1904

IEEE CONTROL SYSTEMS LETTERS, VOL. 9, 2025

Event-Triggered Predeﬁned-Time Distributed
Optimization for Second-Order Multiagent
Systems via Sliding Mode
Tao Jiang , Yan Yan , Shuanghe Yu , Ge Guo , Senior Member, IEEE, and Yi Liu

Abstract—In this letter, the predefined-time distributed
robust optimization is studied for disturbed second-order
multiagent systems under the event-triggering mechanism. To this end, a sliding mode-based hierarchical
control scheme is presented. In the first part, a local
reference output signal with predefined-time convergence
is generated by the proposed time-varying functions
(TVFs)-based event-triggered local-minimization-free zerogradient-sum (LMFZGS) algorithm and evolves to the global
cost function’s minimizer. In the second part, a TVFsbased predefined-time sliding mode tracking controller is
designed to drive the agents’ outputs to track the local
reference output, regardless of lumped disturbances. By
this scheme, all agents’ outputs can reach the global minimizer, characterized by predefined-time convergence, with
only the virtual output signal required to be shared among
agents. Simulation results demonstrate the effectiveness of
our scheme.
Index Terms—Distributed optimization, event-triggering
mechanism, predefined-time convergence, sliding mode,
zero-gradient-sum.

I. I NTRODUCTION
ECENTLY, research on distributed optimization has
garnered increasing attention. The driving force behind
this trend is its wide range of practical applications, such as
traffic light control in mixed intersections [1] and resource
allocation [2]. Distributed optimization is a process in which
multiple agents collaboratively minimize the global cost function by sharing information over a communication network.
To this end, numerous distributed optimization schemes based
on multiagent systems (MASs) have been presented [3], [4],
[5], [6], [7].

R

Received 3 April 2025; revised 4 June 2025; accepted 27 June 2025.
Date of publication 4 July 2025; date of current version 29 July 2025.
This work was supported by the National Natural Science Foundation
of China under Grant 62173054, Grant 62073054, Grant 62173079,
and Grant U1808205. Recommended by Senior Editor A. Loria.
(Corresponding author: Yan Yan.)
Tao Jiang, Yan Yan, Shuanghe Yu, and Yi Liu are with the College
of Marine Electrical Engineering, Dalian Maritime University, Dalian
116026, China (e-mail: sevenjtao@163.com; y.yan@dlmu.edu.cn;
shuanghe@dlmu.edu.cn; liuyi11504027@163.com).
Ge Guo is with the State Key Laboratory of Synthetical Automation of
Process Industries, Northeastern University, Shenyang 110819, China
(e-mail:geguo@yeah.net).
Digital Object Identiﬁer 10.1109/LCSYS.2025.3585954

The zero-gradient-sum (ZGS) algorithm, which offers novel
insights into solving distributed convex optimization problems,
has gained popularity among researchers [8]. Specifically, it
can ensure that the sum of local gradients for all agents is zero
by iteratively updating the states of the agents based solely
on local information and communication with neighboring
agents. For example, ZGS-based and segmented ZGS-based
fixed-time distributed optimization algorithms were proposed
in [9], [10], respectively. However, it should be noted that they
are constrained and conservative, as the former is constrained
by the need to specify initial conditions in advance [9]
while the latter involves local minimization [10]. In view of
this, the authors in [11] proposed a fixed-time robust localminimization-free zero-gradient-sum (LMFZGS) algorithm by
integrating sliding mode technology with the ZGS mechanism.
This algorithm, for the first time, simultaneously addresses the
aforementioned issues, namely, it is free from the constraints
of initial conditions and local minimization. Nevertheless,
it is a result with respect to first-order MASs. Although
work [12] extended the LMFZGS algorithm to second-order
MASs subject to lumped disturbances, the upper bound of the
convergence time involves complex parameter calculations. In
practice, algorithms that enable simply adjusting or calculating
the upper bound are more preferable.
On the other hand, to reduce the communication cost in networked MASs, distributed optimization schemes incorporating
event-triggering mechanisms (ETMs) are frequently considered [13], [14]. In [15], an ETM was combined with the ZGS
algorithm to achieve practical distributed optimization. In [16],
exponential convergence of the event-triggered ZGS algorithm
was achieved. It should be noted that relatively little attention
has been paid to simple adjustments or calculations of the
upper bound of convergence time in existing event-triggered
ZGS-based results, including those mentioned above. In this
regard, predefined-time stability, which allows the adjustment
of convergence time bounds via a single control parameter,
offers a viable solution [17]. In [18], two types of timevarying functions (TVFs) were introduced, and subsequently,
an event-triggered ZGS algorithm with predefined-time convergence and easy implementation was proposed. Regrettably,
it is a segmented algorithm for first-order MASs. Moreover,
in [19], a predefined-time version of the LMFZGS algorithm
was introduced to achieve distributed optimization under the
ETM. However, the realization of predefined-time convergence

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JIANG et al.: EVENT-TRIGGERED PREDEFINED-TIME DISTRIBUTED OPTIMIZATION FOR SECOND-ORDER MASS

in [19] involves the use of fraction-power-based terms [20],
which may incur high computational complexity or singularity
if directly applied to second-order MASs. Besides, the above
results do not concern the robustness to disturbances, which
may lead to unreliable performance.
Integrating the surveys above, this letter aims to present
an event-triggered predefined-time LMFZGS algorithm and
apply it to second-order MASs with lumped disturbances.
Inspired by [5], a sliding mode-based hierarchical control
scheme is proposed to achieve distributed optimization for
second-order MASs. In the first part, we propose a predefinedtime robust LMFZGS algorithm using simple yet effective
TVFs. It can ensure local reference outputs that converge to the
global minimizer, even under the ETM. In the second part, a
TVFs-based predefined-time sliding mode tracking controller
is designed to ensure that each agent’s output tracks the local
reference output. The main contributions are summarized as
follows.
1) Within the sliding mode-based hierarchical optimization
framework, the proposed scheme allows for the predetermination of convergence time bounds through a
single parameter compared to [12], and enhances privacy
protection compared to [21] since the interaction among
agents involves only the virtual output signal. Moreover,
an ETM is utilized to reduce the cost of communication
among agents, which is not discussed in [12], [21].
2) Superior to the ZGS method in [10], [18], the designed
algorithm does not exhibit segmented behaviors. In
addition, unlike the fraction-power-based LMFZGS
method in [19], the proposed TVFs-based LMFZGS
algorithm offers the merits of strong robustness and
simplicity in control design. More importantly, the distributed optimization problem for second-order MASs is
solved.
The organization of this letter is as follows. In Section II,
the preliminaries and problem formulation are presented. In
Section III, the main results are provided. In Section IV,
numerical simulations are conducted to validate the presented
scheme. In Section V, the conclusion is drawn.
II. P RELIMINARIES AND P ROBLEM F ORMULATION
A. Preliminaries
1) Notions: Let the eigenvalues of matrix G ∈ Rn×n be
represented as λ1 (G) ≤ . . . ≤ λn (G), in non-decreasing order.
0n denotes the n-dimensional column vector of all zeros, In
denotes the n × n identity matrix, and ⊗ represents Kronecker
product. signn (·) denotes the norm-normalized sign function.
2) Graph Theory: The communication topology among N
agents can be modeled by an undirected graph T = {N , E, A},
where N = {1, . . . , N} is the set of nodes, E ⊆ N × N is
the set of edges, and A = [aij ] ∈ RN×N is the associated
adjacency matrix with aij = aji . The neighbor set of node i
is given as Vi = {j ∈ N |(j, i) ∈ E}. Define (i, j) as an edge
that connects agent j with agent i. If (i, j) ∈ E, then aij = 1;
otherwise aij = 0. Note that there are no self-loops allowed
here, i.e., aii = 0 for all i ∈ N . Denote
N H = diag{h1 , . . . , hN }
as the degree matrix with hi =
j=1 aij (i ∈ N ). On this
basis, the Laplacian matrix of graph T can be calculated as
L = H − A.

1905

3) Convex Function: Let Q ⊆ Rm denote a convex set.

For a twice differentiable function f : Rm → R, its gradient
is denoted as ∇f (μ) and the Hessian matrix is denoted as
∇ 2 f (μ). Then, a function f : Rm → R is convex if both
f (ς μ + (1 − ς )ν) ≤ ς f (μ) + (1 − ς )f (ν) and its domain is
a convex set Q hold for any μ, ν ∈ Q and ς ∈ [0, 1]. A
twice continuously differentiable convex function f : Rm → R
is θ -strongly convex (θ > 0), if and only if θ2 μ − ν 2 ≤
f (ν) − f (μ) − ∇f (μ)T (ν − μ) and ∇ 2 f (μ) ≥ θ Im , ∀μ, ν ∈ Q.
Besides, for any constant  > 0, the ∇f (μ) of the convex
function f is Lipschitz continuous with , if and only if f (ν)−
f (μ) − ∇f (μ)T (ν − μ) ≤ 2 μ − ν 2 and ∇ 2 f (μ) ≤ Im ,
∀μ, ν ∈ Q.
4) TVFs: In what follows, we will introduce two types of
TVFs. Denote Tp as the pre-given time instant and t = 0 as
the starting running time of the system.
The first TVF is denoted as (t, Tp , σ0 ) and satisfies the
following generalized properties:
• (t) is at least C2 on (0, +∞), in which (t)
˙
is bound
and (t)
˙ = 0 for t ≥ Tp ;
• (0) = σ0 , where σ0 can be predefined;
• (t) → 0 as t → Tp and (t) = 0 for t ≥ Tp .
The second TVF is denoted as α(t) and expressed as
˙ (t)
,
(1)
α(t) =
1 − (t) + κ
where
(t) is an internal TVF (or called as time-basegenerator) and the constant κ is chosen to satisfy 0 < κ
1.
The selection of (t) should satisfy the following generalized
conditions:
•
(t) is at least C2 on (0, +∞);
•
(t) is non-decreasing on [0, Tp ], where (0) = 0 is
the initial value, (Tp ) = 1 is the terminal value, and Tp
is the predefined-time instant;
• ˙ (0) = 0. For t ∈ [Tp , ∞), there are
(t) = 1 and
˙ (t) = 0.
Lemma 1 [22]: For a given Lyapunov function V, if it
satisfies
V̇ ≤ −(ζ + α(t))V + ,

V(0) = V0

(2)

where α(t) is the TVF given in (1), ζ > 0 and > 0 are some
constants, then V will converge into the bounded region
V0 − /ζ
+
(3)
V≤κ
1+κ
ζ
within a predefined time Tp .
Remark 1: If ζ = = 0, then V̇ ≤ −α(t)V. It follows
κ
that V converges to the bounded region V ≤ 1+κ
V0 within a
predefined time Tp [2]. Moreover, α(t) = 0 for t > Tp and if
= 0, then we have V̇ ≤ −ζ V, which means that V is nonκ
increasing. Thus, we ultimately obtain limt→Tp V ≤ 1+κ
V0
and limt→∞ V = 0.
5) ETM: Let z(t) ∈ Rn denote the signal to be triggered,
and let {tk }k∈Z+ represent the sequence of triggering instants.
The measurement error is defined as ξ(t) = z(tk ) − z(t) for
t ∈ [tk , tk+1 ), where z(tk ) denotes the latest broadcast signal.
Then, an event trigger is given as


(4)
tk+1 = inf t > tk : ξ(t) > ρe−βt
where β > 12 and ρ > 0 are constants that construct the
threshold signal for the event trigger.

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1906

IEEE CONTROL SYSTEMS LETTERS, VOL. 9, 2025

consider a virtual MAS consisting of N first-order agents,
where the dynamics of agent i is given by
ẋi = uvi ,
Fig. 1. Control block diagram of the proposed scheme.

B. Problem Formulation
Consider a MAS consisting of N agents, where the communication topology is T and the dynamics of the i-th agent
are given by
ẏi = vi ,
v̇i = bi (ȳi )ui + di

(5)

where ȳi = [yTi , vTi ]T ∈ R2m represent the measurable system
state, bi (ȳi ) = 0 denotes continuous nonlinear function of ȳi ,
di ∈ Rm represents the bounded lumped disturbances, ui ∈ Rm
is the actual control input, and m is the dimension. Especially,
yi ∈ Rm is the output signal of the ith agent.
For agent i, it can only access the local cost function fi (yi )
on its own. Subsequently, the distributed optimization problem
for MAS (5) is proposed as follows:
min
yi

N


fi (yi )

s.t. yi = yj i ∈ N , j ∈ Vi

(6)

i=1

Let y∗ ∈ Rm be the global minimizer of the above optimization
problem. The objective of this letter is to design appropriate
control inputs for the MAS (5) that enable the agents’ outputs
to track y∗ .
Assumption 1: The lumped disturbances di satisfy
di (t) ≤ D for i ∈ N , where D is a constant.
Assumption 2: The graph T is undirected and connected.
Assumption 3: If the i-th agent is triggered, its output signal
is instantaneously broadcast and subsequently employed by
the neighboring agents.
Assumption 4: The local cost function fi (yi ) is twice continuously differentiable, θi -strongly convex, and i -Lipschitz
gradient continuous with respect to yi , i ∈ N .
Definition 1 [2], [17]: The optimization problem (6) is said
to achieve predefined-time convergence if for any yi (0), i ∈ N ,
there exists ε > 0 such that
⎧
∗
⎪
⎨ lim yi − y ≤ ε,
t→To

yi − y∗ ≤ ε, ∀t ≥ To ,
⎪
⎩ lim yi − y∗ = 0,

i∈N

(8)

with xi ∈ Rm being the virtual state and uvi being the virtual
control input. Note that xi denotes the local reference output

(7)

t→∞

where To is a predefined time.
III. M AIN R ESULTS
In this section, a hierarchical robust control scheme is
designed, which consists of a TVFs-based event-triggered
LMFZGS algorithm and a TVFs-based predefined-time sliding
mode tracking controller. The control structure and signal flow
are shown in Fig. 1.
A. TVFs-Based Event-Triggered LMFZGS Algorithm
Design Part
Due to the uniqueness of y∗ , it remains invariant without
concerning the orders of the system [5]. For simplicity,

signal, and the communication topology of the virtual MASs
is assumed to be T . The task here is to design suitable control
law uvi to achieve limt→To1 xi − y∗ ≤ ε1 and limt→∞ xi −
y∗ = 0 for i ∈ N under the ETM (4), where ε1 and To1 will
be given later.
i
), a time-varying sliding
Based on the TVFs, for t ∈ [tki , tk+1
variable and a distributed sliding mode control law for the
virtual MAS (8) are designed as follows
⎧

j
t
⎪
δi = ∇fi (xi ) + 0 k2 (α1 + 1)
aij xi (tki ) − xj (tk ) dτ
⎪
⎪
⎪
j∈Vi
⎪
⎪
⎨
+i (t),
−1 
(9)
⎪
k1 signn (δi ) + ˙ i (t)
uvi = − ∇ 2 fi (xi )
⎪
⎪


⎪
j
⎪
⎪
aij xi (tki ) − xj (tk )
+k2 (α1 + 1)
⎩
j∈Vi

¯

where k1 is the robust gain, k2 > λ2(L) > 0, tki and tk
are the latest event time of agents i and j at time t, and
α1 = α(t, κ1 , Tp2 ) and i (t) is given in Section II-A with
σ0i = i (0) − δi (0). Herein, Tp1 and Tp2 are defined as the
predefined instants in i (t) and α1 , respectively, and the local
cost function fi (xi ) is defined in the same way as that for the
actual agent i in (5).
Theorem 1: Suppose that Assumptions 2-4 hold. For the
virtual MAS (8), if the predefined-time robust LMFZGS
algorithm (9) with the ETM (4) is adopted, then the virtual
output signal xi tracks the minimizer y∗ within a predefined
time, i.e.,
⎧

⎪
∗
W (Tp1 )−∗
⎪
∗
⎪
lim xi − y ≤ ε1 = 2κ1 θ(1+κ
+ 2
⎪
θ ,
⎪
1)
⎨ t→To1

(10)
∗
W (Tp1 )−∗
∗
⎪
xi − y ≤ ε1 = 2κ1 θ(1+κ
+ 2
⎪
θ , ∀t ≥ To1 ,
1)
⎪
⎪
⎪
⎩ lim xi − y∗ = 0,
j

t→∞

where To1 = Tp1 + Tp2 , κ1 > 0, W and ∗ will be given later.
Furthermore, the Zeno behavior can be excluded.
Proof: Choose the Lyapunov function candidate V1i =
1
δi 2 . By differentiating V1i along (9), we can obtain V̇1i ≤
2√
1/2
− 2k1 V1i . According to the second property of the function
(t) and the appointment of σ0i , it can be deduced that δi (t) =
0 for t ≥ 0. In other words, the equation
 t

j
k2 (α1 + 1)
aij xi (tki ) − xj (tk ) dτ = 0 (11)
∇fi (xi ) +
0

j∈Vi

holds within the predetermined time Tp1 . Since Assumptions 2
N

j
t
and 3 hold,
aij (xi (tki ) − xj (tk )) dτ =
i=1 0 k2 (α1 + 1)
 j∈Vi N
0m [19]. Therefore, we have N
i=1 δi =
i=1 ∇fi (xi ) = 0m at
t = Tp1 .
Then, based on the third property of the TVF (t) and (9),
for t ∈ (Tp1 , ∞), we have
−1


j
ẋi = − ∇ 2 fi (xi ) k2 (α1 + 1)
aij xi (tki ) − xj (tk )
(12)
j∈Vi

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JIANG et al.: EVENT-TRIGGERED PREDEFINED-TIME DISTRIBUTED OPTIMIZATION FOR SECOND-ORDER MASS

N
Define Qi = ∇fi (xi ). It follows from (12) that
i=1 Q̇i =
N 
j
aij (xi (tki ) − xj (tk )) = 0m . Thus, we have
−k2 (α1 + 1) i=1
N j∈Vi
N
δ
=
i=1 i
i=1 ∇fi (xi ) = 0m for t ∈ [Tp1 , ∞). It indicates
that the ZGS manifold is ensured within a predetermined time.
Next, take the Lyapunov function candidate as W =
N

(fi (y∗ ) − fi (xi ) − ∇fi (xi )T (y∗ − xi )). Taking the derivative of

divided into two stages, namely t ∈ [0, To1 + T  ] and t ∈
(To1 , ∞), where T  ≥ 0 is an arbitrarily small constant.
The results within time period t ∈ [0, To1 + T  ] are as
follows. Note that ξi (tki ) = 0, and from (8), (9), and the upper
bound on the measurement error ξi given in (4), we have
 t
ξi (t) ≤
(φ ξi (τ ) + i ) dτ
tki




i
≤ t − tki φρe−βtk + i

i=1

W along (12) yields
Ẇ = k2 (α1 + 1)

N 


= k2 (α1 + 1)

i.e., (t − tki ) ≥


T 

aij y∗ − xi xi − xj

bounded by our proof, there exists a constant  > 0 such
ρe−βt
that
≥ . In other words, t − tki ≥  > 0 for
−βti

i = θ (k1 +

i=1 j∈Vi

+ k2 (α1 + 1)

N 


φρe

k2 (α1 + 1)  
aij xi − xj 2
2
N

N 



T
aij xi − xj ξi

N

i=1 j∈Vi

N 


aij ξi 2

(13)

i=1 j∈Vi

where Young’s inequality and Assumption 2 are used for
mathematical derivation [18].
N

Define x̄ = N1
xi and z = [zT1 , zT2 , . . . , zTN ]T with zi =
i=1

xi −x̄. Since

N 


i=1 j∈Vi

aij xi − xj 2 = 2zT (L(T )⊗Im )z, it follows

from (13) that Ẇ ≤ − k2 (α21 +1) zT (L ⊗ Im )z + ∗ , where ∗
is defined as the supremum of  with  = k2 (α1 + 1)ξ T Hξ
and ξ = [ξ1T , . . . , ξNT ]T . On the other hand, it follows from
¯
¯ = maxi∈N {i } [18].
f (y∗ ) ≤ f (x̄) that W ≤ 2 z 2 , where 
Then, one can deduce that
Ẇ ≤ −k2 (α1 + 1)
¯

xi − xj ). Since xi is

k +i

k

k2 (α1 + 1)  
aij xi − xj 2
4

+ k2 (α1 + 1)

, where φ = 2k2 (α1 + 1)θ λn (H) and

j∈Vi

i=1 j∈Vi

≤−

−βti

φρe k +i

˙ i (t) + (α1 + 1)
aij
j∈Vi

Similar to the previous stage, here ξi (tki ) = 0. Then,
t
it yields ξi (t) ≤ ti h−1
dτ ,
f (L ⊗ Im )(x(τ ) + ξ(τ ))

i=1 j∈Vi

− k2 (α1 + 1)

ρe−βt

t ∈ [0, To1 + T  ].
On the other hand, for the time interval t ∈
(To1 , ∞), we can obtain δi = 
0 and α1 = 0,
j
which means uvi = −[∇ 2 fi (xi )]−1 (
aij (xi (tki ) − xj (tk ))).


T 

aij y∗ − xi ξi − ξj

i=1 j∈Vi

=−

(15)


T
j
xi (tki ) − xj (tk )
aij y∗ − xi

i=1 j∈Vi
N 


1907

λ2 (L)
W + ∗ ≤ −(α1 + 1)W + ∗ (14)
¯


where k2 > λ2(L) is used here. From Lemma 1 and α1 = 0 for
t > To1 = Tp1 + Tp2 , it can be concluded that limt→To1 W ≤
W(Tp1 )−∗
κ1 1+κ
+ ∗ and limt→∞ W = 0. Moreover, based on
1
N

θi
∗ 2 [11]. Thus, we
Assumption 4, there are W ≥
2 xi − y
i=1
∗
W(Tp1 )−∗
+ 2
have limt→To1 xi − y∗ ≤ 2κ1 θ(1+κ
θ = ε1 and
1)
limt→∞ xi − y∗ = 0, where θ = mini∈N {θi }.
In what follows, we provide the positive lower bound of
t − tki , which proves the absence of Zeno behavior. Given
the vanishing characteristics of TVFs, the analysis process is

= diag{∇ 2 f1 (x1 ), . . . , ∇ 2 fN (xN )} and x(τ ) =
where h−1
f
[x1 (τ ), . . . , xN (τ )]T . Since Assumption 2 holds, there exists a
positive constants Ľ such that L ≤ Ľ [16]. Based on this
and reviewing the Zeno-free behavior proof of Theorem 3.1
in [16], here is

Ľ t
ξi (t) ≤
x(τ ) − 1N ⊗ y∗ + ξ (τ ) dτ
(16)
θ tki
According to (4) and (14), we have
x(τ ) − 1N ⊗ y∗ + ξ (τ )


√
2W(0) − t
2k2 N
e 2+
H ρe−βt + Nρe−βt
≤
θ
θ
≤ e−βtk
(17)


√
where  = 2W(0)
+ 2kθ2 N H ρ + Nρ and β > 12 is
θ
used here. Substituting (17) into (16), we have ξi (t) ≤
Ľ
−βtki
(t − tki ). Note that the next event will not be triggered
θ e
before ξi (t) = ρe−βt . Thus, define t as the solution of
Ľ
−βt . Clearly, t > 0, and t − ti ≥ t > 0 for
k
θ t = ρe
t ∈ (To1 , ∞).
In conclusion, there exists a positive lower bound on t − tki ,
which avoids the Zeno phenomenon. The proof ends.
i

B. TVFs-Based Predefined-Time Sliding Mode Tracking
Controller Design Part
For the MAS (5), based on the local reference output xi , a
time-varying sliding variable is constructed as follows
α2
si = vi + ei
(18)
2
where ei = yi − xi denotes the error variable and α2 =
α(t, κ2 , Tp3 ) is given in (1). Then, based on virtual output

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1908

IEEE CONTROL SYSTEMS LETTERS, VOL. 9, 2025

signals and measurable original system states, the sliding mode
tracking controller is designed as


α2 ėi + α̇2 ei + α3 si
(19)
ui = −b−1
sign
+
k
(ȳ
)
(s
)
i
d
n i
i
2
where kd is the robust gain and α3 = α(t, κ3 , Tp4 ). Note that
Tp3 and Tp4 are user-defined time instants, and Tp3 should be
selected appropriately such that Tp3 ≥ Tp4 and Tp3 ≥ To1 ,
which will be consistent with the subsequent reasoning logic.
Under the controller (19), the following theorem holds.
Theorem 2: Suppose that Assumptions 1-4 hold. On the
basis of Theorem 1, if the designed tracking controller (19) is
used for the MAS (5), then yi converges to the local reference
output xi within a predefined time, i ∈ N , that is,

⎧

2κ2 
∗
∗
⎪
lim
e
≤
ε
=
⎪
i
2
1+κ2 V3i (0) − ϒi + 2ϒi ,
⎪
⎨ t→Tp3


2κ2 
ei ≤ ε2 = 1+κ
V3i (0) − ϒi∗ + 2ϒi∗ , ∀t ≥ Tp3 , (20)
⎪
2
⎪
⎪
⎩ lim ei = 0,
t→∞

where κ2 > 0, V3 and ϒi∗ will be given later. Ultimately,
predefined-time convergence in the optimization problem (6)
with dynamic MAS (5) is achieved.
Proof: In view of (19), the dynamic of the time-varying
sliding variable si is
α2 ėi + α̇2 ei
+ bi (ȳi )ui + di
2
α3
= di − kd signn (si ) − si
2

s˙i =

(21)

Then, choose the Lyapunov function candidate V2i = 12 si 2 .
Differentiating V2i with respect to time and applying (21), we
obtain
α3
V̇2i = sTi di − kd signn (si ) − si
2
√
1/2
(22)
≤ − 2(kd − di )V2i − α3 V2i
κ3
V2i (0) and
According to Remark 1, limt→Tp4 V2i (t) ≤ 1+κ
3

limt→(Tp4 +T  ) V2i (t) = 0 hold if kd > D is satisfied, where T
is a finite time constant.
On the other hand, Theorem 1 implies that xi − y∗ is
bounded and decays to zero at an exponential rate. Based on
this and for subsequent analysis, we define xi = y∗ + ιi , where
ιi is differentiable with respect to time. Then, (18) can be
rewritten as
α2
(23)
si = ėi + ι̇i + ei
2

Choose a Lyapunov function candidate V3i = 12 ei 2 , whose
derivative along (23) is formulated as
α2
V̇3i = eTi si − ei − ι̇i ≤ −(α2 + 1)V3i + ϒi∗ (24)
2
∗
where ϒi is defined as the supremum of ϒi with ϒi =
1
2
2 si + ι̇i . Recalling (22) and Theorem 1, the term ϒi
will vanish as t → ∞, i.e., limt→∞ ϒi = 0. Hence, by
κ2
utilizing Lemma 1, we have limt→Tp3 V3i ≤ 1+κ
(V3i (0) −
2
∗ ) + ϒ ∗ and lim
ϒ
V
(t)
=
0.
Further,
lim
t→∞ 3i
t→Tp3 ei ≤
i
i
2κ2
∗
∗
1+κ2 (V3i (0) − ϒi ) + 2ϒi

= ε2 and limt→∞ ei

= 0

can be obtained. As a result, according to Definition 1,
the optimization problem (6) is solved and characterized by
predefined-time convergence. The proof ends here.
Remark 2: In this letter, the peculiar second-order dynamics (5) is considered. However, the proposed method may
encounter challenges when applied to generalized secondorder systems. For example, the problems of irreversible gain
bi and mismatched disturbances. This implies a direction for
future research.
Remark 3: From (9) and (19), it can be seen that the
proposed algorithm is simple in design and does not involve
the true information of neighbors, which contributes to theoretical analysis and privacy protection. On the other hand, the
appropriate selection of parameters in the controller is crucial
for practical applications. The sliding mode gain parameters
k1 and k2 in the proposed algorithm (9) need to satisfy
¯
k1 > 0 and k2 > λ2(L) , where the size of k1 should
be sufficient to suppress potential disturbances while being
constrained by the rated output. The selection of the robust
gain parameter kd is similar to that of k1 . That is, kd >
D. The convergence rate of the overall control algorithm is
primarily determined by the time-varying functions , α1 ,
α2 , and α3 , and the selection criteria for their components
have been explained in Section II-A. In addition, considering
the convergence behavior of the sliding mode dynamics
and the fact that system (8) should have faster dynamics
than the tracking control part [5], we have Tp3 ≥ Tp4 and
Tp3 ≥ To1 .
IV. S IMULATION
In this section, a MAS consisting of four agents is considered to solve optimization problems (6). The communication
topology among agents is set to an undirected graph: 1 ↔
2 ↔ 3 ↔ 4 ↔ 1. For ease of calculation, the local cost
functions are given as f1 (y1 ) = 0.1yT1 y1 , f2 (y2 ) = 0.05(y2 −
[1, 1]T )T (y2 − [1, 1]T ), f3 (y3 ) = 0.5((y31 + 1)2 + y232 ), and
f4 (y4 ) = 0.2(y4 −[2, 2]T )T (y4 −[2, 2]T ). The global minimizer
is calculated as y∗ = [y∗1 , y∗2 ]T = [−0.059, 0.529]T .
Next, we will not only verify the effectiveness of the
proposed hierarchical control algorithm, but also highlight the
advantages of the proposed algorithm (9) by comparing it
with the ZGS-based algorithm in [10]. For virtual MASs (8),
the initial conditions are given as x1 (0) = [1, 3]T , x2 (0) =
[−3, 2]T , x3 (0) = [5, −1]T , and x4 (0) = [−3, −6]T , respectively. The control parameters in (9) are set as k1 = 0.01,
k2 = 3, κ1 = 3 × 10−4 , Tp1 = 3 s, and Tp2 = 4 s.
Then, we can get To1 = 7 s. On the other hand, for actual
MASs (5), the initial conditions are given as y1 (0) = [−5, 9]T ,
y2 (0) = [5, 7]T , y3 (0) = [−7, −7]T , and y4 (0) = [3, −8]T ,
respectively. The lumped disturbances suffered by agent i is
assumed as di (t) = [0.5 sin(0.3t), 0.5 sin(0.3t)]T . The control
parameters in (19) are set as kd = 1, η = 0.01, b = 1,
κ2 = 5 × 10−6 , κ3 = 1 × 10−6 , Tp3 = 15 s, and Tp4 = 10
s. The parameter settings in the ETM (4) are ρ = 3 and β =
1. Based on the given parameters, the results are shown in
Figs. 2 and 3. The variation of the signal xi under the proposed
algorithm and the algorithm in [10] is shown in Fig. 2(a-b).
Evidently, xi extremely approaches the global minimizer y∗ at

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JIANG et al.: EVENT-TRIGGERED PREDEFINED-TIME DISTRIBUTED OPTIMIZATION FOR SECOND-ORDER MASS

1909

extending our proposed scheme to tackle constrained timevarying optimization problems along similar lines.
R EFERENCES

Fig. 2. Evolutions of xi and
the algorithm in [10].

4


fi (xi ) under the proposed algorithm and

i=1

Fig. 3. Evolutions of ei and the triggering event tki under the proposed
algorithm.

t = To1 = 7s, and thereafter, xi remains close to the minimizer.
It can be seen from the zoomed-in region of Fig. 2(b-c) that
under the algorithm in [10], xi exhibits segmented behavior
not caused by the ETM, while the proposed algorithm does
not. Furthermore, from Fig. 2(c) and the given Tp1 = 3 s, the
global cost function approaches the optimal value within the
predefined time, and the proposed algorithm exhibits better
convergence performance after reaching the ZGS manifold. On
the other hand, under the proposed algorithm, the predefinedtime convergence of the output yi to the local reference output
signal xi is achieved, as depicted in Fig. 3(a). Besides, the
triggering event tki is shown in Fig. 3(b). To sum up, the
simulation results corroborate the theoretical findings, thereby
confirming the effectiveness and superiority of the proposed
scheme.
V. C ONCLUSION
This letter has provided a TVFs-based robust solution
to the predefined-time distributed optimization for disturbed
second-order MASs. With the proposed scheme, the agents’
outputs have achieved predefined-time convergence toward the
minimizer of the global cost function, even in the presence of
the ETM. Moreover, the controller design process for agents is
simple and uses only their own information and virtual output
information from neighbors, thereby enhancing privacy and
reducing communication costs. Future research will involve

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