# H∞ Filter for Discrete-Time Takagi–Sugeno Fuzzy Systems with Time-Varying Delays via a Novel Wirtinger-Based Inequality

Authors: Wanru Wang; Yonggang Wen; Liankun Sun; Kaifan Ma
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

This paper is devoted to the analysis and design ofH∞ filtering for discrete-time Takagi–Sugeno (TS) fuzzy systems with time-varying delays. By using the delay-partitioning method, more systematic information is introduced, which can reduce the conservatism of the conclusion. By constructing an appropriate Lyapunov functional and combining with a newly Wirtinger-based summation inequality, the existence condition of the H∞ filter is obtained. Thus, utilizing the contract matrix transformation method, the H∞ filter design for discrete-time TS fuzzy systems with time-varying delay based on LMI is proposed. Finally, a few numerical analysis results are given to prove the eﬀectiveness of the method.


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## Body

## 1. Introduction

In recent years, the analysis and synthesis of nonlinear systems has gradually become one of the research focuses in the domain of automatic control and automatic control technology. Nonlinearity is ubiquitous in chemical processes, robotic systems, automotive systems, and many manufacturing processes, which could introduce severe difficulties in the analysis and synthesis of control systems. Although nonlinearity is more accordant with people’s actual production processes, nonlinear systems cannot be simply described by linear differential equations, and the stability of the system not only relies on the structural parameters of the system but also depends on the initial state of the system. These problems have caused considerable dilemmas for researchers to analyze and synthesize the stability of nonlinear systems. With the deepening of research, methods such as output control feedback [1–5], sliding mode control [6–8], optimal control [9, 10], and fuzzy control [11–20] have been proposed. Currently, the TS fuzzy system [21–24] is deemed to be one of the most effective methods for dealing with nonlinear systems.In recent years, the TS fuzzy system has been widely used. The TS fuzzy model is a nonlinear model described by a series of if-then rules, which is arbitrarily smooth and nonlinear. The system can be approximated by the TS fuzzy model without constant term with any specified accuracy. Up to now, the strict LMI stability sufficient conditions for the robust stability of uncertain continuous TS fuzzy descriptor systems are given [25]. By constructing the augmented Lyapunov–Krasovskii functional and utilizing Wirtinger-based integral inequality, an improved condition for stability of the concerned systems was derived in terms of LMIs [26]. Li et al. [27] designed integral switching function to study the fault detection of sliding mode controller for a class of TS fuzzy singular systems. More research results can be found in [28–30] and the references therein.Furthermore, time delays are common in any practical system, such as network systems, power systems, hydraulic systems, and so on. Thus, the problems such as poor system stability and degraded performance are unavoidable. Meanwhile, people have made a lot of analysis and research on the delay-dependent and delay-independent TS fuzzy systems [31–33]. At present, delay-partitioning [34, 35] is considered as one of the most effective methods to process the time delay problems in control systems. By artificially dividing the lower bound dm into m equal-length subintervals, the maximum time delay of the system’s asymptotic stability in the guaranteed interval is obtained, thereby reducing the conservativeness of the stability conditions. Compared with the early Jensen inequality [47] and the free weight matrix method [48–50], the recently proposed Wirtinger-based inequality [36–38], Bessel–Legendre inequality [39–42], and reciprocal convex method [43–46] have many improvements in the derivation of quadratic terms. However, to our knowledge, few results can be found in the literature concerning with the discrete-time TS fuzzy delay systems by the delay-partitioning method and Wirtinger-based inequality. Thus, research on the filtering problem for discrete-time TS fuzzy systems with time-varying delay by employing the delay-partitioning method and Wirtinger-based inequality should be of both theoretical and practical importance, which motivates us to carry out this study.As stated previously, in this paper, we consider the delay-dependentH∞ filter for TS fuzzy discrete-time systems with time-varying delay. Firstly, we will utilize the delay-partitioning method to reduce the effect of system time-varying delays, and the quadratic accumulation term appearing in Lyapunov–Krasovskii function is processed by the improved Wirtinger-based inequality to ensure that the fuzzy system is asymptotically stable. Then, the contract matrix transformation is used to design the H∞ filter. Finally, we will demonstrate the validity and low conservation of our experimental results through some simulation examples.Notations. Throughout this paper, the superscript “T” represents the matrix transposition, ℝn expresses the n-dimensional Euclidean space, I and 0 signify an identity matrix and a zero matrix with appropriate dimension, respectively, and P>0 is used to denote that the matrix P is positively defined and symmetric. Besides, sym A is equivalent to A+AT, diag{…} is a block-diagonal matrix, and colA,B,C=, that is, AT,BT,CTT. It should be noted that in the symmetric block matrices, ∗ represent terms caused by symmetry, e.g.,(1)X1X2∗X3=X1X2X2TX3.


## 2. Model Description and Problem Analysis

The description of the TS fuzzy model is represented by the fuzzy IF-THEN rule of the local linear input-output relationship of the nonlinear system. Then, consider a nonlinear system, which can be described as a discrete-time TS fuzzy time-varying delay model as follows.Rule (i): IFϑ1t is Fi1 and … and ϑgt is Fig, THEN(2)xt+1=Aixt+Adixt−τt+Biωt,yt=Cixt+Cdixt−τt+Diωt,zt=Lixt+Ldixt−τt+Fiωt,xt=φt,t=−τ1,−τ1+1,…,0,i=1,2,…r,where xt∈ℝn expresses the state vector, yt∈ℝq denotes the measured output, zt∈ℝp is the signal to be measured, ωt∈ℝl signifies the noise input vector, which belongs to l20,∞, t=−τ1,−τ1+1,…,0 is given initial condition sequence, and τt express the time-varying delays (we assume that it satisfies 1≤τ2≤τt≤τ1,t=1,2,…); besides, τ2 and τ1 indicate the known minimum and maximum bounds of delays, respectively. Meanwhile, Fij is the fuzzy set, r represents the number of IF-THEN rule, and ϑt=ϑ1t,ϑ2t…ϑgt shows the premise variables vector. Ai,Adi,Bi,Ci,Cdi,Di,Li,Ldi, and Fi are known constant systems matrices, and xt=φt refers to the initial condition.Then, describe the normalized fuzzy-basis function as follows:(3)ℏiϑt=∏j=1gFijϑjt∑i=1r∏j=1gFijϑjt≥0,(4)∑i=1rℏiϑt=1,with Fijϑjt indicating the grade of membership of ϑt in Fij, and define ℏi=ℏ1,…,ℏr. Through product fuzzy inference and center-of-gravity defuzzification, the following discrete time-varying TS fuzzy dynamic model ℵ can be obtained:(5)xt+1=Aℏ,txt+Adℏ,txt−τt+Bℏ,tωtyt=Cℏ,txt+Cdℏ,txt−τt+Dℏ,tωtzt=Lℏ,txt+Ldℏ,txt−τt+Fℏ,tωtxt=φt,t=−τ1,−τ1+1,…,0,where(6)Aℏ,t≜∑i=1rℏiAit,Adℏ,t≜∑i=1rℏiAdit,Bℏ,t≜∑i=1rℏiBit,Cℏ,t≜∑i=1rℏiCit,Cdℏ,t≜∑i=1rℏiCdit,Dℏ,t≜∑i=1rℏiDit,Lℏ,t≜∑i=1rℏiLit,Ldℏ,t≜∑i=1rℏiLdit,Fℏ,t≜∑i=1rℏiFit.Given system (1), we are committed to designing a full-order fuzzy filter for the estimation of zt of general structure described by ℵF:(7)xFt+1=AFℏxFt+BFℏyt,zFt=LFℏxFt+DFℏyt,xF0=0,where xFt∈ℝn is the filter state vector of the system ℵF and AFℏ,BFℏ,LFℏ, and DFℏ are filer matrices to be determined.Next, define the augmented state vectorζt:=colxt,xFt and the estimation error et:=zt−zFt. Thus, we could acquire the filtering error system ℵe as follows:(8)ζt+1=A¯ℏ,tζt+A¯dℏ,tζt−τt+B¯ℏ,tωt,et=L¯ℏ,tζt+L¯dℏ,tζt−τt+F¯ℏ,tωt,ζt=ϕTt,0T,t=−τ1,−τ1+1,…,0,where(9)A¯ℏ,t≜Aℏ,t0BFℏCℏ,tAFℏ,A¯dℏ,t≜Adℏ,tBFℏCdℏ,t,B¯ℏ,t≜Bℏ,tBFℏDℏ,t,L¯ℏ,t≜Lℏ,t−DFℏCℏ,t−CFℏ,L¯dℏ,t≜Ldℏ,t−DFℏCdℏ,t0,F¯ℏ,t≜Fℏ,t−DFℏDℏ,t.Remark 1.
TheH∞ filter is designed as follows. Given a fuzzy system (1), a fuzzy filter such as (6) with filter matrices AFℏ,BFℏ,LFℏ,DFℏ is designed to estimate the signal zt to ensure that the filtering error system ℵe in (7) is asymptotically stable, and zFt is a good estimate of zt with the energy bounded disturbance ωt. That means, for a given γ>0, the filter error system ℵe is asymptotically stable when the H∞ performance index γ is asymptotically stable under the condition of ω=0, satisfying the following requirements: e2<γω2, for all nonzero ω∈l20,∞, where e2≜∑t=0∞eTtet.Remark 2.
Since the control input of the original fuzzy system modelℵ is unknown and the system to be estimated must be asymptotically stable, which is also the precondition of asymptotically stable filter error system in ℵe, we assume that system ℵ is asymptotically stable.
Before entering the main sections, a very vital lemma is introduced for the deduction of our conclusions.Lemma 1 (see [51, 52]).
For a given symmetric positive definite matrixℜ∈ℝn×n, matrices Lp∈ℝ4n×n,p=0,1,2, and any differentiable function ϖ in ρ0,ρ1↦ℝn, the following summation inequality holds:(10)−∑i=ρ0ρ1−1ϖTiℜϖi≤Δ,where(11)Δ=δ¯TΛℓ,ℜ,Lp+symχLp,Ipδ¯,δ¯=colxρ1,xρ0,1ℓ+1Σ0,2ℓ+1ℓ+1Σ1,ℓ=ρ1−ρ0,Σ0=∑i=ρ0ρ1xi,Σ1=∑i=ρ0ρ1∑j=iρ1xj,Λℓ,ℜ,Lp=ℓL0ℜ−1L0T+ℓℓ−13ℓ+1L1ℜ−1L1T+ℓℓ−1ℓ−25ℓ+1ℓ+2L2ℜ−1L2T,χLp,Ip=L0I0+L1I1+L2I2,I0=1−100,I1=11−20,I2=1−16−6.Remark 3.
Introduced by Lemma1, we can find that ℓ−1/ℓ+1<1 and ℓ−1ℓ−2/ℓ+1ℓ+2<1, and inequality (10) could be abbreviated as follows:(12)Λℓ,ℜ,Lp=ℓL0ℜ−1L0T+13L1ℜ−1L1T+15L2ℜ−1L2T.
Furthermore, compared with previous Jensen’s inequality, free-weighting matrices method, and Wirtinger-based inequality with single cumulative term, the new summation inequality reduces the conservativeness of the obtained results more effectively when dealing with the quadratic summation term.


## 3. H∞ Performance Analysis

In this section, based on the fuzzy-basis Lyapunov–Krasovskii function, delay-partitioning technique, and the improved Wirtinger-based inequality, novel delay-dependent stability and stabilization criteria for the fuzzy systemℵ in (4) will be proposed to resolve the problem of robust H∞ filter design mentioned in Section 2. Before beginning, here is an overview of several definitions that will be implemented in this article.Before we start, in order to simplify, define it as follows:(13)τ2≜mμm=1,…,n,τt≜τt,τ¯≜τ2−τ1,ξt≜colxt,xt−μ,…,xt−m−1μ.Theorem 1.
In consideration of system (1) and fuzzy filter ℵF in (6), the filtering error system ℵe in (7) is asymptotically stable with H∞ performance γ, if there exist real symmetric positive-definite matrices Pℏi,t,α1i,1l,2i,20,Ri,v,o,S1z,2z,T1i,2i, constant matrices Nℏj,Mℏj,j=0,1,2, such that for all i, l, o, t, v, z ∈ (1, 2, …, r), and scalar γ > 0. For given integers m > 0 and µ > 0 with satisfying τ2=mμ, the following inequality holds(14)ϒi,l,o,t,v,z<0,i,l,o,t,v,z∈1,2,…,r,where(15)ϒi,l,o,t,v,z≜ϒ0+ϒ1+ϒ2,ϒ0≜−∏˜1TPi∏˜1+∏˜2Tα1i∏˜2−∏˜3Tα1l∏˜3+e1Tα2ie1−e2+m+1Tα20e2+m+1+e1Tτ¯+1Rie1−e2+mTRve2+m−e2+m+1TR0e2+m+1+sym∏˜4TNℏ0∏˜5+∏˜4TNℏ1∏˜6+∏˜4TNℏ2∏˜7+sym∏˜8TMℏ0∏˜9+∏˜8TMℏ1∏˜10+∏˜8TMℏ2∏˜11,ϒ1≜Γ0TPtΓ0+Γ1TμS1z+τ¯S2zΓ1,ϒ2≜μΠ˜4TNℏjT1i−1NℏjTΠ˜4+τ¯Π˜8TMℏjT2i−1MℏjTΠ˜8Γ0≜A¯ℏ,tΠ˜4,A¯dℏ,te2+m,B¯ℏ,tΓ1≜A¯ℏ,te1,Adℏ,te2+m,B¯ℏ,tei≜0n×i−1nIn0n×9+m−in,i∈1,…,9+m,Π˜1≜cole1,e2,Π˜2≜cole1,e3,Π˜3≜cole2+s,s∈1,…,m,Π˜4≜cole1,e3,e2+m+2,e2+m+4,Π˜5≜e1−e3,Π˜6≜e1+e3−2e2+m+1,Π˜7≜e1−e3+6e2+m+2−6e2+m+4,Π˜8≜cole2+m,e2+m+1,e2+m+3,e2+m+5,Π˜9≜e2+m−e2+m+1,Π˜10≜e2+m+e2+m+1−2e2+m+3,Π˜11≜e2+m−e2+m+1+6e2+m+3−6e2+m+5.Proof.
Considering the idea of slack variable, now we construct the following fuzzy L-K candidate functions with time-varying delay-partitioning terms:(16)Vt≜V0t+V1t+V2t+V3t,with(17)V0t≜ζTtPℏζt,V1t≜∑i=t−μt−1ξTiα1iξi+∑i=t−τ1t−1xTiα2ixi,V2t≜∑i=−τ1+1−mμ∑j=t−1+it−1xTjRjxj,V3t≜∑i=−μ+1−1∑j=t+it−1ςdTjSj1zςdj+∑i=−τ1−mμ−1∑j=t+it−1ςdTjSj2zςdj,where ςdt≜xt+1−xt and matrices P>0,α1,2>0,R>0,S>0. Then, we can define ΔVt≜Vt+1−Vt, and along the trajectory of fuzzy error system (7), we have(18)ΔV0t=ζTt+1Pt+1ζt+1−ζTtPtζt=δTtΓ0TPtΓ0−∏˜1TPt∏˜1δT,(19)ΔV1t=ξTtα1tξt−ξTt−μα1t−μξt−μ+xtTα2txt−xTt−τ1α2t−τ1xt−τ1=δTt∏˜2Tα1i∏˜2−∏˜3Tα1l∏˜3+e1Tα2ie1−e5Tα2oδt,(20)ΔV2t=τ1−mμ+1xTtRtxt−xTt−mμRt−mμxt−mμ−xTt−τ1Rt−τ1xt−τ1=δTte1Tτ1−mμ+1Rie1−e4TRve4−e5TRoe5δt,(21)ΔV3t=ςdTtμS1t+τ1−mμS2tςdt−∑i=t−μt−1ςdTiS1iςdt−∑i=t−τ1t−mμ−1ςdTiS2iςdi=δTtΓ1TμS1z+τ1−mμS2zΓ1δt−∑i=t−μt−1ςdTiS1iςdi−∑i=t−τ1t−mμ−1ςdTiS2iςdi,where δt≜colζt,ξt−μ,1/μ+1∑i=t−μtxi,1/τ¯+1∑i=t−τ1t−mμxi,2/μ+1μ+2∑i=t−μt∑j=itxj2/τ¯+1τ¯+2∑i=t−τ1t−mμ∑j=it−mμxj, Here we set δ0t=colxt,xt−μ,1/μ+1∑i=t−μtxi,2/μ+1μ+2∑i=t−μt∑j=itxj. For the summation terms (16) and (17), utilizing (9) and (11) in Lemma 1 and Remark 3, respectively, the following inequality holds:(22)−∑i=t−μt−1ςdTiS1iςdi≤Δ1Δ1≜δTtΠ˜4TΛ1+symχ1Π˜4δt,Λ1≜ℓM0S1−1M0T+13M1S1−1M1T+15M2S1−1M2T,χ1≜M0I0+M1I1+M2I2.
Similarly, applying (9) and (11) and setting δt1=colxt−mμ,xt−τ1,1/τ¯+1∑i=t−τ1t−mμxi,2/τ¯+1τ¯+2∑i=t−τ1t−mμ∑j=it−mμxj, we have(23)−∑i=t−τ1t−mμ−1ςdTiS2iςdi≤Δ2Δ2≜δTtΠ˜8TΛ2+symχ2Π˜8δt,Λ2≜ℓN0S2−1N0T+13N1S2−1N1T+15N2S2−1N2T,χ2≜N0I0+N1I1+N2I2.
Afterwards, by combining (14)–(19), under the assumption that the input is zero noise, i.e., ω(t) = 0, we can obtain the following conclusions:(24)ΔVt=ΔV0t+ΔV1t+ΔV2t+ΔV3t≤δTtϒ0+ϒ1+ϒ2δt=δTtϒi,l,o,t,v,zδt.
According to Schur’s complement theorem, for all nonzeroδ(t), we have that ϒ0+ϒ1+ϒ2τt<0 is equivalent to the LMI in (7), that is, ∆V(t) < 0. This condition of the negative definite ∆V(t) guarantees that the filtering error system ℵ is asymptotically stable.
Furthermore, in order to construct theH∞ performance index, we define J≜∑t=0∞eTt−γ2ωTtωt. Also, under zero initial condition V(0) = 0, V(∞) > 0, and ω(k)≠0, one obtains(25)J≤∑t=0∞eT=∑t=0∞eTtet−γ2ωTtωt+ΔVt≤∑t=0∞δ∗TtΓ+Γ0TPtΓ0+Γ1TΩ˜Γ1+Γ2TΓ2+Γ∗δ∗ttet−γ2ωTtωt+V0−V∞,where(26)δ∗t≜colδt,ωt,Γ≜ϒ00∗−γ2I,Ω˜=μS1z+τ¯S2z,Γ2≜L¯ℏ,tΠ1,L¯dℏ,te2+m+1,F¯ℏ,t,Γ∗≜ΓMT1iΓMT+ΓNT2iΓNT,ΓM≜μΠ˜4TM0ℏjM1ℏjM2ℏj,ΓN≜τ¯Π˜8TN0ℏjN1ℏjN2ℏj.
Then, using the Schur complement for inequality (21), we can get the following formula:(27)ΓΓ0TPtΓ1TΩ˜Γ2TΓMΓN∗−Pt0000∗∗−Ω˜000∗∗∗−I00∗∗∗∗−T1i0∗∗∗∗∗−T2i<0,where(28)T1i≜diagT1i,3T1i,5T1i,T2i≜diagT2i,3T2i,5T2i.
Inequality (20) indicates that J < 0, which is equivalent to the LMI (21). Besides, for all ω(t)∈l2[0, +∞), ||e||2 < γ||ω||2. Then, this completes the proof.Remark 4.
In this section, the stability analysis method of discrete-time TS fuzzy systems with time-varying delay is mainly based on the TS fuzzy logic theory to model a given mathematical model of a linear system and transform complex martingale linear problems into linear problems. Thus, a bounded real lemma for a discrete-time linear system with time delay is proposed. In addition, in the selection of Lyapunov–Krasovskii functionals, we construct a term that includes time-partitioning division and a double summation term. The double summation term appears during the derivative of the Lyapunov–Krasovskii function, using the novel Wirtinger-based inequality for scaling. As for the common Jensen’s inequality, the method of free-weighting matrix, and Wirtinger-based inequality with signal summation term, the system stability conditions obtained by this method are relatively conservative.


## 4. H∞ Filter Design

In this section, we are committed to designing a full-orderH∞ fuzzy filter ℵF for state delay in system (4).Theorem 2.
Consider a discrete-time TS fuzzy system with time-varying delay from (4), and assume that the integers m > 0, µ > 0 satisfy τ2 = mµ. For a full-order filter of form (6), filter error system (7) is asymptotically stable, with H∞ performance γ, if there exist matrices with appropriate dimensions A,B,L,D,W1,W2,Mq,Nq,q=0,1,2, and the following inequalities hold(29)℘>0,g>0,Q1i=α1i1α1i2∗α1i3>0,Q2ii>0,S1z=S1z1S1z2∗S1z3>0,S2z=S2z1S2z2∗S2z3>0,χ=χ1χ2astχ3>0,Ri>0,T˜1i>0,T˜2i>0.Thus, the following linear matrix inequality can be obtained:(30)ϒ˜i,l,o,t,v,z=Γ˜ΓMΓN∗−T˜1i0∗∗−T˜2i<0,where(31)Γ˜=ϒ˜00Ξ˜11∗−γ2IΞ˜12∗∗Ξ˜13,ϒ˜0=ϒ000−χ,Ξ˜11=colΘ˜0Θ˜102×4Θ˜2Θ˜304×4Θ˜402×4,Θ˜0=L0L1L2L3L4L5L6L7L8,L0=AiT℘T+CiTBT,L1=AiTg+CiTBT,L2=AiTW1+CiTBT−W1,L3=AiTW2+CiTBT−W2,L4=AiTW1+CiTBT−W1,L5=AiTW2+CiTBT−W2,L6=AiTW1+CiTBT−W1,L7=AiTW2+CiTBT−W2,L8=LiT−CiTDT,Θ˜1=ATATAT−gTAT−gTAT−gTAT−gTAT−gTAT−gT−LT,Θ˜2=12M0M1M2M3M4M5M6M7M8,M0=AdiT℘T+CdiTBT,M1=AdiTL+CdiTBT,M2=AdiTW1+CdiTBT,M3=AdiTW2+CdiTBT,M4=AdiTW1+CdiTBT,M5=AdiTW2+CdiTBT,M6=AdiTW2+CdiTBT,M7=AdiTW1+CdiTBT,M8=AdiTW2+CdiTBT,Θ˜3=12N0N1N2N3N4N5N6N7N8,N0=AdiT℘T+CdiTBT,N1=AdiTL+CdiTBT,N2=AdiTW1+CdiTBT,N3=AdiTW2+CdiTBT,N4=AdiTW1+CdiTBT,N5=AdiTW2+CdiTBT,N6=AdiTW1+CdiTBDT,N7=AdiTW2+CdiTBT,N8=LdiT−CdiTDT,Θ˜4=τ¯2P0P1P2P3P4P5P6P7P8,P0=AdiT℘T+CdiTBT,P1=AdiTL+CdiTBT,P2=AdiTW1+CdiTBT,P3=AdiTW2+CdiTBT,P4=AdiTW1+CdiTBT,P5=AdiTW2+CdiTBT,P6=AdiTW1+CdiTBT,P7=AdiTW2+CdiTBT,P8=LdiT−CdiTDT,Ξ˜12=Q0Q1Q2Q3Q4Q5Q6Q7Q8,Q0=BiT℘T+DiTBT,Q1=BiTgT+DiTBT,Q2=BiTW1+DiTBT,Q3=BiTW2+DiTBT,Q4=BiTW1+DiTBT,Q5=BiTW2+DiTBT,Q6=BiTW1+DiTBT,Q7=BiTW2+DiTBT,Q8=FiT−DiTDT,Ξ˜13=−℘−g000000∗−gT000000∗∗μ−1S100000∗∗∗τ¯−1S20000∗∗∗∗Φ000∗∗∗∗∗−I00∗∗∗∗∗&lowast;−T˜1i0∗∗∗∗∗∗∗−T˜2i,S1=S1z1−W1−W1TS1z2−W2−g∗S1z3−g−gT,S2=S2z1−W1−W1TS2z2−W2−g∗S2z3−g−gT,Φ=χ1−W1−W1Tχ2−W2−g∗χ3−g−gT,T˜1i≜diagμT1i,3μT1i,5μT1i,T˜2i≜diagτ¯T2i,3τ¯T2i,5τ¯T2i.Then, the fuzzy filter state space parameters can be implemented as(32)AFℏ=g−1A,BFℏ=g−1B,LFℏ=L,DFℏ=D.Proof.
First, define the matrixV=V1V2∗V3, where V1∈ℝn×n,V3∈ℝn×n are positive-definite symmetric matrices. Here, suppose V2∈ℝn×n is a nonsingular matrix, and define the following nonsingular matrix:(33)f=I00V3−1V2T,W=W1W2V2−TV3V2TV3,g≜V2V3−1V2T,℘≜V1,Q1i≜g−TQ˜1ig−1,Q2ii≜g−TQ˜2ig−1,S1z≜g−TS˜1zg−1,S2z≜g−TS˜2zg−1,T˜1i≜g−TT˜1ig−1,T˜2i≜g−TT˜2ig−1,Mq≜g−TMq˜qg−1,Nq≜g−TN˜qg−1,χ≜g−Tχq˜qg−1.
Given thatV > 0, linear matrix inequality (22) can be obtained:(34)AFℏBFℏLFℏDFℏ≜V2−100IAFℏBFℏLFℏDFℏV3−1V2T00I.
Then, perform contract matrix transformation on equation (12):(35)J1Tϒi,l,o,t,v,zJ1>0,where J1 = diag{ f, f, f, f, I, f, f, f, f, I}, and by combining (24) and (25), we can get linear matrix inequality (22). Here, note that equation (22) can be equivalent to equation (27):(36)AFℏBFℏLFℏDFℏ≜V2−100IABLDV2−TV300I≜V2−TV3−1g−100IABLDV2−TV300I.
Therefore, the state space implementationAFℏ,BFℏ,LFℏ,DFℏ of the desired filter as defined in (6) can be obtained from (22). Then, this completes the proof.Remark 5.
Regarding the design process of theH∞ filter, first, expand (21) to form an augmented matrix in the form of an LMI. It is found here that the filter parameter matrices AFℏ,BFℏ,LFℏ,DFℏ can be written as (36),which implies that V2–T V3 can be viewed as a similarity transformation on the state-space realization of the filter. Therefore, the contract matrix transformation method is used to decouple the matrix to be sought from the Lyapunov–Krasovskii matrix, so that the filter parameter matrix can be systematically replaced and obtained.


## 5. Numerical Examples

Example 1.
(see [21, 53]). Firstly, we consider discrete-time system (4) with time-varying delay:(37)A1=−0.291100.95,Ad1=0.0120.01400.015,A2=−0.101−0.2,Ad2=0.0100.010.015,where τt represents time-varying delay and the number of delay-partitioning segment m = 3. Furthermore, the upper bound of the delay is given in combination with the method in Theorem 1. Table 1 shows a detailed comparison. It can be seen that the method proposed in this paper is superior to the results published in the literature [20, 22, 53].Table 1
The upper bound ofτ1 with various τ2 (Example 1).
τ2312[20, Theorem 1]1421[22,Theorem 3]23(m = 3)32(m = 3)[53,Corollary 1]100(m = 3)109(m = 3)Theorem 2110(m = 3)130(m = 3)Example 2.
(see [21]). Then, we consider the following time-varying delay Henon mapping system:(38)x1t+1=−Cx1t+1−Cx1t−τt2+0.3x2+ωt,x2t+1=Cx1t+1−Cx1t−τt,yt=Cx1t+1−Cx1t−τt+ωt,zt=x1t,where ω(t) is the disturbance input and C ∈[0, 1] is a constant representing the delay coefficient. Next, we define ϑt=∐x1t+1−∐x1t−τ, and suppose that ϑt∈−M,M,M>0. By utilizing the same procedure as in paper [54], the nonlinear term ϑ2(t) could be expressed as ϑ2t=h1ϑt−Mϑt+h2ϑtMϑt, where h1ϑt,h2ϑt∈0,1, and h1ϑt+h2ϑt=1. Then, by solving the equations, the membership functions h1 (ϑ(t)) and h2 (ϑ(t)) can be obtained as(39)h1ϑt=121−ϑtM,h2ϑt=121+ϑtM.
It can be concluded from the above expression thath1(ϑ(t)) = 1, h2(ϑ(t)) = 0 when ϑt=−M, and h1(ϑ(t)) = 0, h2(ϑ(t)) = 1 when ϑt=M. Thus, nonlinear system (30) can be expressed as the following approximate TS fuzzy model.
Plant From.Rule 1.
IFϑ(t) is −M, THEN(40)xt+1=A1xt+Ad1xt−τt+B1ωt,yt=C1xt+Cd1xt−τt+D1ωt,zt=C1xt.Rule 2.
IFϑ(t) is M, THEN(41)xt+1=A2xt+Ad2xt−τt+B2ωt,yt=C2xt+Cd2xt−τt+D2ωt,zt=L2xt,where(42)A1=CM0.3C0,Ad1=1−CM01−C0,A2=−CM0.3C0,Ad2=−1−CM01−C0,B1=10,B2=10,C1=C0,Cd1=1−C0,C2=C0,Cd2=1−C0,D1=1,L1=10,D2=0.5,L2=10.
Here,x(t) = col{x1(t), x2(t)}, C=0.8,M=0.2,m=2, and 1 ≤ τ(t) ≤ 3 means time-varying state delay. Then, by solving the conditions in Theorem 2, the minimum upper bound of the H∞ performance level obtained is γ = 0.04574. It can be seen that the result obtained in this section is smaller than the result obtained in reference [53], γ = 2.0403. The parameters of the full-order fuzzy filter are as follows:(43)AFℏ=0.70320.05160.00190.6709,BFℏ=−1.1930−0.0536,LFℏ=−2.0658×10−5−2.5756×10−5,DFℏ=−6.4756×10−5.
Then, in the case where the initial condition is zero, that isx(0) = 0, assume that the disturbance input ω(t) is(44)ωt=3sin0.85t0.55t2+1.
Utilize the member function shown in Figure1. Also, it can be concluded from the simulation results that Figures 2 and 3 reveal the signal z(t) with its estimation zF(t) of the full-order filters and estimation error e(t), respectively.Figure 1
Membership functions (Example2).Figure 2
Signal z(t) and its estimation zF(t) of the filter (Example 2).Figure 3
Estimation errore(t) (Example 2).


## 6. Conclusions

The main content of this paper is the analysis and design ofH∞ filter for discrete-time TS fuzzy systems with time-varying delay. In this paper, the bounded real lemma based on the improved Wirtinger-based inequality proposed in Section 3 is generalized and applied to the analysis and synthesis of nonlinear systems. For a given nonlinear system with time-varying delays and external disturbances, TS fuzzy logic is used to model it, and the nonlinear problem is transformed into a linear system analysis and synthesis problem. Based on the bounded real lemma and Wirtinger-based inequality with quadratic terms, H∞ performance analysis is performed on the discrete-time TS fuzzy system with time-varying delay, and the coupling terms of the filter matrix and the Lyapunov matrix are uniformly replaced using the contract matrix transformation method. Furthermore, the design method of H∞ filter for TS fuzzy system is given, and the bounded real lemma proposed in this paper is successfully extended to the analysis and synthesis of nonlinear systems. Finally, simulation examples are given to prove the validity and low conservativeness of the conclusion.



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