# Robustl2-l∞ Filtering for Discrete-Time Delay Systems

Authors: Chengming Yang; Zhandong Yu; Pinchao Wang; Zhen Yu; Hamid Reza Karimi; Zhiguang Feng
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

The problem of robustl2-l∞ filtering for discrete-time system with interval time-varying delay and uncertainty is investigated, where the time delay and uncertainty considered are varying in a given interval and norm-bounded, respectively. The filtering problem based on the l2-l∞ performance is to design a filter such that the filtering error system is asymptotically stable with minimizing the peak value of the estimation error for all possible bounded energy disturbances. Firstly, sufficient l2-l∞ performance analysis condition is established in terms of linear matrix inequalities (LMIs) for discrete-time delay systems by utilizing reciprocally convex approach. Then a less conservative result is obtained by introducing some variables to decouple the Lyapunov matrices and the filtering error system matrices. Moreover, the robust l2-l∞ filter is designed for systems with time-varying delay and uncertainty. Finally, a numerical example is given to demonstrate the effectiveness of the filter design method.


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## Body

## 1. Introduction

The uncertainty is unavoidable in practical engineering due to the parameter drafting, modeling error, and component aging. The controllers or filtering obtained based on nominal systems cannot be employed to get the desired performance. Therefore, more and more researchers are devoted to robust control or robust filtering problems; see, for instance, [1–4]. On the other hand, time-delay often exists in the practical engineering systems and is the main reason of the instability and poor performance of the systems. Time-delay systems have been widely studied during the past two decades [5–7]. In order to get less conservative results, more and more approaches have been proposed to develop delay-dependent conditions for discrete-time system with time-varying delay. For examples, Jensen's inequality is proposed in [8]; delay-partitioning method is utilized in [9]; improved results are obtained by using convex combination approach in [10].In some practical applications, the peak value of the estimation error is required to be within a certain range and the aim of thel2-l∞ (energy-to-peak) filtering is to minimize the peak values of the filtering error for any bounded energy disturbance, which has received many attention. By using a parameter-dependent approach, the robust energy-to-peak filtering problem is considered in [11]. An improved robust energy-to-peak filtering condition is proposed by increasing the flexible dimensions in the solution space in [12]. The robust L2-L∞ filtering for stochastic systems and the exponential L2-L∞ filtering for Markovian jump systems are investigated in [13, 14], respectively. Compared with the corresponding continuous-time systems, discrete-time systems with time-varying delay have more stronger application background [15]. For discrete-time Markovian jumping systems, the reduced-order filter is designed in [16] such that the filtering error system satisfies an energy-to-peak performance. When time-delay appears, the robust energy-to-peak filtering problem for networked systems is tackled in [17]. For discrete-time switched systems with time-varying delay, an improved robust energy-to-peak filtering design method is proposed in [18].In this paper we consider the problem of robustl2-l∞ filtering for uncertain discrete-time systems with time-varying delay. The filter is designed by employing the reciprocally convex approach proposed in [19] such that the filtering error system is asymptotically stable with an l2-l∞ performance. Firstly, a sufficient condition of the l2-l∞ performance analysis for nominal systems is obtained in terms of LMIs for systems with time-varying delay and uncertainty. Based on this criterion, by introducing some slack matrices, a less conservative result is obtained. Moreover, the desired filter for nominal systems with time-varying delay is obtained by solving a set of LMIs. Then the result is extended to the uncertain systems. A numerical example is given to illustrate the effectiveness of the presented results.Notation. The notation used throughout the paper is given as follows. ℝn is the n-dimensional Euclidean space and P>0 (≥0) denotes that matrix P is real symmetric and positive definite (semidefinite); I and 0 present the identity matrix and zero matrix with compatible dimensions, respectively; ⋆ means the symmetric terms in a symmetric matrix and sym(A) stands for A+AT; l2 means the space of square summable infinite vector sequences; for any real function x∈l2, we define ∥x∥∞=supkxT(k)x(k) and ∥x∥2=∑k=0∞‍xT(k)x(k); ∥·∥ refer to the Euclidean vector norm. Matrices are assumed to be compatible for algebraic operations if their dimensions are not explicitly stated.


## 2. Problem Statement

Consider a class of uncertain discrete-time systems with time-varying delay described by(1)x(k+1)=A(σ)x(k)+Ad(σ)x(k-d(k))+B(σ)w(k),y(k)=C(σ)x(k)+Cd(σ)x(k-d(k))+D(σ)w(k),z(k)=L(σ)x(k)+Ld(σ)x(k-d(k))+G(σ)w(k),x(k)=ϕ(k),k=-d2,-d2+1,…,0,
where x(k)∈ℝn is the state vector; y(k)∈ℝm is the measured output; z(k)∈ℝp represents the signal to be estimated; w(k)∈ℝl is assumed to be an arbitrary noise belonging to l2 and ϕ(k) is a given initial condition sequence; d(k) is a time-varying delay satisfying
(2)1≤d1≤d(k)≤d2<∞,k=1,2,…A(σ), Ad(σ), B(σ), C(σ), Cd(σ), D(σ), L(σ), Ld(σ), and G(σ) are system matrices and satisfy
(3)A(σ)=A+ΔA(σ),Ad(σ)=Ad+ΔAd(σ),B(σ)=B+ΔB(σ),C(σ)=C+ΔC(σ),Cd(σ)=Cd+ΔCd(σ),D(σ)=D+ΔD(σ),L(σ)=L+ΔL(σ),Ld(σ)=Ld+ΔLd(σ),G(σ)=G+ΔG(σ).
Matrices ΔA(σ), ΔAd(σ), ΔB(σ), ΔC(σ), ΔCd(σ), ΔD(σ), ΔL(σ), ΔLd(σ), and ΔG(σ) are unknown time-invariant matrix representing the uncertainty of the system satisfying the following conditions:
(4)[ΔA(σ)ΔAd(σ)ΔB(σ)]=M1Δ1(σ)[NANAdNB],[ΔA(σ)ΔAd(σ)ΔB(σ)]p=M1Δ1(σ)Δ1T(σ)Δ1(σ)≤I,[ΔC(σ)ΔCd(σ)ΔD(σ)]=M2Δ2(σ)[NCNCdND],[ΔC(σ)ΔCd(σ)ΔD(σ)]p=M2Δ2(σ)Δ2T(σ)Δ2(σ)≤I,[ΔL(σ)ΔLd(σ)ΔG(σ)]=M3Δ3(σ)[NLNLdNG],[ΔL(σ)ΔLd(σ)ΔG(σ)].=M3Δ3(σ)Δ3T(σ)Δ3(σ)≤I,
where σ∈Θ and Θ is a compact set in ℝ. The system in (1) is assumed to be asymptotically stable. Our purpose is to design a full order linear filter for the estimate of z(k):
(5)x^(k+1)=Afx^(k)+Bfy(k),x^(0)=0,z^(k)=Cfx^(k)+Dfy(k),
where Af, Bf, Cf, and Df are filter gains to be determined.Let the augmented state vectorx~(k)=[xT(k)x^T(k)]T and z~(k)=z(k)-z^(k). Then the filtering error system is described as
(6)x~(k+1)=A~(σ)x~(k)+A~d(σ)Φx~(k-d(k))+B~(σ)w(k)z~(k)=L~(σ)x~(k)+L~d(σ)Φx~(k-d(k))+G~(σ)w(k)x~(k)=[ϕT(k)0]T,k=-d2,-d2+1,…,0,
where Φ=[I0] and
(7)A~(σ)=[A(σ)0BfC(σ)Af],A~d(σ)=[Ad(σ)BfCd(σ)],B~=[B(σ)BfD(σ)],L~(k)=[L(σ)-DfC(σ)-Cf],L~d(σ)=Ld(σ)-DfCd(σ),G~(σ)=G(σ)-DfD(σ).The nominal system of (6) is system (6) without uncertainty; that is, ΔA(σ)=0, ΔAd(σ)=0, ΔB(σ)=0, ΔC(σ)=0, ΔCd(σ)=0, ΔD(σ)=0, ΔL(σ)=0, ΔLd(σ)=0, and ΔG(σ)=0.The following lemmas and definition will be utilized in the derivation of the main results.Lemma 1 (see [20]).
For any matricesU and V>0, the following inequality holds:
(8)UV-1UT≥U+UT-V.Lemma 2 (see [19]).
Letf1,f2,…,fN:ℝm→ℝ have positive values in a subset D of ℝm. Then, the reciprocally convex combination of fi over D satisfies
(9)min{αi∣αi>0,∑iαi=1}∑i‍1αifi(k)=∑i‍fi(k)+maxgi,j(k)∑i≠j‍gi,j(k)
subject to
(10){gi,j:ℝm⟶ℝ,gj,i(k)=gi,j(k),[fi(k)gi,j(k)gj,i(k)fj(k)]≥0}.Lemma 3.
For any constant matrixM>0, integers a≤b, vector function w: {a,a+1,…,b}→ℝn, then
(11)-(b-a+1)∑i=ab‍wT(i)Mw(i)≤-(∑i=ab‍w(i))TM(∑i=ab‍w(i)).Lemma 4.
Given a symmetric matrixQ and matrices H, E with appropriate dimensions, then
(12)Q+sym(HFE)<0,
for all FTF≤I, if and only if there exists a scalar ɛ>0 such that
(13)Q+ɛETE+ɛ-1HHT<0.Definition 5.
Given a scalarγ>0, the filtering error z~(k) in (6) is said to satisfy the l2-l∞ disturbance attenuation level γ under zero initial state, and the following condition is satisfied:
(14)∥z~∥∞<γ∥w∥2.Our aim is to design a filter in the form of (5) such that the filtering error system in (6) is asymptotically stable and satisfies the l2-l∞ performance defined in Definition 5.


## 3. Main Results

In this section, the sufficientl2-l∞ performance analysis condition is first derived for nominal filtering error system of (6). Then an equivalent result is obtained by introducing three slack matrices. Based on these results, a desired filter is designed to render the nominal system of (6) asymptotically stable with an l2-l∞ performance. Then the result is extended to the uncertain system in (6).


### 3.1.l2-l∞ Performance Analysis

In this subsection, we first give the result ofl2-l∞ performance analysis for nominal system of (6).Theorem 6.
Given a scalarγ>0, the nominal system of (6) is asymptotically stable with an l2-l∞ performance if there exist matrices P>0,  Q3>0, Q¯i>0,   i=1,2, Sj>0,   j=1,2, and M such that the following LMIs hold:
(15)[S2M⋆S2]≥0,(16)[P00L~T⋆Q30L~dT⋆⋆IG~T⋆⋆⋆γ2I]>0,(17)Π~=[Π~11ΦTS1000Π~16S1Π~17S2A~TP⋆Π~22Π~23MT0000⋆⋆Π~33Π~340d1AdTS1d~AdTS2A~dTP⋆⋆⋆Π~440000⋆⋆⋆⋆-Id1BTS1d~BTS2B~TP⋆⋆⋆⋆⋆-S100⋆⋆⋆⋆⋆⋆-S20⋆⋆⋆⋆⋆⋆⋆-P]<0,
where
(18)Π~11=-P+Q1+Q2+(d~+1)ΦTQ3Φ-ΦTS1Φ,Π~22=-S2-Q¯1-S1,Π~23=S2-MT,Π~33=-Q3-2S2+sym(M),Π~34=S2-MT,Π~44=-Q¯2-S2,Π~16=d1ΦT(A-I)T,Π~17=d~ΦT(A-I)T,Qi=diag{Q¯i,Q¯i},i=1,2,d~=d2-d1.Proof.
First, the asymptotic stability of the nominal system of (6) is proved. We denote η~(k)=x~(k+1)-x~(k) and the following Lyapunov functional is chosen:
(19)V(k)=V1(k)+V2(k)+V3(k)+V4(k)+V5(k),
where
(20)V1(k)=x~T(k)Px~(k),V2(k)=∑j=12‍∑i=k-djk-1‍x~T(i)Qix~(i),V3(k)=∑i=k-d(k)k-1‍x~T(i)ΦTQ3Φx~(i)=+∑j=-d2+1-d1‍∑i=k+jk-1‍x~T(i)ΦTQ3Φx~(i),V4(k)=∑j=-d1-1‍∑i=k+jk-1‍d1η~T(i)ΦTS1Φη~(i),V5(k)=∑j=-d2-d1-1‍∑i=k+jk-1‍d~η~T(i)ΦTS2Φη~(i).
Calculating the forward difference of V(k) along the trajectories of filtering error system (6) with w(k)=0 yields
(21)ΔV1(k)=x~T(k+1)Px~(k+1)-x~T(k)Px~(k)=(A~x~(k)+A~dΦx~(k-d(k)))T=×P(A~x~(k)+A~dΦx~(k-d(k)))=-x~T(k)Px~(k),(22)ΔV2(k)=∑j=12‍x~T(k)Qjx~(k)=-∑j=12‍x~(k-dj)Qjx~(k-dj)≤∑j=12‍x~T(k)Qjx~(k)=-∑j=12‍x~(k-dj)ΦTQ¯jΦx~(k-dj),(23)ΔV3(k)=(d~+1)x~T(k)ΦTQ3Φx~(k)=+∑i=k+1-d(k+1)k-1‍x~T(i)ΦTQ3Φx~(i)=-∑i=k+1-d(k)k-1‍x~T(i)ΦTQ3Φx~(i)=-x~T(k-d(k))ΦTQ3Φx~(k-d(k))=-∑i=k-d2+1k-d1‍x~T(i)ΦTQ3Φx~(i)=(d~+1)x~T(k)ΦTQ3Φx~(k)=+∑i=k+1-d1k-1‍x~T(i)ΦTQ3Φx~(i)=+∑i=k+1-d(k+1)k-d1‍x~T(i)ΦTQ3Φx~(i)=-x~T(k-d(k))ΦTQ3Φx~(k-d(k))=-∑i=k+1-d(k)k-1‍x~T(i)ΦTQ3Φx~(i)=-∑i=k-d2+1k-d1‍x~T(i)ΦTQ3Φx~(i)≤(d~+1)x~T(k)ΦTQ3Φx~(k)=-x~T(k-d(k))ΦTQ3Φx~(k-d(k)).
By using Lemma3, we have
(24)ΔV4(k)=d12η~T(k)ΦTS1Φη~(k)=-d1∑i=k-d1k-1‍η~T(i)ΦTS1Φη~(i)≤d12((A-I)Φx~(k)+AdΦx~(k-d(k)))T=×S1((A-I)Φx~(k)+AdΦx~(k-d(k)))=-(Φx~(k)-Φx~(k-d1))T=×S1(Φx~(k)-Φx~(k-d1)).
Since[S2M⋆S2]≥0, the following inequality holds:
(25)[α1α2(x(k-d(k))-x(k-d2))-α2α1(x(k-d1)-x(k-d(k)))]T×[S2M⋆S2]×[α1α2(x(k-d(k))-x(k-d2))-α2α1(x(k-d1)-x(k-d(k)))]≥0,
where α1=(d2-d(k))/d~ and α2=(d(k)-d1)/d~. Then employing Lemma 2, for d1<d(k)<d2, we have
(26)ΔV5(k)=d~2η~T(k)ΦTS2Φη~(k)=-d~∑i=k-d2k-d(k)-1‍η~T(i)ΦTS2Φη~(i)=-d~∑i=k-d(k)k-d1-1‍η~T(i)ΦTS2Φη~(i)≤d~2η~T(k)ΦTS2Φη~(k)=-d~d2-d(k)(∑i=k-d2k-d(k)-1‍Φη~(i))TS2(∑i=k-d2k-d(k)-1‍Φη~(i))=-d~d(k)-d1(∑i=k-d(k)k-d1-1‍Φη~(i))TS2(∑i=k-d(k)k-d1-1‍Φη~(i))≤d~2((A-I)Φx~(k)+AdΦx~(k-d(k)))T=×S2((A-I)Φx~(k)+AdΦx~(k-d(k)))=-[x(k-d(k))-x(k-d2)x(k-d1)-x(k-d(k))]T[S2M⋆S2]=×[x(k-d(k))-x(k-d2)x(k-d1)-x(k-d(k))].
Note that when d(k)=d1 or d(k)=d2, it yields x(k-d1)-x(k-d(k))=0 or x(k-d(k))-x(k-d2)=0. Hence, the inequality in (24) still holds. Combining the conditions from (21) to (24), we have
(27)ΔV(k)=ζT(k)Πsζ(k),
where
(28)ζ(k)=[x~T(k)x~T(k-d1)ΦTx~T(k-d(k))ΦTx~T(k-d2)ΦT]T,(29)Πs=[Π~11ΦTS100⋆Π~22Π~23MT⋆⋆Π~33Π~34⋆⋆⋆Π~44]+[Π~160d1AdT0]S1[Π~160d1AdT0]T=+[Π~170d~AdT0]S2[Π~170d~AdT0]T+[A~T0A~dT0]P[A~T0A~dT0]T.
On the other hand, the following inequality can be obtained from (17):
(30)Πs1=[Π~11ΦTS100Π~16S1Π~17S2A~TP⋆Π~22Π~23MT000⋆⋆Π~33Π~34d1AdTS1d~AdTS2A~dTP⋆⋆⋆Π~44000⋆⋆⋆⋆-S100⋆⋆⋆⋆⋆-S20⋆⋆⋆⋆⋆⋆-P]<0
which is equivalent to Πs<0. Hence, ΔV(k)<0 which implies that the filtering error system in (6) with w(k)=0 is asymptotically stable.
Next, we show thel2-l∞ performance of system (6). To this end, we define
(31)J(k)=V(k)-∑i=0k-1‍wT(i)w(i).
Then under the zero initial condition, that is, x(k)=0,   k=-d2,-d2+1,…,0, it can be shown that for any nonzero w(k)∈l2[0,∞)(32)J(k)=∑i=0k-1‍[ΔV(i)-wT(i)w(i)]=∑i=0k-1‍ξT(i)(Π¯+Π¯1TPΠ¯1+d12Π¯2TS1Π¯2=∑i=0k-1‍5ξT(i)(+d122Π¯2TS2Π¯2)ξ(i),
where
(33)ξ(i)=[x~T(i)x~T(i-d1)ΦTx~T(i-d(i))ΦTx~T(i-d2)ΦTw(i)]T,Π¯=[Π~11ΦTS100Π~15⋆Π~22Π~23MT0⋆⋆Π~33Π~34-L~dTS⋆⋆⋆Π~440⋆⋆⋆⋆Π~55],Π¯1=[A~0A~d0B~],Π¯2=[(A-I)Φ0Ad0B],Π¯3=[L~0L~d0G~].
By using Schur complement equivalence, the inequality in (17) is equivalent to Π¯+Π¯1TPΠ¯1+d12Π¯2TS1Π¯2+d122Π¯2TS2Π¯2<0. Then we have J(k)<0; that is,
(34)V(k)<∑i=0k-1‍wT(i)w(i).
On the other hand, it yields from (16) and (34) that
(35)z~T(k)z~(k)=ηT(k)[L~L~dG~]T[L~L~dG~]η(k)≤γ2ηT(k)[P00⋆Q30⋆⋆I]η(k)≤γ2(V(k)+wT(k)w(k))≤γ2∑i=0k‍wT(i)w(i)≤γ2∑i=0∞‍wT(i)w(i),
where
(36)η(k)=[x~(k)Φx~(k-d(k))w(k)].
Then, we have ∥z~∥∞<γ∥w∥2 by taking the supremum over time k>0. By Definition 5, the filtering error z~(k) satisfies a given l2-l∞ disturbance attenuation level. This completes the proof.Remark 7.
The advantage of the results benefits from utilizing the reciprocally convex combination approach proposed in [19]. For the extensively used Jensen inequality [8], the integral term
(37)-∑i=k-d2k-d1-1(d2-d1)ηT(i)Sη(i)=-∑i=k-d2k-d(k)-1(d2-d1)ηT(i)Sη(i)=-∑i=k-d(k)k-d1-1(d2-d1)ηT(i)Sη(i)
with d1≤d(k)≤d2,   η(i)=x(i+1)-x(i) by the term
(38)-[x(k-d1)-x(k-d(k))]TS[x(k-d1)-x(k-d(k))]-[x(k-d(k))-x(k-d2)]TS[x(k-d(k))-x(k-d2)]
which is a special case of the following term with M=0(39)-[x(k-d(k))-x(k-d1)x(k-d(k))-x(k-d2)]T×[SM⋆S][x(k-d(k))-x(k-d1)x(k-d(t))-x(k-d2)]
with [SM⋆S]≥0, which is one of the advantages of reciprocally convex combination approach. On the other hand, the delay partitioning method is widely applied to reduce the conservatism of the results [9, 21, 22]. Also, the method can be extended to the problem considered in this paper. However, it will rise significant computation cost with the partitioning number increasing. Therefore, the reciprocally convex method needs less decision variables and can be seen as a tradeoff between the conservatism and the computation cost.Then, an equivalent condition of LMI (17) is obtained by introducing three slack matrices H1, H2, and T, which is presented in the following theorem.Theorem 8.
Given a scalarγ>0, the nominal system of (6) is asymptotically stable with an l2-l∞ performance if there exist matrices P>0,    Q¯i>0,  i=1,2,3, Sj>0, Hj, j=1,2, T, and M, such that the following LMIs hold:
(40)[S2M⋆S2]≥0,(41)[P00L~T⋆Q30L~dT⋆⋆IG~T⋆⋆⋆γ2I]>0,(42)Ω~=[Π~11ΦTS1000Π~16H1TΠ~17H2TA~TTT⋆Π~22Π~23MT0000⋆⋆Π~33Π~340d1AdTH1Td~AdTH2TA~dTTT⋆⋆⋆Π~440000⋆⋆⋆⋆-Id1BTH1Td~BTH2TB~TTT⋆⋆⋆⋆⋆S1-H1T-H100⋆⋆⋆⋆⋆⋆S2-H2T-H20⋆⋆⋆⋆⋆⋆⋆P-TT-T]<0,where Π~ii, i=1,…,4, Π~16, Π~17, Π~23, and Π~34 are defined in (17).Proof.
On one hand, if (17) holds, then there exist Hj=HjT=Sj, j=1,2, and T=TT=P such that (42) holds. On the other hand, if (42) holds, we have the following inequality based on Lemma 1:
(43)[Π~11ΦTS1000Π~16H1TΠ~17H2TA~TTT⋆Π~22Π~23MT0000⋆⋆Π~33Π~340d1AdTH1Td~AdTH2TA~dTTT⋆⋆⋆Π~440000⋆⋆⋆⋆-Id1BTH1Td~BTH2TB~TTT⋆⋆⋆⋆⋆-H1S1-1H1T00⋆⋆⋆⋆⋆⋆-H2S2-1H2T0⋆⋆⋆⋆⋆⋆⋆-TP-1TT].
In addition, matrices Hj, j=1,2, and T are nonsingular due to Sj-HjT-Hj<0, j=1,2, and P-TT-T<0. Then, pre- and promultiplying (43) by diag{I,I,I,I,I,S1H1-1,S2H2-1,PT-1} and its transpose yields (17). Therefore, the equivalence between (42) and (17) is proved.


### 3.2. Robust Filter Design

In this subsection, the filter in the form of (5) is firstly designed such that the nominal filtering error system of (6) is asymptotically stable with an l2-l∞ performance. Then the robust filtering problem is solved. Based on the result of Theorem 8, the filter design method for nominal system of (1) is presented in the following theorem.Theorem 9.
Given a scalarγ>0, the nominal system of (6) is asymptotically stable with an l2-l∞ performance if there exist matrices [P1P2⋆P3]>0, Q3>0, Q~l>0, Q¯l>0, l=1,2, Sj>0, Hj, Fj, j=1,2, diagonal matrix N>0, T1, and M such that the following set of LMIs hold:
(44)[S2M⋆S2]≥0(45)Ω=[ΞΓ⋆Λ]<0(46)Υ=[P1P200(L-D¯fC)T⋆P300-C¯fT⋆⋆Q30(Ld-D¯fCd)T⋆⋆⋆I(G-D¯fCd)T⋆⋆⋆⋆γ2I]>0,
where
(47)Ξ=[Ξ11-P2S1CTNCd00⋆Ξ220000⋆⋆Ξ33S2-MTMT0⋆⋆⋆Ξ44S2-MT0⋆⋆⋆⋆Ξ550⋆⋆⋆⋆⋆-I],Γ=[Γ11Γ12ATT1T+CT𝒜s0B¯fTATF1T+CT𝒜s0B¯fT00A¯fTA¯fT0000d1AdTH1Td~AdTH2TAdTT1T+CdT𝒜s0B¯fTAdTF1T+CdiT𝒜s0B¯fT0000d1BTH1Td~BTH2TBTT1T+DT𝒜s0B¯fTBTF1T+DT𝒜s0B¯fT],Λ=[S1-H1-H1T000⋆S2-H2-H2T00⋆⋆P1-T1-T1TP2-F2-F1T⋆⋆⋆P3-F2-F2T],Ξ11=-(P1+S1)+Q¯1+Q¯2+(d~+1)Q3,Ξ22=-P3+Q~1+Q~2,Ξ33=-S2-Q¯1-S1,Ξ44=-Q3+sym(Mi-S2i),Ξ55=-Q¯2-S2,Γ11=d1(A-I)TH1T,Γ12=d~(A-I)TH2T.
Moreover, a suitable l2-l∞ filter is given by
(48)Af=A¯fF2-1,Bf=B¯f,Cf=C¯fF2-1,Df=D¯f.Proof.
Firstly, we introduce four matricesT1, T2, T3, and T4 with T4 invertible and define
(49)J1=[I00T2T4-1],F1=T2T4-1T3,F2=T2T4-TT2T,Q~l=T2T4-1Q¯lT4-TT2T,J=diag{J1,I,I,I,I,I,I,I,J1},[P1P2⋆P3]=J1PJ1T,T=[T1T2T3T4],J2=diag{J1,I,I,I},[A¯fB¯fC¯fD¯f]=[T200I][AfBfCfDf][T4-TT2T00I].
From (45), we have F2+F2T=T2T4-TT2T+T2T4-1T2T>0 which implies that T2 is nonsingular. Hence, J and J2 are nonsingular. The inequality in (46) can be obtained by pre- an promultiplying (34) with J2 and J2T, respectively. Noting that
(50)Ω¯=JΩ~JT
we have Ω~<0. On the other hand, because T2 and T4 cannot be obtained from (45), we cannot determine the filters from (49). However, we can construct an equivalent filter transfer function from y(k) to z~(k):
(51)Tz~y=Cf(zI-Af)-1Bf+Df=C¯fT2-TT4T(zI-T2-1A¯fT2-TT4T)-1T2-1B¯f+D¯f=C¯f(zF2-A¯f)-1B¯f+D¯f=C¯fF2-1(zI-A¯fF2-1)-1B¯f+D¯f.
Therefore, the desired filter can be obtained from (48). This completes the proof.Then the filter design result for uncertain system (6) is presented in the following theorem.Theorem 10.
Given a scalarγ>0, the system in (6) with uncertainty is asymptotically stable with an l2-l∞ performance if there exist matrices [P1P2⋆P3]>0,   Q3>0,  Q~l>0,  Q¯l>0, l=1,2,  Sj>0, Hj, Fj, j=1,2, diagonal matrix N>0, T1, M, and scalars ɛi>0,  i=1,…,4 such that the following set of LMIs hold:
(52)[S2M⋆S2]≥0,(53)[Ω+ɛ3Ω1TΩ1+ɛ4Ω2TΩ2Ω3Ω4⋆-ɛ3I0⋆⋆-ɛ4I]<0,(54)[Υ11P2Υ13Υ14(L-D¯fC)T00⋆P300-C¯fT00⋆⋆Υ33Υ34(Ld-D¯fCd)T00⋆⋆⋆Υ44(G-D¯fCd)T00⋆⋆⋆⋆γ2IM30⋆⋆⋆⋆⋆ɛ1ID¯fM2⋆⋆⋆⋆⋆⋆ɛ2I]>0,
where Ω is defined in (45) and
(55)Ω1=[NA00NAd0NB0000],Ω2=[NC00NCd0ND0000],Ω3=[000000d1M1TH1Td~M1TH2TM1TT1TM1TF1T]T,Ω4=[00000000M2TB~fTM2TB~fT]T,Υ11=P1-ɛ1NLTNL-ɛ2NCTNC,Υ13=-ɛ1NLTNLd-ɛ2NCTNCd,Υ14=-ɛ1NLTNG-ɛ2NCTND,Υ33=Q3-ɛ1NLdTNLd-ɛ2NCdTNCd,Υ34=-ɛ1NLdTNG-ɛ2NCdTND,Υ44=I-ɛ1NGTNG-ɛ2NDTND.
Moreover, a suitable l2-l∞ filter is given by
(56)Af=A¯fF2-1,Bf=B¯f,Cf=C¯fF2-1,Df=D¯f.Proof.
Firstly, replace matricesA, Ad, B, C, Cd, and D in (45) with A+ΔA, Ad+ΔAd, B+ΔB,  C+ΔC, Cd+ΔCd, and D+ΔD, respectively, and the following inequality is obtained:
(57)Ω+sym(Ω1TΔ1TΩ3T)+sym(Ω2TΔ2TΩ4T)<0,
where Ωi, i=1,…,4 are defined in (53). Then by using Lemma 4, the above inequality holds if and only if
(58)Ω+ɛ3Ω1TΩ1+ɛ3-1Ω3Ω3T+ɛ4Ω2TΩ2+ɛ4-1Ω4Ω4T<0.
Then by using Schur complement equivalence, the inequality in (58) is equivalent to (45). Substituting L, Ld, and G in (46) with L+ΔL,  Ld+ΔLd, and G+ΔG, respectively, we can get
(59)Υ+sym(Υ1TΔ3TΥ3T)+sym(Υ2TΔ2TΥ4T)>0,
where
(60)Υ1=[NLNLdNG0],Υ2=[NCNCdND0]Υ3=[000M3T],Υ4=[000M2TD¯fT].
By following the similar line, the equivalence between (59) and (54) can be proved.


## 3.1.l2-l∞ Performance Analysis

In this subsection, we first give the result ofl2-l∞ performance analysis for nominal system of (6).Theorem 6.
Given a scalarγ>0, the nominal system of (6) is asymptotically stable with an l2-l∞ performance if there exist matrices P>0,  Q3>0, Q¯i>0,   i=1,2, Sj>0,   j=1,2, and M such that the following LMIs hold:
(15)[S2M⋆S2]≥0,(16)[P00L~T⋆Q30L~dT⋆⋆IG~T⋆⋆⋆γ2I]>0,(17)Π~=[Π~11ΦTS1000Π~16S1Π~17S2A~TP⋆Π~22Π~23MT0000⋆⋆Π~33Π~340d1AdTS1d~AdTS2A~dTP⋆⋆⋆Π~440000⋆⋆⋆⋆-Id1BTS1d~BTS2B~TP⋆⋆⋆⋆⋆-S100⋆⋆⋆⋆⋆⋆-S20⋆⋆⋆⋆⋆⋆⋆-P]<0,
where
(18)Π~11=-P+Q1+Q2+(d~+1)ΦTQ3Φ-ΦTS1Φ,Π~22=-S2-Q¯1-S1,Π~23=S2-MT,Π~33=-Q3-2S2+sym(M),Π~34=S2-MT,Π~44=-Q¯2-S2,Π~16=d1ΦT(A-I)T,Π~17=d~ΦT(A-I)T,Qi=diag{Q¯i,Q¯i},i=1,2,d~=d2-d1.Proof.
First, the asymptotic stability of the nominal system of (6) is proved. We denote η~(k)=x~(k+1)-x~(k) and the following Lyapunov functional is chosen:
(19)V(k)=V1(k)+V2(k)+V3(k)+V4(k)+V5(k),
where
(20)V1(k)=x~T(k)Px~(k),V2(k)=∑j=12‍∑i=k-djk-1‍x~T(i)Qix~(i),V3(k)=∑i=k-d(k)k-1‍x~T(i)ΦTQ3Φx~(i)=+∑j=-d2+1-d1‍∑i=k+jk-1‍x~T(i)ΦTQ3Φx~(i),V4(k)=∑j=-d1-1‍∑i=k+jk-1‍d1η~T(i)ΦTS1Φη~(i),V5(k)=∑j=-d2-d1-1‍∑i=k+jk-1‍d~η~T(i)ΦTS2Φη~(i).
Calculating the forward difference of V(k) along the trajectories of filtering error system (6) with w(k)=0 yields
(21)ΔV1(k)=x~T(k+1)Px~(k+1)-x~T(k)Px~(k)=(A~x~(k)+A~dΦx~(k-d(k)))T=×P(A~x~(k)+A~dΦx~(k-d(k)))=-x~T(k)Px~(k),(22)ΔV2(k)=∑j=12‍x~T(k)Qjx~(k)=-∑j=12‍x~(k-dj)Qjx~(k-dj)≤∑j=12‍x~T(k)Qjx~(k)=-∑j=12‍x~(k-dj)ΦTQ¯jΦx~(k-dj),(23)ΔV3(k)=(d~+1)x~T(k)ΦTQ3Φx~(k)=+∑i=k+1-d(k+1)k-1‍x~T(i)ΦTQ3Φx~(i)=-∑i=k+1-d(k)k-1‍x~T(i)ΦTQ3Φx~(i)=-x~T(k-d(k))ΦTQ3Φx~(k-d(k))=-∑i=k-d2+1k-d1‍x~T(i)ΦTQ3Φx~(i)=(d~+1)x~T(k)ΦTQ3Φx~(k)=+∑i=k+1-d1k-1‍x~T(i)ΦTQ3Φx~(i)=+∑i=k+1-d(k+1)k-d1‍x~T(i)ΦTQ3Φx~(i)=-x~T(k-d(k))ΦTQ3Φx~(k-d(k))=-∑i=k+1-d(k)k-1‍x~T(i)ΦTQ3Φx~(i)=-∑i=k-d2+1k-d1‍x~T(i)ΦTQ3Φx~(i)≤(d~+1)x~T(k)ΦTQ3Φx~(k)=-x~T(k-d(k))ΦTQ3Φx~(k-d(k)).
By using Lemma3, we have
(24)ΔV4(k)=d12η~T(k)ΦTS1Φη~(k)=-d1∑i=k-d1k-1‍η~T(i)ΦTS1Φη~(i)≤d12((A-I)Φx~(k)+AdΦx~(k-d(k)))T=×S1((A-I)Φx~(k)+AdΦx~(k-d(k)))=-(Φx~(k)-Φx~(k-d1))T=×S1(Φx~(k)-Φx~(k-d1)).
Since[S2M⋆S2]≥0, the following inequality holds:
(25)[α1α2(x(k-d(k))-x(k-d2))-α2α1(x(k-d1)-x(k-d(k)))]T×[S2M⋆S2]×[α1α2(x(k-d(k))-x(k-d2))-α2α1(x(k-d1)-x(k-d(k)))]≥0,
where α1=(d2-d(k))/d~ and α2=(d(k)-d1)/d~. Then employing Lemma 2, for d1<d(k)<d2, we have
(26)ΔV5(k)=d~2η~T(k)ΦTS2Φη~(k)=-d~∑i=k-d2k-d(k)-1‍η~T(i)ΦTS2Φη~(i)=-d~∑i=k-d(k)k-d1-1‍η~T(i)ΦTS2Φη~(i)≤d~2η~T(k)ΦTS2Φη~(k)=-d~d2-d(k)(∑i=k-d2k-d(k)-1‍Φη~(i))TS2(∑i=k-d2k-d(k)-1‍Φη~(i))=-d~d(k)-d1(∑i=k-d(k)k-d1-1‍Φη~(i))TS2(∑i=k-d(k)k-d1-1‍Φη~(i))≤d~2((A-I)Φx~(k)+AdΦx~(k-d(k)))T=×S2((A-I)Φx~(k)+AdΦx~(k-d(k)))=-[x(k-d(k))-x(k-d2)x(k-d1)-x(k-d(k))]T[S2M⋆S2]=×[x(k-d(k))-x(k-d2)x(k-d1)-x(k-d(k))].
Note that when d(k)=d1 or d(k)=d2, it yields x(k-d1)-x(k-d(k))=0 or x(k-d(k))-x(k-d2)=0. Hence, the inequality in (24) still holds. Combining the conditions from (21) to (24), we have
(27)ΔV(k)=ζT(k)Πsζ(k),
where
(28)ζ(k)=[x~T(k)x~T(k-d1)ΦTx~T(k-d(k))ΦTx~T(k-d2)ΦT]T,(29)Πs=[Π~11ΦTS100⋆Π~22Π~23MT⋆⋆Π~33Π~34⋆⋆⋆Π~44]+[Π~160d1AdT0]S1[Π~160d1AdT0]T=+[Π~170d~AdT0]S2[Π~170d~AdT0]T+[A~T0A~dT0]P[A~T0A~dT0]T.
On the other hand, the following inequality can be obtained from (17):
(30)Πs1=[Π~11ΦTS100Π~16S1Π~17S2A~TP⋆Π~22Π~23MT000⋆⋆Π~33Π~34d1AdTS1d~AdTS2A~dTP⋆⋆⋆Π~44000⋆⋆⋆⋆-S100⋆⋆⋆⋆⋆-S20⋆⋆⋆⋆⋆⋆-P]<0
which is equivalent to Πs<0. Hence, ΔV(k)<0 which implies that the filtering error system in (6) with w(k)=0 is asymptotically stable.
Next, we show thel2-l∞ performance of system (6). To this end, we define
(31)J(k)=V(k)-∑i=0k-1‍wT(i)w(i).
Then under the zero initial condition, that is, x(k)=0,   k=-d2,-d2+1,…,0, it can be shown that for any nonzero w(k)∈l2[0,∞)(32)J(k)=∑i=0k-1‍[ΔV(i)-wT(i)w(i)]=∑i=0k-1‍ξT(i)(Π¯+Π¯1TPΠ¯1+d12Π¯2TS1Π¯2=∑i=0k-1‍5ξT(i)(+d122Π¯2TS2Π¯2)ξ(i),
where
(33)ξ(i)=[x~T(i)x~T(i-d1)ΦTx~T(i-d(i))ΦTx~T(i-d2)ΦTw(i)]T,Π¯=[Π~11ΦTS100Π~15⋆Π~22Π~23MT0⋆⋆Π~33Π~34-L~dTS⋆⋆⋆Π~440⋆⋆⋆⋆Π~55],Π¯1=[A~0A~d0B~],Π¯2=[(A-I)Φ0Ad0B],Π¯3=[L~0L~d0G~].
By using Schur complement equivalence, the inequality in (17) is equivalent to Π¯+Π¯1TPΠ¯1+d12Π¯2TS1Π¯2+d122Π¯2TS2Π¯2<0. Then we have J(k)<0; that is,
(34)V(k)<∑i=0k-1‍wT(i)w(i).
On the other hand, it yields from (16) and (34) that
(35)z~T(k)z~(k)=ηT(k)[L~L~dG~]T[L~L~dG~]η(k)≤γ2ηT(k)[P00⋆Q30⋆⋆I]η(k)≤γ2(V(k)+wT(k)w(k))≤γ2∑i=0k‍wT(i)w(i)≤γ2∑i=0∞‍wT(i)w(i),
where
(36)η(k)=[x~(k)Φx~(k-d(k))w(k)].
Then, we have ∥z~∥∞<γ∥w∥2 by taking the supremum over time k>0. By Definition 5, the filtering error z~(k) satisfies a given l2-l∞ disturbance attenuation level. This completes the proof.Remark 7.
The advantage of the results benefits from utilizing the reciprocally convex combination approach proposed in [19]. For the extensively used Jensen inequality [8], the integral term
(37)-∑i=k-d2k-d1-1(d2-d1)ηT(i)Sη(i)=-∑i=k-d2k-d(k)-1(d2-d1)ηT(i)Sη(i)=-∑i=k-d(k)k-d1-1(d2-d1)ηT(i)Sη(i)
with d1≤d(k)≤d2,   η(i)=x(i+1)-x(i) by the term
(38)-[x(k-d1)-x(k-d(k))]TS[x(k-d1)-x(k-d(k))]-[x(k-d(k))-x(k-d2)]TS[x(k-d(k))-x(k-d2)]
which is a special case of the following term with M=0(39)-[x(k-d(k))-x(k-d1)x(k-d(k))-x(k-d2)]T×[SM⋆S][x(k-d(k))-x(k-d1)x(k-d(t))-x(k-d2)]
with [SM⋆S]≥0, which is one of the advantages of reciprocally convex combination approach. On the other hand, the delay partitioning method is widely applied to reduce the conservatism of the results [9, 21, 22]. Also, the method can be extended to the problem considered in this paper. However, it will rise significant computation cost with the partitioning number increasing. Therefore, the reciprocally convex method needs less decision variables and can be seen as a tradeoff between the conservatism and the computation cost.Then, an equivalent condition of LMI (17) is obtained by introducing three slack matrices H1, H2, and T, which is presented in the following theorem.Theorem 8.
Given a scalarγ>0, the nominal system of (6) is asymptotically stable with an l2-l∞ performance if there exist matrices P>0,    Q¯i>0,  i=1,2,3, Sj>0, Hj, j=1,2, T, and M, such that the following LMIs hold:
(40)[S2M⋆S2]≥0,(41)[P00L~T⋆Q30L~dT⋆⋆IG~T⋆⋆⋆γ2I]>0,(42)Ω~=[Π~11ΦTS1000Π~16H1TΠ~17H2TA~TTT⋆Π~22Π~23MT0000⋆⋆Π~33Π~340d1AdTH1Td~AdTH2TA~dTTT⋆⋆⋆Π~440000⋆⋆⋆⋆-Id1BTH1Td~BTH2TB~TTT⋆⋆⋆⋆⋆S1-H1T-H100⋆⋆⋆⋆⋆⋆S2-H2T-H20⋆⋆⋆⋆⋆⋆⋆P-TT-T]<0,where Π~ii, i=1,…,4, Π~16, Π~17, Π~23, and Π~34 are defined in (17).Proof.
On one hand, if (17) holds, then there exist Hj=HjT=Sj, j=1,2, and T=TT=P such that (42) holds. On the other hand, if (42) holds, we have the following inequality based on Lemma 1:
(43)[Π~11ΦTS1000Π~16H1TΠ~17H2TA~TTT⋆Π~22Π~23MT0000⋆⋆Π~33Π~340d1AdTH1Td~AdTH2TA~dTTT⋆⋆⋆Π~440000⋆⋆⋆⋆-Id1BTH1Td~BTH2TB~TTT⋆⋆⋆⋆⋆-H1S1-1H1T00⋆⋆⋆⋆⋆⋆-H2S2-1H2T0⋆⋆⋆⋆⋆⋆⋆-TP-1TT].
In addition, matrices Hj, j=1,2, and T are nonsingular due to Sj-HjT-Hj<0, j=1,2, and P-TT-T<0. Then, pre- and promultiplying (43) by diag{I,I,I,I,I,S1H1-1,S2H2-1,PT-1} and its transpose yields (17). Therefore, the equivalence between (42) and (17) is proved.


## 3.2. Robust Filter Design

In this subsection, the filter in the form of (5) is firstly designed such that the nominal filtering error system of (6) is asymptotically stable with an l2-l∞ performance. Then the robust filtering problem is solved. Based on the result of Theorem 8, the filter design method for nominal system of (1) is presented in the following theorem.Theorem 9.
Given a scalarγ>0, the nominal system of (6) is asymptotically stable with an l2-l∞ performance if there exist matrices [P1P2⋆P3]>0, Q3>0, Q~l>0, Q¯l>0, l=1,2, Sj>0, Hj, Fj, j=1,2, diagonal matrix N>0, T1, and M such that the following set of LMIs hold:
(44)[S2M⋆S2]≥0(45)Ω=[ΞΓ⋆Λ]<0(46)Υ=[P1P200(L-D¯fC)T⋆P300-C¯fT⋆⋆Q30(Ld-D¯fCd)T⋆⋆⋆I(G-D¯fCd)T⋆⋆⋆⋆γ2I]>0,
where
(47)Ξ=[Ξ11-P2S1CTNCd00⋆Ξ220000⋆⋆Ξ33S2-MTMT0⋆⋆⋆Ξ44S2-MT0⋆⋆⋆⋆Ξ550⋆⋆⋆⋆⋆-I],Γ=[Γ11Γ12ATT1T+CT𝒜s0B¯fTATF1T+CT𝒜s0B¯fT00A¯fTA¯fT0000d1AdTH1Td~AdTH2TAdTT1T+CdT𝒜s0B¯fTAdTF1T+CdiT𝒜s0B¯fT0000d1BTH1Td~BTH2TBTT1T+DT𝒜s0B¯fTBTF1T+DT𝒜s0B¯fT],Λ=[S1-H1-H1T000⋆S2-H2-H2T00⋆⋆P1-T1-T1TP2-F2-F1T⋆⋆⋆P3-F2-F2T],Ξ11=-(P1+S1)+Q¯1+Q¯2+(d~+1)Q3,Ξ22=-P3+Q~1+Q~2,Ξ33=-S2-Q¯1-S1,Ξ44=-Q3+sym(Mi-S2i),Ξ55=-Q¯2-S2,Γ11=d1(A-I)TH1T,Γ12=d~(A-I)TH2T.
Moreover, a suitable l2-l∞ filter is given by
(48)Af=A¯fF2-1,Bf=B¯f,Cf=C¯fF2-1,Df=D¯f.Proof.
Firstly, we introduce four matricesT1, T2, T3, and T4 with T4 invertible and define
(49)J1=[I00T2T4-1],F1=T2T4-1T3,F2=T2T4-TT2T,Q~l=T2T4-1Q¯lT4-TT2T,J=diag{J1,I,I,I,I,I,I,I,J1},[P1P2⋆P3]=J1PJ1T,T=[T1T2T3T4],J2=diag{J1,I,I,I},[A¯fB¯fC¯fD¯f]=[T200I][AfBfCfDf][T4-TT2T00I].
From (45), we have F2+F2T=T2T4-TT2T+T2T4-1T2T>0 which implies that T2 is nonsingular. Hence, J and J2 are nonsingular. The inequality in (46) can be obtained by pre- an promultiplying (34) with J2 and J2T, respectively. Noting that
(50)Ω¯=JΩ~JT
we have Ω~<0. On the other hand, because T2 and T4 cannot be obtained from (45), we cannot determine the filters from (49). However, we can construct an equivalent filter transfer function from y(k) to z~(k):
(51)Tz~y=Cf(zI-Af)-1Bf+Df=C¯fT2-TT4T(zI-T2-1A¯fT2-TT4T)-1T2-1B¯f+D¯f=C¯f(zF2-A¯f)-1B¯f+D¯f=C¯fF2-1(zI-A¯fF2-1)-1B¯f+D¯f.
Therefore, the desired filter can be obtained from (48). This completes the proof.Then the filter design result for uncertain system (6) is presented in the following theorem.Theorem 10.
Given a scalarγ>0, the system in (6) with uncertainty is asymptotically stable with an l2-l∞ performance if there exist matrices [P1P2⋆P3]>0,   Q3>0,  Q~l>0,  Q¯l>0, l=1,2,  Sj>0, Hj, Fj, j=1,2, diagonal matrix N>0, T1, M, and scalars ɛi>0,  i=1,…,4 such that the following set of LMIs hold:
(52)[S2M⋆S2]≥0,(53)[Ω+ɛ3Ω1TΩ1+ɛ4Ω2TΩ2Ω3Ω4⋆-ɛ3I0⋆⋆-ɛ4I]<0,(54)[Υ11P2Υ13Υ14(L-D¯fC)T00⋆P300-C¯fT00⋆⋆Υ33Υ34(Ld-D¯fCd)T00⋆⋆⋆Υ44(G-D¯fCd)T00⋆⋆⋆⋆γ2IM30⋆⋆⋆⋆⋆ɛ1ID¯fM2⋆⋆⋆⋆⋆⋆ɛ2I]>0,
where Ω is defined in (45) and
(55)Ω1=[NA00NAd0NB0000],Ω2=[NC00NCd0ND0000],Ω3=[000000d1M1TH1Td~M1TH2TM1TT1TM1TF1T]T,Ω4=[00000000M2TB~fTM2TB~fT]T,Υ11=P1-ɛ1NLTNL-ɛ2NCTNC,Υ13=-ɛ1NLTNLd-ɛ2NCTNCd,Υ14=-ɛ1NLTNG-ɛ2NCTND,Υ33=Q3-ɛ1NLdTNLd-ɛ2NCdTNCd,Υ34=-ɛ1NLdTNG-ɛ2NCdTND,Υ44=I-ɛ1NGTNG-ɛ2NDTND.
Moreover, a suitable l2-l∞ filter is given by
(56)Af=A¯fF2-1,Bf=B¯f,Cf=C¯fF2-1,Df=D¯f.Proof.
Firstly, replace matricesA, Ad, B, C, Cd, and D in (45) with A+ΔA, Ad+ΔAd, B+ΔB,  C+ΔC, Cd+ΔCd, and D+ΔD, respectively, and the following inequality is obtained:
(57)Ω+sym(Ω1TΔ1TΩ3T)+sym(Ω2TΔ2TΩ4T)<0,
where Ωi, i=1,…,4 are defined in (53). Then by using Lemma 4, the above inequality holds if and only if
(58)Ω+ɛ3Ω1TΩ1+ɛ3-1Ω3Ω3T+ɛ4Ω2TΩ2+ɛ4-1Ω4Ω4T<0.
Then by using Schur complement equivalence, the inequality in (58) is equivalent to (45). Substituting L, Ld, and G in (46) with L+ΔL,  Ld+ΔLd, and G+ΔG, respectively, we can get
(59)Υ+sym(Υ1TΔ3TΥ3T)+sym(Υ2TΔ2TΥ4T)>0,
where
(60)Υ1=[NLNLdNG0],Υ2=[NCNCdND0]Υ3=[000M3T],Υ4=[000M2TD¯fT].
By following the similar line, the equivalence between (59) and (54) can be proved.


## 4. Illustrative Example

In this section, the following example is given to demonstrate the effectiveness of the proposed approach.Example
1.Firstly, consider a nominal discrete-time delay system in (1) with the following parameters:
(61)A=[0.1-0.50.20.5],Ad=[0.1000.2],B=[-11],C=[-0.1-1],Cd=[-0.10.6],L=[10.2],Ld=[0.50.6],D=0.1,G=-0.5.
For different delay cases, the different minima of γ can be calculated by solving the LMIs in Theorem 9. When the upper bound of the time-varying delay is 5, that is, d2=5, the minima of γ for a given d1 are listed in Table 1.Table 1
Minimum allowedγ for d2=5.
Methods
d
1
1
2
3
4
Theorem9
1.8371
1.5652
1.3585
1.1861Moreover, whend1=2, d2=5, the corresponding l2-l∞ filter is given as follows:
(62)Af=[2.6553-1.95352.2783-1.5631],Bf=[-1.5405-1.9024],Cf=[-1.32301.0479],Df=0.2378.
When uncertainty appears in the system, Theorem 10 will be used for the desired filter design. The following uncertainty parameters are considered:
(63)M1=[0.350-0.20.1],NA=[0.20.400.5],NAd=[0.1000.2],NB=[0.30.1],M2=0.2,NC=[0.15-0.22],NCd=[-0.30.2],ND=-0.5,M3=-0.4,NL=[-0.25-0.2],NLd=[0.130.32],ND=0.2.
Similarly, the allowed minimal values of γ can be obtained by solving the LMIs in Theorem 10. For d2=5, the different minimum allowed γ are listed in Table 2 for the uncertain system with different d1.Table 2
Minimum allowedγ for d2=5.
Methods
d
1
1
2
3
4
Theorem10
3.4157
2.6514
2.1568
1.8405Moreover, whend1=2,   d2=5, the desired filter is given as follows:
(64)Af=[-0.12610.2147-0.33330.5671],Bf=[-0.1725-0.2622],Cf=[-0.00760.0125],Df=0.1106.


## 5. Conclusions

The robustl2-l∞ filtering has been studied for uncertain discrete-time systems with time-varying delay in this paper. Based on reciprocally convex approach, the sufficient l2-l∞ performance analysis conditions in terms of LMIs have been proposed to render the filtering error systems asymptotically stable with an l2-l∞ performance. Then the desired filter has been designed for the filtering error system with time-varying delay. The results presented in this paper are in terms of strict LMIs which make the conditions more tractable. Finally, a numerical example is given to demonstrate the effectiveness of our methods. For future research topic, the results can be extended to the system with actuator/sensor failures which may lead to unsatisfactory performance and has attracted many researchers’ attention such as faulty diagnosis [23, 24] and fault tolerant control [25].



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*Source: 408941-2013-12-05.xml*