In this paper, a delaydependent
H
_{∞}
filtering problem is investigated for discretetime delayed nonlinear systems which include a more general sector nonlinear function instead of employing the commonly used Lipschitztype function. By using the LyapunovKrasovskii functional approach, a less conservative sufficient condition is established for the existence of the desired filter, and then, the corresponding solvability condition guarantee the stability of the filter with a prescribed
H
_{∞}
performance level. Finally, two simulation examples are given to show the effectiveness of the proposed filtering scheme.
1. Introduction
During the past decades, the filter design problem has been widely studied due to its extensive applications in control systems and signal processing. The purpose of the filtering design is to estimate the unavailable state variables of a given system through noisy measurements. There are basically two approaches to the problem: the Kalman filtering approach and the
H_{∞}
filtering approach. The Kalman filtering approach is based on the assumption that a linear system model is required and all noise terms and measurements have Gaussian distributions
[1]
. However, the prior knowledge of noise may not always be precisely known. In order to overcome this problem,
H_{∞}
filtering method, which provides both a guaranteed noise attenuation level and robustness against unmodeled dynamics, has been proposed. Compared with the Kalman filtering method, the main advantage of
H_{∞}
filtering method is that the noise sources are supposed to be arbitrary signal with bounded energy, and no exact statistics are required to be known. On the other hand, since time delay is commonly encountered in various engineering systems and is frequently a source of instability and poor performance, the problem of
H_{∞}
filtering for time delay systems have been received increasing attention in the last decades
[2

17]
.
It is wellknown that the main objective of
H_{∞}
filtering is to design a suitable filter such that the bound of induced L
_{2}
norm of the operator from the noise signals to the filtering error is less than a prescribed level. In order to get less conservative results, that is, obtain a smaller
H_{∞}
disturbance attenuation lever 𝛾, various methods are utilized. For example, in
[3]
, a new finite sum inequality was employed to get a sufficient condition for the existence of a suitable filter. In
[8]
, the inputoutput(IO) approach is used to get the less conservative result than
[3]
. But some delay terms are neglected for estimating the derivative bound of the constructed Lyapunov functional in the IO approach. Therefore, the results in
[8]
are conservative to some extent, and there is much room to improve the result in
[8]
. Moreover, it should be noted that the results in
[2

9]
are considered for only linear system.
It is well known that nonlinearities exist universally in practical systems, so the
H_{∞}
filtering problem for nonlinear dynamical system have been investigated by many researchers
[10

17
. Xu
[15]
was concerned with the problem of robust
H_{∞}
filtering for a class of discretetime nonlinear systems with state delay and normbounded parameter uncertainty. In
[11]
, a stable full or reduced order filter with the same repeated scalar nonlinearities was designed to guarantee the induced L
_{2}
or generalized
H_{∞}
performance. In
[14]
, the problem of
H_{∞}
filtering for systems with repeated scalar nonlinearities under unreliable communication was investigated. In
[15]
, a robust
H_{∞}
filtering problem for a class of discretetime nonlinear systems was considered. But it should be pointed out that the time delay was not taken into consideration in
[11
,
14
,
15]
, and only constant time delay was considered in
[10]
. To the best of our knowledge, there are few results on the problem of the
H_{∞}
filtering for a nonlinear system with timevarying delay.
In this paper, we consider the
H_{∞}
filtering problem for a class of discretetime systems involving sector nonlinearities and interval timevarying delay. Inspired by the work
[13]
, the sector nonlinearities considered in the paper are more general than usual Lipschitz conditions. By using a new Lyapunov functional and Linear matrix inequality technique, delay dependent conditions are obtained for designing a filter with an
H_{∞}
disturbance attenuation level 𝛾. When the involved LMIs are feasible, a set of the parameters of a desired filter can be obtained. Two numerical examples are provided to show the usefulness and effectiveness of the proposed design method.
Notation: Throughout the paper,
R
^{n}
denotes the
n
dimensional Euclidean space,
R
^{m × n}
denotes the set of
m
by
n
real matrix. For symmetric matrices
X
,
X
＞ 0 and
X
＜0, mean that
X
is a positive/negative definite symmetric matrix, respectively.
I
and 0 denote the identity matrix and zero matrix with appropriate dimension. ★ represents the elements below the main diagonal of a symmetric matrix.
diag
… denotes the diagonal matrix.
refers to the induced matrix 2norm. L
_{2}
means the space of square integral vector functions on [0, ∞) with norm
2. Problem Statements
Consider a discretetime nonlinear system with timevarying delay and disturbance:
where
x
(
k
) ∈ R
^{n}
is the state,
w
(
k
) ∈ R
^{q}
is a disturbance input belongs to
L
_{2}
[0, ∞),
y
(
k
) ∈ R
^{m}
is the signal to be estimated,
f
(.) and
h
(.) are known vectorvalued nonlinear functions
A,A_{d},F,B_{w},C,C_{d},H
_{1}
,
H
_{2}
,
D_{w},L,L_{d},G_{w}
are known constant matrices of appropriate dimensions, and
d
(
k
) is the time varying delay satisfying
d
_{1}
≤d(k)≤d
_{2}
where d
_{1}
＞0 and d
_{2}
＞0 demote the lower and upper bounds of the delay, respectively.
In this paper, without loss of generality, we always assume that
f
(0) = 0,
h
(0) = 0 and for vectorvalued functions
f,h
, we assume
where
U
_{1}
,
U
_{2}
,
V
_{1}
,
V
_{2}
are known real constant matrices, and
U
_{2}
−
U
_{1}
,
V
_{2}
−
V
_{1}
are positive definite matrices.
Remark 1 Eq. (2) and (3) are the socalled sectorbounded conditions
[18]
, which are more general than the Lipschitz conditions, and have been widely adopted in the literature
[19

20]
. The reason is that if we use the Lipschitz condition, the matrix
U
_{1}
,
U
_{2}
,
V
_{1}
,
V
_{2}
are diagonal matrix, it is a special case included in our considered condition.
The objective of this paper is to estimate the system states
x
(
k
). In this paper, we consider a fullorder linear asymptotically stable filter for system (1) with statespace realization of the form
where
is the filter state vector and
A_{F}B_{F},C_{F}D_{F}
are appropriately dimensioned filter gains to be determined.
Denote
Then the following error system is obtained
where
The aim of this paper is to design the
H
_{∞}
filter satisfying that the filtering error system (5) with
w
(
k
) = 0 is asymptotically stable and
H
_{∞}
performance
is guaranteed under zeroinitial conditions.
Lemma 1
[5]
For any matrix
M
＞0, integers 𝛾
_{1}
and 𝛾
_{2}
satisfying 𝛾
_{2}
＞𝛾
_{1}
, and vector function
w
:
N
[𝛾
_{1}
,𝛾
_{2}
]→
R^{n}
such that the sums concerned are well defined, then
Lemma 2
[5]
For any matrix
, scalars
α
_{1}
(
k
) ＞0,
α
_{2}
(
k
) ＞0 satisfying
α
_{1}
(
k
)+
α
_{2}
(
k
)=1, vector functions
δ
_{1}
(
k
) and
δ
_{2}
(
k
) :
N→R^{n}
, the following inequality holds
3. Main Results
In this section, first of all, let us give a sufficient condition, which ensure system (1) to be asymptotically stable with
H
_{∞}
performance level 𝛾. In convenience, we define
The following theorem provides a sufficient condition, which ensures the system (5) to be asymptotically stable with
H
_{∞}
performance 𝛾.
Theorem 1
.
For given
d
_{2}
＞
d
_{1}
＞ 0, 𝛾 ＞ 0 and
matrix
U
_{1}
,
U
_{2}
,
V
_{1}
,
V
_{2}
,
W
_{1}
,
W
_{2}
,
the nonlinear filtering error system (5) is asymptotically stable with H_{∞} performance 𝛾, if there exist positive definite symmetric matrix
P,Q
_{1}
,
Q
_{2}
,
R
_{1}
,
R
_{2}
and appropriate dimension matrix
,
satisfying the following LMIs
Proof. Consider the following LK functional candidate as
where
with the cost function (6).
Calculating the difference of
V
_{1}
(
k
),
V
_{2}
(
k
) and
V
_{3}
(
k
), we have
Since
an upper bound of the difference of
V
_{4}
(
k
) is obtained from Lemma 2
where
Note that if
d
(
k
) = 0 or
d
(
k
) =
d
_{2}
, we have
x
(
k
) −
x
(
k
−
d
(
k
)) = 0 or
x
(
k
−
d
(
k
)) −
x
(
k
−
d
_{2}
) = 0, respectively. Thus Eq.(14) holds based on Lemma 1.
Similar to Eq. (11), the difference of
V
_{5}
(
k
) is
where
Note that if
d
(
k
) =
d
_{1}
or
d
(
k
) =
d
_{2}
, we have
x
(
k
−
d
_{1}
)−
x
(
k
−
d
(
k
)) = 0 or
x
(
k
−
d
(
k
))−
x
(
k
−
d
_{2}
) = 0, respectively. Thus, Eq. (15) holds based on Lemma 1.
It follows readily from Eq. (2) and Eq. (3) that
To establish the
H
_{∞}
performance for the filtering error system (5), if the difference of
V
(
k
) is negative, then
z
(
k
) goes to zero as
k
→∞. Next, assuming zero initial conditions for the filtering error system, the performance index is
If the inequality
e
(
k
)
^{T}
e
(
k
)−𝛾
^{2}
w
(
k
)
^{T}
w
(
k
)+Δ
V
(
k
) ＜ 0 holds, then
V
(
k
) goes to zero as
k
→∞.
Combining with Eq. (10)(18), one can obtain
By Schur complement, inequality (19) is equivalent to
In the inequality (20), the positivedefinite matrix
P
and the filter parameters
A_{f}, B_{f}, C_{f}, D_{f}
, which included in the matrix
are unknown. Hence it should be converted to LMI via proper variable substitution method.
Let us define
V
as
For positive definite matrix
P
^{−1}
and nonzero matrix
V
, it follows that
Combined with the Eq. (21), pre and post multiplying the matrix inequality (20) by the matrix
diag
{I,I,V,I} and
diag
{I,I,V
^{T}
,I}, then one can get the following inequality
By simple matrix calculation, it is straightforward to verify that
Now, define a new set of variables as follows
Note that the inequality (8) implies that
J
＜0 for any nonzero
w
(
k
)∈
L
_{2}
, i.e., the filtering error system has a guaranteed 𝛾 level of disturbance attenuation. This completes the proof.
Remark 2
. If
F= H_{1} = H_{2}
=0, the system (1) is reduced to the following linear system:
Combining the same filter system with Eq. (4), the corresponding error system is obtained as following
In convenience, we define
Based on Theorem 1, the following Corollary provides a sufficient condition, which ensures the system (24) to be asymptotically stable with
H_{∞}
performance 𝛾.
Corollary 1
.
For given
d
_{2}
＞
d
_{1}
＞0, 𝛾＞0,
the linear filtering error system (24) is asymptotically stable with
H_{∞}
performance
𝛾,
if there exist positive definite symmetric matrix
P,Q
_{1}
,
Q
_{2}
,
R
_{1}
,
R
_{2}
and appropriate dimension matrix
,
satisfying the following LMIs
Remark 3
. For any solutions of the LMIs (7}(8) and LMIs (25)(26) in Theorem 1 and Corollary 1, respectively, a corresponding filter of the form (4} can be reconstructed from the relations
Remark 4
. In
[13]
, the timevarying delay term
x
(
k
−
d
(
k
)) was not considered for estimating the bound of
. In order to obtain a less conservative result, Lemma 2 is applied by using the timevarying delay term
x
(
k
−
d
(
k
)) in the Eq. (14) and (15). The following examples will be given to demonstrate the effectiveness of this method.
4. Numerical Examples
In this section, two examples are given to show the effectiveness of our method on the design of the robust
H_{∞}
filter.
Example 1
Consider the following simplified longitudinal flight system
[10]
:
The measurement signal and signal to be estimated are
It is easy to see that the system described by Eqs. (27)(31) which is satisfied Eqs. (2) and (3) and has the form (1) with
When
d
_{1}
= 1,
d
_{2}
= 5,𝛾 = 0.0670, applying Theorem 1 to above system by utilizing MATLAB(with YALMIP 3.0 an SeDuMi 1.3), the corresponding parameter of the filter gains (Remark 3) are obtained by
Fig. 1
shows that the output
z
(
k
) and its estimated output
under the initial condition
z
(
k
) = [1.5−1 1.5]
^{T}
,
=[−0.5 0 0.5]
^{T}
respectively. Also, the estimated error
e
(
k
) is described in
Fig. 2
, where the external disturbance
w
(
k
) = sin(
k
)
e
^{−10k}
. From these simulation results, we can see that the disturbance is effectively attenuated by designed
H_{∞}
filter for the discretetime system (1) with nonlinearities and timevarying delay.
z(k) (dashed) and its estimate (solid) in Example 1.
Estimated error e(k) in Example 1.
Furthermore, when
d
_{1}
=
d
_{2}
=2, the time delay become constant delay, we can obtain the minimal
H_{∞}
performance lever 𝛾=0.0456, which is less conservative than 1.5 derived in
[10]
.
Example 2
. Consider the system (23) with the following parameters:
The allowable
H_{∞}
performance lever 𝛾 obtained by different methods is depicted in
Table 1
. When
d
_{1}
= 1,
d
_{2}
= 4, we can see that the minimal
H_{∞}
performance lever 𝛾 is 3.8447, which is much small than 4.9431. It means that the obtained result in this paper is less conservative than the one derived in
[13]
. Solving the LMIs (25) and (26) by using the MATLAB(with YALMIP 3.0 an SeDuMi 1.3), the corresponding parameter of the filter gains (Remark 3) are given by
Comparison ofH∞performance 𝛾
Comparison of H_{∞} performance 𝛾
5. Conclusions
In this paper, a robust
H_{∞}
filtering problem for a class of discretetime systems with nonlinear sensor and interval timevarying delay has been proposed. Based on the LyapunovKrasovskii functional approach, sufficient conditions have been provided for the stability of the filtering error system with a prescribed
H_{∞}
performance level. Finally, numerical examples have been given to show the usefulness and effectiveness of the proposed filter design method.
Acknowledgements
This research was supported by the Daegu University Research Grants, 2012.
BIO
이 상 문 (李 相 文) 1973년 6월 15일생. 1999년 경북대학교 전자공학과 졸업(공학). 2006년 포항공과 대학교 전기전자공학부 졸업(공박). 현재 대구대학교 전자전기공학부 조교수. Email : moony@daegu.ac.kr
유 아 연 (柳 兒 蓮) She 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.
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