Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2011, Article ID 760878, 20 pages
doi:10.1155/2011/760878
Research Article
Delay-Dependent H∞ Filtering for Singular
Time-Delay Systems
Zhenbo Li1, 2 and Shuqian Zhu3
1
School of Statistics and Mathematics, Shandong Economic University, Jinan 250014, China
Shandong Provincial Key Laboratory of Digital Media Technology, Shandong Economic University,
Jinan 250014, China
3
School of Mathematics, Shandong University, Jinan 250100, China
2
Correspondence should be addressed to Shuqian Zhu, sqzhu@sdu.edu.cn
Received 14 February 2011; Accepted 29 April 2011
Academic Editor: Xue He
Copyright q 2011 Z. Li and S. Zhu. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
This paper deals with the problem of delay-dependent H∞ filtering for singular time-delay
systems. First, a new delay-dependent condition which guarantees that the filter error system has
a prescribed H∞ performance γ is given in terms of linear matrix inequalities LMIs. Then, the
sufficient condition is obtained for the existence of the H∞ filter, and the explicit expression for the
desired H∞ filter is presented by using LMIs and the cone complementarity linearization iterative
algorithm. A numerical example is provided to illustrate the effectiveness of the proposed method.
1. Introduction
Over the past decades, the filtering problem has been widely studied and has found many
applications 1, 2. Current efforts on this topic can be mainly divided into two classes:
the Kalman filtering approach and the H∞ filtering approach. The objective of the latter
one is to find a filter such that the resulting error system is asymptotically stable and the
L2 -induced norm for continuous systems or l2 -induced norm for discrete systems from
the disturbance input to the filtering error output satisfies a prescribed H∞ performance
level. In contrast to the Kalman filtering, the H∞ filtering approach does not require the
exact knowledge of the statistics of the external noise signals, and it is insensitive to the
uncertainties. These features render the H∞ filtering attracting much attention, and many
efforts have been made on this issue 3–6. The filtering problem for singular systems has
also been investigated by many researchers. For example, a necessary and sufficient condition
is obtained in 7 for the solvability of the H∞ filtering problem and the designed filter is
proper with a McMillan degree no more than the exponential modes of the plant, while,
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in 8, a linear normal H∞ filter is obtained for singular systems. Reduced-order H∞ filters
are designed in 9 for both continuous and discrete singular systems. In 10, a reducedorder H∞ filter design approach is developed for a class of discrete singular systems with
lossy measurements.
On the other hand, for many practical control systems, time delays are frequently
encountered and they are often the sources of instability and degradation in control
performance. So, recently, there has been increasing interest in H∞ filtering for time-delay
systems. Existing results can be classified into two types: delay-independent ones 11–14 and
delay-dependent ones 15–23; the former do not include any information on the size of delay
while the latter employ such information. Generally speaking, delay-dependent results are
less conservative than the delay-independent ones, especially when the size of delay is small.
Singular time-delay systems, which are also referred to as implicit time-delay systems,
descriptor time-delay systems, or generalized differential-difference equations, often appear
in various engineering systems, including aircraft attitude control, flexible arm control
of robots, large-scale electric network control, chemical engineering systems, and lossless
transmission lines see, e.g., 24. Since singular time-delay systems are more general, it is of
significance to consider the H∞ filtering problem for them. Recently, some delay-dependent
25–27 and delay-dependent 28–31 results about H∞ filters for such systems have been
obtained. In 28, the delay-independent filter is of the Luenberger observer type and the
decomposition and transformation of the system matrices are involved, which would result
in some numerical problems. A full-order filter is designed in 29 for singular systems with
communication delays, and H∞ filtering problems are concerned in 30, 31 for singular
systems with time-varying delay in a range.
In this paper, the problem of delay-dependent H∞ filtering is investigated for singular
time-delay systems. We consider the case of discrete delay which is assumed to be constant
and known. First, based on the result in 32, we derive a new delay-dependent condition
which guarantees that the filter error system has a prescribed H∞ performance γ; and it can be
seen that this new condition is more “efficient” than that in 32 since no redundant variables
are involved. Then, the sufficient condition for the existence of the full-order H∞ filter, which
is an admissible singular time-delay system, is obtained and the explicit expression for the
desired H∞ filter is given by using LMIs and the cone complementarity linearization iterative
algorithm.
Notations
Rn denotes the n-dimensional Euclidean space and Rn×m denotes the set of all n × m real
matrices, In is the n-dimensional identity matrix, and diag{· · · } is a block-diagonal matrix.
For real symmetric matrix X, the notation X ≥ 0 X > 0 means that the matrix X is positivesemidefinite positive-definite. The superscript T represents the transpose; the symbol ∗ will
be used in some matrix expressions to induce a symmetric structure. L2 0, ∞ refers to the
∞
space of square-integrable vector functions over 0, ∞ with norm f2 : 0 ft2 dt1/2 .
2. Problem Statement
Consider the following singular time-delay system:
Eẋt Axt Aτ xt − τ Bwt,
yt Cxt Cτ xt − τ B1 wt,
Discrete Dynamics in Nature and Society
3
zt Gxt Gτ xt − τ B2 wt,
xt φt,
t ∈ −τ, 0,
2.1
where xt ∈ Rn is the state, wt ∈ Rr is the external disturbance signal that belongs to
L2 0, ∞, yt ∈ Rm is the measurement output, and zt ∈ Rs is the signal to be estimated.
E, A, Aτ , B, C, Cτ , B1 , G, Gτ , and B2 are known real constant matrices with appropriate
dimensions and 0 < rank E p < n. τ > 0 is the known delay constant and φt ∈ Cn,τ is a
compatible vector-valued initial function.
Without loss of generality, we assume that Cτ 0, B1 0, Gτ 0, and B2 0.
Otherwise, system 2.1 can be equivalently changed into
E 0
0 0
ẋt
A
0
xt
0 −Ims ζt
⎡
⎤
⎡ ⎤
Aτ | 0
B
⎢
⎥
⎢−− − −− ⎥
⎢ ⎥
⎢−−⎥
⎢
⎥ xt − τ
⎢ ⎥
⎢
⎥
⎢ ⎥wt,
⎢ Cτ |
⎥
⎢ B1 ⎥
⎢
⎥ ζt − τ
⎢
⎥
⎣ ⎦
0ms ⎦
⎣
B2
Gτ |
⎡
⎤
C |
yt
⎢
⎥ xt
Ims ⎥
⎢
⎣
⎦ ζt .
zt
G |
ζ̇t
2.2
Then in the sequel, we discuss the system model as follows:
Eẋt Axt Aτ xt − τ Bwt,
yt Cxt,
zt Gxt,
xt φt,
2.3
t ∈ −τ, 0.
Throughout this paper, we need the following assumption for system 2.3.
Assumption 2.1. System 2.3 is admissible, that is, when wt ≡ 0, system 2.3 is regular, impulse
free, and asymptotically stable.
Remark 2.2. About the definitions of regularity, absence of impulses and asymptotical stability
for singular time-delay systems, we refer the readers to 33.
For the estimates of zt, we consider the following linear filter with delay:
Aτf xt
− τ Bf yt,
Ext
˙
Af xt
zt Cf xt,
xt
ψt,
t ∈ −τ, 0,
2.4
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Discrete Dynamics in Nature and Society
where xt
∈ Rn and zt ∈ Rs are the state and the output of the filter, respectively. The
constant matrices Af , Aτf , Bf , and Cf are filter parameters to be determined.
Letting
T
et : xT t xT t ,
zt : zt − zt,
2.5
one obtains the filter error system
τ et − τ Bwt,
Eėt Aet
A
zt Get,
T
et φT t ψ T t ,
2.6
t ∈ −τ, 0,
where
E 0
A
0
,
,
A
0 E
Bf C Af
B
Aτ 0
G −Cf .
τ ,
B ,
G
A
0 Aτf
0
E 2.7
Thus, the filtering problem to be addressed is stated as follows.
H∞ Filtering Problem
For a given γ > 0, design a full-order filter with delay of the form of 2.4 such that the filter
error system 2.6 has prescribed H∞ performance γ, that is,
1 system 2.6 is admissible;
2 under zero initial condition, for any nonzero wt ∈ L2 0, ∞, the H∞ performance
zt2 ≤ γwt2 is guaranteed.
Remark 2.3. Similar to 17, it is easy to see that system 2.3 is admissible if the error system
2.6 is admissible. That is why we made Assumption 2.1 on system 2.3.
3. Main Results
At first, we will concentrate our attention on H∞ performance analysis for the error system
2.6. The following lemma is useful in the proof of Theorem 3.2.
Lemma 3.1 see 32. Given a scalar γ > 0, the filter error system 2.6 has a prescribed H∞
> 0, Z
> 0, P , Y , and W
satisfying
performance γ if there exist matrices Q
Discrete Dynamics in Nature and Society
5
ET P T P E ≥ 0,
⎡
Φ1
⎢
⎢∗
⎢
⎢
⎢
⎢∗
Φ⎢
⎢
⎢∗
⎢
⎢
⎢∗
⎣
Φ2 τ Y T
T Z
G
T
P B τ A
∗
∗
∗
∗
⎤
⎥
0⎥
⎥
⎥
⎥
−τ Z
0
0
0⎥
⎥ < 0,
0⎥
⎥
∗
−γ 2 I τ BT Z
⎥
⎥
∗
∗
−τ Z 0 ⎥
⎦
∗
∗
∗
−I
T
Φ3 τ W
∗
3.1
Tτ Z
τA
0
3.2
where
A
T P T Q
− Y T E − ET Y ,
Φ1 P A
τ Y T E − ET W,
Φ2 P A
W
T E ET W.
Φ3 −Q
3.3
Based on Lemma 3.1, we will present a new delay-dependent bounded real lemma
BRL for the performance analysis of system 2.6, which can be shown to be more “efficient”
than Lemma 3.1.
Theorem 3.2. Given a scalar γ > 0, the filter error system 2.6 has a prescribed H∞ performance γ
> 0, Z
> 0 and P satisfying 3.1 and
if there exist matrices Q
⎡
Ω1
⎢
⎢∗
⎢
⎢
⎢
Ω⎢∗
⎢
⎢
⎢∗
⎣
∗
T Z
G
T
Ω2 P B τ A
Ω3
∗
0
Tτ Z
τA
−γ 2 I τ BT Z
∗
∗
−τ Z
∗
∗
∗
⎤
⎥
0⎥
⎥
⎥
⎥
0 ⎥ < 0,
⎥
⎥
0⎥
⎦
3.4
−I
where
A
T P T Q
− 1 ET Z
E,
Ω1 P A
τ
τ 1 ET Z
E,
Ω2 P A
τ
− 1 ET Z
E.
Ω 3 −Q
τ
3.5
> 0, Z
>
Proof. From Lemma 3.1, if we can prove that the feasibility of Ω < 0 for solution Q
0, P is equivalent to that of Φ < 0 for solution Q > 0, Z > 0, P , Y , W, then Theorem 3.2 is
proved.
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Discrete Dynamics in Nature and Society
Similar to Lemma 4 of 34, take
⎡
Ω1
⎢
⎢∗
⎢
⎢
⎢
⎢∗
T
Ψ ΠΦΠ ⎢
⎢
⎢∗
⎢
⎢
⎢∗
⎣
Ω2 τ Y T − ET Z
T ET Z
Ω3 τ W
∗
T Z
G
T
P B τ A
0
Tτ Z
τA
0
0
∗
−τ Z
∗
∗
∗
∗
∗
−τ Z
∗
∗
∗
∗
−γ 2 I τ BT Z
⎤
⎥
0⎥
⎥
⎥
⎥
0⎥
⎥,
⎥
0⎥
⎥
⎥
0⎥
⎦
3.6
−I
with
⎡
⎢I
⎢
⎢
⎢0
⎢
⎢
⎢
Π ⎢0
⎢
⎢0
⎢
⎢
⎢0
⎣
0
⎤
1 T
E 0 0 0⎥
τ
⎥
⎥
1
I − ET 0 0 0⎥
⎥
τ
⎥
0
I
0 0 0⎥
⎥.
⎥
0
0
I 0 0⎥
⎥
⎥
0
0
0 I 0⎥
⎦
0
0
0 0 I
0
3.7
It follows from Schur complement that
> 0,
Φ < 0 ⇐⇒ Ψ < 0 ⇐⇒ Z
⎡ T T ⎤
⎡ T T ⎤T
τY − E Z
τY − E Z
⎢
⎥
⎥
⎢
T
T ⎥
⎢τ W
⎢ T ET Z
⎥
⎢ E Z
⎥ ⎢τ W
⎥
−1 ⎢
⎢
⎥
⎥
⎥ τZ
⎥ < 0.
⎢
Ω⎢
0
0
⎢
⎥
⎥
⎢
⎢
⎥
⎥
⎢
⎢
⎥
⎥
⎢
0
0
⎣
⎦
⎦
⎣
0
3.8
0
> 0, Z
> 0, P , Y , and W
satisfying Φ < 0, from 3.8 it is easy to see
If there exist Q
> 0, Z
>0
that the above Q, Z, P is a feasible solution of Ω < 0. Conversely, if there exist Q
E and W
−1/τZ
E,
Φ < 0 is also
and P such that Ω < 0 holds, via taking Y 1/τZ
Z,
P , Y , W.
This completes the proof.
feasible for the above Q,
The following corollary is easy to be obtained from Theorem 3.2.
> 0, Z
> 0 and P
Corollary 3.3. The filter error system 2.6 is admissible if there exist matrices Q
satisfying 3.1 and
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⎡
⎤
− 1 ET Z
A
A
T P T Q
E P A
τ 1 ET Z
E τ A
T Z
P
⎢
⎥
τ
τ
⎢
⎥
⎢
1
T T ⎥
⎢
∗
−Q − E ZE τ Aτ Z ⎥ < 0.
⎣
⎦
τ
∗
∗
−τ Z
7
3.9
Remark 3.4. Theorem 3.2 can also be proved by employing the relationship of two integral
inequalities concluded in 35. In fact, we can see that Lemma 3.1 is obtained by using
the integral inequality 7 in 35, while using the integral inequality 9 in 35 yields
Theorem 3.2. As shown by 35, the upper bound provided by 9 in 35 is the least
upper bound provided by 7 in 35; therefore introducing more free matrices cannot
reduce the conservativeness. Then, Theorem 3.2 can be obtained from Lemma 3.1, and the
in Lemma 3.1 are redundant variables. Hence, from the
introduced slack variables Y and W
computational point of view, Theorem 3.2 is more “efficient” than Lemma 3.1.
In the sequel, based on Theorem 3.2, we are devoted to the design of the filter parameters Af , Aτf , Bf , and Cf . Noticing that 3.4 is nonlinear about the unknown variables
A
τ , P , and Z,
to reduce the number of the unknown variables, we can do as follows.
A,
From 3.4 we know that
⎡
⎤
− 1 ET Z
E P A
τ 1 ET Z
E
A
A
T P T Q
P
⎢
⎥
τ
τ
< 0.
⎣
1 T ⎦
∗
−Q − E ZE
τ
Multiplying 3.10 by I
I from the left and by I
3.10
IT from the right results in
T
A
τ P T < 0,
A
τ A
P A
3.11
which implies that P is nonsingular. Let
P P P2
P3 P4
,
P ∈ Rn×n , Pi ∈ Rn×n , i 2, 3, 4.
3.12
Without loss of generality, we can assume that P, Pi , i 2, 3, 4, are all nonsingular 36. Then,
from 3.1, we have that
ET P T P E,
ET P3T P2 E,
ET P4T P4 E.
3.13
Taking
T1 I
0
0 P P3−1
,
T2 I
0
0 P2−1 P
,
T3 diag T1 , T1 , I, T2T , I
3.14
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Discrete Dynamics in Nature and Society
and combining with 2.7 and 3.12, we obtain
E
T
T2−1 ET
1
E
0
E
0
E 0
,
0 E
0 P −1 ET P3T P3−T P T
0 P −1 P2 EP3−T P T
P
P
,
P T1 P T2 P P P3−1 P4 P2−1 P
A
0
A
0
−1 T
,
A T2 AT1 P −1 P2 Bf C P −1 P2 Af P3−T P T
Bf C Af
Aτ 0
0
Aτ
−1 T
,
Aτ T 2 A τ T 1 0 P −1 P2 Aτf P3−T P T
0 Aτf
B
−1 B T2 B ,
0
T G −Cf P −T P T G −Cf ,
G GT
1
3
3.15
where
Af P −1 P2 Af P3−T P T ,
Aτf P −1 P2 Aτf P3−T P T ,
Bf P −1 P2 Bf ,
Cf Cf P3−T P T ,
3.16
and denote
T,
Q T1 QT
1
2.
Z T2T ZT
3.17
Premultiplying by T1 and postmultiplying by T1T on both sides of 3.1, we have that
T ≥ 0,
T1 ET T2−T T2T P T T1T T1 P T2 T2−1 ET
1
3.18
that is,
T
T
E P P E ≥ 0.
3.19
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9
Multiplying 3.4 by T3 from the left and by T3T from the right yields
⎡
T T
⎢P A A P
⎢
⎢
⎢
⎢
⎢
Ω⎢
⎢
⎢
⎢
⎢
⎣
⎤
T
T
1 T
1 T
Q − E Z E P Aτ E Z E P B τA Z G ⎥
τ
τ
⎥
⎥
T
1 T
∗
−Q − E Z E
0 τAτ Z 0 ⎥
⎥
τ
⎥
T
⎥ < 0.
2
∗
∗
−γ I τB Z 0 ⎥
⎥
⎥
∗
∗
∗
−τZ 0 ⎥
⎦
∗
∗
∗
∗
−I
3.20
A,
A
τ , B,
G
and E, A, Aτ , B, G are algebraically
It can be seen that the systems E,
equivalent under the r.s.e. restricted system equivalence transformation, where T2−1
and T1T are taken as the row full rank transformation matrix and the coordinate full
rank transformation matrix, respectively, and comparing the coefficient matrices of the
two systems, we can see that the difference between them is just the filter parameters
Af , Aτf , Bf , Cf , and Af , Aτf , B f , Cf . Moreover, in the r.s.e. transformation, the state and
the equation of the filter change while the state and the equation of system 2.3 do not
change. So, in the design of the filter, we can directly substitute Af , Aτf , Bf , Cf for Af ,
Aτf , Bf , Cf . Noticing that
I
0
−P3 P −1 I
P P2
P3 P4
I −P −1 P2
0
I
P
0
0 P4 − P3 P −1 P2
,
3.21
then P4 − P3 P −1 P2 is nonsingular. Let
P P3−1 P4 P2−1 P − P P P3−1 P4 − P3 P −1 P2 P2−1 P S−1 ;
3.22
then P can be written as
P
P
P
P P S−1
3.23
.
Denote
−1
J SP ,
T
T4 T
J −S
I
0
,
−1
T5 T4 P Z .
3.24
Since P S P −1 P S−1 S P P3−1 P4 P2−1 P S is nonsingular, J is also a nonsingular matrix.
From 3.19, we have that,
T
T
T4 E P T4T T4 P ET4T ≥ 0.
3.25
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Noticing 3.23 and 3.24, we derive
T4 P T4 P
ET4T
EJ
E
I 0
P P
,
EJ
E
EJ
E
3.26
,
P EP −T P E
ET P E
P EJ − P EST P E
T T
E
J E
T T T
;
T4 E P T4 ET ET P T
then 3.25 is just
EJ J E ,
T
T
PE E P ,
T
T
EJ
E
ET P E
≥ 0.
3.27
Premultiplying by diag{T4 , T4 , I, T5 , I} and postmultiplying by diag{T4T , T4T , I, T5T , I} on
both sides of 3.20, we have
⎡
T
⎤
T
T4 A P T4T T4 P AT4T
T
T
T⎥
1
T4 E Z ET4T T4 P B τT4 A ZT5T T4 G ⎥
⎥
τ
⎥
⎥
⎥
T
T
1
T
T
T
−T4 QT4 − T4 E Z ET4
0
τT4 Aτ ZT5
0 ⎥
⎥ < 0. 3.28
τ
⎥
T
⎥
T
2
0 ⎥
∗
−γ I τB ZT5
⎥
⎥
T
∗
∗
−τT5 ZT5
0 ⎥
⎦
⎢
⎢
T
1
⎢
⎢ T4 QT4T − T4 E Z ET4T
⎢
τ
⎢
⎢
∗
⎢
⎢
⎢
⎢
∗
⎢
⎢
⎢
∗
⎣
∗
T4 P Aτ T4T ∗
∗
∗
−I
Noticing that
T4 P
AT4T
AJ
A
,
P AJ P Bf CJ − P Af ST P A P Bf C
Aτ
Aτ J
T
,
T4 P Aτ T4 P Aτ J − P Aτf ST P Aτ
⎡
⎤
T
B
J T GT SCf
T
⎦,
T4 P B ,
T4 G ⎣
PB
GT
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T
T
−1
T
T
T
T
T
−1
T
T
T
T
T
−1
T
T4 A ZT5T T4 A Z Z P T4T T4 A P T4T ,
T4 Aτ ZT5T T4 Aτ Z Z P T4T T4 Aτ P T4T ,
T
−1
T
T
B ZT5T B Z Z P T4T B P T4T ,
−1
−1
T
T5 ZT5T T4 P Z Z Z P T4T T4 P Z P T4T ,
3.29
denote
Q
T4 QT4T
Q1 Q2
3.30
,
Q2T Q3
Z1 Z2
−1 T T
,
Z T4 P Z P T 4 Z2T Z3
L P AJ P Bf CJ − P Af ST ,
3.31
Lτ P Aτ J − P Aτf ST ,
WB P B f ,
3.32
3.33
W C C f ST .
T
Since 3.31 implies that Z P T4T Z−1 T4 P , then
T4 E
T
Z ET4T
T
T4 E P
Introduce matrix W W
1
W2T
T
W2
W3
T4T Z−1 T4 P
τW ≤
ET4T
EJ
E
ET P E
Z
−1
EJ
E
ET P E
.
3.34
≥ 0 satisfying
EJ
E
ET P E
Z
−1
EJ
E
ET P E
,
3.35
then
⎡ 1
⎤
T
T
1
T
T
T
T
−
E
Z
ET
E
Z
ET
4
4
−W W
4
4
⎢ τ
⎥
τ
;
⎣
⎦≤
T
1
∗ −W
∗
− T4 E Z ET4T
τ
3.36
Obviously, if there exist matrices Q1 > 0, Q3 > 0, W1 ≥ 0, W3 ≥ 0, Z1 > 0, Z3 >
0, P, J, WB , WC , L, Lτ , Q2 , W2 , and Z2 with P, J being nonsingular, satisfying 3.35 and
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Discrete Dynamics in Nature and Society
⎡
Ξ11 Ξ12 Aτ J W1
⎢
⎢ ∗ Ξ22 Lτ W2T
⎢
⎢
⎢∗
∗ −Q1 − W1
⎢
⎢
⎢∗
∗
∗
⎢
⎢
⎢∗
∗
∗
⎢
⎢
⎢∗
∗
∗
⎢
⎢
⎢∗
∗
∗
⎣
∗
∗
∗
Aτ W 2
B
τJ T AT
τLT
J T GT WcT
τAT
τAT P T τCT WBT
GT
0
P Aτ W 3 P B
−Q2 − W2
0
τJ T ATτ
τLTτ
−Q3 − W3
0
τATτ
τATτ P T
0
∗
−γ I
τB
T
0
∗
∗
−τZ1
−τZ2
0
∗
∗
∗
−τZ3
0
∗
∗
∗
∗
−I
2
T
τB P
T
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥<0
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
3.37
with
Ξ11 AJ J T AT Q1 − W1 ,
Ξ12 A LT Q2 − W2 ,
Ξ22 P A AT P T WB C CT WBT Q3 − W3 ,
3.38
then taking
S J T − P −1 ,
B f P −1 WB ,
Af P −1 P AJ WB CJ − LS−T ,
Cf WC S−T ,
Aτf P −1 P Aτ J − Lτ S−T ,
3.39
one obtains that there are solutions Q > 0, Z > 0, and P to 3.20.
Hence we get the following theorem for the design of the filter 2.4.
Theorem 3.5. Given a scalar γ > 0, if there are matrices Q1 > 0, Q3 > 0, W1 ≥ 0, W3 ≥ 0, Z1 >
0, Z3 > 0, P, J, WB , WC , L, Lτ , Q2 , W2 , Z2 with P, J being nonsingular, satisfying 3.27,
3.35, and 3.37, then the H∞ filter of the form of 2.4 exists and the parameters are given by
3.39.
Remark 3.6. It is worth noting that 3.35 is not an LMI. In order to use the LMI Toolbox in
MATLAB to get the solutions,
we can do as follows.
Ip 0
Assume that E 0 0 ; otherwise, we can find nonsingular matrices M and N such
that MEN I0p 00 . It is worth noting that the feasibility of 3.27, 3.35, and 3.37 is not
affected by the selection of M and N. Then, the matrices P, J satisfying 3.27 are of the forms
P11 P12
P
,
0 P22
J
0
J11
J21 J22
,
P11 ∈ Rp×p , J11 ∈ Rp×p
3.40
with
J11
I
I
P11
≥ 0.
3.41
Discrete Dynamics in Nature and Society
13
Introduce another variable U > 0; then 3.35 can be replaced by
τW ≤
EJ
ET
EJ E
U
,
PE
ET P E
E
3.42
UZ I.
3.43
Write U as
⎡
U11
⎢ T
⎢U
⎢ 12
U⎢ T
⎢U
⎣ 13
T
U14
U12 U13 U14
⎤
⎥
U22 U23 U24 ⎥
⎥
⎥ > 0,
T
U23 U33 U34 ⎥
⎦
T
T
U24
U34
U44
3.44
where
U11 ∈ Rp×p ,
U22 ∈ Rn−p×n−p ,
U33 ∈ Rp×p ,
U44 ∈ Rn−p×n−p .
3.45
Noticing that
⎡
⎤
Π11 0 Π13 0
⎥
⎢
⎢ ∗ 0 0 0⎥
EJ E
EJ E
⎢
⎥
U
⎢
⎥,
⎢ ∗ ∗ Π33 0⎥
ET P E
ET P E
⎦
⎣
∗ ∗ ∗ 0
3.46
where
T
Π11 J11 U11 J11 U13
J11 J11 U13 U33 ,
3.47
T
Π13 J11 U11 U13
J11 U13 P11 U33 P11 ,
T
Π33 U11 P11 U13
U13 P11 P11 U33 P11 ,
we can assume that
⎡
⎤
W11 0 W21 0
⎢
⎥
⎢ 0 0 0 0⎥
⎢
⎥
W ⎢ T
⎥ ≥ 0,
⎢W 0 W31 0⎥
21
⎣
⎦
0 0 0 0
W11 W21
T
W21
W31
> 0.
3.48
14
Discrete Dynamics in Nature and Society
Then 3.42 is just
τ
W11 W21
T
W21
W31
≤
J11
I
I
P11
U11 U13 J11
T
U13
U33
I
I
P11
3.49
.
Invoking Schur complement again, we have that 3.49 is equivalent to
⎡
U11 U13
J11
⎢
⎢ UT U
I
⎢
33
⎢ 13
−1
⎢ J11 I
W11
⎣
τ
T
I P11
W21
⎤
−1
I
⎥
⎥
⎥
≥ 0.
−1 ⎥
⎥
W21
⎦
W31
P11
3.50
Introducing
α
α1 α2
θ
> 0,
αT2 α3
θ1 θ2
θ2T θ3
3.51
> 0,
then 3.50 can be replaced by
J11
I
⎡
⎤
U11 U13 τα1 τα2
⎢
⎥
⎢ ∗ U33 ταT τα3 ⎥
2
⎢
⎥
⎢
⎥ ≥ 0,
⎢ ∗
⎥
∗
τθ
τθ
1
2⎦
⎣
∗
∗
∗ τθ3
I
α1 α2
W11 W21 θ1 θ2
I,
I.
T
P11 αT2 α3
W21
W31 θ2T θ3
3.52
3.53
Therefore, one can consider the H∞ filter design problem as the following cone complementary problems:
Minimize trUZ tr
J11
I
I
P11
α1 α2
αT2 α3
W11 W21 θ1 θ2
T
W21
W31
3.54
θ2T θ3
subject to LMIs 3.30, 3.31, 3.37, 3.40, 3.41, 3.44, 3.48, 3.51, 3.52, and
⎡
Q > 0,
Z > 0,
U I
I Z
≥ 0,
α1 α2
I
0
I
⎢ T
⎢α α3 0
⎢ 2
⎢
⎢ I 0 J11
⎣
I
0
⎤
⎥
I ⎥
⎥
⎥ ≥ 0,
I ⎥
⎦
P11
⎡
θ1 θ2
I
0
⎤
⎢ T
⎥
⎢θ θ3 0
I ⎥
⎢ 2
⎥
⎢
⎥ ≥ 0.
⎢ I 0 W11 W21 ⎥
⎣
⎦
T
0 I W21 W31
3.55
Discrete Dynamics in Nature and Society
15
Then the filer 2.4 can be solved by using the iterative algorithm as 37, in the interests of
economy, which is omitted here.
Remark 3.7. Since the filter 2.4 is designed with parameters 3.39 such that inequality 3.20
holds, we have
⎤
⎡
T T
T
1 T
1 T
P
A
A
P
Q
−
E
Z
E
P
A
E
Z
E
τA
Z
τ
⎥
⎢
τ
τ
⎥
⎢
T ⎥ < 0.
⎢
1 T
⎢
∗
−Q − E Z E τAτ Z ⎥
⎦
⎣
τ
∗
∗
−τZ
By 3.15, 3.16, 3.17, 3.23 and letting Pf : P S−1 , Q Q1 ∈ Rn×n and Z 1 ∈ Rn×n , we can conclude from 3.56 that
Q1 Q2
∗ Q3
, and Z 3.56
⎡
⎤
T T
T
1 T
1 T
⎢Pf Af Af Pf Q3 − τ E Z3 E Pf Aτf τ E Z 3 E τAf Z 3 ⎥
⎢
⎥
T
⎢
⎥
1
⎢
∗
−Q3 − ET Z 3 E τAτf Z3 ⎥ < 0.
⎣
⎦
τ
∗
∗
−τZ 3
Z1 Z2
∗ Z3
with
3.57
In addition, 3.19 implies that
ET PfT Pf E ≥ 0.
3.58
Invoking Corollary 3.3, it is obtained that the designed filter 2.4 is admissible, and then it is
proper and can be realized in practice.
4. Numerical Examples
Example 4.1. Consider the singular time-delay system given in 25 without uncertainties and
distributed delay and with
⎤
⎤
⎡
⎡
−2 0 0.5
1 0 0
⎥
⎥
⎢
⎢
⎥
⎥
A⎢
E⎢
⎣0.1 −0.9 0.2⎦,
⎣0 1 0⎦,
0 0.5 0.3
0 0 0
⎤
⎡
0.2 0 0
⎥
⎢
⎥
C 1 0 0 ,
B⎢
⎣ 0 0.2 0 ⎦,
0 0 0.2
⎤
⎡
0.2 0.1
0
⎥
⎢
Aτ ⎢
0
0.15⎥
⎦,
⎣0.2
0.1 −0.23 0.1
G 1 0.7 0.8 .
4.1
16
Discrete Dynamics in Nature and Society
1.5
1
0.5
0
−0.5
−1
−1.5
−2
−2.5
−3
0
5
10
15
Times (s)
x1
x2
x3
Figure 1: State responses xt of the original system.
1.5
1
0.5
0
−0.5
−1
−1.5
−2
−2.5
−3
0
5
10
Times (s)
xꉱ1
xꉱ2
xꉱ3
Figure 2: State responses xt
of the filter system.
15
Discrete Dynamics in Nature and Society
17
2
1.5
1
0.5
0
−0.5
−1
−1.5
0
5
10
15
Times (s)
z∼
Figure 3: Error estimation signal zt with the designed filter.
1
0.9
0.8
Singular values
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
−10
−8
−6
−4
−2
0
2
4
6
8
10
Frequency
Figure 4: Singular value curve of the filtering error system.
18
Discrete Dynamics in Nature and Society
By Theorem 3.2, for τ 2 and γ 1, after 10 iterations, the corresponding filter is obtained
with the following parameters:
⎤
⎡
⎡
−0.9429 −0.0102 0.3206
0.0983
⎥
⎢
⎢
⎥
⎢
⎢
Aτf ⎣0.1248
Af ⎣−0.1042 −0.8493 0.1904⎦,
0.3750 0.4870 0.3369
0.0749
⎤
⎡
0.6224
⎥
⎢
⎥
Bf ⎢
Cf −0.8379 −0.7382
⎣−0.2077⎦,
0.4488
0.0960 0.0046
⎤
⎥
−0.0071 0.1461⎥
⎦,
−0.2438 0.0966
4.2
−0.9940 .
With this filter, Figures 1, 2, and 3 show the state responses xt of the original system, the
state responses xt
of the filter system, and the error estimation signal zt zt − zt with
the initial condition φt 1 1 − 1.425T , ψt 1 1 − 2.6341T for t ∈ −2, 0 and
the exogenous disturbance input wtdiag{e−0.5t , e−0.5t , e−0.5t }. By connecting the filter to the
original system, the singular value curve of the resulting filtering error system is also plotted
in Figure 4. We can see that all the maximum singular values are less than 1, which illustrate
the effectiveness of the proposed method in this paper.
5. Conclusions and Future Works
In this paper, we have studied the H∞ filtering problem for singular system with a constant
discrete delay. Based on an improved BRL, a delay-dependent sufficient condition for
the existence of the H∞ filter with delay is obtained. Then, by using LMIs and the
cone complementarity linearization iterative algorithm, the H∞ filter is designed, which
guarantees that the resulting error system is regular, impulse-free, internally stable, and the
L2 -induced norm from the disturbance input to the filtering error output satisfies a prescribed
H∞ performance level. It can be seen that the designed filter in this paper is a full-order filter,
that is, the finite mode of the filter is equal to rank E. To study the delay-dependent reducedorder H∞ filtering problem for singular time-delay systems is the key research in the future.
Acknowledgments
The authors would like to thank the Editor, the Associate Editor, and the anonymous
reviewers very much for the valuable comments and good suggestions. This work was
supported by National Natural Science Foundation of P. R. China 61004011.
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