364

IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. 11, NO. 1, MARCH 2024

Distributed Optimal Resource Allocation With
Local Feasibility Constraints for High-Order
Multiagent Systems
Zeli Zhao , Jinliang Ding , Senior Member, IEEE, Jinxi Zhang , Member, IEEE,
and Tianyou Chai , Life Fellow, IEEE

Abstract—This article is concerned with the distributed
optimal resource allocation problem with local feasibility
constraints for high-order multiagent systems (MASs) over
weight-balanced digraphs. Generally, the local feasibility
constraints are addressed based on the feasible direction,
determined by some time-dependent discontinuous functions. This impedes the optimal resource allocation for
high-order MASs, as resulting the discontinuous control
signal yields challenges in the explicit controller design. To
overcome this problem, a novel integrated event-triggered
distributed control strategy is proposed in this article. It
achieves the optimal outputs of the MAS under local feasibility constraints. Moreover, the communication burden
is significantly reduced. On one hand, only one variable
needs to be broadcasted between the neighboring agents.
On the other hand, the frequency of data transmission is
curtailed by the event-triggered communication protocol.
The simulation results illustrate the effectiveness of the
proposed approach.
Index Terms—Distributed resource allocation, eventtriggered communication, high-order multiagent systems
(MASs), local feasibility constraints, weight-balanced
digraphs.

I. INTRODUCTION
ULTIAGENT systems (MASs) are a collection of interacting agents to accomplish a complex task through
communication, cooperation, and competition among agents.
Compared with single-agent systems, MASs show better robustness and flexibility. In recent years, the optimization, control,
and analysis of MASs have become a hot research field [1], [2],

M

Manuscript received 30 November 2022; revised 23 February 2023;
accepted 24 May 2023. Date of publication 13 June 2023; date of current
version 1 March 2024. This work was supported in part by the National
Key R&D Plan Project under Grant 2022YFB3304700, in part by the National Natural Science Foundation of China under Grant 61988101 and
Grant 62103093, in part by the 111 Project 2.0 under Grant B08015, in
part by the Fundamental Research Funds for the Central Universities
of China under Grant N2108003, and in part by the Liaoning Province
Central Leading Local Science and Technology Development Special
Project under Grant 2022JH6/100100055. Recommended by Associate
Editor J. He. (Corresponding author: Jinliang Ding.)
The authors are with the State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang
110819, China (e-mail: zelizhao@163.com; jlding@mail.neu.edu.cn;
zhangjx@mail.neu.edu.cn; tychai@mail.neu.edu.cn).
Digital Object Identifier 10.1109/TCNS.2023.3285887

[3], [4], [5]; in which distributed optimization is an important
research branch [6], [7], [8].
Distributed optimization aims to achieve the optimization of
a global performance index through coordination among agents.
As a critical type of distributed optimization, resource allocation
problems have been widely investigated in many engineering
practices, such as communication networks [9], transportation
systems [10], and power grid systems [11], [12], [13]. Several
gradient-based algorithms were designed to solve the optimal
resource allocation problems (see, e.g., [14], [15], [16], and
references therein). However, they are executed in a discretetime manner and, thus, cannot be applicable for continuous-time
systems. Some continuous-time distributed resource allocation
algorithms were reported [17], [18], [19], [20], [21], [22], [23].
An initialization-free distributed strategy was developed to solve
the economic dispatch problem under varying loads [17]. By
employing the distributed interior point method, a solution to
the resource allocation problem with the capacity constraints
was provided [18]. They both require that each agent should
send its gradient information to the neighboring agents. In fact,
the gradient information of agents is private and not to be shared.
To get rid of this shortcoming, two projection-based distributed
algorithms were designed [19]. In addition, the time-varying
directed networks with delays and the time-varying cost functions were taken into account in the resource allocation problem
in [20] and [21], respectively. For the case of noisy gradients of
local cost functions, a novel push-based distributed algorithm
was proposed to solve the resource allocation problem in smart
grids [22]. The distributed suboptimal resource allocation problem was discussed [23]. It is noted that only resource allocation
problems were investigated in the aforementioned works. Nevertheless, in practical applications, many resource allocation tasks
are implemented with reference to, for example, the behavior
of the physical system, for example, or depend on the system
dynamics [24]. Therefore, the resource allocation problems involved with systems dynamics are gaining increasing attention
in recent years.
Distributed optimal coordination problems integrate traditional distributed optimization problems and dynamic system
control problems. As a subproblem of distributed optimal coordination, the distributed optimal consensus problems have been
nearly well addressed [24], [25], [26], [27], [28], [29], [30],

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ZHAO et al.: DISTRIBUTED OPTIMAL RESOURCE ALLOCATION WITH LOCAL FEASIBILITY CONSTRAINTS

[31], [32]. Another subproblem, distributed optimal resource
allocation, has attracted growing attention more recently [33],
[34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44].
For single-integrator MASs, two distributed algorithms were
proposed to solve the resource allocation problem without disturbances [33] and with disturbances [34]. For second-order
MASs, this problem was tackled in either continuous communication [35], [36], [37], [38]; or event-triggered communication [39]. For high-order MASs composed yet of integrators,
it was handled by Deng [40]. The optimal resource allocation
for the general single-input and single-output (SISO) high-order
MASs was achieved [41]. The fixed-time and predefined-time
resource allocation problems for the general linear MASs were
solved in [42] and [43], respectively. In addition, the external
disturbances were taken into account in the resource allocation
problem for high-order MASs [44]. However, the following
should be noted for the works [40], [41], [42], [43]. First,
the communication topology with an undirected graph was
required. However, the undirected graph is only a special case
of the weight-balanced digraph, and the latter has extensive
applications. Second, the resource allocation problems focused
only on the network resource constraints, omitting local feasibility constraints. In general, the local feasibility constraints
were addressed by using feasible directions, which, however,
are generated by some time-dependent discontinuous functions
associated with the gradient functions of the cost functions [19],
[20], [33], [34], [35], [36]. The discontinuous control signal, together with the virtual control coefficient matrix of the systems,
results in difficulties in the explicit controller design and analysis. Therefore, it is significant and challenging to investigate
the optimal resource allocation problem with local feasibility
constraints for high-order MASs over weight-balanced digraph.
To this end, a novel integrated event-triggered distributed control strategy is proposed in this article to solve the above problem.
Its novelties and superiorities are summarized as follows.
1) It achieves that the high-order MASs autonomously accomplish a resource allocation task, rather than just generate an ideal reference of the optimal allocation [17],
[18], [19], [20], [21], [22], [23].
2) The local feasibility constraints are considered for the first
time in the resource allocation problem for high-order
MASs, unlike [40], [41], [42], and [43]. It is solved
by introducing the differentiated projection operators.
Moreover, the feasibility constraints can be the general
non-box convex sets.
3) The communication burden is significantly alleviated. For each agent, only one variable needs to
be sent to its neighbors in contrast with [19], and
[41]. On this basis, instead of continuous communication [40], [41], [42], an event-triggered communication protocol is employed to reduce the communication
frequency.
The rest of this article is organized as follows. Section II
introduces some preliminaries and formulates the problem under consideration. Section III presents the proposed approach.
Section IV confirms its effectiveness and superiority. Finally,
Section V concludes this article.

365

II. PRELIMINARIES AND PROBLEM STATEMENTS
In this section, we first give some basic concepts of convex
analysis and the graph theory, and then formulate the problem
under consideration.
A. Notations
Rn and Rm×n denote the real vector space with the dimension
n and the real matrix space with the dimension m × n, respectively.  ·  denotes the induced two-norm. ⊗ is the Kronecker
product. In is the n-dimensional identity matrix. 1n is an ndimensional vector in which all the elements are equal to 1. For a
set of vectors z1 , . . . , zn , col(z1 , . . . , zn ) denotes a column vector by stacking them together. For a matrix A, AT and A−1 are
its transpose and inverse, respectively. Let Sym(A) = A + AT .
For a set Ω, int(Ω) and ∂Ω are its interiors set and the boundary
set, respectively. ×i=1,...,N Ωi is the Cartesian product of Ωi ,
i = 1, . . . , N.
B. Graph Theory
A digraph G = (N ,ε,A) is adopted to describe the connectivity of the nodes in the network, where N = {1, 2, . . . , N } is the
node set; ε⊆ N × N is the edge set, i.e., (i, j) ∈ε with i = j
if and only if the ith node receives information of the jth node.
A = [aij ]N ×N is the weighted adjacency matrix, in which aij >
0 if (i, j) ∈ε, and aij = 0 otherwise. The neighbor set of the
0}. The Laplacian
ith node is defined as Ni = {j ∈ N : aij >
of
G
is
given
by
l
=
matrix L = [lij ]N
×N
ii
j=i aij and lij =


−aij , j = i. If j∈N aij = j∈N aji holds for all i ∈ N , G
is weight-balanced. If there is a directed path connecting them
for every pair of nodes, G is strongly connected. For a weightbalanced and strongly connected G, there are L1N = 1TN L = 0;
L + LT is positive semidefinite; the eigenvalues of L denoted
by σi can be ordered as 0 = σ1 < σ2 ≤ σ3 ≤ . . . ≤ σN ; the
eigenvalues of Sym(L) defined by σ̂i can be ordered as 0 = σ̂1 <
σ̂2 ≤ σ̂3 ≤ . . . ≤ σ̂N ; there exists a unitary matrix [r R] with
r = √1N 1N such that 0 < σ̂2 IN −1 ≤ RT Sym(L)R ≤ σ̂N IN −1
and RRT = IN − N1 1N 1TN .
C. Convex Analysis and Projection
For a differentiable function f (·) : Rm → R, two basic properties are given as follows.
1) If f (z1 ) − f (z2 ) ≥ ∇f T (z2 )(z1 − z2 ), ∀z1 , z2 ∈ Rm ,
then f (·) is convex.
2) If there exists θ > 0 such that, for all z1 , z2 ∈ Rm ,
then
(∇f (z1 ) − ∇f (z2 ))T (z1 − z2 ) ≥ θz1 − z2 2 ,
f (·) is θ-strongly convex.
For a vector-valued function ϕ(·) : Rm → Rm , if there exists
ζ > 0 such that ϕ(z1 ) − ϕ(z2 ) ≤ ζz1 − z2 , ∀z1 , z2 ∈ Rm ,
then ϕ(·) is ζ-Lipschitz.
For an arbitrary convex set Ω ⊂ Rn and a vector ς ∈ Ω, the
normal cone of Ω at ς is given by NΩ (ς) = {ν | ν, x − ς ≤
0, ∀x ∈ Ω}. The subset of NΩ (ς) is defined by nΩ (ς) = {ν |
ν = 1, ν, x − ς ≤ 0, ∀x ∈ Ω}. The tangent cone of set Ω
k
at ς is given by TΩ (ς) = {u | u = limk→∞ ς τ−ς
, τk ≥ 0, τk →
k

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0, xk ∈ Ω, ς k → ς}. The projection of a vector z ∈ Rn onto Ω
is PΩ (z) = argminx∈Ω z − x. For arbitrary two vectors ς ∈ Ω
and υ ∈ Rn , the differentiated projection operator is defined by
PΩ (ς + δυ) − ς
.
δ→0
δ

ΠΩ (ς, υ) = lim

Lemma 1: The operator ΠΩ (ς, υ) has the following properties [45].
1) If ς ∈ int(Ω), then

∗
) is the optimal solution of (1).
where y ∗ = col(y1∗ , . . . , yN
The following common assumptions are introduced.
Assumption 1: (See [26]) (Ai , Bi ) is controllable, and


Ci B i
0
rank
= ni + p.
−Ai Bi Bi

Lemma 5 in [26] shows that under Assumption 1, the following linear matrix equations:
0 = −Ai Xi1 + Bi Ui1

ΠΩ (ς, υ) = υ.

0 = Bi Xi2 − Xi1

2) If ς ∈ ∂Ω and maxy∈nΩ (ς) υ, y ≤ 0, then

Ip = Ci Xi1

ΠΩ (ς, υ) = υ.
3) If ς ∈ ∂Ω and maxy∈nΩ (ς) υ, y ≥ 0, then
ΠΩ (ς, υ) = υ − υ, y ∗ y ∗
where y ∗ = argmaxy∈nΩ (ς) υ, y.
Therefore, ΠΩ (ς, υ) is equivalent with the projection of υ onto
TΩ (ς).

have solution triplets (Xi1 , Xi2 , Ui1 ).
Assumption 2: (See [19]) The Slater’s constraint is satisfied
for the resource allocation problem in (1), namely, there
 exist N
points ỹi ∈ int(Ωi ), i = 1, . . . , N , such that P0 = N
i=1 ỹi .
Assumption 3: (See [19] and [36]) For each i ∈ N , fi (·)
is θi -strongly convex with θi > 0; ∇fi (·) is ζi -Lipschitz with
ζi > 0.
III. MAIN RESULTS

D. Problem Statement
Consider the resource allocation problem for N heterogeneous agents over a strong-connected weight-balanced digraph.
The ith agent i ∈ N has a local cost function fi (yi ) : Rp → R,
where yi is the agent output. Meanwhile, the network resource
constraint and the local feasibility constraints are taken into account. Then, the above resource allocation problem is formulated
by
min

N


fi (yi )
N


yi

i=1

yi ∈ Ωi , i = 1, . . . , N

(2)

where xi (t) ∈ Rni is the state; ui (t) ∈ Rmi is the control input;
yi (t) ∈ Rp is the output; and Ai , Bi , and Ci are the real constant
matrices with appropriate dimensions. The control objective is
to steer the outputs of the agents to the optimal solution of the
resource allocation problem in (1).
In summary, the problem treated in this article reads as follows.
Problem 1: Design a distributed control strategy such that
the outputs of the MAS in (2) converge to the optimal solution
of the resource allocation problem in (1), i.e.,
lim yi (t) = yi∗

t→∞

ui = −Ki xi − (Ui1 − Ki Xi1 )zi + Xi2 żi

(4a)

żi = ΠΩi (zi , −fi (yi ) + λi )

(4b)

(1)

where Ωi ⊂ Rp is a closed convex set; P0 is a known constant vector. It is noted that for each agent, fi (yi ) and Ωi ,
i = 1, . . . , N , are private and cannot be shared with others.
The dynamics of the ith agent, i = 1, . . . , N , is described by
ẋi (t) = Ai xi (t) + Bi ui (t)
yi (t) = Ci xi (t)

A. Distributed Controller Design With Continuous
Communication
For each agent, the procedures for controller design are universal, which are given as follows:

i=1

s.t. P0 =

This section gives an integrated event-triggered distributed
control strategy to solve Problem 1. For ease of introduction, we
first consider the case of continuous communication. Then, the
control strategy is upgraded by an event-triggered communication protocol.

(3)

λ̇i = −

N


aij (λi − λj ) − vi + di0 P0 − zi

(4c)

j=1

v̇i =

N


aij (λi − λj )

(4d)

j=1

where i ∈ N ; zi , λi , vi ∈ Rp are the auxiliary
 variables of the
ith agent; vi (0) and zi (0) should satisfy N
i=1 vi (0) = 0 and
zi (0) ∈ Ωi , respectively; Ki ∈ Rm×n is chosen such that Ai −
Bi Ki is Hurwitz; Ui1 , Xi1 , and Xi2 are the solutions of the
linear matrix
equations in Assumption 1; and di0 is a positive
scalar with N
i=1 di0 = 1. The schematic of the controller in (4)
is shown in Fig. 1.
Remark 1: The above control scheme consists of the reference generator in (4b)–(4d) and the tracking controller in
(4a). The former dedicates to producing the references zi that
meet the optimization requirement in (1); the latter is devoted
to generating the control signals that drive the outputs yi to
track zi . Different from the separate designs [48], [49], the
reference generator in (4b)–(4d) depends additionally on the

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ZHAO et al.: DISTRIBUTED OPTIMAL RESOURCE ALLOCATION WITH LOCAL FEASIBILITY CONSTRAINTS

367

By Assumption 1, (6a) and (6e), we have y e = z e . According to
Lemma 1 and (5b), z˙i ∈ TΩi (zi ). Under zi (0) ∈ Ωi , it follows
from [45, Th. 1] and [46, Th. 3.2] that zi (t) ∈ Ωi ∀t ≥ 0. Then,
there holds y e = z e ∈ Ω. Due to 1TN L = 0 in weight-balanced

digraphs, left multiplying (5d) by 1TN ⊗ Ip gives N
i=1 v˙i = 0,
which yields
N


vi (t) =

i=1

Fig. 1.

Integrated control structure in (4).

current outputs, which implies that (4) is an integrated control
strategy. Moreover, every agent broadcasts only λi and does not
share the cost functions or gradient information to its neighbors.
This indicates that the control scheme in (4) is distributed.
Besides, it is noted
 from (4c) that the update of λi depends
on vi instead of N
j=1 aij (vi − vj ) [19], [41]. This excludes the
need to broadcast vi , i ∈ N over the network. Therefore, the
reference generator consumes less communication cost.
Next, we show the effectiveness of the above controller. It
begins with verifying that the outputs at the equilibrium point
of the closed-loop system satisfy the optimization requirement
of the resource allocation problem in (1).
Under Assumption 1, apply the controller in (4) to the MAS
composed of (2). Then, we have the compact form of the closedloop system as follows:

N


vi (0) = 0, t > 0.

(7)

i=1


e
Therefore, N
= 0. Next, left multiplying (6c) by 1TN ⊗
i=1 vi
e
Ip , we have P0 = N
i=1 zi . Note by definition that the nullspace of L is spanned by 1N . From (6d), it is obtained that λe
is included in the null-space of L ⊗ Ip , i.e., there exists ρ ∈ Rp
such that λe = 1N ⊗ ρ. Then, it is inferred from (6b) and Lemma
1 that
(8)
0N p ∈ f (y e ) − 1N ⊗ ρ + NΩ (y e ).
Thus, utilizing the Kaush-Kuhn-Tucker condition in [47, Th.
3.34], y e is the optimal solution of (1), i.e., y e = y ∗ . The proof is
completed.

Then, it is demonstrated that the above equilibrium point is
asymptotically stable.
Theorem 1: Under Assumptions 1–3, applying the controller
in (4) to the MAS composed of (2) solves Problem 1.
Proof: Let
x̄ = 
x − X1 z, z̄ = z − z e , λ̄ = λ − λe , v̄ = v − v e

(9)

λ̂ = [r R]T ⊗ Ip λ̄, v̂ = [r R]T ⊗ Ip v̄

where [r R] is a unitary matrix with r = √1N 1N ; λ̂ =

ẋ = (A − BK)(x − X1 z) + X1 ż

(5a)

col(λ̂1 , λ̂2 ); v̂ = col(v̂1 , v̂2 ); λ̂1 , v̂1 ∈ Rp . From (7), we have
(1N ⊗ Ip )T v̄ = 0, i.e.,

ż = ΠΩ (z, −f (y) + λ)

(5b)

(1N ⊗ Ip )T ([r R] ⊗ Ip ) v̂ = 0.

λ̇ = −(L ⊗ Ip )λ − v + (d0 ⊗ Ip )P0 − z

(5c)

v̇ = (L ⊗ Ip )λ

(5d)

y = Cx

Note by the definition of [r R] that
√
1TN r = N , 1TN R = 01×(N −1) .

(5e)

where x = col(x1 , . . . , xN ); A = blkdiag(A1 , . . . , AN );
f (y) = col(f1 (y1 ), . . . , f2 (y2 )); Ω = ×i∈N Ωi ; the form
of z, λ, v, y, and d0 is the same as that of x; the form of
B, C, K, and X1 is the same as that of A. L is the Laplacian
matrix of G. For ease of notation, the equilibrium point of (5)
is denoted by [xe , z e , λe , v e ]T , at which the output is denoted
by y e .
Lemma 2: Under Assumptions 1–3, y e satisfies the optimization requirement in (1), i.e., y e = y ∗ .
Proof: Note by definition that
0 = (A − BK)(xe − X1 z e )
e

e

(6a)

e

0 = ΠΩ (z , −f (y ) + λ )
e

e

0 = −(L ⊗ Ip )λ − v + (d0 ⊗ Ip )P0 − z
0 = (L ⊗ Ip )λ
e

e

y = Cx .

e

(6b)
e

Then, (10) is rewritten as
 √
N 01×(N −1) ⊗ Ip

v̂1
v̂2

=

√

(10)

N v̂1 = 0.

Thus, v̂1 ≡ 0. In addition, it is obtained from (9) that y − z =
Cx − CX1 z = C x̄. It follows from Lemma 1 that:
ΠΩi (zi , −f (yi ) + λi ) ∈ −f (yi ) + λi − NΩi (zi ).
Then, there exists a vector N̄i (zi ) ∈ NΩi (zi ) such that
ΠΩi (zi , −f (yi ) + λi ) = −f (yi ) + λi − N̄i (zi ).
β(zi ) = N̄i (zi ) ≥ 0. If N̄i (zi ) > 0, n(zi ) =
1
N̄ (z ) ∈ nΩi (zi ); otherwise n(zi ) is selected as
N̄i (zi ) i i

Let

an arbitrary vector in nΩi (zi ). Then, we have

ΠΩi (zi , −f (yi ) + λi ) = −f (yi ) + λi − β(zi )n(zi ).

(6c)

Therefore, (5) is rewritten as

(6d)

x̄˙ = (A − BK)x̄

(11a)

(6e)

z̄˙ = −h + ([r R] ⊗ Ip )λ̂

(11b)

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IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. 11, NO. 1, MARCH 2024

λ̂˙ 1 = −(rT ⊗ Ip )z̄

(11c)

λ̂˙ 2 = −((RT LR) ⊗ Ip )λ̂2 − v̂2 − (RT ⊗ Ip )z̄

(11d)

v̂˙ 2 = ((R LR) ⊗ Ip )λ̂2

(11e)

T

where h = f (y) − f (y e ) + CΩ (z) − CΩ (z e ); CΩ (z) =
col(β(z1 )n(z1 ), . . ., β(zN )n(zN )); the form of CΩ (z e ) is the
same as that of CΩ (z). Next, construct
 φ
T 

φ+1  T
1
z̄ z̄+ λ̂T1 λ̂1 + λ̂T2 λ̂2 +
V1 =
λ̂2 +v̂2
λ̂2 +v̂2
2
2
2

where k1 , k2 > 0, and ζ = max{ζ1 , . . . , ζN }. Substituting the
above inequalities into (16) yields

φ T T
λ̂ (R Sym(L)R) ⊗ Ip λ̂2
2 2

(12)
− v̂2T v̂2 − (φ + 1)v̂2T λ̂2 − v̂2T RT ⊗ Ip z̄.

2

T

z̄ h = z̄ (f (y) − f (z) + f (z) − f (z ))
+ z̄ T (CΩ (z) − CΩ (z e )).

(13)

Under Assumption 3, there holds
z̄ T (f (z) − f (z e )) ≥ θz̄ T z̄

(14)

where θ = min{θ1 , . . . , θN }. From the definition of normal
cone, n(zie ) ∈ nΩi (zie ) and β(zie ) ≥ 0 imply that β(zie )zi −
zie , n(zie ) ≤ 0; n(zi ) ∈ nΩi (zi ) and β(zi ) ≥ 0 mean that
β(zi )zi − zie , n(zi ) ≥ 0. Then, we have
T

e

z̄ (CΩ (z) − CΩ (z )) =

N


β(zi )zi − zie , n(zi )

N


V = x̄T P x̄ + V1

(15)

Substituting (14) and (15) into (13) gives
z̄ T h ≥ θz̄ T z̄ + z̄ T (f (y) − f (z))
based on which (11) is scaled by
φσ̂2 T
λ̂ λ̂2 − v̂2T v̂2
2 2

whose derivative along (11) is

−

θ(φ + 1) k2
−
2
2

− (φ + 1)v̂2T λ̂2 − v̂2T (RT ⊗ Ip )z̄.

(16)

Next, using Young’s inequality and Assumption 3, we have
1
k1 (φ + 1) T
v̂ T v̂2 +
λ̂2 λ̂2
2k1 (φ + 1) 2
2
1 T
k2
v̂2 v̂2 + z̄ T z̄
2k2
2

θ
ζ 2 C2 T
x̄ x̄
z̄ (f (y) − f (z)) ≤ z̄ T z̄ +
2
2θ
T

z̄ T z̄ − 1 −

φσ̂2
k1 (φ + 1)2
−
2
2

1
1
−
2k1
2k2

v̂2T v̂2

λ̂T2 λ̂2 − (l − η1 )x̄T x̄.

It is seen that V̇ ≤ 0 by choosing l > η1 and k1 , k2 <
φσ̂2
2θφσ̂2
1
2 min{ (φ+1)2 , φ+1 , θ(φ + 1)}. Further, V̇ = 0 if and only
if x̄ = z̄ = λ̂2 = v̂2 = 0. Then, (11b) and (11c) reduce to
z̄˙ = (r ⊗ Ip )λ̂1 and λ̂˙ 1 = 0, respectively. It is obtained from
the mean value theorem that λ̂1 = 0. Using LaSalle’s invariance principle, we have limt→∞ x̄ = 0 and limt→∞ z̄ = 0.
Thus, limt→∞ (y − y e ) = limt→∞ (C x̄ + z̄) = 0. According to
Lemma 2, it is obtained that

The proof is completed.

Remark 2: If Ωi , i = 1, . . ., N, are expanded to the whole
Rn space, the resource allocation problem in (1) reduces to that
without local feasibility constraints. In this case, there holds
ΠΩ (zi (t), −f (yi )(t)) + λi (t)) = −f (yi )(t)) + λi (t). It is
trivial to see from the above proof that the controller in (4) is
still feasible.
B. Distributed Controller Design With Event-Triggered
Communication

− (φ + 1)z̄ T (f (y) − f (z))

v̂2T (RT ⊗ Ip )z̄ ≤

(17)

lim yi (t) = y ∗ .

i=1

v̂2T λ̂2 ≤

2

t→∞

β(zie )zi − zie , n(zie ) ≥ 0.

V̇1 ≤ − (φ + 1)θz̄ T z̄ −

λ̂T2 λ̂2 + η1 x̄T x̄

Thus, the Lyapunov function is defined by

i=1

−

v̂2T v̂2

(A − BK)T P + P (A − BK) = −lIn .

V̇ ≤ −

e

1
1
−
2k1
2k2

C
where η1 = (φ+1)ζ
. Since A − BK is Hurwitz, there are
2θ
a positive definite matrix P ∈ Rn×n with n = n1 + . . . + nN
and a scalar l > 0 such that

Rewrite z̄ T h as
T

z̄ T z̄ − 1 −

φσ̂2
k1 (φ + 1)2
−
2
2

−

where φ > 0. Its derivative along (11b)–(11e) is obtained by
V̇1 = − (φ + 1)z̄ T h −

θ(φ + 1) k2
−
2
2

V̇1 ≤ −

The control strategy with continuous communication in (4)
may cause a high communication cost in implementations. To
alleviate the communication burden, an event-triggered communication protocol is introduced on the basis of (4). For ease of
design, a static triggering strategy is chosen, like [26], [50], and
[55]. Instead of (4c) and (4d), the auxiliary variables λi and vi
are updated by
λ̇i = −
v̇i =

N

j=1

N


j=1


aij λ̃i − λ̃j − vi + di0 P0 − zi


aij λ̃i − λ̃j

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ZHAO et al.: DISTRIBUTED OPTIMAL RESOURCE ALLOCATION WITH LOCAL FEASIBILITY CONSTRAINTS

369


λ̂˙ 1 = − rT ⊗ Ip z̄

λ̂˙ 2 = − RT LR ⊗ Ip λ̂2 − v̂2


− RT ⊗ Ip z̄ − RT L ⊗ Ip e


v̂˙ 2 = RT LR ⊗ Ip λ̂2 − RT L ⊗ Ip e

(21c)

(21d)
(21e)

where e = col(e1 , . . . , eN ). For ease of analyzing the convergence of (21), a new variable χ(t) is introduced, which satisfies
Fig. 2.
(19).



Event-triggered control configuration with (4a), (4b), (18), and

ξ
2β

χ̇(t) = −βχ(t), χ(0) = α

where i ∈ N ; λ̃i (t) = λi (tik ), ∀t ∈ [tik , tik+1 ); tik+1 is determined by
tik+1 = inf{t > tik | g(t) ≥ 0}

(19)

−βt

− γyi (t) − zi (t); ei (t) =
where g(t) = ei (t) − αe
λ̃i (t) − λi (t); ti0 = t0 ; α, β, γ ∈ R > 0.
A block diagram illustrating the control scheme configuration
with (4a), (4b), (18), and (19) is given in Fig. 2. If the triggering
condition g(t) ≥ 0 is met, λ̃i (t) in (18) is updated from λi (tik ) to
λi (tik+1 ). Meanwhile, λi (tik+1 ) is sent to the jth agent, i ∈ Nj ,
and ei (t) is reset to zero.
Next, we show the effectiveness of the proposed eventtriggered control strategy. It starts from verifying that the outputs
at the equilibrium point of the closed-loop system satisfy the
optimization requirement of the resource allocation problem
in (1).
Applying (4a), (4b), (18), and (19) to (2) with i = 1, . . . , N
gives
ẋ = (A − BK)(x − X1 z) + X1 ż

where ξ > 0 is to be defined in the sequel. It is easy to obtain that
χ(t) = χ(0)e−βt > 0. Then, construct a Lyapunov function as
follows:
W (, χ) = V + χ2
where  = col(x̄, z̄, λ̂1 , λ̂2 , v̂2 ); V is given by (17). Thus, the
derivative of W along (21) and (22) is
Ẇ (, χ) ≤ −
−

θ(φ + 1) k2
−
2
2

z̄ T z̄ − (l − η1 )x̄T x̄

φσ̂2
k1 (φ + 1)2
−
2
2

− 1−

1
1
−
2k1
2k2

λ̂T2 λ̂2

v̂2T v̂2 − ξα2 e−2βt

− φλ̂T2 ((RT L) ⊗ Ip )e.

(23)

Equation (19) implies that
ei (t) ≤ αe−βt + γyi (t) − zi (t), t > 0.
Then, we have

ż = ΠΩ (z, −f (y) + λ)

e(t) ≤ N αe−βt + N γη2 x̄

λ̇ = −(L ⊗ Ip )λ̃ − v + (d0 ⊗ Ip )P0 − z

where η2 = max{C1 , . . . , CN }. Using Young’s inequality,
there hold

v̇ = (L ⊗ Ip )λ̃
y = Cx

(22)

(20)

where λ̃ = col(λ̃1 , . . . , λ̃N ); the definitions of other variables
are the same as that of variables in (5). For ease of notation, the
equilibrium point and the corresponding output of (20) are also
denoted by [xe , z e , λe , v e ]T and y e , respectively.
Then, it is substantiated that the above equilibrium point is
asymptotically stable.
Lemma 3: Under Assumptions 1–3, y e satisfies the optimization requirement in (1), i.e., y e = y ∗ .
Proof: It is similar to that of Lemma 2, thus omitted.

Theorem 2: Under Assumptions 1–3, applying the control
strategy in (4a), (4b), (18), and (19) to the MAS composed of (2)
solves Problem 1. Besides, the closed-loop system (20) is free
of Zeno behavior.
Proof: By (9) and λ̃i (t) = ei (t) + λi (t), (20) is rewritten as
x̄˙ = (A − BK)x̄

(21a)

z̄˙ = − h + ([r R] ⊗ Ip ) λ̂

(21b)

eT e ≤ 2 N 2 α2 e−2βt + 2 N 2 γ 2 η22 x̄T x̄
λ̂T2 ((RT L) ⊗ Ip )e ≤

σ̂2 T
RT L2 T
e e.
λ̂2 λ̂2 +
4
σ̂2

From the above inequalities and (23), we get
Ẇ (, χ) ≤ −

θ(φ + 1) k2
−
2
2

z̄ T z̄

− (l − η1 − ξγ 2 η22 )x̄T x̄
−

φσ̂2
k1 (φ + 1)2
−
4
2

− 1−
2

T

1
1
−
2k1
2k2

λ̂T2 λ̂2

v̂2T v̂2

2

L
where ξ = 2 N R
used in (22). It implies Ẇ (, χ) ≤ 0 by
σ̂2
setting


φσ̂2
1
θφσ̂2
, θ(φ + 1)
k1 , k2 < min
,
2
2(φ + 1)2 (φ + 1)

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and l > η1 + ξγ 2 η22 . Further, Ẇ = 0 if and only if x̄ = z̄ =
λ̂2 = v̂2 = 0. Then, (21b) and (21c) reduce to z̄˙ = (r ⊗ Ip )λ̂1
and λ̂˙ = 0, respectively. By applying the mean value theorem,
1

we get λ̂1 = 0. By the LaSalle’s invariance principle, we have
limt→∞ x̄ = 0 and limt→∞ z̄ = 0. Then, limt→∞ (y − y e ) = 0.
According to Lemma 3, it is obtained that limt→∞ yi (t) = y ∗
under the proposed event-triggered control strategy.
What remains to be shown is that the closed-loop system in
(20) is free of Zeno behavior. It follows from ei (t) = λ̃i (t) −
λi (t) that ei (t) has an upper right-hand derivative D+ ei (t). For
t ∈ [tik , tik+1 ), there holds
D+ ei (t) =

N



aij λ̃i − λ̃j + vi − di0 P0 + zi .

j=1

Since e(tik ) = 0, the solution of ei (t) is calculated as
⎛
⎞
 t 
N
⎝
ei (t) =
aij (λ̃i (τ ) − λ̃j (τ )) + vi (τ ) + zi (τ )⎠ dτ
tik

j=1

− di0 P0 (t − tik ).
Since zi , λi , and vi are bounded, there are scalars ω1 , ω2 , ω3 > 0
such that λ̃i  ≤ ω1 , vi  ≤ ω2 , and zi  ≤ ω3 , i = 1, . . . , N.
Let  = t − tik and
ς() = (2 | Ni | ω1 + ω2 + ω3 + di0 P0 ).
Then, ei (t) ≤ ς(). The next event will not be triggered until
ei (t) = αe−βt + γyi − zi .
Thus, the interevent interval is greater than or equal to the
i
implicit solution of ς() = αe−β(tk + ) . Following the line in
the proof of [50, Th. 3.2], we can obtain that the solution is
strictly positive. Therefore, the closed-loop system in (20) is
free of Zeno behavior. The proof is completed.

IV. SIMULATION STUDY
In this section, two examples are given to verify the effectiveness of the proposed approach.
Example 1: Consider the resource allocation problem for
a five-agent system over a digraph, as shown in Fig. 3. The
weighted adjacency matrix is
⎡
⎤
0 1 0 0 1
⎢ 0 0 1 0 0⎥
⎢
⎥
⎥
A=⎢
⎢ 1 0 0 1 0⎥ .
⎣ 0 0 0 0 1⎦
1 0 1 0 0
The dynamics of the agents are heterogeneous, given by






3 −1
−1 1
1 0
A1 =
, B1 =
, C1 =
−2 1
−3 2
3 −1






2 −1
−2 3
1 3
A2 =
, B2 =
, C2 =
−3 1
−1 2
0 1

Fig. 3.

Communication topology for the five-agent system.

⎡

⎤
⎡
⎤
3 1 2
1 0 3
A3 = ⎣−1 0 1 ⎦ , B3 = ⎣0 2 −1⎦
1 2 −1
2 1 1
⎡
⎤


−1 0
0
1 0 2
C3 =
, A4 = ⎣ 3 −4 0 ⎦
0 −1 1
0 −1 −1
⎡
⎤


1 0 1
−1 0 1
⎣
⎦
B 4 = 2 1 0 , C4 =
0 1 0
3 0 2
⎡
⎡
⎤
⎤
−1 2
0 3
1 −1 0
⎢−2 1
⎢
⎥
0 0⎥
⎥ , B5 = ⎢0 1 1⎥
A5 = ⎢
⎣0
⎣0 0 1⎦
3 −1 0⎦
1 −1 0 1
2 1 0


1 0 −1 0
.
C5 =
0 1 1 2
Choose the following functions to calculate the local cost of the
five-agent system:
f1 (y1 ) = (y1,1 − 8)2 + (y1,2 − 1)2
f2 (y2 ) =

2
y2,2
y2
 2,1
+ 
+ y2 2
2
2
20 y2,1 + 1 20 y2,2 + 1

f3 (y3 ) =

2
2
y3,2
y3,1
+
2
2 + 2)
80 ln(y3,1 + 2) 80 ln(y3,2

+ y3 − 5 × 12 2

f4 (y4 ) = ln e−0.05y4,1 + e0.05y4,1 + y4 2

+ ln e−0.05y4,2 + e0.05y4,2
f5 (y5 ) =

2
y2
y5,2
 5,1
+ 
+ y5 2 + 1T2 y5 .
2
2
25 y5,1 + 1 25 y5,2 + 1

The control goal is to steer the outputs of the five-agent system
to the optimal allocation of the following resource allocation
problem:
min

5


fi (yi )

i=1

s.t.

5

i=1

yi =



10
8



yi ∈ Ωi , i = 1, . . . , 5

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ZHAO et al.: DISTRIBUTED OPTIMAL RESOURCE ALLOCATION WITH LOCAL FEASIBILITY CONSTRAINTS

Fig. 4.

Trajectories of z1 and z2 .

Fig. 7.

371

Trajectories of y and z.

⎡

Fig. 5.

Trajectories of z3 and z4 .

Fig. 6.

Trajectory of z5 and the network resource constraint gap.

where
Ωi = {yi ∈ R2 | −yi,2 ≤ yi,1 ≤ yi,2 }, i = 1, 2
Ωj = {yj ∈ R2 | −5 ≤ yj,1 ≤ 5, −3 ≤ yj,2 ≤ 3}, j = 3, 4
2
2
+ y5,2
≤ 4}.
Ω5 = {y5 ∈ R2 | y5,1

The boundaries of these local feasibility constraints are depicted
in Figs. 4–6.
Following Theorems 1 and 2, two distributed controllers
with time-triggered and event-triggered communication rules
are obtained, respectively, where




6 −5
−15 11
, K2 =
K1 =
10 −6
−9 7
⎡
⎤
⎡
⎤
−0.3333 0.3333 −1
0 −1 0
K3 = ⎣ 0.2222 1.1111 1 ⎦ , K4 = ⎣3 0 0⎦
1.4444 0.2222 1
0 1 0

⎤
0.6709
0.3745
0.1670
1.5870
K5 = ⎣−1.0893 −1.3659 −0.1283 −0.6074⎦
−0.3421 3.5230
0.5161
0.5455






1 0
−1 1
−5 3
X11 =
, X12 =
, U11 =
3 −1
0 1
−5 4




1 −3
−2 9
X21 =
, X22 =
0 1
−1 5



T
−13 44
1 0 0
U21 =
, X31 =
−8 27
0 −1 0

T
−0.3333 0.2222
0.4444
X32 =
0.3333 −0.5556 −0.1111

T

T
0 0 1
0.5 0 1.5
U31 =
, X41 =
−1 0 0
0.5 1 0.5
⎡
⎤
⎡
⎤
0.5 −0.5
−0.5 −0.5
2 ⎦ , U41 = ⎣ 2.5 −1.5⎦
X42 = ⎣−1
0
1
0
0

T
0.2222 −0.6825 0.7302
X51 =
0.2222 0.1746 0.3016

T
0.3175 0.0952 −0.7778
X52 =
0.1746 −0.0476 0.2222

T
0.7460 0.1429 −1.2698
U51 =
0.4603 −0.5714 0.3016
α = 1.2, β = 0.3, γ = 1, di0 = 0.2, i = 1, . . . , 5.
Apply the above controllers to the five-agent system, and
the simulation results are displayed in Figs. 4–8. It is observed
from Figs. 4–6 that all the reference trajectories evolve within
their respective constrained regions, and the network resource
constraint gap asymptotically converges to 0. This illustrates that
both the local feasible constraints and the network constraint are
satisfied. It is seen from Fig. 7 that the outputs track asymptotically the references. Although the event-triggered communication causes minor fluctuations in the transient phase with respect
to the continuous communication, the output convergence in
different cases are the same after 12 s. Fig. 8 displays that the
communication frequency is significantly reduced, and Zeno

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372

IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. 11, NO. 1, MARCH 2024

TABLE I
SYSTEM PARAMETERS

Fig. 8.

Triggering instants and interevent times.

Fig. 9.

Communication topology for the BESS.
Fig. 10.

behavior is excluded in the event-triggered case. These exhibit
that the event-triggered distributed controller not only solves the
resource allocation problem, but also saves the communication
source with an acceptable convergence rate. Therefore, the simulation results illustrate the effectiveness and the superiority of
the proposed approach.
Example 2: Consider the economic dispatch problem for a
battery energy storage system (BESS) consisting of three battery
packs over a digraph, as shown in Fig. 9 . As discussed in [51] and
[52], in the islanded mode operation, the frequency controllers
can quickly restore the frequency to its nominal value. Therefore,
the battery dynamics can be simplified by neglecting the inner
loops and secondary frequency control [52], given by
−KiE
Pi + u E
Ėi =
i
3600
Ṗi = uP
i
where i = 1, 2, 3; Ei and Pi are the energy and discharging
power of each battery, respectively; KiE is the parameter and
represents the heterogeneity of BESS due to different battery
sizes. The actual operation of a BESS leads to accelerated
battery degradation in terms of capacity fading and increasing
resistance [53], [54]. Thus, the battery degradation cost needs to
be taken into consideration. The local degradation cost function
of the ith battery is given by the following quadratic function:
fi (Pi ) = ai1 + ai2 Pi + ai3 Pi2
where Pi should be kept within the capacity or security bounds
P i ≤ Pi ≤ P̄i . The total power demand of the BESS is P0 .
P
Let xi (t) = col(Ei , Pi ), ui (t) = col(uE
i , ui ), and yi = Pi . The
system matrices of the BESS are given by




KiE
1 0
0
−
3600 , Bi =
Ai =
, Ci = 0 1 .
0 1
0
0
The parameters of this BESS are shown in Table I , together with
P0 = 100. The control goal is to steer the outputs of the BESS
to the optimal allocation of the following resource allocation

Trajectories of y and the network resource constraint gap.

problem:
min

3


fi (yi )

i=1

s.t.

3


yi = 100

i=1

0 ≤ y1 ≤ 45
0 ≤ y2 ≤ 50
0 ≤ y3 ≤ 40.
Following Theorem 2, a distributed controller with eventtriggered communication is obtained, where


1 −0.0003
K 1 = K2 = K3 =
0
2
 
0
X11 = X12 = X21 = X22 = X31 = X32 =
1




−0.3333
−0.3472
−3
−3
, U21 = 10 ×
U11 = 10 ×
0
0


−0.25
U31 = 10−3 ×
0
α = 2, β = 0.05, γ = 1, d10 = d20 = d30 = 1/3.
Apply it to the BESS, and the simulation results are displayed in Figs. 10 and 11. It is observed from Fig. 10 that
the outputs converge asymptotically to the optimal values
and the network resource constraint gap asymptotically converges to 0. Fig. 11 exhibits that the communication frequency is significantly reduced, and Zeno behavior does not
exist. These indicate that the designed event-triggered distributed controller solves the resource allocation problem with
a low communication cost. Therefore, the simulation results
illustrate the effectiveness and advantage of the proposed
approach.

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ZHAO et al.: DISTRIBUTED OPTIMAL RESOURCE ALLOCATION WITH LOCAL FEASIBILITY CONSTRAINTS

Fig. 11.

Triggering instants and interevent times.

V. CONCLUSION
This article presents an integrated event-triggered distributed
control approach for high-order linear MASs over weightbalanced digraphs. It achieves the optimal resource allocation
under local feasibility constraints and network resource constraints. Moreover, the communication burden is significantly
alleviated in the sense that only one signal needs to be broadcasted between the neighboring agents by event-triggered communication. Two examples are given to verify the effectiveness
and show the superiority of the proposed approach. In the future
work, the distributed optimal resource allocation problem for
nonlinear multi-agent MASs will be investigated.
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Zeli Zhao received the M.S. degree in operations research and cybernetics in 2020
from Northeastern University, Shenyang, China,
where she is currently working toward the Ph.D.
degree in control theory and control engineering
with the State Key Laboratory of Synthetical Automation for Process Industry.
Her current research interests include distributed optimization, distributed control, and
multiagent systems.

Jinliang Ding (Senior Member, IEEE) received
the bachelor’s, master’s, and Ph.D. degrees
in control theory and control engineering from
Northeastern University, Shenyang, China, in
2001, 2004, and 2012, respectively.
He is a Professor with the State Key Laboratory of Synthetical Automation for Process
Industry, Northeastern University. He has authored or coauthored more than 200 refereed
journal papers and refereed papers at international conferences. He has also invented or
coinvented more than 50 patents. His research interests include modeling, plant-wide control and optimization for the complex industrial
systems, machine learning, industrial artificial intelligence, and computational intelligence and applications.
Dr. Ding was a recipient of the Young Scholars Science and Technology Award of China in 2016, the National Science Fund for Distinguished
Young Scholars in 2015, the National Technological Invention Award in
2013, and three First-Prize of Science and Technology Awards of the
Ministry of Education in 2006, 2012, and 2018. One of his articles published on Control Engineering Practice was selected for the Best Paper
Award of 2011–2013. He is an Associate Editor for IEEE TRANSACTIONS
ON CIRCUITS AND SYSTEMS-II: EXPRESS BRIEFS.

Jinxi Zhang (Member, IEEE) received the B.S.
degree in automation and the Ph.D. degree
in control theory and control engineering from
Northeastern University, Shenyang, China, in
2014 and 2020, respectively.
From 2019 to 2020, he was a Research
Fellow at the Institute for Intelligent Systems,
Faculty of Engineering and the Built Environment, University of Johannesburg, Johannesburg, South Africa. He is currently a Distinguished Associate Professor with the State Key
Laboratory of Synthetical Automation for Process Industries, Northeastern University. His research interests include nonlinear control, intelligent control, prescribed performance control, multiagent systems, fault
diagnosis and fault-tolerant control, image processing, etc.
Dr. Zhang is an Associate Editor for the International Journal of
Fuzzy Systems. He served as a Guest Editor for IEEE TRANSACTIONS
ON INDUSTRIAL INFORMATICS. He was the recipient of the Young Elite
Scientists Sponsorship Program by China Association for Science and
Technology in 2022.

Tianyou Chai (Life Fellow, IEEE) received the
Ph.D. degree in control theory and engineering
from Northeastern University, Shenyang, China,
in 1985.
In 1988, he became a Professor at the
Northeastern University. From 2010 to 2018, he
served as Director of Department of Information
Science of National Natural Science Foundation
of China. He is the founder and Director of the
Center of Automation, which became a National
Engineering and Technology Research Center
and a State Key Laboratory. He has published 297 peer-reviewed international journal papers. He has developed control technologies with
applications to various industrial processes. His current research interests include modeling, control, optimization, and integrated automation
of complex industrial processes.
Dr. Chai is a Member of the Chinese Academy of Engineering,
and is an IFAC Fellow. His paper titled “Hybrid intelligent control for
optimal operation of shaft furnace roasting process” was selected as
one of three best papers for the Control Engineering Practice Paper
Prize for 2011–2013. For his contributions, he was the recipient of five
prestigious awards of National Natural Science, National Science and
Technology Progress and National Technological Innovation, the 2007
Industry Award for Excellence in Transitional Control Research from
IEEE Multiple-conference on Systems and Control, and the 2017 Wook
Hyun Kwon Education Award from the Asian Control Association.

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