Cooperative Control of Complex Network Systems with Dynamic Topologies

Guanghui Wen, Wenwu Yu, Yuezu Lv, Peijun Wang

• Publisher: CRC Press
• Year: 2021
• ISBN:
  9781000400199
• Edition: 1
• Topic: Technology & Engineering
• Subtopic: Engineering General
• Index: Technology & Engineering

Far from being separate entities, many social and engineering systems can be considered as complex network systems (CNSs) associated with closely linked interactions with neighbouring entities such as the Internet and power grids. Roughly speaking, a CNS refers to a networking system consisting of lots of interactional individuals, exhibiting fascinating collective behaviour that cannot always be anticipated from the inherent properties of the individuals themselves.

As one of the most fundamental examples of cooperative behaviour, consensus within CNSs (or the synchronization of complex networks) has gained considerable attention from various fields of research, including systems science, control theory and electrical engineering. This book mainly studies consensus of CNSs with dynamics topologies - unlike most existing books that have focused on consensus control and analysis for CNSs under a fixed topology. As most practical networks have limited communication ability, switching graphs can be used to characterize real-world communication topologies, leading to a wider range of practical applications.

This book provides some novel multiple Lyapunov functions (MLFs), good candidates for analysing the consensus of CNSs with directed switching topologies, while each chapter provides detailed theoretical analyses according to the stability theory of switched systems. Moreover, numerical simulations are provided to validate the theoretical results. Both professional researchers and laypeople will benefit from this book.

CHAPTER 1

Introduction

This chapter overviews some recent research progress in cooperative control of complex network systems over directed switching communication topologies. Distributed cooperation of complex network systems (CNSs), including synchronization of complex networks and consensus control of multiagent systems (MASs), has been a very active research topic in a wide variety of scientific communities, ranging from applied mathematics to physics, engineering to biology, even sociology. In Section 1.1, CNSs include MASs and complex networks are introduced. In Section 1.2, definitions of synchronization of complex networks and consensus of MASs are given, moreover, some differences between these two topics are briefly summarized. In Section 1.3, the research progress of synchronization of complex networks with switching topologies are presented. In Section 1.4, the research progress of consensus of MASs with switching topologies are presented. In Section 1.5, we conclude this chapter by presenting some future works from our own viewpoint.

1.1 Complex network systems

Far from being separate entities, many natural, social, and engineering systems can be considered as CNSs associated with tight interactions among neighboring entities within them [39, 201, 130, 29, 18, 219, 211, 194, 175, 10, 141, 3, 120, 50, 92]. Roughly speaking, a CNS refers to a networking system that consists of lots of interconnected agents, where each agent is an elementary element or a fundamental unit with detailed contents depending on the nature of the specific network under consideration [175]. For example, the Internet is a CNS of routers and computers connected by various physical or wireless links. The cell can be described by a CNS of chemicals connected by chemical interactions. The scientific citation network is a CNS of papers and books linked by citations among them. The WeChat social network is a CNS whose agents are users and whose edges represent the relationships among users, to name just a few.

With the aid of coordination with neighboring individuals, a CNS can exhibit fascinating cooperative behaviors far beyond the individuals’ inherent properties. Prototypical cooperative behaviors include synchronization [177, 38, 101, 95], consensus [118, 128, 76], swarming [115, 48], flocking [117, 161]. In this book, we focus on the CNSs which include complex networks and MASs as special cases. A lot of new research challenges have been raised about understanding the emergence mechanisms responsible for various collective behaviors as well as global statistical properties of CNSs [3, 178, 114, 15]. Network science, as a strong interdisciplinary research field, has been established at the first several years of the 21st century [110]. It is increasingly recognized that a detailed study on cooperative dynamics of CNSs would not only help researchers understand the evolution mechanism for macroscopical cooperative behaviors, but also prompt the application of network science to solve various engineering problems, e.g., design of distributed sensor networks [135], formation control of multiple unmanned aerial vehicles [37], distributed localization [89], and load assignment of multiple energy storage units in modern power grid [191].

Among the various cooperative behaviors of CNSs, synchronization of complex networks and consensus of MASs are the most fundamental yet most important ones. Synchronization of complex networks exhibits the cooperative behavior that the states of all entities within these networks achieve an agreement on some quantities of interest. Compared with stability analysis of an isolated control plant, synchronization behavior analysis in CNSs are much more challenging as the synchronization process is determined by the evolution of network topology as well as the inherent dynamics of individual units within these network systems [121, 102, 198, 199, 96]. As a topic closely related to synchronization of complex networks, the consensus of MASs has recently gained much attention from various research fields, especially the system science, control theory, and electrical engineering communities [65, 88, 116, 128, 22]. In the remainder of this chapter, we will review some existing results on achieving synchronization of complex networks and consensus of MASs over dynamically changing communication topologies.

1.2 Definitions of synchronization and consensus

Before moving forward, the definition of consensus of MASs is given. Moreover, the synchronization of complex networks can be defined similarly.

Consider an MAS which consists of N agents. Without loss of generality, we label the N agents as agents $1,\dots ,N$, respectively. The dynamics of agent i, $i=1,\dots ,N$, are represented by

$$\begin{array}{l}{\dot{x}}_i(t)=f(t,x_i(t),u_i(t)),\hfill \end{array}$$(1.1)

where $x_i(t)\in {ℝ}^n$ and $u_i(t)\in {ℝ}^m$ represent, respectively, the state and the control input, $f(·,·,·):[t_0,+\infty )\times {ℝ}^n\times {ℝ}^m\mapsto {ℝ}^n$ represents the nonlinear dynamics of agent i. A particular case is the general linear time-invariant MASs with the dynamics of agent i are described by

$$\begin{array}{l}{\dot{x}}_i(t)=A{x}_i(t)+B{u}_i(t),i=1,\dots ,N,\hfill \end{array}$$(1.2)

where $A\in {ℝ}^{n\times n}$ and $B\in {ℝ}^{n\times m}$ represent, respectively, the state matrix and control input matrix. For convenience, throughout this book, we call MAS (1.1) to represent the MAS whose dynamics are described by (1.1).

Definition 1.1 Consensus of the MAS (1.1) is said to be achieved if for arbitrary initial conditions $x_i(t_0)$, $i=1,\dots ,N$,

(1.3)

The definition of consensus for MAS (1.1) given by Eq. (1.3) does not concern about the final consensus states. However, it is sometimes important to make the states of all agents in the considered MASs to finally converge to some predesigned trajectory, especially from the viewpoint of controlling various complex engineering systems. To ensure the states of all agents in MAS (1.1) converge to some desired states, a target system (may be virtual) is introduced to the network (1.1) as

$$\begin{array}{l}\dot{s}(t)=f(t,s(t))\hfill \end{array}$$(1.4)

for some given initial value $s(t_0)\in {ℝ}^n$. Under this scenario, we call agent i whose dynamics are described by (1.1) the follower i, $i=1,\dots ,N$, and call the agent whose dynamics are described by (1.4) the leader.

Definition 1.2 Consensus tracking (or leader following consensus) of the MAS with the followers given by (1.1) and the leader given by (1.4) is said to be achieved if for some given initial conditions $s(t_0)$ and $x_i(t_0)$, $i=1,\dots ,N$,

(1.5)

The existence and uniqueness of the solutions of system (1.1) will be discussed in Chapter 2.

Remark 1.1 The mathematical definitions for synchronization of complex networks and consensus of MASs are precisely similar. However, some differences between these two topics are briefly summarized as follows from our viewpoint.

1. A complex network typically contains a great number of individual nodes (e.g., the Internet) while the scale of an MAS may be relatively quite small (e.g., a team of several robots).
2. The objective of synchronization control is to make the states of a large-scale network achieve state agreement under some given inner linking matrices by selecting only the coupling strength, while the objective of consensus is to make the states of agents achieve state agreement by designing the gain matrices as well as the coupling strength.
3. Significant attention has been paid to revealing the relationship between the statistical properties (e.g., the degree distribution, the average path length, and the symmetry) of network topology and the synchronizability of complex networks within the context of synchronization in complex networks, while in the context of consensus of MASs, much attention has been focused on addressing the relationship between the algebraic properties (e.g., the algebraic connectivity for undirected interaction topology and the general algebraic connectivity for directed interaction topology) of interaction topology and the consensusability.

Without causing any confusion, synchronization of complex networks and consensus of MASs are referred to as consensus of CNSs in this book.

Practical applications of consensus of CNSs: Achieving consensus in CNSs is critical for controlling these CNSs and thus helpful in dealing with various distributed control problems for practical network systems. For instance, reaching consensus of velocities for all individual agents is a precondition in achieving flocking in various second-order CNSs [117]. In another instance, frequency synchronization of multiple generator units within a power system is one of the most important issues in power system stability control [221]. In addition, clock synchronization among sensors within wireless sensor networks is highly desirab...