In this paper, the problem of reliable control of linear systems with timevarying delays, randomly occurring disturbances, and actuator failures is investigated. It is assumed that actuator failures occur when disturbances affect to the systems. Firstly, by using a suitable LyapunovKrasovskii functional and some recent techniques such as Wirtingerbased integral inequality and reciprocally convex approach, stabilization criterion for nominal systems with timevarying delays is derived. Secondly, the proposed method is extended to the reliable
H
_{∞}
controller design for linear dynamic systems with timevarying delays, randomly occurring disturbances, and actuator failures. Since nonlinear matrix inequalities (NLMIs) are involved in proposed results, the cone complementarity algorithm will be introduced. Finally, two numerical examples are included to show the effectiveness of the proposed criteria.
1. Introduction
The stabilization of linear systems with timedelays is an important issue since timedelays occurs in various systems such as physical and chemical systems, industrial and engineering systems, and so on. It is well known that timedelays can lead to oscillation, poor performance or even instability. Therefore, the problem of delaydependent stability and stabilization criteria for systems with timedelays have been received a great deal of efforts by many researchers
[1

4]
.
One of the objectives of delaydependent stabilization for systems with timedelays is to find maximum upperbounds of timedelays which guarantee the asymptotic stability of the concerned. In order to reduce the conservatism of stabilization criteria for systems with timedelays, many researchers have focused on delaydependent criteria than delayindependent ones since delaydependent ones are less conservative than delayindependent ones especially when the sizes of timedelays are small. While delay independent once do not have information about timedelays, delaydependent criteria have ones such as lowbound, upperbounds and bounds of differential of delays.
In the last decade, the Jensen inequality has been intensively used for analysis of systems with timedelays since it plays key roles to derive a stability condition when estimating the timederivative of LyapunovKrasovskii functional. Very recently, in order to reduce the conservatism of stability criteria obtained by utilizing the Jensen Inequality, Wirtingerbased integral inequality
[5]
is introduced for stability analysis based on Fourier analysis. It can lead to less conservative results than Jensen inequality for integral terms since Wirtingerbased integral inequality allows considering a more accurate integral inequality. In this paper, Wirtingerbased integral inequality is used to obtain stabilization criteria.
On the other hand, recently, the problem of designing reliable control systems has been attracted since practical systems often have actuator failures
[6

7]
. It has been known that the class of reliable control systems is to stabilize the systems against actuator failures or to design faulttolerant control systems. In this paper, actuator failure model which consists of a scaling factor with upper and lower bounds to the signal to be measured or to the control action is introduced.
In line with this thinking, disturbances can have an adverse effect on the stability of systems. Thus, to design a controller for the systems considering disturbances is another important issue in control society. For instance, disturbances such as earthquake and typhoon, controllers are required to minimize the effect of disturbances on building or structure systems. The
H
_{∞}
control has objective that is to design the controllers such that the closedloop systems are stable and its
H
_{∞}
norm of the transfer function between the controlled output and the disturbances will not exceed a prescribed level of performance. Therefore, since
H
_{∞}
control
[8]
was introduced firstly, a number of research results on
H
_{∞}
control have been utilized for various systems
[9

13]
.
Recently, a variety of stochastic systems have been researched
[13

15]
. Systems with timedelays and stochastic sampling were considered in
[13]
. Also, Systems with randomly occurring uncertainties have introduced in
[14

15]
. From the idea of randomly occurring concept, it can be extended to reliable control problem since disturbances can bring out the actuator failures. In other words, when randomly occurring disturbances affect to the system, actuator failures occur simultaneously.
With motivations for the above discussions, this paper focused on the problem of the reliable
H
_{∞}
controller design for linear systems with timedelays. Firstly, in Theorem 1, stabilization criterion will be proposed by using the appropriate LyapunovKrasovskii functional with Wirtingerbased integral inequality
[5]
and reciprocally convex approach
[16]
. Secondly, based on the results of Theorem 1, a reliable
H
_{∞}
controller design method for the systems with time delays, randomly occurring disturbances, and actuator failures will be proposed in Theorem 2. Since results in Theorem 1 and Theorem 2 have developed in terms of NLMIs, the cone complementarity algorithm will be introduced which developed solve the NLMIs
[12
,
17]
. Two numerical examples are included to show the effectiveness of the proposed theorems.
Notations
:
R
^{n}
denotes the ndimensional Euclidean space,
R
^{n×m}
is the set of
n
×
m
real matrices.
diag
{⋯} denotes the block diagonal matrix.
L
_{2}
is the space of square integrable functions on [0,∞). For two symmetric matrices
A
and
B
,
A
>(≥)
B
means that
A
−
B
is (semi) positive definite.
A^{T }
denotes the transpose of
A
.
I
_{n}
denotes the
n
×
n
identity matrix. 0
_{n}
and 0
_{n×m}
are denote the
n
×
n
zero matrix and
n
×
m
zero matrix, respectively. If the context allows it, the dimensions of these matrices are often omitted.
L
_{2}
[0,∞) is the space of square integrable vector.
E
{
x
} and
E
{
x

y
} will, respectively, mean the expectation of
x
and the expectation of
x
condition on
y
.
X
_{[f(t)]}
∈
R
^{m×n}
means that the elements of the matrix
X
_{[f(t)]}
includes the value of
f
(
t
); e.g.,
X
_{[f0]}
≡
X
_{[f(t)=f0]}
. Pr{⋅} means the occurrence probability of the event
"
⋅
"
.
2. Problem Statements
Consider the following linear system with timevarying delay:
where
x
(
t
)∈
R
^{n }
is the state vector,
u^{F}
(
t
)∈
R
^{m }
is the vector of controlled input with actuator failures,
z
(
t
)∈
R
^{p}
is the vector of controlled output,
w
(
t
)∈
R
^{l }
is the disturbance input which belongs to
L
_{2}
[0,∞).
A
∈
R
^{n ×n}
,
A
_{d}
∈
R
^{n×n}
,
B
_{w}
∈
R
^{n ×l }
,
B
∈
R
^{n ×m}
and
C
∈
R
^{q×n}
are known real constant matrices.
Also,
h
(
t
) is a timedelay satisfying timevarying continuous function as follows:
where
h_{M}
is a positive scalar and
h_{d}
is any constant value.
In this paper, it is concerned that actuator has behaviour of faulty. The control input of actuator fault can be described as
where
u
(
t
)∈
R
^{m}
is the vector of controlled input and
R
is the actuator fault matrix with
where
and
(
i
=1,2,...,
m
), are given constants. When
r
_{i }
= 0, it means the complete failure of
i
th actuator. If
r
_{i }
= 1, then
i
th actuator is normal.
Let us define
Then, the actuator fault matrix
R
can be rewritten as
where
ΔJ
=
diag
{
j
_{1}
,
j
_{2}
,…,
j
_{m}
}, −1 ≤
j_{i}
≤1.
It is assumed that actuator failure and disturbances occur randomly. In details, if disturbances occur, then it affects to the system and leads to actuator failures. So, it can be seen that disturbances and actuator failures occur simultaneously.
In order to describe the random occurrence, let us define 𝜌(
t
) as a stochastic variable which satisfy a Bernoulli distribution as follows:
Also, 𝜌(
t
) satisfies the following condition
where 0 ≤ 𝜌
_{0}
≤ 1 is a given constant scalar. 𝜌
_{0 }
is the expectation of 𝜌(
t
) and reflects the occurrence probability of disturbances and actuator failures.
With the concept introduced at Eqs. (3)(8), let us consider the following linear system with timevarying delay with randomly occurring disturbances and actuator failures given by
Also, actuator failure model with randomly occurrence can be described as
where term of (1 − 𝜌(
t
))
I
_{m}
reflects normal actuator when 𝜌(
t
) is 0.
The problem under consideration is to design a memoryless state feedback controller of the following form:
where
K
∈
R
^{m × n}
is a gain matrix of the feedback controller.
To develop a delaydependent reliable
H
_{∞}
controller for the system (9) satisfying following conditions:

(i) With zero disturbance, the closed loop system (9) with control inputu(t) is asymptotically stable.

(ii) With zero condition and a given constant γ > 0 the following condition holds:
where γ ≥ 0 is a prescribed scalar. The objective of this paper is to design a state feedback controller (11) such that system (9) is asymptotically stable and an disturbance attenuation level γ is minimize. If the above objective is achieved, controller (11) is said to be a reliable
H
_{∞}
controller.
Before deriving main results, the following lemmas are introduced.
Lemma 1
.
[5]
For a given matrix R
> 0,
the following inequality holds for all continuously differentiable function
𝜔
in
[
a,b
]→
R
^{n}
Lemma 2
.
[16]
For a scalar α in the interval
(0,1)
a given matrix
R
∈
R
^{n×n}
>0,
two matrices
W
_{1}
∈
R
^{n × m }
and
W
_{2}
∈
R
^{n × m}
,
for all vector
ς
∈
R^{m}
,
let us the function
θ
(
α
,
R
)
given by:
Then, if there exists a matrix X
∈
R
^{n× m }
such that
,
then the following inequality holds
Lemma 3
.
[18]
Let
E
,
H
, and
F
(
t
) be real matrices of appropriate dimensions, and let
F
(
t
) satisfy
F
^{T}
(
t
)
F
(
t
) ≤
I
. Then, for any scalar є> 0, the following matrix inequality holds:
EF
(
t
)
H
+
H^{T}F^{T}
(
t
)
E^{T}
≤ є
H^{T}H
+є
^{−1}
EE^{T}
.
3. Main Results
This section consists of two subsections. The goal of first subsection is to design a controller which stabilize the nominal system. Second subsection will introduce a design method of a reliable
H
_{∞}
controller for linear systems with timevarying delays, randomly occurring disturbances, and actuator failures.
 3.1 Controller design for nominal system
In this subsection, a delaydependent stabilization criterion for the nominal system of (9) without disturbances and actuator failures will be introduced. Here, the following nominal system with control input
u
(
t
) is given by
where
h
(
t
) is satisfied with (2) and
u
(
t
) is defined in (11). Now, for simplicity of matrix and vector representation,
e_{i}
(
i
= 1,...,5)∈
R
^{5n× n}
are defined as block entry matrices which will be used. For example,
e
_{1}
= [
I
_{n}
,0
_{n}
,0
_{n}
,0
_{n}
,0
_{n}
]
^{T}
and
e
_{3}
= [0
_{n}
,0
_{n}
,
I
_{n}
,0
_{n}
,0
_{n}
]
^{T}
. The other notations are defined as
Now, the following theorem is given as a stabilization criterion for the system (13).
Theorem 1
.
For given scalars h_{M}
>0,
h_{d}
,
the system (13) is asymptotically stable for
0 ≤
h
(
t
) ≤
h_{M} and
,
if there exist positive definite matrices
X
∈
R
^{n×n}
,
,
,
,
any matrices
and
Y
∈
R
^{m×n} satisfying the following conditions hold:
where
and
are defined in (14). If the above conditions are feasible, a desired controller gain matrix is obtained by K= YX
^{−1}
.
Proof
. For positive definite matrices
P
,
Q
_{1}
,
Q
_{2}
and
N
, let us consider the following the LyapunovKrasovskii functional candidate as:
where
The upperbound of
can be given as follows:
is calculated as
By using Lemma 1, an upperbound of
can be obtained as
where
From Lemma 2, if the inequality for any matrix
M
∈
R
^{2n × 2n}
holds
then, a new upperbound of (21) is can be obtained as
Note that
ϕ
(
t
) satisfies 0 ≤
ϕ
(
t
) ≤ 1. When
h
(
t
) = 0,
x^{T}
(
t
) −
x^{T}
(
t
−
h
(
t
)) = 0 and
Ω
_{1}
= 0 are obtained and when
h
(
t
) =
h_{M}
,
x^{T}
(
t
−
h
(
t
)) −
x^{T}
(
t
−
h_{M}
) = 0 and
Ω
_{2}
= 0 are obtained. Thus, relation (23) still holds.
By combining (18)(23), an upperbound of
is obtained as follows:
By using Schur complement, stabilization criterion for the system (24) is equivalent to the following
Let us define
X=P
^{−1}
,
,
,
,
, and
Y=KX
. Then, following inequalities can be obtained by pre and postmultiplying (25) and (22) by
diag
{
X,X,X,X,X, I_{n}
} and
diag
{
X,X,X,X
}, respectively
where
and
are defined in (14). This proof is completed.
It should be note that the stabilization condition (15) have the nonlinear term
X
X
. A simple way to solve it is to set
=
αX
, where
α
> 0 is a tuning parameter. However, this method is too conservative. To obtain better results, the cone complementarity algorithm can be used with computational effort.
In order to solve NLMIs, the cone complementarity algorithm in
[12
,
17]
is used which involves iteratively solving linear matrix inequalities (LMIs). Let us define a new variable matrix
L
> 0 satisfying
which is equivalent
X
^{−1}
X
^{−1}
≤
L
^{−1}
. Letting
H
=
L
^{−1}
,
G
=
X
^{−1}
,
F
=
^{−1}
and following a similar method in
[12
,
17]
, the problem of finding a feasible solution of (15) and (16) can be converted to a minimization problem involving LMIs:
Minimize Trace (
LH+XG+
F
)
Subject to
The above minimization problem can be solved using the cone complementarity algorithm in
[12
,
17]
.
Algorithm
Let us define
Γ^{k}
as the set of the variables of
and
k
_{max}
as the number of iterations. Then, following
Figure 1
is the flow chart of algorithm for Theorem 1.
Theorem 1을 위한 알고리듬 흐름선도 Fig. 1 Flow chart of Algorithm for Theorem 1
 3.2 Reliable H∞ controller design for randomly occurring disturbances and actuator failures
In this subsection, the reliable
H
_{∞}
controller design for the system (9) will be derived based on Theorem 1. Now, for simplicity of matrix and vector representation,
ē_{i}
(
i
= 1,…,6)∈
R
^{(5n+l)×n}
are defined as block entry matrices which will be used. For an example, ē
_{5}
= [0
_{n}
,0
_{n}
,0
_{n}
,0
_{n}
,
I_{n}
,0
_{n × l}
]
^{T}
. The following notations are defined for simplicity:
where
Φ
_{2}
,
Φ
_{3}
,
_{2}
and
_{3}
are defined in (14).
Now, we have the following theorem.
Theorem 2
.
For given scalars
,
(
i
= 1,…,
m
),
h_{M}
> 0,
h_{d}
,
the system (9) is asymptotically stabilized by reliable H_{∞} control (11) with disturbance attenuation
γ > 0
for
0 ≤
h
(
t
) ≤
h_{M} and ḣ
(
t
) ≤
h_{d}
, if there exist positive definite matrices
X
∈
R
^{n × n}
,
,
,
,
any matrices
,
Y
∈
R
^{m×n}
,
positive scalars
ε
_{1 }
and
ε
_{2}
,
satisfying the following conditions hold:
where
and
are defined in (30). If the above conditions are feasible, a desired reliable H_{∞} controller gain matrix is obtained by K=YX
^{−1}
.
Proof.
Let us consider the same LyapunovKrasovskii candidate functional in (17). By infinitesimal operator
L
in
[13]
, a new upperbound of
LV
(
t
) is obtained by
From the system (9), replacing
ẋ
(
t
) = (
Π
_{1[ρ(t)]}
+
ΔΠ
_{1[ρ(t)]}
)
ς
(
t
) leads to following inequality
Now,
H
_{∞}
performance for the system (9), let us consider the following inequality under the zero initial condition satisfying
V
(0) = 0 and
V
(∞) ≥ 0
When the inequality (36) is satisfied, the system (9) is stable with
H
_{∞}
performance level γ under the obtained controller (11). Inequality (36) is equivalent to
Replacing
z
(
t
) =
Cx
(
t
) and using Schur complement, following inequality can be obtained as
Since
ΔΨ
_{1[ρ(t)]}
=
E
_{1}
ΔJ^{T}H
_{1}
+
H
_{1}
ΔJE
_{1}
and
ΔΨ
_{2[ρ(t)]}
=
E
_{2}
ΔJ^{T}H
_{2}
+
H
_{2}
ΔJE
_{2}
, using Lemma 3 leads a new upperbound of (38) as follows:
By using Schur complement, inequality (35) is equivalent to
Let us define
X
=
P
^{−1}
,
,
,
,
,
Y= KX
. Then, following inequalities can be obtained by pre and postmultiplying (40) and (34) by
diag
{
X,X,X,X,X,I_{l},I_{n},I_{q},I_{m},I_{m}
} and
diag
{
X,X,X,X
}, respectively
With (8), inequality
is equivalent to
< 0. This proof is completed.
It should be noted that the stabilization condition (31) is not LMIs due to the presence of the nonlinear term
X
X
. With similar way in Theorem 1, better results can be obtained by using the cone complementarity algorithm.
Let us define a new variable matrix
L
> 0 satisfying
which is equivalent
X
^{−1}
X
^{−1}
≤
L
^{−1}
. Letting
H
=
L
^{−1}
,
G
=
X
^{−1}
,
F
=
^{−1}
and following a similar method in
[12
,
17]
, the problem of finding a feasible solution of (31) and (32) can be converted to a minimization problem involving LMIs:
Minimize Trace (
LH+XG+
F
) Subject to
The above minimization problem can be solved using complementarity algorithm in
[12
,
17]
.
Algorithm
Let us define ϓ
^{k}
as the set of the variables of
and
k
_{max}
as the number of iterations. Then, following
Figure 2
is the flow chart of algorithm for Theorem 2.
Theorem 2을 위한 알고리듬 흐름선도 Fig. 2 Flow chart of Algorithm for Theorem 2
4. Numerical Examples
In this section, two numerical examples are introduced to demonstrate the effectiveness of the proposed criteria. In examples, MATLAB, YALMIP, SeDuMi 1.3 and Intel(R) Core(TM) i52500 CPU @ 3.30Ghz (4 CPUs) are used to solve LMI problems.
Example 1
. Consider the system (13) with following parameters:
When
ḣ
(
t
) ≤
h_{d}
= 0, Theorem 1 is used to obtain the feedback controller gain
K
which stabilize the system (13) with upperbounds of
h
(
t
) and number of iterations. The maximum allowable upperbound of
h
(
t
) is 4.3 when the number of iterations is 920. Their results are listed in
Table 1
with previous results in
[1]
,
[9]
and
[13]
. Also, in order to confirm the results, the simulation results is illustrated in
Figure 3
with timedelay
h
(
t
) = 4.3 and feedback controller gain
K
= [−19.076, −29.050] .
예제 1에서hd= 0일 때 제어 이득K를 고려한 최대의hM.Table 1 MaximumhMwith controller gainKwhenhd= 0 in Example 1.
예제 1에서 h_{d} = 0일 때 제어 이득 K를 고려한 최대의 h_{M}. Table 1 Maximum h_{M} with controller gain K when h_{d} = 0 in Example 1.
제어 이득 K = [−19.076, −29.050] 를 고려한 h(t) =4.3 일 때 예제 1의 시뮬레이션 Fig. 3 Simulation for Example 1 with controller gain K = [−19.076, −29.050] when h(t) = 4.3
Example 2
. Consider the system (9) with
Moreover, the disturbances are defined as follows:
where
which satisfied with 0 ≤
h
(
t
) ≤
h_{M}
and
ḣ
(
t
) ≤
h_{d}
. By applying Theorem 2, minimum value of γ and controller gain
K
for system (9) when
h_{M}
= 0.2 ,
h_{d}
= 3, and
ρ
_{0}
are 0.1, 0.5 and 0.9 are listed in
Table 2
.
ρ0에 따른 제어 이득K, γmin, 그리고 반복횟수.Table 2 Controller gainK, γminand iterations withρ0.
ρ_{0}에 따른 제어 이득 K , γ_{min}, 그리고 반복횟수. Table 2 Controller gain K , γ_{min} and iterations with ρ_{0}.
Results in
Table 2
show that
ρ
_{0}
increases γ
_{min}
which is minimum of
H
_{∞}
disturbance attenuation level γ. It can be shown that
ρ
_{0}
is increased, then disturbances and actuator failure occur more frequently. Therefore, when
ρ
_{0}
is 0.9, feedback controller gain
K
is obtained as [−242.4149,−68.4737,−201.6272,−611.0830] which is larger than other ones.
Figure 4
shows linear system responses for each case in
Table 2
. From
Figure 4
, the system (9) with the controller gain in
Table 2
is asymptotically stable with
H
_{∞}
disturbance attenuation level γ for any timevarying delay
h
(
t
) satisfying (2). Furthermore, trajectories of
Bu^{F}
(
t
) show that actuator failure more frequently occur as the value of
ρ
_{0}
increases. It can be seen that the system and actuator are more influenced by the disturbances when the value of
ρ
_{0}
increases. As a result, when the effect of disturbances increases, the state responses and controlled output performances become worse.
표 2의 각각의 상황을 고려한 시뮬레이션 Fig. 4 Simulations for each case in Table 2
5. Conclusions
In this paper, the reliable
H
_{∞}
controller design for linear systems with timedelays was presented. Firstly, in Theorem 1, the stabilization criterion was proposed by constructing the appropriate LyapunovKrasovskii functional and utilizing Wirtingerbased integral inequality
[5]
and reciprocally convex approach
[16]
. Secondly, results of Theorem 1 was extended to design the reliable
H
_{∞}
controller for the systems with time delays, randomly occurring disturbances, and actuator failures in Theorem 2. Since results have NLMIs in Theorem 1 and Theorem 2, the cone complementarity algorithm was used to solve the NLMIs
[12
,
17]
with computational effort. To show the effectiveness of the proposed results, two numerical examples were included.
Acknowledgements
This work was supported by the research grant of the Chungbuk National University in 2013.
BIO
김 기 훈 (金 基 勳)
1986년 12월 7일생. 2012년 충북대학교 전기공학과 졸업(공학). 현재 충북대학교 전기공학과 석사과정.
Email : redpkzo@chunbuk.ac.kr
박 명 진 (朴 明 眞)
1982년 4월 7일생. 2009년 충북대학교 전기공학과 졸업(공학). 현재 충북대학교 전기공학과 박사과정.
Email : netgauss@chunbuk.ac.kr
권 오 민 (權 五 珉)
1974년 7월 13일생. 1997년 경북대학교 전자공학과 졸업(공학). 2004년 포항공과 대학교 전기전자공학부 졸업(공박). 현재 충북대학교 전기공학부 부교수.
Email : madwind@chungbuk.ac.kr
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