Complexity

Volume 2019, Article ID 9487574, 10 pages

https://doi.org/10.1155/2019/9487574

## On the Observability of Leader-Based Multiagent Systems with Fixed Topology

^{1}School of Information Engineering, Minzu University of China, Beijing 100081, China^{2}College of Science, North China University of Technology, Beijing 100144, China^{3}School of Artificial Intelligence and Automation, Huazhong University of Science and Technology, Wuhan 430074, China

Correspondence should be addressed to Housheng Su; moc.qq@usgnehsuoh

Received 18 June 2019; Revised 7 September 2019; Accepted 16 September 2019; Published 5 November 2019

Academic Editor: Xianggui Guo

Copyright © 2019 Bo 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

This paper investigates the observability of first-order, second-order, and high-order leader-based multiagent systems (MASs) with fixed topology, respectively. Some new algebraic and graphical characterizations of the observability for the first-order MASs are established based on agreement protocols. Moreover, under the same leader-following framework with the predefined topology and leader assignment, the observability conditions for systems of double-integrator and high-integrator agents are also obtained. Finally, the effectiveness of the theoretical results is verified by numerical examples and simulations.

#### 1. Introduction

Self-organizing behavior of multiple intelligent dynamic agents is widespread in nature, for example, bird flocking flight [1], fish group patrol [2], microbial collective foraging [3], and ant colony cooperation [4]. This self-organizing behavior of such intelligent and dynamical networked system, through transferring and interaction of information between local agents, as a whole, presenting the ability of effective coordination and the level of advanced intelligence, has led to the wide concern and interest in many fields including applied mathematics, ecology, biology, sociology, physics, computer science, control, and engineering. At present, coordinated control of intelligent systems has become a hot research topic in the field of control and is widely used in smart grids, UAV formation, intelligent transportation systems, artificial satellite networks, and so on [5–20].

Observability is a significant problem and becomes the basis of the design of optimal control and optimal estimation in modern control theory, which was put forward by Kalman in 1960. Intuitively speaking, the so-called system’s observability means that the internal states of the system can be correctly known by observing the external variables of the system. In general, for a multiagent network, the observability is the key problem and means that the entire system’s state can be completely reconstructed by only observing a small quantity of intelligent dynamic agents (i.e., leaders). Observability plays a fundamental role in state estimation and system control as well as has recently been widely applied in wireless sensor networks, satellite navigation systems, autonomous underwater vehicles, machine games, and so on.

The concepts of controllability and observability were first tried to apply to networked multiagent systems by Tanner in 2004 [21], where all agents are partitioned into leader agents and follower agents, in which the follower agents are only affected by the neighbor interaction protocol and the leader agents are influenced by the external control input. This structure, called leader-follower structure, can make the whole system as a standard linear time-invariant system so that the classical controllability and observability are all applied into networked multiagent systems. Observability is very important to study distributed estimation and intrusion detection problems in distributed sensor networks [22]. Zelazo and Mesbahi [23] investigated the observability of networked systems with homogeneous and heterogeneous dynamics, respectively, as well as provided an analysis for the observability of networked linear systems. In [24], the study mainly focused on controllability and observability of multiagent networks under a leader-follower framework with the chain and cyclic topologies, respectively. In [25], the authors established a decentralized condition for controllability and observability of networked MASs using decentralized Laplacian spectrum estimation. The observability of a network system was investigated based on average consensus algorithm for path and cycle graphs in [26], graph Cartesian products in [27], regular graphs and distance-regular graphs [28, 29], and equitable partition [30], respectively. Furthermore, the observability of switched linear systems and switched multiagent systems was studied in [31, 32], respectively. Recently, the researchers [33–35] investigated the observability of MASs with time-varying and switching topologies, respectively. In addition, Liu et al. [36] analyzed the observability for complex networks.

It is worthwhile to note that most of the previous research conclusions mainly focused on controllability and observability of first-order networked MASs. However, it is well known that, in control engineering, almost all control systems are high-order systems, that is, higher-order differential/difference equations are used to describe those control systems studied in many areas, such as consensus, controllability, flocking, containment control, and stabilizability. In this paper, the observability of a class of MASs is considered, and a unified framework for the inertia of first-order, second-order, and high-order MASs is given. Most of the existing literatures on the observability of MASs focused only on a single dynamic equation, such as a first-order dynamics, a second-order dynamics, or a high-order dynamics, rather than a unified consideration. In general, it is difficult to bring the three kinds of networks into a unified framework, which is a hard point in modelling and analyzing. This paper aims at the observability of MASs with first-order/second-order/high-order agents based on agreement protocols, respectively. The main contributions of this paper are summarized as follows: (i) Some necessary and sufficient conditions for the observability of first-order MASs are built by Laplacian matrix, which relies only on the information flows within the agents and between the agents from the leaders. (ii) It is proved that the observability of the second-order/high-order MASs is equivalent to that of first-order MASs with the same information of topology structure and the same prescribed leaders. (iii) A unified framework for the observability inertia of first-order, second-order, and high-order MASs is established, which relies only on the information topology of such network.

#### 2. Mathematical Preliminaries

For the convenience of discussing the observability of networked systems, the information communication links among agents can be described by a weighted undirected/directed graph denoted as , in which is a *vertex set* and is an *edge set*, respectively. The weighted adjacency matrix of is marked as with if and only if and , otherwise. Moreover, represents the neighbors’ set of agent . The Laplacian matrix is given as , where is the weighted adjacency matrix and is the diagonal degree matrix.

For a network with leader-follower structure (see Figure 1), the Laplacian matrix *L* can be divided into the following block matrices:where and stand for the communication information links within the followers and the leaders, respectively, stands for the communication information links from the leaders to the followers, and stands for the communication information links from the followers to the leaders, where if graph is undirected.