# An MDADT-Based Approach forL

Authors: Honglei Xu; Xiang Xie; Lilian Shi
Journal: Mathematical Problems in Engineering
 (2016)

---

## Abstract

We study theL
2-gain analysis problem for a class of discrete-time switched systems with time-varying delays. A mode-dependent average dwell time (MDADT) approach is applied to analyze the L
2-gain performance for these discrete-time switched delay systems. Combining a multiple Lyapunov functional method with the MDADT approach, sufficient conditions expressed in form of a set of feasible linear matrix inequalities (LMIs) are established to guarantee the L
2-gain performance. Finally, a numerical example will be provided to demonstrate the validity and usefulness of the obtained results.


---

## Body

## 1. Introduction

Switched systems consist of a finite number of subsystems and a logical law which orchestrates the switching behaviors  between these subsystems. These dynamical systems  can mathematically model many practical engineering applications  with switching characteristics in a variety of disciplines; see,  for example, [1–7].A constrained switching signal can be regarded as a  powerful tool to stabilize and control these switched  systems [8–10]. Among them, the average dwell time (ADT) switching  is the most common and typical one. It guarantees  that the number of types of switching in a finite interval be bounded  and the average time between any two types of consecutive  switching not be less than a positive constant [11, 12]. In recent  years, it has been recognised that ADT is flexible and efficient for dynamics analysis of many switched systems [8, 13–16]. However, the ADT switching’s property that the average time interval between any two types of consecutive switching should be greater than a positive number τ
a makes the  dwell time independent of the system modes. Hence whether the dwelling at some classes of subsystems will deteriorate the disturbance attenuation cannot be predicted.As shown in [17], the minimum of admissible ADT is computed by two mode-independent parameters: the increase coefficient of the Lyapunov-like function and the decay rate of the Lyapunov function, which will cause certain conservativeness. To solve the problem, more recently, a new mode-dependent ADT concept has been introduced in [18]. Two mode-independent parameters can be set in a mode-dependent manner, which will reduce the conservativeness.Even though stability analysis for the switched systems with MDADT has been investigated extensively (see, e.g., [17, 18]), how to solve the L
2-gain problem of the switched systems with MDADT is interesting and worthwhile to study. This has motivated our study in this paper.The rest of the paper is as follows. In Section2, we introduce the class of discrete-time switched system, some necessary definitions, and lemmas. In Section 3, sufficient conditions for ensuring L
2-gain for the discrete-time switched delay system are constructed. In Section 4, a numerical example is presented to illustrate the obtained results. Conclusion remarks are given in Section 5.


## 2. Preliminaries and Problem Statement

Consider a discrete-time switched system with a time-varying delay:(1)
L
i
:
⁡
x
t
+
1
=
A
i
x
t
+
B
i
x
t
-
d
t
+
C
i
w
t
,
x
t
0
l
=
x
t
0
+
l
=
ϕ
l
,
l
=
-
d
M
,
-
d
M
+
1
,
-
d
M
+
2
,
…
,
0
,
z
t
=
D
i
x
t
+
E
i
w
t
,where x
(
t
)
∈
R
n is the system state, z
(
t
)
∈
R
m is the controlled output, ϕ
(
l
) is a vector-valued initial function, t
0 is the initial time, and w
(
t
) is the disturbance input which belongs to L
2
[
0
,
+
∞
). d
(
t
) is the time-varying delay and satisfies 0
<
d
m
<
d
(
t
)
≤
d
M, where d
m and d
M denote the upper and the lower bounds of the delays. i is the switching signal, which takes its values in the finite set S
=
1
,
…
,
M, where M is the number of subsystems. When t
∈
t
i
,
t
i
+
1, i
∈
N, we call the ith subsystem active. A
p, B
p, C
p, D
p, and E
p are constant matrices with appropriate dimension. When i
=
p
=
1
,
…
,
m, it represents thepth subsystem or pth mode of (1).To proceed, we need the following definitions and lemmas.Definition 1 (see [11]).
For anyT
2
>
T
1
≥
0 and any switching signal i, T
1
≤
t
<
T
2, let N
i
(
T
1
,
T
2
) denote the number of types of switching of i over T
1
,
T
2. If N
i
(
T
1
,
T
2
)
≤
N
0
+
T
2
-
T
1
/
T
a holds for N
0
≥
0 and T
a
>
0, then T
a is the average dwell time and N
0 is the chatter bound. Without loss of generality, we choose N
0
=
0.Definition 2 (see [18]).
For a switching signali and any T
≥
t
≥
0, let N
i
p
(
T
,
t
) be the switching numbers in which the pth subsystem is activated over the interval t
,
T and let T
p
(
T
,
t
) denote the total running time of the pth subsystem over the interval t
,
T, p
∈
S. We say that i has a mode-dependent average dwell time (MDADT) τ
a
p if there exist positive numbers N
o
p and τ
a
p such that (2)
N
i
p
T
,
t
≤
N
o
p
+
T
p
T
,
t
τ
a
p
,
∀
T
≥
t
≥
0and we call N
o
p the mode-dependent chatter bounds. Here, we choose N
o
p
=
0 as well.Definition 3.
Forγ
>
0, the switched delay system (1) is said to have L
2-gain property, if, under zero initial condition ϕ
l
=
0, l
∈
t
0
-
d
M
,
t
0, it holds that (3)
∫
0
∞
z
T
s
z
s
d
s
≤
γ
2
∫
0
∞
w
T
s
w
s
d
s
.Lemma 4.
For any given matricesX
,
Y
∈
R
n
×
n, it holds that(4)
X
T
Y
+
Y
T
X
≤
δ
X
T
X
+
δ
-
1
Y
T
Y
,where δ is any given positive constant.Lemma 5 (see [6]).
LetA, D, E, F, and P be real matrices of appropriate dimensions with P
>
0 and F satisfying F
T
F
≤
I. Then for any scalar ε
>
0 satisfying P
-
1
-
ε
-
1
D
D
T
>
0, one has (5)
A
+
D
F
E
T
P
A
+
D
F
E
≤
A
T
P
-
1
-
ε
-
1
D
D
T
-
1
A
+
ε
E
T
E
.Lemma 6 (Schur complement).
LetM, P, and Q be given matrices such that Q
>
0. Then (6)
P
M
∗
-
Q
<
0
⟺
P
+
M
Q
-
1
M
T
<
0
.Lemma 7 (see [13]).
Letϕ
(
k
)
∈
R
n be a vector-valued function. If there exist any matrices R
>
0, G
1, G
2, and a scalar d
≥
0, then the following inequality (7)
-
∑
s
=
k
-
d
k
-
1
N
T
s
R
N
s
≤
η
T
k
G
1
+
G
1
T
-
G
1
T
+
G
2
∗
-
G
2
-
G
2
T
η
k
+
η
T
k
G
1
T
G
2
T
d
R
-
1
G
1
G
2
η
kholds, where N
s
=
ϕ
s
+
1
-
ϕ
(
s
) and η
t
=
ϕ
(
t
)
ϕ
(
t
-
d
).


## 3.L
2-Gain Analysis

Firstly, we will introduce two important lemmas for theL
2-gain analysis of the switched delay system (1). The first lemma will provide the decay estimation of the Lyapunov functional V
i
t along the trajectory of the switched delay system without disturbances.Lemma 8.
Consider the switched delay system (1) with w
(
t
)
=
0. For given positive integers d
M, d
m, and λ
i, suppose that there exist matrices G
1, G
2, Ω
1, Ω
2, and Ω
3 such that
(i)(8)
Ω
3
≤
0
.
(ii)(9)
Ω
1
-
Ω
2
Ω
3
-
1
Ω
2
T
≤
0
,where(10)
Ω
1
=
A
i
T
P
i
A
i
-
P
i
+
λ
i
-
2
Q
i
+
d
M
-
d
m
λ
i
-
2
Q
i
+
λ
i
-
2
d
M
λ
i
2
A
i
T
R
i
A
i
-
λ
i
R
i
A
i
-
λ
i
A
i
T
R
i
+
R
i
+
λ
i
-
2
G
1
+
G
1
T
+
d
M
G
1
T
R
i
-
1
G
1
,
Ω
2
=
A
i
T
P
i
B
i
+
d
M
A
i
T
R
i
B
i
-
λ
i
-
1
R
i
B
i
+
λ
i
-
2
-
G
1
T
+
G
2
+
d
M
G
1
T
R
i
-
1
G
2
,
Ω
3
=
B
i
T
P
i
B
i
-
λ
i
-
2
1
+
d
M
Q
i
+
d
M
B
i
T
R
i
B
i
+
λ
i
-
2
-
G
2
-
G
2
T
+
d
M
G
2
T
R
i
-
1
G
2with P
i, Q
i, and R
i being symmetric positive definite matrices; then the Lyapunov functional V
i
t along the trajectory of the switched delay system (1) will satisfy (11)
V
i
t
≤
λ
i
-
2
t
-
t
0
V
i
t
0
.Proof.
Choose the following Lyapunov functional candidate:(12)
V
i
t
=
V
i
1
t
+
V
i
2
t
+
V
i
3
t
+
V
i
4
t
.Here,(13)
V
i
1
t
=
x
T
t
P
i
x
t
,
V
i
2
t
=
∑
s
=
t
-
d
t
t
-
1
λ
i
2
s
-
t
x
T
s
Q
i
x
s
,
V
i
3
t
=
∑
θ
=
-
d
M
+
2
-
d
m
+
1
∑
s
=
t
-
1
+
θ
t
-
1
λ
i
2
s
-
t
x
T
s
Q
i
x
s
,
V
i
4
t
=
∑
θ
=
-
d
M
+
1
0
∑
s
=
t
-
1
+
θ
t
-
1
λ
i
2
s
-
t
y
T
s
R
i
y
s
,where P
i, Q
i, and R
i are symmetric positive definite matrices, λ
i
>
1 is a given constant, and y
s
=
λ
i
x
s
+
1
-
x
s. Next, we will estimate the difference of V
i
t along the trajectory of the switched delay system (1):(14)
Δ
V
i
1
t
=
V
i
1
t
+
1
-
V
i
1
t
=
x
T
t
+
1
P
i
x
t
+
1
-
x
T
t
P
i
x
t
=
x
T
t
A
i
T
P
i
A
i
x
t
+
x
T
t
-
d
t
B
i
T
P
i
A
i
x
t
+
x
T
t
A
i
T
P
i
B
i
x
t
-
d
t
+
x
T
t
-
d
t
B
i
T
P
i
B
i
x
t
-
d
t
-
x
T
t
P
i
x
t
.Then, we have(15)
Δ
V
i
1
t
=
x
t
x
t
-
d
t
T
A
i
T
P
i
A
i
-
P
i
A
i
T
P
i
B
i
B
i
T
P
i
A
i
B
i
T
P
i
B
i
x
t
x
t
-
d
t
,
(16)
Δ
V
i
2
t
=
V
i
2
t
+
1
-
V
i
2
t
≤
V
i
2
t
+
1
-
λ
i
-
2
V
i
2
t
=
∑
s
=
t
+
1
-
d
t
+
1
t
λ
i
2
s
-
t
-
1
x
T
s
Q
i
x
s
-
∑
s
=
t
-
d
t
t
-
1
λ
i
2
s
-
t
-
1
x
T
s
Q
i
x
s
=
λ
i
-
2
x
T
t
Q
i
x
t
+
∑
s
=
t
+
1
-
d
t
+
1
t
-
1
λ
i
2
s
-
t
-
1
x
T
s
Q
i
x
s
-
∑
s
=
t
-
d
t
t
-
1
λ
i
2
s
-
t
-
1
x
T
s
Q
i
x
s
.Since the delay d
(
t
) satisfies 0
<
d
m
<
d
(
t
)
≤
d
M, we can consider the following two cases.
Whend
m
>
1, it holds that (17)
∑
s
=
t
+
1
-
d
t
+
1
t
-
1
λ
i
2
s
-
t
-
1
x
T
s
Q
i
x
s
≤
∑
s
=
t
+
1
-
d
m
t
-
1
λ
i
2
s
-
t
-
1
x
T
s
Q
i
x
s
+
∑
s
=
t
+
1
-
d
t
+
1
t
-
d
m
λ
i
2
s
-
t
-
1
x
T
s
Q
i
x
s
≤
∑
s
=
t
+
1
-
d
t
t
-
1
λ
i
2
s
-
t
-
1
x
T
s
Q
i
x
s
+
∑
s
=
t
+
1
-
d
M
t
-
d
m
λ
i
2
s
-
t
-
1
x
T
s
Q
i
x
s
.
Whend
m
=
1,(18)
∑
s
=
t
+
1
-
d
t
+
1
t
-
1
λ
i
2
s
-
t
-
1
x
T
s
Q
i
x
s
≤
∑
s
=
t
+
1
-
d
t
t
-
1
λ
i
2
s
-
t
-
1
x
T
s
Q
i
x
s
+
∑
s
=
t
+
1
-
d
M
t
-
d
m
λ
i
2
s
-
t
-
1
x
T
s
Q
i
x
sis satisfied as well.
So from (16) and (17) we can obtain(19)
Δ
V
i
2
t
≤
λ
i
-
2
x
T
t
Q
i
x
t
+
∑
s
=
t
+
1
-
d
t
t
-
1
λ
i
2
s
-
t
-
1
x
T
s
Q
i
x
s
+
∑
s
=
t
+
1
-
d
M
t
-
d
m
λ
i
2
s
-
t
-
1
x
T
s
Q
i
x
s
-
∑
s
=
t
-
d
t
t
-
1
λ
i
2
s
-
t
-
1
x
T
s
Q
i
x
s
=
λ
i
-
2
x
T
t
Q
i
x
t
-
λ
i
2
-
1
-
d
t
x
T
t
-
d
t
Q
i
x
t
-
d
t
+
∑
s
=
t
+
1
-
d
M
t
-
d
m
λ
i
2
s
-
t
-
1
x
T
s
Q
i
x
s
.Since -
λ
i
2
-
1
-
d
t
≤
-
λ
i
2
-
1
-
d
M, we get(20)
Δ
V
i
2
t
≤
x
t
x
t
-
d
t
T
λ
i
-
2
Q
i
0
0
-
λ
i
2
-
1
-
d
M
Q
i
x
t
x
t
-
d
t
+
∑
s
=
t
+
1
-
d
M
t
-
d
m
λ
i
2
s
-
t
-
1
x
T
s
Q
i
x
s
.The derivation process of Δ
V
i
3
t is similar to Δ
V
i
2
t, and then we have(21)
Δ
V
i
3
t
=
V
i
3
t
+
1
-
V
i
3
t
≤
V
i
3
t
+
1
-
λ
i
-
2
V
i
3
t
=
∑
θ
=
-
d
M
+
2
-
d
m
+
1
∑
s
=
t
+
θ
t
-
1
λ
i
2
s
-
t
-
1
x
T
s
Q
i
x
s
+
∑
θ
=
-
d
M
+
2
-
d
m
+
1
λ
i
-
2
x
T
t
Q
i
x
t
-
∑
θ
=
-
d
M
+
2
-
d
m
+
1
∑
s
=
t
-
1
+
θ
t
-
1
λ
i
2
s
-
t
-
1
x
T
s
Q
i
x
s
≤
-
∑
s
=
t
+
1
-
d
M
t
-
d
m
λ
i
2
s
-
t
-
1
x
T
s
Q
i
x
s
+
d
M
-
d
m
λ
i
-
2
x
T
t
Q
i
x
twhich is equal to(22)
Δ
V
i
3
≤
x
t
x
t
-
d
t
T
d
M
-
d
m
λ
i
-
2
Q
i
0
0
0
x
t
x
t
-
d
t
-
∑
s
=
t
+
1
-
d
M
t
-
d
m
λ
i
2
s
-
t
-
1
x
T
s
Q
i
x
s
,
(23)
Δ
V
i
4
t
=
V
i
4
t
+
1
-
V
i
4
t
≤
V
i
4
t
+
1
-
λ
i
-
2
V
i
4
t
≤
∑
θ
=
-
d
M
+
1
0
∑
s
=
t
+
θ
t
-
1
λ
i
2
s
-
t
-
1
y
T
s
R
i
y
s
+
d
M
·
λ
i
-
2
y
T
t
R
i
y
t
-
∑
θ
=
-
d
M
+
1
0
∑
s
=
t
-
1
+
θ
t
-
1
λ
i
2
s
-
t
-
1
y
T
s
R
i
y
s
≤
d
M
·
λ
i
-
2
y
T
t
R
i
y
t
-
∑
s
=
t
-
d
M
t
-
1
λ
i
2
s
-
t
-
1
y
T
s
R
i
y
s
.Since y
s
=
λ
i
x
s
+
1
-
x
s, we substitute it into (23) and obtain(24)
Δ
V
i
4
t
≤
λ
i
-
2
d
M
x
t
x
t
-
d
t
T
λ
i
A
i
T
-
I
R
i
λ
i
A
i
-
I
λ
i
λ
i
A
i
T
-
I
R
i
B
i
B
i
T
R
i
λ
i
A
i
-
I
λ
i
λ
i
2
B
i
T
R
i
B
i
x
t
x
t
-
d
t
-
∑
s
=
t
-
d
M
t
-
1
λ
i
2
s
-
t
-
1
y
T
s
R
i
y
s
,where we apply the transformation ϕ
s
=
λ
i
s
-
t
-
1
x
(
s
). Then we have λ
i
s
-
t
-
1
y
s
=
ϕ
s
+
1
-
ϕ
(
s
); by Lemma 7 we continue to have (25)
-
∑
s
=
t
-
d
M
t
-
1
λ
i
2
s
-
t
-
1
y
T
s
R
i
y
s
≤
ϕ
t
ϕ
t
-
d
t
T
G
1
+
G
1
T
-
G
1
T
+
G
2
∗
-
G
2
-
G
2
T
ϕ
t
ϕ
t
-
d
t
+
ϕ
t
ϕ
t
-
d
t
T
G
1
T
G
2
T
d
M
R
i
-
1
G
1
G
2
ϕ
t
ϕ
t
-
d
t
.Due to the fact that ϕ
t
=
λ
i
-
1
x
t, ϕ
t
-
d
t
=
λ
i
-
d
t
+
1
x
(
t
-
d
t
)
≤
λ
i
-
1
x
t
-
d
(
t
), it holds that(26)
Δ
V
i
4
t
≤
λ
i
-
2
d
M
x
t
x
t
-
d
t
T
λ
i
A
i
T
-
I
R
i
λ
i
A
i
-
I
λ
i
λ
i
A
i
T
-
I
R
i
B
i
B
i
T
R
i
λ
i
A
i
-
I
λ
i
λ
i
2
B
i
T
R
i
B
i
x
t
x
t
-
d
t
+
λ
i
-
2
x
t
x
t
-
d
t
T
G
1
+
G
1
T
-
G
1
T
+
G
2
∗
-
G
2
-
G
2
T
x
t
x
t
-
d
t
+
λ
i
-
2
x
t
x
t
-
d
t
T
G
1
T
G
2
T
d
M
R
i
-
1
G
1
G
2
x
t
x
t
-
d
t
.Let ξ
t
=
x
(
t
)
x
t
-
d
t; then we add (15), (20), (22), and (26) together to yield (27)
Δ
V
i
≤
ξ
T
t
Ω
ξ
t
,where Ω
=
Ω
1
Ω
2
∗
Ω
3,(28)
Ω
1
=
A
i
T
P
i
A
i
-
P
i
+
λ
i
-
2
Q
i
+
d
M
-
d
m
λ
i
-
2
Q
i
+
λ
i
-
2
d
M
λ
i
2
A
i
T
R
i
A
i
-
λ
i
R
i
A
i
-
λ
i
A
i
T
R
i
+
R
i
+
λ
i
-
2
G
1
+
G
1
T
+
d
M
G
1
T
R
i
-
1
G
1
,
Ω
2
=
A
i
T
P
i
B
i
+
d
M
A
i
T
R
i
B
i
-
λ
i
-
1
R
i
B
i
+
λ
i
-
2
-
G
1
T
+
G
2
+
d
M
G
1
T
R
i
-
1
G
2
,
Ω
3
=
B
i
T
P
i
B
i
-
λ
i
-
2
1
+
d
M
Q
i
+
d
M
B
i
T
R
i
B
i
+
λ
i
-
2
-
G
2
-
G
2
T
+
d
M
G
2
T
R
i
-
1
G
2
.By (8) and (9) and Lemma 6, we can obtain(29)
Ω
=
Ω
1
Ω
2
∗
Ω
3
≤
0
.It follows from (27) and (29) that(30)
V
i
t
+
1
≤
λ
i
-
2
V
i
t
.Therefore,(31)
V
i
t
≤
λ
i
-
2
V
i
t
-
1
≤
⋯
≤
λ
i
-
2
t
-
t
0
V
i
t
0
.This completes the proof.Lemma 9.
For given constantsλ
i and γ
0, suppose that there exist matrices Ξ
1, Ξ
2, and Ξ
3 such that
(i)(32)
Ξ
3
≤
0
(ii)(33)
Ξ
1
-
Ξ
2
Ξ
3
-
1
Ξ
2
T
≤
0and γ
0
>
0, ε
1
>
0, and ε
2
>
0 satisfying (34)
γ
0
2
I
≥
ε
1
-
1
I
+
ε
2
-
1
I
+
C
i
T
P
i
C
i
+
d
M
C
i
T
R
i
C
i
+
E
i
T
E
i
;then along the trajectory of system (1), one has (35)
V
i
t
+
1
≤
λ
i
-
2
V
i
t
+
γ
0
2
w
T
t
w
t
-
Z
T
t
Z
t
,where(36)
Ξ
1
=
Ω
1
+
ε
1
φ
1
i
T
φ
1
i
+
D
i
T
D
i
,
Ξ
2
=
Ω
2
,
Ξ
3
=
Ω
3
+
ε
2
φ
2
i
T
φ
2
i
,
φ
1
i
=
C
i
T
P
i
A
i
+
d
M
C
i
T
R
i
A
i
-
λ
i
-
1
C
i
T
R
i
+
E
i
T
D
i
φ
2
i
=
C
i
T
P
i
B
i
+
d
M
C
i
T
R
i
B
i
.Proof.
Using Lemma8 and (1), we have(37)
V
i
t
+
1
-
λ
i
-
2
V
i
t
+
Z
T
t
Z
t
-
γ
0
2
w
T
t
w
t
≤
ξ
T
t
Ω
ξ
t
+
x
T
t
A
i
T
P
i
C
i
+
d
M
A
i
T
R
i
C
i
-
λ
i
-
1
R
i
C
i
+
D
i
T
E
i
w
t
+
w
T
t
C
i
T
P
i
A
i
+
d
M
C
i
T
R
i
A
i
-
λ
i
-
1
C
i
T
R
i
+
E
i
T
D
i
x
t
+
x
T
t
-
d
t
B
i
T
P
i
C
i
+
d
M
B
i
T
R
i
C
i
w
t
+
w
T
t
C
i
T
P
i
B
i
+
d
M
C
i
T
R
i
B
i
x
t
-
d
t
+
x
T
t
D
i
T
D
i
x
t
+
w
T
t
C
i
T
P
i
C
i
+
d
M
C
i
T
R
i
C
i
+
E
i
T
E
i
-
γ
0
2
I
w
t
.Based on Lemmas 4 and 5, it holds that (38)
x
T
t
A
i
T
P
i
C
i
+
d
M
A
i
T
R
i
C
i
-
λ
i
-
1
R
i
C
i
+
D
i
T
E
i
w
t
+
w
T
t
C
i
T
P
i
A
i
+
d
M
C
i
T
R
i
A
i
-
λ
i
-
1
C
i
T
R
i
+
E
i
T
D
i
x
t
≤
ε
1
x
T
t
φ
1
i
T
φ
1
i
x
t
+
ε
1
-
1
w
T
t
w
t
.
x
T
t
-
d
t
B
i
T
P
i
C
i
+
d
M
B
i
T
R
i
C
i
w
t
+
w
T
t
C
i
T
P
i
B
i
+
d
M
C
i
T
R
i
B
i
x
t
-
d
t
≤
ε
2
x
T
t
-
d
t
φ
2
i
T
φ
2
i
x
t
-
d
t
+
ε
2
-
1
w
T
t
w
t
.Then, it follows from (35) and (38) that(39)
V
i
t
+
1
-
λ
i
-
2
V
i
t
+
Z
T
t
Z
t
-
γ
0
2
w
T
t
w
t
≤
ξ
T
t
Ω
1
+
ε
1
φ
1
i
T
φ
1
i
+
D
i
T
D
i
Ω
2
∗
Ω
3
+
ε
2
φ
2
i
T
φ
2
i
ξ
t
+
w
T
t
ε
1
-
1
I
+
ε
2
-
1
I
+
C
i
T
P
i
C
i
+
d
M
C
i
T
R
i
C
i
+
E
i
T
E
i
-
γ
0
2
I
w
t
.Combining (32), (33) with (34) will lead to (40)
V
i
t
+
1
≤
λ
i
-
2
V
i
t
+
γ
0
2
w
T
t
w
t
-
Z
T
t
Z
t
.This completes the proof.Now, ourL
2-gain analysis results can be presented as follows.Theorem 10.
For given constantsλ
i and γ
0, suppose that there exist matrices Ξ
1, Ξ
2, and Ξ
3 such that
(i)(41)
Ξ
3
≤
0
(ii)(42)
Ξ
1
-
Ξ
2
Ξ
3
-
1
Ξ
2
T
≤
0and γ
0
>
0, ε
1
>
0, and ε
2
>
0 satisfying (43)
γ
0
2
I
≥
ε
1
-
1
I
+
ε
2
-
1
I
+
C
i
T
P
i
C
i
+
d
M
C
i
T
R
i
C
i
+
E
i
T
E
i
.Then the switched delay system (1) has a L
2-gain with MDADT τ
a
p
>
τ
a
p
∗
=
ln
⁡
μ
p
/
2
ln
⁡
λ
p, where μ
p
≥
1 satisfying (35) and φ
1
i, φ
2
i, Ξ
1, Ξ
2, and Ξ
3 are defined in Lemma 9.Proof.
Choose the Lyapunov functional candidate (12). From (41) and (42) and Lemma 9, we have (44)
V
i
t
+
1
≤
λ
i
-
2
V
i
t
+
γ
0
2
w
T
t
w
t
-
Z
T
t
Z
t
.Let Γ
(
t
)
=
γ
0
2
w
T
t
w
t
-
Z
T
t
Z
t. From (35), since t
i
-
1
=
t
i
-
1, we have(45)
V
σ
t
t
≤
λ
σ
t
i
-
2
t
-
t
i
V
σ
t
i
t
i
+
∑
j
=
t
i
t
-
1
λ
σ
t
i
-
2
t
-
j
-
1
Γ
j
≤
μ
σ
t
i
λ
σ
t
i
-
2
t
-
t
i
V
σ
t
i
-
1
t
i
+
∑
j
=
t
i
t
-
1
λ
σ
t
i
-
2
t
-
j
-
1
Γ
j
≤
μ
σ
t
i
λ
σ
t
i
-
2
t
-
t
i
λ
σ
t
i
-
1
-
2
t
i
-
t
i
-
1
V
σ
t
i
-
1
t
i
-
1
+
∑
j
=
t
i
-
1
t
i
-
1
λ
σ
t
i
-
1
-
2
t
i
-
j
-
1
Γ
j
+
∑
j
=
t
i
t
-
1
λ
σ
t
i
-
2
t
-
j
-
1
Γ
j
≤
⋯
≤
∏
s
=
1
i
μ
σ
t
s
·
exp
⁡
-
2
∑
s
=
1
i
ln
⁡
λ
σ
t
s
-
1
t
s
-
t
s
-
1
V
σ
t
0
t
0
+
∑
k
=
1
i
∏
s
=
k
i
μ
σ
t
s
·
exp
⁡
-
2
∑
s
=
k
i
ln
⁡
λ
σ
t
s
-
1
t
s
-
t
s
-
1
Γ
t
k
-
1
=
e
x
p
∑
s
=
1
i
ln
⁡
μ
σ
t
s
-
2
ln
⁡
λ
σ
t
s
-
1
t
s
-
t
s
-
1
V
σ
t
0
t
0
+
∑
k
=
1
i
e
x
p
∑
s
=
k
i
ln
⁡
μ
σ
t
s
-
2
ln
⁡
λ
σ
t
s
-
1
t
s
-
t
s
-
1
Γ
t
k
-
1which combined with Definition 2 and the MDADT scheme N
σ
p
T
,
t
≤
T
p
T
,
t
/
τ
a
p yields(46)
V
σ
t
t
≤
e
x
p
∑
p
=
1
m
ln
⁡
μ
p
N
σ
p
t
,
t
0
-
2
ln
⁡
λ
p
T
p
t
,
t
0
V
σ
t
0
t
0
+
∑
k
=
1
i
e
x
p
∑
p
=
1
m
ln
⁡
μ
p
N
σ
p
t
,
t
k
-
1
-
2
ln
⁡
λ
p
T
p
t
,
t
k
-
1
Γ
t
k
-
1
≤
e
-
β
T
p
t
,
t
0
V
σ
t
0
t
0
+
∑
k
=
1
i
e
-
β
T
p
t
,
t
k
-
1
Γ
t
k
-
1
,where β
=
∑
p
=
1
m
(
2
ln
⁡
λ
p
-
ln
⁡
μ
p
/
τ
σ
p
)
>
0.
Under zero initial condition, from (46), one obtains(47)
0
≤
V
σ
t
t
≤
∑
k
=
1
i
e
-
β
T
p
t
,
t
0
Γ
t
k
-
1which implies that(48)
∑
k
=
1
i
e
-
β
T
p
t
,
t
k
-
1
Z
T
t
k
-
1
Z
t
k
-
1
≤
∑
k
=
1
i
e
-
β
T
p
t
,
t
k
-
1
γ
0
2
w
T
t
k
-
1
w
t
k
-
1
.Then, we multiply both sides by e
-
β
T
p
t
k
-
1
,
t
0 to get(49)
∑
k
=
1
i
e
-
β
T
p
t
,
t
0
Z
T
t
k
-
1
Z
t
k
-
1
≤
∑
k
=
1
i
e
-
β
T
p
t
,
t
0
γ
0
2
w
T
t
k
-
1
w
t
k
-
1
.Thus,(50)
∑
k
=
0
i
Z
T
t
k
Z
t
k
≤
∑
k
=
0
i
γ
0
2
w
T
t
k
w
t
k
.This completes the proof.


## 4. A Numerical Example

Consider the switched delay system (1) with the following specifications: (51)
A
1
=
-
0.2
0.3
0.1
-
0.5
,
B
1
=
0.4
0
0.1
-
0.5
,
A
2
=
-
0.1
1
0
-
0.6
,
B
2
=
-
0.7
0.1
1
0.2
,
C
1
=
1
0
0
0
,
C
2
=
0
1
1
0
,
D
1
=
1,1
,
D
2
=
0,1
,
E
1
=
E
2
=
0.2,0.8
,and d
t
=
sin
⁡
t
π
/
2
+
1, so that d
M
=
2, d
m
=
0. The disturbance input is defined as(52)
w
t
=
1
,
0
<
t
≤
20
,
0
,
t
>
20
.Let μ
1
=
μ
2
=
12; by the LMI Control Toolbox and Theorem 10, we obtain (53)
P
1
=
12.5739
5.0613
5.0613
4.8703
,
Q
1
=
2.6676
1.2445
1.2445
0.9452
,
R
1
=
4.4703
-
0.6996
-
0.6996
9.3862
,
P
2
=
12.3445
18.3565
18.3565
30.2222
,
Q
2
=
2.3941
3.7907
3.7907
6.1097
,
R
2
=
16.1813
18.8134
18.8134
22.4268
,and λ
1
=
27.2485, λ
2
=
38.7807, where τ
a
1
∗
=
ln
⁡
μ
1
/
2
ln
⁡
λ
1
=
0.3759 and τ
a
2
∗
=
ln
⁡
μ
2
/
2
ln
⁡
λ
2
=
0.3397. Now, we choose the switching periods τ
a
1
=
2, τ
a
2
=
1 and take the initial state condition ψ
l
=
[
1
;
2
] for all l
=
-
2
,
-
1,0. Then the numerical simulations can be shown in Figure 1.Figure 1
State trajectories of the switched delay system (1) under MDADT switching.It can be seen from Figure1 that under the designed MDADT switching signals the switched delay system can achieve better dynamics performance and disturbance tolerance capability, which shows the potentiality of our results in practice.


## 5. Conclusions

In this paper, the problem ofL
2-gain analysis for discrete-time switched systems with MDADT switching has been investigated. By combining with the multiple Lyapunov function method, sufficient conditions are established to ensure L
2-gain performance for discrete-time switched delay system, and the admissible MDADT switching signals are also designed accordingly. Finally, a numerical example is given to demonstrate the usefulness of the obtained results.



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*Source: 1673959-2016-03-21.xml*