INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL
Online ISSN 1841-9844, ISSN-L 1841-9836, Volume: 17, Issue: 4, Month: August, Year: 2022
Article Number: 4800, https://doi.org/10.15837/ijccc.2022.4.4800

CCC Publications 

Distributed Adaptive Control for Nonlinear Heterogeneous
Multi-agent Systems with Different Dimensions and Time Delay

Bo Qin, Yongqing Fan, Shuo Yang

Bo Qin*
School of Automation, and Xi’an Key Laboratory of Advanced Control and Intelligent Process,
Xi’an University of Posts and Telecommunications, Xi’an, China.
Chang’an West St. Chang’an District, Xi’an, 710121, China
*Corresponding author: qinbo123@xupt.edu.cn

Yongqing Fan
School of Automation, and Xi’an Key Laboratory of Advanced Control and Intelligent Process,
Xi’an University of Posts and Telecommunications, Xi’an, China.
Chang’an West St. Chang’an District, Xi’an, 710121, China
fanyongqing@xupt.edu.cn

Shuo Yang
Beijing Aerospece Institute for Metrology and Measurement Technology,
No. 1, Nandahongmen Road, Fengtai District, Beijing, 100076, China.
tju1105@163.com

Abstract

A distributed neural network adaptive feedback control system is designed for a class of nonlinear
multi-agent systems with time delay and nonidentical dimensions. In contrast to previous works on
nonlinear heterogeneous multi-agent with the same dimension, particular features are proposed for
each agent with different dimensions, and similar parameters are defined, which will be combined
parameters of the controller. Second, a novel distributed control based on similarity parameters
is proposed using linear matrix inequality (LMI) and Lyapunov stability theory, establishing that
all signals in a closed loop system are eventually ultimately bounded. The consistency tracking
error steadily decreases to a field with a small number of zeros. Finally, simulated examples with
different time delays are utilized to test the effectiveness of the proposed control technique.

Keywords: Multi-agent systems, consensus, neural network, nonlinear, time-delay.

1 Introduction
In the recent decade, multi-agent systems have been widely developed in a range of industries.

Examples include unmanned aerial vehicles[1], robot cooperative systems, [2, 3], aerospace systems[4],
and so on. As a result, distributed cooperative control has become a topic interesting research topic
by many scholars. One of the most important characteristics of this design style is that it is capable of



https://doi.org/10.15837/ijccc.2022.4.4800 2

changing the state based on the data provided by each agent and its neighbors. Consistency is one of
the most essential subjects, as evidenced by references [5, 6, 7], which indicate consistency algorithm
studies that are ground-breaking works on multi-agent system consistency. In [8, 9, 10], authors
suggested a number of excellent distributed optimal control research methodologies. The formation
control was proposed in [11] , and [12, 13, 14] explored a wide range of issues. The synchronous
conditions for multi-agent systems were proposed in [15]. The global asymptotic stability and global
robust stability of neural networks with delay were investigated in [16, 17] by using Lyapunov theory
and linear matrix inequality. Because neural networks are a broad approximation strategy for existing
unknown nonlinear functions or uncertain models in many challenging areas, and a large number
of control algorithms for nonlinear multi-agent systems have been proposed in the last few years in
[18, 19, 20].

Regrettably, these findings were limited to homogeneous or heterogeneous nonlinear multi-agent
systems with the same dimension, so the control approaches for multi-agent systems with different
dimensions would be ineffective. It is necessary to seek some unique control technologies to meet
the class of nonlinear heterogeneous multi-agent systems with nonidentical dimensions. Up till now,
the uniformly bounded stability problem of nonlinear multi-agent systems with various nodes and
dimensions is solved in [21, 22, 23], and a novel distributed adaptive control with similar parameters
is proposed. In reference [24], the synchronization problem of dynamic networks with varied nodes
was investigated. However, none of these previous studies take into account time delays, which could
compromise accuracy and render the system unstable.

By above mentioned analyses, in order to solve the consensus of each nonlinear multi-agent system
with different dimensions and time delay dependent, we aim to create a novel distributed neural
network adaptive control in this research. To cope with unknown nonlinear factors, neural networks
are employed to approximate the uncertainties of systems, and then a distributed feedback adaptive
with similar parameters is performed by solving the linear matrix inequality (LMI).

The remainder of the paper is structured as follows: Section II provides some background infor-
mation, a description of the system, and characteristics of similar composite structures and neural
network systems. A neural network adaptive control and its stability analysis are described in Section
III. A simulated example of the consistency of nonlinear heterogeneous multi-agent systems with time
delay is presented in section IV. Section V summarizes the conclusion.

2 The Model of Dynamical Network and Assumptions
A weighted digraph is denotes as G = {V,E,A},where V = [V1,V2, ...,VN ] represents the non-

empty set of each node, edge set E contains one edge (Vj,Vi), adjacency matrix A = [aij] ⊆ RN×N
represents that node Vj can send the information to node Vi. The value aij > 0, if (Vj,Vi) ∈ E,
otherwise aij = 0. In addition, for all i ∈ {1, 2, ...,N}, there exists aii = 0. Diagonal matrix
D = diag{din} ∈ RN×N, where din =

∑N
j=1 aji is degree matrix. Laplacian matrix is defined as

L = D − A. The Kronecker product of matrices A and P is denoted by A ⊗ P , I represents the
appropriate n-dimensional identity matrix. P > 0(P < 0) means P is a positive (negative) matrix,
‖·‖ represents the 2 norm of vector. AT and A−1 denote the transpose matrix and inverse matrix of
A, respectively.

Consider the following dynamical network with nonlinear multiagent system and time delay:

ẋi(t) = Ai1xi(t) + Ai2xi(t− τ(t)) + Bi[ui(t) + fi(xi)], (1)

where xi ∈ Rni is the state of the ith node. Ai ∈ Rni×ni and Bi ∈ Rni×mi are some known matrices.
ui denotes the input vector, τ is vary time delay and satisfies 0 ≤ τ(t) ≤ h < ∞, fi(xi) represents the
unknown nonlinear function.

Assumption 1. Consider the network (1) with N nodes, there exist N matrices, if Fi ∈ Rn0×ni (Fi 6=



https://doi.org/10.15837/ijccc.2022.4.4800 3

0),Ki ∈ Rmi×ni and a known matrix Ji ∈ R1×pi satisfies the following condition

Fi(Ai1 + BiKi1) = (Aj1 + BjKj1Fi).
Fi(Ai2 + BiKi2) = (Aj2 + BjKj2Fi).
FiBi = BjJj.

(2)

Assumption 2. Based on the good approximation performance of the radial basis function neural
network(RBFNN). The RBFNN is used to approximate the unknown nonlinear function. In this
paper, the RBFNN is described as the following form:

fi(xi) = WTi ϕi(xi) + εi(t),∀xi ∈ Ω, (3)

where Wi means the ideal weight matrix that will be designed automatically, εi is denoted as approxima-
tion error satisfying ‖ εi ‖< ε̄i , where ε̄i is a known positive constant. ϕi(xi) = [ϕi1(xi),ϕi2(xi), ...,ϕisi
(xi)] ∈ Rsi×1 characterizes RBFNN vector, in which si is the number of basic functions. Define the
Gaussian function as follows:

ϕi(xi) = exp(−
‖ xi −µisi ‖

2

2σ2isi
), (4)

where µis = [µis1,µis2, ...,µisni ]T represents the center of RBFNN, σisi is the width of function.

Lemma 1. [25]For any given two vectors x,y ∈ Rn×n and a positive matrix H ∈ Rn×n , the following
inequality holds:

2xTy ≤ xTHx + yTH−1y. (5)

Lemma 2. [26]The LMI
[
Q(x) S(x)
ST (x) R(x)

]
> 0 is equavalent to one of the following conditions:

1)Q(x) > 0,R(x) −ST (x)Q−1(x)S(x) > 0; 2)R(x) > 0,Q(x) −ST (x)R−1(x)S(x) > 0, (6)

where Q(x) = QT (x),R(x) = RT (x).

Lemma 3. There exist positive definite matrices S > 0, T11 > 0, non-negative definite matrices
T22 = TT22 ≥ 0, and any matrices of appropriate dimensions Y1,Y2,T12, such that the following LMIs
are feasible: 

 S Y1 Y2∗ T11 T12
∗ ∗ T22


 ≥ 0. (7)

The following inequality holds:

−
∫ t
t−τ

żT (α)Sż(α)dα ≤ ηT (t)Ψη(t) + τδT (t)Tδ(t), (8)

where ηT (t) =
[
zT (t) żT (t) zT (t− τ)

]
, δT (t) =

[
zT (t) żT (t)

]
, Ψ =


 Y

T
1 + Y1 Y2 −Y T1
∗ 0 −Y T2
∗ ∗ 0


,

T =
[
T11 T12
∗ T22

]
.

Proof. For ∀H1,H1 ∈ Rn×n, the following inequality holds

0 = 2[zT (t)HT1 + ż
T (t)HT2 ][z(t) −z(t− τ) −

∫ t
t−τ

ż(α)dα]

= 2zT (t)HT1 z(t) − 2z
T (t)HT1 z(t− τ) + 2ż

T (t)HT2 z(t) − 2ż
T (t)HT2 z(t− τ)

− 2[zT (t)HT1 + ż
T (t)HT2 ]

∫ t
t−τ

ż(α)dα. (9)



https://doi.org/10.15837/ijccc.2022.4.4800 4

Let H =
[
H1 H2

]
, Y =

[
Y1 Y2

]
, T =

[
T11 T12
∗ T22

]
, so it knows that

− 2δT (t)HT
∫ t
t−τ

ż(α)dα ≤
∫ t
t−τ

[
ż(α)
δ(t)

]T [
X Y −H
∗ T

] [
ż(α)
δ(t)

]
dα. (10)

The right of the above inequality is equal to∫ t
t−τ

ż(α)Xż(α)dα + 2
∫ t
t−τ

δT (t)(Y T −HT )ż(α)dα +
∫ t
t−τ

δT (t)Tδ(t)dα

=
∫ t
t−τ

ż(α)Xż(α)dα + ηT (t)


 Y

T
1 + Y1 −HT1 −H1 Y2 −H2 −Y T1 + HT1

∗ 0 −Y T1 + HT2
∗ ∗ 0


 η(t) + τδT (t)Tδ(t).

(11)
Taking (10) into (11), inequality (8) can be guaranteed. The proof of Lemma 3 is completed.

3 The Control Design
In this section, a decentralized controller with similar parameter is proposed for the consensus of

multi-agent system with different dimension and time-delay.
The following controller is designed:

ui = Ki1xi(t) + Ki2xi(t− τ(t)) + K̄1[Fixi(t) −Fjxj(t)] + K̄2[Fixi(t− τ(t)) −Fjxj(t− τ(t))]

+ ciK̄i
N∑
j=1

aij[Fixi(t) −Fjxj(t)] + ciK̄2
N∑
j=1

aij[Fixi(t− τ(t)) −Fjxj(t− τ(t))] −W̄Ti ϕi(xi), (12)

where W̄i is the estimated value of Wi, W̃i = W̄i −Wi is the estimated error. Here, the adaptive law
of W̄i is designed as follows:

˙̄
Wi = −γwiW̄i + ηwiϕi(xi)(PBjJj)Tei(t). (13)

The adaptive law of coupling weight ci is designed as:

ċi = −γcici −βc1ie
T
1 (t)(PBjJjK̄1)e1(t) −βc2ie

T
2 (t)(PBjJjK̄2)e2(t). (14)

We define x̄i(t) = Fixi(t), denote zij(t) = xi(t) −xj(t) and zij(t− τ(t)) = xi(t− τ(t)) −xj(t− τ(t)),
then get ˙̄xi(t) = Fiẋi(t). By using controller (12), adaptive laws (13)-(14), the following dynamic
equation holds:

˙̄xi(t) = Fi{Ai1xi(t) + Ai2xi(t− τ(t)) + BiKi1xi(t) + BiKi2xi(t− τ(t)) + BiK̄1[Fixi(t) −Fjxj(t)]

+BiK̄2[Fixi(t− τ(t)) −Fjxj(t− τ(t))] + ciBiK̄1
N∑
j=1

aij[Fixi(t) −Fjxj(t)]

+ciBiK̄2
N∑
j=1

aij[Fixi(t− τ(t)) −Fjxj(t− τ(t))] + Bi[fi(xi) −ŴTi ϕi(xi)]}

= Fi[Ai1 + BiKi1]xi(t) + Fi[Ai2 + BiKi2]xi(t− τ(t)) + FiBiK̄1zij(t) + FiBiK̄2zij(t− τ(t))

+ciFiBiK̄1
N∑
j=1

aij[Fixi(t) −Fjxj(t)] + ciFiBiK̄2
N∑
j=1

aij[Fixi(t− τ(t)) −Fjxj(t− τ(t))]



https://doi.org/10.15837/ijccc.2022.4.4800 5

+FiBi[fi(xi) −W̄Ti ϕi(xi)]. (15)

According to Assumption 1, it gets

˙̄xi(t) = (Aj1 + BjKj1)Fixi(t) + (Aj2 + BjKj2)Fixi(t− τ(t)) + BjJjK̄1zij(t) + BjJjK̄2zij

(t− τ(t)) + ciBjJjK̄1ei(t) + ciBjJjK̄2ei(t− τ(t)) + BjJj[−W̃Ti ϕi(xi) + εi]. (16)

At the same time, the jth agent is the tracked system which can be rewritten as:

ẋj(t) = Aj1xj(t) + Aj2xj(t− τi) + Bj[uj + fj(xj)]. (17)

Define the consensus tracking error as:

żij(t) = ˙̄xi(t) − ˙̄xj(t), (18)

then the dynamical of tracking error equation is deduced as:

żij(t) = (Aj1 + BjKj1 + BjJjK̄1)zij(t) + (Aj2 + BjKj2 + BjJjK̄2)zij(t− τ(t))

+ciBjJjK̄1ei(t) + ciBjJjK̄2eij(t− τ(t)) + BjJj[−W̃Ti ϕi(xi,xj) + εij], (19)

for simplicity, (19) can be further transformed as:

ż = [IN ⊗ (Aj1 + BjKj1 + BjJjK̄1)]z(t) + cH ⊗ (BjJjK̄1)z(t) + [IN ⊗ (Aj2 + BjKj2 + BjJjK̄2)]

z(t− τ(t)) + cH ⊗ (BjJjK̄2)z(t− τ(t)) + {IN ⊗ (BjJj)
N∑
i=1

[−W̃Tijϕij(xi,xj) + εij]}, (20)

where z(t) = [zT1 (t),zT2 (t), · · · ,zTN (t)]
T with zTi (t) = [zTi1(t),zTi2(t)], · · · ,zTij(t)]]T , z(t− τ(t)) = [zT1 (t−

τ(t)),zT2 (t−τ(t)), · · · ,zTN (t−τ(t))]
T , c = [cT1 ,cT2 , · · · ,cTN ]

T . Hence (20) is equal to the following form:

ż(t) = [IN ⊗(Aj1 +BjKj1 +BjJjK̄1)]z(t) + [IN ⊗(Aj2 +BjKj2 +BjJjK̄2)]z(t−τ(t)) + [cL⊗(BjJjK̄1)]

z(t) + [cL⊗ (BjJjK̄2)]z(t− τ(t)) + {IN ⊗ (BjJj)
N∑
i=1

BjJj[−W̃Tijϕ(xi,xj) + εij]}, (21)

where matrices Fi and Ki are similar parameters that are defined in Assumption 1, K̄1 and K̄2 are
two the gain matrices to be designed, which can be obtained by solving the following LMIs:

φ̄ =


 φ̄11 φ̄12 φ̄13∗ φ̄22 φ̄23
∗ ∗ φ̄33


 < 0, (22)


 S̄ Ȳ1 Ȳ2∗ T̄11 T̄12
∗ ∗ T̄22


 ≥ 0, (23)

in which φ̄11 = V ATj1 + Aj1V + V MT1 + M1V + GT1 NT + NG1 + Q̂ + R̂ + Ȳ T1 + Ȳ1 + τT̄11 + Q̂11,
φ̄12 = Ȳ2 +τT̄12, φ̄13 = Aj2V +M2V +NG2−Ȳ T1 , φ̄22 = τS̄+τT̄22 +Q̂22, φ̄23 = −Ȳ T2 , φ̄33 = −Q̂++Q̂33,
K̄1 = G1V −1, K̄2 = G2V −1.

Theorem 1. Suppose Assumption 1-2 are satisfied, if there exist some positive definite matrices V > 0,
Q̂ > 0, R̂ > 0, Q̂11 > 0, Q̂22 > 0, Q̂33 > 0, S̄ > 0, and proper dimension matrices M1, M1, G1, G2,
Ȳ1, Ȳ2, T̄11, T̄12, T̄22 such that (22) and (23) can be satisfied, then the multi-agent system (1) in a
connected graph can be guaranteed to consensus by applying the feedback neural network control (12)
with adaptive laws (13) and (14), and all the signals in closed-loop system are bounded.



https://doi.org/10.15837/ijccc.2022.4.4800 6

Proof: The following candidate Lyapunov function is considered:

V (t) = zT (t)(IN ⊗P)z(t) +
∫ t
t−τ

zT (α)(IN ⊗Q)z(α)dα +
∫ 0

−τ

∫ t
t+β

żT (α)(IN ⊗S)ż(α)dαdβ

+
1
2
β−1c c

Tc +
1
2
η−1w tr(W̃

TW̃). (24)

Denoting Ā1 = Aj1 + BjKj1 + BjJjK̄1 and Ā2 = Aj2 + BjKj2 + BjJjK̄2, By using (13) and (14),
then it has

V̇ (t) = żT (t)(IN ⊗P)z(t) + zT (t)(IN ⊗P)ż(t) + żT (t)(IN ⊗Q)z(t) −zT (t− τ)(IN ⊗Q)z(t− τ)

+τżT (t)(IN ⊗S)ż(t) −
∫ t
t−τ

żT (α)(IN ⊗S)ż(α)dα−
N∑
i=1

γci
βci

c2i

+2zT (t)(IN ⊗P)[IN ⊗ (BjJj)]
N∑
i=1

[−W̃Tijϕij(xi,xj) + εij] +
1
ηw
tr(W̃T ˙̄W)

= zT (t)[IN ⊗ (ĀT1 P + PĀ1)]z(t) + z
T (t)[IN ⊗ (ĀT2 P + PĀ2)]z(t− τ) − 2z

T (t)[cL⊗ (PBJK̄1)]z(t)

−2zT (t)[cL⊗ (PBJK̄2)]z(t− τ) + 2zT (t)[I ⊗ (PBJ)]
N∑
i=1

[−W̃Tij (t)ϕij(xi,xj) + εij(t)]

+zT (t)(IN ⊗Q)z(t) −zT (t− τ)(IN ⊗Q)z(t− τ) + τżT (t)(I ⊗S)ż(t)

−
∫ t
t−τ

żT (α)(I ⊗S)ż(α)dα−
N∑
i=1

γwi
ηwi

W̃TijW̄ij −
N∑
i=1

γci
βci

c2i . (25)

According to Lemma 3, it becomes

V̇ (t) ≤ zT (t)[IN ⊗ (ĀT1 P + PĀ1)]z(t) + z
T (t)[IN ⊗ (PĀ2 + ĀT2 P)]z(t− τ)

−2zT (t)[cL⊗ (PBJK̄1)]z(t) − 2zT (t)[cL⊗ (PBJK̄2)]z(t− τ)

+2zT (t)[IN ⊗ (PBJ)]
N∑
i=1

[−W̃Tij (t)ϕij(xi,xj) + εij(t)] + z
T (t)(IN ⊗Q)z(t)

−zT (t− τ)(IN ⊗Q)z(t− τ) + τżT (t)(IN ⊗S)ż(t) −
∫ t
t−τ

żT (α)(IN ⊗S)ż(α)dα

+ ηT (t)(IN ⊗ Ψ)η(t) + τzT (t)(IN ⊗T)z(t) −
N∑
i=1

γwi
ηwi

W̃TijW̄ij −
N∑
i=1

γci
βci

c2i . (26)

Denoting
n∑
i=1

εij = ε̄j, because the following two inequalities hold

2zT (t)[IN ⊗ (PBjJj)ε̄j] ≤ zT (t)(IN ⊗R)z(t) + ε̄Tj [IN ⊗ (J
T
j B

T
j P

TR−1PBjJj)]ε̄j. (27)

−
γwi
ηwi

W̃TijW̄ij ≤−
γwi
ηwi

W̃TijW̃ij +
γwi
2ηwi

W̃TijW̃ij +
γwi
2ηwi

WTijWij = −
γwi
2ηwi

W̃TijW̃ij +
γwi
2ηwi

WTijWij. (28)

Combining with (14) and (15), the following inequality is obtained:

V̇ ≤ ηT (t)φη(t)−
γw
2ηw

tr(W̃TW̃) +
γw
2ηw

tr(WTW)−
γc
2βc

cTc + ε̄Tj [IN ⊗(J
T
j B

T
j P

TR−1PBjJj)]ε̄j, (29)

where φ =


 φ11 φ12 φ13∗ φ22 φ23
∗ ∗ φ33


 with φ11 = ĀT1 P + PĀ1 + Q + R + Y T1 + Y1 + τT11, φ12 = Y2 + τT12,

φ13 = PĀ2 −Y T1 , φ22 = τ(S + T22), φ23 = −Y T2 , φ33 = −Q.



https://doi.org/10.15837/ijccc.2022.4.4800 7

Let φ < −Ω, in which Ω = diag{Q̄1,Q̄2,Q̄3}, multiplying diag{P −1,P −1,P −1} on both side of
this inequality, define P −1 = V , then one has

φ̃ =


 φ̃11 φ̃12 φ̃13∗ φ̃22 φ̃23
∗ ∗ φ̃33


 < 0, (30)

where φ̃11 = V ĀT1 +Ā1V +V QV +V RV +V Y T1 V +V Y1V +τV T11V +V Q̄1V , φ̃12 = V Y2V +τV T12V ,
φ̃13 = Ā2V −V Y T1 V , φ̃22 = τ(V SV + V T22V ) + V Q̄2V , φ̃23 = −V Y T2 V , φ̃33 = −V QV + V Q̄3V .

Now, we let BjK̄j1 = M1, BjJj = N, K̄1V = G1, V Y1V = Ȳ1, V Y2V = Ȳ2, V T11V = T̄11,
V QV = Q̂, V RV = R̂, V SV = S̄, BjK̄j2 = M2, K̄2V = G2, then (29) is equal to (22).

Similarly, multiplying diag{P −1,P −1,P −1} on the right and left of (7), so it becomes inequality
(23).

Based on the inequality (30), let δ = 12tr(W
TW) + ε̄Tj [IN ⊗(JTj BTj PTR−1PBjJj)]ε̄j, (29) becomes

the following result:
V̇ ≤−ζV (t) + δ, (31)

inequality (31) means that all the signals in closed-loop multi-agent system are bounded.

4 Experimental Results and Analysis
In this section, a simulation example is given to prove the effectiveness of the proposed control

method.We consider an multi-agent systems with eight subsystem, which consisting of a leader labeled
1 and seven followers labeled 2, 3, 4, 5, 6, 7, 8, that are shown as Figure 1.

Figure 1: network topology with eight agents

The adjacency matrix is: 


0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 1 0 0 1 0 0 0
0 0 0 0 0 1 0 1
0 0 1 0 0 0 0 0
0 0 1 1 0 0 0 0
0 0 1 0 0 1 0 1
0 0 1 1 1 0 0 0



. (32)

In this multi-agent system, the matrices are provided with different dimensions as: A11 =
[

2 1
−1 −2

]
,



https://doi.org/10.15837/ijccc.2022.4.4800 8

A21 =


 0.1 −0.3 −0.40.2 0.1 0.5

0 0 −0.5


, A31 =




0.1 −0.3 −0.4 0.1
0.2 0.1 0.5 0.3
0 0 −0.5 0
0 0 0 −0.3


,

A41 =




0.1 −0.3 −0.4 0.1 0.2
0.2 0.1 0.5 0.3 −0.1
0 0 −0.5 0 0
0 0 0 −0.3 0
0 0 0 0 −0.2


, A51 =




0.1 −0.3 −0.4 0.1 0.2 0.2
0.2 0.1 0.5 0.3 −0.1 0.2
0 0 −0.5 0 0 0
0 0 0 −0.3 0 0
0 0 0 0 −0.2 0
0 0 0 0 0 −0.5



,

A61 =




0.1 −0.3 −0.4 0.1 0.2 0.2 −0.2
0.2 0.1 0.5 0.3 −0.1 0.2 −0.5
0 0 −0.5 0 0 0 0
0 0 0 −0.3 0 0 0
0 0 0 0 −0.2 0 0
0 0 0 0 0 −0.5 0
0 0 0 0 0 0 −0.3



,

A71 =




0.1 −0.3 −0.4 0.1 0.2 0.2 −0.2 0.1
0.2 0.1 0.5 0.3 −0.1 0.2 −0.5 0.5
0 0 −0.5 0 0 0 0 0
0 0 0 −0.3 0 0 0 0
0 0 0 0 −0.2 0 0 0
0 0 0 0 0 −0.5 0 0
0 0 0 0 0 0 −0.3 0
0 0 0 0 0 0 0 −0.4



,

A81 =




0.1 −0.3 −0.4 0.1 0.2 0.2 −0.2 0.1 0.2
0.2 0.1 0.5 0.3 −0.1 0.2 −0.5 0.5 0.4
0 0 −0.5 0 0 0 0 0 0
0 0 0 −0.3 0 0 0 0 0
0 0 0 0 −0.2 0 0 0 0
0 0 0 0 0 −0.5 0 0 0
0 0 0 0 0 0 −0.3 0 0
0 0 0 0 0 0 0 −0.4 0
0 0 0 0 0 0 0 0 −0.2



. A11 =

[
1 0.5
−0.5 0

]
,

A12 =
[

1 1
1 2

]
, A22 =


 0.2 −0.1 −0.40.3 0.2 0.5

0 0 −0.3


, A32 =




0.2 −0.1 −0.4 −0.1
0.3 0.2 0.5 0.2
0 0 −0.3 0
0 0 0 −0.4


,

A42 =




0.2 −0.1 −0.4 −0.1 0.2
0.3 0.2 0.5 0.2 −0.1
0 0 −0.3 0 0
0 0 0 −0.4 0
0 0 0 0 −0.1


, A52 =




0.2 −0.1 −0.4 −0.1 0.2 0.3
0.3 0.2 0.5 0.2 −0.1 −0.3
0 0 −0.3 0 0 0
0 0 0 −0.4 0 0
0 0 0 0 −0.1 0
0 0 0 0 0 −0.2



,

A62 =




0.2 −0.1 −0.4 −0.1 0.2 0.3 −0.1
0.3 0.2 0.5 0.2 −0.1 −0.3 0.2
0 0 −0.3 0 0 0 0
0 0 0 −0.4 0 0 0
0 0 0 0 −0.1 0 0
0 0 0 0 0 −0.2 0
0 0 0 0 0 0 −0.5



,



https://doi.org/10.15837/ijccc.2022.4.4800 9

A72 =




0.2 −0.1 −0.4 −0.1 0.2 0.3 −0.1 0.5
0.3 0.2 0.5 0.2 −0.1 −0.3 0.2 0.1
0 0 −0.3 0 0 0 0 0
0 0 0 −0.4 0 0 0 0
0 0 0 0 −0.1 0 0 0
0 0 0 0 0 −0.2 0 0
0 0 0 0 0 0 −0.5 0
0 0 0 0 0 0 0 −0.3



,

A82 =




0.2 −0.1 −0.4 −0.1 0.2 0.3 −0.1 0.5 0.3
0.3 0.2 0.5 0.2 −0.1 −0.3 0.2 0.1 −0.1
0 0 −0.3 0 0 0 0 0 0
0 0 0 −0.4 0 0 0 0 0
0 0 0 0 −0.1 0 0 0 0
0 0 0 0 0 −0.2 0 0 0
0 0 0 0 0 0 −0.5 0 0
0 0 0 0 0 0 0 −0.3 0
0 0 0 0 0 0 0 0 −0.1



,

B1 =
[

1 0
0 1

]T
, B2 =

[
1 0 0
0 1 0

]T
, B3 =

[
1 0 0 0
0 1 0 0

]T
, B4 =

[
1 0 0 0 0
0 1 0 0 0

]T
,

B5 =
[

1 0 0 0 0 0
0 1 0 0 0 0

]T
, B6 =

[
1 0 0 0 0 0 0
0 1 0 0 0 0 0

]T
, B7 =

[
1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0

]T
,

B8 =
[

1 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0

]T
, F1 =

[
1 0
0 1

]
, F2 =

[
1 0 0
0 1 0

]
, F3 =

[
1 0 0 0
0 1 0 0

]
,

F4 =
[

1 0 0 0 0
0 1 0 0 0

]
, F5 =

[
1 0 0 0 0 0
0 1 0 0 0 0

]
, F6 =

[
1 0 0 0 0 0 0
0 1 0 0 0 0 0

]
,

F7 =
[

1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0

]
, F8 =

[
1 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0

]
, K11 =

[
−6 −3
−1 0

]
,

K21 =
[
−4.1 −1.7 0.4
−2.2 −2.1 −0.5

]
, K31 =

[
−4.1 −1.7 0.4 −0.1
−2.2 −2.1 −0.5 −0.3

]
,

K41 =
[
−4.1 −1.7 0.4 −0.1 −0.2
−2.2 −2.1 −0.5 −0.3 0.1

]
, K51 =

[
−4.1 −1.7 0.4 −0.1 −0.2 −0.2
−2.2 −2.1 −0.5 −0.3 0.1 −0.2

]
,

K61 =
[
−4.1 −1.7 0.4 −0.1 −0.2 −0.2 0.2
−2.2 −2.1 −0.5 −0.3 0.1 −0.2 0.5

]
,

K71 =
[
−4.1 −1.7 0.4 −0.1 −0.2 −0.2 0.2 −0.1
−2.2 −2.1 −0.5 −0.3 0.1 −0.2 0.5 −0.5

]
,

K81 =
[
−4.1 −1.7 0.4 −0.1 −0.2 −0.2 0.2 −0.1 0.2
−2.2 −2.1 −0.5 −0.3 0.1 −0.2 0.5 −0.5 −0.4

]
. K12 =

[
−2 1
−2 −2

]
,

K22 =
[
−1.2 2.1 0.4
−1.3 −0.2 −0.5

]
, K32 =

[
−1.2 2.1 0.4 0.1
−1.3 −0.2 −0.5 −0.2

]
,

K42 =
[
−1.2 2.1 0.4 0.1 −0.2
−1.3 −0.2 −0.5 −0.2 0.1

]
,

K52 =
[
−1.2 2.1 0.4 0.1 −0.2 −0.3
−1.3 −0.2 −0.5 −0.2 0.1 0.3

]
,

K62 =
[
−1.2 2.1 0.4 0.1 −0.2 −0.3 0.1
−1.3 −0.2 −0.5 −0.2 0.1 0.3 −0.2

]
,

K72 =
[
−1.2 2.1 0.4 0.1 −0.2 −0.3 0.1 −0.5
−1.3 −0.2 −0.5 −0.2 0.1 0.3 −0.2 −0.1

]
,

K82 =
[
−1.2 2.1 0.4 0.1 −0.2 −0.3 0.1 −0.5 −0.3
−1.3 −0.2 −0.5 −0.2 0.1 0.3 −0.2 −0.1 0.1

]
. By solving the LMIs (19) and



https://doi.org/10.15837/ijccc.2022.4.4800 10

(20), time delay, the positive matrix and control gain matrices are obtained respectively as: τ = 0.2

P =
[

0.2399 −0.1842
−0.1842 0.6083

]
, K̄1 =

[
−0.4473 5.7975
4.4031 −7.4369

]
, K̄2 =

[
0.2569 0.4542
0.0320 0.7431

]
.

The nonlinear functions are chosen as:


f(x1) = [−x11sin(x12) + x11sin(x11)cos(x12); x11sin(x12)].
f(x2) = [−x21sin(x22) + x22sin(x23)cos(x22); x21sin(x22)].
f(x3) = [−x31sin(x32) + x32sin(x33)cos(x32); x31sin(x34)].
f(x4) = [−x41sin(x42) + x43sin(x41)cos(x44); x41sin(x45)].
f(x5) = [−x51sin(x52) + x53sin(x54)cos(x52); x55sin(x56)].
f(x6) = [−x61sin(x62) + x63sin(x64)cos(x65); x66sin(x67)].
f(x7) = [−x71sin(x72)cos(x74) + x73sin(x75)cos(x76); x77sin(x78)].
f(x8) = [−x81sin(x82)cos(x83) + x84sin(x85)cos(x86); x87sin(x88)cos(x89)].

(33)

The initial values of the states in the multi-agent system are chosen as: x1(0) =
[

0.2 0.18
]
, x2(0) =[

0.3 −0.16 0.1
]
, x3(0) =

[
0.19 0.24 0.15 0.2

]
, x4(0) =

[
0.11 0.29 −0.1 0.16 0.24

]
,

x5(0) =
[
−0.9 0.6 0.21 −0.11 −0.13 0.22

]
, x6(0) =

[
0.25 0.2 0.1 0.24 0.2 0.11 0.2

]
,

x7(0) =
[

0.18 0.21 −0.1 0.16 0.21 0.1 0.14 −0.11
]
,

x8(0) =
[

0.24 0.2 0.11 0.1 0.1 0.14 0.21 −0.1 −0.12
]
. The initial values of adaptive pa-

rameters W̄i(t) are given as:
W̄1(0) =

[
0.49 0.45 0.41 0.46 0.39

]T
, W̄2(0) =

[
0.9 0.22 0.10 0.10 0.13

]T
,

W̄3(0) =
[

0.10 0.27 0.31 0.11 0.21
]T

, W̄4(0) =
[

0.19 0.15 0.17 0.14 0.21
]T

,

W̄5(0) =
[

0.15 0.12 0.25 0.21 0.31
]T

, W̄6(0) =
[

0.6 0.3 0.7 0.2 0.9
]T

,

W̄7(0) =
[

0.17 0.19 0.15 0.17 0.14
]T

, W̄8(0) =
[

0.4 0.5 0.7 0.3 0.2
]T

. The simulation
results are illustrated in Fig.2-4.

Figure 2: Trajectiories of the state of xi1,xi2,xi3,xi4,xi5,xi6,xi7,xi8,xi9



https://doi.org/10.15837/ijccc.2022.4.4800 11

Figure 3: Time response of the adaptive estimation parameters W̄i

Figure 4: Time response of coupling laws ci

The states of multi-agents with diverse dimensions and time delays can attain uniform stability
using the developed distributed feedback neural network adaptive control, as shown in subfigures (a)-
(i) in Fig. 2. Fig. 3 and Fig. 4 show the time responses of RBFNN estimate parameters and coupling
weights, which are guaranteed to be semi-globally uniformly ultimately bounded.

5 Conclusion
This research focus on the consensus control of nonlinear heterogeneous multi-agent systems with

time delays and nonidentical dimensions. A distributed neural network adaptive control based on
similar parameters and time delay is built, with the goal of achieving the consensus of all the agents,
and the neural network systems are used to approximate the unknown nonlinear functions in the
subsystems. Using the suggested control approach, the states of each follower system can track the
state of the leader system. The control technique can be used to control both homogeneous multi-agent
systems of the same dimension and agents of different dimensions.

Funding

This work is supported by the National Natural Science Foundation of China under Grant (62003262,
61903298), National Natural Science Foundation of Shaanxi under Grant (2022JM-389, 2019JQ-
341),Shaanxi Provincial Department of Science and Technology key project in the field of industry
(2018ZDXM-GY-039), Shaanxi Provincial Scientific and Technological Activities for Overseas Staff
Preferential Projects under Grant 35.



https://doi.org/10.15837/ijccc.2022.4.4800 12

Author contributions

The authors contributed equally to this work.

Conflict of interest

The authors declare no conflict of interest.

References
[1] Seo, J. and Kim, Y. and Kim, S. and Tsourdos, A. (2017). Collision Avoidance Strategies for

Unmanned Aerial Vehicles in Formation Flight, IEEE Transactions on Aerospace and Electronic
Systems, 53(6), 2718-2734, 2017.

[2] Zhang, B. and Shang, W. and Cong, S. and Li, Z. (2021). Coordinated Dynamic Control in the
Task Space for Redundantly Actuated Cable-Driven Parallel Robots, IEEE/ASME Transactions
on Mechatronics, 26(5), 2396-2407, 2021.

[3] Sieber, D. and Hirche, S. (2019). Human-Guided Multirobot Cooperative Manipulation,
IEEE/ASME Transactions on Control Systems Technology, 27(4), 1492-1509, 2019.

[4] Sharf, M. and Zelazo, D. (2019). Analysis and Synthesis of MIMO Multi-Agent Systems Using
Network Optimization, arXiv e-prints, 64(11), 4512-4524, 2019.

[5] Reynolds, C. W. and Flocks, H. (1987). And Schools: A Distributed Behavioral Model ACM
SIGGRAPH Computer Graphics, 21(4), 25-34, 1987.

[6] Vicsek, T. and Czirok, A. and Ben-Jacob, E. and Cohen, I. and Sochet, O. (1995). Novel Type of
Phase Transition in a System of Self-Driven Particles, Physical Review Letters, 75(6), 1226-1229,
1995.

[7] Jadbabaie, A. and Jie, L. and Morse, A. S. (2003). Coordination of groups of mobile autonomous
agents using nearest neighbor rules, IEEE Transactions on Automatic Control, 48(6), 988-1001,
2003.

[8] Liu, D. and Wang, D. and Li, H. (2014). Decentralized Stabilization for a Class of Continuous-
Time Nonlinear Interconnected Systems Using Online Learning Optimal Control Approach, IEEE
Transactions on Neural Networks and Learning Systems, 25(2), 418-428, 2014.

[9] Han, X. and Mandal, S. and Pattipati, K. R. and Kleinman, D. L. and Mishra, M. (2017). An
Optimization-Based Distributed Planning Algorithm: A Blackboard-Based Collaborative Frame-
work, IEEE Transactions on Systems Man and Cybernetics Systems, 44(6), 673-686, 2017.

[10] Zhang, H. and Feng, T. and Yang, G. H. and Liang, H. (2010). Distributed Cooperative Opti-
mal Control for Multiagent Systems on Directed Graphs: An Inverse Optimal Approach, IEEE
Transactions on Cybernetics, 45(7), 1315-1326, 2015.

[11] Lin, J. and Hwang, K. and Wang, Y. (2014). A Simple Scheme for Formation Control Based
on Weighted Behavior Learning, IEEE Transactions on Neural Networks and Learning Systems,
25(6), 1033-1044, 2014.

[12] Loia, V. and Vaccaro, A. (2014). Decentralized Economic Dispatch in Smart Grids by Self-
Organizing Dynamic Agents, IEEE Transactions on Systems Man and Cybernetics Systems,
44(4), 397-408, 2014.

[13] Perez, I. J. and Cabrerizo, F. J. and Alonso, S. and Herrera-Viedma, E. (2017). A New Consensus
Model for Group Decision Making Problems With Non-Homogeneous Experts, IEEE Transactions
on Systems Man and Cybernetics Systems, 44(4), 494-498, 2017.



https://doi.org/10.15837/ijccc.2022.4.4800 13

[14] Owliya, M. and Saadat, M. and Jules, G. G. and Goharian, M. and Anane, R. (2013). Agent-Based
Interaction Protocols and Topologies for Manufacturing Task Allocation, IEEE Transactions on
Systems, Man, and Cybernetics: Systems, 43(1), 38-52, 2013.

[15] Duan, Z. and Wang, Q. and Feng, Z. and Chen, G. (2008). Estimation of Delay on Synchronization
Stability in a Class of Complex Systems with Coupling Delays, Taiwanese Journal of Mathematics,
12(8), 2141-2154, 2008.

[16] Cao, Y. J. and Wu, Q. H. (1996). A note on stability of analog neural networks with time delays,
IEEE Transactions on Neural Networks, 7(6), 1533-1535, 1996.

[17] Chen, T. (2001). Global convergence of delayed dynamical systems, IEEE Transactions on Neural
Networks, 12(6), 1532-1536, 2001.

[18] Peng, Z. and Wang, D. and Zhang H. and Sun G. (2014). Distributed Neural Network Control
for Adaptive Synchronization of Uncertain Dynamical Multiagent Systems, IEEE Transactions
on Neural Networks and Learning Systems, 25(8), 1508-1519, 2014.

[19] Howard A. (2006). Experiments with a Large Heterogeneous Mobile Robot Team: Exploration,
Mapping, Deployment and Detection, International Journal of Robotics Research, 25(5), 431-447,
2006.

[20] Chung, S. J. and Slotine, J. (2009). Cooperative Robot Control and Concurrent Synchronization
of Lagrangian Systems, IEEE Transactions on Robotics, 25(3), 686-700, 2009.

[21] Fan, Y. and Xiao, T. and Li, Z. (2020). Distributed Fuzzy Adaptive Control for Heteroge-
neous Nonlinear Multiagent Systems with Similar Composite Structure, Complexity, https://doi:
10.1155/2020/4081904, 2020.

[22] Qin, B. and Fan, Y. and Xiao, T. and Li, Z. (2021). Distributed type-2 fuzzy adap-
tive control for heterogeneous nonlinear multiagent systems, Asian Journal of Control,
https://doi:org/10.1002/asjc.2566, 2021.

[23] Zhao, J. and Hill, D. J. and Tao, L. (2011). Synchronization of Dynamical Networks With Non-
identical Nodes: Criteria and Control, IEEE Transactions on Circuits and Systems I: Regular
Papers, 58(3), 584-594, 2011.

[24] Lv, Y. and Li, Z. and Duan, Z. and Feng, G. (2015). Novel Distributed Robust Adaptive Consen-
sus Protocols for Linear Multi-agent Systems with Directed Graphs and External Disturbances,
International Journal of Control, 90(2), 137-147, 2015.

[25] Tong, S. and Wang, T. and Li, Y. (2014). Fuzzy Adaptive Actuator Failure Compensation Control
of Uncertain Stochastic Nonlinear Systems With Unmodeled Dynamics, IEEE Transactions on
Fuzzy Systems, 22(3), 563-574, 2014.

[26] Last, E. (2014). Linear Matrix Inequalities in System and Control Theory, Proceedings of the
IEEE, 86(12), 2473-2474, 1994.



https://doi.org/10.15837/ijccc.2022.4.4800 14

Copyright ©2022 by the authors. Licensee Agora University, Oradea, Romania.
This is an open access article distributed under the terms and conditions of the Creative Commons
Attribution-NonCommercial 4.0 International License.
Journal’s webpage: http://univagora.ro/jour/index.php/ijccc/

This journal is a member of, and subscribes to the principles of,
the Committee on Publication Ethics (COPE).

https://publicationethics.org/members/international-journal-computers-communications-and-control

Cite this paper as:
Qin, B.; Fan, Y.; Yang, S. (2022). (2022). Distributed Adaptive Control for Nonlinear Heterogeneous
Multi-agent Systems with Different Dimensions and Time Delay, International Journal of Computers
Communications & Control, 17(4), 4800, 2022.

https://doi.org/10.15837/ijccc.2022.4.4800


	Introduction
	The Model of Dynamical Network and Assumptions
	The Control Design
	Experimental Results and Analysis
	Conclusion