# A Novel Wirtinger-Based Inequality toH∞ Filtering for Discrete-Time Systems with Time-Varying Delay

Authors: Kaifan Ma; Zhangang Wang; Fengdong Shi; Liankun Sun
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

This article is committed toH∞ filtering for linear discrete-time systems with time-varying delay. The novelty of the paper comes from the consideration of the new Wirtinger-based inequality with double accumulation terms and the idea of delay-partitioning, which guarantees a better asymptotic stability and is less conservative than the celebrated free-weighting matrix or Jensen’s inequality methods. In combination with the improved Wirtinger-based inequality to handle the modified Lyapunov-Krasovskii (L-K) functionals, a new delay-dependent bound real lemma (BRL) is gained. In the light of the derived H∞ performance analysis results, the H∞ filter will be designed in response to linear matrix inequality (LMI). The validness of the proposed methods will be expressed via some numerical examples by the comparison of existing results.


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## Body

## 1. Introduction

Concerning the design of theH∞ filter for control systems, a large number of results were investigated in the literature (see, for example, [1–13] and reference therein). As we all know, one of its goals is to design a more suitable and stable filter to make the maximum ratio of noise signal to estimated error less than a certain positive value, thus ensuring that the H∞ norm from the interference input to the estimated error output is minimized. Because the classical Kalman filter provides the optimal estimation with the minimum root mean square error as the objective function and makes idealized assumptions in the filtering process, for example, the system model or system disturbance is available. However, in practical engineering applications, when problems such as model uncertainty and noise assumptions are not clear, the Kalman filter becomes incapable. Therefore, in order to cope with model uncertainty and parameter uncertainty, robust Kalman filtering and adaptive Kalman filtering have emerged, respectively. But the results are not satisfactory. Afterwards, researchers began to explore more realistic filters from the perspective of robustness, including the H∞ filter we have mentioned earlier. Compared with H2 and Kalman filter, the H∞ filter merely requires the noise energy to be limited without statistical information or the prior knowledge of noise. Obviously, H∞ filter has better robustness and it is more suitable for practical engineering applications.In addition, time delay is common and inevitable phenomenon existing in control systems, which often makes our systems property decline or even course instability. However, the results show that the stability condition of delay-dependent filter is less conservative than delay-independent filter [14–18]. In order to reduce the conservativeness of the results obtained, many pieces of literature have studied the bounding techniques for cross-terms and model transformation and selected the appropriate Lyapunov-Krasovskii (L-K) functionals deeply in [19–28]. As mentioned in [29, 30], for delay-dependent H∞ filter design for discrete-time systems, the authors had proposed using delay-partitioning idea [30, 31] to divide the constant lower bound dm into m partial intervals. As a result, this delay-partitioning method was included in the newly defined L-K functionals, which means the results acquired also rely on the number of divisions m for the lower bound dm. Therefore, the idea of delay-partitioning plays an important role in reducing the conservativeness of time-delay systems. Furthermore, in combination with some conventional methods, such as free-weighting matrix [32], Jensen’s inequality [33], and Wirtinger-based integral inequality with single summation [34], to deal with the chosen L-K functionals, we gained efficient stability analysis criteria for discrete-time systems with time-delay. In the latest article, researchers have proposed Wirtinger-based integral inequality with double summation [35] and reciprocally convex combination inequality [36], which have been applied in practice. And we note that the novel Wirtinger-based integral inequality also has a significant effect in dealing with some nonlinear systems. For example, in [37], Wei and Qiu skillfully combine the new Wirtinger-based integral inequality with the improved reciprocally convex approach and S-processes to obtain a novel H∞ performance analysis criterion for the underlying closed-loop system. Thus, it effectively reduces the conservativeness of the original results. As the same, for the nonlinear systems with time-varying delays, choosing the appropriate L-K functional and combining an improved Wirtinger-based inequality to handle with the quadratic accumulation term encountered in the derivation of L-K functional are equally effective, as in [38]. Accordingly, this novel summation inequality is more effective than the routine methods—free-weighting matrix, Jensen’s inequality, and Wirtinger-based inequality with single summation, which has introduced more system elements while adding a quadratic accumulation term. Thus, we can grasp more systematic information to achieve the purpose of reducing the conservativeness of the original results.Inspired by the abovementioned, the main contribution of this paper is to further improve the stability criteria for discrete-time systems with time-varying delay and we work on reducing the conservativeness of the stability results. By introducing the new quadratic summation term, more system information is added to our consideration as much as possible in order to make full use of the novel Wirtinger-based inequality to deal with the integral term in L-K functional. For the effective means delay-partitioning and a novel summation inequality, an improved delay-dependentH∞ filter design for discrete-time systems with a state delay will be presented, importantly, utilizing the new Wirtinger-based inequality with double summation to further process the modified L-K functional, which can acquire a lower conservativeness contrast to what we have mentioned above. Finally, some elaborate numerical examples will be given to prove the effectiveness and the advantage of our ways.Throughout our paper, the superscript “T” represents the matrix transposition, N denotes the set of nonnegative, Rn expresses the n-dimensional Euclidean space, Rm×n means the set of all m×n real matrices, 0m×n stands for m×n zero matrix,I and 0 indicate an identity matrix and a zero matrix with appropriate dimension, respectively, and P>0(≥0) is used to denote the matrix P is positively defined and symmetric. Moreover, the sym{X} shows X+XT, and diag{…} means a block-diagonal matrix. And in the symmetric matrix, the symmetric term is defined as, e.g.,(1)AB∗C=ABBTC.


## 2. System Description and Problem Analysis

Define the following linear time-invariant discrete-time system with a state delay:(2)xt+1=Axt+Adxt-τt+Bωt,yt=Cxt+Cdxt-τt+Dωt,zt=Hxt+Hdxt-τt+Lωt,xt=ϕt,t=-τ¯,-τ¯+1,…,0,where x(t)∈Rn stands for the state vector; y(t)∈Rq represents the measured output; z(t)∈Rp means the signal to be measured; ω(t)∈Rl is the noise input content ω≜{ω(t)}∈l2[0,∞); x(t) = ϕ(t), t =-τ¯,-τ¯+1,…,0, is a given initial condition sequence; τ(t) expresses the time delay and we assume that it satisfies(3)1≤τ_≤τt≤τ¯,t=1,2,…,and besides, τ_ and τ¯ indicate the known lower bounds and upper bounds of time delay, respectively. At the same time, A,Ad,B,C,Cd,D,H,Hd, and L are known constant systems matrices.Given system (2), we are committed to design a linear time-invariant filter to guarantee the zF(t) (output of the filter) tracks the original z(t) to be estimated. Consider a linear full-order filter described by(4)xFt+1=AFxFt+BFyt,xF0=0,zFt=CFxFt+DFyt,where xF(t)∈Rn is the filter state, and AF,BF,CF, and DF are filer matrices to be determined.Defining the augmented state vectorζ(t)=col[x(t),xF(t)], and the estimation error et=z(t)-zF(t), we receive the following filtering error system:(5)ζt+1=A¯ζt+Ad¯Kζt-τt+B¯ωt,et=C¯ζt+Cd¯Kζt-τt+D¯ωt,ζt=ϕTt,0T,t=-τ¯,-τ¯+1,…,0,where(6)A¯=A0BFCAF,Ad¯=AdBFCd,B¯=BBFD,C¯=H-DFC-CF,Cd¯=Hd-DFCd,D¯=L-DFD,K=I0.Owing to the fact that the original system model has no control input, to prove the asymptotic stability of the filtering error system, we need to assume that system (2) is asymptotically stable.The propose of our paper is to design a stable full-order filter of the form (4) and meet the following two points: Firstly, the filter error system (5) is asymptomatically stable and secondly, the filtering error system in (5) has a prescribed level γ of H∞ noise attenuation, and under zero-initial condition, e2≤γω2 is satisfied for all nonzero ω∈l2[0,∞).Before entering the next section, a vital lemma will be introduced to be used in the derivation of our conclusion. Then, for any sequence of discrete-time variablex in [α,β]↦Rn, we define l=β-α, κ1=(1/(l+1))∑i=αβx(i), κ2=(2/(l+1)(l+2))∑i=αβ∑j=iβx(j).Lemma 1 (see [35, 39]).
For any sequence of discrete-time variablex in [α,β]↦Rn, matrices R>0, Li∈R4n×n, i∈{0,1,2}, and an arbitrary vector δ, the following summation inequality holds:(7)-∑i=αβ-1yTiRyi≤δTψ0δ+symδTϜ,where(8)δ≜colxβ,xα,κ1,κ2,ψ0≜σ0L0R-1L0T+σ1L1R-1L1T+σ2L2R-1L2T,Ϝ≜L0Ϝ0+L1Ϝ1+L2Ϝ2,σ0=l,σ1=ll-13l+1,σ2=ll-1l-25l+1l+2,Ϝ0=xβ-xα,Ϝ1=xβ+xα-2κ1,Ϝ2=xβ-xα+6κ1-6κ2.Remark 2.
As Lemma1 described, if (l-1)/(l+1)<1 and (l-1)(l-2)/(l+1)(l+2)<1, inequality (7) could be represented into the following inequality:(9)-∑i=αβ-1yTiRyi≤δTψ1δ+symδTϜ,where(10)ψ1≜σ0L0R-1L0T+13L1R-1L1T+15L2R-1L2T.


## 3.H∞ Performance Analysis

In this part, we will conductH∞ filtering stability analysis by using aforementioned Lemma 1. Before we begin, considering the idea of the delay-partitioning, we note that the lower bound τ_ of time-varying delay will be described as τ_=μλ, where μ and λ are positive integers. Meanwhile, several definitions are as follows:(11)τt≜τt,τ=τ¯-τ_,τ1≜τt-τ_,τ2≜τ¯-τt,Γt≜colxt,xt-λ,…,xt-μ-1λ,η1t≜colζt,Γt-λ,xt-τt,xt-τ¯,η2t≜col1λ+1∑i=t-λtxi,1τ1+1∑i=t-τtt-τ_xi,1τ2+1∑i=t-τ¯t-τtxi,η3t≜col2λ+1λ+2∑i=t-λt∑j=itxj,2τ1+1τ1+2∑i=t-τtt-τ_∑j=it-τ_xj,2τ2+1τ2+2∑i=t-τ¯t-τt∑j=it-τtxj,ηt≜colη1t,η2t,η3t,η∗t≜colηt,ωt,ei≜0n×i-1nIn0n×10+μ-in,i∈1,…,10+μ,Π1≜cole1,e2,Π2≜cole1,e3,Π3≜cole2+j,j∈1,…,μ,Π4≜cole1,e3,e2+μ+3,e2+μ+6,Π5≜e1-e3,Π6≜e1+e3-2e2+μ+3,Π7≜e1-e3+6e2+μ+3-6e2+μ+6,Π8≜cole2+μ,e2+μ+1,e2+μ+4,e2+μ+7,Π9≜e2+μ-e2+μ+1,Π10≜e2+μ+e2+μ+1-2e2+μ+4,Π11≜e2+μ-e2+μ+1+6e2+μ+4-6e2+μ+7,Π12≜cole2+μ+1,e2+μ+2,e2+μ+5,e2+μ+8,Π13≜e2+μ+1-e2+μ+2,Π14≜e2+μ+1+e2+μ+2-2e2+μ+5,Π15≜e2+μ+1-e2+μ+2+6e2+μ+5-6e2+μ+8.Theorem 3.
In view of system (2), we assume integers μ>0 and λ>0 with satisfying τ_=μλ. For an admissible H∞ filter and a prescribed scalar γ>0, the filtering error system (5) is asymptomatically stable if there exist real matrices(12)P=PT>0,Q=Q1Q2∗Q3>0,Rj=RjT>0,Sj=SjT>0,j=1,2,G,F,Lq,Mq,Nq,q=0,1,2,such that following LMI(13)ϕ4ϕ1TGϕ2TFϕ3TϕLϕMτtϕNτt∗ϕ500000∗∗ϕ60000∗∗∗-I000∗∗∗∗-S1t00∗∗∗∗∗-S2t0∗∗∗∗∗∗-S3t<0holds for τt∈{τ_,τ¯}, where(14)ϕ1=A¯Π1Ad¯e2+μ+1B¯,ϕ2=A-Ie1Ade2+μ+1B,ϕ3=C¯Π1Cd¯e2+μ+1D¯,ϕ4=Ξ00∗-γ2I,ϕ5=P-X-XT,ϕ6=Ω-F-FT,Ω=λS1+τS2,ϕL=λΠ4TL0L1L2,ϕMτt=τ1Π8TM0M1M2,ϕNτt=τ2Π12TN0N1N2,Ξ0=-Π1TPΠ1+syme1TQ2e3+e1TQe1+e3TQ3e3-Π3TQΠ3+e1TR1+τ+1R2e1-e2+μ+2TR1e2+μ+2-e2+μ+1TR2e2+μ+1+symΠ4TL0Π5+Π4TL1Π6+Π4TL2Π7+symΠ8TM0Π9+Π8TM1Π10+Π8TM2Π11+symΠ12TN0Π13+Π12TN1Π14+Π12TN2Π15,S~1=diagS1,3S1,5S1,S~2=diagS2,3S2,5S2,S1t=diagλS1,3λS1,5λS1,S2tτt=diagτ1S2,3τ1S2,5τ1S2,S3tτt=diagτ2S2,3τ2S2,5τ2S2.Proof.
Now, considering the idea of a slack-variable [40] and using the similar lines as in [29], the following LMI (15) is equivalent to (13):(15)ϕ4ϕ1TPϕ2TΩϕ3TϕLϕMτtϕNτt∗-P00000∗∗-Ω0000∗∗∗-I000∗∗∗∗-S1t00∗∗∗∗∗-S2t0∗∗∗∗∗∗-S3t<0.Then, we consider the modified L-K functional candidate which contains the delay partitioning items:(16)Vt≜V0t+V1t+V2t+V3t,with(17)V0t≜ζTtPζt,V1t≜∑i=t-λt-1ΓTiQΓi,V2t≜∑i=t-τ¯t-1xTiR1xi+∑j=-τ¯+1-τ_+1∑i=t-1+jt-1xTiR2xi,V3t≜∑i=-λ-1∑j=t+it-1θTjS1θj+∑i=-τ¯-τ_-1∑j=t+it-1θTjS2θj.And we note that θ(t)≜x(t+1)-x(t)=ϕ2η∗(t), P>0, Q>0, Rj>0, Sj>0, j=1,2. Then utilizing the previously described V(t), we will obtain the equation as follows:(18)ΔV0t=ζTt+1Pζt+1-ζTtPζt=ηTtΠ1TA¯TPA¯-PΠ1+symΠ1TA¯TPAd¯e2+μ+1+e2+μ+1TAd¯TPAd¯e2+μ+1ηt,(19)ΔV1t=ΓTtQΓt-ΓTt-λQΓt-λ=ηTtΠ2TQΠ2-Π3TQΠ3ηt,(20)ΔV2t=xTtR1xt-xTt-τ¯R1xt-τ¯+∑j=-τ¯+1-τ_+1xTtR2xt-∑j=t-τ¯t-τ¯xTjR2xj≤ηTte1TR1+τ+1R2e1-e2+μ+2TR1e2+μ+2-e2+μ+1TR2e2+μ+1ηt,(21)ΔV3t=θTtλS1+τS2θt-∑i=t-λt-1θTiS1θi-∑i=t-τ¯t-τ_-1θTiS2θi=ηTte1TA-ITΩA-Ie1+syme1TA-ITΩAde2+μ+1+e2+μ+1TAdTΩAde2+μ+1ηt-∑i=t-λt-1θTiS1θi-∑i=t-τtt-τ_-1θTiS2θi-∑i=t-τ¯t-τt-1θTiS2θi.Applying inequality (9), to the summation terms in (21), and we can set δ = col{xt,xt-λ,(1/(λ+1))∑i=t-λtxi,(2/(λ+1)(λ+2))∑i=t-λt∑j=itx(j)}, the following inequalities hold:(22)-∑i=t-λt-1θTiS1θi≤ηTtλΠ4TLjS~1-1LjTΠ4+symΠ4TL0Π5+Π4TL1Π6+Π4TL2Π7ηt.Also, apply inequality (9), and set δ = col{x(t-τ_),x(t-τt), (1/(τ1+1))∑i=t-τtt-τ_x(i), (2/(τ1+1)(τ1+2))∑i=t-τtt-τ_∑j=it-τ_x(j)}; we have(23)-∑i=t-τtt-τ_-1θTiS2θi≤ηTtτ1Π8TMjS~2-1MjTΠ8+symΠ8TM0Π9+Π8TM1Π10+Π8TM2Π11ηt.Similarly, apply inequality (9), and set δ = col{x(t-τt),x(t-τ¯),(1/(τ2+1))∑i=t-τ¯t-τtxi,(2/(τ2+1)(τ2+2))∑i=t-τ¯t-τt∑j=it-τtx(j)}; then we have(24)-∑i=t-τ¯t-τt-1θTiS2θi≤ηTtτ2Π12TNjS~2-1NjTΠ12+symΠ12TN0Π13+Π12TN1Π14+Π12TN2Π15ηt.
Then, by combining (18)–(24), and we suppose that the input has zero noise, i.e., ω(t)=0, and we have(25)ΔVt=ΔV0t+ΔV1t+ΔV2t+ΔV3t≤ηTtΞ0+Ξ1+Ξ2τtηt,where(26)Ξ1=Π1TA¯TPA¯Π1+symΠ1TA¯TPAd¯e2+μ+1+e2+μ+1TAd¯TPAd¯e2+μ+1+e1TA-ITΩA-Ie1+syme1TA-ITΩAde2+μ+1+e2+μ+1TAdTΩAde2+μ+1,Ξ2τt=λΠ4TLS~1-1LTΠ4+τ1Π8TMS~2-1MTΠ8+τ2Π12TNS~2-1NTΠ12.
According to the Schur complement, for all nonzeroζ(t), Ξ0+Ξ1+Ξ2(τt)<0 is equivalent to the LMI in (13). Thus, the negative definition of the ΔV(t) will guarantee the filtering error system (5) is asymptotically stable.
Furthermore, in order to construct theH∞ performance index, we define J≜∑t=0∞[eT(t)e(t)-γ2ωT(t)ω(t)]. And under zero initial condition V(0)=0, V(∞)>0, and ω(t)≠0, one obtains(27)J≤∑t=0∞eTtet-γ2ωTtωt+V0-V∞=∑t=0∞eTtet-γ2ωTtωt+ΔVt≤∑t=0∞η∗Ttϕ4+ϕ1TPϕ1+ϕ2TΩϕ2+ϕ3Tϕ3+ϕ⋆η∗t,where ϕ⋆=ϕLS~1ϕLT+ϕMS~2ϕMT+ϕNS~2ϕNT. By using the Schur complement, inequality (27) indicates that J<0, which is equivalent to the LMI (15). Besides, for all ωt∈l20,+∞, e2<γω2. Then, this completes the proof.Remark 4.
Different from the previous methods where free-weighting matrix and Jensen’s inequality are used to handle the double summation functionΔV3(t) in the proof of Theorem 3, the novel Wirtinger-based inequality with double summation could carry out an accurate summation inequality to bound the signal summation term and give rise to the derivative of the L-K functional to get vital in reducing the conservatism.


## 4.H∞ Filter Design

In this section, we will focus on the design of full-orderH∞ filter for state delay in system (2).Theorem 5.
In view of system (2), we assume integers μ>0 and λ>0 with satisfying τ_=μλ. For an admissible filter of form (4) ensure a prescribed H∞ norm bound γ and the asymmetric stability of the filtering error system (5) exist if there exist matrices V1,V2,V3,F,A¯F,B¯F,C¯F,D¯F of appropriate dimensions and(28)P¯=P¯1P¯2∗P¯3>0,Q=Q1Q2∗Q3>0,Rj=RjT,Sj=SjT,j=1,2,G,F,Lq,Mq,Nq,q=0,1,2,such that following LMIs are satisfied:(29)Ξ=ϕϕLϕMτtϕNτt∗-S1t00∗∗-S2t0∗∗∗-S3t<0,where(30)ϕ=Ξ00Ξ11∗-γ2IΞ12∗∗Ξ13,Ξ11=colΘ0Θ102×4Θ207×4,Θ0=ATV1+CTB¯FTATV3+CTB¯FTA-ITFHT-CTD¯FT,Θ1=A¯FTA¯FT0-C¯FT,Θ2=AdTV1+CdTB¯FTAdTV3+CdTB¯FTAdTFHdT-CdTD¯FT,Ξ12=BTV1+DTB¯FTBTV3+DTB¯FTBTFLT-DTD¯FT,Ξ13=P¯1-V1-V1TP¯2-V2-V300∗P¯3-V2-V2T00∗∗ϕ60∗∗∗-I.
Then, the filtering problem can be solved. In addition, a suitable filter state space realization is presented by(31)AF=V2-1A¯F,BF=V2-1B¯F,CF=C¯F,DF=D¯F.Proof.
Due toP¯>0 and LMI in (29), it follows that P¯3>0 and -P¯3+V2+V2T>0, respectively, which means V2+V2T>0. Thus, we can conclude that V2 is nonsingular and for arbitrary square and nonsingular matrices X3 and X4, meeting V2=X3TX4-1X3. Then, we define the matrices:(32)J1=I00X4-1X3>0,X2=V3X3-1X4,X=V1X2X3X4,P=J1-TP¯1P¯2∗P¯3J1-1,J2=diagJ1,I⋯I,J1,I⋯I,(33)AF=X3-TA¯FX3-1X4,BF=X3-TB¯F,CF=C¯FX3-1X4,DF=D¯F.
Then, matricP¯ and LMI (29) can be expressed as(34)J1TPJ1>0,(35)J2TΞJ2<0,So, by way of congruent transformation to LMI (34) and (35), respectively, with J1-1 and J2-1, it is clear that LMI in (29) is equivalent to (13). Consequently, from Theorem 5, we could draw a conclusion that the filtering error system in (5) is asymptotically stable with a H∞ norm bound γ.
The next job is to get the filer state-space realization as (31). Through some conventional matrix processing in [10], substituting V2=X3TX4-1X3 into (33), then(36)AFBFCFDFcould be directly rewritten as(37)X3-1X400I-1V2-1A¯FV2-1B¯FC¯FD¯FX3-1X400I.
Due to the fact that matricesX3 and X4 are nonsingular, we can infer that(38)X3-1X400Iis also nonsingular. And this makes the point that the following systems are algebraically equivalent:(39)AFBFCFDF⇔V2-1A¯FV2-1B¯FC¯FD¯F,and thus, a state-space realization (AF,BF,CF,DF), which is defined in (4), of the desired filter could be acquired from (39). This completes the proof.


## 5. Numerical Examples

Example 1 (see [29]).
Firstly, we consider system (2) with varying-delay:(40)A=0.800.10.9,Ad=-0.10.15-0.1-0.15,B=01,C=11,Cd=0.40.6,D=1,H=12,Hd=0.50.6,L=-0.5,and τ_=6, τ¯=8, μ=2, by processing of Theorem 5, the optimal γ∗=9.9691, which is much less than γ∗=13.8580[1], 8.7038[29], 10.3752[29], respectively. Our achieved results have lower conservativeness than presently existing methods. And when τ¯=9 the H∞ filters parameters(41)
A
F
B
F
C
F
D
Fare given as follows:(42)
0.1250
0.1086
-
0.2255
0.1380
0.3049
-
0.3008
1.4573
×
1
0
-
5
-
1.8674
×
1
0
-
5
1.4699
.
More detailed data will be listed in Table1, which content the minimum γ∗ with different upper bounds τ¯, and set τ_=6.Table 1
The minimumγ∗ with different τ¯ when τ_=6.
τ¯ 7 8 9  [1] 7.5050 13.8580 ∞  [29]Theorem 4 6.0430 8.7038 13.9799  [29]Theorem 5 6.2928 10.3752 29.9862  Theorem5 5.7470 7.5334 9.9691Example 2 (see [10]).
Then, we consider system (2) with(43)A=0.850.1-0.10.7,Ad=0.20-0.20.1,B=0.10.4,C=0.22.5,Cd=-0.50.5,D=-1,H=02.2,Hd=1.5-0.4,L=-0.1,and let μ=1,μ=2, respectively, with different lower and upper bounds of time-delay. Thus, we could have the clear discovery that when μ=2,τ_=2, and τ¯=6, by processing of Theorem 5, we could acquire γ∗=3.9910, which is much less than γ∗=6.7185,6.7581, respectively, in [10, 29]. The H∞ filters parameters are given in (44)(44)
1.0252
1.4341
0.3874
-
0.8278
-
1.1579
-
0.0675
-
0.3177
-
0.4441
0.8472
.
Similarly, more specific data will be listed in Table2, which content the minimum γ∗ with different delay partitioning number μ and delay boundary.Table 2
The minimumγ∗ with different μ,τ¯ and τ_.
μ 1 1 2 2  τ_ 1 1 2 2  τ¯ 4 5 5 6  [10] 4.9431 6.1608 5.3551 6.7581  [29]Theorem 4 3.6545 4.6494 5.3458 6.7185  Theorem5 1.7342 2.2883 3.9551 3.9910


## 6. Conclusions

By this we have completed the study for the delay-dependentH∞ filter design for discrete-time system with varying-delay. In the research process, by introducing the novel Wirtinger-based inequality to further handle the quadratic accumulative items coming from the modified Lyapunov-Krasovskii functionals and taking advantage of the delay-partitioning idea, a new BRL for the filtering system has been gained. Furthermore, the achieved result has lower conservativeness than presently existing methods, for instance, Jensen’s inequality and free-weighting matrix methods, etc. Eventually, a numerical example has been provided to verify the validness of our method.



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