# An Improved Finite-Time Stability and Stabilization of Linear System with Constant Delay

Authors: Thaned Rojsiraphisal; Jirapong Puangmalai
Journal: Mathematical Problems in Engineering
 (2014)

---

## Abstract

Practical systems in engineering fields often require that values of state variables, during the finite-time interval, must not exceed a certain value when the initial values of state are given. This leads us to investigate the finite-time stability and stabilization of a linear system with a constant time-delay. Sufficient conditions to guarantee the finite-time stability and stabilization are derived by using a new form of Lyapunov-Krasovskii functional and a desired state-feedback controller. These conditions are in the form of LMIs and inequalities. Two numerical examples are given to show the effectiveness of the proposed criteria. Results show that our proposed criteria are less conservative than previous works in terms of versatility of minimum bounds and larger bounds of time-delay.


---

## Body

## 1. Introduction

In the past decades, researchers have paid much attention to asymptotic stability which concerns behaviors of state variables over an infinite time interval. One disadvantage of the asymptotic stability behavior is that large values of state variables may present during transient period. In practical system, the presence of large values should not exceed its limit, for example, the presence of saturations or the excitation of nonlinear dynamics [1, 2]. This leads us to a concept called finite-time stability, introduced back in 1960s. This concept is focusing on the behavior of state variables during the transient period which must not exceed a certain value when the upper bound of initial condition is given (see [1, 3–5]). Researchers have studied finite-time stability on various systems such as linear system, impulsive system, neural networks, and switched systems (see [1–18] and references therein) and proposed sufficient conditions to guarantee finite-time stability in the forms of linear matrix inequality (LMI), Lyapunov differential matrix equation, or algebraic inequality, and so forth.Time-delay often occurs in practical systems such as biological systems, chemical systems, electrical networks, and engineering fields. It is known that the small change of time-delay can cause instability and poor performance of such systems. Considering the broad applications related to the time-delay and the suitable values of state variables during the transient, it is important to investigate the finite-time stability and stabilization of systems with time-delay. From literature, the studies of finite-time stability and finite-time stabilization on time-delay system are not many [9, 12–16, 18–21] and only few studies are on linear system with time-delay [9, 11, 16].In this paper, we propose new sufficient conditions on finite-time stability and stabilization for a linear system with constant time-delay in the form of LMIs. The proposed conditions are formulated using a new form Lyapunov-Krasovskii functional. To illustrate the efficiency of the proposed conditions, two numerical examples are presented at the end.


## 2. Preliminaries

The following notations will be used in this paper.R
n denotes the n-dimensional space with the scalar product x
T
y and the vector norm ·; R
n
×
n denotes an n
×
n matrix with real value elements; A
T denotes the transpose of the matrix A; λ
(
A
) denotes eigenvalues of A; λ
max
⁡
(
A
) and λ
min
⁡
(
A
) represent the maximum and minimum of real part of λ
(
A
), respectively. x
t
∶
=
{
x
(
t
+
s
)
:
s
∈
[
-
τ
,
0
]
}; x
t
∶
=
sup
⁡
s
∈
[
-
τ
,
0
]
{
x
(
t
+
s
)
,
x
˙
(
t
+
s
)
}; A
>
0  (
A
<
0
) means A is positive (negative) definite; A
>
B is equivalent to A
-
B
>
0. Entries * in a matrix represent the symmetric elements of the symmetric matrix.Consider a linear system with constant time-delay(1)
x
˙
t
=
A
0
x
t
+
A
1
x
t
-
τ
+
B
u
t
,
where t
>
0 and x
(
t
)
∈
R
n is the state vector of the system. u
(
t
)
∈
R
m is the control input. A
0
,
A
1
∈
R
n
×
n and B
∈
R
n
×
m are known constant matrices. The delay τ is a positive constant.We assume that the initial condition isx
(
t
+
θ
)
=
ϕ
(
θ
), ∀
θ
∈
[
-
τ
,
0
], and ϕ
(
·
) is a differentiable vector-valued initial function with the norm ϕ
∶
=
sup
⁡
t
∈
[
-
τ
,
0
]
{
ϕ
(
t
)
,
ϕ
˙
(
t
)
}.Here we choose a state-feedback control law of the form(2)
u
t
=
-
0.5
B
T
P
x
t
,
where P is a design parameter to be determined.To formulate sufficient conditions of finite-time stability and stabilization of the linear system (1), the following definition and lemmas are used.Definition 1.
System (1) is said to be finite-time stable with respect to (
c
1
,
c
2
,
T
), where 0
≤
c
1
<
c
2, if
(3)
ϕ
2
≤
c
1
⟹
x
t
2
≤
c
2
,
∀
t
∈
0
,
T
.Lemma 2 (see [22]).
For any symmetric positive-definite matrixM
∈
R
n
×
n, scalar σ
≥
0, and vector function w
:
[
0
,
σ
]
→
R
n such that the concerned integrations are well defined, then
(4)
∫
0
σ
w
s
d
s
T
M
∫
0
σ
w
s
d
s
≤
σ
∫
0
σ
w
T
s
M
w
s
d
s
.Lemma 3 (see [23], Schur complement).
Given constant matricesX, Y, Z with appropriate dimensions satisfying X
=
X
T, Y
=
Y
T
>
0, then X
+
Z
T
Y
-
1
Z
<
0 if and only if
(5)
X
Z
T
Z
-
Y
<
0
o
r
-
Y
Z
Z
T
X
<
0
.


## 3. Main Results

In this section, we first formulate the finite-time stability of the linear system (1) without controller, that is, u
(
t
)
=
0. We then provide proof of the finite-time stabilization of the linear system (1) with a feedback controller defined in (2). The formulations are as follows.Theorem 4.
Linear system (1) with  u
(
t
)
=
0; time-delay τ is finite-time stable with respect to (
c
1
,
c
2
,
T
), 0
≤
c
1
<
c
2, if there exist nonnegative scalar α, symmetric positive-definite matrices P, Q, R, and positive scalars β
1, β
2, β
3, β
4 such that
(6)
β
1
I
<
P
<
β
2
I
,
0
<
Q
<
β
3
I
,
0
<
R
<
β
4
I
,
(7)
Ω
=
-
β
1
c
2
e
-
α
T
β
2
c
1
β
3
c
1
τ
β
4
0.5
c
1
τ
3
*
-
β
2
0
0
*
*
-
β
3
0
*
*
*
-
β
4
<
0
,
(8)
Ψ
=
Ψ
11
P
A
1
+
τ
2
A
0
T
R
A
1
+
R
*
τ
2
A
1
T
R
A
1
-
Q
-
R
<
0
,
where
(9)
Ψ
11
=
A
0
T
P
+
P
A
0
+
Q
+
τ
2
A
0
T
R
A
0
-
R
-
α
P
.Proof.
Let us consider the following Lyapunov-Krasovskii functional:(10)
V
x
t
=
e
α
t
x
T
t
P
x
t
+
e
α
t
∫
t
-
τ
t
‍
x
T
s
Q
x
s
d
s
+
τ
e
α
t
∫
-
τ
0
‍
∫
t
+
s
t
‍
x
˙
T
θ
R
x
˙
θ
d
θ
d
s
.
The derivative of V
(
x
(
t
)
) along solution (1) with u
(
t
)
=
0 is
(11)
V
˙
=
2
e
α
t
x
T
t
P
x
˙
t
+
e
α
t
x
T
t
Q
x
t
-
x
T
t
-
τ
Q
x
t
-
τ
+
τ
e
α
t
τ
x
˙
T
t
R
x
˙
t
-
∫
t
-
τ
t
‍
x
˙
T
s
R
x
˙
s
d
s
+
α
V
=
2
e
α
t
x
T
t
P
A
0
x
t
+
A
1
x
t
-
τ
+
e
α
t
x
T
t
Q
x
t
-
x
T
t
-
τ
Q
x
t
-
τ
+
τ
e
α
t
τ
A
0
x
t
+
A
1
x
t
-
τ
T
∫
t
-
τ
t
‍
x
˙
T
s
R
x
˙
s
d
s
×
R
A
0
x
t
+
A
1
x
t
-
τ
-
∫
t
-
τ
t
‍
x
˙
T
s
R
x
˙
s
d
s
+
α
V
.
Applying Lemma2 and the Newton-Leibniz formula, we obtain
(12)
V
˙
≤
2
e
α
t
x
T
t
P
A
0
x
t
+
A
1
x
t
-
τ
+
e
α
t
x
T
t
Q
x
t
-
x
T
t
-
τ
Q
x
t
-
τ
+
τ
2
e
α
t
A
0
x
t
+
A
1
x
t
-
τ
T
×
A
0
x
t
+
A
1
x
t
-
τ
T
R
A
0
x
t
+
A
1
x
t
-
τ
-
e
α
t
x
t
-
x
t
-
τ
T
×
R
x
t
-
x
t
-
τ
+
α
V
.
Since α
>
0, P
>
0, and V
>
0, thus we have
(13)
V
˙
(
x
(
t
)
)
-
2
α
V
x
t
≤
e
α
t
x
T
t
×
A
0
T
P
+
P
A
0
+
Q
+
τ
2
A
0
T
R
A
0
-
R
-
α
P
x
t
+
2
x
T
t
P
A
1
+
τ
2
A
0
T
R
A
1
+
R
x
t
-
τ
+
x
T
t
-
τ
τ
2
A
1
T
R
A
1
-
Q
-
R
x
t
-
τ
=
η
T
t
Ψ
η
t
,
where η
(
t
)
=
[
x
(
t
)
,
x
(
t
-
τ
)
].
BecauseΨ
<
0, thus inequality (13) leads to V
˙
(
x
(
t
)
)
-
2
α
V
(
x
(
t
)
)
<
0. Multiplying this inequality by e
-
2
α
t and integrating from 0 to t, with t
∈
[
0
,
T
], we obtain
(14)
e
α
t
λ
min
⁡
P
x
t
2
≤
V
x
t
<
e
2
α
t
V
x
0
,
with
(15)
V
x
0
=
x
T
0
P
x
0
+
∫
-
τ
0
‍
x
T
s
Q
x
s
d
s
+
τ
∫
-
τ
0
‍
∫
s
0
‍
x
˙
T
θ
R
x
˙
θ
d
θ
d
s
≤
λ
max
⁡
P
+
τ
λ
max
⁡
Q
+
0.5
τ
3
λ
max
⁡
R
ϕ
2
.
Now we define
(16)
Π
:
≡
c
1
β
2
+
τ
β
3
+
0.5
τ
3
β
4
-
c
2
β
1
e
-
α
T
<
0
,
where β
1, β
2, β
3, β
4 are positive and satisfy (6). Applying Schur compliment, it is easy to see that Π
<
0 is equivalent to Ω
<
0. Because ϕ
2
≤
c
1, thus the relations (14)-(15) lead to
(17)
x
t
2
≤
c
1
e
α
T
λ
min
⁡
P
λ
max
⁡
P
+
τ
λ
max
⁡
Q
+
0.5
τ
3
λ
max
⁡
R
.
With P, Q, R satisfying (6), we have
(18)
x
(
t
)
2
≤
c
1
e
α
T
β
1
β
2
+
τ
β
3
+
0.5
τ
3
β
4
<
c
2
.
The proof is complete.Next consider the linear system (1) with a feedback controller of the form u
(
t
)
=
-
0.5
B
T
P
x
(
t
) as defined in (2). This equation can be rewritten as
(19)
x
˙
t
=
A
^
0
x
t
+
A
1
x
t
-
τ
,
where A
^
0
=
A
0
-
0.5
B
B
T
P.Remark 5.
One can notice that the finite-time stabilization of (1) with the feedback controller above is equivalent to finite-time stability of (19). Thus we can formulate the finite-time stabilization as in the following theorem.Theorem 6.
The linear system (1) with feedback controller (2) is finite-time stabilization with respect to (
c
1
,
c
2
,
T
), 0
≤
c
1
<
c
2, if there exist a scalar α
>
0, symmetric positive-definite matrices X, Y, Z, R, W
1, W
2, and matrices W
3, W
4, W
5 such that
(20)
Σ
=
Σ
11
Σ
12
*
Σ
22
<
0
,
where
(21)
Σ
11
=
X
A
0
T
+
A
0
X
+
Y
-
Z
+
τ
2
W
1
-
τ
2
B
B
T
W
4
+
0.25
τ
2
B
B
T
R
B
B
T
-
B
B
T
-
α
X
,
Σ
12
=
A
1
X
+
Z
+
τ
2
W
3
-
0.5
τ
2
B
B
T
W
5
,
Σ
22
=
τ
2
W
2
-
Y
-
Z
,
(22)
c
1
e
α
T
λ
max
⁡
X
1
λ
min
⁡
X
+
τ
λ
min
⁡
X
Y
-
1
X
0.5
τ
3
λ
min
⁡
X
Z
-
1
X
+
0.5
τ
3
λ
min
⁡
X
Z
-
1
X
<
c
2
.Proof.
ReplacingA
0 in the LMI (8) with A
^
0
=
A
0
-
0.5
B
B
T
P, we obtain
(23)
Ψ
^
=
Ψ
^
11
Ψ
^
12
*
Ψ
^
22
<
0
,
where
(24)
Ψ
^
11
=
A
0
T
P
+
P
A
0
+
Q
+
τ
2
A
0
T
R
A
0
-
R
-
α
P
-
P
B
B
T
P
+
0.25
τ
2
P
B
B
T
R
B
B
T
P
-
τ
2
P
B
B
T
R
A
0
,
Ψ
^
12
=
P
A
1
+
R
+
τ
2
A
0
T
R
A
1
-
0.5
τ
2
P
B
B
T
R
A
1
,
Ψ
^
22
=
τ
2
A
1
T
R
A
1
-
Q
-
R
.
Pre- and postmultiplying the above inequality bydiag
⁡
{
P
-
1
,
P
-
1
}, we obtain
(25)
Σ
^
11
Σ
^
12
*
Σ
^
22
<
0
,
where
(26)
Σ
^
11
=
P
-
1
A
0
T
+
A
0
P
-
1
+
P
-
1
Q
P
-
1
+
τ
2
P
-
1
A
0
T
R
A
0
P
-
1
-
P
-
1
R
P
-
1
-
α
P
-
1
-
B
B
T
-
τ
2
B
B
T
R
A
0
P
-
1
+
0.25
τ
2
B
B
T
R
B
B
T
,
Σ
^
12
=
A
1
P
-
1
+
P
-
1
R
P
-
1
+
τ
2
P
-
1
A
0
T
R
A
1
P
-
1
-
0.5
τ
2
B
B
T
R
A
1
P
-
1
,
Σ
^
22
=
τ
2
P
-
1
A
1
T
R
A
1
P
-
1
-
P
-
1
Q
P
-
1
-
P
-
1
R
P
-
1
.
Set X
=
P
-
1, Y
=
P
-
1
Q
P
-
1, Z
=
P
-
1
R
P
-
1, W
1
=
P
-
1
A
0
T
R
A
0
P
-
1, W
2
=
P
-
1
A
1
T
R
A
1
P
-
1, W
3
=
P
-
1
A
0
T
R
A
1
P
-
1, W
4
=
R
A
0
P
-
1, and W
5
=
R
A
1
P
-
1. Thus, LMI (25) is equivalent to (20).
With the setting relations above, inequalities (17) in Theorem 4 can be bounded by
(27)
x
t
2
≤
c
1
e
α
T
λ
min
⁡
P
λ
max
⁡
P
+
τ
λ
max
⁡
Q
+
0.5
τ
3
λ
max
⁡
R
=
c
1
e
α
T
λ
max
⁡
X
1
λ
min
⁡
X
+
τ
λ
min
⁡
X
Y
-
1
X
0.5
τ
3
λ
min
⁡
X
Z
-
1
X
+
0.5
τ
3
λ
min
⁡
X
Z
-
1
X
<
c
2
.
The proof is complete.


## 4. Numerical Examples

In this section, we give two numerical examples to show the effectiveness of our main results by investigating the linear system of the form(28)
x
˙
t
=
A
0
x
t
+
A
1
x
t
-
τ
+
B
u
t
,
where
(29)
A
0
=
-
1.7
1.7
0
1.3
-
1
0.7
0.7
1
-
0.6
,
A
1
=
1.5
-
1.7
0.1
-
1.3
1
-
0.3
-
0.7
1
0.6
,
B
=
-
1
10
20
,
τ
=
0.2
.Remark 7.
The linear system (28) with u
(
t
)
=
0 is not asymptotically stable with initial condition ϕ
T
(
t
)
=
[
0.7,0
,
0
] as shown in Figure 1. The figure reveals that the state variables x
i
(
t
)
→
∞, i
=
1,2
,
3, as t
→
∞. In the next example we will show that this system is finite-time stable.State trajectories (a) and norm of the state trajectories (b) of the linear system (28) with initial condition ϕ
T
(
t
)
=
[
0.7,0
,
0
].
(a)
(b)Example 1.
Consider the finite-time stability of linear system (28) with respect to (
c
1
,
c
2
,
T
)
=
(
0.55,100,2
) and ϕ
T
(
t
)
=
[
0.7,0
,
0
] without the controller; that is, u
(
t
)
=
0. Note that ϕ
T
(
t
)
ϕ
(
t
)
=
0.49
<
0.55
=
c
1. For fixed α
=
1.95, we solve inequalities (6) and LMIs (7) and (8) using MATLAB control toolbox. The feasible solutions guaranteeing finite-time stability of the linear system (28) are β
1
=
4.5462, β
2
=
11.1083, β
3
=
26.8644, β
4
=
18.9680, and
(30)
P
=
7.1646
0.3981
0.0289
0.3981
10.8567
-
0.5974
0.0289
-
0.5974
4.6536
,
Q
=
18.1534
-
7.8255
-
4.4657
-
7.8255
18.5720
-
2.9397
-
4.4657
-
2.9397
2.8834
,
R
=
12.4769
-
1.6625
2.1735
-
1.6625
6.3204
-
2.6576
2.1735
-
2.6576
1.8071
.
We further investigate the finite-time stability of the linear system by comparing the smallest eligible value ofc
2 and the largest eligible value of delay τ when α varies between condition given in [16] and our condition in Theorem 4. Comparing results are plotted in Figure 2. From this figure, one can clearly see that our condition allows smaller value of c
2 (Figure 2(a)) and larger value of delay τ (Figure 2(b)) for all values of α. Moreover, we observe that the optimal values of c
2 and τ are obtained when α
∈
[
2
.
05,2
.
1
]. For α
=
2.05 is fixed, Theorem 4 allows smaller value of c
2 by 17.1
% and larger value of τ by 11.2
%.
In addition, we compare the smallest eligible value ofc
2 for two different time-delays (
τ
=
0.2,0.5
) with α
=
2.1. Results, as shown in Table 1, reveal that Theorem 4 allows smallest value of c
2 compared with the others. Our lower bounds of c
2 are smaller than those given in [16] by 17.1
% and 6.8
% for τ
=
0.2,0.5, respectively. In other words, our proposed condition for finite-time stability of the linear system (28) would be more tolerant for smaller values of c
2 than conditions given in [4, 12, 16]. Note that results obtained in [4, 12] do not use LMI technique.Table 1
The smallest value ofc
2 for c
1
=
0.55
,
T
=
2, and α
=
2.1.
τ
0.2
0.5
[4] (Theorem 6)
1.6
×
1
0
5
2.01
×
1
0
5
[12] (Theorem 3)
2.46
×
1
0
10
2.46
×
1
0
10
[16] (Corollary 1)
95.59
145.04
Theorem4
79.18
135.08Comparing the values ofc
2 (a) and the time-delay τ (b) when α is varied guaranteeing FTS of the linear system (28).
(a)
(b)Example 2.
Consider the finite-time stabilization of linear system (28) with a nonzero feedback controller defined in (2) with respect to (
c
1
,
c
2
,
T
)
=
(
0.55,5
,
10
). Solving LMI (20) and inequality (22), for α
=
1.1
×
1
0
-
5, we obtain the feasible solutions as follows:
(31)
X
=
26.3675
12.3498
-
5.8926
12.3498
22.4784
5.2751
-
5.8926
5.2751
34.4024
,
Y
=
26.0383
-
2.6851
1.4135
-
2.6851
26.0587
-
0.8541
1.4135
-
0.8541
25.1168
,
Z
=
23.1998
2.6851
-
1.4135
2.6851
23.1793
0.8541
-
1.4135
0.8541
24.1213
,
R
=
36.1954
0.0000
-
0.0000
0.0000
36.1954
0.0000
-
0.0000
0.0000
36.1954
,
W
1
=
36.2521
-
0.1074
0.0565
-
0.1074
36.2530
-
0.0342
0.0565
-
0.0342
36.2153
,
W
2
=
36.6584
-
0.0000
0.0000
-
0.0000
36.6584
-
0.0000
0.0000
-
0.0000
36.6584
,
W
3
=
-
996.5122
365.9789
332.9969
110.2025
-
241.3045
528.2569
-
376.9375
472.0982
-
124.1091
,
W
4
=
1
0
3
×
0.0678
0.4369
3.3416
-
0.4318
4.3559
9.1266
0.0011
0.1027
0.1123
,
W
5
=
1
0
4
×
6.8609
7.9208
0.0197
0.1372
0.1584
0.0004
0.2744
0.3168
0.0008
,
c
1
e
α
T
λ
max
⁡
X
1
λ
min
⁡
X
+
τ
λ
min
⁡
X
Y
-
1
X
0.5
τ
3
λ
min
⁡
X
Z
-
1
X
+
0.5
τ
3
λ
min
⁡
X
Z
-
1
X
=
3.59
<
5
=
c
2
.
Here, the feedback controller guarantees that finite-time stabilization of the linear system is designed byu
(
t
)
=
-
0.5
B
T
X
-
1
=
[
0.0526
,
-
0.1921
,
-
0.2522
]. Figure 3 shows the trajectories of states (Figure 3(a)) and its norm (Figure 3(b)) with initial condition ϕ
T
(
t
)
=
[
0.7,0
,
0
]. It can be observed that each trajectory is bounded between 0 and 0.7 for all t
∈
[
0,10
] and its norm x
T
(
t
)
x
(
t
) is bounded by c
1
=
0.55. In fact, we further extend the time interval and observe that the norm x
T
(
t
)
x
(
t
)
<
0.55 for all t
>
0, that is, the linear system (28) with the desired controller, is finite-time stabilization with respect to (
0.55,5
,
T
) for all T
>
0.State trajectories (a) and norm of the state trajectories (b) of the linear system (28) with controller u
(
t
)
=
[
0.0526
,
-
0.1921
,
-
0.2522
] and initial condition ϕ
T
(
t
)
=
[
0.7,0
,
0
].
(a)
(b)Remark 8.
The derivations of the main theorem above are based upon the Lyapunov-Krasovskii approach. Thus, these conditions are not only finite-time stable, but also asymptotically stable. This behavior can be seen when the time domain in Figure3 is extended.


## 5. Conclusion

In this paper, the finite-time stability and stabilization conditions of the linear system with constant delay are obtained. The sufficient conditions are formulated using a new form of Lyapunov-Krasovskii functional. Results from both examples illustrate that our proposed criteria are less conservative than other existing works.



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*Source: 154769-2014-12-28.xml*