Nonlinear Dyn (2025) 113:26417–26432
https://doi.org/10.1007/s11071-025-11478-5

RESEARCH

Distributed optimization-based fixed-time formation control
with external disturbances: an event-based intermittent
approach
Lanlan Ma · Cheng Hu · Juan Yu · Shiping Wen ·
Leimin Wang

Received: 9 April 2025 / Accepted: 9 June 2025 / Published online: 17 June 2025
© The Author(s), under exclusive licence to Springer Nature B.V. 2025

Abstract The problem of fixed-time distributed optimization based formation control is discussed for
multi-agent systems with external disturbances. To
begin with, by introducing an integral sliding-mode
strategy and two auxiliary functions with fixed-time
convergence, a decentralized event-based intermittent
control algorithm is developed to suppress the external
perturbation. Subsequently, a gradient-based decentralized power-law control protocol is developed to guide
each agent towards the minimum of its local cost function within a fixed time. Thirdly, a fixed-time distributed algorithm is presented and the formation is
established in a fixed time around the minimum point
of the global cost function. Finally, the proposed formation control scheme is demonstrated through several
numerical analysis.

L. Ma · C. Hu (B) · J. Yu
College of Mathematics and System Science, Xinjiang University, Urumqi 830017, China
e-mail: hucheng@xju.edu.cn
S. Wen
Centre for Artificial Intelligence, University of Technology Sydney, Sydney 2007, Australia
e-mail: shiping.wen@uts.edu.au
L. Wang
School of Automation, China University of Geosciences, Wuhan
430074, China
e-mail: wangleimin@cug.edu.cn

Keywords Distributed algorithm · Formation control ·
Fixed-time optimization · Event-based intermittent
control · Multi-agent system

1 Introduction
As a vital aspect of coordinated control, formation control has emerged as a significant research
area characterized by substantial advancements, such
as aerospace exploration, intelligent transport, disaster relief, and military [1–3]. Consequently, it has
been widely employed in various multi-agent system
(MAS), including multi mobile robots [4], unmanned
surface vessels [5], and Euler-Lagrange system [6].
The main task of distributed formation control is
to develop effective control protocols to facilitate the
collective achievement of specific formation configurations, while maintaining the geometric properties of
the collection, including shape, distance and orientation
[7–10]. Notably, consensus-based distributed formation control has garnered considerable attention [11–
14]. However, when considering practical constraints
and requirements, it may be insufficient to rely solely on
consensus constraints. Indeed, optimization is a crucial
aspect of various formation tasks, including the minimization of travel distance, energy consumption, and
search rescue operations [15–17]. How to ensure optimal control inputs and minimize energy consumption
or execution time? This poses a challenge of distributed
optimal formation control.

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The core challenge in the above scenarios is to
develop feasible control protocols to form a specified
formation for all agents while simultaneously minimizing a global objective function. To address this difficulty, the distributed optimization-based (DOB) formation problem was developed [18,19]. Compared to
traditional formation control, DOB formation control
provides enhanced flexibility, adaptability robustness
and security. Consequently, there has been a notable
increase in the research of DOB formation control [20–
23]. It is notable that the existing research on DOB formation control predominantly emphasizes asymptotic
convergence, however, practical engineering applications require tasks to be completed within a finite time
framework. Therefore, there is a critical requirement to
accelerate the convergence rate of the algorithms utilized in the control of DOB formation.
It is essential to recognize that the settling time of
finite-time algorithms relies on system’s initial states
[24,25]. Nevertheless, acquiring these initial values is
challenging due to the presence of disturbances, information loss, and various practical considerations. In
light of these challenges, fixed-time (FXT) stability theory was introduced [26,27], wherein the upper bound
of the convergence time is exclusively linked to system parameters. Following the FXT stability theory, the
FXT distributed optimization approaches [28–32] and
FXT formation control strategies [33–36] were studied.
To the authors’ knowledge, the research on fixed-time
DOB formation control for MAS remains limited, and
there is a lack of sufficient enhancement of the estimate
accuracy of the optimal formation control time. Hence,
how to guarantee both of the desired formation performance and optimization objective within a fixed time
is the first motivation of this paper.
In light of the need for resource conservation
and sustainable development, intermittent control has
become a significant research topic within the realm of
discontinuous control strategies [37–41]. Distinct from
traditional intermittent control schemes, event-based
intermittent (EBI) control [42–46], which is updated
by specific state-dependent events, does not require
prior knowledge of the switching time. Although the
aforementioned studies have well investigated various
issues with EBI control schemes, nonperiodic EBI control strategies limited by the fixed-time convergence
framework have not been fully investigated so far. Consequently, developing a novel algorithm with EBI control scheme to address DOB formation problem with

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L. Ma et al.

external disturbance under the fixed-time framework
is a significant research direction, this constitutes the
second motivation for the current study.
In light of the above considerations, an innovative
three-component algorithm is proposed to tackle fixedtime DOB formation control issue for MAS. Firstly,
external disturbances are addressed through the implementation of an integral sliding-mode scheme, which
is designed with the EBI control strategy. Secondly,
local optimization is performed via the gradient method
within a fixed time. Thirdly, the global optimization and
formation tasks are completed, where the formation
center is the optimum point of the global cost function.
The main innovations of this study can be summarized
as follows:
(1) Distinct from the formation control by tracking a
reference point investigated in [17,21], the preset
formation concerned here is around the optimal
point of the global cost function and is ensured
to be formed concurrently with the solution of
optimization problem. Moreover, the formation
results obtained here can be extended to the general distributed optimization problems with consensus constraints.
(2) Considering the presence of external disturbances
within practical systems [45,46], this study incorporates external disturbances into the analytical framework. To address this issue, a fully
decentralized fixed-time EBI control algorithm
is developed by combining the integral sliding
mode method. Specifically, a decentralized eventtrigger mechanism is proposed through introducing two auxiliary functions with fixed-time convergence, and the switch time instants of control
and rest are automatically determined. Evidently,
the presented fixed-time EBI control strategy is
more intelligent and more advantageous in saving
communication resources in comparison to timedependent intermittent control presented in [38–
40] and event-based intermittent control with predefined control or rest time instants in [42–44].
(3) To realize fixed-time DOB formation control, a
three-component distributed control protocol is
presented based on the FXT Lyapunov stability theory. The fixed-time convergence of the
DOB formation control discussed herein facilitates rapid convergence, enhancing system robustness and efficiency in practical applications, com-

Distributed optimization-based fixed-time formation control…

26419

pared with the existing studies on DOB formation control in sense of asymptotic convergence
[20–23] and finite-time convergence [24,25]. In
theoretical analysis, the Lyapunov functions are
constructed with a decentralized manner, which
relies only on the local information and provides
better robustness and flexibility, and is more effective in addressing large-scale or complex systems
compared with the previous Lyapunov functions
with the summation form in [46].

nected, iff for any l, k ∈ V, there has a directed path
from l to k. The digraph G is detailed balanced with
respect to the weight vector ς = (ς1 , ς2 , . . . , ς N )T if
ςi ai j = ς j a ji for all ςi > 0, i, j ∈ V.

The structure of this paper is as follows. Section 2
reviews the essential preliminaries. In Section 3, a
type of fixed-time DOB formation control algorithm is
developed to address optimization and formation control simultaneously. Section 4 presents several numerical results to demonstrate the efficacy of proposed
methods. Finally, this paper is summarized in Section 5.
Notations: Throughout the article, R, R + , N , and
m
R denote the sets of real numbers, positive real
numbers, natural numbers and the m-dimensional real
space, respectively. The m-order identity matrix is represented by Im , the all-zero column vector with mdimension is denoted as 0m , col(s1 , s2 , . . . , s N ) represents a column vector composed of vectors [s1 , . . . , s N ]
with appropriate dimensions. s represents the
Euclidean norm of s ∈ R m , sigα (s) = sign(s)sα
for α > 0, in which sign(s) = (sign(s1 ), sign(s2 ), . . . ,
sign(sm ))T , and for any si ∈ R, i = 1, . . . , m, sign(si )
is the sign function of si . For function h(s), ∇h(s) and
∇ 2 h(s) represent separately its gradient and Hessian
matrix.

2.2 Problem formulation
Consider a MAS with N individuals described as
ẋi (t) = u i (t) + ωi (t), i ∈ V,

(1)

here xi (t), u i (t), ωi (t) ∈ R m are the state, control input
and bounded external disturbance of agent i, respectively.
Definition 1 [37] The MAS (1) achieves FXT formation provided that there exist time instants T̂ (X (0)) ≥
0 and Tmax > 0 satisfying T̂ (X (0)) ≤ Tmax for any
X (0) ∈ R N m , and
⎧
N

⎪
⎪
⎪
lim
di (t) − c(t) = 0,
⎪
⎨ t→T̂ (X (0))
i=1
(2)
N
⎪

⎪
⎪
⎪
di (t) − c(t) = 0, for all t ≥ T̂ (X (0)),
⎩
i=1

in which X = col(x1 , x2 , . . . , x N ), di (t) = xi (t) − z i
is the distance offset for agent i, z i ∈ R m is formation
configuration of agent i that is defined by users, and
c(t) ∈ R m is the formation center function. The time
N


T (X (0))=inf T̂ (X (0)) ≥ 0,
di (t) − c(t)=0,
i=1

t ≥ T̂ (X (0))

2 Preliminaries and problem description
2.1 Graph Theory
Consider a weighted directed topology G = (V, E, A),
in which the vertex set V = {1, 2, . . . , N }, the edge
set E = {(i, j), i, j ∈ V, i = j} ⊆ V × V. If
there has a link from the agent i to the agent j, it is
denoted by (i, j) ∈ E. The set of neighbors of the ith
agent is defined as Ni = { j ∈ V, ( j, i) ∈ E, j = i}.
A = [ai j ] N ×N is the weighted adjacency matrix with
ai j > 0 if ( j, i) ∈ E, otherwise ai j = 0, and aii = 0,
which implies that the graph G has no self-loops. The
Laplacian matrix L = [li j ] N ×N , where li j = −ai j for

ai j . Digraph G is strongly coni = j and lii =
j∈Ni

is the settling time to achieve the FXT formation.
Considering the following distributed optimization
problem
min F(X ) =

N


f i (xi ),

i=1

s.t.

xi = x j ,

i, j ∈ V,

(3)

here xi ∈ R m , f i : R m → R is the local cost function
of the ith agent, which satisfies the following condition.
Assumption 1 [47] For any agent i ∈ V, f i (xi ) is
twice continuously differentiable, θi -strongly convex

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L. Ma et al.

with θi > 0, and Θi -smooth with Θi > 0, that is,
θi Im ≤ ∇ 2 f i (u) ≤ Θi Im , or equivalently,

Laplacian matrix LG of a connected undirected graph
G, it has

θi
u − v2 ≤ f i (v) − f i (u) − (∇ f i (u))T (v − u)
2
Θi
u − v2 ,
≤
2

λ2 (LG )x T LḠ ⊗ Im x ≤ N x T LG ⊗ Im x,
here x ∈ R N m .

here 0 < θi ≤ Θi , i ∈ V.
By Assumption 1, there exists a unique number xi∗ ∈
m
R such that ∇ f i (xi∗ ) = 0m .
Definition 2 [34] For the MAS (1), it achieves fixedtime DOB formation with the optimal solution x ∗ of the
optimization problem (3), if c(t) ≡ x ∗ in Definition 1.
Given the predefined formation z = col(z 1 , z 2 , . . . ,
z N ), let f˜i (di (t)) = f i (di (t) + z i ) for i ∈ V, then the
fixed-time DOB formation problem is depicted by
min F̃(d) =

N


N


μ
ξi ≥

i=1

N


μ

ξi

,

i=1

N


ξiν ≥ N 1−ν

i=1

N


ν

ξi

.

i=1

Lemma 4 [27] For system ẋ = f (x) satisfying
f (0) = 0 and a positive-definite and radially
unbounded function V (x) : R m → R, if there exist
α1 > 0, α2 > 0, 0 < κ1 < 1 and κ2 > 1 satisfying
d
V (x(t)) ⩽ −α1 V κ1 (x(t)) − α2 V κ2 (x(t)),
dt

f˜i (di ),

i=1

s.t.

Lemma 3 [50] Suppose that ξi ≥ 0 for i ∈ V, 0 ≤
μ ≤ 1 and ν > 1, then

⎧
N

⎪
⎪
⎪
lim
di (t) − x ∗  = 0,
⎪
⎨ t→T̂ (X (0))

then the origin is FXT stable within the time

i=1

T =

N
⎪

⎪
⎪
⎪
di (t) − x ∗  = 0, for all t ≥ T̂ (X (0)),
⎩

π
α1 (κ2 − κ1 )



α1
α2

ζ

csc (ζ π ) , ζ =

1 − κ1
.
κ2 − κ1

i=1

(4)
here d = col(d1 , d2 , . . . , d N ).
Remark 1 As indicated in [24], the problem of formation efficiency can be effectively solved by taking the
optimal point x ∗ as the center of formation. However, it
is generally unknown a priori. Therefore, the objective
of this article is to design a proper algorithm to achieve
DOB formation by accessing the optimal value x ∗ in a
distributed manner.
Lemma 1 [48] The eigenvalues of the Laplacian
matrix L, for a connected and undirected graph, can
be ordered by 0 = λ1 < λ2 ≤ . . . ≤ λ N , and for any
positive semi-definite matrix Λ ∈ R m×m ,
x T (L ⊗ Λ) x ≥ λ2 (L)x T (I N ⊗ Λ) x,
here x = col(x1 , . . . , x N ) satisfying
xi ∈ R m .

N

i=1 x i = 0 with

Lemma 2 [49] For Laplacian matrix LḠ ∈ R N ×N of
the unweighted complete undirected graph Ḡ, and the

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3 Fixed-time distributed optimization-based
formation
In this section, the DOB formation problem (4) is
addressed. At this stage, the following assumptions are
introduced.
Assumption 2 [29] The digraph G is detail-balanced
and strongly connected.
Assumption 3 [36] For i ∈ V, the disturbance ωi (t)
is bounded by a known constant ω̄ ∈ R + , that is,
ωi (t) ≤ ω̄.
Remark 2 Assumption 1 is a reasonable and standard
condition required in many researches about distributed
optimization and DOB formation problems [32–35],
which guarantees a unique optimization solution x ∗
to the optimization problem (4). And some special
cost functions satisfy Assumption 1, such as quadratic
cost functions in economic dispatch [45], and squared
distance function in the optimal formation of multiple mobile robots [15]. Assumption 2 regarding the

Distributed optimization-based fixed-time formation control…

26421

directed communication topology has been extensively
employed in existing researches [29,32] to address the
distributed optimization problems. Assumption 3 contributes to the design of the control parameters that mitigates the disruptive effects of the disturbances. There
has a range of practical disturbances meet Assumption
3, such as sinusoidal and constant disturbances [29,36].

are the control activated time instants and the control
rest time instants.
To realize DOB formation problem with EBI control, two nonnegative and differentiable auxiliary functions, V1i (t) and V2i (t), are introduced in what follows
⎧
1+ι
3−ι
⎪
V̇1i (t) = −k1 V1i (t) 2 − k2 V1i (t) 2 ,
⎪
⎪
⎪
⎪
⎨ V1i (0) = 1 si (0)2 ,
2
(9)
3−ι
⎪ V̇2i (t) = −γ1 V2i (t) 1+ι
2 − γ2 V2i (t) 2 ,
⎪
⎪
⎪
⎪
⎩ V2i (0) = σ si (0)2 ,
2
where parameters are designed as 0 < k1 ≤ γ1 , 0 <
k2 ≤ γ2 , and 0 < σ < 1. According to Lemma 4,
V1i (t) and V2i (t) reach to the origin within the time
T1 = √k kπ(1−ι) and T1 = √γ1 γπ2 (1−ι) , respectively.
1 2
Based on the functions V1i (t) and V2i (t), the intermittent switching instants {tki } and {τki } in the control
protocol (6) are determined by the following triggered
mechanism:
1
τki = inf {t|t > tki , V2i (t) − siT (t)si (t) ≥ 0},
2
1
i
tk+1
= inf {t|t > τki , V1i (t) − siT (t)si (t) ≤ 0},
2
(10)

To solve the fixed-time DOB formation (4), the following distributed control protocol is proposed
u i (t) = u 1i (t) + u 2i (t),
where



u 1i (t) =

−c1 sigι (si (t)) − c2 sig2−ι (si (t))
−c3 sign(si (t)) − c0 si (t),
t ∈ [tki , τki ),
i
i
0,
t ∈ [τk , tk+1 ),

(5)

(6)

and


⎧ 2
(∇ f i (di (t)))−1 − a1 sig p1 (∇ f i (di (t)))
⎪
⎪
⎪

⎪
⎪
⎪ −a2 sig p2 (∇ f i (di (t))) ,
⎪
⎪
⎪
⎪
⎪
t ∈ [0, T2 ),
⎪
⎪
⎡
⎨

u 2i (t) =
ςi ai j sigq1 (d ji (t))
(∇ 2 f i (di (t)))−1 ⎣b1
⎪
⎪
⎪
⎪
j∈
N
⎪
i
⎪
⎤
⎪
⎪
⎪

⎪
⎪
q
2
⎪ +b2
ςi ai j sig (d ji (t))⎦ ,
t ≥ T2 ,
⎪
⎩

(7)

where k ∈ N and t0i = 0 for all i ∈ V.

j∈Ni

in which
si (t) = xi (t) −

 t

u 2i (τ )dτ,

0

d ji (t) = d j (t) − di (t), i ∈ V,

(8)

and 0 < ι, p1 , q1 < 1, p2 , q2 > 1, a1 , a2 , b1 , b2 ,
c0 , c1 , c2 , c3 are positive constants, and the estimate of
the settling time is
  1− p1
a1 p2 − p1
π
T2 = T1 +
a 1 ( p2 − p1 ) a 2


1 − p1
× csc
π
p2 − p1

T1 = √

π
,
k1 k2 (1 − ι)

here k1 , k2 are positive numbers to be designed later,
they are related to the auxiliary functions, {tki } and {τki }

Remark 3 Compared to event-dependent intermittent
control methods [42–44], in which intermittent control
intervals are specified in advance, this paper defines
the triggering instants as the intermittent switching
instants, which is more intelligent, and in line with the
actual needs. Specifically, from control protocol (6) and
triggered scheme (10), it can be observed that the operational time of the control protocol u 1i (t) is governed
by the relationship among Vli (t), l ∈ {1, 2}, and V3i (t),
in which V3i (t) is the Lyapunov function to be constructed and V3i (0) = V1i (0). The EBI control scheme
is outlined as follows:
(1) If V3i (t) ≥ V1i (t), the control protocol u 1i (t) is
activated at time t and works continuously until
V3i (t) = V2i (t).
(2) If V2i (t) < V3i (t) < V1i (t) and the control protocol u 1i (t) is acted at time t − , then u 1i (t) continues to operate after time t, like the curve A to B
in Fig. 1.
(3) If V2i (t) < V3i (t) < V1i (t), and u 1i (t) = 0 at
time t − , then the control protocol u 1i (t) remains
at zero after time t, like the curve B to C in Fig. 1.

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L. Ma et al.

Fig. 2 Time distribution of DOB formation control

Fig. 3 EBI control interval

Fig. 1 The relationship among V1i (t), V2i (t), and V3i (t)

(4) If V3i (t) ≤ V2i (t), then u 1i (t) = 0 at time t and
remains zero until V3i (t) = V1i (t).

Algorithm 1: The control process of the DOB formation.
1: Input: initial state, topology structure, local objective
function and parameters
2: for t ∈ [0, T1 ] do
suppress external disturbances ωi (t) with the EBI
control (6)
end for
output: ẋi (t) = u 2i (t)
3: for t ∈ [0, T2 ] do
local DOB formation with the first part of control
protocol (7)
end for
output: di (t) = xi∗
4: for t ∈ [T2 , +∞) do
global DOB formation with the second part of control
protocol (7)
end for
output: di (t) = x ∗

under the control scheme u 1i (t), T2 represents the duration time to achieve local optimization under the first
part of u 2i (t), and T3 is the time for realizing global
optimization and formation control based on the second
part of control protocol u 2i (t). In addition, a detailed
illustration of switched instants in u 1i (t) is provided in
Fig. 3, in which blue and purple regions severally represent the control and rest intervals, indicating that the
intermittent switching points correspond to the triggering points defined by the triggered condition.
Based on the above preparation, the first step in this
paper is to explain that external disturbances can be
suppressed with EBI control strategy within a designed
time.
Theorem 1 Under Assumptions 2-3 and control protocol (5), the local disturbance ωi (t) is rejected within
the time T1 for system (1), if
1+ι

c1 2 2 ≥ k1 ,

123

c3 ≥ ω̄.

(11)

Proof Construct a Lyapunov function as follows
V3i (t) =

Remark 4 To realize fixed-time DOB formation control, a two-component distributed control protocol (5) is
presented: an EBI control protocol u 1i (t) with integral
sliding mode term, which is designed to suppress external disturbances ωi (t) in the MAS (1) for t ∈ [0, T1 ),
and a piecewise control protocol u 2i (t), which is developed to realize DOB formation by first achieve local
optimization for t ∈ [0, T2 ), and then achieve global
DOB formation for t ∈ [T2 , T3 ). In order to clearly
show the control process of this paper, the control
scheme of fixed-time DOB formation is outlined in
Algorithm 1. Additionally, Fig. 2 is given to show the
completion time of the three tasks, in which T1 denotes
the time allocated for suppressing external disturbances

3−ι

c2 2 2 ≥ k2 ,

1 T
s (t)si (t).
2 i

Obviously, V3i (0) = V1i (0), and t0i = 0 is the first
trigger instant.
(1) If V2i (t) < V3i (t) for all t ≥ t0i , according to the
triggered conditions, there doesn’t exist rest time
instant series {τki }. Hence,
V̇3i (t) =siT (t)(ẋi (t) − u 2i (t)) = siT (t)(ωi (t) + u 1i (t))
= − c0 siT (t)si (t) − c1 siT (t)sigι (si (t)) + siT (t)ωi (t)
− c2 siT (t)sig2−ι (si (t)) − c3 siT (t)sign(si (t)),
≤ − c1 si (t)1+ι − c2 si (t)3−ι − c3 si (t)
+ ω̄si (t)
1+ι

1+ι

3−ι

3−ι

≤ − c1 2 2 V3i (t) 2 − c2 2 2 V3i (t) 2 .

Distributed optimization-based fixed-time formation control…

26423

From condition (11), it has
V̇3i (t) ≤ −k1 V3i (t)

1+ι
2

− k2 V3i (t)

3−ι
2

.

By comparison principle, V3i (t) ≤ V1i (t) for t ≥
0. Note that, lim V1i (t) = 0 and V1i (t) ≡ 0 for
t→T1

t ≥ T1 . Thus, lim V3i (t) = 0 and V3i (t) ≡ 0 for

the proposed EBI control strategy streamlines the convergence analysis by dispensing with the requirement
that the derivative of the Lyapunov function must be
non-positive during uncontrolled periods.
Subsequently, the exclusion of Zeno behavior for the
EBI protocol (5) is demonstrated.

t→T1

t ≥ T1 , that is, lim si (t) = 0 and si (t) ≡ 0 for
t→T1

t ≥ T1 . Therefore, ṡi (t) = 0 for t ≥ T1 , which
indicates that ẋi (t) = u 2i (t) for t ≥ T1 .
(2) If there exists a time instant τ0i > t0i , such that
V3i (τ0i ) = V2i (τ0i ), according to the intermittent
switching scheme, the control protocol u 1i (t) will
switch to zero.
If V3i (t) < V1i (t) always holds for all t > τ0i , then,
lim V3i (t) = 0 and V3i (t) ≡ 0 for t ≥ T1 , since V1i (t)

t→T1

Theorem 2 The Zeno behavior does not exists in EBI
control algorithm (5) with triggered condition (10).
Proof According to the definition of auxiliary functions and the analysis of Theorem 1, there exists M >
i ),
0, such that Vki ≤ M, k = 1, 2, 3. For t ∈ [τki , tk+1
1+ι

define h i (t) = V1i (t) − V3i (t), and ε = k1 M 2 +
√
3−ι
1
i−
) = 0 and
k2 M 2 + 2ω̄M 2 . Obviously, h i (tk+1
i
h i (τk ) = Γ > 0. From equation (9) and inequality
(12), it has

reaches to the origin within T1 . As a result, lim si (t) =
t→T1

0 and si (t) = 0 for t ≥ T1 , which also implies that
ṡi (t) = 0 and ẋi (t) = u2i (t) should be u 2i (t). for t ≥
T1 .
Conversely, if there exists t1i > τ0i , such that
V3i (t1i ) = V1i (t1i ), then, from Assumption 3 and control protocols (5)-(8) for t ∈ (τ0i , t1i ), one has
V̇3i (t) = siT (t)(ẋi (t) − u 2i (t))
≤ ω̄si (t) =

√
1
2ω̄V3i (t) 2 .

(12)

If V3i (t) > V2i (t) for all t > t1i , then there doesn’t
exist time instant series {τki } \ {τ0i }. Similar to the analysis of case (1), si (t) = 0 and ẋi (t) = u 2i (t) for all
t ≥ T1 . If there exists a time instant τ1i > t1i , similar to the above analysis, si (t) = 0, ṡi (t) = 0 and
ẋi (t) = u 2i (t) for all t ≥ T1 .
Repeat the above analysis process, it can conclude
that si (t) = 0, ṡi (t) = 0 and ẋi (t) = u 2i (t) for all
t ≥ T1 .
Therefore, the external disturbances are suppressed
under the control protocol (5) within the time T1 , that
is, si (t) = 0, ṡi (t) = 0 and ẋi (t) = u 2i (t) for all


t ≥ T1 .
Remark 5 Note that, in the convergence analysis, a
decentralized form of the Lyapunov function is introduced in this study, contrasting with the summation
form employed in [46], this construction enhances
robustness against external perturbations. Furthermore,

ḣ i (t) = V̇1i (t) − V̇3i (t)

√
1+ι
3−ι
1
≥ −k1 V1i (t) 2 − k2 V1i (t) 2 − 2ω̄V3i (t) 2
√
1+ι
3−ι
1
≥ −k1 M 2 − k2 M 2 − 2ω̄M 2
= −ε.

(13)

Integrate both sides of inequality (13) from τki to t,
h i (t) − h i (τki ) ≥ −ε(t − τki ).

(14)

i , from the definition of h (t),
For t = tk+1
i
i
i
i
h i (tk+1
) = V1i (tk+1
) − V3i (tk+1
) = 0.

(15)

Then, from the formulas (14) and (15), one has
i
i
h i (tk+1
) − h i (τki ) ≥ −ε(tk+1
− τki ),

which indicates that
i
tk+1
− τki ≥

Γ
.
ε

Since τki − tki ≥ 0, it has
i
− tki ≥
tk+1

Γ
.
ε

From the above analysis, the Zeno behavior is avoided.



123

26424

L. Ma et al.

From Theorems 1-2, the external disturbances are
effectively suppressed within the time T1 , and ẋi (t) =
u 2i (t) for t ≥ T1 . Subsequently, the fixed-time DOB
formation control problem will be examined.
For the graphs GA and GB with edge weights
2

According to Lemma 4, lim ∇ f i (di (t)) = 0, i.e.,
t→T2

lim di (t) = xi∗ and di (t) ≡ xi∗ when t ≥ T2 .

t→T2

Step 2: For t ≥ T2 , from the MAS (1) with control
scheme (7), one has

2

(ςi ai j ) 1+q1 and (ςi ai j ) 1+q2 , the Laplacian
  matrices
liAj

are severally represented as LA =
 
, here
LB = liBj

N ×N

and

ẋi (t) =

∇ 2 f i (di (t))
−b2

N ×N

i=1,i= j

b1 (λ2 (LA ))

×⎝

b2 N (N − 1)


csc
q +1
1
2

1 − q1
π
q2 − q1

2 (LB ))

q2 +1
2

⎠

j∈Ni

.

i=1

Step 2.
Step 1: Choose a Lyapunov candidate function as
follows
1
V4i (di (t)) = ∇ f i (di (t))2 .
2
From Assumption 1,

V̇4i (di (t)) = (∇ f i (di (t)))T − a1 sig p1 (∇ f i (di (t)))

− a2 sig p2 (∇ f i (di (t)))
p1 +1
2

≤ −a1 2

123

(V4i (t))

− a2 ∇ f i (di (t))

p2 +1

p2 +1
2

p2 +1
2

− a2 2

(16)

N

t→T2
is proved in Step 1 and lim di (t) = x ∗ is proved in
t→T3

p1 +1
2


ςi ai j sigq2 d̃i (t) − d̃ j (t) .

d 
∇ f i d̃i (t) + x ∗ = 0,
dt

q2 −q1

Proof The proof includes two steps: lim di (t) = xi∗

≤ −a1 ∇ f i (di (t))

j∈Ni

Furthermore, from equation (16),

Theorem 3 Under Assumptions 1-2, the DOB formation problem (4) is addressed with the control protocol
(5) within the time T3 .

p1 +1





⎞ 1−q1

q1 +1
2

1−q2
2 (λ


ςi ai j sigq2 di (t) − d j (t) , i ∈ V .

− d̃ j (t) − b2

Denote Θ0 = max{Θi , i ∈ V} and the estimate of the
settling time is

⎛

ςi ai j sigq1 di (t) − d j (t)

j∈Ni



−1
− b1
ςi ai j sigq1 d̃i (t)
d̃˙i (t) = ∇ 2 f i (d̃i (t) + x ∗ )

i=1,i= j

b1 (q2 − q1 ) λ2 (LA )



− b1



Let d̃(t) = col d̃1 (t), . . . , d̃ N (t) and d̃i (t) = di (t)−
x ∗ , for each i ∈ V, then

⎧
2
q +1
⎪
⎪
⎨ −(ςi ai j ) 2 , i = j,
N

2
liBj =
⎪
(ςi ai j ) q2 +1 , i = j.
⎪
⎩

2π Θ0



j∈Ni

⎧
2
q +1
⎪
⎪
⎨ −(ςi ai j ) 1 , i = j,
N

2
liAj =
⎪
(ςi ai j ) q1 +1 , i = j,
⎪
⎩

T3 =T2 +



−1

(V4i (t))

.

which indicates that there has a constant vector Cm
satisfying
N


∇ f i d̃i (t) + x ∗ = Cm .

i=1

Note that
N


∇ f i (d̃i (T2 ) + x ∗ ) =

i=1

N


∇ f i (xi∗ ) = 0m ,

i=1

therefore,
N


∇ f i (d̃i (t) + x ∗ ) ≡ 0m ,

i=1

that is, the zero-gradient-sum is satisfied.
Choose a Lyapunov candidate function as follows

Distributed optimization-based fixed-time formation control…

V (d̃(t)) =

N 


= − d̄(t) − x ∗

f i (x ∗ ) − f i d̃i (t) + x ∗

i=1

+ ∇ f i (d̃i (t) + x ∗ )

T

V (d̃(t)) ≥

i=1

d̃i (t) .

θi
d̃i (t)2 .
2

i=1

− ∇ f i (d̃i (t) + x ∗ )

⎞ q1 +1
2

i=1 j∈Ni

i=1

⎞ q2 +1

N

= − b1 2

q1 −1
2

1−q2
2

− b2 2

d̃ T (t)(LB ⊗ Im )d̃(t)

≤

q2 −1
2

q2 +1
2

.

(17)

j=1

j=1

Since f d̄(t)

≥ f (x ∗ ) and

∇ f i d̃i (t) + x ∗

i=1

for t ≥ T2 , then,
N 


N


Θ0 T
=
d̃ (t) LḠ ⊗ Im d̃(t)
N
Θ0
≤
d̃ T (t) LG ⊗ Im d̃(t),
λ2 (LG )

here LG is the Laplacian matrix of any connected undirected topology. Particulary,

1 
1 
d̃ j (t) + x ∗ =
d j (t).
d̄(t) =
N
N
N

Θ0
d̃ T (t) LA ⊗ Im d̃(t),
λ2 (LA )
Θ0
V (d̃(t)) ≤
d̃ T (t) LB ⊗ Im d̃(t).
λ2 (LB )

V (d̃(t)) ≤

≡ 0m

T

i=1

+ f i d̄(t) − f i d̃i (t) + x ∗

− V (d̃(t))

b1 (λ2 (LA ))
2

d̄(t) − (d̃i (t) + x ∗ )


(18)
(19)

Combining the inequalities (17), (18) and (19), one has
V̇ (d̃(t)) ≤ −

− ∇ f i (d̃i (t) + x ∗ )

N

i=1 j=1

Denote
N

j=1

2
Θ0  
d̃i (t) − d̃ j (t)
2N
N

d̃ T (t)(LA ⊗ Im )d̃(t)

× N (N − 1)

i=1

1−q2
2
q1 +1
2

2

N
N
Θ0  
=
d̃i (t) + x ∗ − (d̃ j (t) + x ∗ )
2N 2

2

i=1 j∈Ni

× N (N − 1)

Θ0 
2
d̃i (t) + x ∗ − d̄(t)
2
N

V (d̃(t)) ≤

2
b2 ⎝ 
(ςi ai j ) q2 +1 d̃i (t) − d̃ j (t)2 ⎠
2

d̄(t) − (d̃i (t) + x ∗ )

T

It follows from Lemma 2 that

i=1 j∈Ni
N


d̄(t) − (d̃i (t) + x ∗ ) .

+ f i d̄(t) − f i d̃i (t) + x ∗
Θ0
≤ d̃i (t) + x ∗ − d̄(t)2 .
2

i=1 j∈Ni

2
b1 ⎝ 
(ςi ai j ) q1 +1 d̃i (t) − d̃ j (t)2 ⎠
2

T

By Assumption 1, it has

N
b2  
T
d̃i (t) − d̃ j (t) ςi ai j sigq2 d̃i (t) − d̃ j (t)
−
2

−

f i (d̄(t)) − f i (d̃i (t) + x ∗ )

− ∇ f i (d̃i (t) + x ∗ )

b1  
T
d̃i (t) − d̃ j (t) ςi ai j sigq1 d̃i (t) − d̃ j (t)
2

⎛

N



V (d̃(t)) ≤

N

≤−

∇ f i d̃i (t) + x ∗ − f (x ∗ ) + f (d̄(t))

Therefore,

V̇ (d̃(t))

⎛

N


≥0.

1+q2
1
From Lemma 3, 0 < 1+q
2 < 1 and 2 > 1,

=−

T

i=1



According to Assumption 1, one has
N


26425

−

1−q1
2

q1 +1
2

q1 +1
2

V (d̃(t))

q1 +1
2

Θ0

b2 N (N − 1)
2

1−q2
2

1−q2
2

(λ2 (LB ))
q2 +1
2

q2 +1
2

V (d̃(t))

q2 +1
2

.

Θ0

123

26426

L. Ma et al.

By Lemma 4, d̃i (t) ≡ 0 for t ≥ T3 , and then di (t) ≡
x ∗ , i ∈ V for t ≥ T3 . From the above derivation, the
DOB formation is finally solved within fixed time T3 .


Remark 6 Although the control protocols u 1i (t) and
u 2i (t) are activated at the same time, the local optimization control is actually performed after the disturbance
suppression. It is worth noting that, if T2 − T1 is smaller
than T1 , the local optimization is completed at the same
time as the disturbances are suppressed. Although this
situation may exist in the reality, there is a challenge
for us in the specific theoretical analysis. Nevertheless,
this issue will be considered in future work.
Remark 7 In recent researches [33,34], the DOB formation control problem has been studied, which
addressed local optimization xi (t) = xi∗ , prior to global
DOB formation di (t) = x ∗ , by using the zero-gradientsum property. However, in theoretical analysis, the use
of zero-gradient-sum property is questionable. To avoid
this issue, this paper adopts a distinct approach that
prioritizes local formation optimization di (t) = xi∗ ,
followed by global DOB formation di (t) = x ∗ with
zero-gradient-sum method. Additionally, the formation
is determined by z i , which is not directly used in theoretical proof. Therefore, the algorithm designed in this
paper can also be applied to other formation structures.
Remark 8 Up to now, the asymptotic [20–23] and
finite-time [24,25] DOB formation control has been
thoroughly studied, in which, the achievement of these
formations is contingent upon the initial states of the
individuals. It is recognized that obtaining accurate initial values is challenging, due to external perturbations,
information loss, and other practical constraints. Consequently, this paper investigates the DOB formation
control within a fixed time, which enhances robustness
and practicality.
According to Definition 1, if z i ≡ 0, the results in
Theorems 1-3 can be degenerated into the consensusbased optimization issue depicted as follows.

Fig. 4 Detail-balanced communication digraph

are not considered, such as minimization of travel distance and minimization of energy consumption. With
this in mind, the DOB formation scheme presented in
this study addresses both the distributed optimization
problem and formation control simultaneously. Furthermore, the result of this research is not only applicable to the DOB formation control, it is also applicable to
the general distributed optimization problem under the
consensus constraints, this process is consistent with
the results in [28].

4 Numerical simulations
Take source localization problem [33] into account,
the task is to achieve a specified formation around the
global minimum point x ∗ . The proposed algorithm will
be verified through the following numerical example.
Example 1 (Optimization-Based Formation Control)
Consider the MAS (1) 6 nodes depicted in Fig. 4. Obviously, the topology is detailed balanced and strongly
connected with ς = (4, 1, 4, 2, 0.8, 0.6)T . The agents’
initial states are randomly selected in [−3, 3]. Consider
the local cost functions as follows

Corollary 1 Under Assumptions 1-3, the distributed
optimization issue (3) is resolved within the time T3 for
system (1), if z i ≡ 0 in DOB formation problem (4).

f i (xi ) = (xi1 − 0.2i)2 + cos(xi1 + i)

Remark 9 In [12–14], the consensus-based formation
control was investigated, unfortunately, some problems that need to be optimized in formation control

where xi = (xi1 , xi2 )T . The corresponding optimization
problem is depicted as

123

+(xi2 − 0.2i)2 + e−(xi ) ,
2 2

Distributed optimization-based fixed-time formation control…

min F(X ) =

6


26427

f i (xi ),

i=1

s.t. xi = x j , i = 1, 2, . . . , 6,

(20)

in which X = col(x1 , x2 , . . . , x6 ). By calculation,
x1∗ = [0.6961, 0.6225]T , x2∗ = [0.6403, 0.8188]T ,

x3∗ = [0.4488, 0.9764]T , x4∗ = [0.3352, 1.1196]T ,

x5∗ = [0.7427, 1.2583]T , x6∗ = [1.6936, 1.398]T ,

and x ∗ = [0.6789, 1.049]T , Θ0 = 1.932, F(X ∗ ) =
4.3392, where X ∗ = col(x ∗ , x ∗ , . . . , x ∗ ). In addition,
!
"#
$

Fig. 5 Evolution of the integral sliding mode si (t)

6

denote X̃ ∗ = col(x1∗ , x2∗ , . . . , x6∗ ).
Accordingly, the DOB formation problem is
depicted as
min F̃(d) =

6


f˜i (di ),

i=1

in which d = col(d1 , d2 , . . . , d6 ), di = (di1 , di2 )T , and
f˜i (di ) =(di1 + z i1 − 0.2i)2 + cos(di1

Fig. 6 Evolution of the control protocol (6)

+ z i1 + i) + (di2 + z i2 − 0.2i)2 + e−(di +zi ) ,
2

2 2

here the desired formation pattern is a regular hexagon
embedded with an external circle of radius r = 0.2,
and
z i = (z i1 , z i2 )T


&
%

(i − 1)π T
(i − 1)π
, sin
= r cos
.
3
3
Additionally, the external disturbance is depicted as
ωi (t) =

3
sin(xi (t)), i = 1, 2, . . . , 6.
5

(21)

Evidently, Assumption 3 is satisfied with ω̄ = 0.6.
Choose ι = 0.5, c0 = 40, c1 = 10, c2 = 8,
c3 = 0.7, k1 = 0.1, k2 = γ1 = 0.2, γ2 = 0.3, then
the conditions of Theorem 1 are satisfied. By Theorem
1, the external disturbance ωi (t) is suppressed within
should be T1 = 44.43s, and the evolution of integral
sliding mode si (t) is displayed in Fig. 5. The evolution
of the algorithm u 1i (t) is illustrated in Fig. 6, which
shows that the control to suppress external disturbances

is intermittent, which is consistent with the theoretical
analysis, and the relationship among Lyapunov functions V3i (t) with auxiliary functions V1i (t) and V2i (t)
is demonstrated in Fig. 7.
Next, the exclusion of Zeno behavior for any agent in
system (1) is discussed by Theorem 2. The simulation
result of trigger time series based on the algorithm (6)
is given in Fig. 8, which shows that there is no Zeno
behavior.
Choose p1 = 0.1, p2 = 1.2, q1 = 0.2, q2 = 1.1,
a1 = 1.2, a2 = 0.9, b1 = 3.8, b2 = 4. By calculation, λ2 (LA ) = 1.1129, λ2 (LB ) = 1.1228. From
Theorem 3, the fixed-time DOB formation is realized
under the control protocol (7) within T3 = 60.28s.
For DOB formation problem (4), the simulations are
shown in Figs. 9, 10, 11, 12 and 13. As shown in the
Fig. 9, the first part of EBI control protocol (7) is used
to reach local optimization within the time T2 = 50s,
and the first part is used to realize DOB formation
within the time T2 = 60.28s. By utilizing the above
control protocol (7), as depicted in Fig. 10, the variable
di (t) converges to local optimal point xi∗ within the
time T2 = 50s first, and then converges to global opti-

123

26428

L. Ma et al.

Fig. 9 Evolution of control protocol (7)

Fig. 7 The relationship among V1i (t), V2i (t), and V3i (t), i =
1, . . . , 6.

Fig. 8 The trigger time series of agents with algorithm (6)

mum x ∗ within the time T3 = 60.28s, this is consistent
with Theorem 3. In Fig. 11, the time evolution of cost
function F̃(d) is given. To provide a more intuitive
understanding of the optimization process, the phase
trajectory of variable di (t) and the trajectory of F̃(d)
about the variable di (t) under algorithm (7) is shown
in Fig. 12 and Fig. 13.
In addition, the phase trajectories of all agents and
the trajectory of F(X ) about the agents’ states are
depicted respectively in Fig. 14 and Fig. 15, it follows

123

Fig. 10 Evolution of the variables di (t)

Fig. 11 Evolution of global cost function F̃(d(t))

from them that the preset regular hexagonal formation
is finally achieved in a fixed time.
Example 2 (Consensus-Based Optimization Control)
Especially, consider the scenario where z i = 0. In
this case, the fixed-time DOB formation control problem is reduced to a general distributed optimization
problem subject to the consensus constraint. Following the parameter selection detailed in the above DOB
formation control, the distributed optimization prob-

Distributed optimization-based fixed-time formation control…

Fig. 12 The phase trajectory of variables di

26429

Fig. 15 Trajectory of cost function F(X ) about xi

Fig. 13 Trajectory of cost function F̃(d) about di

Fig. 16 The relationship among V1i (t), V2i (t), and V3i (t), i =
1, . . . , 6
Fig. 14 The phase trajectory of each agent

lem (3) is solved under the control protocol (5) within
the time should be T3 = 60.28s by Corollary 1.
Figs. 16 and 17 illustrate that the designed EBI control strategy can also be implemented in general distributed optimization problem without Zeno behavior.
Fig. 18 depicts that all agents achieve local optimization
xi (t) = xi∗ with the first part of control u 2i (t) within the
time T2 = 50s, and then achieve global optimization
xi (t) = x ∗ with the second part of control u 2i (t) within

the time T3 = 60.28s. Fig. 19 shows the time evolution of global cost function F(X ). In order to more
clearly show the trajectories of agents’ states and the
global cost function along with agents’ states, Figs. 20
and 21 are given respectively, from which the global
fixed-time distributed optimization is addressed.

123

26430

L. Ma et al.

Fig. 20 The phase trajectory of agents’ states xi .
Fig. 17 The trigger time series of agents

Fig. 18 Evolution of agents’ states xi (t)

Fig. 19 Time evolution of cost function F(X (t))

5 Conclusion
This paper presented a novel DOB formation control
scheme for MAS in the presence of external disturbances. The proposed scheme was comprised of three
critical steps. Firstly, a new EBI control scheme based
on integral sliding mode technique was developed as a
means of suppressing the external disturbances. Note

123

Fig. 21 The trajectory of F(X ) with xi

that the time instants of work and rest are fully determined by preset event-trigger conditions, the control
cost is effectively reduced and the Zeno behavior is also
eliminated. Secondly, the local optimal solution to the
fixed-time distributed optimization issue was derived.
Thirdly, a fixed-time DOB formation control algorithm was presented to ensure that the state of each
agent tends to the desired formation around the global
optimum.
Nevertheless, a number of theoretical and technical challenges remain to be addressed such as target search and cooperative reconnaissance [51,52],
which requires time-varying formation control. Therefore, investigating the time-varying DOB formation
control problem with the FXT distributed algorithm
presents a promising and challenging direction for
future research.
Funding This work was supported by National Natural Science
Foundation of China (Grant Nos. 62373317), Tianshan Talent
Training Program (2022TSYCCX0013), Key Project of Natural Science Foundation of Xinjiang Uygur Autonomous Region
(2021D01D10, 2024D01D04), the Excellent Doctoral Innova-

Distributed optimization-based fixed-time formation control…

26431

tion
Program
of
Xinjiang
University
(XJU
2023BS018), the Basic Research Foundation for Universities of
Xinjiang (XJEDU2023P023), and Intelligent Control and Optimization Research Platform in Xinjiang University.

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Data Availability Statement The datasets generated during
and/or analysed during the current study are available from the
corresponding author on reasonable request.
Declarations
Conflict of interest The authors declare that they have no Conflict of interest.

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