Filtering for Singular Time-Delay Systems

(1)

Discrete Dynamics in Nature and Society Volume 2011, Article ID 760878,20pages doi:10.1155/2011/760878

Research Article

Delay-Dependent H

∞

Filtering for Singular Time-Delay Systems

Zhenbo Li

^{1, 2}

and Shuqian Zhu

³

1School of Statistics and Mathematics, Shandong Economic University, Jinan 250014, China

2Shandong Provincial Key Laboratory of Digital Media Technology, Shandong Economic University, Jinan 250014, China

3School of Mathematics, Shandong University, Jinan 250100, China

Correspondence should be addressed to Shuqian Zhu,sqzhu@sdu.edu.cn Received 14 February 2011; Accepted 29 April 2011

Academic Editor: Xue He

Copyrightq2011 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 prescribedH∞performanceγis given in terms of linear matrix inequalitiesLMIs. Then, the suﬃcient condition is obtained for the existence of theH∞filter, and the explicit expression for the desiredH∞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 normfor continuous systemsorl2-induced normfor discrete systemsfrom 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 issue3–6. The filtering problem for singular systems has also been investigated by many researchers. For example, a necessary and suﬃcient condition is obtained in 7for 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)

in8, a linear normalH∞ filter is obtained for singular systems. Reduced-orderH∞filters are designed in 9 for both continuous and discrete singular systems. In10, a reduced- orderH∞ 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 inH∞filtering for time-delay systems. Existing results can be classified into two types: delay-independent ones11–14and delay-dependent ones15–23; 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 linessee, e.g.,24. Since singular time-delay systems are more general, it is of significance to consider theH∞filtering problem for them. Recently, some delay-dependent 25–27 and delay-dependent28–31results aboutH_∞ 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 in29for 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-dependentH_∞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 in32, we derive a new delay-dependent condition which guarantees that the filter error system has a prescribedH_∞performanceγ; and it can be seen that this new condition is more “eﬃcient” than that in32since no redundant variables are involved. Then, the suﬃcient condition for the existence of the full-orderH∞filter, which is an admissible singular time-delay system, is obtained and the explicit expression for the desiredH_∞filter is given by using LMIs and the cone complementarity linearization iterative algorithm.

Notations

Rⁿ denotes the n-dimensional Euclidean space and R^n×m denotes the set of alln×mreal matrices,In is then-dimensional identity matrix, and diag{· · · }is a block-diagonal matrix.

For real symmetric matrixX, the notationX≥0 X >0means that the matrixXis positive- semidefinitepositive-definite. The superscriptTrepresents the transpose; the symbol∗will be used in some matrix expressions to induce a symmetric structure.L20,∞refers to the space of square-integrable vector functions over0,∞with normf2: _∞

0 ft²dt^1/2.

2. Problem Statement

Consider the following singular time-delay system:

Ext ˙ Axt Aτxt−τ Bwt, yt Cxt Cτxt−τ B1wt,

(3)

zt Gxt Gτxt−τ B2wt, xt φt, t∈−τ,0,

2.1 where xt ∈ Rⁿ is the state,wt ∈ R^r is the external disturbance signal that belongs to L20,∞, yt∈ R^mis the measurement output, andzt ∈ R^s is the signal to be estimated.

E, A, Aτ, B, C, Cτ, B1, G, Gτ, andB2 are known real constant matrices with appropriate dimensions and 0 <rankE p < n.τ > 0 is the known delay constant andφt∈Cn,τ is a compatible vector-valued initial function.

Without loss of generality, we assume that Cτ 0, B1 0, Gτ 0, andB2 0.

Otherwise, system2.1can be equivalently changed into E 0

0 0 xt˙

ζt˙

A 0 0 −Im s

xt ζt

⎡

⎢⎢

⎢⎣

Aτ | 0

−− − −−

Cτ | 0_{m s} Gτ |

⎤

⎥⎥

⎥⎦

xt−τ ζt−τ

⎡

⎢⎢

⎢⎣ B

−−

B1

B2

⎤

⎥⎥

⎥⎦wt,

yt zt

⎡

⎢⎢

⎣ C |

I_{m s} G |

⎤

⎥⎥

⎦ xt

ζt

.

2.2

Then in the sequel, we discuss the system model as follows:

Ext ˙ Axt Aτxt−τ Bwt, yt Cxt,

zt Gxt, xt φt, t∈−τ,0.

2.3

Throughout this paper, we need the following assumption for system2.3.

Assumption 2.1. System2.3is admissible, that is, whenwt≡0, system2.3is 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 to33.

For the estimates ofzt, we consider the following linear filter with delay:

Ext ˙ Afxt Aτfxt −τ Bfyt,

zt Cfxt,

xt ψt, t∈−τ,0,

2.4

(4)

wherext ∈ Rⁿ and zt ∈ R^s are the state and the output of the filter, respectively. The constant matricesAf, Aτf, Bf, andCf are filter parameters to be determined.

Letting

et:

x^Tt x^TtT

, zt :zt−zt, 2.5

one obtains the filter error system

Eet ˙ Aet Aτet−τ Bwt, zt Get,

et

φ^Tt ψ^TtT

, t∈−τ,0,

2.6

where

E E 0

0 E

, A

A 0 BfC Af

,

Aτ

Aτ 0 0 Aτf

, B B

0

, G G −Cf

.

2.7

Thus, the filtering problem to be addressed is stated as follows.

H∞ Filtering Problem

For a givenγ >0, design a full-order filter with delay of the form of2.4such that the filter error system2.6has prescribedH∞performanceγ, that is,

1system2.6is admissible;

2under zero initial condition, for any nonzerowt∈L20,∞, theH∞performance zt2≤γwt2is guaranteed.

Remark 2.3. Similar to17, it is easy to see that system2.3is admissible if the error system 2.6is admissible. That is why we made Assumption2.1on system2.3.

3. Main Results

At first, we will concentrate our attention onH∞performance analysis for the error system 2.6. The following lemma is useful in the proof of Theorem3.2.

Lemma 3.1 see32. Given a scalar γ > 0, the filter error system2.6 has a prescribed H_∞ performanceγif there exist matricesQ > 0, Z > 0, P , Y, andWsatisfying

(5)

E^TP^T PE≥0, 3.1

Φ

⎡

⎢⎢

⎣

Φ1 Φ2 τY^T PB τ A^TZ G^T

∗ Φ3 τW^T 0 τA^T_τZ 0

∗ ∗ −τZ 0 0 0

∗ ∗ ∗ −γ²I τB^TZ 0

∗ ∗ ∗ ∗ −τZ 0

∗ ∗ ∗ ∗ ∗ −I

⎤

⎥⎥

⎦

<0, 3.2

where

Φ1 PA A^TP^T Q−Y^TE−E^TY , Φ2PAτ Y^TE−E^TW, Φ3−Q W^TE E^TW.

3.3

Based on Lemma3.1, we will present a new delay-dependent bounded real lemma BRLfor the performance analysis of system2.6, which can be shown to be more “eﬃcient”

than Lemma3.1.

Theorem 3.2. Given a scalarγ >0, the filter error system2.6has a prescribedH∞performanceγ if there exist matricesQ > 0, Z > 0 andPsatisfying3.1and

Ω

⎡

⎢⎢

⎢⎣

Ω1 Ω2 PB τ A^TZ G^T

∗ Ω3 0 τA^T_τZ 0

∗ ∗ −γ²I τB^TZ 0

∗ ∗ ∗ −τZ 0

∗ ∗ ∗ ∗ −I

⎤

⎥⎥

⎥⎦

<0, 3.4

where

Ω1PA A^TP^T Q−1

τE^TZE, Ω2PAτ

1

τE^TZE, Ω3−Q −1

τE^TZE. 3.5

Proof. From Lemma3.1, if we can prove that the feasibility ofΩ<0 for solutionQ > 0,Z >

0,P is equivalent to that ofΦ< 0 for solutionQ > 0,Z > 0,P , Y , W, then Theorem 3.2is proved.

(6)

Similar to Lemma 4 of34, take

Ψ ΠΦΠ^T

⎡

⎢⎢

⎣

Ω1 Ω2 τY^T−E^TZ PB τ A^TZ G^T

∗ Ω3 τW^T E^TZ 0 τA^T_τZ 0

∗ ∗ −τZ 0 0 0

∗ ∗ ∗ −γ²I τB^TZ 0

∗ ∗ ∗ ∗ −τZ 0

∗ ∗ ∗ ∗ ∗ −I

⎤

⎥⎥

⎦

, 3.6

with

Π

⎡

⎢⎢

⎢⎣ I 0 1

τE^T 0 0 0 0 I −1

τE^T 0 0 0

0 0 I 0 0 0

0 0 0 I 0 0

0 0 0 0 I 0

0 0 0 0 0 I

⎤

⎥⎥

⎥⎦

. 3.7

It follows from Schur complement that

Φ<0⇐⇒Ψ<0⇐⇒Z > 0, Ω

⎡

⎢⎢

⎣

τY^T−E^TZ τW^T E^TZ

0 0 0

⎤

⎥⎥

⎦

τZ₋₁

⎡

⎢⎢

⎣

τY^T−E^TZ τW^T E^TZ

0 0 0

⎤

⎥⎥

⎦

T

<0. 3.8

If there existQ > 0, Z > 0,P , Y, andWsatisfyingΦ< 0, from3.8it is easy to see that the aboveQ, Z, P is a feasible solution ofΩ<0. Conversely, if there existQ > 0, Z > 0 andP such thatΩ < 0 holds, via takingY 1/τZE andW −1/τZE, Φ < 0 is also feasible for the aboveQ, Z, P , Y , W. This completes the proof.

The following corollary is easy to be obtained from Theorem3.2.

Corollary 3.3. The filter error system2.6is admissible if there exist matricesQ > 0, Z > 0 andP satisfying3.1and

(7)

⎡

⎢⎢

⎣

PA A^TP^T Q− 1

τE^TZE PAτ

1

τE^TZE τ A^TZ

∗ −Q −1

τE^TZE τ A^T_τZ

∗ ∗ −τZ

⎤

⎥⎥

⎦<0. 3.9

Remark 3.4. Theorem3.2can 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, Theorem3.2can be obtained from Lemma3.1, and the introduced slack variablesY andWin Lemma3.1are redundant variables. Hence, from the computational point of view, Theorem3.2is more “eﬃcient” than Lemma3.1.

In the sequel, based on Theorem3.2, 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, Aτ, P, and Z, to reduce the number of the unknown variables, we can do as follows.

From3.4we know that

⎡

⎢⎣

PA A^TP^T Q− 1

τE^TZE PAτ 1 τE^TZE

∗ −Q−1

τE^TZE

⎤

⎥⎦<0. 3.10

Multiplying3.10byI Ifrom the left and byI I^Tfrom the right results in

P A Aτ

A Aτ

_T

P^T<0, 3.11

which implies thatPis nonsingular. Let

P P P2

P3 P4

, P ∈R^n×n, Pi∈R^n×n, i2,3,4. 3.12

Without loss of generality, we can assume thatP, Pi, i2,3,4, are all nonsingular36. Then, from3.1, we have that

E^TP^TP E, E^TP₃^T P2E, E^TP₄^TP4E. 3.13

Taking

T1 I 0

0 P P₃⁻¹

, T2 I 0

0 P₂⁻¹P

, T3diag

T1, T1, I, T₂^T, I

3.14

(8)

and combining with2.7and3.12, we obtain

ET₂⁻¹ET ₁^T

E 0 0 P⁻¹P2EP₃^−TP^T

E 0 0 P⁻¹E^TP₃^TP₃^−TP^T

E 0 0 E

,

P T1P T 2

P P P P P₃⁻¹P4P₂⁻¹P

,

AT₂⁻¹AT ₁^T

A 0

P⁻¹P2BfC P⁻¹P2AfP₃^−TP^T

A 0 BfC Af

,

Aτ T₂⁻¹AτT₁^T

Aτ 0 0 P⁻¹P2AτfP₃^−TP^T

Aτ 0 0 Aτf

,

BT₂⁻¹B B

0

,

GGT ₁^T

G −CfP₃^−TP^T

G −Cf

,

3.15

where

Af P⁻¹P2AfP₃^−TP^T, Aτf P⁻¹P2AτfP₃^−TP^T, Bf P⁻¹P2Bf, Cf CfP₃^−TP^T, 3.16

and denote

QT1QT ₁^T, ZT₂^TZT 2. 3.17

Premultiplying byT1and postmultiplying byT₁^T on both sides of3.1, we have that

T1E^TT₂^−TT₂^TP^TT₁^T T1P T 2T₂⁻¹ET ₁^T≥0, 3.18

that is,

E^TP^TP E≥0. 3.19

(9)

Multiplying3.4byT3from the left and byT₃^T from the right yields

Ω

⎡

⎢⎢

⎢⎣

P A A^TP^T Q−1

τE^TZ E P Aτ

1

τE^TZ E P B τA^TZ G^T

∗ −Q− 1

τE^TZ E 0 τA^T_τZ 0

∗ ∗ −γ²I τB^TZ 0

∗ ∗ ∗ −τZ 0

∗ ∗ ∗ ∗ −I

⎤

⎥⎥

⎥⎦

<0. 3.20

It can be seen that the systemsE, A, Aτ,B, G and E, A, Aτ, B, G are algebraically equivalent under the r.s.e. restricted system equivalence transformation, where T₂⁻¹ and T₁^T 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, andAf, Aτf, Bf, 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

−P3P⁻¹ I

P P2

P3 P4

I −P⁻¹P2

0 I

P 0 0 P4−P3P⁻¹P2

, 3.21

thenP4−P3P⁻¹P2is nonsingular. Let

P P₃⁻¹P4P₂⁻¹P−PP P₃⁻¹

P4−P3P⁻¹P2

P₂⁻¹PS⁻¹; 3.22

thenPcan be written as

P

P P P P S⁻¹

. 3.23

Denote

J^T S P⁻¹, T4

J^T −S I 0

, T5 T4P Z⁻¹. 3.24

SincePS P⁻¹ P S⁻¹SP P₃⁻¹P4P₂⁻¹P Sis nonsingular,Jis also a nonsingular matrix.

From3.19, we have that,

T4E^TP^TT₄^TT4P ET₄^T ≥0. 3.25

(10)

Noticing3.23and3.24, we derive

T4P I 0

P P

,

T4P ET₄^T

EJ E P EJ−P ES^T P E

EJ E P EP^−T P E

EJ E E^T P E

,

T4E^TP^TT₄^T

J^TE^T E E^T E^TP^T

;

3.26

then3.25is just

EJJ^TE^T, P EE^TP^T,

EJ E E^T P E

≥0. 3.27

Premultiplying by diag{T4, T4, I, T5, I}and postmultiplying by diag{T₄^T, T₄^T, I, T₅^T, I}on both sides of3.20, we have

⎡

⎢⎢

⎣

T4A^TP^TT₄^T T4P AT₄^T T4QT₄^T− 1

τT4E^TZ ET₄^T T4P AτT₄^T 1

τT4E^TZ ET₄^T T4P B τT4A^TZT₅^T T4G^T

∗ −T4QT₄^T−1

τT4E^TZ ET₄^T 0 τT4A^T_τZT₅^T 0

∗ ∗ −γ²I τB^TZT₅^T 0

∗ ∗ ∗ −τT5ZT₅^T 0

∗ ∗ ∗ ∗ −I

⎤

⎥⎥

⎦

<0. 3.28

Noticing that

T4P AT₄^T

AJ A

P AJ P BfCJ−P AfS^T P A P BfC

,

T4P AτT₄^T

AτJ Aτ

P AτJ−P AτfS^T P Aτ

,

T4P B B

P B

, T4G^T

⎡

⎣J^TG^T SC^T_f G^T

⎤

⎦,

(11)

T4A^TZT₅^T T4A^TZ Z⁻¹P^TT₄^T T4A^TP^TT₄^T, T4A^T_τZT₅^T T4A^T_τZ Z⁻¹P^TT₄^T T4A^T_τP^TT₄^T, B^TZT₅^TB^TZ Z⁻¹P^TT₄^T B^TP^TT₄^T, T5ZT₅^T T4P Z⁻¹Z Z⁻¹P^TT₄^T T4P Z⁻¹P^TT₄^T,

3.29 denote

QT4QT₄^T

Q1 Q2

Q^T₂ Q3

, 3.30

ZT4P Z⁻¹P^TT₄^T

Z1 Z2

Z^T₂ Z3

, 3.31

LP AJ P BfCJ−P AfS^T, Lτ P AτJ−P AτfS^T, 3.32

WBP Bf, WC CfS^T. 3.33

Since3.31implies thatZP^TT₄^TZ⁻¹T4P, then

T4E^TZ ET₄^T T4E^TP^TT₄^TZ⁻¹T4P ET₄^T

EJ E E^T P E

Z⁻¹

EJ E E^T P E

. 3.34

Introduce matrixW _W

1 W2

W₂^T W3

≥0 satisfying

τW ≤

EJ E E^T P E

Z⁻¹

EJ E E^T P E

, 3.35

then

⎡

⎢⎣−1

τT4E^TZ ET₄^T 1

τT4E^TZ ET₄^T

∗ −1

τT4E^TZ ET₄^T

⎤

⎥⎦≤

−W W

∗ −W

; 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, andZ2withP, Jbeing nonsingular, satisfying3.35and

(12)

⎡

⎢⎢

⎢⎣

Ξ11 Ξ12 AτJ W1 Aτ W2 B τJ^TA^T τL^T J^TG^T W_c^T

∗ Ξ22 Lτ W₂^T P Aτ W3 P B τA^T τA^TP^T τC^TW_B^T G^T

∗ ∗ −Q1−W1 −Q2−W2 0 τJ^TA^T_τ τL^T_τ 0

∗ ∗ ∗ −Q3−W3 0 τA^T_τ τA^T_τP^T 0

∗ ∗ ∗ ∗ −γ²I τB^T τB^TP^T 0

∗ ∗ ∗ ∗ ∗ −τZ1 −τZ2 0

∗ ∗ ∗ ∗ ∗ ∗ −τZ3 0

∗ ∗ ∗ ∗ ∗ ∗ ∗ −I

⎤

⎥⎥

⎥⎦

<0 3.37

with

Ξ11 AJ J^TA^T Q1−W1, Ξ12 A L^T Q2−W2,

Ξ22 P A A^TP^T WBC C^TW_B^T Q3−W3, 3.38

then taking

SJ^T−P⁻¹, Bf P⁻¹WB, Cf WCS^−T, Af P⁻¹P AJ WBCJ−LS^−T, Aτf P⁻¹P AτJ−LτS^−T,

3.39

one obtains that there are solutionsQ >0, Z >0, andPto3.20.

Hence we get the following theorem for the design of the filter2.4.

Theorem 3.5. Given a scalarγ >0, if there are matricesQ1 >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, and3.37, then theH∞ filter of the form of 2.4exists and the parameters are given by 3.39.

Remark 3.6. It is worth noting that3.35is not an LMI. In order to use the LMI Toolbox in MATLAB to get the solutions, we can do as follows.

Assume thatE

Ip 0 0 0

; otherwise, we can find nonsingular matricesMandNsuch

thatMEN

Ip0 0 0

. It is worth noting that the feasibility of3.27,3.35, and3.37is not aﬀected by the selection ofMandN. Then, the matricesP, Jsatisfying3.27are of the forms

P

P11 P12

0 P22

, J

J11 0 J21 J22

, P11 ∈R^p×p, J11∈R^p×p 3.40

with

J11 I I P11

≥0. 3.41

(13)

Introduce another variableU >0; then3.35can be replaced by

τW ≤

EJ E E^T P E

U

EJ E E^T P E

, 3.42

UZI. 3.43

WriteUas

U

⎡

⎢⎢

⎢⎣

U11 U12 U13 U14

U^T₁₂ U22 U23 U24

U^T₁₃ U^T₂₃ U33 U34

U^T₁₄ U^T₂₄ U^T₃₄ U44

⎤

⎥⎥

⎥⎦>0, 3.44

where

U11 ∈R^p×p, U22∈R^n−p×n−p, U33∈R^p×p, U44∈R^n−p×n−p. 3.45

Noticing that

EJ E E^T P E

U

EJ E E^T P E

⎡

⎢⎢

⎢⎣

Π11 0 Π13 0

∗ 0 0 0

∗ ∗ Π33 0

∗ ∗ ∗ 0

⎤

⎥⎥

⎥⎦, 3.46

where

Π11J11U11J11 U^T₁₃J11 J11U13 U33, Π13J11U11 U^T₁₃ J11U13P11 U33P11, Π33U11 P11U^T₁₃ U13P11 P11U33P11,

3.47

we can assume that

W

⎡

⎢⎢

⎢⎣

W11 0 W21 0

0 0 0 0

W₂₁^T 0 W31 0

0 0 0 0

⎤

⎥⎥

⎥⎦≥0,

W11 W21

W₂₁^T W31

>0. 3.48

(14)

Then3.42is just

τ

W11 W21

W₂₁^T W31

≤

J11 I I P11

U11 U13

U₁₃^T U33

J11 I I P11

. 3.49

Invoking Schur complement again, we have that3.49is equivalent to

⎡

⎢⎢

⎢⎣

U11 U13

U^T₁₃ U33

J11 I I P11

−1

J11 I I P11

−1 τ

W11 W21

W₂₁^T W31

−1

⎤

⎥⎥

⎥⎦≥0. 3.50

Introducing

α

α1 α2

α^T₂ α3

>0, θ

θ1 θ2

θ₂^T θ3

>0, 3.51

then3.50can be replaced by

⎡

⎢⎢

⎢⎣

U11 U13 τα1 τα2

∗ U33 τα^T₂ τα3

∗ ∗ τθ1 τθ2

∗ ∗ ∗ τθ3

⎤

⎥⎥

⎥⎦≥0, 3.52

J11 I I P11

α1 α2

α^T₂ α3

I,

W11 W21

W₂₁^T W31

θ1 θ2

θ^T₂ θ3

I. 3.53

Therefore, one can consider theH∞filter design problem as the following cone complemen- tary problems:

Minimize

trUZ tr

J11 I I P11

α1 α2

α^T₂ α3

W11 W21

W₂₁^T W31

θ1 θ2

θ^T₂ θ3

3.54 subject to LMIs3.30,3.31,3.37,3.40,3.41,3.44,3.48,3.51,3.52, and

Q >0, Z >0,

U I I Z

≥0,

⎡

⎢⎢

⎢⎣

α1 α2 I 0 α^T₂ α3 0 I I 0 J11 I 0 I I P11

⎤

⎥⎥

⎥⎦≥0,

⎡

⎢⎢

⎢⎣

θ1 θ2 I 0 θ^T₂ θ3 0 I I 0 W11 W21

0 I W₂₁^T W31

⎤

⎥⎥

⎥⎦≥0.

3.55

(15)

Then the filer2.4can be solved by using the iterative algorithm as37, in the interests of economy, which is omitted here.

Remark 3.7. Since the filter2.4is designed with parameters3.39such that inequality3.20 holds, we have

⎡

⎢⎢

⎣

P A A^TP^T Q− 1

τE^TZ E P Aτ

1

τE^TZ E τA^TZ

∗ −Q− 1

τE^TZ E τA^T_τZ

∗ ∗ −τZ

⎤

⎥⎥

⎦<0. 3.56

By3.15,3.16,3.17,3.23and lettingPf :P S⁻¹,Q _Q

1Q₂

∗ Q₃

, and Z

Z1Z2

∗ Z3

with

Q₁∈R^n×nandZ1 ∈R^n×n, we can conclude from3.56that

⎡

⎢⎢

⎣

PfAf A^T_fP_f^T Q₃−1

τE^TZ3E PfAτf

1

τE^TZ3E τA^T_fZ3

∗ −Q₃− 1

τE^TZ3E τA^T_τfZ3

∗ ∗ −τZ3

⎤

⎥⎥

⎦<0. 3.57

In addition,3.19implies that

E^TP_f^T PfE≥0. 3.58

Invoking Corollary3.3, it is obtained that the designed filter2.4is 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 in25without uncertainties and distributed delay and with

E

⎡

⎢⎢

⎣ 1 0 0 0 1 0 0 0 0

⎤

⎥⎥

⎦, A

⎡

⎢⎢

⎣

−2 0 0.5 0.1 −0.9 0.2 0 0.5 0.3

⎤

⎥⎥

⎦, Aτ

⎡

⎢⎢

⎣

0.2 0.1 0 0.2 0 0.15 0.1 −0.23 0.1

⎤

⎥⎥

⎦,

B

⎡

⎢⎢

⎣

0.2 0 0 0 0.2 0 0 0 0.2

⎤

⎥⎥

⎦, C 1 0 0

, G

1 0.7 0.8 .

4.1

(16)

−3

−2.5

−2

−1.5

−1

−0.5 0 0.5 1 1.5

0 5 10 15

Times(s) x1

x2

x3

Figure 1: State responsesxtof the original system.

ꉱ x1

ꉱ x2

ꉱ x3

−3

−2.5

−2

−1.5

−1

−0.5 0 0.5 1 1.5

0 5 10 15

Times(s)

Figure 2: State responsesxt of the filter system.

(17)

−1.5

−1

−0.5 0 0.5 1 1.5 2

∼ z

0 5 10 15

Times(s)

Figure 3: Error estimation signalzt with the designed filter.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−10 −8 −6 −4 −2 0 2 4 6 8 10

Frequency

Singularvalues

Figure 4: Singular value curve of the filtering error system.

(18)

By Theorem3.2, forτ 2 andγ 1, after 10 iterations, the corresponding filter is obtained with the following parameters:

Af

⎡

⎢⎢

⎣

−0.9429 −0.0102 0.3206

−0.1042 −0.8493 0.1904 0.3750 0.4870 0.3369

⎤

⎥⎥

⎦, Aτf

⎡

⎢⎢

⎣

0.0983 0.0960 0.0046 0.1248 −0.0071 0.1461 0.0749 −0.2438 0.0966

⎤

⎥⎥

⎦,

Bf

⎡

⎢⎢

⎣ 0.6224

−0.2077 0.4488

⎤

⎥⎥

⎦, Cf

−0.8379 −0.7382 −0.9940 .

4.2

With this filter, Figures1,2, and3show the state responsesxtof the original system, the state responsesxt of the filter system, and the error estimation signalzt zt−zt with the initial conditionφt 1 1 −1.425^T,ψt 1 1 −2.6341^T for t ∈ −2,0and the exogenous disturbance inputwtdiag{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 Figure4. 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 theH∞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 rankE. To study the delay-dependent reduced- orderH_∞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. China61004011.

References

1 B. D. O. Anderson and J. B. Moore, Optimal Filtering, Prentice Hall, New York, NY, USA, 1979.

2 J. O’Reilly, Observers for Linear Systems, vol. 170 of Mathematics in Science and Engineering, Academic Press, Orlando, Fla , USA, 1983.

3 K. M. Nagpal and P. P. Khargonekar, “Filtering and smoothing in anH∞setting,” Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, vol. 36, no. 2, pp. 152–166, 1991.

(19)

4 M. Ariola and A. Pironti, “Reduced-order solutions for the singularH∞filtering problem,” Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, vol. 48, no. 2, pp. 271–275, 2003.

5 H.-C. Choi, D. Chwa, and S.-K. Hong, “An LMI approach to robust reduced-orderH∞filter design for polytopic uncertain systems,” International Journal of Control, Automation and Systems, vol. 7, no. 3, pp. 487–494, 2009.

6 P. Park and T. Kailath, “H∞filtering via convex optimization,” International Journal of Control, vol. 66, no. 1, pp. 15–22, 1997.

7 S. Xu, J. Lam, and Y. Zou, “H∞filtering for singular systems,” Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, vol. 48, no. 12, pp. 2217–2222, 2003.

8 Y.-P. Chen, Z.-D. Zhou, C.-N. Zeng, and Q.-L. Zhang, “H∞ filtering for descriptor systems,”

International Journal of Control, Automation and Systems, vol. 4, no. 6, pp. 697–704, 2006.

9 S. Xu and J. Lam, “Reduced-orderH∞filtering for singular systems,” Systems & Control Letters, vol. 56, no. 1, pp. 48–57, 2007.

10 R. Lu, H. Su, J. Chu, S. Zhou, and M. Fu, “Reduced-orderH∞filtering for discrete-time singular systems with lossy measurements,” IET Control Theory & Applications, vol. 4, no. 1, pp. 151–163, 2010.

11 A. W. Pila, U. Shaked, and C. E. de Souza, “H∞ filtering for continuous-time linear system with delay,” Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, vol. 44, no. 7, pp. 1412–1417, 1999.

12 C. E. de Souza, R. M. Palhares, and P. L. D. Peres, “RobustH∞filter design for uncertain linear systems with multiple time-varying state delays,” IEEE Transactions on Signal Processing, vol. 49, no. 3, pp. 569–

576, 2001.

13 R. M. Palhares, C. E. de Souza, and P. L. D. Peres, “RobustH∞filtering for uncertain discrete-time state-delayed systems,” IEEE Transactions on Signal Processing, vol. 49, no. 8, pp. 1696–1703, 2001.

14 Z. Wang and F. Yang, “Robust filtering for uncertain linear systems with delayed states and outputs,”

IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 49, no. 1, pp. 125–

130, 2002.

15 E. Fridman and U. Shaked, “A newH∞filter design for linear time delay systems,” IEEE Transactions on Signal Processing, vol. 49, no. 11, pp. 2839–2843, 2001.

16 E. Fridman, U. Shaked, and L. Xie, “RobustH∞filtering of linear systems with time-varying delay,”

Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, vol. 48, no. 1, pp. 159–

165, 2003.

17 H. Gao and C. Wang, “A delay-dependent approach to robustH∞filtering for uncertain discrete-time state-delayed systems,” IEEE Transactions on Signal Processing, vol. 52, no. 6, pp. 1631–1640, 2004.

18 S. Xu, J. Lam, T. Chen, and Y. Zou, “A delay-dependent approach to robustH∞filtering for uncertain distributed delay systems,” IEEE Transactions on Signal Processing, vol. 53, no. 10, part 1, pp. 3764–3772, 2005.

19 X.-M. Zhang and Q.-L. Han, “Delay-dependent robust H∞ filtering for uncertain discrete-time systems with time-varying delay based on a finite sum inequality,” IEEE Transactions on Circuits and Systems II, vol. 53, no. 12, pp. 1466–1470, 2006.

20 X.-M. Zhang and Q.-L. Han, “Stability analysis andH∞ filtering for delay diﬀerential systems of neutral type,” IET Control Theory & Applications, vol. 1, no. 3, pp. 749–755, 2007.

21 X. Jiang and Q.-L. Han, “Delay-dependentH∞ filter design for linear systems with interval time- varying delay,” IET Control Theory & Applications, vol. 1, no. 4, pp. 1131–1140, 2007.

22 X.-M. Zhang and Q.-L. Han, “RobustH∞filtering for a class of uncertain linear systems with time- varying delay,” Automatica, vol. 44, no. 1, pp. 157–166, 2008.

23 X.-M. Zhang and Q.-L. Han, “A less conservative method for designingH∞filters for linear time- delay systems,” International Journal of Robust and Nonlinear Control, vol. 19, no. 12, pp. 1376–1396, 2009.

24 J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional-Diﬀerential Equations, vol. 99 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1993.

25 D. Yue and Q.-L. Han, “RobustH∞filter design of uncertain descriptor systems with discrete and distributed delays,” IEEE Transactions on Signal Processing, vol. 52, no. 11, pp. 3200–3212, 2004.

26 S. Xu and J. Lam, Robust Control and Filtering of Singular Systems, vol. 332 of Lecture Notes in Control and Information Sciences, Springer, Berlin, Germany, 2006.

27 B. Zhang, S. Zhou, and D. Du, “RobustH∞filtering of delayed singular systems with linear fractional parametric uncertainties,” Circuits, Systems, and Signal Processing, vol. 25, no. 5, pp. 627–647, 2006.

28 E. Fridman and U. Shaked, “H∞-control of linear state-delay descriptor systems: an LMI approach,”

Linear Algebra and Its Applications, vol. 351-352, pp. 271–302, 2002.

(20)

29 R. Lu, Y. Xu, and A. Xue, “H∞filtering for singular systems with communication delays,” Signal Processing, vol. 90, no. 4, pp. 1240–1248, 2010.

30 Z. Wu, H. Su, and J. Chu, “H∞filtering for singular systems with time-varying delay,” International Journal of Robust and Nonlinear Control, vol. 20, no. 11, pp. 1269–1284, 2010.

31 X. Zhu, Y. Wang, and Y. Gan, “H∞filtering for continuous-time singular systems with time-varying delay,” International Journal of Adaptive Control and Signal Processing, vol. 25, no. 2, pp. 137–154, 2011.

32 S. Xu, J. Lam, and Y. Zou, “An improved characterization of bounded realness for singular delay systems and its applications,” International Journal of Robust and Nonlinear Control, vol. 18, no. 3, pp.

263–277, 2008.

33 S. Zhu, C. Zhang, Z. Cheng, and J. Feng, “Delay-dependent robust stability criteria for two classes of uncertain singular time-delay systems,” Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, vol. 52, no. 5, pp. 880–885, 2007.

34 X.-L. Zhu and G.-H. Yang, “Jensen integral inequality approach to stability analysis of continuous- time systems with time-varying delay,” IET Control Theory & Applications, vol. 2, no. 6, pp. 524–534, 2008.

35 X.-M. Zhang and Q.-L. Han, “A new stability criterion for a partial element equivalent circuit model of neutral type,” IEEE Xplore—Circuits and Systems II, vol. 56, no. 10, pp. 798–802, 2009.

36 I. Masubuchi, Y. Kamitane, A. Ohara, and N. Suda, “H∞control for descriptor systems: a matrix inequalities approach,” Automatica, vol. 33, no. 4, pp. 669–673, 1997.

37 L. El Ghaoui, F. Oustry, and M. AitRami, “A cone complementarity linearization algorithm for static output-feedback and related problems,” Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, vol. 42, no. 8, pp. 1171–1176, 1997.