Complexity

Volume 2018, Article ID 8265637, 9 pages

https://doi.org/10.1155/2018/8265637

## Stabilization of Heterogeneous Multiagent Systems via Harmonic Control

^{1}Institute of Complexity Science, College of Automation and Electrical Engineering, Qingdao University, Qingdao 266071, China^{2}School of Mathematics and Statistics, Qingdao University, Qingdao 266071, China^{3}College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China

Correspondence should be addressed to Zhijian Ji; nc.gro.ukp@naijihzij

Received 11 April 2018; Revised 25 July 2018; Accepted 8 August 2018; Published 30 September 2018

Academic Editor: Zidong Wang

Copyright © 2018 Xianzhu Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Stabilizing multiagent systems including unstable agents shows the advantage of cooperation. This paper addresses the problem of stabilization of heterogeneous multiagent systems. Under cycle graphs, a sufficient condition for the stabilization problem via harmonic control is provided and an algorithm of designing the interconnection gains is presented. Furthermore, a sufficient and necessary condition for stabilization problem via harmonic control under cycle graphs is first given when the graph contains less than 5 nodes.

#### 1. Introduction

In recent years, multiagent systems have broad applications in science and engineering areas such as consensus [1–11], controllability [12–22], and optimal control [23–25]. Multiagent systems are concerned chiefly with structures of networks and local information feedback including self-state feedback and neighbor-state feedback. Designing decentralized controller with local information to realize stability is a basic problem in large-scale systems and multiagent systems [26–38]. Kim et al. [27] proposed a problem of stabilizability for multiagent systems with single-integrator dynamics by using external control inputs. Guan et al. extended the study to multiagent systems with general linear dynamics [28]. It is noted that [27, 28] only consider the identical agents. But in practice, many multiagent systems have different subsystems. These systems are called heterogeneous multiagent systems. For example, agents of flocks or satellite clusters might have different dynamic equations from each other due to their different masses or different structures [31]. In [32], a sufficient algebraic condition was provided for stabilization of heterogeneous multiagent systems in the case of static topology. The basic requirement of the sufficient condition in [32] is that each agent is stable or can be stabilized through self-state feedback. In the case where there exists an unstable agent which cannot be stabilized via self-state feedback, how to use neighbor-state feedback to stabilize multiagent systems is an important issue. Designing the gains of neighbor-state feedback has been applied in the plague control of some power networks [31, 35]. However, there are rarely interesting results on such problems until recent years. To this day, the results for stabilization of heterogeneous multiagent systems are limited to cycle topologies, and only sufficient conditions are obtained. Reference [30] presented a sufficient condition for the stabilization problem, in which designing interconnection gains is called harmonic control and the system studied is composed of two subsystems. In [31], Zhu provided a sufficient condition for the stabilization of heterogeneous multiagent systems under directed cycle graphs. It will be challenging to explore necessary and sufficient conditions for stabilization of heterogeneous multiagent systems. The main reason is that heterogeneousness and complicated interconnections make the problem tricky. Therefore, what conditions can stabilize heterogeneous multiagent systems via harmonic control is still an open problem.

This paper studies the stabilization of heterogeneous multiagent systems under directed communication topologies. The graphs include not only cycles but also paths, stars, and trees. We extended the results of [31]. The contributions are twofold: (i)A sufficient condition for the stabilization problem via harmonic control is provided under cycle graphs. This condition is more general than the sufficient condition given in [31]. Besides, an approach is introduced to design the interconnection gains(ii)When the multiagent system contains less than 5 agents, a necessary and sufficient algebraic condition is presented for this stabilization problem under cycle graphs. To the best of our knowledge, it is the first time to provide necessary and sufficient condition for stabilization problem of heterogeneous multiagent systems via harmonic control under cycle graphs

The structure of this paper is as follows. Section 2 presents some preliminaries and formulates the stabilization problem of multiagent systems. Section 3 provides the main results. Two numerical examples are given in Section 4 to show the applicability of the obtained results. Finally, conclusion is summarized in Section 5.

#### 2. Preliminaries

Throughout this paper, the set of integers is denoted by . With vertices representing agents and edges indicating the interconnections between them, graph theory proves to be a natural framework for modeling and treatment of multiagent systems. We consider directed graph rather than undirected graph. A directed graph is denoted by , where and represent the vertex and edge set, respectively. An edge is represented by an arrow tailed at the node and headed toward the node , which means node can receive information from . The set of neighbors of node is denoted by . The indegree of a vertex is the number of edges with head . And the outdegree of a vertex is the number of edges with tail . If every possible edge exists, the graph is said to be complete. A path of length from to is an ordered set of distinct vertices such that for all . An -cycle is a path except for which , meaning the path rejoins itself. is the adjacency matrix whose entry is 1 if is one of ’s edges and 0 otherwise. A tree graph with root is a graph that for each node other than , there exists one and only one path from to this node. A node is called a leaf if its outdegree is zero, and two nodes are said to be in different branches when there is no path from any one of them to the another. A graph is said to contain a spanning tree if there exists a tree whose nodes are all those in and edges in the tree are also in . A star graph is a kind of special tree graph whose root is a neighbor of all nodes rest.

Let us consider a group of linear agents with information flow among them described by graph with , whose linear dynamics is where is the state vector of the th agent, and , , and are real matrices. is the self-state feedback law described by where . is the neighbor-state feedback law described by where . The closed loop system is , where , where is the entry of the adjacent matrix of the graph .

*Definition 1 [30]. *The stabilization of a multiagent system is said to be solvable if there exists feedback law (2) and (3) such that the closed loop system (4) is stable.

In order to investigate more deeply the influence of neighbor-state feedback on the stabilization problem, the heterogeneous multiagent system (1) without self-state feedback gains is simplified to

Lemma 1 (Routh-Hurwitz criterion) [39]. *A necessary and sufficient condition for polynomial
to be stable is that the determinants are all positive, where
it is being understood that in each determinant, all the with subscripts that are either negative or greater than are to be replaced by zero. is named as the Hurwitz determinant.*

#### 3. Main Results

In the section, we mainly investigate how to use neighbor-state feedback to solve the stabilization problem when the multiagent systems are still unstable after using their self-state feedback.

##### 3.1. The Case without Self-State Feedback

Directed paths and directed cycles are basic ingredients for the investigation of stabilization of directed graphs. The analysis on path and cycle graphs is expected to provide insights for that of more complex structures. According to Appendix, for path graphs, star graphs, and tree graphs, the stabilizability is not affected by the neighbor-state feedback. That is to say, for path graphs, star graphs, or tree graphs, the stabilization problem is solvable if and only if each individual is stable or can be stabilized by its self-state feedback. Now, we consider the directed cycle graphs. The following assumption will be taken into account for the cycle graphs.

*Assumption 1. * is controllable, where .

For the cycle graph which is shown in Figure 1, the system matrix of the closed loop is where , .