多智能体系统协同控制：一致性区域方法

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"《Cooperative Control of Multi-Agent Systems: A Consensus Region Approach》是由北大教授编写的关于多智能体一致性控制的学术书籍，是自动化与控制工程系列参考书目之一。书中深入探讨了多智能体系统协同控制的共识区域方法。" 这本书详细介绍了多智能体系统合作控制的理论和实践，特别关注如何通过共识区域的方法实现多个智能体之间的协调与一致性。在多智能体系统中，每个个体都有其特定的目标和任务，而一致性控制旨在确保所有个体在行为、状态或决策上达成一致，这在分布式计算、网络控制、群体动态等领域具有广泛的应用。一致性问题在多智能体系统中至关重要，因为它允许各个智能体通过局部通信来实现全局目标。书中可能涵盖了以下几个关键知识点： 1. **多智能体系统理论**：包括多智能体系统的建模，如线性与非线性模型，以及多智能体相互作用的拓扑结构分析。 2. **共识协议设计**：讨论如何设计有效的共识协议，使得智能体在有限的信息交换下逐渐趋近于一个共同的状态或行为。 3. **分布式算法**：介绍用于实现一致性控制的分布式算法，这些算法通常依赖于邻域通信，即智能体只与其邻居进行信息交换。 4. **鲁棒性和稳定性分析**：分析在存在不确定性、通信延迟和外界干扰的情况下，多智能体系统的稳定性和一致性性能。 5. **应用实例**：可能包含多个实际应用场景，如无人机群控制、传感器网络、社交网络动力学等，以展示理论在实际问题中的应用。 6. **MATLAB®工具**：可能利用MATLAB®进行控制系统的设计和仿真，帮助读者更好地理解和应用所学理论。 7. **抗干扰控制**：探讨如何设计抗干扰策略，使系统能够在遭受外部扰动时仍能保持一致性。本书对于理解多智能体系统的协同控制机制以及如何在实际工程问题中应用这些机制提供了深入的见解。无论是对研究人员、研究生还是高级工程师，都能从中受益，提升他们在多智能体系统一致性控制领域的理论水平和实践能力。

4 Cooperative Control of Multi - Agent Systems

the agents are generally dynamically decoupled from each other, which im-

plies the necessity of cooperation in terms of information exchange between the

agents to achieve collective behavior. Speciﬁcal ly, each age nt needs to receive

information from other agents via a direct sensing or communication network.

The interaction (information exchange) topology among the agents is usually

represented as a graph, where each agent is a node and the information ex -

changes between t h e agents are represented as edges. Detailed knowledge of

the graph theory is summarized in Section 1.3. An edge in a graph implies

that there exists informat i on ﬂow between the nodes on this edge. The infor-

mation ﬂows among the agents can be directed or undirected. The interaction

graph among the agents can be static or dynami c . For static graphs, the edges

are not time varyin g, i .e . , th e grap h re mai n s unchanged t hr ou ghou t t he wh ole

process. In contrast, the edges in dynamic graphs are time-varying, due to

disturbances and/or subject to communication range limitations.

For the cooperative control problem, the main task is to design appropriate

controllers t o achieve a desired coordination objective. For cooperative control

design, two approaches, namely, the centralized app r oach and the distributed

approach, are commonly adopted. Th e centralized approach assumes that at

least a central hub (which can be one of the agents) is available an d has the

ability to collect information from and send control signals to all the other

agents. The di s t ri b ut e d approach is based on local interactions only, i.e., each

agent exchanges information with its neighbors. Due to the large number of

agents, the spatial distribution of actuators, limited sensing capability of sen-

sors, and short wirel e s s communication ranges, it is c ons i de re d too expensive

or even infeasible in practi ce to impl eme nt centralized controllers. Thus, dis-

tributed control, depe nd i ng only on local information of the agents and their

neighbors, appears to be a promising resolution for multi-agent systems. In

some cases, only the relative states or output information between neighboring

agents, rather than the absolute states or outputs of t h e agents, is accessible

for controllers design. One typical case is that the agents are equipped with

only ultrasonic range sensors. Another instance is deep-space formation ﬂying,

where the measu r eme nt of the absolute position of the spacecraft is accurate

only in the order of kilometers, which is thus use l ess for control purposes [155].

In the past decade, cooperative control of multi-agent systems has been

extensively studied and quite a large amount of r es ear ch works has been

reported; see the recent surveys [4, 20, 115, 120, 134], the monographs

[8, 74, 112, 127, 138, 139], and the references therein. In the following sub-

sections, we will brieﬂy introduce some typical cooperative control problems:

consensus, f orm ati on control, and ﬂocking.

1.1.1 Consensus

In the are a of cooperative control of multi-agent systems, consensus is an

important and fundamental problem, which is closely related to formation

control, ﬂocking, distributed estimation, and so on. Consensus means that a

Introduction and Mathematical Background 5

team of agents reaches an agreement on a common value by interacting with

each other via a sensing or communication networ k.

Consensus problems have a long history in computer sci en ce and form the

foundation of distributed computing. Pioneering works on the consensus prob-

lems for distributed decision mak i ng and parallel computing includ e [14, 170].

The emergi n g interest in consensus control in the last decade is mainly stim-

ulated by [65] and [121]. In particul ar, a theoretical explanation was provided

in [65] for the alignment behavior observed in the Vicsek model [174] and a

general framework of the consensus problem for networks of integrators was

proposed in [121]. Since then, a large body of works has been reported on the

consensus prob l em s; refer to the surveys [4, 20, 120, 134].

In the following, we brieﬂy introduce some recent works on consensus. In

[133], a suﬃcient condition was deri ved to achieve consensus for ﬁrst-order

integrator multi-agent systems with jointly connected communication graphs.

Consensus of networks of second- and high-order integrators was studied in

[68, 97, 130, 132, 135, 202]. The consensus problem of multi-agent systems with

general linear dynamics was addressed in [83, 86, 93, 106, 146, 173, 185, 195].

Consensus of multiple rigid bodies or spacecraft described as Euler-Lagrange

systems was investigated in [7, 24, 25, 79, 109, 142, 176]. In [90, 156, 189],

conditions were derived for achieving consensus for multi-agent systems with

Lipschitz-type nonlinearity. Consensus algorithms were designed in [22, 75]

for multi-agent systems wi t h quantized communication li n ks . The eﬀect of

various communication delays an d input delays on consensu s was considered

in [13, 97, 114, 122, 166]. Sampled-data control protocols were proposed to

achieve consensus for ﬁxed and switching agent networks in [17, 47]. Distribut-

ed consensus protocols with event-triggered c ommunications were designed in

[34, 149]. Distributed H

∞

consensus and control problems were investigated

in [85, 99] for networks of agents subject to external disturbances.

Existing consensus algorithms can be roughly categorized into two cl asses,

namely, consensus without a leade r (i.e., leaderless consensus) and consensus

with a leader. The latter is also called leader-follower consensus or distributed

tracking. The authors in [56, 57] designed a distributed neighbor-based esti ma-

tor to track an active leade r. Di st r i but e d tracking algorithms were proposed in

[131] for a network of agents with ﬁrst-order dynamics. In [21], discontinuous

controllers were studied in the absence of velocity or acceleration measure-

ments. The authors in [109] addressed the distributed coordinated tracking

problem for multiple Euler-Lagrange systems with a dynami c leader. The dis-

tributed tracking of general linear multi-agent systems with a leader of nonzero

control input was addressed in [92, 95].

The dist r i bu t ed tracking problem deals with only one leader. However, in

some practical applicat i ons , there might exist more than one leader in agent

networks. In the presence of multiple leaders, the containment control prob-

lem arises, where the followers are to be driven into a given geometric space

spanned by the leaders [67]. The st u dy of containment control has been mo-

tivated by many potential applications. For instance, a group of auton omou s

6 Cooperative Control of Multi - Agent Systems

vehicles (designated as leaders) equipped with necessary s en sor s to detect the

hazardous obstacles can be used to safely maneuver anothe r group of vehicles

(designated as followers) from one t ar get to another, by ensuring t hat the

followers are contained within the moving safety area formed by the leaders

[19, 67]. A hybrid containment control law was proposed in [67] to drive the

followers into the convex hull span n ed by the lead er s . Distributed contain-

ment control problems were studied in [16, 18, 19] for a group of ﬁrst-order or

second-order integrator agents u n de r ﬁxed or switching directed communica-

tion topologies. The containment control was considered in [104] for second-

order multi-agent systems with random switching topologies. A hybrid model

predictive control scheme was proposed in [46] to solve the containment and

distributed sensing problems in leader-follower multi-agent systems. The au-

thors in [36, 110, 111] studied the containment control problem for a collecti on

of Euler-Lagrange systems. The containment control for multi-agent systems

with general linear dynamics was considered in [ 88, 94].

1.1.2 Formation Control

Compared with the consensus problem where the ﬁnal states of all agents typ-

ically become a singleton, the ﬁnal states of all agents can be more diversiﬁed

under th e formation control scenario. The objective of formation control is t o

stabilize the relative distances/positi ons among the agents to prescribed val-

ues. Formation control ﬁnds natural applications in satellit e formati on ﬂying,

cooperative transportation, sensor networks, and so forth. Roughly speaking,

formation control can be categorized into formation producing and formation

tracking. Formation producing refers to the algorithm design for a group of

agents to reach some pre-desired geometric pattern in the absence of a group

reference while formation tracking refers to the same task following a pre-

designated group reference [20]. Due to the existence of the group ref erence,

formation tracking is usually much more challenging than formation produc-

ing.

The consen su s algorithms lay a ﬁrm basis for the formation control prob-

lem. By adding appropriate oﬀset variables, the consensus protocols can be

modiﬁed to produce desired formation structures [44, 86, 136]. Also, as an

extension of the consensus algorithms, some coupling matrices were intro-

duced [100, 123, 137] for systems with single-integrator kinematics and double-

integrator dynamics to oﬀset the corresponding control inputs by some an-

gles. The collective motions for nonholonomic mobil e robots were studied in

[147, 148] for both all-to-all and limited communications. The inverse agree-

ment problem was studied in [35] and was used to force a team of agents

to disperse in space. Motivated by the graph rigidity, research has been con-

ducted in [45, 54, 188] to drive a group of agents to a desired inter-agent

conﬁguration by ensuring that a certain number of edge distances are identi-

cal to the desired ones. Compared with other formation producing algorithms

which requi r e edge vector information, the formation control algor i t hm s based

Introduction and Mathematical Background 7

on rigidity only require edge distance informat ion . As a tr ade oﬀ, some unst a-

ble equilib r ia, rather than the desired inter-agent conﬁguration, might ex is t .

Formation control in the presence of a group reference was stud ie d in [38, 40]

for a group of nonholonomic mobile robots and in [141] for multiple rigid

bodies. More works on formation control were reported in [20].

1.1.3 Flocking

Flocking is a typical collective behavior in autonomous multi-agent systems,

which can be widely observed in nature in the scenes of grouping of birds,

schooling of ﬁsh, swarming of bacteria, and so on. In practice, understanding

the mechanisms responsible for the emergence of ﬂocking in animal groups can

help develop many artiﬁcial autonomous systems such as formation control of

unmanned air vehicles, motion planning of mobile robots, and scheduling of

automated highway systems.

The classical ﬂocking model proposed by Reynolds [ 140] in the 1980s con-

sists of three heuristic rules: 1) c oll i si on avoidance: at t em pt to avoid collisions

with nearby ﬂockmates; 2) velo ci ty matching: attempt to match velocity wit h

nearby ﬂockmates; and 3) ﬂock centering: attempt to stay close to nearby

ﬂockmates. These t h r ee rules are usually known as separation, alignment, and

cohesion, respectively. In 1995, Vi cs ek et al. proposed a simple model for phase

transition of a group of self-dr i ven particles and numerically observed th e col-

lective behavior of synchronization of multiple agents’ headi ngs [174]. In a

sense, Vicsek’ s model c an be vi ewed as a special case of Reynolds’ ﬂocking

model, because only velocity matching is considered in the forme r. To sys-

tematically i nvestigate the ﬂocking control problem in multi-agent systems,

distributed algorithms consisting of a local artiﬁcial potential function and a

velocity consensus component were p roposed in [119, 165] . The local artiﬁci al

potential function guarantees separation and cohesion where as the velocity

consensus comp on ent ensures alignment in such multi-agent systems. This

kind of three-rule combined ﬂocking algorithms were then extended to deal

with obstacles avoidance in [28]. The ﬂocking control problem of multiple n on -

holonomic mobile robots was investigated in [39] by using tools from nonlinear

control theory. Distributed protocols to achieve ﬂocking in the presence of a

single or multipl e virtual leaders were designed in [160, 161, 162]. The ﬂocking

control problem of multi-agent systems with n onl i n ear velocity couplings was

investigated in [52, 179] . Distrib ut e d hybrid controllers were proposed in [194],

which can ensure that the ﬂocking behavior emerges and meanwhile retain th e

network connectivity.

8 Cooperative Control of Multi - Agent Systems

1.2 Overview of This Monograph

The obj ective of t hi s monograph is to addre ss typical cooperative con-

trol problems of multi-agent systems, incl ud i ng consensus, di s tributed track-

ing, and containment control. Although there exist several monographs

[8, 74, 112, 127, 138, 139] on recent advances in cooperative control of multi-

agent systems, it is worth noting that the agent dynamics considered in those

monographs are assumed to be ﬁrst-order, second -or de r integrators, Euler-

Lagrange systems, or systems with passivity properties. These assumptions

might be too restrictive under many circumstances. This motivates us to write

this monograph , which focuses on cooperative control of multi-agent systems

with more general and complex agent dy nami c s.

In this monograph, we address the consensus, distributed tracking and

containment problems for multi-agent systems with general linear agent dy-

namics, linear agent dynamics with uncertainties, or Lipschitz nonlinear agent

dynamics. Diﬀering from the existing works, in th i s monograph we present

a systematic consensus region approach to designing distribu t ed cooperative

control laws, which is a quite challenging task due to the interplay of the agent

dynamics, the interaction graph among agents, and the cooperative control

laws. The proposed consensus region decouples the d es i gn of the feedback

gain matrices of the consensu s protocols from the communication graph and

can serve in a certain sense as a measure for the robustness of the consensus

protocols to variations of the com munication graph. By exploiting the decou-

pling feature of the proposed consensus region approach, di s tr ib ut ed adaptive

consensus protocols are proposed, whi ch adaptively tune the weights on the

communication graph and can be designed and implemented by each agent in

a fully distributed fashion. This is one main contribution of this monograph.

The results in this monograph also extend many existing results in the area

of cooperative control.

This monograph consists of nine chapters. A brief introduction of these

chapters is given as follows.

• Chapter 1 introduces the background of cooperative control and reviews

the basic graph theory and some t e chnical tools which will be used later.

• Chapter 2 i s concerned with the consensus control problem of multi-agent

systems with general continuous-time linear agent dynamics. The notion

of continuous-time consensus region is introduced. Algorithms based on

the consensus region will be presented to design several consensus pro-

tocols based on relative states or relative outputs of neighboring agents.

Extensions of consensus algorithms to formation control are also discussed.

• Chapter 3 presents the discrete-time counterpart of Chapter 2. The notion

of discrete-time consensus region is introduced and algorithms are present-

ed to design consensu s protocols based on relative states or relative outputs

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