"2020期刊：二阶符号网络时滞控制器设计和分析"
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The article "Distributed Controller Design and Analysis of Second-Order Signed Networks With Communication Delays" published in the IEEE Transactions on Neural Networks and Learning Systems focuses on the development of a distributed control system for second-order signed networks with communication delays. The research, conducted by Mingjun Du, Deyuan Meng, and Zheng-Guang Wu, addresses the challenges of controlling complex networks with delays in communication.
The study proposes a distributed controller design approach to effectively manage the dynamics of second-order signed networks. By incorporating communication delays into the control system, the researchers aim to improve the stability and performance of the network. The analysis presented in the article demonstrates the effectiveness of the proposed controller design in mitigating the impact of delays on network behavior.
Overall, this research contributes to the advancement of distributed control systems for complex networks, offering insights into the design and analysis of second-order signed networks with communication delays. The findings of this study have the potential to enhance the efficiency and reliability of networked systems in various applications.
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.
DU et al.: DISTRIBUTED CONTROLLER DESIGN AND ANALYSIS OF SECOND-ORDER SIGNED NETWORKS 3
III. CONSENSUS RESULTS OF SECOND-ORDER SNSS
A distributed control protocol is given by
u
i
(t) =−k
1
j ∈N (i)
|a
ij
|[x
i
(t) − sgn(a
ij
)x
j
(t)]
−k
2
j ∈N (i)
|a
ij
|[ν
i
(t) − sgn(a
ij
)ν
j
(t)], i ∈ F
n
(2)
where k
1
and k
2
are control gains. Let
x(t) =[x
1
(t), x
2
(t),...,x
n
(t)]
T
and ν(t) =
[ν
1
(t), ν
2
(t),...,ν
n
(t)]
T
denote the position state vector
and the velocity state vector, r espectively. With L, we can
rewrite (1) and (2) as a compact form of
˙x (t)
˙ν(t)
=
0
n×n
I
n
αI
n
− k
1
L β I
n
− k
2
L
x(t)
ν(t)
x(t)
ν(t)
. (3)
Next, we explore the behavior analysis of the system (3). Let
λ
1
,λ
2
,...,λ
n
represent the eigenvalues o f L. Without loss of
generality, it follows from [28] that the following statements
hold.
1) λ
1
= 0andλ
2
,λ
3
,...,λ
n
have positive real parts when
G is strongly connected and structurally balanced.
2) All eig envalues of L have positive real parts when G is
strongly connected and structurally unbalanced.
When G is structurally balanced, we will introduce
a tree-type transformation for second-order SNSs. Based
on this tree-type transfo rmation, a reduced-order system
can be derived, in which bipartite consensus pr oblems of
second-order SNSs are equivalently converted into stability
problems of the reduced-order system. To be specific, since G
is structurally balanced, a matrix D
n
= diag{σ
1
,σ
2
,...,σ
n
}∈
D
n
satisfying L = D
n
LD
n
exists. Define ˆx = Qx and
ˆν = Qν,where
Q =
⎡
⎢
⎢
⎢
⎣
σ
1
00··· 0
σ
1
−σ
2
0 ··· 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
σ
1
00··· −σ
n
⎤
⎥
⎥
⎥
⎦
C ∈ R
1×n
E ∈ R
(n−1)×n
.
Obviously, Q is invertible and
Q
−1
=
⎡
⎢
⎢
⎢
⎣
σ
1
00··· 0
σ
2
−σ
2
0 ··· 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
σ
n
00··· −σ
n
⎤
⎥
⎥
⎥
⎦
[
D
n
1
n
F ∈ R
n×(n−1)
].
Substituting ˆx = Qx and ˆν = Qν into (3) can lead to
˙
ˆx (t)
˙
ˆν(t)
=
0
n×n
I
n
αI
n
− k
1
QLQ
−1
β I
n
− k
2
QLQ
−1
×
ˆx(t)
ˆν(t)
=
0
n×n
I
n
αI
n
− k
1
H β I
n
− k
2
H
ˆx (t)
ˆν(t)
(4)
where H =
0 CLF
0
n −1
ELF
,andLD
n
1
n
= 0
n
is inserted.
Denote ˆx
1
(t) = Cx(t) = σ
1
x
1
(t), ˆν
1
(t) = Cν(t) = σ
1
ν
1
(t),
ˆx
2
(t) = Ex(t) ∈ R
n−1
,andˆν
2
(t) = Eν(t) ∈ R
n−1
. Thus,
we revisit (4) and can derive two subsystems as
˙
ˆx
1
(t)
˙
ˆν
1
(t)
=
01
αβ
ˆx
1
(t)
ˆν
1
(t)
−
0
T
n−1
0
T
n−1
k
1
CLF k
2
CLF
ˆx
2
(t)
ˆν
2
(t)
(5)
and
˙
ˆx
2
(t)
˙
ˆν
2
(t)
=
0
(n−1)×(n−1)
I
n−1
αI
n−1
− k
1
ELF β I
n−1
− k
2
ELF
×
ˆx
2
(t)
ˆν
2
(t)
ˆx
2
(t)
ˆν
2
(t)
. (6)
From ˆx
2
(t) = Ex(t) and ˆν
2
(t) = Eν(t), it is immediate to
develop that the system (3) can achieve the bipartite consensus
if and only if the reduced-order system (6) converges to zero.
By the fact of
QLQ
−1
=
0 CLF
0
n−1
ELF
we realize that the matrix ELF is positive stable (namely,
all the eigenvalues o f it have positive real parts). To proceed,
we calculate the characteristic polynomial of as
det(sI
2n−2
− )
=
sI
n−1
−I
n−1
−αI
n−1
+ k
1
ELF sI
n−1
− β I
n−1
+ k
2
ELF
=|s
2
I
n−1
− β I
n−1
s + k
2
ELFs − αI
n−1
+ k
1
ELF|.
Clearly, there exist two invertible matrices P ∈ R
(n−1)×(n−1)
and P
−1
∈ R
(n−1)×(n−1)
such that
P
−1
ELFP =
⎡
⎢
⎢
⎢
⎢
⎣
λ
2
∗
λ
3
.
.
.
.
.
.
∗
λ
n
⎤
⎥
⎥
⎥
⎥
⎦
where the element ∗ may be 0 or 1. We can further deduce
det(sI
2n−2
− )
=|s
2
I
n−1
− β I
n−1
s + k
2
s − αI
n−1
+ k
1
|
=
n
i=2
f
i
(s)
where f
i
(s) = s
2
−(β −k
2
λ
i
)s +k
1
λ
i
−α. Therefore, 2n −2
eigenvalues of the matrix can be developed by solving the
following n − 1 characteristic equations:
s
2
− (β −k
2
λ
i
)s + k
1
λ
i
− α = 0 ∀i ∈{2, 3,...,n}. (7)
When G is structurally unbalanced, all eigenvalues λ
1
,
λ
2
,...,λ
n
of L have positive real parts. Similarly, one can
derive the characteristic polynomial of as follows:
det(sI
2n
− ) =
n
i=1
f
i
(s)
where f
i
(s) = s
2
− (β − k
2
λ
i
)s + k
1
λ
i
− α.
Authorized licensed use limited to: Qingdao University. Downloaded on September 15,2020 at 09:06:07 UTC from IEEE Xplore. Restrictions apply.
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