# Network-BasedH∞ Filter Design for Linear System with Random Delays

Authors: Ruixia Yan; Zhong Wu; Jinliang Liu
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper investigates the filtering problem for a class of network-based systems with random network-induced delays. The considered random delay between the sensor and the filter is assumed to be satisfying a certain stochastic characteristic. Considering the probability of the delay taking value in different intervals, a stochastic variable satisfying Bernoulli random binary distribution is introduced and a new system model is established by employing the information of the probability distribution. By using a properly constructed Lyapunov function, sufficient conditions for the existence of theH∞ filters are presented in terms of linear matrix inequalities, which are dependent on the occurrence probability of both the random communication delays. The filter parameter is then characterized by the solution to a set of LMIs. A simulation example is employed to show the effectiveness of the proposed method.


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## Body

## 1. Introduction

With the wide application of networks in the complex dynamical processes such as advanced aircraft and manufacturing processes, the networked control systems (NCSs) have attracted great attention during the past few years, see [1–5]. However, they also present some challenging problems arising from the fact that data typically travel through the communication networks from sensors to the controller and from controller to the actuators. In particular, due to the finite bandwidth for data transmission over networks, random delay and packets loss are inevitable in networked systems where a common medium is used for data transfers. In many situations, they can even destabilize the system. Hence, the uncertainties in the measurement transmission have attracted increasing interest.In recent years, lots of outstanding results have been published on sensor delays, packet dropouts, and missing measurements, see [6–8]. In [6], based on a Markov chain, the authors considered mixed uncertainties in the measurement transmission and developed a unified/combined model to accommodate random delay, packet dropouts and missing measurements. The authors in [7] invested a class of linear uncertain discrete-time stochastic systems with randomly varying sensor delay. The authors in [8] studied a delay-dependent H∞ filtering for Markovian jump systems with time-varying delays, and criterion was derived for the H∞ performance analysis of the filtering-error systems, which could lead to much less conservative analysis results. The filtering problem for network-based systems with random delay and packets loss has attracted increasing attention during the past years, see [9–14]. For example, in [9], the filtering problem of network-based system with long time-varying delay was considered. The authors in [12] studied the robust filtering problem for a class of uncertain discrete time-delay systems with missing measurements. In [13], the authors designed H∞ filter for nonlinear systems with time-delay via a T-S fuzzy model approach.However, to the best of our knowledge, the filtering problems for NCSs with random communication delays and missing measurements has not been fully investigated yet, which still remains as a challenging research issue in the literature. Moreover, the filtering problem for network-based systems with random delay and event triggered scheme has not been investigated in the existing literature, which motivates our present work.In this paper, the filtering problem is addressed for communication delays and missing measurements. The delay is modeled as a Bernoulli process. By using the Lyapunov method and the LMI technique, a sufficient condition to guarantee the filtering error system exponentially mean-square stable is derived. A convex optimization problem is also formulated to design the optimalH∞ filter. A simulation example is used to demonstrate the effectiveness of the proposed design procedures.Notation. ℝn and ℝn×m denote the n-dimensional Eculidean space, and the set of n×m real matrices; the superscript “T” stands for matrix transposition; I is the identity matrix of appropriate dimension; ∥·∥ stands for the Euclidean vector norm or the induced matrix 2-norm as appropriate; the notation X>0 (resp., X≥0), for X∈ℝn×n means that the matrix X is real symmetric positive definite (resp., positive semidefinite). When x is a stochastic variable, 𝔼{x} stands for the expectation of x. For a matrix B and two symmetric matrices A and C, [A*BC] denotes a symmetric matrix, where * denotes the entries implied by symmetry.


## 2. System Description

In this paper, we consider the following system:(1)x(k+1)=Ax(k)+Bw(k),(2)y(k)=Cx(k),(3)z(k)=Lx(k),
where x(k)∈ℝn, y(k)∈ℝm, and z(k)∈ℝq denote the state vector, measurement vector, and the signal vector to be estimated, respectively; A, B, C, and L are parameter matrices with appropriate dimensions. We consider the following full order linear dynamic filter:
(4)xf(k+1)=AFxf(k)+BFy^(k),zf(k)=CFxf(k)+DFy^(k),
where xf(k)∈ℝn is the filter state, y^(k)∈ℝm is the filter input, and zf(k)∈ℝn is the estimated signal, and Af, Bf, Cf, and Lf are filter parameters to be determined.We useτ(k) to denote the time-delay in the network between the sensor and the filter, and it is assumed that τ(k) takes values in [τ1,τ2].Assumption 1.
τ
(
k
) changes randomly and for a constant τ0∈[τ0,τ2], the probability of τ(k)∈[τ1,τ0] and τ(k)∈[τ0,τ2] can be known.Define two sets as follows:(5)Ω1={τ(k)∈[τ1,τ0]},Ω2={τ(k)∈[τ0,τ2]}.Furthermore, two functions are defined as follows:(6)τ1(k)={τ(k),forτ(k)∈Ω10,forτ(k)∉Ω1τ2(k)={τ(k),forτ(k)∈Ω20,forτ(k)∉Ω2.Obviously,Ω1⋃‍Ω2=ℝ+ and Ω1⋂‍Ω2=Φ (empty set).From the definitions ofΩ1 and Ω2, it can be seen that τ(k)∈Ω1 means the event τ(k)∈[τ1,τ0] occurs and τ(k)∈Ω2 means the event τ(k)∈[τ0,τ2] occurs. Corresponding to τ(k) taking values in different intervals, a stochastic variable β(k) is defined as
(7)β(k)={1,τ(k)∈Ω10,τ(k)∈Ω2.Assumption 2.
β
(
k
) is a Bernoulli distributed sequence with
(8)Prob{β(k)=1}=𝔼{β(k)}=β0,Prob{β(k)=0}=1-𝔼{β(k)}=1-β0,
where 0≤β0≤1 is a constant.Remark 3.
The introduction ofβ(k) is motivated by [7, 12, 15], where the Bernoulli distributed sequence β(k) is used to model the missing message of the systems. Different from [7, 12, 15], β(k) is used in this paper to describe the time-varying delay taking values in different intervals.Remark 4.
From Assumption2, it can be shown that 𝔼{β(k)-β0}=0 and 𝔼{(β(k)-β0)2}=β0(1-β0). Since Prob{τ(k)∈[τ1,τ0]}=Prob{β(k)=1} and Prob{τ(k)∈[τ0,∞]}=Prob{β(k)=0}, β0 and 1-β0, respectively, denote the probability of τ(k) taking values in [τ1,τ0] and [τ0,τ2].By using the new functionsτi(k)(i=1,2) and β(k), the filter input y^ can be expressed as
(9)y^(k)=y(k-τ(k))=β(k)Cx(k-τ1(k))+(1-β(k))Cx(k-τ2(k)).Then, (4) can be rewritten as follows:
(10)xf(k+1)=AFxf(k)+β(k)BFCx(k-τ1(k))+(1-β(k))BFCx(k-τ2(k))zf(k)=CFxf(k)+β(k)DFCx(k-τ1(k))+(1-β(k))DFCx(k-τ2(k)).Definex-(k)=[x(k)xf(k)] and e(k)=z(k)-zf(k), combining (1), (3), and (10), we can obtain the filtering-error system as follows:
(11)x-(k+1)=A-x-(k)+β(k)C-x(k-τ1(k))+(1-β(k))C-x(k-τ2(k))+B-w(k)(12)e(k)=L-x-(k)-β(k)DFCx(k-τ1(k))-(1-β(k))DFCx(k-τ2(k)),
where A-=[A00Af], C-=[0BfC], B-=[B0], and  L-=[L-CF].Before giving the main result, we first introduce the following useful definitions and lemmas.Definition 5.
The system (1)–(3) are said to be exponentially stable in the mean-square sense (ESMSS), if there exist constants α>0, λ>0, such that t>0(13)𝔼{∥x(t)∥2}≤αe-λtsup-τM<s<0{∥ϕ(s)∥2}.Definition 6.
System (11)-(12) are said to be asymptotically stable with an H∞ norm bound γ, if the following conditions hold:(i)
The filtering error system (11)-(12) with w(t)=0 is asymptotically stable.(ii)
Under the assumption of zero initial condition, the controlled outpute(t) satisfies ∥e(t)∥2≤γ∥w(t)∥2 for any nonzero w(t)∈ℒ2[t0,∞).We recall the following lemmas that will be needed in the proof of our main results.Lemma 7 (see [16]).
For any vectorsx, y∈Rn, and positive definite matrix Q∈Rn×n, the following inequality holds:
(14)2xTy≤xTQx+yTQ-1y.Lemma 8 (see [17]).
For given matrixQ>0 and τk, x(k), y(k)(k=1,2,…), let δ(k)=x(k+1)-x(k), then the following four inequalities holds:
(15)-τk∑i=k-τkk-1‍δT(i)Qδ(i)≤[x(k)x(k-τk)]T[-QQQ-Q][x(k)x(k-τk)].Lemma 9 (see [18]).
Ξ
11, Ξ12, Ξ21, Ξ22, and Ω are constant matrices of appropriate dimensions, and τi(k) is function of k and satisfies τ1≤τ1(k)≤τ0≤τ2(k)≤τ2, i=1,2, then
(16)[(τ1(k)-τ1)Ξ11+(τ0-τ1(k))Ξ21]+[(τ2(k)-τ0)Ξ12+(τ2-τ2(k))Ξ22]+Ω<0
if and only if the following four inequalities hold:
(17)(τ0-τ1)Ξ11+(τ2-τ0)Ξ12+Ω<0(τ0-τ1)Ξ11+(τ2-τ0)Ξ22+Ω<0(τ0-τ1)Ξ21+(τ2-τ0)Ξ12+Ω<0(τ0-τ1)Ξ21+(τ2-τ01)Ξ22+Ω<0.Our purpose is to develop filters of the form (4) such that the filtering error system (11) and (12) are asymptotically stable in the mean-square sense with an H∞ disturbance attenuation level γ.


## 3. Main Results

The following theorem can be obtained for system (11) and (12).Theorem 10.
For some given constantsτ1≤τ0≤τ2 and γ, the system (11) and (12) are ESMSS if there exist matrices P>0, Q1>0, Q2>0, Q3>0, R1>0, R2>0, R3>0 and matrices N, M, T, S of appropriate dimensions, for s=1,2,3,4, such that
(18)[Ξ11+Γ+ΓT**Ξ21Ξ22*Ξ31(s)0Ξ33]<0,
where (19)Ξ11=[Υ1******R1H-Q1-R1*****β0C-TP00****000-Q3***(1-β0)C-TP0000**00000-Q2*B-TP00000-γ2IX]Γ=[0N-N+M-M+T-T+S-S0]Ξ21=[β0PA-0β0PC-000β0PB-1-β0PA-0001-β0PC-01-β0PB-β0R~H(A--I)0β0R~HC-000β0R~HB-1-β0R~H(A--I)0001-β0R~HC-01-β0R~HB-β0L-0-β0DFC00001-β0L-000-1-β0DFC00]Ξ22=daig{-P,-P,-R~,-R~,-I,-I}Ξ31(1)=[τ0-β1NTτ2-β0TT],Ξ31(2)=[τ0-β1NTτ2-β0ST],Ξ31(3)=[τ0-β1MTτ2-β0ST],Ξ31(4)=[τ0-β1MTτ2-β0TT]Ξ33=daig{-R2,-R3}Υ1=PA-+A-TP+HT(Q1+Q2+Q3-R1)HR~=τ12R1+(τ0-τ1)R2+(τ2-τ0)R3,H=[I0].Proof.
Define(20)δ(k)=x(k+1)-x(k)=H[x-(k+1)-x(k)]=H[(A--I)x-(k)+β(k)C-x(k-τ1(k))+(1-β(k))C-x(k-τ2(k))+B-w(k)].
Choose the following Lyapunov functional candidate as(21)V(k)=V1(k)+V2(k)+V3(k),
where
(22)V1(k)=x-T(k)Px-(k)V2(k)=∑i=k-τ1k-1‍xT(i)Q1x(i)+∑i=k-τ2k-1‍xT(i)Q2x(i)+∑i=k-τ0k-1‍xT(i)Q3x(i)V3(k)=∑i=-τ1-1‍∑j=k+ik-1‍δT(j)R1δ(j)+∑i=-τ0-τ1-1‍∑j=k+ik-1‍δT(j)R2δ(j)+∑i=-τ2-τ0-1‍∑j=k+ik-1‍δT(j)R3δ(j).
Let ΔVi(k)=Vi(k+1)-Vi(k), i=1,2,3. Then along the system (11), we have
(23)𝔼ΔV1(k)=𝔼{2x-T(k)Px-(k)+x-T(k)Px-(k)}=2xT(k)P[A-x-(k)+β0C-x(k-τ1(k))2xT(k)P+(1-β0)C-x(k-τ2(k))+B-w(k)]+β0[A-x-(k)+C-x(k-τ1(k))+B-w(k)]T×P[A-x-(k)+C-x(k-τ1(k))+B-w(k)]+(1-β0)[A-x-(k)+C-x(k-τ2(k))+B-w(k)]T×P[A-x-(k)+C-x(k-τ2(k))+B-w(k)](24)𝔼ΔV2(k)=x-T(k)[HT(Q1+Q2+Q3)H]x-(k)-xT(k-τ1)Q1x(k-τ1)-xT(k-τ2)Q2x(k-τ2)-xT(k-τ0)Q3x(k-τ0)(25)𝔼ΔV3(k)=𝔼{δT(k)R~δ(k)}-τ1∑i=k-τ1k-1‍δT(i)R1δ(i)-∑i=k-τ0k-τ1-1‍δT(i)R2δ(i)-∑i=k-τ2k-τ0-1‍δT(i)R3δ(i),
where P>0, Qi>0 and Ri>0, i=1,2,3.
Note that(26)𝔼{δT(k)R~δ(k)}=β0Λ1THTR~HΛ1+(1-β0)Λ2THTR~HΛ2,
where
(27)Λ1=(A--I)x-(k)+C-x(k-τ1(k))+B-w(k),Λ2=(A--I)x-(k)+C-x(k-τ2(k))+B-w(k).
By Lemma 8, we get
(28)-τ1∑i=k-τ1k-1‍δT(i)R1δ(i)≤[x-(k)x(k-τ1)]T[-HTR1HHTR1R1H-R1][x-(k)x(k-τ1)].
Employing the free-weighting matrices method [3], we have
(29)2ξT(k)N[x(k-τ1)-x(k-τ1(k))-∑i=k-τ1(k)k-τ1-1‍δ(i)]=02ξT(k)M[x(k-τ1(k))-x(k-τ0)-∑i=k-τ0k-τ1(k)-1‍δ(i)]=02ξT(k)T[x(k-τ0)-x(k-τ2(k))-∑i=k-τ2(k)k-τ0-1‍δ(i)]=02ξT(k)S[x(k-τ2(k))-x(k-τ2)-∑i=k-τ2k-τ2(k)-1‍δ(i)]=0,
where(30)ξT(k)=[x-T(k)xT(k-τ1)xT(k-τ1(k))xT(k-τ0)xT(k-τ2(k))xT(k-τ2)wT(k)]NT=[N1TN2TN3TN4TN5TN6TN7T]MT=[M1TM2TM3TM4TM5TM6TM7T]TT=[T1TT2TT3TT4TT5TT6TT7T]ST=[S1TS2TS3TS4TS5TS6TS7T].By Lemma 7, there exist R2 and R3, such that
(31)-2ξT(k)N∑i=k-τ1(k)k-τ1-1‍δ(i)≤(τ1(k)-τ1)ξT(k)NR2-1NTξ(k)+∑i=k-τ1(k)k-τ1-1‍δT(i)R2δ(i)-2ξT(k)M∑i=k-τ0k-τ1(k)-1‍δ(i)≤(τ0-τ1(k))ξT(k)MR2-1MTξ(k)+∑i=k-τ0k-τ1(k)-1‍δT(i)R2δ(i)-2ξT(k)T∑i=k-τ2(k)k-τ0-1‍δ(i)≤(τ2(k)-τ0)ξT(k)TR3-1TTξ(k)+∑i=k-τ2(k)k-τ0-1‍δT(i)R3δ(i)-2ξT(k)S∑i=k-τ2k-τ2(k)-1‍δ(i)≤(τ2-τ2(k))ξT(k)SR3-1STξ(k)+∑i=k-τ2k-τ2(k)-1‍δT(i)R3δ(i).By computing, we obtain
(32)eT(k)e(k)=β0(L-x-(k)-DFCx(k-τ1(k)))T×(L-x-(k)-DFCx(k-τ1(k)))+(1-β0)(L-x-(k)-DFCx(k-τ2(k)))T×(L-x-(k)-DFCx(k-τ2(k))).
Combining (23)–(32), we have
(33)𝔼{ΔV(k)-γ2wT(k)w(k)+eT(k)e(k)}≤2xT(k)P[A-x-(k)+β0C-x(k-τ1(k))≤xT(k)P+(1-β0)C-x(k-τ2(k))+B-w(k)]+β0[A-x-(k)+C-x(k-τ1(k))+B-w(k)]T×P[A-x-(k)+C-x(k-τ1(k))+B-w(k)]+(1-β0)[A-x-(k)+C-x(k-τ2(k))+B-w(k)]T×P[A-x-(k)+C-x(k-τ2(k))+B-w(k)]+x-T(k)[HT(Q1+Q2+Q3)H]x-(k)-xT(k-τ1)Q1x(k-τ1)-xT(k-τ2)Q2x(k-τ2)-xT(k-τ0)Q3x(k-τ0)+β0Λ1THTR~HΛ1+(1-β0)Λ2THTR~HΛ2+[x-(k)x(k-τ1)]T[-HTR1HHTR1R1H-R1][x-(k)x(k-τ1)]+2ξT(k)N[x(k-τ1)-x(k-τ1(k))]+2ξT(k)M[x(k-τ1(k))-x(k-τ0)]+2ξT(k)T[x(k-τ0)-x(k-τ2(k))]+2ξT(k)S[x(k-τ2(k))-x(k-τ2)]+(τ1(k)-τ1)ξT(k)NR2-1NTξ(k)+(τ0-τ1(k))ξT(k)MR2-1MTξ(k)+(τ2(k)-τ0)ξT(k)TR3-1TTξ(k)+(τ2-τ2(k))ξT(k)SR3-1STξ(k)-γ2wT(k)w(k)+β0(L-x-(k)-DFCx(k-τ1(k)))T×(L-x-(k)-DFCx(k-τ1(k)))+(1-β0)(L-x-(k)-DFCx(k-τ2(k)))T×(L-x-(k)-DFCx(k-τ2(k))).
Then, by using well-known Schur complement and Lemma9, from (33), one can easily see that (18) with s=1,2,3,4 can lead 𝔼{ΔV(k)-γ2wT(k)w(k)+eT(k)e(k)}<0. The remaining part of the proof is similar to those in [19] and so omitted here for simplicity. The proof is complete.In the following, based on Theorem10, we will design the filter parameters AF, BF, CF, DF, in (4).Theorem 11.
For some given constantsτ1≤τ0≤τ2 and γ, the system (11) and (12) are ESMSS if there exist matrices P>0, P3>0, Q1>0, Q2>0, Q3>0, R1>0, R2>0, R3>0 and matrices A-f, B-f, C-f, D-f, N-10, N-11, M-10, M-11, T-10, T-11, S-10, S-11, Ni, Mi, Ti, Si, (i=1,2,…,7) of appropriate dimensions such that the following LMIs hold:
(34)[Ξ-11**Ξ-21Ξ-22*Ξ-31(s)0Ξ33]<0(35)P1-P-3>0(36)Ξ-11=[Γ-1*******P-3A+A-FTA-F+A-FT******R1+N10TN11TΓ-2*****Γ-3Γ-4N3-N2T+M2TΓ-5****-M10T+T10T-M11T+T11TN4-M2T+T2TΓ-6Γ-7***Γ-8Γ-9N5-T2T+S2TΓ-10Γ-11Γ-12**-S10T-S11TN6-S2T-N6+M6-S3TΓ-13Γ-14Γ-15*BTP1BTP-3TN7-N7+M7-M7+T7-T7+S7-S7-γ2I]Ξ-21=[β0P1Aβ0A-F0β0B-FC000β0P1Bβ0P-3Aβ0A-F0β0B-FC000β0P-3B1-β0P1Aβ0A-F0001-β0B-FC01-β0P1B1-β0P-3Aβ0A-F0001-β0B-FC01-β0P-3Bβ0R~(A-I)000000β0R~B1-β0R~(A-I)000000β0R~Bβ0L-β0C-F0-β0D-FC00001-β0L-1-β0C-F000-1-β0D-FC00](37)Ξ-31(1)=[τ0-β1N-Tτ2-β0T-T],Ξ-31(2)=[τ0-β1N-Tτ2-β0S-T],Ξ-31(3)=[τ0-β1M-Tτ2-β0S-T],Ξ-31(4)=[τ0-β1M-Tτ2-β0T-T]Ξ-33=diag{-P~,-P~,-R~,-R~,-I,-I}Γ-1=P1A+ATP1+Q1+Q2+Q3-R1,Γ-2=-Q1-R1+N2+N2TΓ-3=β0CTB-FT-N10T+M10T,Γ-4=β0CTB-FT-N11T+M11TΓ-5=-N3-N3T+M3+M3T,Γ-6=-N4+M4-M3T+T3TΓ-7=-Q3-M4-M4T+T4+T4T,Γ-8=(1-β0)CTB-FT-T10T+S10TΓ-9=(1-β0)CTB-FT-T11T+S11T,Γ-10=-N5+M5-T3T+S3TΓ-11=-M5+T5-T4T+S4T,Γ-12=-T5-T5T+S5+S5TΓ-13=-M6+T6-S4T,Γ-14=-T6+S6-S5T,Γ-15=-Q2-S6-S6TN-T=[N10TN11TN2TN3TN4TN5TN6TN7T]M-T=[M10TM11TM2TM3TM4TM5TM6TM7T]T-T=[T10TT11TT2TT3TT4TT5TT6TT7T]S-T=[S10TS11TS2TS3TS4TS5TS6TS7T]P~=[-P1*-P-3-P-3].Moreover, if the above conditions are feasible, the parameter matrices of the filter are given by(38)AF=A-FP-3-1,BF=B-F,CF=C-FP-3-1,DF=D-F.Proof.
SinceP-3>0, there exist P2 and P3>0 satisfying P-3=P2TP3-1P2.
Define(39)P=[P1P2TP2P3],J=[I00P2TP3-1],Π=dig{J,I,…,I︸6,J,J,I,…,I︸6}.
By schur complement, it is easy to see thatP>0 is equivalent to P1-P-3=P1-P2TP3-1P2>0.
Pre- and postmultiplying (18) by Π and its transpose, respectively, and let
(40)A-F=A^FP-3-1,A^F=P2TAFP2-TB-F=P2TBFC-F=C^FP-3-1,C^F=CFP2-TD-F=DFN1TJT=[N-10TN-11T]M1TJT=[M-10TM-11T]T1TJT=[T-10TT-11T]S1TJT=[S-10TS-11T].
Then, (18) for s=1,2,3,4 are equivalent to (34) for s=1,2,3,4, respectively.
Next, we will show that, if (34) and (35) are solvable for A-F, B-F, C-F, D-F and P-3, then the parameter matrices of the filter (4) can be chosen as in (38).
Replacing(A-F,B-F,C-F,D-F) by (P2-TA^FP2T,P2-TB-F,C^FP2T,D-F) in (34) and then pre- and post-multiplying them with Π and ΠT, we can also obtain (34). Obviously, (P2-TA^FP2T,P2-TB-F,C^FP2T,D-F) can be chosen as the filter parameters. That is, the following filter
(41)x~(k+1)=P2-TA^FP2Tx~(k)+P2-TB-Fy~(k)r~(k)=C^FP2Tx~(k)+D-Fy~(k)
can guarantee that the filter system (11) and (12) are asymptotically stable with the H∞ performance bound γ. Defining x^(k)=P2Tx~(k), (41) becomes
(42)x^(k+1)=A^Fx^(k)+B-Fy~(k),r(k)=C^Fx^(k)+D-Fy~(k).
Then, from (40) and (42) we can obtain (38). This completes the proof.


## 4. Simulation Examples

Consider the system (1)–(3) with parameter matrices
(43)A=[0.20.40.40.2],B=[-11],C=[1-1],L=[-11].When1≤τ(k)≤4, τ0=2, β0=7, we can obtain the parameter matrices of the filter as follows:
(44)A=[-0.41270.4231-1.23740.6783],BF=[-0.01290.0237],CF=[-0.6778-0.2837],DF=-0.3427.To illustrate the performance of the designed filter, choose the disturbance function as follows:(45)w(k)={-1,5s<t<8s1,15s<t<18s0,otherwise.Figure1 shows the error-estimation signal of e(k), and Figure 2 shows the state of filtering-error system with the initial values xT(0)=[0.5-0.5].Figure 1
Estimated signals errore(k)=z(k)-zf(k).Figure 2
The state curves of filtering error system.


## 5. Conclusion

This paper has investigated the problem of filtering design for linear time-delay system. The probability distribution of the delay taking values in some intervals is assumed to be known a priori. Corresponding to the probability of the delay taking value in different intervals, a stochastic variable satisfying Bernoulli random binary distribution has been introduced and a new modeling method is presented to describe the overall filtering error system. Sufficient conditions are derived to guarantee the stochastic stability and aH∞ performance level for the filtering error system. An optimal filter design problem is also provided by optimizing the filtering performances. Simulations are conducted to evaluate and demonstrate the performance of the proposed approach.



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