# Delay-DependentH∞ Filtering for Singular Time-Delay Systems

Authors: Zhenbo Li; Shuqian Zhu
Journal: Discrete Dynamics in Nature and Society
 (2011)

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## Abstract

This paper deals with the problem of delay-dependentH∞ 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.


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## Body

## 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, 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 reduced-order 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 inH∞ 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-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 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 positive-semidefinite (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 ∥f∥2∶=(∫0∞∥f(t)∥2dt)1/2.


## 2. Problem Statement

Consider the following singular time-delay system:(2.1)Eẋ(t)=Ax(t)+Aτx(t-τ)+Bw(t),y(t)=Cx(t)+Cτx(t-τ)+B1w(t),z(t)=Gx(t)+Gτx(t-τ)+B2w(t),x(t)=ϕ(t),t∈[-τ,0],
where x(t)∈Rn is the state, w(t)∈Rr is the external disturbance signal that belongs to L2[0,∞),y(t)∈Rm is the measurement output, and z(t)∈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<rankE=p<n. τ>0 is the known delay constant and ϕ(t)∈Cn,τ is a compatible vector-valued initial function.Without loss of generality, we assume thatCτ=0,B1=0,Gτ=0, and B2=0. Otherwise, system (2.1) can be equivalently changed into(2.2)[E000][ẋ(t)ζ̇(t)]=[A00-Im+s][x(t)ζ(t)]+[Aτ∣0-----Cτ∣0m+sGτ∣][x(t-τ)ζ(t-τ)]+[B--B1B2]w(t),[y(t)z(t)]=[C∣Im+sG∣][x(t)ζ(t)].
Then in the sequel, we discuss the system model as follows:(2.3)Eẋ(t)=Ax(t)+Aτx(t-τ)+Bw(t),y(t)=Cx(t),z(t)=Gx(t),x(t)=ϕ(t),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 w(t)≡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 ofz(t), we consider the following linear filter with delay:(2.4)Ex̂̇(t)=Afx̂(t)+Aτfx̂(t-τ)+Bfy(t),ẑ(t)=Cfx̂(t),x̂(t)=ψ(t),t∈[-τ,0],
where x̂(t)∈Rn and ẑ(t)∈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(2.5)e(t):=[xT(t)x̂T(t)]T,z̃(t)∶=z(t)-ẑ(t),
one obtains the filter error system(2.6)Ẽė(t)=Ãe(t)+Ãτe(t-τ)+B̃w(t),z̃(t)=G̃e(t),e(t)=[ϕT(t)ψT(t)]T,t∈[-τ,0],
where (2.7)Ẽ=[E00E],Ã=[A0BfCAf],Ãτ=[Aτ00Aτf],B̃=[B0],G̃=[G-Cf].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 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 nonzerow(t)∈L2[0,∞), the H∞ 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 (2.3).


## 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 Theorem 3.2.Lemma 3.1 (see [32]).
Given a scalarγ>0, the filter error system (2.6) has a prescribed H∞ performance γ if there exist matrices Q̃>0,Z̃>0,P̃,Ỹ, and W̃ satisfying
(3.1)ẼTP̃T=P̃Ẽ≥0,(3.2)Φ=[Φ1Φ2τỸTP̃B̃τÃTZ̃G̃T*Φ3τW̃T0τÃτTZ̃0**-τZ̃000***-γ2IτB̃TZ̃0****-τZ̃0*****-I]<0,
where
(3.3)Φ1=P̃Ã+ÃTP̃T+Q̃-ỸTẼ-ẼTỸ,Φ2=P̃Ãτ+ỸTẼ-ẼTW̃,Φ3=-Q̃+W̃TẼ+ẼTW̃.Based on Lemma3.1, 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 scalarγ>0, the filter error system (2.6) has a prescribed H∞ performance γ if there exist matrices Q̃>0,Z̃>0 and P̃ satisfying (3.1) and
(3.4)Ω=[Ω1Ω2P̃B̃τÃTZ̃G̃T*Ω30τÃτTZ̃0**-γ2IτB̃TZ̃0***-τZ̃0****-I]<0,
where
(3.5)Ω1=P̃Ã+ÃTP̃T+Q̃-1τẼTZ̃Ẽ,Ω2=P̃Ãτ+1τẼTZ̃Ẽ,Ω3=-Q̃-1τẼTZ̃Ẽ.Proof.
From Lemma3.1, if we can prove that the feasibility of Ω<0 for solution (Q̃>0,Z̃>0,P̃) is equivalent to that of Φ<0 for solution (Q̃>0,Z̃>0,P̃,Ỹ,W̃), then Theorem 3.2 is proved.
Similar to Lemma 4 of [34], take
(3.6)Ψ=ΠΦΠT=[Ω1Ω2τỸT-ẼTZ̃P̃B̃τÃTZ̃G̃T*Ω3τW̃T+ẼTZ̃0τÃτTZ̃0**-τZ̃000***-γ2IτB̃TZ̃0****-τZ̃0*****-I],
with
(3.7)Π=[I01τẼT0000I-1τẼT00000I000000I000000I000000I].
It follows from Schur complement that
(3.8)Φ<0⟺Ψ<0⟺Z̃>0,Ω+[τỸT-ẼTZ̃τW̃T+ẼTZ̃000](τZ̃)-1[τỸT-ẼTZ̃τW̃T+ẼTZ̃000]T<0.
If there existQ̃>0,Z̃>0, P̃,Ỹ, and W̃ satisfying Φ<0, from (3.8) it is easy to see that the above (Q̃,Z̃,P̃) is a feasible solution of Ω<0. Conversely, if there exist Q̃>0,Z̃>0 and P̃ such that Ω<0 holds, via taking Ỹ=(1/τ)Z̃Ẽ and W̃=-(1/τ)Z̃Ẽ,Φ<0 is also feasible for the above (Q̃,Z̃,P̃,Ỹ,W̃). This completes the proof.The following corollary is easy to be obtained from Theorem3.2.Corollary 3.3.
The filter error system (2.6) is admissible if there exist matrices Q̃>0,Z̃>0 and P̃ satisfying (3.1) and
(3.9)[P̃Ã+ÃTP̃T+Q̃-1τẼTZ̃ẼP̃Ãτ+1τẼTZ̃ẼτÃTZ̃*-Q̃-1τẼTZ̃ẼτÃτTZ̃**-τZ̃]<0.Remark 3.4.
Theorem3.2 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 W̃ 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 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 Ã,Ãτ,P̃, and Z̃, to reduce the number of the unknown variables, we can do as follows.From (3.4) we know that(3.10)[P̃Ã+ÃTP̃T+Q̃-1τẼTZ̃ẼP̃Ãτ+1τẼTZ̃Ẽ*-Q-1τẼTZ̃Ẽ]<0.
Multiplying (3.10) by [II] from the left and by [II]T from the right results in(3.11)P̃(Ã+Ãτ)+(Ã+Ãτ)TP̃T<0,
which implies that P̃ is nonsingular. Let(3.12)P̃=[PP2P3P4],P∈Rn×n,Pi∈Rn×n,i=2,3,4.
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(3.13)ETPT=PE,ETP3T=P2E,ETP4T=P4E.
Taking(3.14)T1=[I00PP3-1],T2=[I00P2-1P],T3=diag⁡{T1,T1,I,T2T,I}
and combining with (2.7) and (3.12), we obtain (3.15)E¯=T2-1ẼT1T=[E00P-1P2EP3-TPT]=[E00P-1ETP3TP3-TPT]=[E00E],P¯=T1P̃T2=[PPPPP3-1P4P2-1P],A¯=T2-1ÃT1T=[A0P-1P2BfCP-1P2AfP3-TPT]=[A0B¯fCA¯f],A¯τ=T2-1ÃτT1T=[Aτ00P-1P2AτfP3-TPT]=[Aτ00A¯τf],B¯=T2-1B̃=[B0],G¯=G̃T1T=[G-CfP3-TPT]=[G-C¯f],
where(3.16)A¯f=P-1P2AfP3-TPT,A¯τf=P-1P2AτfP3-TPT,B¯f=P-1P2Bf,C¯f=CfP3-TPT,
and denote(3.17)Q¯=T1Q̃T1T,Z¯=T2TZ̃T2.
Premultiplying by T1 and postmultiplying by T1T on both sides of (3.1), we have that (3.18)T1ẼTT2-TT2TP̃TT1T=T1P̃T2T2-1ẼT1T≥0,
that is,(3.19)E¯TP¯T=P¯E¯≥0.
Multiplying (3.4) by T3 from the left and by T3T from the right yields(3.20)Ω¯=[P¯A¯+A¯TP¯T+Q¯-1τE¯TZ¯E¯P¯A¯τ+1τE¯TZ¯E¯P¯B¯τA¯TZ¯G¯T*-Q¯-1τE¯TZ¯E¯0τA¯τTZ¯0**-γ2IτB¯TZ¯0***-τZ¯0****-I]<0.It can be seen that the systems(Ẽ,Ã,Ãτ,B̃,G̃) and (E¯,A¯,A¯τ,B¯,G¯) are algebraically 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 A¯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 Af,Aτf,Bf,Cf. Noticing that(3.21)[I0-P3P-1I][PP2P3P4][I-P-1P20I]=[P00P4-P3P-1P2],
then P4-P3P-1P2 is nonsingular. Let(3.22)PP3-1P4P2-1P-P=PP3-1(P4-P3P-1P2)P2-1P=S-1;
then P¯ can be written as(3.23)P¯=[PPPP+S-1].Denote(3.24)JT=S+P-1,T4=[JT-SI0],T5=T4P¯Z¯-1.
Since P(S+P-1)=(P+S-1)S=PP3-1P4P2-1PS is nonsingular, J is also a nonsingular matrix.From (3.19), we have that,(3.25)T4E¯TP¯TT4T=T4P¯E¯T4T≥0.
Noticing (3.23) and (3.24), we derive (3.26)T4P¯=[I0PP],T4P¯E¯T4T=[EJEPEJ-PESTPE]=[EJEPEP-TPE]=[EJEETPE],T4E¯TP¯TT4T=[JTETEETETPT];
then (3.25) is just(3.27)EJ=JTET,PE=ETPT,[EJEETPE]≥0.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(3.28)[T4A¯TP¯TT4T+T4P¯A¯T4T+T4Q¯T4T-1τT4E¯TZ¯E¯T4TT4P¯A¯τT4T+1τT4E¯TZ¯E¯T4TT4P¯B¯τT4A¯TZ¯T5TT4G¯T*-T4Q¯T4T-1τT4E¯TZ¯E¯T4T0τT4A¯τTZ¯T5T0**-γ2IτB¯TZ¯T5T0***-τT5Z¯T5T0****-I]<0.
Noticing that (3.29)T4P¯A¯T4T=[AJAPAJ+PB¯fCJ-PA¯fSTPA+PB¯fC],T4P¯A¯τT4T=[AτJAτPAτJ-PA¯τfSTPAτ],T4P¯B¯=[BPB],T4G¯T=[JTGT+SC¯fTGT],T4A¯TZ¯T5T=T4A¯TZ¯Z¯-1P¯TT4T=T4A¯TP¯TT4T,T4A¯τTZ¯T5T=T4A¯τTZ¯Z¯-1P¯TT4T=T4A¯τTP¯TT4T,B¯TZ¯T5T=B¯TZ¯Z¯-1P¯TT4T=B¯TP¯TT4T,T5Z¯T5T=T4P¯Z¯-1Z¯Z¯-1P¯TT4T=T4P¯Z¯-1P¯TT4T,
denote(3.30)Q=T4Q¯T4T=[Q1Q2Q2TQ3],(3.31)Z=T4P¯Z¯-1P¯TT4T=[Z1Z2Z2TZ3],(3.32)L=PAJ+PB¯fCJ-PA¯fST,Lτ=PAτJ-PA¯τfST,(3.33)WB=PB¯f,WC=C¯fST.Since (3.31) implies that Z¯=P¯TT4TZ-1T4P¯, then(3.34)T4E¯TZ¯E¯T4T=T4E¯TP¯TT4TZ-1T4P¯E¯T4T=[EJEETPE]Z-1[EJEETPE].Introduce matrixW=[W1W2W2TW3]≥0 satisfying(3.35)τW≤[EJEETPE]Z-1[EJEETPE],
then(3.36)[-1τT4E¯TZ¯E¯T4T1τT4E¯TZ¯E¯T4T*-1τT4E¯TZ¯E¯T4T]≤[-WW*-W];
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(3.37)[Ξ11Ξ12AτJ+W1Aτ+W2BτJTATτLTJTGT+WcT*Ξ22Lτ+W2TPAτ+W3PBτATτATPT+τCTWBTGT**-Q1-W1-Q2-W20τJTAτTτLτT0***-Q3-W30τAτTτAτTPT0****-γ2IτBTτBTPT0*****-τZ1-τZ20******-τZ30*******-I]<0
with (3.38)Ξ11=AJ+JTAT+Q1-W1,Ξ12=A+LT+Q2-W2,Ξ22=PA+ATPT+WBC+CTWBT+Q3-W3,
then taking(3.39)S=JT-P-1,B¯f=P-1WB,C¯f=WCS-T,A¯f=P-1(PAJ+WBCJ-L)S-T,A¯τf=P-1(PAτJ-Lτ)S-T,
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 scalarγ>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.
Assume thatE=[Ip000]; otherwise, we can find nonsingular matrices M and N such that MEN=[Ip000]. 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
(3.40)P=[P11P120P22],J=[J110J21J22],P11∈Rp×p,J11∈Rp×p
with
(3.41)[J11IIP11]≥0.
Introduce another variable U>0; then (3.35) can be replaced by
(3.42)τW≤[EJEETPE]U[EJEETPE],(3.43)UZ=I.
Write U as
(3.44)U=[U11U12U13U14U12TU22U23U24U13TU23TU33U34U14TU24TU34TU44]>0,
where
(3.45)U11∈Rp×p,U22∈R(n-p)×(n-p),U33∈Rp×p,U44∈R(n-p)×(n-p).
Noticing that
(3.46)[EJEETPE]U[EJEETPE]=[Π110Π130*000**Π330***0],
where
(3.47)Π11=J11U11J11+U13TJ11+J11U13+U33,Π13=J11U11+U13T+J11U13P11+U33P11,Π33=U11+P11U13T+U13P11+P11U33P11,
we can assume that
(3.48)W=[W110W2100000W21T0W3100000]≥0,[W11W21W21TW31]>0.
Then (3.42) is just
(3.49)τ[W11W21W21TW31]≤[J11IIP11][U11U13U13TU33][J11IIP11].
Invoking Schur complement again, we have that (3.49) is equivalent to
(3.50)[[U11U13U13TU33][J11IIP11]-1[J11IIP11]-1(τ[W11W21W21TW31])-1]≥0.
Introducing
(3.51)α=[α1α2α2Tα3]>0,θ=[θ1θ2θ2Tθ3]>0,
then (3.50) can be replaced by
(3.52)[U11U13τα1τα2*U33τα2Tτα3**τθ1τθ2***τθ3]≥0,(3.53)[J11IIP11][α1α2α2Tα3]=I,[W11W21W21TW31][θ1θ2θ2Tθ3]=I.
Therefore, one can consider the H∞ filter design problem as the following cone complementary problems:
(3.54)Minimize{tr⁡(UZ)+tr⁡([J11IIP11][α1α2α2Tα3]+[W11W21W21TW31][θ1θ2θ2Tθ3])}
subject to LMIs (3.30), (3.31), (3.37), (3.40), (3.41), (3.44), (3.48), (3.51), (3.52), and
(3.55)Q>0,Z>0,[UIIZ]≥0,[α1α2I0α2Tα30II0J11I0IIP11]≥0,[θ1θ2I0θ2Tθ30II0W11W210IW21TW31]≥0.
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
(3.56)[P¯A¯+A¯TP¯T+Q¯-1τE¯TZ¯E¯P¯A¯τ+1τE¯TZ¯E¯τA¯TZ¯*-Q¯-1τE¯TZ¯E¯τA¯τTZ¯**-τZ¯]<0.
By (3.15), (3.16), (3.17), (3.23) and letting Pf:=P+S-1, Q¯=[Q¯1Q¯2*Q¯3], and Z¯=[Z¯1Z¯2*Z¯3] with Q¯1∈Rn×n and Z¯1∈Rn×n, we can conclude from (3.56) that
(3.57)[PfA¯f+A¯fTPfT+Q¯3-1τETZ¯3EPfA¯τf+1τETZ¯3EτA¯fTZ¯3*-Q¯3-1τETZ¯3EτA¯τfTZ¯3**-τZ¯3]<0.
In addition, (3.19) implies that
(3.58)ETPfT=PfE≥0.
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
(4.1)E=[100010000],A=[-200.50.1-0.90.200.50.3],Aτ=[0.20.100.200.150.1-0.230.1],B=[0.20000.20000.2],C=[100],G=[10.70.8].
By Theorem 3.2, for τ=2 and γ=1, after 10 iterations, the corresponding filter is obtained with the following parameters:
(4.2)Af=[-0.9429-0.01020.3206-0.1042-0.84930.19040.37500.48700.3369],Aτf=[0.09830.09600.00460.1248-0.00710.14610.0749-0.24380.0966],Bf=[0.6224-0.20770.4488],Cf=[-0.8379-0.7382-0.9940].
With this filter, Figures 1, 2, and 3 show the state responses x(t) of the original system, the state responses x̂(t) of the filter system, and the error estimation signal z̃(t)=z(t)-ẑ(t) with the initial condition ϕ(t)=[11-1.425]T, ψ(t)=[11-2.6341]T for t∈[-2,0] and the exogenous disturbance input w(t)=diag{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.Figure 1
State responsesx(t) of the original system.Figure 2
State responsesx̂(t) of the filter system.Figure 3
Error estimation signalz̃(t) with the designed filter.Figure 4
Singular value curve of the filtering error system.


## 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 rank E. To study the delay-dependent reduced-order H∞ filtering problem for singular time-delay systems is the key research in the future.



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*Source: 760878-2011-06-30.xml*