weights can better characterize the interactions among agents in many scenarios, especially in some complex social networks and biological systems, which is a natural extension of scalar weights. The weight of the edge can be defined either by a scalar or by a matrix. For scalar weights, weights can only match multi-dimensional states by using Knonecker product, and states are independent of each other. Comparing to scalar weights, the state coupling relationship between agents can be more accurately depicted using matrix weights. For example, the change of viewpoints of people is very complex when people discuss a certain viewpoint, especially when they discuss some related viewpoints together, in which the dependencies among these related topics can be depicted by matrix weights. Besides, the emotion modeling and behavior decision-making of a multi-emotion robot system rely on the interrelated factors among individuals and all aspects of individuals; and the interactions between electrical oscillators can be described by a matrix-weighted Laplacian. Thus, the matrix interaction mode provides a mechanism to capture more elaborate correlations in complex networks, which extends numerical weights to a correlation to characterize the interdependence between multi-dimensional states of neighboring agents. Tuna respectively investigated the synchronization and observability problems of matrix-weighted coupled LC oscillators in Tuna [21,22]. Recently, consensus problems of MASs over matrix-weighted networks have gradually been concerned and studied (e.g. [23], [24], [25] and the references therein). Trinh et al. [23] gave some algebraic conditions for achieving a consensus for MASs based on matrix weighted Laplacian. Pan et al. [24] established sufficient and necessary condition for the bipartite consensus of matrix weighted networks. Su et al. explored the bipartite consensus for MASs with matrix-weight-based signed network with structurally balanced graph and structurally unbalanced graph in Su et al. [25]. More recently, structural controllability was investigated for diffusive networks with vector-weighted edges and undirected graph in Zhang et al. [26], as well as controllability on matrix-weighted networks was also discussed by graph partitions [27].
In the traditional control theory (e.g., Refs. [28], [29]), the controllability emphasizes the intrinsic dynamic characteristics of a single high-dimensional system, corresponding to the microscopic individual dynamics. Today, our lives have been inseparable from the network. The emergence of large-scale complex networks such as internet, power network, intelligent transportation network, biological neural network and gene regulation network, etc., makes people pay more and more attention to the influence of network topology (i.e., the couplings between agents) on the whole behavior of the system. The controllability of networked systems in this paper is studied from the interactions between multiple subsystems, corresponding to the macroscopic group dynamics behavior. The work in Kalman [28] mainly emphasized the external whole of a system, but this paper emphasizes the internal structure of the network. Moreover, the controllability of the linear dynamical system with an impulse response matrix was studied in Kalman [28], but this paper considers the controllability of MASs with an adjacency matrix (or Laplacian matrix). The use of a matrix to describe edge weights between agents in multi-agent networks can be more efficient, accurate, complete and beneficial to examine the properties of general scalar networks from higher dimensions. More importantly, high-dimensional individual states can also be viewed as multi-layer networks, providing new perspectives and tools to research more complex networks such as multi-layer networks, cross-layer networks and hyper-networks. Matrix coupling networks are widely available in reality, therefore, the modeling, analysis and control of the matrix coupling networks are of great theoretical significance and practical value. Up to now, the controllability of MASs with matrix coupling weights is still in the exploratory stage, facing many complicated factors such as nodal dynamics, evolutionary behaviors, system dimensions, etc., so the controllability presents new features and brings greater difficulties to analyze because of the limited understanding in matrix weights. In particular, a recent paper [27] attempted to make some simple analysis for the controllability of MASs with matrix coupling weights from graph partition. But our work establishes a MAS model with matrix-weight-based signed networks and mainly studies the controllability of MASs with matrix coupling weights from the angle of algebra, especially some the controllable conditions are achieved under structural balanced networks by choosing leaders, which is different from existing analytical methods (e.g. [27]) and shows theoretical difficulties. This paper studies the controllability of MASs based on a matrix coupling network, aiming to solve several key problems in the theoretical level and reveal the qualitative and quantitative relationship between dynamics of agents, topology structure of matrix coupling networks and the system controllability.
To the best of our knowledge, so far very few results of the controllability for multi-agent systems with matrix-weight-based signed network have been available. Different from existing works with scalar weights and the recent work with matrix weights in Pan et al. [27], this study focuses on the controllability of MASs with signed networks. The main contributions of this work can be devoted to the following points: (i) It is the first attempt to study the controllability of matrix-weighted MASs from an algebra point of view, which is a pioneering work. (ii) A controllable model is established under the signed network. (iii) The controllability with antagonistic interactions and the controllability with cooperative interactions are compared. (iv) The algebra-theoretic necessary and sufficient conditions for controllability are given under proposed by virtue of almost equitable partition of directed weighted over structurally balanced graphs. (v) The uncontrollable condition can also be obtained whether structural balance or structural unbalance.
Section snippets
Mathematical preliminaries
is said to be a fixed, undirected, signed and matrix-weight-based graph, whose underlying graph is denoted as and is a signal mapping; and are the vertex and edge sets, respectively; is the neighboring set of agent ; represents the set of matrix weights with be the matrix weights dimension, where means that matrix is semi-positive definite
Controllability analysis
When the leaders are predetermined, ’s Laplacian can be partitioned intowhere is the leaders’ index and is the followers’ index, respectively. Definition 1 is said to be connected to if there exists a path from each agent to at least one of the leaders in . Lemma 1 Invertibility Let be a structurally balanced matrix-weight-based signed network. Then the matrices and of are both positive semi-definite. Moreover, is positive definite if and only if is connected to . Proof Let
Examples and simulations
The study of MASs’ controllability has important theoretical value and practical significance. First of all, from the point of view of natural phenomena, a small number of simple individuals under external control or influence can present a certain swarm intelligence. For example, a small number of ants guide the entire ant colony to carry food in the shortest path by releasing their own chemical hormones; a small number of shepherd dogs help owners drive sheep to a designated place to eat
Conclusion
This study has discussed the controllability for MASs with the matrix-weight-based signed graph. Due to the diversity of geometric shapes of complex space objects and the randomness of multidimensional positions, spatial clustering is widespread. Some novel algebraic conditions on controllability for MASs have been established, which completely relied on the topology structure and leaders’ selections. Moreover, uncontrollability can also been analyzed.
At present, the research on the
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grant nos. 61773023, 61991412, 61773416, and 61873318, the Program for HUST Academic Frontier Youth Team under Grant no. 2018QYTD07, and the Frontier Research Funds of Applied Foundation of Wuhan under Grant no. 2019010701011421. The authors declare that there is no conflict of interest regarding the publication of this paper.
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