多智能体分布式控制详解：理论与算法深度解析

多智能体

分布式控制

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多智能体分布式控制是现代信息技术领域中的一个重要研究分支，它主要探讨如何通过多个智能实体（agent）之间的协同作用实现复杂系统的控制和优化。在DISC课程中，这一主题涵盖了丰富的理论基础和算法设计，核心概念包括图论、矩阵理论以及多智能体一致性算法。首先，图论在多智能体系统中扮演了关键角色。一个矩阵A与它的直接图Γ(A)相关联，这表示了实体间的关系结构，如邻接矩阵用于描述每个节点与其邻居的连接。通过分析图的性质，如路径长度、连通性等，可以影响智能体间的通信策略和决策制定。矩阵轮是一种常用的技术，其中智能体按照特定的顺序轮流执行控制任务，这种调度方式有助于确保信息的有效传递和全局协调。矩阵的正定性（A>0或A≥0）和谱理论（如spectral radius ρ(A)和eigenvalues λi(A)）在此背景下非常重要，因为它们直接影响系统的稳定性、性能和收敛速度。系统论的引入为理解多智能体系统的整体行为提供了框架。例如，通过定义最大值和最小值运算（max和min），可以确定一组智能体的集体行为范围。此外，极限运算（→和=》）和逻辑蕴含符号（⇐和⇒）被用来表达系统随着时间和交互的演化过程中的条件和结果。一致性算法是多智能体分布式控制的核心部分。这些算法旨在达成群体内部的共识，例如，让所有智能体达到相同的状态或者共享的信息一致。常见的算法可能涉及到迭代过程，如迭代求解线性方程组，利用矩阵的乘法（如P和Q表示求和和乘积）和Kronecker积（⊗）来更新状态信息。同时，矩阵的对角化（diag和block diagonal）有助于简化计算并保持局部性和隐私性。在数学工具方面，我们看到使用了如行列式（det(A)）来衡量矩阵的重要性，以及谱分解和特征向量（σ(A)、λi(A)、λmax(A)和λmin(A)）来分析系统的动力学特性。此外，集合论的符号（∈、⊂和⊆）用于描述元素与集合的关系，以及极限运算和逻辑关系的表达。多智能体分布式控制讲义涵盖了广泛的数学理论和技术，从基础的图论和矩阵运算到高级的系统建模和算法设计，为理解、设计和优化复杂的分布式系统提供了坚实的理论基础。通过深入学习这些内容，研究人员和工程师能够开发出更为高效、可靠且具有自适应性的智能系统。

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3 Fundamental Results in Distributed Consensus for

Agents with Single-integrator Dynamics

The reading materials for this subsection are [3, 5, 11, 17, 20, 26, 31, 36].

Consider information states with single-integrator dynamics given by

= u

, i = 1, . . . , n, (5)

where ξ

∈ R

is the information state and u

∈ R

is the information control input of the

ith agent. A continuous-time consensus algorithm is given by

= −

j=1

(t)(ξ

− ξ

), i = 1, . . . , n, (6)

where a

(t) is the (i, j) entry of the adjacency matrix A

(t) ∈ R

n×n

at time t. Note that

(t) > 0 if (j, i) ∈ E

at time t and a

(t) = 0 otherwise, ∀j 6= i. Intuitively, the information

state of each agent is driven toward the information states of its neighbors. Equations (5)

and (6) can be written in matrix form as

ξ = −[L

(t) ⊗ I

]ξ, (7)

where ξ = [ξ

, . . . , ξ

]

, L

(t) ∈ R

n×n

is the nonsymmetrical Laplacian matrix at time t and

⊗ denotes the Kronecker product. With (6), consensus is achieved or reached by the team

of agents if, for all ξ

(0) and all i, j = 1, . . . , n, kξ

(t) − ξ

(t)k → 0, as t → ∞.

When interaction among agents occurs at discrete instants, the information state is up-

dated using a diﬀerence equation. A discrete-time consensus algorithm is given by

[k + 1] =

j=1

[k]ξ

[k], i = 1, . . . , n, (8)

where k ∈ {0, 1, . . .} is the discrete-time index; d

[k] is the (i, j) entry of a row-stochastic

matrix D

= [d

] ∈ R

n×n

at the discrete-time index k with the additional assumption that

[k] > 0 for all i = 1, . . . , n and d

[k] > 0, ∀i 6= j, if (j, i) ∈ E

and d

[k] = 0 otherwise.

Intuitively, the information state of each agent is updated as the weighted average of its

current state and the current states of its neighbors. Equation (8) can be written in matrix

form as

ξ[k + 1] = (D

[k] ⊗ I

)ξ[k]. (9)

With (8), consensus is achieved or reached by the team of agents if, for all ξ

[0] and all

i, j = 1, . . . , n, kξ

[k] − ξ

[k]k → 0, as k → ∞.

The main results of [31, 36] will be introduced by the instructor in the class.

Key result from [11]:

(1) Main result: Algorithm (9) achieves average consensus if undirected graph G

(t) is con-

nected jointly.

(2) Approach: algebraic graph theory, matrix theory

Key property used in the proof: Let S

, S

, ··· , S

∈ R

n×n

be a ﬁnite set of SIA matrices

with the property that for each sequence S

, S

, ··· , S

of positive length, the matrix prod-

uct S

j−1

···S

is SIA. Then, for each inﬁnite sequence S

, S

, ···, there exists a column

vector y such that lim

j→∞

j−1

···S

= 1

(see [37]). More detailed proof of the main

result of [11] will be shown by the instructor in the class.

Key result from [20]:

(1) Main result: Suppose that A

is constant. Algorithm (6) achieves average consensus if

and only if directed graph G

is strongly connected and balanced or undirected graph G

is connected. Suppose that A

(t) is switching. Algorithm (6) achieves average consensus if

directed graph G

is strongly connected and balanced at each time instant.

(2) Approach: algebraic graph theory, Lyapunov theory, matrix theory

Key prop erties used: If directed graph G

is strongly connected and balanced, then L

has a simple zero eigenvalue with an associated right eigenvector 1

and an associated left

eigenvector 1

and all other eigenvalue have positive real parts. More detailed proof of the

main result of [20] will be shown by the instructor in the class.

Key result from [17]:

(1) Main result: Algorithm (9) achieves consensus if and only if there exists K ≥ 0 such

that the union of directed graphs G

(t) has a directed spanning tree across each interval of

length Kh, whereh is the sample time.

(2) Approach: algebraic graph theory, set-valued Lyapunov theory

Key technique used in the proof: A set-valued Lyapunov function V is deﬁned as V (x

, . . . , x

) =

(conv{x

, . . . , x

})

, where conv{x

, . . . , x

} denotes the convex hull of {x

, . . . , x

}, and

= X × ··· × X. More detailed proof of the main result of [17] will be shown by the

instructor in the class.

Key result from [26]:

(1) Main result: Suppose that A

is constant. Algorithm (6) achieves consensus if and

only if directed graph G

has a directed spanning tree. Suppose that A

(t) is switching.

Algorithm (6) achieves consensus if directed graph G

(t) has a directed spanning tree jointly.

(2) Approach: algebraic graph theory, matrix theory

Key technique used in the proof: The nonsymmetric Laplacian matrix L

has a simple zero

eigenvalue and all other eigenvalues have positive real parts if and only if G

has a directed

spanning tree. In addition, there exist 1

satisfying L1

= 0 and a nonnegative vector

p ∈ R

satisfying p

= 0 and p

= 1. That is, 1

and p are, respectively, right and

left eigenvectors of L

associated with the zero eigenvalue. If G has a directed spanning

tree, then p is unique. More detailed proof of the main result of [26] will be shown by the

instructor in the class.

Key result from [5]:

(1) Main result: Algorithm (9) achieves consensus if and only if there exists K ≥ 0 such that

the composition of directed graphs G

(t) has a directed spanning tree across each interval of

length Kh, whereh is the sample time.

(2) Approach: A graph composition op eration is deﬁned which is used to discuss the con-

nectivity of a sequence of graphs.

More detailed proof of the main result of [5] will be shown by the instructor in the class.

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