This paper presents new results on delaydependent global exponential stability for uncertain linear systems with interval timevarying delay. Based on LyapunovKrasovskii functional approach, some novel delaydependent stability criteria are derived in terms of linear matrix inequalities (LMIs) involving the minimum and maximum delay bounds. By using delaypartitioning method and the lower bound lemma, less conservative results are obtained with fewer decision variables than the existing ones. Numerical examples are given to illustrate the usefulness and effectiveness of the proposed method.
1. Introduction
Time delays are frequently encountered in many fields such as chemical engineering system, vehicles, biological modeling, economy and other fields. Thus the stability analysis of linear time delay systems has been received considerable attention during the last decades. Also, it is well known that the existence of time delay can lead oscillation, instability or divergence in system performance
[1

2]
. Thus, the stability analysis and synthesis of systems with time delay have been one of the hottest topic in control society. For recent trends of the topic, see
[3

19]
and references therein.
More recently, the stability analysis of linear systems with interval timevarying delay, which the lower bound is not restricted to be zero, has been extensively studied by many researchers
[3

11
,
18
,
19]
. A real example of such systems is networked control systems
[13]
. In general, the stability analysis of timedelay systems can be classified into two categories: delaydependent
[6

8]
and delayindependent approach
[12
,
14]
. It is well known that delaydependent criteria are less conservative than the delayindependent ones when the size of timedelay is small. Hence, more attentions have been paid to the study for checking the conservatism of delaydependent conditions. In the field of delaydependent stability analysis, an important issue is to enlarge the feasibility region of stability criteria or to obtain a maximum allowable upper bound of time delays as large as possible. In this regard, Shao
[7]
provided a new delaydependent stability criterion for linear systems with interval timevarying delay by utilizing the convex combination method. In
[9]
, less conservative results were derived with much fewer decision variables by introducing the lower bound lemma. Liu
[11]
constructed a new Lyapunov functional which makes the results less conservative than the results of
[7
,
9]
. It should be noted that these results only focused their effort on asymptotic stability.
In practice, some uncertainties in the systems are unavoidable because it is very difficult to obtain an exact mathematical model due to the environmental noise, uncertain or slowly varying parameters. Thus, it is natural to consider the parameter uncertainties in a mathematical model. In addition, fast convergence of a system is essential for realtime computation, and the exponential convergence rate is generally used to determine the speed of computations. Therefore, the exponential stability analysis for systems with time delays has received deep concern in very recent years
[14

19]
. However, to the best of authors’ knowledge, there are few results about the exponential stability of uncertain linear system with interval time varying delays
[18
,
19]
. In
[18]
, some delaydependent sufficient conditions for the exponential stabilization of the systems are established in terms of LMIs by the construction of improved Lyapunov Krasovskii functional combined with Leibniz Newton’s formula. In
[19]
, the authors dealt with the same problem using two novel integral equalities. But, these results have two drawbacks. One is that the results cannot be applied when the time delay is differentiable
[18
,
19]
, the other is that it involves many matrix variables
[19]
, which increase computation burden.
With this motivation, we revisit the problem of exponential stability of uncertain linear systems with interval timevarying delay. By constructing a new LyapunovKrasovskii functional, novel delaydependent stability criteria with an exponential convergence rate are derived. Our results can be applied when the delay is differentiable. Less conservative results are obtained with fewer matrix variables than
[19]
by using delaypartitioning technique and the lower bound lemma. Finally, three numerical examples are shown to confirm the superiority of our results
Notations:
Throughout this paper,
I
demotes the identity matrix with appropriate dimensions,
R^{n}
denotes the
n
dimensional Euclidean space, and
R
^{m×n}
is the set of all
m
×
n
real matrices, ‖ ⋅ ‖ refers to the Euclidean vector norm and the induced matrix norm. For symmetric matrices
A
and
B
, the notation
A
>
B
(respectively,
A
≥
B
) means that the matrix
A
−
B
is positive definite (respectively, nonnegative),
λ
_{max}
(⋅) and
λ
_{min}
(⋅) stand for the largest and smallest eigenvalue of given square matrix, respectively. d
iag
{···} denotes the block diagonal matrix. ‖
Φ
‖ = sup −
h_{M}
≤
s
≤0 {‖
x
(
t
+
s
)‖, ‖
ẋ
(
t
+
s
) ‖}, where
h_{M}
> 0 is some constant.
2. Problem Statement
Construction the following uncertain linear systems with time varying delay:
where
x
(
t
)∈
Rn
is the state,
A
and
A_{1}
are known real constant matrices with appropriate dimensions;
φ
(
t
) is the initial condition of the system. The time varying delay
h
(
t
) is differentiable function satisfying
where the bounds
h_{m}, h_{M}, μ
are known positive scalars.
The uncertainties satisfy the following conditions:
where
D, E, E
_{1}
are known constant matrices,
F
(
t
) ∈
R^{n×n}
is the unknown real time varying matrices with Lebesgue measurable elements bounded by
Therefore, system (1) with uncertainties satisfying (4) and (5) can be written in the following form:
We need the following definition and Lemmas for deriving the main results.
Definition 2.1 [16]
For a given positive scalar
k
, the zero solution of (6) is exponentially stable if there exist a positive scalar
γ
such that every solution
x
(
t
) of (6) satisfies the following condition
‖ x (t)‖ ≤ γ e ^{−kt} ‖φ‖ .
Lemma 2.1 [7]
For any constant positive definite matrix
M
∈
R^{n}
and
β
≤
s
≤
α
, the following inequalities hold
Lemma 2.2 [9]
(Lower bounds lemma) Let
f
_{1}
,
f
_{2}
,···,
fN : R^{m}
➔
R
have positive values in an open subset
D
of
R^{m}
. Then, the reciprocally convex combination of
fi
over
D
satisfies
Subject to
3. Main Results
In this section, new stability criteria for system (1) will be derived by use of Lyapunov method and LMI framework
 3.1 Exponential stability for nominal systems with interval timevarying delay
First, we present a delay dependent exponential stability condition for the following nominal interval timevarying delay systems with Δ
A
(
t
) = 0, Δ
A
_{1}
(
t
) = 0 .
By introducing augmented LyapunovKrasovskii functional, a less conservative delaydependent stability criterion for system (7) will be proposed. In convenience, let us define
The corresponding block entry matrices are defined as
e_{i}
∈
R
^{6n×n}
(
i
=1, 2, ⋯, 6) e.g.,
∈
R
^{6n×n}
. Defining
, then (7) can be written as
.
Denote
Theorem 3.1
For given scalars
h_{m}
,
h_{M}
(
h_{M}
>
h_{m}
),
μ
,
k
≥ 0 , system (7) is globally exponentially stable if there exist symmetric positive definite matrices ℚ
_{1}
∈ ℝ
^{2n×2n}
, ℚ
_{2}
∈ ℝ
^{2n×2n}
,
n
×
n
dimensional positive symmetric matrices
P
,
R
,
U
,
S
,
Z
_{1}
,
Z
_{2}
, and appropriate dimension matrices
T
,
T
_{1}
,
N
satisfying the following LMIs:
Proof.
Consider the following LyapunovKrasoskii functional
where
Calculating the timederivative of
V
(
t
) , we have
Here, using Lemma 2.1, it can be obtained that
Also, the following inequality is obtained from Lemma 2.2,
It should be noted that when
h
(
t
) =
h_{m}
or
h
(
t
)=
h_{M}
, we have
or
, respectively. So the relation (15) still holds.
Form (13)(15), we obtain
Inspired by the work of
[3]
, the following two zero inequalities hold for any appropriate dimension matrix
N
,
From the above zero equalities, one can obtain
Here, using Lemma 2.1, we have
In the following discussions, the upper bound of
_{6}
is derived by considering two different cases for (i)
h_{m}
≤
h
(
t
) ≤
h
_{2}
and (ii)
h
_{2}
≤
h
(
t
) ≤
h_{M}
.
When
h_{m}
≤
h
(
t
) ≤
h
_{2}
, the following inequality is satisfied by Lemma 2.1
By using the similar methods in (15), we obtain
From (18)(21), in this cases
If (9) hold, the following inequality is satisfied with (11)(22) by Sprocedure
[1]
,
When
h_{m}
≤
h
(
t
) ≤
h
_{2}
, the following inequality is satisfied by Lemma 2.1
By using the similar methods in (15), we obtain
From (18)(21), in this cases
If (9) hold, the following inequality is satisfied with (11)(19) and (26) by using Sprocedure
[1]
,
Now we can conclude that if condition (8) and (9) are satisfied, then
(
x
)(
t
)) ≤ 0. Thus,
V
(
x
(
t
)) ≤
V
(
x
(0)). Furthermore, from the definition of
V
(
x
(
t
)) and (10), we can derive the following inequalities:
V(x(0)) ≤ a φ^{2},
where
On the other hand, we have
Hence,
Then, the proof is completed by the Lyapunov stability theorem.
Remark 3.1
When 0 ≤
h_{m}
≤
h
(
t
) ≤
h_{M}
, the most attractive contribution is that we have made the best use of the lower bound (network induced delay) of the interval timevarying delay. In fact, in order to derive the less conservative stability criterion, we employ a new LyapunovKrasovskii functional (10), which is mainly based on the information about
,
Remark 3.2
When
μ
is known, Theorem 3.1 can be applied while
[8
,
16
,
17]
fails to work. If
μ
is unknown or
h
(
t
) is not differentiable, then the following result can be obtained from Theorem 3.1 by setting
R
= 0 , which will be introduced as Corollary 3.1.
Corollary 3.1
For given scalars
h_{m}
,
h_{M}
(
h_{M}
>
h_{m}
),
k
≥ 0 , system (7) is globally exponentially stable if there exist symmetric positive definite matrices ℚ
_{1}
∈ ℝ
^{2n×2n}
, ℚ
_{2}
∈ ℝ
^{2n×2n}
,
n
×
n
dimensional positive symmetric matrices
P
,
U
,
S
,
Z
_{1}
,
Z
_{2}
, and appropriate dimension matrices
T
,
T
_{1}
,
N
satisfying the following LMIs:
where
_{1} = Σ_{1} + Σ_{3} + Σ_{4} + Σ_{5} + Σ_{6} ,
_{2} = Σ_{1} + Σ_{3} + Σ_{4} + Σ_{5} + Σ_{7}.
 3.2 Robust exponential stability for uncertain systems with interval time varying delay
Based on the result of Theorem 3.1, the following theorem provides a robust exponential stability condition of uncertain linear systems with interval timevarying delay (6).
For simplicity in Theorem 3.2,
e_{i}
∈
R
^{7n×n}
(
i
=1,2,···, 7) are defined as block matrices. (for example,
=[0 0 0
I
0 0 0]
^{T}
∈
R
^{7n×n}
). The other notations for some vectors and matrices are defined as:
Theorem 3.2
For given scalars
h_{m}
,
h_{M}
(
h_{M}
>
h_{m}
),
μ
,
k
≥ 0 , system (6) is globally exponentially stable if there exist symmetric positive definite matrices ℚ
_{1}
∈ ℝ
^{2n×2n}
, ℚ
_{2}
∈ ℝ
^{2n×2n}
,
n
×
n
dimensional positive symmetric matrices
P
,
R
,
U
,
S
,
Z
_{1}
,
Z
_{2}
, and appropriate dimension matrices
T
,
T
_{1}
,
N
satisfying the following LMIs:
Proof.
By considering the same LyapunovKrasovskii functional in Theorem 3.1, the upper bounds of
(
t
) are obtained as
On the other hand, the following inequality holds From (4)(6)
By Sprocedure
[1]
, we can derive inequalities (32) with a positive scalar
ε
such that
Therefore, the uncertain system (6) is exponentially stable, if LMIs (3031) hold. This completes our proof.
The following result is obtained from Theorem 3.2 when
μ
is unknown or
h
(
t
) is not differentiable.
Corollary 3.2
For given scalars
h_{m}
,
h_{M}
(
h_{M}
>
h_{m}
)
k
≥ 0 , system (7) is globally exponentially stable if there exist symmetric positive definite matrices ℚ
_{1}
∈ ℝ
^{2n×2n}
, ℚ
_{2}
∈ ℝ
^{2n×2n}
n
×
n
dimensional positive symmetric matrices
P
,
U
,
S
,
Z
_{1}
,
Z
_{2}
, and appropriate dimension matrices
T
,
T
_{1}
,
N
satisfying the following LMIs:
where
Remark 3.3
For system (7) with the routine delay case described by 0 ≤
h
(
t
) ≤
h_{M}
≤
h_{M}
(
h
(
t
) ≤
μ
), the corresponding LyapunovKrasovskii reduces to
Similar to the proof of Theorem 3.1(Corollary 3.1) and Theorem 3.2(Corollary 3.2), one can easily derive less conservative results than some existing ones.
Remark 3.4
In most of the applications of neural networks there is a shared requirement of raising the networks convergence speed in order to cut down the time of neural computing. Since the exponential convergence rate could be used to determine the speed of neural computation
[20
,
21]
. On the other hand, from the Dynamic simulation systems
[22]
and Realtime computing
[23]
, it can be confirmed that, fast convergence of a system is essential for realtime computation, and the exponential convergence rate is generally used to determine the speed of computations.
Remark 3.5
By constructing a new augmented LyapunovKrasosvskii functional and using lower bound lemma, exponential stability criteria have been obtained which are expected to be less conservative than the results discussed in the recent literature
[5
,
8
,
11]
. The effectiveness of the proposed methods has been shown elaborately through the following numerical examples.
4. Numerical examples
Example 4.1
Consider the system given in (7) with following parameters
Case 1: When
h_{m}
= 0.5,
h_{M}
= 1.
Table 1
gives the allowable of the maximum exponential convergence rate k for different
μ
. For this case, the exponential stability criteria in
[8
,
11]
are not applicable because the criteria are only for asymptotic stability.
Case 2: For various
μ
and unknown
μ
, the allowable bound
h_{M}
, which guarantee the asymptotic stability of system for given lower bounds
h_{m}
are provided in
Table 2
. It is easy to see that our method gives improved results than the existing ones.
Example 4.2
Consider the system given in (6) with following parameters
For 0 ≤
h_{m}
≤
h
(
t
) ≤
h_{M}
,
Table 3
presents the allowance of the upper bound when
μ
is unknown and the convergence
k
= 0 . It is obvious that our results are better than
[5]
.
The maximum exponential convergence ratekfor variousμ
The maximum exponential convergence rate k for various μ
The maximum boundhMwith givenhmfor differentμ
The maximum bound h_{M} with given h_{m} for different μ
The maximum boundhMwith givenhmfor unknownμ
The maximum bound h_{M} with given h_{m} for unknown μ
hm= 0.5,μ= 0.5, The maximum boundhMfor variousk
h_{m} = 0.5, μ = 0.5, The maximum bound h_{M} for various k
Example 4.3
Consider the system given in (6) with following parameters
Given
h_{m}
= 0.5,
μ
= 0.5,
Table 4
gives the maximum allowable value of
h_{M}
for different convergence rate
k
. For this cases, it is noted that the criteria in
[5
,
8
,
11]
are not applicable for exponential stability. Therefore, our work is more general cases than in the existing results
5. Conclusion
In this paper, the problem of delaydependent exponential stability of timedelay systems has been investigated. We have considered the timevarying delay in a range for which the lower bound is not restricted to be zero. By introducing a different Lyapunov functional, new delaydependent criteria have been derived in terms of LMIs. It is shown via numerical examples that our proposed criteria are less conservative than existing ones.
Acknowledgements
This research was supported by the Daegu University Research Grants, 2011.
BIO
Yajuan Liu received her B.S degree in mathematics and applied mathematics from shanxi nominal university, Linfen, China, in 2010, and M.S degree in applied mathematics from University of Science and Technology Beijing, Beijing, China, in 2012. She is currently working toward the PH.D degree in Electronic Engineering from Daegu University, Korea. Her current research interests include time delay systems, robust control and filtering, model predictive control.
SangMoon Lee received the B.S. degree in Electronic Engineering from Gyungbuk National University, and M. S and PH. D degrees at Department of Electronic Engineering from POSTECH， Korea Currently, he is an Assistant Professor at Division of Electronic Engineering in Daegu University. His main research interests include robust control theory, nonlinear systems, model predictive control and its industrial applications.
OhMin Kwon received B.S degree in Electronic Engineering from Gyungbuk National University, Daegu, Korea in 1997, and PH.D degree in Electrical and Engineering from Pohang University of Science and Technology, Pohang, Korea, in 2004. From February 2004 to January 2006, he was a senior Researcher in Mechatronics Center of Samsung Heavy Industries. He is currently working as an Associate Professor in School of Electrical Engineering, Chungbuk National University. His research interests include timedelay systems, cellular neural networks, robust control and filtering, largescale systems, secure communication through synchronization between two chaotic systems, complex dynamical networks, multiagent systems, and so on. He has presented a number of papers in these areas. He is a member of KIEE, ICROS, and IEEK. He had joined an editorial member of KIEE from 2011 to 2012 and also served as an editorial member in Nonlinear Analysis: Hybrid Systems from 2011 and Scientific World Journal from 2013.
Ju H. Park received the B.S and M.S degrees in Electronics Engineering from Kyungpook National University, Daegu, Republic of Korea, in1990 and 1992 and the Ph.D. degree in Electronics and Electrical Engineering from POSTECH, Pohang, Republic of Korea, in 1997. From May 1997 to February 2000, he was a Research Associate in ERCARC, POSTECH. In March 2000, he joined Yeungnam University, Kyongsan, Republic of Korea, where he is currently a Full Professor. From December 2006 to December 2007, he was a Visiting Professor in the Department of Mechanical Engineering, Georgia Institute of Technology. Prof Park’s research interests include robust control and filtering, neural networks, complex networks, and chaotic systems. He has published a number of papers in these areas. Prof. Park severs as Editor of International Journal of Control, Automation and Systems. He is also an Associate Editor/ Editorial Board member for several international journals, including Applied Mathematics and Computation, Journal of The Franklin Institute, Journal of Applied Mathematics and Computing, etc.
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