644

IEEE SYSTEMS JOURNAL, VOL. 18, NO. 1, MARCH 2024

Nonsmooth Distributed Resource Allocation Over
Second-Order Nonlinear Multiagent Systems
Xiasheng Shi , Member, IEEE, Yanxu Su , Member, IEEE, Chaoxu Mu , Senior Member, IEEE,
and Changyin Sun , Senior Member, IEEE

Abstract—The rapid development of cyber-physical systems
technique has promoted the development of autonomous systems.
To perform the distributed task autonomously, this study delves
into the problem of distributed resource allocation over secondorder multiple autonomous physical systems. It aims to minimize
the global cost function while allocating plenty of resources to finite
distributed second-order agents. Since exogenous disturbances in
the control channel are unavoidable, and the nonlinear dynamical system can more accurately describe the physical behavior
of the system, i.e., Euler–Lagrange systems. The nonlinear term
in the control channel is considered, including the exogenous disturbances and nonlinear dynamics. Unlike most existing results,
the nonlinear term is assumed to be bounded by a known constant. To this end, the bounded nonlinear term is handled using a
sign function-based discontinuous nonlinear control scheme. Based
on this, a distributed method with an adjustable dual Lagrange
multiplier is designed using the modified primal-dual scheme for
achieving the optimal allocation result. In addition, its convergence
proof is completed in view of the fixed-time stability and the setvalued LaSalle invariance principle. With our designed method, the
agent state asymptotically converges to the exact solution for the
undirected and weighted-balanced directed connected networks.
Last, we adopt some case studies to certify the effectuality of the
developed analytical outcomes.
Index Terms—Adaptive method, distributed resource allocation,
nonlinear systems, nonsmooth cost function, sliding mode control
(SMC).

I. INTRODUCTION
ECENTLY, the research on distributed resource allocation
problems over multiagent systems has garnered significant
consideration on account of its comprehensive uses in various
engineering fields, including the economic dispatch in smart
grids, the channel resource scheduling and allocation issues

R

Manuscript received 31 May 2023; revised 7 October 2023 and 30 November
2023; accepted 14 January 2024. Date of publication 5 February 2024; date
of current version 15 March 2024. This work was supported by the National
Natural Science Foundation of China under Grant 61921004, Grant 62236002,
Grant 62203001, and Grant 62303009. (Corresponding author: Chaoxu Mu.)
Xiasheng Shi, Yanxu Su, and Changyin Sun are with the School of Artificial Intelligence, Anhui University, Hefei 230039, China, also with the
Engineering Research Center of Autonomous Unmanned System Technology,
Ministry of Education, Anhui University, Hefei 230039, China, and also with
the Anhui Provincial Engineering Research Center for Unmanned System
and Intelligent Technology, Anhui University, Heifei 230039, China (e-mail:
shixiasheng@zju.edu.cn; suyanxu0616@gmail.com; cysun@seu.edu.cn).
Chaoxu Mu is with the School of Electrical and Information Engineering,
Tianjin University, Tianjin 300072, China, and also with the Engineering
Research Center of Autonomous Unmanned System Technology, Ministry of
Education, Anhui University, Hefei 230039, China (e-mail: cxmu@tju.edu.cn).
Digital Object Identifier 10.1109/JSYST.2024.3356576

in mobile communication systems, and the spectrum resource
allocation issues in the vehicle to all communication [1]. The
primary objective of the distributed resource allocation problem
is to allocate an amount of resources while minimizing the
global cost function [2]. The resources can be energy, information, materials, factory production capacity, and the number
of machines. As an essential area of research in distributed optimization, a large amount of discrete-time and continuous-time
distributed resource allocation approaches have been designed
based on the consensus scheme of multiagent systems. The
most apparent difference between the two is that the former is a
difference equation, while the latter is a differential equation. In
the case of discrete-time mode, some existing linear or sublinear
distributed optimization approaches are proved by the matrix
contraction theory or the small gain theory. For instance, the
surplus-based approach in [3] adopted a nonlinear scheme with
fully distributed uncoordinated step size to achieve decoupling
of control parameters. Lu et al. [4] introduced a momentum
term in updating the dual variable to accelerate the convergence
performance. A dynamic event-triggered scheme was developed
for the dual variable in [5] to reduce the communication burden.
A dynamic quantization-based scheme was designed in [1] to
prevent sensitive information leakage. Further arguments on
these algorithms can be found in [6].
On the other hand, since the continuous-time algorithm can
establish more robust theoretical conditions via several new
techniques, such as the differential inclusion and the fixed-time
stability criterion [7], [8], [9]. As a result, a considerable number of continuous-time distributed resource allocation methods
have been rapidly proposed recently. For instance, a distributed
projection-based method without initialization was presented
in [10] by virtue of the primal-dual gradient flow scheme over
undirected networks. The local constraint was tackled by the
projection operator. In the case of a weighted balanced directed
network, the differentiated projection operator was utilized to
ensure local constraint satisfaction in [11]. However, the projection and differentiated projection operators may not be suitable
for dealing with complex constraints. To address this limitation,
some adaptive distributed resource allocation algorithms were
proposed [12], [13], [14] by virtue of the Karush–Kuhn–Tucker
(KKT) condition. However, the results above are limited to firstorder systems and cannot be straightly extended to the secondorder or higher-order systems. Furthermore, the presence of local constraints leads the resource allocation problem to be much
more challenging. It is notable that the second-order systems

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SHI et al.: NONSMOOTH DISTRIBUTED RESOURCE ALLOCATION OVER SECOND-ORDER NONLINEAR MULTIAGENT SYSTEMS

can be seen in several practical engineering experiences, i.e.,
thermal power generation systems and automotive mechanical
systems [15]. Consequently, the distributed resource allocation
problem over second-order systems has attracted much attention
nowadays.
A. Related Work
For the distributed resource allocation problem over secondorder systems, two asymptotical and exponential convergent
distributed algorithms were developed in [16] regarding the
proportional–integral (PI) structure. An accelerated distributed
algorithm with Nesterov’s momentum strategy was designed
in [17] to improve the convergence rate. However, only the
distributed resource allocation problem with global equality
constraints is taken into consideration in [16] and [17]. A series of differential projection-based and penalty-function-based
distributed optimal resource allocation approaches were proposed in [18] and [19] to address distributed resource allocation
problems with local and global constraints. For some certain
second- or high-order systems, two distributed optimal resource
allocation approaches were designed in [20] and [21] using the
state-based feedback control idea. It is worth pointing out that
the linear double-integrator system is researched in [16], [17],
[18], [19]. In industrial systems, nonlinear terms and disturbances will inevitably appear in the control channel, such as
the Euler–Lagragian system [22], which can significantly affect
the system stability [23], [24]. There are three main methods to
deal with nonlinear terms: State-based estimator methods [25],
[26], internal model control (IMC) methods [27] and slidingmode control (SMC) methods [28], [29]. However, the results
above have mainly been investigated for distributed convex
optimization problems over first-order systems. The distributed
resource allocation problem over second-order systems nonlinear terms or disturbances has received relatively less attention.
For example, a distributed optimization method with an internal
model observer was developed in [30] for handling the nonlinear
term. In the case of agents with known disturbances, an internal
model approach was designed for distributed resource allocation
problems [31]. For more general partial unknown disturbances
cases, a reduced-order extended state observer-based distributed
algorithm was developed [32].
However, it is worth noting that except for [18], the local constraints are not considered in these second-order algorithms [16],
[17], [30], [31], [32]. The local constraints are critical in practical
applications, for instance, the generation output limitation in
the economic dispatch question and the safety range of robot
actuators [16]. In contrast to the prior works of [16], [17], recent
research has been applied to distributed resource allocation problems with L-smooth-based global cost function over secondorder systems [18], [30], [31], [32]. This implies that the cost
function has a quadratic upper bound, limiting the applicability
of algorithms. In addition, the nonlinear term is required to be
Lipschitz continuous in [30]. In addition, the nonlinearities of
the gradient-based method for distributed resource allocation
problems further increase the difficulty of the algorithm convergence analysis. Hence, studies on distributed resource allocation

645

problems over second-order systems combined with nonlinear
dynamics and disturbances are scarce, which motivates this
study.
B. Motivations
From the above statements, we can summarize the existing
distributed resource allocation algorithms in Table I. Existing
results are less subject to local constraints [16], [17], [21], [30],
[31], [32]. However, local constraints are essential for some
resource allocation problems. For instance, the generation of
power systems has its local upper and lower limitations [4]. The
local constraints rely on the position variable, and the control
input directly affects the velocity variable for second-order
physical systems. Therefore, the smooth cost function is required
in the existing literature to ensure the optimality analysis [32].
However, the nonsmooth cost function is quite common in practical engineering applications. For example, the fuel objective
function in the smart grid is nonsmooth due to the existence of
valve point loading effects [20]. In addition, the nonlinear terms
are unavoidable [30]. Nevertheless, the nonlinear term is assumed to be Lipschtiz continuous in the existing outcomes [30],
[31], [32]. The exponential convergence property is required for
the primal algorithm when nonlinear terms are not considered.
Therefore, the state-based estimator is ineffective when considering the local constraints for the resource allocation problem
over second-order multiple physical systems. Eventually, how
to develop the distributed optimization method for the resource
allocation problem with local constraints and nonsmooth cost
function over nonlinear second-order physical systems is still
challenging.
C. Statement of Contributions
Inspired by the above discussion, this study provides an innovative distributed method of the nonsmooth resource allocation
problem over second-order nonlinear systems, where the SMC
and modified primal-dual scheme are used. Furthermore, the
contributions of this article include:
1) Both local and global constraints are considered in our
work, which distinguishes our work from prior works such
as [16], [17], [30], [31], [32]. The global cost function is
not required to be differentiable and Lipschtiz smooth, and
contrary to the differentiable assumption made in [16],
[17], [18], [30], [31], [32].
2) In contrast to the results in [16], [17], [18], [32], the
nonlinear terms are considered in our work and addressed
by using a SMC scheme. Compared to the state-based
estimator, the sliding model control scheme makes the
local constraints solvable. Employing this scheme, the
nonlinear term in this study is assumed to be bounded
and weaker than the time-dependent Lipschitz-like or
Lipschitz continuous assumption made in [30].
3) To address the local constraint of each agent, we propose
an adaptive scheme based on the KKT conditions and
max operator. This approach is more straightforward to
calculate than the penalty function approach in [18] and the

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IEEE SYSTEMS JOURNAL, VOL. 18, NO. 1, MARCH 2024

TABLE I
COMPARISONS OF EXISTING DISTRIBUTED RESOURCE ALLOCATION ALGORITHMS OVER SECOND-ORDER MULTIAGENT SYSTEMS

with suitable dimensions. Given m numbers s1 , s2 , . . ., sm ,
let col(s) = [s1 , s2 , . . ., sm ]T ∈ Rm represent their aggregate
vector. Let span{1m } = {s|s = s0 ⊗ 1m , s0 ∈ R} indicate the
expand space of vector 1m . Given a matrix A, the transpose of
A is denoted by AT . The 2-norm is expressed as  · . Consider
a vector s = [s1 , s2 , . . ., sm ]T ∈ Rm , s ≤ 0 means that all
components of s are less than zero. Symbols <, ≥, > have
the same property. Given a positive number μ, let sigμ (s) =
[sign(s1 )s1 μ , sign(s2 )s2 μ , . . . , sign(sm )sn μ ]T ∈ Rm
and (s)+ = [max(0, s1 ), max(0, s2 ), . . ., max(0, sm )]T ∈
Rm . Given a function f (s), let ∇f (s) and ∂f (s) represent its
gradient and subgradient, respectively.
II. PRELIMINARIES
We provide a number of preparations about the communication network and the differential inclusion in this section.
A. Graph Theory

Fig. 1.

Schematic block diagram of this article.

projection-based approach in [19]. By dint of the proportional integral control approach and gradient flow method,
a distributed scheme with self-adjusting communication
weight is designed to tackle the global equality constraint.
In addition, the asymptotical convergence property is
proved by using the fixed-time stability and the set-valued
LaSalle invariance principle.
The rest os this article is organized as follows. The preliminaries about the graph theory and the differential inclusion are
introduced in Section II. Section III outlines the problem formulation and algorithm design of this study. Section IV presents
the optimality analysis and extends the developed approach to
the weight-balanced directed networks. Section V gives two
examples of the designed method. Finally, Section VI concludes
this article. The schematic block diagram of this article can be
seen in Fig. 1.
Notation: Let R and R≥0 , respectively, represent the real
and nonnegative real number sets. Let Im , 1m , and 0m be
the identity matrix, all one vector and all zero vector with
m-dimension. Moreover, 0m is generally abbreviated as 0

A network combined by n autonomous agents is considered
in this article, which means that information can be transferred
between any two connected agents. In graph theory, the above
network can be denoted by the symbol G = (V, E, A), where
the node set is denoted by V := {1, 2, . . ., n}, the edge set
is indicated by E ⊂ V × V, and the corresponding weighted
adjacency matrix is represented by A := [aij ]n×n . Specifically,
we have aij > 0 when there exists a communication link from
network G
agent j to agent i, and aij = 0 otherwise. The 
is weighted-balanced (or undirected) if we have nj=1 aij =
n
We determine the
j=1 aji ∀i, j ∈ V(or aij = aji ∀i, j ∈ V).
n
Laplacian matrix LA := [lij ]n×n by lii = j=1 aij ∀i ∈ V and
lij = −aij , i = j ∀i, j ∈ V. If the connected G is undirected
or weighted-balanced directed, the eigenvalues of L̂A can
be ordered as 0 = ρ1 < ρ2 ≤ · · · ≤ ρn , where L̂A = 12 (LA +
LTA ). More information about graph theory can be referred
to [33].
Lemma 1 ([34]): Consider the following closed system:
ṡ = Q(s, t), s(0) = s0

(1)

in which Q : Rm → Rm is a nonlinear function, and the origin is the equilibrium point. For a given continuous radially
unbounded function W : Rm → R≥0 . If we have W (s) = 0 ⇔
s = 0 and Ẇ (s(t)) ≤ −ι1 W κ1 (s(t)) − ι2 W κ2 (s(t)) with 0 <
κ1 < 1, 1 < κ2 , ι1 > 0, ι2 > 0, then s achieves the origin at a
fixed time T (s0 ). Furthermore, T (s0 ) is bounded by T (s0 ) ≤
1
1
ι1 (1−κ1 ) + ι2 (κ2 −1) .

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SHI et al.: NONSMOOTH DISTRIBUTED RESOURCE ALLOCATION OVER SECOND-ORDER NONLINEAR MULTIAGENT SYSTEMS

B. Differential Inclusion

The multiagent systems with subsystem (4) aim to find a
suitable control input ui in order that the final state xi is the
exact solution to the following optimization problem:

For the following differential inclusion system:
ṡ = Q(s), s(0) = s0

(2)

where Q := Rm → Rm represents a set-valued map. Let s :
[0, T ] → Rm be an absolutely continuous map so that system
(2) stands up for almost all t ∈ [0, T ]. Then, we call s a right
maximal solution of (2) under the condition that s can not be
expanded forward in time. If the set M contains a maximal
solution (or all maximal solutions) of (2) from any start point in
M, then M is a weakly invariant(or strongly invariant) of (2).
For (2), the following lemma holds.
Lemma 2 ([35]): If Q is an upper semicontinuous and
locally bounded set-value map, and it takes nonempty, compact, and convex values, then the system (2) is solvable under
Carathéodory solution from any initial start point.
For system (2), the set-valued Lie derivative of a continuous
differential function W : Rm → R is given by
LQ W = {p ∈ R|∃δ ∈ Q(s) s.t. δ T ∇W (s) = p}.

(3)

Then, the following set-valued LaSalle invariance principle ensures the stability of (2):
Lemma 3 ([36]): Let W : Rm → R be the candidate continuously differentiable function, and S ∈ Rm be a compact and
strongly positively invariant set for (2). The Carathéodory solutions of (2) are bounded under the condition that max LQ W ≤ 0
or LQ W is empty for all s ∈ S. Moreover, from any initial
point in S, the corresponding solution y converges to the largest
weakly invariant set S ∩ {s ∈ Rm |0 ∈ LQ W (s)}.
III. PROBLEM FORMULATION AND ALGORITHM DESIGN
First, the distributed resource allocation problem over nonlinear second-order system is designed. Second, we will provide an
adaptive method based on the KKT condition and the maximal
operator for the undirected systems.
A. Problem Formulation
For each agent i, ∀i ∈ V, different from the linear double
integrator system, the nonlinear characteristic is formulated as
follows:
ẋi (t) = vi (t)
v̇i (t) = hi (xi (t), vi (t), t) + ui (t)

647

(4)

in which xi (t) ∈ RN indicates the state information, vi (t) ∈ RN
presents the corresponding velocity, ui (t) ∈ RN symbolizes the
corresponding control input, and hi (xi (t), vi (t), t) ∈ RN is the
nonlinear term. From (4), it can be found that vi is not only
controlled by ui but also affected by the term hi (xi (t), vi (t), t),
which is more challenging than the linear double integrator system. Furthermore, hi (xi (t), vi (t), t) is required to be limited by a
positive constant Hi , in mathematically, hi (xi (t), vi (t), t) ≤
Hi . In other words, hi (xi (t), vi (t), t) is bounded by the known
constant Hi . For convenience, we omit the time variable t in the
following analysis.

min f (x) =

x∈RnN

n


fi (xi )

i=1

s.t. 1TnN x = 1TnN d
gi (xi ) ≤ 0 ∀i ∈ V.

(5)

In (5), fi (xi ) and gi (xi ) = [gi1 (xi ), . . ., giqi (xi )]T ≤ 0 denote
the local cost function and local convex function of agent i,
respectively. f (x) and 1TnN x = 1TnN d, respectively, indicate the
global cost function and the global equality resource constraint.
Moreover, di denotes the private demand resource, and qi represents the private convex function number of agent i. The
objective of problem (5) attempts to allocate a certain amount
of wealth to n agents while minimizing the total cost function
considering several local convex inequality constraints. Besides,
the private information di , xi , and gi (xi ) cannot be transmitted
to neighboring agents for information security. The resource
allocation problem has emerged in many applications [5], [19],
such as the economic dispatch, in which all generators seek to
fulfill load-side electricity demand within their respective power
generation capabilities while minimizing the total generation
cost. Furthermore, the resource indicates the power generation
plan or demand power generation in the economic dispatch. For
simplicity, let N = 1 in the subsequent analysis. The higher
dimensional case (N > 1) can be similarly deduced by using
the Kronecker product.
The following mild assumptions are presented to guarantee
that the developed method achieves optimal results and to simplify the convergence analysis.
Assumption 1: The private constraint function gi (xi ) ∀i ∈
V is convex. The total cost function f (x) is required to
be ω−strongly convex(ω > 0). That is, f (y) ≥ f (s) + (y −
s)T ∂f (s) + ω2 y − s2 ∀y, z ∈ Rn , ω > 0.
Assumption 2: At least one point x̃ exists in order that 1Tn x̃ =
T
1n d and g(x̃) < 0.
Remark 1: The convex property of function gi (xi ) indicates
that the local constraint Ωi = {xi |gi (xi ) ≤ 0} is a convex set.
The strongly convex assumption for the function f (x) means
a quadratic function exists as its lower bound. Assumption 2,
also called Slater’s condition, states that the feasible region for
problem (5) must have an interior point and guarantees the
zero duality gap between the primal problem (5) and its dual
problems. The above two assumptions are introduced to ensure
the existence and the uniqueness of optimal point and the strong
duality of problem (5), which have been widely used in existing
distributed constrained optimization algorithms [11], [14], [18],
[30], [31], [32].
Lemma 4 ([12]): With the help of Assumptions 1 and 2, x∗
is the optimal solution of (5) if there exist vectors (λ0 , c∗ ) ∈
R × Rq in order that the following conditions hold:
ξ ∗ + (ζ ∗ )T c∗ + λ0 ⊗ 1n = 0
1Tn x∗ = 1Tn d

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IEEE SYSTEMS JOURNAL, VOL. 18, NO. 1, MARCH 2024

(c∗ )T g(x∗ ) = 0, c∗ ≥ 0, g(x∗ ) ≤ 0
(6)

in which ξ ∗ ∈ ∂f (x∗ ), ζ ∗ ∈ ∇g(x∗ ), and q = ni=1 qi .
There are four parts to the KKT conditions. 1) Primary feasibility. 1Tn x= 1Tn d and g(x∗ ) ≤ 0; 2) Stationarity. ξ ∗ + (ζ ∗ )T c∗ +
λ0 ⊗ 1n = 0; 3) Dual feasibility for inequality constraints. c∗ ≥
0. 4) Complementary slackness. (c∗ )T g(x∗ ) = 0. The primary
and dual feasibility conditions imply that the constraints must
be held at optimal conditions. The stationarity condition ensures
that there is no feasible direction to improve the cost function.
The complementarity condition is applied to the inequality
constraints and enforces a positive (or zero) Lagrange multiplier
when the constraint is equal to (or less than) zero.

Lemma 4 shows that dual variables λ0 and c∗ are the key to
solving problem (5). Then, inspired by the primal-dual method
in [10], we design the following adaptive modified primal-dual
optimization approach to cope with the distributed optimization
problem (5) over the second-order nonlinear systems (4). The
most crucial idea is to allocate one local variable λi to each
agent. Then, we design the following robust distributed resource
allocation of second-order systems (RDRAoSOS) by virtue of
the consensus technique
ui = uoi + uri

(7a)

uoi = − ki vi − α(ξi + ζiT (ci + gi (xi + vi ))+ ) − λi

(7b)

uri = − sigμ (si ) − sigν (si ) − γi sig0 (si )

(7c)

λ̇i = −

aij βij (λi − λj ) + xi + vi − di − zi

(7d)

j=1

żi =

n


vi (ι) = vi (ι − 1) + hi + ui × stepsize,
xi (ι) = xi (ι − 1) + vi (ι) × stepsize.
Step 4: Agent i sends λi (ι) to its neighbors.
end for

B. Algorithm Design

n


Algorithm 1: Step-By-Step of Algorithm RDRAoSOS.
Initialization: Set
1
,
0 < μ < 1, ν > 1, ki > 1, τij = 1, γi > Hi , α > 2ω
N
N
xi (0) ∈ R , vi (0) ∈ R , λi (0) = 0, zi (0) = 0, ci (0) =
0, βij = 0, si (0) = 0 and stepsize.
for ι = 1, . . .,do:
Step 1: Agent i receives its neighbor’s λj (ι − 1).
Step 2: Agent i calculates ui (ι) according to (7).
Step 3: Agent i updates its state xi (ι), vi (ι):

aij βij (λi − λj )

(7e)

j=1
2

are equivalent to the result c∗ = (c∗ + g(x∗ ))+ . Thus, the
dynamic (7g) is created. In addition, we have ci ≥ 0 ∀t ≥
0 due to the definition of (·)+ and nonnegative initial state
ci (0).
3) Nonlinear term: The control input ui has two parts, uoi and
uri . Specifically, uoi ensures the optimality, and uri offsets
the nonlinear dynamics. Depending on the discontinuity
of the sign function, the system’s nonlinear term is suppressed.
The control parameters satisfy that 0 < μ < 1, 1 < ν, α >
1
2ω , ki > 1, γi ≥ Hi , τij > 0 ∀i, j ∈ V, where ω is the strongly
convex coefficient in Assumption 1. The initial value of variables
xi , vi , λi , and si can be arbitrarily selected. Without loss of
generality, we set x(0) = 0, v(0) = 0, λ(0) = 0 and s(0) = 0.
Let x, v, λ, z, uo , ur , d, c, g denote their aggregative vectors,
respectively. Then, we rewrite the compact formulation of (4)
with controller (7) by the following dynamics:
ẋ = v

β̇ij = τij aij λi − λj 

(7f)

v̇ = uo + ur + h(x, v, t)

ċi = − ci + (ci + gi (xi + vi ))+

(7g)

λ̇ = − Lβ λ + x + v − d − z

in which ξi ∈ ∂fi (xi + vi ), ζi ∈ ∂gi (xi + vi ), and si = vi −
t
vi (0) − 0 uoi (τ )dτ . The three parts in the algorithm (7) are
given as follows:
1) Global equality constraint: Different from the centralized
methods, in Algorithm (7), only the neighboring information λj ∀(i, j) ∈ E is used. The auxiliary variable zi with
zero initialization is utilized to make complete between
the real
value xi and the demanding value di . Furthermore,
zi and nj=1 aij βij (λi − λj ) constitute proportional integral control of variable λi . The variable βij ∀i, j ∈ V is
designed in order that the eigenvalues of LA are not required for ensuring the convergence domain. Compared to
the primal-dual method, this modified primal-dual method
does not require the information exchange of variable zi .
Thus leading to less communication burden.
2) Local equality constraint: Penalty factor ci ensures that
the stable state x∗i satisfies the local constraint gi (x∗i ) ≤ 0.
From Lemma 4, the conditions (c∗ )T g(x∗ ) = 0 and c∗ ≥ 0

ż = Lβ λ
(8)
ċ = − c + (c + g(x + v))+
n
where Lβ = [lβ,ij ]n×n is denoted as lβ,ii = j=1 aij βij ∀i ∈
V and lβ,ij = −aij βij , i = j, ∀i, j ∈ V. The step-by-step process of designed approach (8) is given in the following Algorithm
1, where the stepsize is a small positive step size. From (7), only
the Lagrange multiplier λj (t) is needed when updating the control input ui . Thus, the communication complexity of each agent
is given O(N Ni ), where Ni is the number of neighbor agents
of agent i ∀i ∈ V. In addition, the computational complexity of
each agent i is given by O(N (Ni + mi )).
Remark 2: Different from the projection operator in [10],
the differentiated projection operator in [11], and the penalty
function method in [18], an adaptive approach is designed in
controller (7) based on the max operator and KKT condition
to address the local constraints. The penalty factor in controller
(7) can be adaptively changed based on the state information,

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SHI et al.: NONSMOOTH DISTRIBUTED RESOURCE ALLOCATION OVER SECOND-ORDER NONLINEAR MULTIAGENT SYSTEMS

which is more flexible than the adaptive approach in [13] with
a nondecreasing penalty factor. Thus, the proposed algorithm
(7) can deal with time-varying distributed resource allocation
problems without reinitialization.
Remark 3: From (4), it is not difficult to find that existing
first-order algorithms cannot be straightly applied to secondorder systems, especially to second-order nonlinear systems.
For second-order systems in [16], [18], [30], [31], and [32],
existing gradient-based algorithms calculating its gradient or
subgradient at x. The L-smooth property is of vital importance
to the proof of system stability. In contrast, we calculate the gradient or subgradient at x + v instead. Accordingly, the L-smooth
assumption is removed in our work. That is, ∂f (y) − ∂f (s) ≤
Ly − s ∀y, s ∈ Rn . Besides, we add one control parameter
α to change the final state λ∗ , and thus the convergence rate
of algorithm (7) can be improved. Furthermore, we design one
fully distributed control parameter ki for term −ki vi to adjust
the convergence rate.
Remark 4: To deal with the local convex function inequalities, the penalty function method is used in [18] so that the
local convex function can be transformed to the private cost
function. However, it is not easy to estimate the suitable penalty
factor and a poor penalty factor may cause the algorithm to
oscillate back and forth. On the contrary, the differential projection operator in [19] is difficult to calculate for the complex
case. Moreover, the provided algorithm in [19] depends on the
virtual second-order system and the projection-based distributed
resource allocation method of first-order systems in [10] cannot
be directly extended to second-order systems.
Remark 5: There are three main methods to deal with nonlinear terms in agent dynamics: The state-based estimator method,
the IMC method, and the SMC method. Among them, the statebased estimator and IMC methods have the following characteristics: 1) both of them require nonlinear terms to satisfy certain
constraints, including smoothness assumptions, boundedness
assumptions, and slow change rate assumptions, etc. 2) both
methods require that the designed algorithm has exponential
convergence properties when nonlinear terms are considered.
The above two methods do not easily to analyze its convergence
property when considering the local constraints. For instance,
the state-based estimator is designed in [30] to address nonlinear
terms. However, it could only address the distributed resource
allocation problem without local constraints due to its exponential convergence requirement for the double-integrator systems.
Besides, the Lipschitz continuous or Lipschitz-like assumption
and time-independent property on the nonlinear term is much
stronger than the bounded property required in our work.

649

by means of the fixed-time stability and the set-value LaSalle
invariance principle.
Theorem 1: Considering that Assumptions 1 and 2 hold, if
1
, the
the algorithm parameters satisfy ki > 1 ∀i ∈ V, α > 2ω
proposed control method (7) ensures that the second-order
nonlinear MASs (4) with connected and undirected network
asymptotically achieve the optimal results of problem (5).
Proof: The convergence proof consists of two steps. I) It will
show that si achieves zero at a fixed time; II) The state variable
x under controller (7) achieves the optimal results of problem
(5) by virtue of the set-valued LaSalle invariance principle.
Step I: From the definition of the variable si , we have
ṡi = −sigμ (si ) − sigν (si ) − γi sig0 (si ) + hi (xi , vi , t). (9)
Let V1 = si 2 ∀i ∈ V. Then, it is obvious that V1 is a semipositive definite radially unbounded function. The time derivative of V1 along system (9) is given as
V̇1 = 2sTi (−sigμ (si )−sigν (si )−γi sig0 (si )+hi (xi , vi , t))
= − 2si 1+μ − 2si 1+ν
− 2γi si  + 2sTi hi (xi , vi , t)
1+μ

1+ν

≤ − 2(si 2 ) 2 − 2(si 2 ) 2
1+μ

1+ν

= − 2V1 2 − 2V1 2

(10)

where the fact −γi si  + sTi hi (xi , vi , t) ≤ 0 with γi ≥ Hi has
been used in the first inequality. The results sTi sigμ (si ) =
sTi sign(si )si μ = si 1+μ , sTi sigν (si ) = sTi sign(si )si μ =
si 1+ν , and sTi sig0 (si ) = sTi sign(si )si 0 = si  are used in
the second equality. Furthermore, based on Lemma 1, si ∀i ∈ V
achieves zero within fixed time T (s0 ) under the initial value s0 .
In addition, T (s0 ) is limited by
T (s0 ) ≤

1
1
+
.
1−μ ν−1

(11)

Step II: If t ≥ T (s0 ), we have
ẋ = v
v̇ = − Kv − α(ξ + ζ T (c + g(x + v))+ ) − λ
λ̇ = − Lβ λ + x + v − d − z
ż = Lβ λ
ċ = − c + (c + g(x + v))+

(12)

where K = diag{k1 , k2 , . . ., kn } ∈ Rn×n . Denote the equilibrium point of system by (x∗ , v ∗ , λ∗ , z ∗ , c∗ ), which yields
0 = v∗
0 = − Kv ∗ − α(ξ ∗ + (ζ ∗ )T (c∗ + g(x∗ + v ∗ ))+ ) − λ∗
0 = − Lβ λ ∗ + x ∗ + v ∗ − d − z ∗

IV. OPTIMALITY ANALYSIS AND EXTENSION
The optimality analysis is given by virtue of the LaSalle
invariance principle. Finally, we modify the proposed adaptive
method to solve the problem over a weighted-balanced digraph.
Now, the optimality derivation for system (8) is given as follows

0 = Lβ λ ∗
0 = − c∗ + (c∗ + g(x∗ + v ∗ ))+ .
+

(13)
∗

∗

∗

In view of the definition of (·) , c = (c + g(x + v ))+ ,
and c∗ ≥ 0, v ∗ = 0, we have the results (c∗ )T g(x∗ ) = 0 and

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∗

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IEEE SYSTEMS JOURNAL, VOL. 18, NO. 1, MARCH 2024

g(x∗ ) ≤ 0. Namely, the local constraints in problem (5) hold.
Since G is undirected and connected, the outcome 1Tn ż = 0 is
obtained based on (7e). Therefore, it is shown that 1Tn z ∗ = 0
due to the zero initialization of variable zi . By means of the
results −Lβ λ∗ + x∗ + v ∗ − d − z ∗ = 0 and v ∗ = 0, the global
equality constraint 1Tn x∗ = 1Tn d holds. Similarly, we can obtain
that λ∗ = λ0 ⊗ 1n , λ0 ∈ R, which implies that (6) in Lemma 4
holds. Eventually, x∗ is the global exact result of problem (5) in
light of Lemma 4.
Next, we prove that the state x generated under controller (7)
asymptotically achieves the optimal solution x∗ . Let x̄ = x −
x∗ , v̄ = v − v ∗ , λ̄ = λ − λ∗ , and z̄ = z − z ∗ , then we define W2
as
1
1
V2 = x̄T (K − In )x̄ + x̄ + v̄2
2
2
1
α
1
+ λ̄2 + c − c∗ 2 + z̄ + λ̄2
4
2
4
n
n


(βij − σ)2
+
(14)
8τij
i=1
j=1,i=j

where σ > ρ22 . The set-valued Lie derivative of V2 along system
(13) is denoted as L(13) V2 . For any φ ∈ L(13) V2 , we establish
that
φ = − α(x̄ + v̄)T (ξ − ξ ∗ ) − v̄ T (K − In )v̄
1
σ
+ (x̄ + v̄)T z̄ − λ̄T LA λ̄
2
2
+ α(c − c∗ )T (−c + c̄) − λ̄T z̄
1
− z̄2 − α(x̄ + v̄)T (ζ T c̄ − (ζ ∗ )T c̄∗ )
(15)
2
where c̄ = (c + g(x + v))+ and c̄∗ = (c∗ + g(x∗ + v ∗ ))+ . In
the following, we simplify the term ω = (c − c∗ )T (−c + c̄) −
(x + v − x∗ − v ∗ )T (ζ T c̄ − (ζ ∗ )T c̄∗ ) in (15). Replacing c − c∗
with c − c̄ + c̄ − c∗ in term ϑ, it yields
ϑ = − (x̄ + v̄)T (ζ T c̄ − (ζ ∗ )T c̄∗ )
− c − c̄2 + (c̄ − c̄∗ )T (−c + c̄)
= − c − c̄2 + c̄T (−c + c̄ − ζ(x̄ + v̄))
− (c̄∗ )T (−c + c̄ − ζ ∗ (x̄ + v̄))
= − c − c̄2
+ c̄T (g(x + v) − g(x∗ + v ∗ ) − ζ(x̄ + v̄))
+ c̄T (g(x∗ + v ∗ ) + (−c − g(x + v)+ ))
− (c̄∗ )T (g(x + v) − g(x∗ + v ∗ ) − ζ ∗ (x̄ + v̄))
− (c̄∗ )T (g(x∗ + v ∗ ) + (−c − g(x + v))+ )

(16)

where we have used the result −c + c̄ = g(x + v) + (−c −
g(x + v))+ in the third equality. Due to the facts c̄ ≥ 0, c̄∗ ≥ 0
and the convex function g(x + v), it yields that c̄T (g(x + v) −
g(x∗ + v ∗ ) − ζ(x̄ + v̄)) ≤ 0 and −(c̄∗ )T (g(x + v) − g(x∗ +
v ∗ ) − ζ ∗ (x̄ + v̄)) ≤ 0. From Lemma 4, we have g(x∗ + v ∗ ) ≤
0. Then, the result c̄T (g(x∗ + v ∗ ) ≤ 0 holds. From the definition

of (·)+ , The results c̄T (−c − g(x + v))+ = 0, −(c̄∗ )T g(x∗ +
v ∗ ) = 0 and −(c̄∗ )T (−c − g(x + v))+ ≤ 0 hold. Thus, we can
conclude that ϑ ≤ −c − c̄2 . Then, it follows (15) that:
σ
φ ≤ − λ̄T LA λ̄ − v̄ T (K − In )v̄
2
− α(x̄ + v̄)T (ξ − ξ ∗ ) − αc − c̄2
1
1
+ (x̄ + v̄)T z̄ − λ̄T z̄ + z̄2 .
(17)
2
2
As the undirected communication network is connected, we
obtain that λ̄T LA λ̄ ≥ ρ2 λ̄⊥ 2 , where λ̄ − λ̄⊥ is the element
of set span{1n } and (λ̄⊥ )T (λ̄ − λ̄⊥ ) = 0. Besides, based on
the Young inequality, it yields that λ̄T z̄ ≤ 14 z̄2 + λ̄⊥ 2 and
(x̄ + v̄)T z̄ ≤ 14 z̄2 + x̄ + v̄2 . Thus, substituting the above
results into (17) yields


1
φ ≤ −v̄ T (K − In ) v̄ − αω −
x̄ + v̄2
2

 σρ
1
2
− 1 λ̄⊥ 2 − z̄2 .
− αc − c̄2 −
(18)
2
8
Based on the arbitrariness of φ, we have max L(13) V2 ≤ 0.
From the definition of V2 , it yields that V2 ≥ 12 (x − x∗ )T (K −
In )(x − x∗ ) + 12 x + v − x∗ − v ∗ 2 + 12 λ − λ∗ 2 + 2ρ1n
z − z ∗ 2 + α2 c − c∗ 2 ≥ 0 and V2 (x, v, λ, z, c) = 0 iff
(x, v, λ, z, c) = (x∗ , v ∗ , λ∗ , z ∗ , c∗ ). For a given positive
bounded constant M , let M = {(x, v, λ, z, c) ∈ Rn × Rn ×
Rn × Rn × Rq |V2 (x, v, λ, z, c) ≤ M }. Then, M is a strongly
positive invariant set based on the results (18). Therefore,
from any start point, it is ensured that the trajectories
(x(t), v(t), λ(t), z(t), c(t)) ∀t ≥ 0 and βij generated by (13)
are bounded. Note that βij is monotonously nondecreasing,
then limt→∞ βij exists. Thus, with the help of Lemma 3, it is
shown that x(t) generated from (13) achieves the exact result
of problem (5).
In summary, for the second-order nonlinear systems (4), it
can be obtained that the state value x achieves the exact result
of problem (5) under the controller (7).

Remark 6: The term sigν (si ) accelerates the algorithm when
si  is much greater than 1 and the term sigμ (si ) accelerates
the algorithm when si  is much less than 1. It also can be
seen from (11) that the closer the parameters μ and ν are to
1, the greater the upper bound of the algorithm’s convergence
time T (s0 ). However, it should be noted that the SMC variable
si is introduced to address the nonlinear term hi (xi , vi , t). It
has no obvious impact on the algorithms convergence rate. The
algorithms convergence rate depends on the subsystem (12).
Remark 7: The coordinated control parameter k in [18], [30],
[31], [32] is related to the smallest positive eigenvalue ρ2 , which
is not easy to obtain in a distributed manned. Similarly, from
(14), the condition σ > ρ22 is needed for the convergence analysis. To remove the knowledge requirement of ρ2 , an adaptive
communication weight control scheme is developed in (7). In
addition, we establish the uncoordinated control parameter ki
for each agent in (7).
Remark 8: Theorem 1 proves the asymptotical convergent
property of our created approach (7). The convergence rate is

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SHI et al.: NONSMOOTH DISTRIBUTED RESOURCE ALLOCATION OVER SECOND-ORDER NONLINEAR MULTIAGENT SYSTEMS

an important indicator of algorithm performance. Some existing
literature has designed different fast-distributed optimization approaches by means of the time-based generator, signal functionbased discontinuous control, and aperiodic sampling control
techniques. For instance, Nesterov accelerated primal-dual [17],
finite-time [29], Fixed-time [38], and predefined-time [39] distributed approaches. However, the second-order dynamics, and
local constraints for each local agent and the nonsmooth global
cost function assumed in this article increase the difficulty of
the design and analysis of the accelerated algorithm. Motivated
by the existing faster-distributed optimization, we will study
the accelerated distributed resource allocation algorithm in the
future, including the finite-time sliding mode observer in [29].
Theorem 1 shows the optimality of algorithm (7) under the
condition that graph G is undirected and connected. To solve
the problem (5) for the MASs with strongly connected weightbalanced digraph, we modify the algorithm (7) by
ui = uoi + uri

(19a)

uoi =

(19b)

− ki vi − α(ξi + ζiT (ci + gi (xi + vi ))+ ) − λi

uri = − sigμ (si ) − sigν (si ) − γi sig0 (si )
λ̇i = − β

n


aij (λi − λj ) + xi + vi − di − zi

(19c)

TABLE II
PARAMETERS SET FOR EXAMPLE 1

edge-based adaptive algorithms (7) can not be extended to the
weight-balanced network.
V. SIMULATIONS
In the following, we provide three simulation examples for
illustrating the usefulness and validity of our designed method.
In addition, the case simulation for developed continuous-time
approaches is completed by the MATLAB R2020b and the
fourth-order Runger–Kutta methods.
Example 1: The resource allocation problem over the turbinegenerator system with ring undirected network is provided [31],
[37]. The resource allocation problem is given by
min

(19d)

Pi ∈[Pimin ,Pimax ]

j=1

żi = β

n


aij (λi − λj )

(19e)

ċi = − ci + (ci + gi (xi + vi ))+

(19f)

j=1

where β > ρ22 and ρ2 denotes the smallest eigenvalue of matrix

L̂A = 12 (LA + LTA ). Then, we will prove that the modified algorithm (19) can effectively deal with the problem (5) for the
multiagent systems with a strongly connected digraph. To this
end, we provide the following corollary.
Corollary 1: Considering that Assumptions 1 and 2 hold, if
1
,β >
the algorithm parameters satisfy ki > 1 ∀i ∈ V, α > 2ω
2
,
the
proposed
control
method
(7)
ensures
that
the
secondρ2
order nonlinear systems (4) with strongly connected digraph
asymptotically achieve the optimal results of problem (5).
Proof: We prove the above claim using the same approach
in Theorem 1. It is noted that the sum of rows and columns are
still equal to zero, then the equilibrium point x∗ in (12) is still
the global exact result of problem (5). Let W2 be the same as
Theorem 1. Then, we only need to modify the item − σ2 λ̄T LA λ̄ in
(17) to − β2 λ̄T L̂A λ̄, where L̂A = 12 (LA + LTA ). Thus, the result
in (18) holds by similar analysis. Finally, we prove that algorithm
(7) is effective for the strongly connected digraph.
Remark 9: The strongly connected digraph implies that each
agent has at least one in-neighbor and out-neighbor agent. This
assumption ensures that the global constant 1Tn x = 1Tn d can be
obtained in a distributed way. In addition, the communication
links are asymmetric in the directed network. Furthermore, the
properties 1Tn Lβ = 0 and Lβ 1n = 0 cannot be guaranteed in the
weight-balanced directed topology. In other words, the coupled
constraint 1Tn x∗ = 1Tn d no longer holds. Therefore, the proposed

651

f (P ) =

n


fi (Pi )

i=1

s.t. 1Tn P = 1Tn d

(20)

where Pi denotes the output power of generator i, P and d
represent the aggregative formula of variables Pi and di , and
Ωi = [Pimin , Pimax ] is the corresponding safe power generation range with equivalent type gi (Pi ) = (Pi − Pimin )(Pi −
Pimax ) ≤ 0. Particularly, the generation cost function is defined
as fi (Pi ) = αi1 + αi2 |Pi − αi3 | + αi4 Pi2 with constant parameters αi1 , αi2 , αi3 , αi4 . The private generation cost function
fi (Pi ) is nondifferentiable due to the existence of the absolute
value function. For any generator i ∈ V, we have
P̈i = −

1
Kmi
Tmi + Tei
Ṗi −
Pi +
ui + i
Tmi Tei
Tmi Tei
Tmi Tei

(21)

where i = Ai sin(t + ℵi ) is the disturbance, and Tmi ,
Tei , Kmi , Ai , , ℵi are the parameters of each generator. From
(21), it is worth pointing out that i is time-dependent, which is
more complex than the time-independent case in [30]. Furthermore, we can obtain that Hi = Ai + maxPi |(Tmi + Tei )Ṗi | +
maxPi |Pi |. Therefore, the state-based observer in [30] is not
applicable. Eventually, the parameters are selected as Tmi =
0.5, Tei = 0.2, Kmi = 1 ∀i ∈ V,  = 1 and the other parameters are set as in Table II. Therefore, we have the result ω = 4
in Assumption 1.
Let P (0) = d = [45, 40, 25, 35, 30, 40]T ∈ R6 , v(0) = 0,
λ(0) = 0, z(0) = 0, c(0) = 0, s(0) = 0 and α = 1, ki = 2,
∀i ∈ V, γi = 1000 ∀i ∈ V, τij = 1 ∀i, j ∈ V, the detailed
curve of variables Pi and ci ∀i ∈ V is described in Figs. 2
and 3. It is shown that the generation outputs of the agents 2
and 3 achieve their limitations in Fig. 3. Besides, Fig. 2 verifies
the nonnegative property of variable ci ∀i ∈ V. It can be seen
that the solution P ∗ = [23.50, 34.96, 49.97, 31, 43.75, 31.83]T

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IEEE SYSTEMS JOURNAL, VOL. 18, NO. 1, MARCH 2024

Fig. 2.

Convergence curve of Pi ∀i ∈ V in Example 1.

Fig. 3.

Convergence curve of ci ∀i ∈ V in Example 1.

generated by controller (7) is same with [37] in Fig. 2. Since
nonlinear terms and external disturbances exist in (21), the
real output may not be the same as the theoretical output.
However, the output error is small enough, for example, 0.1%
for agent 2 and 0.004% for the total demand generation. The
control performance of parameter α is 
depicted in Fig. 4,
where the error e is defined as e(t) = 12 ni=1 (xi (t) − x∗i )2 .
Smaller α leads to bigger λ∗ . Therefore, it will take longer
when the variable λi converges to λ∗ with zero initialization.
On the other hand, smaller α leads to smaller feedback quantity
αξi and thus slows down the convergence rate. Thus, from
the perspective of convergence rate, we make α close to 1.
As shown in Fig. 4, smaller γi results in higher convergence
accuracy due to nonlinear term hi (xi , vi , t), and bigger γi
leads to stronger chattering. It also can be obtained from
Fig. 4 that there is a similar convergence rate for different
control parameters ki . Therefore, we recommend setting it to
α = 1, ki = 1.1, γi = 2Hi ∀i ∈ V under the conditions that
1
and ki > 1, γi > Hi ∀i ∈ V.
α > 2ω
Next, we check the usefulness of the Corollary 1 based on
Example 1. Let a12 = a23 = a34 = a45 = a56 = a61 = 1, β =
100 and other variables and control parameters are same with
Example 1, then, the detailed curve of variable Pi ∀i ∈ V is

Fig. 4. Convergence curve of e(t) with different control parameters α and ki
in Example 1.

Fig. 5. Convergence curve of Pi ∀i ∈ V with adaptive control parameter γi
in Example 1.

described in Fig. 5. Fig. 5 shows that the designed method (19)
effectively solves the distributed resource allocation problem
(5) over a strongly connected digraph. The projection-based
approach in [19] and the penalty function-based approach in [18]
are adequate for the resource allocation problem over doubleintegrator systems, in which the nonlinear dynamic hi (xi , vi , t)
is not considered. Therefore, the equilibrium of [19] and [18]
over second-order nonlinear system (4) satisfies the condition
h(x∗ , v ∗ , t) + ξ ∗ + (ζ ∗ )T c∗ + λ0 ⊗ 1n = 0. This means that the
first equality in (6) of Lemma 4. Then, x∗ is not the optimal
solution to the problem (5) except that h(x∗ , v ∗ , t) = 0. The
simulation results are shown in Figs. 6 and 7, which indicates
that they can address the resource allocation problems over
double-integrator systems, but cannot accurately address the resource allocation problems over general nonlinear second-order
systems.
Example 2: We use the simulation example with ring network
in [11] to indicate the usefulness of our developed approach
for the higher-dimensional distributed resource allocation problems. Specifically, let V = {1, 2, 3, 4} and aij = 1 ∀(i, j) ∈ E.

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SHI et al.: NONSMOOTH DISTRIBUTED RESOURCE ALLOCATION OVER SECOND-ORDER NONLINEAR MULTIAGENT SYSTEMS

Fig. 6. Convergence curve of Pi ∀i ∈ V by the approach in [19]. (a) Without
nonlinear term. (b) With nonlinear term.

Fig. 8.

Convergence curve of xi ∀i ∈ V in Example 2.

Fig. 9.

Convergence curve of βij ∀(i, j) ∈ E in Example 2.

Fig. 10.

Convergence curve of si ∀i ∈ V in Example 2.

Fig. 7. Convergence curve of Pi ∀i ∈ V by the approach in [18]. (a) Without
nonlinear term. (b) With nonlinear term.

The individual cost functions of all agents are given by
f1 (x1 ) = x211 + x212 +
f2 (x2 ) = x221 + x222 +

(x11 − 2)2 + (x12 − 2)2
x221
20

x221 + 1

+

x222
20

x222 + 1

f3 (x3 ) = (x31 − 2)2 + (x32 − 3)2
f4 (x4 ) = ln(exp(−0.05x41 ) + exp(0.05x41 )) + x4 2
+ ln(exp(−0.05x42 )+exp(0.05x42 ))

(22)

in which xi = [xi1 , xi2 ]T ∈ R2 ∀i ∈ V is the resource
variable. The nonlinear terms are set as h1 =
[x11 sin(t), x12 sin(t)], h2 = [x221 cos(t), x222 cos(t)], h3 =
[cos(2t), cos(2t)], and h4 = [sin(2x41 ), sin(2x42 )].
Let x(0) = [2, 0, 1.5, 0.5, 1, 1, 4, 6] ∈ R8 , d(0) = [2, 2, 2, 1,
1, 3, 4, 5]T ∈ R8 , λ(0) = 0, z(0) = 0, v(0) = 0, c(0) = 0 and
α = 1, ki = 10 ∀i ∈ V, γi = 10 ∀i ∈ V, τij = 1 ∀i, j ∈ V, then
the detailed strategies of variables x and βij ∀i, j ∈ V are
shown in Figs. 8 and 9, respectively. The exact result x∗ =
[(1.89, 3.47), (1.82, 1), (1.44, 4.56), (1.85, 3.97)]T ∈ R8 is described as a red star ∗ in Fig. 8 and same with [11]. Fig. 9 shows
that the adaptive weight βij ∀i, j ∈ V keeps rising until the Lagrange multiplier λ reaches consensus. In addition, the detailed
convergence curve of the auxiliary variable si is depicted in
Fig. 10. As shown in Fig. 10, the auxiliary variable si oscillates
back and forth near zero. These nonlinear and discontinuous
characteristics ensure that the developed algorithm can suppress

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653

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IEEE SYSTEMS JOURNAL, VOL. 18, NO. 1, MARCH 2024

resource allocation problem for the higher-order systems with
both matched and unmatched nonlinear terms.

REFERENCES

Fig. 11.

Convergence curve of xi ∀i ∈ V in Example 3.

the system’s nonlinear term hi (xi , vi , t). Furthermore, the super
twisting-based ISM in [29] can be adopted to avoid chattering.
Thus, the designed distributed adaptive method can effectively
solve the resource allocation problems of second-order nonlinear
systems.
Example 3: We use the simulation example with 100 agents
to indicate the usefulness of our developed approach. The local cost function is defined as fi (xi ) = ai1 x2i + 2ai2 xi with
ai,1 and ai2 randomly selected from [0.5, 2]. the nonlinear
i
sin(xi +
term for each agent is given by hi (xi , vi , t) = 100
vi ) + exp(−xi t) ∀i ∈ V. The local demand resource is all set
as di = 0.5 ∀i ∈ V. Let μ = 0.8, ν = 1.2, γij=1 ∀i, j ∈ V, ki =
1.1 ∀i ∈ V, α = 1, and γ = 30. Then, the detailed convergence
curve is depicted in Fig. 11, which shows that the proposed
approach can achieve the optimal result. However, it can also
be seen that the convergence time becomes bigger when the
network scale increases. From (7d), the convergence rate is
related to the connectivity of the considered network. Therefore,
we can conclude that the lower connectivity and larger scale
network size leads to longer convergence time.
VI. CONCLUSION
This article researched the distributed nonsmooth resource
allocation problem over second-order nonlinear systems. Different from the existing time-independent Lipschitz-like nonlinear
terms, the time-dependent bounded nonlinear terms considered
in this article are more general. To remove the knowledge
requirement of the smallest positive eigenvalue, an adaptive
distributed algorithm via a SMC approach was designed to
address this problem. Then, we proved that the provided algorithm asymptotically achieves the optimal result by virtue of the
fixed-time stability and the set-value LaSalle invariance principle. The economic dispatch problem in the smart grid with six
agents, 2-D resource allocation problems with four agents, and
1-D resource allocation problem with 100 agents are provided
to illustrate the effectiveness of our proposed approaches. The
experimental results show that the lower connectivity and larger
scale network size lead to longer convergence time and lower
accuracy. Besides, only the matched nonlinear term is researched
here. In the future, we will investigate the distributed nonsmooth

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655

Xiasheng Shi (Member, IEEE) received the B.S.
degree in automation from Hefei University of Technology, Hefei, China, in 2014, and the Ph.D. degree in
control theory and control engineering from Zhejiang
Univeristy, Hangzhou, China, in 2020.
He is currently a Postdoctor with the School of Artificial Intelligence, Anhui University, Hefei, China.
His research interest includes the distributed optimization, control, and estimation.

Yanxu Su (Member, IEEE) received the B.E. and
M.E. degrees in control engineering from the College of Automation Engineering, Nanjing University
of Aeronautics and Astronautics, Nanjing, China, in
2012 and 2015, respectively, and the Ph.D. degree
in control science and engineering from Southeast
University, Nanjing, China, in 2021.
He was a Visiting Ph.D. Student with the Department of Mechanical Engineering, University of
Victoria, Victoria, BC, Canada, from 2018 to 2019.
He is currently an Assistant Professor with the School
of Artificial Intelligence, Anhui University, Hefei, China. His research interest
includes the distributed optimization and model predictive control.

Chaoxu Mu (Senior Member, IEEE) received the
Ph.D. degrees in control science and engineering
from the School of Automation, Southeast University,
Nanjing, China, in 2012.
She was a Visiting Ph.D. student with the Royal
Melbourne Institute of Technology University, Melbourne, VIC, Australia, from 2010 to 2011. From
2014, She is currently a Professor with the School
of Electrical and Information Engineering, Tianjing
University, Tianjin, China. Her current research interest include nonlinear system control and optimization
and adaptive and learning system.
Changyin Sun (Senior Member, IEEE) received the
B.S. degree in applied mathematics from Sichuan
University, Chengdu, China, in 1996, and the M.S.
and Ph.D. degrees in electrical engineering from
Southeast University, Nanjing, China, in 2001 and
2004, respectively.
He is currently a Professor with the School of Artificial Intelligence, Anhui University, Hefei, China.
His research interests include intelligent control,
flight control, and optimal theory.
Dr. Sun is an Associate Editor for the IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, NEURAL PROCESSING LETTERS, AND IEEE/CAA JOURNAL OF AUTOMATICA SINICA.

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