I. INTRODUCTION

Global Optimal Consensus for
UAV Swarms With
Time-Varying Objective
Functions and Bounded Input
Constraints

AIWU YANG
XIAOLONG LIANG
JIAQIANG ZHANG
YUEQI HOU
NING WANG
Air Force Engineering University, Xi’an, China

This article investigates a distributed time-varying optimization
problem concerning unmanned aerial vehicle (UAV) swarm systems
with bounded inputs. The objective is to achieve state consensus
and minimize the sum of local time-varying objective functions in a
distributed manner, while considering the constraints on inputs. To
address this problem, we propose a distributed optimization protocol utilizing the projection operator and the prediction–correction
method. This protocol enables the UAVs to track the time-varying
optimal solution with an asymptotically diminishing error. It is shown
that the proposed protocol achieves approximate global optimal consensus for UAVs with different connected communication topologies,
including undirected, directed detail-balanced, and switching topologies. Consequently, the UAV swarm, equipped with the proposed protocol, can establish a predetermined formation and effectively track
moving targets. To validate the theoretical findings, we present three
applications: a scalar example, a cooperative hunting task, and swarm
tracking behavior.

Manuscript received 31 August 2023; revised 19 December 2023; accepted
16 February 2024. Date of publication 23 February 2024; date of current
version 9 August 2024.
DOI. No. 10.1109/TAES.2024.3368996
Refereeing of this contribution was handled by Sean Phillips.
Authors’ address: Aiwu Yang, Xiaolong Liang, Jiaqiang Zhang,
Yueqi Hou, and Ning Wang are with the Air Traffic Control and
Navigation School and the Shaanxi Key Laboratory of Meta-Synthesis
for Electronic and Information System, Air Force Engineering University,
Xi’an 710051, China, E-mail: (ai_five@163.com; afeu_lxl@sina.com;
jiaqiang-z@163.com; afeu_hyq@163.com; wnlearning@163.com).
(Corresponding author: Xiaolong Liang.)
0018-9251 © 2024 IEEE

3822

Distributed optimization has attracted considerable attention recently due to its potential for applications in
various multiagent domains, including microgrids [1], [2],
economic dispatch [3], [4], [5], and robot navigation [6],
[7]. An unmanned aerial vehicle (UAV) swarm, as a typical
multiagent system, has advantages of autonomy and cost
effectiveness when performing specific tasks, where UAVs
collaborate in a distributed and optimized manner [8], [9],
[10], [11]. Consequently, the application of distributed optimization technology in the UAV swarm cooperative tasks
is of great significance [12].
Consider the cooperative tasks executed by a swarm of
UAVs, such as swarm tracking behavior. These tasks are
characterized by: 1) a dynamic process where the objective function is time varying, leading to varying optimal
solutions; 2) coordination that UAVs in the swarm perform
the tasks in a predesigned formation, and achieving state
consensus is crucial for successful task completion; and 3)
executable inputs that the actuators of UAVs are subject
to saturation due to the physical limitations, resulting in
bounded input constraints. Considering these characteristics, the problem in cooperative tasks can be summarized
as the global optimal consensus problem with time-varying
objective functions and bounded input constraints, which
has not been investigated in the existing literature.
Distributed optimization methods with continuous-time
frameworks [13], [14], [15] exhibit substantial promise
for solving the optimization problem mentioned above,
as the well-developed theory of continuous-time stability
facilitates convergence analysis [16], [17]. Specifically, the
local objective functions in distributed optimization problems vary with time, resulting in an optimal trajectory that
changes over time. One way to address the challenge is to
periodically sample the objective function and constraints
and process them using time-invariant methods, as proposed
in [18]. However, this scheme may suffer from steady-state
optimality errors. The prediction–correction scheme is a
promising alternative for solving the time-varying problem.
In this scheme, it will predict the drift of the optimal
solution and then correct the prediction by an optimization
algorithm, as detailed in [19] and [20]. More challenging
is incorporating various constraints into the prediction–
correction scheme, such as consensus or set constraints, as
presented in the problems of [15] and [21].
The state consensus constraint is a fundamental and
essential constraint for multiagent systems. Various distributed consensus control schemes provide different feasible solutions for handling consensus constraints during the
optimization process [16], [17]. In contrast to the asymptotic or exponential consensus convergence, time-bounded
consensus schemes, such as finite-time consensus [22],
[23], fixed-time consensus [12], [24], [25], and predefinedtime consensus [26], [27], [28], have been introduced and
implemented in distributed optimization. These schemes
offer distinct advantages in scenarios that demand rapid

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formation establishment. However, it is worth noting that,
in order to expedite the consensus process, time-bounded
consensus schemes often incorporate specialized power-law
structures or signum functions, which can potentially give
rise to issues, such as input saturation or input chattering.
These issues should not be overlooked in practical applications, as a significant input magnitude may cause damage
to the UAV’s actuator or compromise the overall system
performance.
Alternatively, one can utilize the knowledge of the
bounded input control to limit the amplitude of the input [29], [30]. However, in most literature, the box constraints are commonly imposed on the state variables to
limit their amplitude based on practical optimization problems. Subsequently, projection mapping or δ-exact penalty
techniques are applied to address these box constraints, as
detailed in [1], [21], [25], and [31]. Nevertheless, the distributed time-invariant optimization problem with bounded
input has been explored in [32], focusing on both the singleand double-integrator multiagent systems. Furthermore, the
findings in [32] have been extended to the discrete-time multiagent systems in [33] and higher order multiagent systems
in [34]. These contributions have advanced the application
of distributed optimization technology in input-bounded
problems. It is important to note that the global optimal consensus problem, considering time-varying objective
functions and bounded input constraints, warrants further
investigation. In addition, in [15], [17], [21], and [28], the
communication topology of the multiagent system is limited
to an undirected and connected network, which may not
accurately represent real-world communication dynamics.
Thus, exploring more complex directed or switching topologies becomes crucial to addressing various scenarios.
Motivated by the observations above, we investigate distributed optimization problems characterized by: 1) timevarying objective functions; 2) consensus and bounded input constraints; and 3) the cases of undirected and connected
topology, directed detail-balanced topology, and switching
topology. The contributions of this study are threefold.
1) We extend the distributed optimization method to
address cooperative tasks involving the UAV swarm,
which have received limited attention in the literature. Compared to the distributed time-invariant
optimization problem in [25], [32], [33], and [34],
where the optimal solution is a fixed point, we
propose a global optimal consensus problem with
time-varying quadratic objective functions, and a
prediction–correction scheme is incorporated into
the proposed protocol to track the optimal solution
trajectory.
2) Compared to the consensus constraint of optimization problems in [12], [21], and [28], we consider
an additional bounded input constraint that is more
conducive for the execution actuator of the UAV
system. The projection operator is utilized to address
this constraint, and the approximate global optimal
solution is then obtained.

3) Compared to the undirected topology setting discussed in [15], [17], [21], and [28], or the directed
detail-balanced topology setting discussed in [25],
[27], and [32], the proposed protocol is applicable
to systems characterized by the undirected and connected topology, directed detail-balanced topology,
and switching topology.
The rest of this article is organized as follows. In Section II, we recall some basic definitions and lemmas in
graph theory and the projection operator. In Section III,
we formulate the global optimal consensus problem with
time-varying quadratic objective functions and bounded
input constraints. To address this problem, a well-designed
distributed protocol is proposed in Section IV for the undirected and connected communication topology. We then
consider the particular cases of the switching topology
and the directed detail-balanced topology. In Section V,
we present two comparative evaluations and discuss the
practical application of the UAV swarm to illustrate our
results. Finally, Section VI concludes this article.
Notations: The sets of real and nonnegative numbers
are, respectively, denoted by R and R+ . The p-norm of a
column vector x ∈ Rn is represented by ||x|| p, where p >
0. The gradient of a function f (x, t ) : Rn × R+ → R with
respect to x ∈ Rn is defined as ∇x f (x, t ). In addition, the
partial derivatives of ∇x f (x, t ) with respect to x ∈ Rn and
t ∈ R+ are denoted as H (x, t ) and ∇xt f (x, t ), respectively.
The variable x is time dependent, with the omission of the
time index for the sake of convenience in its representation.
II. PRELIMINARIES
A. Graph Theory

The communication topology of the UAV swarm is modeled by a graph G = {V, E, A}, where V = {v1 , v2 , . . . , vN }
is a set of nodes indexed by the associated UAV set
I = {1, 2, . . . , N} and E = {(vi , v j )|vi , v j ∈ V} represents
the communication links. The neighbor set for UAV i is
defined by Ni = {v j ∈ V|(v j , vi ) ∈ E} as a subset of V. The
graph without self-loops is described by the weighted adjacency matrix A = [ai j ]N×N and the Laplacian matrix L =
[li j ]N×N , where ai j > 0 if ( j, i ) ∈ E; otherwise, ai j = 0, and

 j and lii = Nj=1, j=i ai j . In an undirected
li j = −ai j for i =
graph, ai j = a ji , while for a directed detail-balanced graph
with weight θ = [θ1 , θ2 , . . . , θN ]T , θi ai j = θ j a ji , i, j ∈ I.
If the graph without self-loops is both undirected and
connected, the ensuing lemma holds.
LEMMA 1 (SEE [35]) For an
and connected
 undirected

graph G, we have xT Lx = 21 Ni=1 j∈Ni (xi − x j )2 , where
x = [x1 , x2 , . . . , xN ]T is a stacked column vector, and if
1TN x = 0, then xT Lx ≥ λ2 (L )xT x. Here, λ2 (L ) is the second
smallest eigenvalue of the Laplacian matrix L.
Assumption 1 The communication topology of the UAV
swarm is undirected and connected.

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B. Projection Operator

Given a closed convex set  ⊂ R , the distance from
x ∈ Rn to  is defined as |x| = minω∈ x − ω, where
 ·  is the Euclidean norm. The projector P (x ) of x onto 
is the unique element within  that satisfies x − P (x ) =
|x| . The function |x|2 is continuously differentiable with
respect to x and obeys
n

∇|x|2 = 2 (x − P (x )) .

(1)

LEMMA 2 (SEE [36]) Consider a closed convex set  in Rn ,
and x ∈ Rn and y ∈ . Then


(P (x ) − x )T P (x ) − y ≤ 0 ∀x ∈ Rn ∀y ∈ . (2)
Before proceeding further, some useful lemmas are
listed for convergence derivation.
LEMMA 3 (SEE [12]) Letting scalar constants l > r > 0, for
any vector z ∈ Rn , we have
zl ≤ zr ≤ n r − l zl .
(3)
 t −β (t−τ )
LEMMA 4 (SEE [37]) Define χ (t ) = t0 e
ω (τ )dτ ,
where β is a positive constant, ω (t ) is bounded, and
limt→∞ ω (t ) = ω0 ; then, limt→∞ χ (t ) = ωβ0 .
1

1

III. PROBLEM FORMULATION

Consider the following autonomous UAV system,
whose dynamics is modeled by
ẋi = ui , i ∈ I

(4)

where xi ∈ Rd represents the system state of the ith
UAV. ui ∈  is the bounded control inputs of the ith
UAV, where  = {s ∈ Rd |sk = sgn(sk ) min{|sk |, k }, k =
1, . . . , d} represents the compact input set, and k > 0
denotes the upper bound of the kth input component.
We introduce the following distributed convex optimization problem for the UAVs with dynamics (4). This problem
incorporates time-varying objective functions, consensus
constraints, and bounded input constraints
min f (x, t ) =

x∈RNd

N


f i ( xi , t )

i=1

subject to ẋi ∈ , xi = x j ∀i, j ∈ I.

(5)

REMARK 1 The problem (5) integrating the dynamics (4)
can be characterized as a global optimal consensus problem
with time-varying objective functions and bounded input
constraints. It is important to highlight that the bounded
input constraint, which aligns with the practical characteristics of UAVs, distinguishes itself from the box constraint
imposed on state variables in [12], [21], and [28]. This problem is closely related to the task of swarm tracking behavior
with a predesigned formation using a swarm of UAVs, and
it can be employed to depict the collaborative tasks performed by various other multiagent systems. The solution
to the problem (5) in certain practical higher order nonlinear systems showcases its potential utility as an upperlayer optimal trajectory within a hierarchical framework,
3824

where distributed optimization and complex nonlinear control are effectively decoupled, as illustrated in [38] and [39].

In problem (5), each UAV has its own objective function
fi (xi , t ) : Rd × R+ → R, which is known only to itself. The
objective is to determine the control input (i.e., ẋi ∀i ∈ I)
and find an optimal trajectory (i.e., xi → x ∗ ∀i ∈ I) over
time that enables the UAVs with dynamics (4) to track the
varying optimal solution and perform some tasks in a distributed manner. Then, we make the following assumption
on these objective functions.

Assumption 2 (see [12]) The local objective function
fi (xi , t ) is quadratic and at least twice continuously differentiable. It follows the form of fi (xi , t ) = (ai xi + gi (t ))2 ,
where ai > 0 is a positive constant and gi (t )2 is bounded.
We have that ∇x fi (xi , t ) and Hi−1 (xi , t ) are bounded.
Let F̄1 > supt ∇x fi (xi , t )∞ , h̄l = maxi {1/(2ai )}, and h̄r =
maxi {2ai } ∀i ∈ I.

Assumption 3 (see [15]) The first- and second-order partial
derivatives of the gradient ∇x fi (xi , t ) with respect to t are
both bounded. We define F̄2 > supt ∇xt fi (xi , t )∞ , F̄3 >
supt (∂/∂t )∇xt fi (xi , t )∞ , F̄2,k = supt |∇xt fik (xi , t )|, and
F̄3,k = supt |(∂/∂t )∇xt fik (xi , t )| ∀i ∈ I, k = 1, 2, . . . , d.
Here, ∇x fik (xi , t ) and (∂/∂t )∇xt fik (xi , t ) denote their
respective kth component.

REMARK 2 Assumption 2 ensures the existence of the
unique continuous solution trajectory x ∗ for solving problem (5). It is weaker than [12, Assumption 2], which imposes the condition of equal Hessians. Meanwhile, it does
indicate that fi (xi , t ) belongs to a class of quadratic convex
functions. Practical instances of such functions can be found
in various domains, including economic dispatch of smart
grids [2], [5], [12], swarm tracking behavior [20], [28],
vehicle trajectory tracking [38], [39], and multirobot navigation [40]. Moreover, the bounded conditions specified
in Assumptions 2 and 3 are equivalent to the conditions
that gi (t )2 , ġi (t )2 , and g̈i (t )2 are bounded. Although
these conditions may appear restrictive, they are still applicable to a wide range of time-varying functions, such
as sin(t ), e−t cos(t ), 1/(1 + t ), and tanh(t )[15]. In practical
applications involving target tracking, these conditions can
be interpreted as upper bounds for the target’s position,
velocity, and acceleration. They hold when considering the
physical constraints imposed on the target’s dynamics.

LEMMA 5 (SEE [41]) Suppose that f0 (x ) : Rn → R is a
continuously differentiable convex function. The minimum value of f0 (x ) is attained at x ∗ ∈ Rn if and only if
∇x f0 (x ∗ ) = 0.

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Algorithm 1: Distributed Discrete Approximation
Execution.

the trajectory of each auxiliary variable yi is bounded under
the protocol (6) and follows:
lim sup |yi (t )| = 0 ∀i ∈ I.

(8)

t→∞

PROOF Let di (t ) = |yi |2 , where |yi | represents the distance
between yi and its projector onto . Its derivative along (6)
satisfies
ḋi (t ) = 2 yi − P (yi ), ẏi 


= − 2α
yi − P (yi ), P (yi ) − P (y j )
j∈Ni


− 2 yi − P (yi ), φ̇i .

(9)

Invoking Lemma 2, we have


yi − P (yi ), P (yi ) − P (y j ) ≤ 0 ∀i ∈ I.
−2 α
j∈Ni

(10)
IV. MAIN RESULTS
A. Distributed Protocol Design

For each UAV i, i ∈ I, we construct a distributed
protocol as follows:
⎧
(6a)
⎪
⎨ẋi = P (yi ) 


yi = −φi − α j∈Ni xi − x j
(6b)
⎪
⎩
(6c)
φi = Hi−1 (xi , t ) β∇x fi (xi , t ) + ∇xt fi (xi , t )
where yi is an auxiliary variable for estimating the unconstrained input. α and β are some positive parameters to be
determined.
The designed protocol (6) consists of three terms: 1) a
projection map term P (yi ) ensures that the input
 remains
within the bounded set ; 2) a consensus term α j∈Ni (xi −
x j ) aims to achieve consensus among the UAV’s state xi ; and
3) a prediction–correction term φi utilizes the optimization
method to correct the drift in the prediction of the optimal
solution and guide the state xi toward the optimal solution.
The distributed discrete approximation execution method
of the designed protocol (6) is shown in Algorithm 1.
Next, we aim to establish the uniform convergence
of the auxiliary variable trajectory yi toward the bounded
input set , while ensuring that the state variable xi attains an approximate global consistency and optimality, i.e.,
lim supt→∞ xi (t ) − x ∗ (t )2 ≤ ε ∀i ∈ I, where ε represents
the convergence error.

It follows from (6c) that the second term of (9) can be
expressed as:

− 2 yi − P (yi ), φ̇i

d
H −1 (xi , t ) ∇x fi (xi , t )
= −2 yi − P (yi ), β
dt i
d
H −1 (xi , t ) ∇xt fi (xi , t )
+
dt i

d
−1
+ Hi (xi , t )
β∇x fi (xi , t ) + ∇xt fi (xi , t )
dt

d
H −1 (xi , t ) ∇x fi (xi , t )
= −2 yi − P (yi ), β
dt i
d
H −1 (xi , t ) ∇xt fi (xi , t )
+
dt i


+ Hi−1 (xi , t ) β∇xx fi (xi , t ) ẋi + β∇xt fi (xi , t )

∂
∂
+
∇xt fi (xi , t ) ẋi + ∇xt fi (xi , t ) .
(11)
∂x
∂t

We first give a proposition to show that the trajectory of
the auxiliary variable yi converges to the bounded input set
 under specific conditions.

Under Assumption 2, it can be observed that
(d/dt )[Hi−1 (xi , t )] = 0
and
(∂/∂x )∇xt fi (xi , t )ẋi = 0.
Thus, (11) can be further simplified as

− 2 yi − P (yi ), φ̇i


= −2 yi − P (yi ), Hi−1 (xi , t ) β∇xx fi (xi , t ) ẋi

∂
+ β∇xt fi (xi , t ) + ∇xt fi (xi , t )
∂t

= −2β yi − P (yi ), P (yi ) + Hi−1 (xi , t ) ∇xt fi (xi , t )

∂
+ β −1 Hi−1 (xi , t ) ∇xt fi (xi , t ) .
(12)
∂t

PROPOSITION 1 Consider the UAV system (4) with bounded
inputs, and suppose that Assumptions 2 and 3 hold. If the
upper bound of the input component k satisfies


−1
(7)
k ≥ h̄l F̄2,k + β F̄3,k , k = 1, 2, . . . , d

For each component of the auxiliary variable yik , where
k = 1, 2, . . . , d, we will discuss (12) in the following cases.
Case 1: If yik > P (yik ), then P (yik ) = k > 0. Based
on condition (7), we have that k + ∇xt fik (xi , t )/(2ai ) +
(∂/∂t )∇xt fik (xi , t )/(2ai β ) ≥ 0. Thus, it follows that
−2(yik − P (yik ))φ̇ik ≤ 0.

B. Consensus Analysis

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3825

Case 2: If yik = P (yik ), it is evident that −2(yik −
P (yik ))φ̇ik = 0.
Case 3: If yik < P (yik ), then P (yik ) = − k < 0. Using condition (7), we find that − k + ∇xt fik (xi , t )/(2ai ) +
(∂/∂t )∇xt fik (xi , t )/(2ai β ) ≤ 0, and it also follows that
−2(yik − P (yik ))φ̇ik ≤ 0.
In all the cases, we can assert that the following inequality holds:

−2 yi − P (yi ), φ̇i ≤ 0 ∀i ∈ I.
(13)


Define the state error ei = xi − N1 Nj=1 x j . Consider a Lya
punov candidate as V (t ) = 21 Ni=1 eTi ei . Invoking (6) and
(17), the time derivative of V (t ) can be written as
V̇ (t ) =
=

REMARK 3 Condition (7) can be interpreted as a velocity
constraint for the UAV relative to the target in certain
target tracking problems. It is evident that in distributed
time-invariant optimization problems with bounded input
constraints, both ∇xt fi (xi , t ) and (∂/∂t )∇xt fi (xi , t ) are equal
to zero. Consequently, this convergence condition is no
longer necessary, as demonstrated in [32]. Furthermore,
it is possible for the positive upper bound ¯ k and the
negative lower bound k to have different absolute values.
In order to achieve this, condition (7) should be modified as
follows: ¯ k ≥ h̄l (F̄2,k + β −1 F̄3,k ) ≥ 0 and k ≤ h̄l (F 2,k +
β −1 F 3,k ) ≤ 0. In addition, Proposition 1 does not impose
any restrictions on the communication graph G.
Next, we will prove that all the UAVs’ states xi , i ∈ I,
achieve an approximate global consensus.
THEOREM 1 Consider the UAV system (4) with bounded
inputs, and suppose that Assumptions 1–3 hold. If condition
(7) is met and the parameter α satisfies
α > 1/ (2λ2 (L ))

(15)

the approximate state consensus is achieved under the
protocol (6), i.e.,
lim sup xi (t ) − x j (t )2 ≤ ε1 ∀i, j ∈ I

(16)

t→∞

where the consensus error ε1 = Nρ (2αλ2 (L ) − 1)− 2 , and
ρ = h̄l (β F̄1 + F̄2 ) + ε0 .
1

PROOF Proposition 1 has guaranteed the boundedness of the
auxiliary variable yi . Thus, we can denote P (yi ) = yi + δi ,
where δi represents the time-varying projection error between yi and its projector onto . Based on Proposition 1, we
can find a specific time τs1 such that lim supt→τs1 δi (t )2 ≤
ε0 , where ε0 denotes the upper bound of the projection error.
The proposed protocol (6a) can be rewritten as follows:
ẋi = yi + δi .
3826

(17)

N

i=1

N

N 
N

 1 
eTi yi + δi −
eTi ẋ j
N
i=1
i=1 j =1

= −

N


eTi φi − α

i=1

N 




eTi xi − x j

i=1 j∈Ni

N


+

1  T
e ẋ j .
N i=1 j =1 i
N

eTi δi −

i=1

Due to

1  T
e ẋ j
N i=1 j =1 i
N

eTi ẋi −

N


(14)

For all i ∈ I, the time derivative of di (t ) is nonpositive. In
light of the LaSalle invariance principle, the trajectory of
yi converges to a compact set defined by |yi | remaining
bounded. In other words, lim supt→∞ |yi (t )| = 0 holds for
all i ∈ I.


eTi ėi =

i=1

In a combination with (10) and (13), we finally conclude
that
ḋi (t ) ≤ 0 ∀i ∈ I.

N


N

(18)

N

T
i=1 ei = 0, we have

1  T
e ẋ j = 0.
N i=1 j =1 i
N

−

N

(19)

Define the stacked column vector e = [e1 , e2 , . . . , eN ]T . By
applying Lemma 1, the second term in (18) can be expressed
as
−α

N 


N 





eTi xi − x j = −α
eTi ei − e j

i=1 j∈Ni

=−

i=1 j∈Ni

N 



2
α
ei − e j = −αeT Le
2 i=1
j∈N
i

≤ −αλ2 (L )eT e = −αλ2 (L )e22 .

(20)

Invoking Lemma 3, when t > τs1 , the remainder terms of
(18) can be denoted as
−

N

i=1

eTi φi +

N

i=1

eTi δi ≤

N


ei 2 (φi 2 + δi 2 )

i=1

N
√
 


≤ h̄l β F̄1 + F̄2 + ε0
ei 2 ≤ ρ Ne2

(21)

i=1

where ρ = h̄l (β F̄1 + F̄2 ) + ε0 .
In terms of (19)–(21), (18) can be rewritten as
√
V̇ (t ) ≤ −αλ2 (L )e22 + ρ Ne2


1
Nρ 2
≤ − αλ2 (L ) −
e22 +
2
2



Nρ 2
1
= − αλ2 (L ) −
e22 −
.
2
2αλ2 (L ) − 1
(22)
Since α > 1/(2λ2 (L )), it can be deduced that V̇ (t ) < 0 for
all e22 > Nρ 2 (2αλ2 (L ) − 1)−1 . Therefore, the 2-norm of
the stacked vector for the state error is uniformly
 ultimately
bounded with the ultimate bound e22 = Ni=1 ei 22 <
Nρ 2 (2αλ2 (L ) − 1)−1 . By applying the triangle inequality

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Based on Lemma 4, the following holds:
 t
lim sup
e−β (t−τ ) ω (τ )dτ ≤ N h̄r β −1 (αNε1 + ε0 ) .

and the Cauchy–Schwarz inequality, it follows that:
xi − x j 2 = ei − e j 2 ≤ ei 2 + e j 2

N

N
√
≤
ei 2 ≤ N
ei 22 ∀i, j ∈ I.
i=1

i=1

t→∞

Consequently, the trajectory of the consensus error xi −
x j remains bounded and satisfies lim supt→∞ xi (t ) −
1
x j (t ) ≤ ε1 = Nρ (2αλ2 (L ) − 1)− 2 ∀i, j ∈ I. The upper
bound of the consensus error can be adjusted to be arbitrarily
small by increasing the parameter α.


Then, we will prove that all agents’ states asymptotically
converge to the neighborhood of the optimal solution of the
optimization problem (5).
THEOREM 2 Consider the UAV system (4) with bounded
inputs, and suppose that Assumptions 1–3 hold. If conditions (7) and (15) are satisfied, the approximate global
optimization for the problem (5) is asymptotically achieved
under the protocol (6). In other words, all the states of UAVs
converge uniformly to a neighborhood of the time-varying
optimal solution x ∗ . Then, there exists ε2 such that
∗

lim sup xi (t ) − x (t ) ≤ ε2

(24)

t→∞

=

N


⎛
Hi (xi , t ) ⎝−φi − α

i=1

+

=

⎛

Hi (xi , t ) ⎝−α

i=1

−β

⎞
( x i − x j ) + δi ⎠

∇xt fi (xi , t )

i=1
N



j∈Ni

N




⎞
( x i − x j ) + δi ⎠

j∈Ni
N


∇x fi (xi , t )

i=1

= −βχ (t ) + ω (t )

where
ω (t ) =

N


⎛
Hi (xi , t ) ⎝−α

i=1

χ (t ) = e−β (t−t0 ) χ (t0 ) +

 t

e−β (t−τ ) ω (τ )dτ.

(29)

t0

By combining (25)–(29), we have
lim sup χ (t ) ≤ N h̄r β −1 (αNε1 + ε0 ) .

(30)

(25)



Define p = N1 Ni=1 xi . Due to the convexity of the
N
objective function, we denote that
i=1 ∇x f i ( p, t ) =
N
( i=1 Hi (q, t ))( p − x ∗ ), with q being a convex combination of p and x ∗ . Then, we have h̄l = maxi {1/(2ai )} =
(mini {2ai } )−1 , and


N


∇x fi ( p, t )2 ≥ h̄l−1 Np − x ∗ 2 .

(31)

i=1

It follows from (30) and (31) that:
lim sup p(t ) − x ∗ (t )2 ≤ h̄l h̄r β −1 (αNε1 + ε0 ) .

(32)

t→∞

lim sup xi (t ) − p(t )2 ≤ ε1 .

(33)

t→∞

In view of (32), it follows that:
lim sup xi (t ) − x ∗ (t )2 ≤ ε2

(34)

t→∞



Hi (xi , t )ẋi + ∇xt fi (xi , t )

i=1

It follows from (25) that:

Considering the consensus error in (16), we have

where ε2 = ε1 + h̄l h̄r β −1 (αNε1 + ε0 ). ε1 and ε0 are defined
in (16).

PROOF When t > τs1 , we define χ (t ) = Ni=1 ∇x fi (xi , t ). It
follows from (6) and (17) that:
χ̇ (t ) =

(28)

t→∞

C. Convergence Analysis

N


t0

(23)

⎞
( x i − x j ) + δi ⎠ .

(26)

j∈Ni

Proposition 1 and Theorem 1 have guaranteed the boundedness of δi and xi − x j . It, thus, follows that ω (t ) is bounded
and satisfies:
lim sup ω (t )2 ≤ N h̄r (αNε1 + ε0 ) .

(27)

t→∞

where ε2 = ε1 + h̄l h̄r β −1 (αNε1 + ε0 ).
In summary, the approximate global optimal consensus
of the problem (5) has been guaranteed through analysis.
REMARK 4 This article utilizes a “piecewise” analysis
method, as introduced in [21], to establish the approximate
global optimal consensus. First, we establish the asymptotic
convergence of the auxiliary variable yi to the bounded
input set  under the input upper bound condition (7).
Subsequently, we demonstrate that the state of all the UAVs
achieves the approximate global consensus under the consensus condition (15). By combining these findings, we
ultimately conclude that all the states uniformly converge
to the neighborhood of the optimal solution.
REMARK 5 It is worth noting that the convergence rate of the
proposed algorithm depends on the upper bound of inputs
, as well as the parameters α and β. Both the consensus
error ε1 and the convergence error ε2 are influenced by the
parameters α and β. Specifically, the convergence rate is
significantly affected by
due to the establishment of
the global optimal consensus based on the convergence
of the projection distance. Increasing
(i.e., relaxing the
bounded input constraints) can accelerate the algorithm’s
convergence. Moreover, increasing α can expedite state
consensus and decrease the consensus error ε1 , but it may
also reduce the convergence rate of the gradients, as shown

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3827

in (22) and (34). Furthermore, increasing β can expedite
gradient convergence for this continuous-time system, as
the prediction–correction term φ can drive the exponential
convergence of the sum of gradients with respect to β,
as demonstrated in (29) and [20]. However, a high value
of β will affect the discrete execution of the proposed
algorithm, as it determines the maximum allowable step
size that ensures stability and convergence. In other words,
it reduces the step size such that the effective step β t is
bounded by one [19], [20], where t denotes the constant
sampling period.
D. Extension of the Protocol

Due to uncertainties in the environment, there may exist
some communication link failures among networked UAVs.
In what follows, we will investigate the cases of switching
topology and directed detail-balanced topology. The results
presented in Theorems 1 and 2 are extended to encompass
the following defined cases of topology.
Let G = {G1 , G2 , . . . , Gn } be a set of undirected and
connected graphs. A switching signal function ρ (t ) : R+ →
G determines the graph property at time t, specifically
G (t ) = Gρ (t ) . We assume that the graph G (t ) remains unchanged among successive switches and belongs to the set
G. Moreover, it is noted that the eigenvalues λ(L(t )) of the
corresponding Laplacian matrix L(t ) are real at any time t.

provides a construction methodology for Gd . The corresponding Laplacian matrix is denoted as Ld , and it possesses real eigenvalues because it can be transformed into
a symmetric positive-semidefinite matrix.
COROLLARY 2 For the UAV system (4) with bounded inputs, the communication topology Gd is directed and detailed balanced, and Assumptions 2 and 3 hold. If condition
(7) is met and the parameter α satisfies α > 1/(2ϑλ2 (L )),
where ϑ = 1/(Nθ22 ), the approximate global optimal
consensus for the problem (5) is asymptotically achieved
under the protocol (6).
PROOF For the case of the directed detail-balanced topology,
another Lyapunov candidate as V1 (t ) =
Nwe consider
1
T
θ
e
e
.
The
derivative of V1 (t ) is
i
i
i
=1
i
2
V̇1 (t ) =

N


θi eTi ėi =

i=1

=

1  T
θi e ẋ j
N i=1 j =1 i
N

θi eTi ẋi −

i=1

N

N


N 
N

 1 
θi eTi yi + δi −
θi eTi ẋ j
N
i=1
i=1 j =1

= −

N


θi eTi φi − α

i=1

+

N


N 
N




θi ai j eTi xi − x j

i=1 j =1

1  T
θi e ẋ j .
N i=1 j =1 i
N

θi eTi δi −

i=1

COROLLARY 1 For the UAV system (4) with bounded inputs, the communication topology G switches among G, and
Assumptions 2 and 3 hold. If condition (7) is met and the
parameter α satisfies α > 1/(2λ∗2 (L(t ))), the approximate
global optimal consensus for the problem (5) is asymptotically achieved under the protocol (6).

N


N

(36)

N
For the second term in (36), due to
i=1 θi = 1 and
θi ai j = θ j a ji , we have the following inequality based on [42,
Lemma 3]:
−α

N 
N




θi ai j eTi xi − x j

i=1 j =1

PROOF For the case of the switching topology, Gρ (t ) ∈ G at
time t, (20) can be expressed as

−α

N 




eTi xi − x j = −αeT L(t )e

i=1 j∈Ni

≤ −αλ∗2 (L(t )) eT e = −αλ∗2 (L(t ))e22

(35)

where λ∗2 (L(t )) = minGρ (t ) ∈G {λ2 (L(t ))}.
By substituting (35) into (18), the resulting expression
for V̇ (t ) is formally equivalent to (22). The function V (t )
can be considered as a common Lyapunov function for
any Gρ (t ) ∈ G. In light of the independence of Lyapunov
functions V (t ) from topology, Theorems 1 and 2 can be
easily extended to the case of the switching topology. The
only requirement is to modify the consensus condition. 
In addition, we consider a stronglyconnected directed
graph Gd that satisfies the conditions Ni=1 θi = 1, θi ai j =
θ j a ji , and θ T Ld = 0N . In [25], the proposed Algorithm 1
3828

=−


2
α 
θi ai j ei − e j = −αeT Ld e
2 i, j =1

≤−

αθ21 λ2 (Ld ) T
e e = −αϑλ2 (Ld )e22
Nθ22

N

(37)

where ϑ = 1/(Nθ22 ).
Then, we have

√
V̇1 (t ) ≤ −αϑλ2 (Ld )e22 + ρ Nθmax e2


1
Nθmax ρ 2
≤ − αϑλ2 (L ) −
e22 +
2
2



Nθmax ρ 2
1
2
= − αϑλ2 (L ) −
e2 −
2
2αλ2 (L ) − 1
(38)

where θmax = maxi {θi }.
Given that α > 1/(2ϑλ2 (L )), it can be inferred that
V̇1 (t ) < 0 for all e22 > Nθmax ρ 2 (2αλ2 (L ) − 1)−1 . Subsequently, by following the same lines in (23) and (25), the
approximate global optimal consensus can be guaranteed
with different convergence errors.


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
f4 (x4 , t ) = (x41 − e−t cos(t ))2 + x42 −

f5 (x5 , t ) = (x51 − sin(t )) + x52 +
2

1
t +1
2

2

1
t +1

f6 (x6 , t ) = (x61 − arctan(t ))2 + (x62 − cos(t ))2
f7 (x7 , t ) = (x71 + arctan(t ))2 + (x72 − sin(t ))2
f8 (x8 , t ) = (x81 − 4 sin(t ))2 + (x82 − 2 cos(t ))2 .
Fig. 1. Designed undirected and connected communication topology of
the multiagent system.

REMARK 6 Note that the dynamics of the UAV in this
article are simplified as a single integrator, and future work
will focus on the high-order nonlinear systems. A promising approach is to directly model distributed optimization
problems on these high-order nonlinear dynamics. This
approach provides a straightforward algorithm structure and
control inputs, while also introducing complex convergence
analysis, as explored in [43] and [44]. Another feasible
approach is to decouple distributed optimization problems
from complex dynamics using a hierarchical framework.
This framework involves conducting distributed optimization at the upper layer to determine the optimal trajectory
or waypoints. Meanwhile, the lower layer’s control inputs
are obtained by tracking the trajectory or waypoints using
a practical controller designed for the complex dynamics,
as illustrated in [38] and [39]. It is worth emphasizing that
the practical physical constraints need to be considered at
the upper layer to ensure effective trajectory or waypoint
tracking by the UAV. This highlights the significance of
studying the first-order derivative constraint of the state
variable, as explored in this article. Furthermore, more
practical constraints, such as the second-order derivative
constraint and coupled inequality constraint of the state
variable, require further investigation.
V. SIMULATION VALIDATION

In this section, comparison and application simulations
are performed to verify the effectiveness of the proposed
distributed protocol (6).
A. Comparison Verification

For the multiagent system (4) with eight agents
and their communication topology designed as shown
in Fig. 1,the global optimization problem is minx∈RNd
f (x, t ) = Ni=1 fi (xi , t ), where
f1 (x1 , t ) = (x11 − sin(t ))2 + (x12 − cos(t ))2
1 2
f2 (x2 , t ) = x11
+ (x22 − cos(t ))2
2
1 2
f3 (x3 , t ) = (x31 − sin(t ))2 + x32
2

Clearly, Assumptions 1–3 hold, and the Hessian matrix
of the local time-varying objective functions is not entirely
consistent. To clarify the bounded input characteristics of
the proposed protocol (6), we compare it with [15, protocol
(9)] using this designed scalar example. In [15], a distributed
protocol (9) with varying gains has been proposed to solve
the problem of global optimal consensus. The difference is
that a practical constraint on inputs is integrated into the
problem in this study.
The initial states are set as follows: x1 (0) = (−1,
−1.5)T , x2 (0) = (2, 12)T , x3 (0) = (3, 3)T , x4 (0) =
(−4, −4)T , x5 (0) = (2, 2)T , x6 (0) = (3, 3)T , x7 (0) =
(−3, 2)T , and x8 (0) = (−2, −1)T . In the proposed protocol
(6), we choose the parameter β = 0.5, 1, 3. For the
convenience of parameter analysis, we replace k with the
same , where
= maxk { k } = 4/2 + 4/(2 × 0.5) = 6
by solving condition (7). By satisfying condition (15), we
find that α > 1/(2 × 0.586) = 0.854. In order to reduce the
consensus error, we choose the parameter α = 10, 30, 50.
Set the simulation step size t = 0.001.
Fig. 2 shows the comparison results of the global objective function values under different solving methods and
parameter configurations. It is evident that the proposed
protocol (6) asymptotically achieves an approximate global
optimal consensus while considering bounded input constraints. Some phenomena that align with the analysis presented in Remark 5 can be observed. Specifically, relaxing
the constraints on the upper bound of inputs can increase
the convergence rate of the proposed algorithm. In fact, by
further relaxing this constraint and increasing the parameter
β, the algorithm’s convergence rate can even surpass the
result obtained by [15, protocol (9)]. In addition, higher
values of α, given the same β and , have an impact on
the convergence rate
of the gradients. This is because the
consensus term −α j∈Ni (xi − x j ) dominates the protocol.
Conversely, for the same α and , higher values of β can
expedite gradient convergence.
The comparison results of states and inputs are presented
in Fig. 3. It is worth noting that input chattering occurs during the optimization process of [15, protocol (9)] due to the
introduction of the signum function. We utilize the moving
average method with a window size of 100 to process these
input data, and it is then utilized to generate Fig. 3(b). As
depicted in Fig. 3, the control inputs in Fig. 3(d) and (f)
are constrained within the predetermined upper bound
compared to Fig. 3(b), and the high-frequency chattering is
avoided. This is beneficial for the agent’s actuator to track

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3829

Fig. 2. Evolutions of the value of the global objective function

N

i=1 f i (xi , t ) with different parameter configurations.

Fig. 3. Comparison simulation results with [15, protocol (9)] for a scalar example. (a) Evolutions of the states xi (t ) under [15, protocol (9)].
(b) Evolutions of the inputs ui (t ) under [15, protocol (9)], smoothed using the moving average method with a window size of 100. (c) Evolutions of the
states xi (t ) under the proposed protocol (6) with α = 30, β = 0.5, and = 6. (d) Evolutions of the bounded inputs ui (t ) under the proposed protocol
(6) with α = 30, β = 0.5, and = 6. (e) Evolutions of the states xi (t ) under the proposed protocol (6) with α = 10, β = 0.5, and = 6.
(f) Evolutions of the bounded inputs ui (t ) under the proposed protocol (6) with α = 30, β = 1, and = 6.

the optimal solution trajectory of the time-varying objective
function.
The results in Fig. 3 provide further evidence to support
parameter analysis. In particular, higher values of α can
expedite state consensus and reduce the consensus error,
as depicted in Fig. 3(c) and (e). Although higher values
of β can increase the gradient convergence rate, too large
values would eventually destabilize the solution and even
cause algorithm divergence due to the significant gradient
updates, as shown in Fig. 3(d) and (f) and [20]. The selection

3830

of parameters in the proposed algorithm could consider
the acceptable consensus error of problems, the desired
convergence rate of the gradient, the sampling period for
discrete execution, and the allowable input fluctuations of
the actuator, while satisfying conditions (7) and (15).
Then, we will explore a practical application scenario
for the UAV swarm, as discussed in [28]. In this scenario,
multi-UAVs collaborate to hunt down an evader swarm. The
hunting task is formulated as a distributed time-varying
optimization problem with consensus constraints, where

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Fig. 4. Comparison simulation results with [28, protocol (9)] for the cooperative hunting task using the UAV swarm. The initial positions of the
hunter UAVs are depicted as red star dots, while the formation of the evader swarm is represented by the black dashed line. The teal dashed line
indicates
that the predesigned encirclement formation has formed. (a) Evolutions of the positions pi (t ), the control inputs ui (t ), and the sum of
gradients 5i=1 ∇ fi (xi (t ), t ) under [28, protocol (9)] with prescribed time T = 4 s. (b) Evolutions of the positions pi (t ), the control inputs ui (t ), and

the sum of gradients 5i=1 ∇ fi (xi (t ), t ) under the protocol (6) with α = 200, β = 20, and k = 2.

Fig. 5. Communication topology definition for the interception UAV
swarm. The graph G2 is specifically applied to the case of undirected
topology. In the switching topology case, a set of graphs
G = {G1 , G2 , G3 } is employed, starting with G1 and switching every 5 s.
In addition, the graph G4 is applied in the directed detail-balanced case.

the objective is to achieve a prescribed-time solution. To
address this problem, a proportional–integral-based distributed protocol with a time-varying gain has been devised

in [28] from the perspective of time-domain transformation.
The proposed protocol (6) differs from [28, protocol (9)].
The key distinction lies in the use of projection operators
in the proposed protocol, which ensures the boundedness
of the control inputs of UAVs. This improvement enhances
the practicality of the protocol.
Regarding the scenario settings, which include the local
objective function fi (qi (t ), t ) for each hunter UAV i, the
information flow topology, and the dynamics and movement
strategy of the evader swarm, they remain consistent with
those outlined in [28]. In addition, we choose the parameters α = 200, β = 20, and 1 = 2 = 2 for the proposed
protocol (6) in order to achieve a small consensus error,
while simultaneously ensuring a rapid convergence rate of
the gradient. It is evident that this configuration satisfies the
.5×(π /3)2
conditions that k ≥ 2+0.52×π /3 + 2×π /3+0
= 1.33
2×20
and α > 1/(2λ2 (L )) = 1/(2 × 1.38) = 0.36. We set the

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3831

Fig. 6. Trajectories of the interception UAV swarm pi (t ) and the incoming target swarm τk (t ) in the case of the undirected and connected topology
G2 . The teal dashed line represents the predesigned interception formation, while the black dashed line represents the penetration formation of the
target swarm.

prescribed time as T = 4 s, and the simulation step size
as t = 0.001 s. With these settings, we obtain the comparison results using two different protocols, as depicted in
Fig. 4.
The curves of trajectories in Fig. 4 clearly demonstrate
that the hunter UAVs, regardless of the protocol used,
can effectively establish a predetermined encirclement formation and track the evader swarm. However, there is a
distinction in the process. Hunter UAVs employing the proposed protocol (6) tend to first form the formation and then
track the evader swarm, which is a result of the parameter
configuration. On the other hand, this process is almost
simultaneous when using [28, protocol (9)]. Consequently,
the time required for achieving state consensus with an
error of 0.05, indicating the completion of the encirclement
formation, is 2.51 and 3.54 s, as depicted by the teal dashed
line in Fig. 4.
Furthermore, the control inputs generated by the proposed protocol (6) can be effectively limited within the
upper bound k , as illustrated by the control input curves in
Fig. 4. It should be noted that the control inputs exhibit large
values near the vicinity of the designed prescribed time,
which is a result of the specific structure of the time-varying
T
gain (T −t
when utilizing [28, protocol (9)]. However,
)2
this issue can be mitigated by employing the proposed
protocol (6). Moreover, the convergence of the sum of
local objective function gradients (equivalent to the gradient
of the global objective function) to small values provides
supportive evidence for Theorem 2. It is evident that the
error in gradients generated by (6) is smaller compared to
those generated by the protocol in [28], as depicted by the
sum of gradient curves in Fig. 4. This discrepancy arises
from the presence of nonexecutable control inputs near the
vicinity of the designed prescribed time. In conclusion,
3832

the proposed protocol offers advantages in scenarios that
necessitate rapid encirclement formation while considering
bounded input constraints.
B. Performance Analysis

In the following, we present a cooperative interception
scenario involving various topology configurations for a
UAV swarm. This scenario can be considered as a form
of swarm tracking behavior, specifically focusing on an
interception swarm consisting of four UAVs working collaboratively to intercept a swarm of three target UAVs. The
primary objective of the interception swarm is to establish a
predesigned formation and subsequently track the incoming
target swarm. The challenge of swarm tracking behavior can
be viewed as a distributed global optimal consensus problem, taking into account time-varying objective functions
and bounded input constraints.
Consider the existence of three communication link
cases among UAVs in the swarm: undirected and connected
topology G2 , switching topology G, and directed detailbalanced topology G4 , as illustrated in Fig. 5. It is worth
noting that in the case of switching topology, a finite set of
graphs denoted as G = {G1 , G2 , G3 } is defined. The graph sequence starts at G1 and switches every 5 s. Furthermore, [25,
Algorithm 1] is employed to construct the graph G4 , which
possesses a weight vector θ = [ 18 , 14 , 38 , 14 ]T .
The global objective function
 in the designed interception scenario is denoted as 4i=1 fi (xi (t ), t ), with its local
components defined as

xi (t ) − τi (t )2 , i = 1, 2, 3
fi (xi (t ), t ) =
i=4
0,
where the state xi (t ) represents the difference between the
current position pi (t ) of the ith UAV and the displacement

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Fig. 7. Control inputs under the proposed protocol (6) with parameters α = 100 and β = 10, upper bound = 60 m/s, and terminal time T = 40 s
in scenario of the cooperative interception using the UAV swarm. (a) Evolutions of the control inputs ui (t ) in the case of the undirected topology G2 .
(b) Evolutions of the control inputs ui (t ) in the case of the switching topology G = {G1 , G2 , G3 }. (c) Evolutions of the control inputs ui (t ) in the case of
the directed and detail-balanced topology G4 .

Fig. 8. Sum of gradients in three different cases. Case 1: undirected and
connected topology. Case 2: switching topology. Case 3: directed and
detail-balanced topology.

vector di of the interception formation center. Mathematically, we can express the state as xi (t ) = pi (t ) − di . To establish the interception formation, we consider the number

of participating UAVs and utilize the displacement vectors
di = R[cos( π3 i + π6 ), sin( π3 i + π6 )], where the interception
radius R = 150 m. Consensus among the states xi (t ) signifies the establishment of the interception formation. In
addition, the ith UAV can solely access the real-time position of the corresponding ith target τi (t ), i = 1, 2, 3. The
fourth UAV lacks target information and relies exclusively
on the state information of its neighboring UAVs.
The initial positions of the interception UAVs
are as follows: p1 (0) = [100 m, 1000 m]T , p2 (0) = [0 m,
1000 m]T , p3 (0) = [0 m, 900 m]T , and p4 (0) = [100 m,
900 m]T . The center position and velocity of the incoming target swarm formation are initialized as τ0 (0) =
[3000 m, 1000 m]T and vt (0) = [−10 m/s, 0 m/s]T , respectively. The formation radius of the target swarm is
set to 100 m. The targets follow the dynamics of a
double integrator, and their control inputs are given by

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[−1 m/s2 , (20 cos( 2t ) − 1)m/s2 ]T . In addition, we choose
the parameters α = 100 and β = 10 by considering the
tradeoff between the error in interception formation and the
rate of gradient convergence. The maximum speed of the
UAV is set to 60 m/s, while the simulation terminal time T
is chosen as 40 s, and the simulation step size is defined as
t = 0.001 s. It is evident that, in all three cases, both the
input upper bound condition and the consensus condition
are satisfied within the time T .
The simulation results for the undirected and connected
topology scenario are presented in Fig. 6. Due to the slight
difference in trajectory curves between the other two cases,
we omit the presentation of these figures in this article. The
trajectory curves clearly demonstrate that the interception
UAVs effectively form the predesigned formation and track
the target swarm. The time required for state consensus,
with an error of 15 m, in the three cases is as follows:
32.31, 32.05, and 32.29 s, respectively. Accordingly, the
proposed protocol (6) holds potential for application in
various scenarios involving swarm tracking behavior with
a predesigned formation, such as tracking ground vehicles,
robot navigation, and more.
As depicted in Fig. 7, the control inputs of UAVs in
all three cases can be effectively controlled within the
upper bound . Furthermore, the proposed protocol (6)
successfully avoids the issues of continuous chattering phenomenon and infinite inputs. The control inputs obtained
through this protocol are more suitable for practical execution of UAVs. In Fig. 7(b), the control inputs exhibit
transitory chattering during the topology switching, which
is caused by the large gradient update resulting from the
high value of parameter β. However, this phenomenon has
little impact on the execution of the UAV control commands,
and the gradients can still converge to the small values, as
illustrated in Fig. 8. It is worth noting that reducing the
value of parameter β can eliminate this chattering effect,
but it will also affect the convergence rate of the global
objective function. Therefore, it is crucial to select appropriate parameter configurations based on the specific practical
situation.
The convergence of the gradients of the global function
supports Theorem 2, Corollarys 1, and 2, as depicted in
Fig. 8. In conclusion, the proposed protocol (6) provides
a viable solution to global optimal consensus problems
with bounded input constraints. It can also be extended
to scenarios involving switching topology and directed
detail-balanced topology. However, it must be acknowledged that unlike fixed-time or predefined-time consensus
schemes [25], [28], the consensus time cannot be precisely predetermined and is influenced by the parameters.
In addition, Assumptions 2 and 3 impose limitations on
the local objective function. Thus, the proposed protocol (6) may not be suitable for scenarios where the incoming target exhibits superior maneuvering performance.
Future research will focus on relaxing these limitations
while building upon the proposed protocol (6) and incorporating higher order or nonlinear systems into the
study.
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VI. CONCLUSION

In this article, we have investigated the global optimal consensus problem, specifically focusing on scenarios
involving bounded input constraints and individual timevarying quadratic objective functions exclusive to each
UAV. Our investigation has led to the development of a
distributed protocol that integrates a projection operator
and a prediction–correction method. This novel protocol
has the potential to achieve asymptotic approximate global
optimal consensus for UAV swarms operating within the
undirected and connected, directed and detail-balanced, and
switching communication topologies. Furthermore, the implementation of the proposed protocol has demonstrated the
UAV swarm’s capability to form predetermined formations
and effectively track moving targets, thereby facilitating the
application in a variety of cooperative tasks. In future work,
we will explore the migration of this problem to higher
order or nonlinear UAV systems and the optimization of the
algorithm parameters.
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Aiwu Yang received the B.S. and M.S. degrees
in armament science and technology from the
School of Aeronautics and Astronautics Engineering, Air Force Engineering University, in
2018 and 2020, respectively. He is currently
working toward the Ph.D. degree in control science and engineering with the Air Traffic Control and Navigation School, Air Force Engineering University, Xi’an, China.
His research interests include distributed control and optimization with application in multiagent systems and unmanned autonomous systems.

Xiaolong Liang received the B.S. and M.S.
degrees in communications engineering and the
Ph.D. degree in armament science and technology from Air Force Engineering University,
Xi’an, China, in 2005, 2007, and 2010, respectively.
He is currently a Professor with the Air Traffic
Control and Navigation School, Air Force Engineering University, where he is also with the
Shaanxi Key Laboratory of Meta-Synthesis for
Electronic and Information System. His research
interests include aircraft swarm technology, airspace management intelligence, and intelligent aviation systems.

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Jiaqiang Zhang received the M.S. degree in
weapon system and application engineering and
the Ph.D. degree in armament science and technology from the School of Aeronautics and Astronautics Engineering, Air Force Engineering
University, Xi’an, China, in 2009 and 2012,
respectively.
He is currently a Lecturer with the School of
Air Traffic Control and Navigation College, Air
Force Engineering University. He has authored
or coauthored more than 20 journal papers and
finished more than ten projects. His research interests include aircraft
swarm technology and airspace management intelligence.

Ning Wang received the B.S. and M.S. degrees in communications engineering, in 2019
and 2021, respectively, from Air Traffic Control
and Navigation School, Air Force Engineering
University, Xi’an, China, where he is currently
working toward the Ph.D. degree in control science and engineering.
His research interests include intelligent decision making and cooperative control of crossdomain unmanned swarms.

Yueqi Hou received the B.S., M.S., and Ph.D.
degrees in control science and engineering from
Air Traffic Control and Navigation School, Air
Force Engineering University, Xi’an, China, in
2017, 2019, and 2023, respectively.
His research interests include deep reinforcement learning, multiagent reinforcement learning, and intelligent decision making with applications in air confrontation of aircraft swarm.

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