分布式固定时间多智能体系统优化算法

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"Yu2021_Article_DistributedFixed-timeOptimizat.pdf" 这篇论文“Distributed fixed-time optimization for multi-agent systems over a directed network”由Zhiyong Yu、Shuzhen Yu、Haijun Jiang和Xuehui Mei撰写，发表在Nonlinear Dyn期刊2021年的第103期，页码为775-789。该研究主要探讨了多智能体系统中的分布式固定时间优化问题，特别是带有等式约束的多目标优化问题。在本文中，作者首先构建了一个与等式约束相关的有向网络，以此来转化受限的优化问题为一个无约束的问题。这种转化方法使得原本复杂的优化问题变得更容易处理。接着，他们设计了一种基于局部信息的连续算法，这个算法能够在预设的固定时间内使目标函数收敛到全局最优解。这意味着无论初始条件如何，算法都能在确定的时间内找到最优解，提高了优化效率和系统的稳定性。此外，为了降低通信成本，研究者还提出了一种基于事件触发机制且包含符号函数的算法。这种事件触发机制只在必要的时候进行通信更新，从而减少了不必要的信息交换，同时保持了算法的固定时间收敛性。研究表明，即使采用这种降低通信频率的方法，系统仍然能在预设的固定时间内达到最优值。固定时间优化是控制理论中的一个重要概念，它与传统的渐近稳定或最终稳定不同，其特点是算法能在确定的时间内完成任务，这对于实时性和安全性要求高的应用尤其关键。在多智能体系统中，每个智能体仅需依赖本地信息和有限的邻接节点交互，就能实现全局优化，这在大规模网络系统中具有广泛的应用前景，如协同控制、分布式感知和自主机器人系统等。这篇论文为解决多智能体系统中的优化问题提供了一种创新的固定时间算法，并通过引入事件触发机制优化了通信效率，对于分布式计算和控制领域有着重要的理论价值和实践意义。

Distributed ﬁxed-time optimization for multi-agent systems 777

stability theory. By using improved ﬁxed-time sta-

bility theory [27], a high-precision estimation of the

settling-time is obtained.

3. In order to reduce the communication burden, two

event-triggered protocols are designed by using

sign function and saturation function. For the event-

triggered protocol with sign function, although the

optimal value can be achieved theoretically, there

exists a high frequency chattering. To overcome this

disadvantage, the saturation function is employed

and an improved event-triggered protocol is pre-

sented. It is found that the optimal value also can

be reached in a ﬁxed-time interval and the high-

frequency chattering can be eliminated.

The rest of this paper is organized as follows. Some

preliminaries including graph theory, ﬁxed-time stabil-

ity theory, and problem statement are given in Sect. 2.

The main results are presented in Sect. 3. In Sect. 4,

an example is given to illustrate the correctness of the

results. Section 5 gives a conclusion of t his paper.

Notations In this paper, R and R

denote real num-

ber set and n-dimensional Euclidean space, respec-

tively. For a vector y (or matrix A), let y

) rep-

resents its transpose. Let y =[y

, y

,...,y

]

be an

n-dimensional column vector, and F(y) is a twice dif-

ferentiable function. Then,

∂ F(y)

∂y

denotes the partial

derivative of F(y) with respect to y

(i = 1, 2,...,n).

∇F(y) and ∇

F(y) represent its gradient and Hes-

sian matrix, respectively. y represents the 2-norm of

y.Letsig(y)

=[sig(y

)

, sig(y

)

,...,sig(y

)

]

with sig(y

)

= sign(y

)|y

(i = 1, 2,...,n),in

which sign(·) represents the sign function. N denotes

the natural number set. diag(·) represents the diagonal

matrix.

2 Preliminaries

2.1 Graph theory

Let G = (V, E, A) represent a directed graph, in which

V ={v

,...,v

} denotes the nodes set, E ⊆ V × V

denotes the edges set, and A =[a

]

N ×N

denotes the

weighted adjacency matrix. An edge is described by

an ordered pair of distinct nodes. For an edge e

), we refer to v

and v

as the tail node and head

node, respectively. The weighted adjacency matrix A =

]

N ×N

is deﬁned by a

= 0, a

> 0ife

∈ E

and a

= 0, otherwise. The neighbors of node v

are

denoted by N

={v

∈ V:(v

) ∈ E }. The graph G

is called strongly connected if there exists an directed

path between any different nodes. The Laplacian matrix

L =[l

]

N ×N

of G is deﬁned as l



j=1, j=i

and l

=−a

for i = j.

2.2 Fixed-time stability theory

Consider the following differential equation

˙x(t) = f (x(t)), x(0) = x

, (1)

where x(t) ∈ R

denotes the state variable and

f : R

→ R

is a nonlinear function. Assume that the

origin is the equilibrium point of system (1).

Deﬁnition 1 [26].Theoriginofsystem(1)issaidtobe

globally uniformly ﬁnite-time stable if for any solution

x(t, t

, x

) with x

∈ R

, there exists a positive number

T (x

) such that x(t, t

, x

) = 0 for all t ≥ t

+T (x

The positive number T (x

) is called the settling time.

Moreover, if the positive number T (x

) independent of

initial value x

, the origin is said to be globally ﬁxed-

time stable.

Lemma 1 [27] Consider the differential equation (1).

If there exists a regular, positive deﬁnite and radially

unbounded function V (x): R

→ R such that any solu-

tion of (1) satisﬁes the inequality

V (x(t)) ≤−



(x(t)) + bV

(x(t))



x(t) ∈ R

\0,

where a, b, p, k > 0,q ≥ 0 and pk > 1,qk < 1,

then the origin of system (1) is ﬁxed-time stable, and

the settling time T (x

) is estimated by

T (x

) ≤





1−qk

p−q



1 − qk

pk − 1



Lemma 2 [29] Let θ

,θ

,...θ

≥ 0. Then,



i=1

≥





i=1



, if 0 < a ≤ 1



i=1

≥ n

1−a





i=1



, if 1 < a ≤∞.

123

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分布式固定时间多智能体系统优化算法

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