2820

IEEE TRANSACTIONS ON CYBERNETICS, VOL. 55, NO. 6, JUNE 2025

Distributed Practical Fixed-Time Resource
Allocation Algorithm for Disturbed Multiagent
Systems: An Integrated Framework
Qingxiang Ao , Cheng Li , Ben Niu , Zhiliang Zhao , Jiaxin Yuan, Sen Chen , and Xiaole Yang

Abstract—The practical fixed-time resource allocation problem
is investigated for multi-input–multi-output nonlinear uncertain
multiagent systems with disturbed dynamics, subject to global
equality and local inequality constraints. Due to the coexistence
of distributed high-order dynamics system within agents and
decision-making constraints, decision variables in resource allocation optimization problems cannot be directly obtained from
the system. Existing strategies are insufficient to solve such
complex fixed-time optimization control problems with coupled
decision-making constraints. To address these challenges, a novel
integrated framework is proposed, fusing symbolic-functionbased fixed-time control theory with gradient consistency. The
proposed algorithm is implemented through an output-feedback
backstepping design process, which involves two stages. First,
in the output-feedback design stage, a fixed-time high-order
extended state observer estimates the uncertain dynamics and
disturbances. Second, in the backstepping design stage, a timeswitching controller is developed. This controller’s virtual control
law has two components: the first employs the proportional–
integral control method to satisfy the equality constraints,
while the second uses gradient information from the -exact
penalty function to address the inequality constraints. Using the
Lyapunov stability criterion, the proposed algorithm can ensure
that all signals remain practical fixed-time stable, and that the
error between the outputs of all agents and the optimal solution
is maintained within a neighborhood of the origin. Finally,
simulations are presented to demonstrate the effectiveness of the
approach.
Index Terms—Fixed-time convergence, high-order extended
state observer, multiagent systems (MASs), resource allocation.

I. I NTRODUCTION
S A CRITICAL branch of distributed optimization,
resource allocation has been extensively applied in various fields [1], [2], [3], including the generation scheduling

A

Received 21 December 2024; revised 7 March 2025; accepted 3
April 2025. Date of publication 18 April 2025; date of current version
16 May 2025. This article was recommended by Associate Editor C.-Y. Su.
(Corresponding author: Cheng Li.)
Qingxiang Ao, Cheng Li, Jiaxin Yuan, and Xiaole Yang are with the College
of Air Transportation, Shanghai University of Engineering Science, Shanghai
201620, China (e-mail: m385123208@sues.edu.cn; Licheng@sues.edu.cn;
yuanke2964@sjtu.edu.cn; m385121103@sues.edu.cn).
Ben Niu is with the School of Control Science and Engineering, Dalian
University of Technology, Dalian 116024, Liaoning, China (e-mail: niuben@
dlut.edu.cn).
Zhiliang Zhao and Sen Chen are with the School of Mathematics and
Information Science, Shaanxi Normal University, Xi’an 710119, Shaanxi,
China (e-mail: zhiliangzhao@snnu.edu.cn; senchen@snnu.edu.cn).
Color versions of one or more figures in this article are available at
https://doi.org/10.1109/TCYB.2025.3558787.
Digital Object Identifier 10.1109/TCYB.2025.3558787

problem in power systems [1], power loss optimization [2],
and traffic allocation in communication networks [3]. Resource
allocation optimization aims to design distributed algorithms
for multiagent systems (MASs) to optimize a global cost
function under network constraints. Given the complexity
of networked systems and the limitations of communication
resources, each agent must rely solely on its own and its
neighbors’ information, along with its local cost function. This
leads to a challenge for agents to adhere to local constraints.
Simultaneously, they must collaborate with their neighbors to
achieve a globally optimal solution under global constraints,
maximizing overall system performance in dynamic environments.
Several optimization algorithms have been proposed to
address the diverse challenges in multiagent resource allocation systems [4], [5], [6], [7], [8], [9], [10], [11]. For instance,
a federated learning-based resource allocation algorithm was
developed in [4] to solve the resource allocation problem
(RAP) for first-order MASs. Similarly, in [5] and [6], the RAP
was optimized through projected gradient and approximate
gradient exchanges among agents, respectively. Building on
gradient descent, output regulation theory was employed to
address a class of time-varying RAP, as demonstrated in [7].
Additionally, the approach of converting the RAP into a
dual problem, allowing agents to exchange dual factors for
optimization, was employed in [8], [9], and [10]. In most
cases, each agent should not be assumed to have unbounded
resources. To handle agents’ local constraints, an -exact
penalty function was introduced in [11], ensuring that agents
meet their constraints while still achieving an optimal solution.
However, these algorithms face limitations when applied to
second-order agents, primarily because they cannot directly
control their outputs through control inputs. To overcome
this challenge, many studies have adopted a hierarchical
approach [12], [13], [14]. In [12], for example, the RAP
was solved using a game-theoretic approach to generate an
optimal resource allocation signal, followed by a controller
design based on Euler–Lagrange systems to ensure the system
outputs the optimal signal. Similarly, hierarchical resource
allocation algorithms based on gradient descent and state
feedback were explored in [13] and [14]. It is noteworthy that
some researchers have extended their focus beyond secondorder agents, delving into high-order RAP in complex systems,
leading to significant progress in the study of high-order
multiagent RAP.

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AO et al.: DISTRIBUTED PRACTICAL FIXED-TIME RESOURCE ALLOCATION ALGORITHM

In line with the hierarchical approach used in solving
second-order multiagent RAP, most algorithms for highorder MASs adopt a layered framework [15], [16], [17],
[18], [19], [20], [21], [22], [23]. For example, linear multiinput–multi-output (MIMO) high-order MASs were explored
in [15], [16], and [17], where controllers were designed to
enable the system to track signals generated by the optimal
signal generator, thereby achieving the optimal solution of the
RAP. In more complex nonlinear systems, dynamic gain techniques and neural network methods were incorporated into the
controllers in [18] and [19], respectively, to mitigate the effects
of nonlinear terms, facilitating the tracking of signals from the
optimal signal generator. In certain complex engineering fields,
external disturbances are inevitable, especially in MASs,
where disturbances may arise from factors, such as environmental changes, sensor noise, or communication delays,
significantly affecting system stability and performance. To
address these disturbances, state- and output-feedback control
strategies were implemented in controller designs, as discussed
in [20], to suppress their effects, ensuring that the system
maintains stability and good performance even in the presence
of disturbances. Similarly, in [21], the internal model principle
was integrated into the controller, automatically generating
additional signals to counteract disturbances, thereby ensuring
the system could effectively track the optimal signal and
enhancing robustness against disturbances. However, simple
feedback control may not be sufficient when dealing with
more complex coupled disturbances. To address this challenge,
an extended state observer based on an active disturbance
rejection strategy was designed in [22], which estimates
unknown coupled disturbances in real time, complementing
the controller and optimal signal generator to effectively
manage complex coupled disturbances, thus ensuring proper
system operation in uncertain environments. Despite significant progress in resource allocation for high-order MASs,
existing studies primarily achieve asymptotic or exponential
convergence [15], [17], [18], [19], [20], [21], [22], meaning
that agents gradually approach the optimal solution over an
infinite time horizon. This characteristic may not be suitable
for many practical applications, especially those requiring fast
responses. Notably, [23] considered finite-time convergence
rates, while [16] focused on fixed-time convergence rates.
However, the latter study concentrated solely on linear agents
without disturbances, leaving a gap in research on solving resource allocation problems for high-order MASs with
unknown coupled disturbances within a fixed time. Therefore,
addressing how to effectively deal with unknown coupled
disturbances and unmeasurable states in resource allocation for
high-order MASs, while achieving stable convergence within
a fixed time and ensuring high-performance optimal solutions,
remains an important area for further investigation.
Existing research on resource allocation for high-order
MASs typically follows a two-step approach: 1) designing
a signal generator to produce the optimal resource signal
and 2) designing a controller to ensure that the system
effectively tracks this signal. This hierarchical design may
fail to fully account for the system’s dynamic characteristics during the optimization process in high-order systems,

2821

leading to significant tracking errors when optimization and
control are decoupled, especially when control inputs are not
directly related to optimization variables. In contrast, this study
proposes an integrated control framework to address fixedtime resource allocation for high-order MASs with coupled
disturbances and MIMO dynamics. The framework consists
of two stages: first, a fixed-time high-order extended state
observer (FHESO) is developed to estimate unmeasurable
states; second, an integrated control scheme is introduced that
eliminates the need for a separate optimal signal generator.
Each agent tightly couples its objective function with the
communication topology, allowing for direct resolution of
the optimization problem and generation of control output.
Finally, the fixed-time convergence of the closed-loop system
is rigorously proven based on fixed-time theory. The main
contributions of this article are threefold.
1) Unlike the fixed-time extended state observers for
second-order systems studied in [24], [25], and [26],
this article develops an FHESO based on active disturbance rejection control, tailored for high-order systems.
The closed-loop stability of the high-order error system
is rigorously established using homogeneity theory.
2) Unlike the resource allocation algorithms for firstand second-order MASs discussed in [8], [11], [12],
[13], and [14] and those targeting high-order systems
in [15], [16], and [17], this study investigates highorder MASs with unknown states and coupled uncertain
disturbances. An integrated resource allocation scheme
based on active disturbance rejection control is proposed
for this class of systems.
3) Compared to the asymptotically stable or finite-time stable resource allocation algorithms for high-order MASs
proposed in [15], [17], [18], [19], [20], [21], [22],
and [23], this article introduces a fixed-time distributed
resource allocation framework, which ensures that the
global signals of the closed-loop system stabilize within
a fixed time and achieves an optimal solution with
sufficiently small error.
Section II presents the preliminaries and formulates the
resource allocation problem under consideration. Section III
presents the resource allocation control framework and its
convergence analysis. In Section IV, the effectiveness of the
scheme is validated through two simulation examples. Finally,
the conclusions of this article are summarized in Section V.
The technical framework of this article is illustrated in
Fig. 1.
Notations: R denotes the real number field, and RN represents the N-dimensional real vector space. The identity matrix
of size n × n is In , while 1n = [1, . . . , 1] ∈ Rn are
the all-ones vector. For k ∈ (0, +∞), the function xk is
defined as sign(x)|x|k , where sign(x) is the sign function.
The ith eigenvalue of matrix A is λi (A), ordered ascendingly.
The symbol ⊗ denotes the Kronecker product, and ∇f (x)
represents the gradient of the function f at the point x. For an
m × n matrix P = {pij }m×n , and Po = {pij o }m×n , for some
constant o. tr(A) represents the trace of matrix A, which is
the sum of its diagonal elements. · represents the standard
Euclidean norm.

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2822

IEEE TRANSACTIONS ON CYBERNETICS, VOL. 55, NO. 6, JUNE 2025

()

min ( )

()

( ( ))

0

2

1

()
..

( ))

()
1

( ( ))

( ( ))

1

*

Yes

No

1

,1

( )

,

( )

,

1

2

( )

,1

()

,

()

,

1



()



(4)

()

()

,2

()

, 1

( ),

0

Yes

*

1

,

1

No

,

(

()

If both functions are homogeneous of degrees k1 > 0 and
k2 > 0, then there exists a set  = {z | V1 (z) = 1} for any x
such that




k2
k2
k
min 2 (z) [1 (x)] 1 ≤ 2 (x) ≤ max 2 (z) [1 (x)] k1 .

( ))

()

,1

()
( ),

( ( ))
( ( ))
( ( )) (

,1

()
,

(

,1

,

,

,

,

( ),

,

( ),

( ))

( ))

Fig. 1. Schematic of the resource allocation framework for high-order MASs.

II. P RELIMINARIES AND F ORMULATION
A. Preliminaries
Consider an undirected graph G = (V, E, A), where V =
{1, 2, . . . , N} represents N agents, and E = {(i, j) | i, j ∈
V, i = j} is the set of edges denoting connections between
agents. An edge (i, j) ∈ E indicates a communication link
between agents i and j. The weighted adjacency matrix A =
[aij ] ∈ RN×N satisfies aij > 0 if (i, j) ∈ E, and aij = 0
otherwise. The neighbors of agent i, denoted Ni , are given by
Ni = {j | (i, j) ∈ E}. The Laplacian matrix L = [lij ] ∈ RN×N
is defined such that lii is the sum
 of the weights of all edges
connected to node i (i.e., lii = N
j=1 aij ), and for i = j, lij =
−aij if (i, j) ∈ E; otherwise, lij = 0.
Definition 1 [27]: A set  ⊆ Rn is convex if, for any x, y ∈
 and 0 ≤ a ≤ 1, the point ax + (1 − a)y also lies in .
A function f : Rn → R is convex if f (ax + (1 − a)y) ≤
af (x) + (1 − a)f (y) holds for all x, y ∈  and 0 ≤ a ≤ 1. A
twice continuously differentiable function f : Rn → R is strongly convex, with  > 0, if f (y) − f (x) − ∇f (x) (y − x) ≥
(/2) y − x 2 and ∇ 2 f (x) ≥ In for all x, y ∈ .
Definition 2 [28]: Consider the nonlinear system
dx(t)
= f (t, x), f (0) = 0
(1)
dt
where f (t, x) is a continuous nonlinear function, x(t) ∈ Rn is
the system state, and x(0) = x0 . If there exists a bound Tmax ,
independent of the initial state, such that for any t > Tmax ,
x(x0 , t) ≤ ε (with ε > 0), then the equilibrium point of
system (1) is considered practical fixed-time stable (PFTS).
Definition 3 [25]: Consider x = [x1 , x2 , . . . , xn ] ∈ Rn and
a continuous function f (x) : Rn → R. There exist constants
μ > 0 and k > − min{ri } for i = 1, 2, . . . , n such that if f (x)
satisfies


(2)
f μr1 x1 , μr2 x2 , . . . , μrn xn = μk f (x)
then f (x) is considered homogeneous of degree k with respect
to the dilation operator (μr1 x1 , μr2 x2 , . . . , μrn xn ). Next, take
a vector field (x) = [1 (x), 2 (x), . . . , n (x)] : Rn → Rn ,
and for all μ > 0, if (x) satisfies


i μr1 x1 , μr2 x2 , . . . , μrn xn = μk+ri i (x)
(3)
the vector field (x) is homogeneous of degree k with respect
to the dilation operator (μr1 x1 , μr2 x2 , . . . , μrn xn ).
Lemma 1 [29]: Consider two continuous functions, 1 (x)
and 2 (x), with 1 (x) being positive definite and x ∈ Rn .

Lemma 2 [24]: Let (x) be a continuous function with
x ∈ Rn . If (x) is differentiable with respect to xn and
homogeneous of degree k, then its partial derivative with
respect to xn satisfies
μrn

∂(μr1 x1 , μr2 x2 , . . . , μrn xn )
∂(x1 , x2 , . . . , xn )
= μk
.
∂xn
∂xn
(5)

Lemma 3 [30]: For a connected undirected graph, the
eigenvalues of the Laplacian matrix L are ordered as 0 = λ1 ≤
λ2 ≤ · · · ≤ λN , where the second smallest eigenvalue λ2 is
defined as λ2 = minx=0N ,1 x=0 (x Lx/x x).
N
Lemma 4 [31]: Consider a positive-definite continuous
function V(x) : Rn → R satisfying
dV(x)
≤ −γ1 V α (x) − γ2 V β (x)
(6)
dt
where γ1 > 0, γ2 > 0, α > 1, and 0 < β < 1. The system’s
states achieve fixed-time stability, where the settling time
T ≤ Tmax =

1
1
+
.
γ1 (α − 1) γ2 (1 − β)

(7)

If the function V(x) satisfies the perturbed inequality
dV(x)
≤ −γ1 V α (x) − γ2 V β (x) + ϑ
(8)
dt
where ϑ > 0, there exists a constant 0 < τ <
1 such that the state x converges within fixed
time T to a neighborhood Ξ = {x | V(x) ≤
The
min([1/γ1 ](ϑ/1 − τ )(1/α) , [1/γ2 ](ϑ/1 − τ )[1/β] )}.
system (1) is PFTS, with T bounded by
T ≤ Tmax =

1
1
+
.
γ1 τ ϑ(α − 1) γ2 τ ϑ(1 − β)

(9)

Lemma 5 [32]: For real variables s and t, and any positive
constants ρ, τ , and  , the following inequality holds:
−ρ
ρ
τ
|s|ρ |t|τ ≤
 |s|ρ+τ +
 τ |t|ρ+τ .
(10)
ρ+τ
ρ+τ
Lemma 6 [33]: For any x, y ∈ Rn that satisfy
nl
1
x l+
y m
l
mnm
where l > 1, m > 1, n > 0, and (l − 1)(m − 1) = 1.
Lemma 7 [34]: Given ∀x1 , . . . , xm ∈ R
 m
q
m

|xi |q
if 0 ≤ q ≤ 1
i=1 
|xi | ≤
m
q−1
q if q > 1.
m
|x
|
i=1 i
x y ≤

(11)

(12)

i=1

Lemma 8 [35]: Given the high-gain TD as
dρ1 (t)
= ρ2 (t)
dt
1
ρ2 (t)
dρ2 (t)
= −ζ 2 1 ρ1 (t) − z(t) 2 + 2
dt
ζ

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2
3


(13)

AO et al.: DISTRIBUTED PRACTICAL FIXED-TIME RESOURCE ALLOCATION ALGORITHM

if z(t) satisfies supt∈[0,∞) |z(k) (t)| < ∞ for k = 0, 1, 2, . . .
and 1 > 0, 2 > 0, ζ > 0, then as ζ → ∞, there exist
ς1∗ > 0 and ς2∗ > 0 such that |ρ1 (t) − z(t)| ≤ ς1∗ and
|ρ2 (t) − [dz(t)/dt]| ≤ ς2∗ .
B. Problem Formulation
Consider a class of MASs with external disturbances, where
the dynamics of the agents are given by the following:
⎧


ẋi,1 (t) = xi,2 (t) + gi,1 x i,1 (t) 
⎪
⎪
⎨
ẋi,l (t) = xi,l+1 (t) + gi,l xi,l (t)


 (14)
ẋi,n (t) = ui (t) + gi,n xi,n (t) + qi xi,n (t), ωi (t)
⎪
⎪
⎩
yi (t) = xi,1 (t)
where i = 1, . . . , N, l = 1, . . . , n − 1. xi,h (t) ∈ Rm denotes
the system state for h = 1, . . . , n, xi,h (t) = [xi,1 (t) , . . . ,
xi,h (t) ] is the state vector. gi,h (xi,h (t)):Rm → Rm is the
nonlinear smooth function, ui (t) ∈ Rm is the control input
and yi (t) ∈ Rm is the system output. ωi (t) ∈ Rm represents
the external disturbance, and the unknown nonlinear function
qi (·) : R(n+1)×m → Rm represents the “total disturbance.”
Remark 1: This article examines a system with two types of
nonlinearities: 1) deterministic and 2) uncertain. Nonlinearities
in other channels can be transformed into the control channel [36], concentrating the uncertainties in (14).
Assumption 1:
1) There exists a constant  > 0 such
that: supt∈[0,∞) ||ωi (t)|| + ||[dωi (t)/dt]|| =  < ∞.
2) There exist continuous functions Ψi ∈ C(Rm , [0, ∞))
and Φi ∈ C(Rnm , [0, ∞)) such that, for constants δ > 0
and K > 0, if z ≤ δ, then Φi (z) ≤ K z . These
functions satisfy: [∂qi (z, ωi )/∂ωi ] ≤ Ψi (ωi )Φi (z),
where ωi ∈ Rm , z = col(z1 , . . . , zm ) ∈ Rmn , and zi =
(zi1 , zi2 , . . . , zin ) ∈ Rn .
Assumption 2: There exist a constant θ ∈ (0, 1] and continuous functions ψ such that: |gij (zi1 , 0, . . . , 0)| ≤ ψ(zi1 )|zi1 |θ
and |gij (zi1 , zi2 , . . . , zij ) − gij (zi1 , ẑi2 , . . . , ẑij )| ≤ , where
j = 1, 2, . . . , n and  is a constant.
Remark 2: Condition 1) of Assumption 1 addresses standard assumptions regarding unknown disturbances, without
it, the control input could become unbounded. Condition
2) relates to unknown nonlinearities and is significantly weaker
than the conditions found in existing literature on continuous
output-feedback stabilization [37]. In Assumption 2, to prove
the fixed-time convergence of the designed ESO, the same
settings as in [26] are adopted.
Assumption 3: The communication between the agents
in (14) is governed by the undirected and connected graph G.
We consider the MASs described by (14) to address an RAP
as follows:
min

y(t)∈RNm

s.t.

N




1
m fi yi (t)

i=1
ymin
≤ yi (t) ≤ ymax
i

iN
N
y
(t)
=
i=1 i
i=1 di

(15)

where fi (yi (t)) : Rm → Rm is the local cost function. ymin
and
i
are
the
lower
bound
and
upper
bound
of
y
(t).
d
∈
Rm
ymax
i
i
i
represents the local demand resource of agent i.

2823

Assumption 4: The local objective functions fi (yi (t)) are
continuously differentiable, -strongly convex, and have
locally Lipschitz gradients.
max ], ∇f (y (t)) is
Assumption 5 [38]: For yi (t) ∈ [ymin
i i
i , yi
bounded, that is, |∇fi (yi (t))| ≤ Bi .
Remark 3: Assumption 4 ensures a unique solution for
RAP (15). Since this article focuses on high-order dynamical
systems (14) to perform the resource allocation task (15), the
dynamics (14) only need to ensure compliance with constraints
after the system has stabilized [17].
To simplify the analysis of the problem with inequality
constraints, we use an -exact penalty function [23]
⎧


if hi yi (t) < 0
 ⎨ 0, 2 
 

ℵi hi yi (t) = ηi h i yi (t) /(2),  if 0 ≤ hi yi (t) ≤ 
⎩
ηi hi (yi (t)) − /2 , if hi yi (t) > 
(16)
where ηi is the penalty parameter and hi (yi (t)) = (ymin
−
i
−
y
(t))
+

with

>
0.
Then,
the
problem
(15)
yi (t)) (ymax
i
i
is reconstructed as
N
N
 


 




min
1
Y
y
(t)
=
min
1
i
m
m fi yi (t) + ℵi hi yi (t)
y(t)

s.t.

i=1
N

i=1

y(t)

yi (t) =

N


i=1

di .

(17)

i=1

Denote the gradient of Y(yi (t)) as ∇Y(yi (t))
=
 ∈ Rm and y(t) =
(t))]
[∇Y(y1i (t)), ∇Y(y2i (t)), . . . , ∇Y(ym
i
[y1 (t), y2 (t), . . . , yN (t)] ∈ RNm .
Remark 4: Based on Assumptions 4 and 5, the new objective function Y(yi (t)) is also -strongly convex. Furthermore,
there exists a constant Bi > 0 such that ∇Y(yi (t)) ≤ Bi .
III. D ISTRIBUTED F IXED -T IME R ESOURCE A LLOCATION
A LGORITHM AND C ONVERGENCE A NALYSIS
In this section, a resource allocation algorithm framework is
presented, which includes an FHESO for estimating unknown
states and coupled uncertainties, as well as an integrated controller for solving the problem and ensuring system stability. It
is then proven in Sections III-B that both the designed FHESO
and the integrated controller achieve fixed-time convergence.
A. Control Framework Design
An adaptive disturbance rejection algorithm for distributed
resource allocation is proposed for MASs in (14)
⎧




2− p1
y 
p2
⎪
−ci1 j∈Ni aij ∇ Y yi (t) − ∇ Y yj (t)
⎪
⎪
⎪




 p1
⎪
y 
⎪
p
⎪
⎪ −ci2 j∈Ni aij ∇ Y yi (t) − p∇ Y yj (t) 2 p
⎪
⎪
1
1


2−
⎪
e
e
p
⎪
−gi,1 
xi,1 (t) − ci1 ei (t) 2 − ci2 ei (t) p2
⎪
⎪
⎨ 1
t ∈ (0, T1 ]
x∗2 (t) = − 2 ei (t),
⎪
⎪
⎪
⎪




2− p1

⎪
⎪
p2
⎪ −cyi1 j∈Ni aij ∇ Y yi (t) − ∇ Y yj (t)
⎪
⎪
⎪




 p1

⎪
y
⎪
⎪ −ci2 j∈Ni aij ∇ Y yi (t) − ∇ Y yj (t) p2
⎪


⎩
−gi,1 
xi,1 (t) ,
t ∈ (T1 , T2 ]



 3
2− p1
p2
x∗l+2 (t) = −gi,l+1 
xi,l+1 (t) − si,l (t) − csil1 si,l (t)
2

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2824

IEEE TRANSACTIONS ON CYBERNETICS, VOL. 55, NO. 6, JUNE 2025


 p1
−csil2 si,l (t) p2 + ρ i,l,2 (t),
 t
 
2− p1
y
p2
ei (t) =
ci1
aij ∇ Y (yi (x)) − ∇ Y (yj (x))
+ yi (t)
0

j∈Ni
y

+ci2




 p1
aij ∇ Y (yi (x)) − ∇ Y (yj (x)) p2 dx − di

j∈Ni



ui (t) = −gi,n 
xi,n (t) − 
xi,n+1 (t) − si,n−1 (t) + ρ i,n−1,2 (t)


2− p1
 p1
s
p2
− csi(n−1)2 si,n−1 (t) p2
−ci(n−1)1 si,n−1 (t)

(18)
y

y

where l = 1, . . . , n − 2, p1 , p2 , ci1 , ci2 , cei1 , cei2 , csil1 , and csil2
are positive design parameters, T1 and T2 are time constants
to be defined, and the gradient differences exchanged between
agents are used to find the optimal resource allocation. Both
p1 and p2 are positive odd numbers with p1 < p2 . The term

xi,l (t) = [
xi,1 (t),
xi,2 (t), . . . ,
xi,l (t)] represents the unknown
observation of the state by the FHESO (19). x∗l+1 (t) is the
virtual control law, and ui (t) is the control input. The term
ei (t) represents the error related to the equality constraints.
Remark 5: Unlike the approaches in [18], [19], and [20],
which use coordinators or signal generators to estimate the
optimal signal, the virtual control law x∗2 (t) in the controller (18) directly searches for the optimal solution while
ensuring system stability by leveraging communication topology and local gradient information.
To observe the internal information of the system and
mitigate the impact of disturbances on solving the system’s
optimization problem, inspired by [24] and [25], we designed
an FHESO


⎧
x˙ i,1 (t) = 
xi,2 (t) + πi1 Παi1 + ri1 Πβi1 + gi,1 yi (t) 
⎪
⎪
⎪
⎪
xi,3 (t) + πi2 Παi2 + ri2 Πβi2 + gi,2 
xi,2 (t)
x˙ i,2 (t) = 
⎪
⎪
⎪
⎨ ..
.
(19)
⎪
˙ i,n (t) = 
Παin + rin Πβin
xi,n+1 (t)
+ πin

x
⎪
⎪


⎪
⎪
⎪
+gi,n 
xi,n (t) + ui (t)
⎪
⎩˙

xi,n+1 (t) = πi(n+1) Παi(n+1) + ri(n+1) Πβi(n+1)
where Π = yi (t) − 
xi,1 (t), αi = (1/βi ) > 1, βi ∈
([n/n + 1], 1), αij = αi + (j − 1)(βi − 1), and βij = jβi − (j − 1)
with j = 1, 2, . . . , n+1. The term 
xi,j (t) represents the estimate
xi,n+1 (t) estimates qi (xi (t), ωi (t)). The
of the state xi,j (t), while 
constants πij and rij are chosen so that the following matrix
is Hurwitz:
⎤
⎤
⎡
⎡
−πi1 1 · · · 0
−ri1 1 · · · 0
⎢
⎢
..
.. . . .. ⎥
..
.. . . .. ⎥
⎢
⎢
.
. . .⎥
.
. . .⎥
πi = ⎢
⎥ri = ⎢
⎥. (20)
⎣ −πin 0 · · · 1⎦
⎣ −rin 0 · · · 1⎦
−πi(n+1) 0 · · · 0
−ri(n+1) 0 · · · 0
Remark 6: The FHESO designed in this article (19) enables
fixed-time approximation of coupled disturbances and precise
observation of the internal states in high-order systems, relying
solely on the system’s inputs and outputs. Its uniqueness
lies in its tailored design for high-order systems, while its
versatility ensures effective applicability to first- and secondorder systems as well. Notably, the FHESO is capable of
addressing state-coupled disturbances, making it a robust
tool. Consequently, the proposed FHESO not only offers an
effective solution to resource allocation problems in high-order

systems but also demonstrates significant potential for broader
engineering applications.
Theorem 1: Assumptions 1–5 hold, there exist constants
αij∗ > 1, βij∗ ∈ (0, 1), π ∗i > 0, and r∗i > 0 such that the
observation errors in the FHESO (19) converge within a small
neighborhood in fixed time T . In the controller (18), positive
y∗ y∗
e∗ s∗
s∗
constants ci1 , ci2 , ce∗
i1 , ci2 , cil1 , and cil2 ensure that the system
output meets the RAP (15) equality constraints within time T1
and reaches the optimal solution within time T2 . Additionally,
all signals in the closed-loop system (14) maintain PFTS. The
times T , T1 , and T2 are defined as follows:

1
1
+
T ≤ max
i∈V ζi1 (1 − k1 )
ζi2 (k2 − 1)(1 − ν1 − ν2 )
1
4p2
1
+
T1 ≤ T +
τ (p2 − p1 )Δ1 ι1
ι2
1
4p2
1
T2 ≤ T1 +
(21)
+
τ (p2 − p1 )Δ2 κ1
κ2
 n−1 

where Δ = (1/2) N
ς ς ik2∗ , Δ2 = (1/2) N
i=1
i=1
n−1 1
N k=1  ik2∗
e
s
2
i=1 (Bi ) , ι1 = min{2ci1 , 2cij1 },
k=1 ς ik2∗ ς ik2∗ + m2
y
ι2 = min{2cei2 , 2csij2 }, κ1 = min{2(2p2 −p1 /p2 ) ci1 (mN)(p2 −p1 /2p2 )
y
(λ2 (La ))[3p2 −p1 /2p2 ] , 2csij1 }, κ2
=
min{2[p1 /p2 ] ci2 (λ2
(Lb ))[p2 +p1 /2p2 ] , 2csij2 } and j = 1, . . . , n − 1.
Remark 7: In PFTS scenarios, the difficulty in accurately
determining settling time makes it challenging to switch
control inputs based on time, as noted in [39]. The two critical
time points T1 and T2 in this study are theoretical constructs.
Analysis of (18) shows that the difference in x∗2 (t) at times T1
and T2 is solely due to ei (t). Subsequent convergence analysis
indicates that there exists a theoretical time T1 after which
ei (t) converges to zero, eliminating the need for compensation
of x∗2 (t) at T2 .
B. Stability Analysis
Proof of Theorem 1: To demonstrate the effectiveness of
the control framework designed in this article, we divide the
proof of Theorem 1 into two main parts. Under the assumption
of bounded initial states, Part 1 proves that the observer
error of the FHESO can converge to a small neighborhood
within a fixed time, ensuring the availability of signals in each
channel. Then, in Part 2, we demonstrate the boundedness and
convergence of all signals within the control framework to
ensure that the output signals of the entire closed-loop system
achieve the optimal solution of the RAP.
Suppose there exists a compact set Ai ⊂ Rn such that the
initial state of the system (14) satisfies: {xi,j (0)} ⊂ Ai .
xi,j (t) for j = 1, . . . , n, and
Part 1: Let ξ i,j (t) = xi,j (t) − 
xi,n+1 (t) as the observer
define ξ i,n+1 (t) = qi (xi,n (t), ωi (t)) − 
error. The closed-loop error system is


α
β
ξ̇ i,1 (t) = ξ i,2 (t) − πi1 ξ i,1 (t) i1 − ri1 ξ i,1 (t) i1


α
β
ξ̇ i,l (t) = ξ i,l+1 (t) − πil ξ i,1 (t) il − ril ξ i,1 (t) il + Gi,l (t)


α
β
ξ̇ i,n+1 (t) = −πi(n+1) ξ i,1 (t) i(n+1) − ri(n+1) ξ i,1 (t) i(n+1)
+i (t)
(22)
where l = 2, . . . , n, i (t) = [dqi (xi,n (t), ωi (t))/dt], and
xi,l (t)). The error vector is
Gi,l (t) = gi,l (xi,l (t)) − gi,l (

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AO et al.: DISTRIBUTED PRACTICAL FIXED-TIME RESOURCE ALLOCATION ALGORITHM

ξ i = [ξ i,1 (t) , ξ i,2 (t) , . . . , ξ i,n+1 (t) ] ∈ R(n+1)m , and the
system is described by the following vector field:
⎧

βi1
⎪
⎨ ξ̇ i,1 (t) = ξ i,2 − ri1 ξ i,1 (t) 
β
(23)
ξ̇ i = β (ξ i ) =
ξ̇ (t) = ξ i,l+1 − ril ξ i,1 (t) il
⎪ i,l
βi(n+1)

⎩
ξ̇ i,n+1 (t) = −ri(n+1) ξ i,1 (t)
.
According to Definition 3, (23) is homogeneous of degree βi −
1 < 0 with respect to i = (1, βi , 2βi − 1, . . . , nβi − n + 1).
We define the Lyapunov function as follows:
Vβ (ξ i ) =

1 
ε Pi ε i
2 i

(24)

(1/i ) , ξ  (1/βi1 i ) , . . . , ξ 
(1/βin i ) ]
where εi = [ξ 
i,1 
i,2
i,n+1 
with i = βi1 βi2 . . . βi(n+1) . The matrix Pi ∈ R(n+1)m×(n+1)m
is a positive-definite symmetric solution to the Lyapunov
equation M
i Pi + Pi M i = −Qi , with Qi being a positivedefinite symmetric matrix of size ((n + 1)m) × ((n + 1)m).
The matrix Mi is defined as follows:
⎡
⎤
−ri1 Im Im . . . 0
⎢
..
.. . . .. ⎥
⎢
.
. . .⎥
(25)
Mi = ⎢
⎥.
⎣ −rin Im 0 . . . Im ⎦
−ri(n+1) Im 0 . . . 0

The term L Vβ (ξ i ) denotes the Lie derivative of Vβ (ξ i ) along
the vector field β (ξ i ). The parameter ζi1 is defined as ζi1 =
− maxξi ∈[Vβ (ξ i )] L Vβ (ξ i ), and the following inequality holds:
L Vβ (ξ i ) ≤ −ζi1 Vβk1 (ξ i )

(26)

where k1 = 1 + (i (βi − 1)/2) < 1.
Similarly, we have the following vector field system:
⎧

αi1
⎨ ξ̇ i,1 (t) = −πi1 ξ i,1 (t)
α
(27)
ξ̇ i = α (ξ i ) = ξ̇ i,l (t) = −πil ξ i,1 (t) il
α
⎩
ξ̇ i,n+1 (t) = −πi(n+1) ξ i,1 (t) i(n+1)
where (27) is homogeneous of degree αi − 1 > 0 relative to
the weight vector i . A Lyapunov function is defined as
1
Pα,i ε α,i
(28)
Vα (ξ i ) = ε 
2 α,i
where ε α,i = ε i and Pα,i = Pi . The term L Vα (ξ i ) denotes the
Lie derivative of Vα (ξ i ) along the vector field α (ξ i ). Both
Vα (ξ i ) and L Vα (ξ i ) are homogeneous of degrees (2/i )
and [2/i ] + αi − 1, respectively, with respect to the weight
vector i . Applying Lemma 1, we obtain a positive parameter
ζi2 = − maxξ i ∈[Vα (ξ i )] L Vα (ξ i ), which leads to the following
inequality:
L Vα (ξ i ) ≤ −ζi2 Vαk2 (ξ i )

(29)

where k2 = 1 + [i (αi − 1)/2] > 1. Construct a Lyapunov
function as follows:
1
P,i ε ,i
(30)
V (ξ i ) = ε 
2 ,i
where ε,i = ε i and P,i = Pi . The first-order derivative of
V (ξ i ) with respect to the vector field in (22) is given by
V̇ (ξ i ) = L Vβ (ξ i ) + L Vα (ξ i )
∂V (ξ i )
∂V (ξ i )
+
+
i (t)
∂ξ i
∂ξ i,n+1

(31)

2825

where  = [0, Gi,2 (t) , Gi,3 (t) , . . . , Gi,n (t) ] .
Applying Lemmas 1 and 2 yields the following:
∂V (ξ i )
k
≤ ζi4 V3
∂ξ i
∂V (ξ i )
k4
k4
ζi5 V
≤
≤ ζi6 V
.
∂ξ i,n+1
k

ζi3 V3 ≤

(32)

According to Lemma 1, the parameters are given as k3 =
[i /2]([2/i ] − βi ) and k4 = [i /2]([2/i ] − 2βi + 1).
Additionally, we have ζi3 = minξ i : V (ξ i )=1 (∂V (ξ i )/∂ξ i ),
=
maxξ i :V (ξ i )=1 [∂V (ξ i )/∂ξ i ],
ζi5
=
ζi4
minξ i :V (ξ i )=1 [∂V (ξ i )/∂ξ i,n+1 ], and ζi6 = maxξ i :V (ξ i )=1
[∂V (ξ i )/∂ξ i,n+1 ]. Using (26), (29), and (32), (31) can be
rewritten as follows:
k

k1
k2
V̇ (ξ i ) ≤ −ζi1 V
(ξ i ) − ζi2 V
(ξ i ) + ζ1 V3 (ξ i ) 
k4
+ ζ2 V
(ξ i )χ

(33)

where ζ1 = max{|ζi3 |, |ζi4 |}, ζ2 = max{|ζi5 |, |ζi6 |}, and
χ = δK, derived from Assumption 1 and the condition
{xi,j (0)} ⊂ Ai .
Remark 8: Based on Conditions 1) and 2) of Assumption 1,
and given that ωi is bounded, it follows that a continuous
function Ψi (ωi ) has an upper bound . Therefore, we have
i (t) ≤ δK.
Then, (33) yields

 k2
k1
(ξ i ) − ζi2 1 − (ν1 + ν2 ) V
(ξ i )
V̇ (ξ i ) ≤ −ζi1 V


k3
k2 −k3
− V (ξ i ) ν1 ζi2 V (ξ i ) − ζ1 


k4
k2 −k4
− V
(ξ i ) ν2 ζi2 V
(ξ i ) − ζ2 χ
(34)
where ν1 and ν2 are defined as 0 < ν1 + ν2 < 1. To guarantee
V̇ (ξ i ) ≤ 0, the following inequalities must be satisfied:
!
k −k
ν1 ζi2 V2 3 (ξ i ) ≥ ζ1 
(35)
k2 −k4
ν2 ζi2 V
(ξ i ) ≥ ζ2 χ .
V (ξ i (0)) will converge to V  = min{V 1 , V 2 }
⎧
 1

⎪
⎨ V < V  = ζ1  k2 −k3
1
ν1 ζi2
 1

⎪
⎩ V < V = ζ2 χ k2 −k4 .

2
ν2 ζi2

(36)

Based on (34), (36), and Lemma 4, the settling time for V (ξ i )
to converge from V (ξ i (0)) to V  is determined as follows:
Ti ≤

1
1
+
.
ζi1 (1 − k1 ) ζi2 (k2 − 1)(1 − ν1 − ν2 )

(37)

From (36) and (37), it follows that for t > T , there exists
||ξi || ≤ (V /λmin (Pi ))1/2 . Based on Lemma 7, the upper
bound of the observer error can be derived as
n

 i (jβi2−j+1)  i (jβi −j+1)
2
ξ i,j+1
||ξ i || ≤ 1
m
j=0

≤
≤

n

j=0
n


m1−

i (jβi −j+1)
2

"" i (jβi1−j+1) ""i (jβi −j+1)
""ξ
""

m1−

i (jβi −j+1)
2

"" ""i (jβi −j+1)
""ε i ""
.

i,j+1

j=0

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(38)

2826

IEEE TRANSACTIONS ON CYBERNETICS, VOL. 55, NO. 6, JUNE 2025

From (38), the estimation error of the fixed-time observer
converges to the residual set Ωi , where
"
"
Ωi = ξ i "||ξ i || ≤

n


1−

m

i (jβi −j+1)
2

j=0



V 
λmin (Pi )

i (jβi −j+1)
2

#
.
(39)

Remark 9: Since the boundedness and convergence of the
ith FHESO have been established, a similar proof can be
applied to the remaining agents in the network G. It is
worth noting that the proposed algorithm framework operates synchronously across all agents in the network. As a
result, the convergence time of the proposed FHESO for
the entire network corresponds to the maximum fixed time
among the agents in the set V of the network G. According
to (37), the maximum convergence time of FHESO in the
agent network is given by: T ≤ maxi∈V {[1/ζi1 (1 − k1 )] +
[1/ζi2 (k2 − 1)(1 − ν1 − ν2 )]}. Furthermore, the maximum
error bound is Ω = maxi∈V {Ωi }, where Ωi is defined in (39).
Part 2: The convergence of the control framework will be
proven in two steps. First, we will demonstrate that the system
output satisfies the equality constraint of the RAP within time
T1 . Second, we will prove that the agents’ outputs achieve
optimal resource allocation within time T2 . This is crucial for
verifying the effectiveness of the proposed control scheme,
as it ensures the convergence of two key error signals in the
algorithm: 1) the global equality error and 2) the optimal
solution error. It is worth noting that due to the dynamic
characteristics of the system, the tracking error of the internal
system states must also be considered while proving the
convergence of these two key errors. To this end, we define
the tracking errors for each signal as follows:
xi,l+1 (t) − x∗i,l+1 (t)
si,l (t) = 

(40)

According to Lemma 5, we can derive
ei (t) si,1 (t) ≤

1
1
ei (t) ei (t) + si,1 (t) si,1 (t).
2
2

(42)

Substituting (42) into (41), we get
p
dV1 (t)  1
3− p1
2
≤
si,1 (t) si,1 (t) − cei1 1
m ei (t)
dt
2
i=1
p 
1+ p1
2 .
− cei2 1
m ei (t)

N

(43)

For l = 1, 2, . . . , n
− 2, we choose the Lyapunov function as

Vl+1 = Vl + (1/2) N
i=1 si,l (t) si,l (t). We get
dVl+1 (t)
dVl (t) 
si,l (t) si,l+1 (t) + x∗i,l+2
=
+
dt
dt
i=1

 dx∗i,l+1 (t) 
.
(44)
+ gi,l+1 
xi,l+1 (t) −
dt
N

To avoid “differential explosion” caused by high-order derivatives, a high-gain TD is used to track [dx∗i,l+1 (t)/dt] according
to Lemma 8, yielding
dx∗i,l+1 (t)
dt

= ρ i,l,2 (t) + ς i,l,2 (t)

(45)

where ς i,l,2 (t) represents the tracking error of the high-gain
TD and satisfies |ς i,l,2 (t)| ≤ ς il2∗ . Based on Lemma 5
and (45), we obtain
1
1
si,l (t) si,l (t) + si,l+1 (t) si,l+1 (t)
2
2
1
1
−si,l (t) ς i,l,2 (t) ≤ si,l (t) si,l (t) + ς 
ς .
(46)
2
2 il2∗ il2∗
si,l (t) si,l+1 (t) ≤

Combining (18) and (44)–(46), we obtain
dVl (t)  1
dVl+1 (t)
1
≤
+
si,l+1 (t) si,l+1 (t) + ς 
ς
dt
dt
2
2 il2∗ il2∗
i=1


3− p1
1+ p1 
s 
p2
p2
.
s
s
− csil1 1
(t)
−
c
1
(t)
i,l
i,l
m
il2 m
N

where x∗i,l+1 (t) denotes an intermediate virtual control variable
that will be designed later, with l = 1, 2, . . . , n − 1.
Step 1: Define the errors yi (t) and di as ei (t) in (18).
We 
choose a Lyapunov function as follows: V1 (t) =

1/2 N
i=1 ei (t) ei (t). When t < T1 , taking the derivative and
combining with (18), we obtain
dV1 (t)  dei (t)
=
ei (t)
dt
dt
N

=

i=1
N


y

+ ci1

N

i=1





si,1 (t) + x∗i,2 (t) + gi,1 yi (t)





aij ∇Y(yi (t)) − ∇Y(yj (t))

j∈Ni
y
+ ci2



=

ei (t)

p
2


1
− ei (t) ei (t) .
2

(48)

When l = n − 1, 
we choose the Lyapunov function as Vn (t) =

Vn−1 (t) + (1/2) N
i=1 si,n−1 (t) si,n−1 (t), we get

p

1+ p1


3− p1
p2
csik1 1
m si,k (t)

k=1

3− p1
2
si,1 (t) − cei1 1
m ei (t)

− cei2 1
m ei (t)

n−2 

k=1


 p1 
aij ∇Y(yi (t)) − ∇Y(yj (t)) p2

i=1

−

n−2

1+ p1  1 

p2
+ csik2 1
(t)
ς
+
s
i,k
m
ik2∗ ς ik2∗ .
2

p
2− p1
2



p
2

1+ p1

− cei2 1
m ei (t)

j∈Ni
N


From the above derivation, we obtain
p
 1
dVn−1 (t)
3− p1
2
≤
si,n−1 (t) si,n−1 (t) − cei1 1
m ei (t)
dt
2



ei (t)

i=1

(47)

(41)

N


dVn−1 (t) 
dVn (t)
=
+
si,n−1 (t) ui (t) + gi,n 
xi,n (t)
dt
dt
i=1
dx∗i,n (t) 
.
(49)
+
xi,n+1 (t) −
dt

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AO et al.: DISTRIBUTED PRACTICAL FIXED-TIME RESOURCE ALLOCATION ALGORITHM

Similar to (45) and in conjunction with Lemma 5, By
combining (18), (48), and (49), we have
p
p
dVn (t) 
3− p1
1+ p1
2 − ce 1 ei (t)
2
≤
− cei1 1
m ei (t)
i2 m
dt

−

p
3− p1
2

csik1 1
m si,k (t)

−

k=1
n−1


1
+
2

p −p

≤−
p
1+ p1
2

+ csik2 1
m si,k (t)





ς
ik2∗ ς ik2∗ .

(50)

k=1

Similar to the derivation approach in [11, eqs. (15) and (16)],
we obtain (50) in this article
N


 3p2 −p1
 p2 +p1
dVn (t) 
≤
− cei1 ei (t) ei (t) 2p2 − cei2 ei (t) ei (t) 2p2
dt
i=1
n−1 



 3p2 −p1
csik1 si,k (t) si,k (t) 2p2

−

n−1


 p2 +p1  1 
+ csik2 si,k (t) si,k (t) 2p2 +
ς
ς
ik2∗ ik2∗
2

≤ −ι1 Vn (t)

− ι2 Vn (t)

+ Δ1

(51)

where ([3p2 − p1 ]/2p2 ) > 1, (1/2) < ([p2 + p1 ]/2p2 ) <
1, ι1 = min{2cei1 , 2csij1 }, ι2 = min{2cei2 , 2csij2 }, and Δ1 =
 n−1 
(1/2) N
i=1
k=1 ς ik2∗ ς ik2∗ with j = 1, 2, . . . , n−1. Through
Lemma 4, a fixed-time T1 exists such that the MASs’ output
meets the optimization problem’s equality constraints. The
upper bound of T1 is
1
4p2
1
T1 ≤ T1,max =
+
.
(52)
τ (p2 − p1 )Δ1 ι1
ι2

dt

=

N


(2p /[p +p ])

aij 2 2 1 ,
respectively,
where
∇Y(y(t))
=

N×m
∈ R
. Since
[∇Y(y1 (t)), ∇Y(y2 (t)), . . . , ∇Y(yN (t))]
Y(yi (t)) is -strongly convex, for yi (t), it holds that
∗
∗
≥ 1
1
N ∇Y(y(t))(y − y(t)) +
N (Y(y ) − Y(y(t)))1m

∗
2
∗
(/2)1m (y − y(t)) . By Lemma 6, we have: 1
N (Y(y ) −

2
Y(y(t)))1m ≥ −(1/2)1N (∇Y(y(t))) 1m . This yields


∗

2
21
(56)
N Y(y(t)) − Y(y ) 1m ≤ 1N (∇Y(y(t))) 1m .

y

− ci2



aij Υ  p2

p2 −p1

i=1

− Y(y∗i )
− Y(y∗i )
≤

(53)

y

i=1

=−

1
2

p
2

2− p1

aij Υ 

y

+ ci2

j∈Ni

N


1
m ci1

i=1
p −p

≤−



2 1
y
ci1 N 2p2

2

y



p
2

y

+ ci2

j∈Ni
N 

i=1 j∈Ni



aij

p1

aij Υ  p2

Υ Υ



3p2 −p1
2p2

p
2

1+ p1

aij Υ 

j∈Ni
2p2
3p2 −p1

(54)

j∈Ni

3− p1

aij Υ 



% p2 +p1 

−2

p1

y $
− 2 p2 ci2 λ2 (Lb )1
m Y(yi (t))

2p2

2p2 −p1
p2

p2 −p1

3p2 −p1

ci1 (mN) 2p2 (λ2 (La )) 2p2 1
m
y

Then

m
1
∇Y(yi (t)) si,1 (t) ≤ (Bi )2 + si,1 (t) si,1 (t).
2
2
The last two terms of (53) are


∇Y(yi (t)) ci1

2p2

i=1

by Assumption 5 and Lemma 5, we have

N


N


% 3p2 −p1

p1
p2 +p1

 3p2 −p1
y
Y(yi (t)) − Y(y∗i ) 2p2 − 2 p2 ci2 (λ2 (Lb )) 2p2

 p2 +p1 
∗
2p2
.
(57)
1
m Y(yi (t)) − Y(yi )

j∈Ni

−

3p −p

y
2 −p1
  3p2p

c N 2p2
∗
2
4λ2 (La )1
≤ − i1
)
1
Y(y(t))
−
Y(y
m
N
2
y
  p22p+p2 1

c
∗
− i2 4λ2 (Lb )1
)
1m
Y(y(t))
−
Y(y
N
2
N
2p2 −p1
p2 −p1 $


y
− 2 p2 ci1 N 2p2 λ2 (La )1
≤
m Y(yi (t))

j∈Ni

p1 

p2 −p1

 2 1
c N 2p2
2p2
2
(55) ≤ − i1
2λ2 (La )1
(∇Y(y(t)))
1
m
N
2
y
 p2 +p1
ci2
2p2

2
2λ2 (Lb )1N (∇Y(y(t))) 1m
−
2

p

2− 1
y
∇Y(yi (t)) si,1 (t) − ci1
aij Υ  p2

i=1

(55)

where La and Lb have the same structure as
(2p /[3p −p ])
L, with aij in L replaced by aij 2 2 1
and

Step 2: Let y∗i ∈ Rm be the optimal solution of (17)
for agent
define the following Lyapunov function:
 i. We
 (Y(y (t)) − Y(y∗ (t))). Let Υ = ∇Y(y (t)) −
1
V1 (t) = N
i
i
i=1 m
i
∇Y(yj (t)) ∈ Rm , we have
dV1 (t)


 3p2 −p1
2p2
2tr ∇Y(y(t)) (La ⊗ Im )∇Y(y(t))

2
y

 p2 +p1
ci2
2p2
2tr ∇Y(y(t)) (Lb ⊗ Im )∇Y(y(t))
−
2

y

k=1

p2 +p1
2p2

2 1
y
ci1 N 2p2

According to Lemma 3, due to λ2 (La ) = λ2 (La ⊗ Im ) and
λ2 (Lb ) = λ2 (Lb ⊗ Im ), we know that

k=1

3p2 −p1
2p2

p +p
y
N
2p
ci2   p2 +p2 1   22p2 1
aij
Υ Υ
2

i=1 j∈Ni

N

i=1
n−1 


2827





dV1 (t)
dt
N
2p2 −p1
p2 −p1

m  2 1
y
(Bi ) + si,1 (t) si,1 (t) − 2 p2 ci1 (mN) 2p2
≤
2
2
i=1

3p2 −p1

 3p2 −p1
∗
2p2
(λ2 (La )) 2p2 1
m Y(yi (t)) − Y(yi )
p1
p2 +p1

 p2 +p1 
y
∗
2p2
. (58)
− 2 p2 ci2 (λ2 (Lb )) 2p2 1
m Y(yi (t)) − Y(yi )

Next, we will discuss the process for l = 1, 2, . . . , n − 2. The
relevant proof process is similar to (44)–(50) and will not be
repeated. Combining (55), when l = n − 1, we obtain

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2828

Fig. 2.

IEEE TRANSACTIONS ON CYBERNETICS, VOL. 55, NO. 6, JUNE 2025

Communication topology for Example 1.

Fig. 3.

Communication topology for Example 2.
TABLE I
PARAMETERS OF G ENERATORS

dVn (t)
dt
N
2p2 −p1
p2 −p1
3p2 −p1


y
≤
− 2 p2 ci1 (mN) 2p2 (λ2 (La )) 2p2 1
m Y(yi (t))
i=1

− Y(y∗i )
− Y(y∗i )

 3p2 −p1

p1
p2 +p1

y
− 2 p2 ci2 (λ2 (Lb )) 2p2 1
m Y(yi (t))

2p2

 p2 +p1
2p2

−

n−1

 3p2 −p1
$ s 
cik1 1m si,k (t) p2
k=1

n−1
 p2 +p1 % 1 

m  2

p2
(B ) . (59)
+ csik2 1
(t)
ς
ς
+
s
+
m i,k
ik2∗ ik2∗
2
2 i
k=1

min
P(t)

Then, we obtain
dVn (t)

topology of the generators, depicted in Fig. 2, has communication weights set to aij = 1. The optimization problem is
simplified as follows [11]:

3p2 −p1
2p2

p2 +p1
2p2

≤ −κ1 Vn (t)
− κ2 Vn (t)
+ Δ2 (60)
dt
where ([3p2 − p1 ]/2p2 ) > 1, (1/2) < ([p2 + p1 ]/2p2 ) <
y
=
min{2(2p2 −p1 /p2 ) ci1
(mN)[(p2 −p1 )/2p2 ]
1,
κ1
y
s
([3p
−p
]/2p
)
2
1
2
(λ2 (La ))
, 2cij1 }, and κ2 = min{2[p1 /p2 ] ci2
s
[(p
+p
)/2p
]
(λ2 (Lb )) 2 1 2 , 2cij2 }, with j = 1, 2, . . . , n − 1, and
 n−1 
N
 2
Δ2 = (1/2) N
i=1
i=1 (Bi ) . By
k=1 ς ik2∗ ς ik2∗ + (m/2)
Lemma 4, within the fixed time T2 , the output of the MAS will
converge to a small neighborhood of the optimal solution of
the RAP, where T2 is given by
1
4p2
1
+
(61)
T2 ≤ T1 +
τ (p2 − p1 )Δ2 κ1
κ2
where τ ∈ (0, 1). The residual set of the solution of the
system is
(
)1
&
'
Δ1 α
1

lim max |Vn (t)|, |Vn (t)| ≤ min
t→T2
ι1 1 − τ
(
(
(
)1
)1
) 1 ##
Δ1 β 1
Δ2 α 1
Δ2 β
1
.
(62)
,
,
ι2 1 − τ
κ1 1 − τ
κ2 1 − τ
From (62), we know that Y(yi (t)) − Y(y∗i ), si,k (t), and ei (t)
are all bounded. Consequently, the error between the agents’
outputs and the optimal solution to the distributed RAP will
converge to an arbitrarily small neighborhood around the
origin within a fixed time.
IV. S IMULATION R ESULTS
In this section, we present two numerical examples to
demonstrate the effectiveness of the proposed framework.
Example 1: We consider a smart grid consisting of four
generators that supply power to specific areas. The network

4


1
2 fi ( Pi (t)) =

i=1

s.t.

4


$
%
2
1
2 bi,1 + bi,2 Pi (t) + bi,3 (Pi (t))

i=1
min
Pi ≤ Pi (t) ≤ Pmax
4
4i
P
(t)
=
i
i=1
i=1 di .

(63)

Let P(t) = col(P1 (t), . . . , P4 (t)) (p.u.), where Pi (t) ∈ R2
denotes the output power of the ith generator. The parameter
di ∈ R2 represents the power demand in the ith region,
 and
the total power demand across all regions is given by 4i=1 di
(p.u.). The minimum and maximum generating capacities of
and Pmax
, respectively.
the ith generator are denoted by Pmin
i
i
2
2
The function fi (Pi (t)):R → R (p.u.) represents the cost of
the ith generator, which is typically modeled as a quadratic
function, as described in [40]. Table I provides the relevant
parameters. The multiagent model analyzed in this article is
given by
⎧


⎨ ẋi,1 (t) = xi,2 (t) + gi,1
 xi,1 (t)



ẋi,2 (t) = ui (t) + gi,2 xi,2 (t) + qi xi,2 (t), ωi (t)
(64)
⎩
Pi (t) = xi,1 (t)
where qi (xi (t), ωi (t)) is the unknown coupling term, ωi (t) =
1 (t), x2 (t)]
[ωi1 (t), ωi2 (t)] is the disturbance, xi,1 (t) = [xi,1
i,1
1
2
2

2
∈ R and xi,2 (t) = [xi,2 (t), xi,2 (t)] ∈ R represent the system
states, Pi (t) is the system output, and ui (t) ∈ R2 is the control
input. The functions gi,1 (xi,1 (t)) and gi,2 (xi,2 (t)) are nonlinear
functions defined as follows:

 $ 1
%
1
2
2
(t) + cos(x1,1
(t)); x1,1
(t) + sin(x1,1
(t))
g1,1 x1,1 (t) = x1,1

 $ 1
%
1
1
2
2
g2,1 x2,1 (t) = x2,1
(t) + sin(2x2,1
); x2,1
(t) + cos2 (x2,1
(t))
2

 $
%
1
2
(t)); cos(x3,1
(t))
g3,1 x3,1 (t) = sin(x3,1

 $
%
1
1
2
(t)) cos2 (x4,1
(t)); sin(x4,1
(t))
g4,1 x4,1 (t) = sin(x4,1

 $
%
1
1
2
2
(t)) sin(x1,2
(t)); x1,1
(t) cos(x1,2
(t))
g1,2 x1,2 (t) = cos(x1,1

 $
%
1
1
2
2
(t)x2,2
(t)); sin(x2,2
(t) − 2x2,1
(t))
g2,2 x2,2 (t) = sin(x2,1

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AO et al.: DISTRIBUTED PRACTICAL FIXED-TIME RESOURCE ALLOCATION ALGORITHM

2829

Fig. 4.

Trajectory of Pi (t).

Fig. 6.

Trajectory of total mismatch.

Fig. 5.

Trajectory of ei (t).

Fig. 7.

Trajectory of four agents.

Fig. 8.

Trajectory of xi (t) − x̂i (t).

Fig. 9.

Trajectory of qi (t) − x̂i (t).


 $
%
1
1
2
2
g3,2 x3,2 (t) = cos(x3,1
(t))x3,2
(t); cos(x3,1
(t)) + sin(x3,2
(t))
4
4


5
5
1
1
2
2
(t) + x4,2
(t) ; x4,1
(t) + x4,2
(t)
g4,2 x4,2 (t) = x4,2
(65)
and the function qi (xi,2 (t), ωi (t)) is represented as follows:

1 (t)) + ω1 (0.02t) + 2


sin(0.01x1,1
1
q1 x1,2 (t), ω1 (t) =
2 (t)) + ω2 (t) + 2
cos(0.5x2,1
1

1 (t))ω1 (0.1t) + 3


cos(0.1x2,1
2
q2 x2,2 (t), ω2 (t) =
2 (t))ω2 (t) + 1
cos(0.25x2,1
2

1 (t)) + ω1 (t) + 2 


sin(0.1x3,2
3
q3 x3,2 (t), ω3 (t) =
2 (t))ω2 (t) + 2
0.5 cos(0.05x3,1
3

1 (t) + ω1 (t))) + 3


sin(0.05(x4,1
4
q4 x4,2 (t), ω4 (t) =
(66)
2 (t) + ω2 (t))) + 2
cos(0.1(x4,1
4
where ωi (t) is defined as follows:
ω1 (t) = [0.5 cos(t); cos(0.05t)]ω2 (t) = [sin(t); sin(0.05t)]
ω3 (t) = [cos(0.01t); cos(0.25t)]ω4 (t) = [t; t].
(67)
In the simulation, the proposed distributed fixed-time output
feedback allocation framework in (18) is employed to address
the optimal problem in (63). The parameters are configured
y
y

as follows: p1 = [9, 7] , p2 = [11, 9]
 , c1 = c2 = 12×4 ,
41×4
. The
ce1 = ce2 = 12×4 , and cs11 = cs12 =
5 4.5 4.4 4
penalty
function
 parameters from

 (16) are defined as η2×4 =
11×4
0.021×4
and 2×4 =
. The
1 4 3.5 1
0.1 0.04 0.02 0.1
parameters for the FHESO are set as α1 = 1.22×4 , α2 =
1.42×4 , α3 = 1.62×4 , β1 = 0.82×4 , β2 = 0.62×4 , β3 = 0.42×4 ,
π i1 = ri1 = 52×4 , π i2 = ri2 = 152×4 , and π i3 = ri3 =
102×4 . Finally, the high-gain TD parameters are specified as
ζ = 12×4 , 1 = 0.012×4 , and 2 = 0.12×4 .
Figs. 4 and 5 show the simulation results for solving the
optimization problem (63) of the system (64). Specifically,

Fig. 4 displays the output power of the ith generator in
the 2-D case, while Fig. 5 illustrates the trajectory of the
signal ei (t), which is used to ensure the error variables of
the equality constraints in the optimization problem. The
simulation clearly indicates that, at time t ≥ 4.617 s,
the output power of the ith generator converges to the
optimal solution of the optimization problem, and the signal
ei (t) simultaneously converges to a neighborhood of zero.
Additionally, it can be noted that the global equality constraint
is not satisfied initially, but only after the system stabilizes,
which is due to the impact of internal coupling disturbances and tracking errors, consistent with the discussion in
Remark 3.
Figs. 6 and 7 depict the total mismatch and decision trajectories of the four agents, demonstrating that the final decisions
comply with the network resource constraints and confirming

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2830

IEEE TRANSACTIONS ON CYBERNETICS, VOL. 55, NO. 6, JUNE 2025

Fig. 10.

Trajectories of output powers.

Fig. 12.

Total output power.

Fig. 11.

Trajectories of Pi (t) − P∗i .

Fig. 13.

Trajectories of ∇fi (Pi (t)).

Fig. 14.

Trajectories of P̂i,5 (t) − qi (P(t), ω(t)).

the effectiveness of the proposed framework. Figs. 8 and 9
highlight the extended observer’s capability to accurately
estimate system states and approximate unknown nonlinear
disturbances. The simulation results show that FHESO can
detect the system’s internal states and external disturbances
within a fixed time, with the error remaining consistently
within the neighborhood of zero. This is consistent with the
results derived in Part 1.
Example 2: This example examines the constrained economic dispatch problem in smart grids using the IEEE-118-bus
system [17], which includes 54 boiler–turbine generation units.
For the ith generation unit, when the governor valve position
remains constant, its dynamics are modeled by a fourth(4)
order equation Pi,1 (t) = ui (t) + qi (P(t), ω(t)), where Pi (t) =
Pi,1 (t). Here, qi (P(t), ω(t)) represents the coupling disturbance
associated with the system’s internal states. Moreover, these
54 generation units are tasked with solving an RAP, and
their generation costs are approximated using the following
quadratic functions [40]:
min

Pi (t)∈R

s.t.

54

i=1

fi (Pi (t)) =

54


ai + bi Pi (t) + ci (Pi (t))2

i=1

54
i=1 Pi (t) =
i=1 di
max , i ∈ {1, 2, . . . , 54}
Pmin
≤
P
(t)
≤
P
i
i
i
54

(68)

where fi : R → R and Pi (t) ∈ R denote the cost function and
power output of the ith generator, respectively. The parameter
and Pmax
di represents the local load demand, while Pmin
i
i
specify the lower and upper bounds of the ith generator’s
power output. Additionally, ai , bi , and ci are characteristic
parameters of the ith generator system.
Following the parameter configuration in [17], these parameters are randomly chosen within the specified intervals: ai ∈
[6.78, 74.33] (M$), bi ∈ [8.3391, 37.6968] (M$/MW), and
ci ∈ [0.24, 0.679] (M$/MW2 ). Similarly, the power output lim∈ [0, 20] (MW) and Pmax
∈ [70, 110]
its are defined as Pmin
i
i
(MW). The demand di is randomly selected from [50, 60]
(MW) when t ∈ [0, 150) s and from [40, 50] (MW) when

t ∈ [150, 300] s. The initial power outputs Pi (0) are randomly
assigned within [40, 50] (MW). The communication topology,
as shown in Fig. 3, forms a ring structure among the generators, with additional edges {1, 27}, {2, 28}, {3, 29}, {10, 27}.
The coupling disturbance is expressed as qi (P(t), ω(t)) =
cos(e−Pi,1 (t) ω(t)) + Pi,2 (t) + 2, where the external disturbance
is ω(t) = 0.02t. The system uses the following parameters:
y
y
(p1 /p2 ) = (99/101), control gains ci1 = ci2 = 2, cei1 = cei2 =
s
s
s
s
s
1, ci11 = ci12 = 5, ci21 = ci22 = 20, and ci31 = csi32 = 70.
The simulation results are shown in Figs. 10–14.
Specifically, Fig. 10 illustrates the evolution of the output
powers of the 54 generators. Fig. 11 shows the evolution
of Pi (t) − P∗i , where it can be observed that the outputs of
all generators converge to the optimal solution. Even with
changes in demand, the algorithm ensures that the output
reaches the optimal solution.
Fig. 12 demonstrates that the total output power can meet
the total load demand. Fig. 13 presents the gradient evolution,
where it can be seen that, under two different demand scenarios, the proposed algorithm guarantees the consistency of each
local objective function gradient, which corresponds to the
gradient consistency term designed in our algorithm in (18).
Additionally, in Fig. 14, we demonstrate the approximation

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AO et al.: DISTRIBUTED PRACTICAL FIXED-TIME RESOURCE ALLOCATION ALGORITHM

capability of the FHESO (19) for the fourth-order boiler–
turbine generation system with coupling disturbances. When
local demand changes, the corresponding power output adjusts
accordingly, leading to fluctuations in the coupling disturbance
as the system stabilizes, particularly around 150 s. Despite
these fluctuations, the proposed FHESO effectively maintains
system stability during such disturbance variations.
V. C ONCLUSION
This article has proposed a novel integrated framework
to address the fixed-time RAP for MIMO nonlinear MASs
under disturbances. The framework ensures agents achieve
the desired formation shape while meeting global equality
and local inequality constraints, minimizing the global cost.
It primarily comprises an FHESO and a time-switching
controller, where the FHESO estimates the agents’ internal
states and coupled disturbance information, which the timeswitching controller then uses to guide each agent toward
the optimal RAP solution. Theoretical analysis has confirmed
that the framework not only guarantees PFTS for each
agent but also provides definitive upper bounds for all error
signals. The effectiveness of the proposed framework has
been demonstrated through two detailed simulations. Future
research could further extend these findings to unbalanced
time-varying switching topologies and time-varying resource
allocation problems.
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Qingxiang Ao received the B.S. degree in traffic
engineering from Jiangxi University of Finance
and Economics, Nanchang, China, in 2023. He is
currently pursuing the M.S. degree in traffic and
transportation engineering with Shanghai University
of Engineering Science, Shanghai, China.
His research interests include distributed
optimization, adaptive control, and nonlinear
systems control.

Cheng Li received the B.S. and M.S. degrees
from the East China University of Science and
Technology, Shanghai, China, in 2002 and 2005,
respectively, and the Ph.D. degree from Donghua
University, Shanghai, in 2015.
He is a Professor and the Dean of the College
of Air Transportation (College of Flying), Shanghai
University of Engineering Science, Shanghai. His
research interests include algorithm optimization,
transportation planning, and forecasting methods.

Ben Niu received the Ph.D. degree in control
theory and control engineering from Northeastern
University, Shenyang, China, in 2013.
He is currently a Professor with the School of
Control Science and Engineering, Dalian University
of Technology, Dalian, China. His research interests
include switched systems, stochastic systems,
adaptive control, intelligent control, and their
applications.

Zhiliang Zhao received the B.S. degree in mathematics from Shaanxi Normal University, Xi’an,
China, in 2003, the M.S. degree in mathematics from
the Huazhong University of Science and Technology,
Wuhan, China, in 2007, and the Ph.D. degree in
mathematics from the University of Science and
Technology, Hefei, China, in 2012.
He is currently a Professor with the School
of Mathematics and Statistics, Shaanxi Normal
University. His research interests include stability
theory, nonlinear systems and control, and active
disturbance rejection control.

Jiaxin Yuan received the B.S. degree from
Zaozhuang University, Zaozhuang, China, the M.S.
degree from Xi’an Jiaotong University, Xi’an, China,
in 2013, and the Ph.D. degree from Shanghai Jiao
Tong University, Shanghai, China, in 2018.
He is currently an Associate Professor with
the College of Air Transportation, Shanghai
University of Engineering Science, Shanghai.
His research interests include adaptive control,
distributed optimization, and UAV formation.

Sen Chen received the B.S. degree in mathematics
from Beihang University, Beijing, China, in 2014,
and the Ph.D. degree in operational research and
cybernetics from the Academy of Mathematics and
Systems Science, Chinese Academy of Sciences,
Beijing, in 2019.
Since 2019, he has been an Assistant Professor
with the School of Mathematics and Statistics,
Shaanxi Normal University, Xi’an, China. His
research interests include active disturbance
rejection control and reinforcement learning.

Xiaole Yang received the B.S. degree from Jiangsu
Normal University, Xuzhou, China, in 2021, and
the M.S. degree from Shanghai University of
Engineering Science, Shanghai, China, in 2024.
He is currently pursuing the Ph.D. degree in
control science and engineering with the College
of Information Science and Technology, Beijing
University of Chemical Technology, Beijing, China.
His current research interests include distributed
optimization, multiagent systems, pursuit–evasion
games, and reinforcement learning.

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