# Decentralized Finite-TimeH

Authors: Xiaohua Li; Yang Liu; Xiaoping Liu
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

Decentralized finite-timeH
∞ connective control problem for a class of large-scale interconnected systems is studied in this paper. The research aims at two structural forms, namely, the interconnected structure and the one with expanding construction. A new method is proposed to design a decentralized state feedback control law for a large-scale interconnected system so that the closed-loop system is finite-time H
∞ connectively bounded. The sufficient conditions for the existence of such a decentralized control law are deduced by using LMI method. Another method is presented for a large-scale interconnected system with expanding construction which can be used without changing the decentralized state feedback control law of the original system to design a controller for the newly added subsystem so that both the new subsystem and the resulting expanded system are finite-time H
∞ connectively bounded. The feasibility and effectiveness of the proposed method are verified by some simulation results.


---

## Body

## 1. Introduction

In many practical applications, the operating time of a given dynamic system is finite. It is natural to study finite-time stability rather than the classical Lyapunov asymptotic stability. The concept of finite-time stability was first presented in [1], which was extended to finite-time boundedness in [2]. Nowadays, finite-time control problem has been paid close attention by many scholars and a large number of results have been reported in the literature [3–6].Due to the development ofH
∞ control theory, finite-time H
∞ control problem has been well investigated in [7–12]. Therein, the finite-time H
∞ control problems of a variety of systems have been studied, such as linear continuous systems, linear stochastic systems, uncertain discrete-time singular systems, switched systems with time-varying delay, linear time-invariant and time-varying systems, and a class of nonlinear time-delay Hamiltonian systems. But all the above results are based on the single system. In [13], decentralized finite-time H
∞ control method is applied to interconnected linear systems. However, so far, there is no research report on decentralized finite-time H
∞ connective control problem of large-scale interconnected systems and the one with expanding construction.In this paper, connective boundedness and finite-timeH
∞ performance index are applied to a class of large-scale interconnected systems and the concepts of finite-time connective stability, finite-time connective boundedness, and finite-time H
∞ connective boundedness are given. A decentralized state feedback control law is designed for a large-scale interconnected system so that the corresponding closed-loop system is finite-time H
∞ connectively bounded. Furthermore, without changing the decentralized state feedback control law of the original system, a control law is constructed for the newly added subsystem so that both the new subsystem and the resulting expanded system are finite-time H
∞ connectively bounded.The paper is organized as follows. In Section2, some definitions and preliminary results are provided and the mathematical models of large-scale interconnected systems and the one with expanding construction are precisely stated. Sections 3 and 4 present the sufficient conditions for the existence of decentralized state feedback control laws for finite-time H
∞ connective boundedness of large-scale interconnected systems and the expanded systems. To illustrate the feasibility of the proposed method, one example is provided in Section 5.


## 2. Mathematical Description and Preliminary Results

Consider a large-scale interconnected system and the one with expanding construction as in [14]. The basic structure of these systems is shown in Figure 1.Figure 1
The basic structure of an expanded system.In Figure1, S
¯
N
-
1 is the original system composed of (
N
-
1
) subsystems, which are described by(1)
x
˙
i
t
=
A
i
x
i
t
+
B
i
u
i
t
+
Γ
i
ω
i
t
+
G
i
v
i
t
,
z
i
t
=
C
i
x
i
t
,
w
i
t
=
H
i
x
i
t
,
v
i
t
=
∑
j
=
1
,
j
≠
i
N
-
1
E
i
j
w
j
t
,where i
=
1,2
,
…
,
N
-
1, x
i
∈
R
n
i, u
i
∈
R
s
i, z
i
∈
R
p
i, and ω
i
∈
R
l
i are the state, control input, output, and external disturbance vectors of the ith subsystem, respectively; v
i represents the impact on the ith subsystem from the other subsystems; w
i denotes the impact on the other subsystems by the ith subsystem. A
i, B
i, C
i, Γ
i, G
i, and H
i are the constant matrices with appropriate dimensions; E
i
j represents the interconnection term from the jth subsystem to the ith subsystem. It is clear that E
i
j
=
1 represents the fact that there is an interconnection and E
i
j
=
0 means that there is no interconnection. For simplification, system (1) can be rewritten as(2)
X
¯
˙
N
-
1
t
=
A
¯
N
-
1
X
¯
N
-
1
t
+
B
¯
N
-
1
u
¯
N
-
1
t
+
Γ
¯
N
-
1
ω
¯
N
-
1
t
+
G
¯
N
-
1
v
¯
N
-
1
t
,
z
¯
N
-
1
t
=
C
¯
N
-
1
X
¯
N
-
1
t
,
w
¯
N
-
1
t
=
H
¯
N
-
1
X
¯
N
-
1
t
,
v
¯
N
-
1
t
=
E
¯
N
-
1
w
¯
N
-
1
t
,where(3)
X
¯
N
-
1
=
x
1
T
,
x
2
T
,
…
,
x
N
-
1
T
T
,
u
¯
N
-
1
=
u
1
T
,
u
2
T
,
…
,
u
N
-
1
T
T
,
ω
¯
N
-
1
=
ω
1
T
,
ω
2
T
,
…
,
ω
N
-
1
T
T
,
z
¯
N
-
1
=
z
1
T
,
z
2
T
,
…
,
z
N
-
1
T
T
,
w
¯
N
-
1
=
w
1
,
w
2
,
…
,
w
N
-
1
T
,
v
¯
N
-
1
=
v
1
,
v
2
,
…
,
v
N
-
1
T
,
A
¯
N
-
1
=
diag
⁡
A
1
,
A
2
,
…
,
A
N
-
1
,
B
¯
N
-
1
=
diag
⁡
B
1
,
B
2
,
…
,
B
N
-
1
,
C
¯
N
-
1
=
diag
⁡
C
1
,
C
2
,
…
,
C
N
-
1
,
Γ
¯
N
-
1
=
diag
⁡
Γ
1
,
Γ
2
,
…
,
Γ
N
-
1
,
G
¯
N
-
1
=
diag
⁡
G
1
,
G
2
,
…
,
G
N
-
1
,
H
¯
N
-
1
=
diag
⁡
H
1
,
H
2
,
…
,
H
N
-
1
,
E
¯
N
-
1
=
0
E
12
⋯
E
1
,
N
-
1
E
21
0
⋯
E
2
,
N
-
1
⋮
⋮
⋱
⋮
E
N
-
1,1
E
N
-
1,2
⋯
0
.In Figure1, S
N is the Nth subsystem newly added to the original system, which is modeled by(4)
x
˙
N
t
=
A
N
x
N
t
+
B
N
u
N
t
+
Γ
N
ω
N
t
+
G
N
v
N
t
,
z
N
t
=
C
N
x
N
t
,
w
N
t
=
H
N
x
N
t
,where v
N
(
t
)
=
E
¯
N
,
N
-
1
w
¯
N
-
1
(
t
), which represents the impact on the Nth subsystem by the original (
N
-
1
) subsystems. After the Nth subsystem is added, the impact on the original (
N
-
1
) subsystems by the Nth subsystem is v
¯
N
-
1
=
E
¯
N
-
1
,
N
w
N. E
¯
N
-
1
,
N
=
E
1
N
,
E
2
N
,
…
,
E
N
-
1
,
N
T and E
¯
N
,
N
-
1
=
E
N
1
,
E
N
2
,
…
,
E
N
,
N
-
1. The large-scale system with N subsystems becomes(5)
X
¯
˙
N
-
1
t
=
A
¯
N
-
1
X
N
-
1
t
+
B
¯
N
-
1
u
¯
N
-
1
t
+
Γ
¯
N
-
1
ω
¯
N
-
1
t
+
h
¯
t
,
X
¯
N
-
1
,
x
N
,
E
,
x
˙
N
t
=
A
N
x
N
t
+
B
N
u
N
t
+
Γ
N
ω
N
t
+
h
t
,
X
¯
N
-
1
,
x
N
,
E
,where (6)
h
¯
t
,
X
¯
N
-
1
,
x
N
,
E
=
G
¯
N
-
1
E
¯
N
-
1
H
¯
N
-
1
X
¯
N
-
1
t
+
G
¯
N
-
1
E
¯
N
-
1
,
N
H
N
x
N
t
,
h
t
,
X
¯
N
-
1
,
x
N
,
E
=
G
N
E
¯
N
,
N
-
1
H
¯
N
-
1
X
¯
N
-
1
t
.Equation (5) can be written as the following compact form:(7)
x
^
˙
t
=
A
^
x
^
t
+
B
^
u
^
t
+
Γ
^
ω
^
t
+
h
^
t
,
x
^
t
,
E
,
z
^
t
=
C
^
x
^
t
,where(8)
A
^
=
A
¯
N
-
1
0
0
A
N
,
B
^
=
B
¯
N
-
1
0
0
B
N
,
Γ
^
=
Γ
¯
N
-
1
0
0
Γ
N
,
C
^
=
C
¯
N
-
1
0
0
C
N
,
u
^
t
=
u
¯
N
-
1
t
0
0
u
N
t
,
z
^
t
=
z
¯
N
-
1
t
0
0
z
N
t
,
ω
^
t
=
ω
¯
N
-
1
t
0
0
ω
N
t
,
h
^
t
,
x
^
t
,
E
=
h
¯
T
t
,
X
¯
N
-
1
,
x
N
,
E
,
h
T
t
,
X
¯
N
-
1
,
x
N
,
E
T
,
x
^
t
=
X
¯
N
-
1
t
x
N
t
=
x
1
T
t
,
x
2
T
t
,
…
,
x
N
-
1
T
t
,
x
N
T
t
T
.It is assumed that ω
^
(
t
) satisfies(9)
∫
0
∞
ω
^
T
t
ω
^
t
d
t
<
d
2
.The following definitions for the large-scale systems with expanding construction are given.Definition 1 (finite-time connective stability).
The large-scale system described by (7) with u
^
(
t
)
=
ω
^
(
t
)
=
0 is said to be finite-time connectively stable with respect to c
1
,
c
2
,
T
,
R with 0
<
c
1
<
c
2 and R
>
0 if x
^
0
T
R
x
^
0
≤
c
1; then (10)
x
^
T
t
R
x
^
t
<
c
2
,
∀
t
∈
0
,
T
,for any E
i
j, i
=
1
,
…
,
N, j
=
1
,
…
,
N, and x
^
(
0
)
=
x
^
0.Definition 2 (finite-time connective boundedness).
The large-scale system described by (7) with u
^
(
t
)
=
0 is said to be finite-time connectively bounded with respect to c
1
,
c
2
,
T
,
R with 0
<
c
1
<
c
2 and R
>
0 if x
^
0
T
R
x
^
0
≤
c
1; then (11)
x
^
T
t
R
x
^
t
<
c
2
,
∀
t
∈
0
,
T
,for any E
i
j, i
=
1
,
…
,
N, j
=
1
,
…
,
N, and any nonzero ω
^
(
t
) satisfying (9).Definition 3 (finite-timeH
∞ connective boundedness).
The large-scale system described by (7) with u
^
(
t
)
=
0 is said to be finite-time H
∞ connectively bounded if(1)
equation (7) with u
^
(
t
)
=
0 is finite-time connectively bounded, and
(2)
with zero initial condition(12)
∫
0
t
z
^
s
2
d
s
<
e
α
T
γ
2
∫
0
t
ω
^
s
2
d
sfor any E
i
j, i
=
1
,
…
,
N, j
=
1
,
…
,
N, and any nonzero ω
^
(
t
) satisfying (9), where γ is a positive constant and t
∈
[
0
,
T
].The decentralized state feedback control law of the form(13)
u
^
t
=
K
^
x
^
twith K
^
=
d
i
a
g
(
K
1
,
…
,
K
N
) is introduced to control system (7). Then the closed-loop system becomes(14)
x
^
˙
t
=
A
^
+
B
^
K
^
x
^
t
+
Γ
^
ω
^
t
+
h
^
t
,
x
^
t
,
E
=
A
¯
^
x
^
t
+
Γ
^
ω
^
t
+
h
^
t
,
x
^
t
,
E
,
z
^
t
=
C
^
x
^
t
,where(15)
A
¯
^
=
A
^
+
B
^
K
^
=
A
¯
N
-
1
+
B
¯
N
-
1
K
¯
N
-
1
0
0
A
N
+
B
N
K
N
=
A
^
N
-
1
0
0
A
^
N
.Definition 4.
The decentralized finite-timeH
∞ connective control problem is said to be solvable if there exists a decentralized state feedback control law (13) so that the closed-loop system (14) is finite-time H
∞ connectively bounded.In order to obtain the main results in this paper, the following assumption and lemmas are proposed.Assumption 5.
The interconnection itemh
^
(
t
,
x
^
(
t
)
,
E
) is uncertain and satisfies the following inequality:(16)
h
^
T
t
,
x
^
t
,
E
h
^
t
,
x
^
t
,
E
≤
β
^
2
x
^
T
t
Q
^
T
Q
^
x
^
t
,where β
^
>
0 is bounding parameter and Q
^ is the interconnected constraint matrix.Lemma 6.
System (7) with u
^
(
t
)
=
0 is finite-time connectively bounded if there exist two scalars, α
≥
0 and γ
>
0, and a symmetric positive definite matrix P such that(17)
A
^
T
P
^
+
P
^
A
^
-
α
P
^
+
β
^
2
Q
^
T
Q
^
P
^
Γ
^
P
^
Γ
^
T
P
^
-
γ
2
I
0
P
^
0
-
I
<
0
,
(18)
c
1
λ
m
i
n
P
+
d
2
γ
2
<
c
2
e
-
α
T
λ
m
a
x
Pwith P
^
=
R
1
/
2
P
-
1
R
1
/
2 and λ
m
i
n
(
P
) and λ
m
a
x
(
P
) being the minimum and maximum eigenvalues of P, respectively.Proof.
LetV
(
x
^
(
t
)
)
=
x
^
T
(
t
)
P
^
x
^
(
t
); then(19)
V
˙
x
^
t
=
x
^
T
t
A
^
T
P
^
+
P
^
A
^
x
^
t
+
ω
^
T
t
Γ
^
T
P
^
x
^
t
+
x
^
T
t
P
^
Γ
^
ω
^
t
+
h
^
T
t
,
x
^
t
,
E
P
^
x
^
t
+
x
^
T
t
·
P
^
h
^
t
,
x
^
t
,
E
≤
x
^
T
t
A
^
T
P
^
+
P
^
A
^
x
^
t
+
ω
^
T
t
Γ
^
T
P
^
x
^
t
+
x
^
T
t
P
^
Γ
^
ω
^
t
+
h
^
T
t
,
x
^
t
,
E
·
P
^
x
^
t
+
x
^
T
t
P
^
h
^
t
,
x
^
t
,
E
-
h
^
T
t
,
x
^
t
,
E
·
h
^
t
,
x
^
t
,
E
+
β
^
2
x
^
T
t
Q
^
T
Q
^
x
^
t
=
x
^
t
ω
^
t
h
^
t
T
·
A
^
T
P
^
+
P
^
A
^
+
β
^
2
Q
^
T
Q
^
P
^
Γ
^
P
^
Γ
^
T
P
^
0
0
P
^
0
-
I
x
^
t
ω
^
t
h
^
t
.From (17) and (19), we have(20)
V
˙
x
^
t
<
α
V
x
^
t
+
γ
2
ω
^
T
t
ω
^
t
.Multiplying (20) by e
-
α
t, we can obtain(21)
e
-
α
t
V
˙
x
^
t
-
e
-
α
t
α
V
x
^
t
<
e
-
α
t
γ
2
ω
^
t
T
ω
^
t
,which is equivalent to(22)
d
d
t
e
-
α
t
V
x
^
t
<
e
-
α
t
γ
2
ω
^
t
T
ω
^
t
.Integrating (22) from 0 to t gives(23)
e
-
α
t
V
x
^
t
-
V
x
^
0
<
∫
0
t
e
-
α
s
γ
2
ω
^
T
s
ω
^
s
d
s
.Noting α
≥
0, then(24)
V
x
^
t
<
e
α
t
V
x
^
0
+
e
α
t
∫
0
t
e
-
α
s
γ
2
ω
^
T
s
ω
^
s
d
s
<
e
α
t
V
x
^
0
+
e
α
t
γ
2
∫
0
t
ω
^
T
s
ω
^
s
d
s
<
e
α
t
V
x
^
0
+
d
2
γ
2
.Due to P
^
=
R
1
/
2
P
-
1
R
1
/
2, it follows from (24) that(25)
x
^
T
t
R
1
/
2
P
-
1
R
1
/
2
x
^
t
<
e
α
t
x
^
T
0
R
1
/
2
P
-
1
R
1
/
2
x
^
0
+
d
2
γ
2
.Now, we have(26)
x
^
T
t
R
1
/
2
P
-
1
R
1
/
2
x
^
t
≥
λ
m
i
n
P
-
1
x
^
T
t
R
x
^
t
,
e
α
t
x
^
T
0
R
1
/
2
P
-
1
R
1
/
2
x
^
0
+
d
2
γ
2
≤
e
α
T
λ
m
a
x
P
-
1
x
^
T
0
R
x
^
0
+
d
2
γ
2
.Putting (25) and (26) together, we have(27)
x
^
T
t
R
x
^
t
<
e
α
T
λ
m
a
x
P
-
1
x
^
T
0
R
x
^
0
+
d
2
γ
2
λ
m
i
n
P
-
1
=
e
α
T
λ
m
a
x
P
-
1
c
1
+
d
2
γ
2
λ
m
i
n
P
-
1
.Condition (18) implies that, for all t
∈
[
0
,
T
], x
^
T
(
t
)
R
x
^
(
t
)
<
c
2. Because (16) holds for any E
i
j, then system (7) with u
^
(
t
)
=
0 is connectively bounded. The proof is completed.Lemma 7.
System (14) is finite-time H
∞ connectively bounded if there exist two scalars, α
≥
0 and γ
>
0, and a symmetric positive definite matrix P such that(28)
A
¯
^
T
P
^
+
P
^
A
¯
^
-
α
P
^
+
C
^
T
C
^
+
β
^
2
Q
^
T
Q
^
P
^
Γ
^
P
^
Γ
^
T
P
^
-
γ
2
I
0
P
^
0
-
I
<
0
,
(29)
c
1
λ
m
i
n
P
+
d
2
γ
2
<
c
2
e
-
α
T
λ
m
a
x
Pwith P
^
=
R
1
/
2
P
-
1
R
1
/
2.Proof.
Note that(30)
C
^
T
C
^
0
0
0
0
0
0
0
0
=
C
^
T
0
0
C
^
0
0
≥
0
.Therefore, condition (28) implies that(31)
A
¯
^
T
P
^
+
P
^
A
¯
^
-
α
P
^
+
β
^
2
Q
^
T
Q
^
P
^
Γ
^
P
^
Γ
^
T
P
^
-
γ
2
I
0
P
^
0
-
I
<
0
.From Lemma 6, conditions (29) and (31) guarantee that system (14) is finite-time connectively bounded. Now, we need to proof that (12) holds.
LetV
(
x
^
(
t
)
)
=
x
^
T
(
t
)
P
^
x
^
(
t
); then(32)
V
˙
x
^
t
=
x
^
T
t
A
¯
^
T
P
^
+
P
^
A
¯
^
x
^
t
+
ω
^
T
t
Γ
^
T
P
^
x
^
t
+
x
^
T
t
P
^
Γ
^
ω
^
t
+
h
^
T
t
,
x
^
t
,
E
P
^
x
^
t
+
x
^
T
t
·
P
^
h
^
t
,
x
^
t
,
E
≤
x
^
T
t
A
¯
^
T
P
^
+
P
^
A
¯
^
x
^
t
+
ω
^
T
t
Γ
^
T
P
^
x
^
t
+
x
^
T
t
P
^
Γ
^
ω
^
t
+
h
^
T
t
,
x
^
t
,
E
·
P
^
x
^
t
+
x
^
T
t
P
^
h
^
t
,
x
^
t
,
E
-
h
^
T
t
,
x
^
t
,
E
·
h
^
t
,
x
^
t
,
E
+
β
^
2
x
^
T
t
Q
^
T
Q
^
x
^
t
=
x
^
t
ω
^
t
h
^
t
T
·
A
¯
^
T
P
^
+
P
^
A
¯
^
+
β
^
2
Q
^
T
Q
^
P
^
Γ
^
P
^
Γ
^
T
P
^
0
0
P
^
0
-
I
x
^
t
ω
^
t
h
^
t
.It follows from (28) and (32) that(33)
V
˙
x
^
t
<
α
V
x
^
t
+
γ
2
ω
^
T
t
ω
^
t
-
z
^
T
t
z
^
twhich implies that(34)
e
-
α
t
V
˙
x
^
t
-
α
e
-
α
t
V
x
^
t
<
e
-
α
t
γ
2
ω
^
T
t
ω
^
t
-
z
^
T
t
z
^
tthat is,(35)
d
d
t
e
-
α
t
V
<
e
-
α
t
γ
2
ω
^
T
t
ω
^
t
-
z
^
T
t
z
^
t
.Taking x
^
(
0
)
=
0 into consideration, integrating (35) from 0 to t produces(36)
e
-
α
t
V
x
^
t
<
∫
0
t
e
-
α
s
γ
2
ω
^
T
s
ω
^
s
-
z
^
T
s
z
^
s
d
s
,which is equivalent to(37)
∫
0
t
e
-
α
s
z
^
T
s
z
^
s
d
s
<
γ
2
∫
0
t
e
-
α
s
ω
^
T
s
ω
^
s
d
s
.Due to e
-
α
T
<
e
-
α
s for s
∈
[
0
,
T
) and e
-
α
s
<
1 for s
∈
(
0
,
T
), it follows that(38)
e
-
α
T
∫
0
t
z
^
T
s
z
^
s
d
s
<
∫
0
t
e
-
α
s
z
^
T
s
z
^
s
d
s
,
γ
2
∫
0
t
e
-
α
s
ω
^
T
s
ω
^
s
d
s
<
γ
2
∫
0
t
ω
^
T
s
ω
^
s
d
s
.Equation (12) can be obtained from (37)-(38). Because (16) holds for any E
i
j, then system (14) with u
^
(
t
)
=
0 is connectively bounded. The proof is completed.


## 3. Decentralized Finite-TimeH
∞ Connective Control

The main result is given in Theorem8 which is based on Lemmas 6 and 7 in Section 2.Theorem 8.
The decentralized finite-timeH
∞ connective control problem for (14) including N subsystems is solvable with feedback control laws u
^
(
t
)
=
K
^
x
^
(
t
) if there exist two scalars, α
≥
0 and γ
>
0, matrix M
^, and the interconnected constraint matrix, Q
^, as well as symmetric positive definite matrices Y
^ and P such that(39)
min
⁡
γ
2
+
ε
^
+
c
2
,
(40)
Ψ
^
Γ
^
I
Y
^
Q
^
T
Y
^
C
^
T
Γ
^
T
-
γ
2
I
0
0
0
I
0
-
I
0
0
Q
^
Y
^
0
0
-
ε
^
I
0
C
^
Y
^
0
0
0
-
I
<
0
,
(41)
c
1
λ
m
i
n
P
+
d
2
γ
2
<
c
2
e
-
α
T
λ
m
a
x
P
,where(42)
Ψ
^
=
Y
^
A
^
T
+
A
^
Y
^
+
M
^
T
B
^
T
+
B
^
M
^
-
α
Y
^
,
ε
^
=
1
β
^
2
,
Y
^
=
P
^
-
1
,
P
=
R
1
/
2
P
^
-
1
R
1
/
2with λ
m
i
n
(
P
) and λ
m
a
x
(
P
) being the minimum and maximum eigenvalues of P, respectively. And the control law can be determined by K
^
=
M
^
Y
^
-
1
.Proof.
According to Lemmas6 and 7, let M
^
=
K
^
Y
^. Then (40) is equivalent to the following inequality:(43)
Ψ
^
Γ
^
I
Y
^
Q
^
T
Y
^
C
^
T
Γ
^
T
-
γ
2
I
0
0
0
I
0
-
I
0
0
Q
^
Y
^
0
0
-
ε
^
I
0
C
^
Y
^
0
0
0
-
I
<
0with Ψ
^
=
Y
^
A
^
T
+
A
^
Y
^
+
Y
^
K
^
T
B
^
T
+
B
^
K
^
Y
^
-
α
Y
^.
Using Schur complement lemma, then (43) can be changed to(44)
Ψ
~
Γ
^
I
Y
^
Q
^
T
Γ
^
T
-
γ
2
I
0
0
I
0
-
I
0
Q
^
Y
^
0
0
-
ε
^
I
<
0with Ψ
~
=
Y
^
A
^
T
+
A
^
Y
^
+
Y
^
K
^
T
B
^
T
+
B
^
K
^
Y
^
-
α
Y
^
+
Y
^
C
^
T
C
^
Y
^.
SetΘ
=
d
i
a
g
(
Y
^
-
1
,
I
,
I
,
I
) and Y
^
-
1
=
P
^. Then, pre- and postmultiplying (44) by Θ yields(45)
Ψ
¯
P
^
Γ
^
P
^
Q
^
T
Γ
^
T
P
^
-
γ
2
I
0
0
P
^
0
-
I
0
Q
^
0
0
-
ε
^
I
<
0with Ψ
¯
=
A
^
T
P
^
+
P
^
A
^
+
K
^
T
B
^
T
P
^
+
P
^
B
^
K
^
-
α
P
^
+
C
^
T
C
^
=
A
¯
^
T
P
^
+
P
^
A
¯
^
-
α
P
^
+
C
^
T
C
^ due to A
¯
^
=
A
^
+
B
^
K
^.
Using Schur complement lemma again, it can be proved that (45) is equivalent to (28), which completes the proof. When E
i
j varies from 1 to 0 or from 0 to 1, the change of interconnection is still within limited range. So the control law K
^ can make the system stable and robustly connectively stable.It is obvious that (41) is not an LMI. However, it is guaranteed by imposing the conditions(46)
λ
1
I
<
P
<
λ
2
I
,
(47)
c
1
λ
1
+
d
2
γ
2
<
c
2
e
-
α
T
λ
2for two positive numbers λ
1 and λ
2. Using Schur complements, inequality (47) can be converted to the following LMI:(48)
I
c
1
d
γ
c
2
e
-
α
T
/
2
c
1
λ
1
0
0
d
γ
0
I
0
c
2
e
-
α
T
/
2
0
0
-
λ
2
>
0
.Therefore, (41) holds if LMIs (46) and (48) are true. From a computational point of view, it is important to notice that, for a given α, the feasibility of the conditions stated in Theorem 8 can be turned into the following feasibility problem of LMIs.LMI Feasibility Problem 1 (from Theorem8). For system (14), given c
1
,
T
,
R, and α
>
0, let P
^
=
R
1
/
2
P
-
1
R
1
/
2; find a symmetric positive definite matrix Y
^ and two positive scalars, λ
1 and λ
2, satisfying LMIs (39), (40), (46), and (48). If the problem is feasible, the decentralized finite-time H
∞ connective control problem can be solved by K
^
=
M
^
Y
^
-
1.


## 4. Decentralized Finite-TimeH
∞ Connective Control for Large-Scale Systems with Expanding Construction

Section3 has introduced a method for constructing a decentralized state feedback control law to solve the finite-time H
∞ connective control problem for large-scale systems. This section will propose a control design method for the finite-time H
∞ connective control problem of large-scale systems with expanding construction. The technical difficulty of dealing with this kind of systems is that a control law is constructed for the newly added subsystem so that both the new subsystem and the resulting expanded system are finite-time H
∞ connectively bounded without changing the decentralized state feedback control law of the original system. To this end, the following assumption is made.Assumption 9.
The decentralized finite-timeH
∞ connective control problem for the original system (1) can be solved by the control law u
¯
N
-
1
(
t
)
=
K
¯
N
-
1
X
¯
N
-
1
(
t
).The main result is given in Theorem10.Theorem 10.
The decentralized finite-timeH
∞ connective control problem for the Nth subsystem in (14) is solvable with feedback control law u
N
(
t
)
=
K
N
x
N
(
t
), if there are positive constants, α and γ, as well as symmetrical positive definite matrices, Y
N and P
N, and matrix M
N, so that the following inequalities are feasible:(49)
min
⁡
γ
2
+
ε
^
+
c
2
,
(50)
Λ
¯
N
-
1
0
P
¯
N
-
1
Γ
¯
N
-
1
0
P
¯
N
-
1
0
q
^
1
T
q
^
3
T
0
0
Λ
~
~
N
0
Γ
¯
N
0
I
Y
N
q
^
2
T
0
Y
N
C
N
T
Γ
¯
N
-
1
T
P
¯
N
-
1
0
-
γ
2
I
0
0
0
0
0
0
0
Γ
¯
N
T
0
-
γ
2
I
0
0
0
0
0
P
¯
N
-
1
0
0
0
-
I
0
0
0
0
0
I
0
0
0
-
I
0
0
0
q
^
1
q
^
2
Y
N
0
0
0
0
-
ε
^
I
0
0
q
^
3
0
0
0
0
0
0
-
ε
^
I
0
0
C
N
Y
N
0
0
0
0
0
0
-
I
<
0
,
(51)
c
1
λ
m
i
n
P
N
+
d
2
γ
2
<
c
2
e
-
α
T
λ
m
a
x
P
N
,where q
^
1, q
^
2, and q
^
3 are the elements of the interconnected constraint matrix Q
^ and(52)
Λ
¯
N
-
1
=
A
^
N
-
1
T
P
¯
N
-
1
+
P
¯
N
-
1
A
^
N
-
1
-
α
P
¯
N
-
1
+
C
¯
N
-
1
T
C
¯
N
-
1
,
Λ
~
~
N
=
Y
N
A
N
T
+
A
N
Y
N
+
M
N
T
B
N
T
+
B
N
M
N
-
α
Y
N
,
ε
^
=
1
β
^
2
,
Y
N
=
P
^
N
-
1
,
P
N
=
R
1
/
2
P
^
N
-
1
R
1
/
2
,
P
¯
N
-
1
=
diag
⁡
P
1
,
P
2
,
…
,
P
N
-
1
,
Q
^
=
q
^
1
q
^
2
q
^
3
0with λ
m
i
n
(
P
N
) and λ
m
a
x
(
P
N
) being the minimum and maximum eigenvalues of P
N, respectively. And the control law can be determined by K
N
=
M
N
Y
N
-
1
.Proof.
The result of Theorem10 is based on Lemmas 6 and 7 and Theorem 8. Let M
N
=
K
N
Y
N. Then, (50) is equivalent to(53)
Λ
¯
N
-
1
0
P
¯
N
-
1
Γ
¯
N
-
1
0
P
¯
N
-
1
0
q
^
1
T
q
^
3
T
0
0
Λ
~
~
N
0
Γ
N
0
I
Y
N
q
^
2
T
0
Y
N
C
N
T
Γ
¯
N
-
1
T
P
¯
N
-
1
0
-
γ
2
I
0
0
0
0
0
0
0
Γ
N
T
0
-
γ
2
I
0
0
0
0
0
P
¯
N
-
1
0
0
0
-
I
0
0
0
0
0
I
0
0
0
-
I
0
0
0
q
^
1
q
^
2
Y
N
0
0
0
0
-
ε
^
I
0
0
q
^
3
0
0
0
0
0
0
-
ε
^
I
0
0
C
N
Y
N
0
0
0
0
0
0
-
I
<
0
,where(54)
Λ
~
~
N
=
Y
N
A
N
T
+
A
N
Y
N
+
Y
N
K
N
T
B
N
T
+
B
N
K
N
Y
N
-
α
Y
N
.Using Schur complement lemma results in(55)
Λ
¯
N
-
1
0
P
¯
N
-
1
Γ
¯
N
-
1
0
P
¯
N
-
1
0
q
^
1
T
q
^
3
T
0
Λ
~
N
0
Γ
N
0
I
Y
N
q
^
2
T
0
Γ
¯
N
-
1
T
P
¯
N
-
1
0
-
γ
2
I
0
0
0
0
0
0
Γ
¯
N
T
0
-
γ
2
I
0
0
0
0
P
¯
N
-
1
0
0
0
-
I
0
0
0
0
I
0
0
0
-
I
0
0
q
^
1
q
^
2
Y
N
0
0
0
0
-
ε
^
I
0
q
^
3
0
0
0
0
0
0
-
ε
^
I
<
0with Λ
~
N
=
Y
N
A
N
T
+
A
N
Y
N
+
Y
N
K
N
T
B
N
T
+
B
N
K
N
Y
N
-
α
Y
N
+
Y
N
C
N
T
C
N
Y
N.
SetΞ
=
d
i
a
g
(
I
,
Y
N
-
1
,
I
,
I
,
I
,
I
,
I
,
I
) and Y
N
-
1
=
P
^
N. Then, the following inequality follows from pre- and postmultiplying (55) by Ξ:(56)
Λ
¯
N
-
1
0
P
¯
N
-
1
Γ
¯
N
-
1
0
P
¯
N
-
1
0
q
^
1
T
q
^
3
T
0
Λ
N
0
P
^
N
Γ
N
0
P
^
N
q
^
2
T
0
Γ
¯
N
-
1
T
P
¯
N
-
1
0
-
γ
2
I
0
0
0
0
0
0
Γ
N
T
P
^
N
0
-
γ
2
I
0
0
0
0
P
¯
N
-
1
0
0
0
-
I
0
0
0
0
P
^
N
0
0
0
-
I
0
0
q
^
1
q
^
2
0
0
0
0
-
ε
^
I
0
q
^
3
0
0
0
0
0
0
-
ε
^
I
<
0
,where(57)
Λ
N
=
A
N
T
P
^
N
+
P
^
N
A
N
+
K
N
T
B
N
T
P
^
N
+
P
^
N
B
N
K
N
-
α
P
^
N
+
C
N
T
C
N
.Set P
^
=
P
¯
N
-
1
0
0
P
^
N. Due to (58)
A
¯
^
T
P
^
+
P
^
A
¯
^
-
α
P
^
+
C
^
T
C
^
=
Λ
¯
N
-
1
0
0
Λ
N
,then (56) is equivalent to(59)
A
¯
^
T
P
^
+
P
^
A
¯
^
-
α
P
^
+
C
^
T
C
^
P
^
Γ
^
P
^
Q
^
T
Γ
^
T
P
^
-
γ
2
I
0
0
P
^
0
-
I
0
Q
^
0
0
-
ε
^
I
<
0
.
Applying Schur complement lemma to (59) shows that (28) is true. It follows from Lemma 7 that Theorem 10 holds. Similar to Theorem 8, the control law K
N can make the system stable and robustly connectively stable and the following feasibility problem of LMIs is given.LMIs Feasibility Problem 2 (from Theorem10). For system (14), given c
1
,
T
,
R, and α
>
0, let P
^
N
=
R
1
/
2
P
N
-
1
R
1
/
2; find a symmetric positive definite matrix, Y
N, matrix M
N, and two positive scalars, λ
1 and λ
2, satisfying LMIs (46), (48), (49), and (50). If the problem is feasible, the decentralized finite-time H
∞ connective control problem for the expanded subsystem can be solved by K
N
=
M
N
Y
N
-
1.


## 5. Simulation Example

Consider a class of multiarea interconnected power systems, in which each area includes a hydroelectric power unit and a thermal power unit. The mathematical model, state variables, and output variables can be found in [15, 16]. This is a deviation model of automatic generation control (AGC). The ith area-subsystem model S
i can be described as(60)
x
˙
i
=
A
i
i
x
i
+
B
i
u
i
+
Γ
i
ω
i
+
∑
j
=
1
,
j
≠
i
N
G
i
E
i
j
H
j
x
j
,
y
i
=
C
i
x
i
,where x
i
∈
R
n
i
,
u
i
∈
R
m
i
,
y
i
∈
R
l
i, and ω
i
∈
R
m
i are the state, control input, output, and uncertain disturbance input of subsystems, respectively. And the matrices in (60) are, respectively,(61)
A
i
i
=
A
i
0
a
t
i
d
i
T
0
1
σ
1
i
∑
j
=
1
,
j
≠
i
N
m
i
j
T
0
0
,
B
i
=
b
i
0
0
,
Γ
i
=
f
i
0
0
,
C
i
=
c
i
0
0
0
1
0
0
0
1
,
G
i
=
0
0
m
i
j
T
T
,
H
i
=
0
0
0
0
0
0
0
-
1
0
0
.The specific parameters of (61) can be found in [15, 16]. And other parameters are given as follows:(62)
Γ
1
=
Γ
2
=
Γ
3
=
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
T
,
R
1
=
R
2
=
R
3
=
I
10
,
G
1
=
G
2
=
G
3
=
0
0
0
0
0
0
0
0
0
22.2144
T
,
H
1
=
H
2
=
H
3
=
0
0
0
0
0
0
0
-
1
0
0
,
ω
1
=
ω
2
=
ω
3
=
0.1
,
α
=
20
,
T
=
6
,
d
=
1
,
c
1
=
1
.Suppose that the original system consists of two subsystems. The initial values are x
^
10
=
x
^
20
=
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
T. The connective relations are E
¯
=
0
1
1
0. According to Theorem 8, the corresponding control gain matrices K
1, and K
2 are obtained as(63)
K
1
=
-
0.2372
-
0.0634
0.0042
0.0061
-
0.6209
-
0.5948
0.1474
0.0028
-
0.0098
-
0.0699
,
K
2
=
-
0.3613
-
0.0974
0.0052
0.0076
-
0.6277
-
0.6034
0.1416
0.0041
-
0.0091
-
0.0664and m
i
n
(
c
2
)
=
37.6856. θ
=
∑
i
=
1
2
‍
x
^
i
T
t
R
i
x
^
i
t is plotted as shown in Figure 2. It can be easily seen from Figure 2 that θ
<
m
i
n
(
c
2
) and t
∈
0,6, which means that the decentralized finite-time H
∞ connective control problem is solvable for the original system. In order to check connective stability of the system, we cut off the interconnection E
12 between two subsystems. The result is shown in Figure 3, which shows that the closed-loop system is finite-time H
∞ connectively bounded.Figure 2
The simulation curve of the original system.Figure 3
The simulation curve of the original system whenE
12 varies from 1 to 0.Now the third subsystem is added to the original system with(64)
x
^
30
=
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
T
.The connective relations are given by(65)
E
=
0
1
1
1
0
1
1
1
0
.According to Theorem 10, the control gain matrix K
3 is obtained as (66)
K
3
=
-
0.3093
-
0.0954
0.0048
0.0071
-
0.5086
-
0.5855
0.1406
0.0113
0.0254
0.0479and m
i
n
(
c
2
)
=
36.7686. θ
1
=
∑
i
=
1
3
‍
x
^
i
T
t
R
i
x
^
i
t is shown in Figure 4. It is obvious that θ
1
<
m
i
n
(
c
2
) and t
∈
0,6, which implies that the decentralized finite-time H
∞ connective control problem is solvable for the expanded system. In order to check the connective stability of the expanded system, the connection E
23 varies from 1 to 0 and the result is shown in Figure 5.Figure 4
The simulation curve of the expanded system.Figure 5
The simulation curve of the expanded system whenE
23 varies from 1 to 0.From the figures, it can be known that both the original system and the resulting expanded system are finite-timeH
∞ connectively bounded.


## 6. Conclusion

Decentralized finite-timeH
∞ connective control problem for a class of large-scale systems is studied in this paper. The large-scale systems include the original structural interconnected system and the systems with expanding construction. Based on state feedback, the sufficient conditions of decentralized finite-time H
∞ connective boundedness for large-scale systems are deduced by using LMI method. The design methods for the decentralized finite-time H
∞ connective control problem are given. The simulation examples verify the feasibility and effectiveness of the proposed method. In particular, the paper proposes a method for the structure expansion of large-scale systems. A controller can be designed for the newly added subsystem on the basis of keeping the decentralized state feedback control laws of the original construction systems unchanged so that both the new subsystem and the resulting expanded system are finite-time H
∞ connectively bounded. Therefore, this paper can be used as the theoretical basis for expansion of large-scale interconnected system online.



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*Source: 132517-2015-06-28.xml*