This paper investigates the problem of stability analysis and robust
H
_{∞}
controller for timedelayed systems with parameter uncertainties and stochastic disturbances. It is assumed parameter uncertainties are norm bounded and mean and variance for disturbances of them are known. Firstly, by constructing a newly augmented LyapunovKrasovskii functional, a stability criterion for nominal systems with timevarying delays is derived in terms of linear matrix inequalities (LMIs). Secondly, based on the result of stability analysis, a new controller design method is proposed for the nominal form of the systems. Finally, the proposed method is extended to the problem of robust
H
_{∞}
controller design for a timedelayed system with parameter uncertainties and stochastic disturbances. To show the validity and effectiveness of the presented criteria, three examples are included.
1. Introduction
During the past several decades, the topic of stability analysis and stabilization for timedelayed systems has been tackled by many researchers since time delays occur in many practical systems. For examples, most sensors in systems usually have timedelays since they take some times to measure. Also, it is unavoidable that communications have timedelays since processes of modulation and demodulation require some times. Moreover, in digital systems, timedelays occur in some processes like a sampling. It is well known that occurrence of time delays cause poor performance or even instability. Therefore, a great deal of efforts has been done in stability analysis and stabilization for timedelayed systems
[1

26]
based on delaydependent approach since delaydependent criteria are less conservative than delayindependent ones especially when the sizes of timedelays are small.
One of the main issue in the field of delaydependent stability analysis and stabilization is to find larger delay bounds for guaranteeing the asymptotic stability of the concerned dynamic system. Therefore, the size of maximum delay bound obtained by stability or stabilization criteria has been the main index for comparing the superiority of stability or stabilization criteria. With this regard, to enhance the feasible region of stability or stabilization, some new LyapunovKrasovskii functional such as triple or quadruple integral form were introduced in
[3

7
,
16
,
18
,
20]
, and
[22]
. The application of freeweighting matrix techniques to reduce the conservatism of stability criteria for various systems can be found in
[1
,
2
,
5
,
13
,
15
,
17
,
18
,
21
,
23]
, and
[24]
. In
[5]
, delaypartitioning techniques with a newly augmented LyapunovKrasovskii functional were utilized in stability analysis for uncertain neutral systems with timevarying delays. Zhu et al.
[12]
tried to reduce the decision number of LMI variable by not introducing slack variables in delaydependent stability analysis of timedelay systems. The model transformation such as a descriptor system was utilized in
[14
,
25]
, and
[26]
. To obtain more tighter upper bound of timederivative of LyapunovKrasovskii functional, Jensen’s inequality
[10]
, reciprocally convex optimization approach
[8]
, and Wirtingerbased integral inequality
[9]
are well recognized in reducing the conservatism of delaydependent stability analysis. The application of reciprocally convex optimization approach
[8]
in discretetime systems with timevarying delays can be found in
[11]
and
[19]
. From the analysis of delaydependent stability and stabilization for dynamic systems with timedelays mentioned before, one can see that how to construct LyapunovKrasovskii functional and estimate its timederivative value with some techniques play key roles to increase maximum delay bounds.
In practical systems, there exist some uncertainties because it is very difficult to be obtain an exact mathematical model due to environment noise, system complexities, friction, uncertain or slowly varying parameters, and so on. Hence, considerable efforts in
[20

24]
,
[27

29]
have been devoted to the stability and stabilization for uncertain dynamics with normbounded parameter uncertainties.
On the other hand, disturbances can have an adverse effect on the stability of systems. Thus, it is important to design a controller for the systems with disturbances. For an example, when disturbances like earthquake occurred, building and structure systems require controllers that minimize the effect of external disturbances as well as stabilize the system. Therefore, an
H
_{∞}
control
[30]
has been used to minimize the effect of the disturbances because the goal of the
H
_{∞}
control is to design the controllers such that the closedloop systems are stable and their
H
_{∞}
norm of the transfer functions between the controlled output and the disturbances will not exceed prescribed level of performances. With this regard, a number of research results on
H
_{∞}
control has been addressed for various systems such as timedelayed systems in
[25

28]
, singular systems
[29]
, and so on.
Recently, a variety of stochastic systems have been researched such as systems with stochastic sampling and stochastic systems with missing measurement in
[31
,
32]
. Moreover, systems with randomly occurring uncertainties have been introduced in
[33
,
34]
. In
[31]
, it is assumed that the timedelay has a stochastic characteristic to describe the systems which has sampling period varying by stochastic characteristic. In
[32]
, it was assumed that the system output has the stochastic variables for probabilistic missing data. Also, the parameter uncertainties are multiplied by the stochastic variables in
[33
,
34]
.
However, it is natural to assume that stochastic characteristic exist not only in the mentioned systems in
[31

34]
but also in disturbances since the disturbances are affected by random change of environment. For an example, aircraft is affected by disturbances when the wind is strong like a storm. Other existing literatures in
[31

34]
, even though stochastic variables are used, variance is not considered. Thus, it is desirable and realistic that the stochastic data of the disturbances such as mean and variance is utilized in stability analysis and stabilization for systems with stochastic disturbances. However, the systems with stochastic disturbances have not been fully investigated yet.
With motivations mentioned in above discussions, this paper focuses on the problem of the
H
_{∞}
controller design for linear timedelayed systems with stochastic disturbances. The main contribution of this research lies in two aspects.
• Some new augmented LyapunovKrasovskii functional are introduced in stability and stabilization problem for timedelayed systems with parameter uncertainties.
• The problem of designing a robust
H
_{∞}
control for the systems with both stochastic disturbance and parameter uncertainties is investigated for the first time.
First, by construction of a newly LyapunovKrasovskii functional and utilization of reciprocally convex approach [8] with some new zero equalities, a new stability criterion for the nominal form of systems with timevarying delays is derived in Theorem 1 with the LMI framework, which can be formulated as convex optimization algorithms which are amenable to computer solution
[35]
. Secondly, based on the results of Theorem 1, a new controller design method for the nominal form of systems with timevarying delays will be proposed in Theorem 2. Finally, Theorem 2 will be extended to Theorem 3 which deals with the robust
H
_{∞}
controller design methods for the systems with both stochastic disturbance and parameter uncertainties. Through three numerical examples, the advantage and effectiveness of the proposed theorems will be shown.
Notation
:
R
^{n}
is the
n
dimensional Euclidean space,
R
^{m×n}
denotes the set of
m
×
n
real matrix. For symmetric matrices
X
and
Y
,
X
>
Y
(respectively,
X
≥
Y
) means that the matrix
X
−
Y
is positive definite (respectively, nonnegative).
X^{T}
denotes the transposition of
X
.
I_{n}
denotes the
n
×
n
identity matrix.
0_{n}
and
0
_{n×m}
denote the
n
×
n
zero matrix and the
n
×
m
zero matrix, respectively.  .  refers to the Euclidean vector norm and the induced matrix norm.
diag
{…} denotes the block diagonal matrix, respectively. * represents the elements below the main diagonal of a symmetric matrix.
L
_{2}
[0, ∞) is the space of square integrable vector.
E
{
x
} and
E
{
x

y
} denote the expectation of
x
and the expectation of
x
conditional on
y
, respectively.
X
_{[f(t)]}
means that the elements of the matrix
X
_{[f(t)]}
includes the value of
f
(
t
); e.g., X
_{[a]}
= X
_{[f(t)=a]}
.
Pr
{
x
} means the varying probability of the event
x
.
2. Problem statements
Consider the linear system with uncertain parameters and a timevarying delay:
where
x
(
t
) ∈
R
^{n}
is the state vector,
u
(
t
) ∈
R
^{m}
is the control input,
w
(
t
) ∈
R
^{p}
is the disturbance input which belongs to
L
_{2}
[0, ∞) ,
z
(
t
) ∈
R
^{q}
is the vector of controlled output,
A
,
A_{d}
∈
R
^{n×n}
,
B
∈
R
^{n×m}
,
B_{w}
∈
R
^{n×q}
and
C
∈
R
^{q×n}
are known real constant matrices, Δ
A
(
t
) and Δ
A_{d}
(
t
) are the parameter uncertainties of system matrices of the form
in which
D
∈
R
^{n×s}
,
E_{a}
∈
R
^{r×n}
and
E_{d}
∈
R
^{r×n}
are known constant matrices and
F
(
t
) ∈
R
^{s×r}
is norm bounded with
F^{T}
(
t
)
F
(
t
) ≤
I_{r}
. Also,
h
(
t
) is a timedelay satisfying timevarying continuous function as follows:
where
h_{M}
is a known positive scalar and
h_{d}
is any constant one.
In this paper, it is assumed that the disturbance has stochastic properties. Let us define the stochastic variable
ρ
(
t
) which satisfies the following conditions:
where
ρ
_{0}
and
σ
^{2}
are mean and variance of
ρ
(
t
) , respectively. From (4), the expectation of
ρ
^{2}
(
t
) can be obtained as
For the assumption about the stochastic disturbances, in this paper, the term
B_{w}
w
(
t
) in Eq. (1) is multiplied by
ρ
(
t
) . Thus, in this paper, the following model is considered:
This model has a form of
ρ
(
t
)
B_{w}
w
(
t
) which obtain the stochastic variable
ρ
(
t
) and reflects a stochastic character of disturbances. For the various disturbances, even if means of disturbances are same, the disturbances can be different each other because of variances. If the variance of
ρ
(
t
) is 0,
ρ
(
t
) can be assumed by the constant value
ρ
_{0}
.
Remark 1.
It is supposed that the
ρ
(
t
) has the stochastic characteristic and its mean and variance are known. Thus, the disturbances considered in Eq. (6) can be characterized by stochastic properties. Since the disturbances are affected by random change of environment, considering mean and variance of
ρ
(
t
) is reasonable. For the aircraft as an example, if wind is strong like storm, then the flight performance of the aircraft will be deteriorated since the strength of the disturbances is intense. That is, mean of
ρ
(
t
) will be increased. Also, if wind is fluctuating sharply, variance of
ρ
(
t
) will be increased. Thus, if the stochastic information of
ρ
(
t
) when a controller is designed is utilized, then it may be expected that more practical controllers can be provided to the system (6) than other ones in existing literatures.
Let us consider a memoryless state feedback controller:
where
K
∈
R
^{m×n}
is a gain matrix of the feedback controller. In order to develop a delaydependent
H
_{∞}
controller for the system (6), the following conditions are satisfied:
(i) With
w
(
t
) = 0 , the closed loop system (6) with control input is asymptotically stable.
(ii) Under zero initial condition, the closedloop system satisfies
where
γ
> 0 is a prescribed scalar. Under these condition, the system (6) is said to be stabilizable with an
H
_{∞}
disturbance attenuation level
γ
. Then, the obtained controller
u
(
t
) is said to be an
H
_{∞}
stabilization controller.
The objective of this paper is to design a state feedback controller (7) such that system (6) is asymptotically stable and an
H
_{∞}
disturbance attenuation level
γ
is minimized. To derive main results, the following lemmas are utilized in deriving the main results.
Lemma 1 [4].
For a positive matrix
Z
, scalars h
_{2}
>
h
_{1}
>0 such that the integrations are well defined, then
Lemma 2 [8].
For a scalar
α
(0 <
α
< 1) , a given matrix
R
∈
R
^{n×n}
> 0 , two matrices
W
_{1}
∈
R
^{n×m}
and
W
_{2}
∈
R
^{n×m}
, any vector
ξ
∈
R
^{m}
, let us define the function Θ(
α
,
R
) given by:
Then, if there exists a matrix
X
∈
R
^{n×n}
such that
, then the following inequality holds
Lemma 3 (Finsler's lemma) [36].
Let
ζ
∈
R
^{n}
, Φ = Φ
^{T}
∈
R
^{n×n}
and
B
∈
R
^{m×n}
such that
rank
(
B
) <
n
. The following statements are equivalent:
(
B
^{⊥}
)
^{T}
Φ
B
^{⊥}
< 0, where
B
^{⊥}
is right orthogonal complement of
B
.
Lemma 4 [11].
For a positive matrix Λ , a symmetric matrix Ξ , and a matrix Γ , two following statements are equivalent:
(1) Ξ − Γ
^{T}
ΛΓ < 0,
(2) There exists a matrix
U
of appropriate dimension such that
Lemma 5 [37].
Let
E
,
H
, and
F
(
t
) be real matrices of appropriate dimensions, and let
F
(
t
) satisfy
F^{T}
(
t
)
F
(
t
) ≤
I
for all
t
. Then, for a positive scalar
ε
, the following inequality holds:
3. Main results
This section consists of three subsections. The goal of first subsection is stability analysis of a nominal system. The second subsection will be extended to a stabilization. And the final section will introduce a design method of a robust
H
_{∞}
controller for uncertain linear systems with timevarying delays and stochastic disturbances.
 3.1 Stability analysis
Firstly, in this subsection, a delaydependent stability criterion for the nominal system of (6) with
w
(
t
) = 0 ,
u
(
t
) = 0 and without parameter uncertainties is derived. Here, the following nominal system of (6) is given by
Now, for simplicity of matrix and vector representation,
e_{i}
∈
R
^{9n×n}
(
i
= 1,...,9) are defined as block entry matrices which will be used in Theorem 1. The other notations are defined as
Now, the following theorem is given as a delaydependent stability criterion for the system (16).
Theorem 1.
For given scalars 0≤
h_{M}
and
h_{d}
, the system (16) is asymptotically stable for 0 ≤
h
(
t
) ≤
h_{M}
,
ḣ
(
t
) ≤
h_{d}
, if there exist positive definite matrices
R
∈
R
^{5n×5n}
,
N
∈
R
^{4n×4n}
,
G
∈
R
^{3n×3n}
,
Q
_{1}
∈
R
^{2n×2n}
,
Q
_{2}
∈
R
^{2n×2n}
,
Q
_{3}
∈
R
^{n×n}
, symmetric matrices
P
_{1}
∈
R
^{n×n}
,
P
_{2}
∈
R
^{n×n}
, any matrices
S
_{1}
∈
R
^{2n×2n}
,
S
_{2}
∈
R
^{2n×2n}
and
U
∈
R
^{4n×8n}
satisfying the following LMIs:
Proof.
Let us consider the following LyapunovKrasovskii functional candidate as
where
Now, the timederivative of
V
_{1}
(
t
) is
Also,
is calculated as
The upperbound of
can be given as follows:
Moreover,
can be derived as:
Inspired by the work in
[13]
, for symmetric matrices
P
_{1}
and
P
_{2}
, let us consider the following zero equalities:
By adding (26) into (25) and using Lemma 1, an upper bound of
is still same as
where
ϕ
_{1}
(
t
) =
h
(
t
) /
h_{M}
which satisfies 0 <
ϕ
_{1}
(
t
) < 1 when 0 <
h
(
t
) <
h_{M}
. It should be pointed that when
h
(
t
)=0, equalities
is obtained, and when
h
(
t
)=
h_{M}
, equalities
is obtained. Thus, equality (27) still holds when 0 ≤
h
(
t
) ≤
h_{M}
.
From Lemma 2 with Λ
_{1}
≥ 0 which is defined in (17), a new upperbound of
calculated by
In succession, through Lemma 1,
can be obtained as
where
,
ϕ
_{2}
(
t
) = (
h
(
t
) /
h_{M}
)
^{2}
which satisfies 0 <
ϕ
_{2}
(
t
) < 1 when 0 <
h
(
t
) <
h_{M}
. Note that when
h
(
t
)=0, equalities
is obtained, and when
h
(
t
)=
h_{M}
, equalities
is obtained. Thus, equality (29) still holds when 0 ≤
h
(
t
) ≤
h_{M}
.
Using Lemma 2 with Λ
_{2}
≥ 0 defined in (17) yields the following inequality:
where
.
Moreover, from Lemma 1,
has a new upperbound given by
By combining (22)(31), an upperbound of
is obtained as follows:
From the dynamic equation of the system (16), it is true that 0 = Θ
ζ
(
t
). Therefore, by using Lemma 3, a stability criterion for system (16) can be obtained as
For any matrix
U
and by using Lemma 4, a new upperbound of (33) is given by
Note that inequality (34) is affinely dependent on
h
(
t
) where 0 ≤
h
(
t
) ≤
h_{M}
. So, ϒ
_{[h(t)]}
< 0. is satisfied if and only if LMIs (18) and (19). This completes our proof. ■
Remark 2.
Compared with the existing works, the main differences of LyapunovKrasovskii functionals are
V
_{1}
(
t
),
V
_{2}
(
t
) and
V
_{3}
(
t
). They are used to reduce the conservatism of stability and stabilization criteria for the timedelayed system since the utilized augmented vector in
V
_{1}
(
t
) includes not only single and double integral terms of states but also triple integral term
. In the proposed methods, more information on states were utilized by using
V
_{1}
(
t
) with this vector as one term of the LyapunovKrasovkii functional in (21). By calculating the timederivative of
V
_{2}
(
t
) and
V
_{3}
(
t
), some cross terms such as
and
are obtained and utilized in estimating the timederivative of
V
(
t
) by including the integral terms of derivative of state as the integrands of
V
_{2}
(
t
) and
V
_{3}
(
t
). In next section, it will be shown Theorem 1 can provide larger delay bounds than the works in other literature by comparing maximum delay bounds for a system which has been utilized in many literature to check the conservatism of delaydependent stability criteria.
 3.2 Controller design
Let us consider the following system
where
h
(
t
) is satisfied with (3) and
u
(
t
) is the control input which is defined in (7). In this section, a stabilization method for the system (35) will be introduced based on the result of Theorem 1. For simplicity of matrix representation, some notations are defined as
where other terms were defined in (17).
Now, the following theorem is given by the second result.
Theorem 2.
For given scalars 0≤
h_{M}
,
h_{d}
and
α
> 0 the system (35) is asymptotically stable for 0 ≤
h
(
t
) ≤
h_{M}
and
ḣ
(
t
) ≤
h_{d}
, if there exist positive definite matrices
,
,
,
,
,
, symmetric matrices
,
, any matrices
,
,
,
X
∈
R
^{n×n}
and
Y
∈
R
^{m×n}
satisfying the following LMIs:
Then, if the above conditions are feasible, a desired controller gain matrix is obtained by
K
=
YX
^{−1}
.
Proof.
For the same LyapunovKrasovskii candidate functional in (21) and by (32), the upperbound of
is given as follows:
with the inequalities (20).
Then, for any matrices
Z
_{1}
and
Z
_{2}
, the following zero equality is considered:
where
.
For
R
= [
R_{ij}
]
_{5×5}
∈
R
^{5n×5n}
and a given scalar
α
> 0 , let us define
Z
_{1}
=
αR
_{11}
and
Z
_{2}
=
R
_{11}
. By adding (41) to (40), a new upperbound of
is given by
Let us define
and
. For examples, Φ
_{1}
=
X
and Φ
_{2}
=
diag
{
X
,
X
} . Then, to derive more simply, let us define following matrices as
,
,
,
,
,
,
,
,
,
and
Y
=
KX
. Then, the following inequalities can be obtained by multiplying Φ
_{9}
to pre and post of (42)
At this time, the conditions are attended as follows:
.
Also, by Lemma 4 with any matrix
Û
, an inequality (43) is equivalent to
Note that inequality (44) is affinely dependent on 0 ≤
h
(
t
) ≤
h_{M}
. So,
is satisfied if and only if (37) and (38). This completes our proof. ■
 3.3 RobustH∞controller design for stochastic disturbances
In this subsection, the robust
H
_{∞}
controller design for the system (6) will be derived based on Theorem 2. The notations are defined as
where other terms were defined in (17) and (36).
Theorem 3.
For given scalars 0≤
h_{M}
,
h_{d}
and
α
> 0 the system (35) is asymptotically stable for 0 ≤
h
(
t
) ≤
h_{M}
and
ḣ
(
t
) ≤
h_{d}
, if there exist positive definite matrices
,
,
,
,
,
, symmetric matrices
,
, any matrices
,
,
,
X
∈
R
^{n×n}
,
Y
∈
R
^{m×n}
and a positive scalar
λ
satisfying the following LMIs:
Then, if the above conditions are feasible, a desired controller gain matrix is obtained by
K
=
YX
^{−1}
.
Proof.
Let us consider the same LyapunovKrasovskii candidate functional in (21). By infinitesimal operator
L
in
[31]
, a new upperbound of
L
V
(
t
) is obtained by
with the inequalities (20).
Then, the following equation is obtained for any matrix
Z
_{1}
and
Z
_{2}
:
where
Let us define
Z
_{1}
=
αR
_{11}
and
Z
_{2}
=
R
_{11}
. By adding (50) into (49), a new upperbound of
L
V
(
t
) can be obtained as follows:
The expectation of (51) can be obtained with (4) and (5) as follows:
Since
V
(0)=0 and
V
(∞)≥0 are satisfied under the zero initial condition, the following inequality can be obtained from
H
_{∞}
performance index in (8) as
If the inequality (53) is satisfied, the system (6) is stable with
H
_{∞}
performance level
γ
under the obtained controller (7) . With
z^{T}
(
t
)
z
(
t
)=
x^{T}
(
t
)
C^{T}
Cx
(
t
), (53) is equivalent to the following inequality
Using Lemma 5, an upperbound of (54) can be obtained for
λ
>0 as follows:
where
Therefore, inequality (55) is equivalent to the following condition
Let us define
,
,
,
,
,
,
,
,
,
,
,
and
Y
=
KX
. Then, the following inequalities can be obtained by pre and postmultiplying (56) by
diag
{Φ
_{9}
,
I_{p}
}
where
At this time, the conditions are attended as follows:
.
Using Lemma 4 with any matrix
Û
, (57) is equivalent to
Note that inequality (58) is affinely dependent on
h
(
t
) where 0 ≤
h
(
t
) ≤
h_{M}
. So,
is satisfied if and only if (46) and (47). This completes our proof. ■
4. Numerical examples
In this section, three numerical examples demonstrate the effectiveness of the proposed criteria.
Example 1.
Consider the system (16) with following parameters:
For different
h_{d}
, the maximum delay bounds for guaranteeing the asymptotic of system (16) with above parameters are listed in
Table 1
which conducts the comparison of the obtained results by Theorem 1 with the previous results.
Table 1
shows that when
h_{d}
is bigger, the maximum upperbound of
h
(
t
) is smaller. It means that
h_{d}
affects the stability region of the system. Moreover, it can be seen that Theorem 1 is less conservative than those of the existing results in
[1

3]
,
[6]
,
[8]
and
[12]
. Thus, it can be confirm that the proposed LayponoveKrasovskii functional and some utilized techniques in Theorem 1 are effective in reducing the conservatism of stability criterion. It should be noted that when timedelay is constant, the maximum delay bound for guaranteeing the asymptotic stability of system (16) with above parameters is 6.1725.
The maximumhMin Example 1.
The maximum h_{M} in Example 1.
Example 2.
Consider the system (35) with following parameters:
When
h_{d}
= 0, by applying Theorem 2 to the system (35) with the above parameters, the obtained maximum delay bounds and the corresponding controller gain are listed and comparison with the previous results are conducted in
Table 2
. The obtained maximum allowable delay bounds of
h
(
t
) is 50.01 and the controller gain is
K
= 10
^{4}
×[−1.7155, −1.7331] . At this time, the tuning parameter
α
is 0.0334 when
h_{M}
=50.01. In order to confirm one of the results, the simulation result with the time delay
h
(
t
)=50.01 and the designed controller gain
K
is illustrated in
Fig. 1
which shows the state trajectories goes to zero as time increases. To compare the obtained results by Theorem 2 with some other ones, the various
h_{M}
and corresponding controller gain are listed in
Table 2
when
h_{M}
are 1, 2, 5, 10 and 11. From
Table 2
, one can confirm that the feasible region of Theorem 2 is much larger than the previous literature
[14

16]
,
[24

26]
.
The maximumhMwithhd=0 in Example 2.
The maximum h_{M} with h_{d} =0 in Example 2.
Simulation for Example 2 with the controller gain K when h_{M} =50.01 and h_{d} =0.
Example 3.
Consider the system (6) with following parameters:
Also, let us define the disturbances as
where
Moreover,
ρ
(
t
) is a stochastic variable of which mean and variance are
ρ
_{0}
and
σ
^{2}
, respectively.
By applying Theorem 3, the feedback controller gains
K
can be obtained with minimum
γ
when
h_{M}
=2,
h_{d}
=1,
α
=4,
ρ
_{0}
=1 and various
σ
^{2}
in
Table 3
. More specifically, in order to show the effectiveness of the variance for the system, the results are obtained when
σ
^{2}
are 0, 1, 2 and 3. Even though the means of the disturbances are same, minimum values of
γ
can be obtained differently which are increased as
σ
^{2}
is increased. The results are dependent on variance and are obtained by Theorem 3 which is derived for our new system model. Also,
Fig. 2
is illustrated for state response of the system (6) with stochastic disturbances and the timedelay
h
(
t
) = 0.5
h_{M}
(cos(2
h_{d}
t
/
h_{M}
) + 1) by simulation with parameters listed in
Table 3
. It shows that the closedloop system (6) is asymptotically stable with
H
_{∞}
disturbance attenuation level
γ
for any timevarying delay
h
(
t
) satisfying (3). Even if the mean of disturbances are same, the effect of disturbances increases when the variance
σ
^{2}
becomes large.
Minimum value ofγwithρ0=1,hM=2,hd=1,α=4 and variousσ2in Example 3.
Minimum value of γ with ρ_{0} =1, h_{M} =2, h_{d}=1, α =4 and various σ^{2} in Example 3.
Simulation for Example 3 with parameters in Table 3.
In addition, in
Table 4
, minimum values of
γ
and controller gains
K
which dependent on mean of
ρ
(
t
) are listed when
σ
^{2}
=1,
h_{M}
=2,
h_{d}
=1,
α
=4 and various
ρ
_{0}
. It can be shown that the minimum of
γ
is increased as
ρ
_{0}
is increased which means that the system is influenced more by the disturbances. It is natural that the results are worse when the mean of disturbances increases because disturbances cause poor performances.
Fig. 3
shows the simulation results with parameters listed in
Table 4
and the timedelay
h
(
t
) = 0.5
h_{M}
(cos(2
h_{d}t
/
h_{M}
) + 1) . It shows the state responses of the system (6) when the stochastic disturbances occurred with various
ρ
_{0}
. From
Fig. 3
, the closedloop system (6) with the controller gain in
Table 4
is asymptotically stable with
H
_{∞}
disturbance attenuation level
γ
for any timevarying delay
h
(
t
) satisfying (3). Also, when
ρ
_{0}
increases, the signal of disturbance become large. This means that system is more affected by disturbance and thus the performance of the closedloop system (6) is deteriorated.
Minimum value ofγwithσ2=1,hM=2,hd=1,α=4 and variousρ0in Example 3.
Minimum value of γ with σ^{2} =1, h_{M} =2, h_{d}=1, α =4 and various ρ_{0} in Example 3.
Simulation for Example 3 with parameters in Table 4.
5. Conclusion
In this paper, the problems of stability analysis, stabilization, and design the delaydependent robust
H
_{∞}
controller for timedelayed systems with parameter uncertainties and stochastic disturbances were investigated. It was assumed that the parameter uncertainties were norm bounded. In order to use stochastic characteristic of the disturbances, the mean and variance for disturbances were utilized. Main results were three separated subsections which were the stability analysis criterion, controller design criterion and robust
H
_{∞}
controller design criterion. Firstly, in Theorem 1, the stability criterion for the nominal systems with timevarying delays was derived by utilizing the newly augmented LyapunovKrasovskii functional. Secondly, based on the results of Theorem 1, the new stabilization criterion for the nominal form of the systems was established in Theorem 2. Finally, Theorem 2 was extended to the problem of robust
H
_{∞}
controller design for the timedelayed systems with parameter uncertainties and stochastic disturbances in Theorem 3. Moreover, three examples were given to illustrate the effectiveness of the presented criterion.
Acknowledgements
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (20110009273). This work was also supported by the Human Resources Development of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea government Ministry of Trade, Industry & Energy (no. 20144030200450).
BIO
KiHoon Kim He received B.S. and M.S. degrees both in Electrical Engineering from Chungbuk National University, Cheongju, Korea, in 2012 and 2014, respectively. His research interests are timedelay systems.
MyeongJin Park He received B.S. and Ph.D. degrees both in Electrical Engineering from Chungbuk National University, Cheongju, Korea, in 2009 and 2015, respectively. His current research interests include consensus of multiagent systems and control of timedelay systems.
OhMin Kwon He received B.S. degree in Electronic Engineering from Kyungbuk National University, Daegu, Korea, in 1997, and Ph.D. degree in Electrical and Electronic Engineering from POSTECH, 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 more than 130 international papers in these areas. He is a member of KIEE, ICROS, and IEEK. Currently, he serves as an editorial member of ICROS, Nonlinear Analysis: Hybrid Systems, and The Scientific World Journal.
SangMoon Lee He received B.S. degree in Electronic Engineering from Kyungpook 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.
EunJong Cha He received B.S. degree in Electronic Engineering from the Seoul National University, Seoul, Korea, in 1980, and Ph.D. degree in Biomedical Engineering from the University of Southern California, LosAngeles, USA, in 1987. He founded a venture company, CK International Co., in 2000 and is serving as the president since then. In 20052006, he served as the Director of Planning and Management of the Chungbuk National University. He is currently appointed as a Professor and the Chair of the Biomedical Engineering Department, Chungbuk National University, Cheongju, Korea. His research interest includes biomedical transducer, cardiopulmonary instrumentation, and intelligent biomedical system. He serves as a Member of KOSOMBE, KSS, KOSMI, IEEK, KIEE, and IEEE. He has also been serving the Korean Intellectual Patent Society as the Vice President since 2004.
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