Research Article Delay-Dependent H Filtering for Singular Time-Delay Systems

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1 Discrete Dynamics in Nature and Society Volume 211, Article ID 76878, 2 pages doi:1.1155/211/76878 Research Article Delay-Dependent H Filtering for Singular Time-Delay Systems Zhenbo Li 1, 2 and Shuqian Zhu 3 1 School of Statistics and Mathematics, Shandong Economic University, Jinan 2514, China 2 Shandong Provincial Key Laboratory of Digital Media Technology, Shandong Economic University, Jinan 2514, China 3 School of Mathematics, Shandong University, Jinan 251, China Correspondence should be addressed to Shuqian Zhu, sqzhu@sdu.edu.cn Received 14 February 211; Accepted 29 April 211 Academic Editor: Xue He Copyright q 211 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 L 2 -induced norm for continuous systems or l 2 -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,

2 2 Discrete Dynamics in Nature and Society 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 1, 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 and delay-dependent ones ; 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 and delay-dependent 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 3, 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 R n denotes the n-dimensional Euclidean space and R n m denotes the set of all n m real matrices, I n is the n-dimensional identity matrix, and diag{ } is a block-diagonal matrix. For real symmetric matrix X, the notation X X > 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. L 2, refers to the space of square-integrable vector functions over, with norm f 2 : f t 2 dt 1/2. 2. Problem Statement Consider the following singular time-delay system: Eẋ t Ax t A τ x t τ Bw t, y t Cx t C τ x t τ B 1 w t,

3 Discrete Dynamics in Nature and Society 3 z t Gx t G τ x t τ B 2 w t, x t φ t, t τ,, 2.1 where x t R n is the state, w t R r is the external disturbance signal that belongs to L 2,,y t R m is the measurement output, and z t R s is the signal to be estimated. E, A, A τ, B, C, C τ,b 1, G, G τ,andb 2 are known real constant matrices with appropriate dimensions and < rank E p<n. τ> is the known delay constant and φ t C n,τ is a compatible vector-valued initial function. Without loss of generality, we assume that C τ, B 1, G τ, and B 2. Otherwise, system 2.1 can be equivalently changed into [ẋ t ] E A x t ζ t I m s ζ t A τ B x t τ C τ ζ t τ B 1 w t, m s B 2 G τ C y t x t z t I m s. ζ t G 2.2 Then in the sequel, we discuss the system model as follows: Eẋ t Ax t A τ x t τ Bw t, y t Cx t, z t Gx t, x t φ t, t τ,. 2.3 Throughout this paper, we need the following assumption for system 2.3. Assumption 2.1. System 2.3 is admissible, that is, when w t, 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 z t, we consider the following linear filter with delay: E x t A f x t A τf x t τ B f y t, ẑ t C f x t, x t ψ t, t τ,, 2.4

4 4 Discrete Dynamics in Nature and Society where x t R n and ẑ t R s are the state and the output of the filter, respectively. The constant matrices A f,a τf,b f,andc f are filter parameters to be determined. Letting e t : [ x T t x T t ] T, z t : z t ẑ t, 2.5 one obtains the filter error system Ẽė t Ãe t à τ e t τ Bw t, z t Ge t, 2.6 e t [ φ T t ψ T t ] T, t τ,, where E A Ẽ, Ã, E B f C A f [ 2.7 Aτ B à τ, B ], G G C f. A τf Thus, the filtering problem to be addressed is stated as follows. H Filtering Problem For a given γ>, 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, thatis, 1 system 2.6 is admissible; 2 under zero initial condition, for any nonzero w t L 2,, theh performance z t 2 γ w t 2 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 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 >, the filter error system 2.6 has a prescribed H performance γ if thereexist matrices Q >, Z >, P, Ỹ, and W satisfying

5 Discrete Dynamics in Nature and Society 5 Ẽ T P T PẼ, 3.1 Φ 1 Φ 2 τỹ T P B τã T Z G T Φ 3 τ W T τã T Z τ τ Z Φ γ 2 I τ B T <, 3.2 Z τ Z I where Φ 1 PÃ Ã T P T Q Ỹ T Ẽ Ẽ T Ỹ, Φ 3 Q W T Ẽ Ẽ T W. Φ 2 PÃ τ Ỹ T Ẽ Ẽ T W, 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>, the filter error system 2.6 has a prescribed H performance γ if there exist matrices Q >, Z > and P satisfying 3.1 and Ω 1 Ω 2 P B τã T Z G T Ω 3 τã T Z τ Ω γ 2 I τ B T Z <, 3.4 τ Z I where Ω 1 PÃ Ã T P T Q 1 τ ẼT ZẼ, Ω 2 PÃ τ 1 τ ẼT ZẼ, Ω 3 Q 1 τ ẼT ZẼ. 3.5 Proof. From Lemma 3.1, if we can prove that the feasibility of Ω < for solution Q >, Z >, P is equivalent to that of Φ < for solution Q >, Z >, P,Ỹ, W, then Theorem 3.2 is proved.

6 6 Discrete Dynamics in Nature and Society Similar to Lemma 4 of 34, take Ω 1 Ω 2 τỹ T Ẽ T Z P B τã T Z G T Ω 3 τ W T Ẽ T Z τã T Z τ τ Z Ψ ΠΦΠ T γ 2 I τ B T, 3.6 Z τ Z I with 1 I τ ẼT I 1 τ ẼT Π I. 3.7 I I I It follows from Schur complement that Φ < Ψ < Z >, τỹ T Ẽ T Z τ W T Ẽ T Z ( Ω τ Z ) 1 τỹ T Ẽ T Z τ W T Ẽ T Z T <. 3.8 If there exist Q >, Z >, P, Ỹ, and W satisfying Φ <, from 3.8 it is easy to see that the above Q, Z, P is a feasible solution of Ω <. Conversely, if there exist Q >, Z > and P such that Ω < holds, via taking Ỹ 1/τ ZẼ and W 1/τ ZẼ, Φ < isalso feasible for the above Q, Z, P,Ỹ, W. This completes the proof. The following corollary is easy to be obtained from Theorem 3.2. Corollary 3.3. The filter error system 2.6 is admissible if there exist matrices Q >, Z > and P satisfying 3.1 and

7 Discrete Dynamics in Nature and Society 7 PÃ Ã T P T Q 1 τ ẼT ZẼ PÃ τ 1 τ ẼT ZẼ τã T Z Q 1 τ ẼT ZẼ τã T Z τ <. τ Z 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 introduced slack variables Ỹ and W in Lemma 3.1 are redundant variables. Hence, from the 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 A f, A τf,b f,andc f. Noticing that 3.4 is nonlinear about the unknown variables Ã, Ã τ, P,and Z, to reduce the number of the unknown variables, we can do as follows. From 3.4 we know that PÃ Ã T P T Q 1 τ ẼT ZẼ PÃ τ 1 τ ẼT ZẼ Q 1 <. τ ẼT ZẼ 3.1 Multiplying 3.1 by I I from the left and by I I T from the right results in ) ) T P (Ã Ãτ (Ã Ãτ P T <, 3.11 which implies that P is nonsingular. Let P P2 P, P R n n, P i R n n,i 2, 3, P 3 P 4 Without loss of generality, we can assume that P, P i,i 2, 3, 4, are all nonsingular 36. Then, from 3.1, we have that E T P T PE, E T P T 3 P 2E, E T P T 4 P 4E Taking I I } T 1, T PP P 1 2 P, T 3 diag {T 1,T 1,I,T T 2,I 3.14

8 8 Discrete Dynamics in Nature and Society and combining with 2.7 and 3.12,weobtain E E E E T 1 2 ẼT T 1 P 1 P 2 EP T 3 P T P 1 E T P T 3 P T 3 P, T E P P P T 1 PT 2 P PP 1 3 P 4P 1 2 P, [ ] A T 1 2 ÃT T 1 A A P 1 P 2 B f C P 1 P 2 A f P T 3 P, T B f C A f Aτ A τ T 1 2 Ã τ T T 1 Aτ P 1 P 2 A τf P T 3 P, T A τf B B T 1 2 B, G GT T 1 [ G C f P T 3 P T] ] [G C f, 3.15 where A f P 1 P 2 A f P T 3 P T, A τf P 1 P 2 A τf P T 3 P T, B f P 1 P 2 B f, C f C f P T 3 P T, 3.16 and denote Q T 1 QT T 1, Z T T 2 ZT Premultiplying by T 1 and postmultiplying by T T 1 on both sides of 3.1, we have that T 1 Ẽ T T T 2 T T P T 2 T T 1 T 1 PT 2 T 1 2 ẼT T 1, 3.18 that is, E T P T P E. 3.19

9 Discrete Dynamics in Nature and Society 9 Multiplying 3.4 by T 3 from the left and by T T 3 from the right yields P A A T P T Q 1 τ ET Z E P A τ 1 τ ET Z E P B τa T Z G T Q 1 τ ET Z E τa T τ Z Ω γ 2 I τb T <. 3.2 Z τz I It can be seen that the systems Ẽ, Ã, Ã τ, B, G and E, A, A τ, B, G are algebraically equivalent under the r.s.e. restricted system equivalence transformation, where T 1 2 and T T 1 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 A f,a τf,b f,c f,anda f, A τf, B f, C f. 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 A f, A τf, B f, C f for A f, A τf,b f, C f. Noticing that [ I P P2 I P 1 ] P 2 P, 3.21 P 3 P 1 I P 3 P 4 I P 4 P 3 P 1 P 2 then P 4 P 3 P 1 P 2 is nonsingular. Let ) 3 P 4P 1 2 P P PP 1 3 (P 4 P 3 P 1 P 2 P 1 2 P S 1 ; 3.22 PP 1 then P can be written as P P P P P S 1 Denote J T S P 1, [ J T ] S T 4, T 5 T 4 P Z I Since P S P 1 P S 1 S PP 1 3 P 4P 1 2 PS is nonsingular, J is also a nonsingular matrix. From 3.19, we have that, T 4 E T P T T T 4 T 4P ET T

10 1 Discrete Dynamics in Nature and Society Noticing 3.23 and 3.24, we derive I T 4 P, P P [ ] T 4 P ET T 4 EJ E EJ E EJ E, PEJ PES T PE PEP T PE E T PE [ T 4 E T P T J T T T 4 E T ] E ; E T E T P T 3.26 then 3.25 is just EJ J T E T, PE E T P T, EJ E. PE E T 3.27 Premultiplying by diag{t 4,T 4,I,T 5,I} and postmultiplying by diag{t T 4,TT 4,I,TT 5,I} on both sides of 3.2, we have T 4 A T P T T T 4 T 4P AT T 4 T 4 QT T 4 1 T τ T 4E T 4 P A τ T T Z ET T 4 1 τ T 4E T Z ET T 4 T 4 P B τt 4 A T ZT T 5 T 4 G T 4 T 4 QT T 4 1 τ T 4E T Z ET T 4 τt 4 A T τ ZT T 5 < γ 2 I τb T ZT T 5 τt 5 ZT T 5 I Noticing that T 4 P AT T 4 [ AJ PAJ PB f CJ PA f S T ] A, PA PB f C [ ] T 4 P A τ T T 4 A τ J A τ, PA τ J PA τf S T PA τ B T 4 P B, T 4 G T JT G T SC T f, PB G T

11 Discrete Dynamics in Nature and Society 11 T 4 A T ZT T 5 T 4A T Z Z 1 P T T T 4 T 4A T P T T T 4, T 4 A T τ ZT T 5 T 4A T τ Z Z 1 P T T T 4 T 4A T τ P T T T 4, B T ZT T 5 BT Z Z 1 P T T T 4 BT P T T T 4, T 5 ZT T 5 T 4P Z 1 Z Z 1 P T T T 4 T 4P Z 1 P T T T 4, 3.29 denote Q1 Q T 4 QT T 4 Q 2, Q T 2 Q Z T 4 P Z 1 P T Z1 T T 4 Z 2, Z T 2 Z L PAJ PB f CJ PA f S T, L τ PA τ J PA τf S T, 3.32 W B PB f, W C C f S T Since 3.31 implies that Z P T T T 4 Z 1 T 4 P, then [ T 4 E T Z ET T 4 T 4E T P T EJ E EJ T T 4 Z 1 T 4 P ET T 4 Z 1 E T PE E T ] E PE Introduce matrix W [ W1 W 2 W T 2 W 3 ] satisfying [ EJ E EJ τw Z 1 PE E T E T ] E, 3.35 PE then 1 τ T 4E T Z ET T 1 4 τ T 4E T Z ET T 4 W W 1 ; 3.36 τ T 4E T Z ET T W 4 Obviously, if there exist matrices Q 1 >, Q 3 >,W 1, W 3,Z 1 >, Z 3 >,P, J, W B, W C,L,L τ,q 2,W 2,andZ 2 with P, J being nonsingular, satisfying 3.35 and

12 12 Discrete Dynamics in Nature and Society Ξ 11 Ξ 12 A τ J W 1 A τ W 2 B τj T A T τl T J T G T W T c Ξ 22 L τ W T 2 PA τ W 3 PB τa T τa T P T τc T W T B G T Q 1 W 1 Q 2 W 2 τj T A T τ τl T τ Q 3 W 3 τa T τ τa T τ P T < 3.37 γ 2 I τb T τb T P T τz 1 τz 2 τz 3 I with Ξ 11 AJ J T A T Q 1 W 1, Ξ 12 A L T Q 2 W 2, Ξ 22 PA A T P T W B C C T W T B Q 3 W 3, 3.38 then taking S J T P 1, B f P 1 W B, C f W C S T, A f P 1 PAJ W B CJ L S T, A τf P 1 PA τ J L τ S T, 3.39 one obtains that there are solutions Q>, Z>, and P to 3.2. Hence we get the following theorem for the design of the filter 2.4. Theorem 3.5. Given a scalar γ>, if there are matrices Q 1 >, Q 3 >, W 1, W 3, Z 1 >, Z 3 >, P,J,W B, W C,L,L τ,q 2,W 2,Z 2 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 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. Assume that E Ip ; otherwise, we can find nonsingular matrices M and N such that MEN Ip. 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 P 12 J11 P, J, P 11 R p p, J 11 R p p 3.4 P 22 J 21 J 22 with J11 I. I P

13 Discrete Dynamics in Nature and Society 13 Introduce another variable U>; then 3.35 can be replaced by EJ E EJ E τw U, 3.42 PE E T PE E T UZ I Write U as U 11 U 12 U 13 U 14 U T 12 U U 22 U 23 U 24 U T 13 UT 23 U >, U 34 U T 14 UT 24 UT 34 U 44 where U 11 R p p, U 22 R n p n p, U 33 R p p, U 44 R n p n p Noticing that Π 11 Π 13 EJ E EJ E U E T PE E T PE Π 33, 3.46 where Π 11 J 11 U 11 J 11 U T 13 J 11 J 11 U 13 U 33, Π 13 J 11 U 11 U T 13 J 11U 13 P 11 U 33 P 11, 3.47 Π 33 U 11 P 11 U T 13 U 13P 11 P 11 U 33 P 11, we can assume that W 11 W 21 W W T 21 W, 31 W11 W 21 W T 21 W >

14 14 Discrete Dynamics in Nature and Society Then 3.42 is just W11 W 21 J11 I U11 U 13 J11 I τ W T 21 W 31 I P 11 U T 13 U I P 11 Invoking Schur complement again, we have that 3.49 is equivalent to [ U11 U 13 J11 I U T 13 U 33 I P 11 1 ( [ J11 I W11 W 21 τ I P 11 W T 21 W 31 ] 1 ]) Introducing α1 α 2 θ1 θ 2 α >, θ >, 3.51 α T 2 α 3 θ T 2 θ 3 then 3.5 can be replaced by U 11 U 13 τα 1 τα 2 U 33 τα T 2 τα 3, τθ 1 τθ 2 τθ 3 J11 I α1 α 2 W11 W 21 θ1 θ 2 I, I P 11 α T 2 α 3 W T 21 W I. 31 θ T 2 θ Therefore, one can consider the H filter design problem as the following cone complementary problems: { ( )} J11 I α1 α 2 W11 W 21 θ1 θ 2 Minimize tr UZ tr I P 11 α T 2 α 3 W T 21 W 31 θ T 2 θ subject to LMIs 3.3, 3.31, 3.37, 3.4, 3.41, 3.44, 3.48, 3.51, 3.52, and Q>, Z >, [ U ] I I Z, α 1 α 2 I α T 2 α 3 I I J 11 I, θ 1 θ 2 I θ T 2 θ 3 I. I W 11 W 21 I I P 11 I W T 21 W

15 Discrete Dynamics in Nature and Society 15 Then the filer 2.4 can be solved by using the iterative algorithm as 37, intheinterestsof economy, which is omitted here. Remark 3.7. Since the filter 2.4 is designed with parameters 3.39 such that inequality 3.2 holds, we have P A A T P T Q 1 τ ET Z E P A τ 1 τ ET Z E τa T Z Q 1 τ ET Z E τa T τ Z <. τz 3.56 By 3.15, 3.16, 3.17, 3.23 and letting P f : P S 1, Q Q 1 R n n and Z 1 R n n, we can conclude from 3.56 that Q1 Q 2,and Z Z1 Z 2 with Q 3 Z 3 P f A f A T f P T f Q 3 1 τ ET Z 3 E P f A τf 1 τ ET Z 3 E τa T f Z 3 Q 3 1 τ ET Z 3 E τa T τfz 3 <. τz In addition, 3.19 implies that E T P T f P fe 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 E 1, A.1.9.2, A τ.2.15, B.2, C [ 1 ], G [ ]

16 16 Discrete Dynamics in Nature and Society Times (s) x 1 x 2 x 3 Figure 1: State responses x t of the original system Times (s) ꉱx 1 ꉱx 2 ꉱx 3 Figure 2: State responses x t of the filter system.

17 Discrete Dynamics in Nature and Society Times (s) z Figure 3: Error estimation signal z t with the designed filter Singular values Frequency Figure 4: Singular value curve of the filtering error system.

18 18 Discrete Dynamics in Nature and Society By Theorem 3.2, forτ 2andγ 1, after 1 iterations, the corresponding filter is obtained with the following parameters: A f , A τf , B f.277, C f [ ] With this filter, Figures 1, 2, and3 show the state responses x t of the original system, the state responses x t of the filter system, and the error estimation signal z t z t ẑ t with the initial condition φ t T, ψ t T for t 2, and the exogenous disturbance input w t diag{e.5t,e.5t,e.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 L 2 -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 References 1 B. D. O. Anderson and J. B. Moore, Optimal Filtering, Prentice Hall, New York, NY, USA, J. O Reilly, Observers for Linear Systems, vol. 17 of Mathematics in Science and Engineering, Academic Press, Orlando, Fla, USA, K. M. Nagpal and P. P. Khargonekar, Filtering and smoothing in an H setting, Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, vol. 36, no. 2, pp , 1991.

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