Robust 𝑙₂-𝑙_∞ Filtering for Discrete-Time Delay Systems

Abstract

The problem of robust 𝑙₂-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.

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 the 𝑙₂-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 robust 𝑙₂-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. ℝ^𝑛 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+𝐴^𝑇; 𝑙₂ means the space of square summable infinite vector sequences; for any real function 𝑥∈𝑙₂, we define ‖𝑥‖_∞=sup_𝑘𝑥^𝑇(𝑘)𝑥(𝑘) and ‖𝑥‖₂=∑^∞_𝑘=0𝑥^𝑇(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 𝑥(𝑘+1)=𝐴(𝜎)𝑥(𝑘)+𝐴_𝑑(𝜎)𝑥((𝑘(𝜎)𝑤(𝑘),𝑦(𝑘)(𝜎)𝑥(𝑘)𝑘−𝑑))+𝐵=𝐶+𝐶_𝑑(𝜎)𝑥((𝑘(𝜎)𝑤(𝑘),𝑧(𝑘)(𝜎)𝑥(𝑘)𝑘−𝑑))+𝐷=𝐿+𝐿_𝑑(𝜎)𝑥((𝑘(𝜎)𝑤(𝑘),𝑥(𝑘)(𝑘)𝑘−𝑑))+𝐺=𝜙,𝑘=−𝑑₂,−𝑑₂+1,…,0, (1) where 𝑥(𝑘)∈ℝ^𝑛 is the state vector; 𝑦(𝑘)∈ℝ^𝑚 is the measured output; 𝑧(𝑘)∈ℝ^𝑝 represents the signal to be estimated; 𝑤(𝑘)∈ℝ^𝑙 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 1≤𝑑₁≤𝑑(𝑘)≤𝑑₂<∞,𝑘=1,2,… (2)𝐴(𝜎), 𝐴_𝑑(𝜎), 𝐵(𝜎), 𝐶(𝜎), 𝐶_𝑑(𝜎), 𝐷(𝜎), 𝐿(𝜎), 𝐿_𝑑(𝜎), and 𝐺(𝜎) are system matrices and satisfy 𝐴(𝜎)=𝐴+Δ𝐴(𝜎),𝐴_𝑑(𝜎)=𝐴_𝑑+Δ𝐴_𝑑(𝜎),𝐵(𝜎)(𝜎),𝐶(𝜎)(𝜎)=𝐵+Δ𝐵=𝐶+Δ𝐶,𝐶_𝑑(𝜎)=𝐶_𝑑+Δ𝐶_𝑑(𝜎),𝐷(𝜎)(𝜎),𝐿(𝜎)(𝜎)=𝐷+Δ𝐷=𝐿+Δ𝐿,𝐿_𝑑(𝜎)=𝐿_𝑑+Δ𝐿_𝑑(𝜎),𝐺(𝜎)=𝐺+Δ𝐺(𝜎). (3) Matrices Δ𝐴(𝜎), Δ𝐴_𝑑(𝜎), Δ𝐵(𝜎), Δ𝐶(𝜎), Δ𝐶_𝑑(𝜎), Δ𝐷(𝜎), Δ𝐿(𝜎), Δ𝐿_𝑑(𝜎), and Δ𝐺(𝜎) are unknown time-invariant matrix representing the uncertainty of the system satisfying the following conditions: Δ𝐴(𝜎)Δ𝐴_𝑑(𝜎)Δ𝐵(𝜎)=𝑀₁Δ₁(𝜎)𝑁_𝐴𝑁_𝐴𝑑𝑁_𝐵,Δ^𝑇₁(𝜎)Δ₁(𝜎)(𝜎)≤𝐼,Δ𝐶Δ𝐶_𝑑(𝜎)Δ𝐷(𝜎)=𝑀₂Δ₂(𝜎)𝑁_𝐶𝑁_𝐶𝑑𝑁_𝐷,Δ^𝑇₂(𝜎)Δ₂(𝜎)(𝜎)≤𝐼,Δ𝐿Δ𝐿_𝑑(𝜎)Δ𝐺(𝜎)=𝑀₃Δ₃(𝜎)𝑁_𝐿𝑁_𝐿𝑑𝑁_𝐺,Δ^𝑇₃(𝜎)Δ₃(𝜎)≤𝐼, (4) 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): 𝑥(𝑘+1)=𝐴_𝑓𝑥(𝑘)+𝐵_𝑓𝑦(𝑘),𝑥(0)𝑧(𝑘)=0,=𝐶_𝑓𝑥(𝑘)+𝐷_𝑓𝑦(𝑘), (5) where 𝐴_𝑓, 𝐵_𝑓, 𝐶_𝑓, and 𝐷_𝑓 are filter gains to be determined.

Let the augmented state vector 𝑥(𝑘)=𝑥^𝑇(𝑘)𝑥^𝑇(𝑘)^𝑇 and 𝑧(𝑘)=𝑧(𝑘)−𝑧(𝑘). Then the filtering error system is described as 𝑥(𝑘+1)=𝐴(𝜎)𝑥(𝑘)+𝐴_𝑑(𝜎)Φ𝑥((𝑘+𝐵(𝜎)𝑤(𝑘)𝑧(𝑘)=𝐿(𝜎)𝑥(𝑘)+𝐿𝑘−𝑑))_𝑑(𝜎)Φ𝑥((𝑘+𝐺(𝜎)𝑤(𝑘)𝑥(𝑘)=𝜙𝑘−𝑑))_𝑇(𝑘)0^𝑇,𝑘=−𝑑₂,−𝑑₂+1,…,0, (6) where Φ=𝐼0 and 𝐴(𝜎)=𝐴(𝜎)0𝐵_𝑓𝐶(𝜎)𝐴_𝑓,𝐴_𝑑(𝜎)=𝐴_𝑑(𝜎)𝐵_𝑓𝐶_𝑑(𝜎),𝐵(𝜎)𝐵𝐵=_𝑓𝐷(𝜎),𝐿(𝑘)=𝐿(𝜎)−𝐷_𝑓𝐶(𝜎)−𝐶_𝑓,𝐿_𝑑(𝜎)=𝐿_𝑑(𝜎)−𝐷_𝑓𝐶_𝑑(𝜎),𝐺(𝜎)=𝐺(𝜎)−𝐷_𝑓𝐷(𝜎). (7)

The nominal system of (6) is system (6) without uncertainty; that is, Δ𝐴(𝜎)=0, Δ𝐴_𝑑(𝜎)=0, Δ𝐵(𝜎)=0, Δ𝐶(𝜎)=0, Δ𝐶_𝑑(𝜎)=0, Δ𝐷(𝜎)=0, Δ𝐿(𝜎)=0, Δ𝐿_𝑑(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 matrices 𝑈 and 𝑉>0, the following inequality holds: 𝑈𝑉⁻¹𝑈^𝑇≥𝑈+𝑈^𝑇−𝑉. (8)

Lemma 2 (see [19]). Let 𝑓₁,𝑓₂,…,𝑓_𝑁∶ℝ^𝑚→ℝ have positive values in a subset 𝐷 of ℝ^𝑚. Then, the reciprocally convex combination of 𝑓_𝑖 over 𝐷 satisfies min_{{𝛼𝑖∣𝛼𝑖>0,∑𝑖𝛼𝑖=1}𝑖}1𝛼_𝑖𝑓_𝑖(𝑘)=_𝑖𝑓_𝑖(𝑘)+max_{𝑔𝑖,𝑗(𝑘)𝑖≠𝑗}𝑔_𝑖,𝑗(𝑘) (9) subject to 𝑔_𝑖,𝑗∶ℝ^𝑚⟶ℝ,𝑔_𝑗,𝑖(𝑘)=𝑔_𝑖,𝑗(𝑘),𝑓_𝑖(𝑘)𝑔_𝑖,𝑗(𝑘)𝑔_𝑗,𝑖(𝑘)𝑓_𝑗(𝑘)≥0. (10)

Lemma 3. For any constant matrix 𝑀>0, integers 𝑎≤𝑏, vector function 𝑤: {𝑎,𝑎+1,…,𝑏}→ℝ^𝑛, then −(𝑏−𝑎+1)^𝑏_𝑖=𝑎𝑤^𝑇(𝑖)𝑀𝑤(𝑖)≤−^𝑏_𝑖=𝑎𝑤(𝑖)^𝑇𝑀^𝑏_𝑖=𝑎𝑤(𝑖). (11)

Lemma 4. Given a symmetric matrix 𝑄 and matrices 𝐻, 𝐸 with appropriate dimensions, then 𝑄+sym(𝐻𝐹𝐸)<0, (12) for all 𝐹^𝑇𝐹≤𝐼, if and only if there exists a scalar 𝜀>0 such that 𝑄+𝜀𝐸^𝑇𝐸+𝜀⁻¹𝐻𝐻^𝑇<0. (13)

Definition 5. Given a scalar 𝛾>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: ‖𝑧‖_∞<𝛾‖𝑤‖₂. (14)

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 sufficient 𝑙₂-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 of 𝑙₂-𝑙_∞ performance analysis for nominal system of (6).

Theorem 6. Given a scalar 𝛾>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, 𝑆_𝑗>0, ??𝑗=1,2, and 𝑀 such that the following LMIs hold: 𝑆₂𝑀⋆𝑆₂≥0,𝑃00𝐿^𝑇⋆𝑄₃0𝐿^𝑇_𝑑⋆⋆𝐼𝐺^𝑇⋆⋆⋆𝛾²𝐼>0,ΠΠ=¹¹Φ^𝑇𝑆₁000Π₁₆𝑆₁Π₁₇𝑆₂𝐴^𝑇𝑃⋆Π₂₂Π₂₃𝑀^𝑇Π0000⋆⋆₃₃Π₃₄0𝑑₁𝐴^𝑇_𝑑𝑆₁𝑑𝐴^𝑇_𝑑𝑆₂𝐴^𝑇_𝑑𝑃⋆⋆⋆Π₄₄0000⋆⋆⋆⋆−𝐼𝑑₁𝐵^𝑇𝑆₁𝑑𝐵^𝑇𝑆₂𝐵^𝑇𝑃⋆⋆⋆⋆⋆−𝑆₁00⋆⋆⋆⋆⋆⋆−𝑆₂0⋆⋆⋆⋆⋆⋆⋆−𝑃<0, (15)(16)(17) where Π₁₁=−𝑃+𝑄₁+𝑄₂+𝑑+1Φ^𝑇𝑄₃Φ−Φ^𝑇𝑆₁ΠΦ,₂₂=−𝑆₂−𝑄₁−𝑆₁,Π₂₃=𝑆₂−𝑀^𝑇,Π₃₃=−𝑄₃−2𝑆₂+sym(𝑀),Π₃₄=𝑆₂−𝑀^𝑇,Π₄₄=−𝑄₂−𝑆₂,Π₁₆=𝑑₁Φ^𝑇(𝐴−𝐼)^𝑇,Π₁₇=𝑑Φ^𝑇(𝐴−𝐼)^𝑇,𝑄_𝑖=diag𝑄_𝑖,𝑄_𝑖,𝑖=1,2,𝑑=𝑑₂−𝑑₁. (18)

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: 𝑉(𝑘)=𝑉₁(𝑘)+𝑉₂(𝑘)+𝑉₃(𝑘)+𝑉₄(𝑘)+𝑉₅(𝑘), (19) where 𝑉₁(𝑘)=𝑥^𝑇(𝑘)𝑃𝑥(𝑘),𝑉₂(𝑘)=²_𝑗=1^𝑘−1_{𝑖=𝑘−𝑑𝑗}𝑥^𝑇(𝑖)𝑄_𝑖𝑥(𝑖),𝑉₃(𝑘)=^𝑘−1_{𝑖=𝑘−𝑑(𝑘)}𝑥^𝑇(𝑖)Φ^𝑇𝑄₃Φ𝑥(𝑖)+^−𝑑1_{𝑗=−𝑑2+1}^𝑘−1_{𝑖=𝑘+𝑗}𝑥^𝑇(𝑖)Φ^𝑇𝑄₃Φ𝑥(𝑖),𝑉₄(𝑘)=⁻¹_{𝑗=−𝑑1}^𝑘−1_{𝑖=𝑘+𝑗}𝑑₁𝜂^𝑇(𝑖)Φ^𝑇𝑆₁Φ𝜂(𝑖),𝑉₅(𝑘)=^−𝑑1−1_{𝑗=−𝑑2}^𝑘−1_{𝑖=𝑘+𝑗}𝑑𝜂^𝑇(𝑖)Φ^𝑇𝑆₂Φ𝜂(𝑖). (20) Calculating the forward difference of 𝑉(𝑘) along the trajectories of filtering error system (6) with 𝑤(𝑘)=0 yields Δ𝑉₁(𝑘)=𝑥^𝑇(𝑘+1)𝑃𝑥(𝑘+1)−𝑥^𝑇(𝑘)𝑃𝑥(𝑘)=𝐴𝑥(𝑘)+𝐴_𝑑Φ𝑥(𝑘−𝑑(𝑘))^𝑇×𝑃𝐴𝑥(𝑘)+𝐴_𝑑Φ𝑥((𝑘−𝑥𝑘−𝑑))_𝑇(𝑘)𝑃𝑥(𝑘),Δ𝑉₂(𝑘)=²_𝑗=1𝑥^𝑇(𝑘)𝑄_𝑗𝑥(𝑘)−²_𝑗=1𝑥𝑘−𝑑_𝑗𝑄_𝑗𝑥𝑘−𝑑_𝑗≤²_𝑗=1𝑥^𝑇(𝑘)𝑄_𝑗𝑥(𝑘)−²_𝑗=1𝑥𝑘−𝑑_𝑗Φ^𝑇𝑄_𝑗Φ𝑥𝑘−𝑑_𝑗,Δ𝑉₃(𝑘)=𝑑+1𝑥^𝑇(𝑘)Φ^𝑇𝑄₃Φ𝑥(𝑘)+^𝑘−1_{𝑖=𝑘+1−𝑑(𝑘+1)}𝑥^𝑇(𝑖)Φ^𝑇𝑄₃Φ𝑥(𝑖)−^𝑘−1_{𝑖=𝑘+1−𝑑(𝑘)}𝑥^𝑇(𝑖)Φ^𝑇𝑄₃Φ𝑥(𝑖)−𝑥^𝑇(𝑘−𝑑(𝑘))Φ^𝑇𝑄₃Φ𝑥((𝑘−𝑘−𝑑))_{𝑘−𝑑1𝑖=𝑘−𝑑2+1}𝑥^𝑇(𝑖)Φ^𝑇𝑄₃Φ𝑥(𝑖)=𝑑+1𝑥^𝑇(𝑘)Φ^𝑇𝑄₃Φ𝑥(𝑘)+^𝑘−1_{𝑖=𝑘+1−𝑑1}𝑥^𝑇(𝑖)Φ^𝑇𝑄₃Φ𝑥(𝑖)+^𝑘−𝑑1_{𝑖=𝑘+1−𝑑(𝑘+1)}𝑥^𝑇(𝑖)Φ^𝑇𝑄₃Φ𝑥(𝑖)−𝑥^𝑇(𝑘−𝑑(𝑘))Φ^𝑇𝑄₃Φ𝑥((𝑘−𝑘−𝑑))_{𝑘−1𝑖=𝑘+1−𝑑(𝑘)}𝑥^𝑇(𝑖)Φ^𝑇𝑄₃Φ𝑥(𝑖)−^𝑘−𝑑1_{𝑖=𝑘−𝑑2+1}𝑥^𝑇(𝑖)Φ^𝑇𝑄₃Φ𝑥(𝑖)≤𝑑+1𝑥^𝑇(𝑘)Φ^𝑇𝑄₃Φ𝑥(𝑘)−𝑥^𝑇(𝑘−𝑑(𝑘))Φ^𝑇𝑄₃Φ𝑥(𝑘−𝑑(𝑘)). (21)(22)(23)
By using Lemma 3, we have Δ𝑉₄(𝑘)=𝑑²₁𝜂^𝑇(𝑘)Φ^𝑇𝑆₁Φ𝜂(𝑘)−𝑑₁^𝑘−1_{𝑖=𝑘−𝑑1}𝜂^𝑇(𝑖)Φ^𝑇𝑆₁Φ𝜂(𝑖)≤𝑑²₁(𝐴−𝐼)Φ𝑥(𝑘)+𝐴_𝑑Φ𝑥(𝑘−𝑑(𝑘))^𝑇×𝑆₁(𝐴−𝐼)Φ𝑥(𝑘)+𝐴_𝑑Φ𝑥((𝑘−Φ𝑥(𝑘)𝑥𝑘−𝑑))−Φ𝑘−𝑑₁^𝑇×𝑆₁Φ𝑥(𝑘)−Φ𝑥𝑘−𝑑₁. (24)
Since ^𝑆2𝑀_⋆𝑆2≥0, the following inequality holds: 𝛼₁𝛼₂𝑥(𝑘−𝑑(𝑘))−𝑥𝑘−𝑑₂−𝛼₂𝛼₁𝑥𝑘−𝑑₁−𝑥(𝑘−𝑑(𝑘))^𝑇×𝑆₂𝑀⋆𝑆₂×𝛼₁𝛼₂𝑥(𝑘−𝑑(𝑘))−𝑥𝑘−𝑑₂−𝛼₂𝛼₁𝑥𝑘−𝑑₁−𝑥(𝑘−𝑑(𝑘))≥0, (25) where 𝛼₁=(𝑑₂−𝑑(𝑘))/𝑑 and 𝛼₂=(𝑑(𝑘)−𝑑₁)/𝑑. Then employing Lemma 2, for 𝑑₁<𝑑(𝑘)<𝑑₂, we have Δ𝑉₅(𝑘)=𝑑²𝜂^𝑇(𝑘)Φ^𝑇𝑆₂Φ𝜂(𝑘)−𝑑^{𝑘−𝑑(𝑘)−1}_{𝑖=𝑘−𝑑2}𝜂^𝑇(𝑖)Φ^𝑇𝑆₂Φ𝜂(𝑖)−𝑑^{𝑘−𝑑1−1}_{𝑖=𝑘−𝑑(𝑘)}𝜂^𝑇(𝑖)Φ^𝑇𝑆₂Φ𝜂(𝑖)≤𝑑²𝜂^𝑇(𝑘)Φ^𝑇𝑆₂Φ𝜂(𝑘)−𝑑𝑑₂−𝑑(𝑘)^{𝑘−𝑑(𝑘)−1}_{𝑖=𝑘−𝑑2}Φ𝜂(𝑖)^𝑇𝑆₂^{𝑘−𝑑(𝑘)−1}_{𝑖=𝑘−𝑑2}Φ𝜂(𝑖)−𝑑𝑑(𝑘)−𝑑₁^{𝑘−𝑑1−1}_{𝑖=𝑘−𝑑(𝑘)}Φ𝜂(𝑖)^𝑇𝑆₂^{𝑘−𝑑1−1}_{𝑖=𝑘−𝑑(𝑘)}Φ𝜂(𝑖)≤𝑑²(𝐴−𝐼)Φ𝑥(𝑘)+𝐴_𝑑Φ𝑥(𝑘−𝑑(𝑘))^𝑇×𝑆₂(𝐴−𝐼)Φ𝑥(𝑘)+𝐴_𝑑Φ𝑥((𝑘−𝑘−𝑑))𝑥(𝑘−𝑑(𝑘))−𝑥(𝑘−𝑑₂)𝑥(𝑘−𝑑₁)−𝑥(𝑘−𝑑(𝑘))^𝑇𝑆₂𝑀⋆𝑆₂×𝑥(𝑘−𝑑(𝑘))−𝑥𝑘−𝑑₂𝑥𝑘−𝑑₁((𝑘.−𝑥𝑘−𝑑)) (26) 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 Δ𝑉(𝑘)=𝜁^𝑇(𝑘)Π_𝑠𝜁(𝑘), (27) where 𝜁(𝑘)=𝑥^𝑇(𝑘)𝑥^𝑇𝑘−𝑑₁Φ^𝑇𝑥^𝑇(𝑘−𝑑(𝑘))Φ^𝑇𝑥^𝑇𝑘−𝑑₂Φ^𝑇𝑇,Π_𝑠=Π₁₁Φ^𝑇𝑆₁⋆Π00₂₂Π₂₃𝑀^𝑇⋆⋆Π₃₃Π₃₄⋆⋆⋆Π₄₄+Π₁₆0𝑑₁𝐴^𝑇_𝑑0𝑆₁Π₁₆0𝑑₁𝐴^𝑇_𝑑0^𝑇+Π₁₇0𝑑𝐴^𝑇_𝑑0𝑆₂Π₁₇0𝑑𝐴^𝑇_𝑑0^𝑇+𝐴^𝑇0𝐴^𝑇_𝑑0𝑃𝐴^𝑇0𝐴^𝑇_𝑑0^𝑇. (28)(29) On the other hand, the following inequality can be obtained from (17): Π_𝑠1=Π₁₁Φ^𝑇𝑆₁00Π₁₆𝑆₁Π₁₇𝑆₂𝐴^𝑇𝑃⋆Π₂₂Π₂₃𝑀^𝑇Π000⋆⋆₃₃Π₃₄𝑑₁𝐴^𝑇_𝑑𝑆₁𝑑𝐴^𝑇_𝑑𝑆₂𝐴^𝑇_𝑑𝑃⋆⋆⋆Π₄₄000⋆⋆⋆⋆−𝑆₁00⋆⋆⋆⋆⋆−𝑆₂0⋆⋆⋆⋆⋆⋆−𝑃<0 (30) which is equivalent to Π_𝑠<0. Hence, Δ𝑉(𝑘)<0 which implies that the filtering error system in (6) with 𝑤(𝑘)=0 is asymptotically stable.
Next, we show the 𝑙₂-𝑙_∞ performance of system (6). To this end, we define 𝐽(𝑘)=𝑉(𝑘)−^𝑘−1_𝑖=0𝑤^𝑇(𝑖)𝑤(𝑖). (31) 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,∞)𝐽(𝑘)=^𝑘−1_𝑖=0Δ𝑉(𝑖)−𝑤^𝑇(𝑖)𝑤(𝑖=_{𝑘−1𝑖=0})𝜉^𝑇(𝑖)Π+Π^𝑇₁𝑃Π₁+𝑑²₁Π^𝑇₂𝑆₁Π₂+𝑑²₁₂Π^𝑇₂𝑆₂Π₂𝜉(𝑖), (32) where 𝜉(𝑖)=𝑥^𝑇(𝑖)𝑥^𝑇𝑖−𝑑₁Φ^𝑇𝑥^𝑇(𝑖−𝑑(𝑖))Φ^𝑇𝑥^𝑇𝑖−𝑑₂Φ^𝑇𝑤(𝑖)^𝑇,ΠΠ=¹¹Φ^𝑇𝑆₁00Π₁₅⋆Π₂₂Π₂₃𝑀^𝑇0⋆⋆Π₃₃Π₃₄−𝐿^𝑇_𝑑𝑆⋆⋆⋆Π₄₄0⋆⋆⋆⋆Π₅₅,Π₁=𝐴0𝐴_𝑑0𝐵,Π₂=(𝐴−𝐼)Φ0𝐴_𝑑0𝐵,Π₃=𝐿0𝐿_𝑑0𝐺. (33) By using Schur complement equivalence, the inequality in (17) is equivalent to Π+Π^𝑇₁𝑃Π₁+𝑑²₁Π^𝑇₂𝑆₁Π₂+𝑑²₁₂Π^𝑇₂𝑆₂Π₂<0. Then we have 𝐽(𝑘)<0; that is, 𝑉(𝑘)<^𝑘−1_𝑖=0𝑤^𝑇(𝑖)𝑤(𝑖). (34) On the other hand, it yields from (16) and (34) that 𝑧^𝑇(𝑘)𝑧(𝑘)=𝜂^𝑇(𝑘)𝐿𝐿_𝑑𝐺^𝑇𝐿𝐿_𝑑𝐺𝜂(𝑘)≤𝛾²𝜂^𝑇(𝑘)𝑃00⋆𝑄₃0𝜂(𝑘)⋆⋆𝐼≤𝛾²𝑉(𝑘)+𝑤^𝑇(𝑘)𝑤(𝑘)≤𝛾^2𝑘_𝑖=0𝑤^𝑇(𝑖)𝑤(𝑖)≤𝛾^2∞_𝑖=0𝑤^𝑇(𝑖)𝑤(𝑖), (35) where 𝜂(𝑘)=𝑥(𝑘)Φ𝑥((𝑘𝑤(𝑘)𝑘−𝑑)). (36) 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 −^{𝑘−𝑑1−1}_{𝑖=𝑘−𝑑2}𝑑₂−𝑑₁𝜂^𝑇(𝑖)𝑆𝜂(𝑖)=−^{𝑘−𝑑(𝑘)−1}_{𝑖=𝑘−𝑑2}𝑑₂−𝑑₁𝜂^𝑇(𝑖)(𝑖)−𝑆𝜂_{𝑘−𝑑1−1𝑖=𝑘−𝑑(𝑘)}𝑑₂−𝑑₁𝜂^𝑇(𝑖)𝑆𝜂(𝑖) (37) with 𝑑₁≤𝑑(𝑘)≤𝑑₂, ??𝜂(𝑖)=𝑥(𝑖+1)−𝑥(𝑖) by the term −𝑥(𝑘−𝑑₁)−𝑥(𝑘−𝑑(𝑘))^𝑇𝑆𝑥𝑘−𝑑₁((𝑘−−𝑥𝑘−𝑑))𝑥(𝑘−𝑑(𝑘))−𝑥(𝑘−𝑑₂)^𝑇𝑆𝑥(𝑘−𝑑(𝑘))−𝑥𝑘−𝑑₂ (38) which is a special case of the following term with 𝑀=0−𝑥(𝑘−𝑑(𝑘))−𝑥(𝑘−𝑑₁)𝑥(𝑘−𝑑(𝑘))−𝑥(𝑘−𝑑₂)^𝑇×𝑆𝑀𝑥((𝑘⋆𝑆𝑘−𝑑))−𝑥𝑘−𝑑₁𝑥(𝑘−𝑑(𝑡))−𝑥𝑘−𝑑₂ (39) 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 scalar 𝛾>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, 𝑆_𝑗>0, 𝐻_𝑗, 𝑗=1,2, 𝑇, and 𝑀, such that the following LMIs hold: 𝑆₂𝑀⋆𝑆₂≥0,𝑃00𝐿^𝑇⋆𝑄₃0𝐿^𝑇_𝑑⋆⋆𝐼𝐺^𝑇⋆⋆⋆𝛾²𝐼>0, (40)(41)ΠΩ=¹¹Φ^𝑇𝑆₁000Π₁₆𝐻^𝑇₁Π₁₇𝐻^𝑇₂𝐴^𝑇𝑇^𝑇⋆Π₂₂Π₂₃𝑀^𝑇Π0000⋆⋆₃₃Π₃₄0𝑑₁𝐴^𝑇_𝑑𝐻^𝑇₁𝑑𝐴^𝑇_𝑑𝐻^𝑇₂𝐴^𝑇_𝑑𝑇^𝑇⋆⋆⋆Π₄₄0000⋆⋆⋆⋆−𝐼𝑑₁𝐵^𝑇𝐻^𝑇₁𝑑𝐵^𝑇𝐻^𝑇₂𝐵^𝑇𝑇^𝑇⋆⋆⋆⋆⋆𝑆₁−𝐻^𝑇₁−𝐻₁00⋆⋆⋆⋆⋆⋆𝑆₂−𝐻^𝑇₂−𝐻₂0⋆⋆⋆⋆⋆⋆⋆𝑃−𝑇^𝑇,−𝑇<0 (42)where Π_𝑖𝑖, 𝑖=1,…,4, Π₁₆, Π₁₇, Π₂₃, and Π₃₄ are defined in (17).

Proof. On one hand, if (17) holds, then there exist 𝐻_𝑗=𝐻^𝑇_𝑗=𝑆_𝑗, 𝑗=1,2, and 𝑇=𝑇^𝑇=𝑃 such that (42) holds. On the other hand, if (42) holds, we have the following inequality based on Lemma 1: Π₁₁Φ^𝑇𝑆₁000Π₁₆𝐻^𝑇₁Π₁₇𝐻^𝑇₂𝐴^𝑇𝑇^𝑇⋆Π₂₂Π₂₃𝑀^𝑇Π0000⋆⋆₃₃Π₃₄0𝑑₁𝐴^𝑇_𝑑𝐻^𝑇₁𝑑𝐴^𝑇_𝑑𝐻^𝑇₂𝐴^𝑇_𝑑𝑇^𝑇⋆⋆⋆Π₄₄0000⋆⋆⋆⋆−𝐼𝑑₁𝐵^𝑇𝐻^𝑇₁𝑑𝐵^𝑇𝐻^𝑇₂𝐵^𝑇𝑇^𝑇⋆⋆⋆⋆⋆−𝐻₁𝑆₁⁻¹𝐻^𝑇₁00⋆⋆⋆⋆⋆⋆−𝐻₂𝑆₂⁻¹𝐻^𝑇₂0⋆⋆⋆⋆⋆⋆⋆−𝑇𝑃⁻¹𝑇^𝑇. (43) In addition, matrices 𝐻_𝑗, 𝑗=1,2, and 𝑇 are nonsingular due to 𝑆_𝑗−𝐻^𝑇_𝑗−𝐻_𝑗<0, 𝑗=1,2, and 𝑃−𝑇^𝑇−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 scalar 𝛾>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, 𝑆_𝑗>0, 𝐻_𝑗, 𝐹_𝑗, 𝑗=1,2, diagonal matrix 𝑁>0, 𝑇₁, and 𝑀 such that the following set of LMIs hold: 𝑆₂𝑀⋆𝑆₂≥0Ω=ΞΓ⋆Λ<0𝑃Υ=¹𝑃₂00𝐿−𝐷_𝑓𝐶^𝑇⋆𝑃₃00−𝐶^𝑇_𝑓⋆⋆𝑄₃0𝐿_𝑑−𝐷_𝑓𝐶_𝑑^𝑇⋆⋆⋆𝐼𝐺−𝐷_𝑓𝐶_𝑑^𝑇⋆⋆⋆⋆𝛾²𝐼>0, (44)(45)(46) where ΞΞ=¹¹−𝑃₂𝑆₁𝐶^𝑇𝑁𝐶_𝑑00⋆Ξ₂₂0000⋆⋆Ξ₃₃𝑆₂−𝑀^𝑇𝑀^𝑇0⋆⋆⋆Ξ₄₄𝑆₂−𝑀^𝑇0⋆⋆⋆⋆Ξ₅₅0,Γ⋆⋆⋆⋆⋆−𝐼Γ=¹¹Γ₁₂𝐴^𝑇𝑇^𝑇₁+𝐶^𝑇𝒜_𝑠0𝐵^𝑇_𝑓𝐴^𝑇𝐹^𝑇₁+𝐶^𝑇𝒜_𝑠0𝐵^𝑇_𝑓00𝐴^𝑇_𝑓𝐴^𝑇_𝑓𝑑0000₁𝐴^𝑇_𝑑𝐻^𝑇₁𝑑𝐴^𝑇_𝑑𝐻^𝑇₂𝐴^𝑇_𝑑𝑇^𝑇₁+𝐶^𝑇_𝑑𝒜_𝑠0𝐵^𝑇_𝑓𝐴^𝑇_𝑑𝐹^𝑇₁+𝐶^𝑇_𝑑𝑖𝒜_𝑠0𝐵^𝑇_𝑓𝑑0000₁𝐵^𝑇𝐻^𝑇₁𝑑𝐵^𝑇𝐻^𝑇₂𝐵^𝑇𝑇^𝑇₁+𝐷^𝑇𝒜_𝑠0𝐵^𝑇_𝑓𝐵^𝑇𝐹^𝑇₁+𝐷^𝑇𝒜_𝑠0𝐵^𝑇_𝑓,𝑆Λ=¹−𝐻₁−𝐻^𝑇₁000⋆𝑆₂−𝐻₂−𝐻^𝑇₂00⋆⋆𝑃₁−𝑇₁−𝑇^𝑇₁𝑃₂−𝐹₂−𝐹^𝑇₁⋆⋆⋆𝑃₃−𝐹₂−𝐹^𝑇₂,Ξ₁₁=−𝑃₁+𝑆₁+𝑄₁+𝑄₂+𝑑+1𝑄₃,Ξ₂₂=−𝑃₃+𝑄₁+𝑄₂,Ξ₃₃=−𝑆₂−𝑄₁−𝑆₁,Ξ₄₄=−𝑄₃+sym𝑀_𝑖−𝑆_2𝑖,Ξ₅₅=−𝑄₂−𝑆₂,Γ₁₁=𝑑₁(𝐴−𝐼)^𝑇𝐻^𝑇₁,Γ₁₂=𝑑(𝐴−𝐼)^𝑇𝐻^𝑇₂. (47) Moreover, a suitable 𝑙₂-𝑙_∞ filter is given by 𝐴_𝑓=𝐴_𝑓𝐹₂⁻¹,𝐵_𝑓=𝐵_𝑓,𝐶_𝑓=𝐶_𝑓𝐹₂⁻¹,𝐷_𝑓=𝐷_𝑓. (48)

Proof. Firstly, we introduce four matrices 𝑇₁, 𝑇₂, 𝑇₃, and 𝑇₄ with 𝑇₄ invertible and define 𝐽₁=𝐼00𝑇₂𝑇₄⁻¹,𝐹₁=𝑇₂𝑇₄⁻¹𝑇₃,𝐹₂=𝑇₂𝑇₄^−𝑇𝑇^𝑇₂,𝑄_𝑙=𝑇₂𝑇₄⁻¹𝑄_𝑙𝑇₄^−𝑇𝑇^𝑇₂,𝐽=diag𝐽₁,𝐼,𝐼,𝐼,𝐼,𝐼,𝐼,𝐼,𝐽₁,𝑃₁𝑃₂⋆𝑃₃=𝐽₁𝑃𝐽^𝑇₁𝑇,𝑇=¹𝑇₂𝑇₃𝑇₄,𝐽₂=diag𝐽₁,𝐼,𝐼,𝐼,𝐴_𝑓𝐵_𝑓𝐶_𝑓𝐷_𝑓=𝑇₂0𝐴0𝐼^𝑓𝐵_𝑓𝐶_𝑓𝐷_𝑓𝑇₄^−𝑇𝑇^𝑇₂0.0𝐼 (49) From (45), we have 𝐹₂+𝐹^𝑇₂=𝑇₂𝑇₄^−𝑇𝑇^𝑇₂+𝑇₂𝑇₄⁻¹𝑇^𝑇₂>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 𝐽^𝑇₂, respectively. Noting that Ω=𝐽Ω𝐽^𝑇 (50) 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): 𝑇𝑧_𝑦=𝐶_𝑓𝑧𝐼−𝐴_𝑓⁻¹𝐵_𝑓+𝐷_𝑓=𝐶_𝑓𝑇₂^−𝑇𝑇^𝑇₄𝑧𝐼−𝑇₂⁻¹𝐴_𝑓𝑇₂^−𝑇𝑇^𝑇₄⁻¹𝑇₂⁻¹𝐵_𝑓+𝐷_𝑓=𝐶_𝑓𝑧𝐹₂−𝐴_𝑓⁻¹𝐵_𝑓+𝐷_𝑓=𝐶_𝑓𝐹₂⁻¹𝑧𝐼−𝐴_𝑓𝐹₂⁻¹⁻¹𝐵_𝑓+𝐷_𝑓. (51) 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 scalar 𝛾>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, ?𝑆_𝑗>0, 𝐻_𝑗, 𝐹_𝑗, 𝑗=1,2, diagonal matrix 𝑁>0, 𝑇₁, 𝑀, and scalars 𝜀_𝑖>0,??𝑖=1,…,4 such that the following set of LMIs hold: 𝑆₂𝑀⋆𝑆₂≥0,Ω+𝜀₃Ω^𝑇₁Ω₁+𝜀₄Ω^𝑇₂Ω₂Ω₃Ω₄⋆−𝜀₃𝐼0⋆⋆−𝜀₄𝐼<0,Υ₁₁𝑃₂Υ₁₃Υ₁₄𝐿−𝐷_𝑓𝐶^𝑇00⋆𝑃₃00−𝐶^𝑇_𝑓00⋆⋆Υ₃₃Υ₃₄𝐿_𝑑−𝐷_𝑓𝐶_𝑑^𝑇00⋆⋆⋆Υ₄₄𝐺−𝐷_𝑓𝐶_𝑑^𝑇00⋆⋆⋆⋆𝛾²𝐼𝑀₃0⋆⋆⋆⋆⋆𝜀₁𝐼𝐷_𝑓𝑀₂⋆⋆⋆⋆⋆⋆𝜀₂𝐼>0, (52)(53)(54) where Ω is defined in (45) and Ω₁=𝑁_𝐴00𝑁_𝐴𝑑0𝑁_𝐵,Ω0000₂=𝑁_𝐶00𝑁_𝐶𝑑0𝑁_𝐷,Ω0000₃=000000𝑑₁𝑀^𝑇₁𝐻^𝑇₁𝑑𝑀^𝑇₁𝐻^𝑇₂𝑀^𝑇₁𝑇^𝑇₁𝑀^𝑇₁𝐹^𝑇₁^𝑇,Ω₄=00000000𝑀^𝑇₂𝐵^𝑇_𝑓𝑀^𝑇₂𝐵^𝑇_𝑓^𝑇,Υ₁₁=𝑃₁−𝜀₁𝑁^𝑇_𝐿𝑁_𝐿−𝜀₂𝑁^𝑇_𝐶𝑁_𝐶,Υ₁₃=−𝜀₁𝑁^𝑇_𝐿𝑁_𝐿𝑑−𝜀₂𝑁^𝑇_𝐶𝑁_𝐶𝑑,Υ₁₄=−𝜀₁𝑁^𝑇_𝐿𝑁_𝐺−𝜀₂𝑁^𝑇_𝐶𝑁_𝐷,Υ₃₃=𝑄₃−𝜀₁𝑁^𝑇_𝐿𝑑𝑁_𝐿𝑑−𝜀₂𝑁^𝑇_𝐶𝑑𝑁_𝐶𝑑,Υ₃₄=−𝜀₁𝑁^𝑇_𝐿𝑑𝑁_𝐺−𝜀₂𝑁^𝑇_𝐶𝑑𝑁_𝐷,Υ₄₄=𝐼−𝜀₁𝑁^𝑇_𝐺𝑁_𝐺−𝜀₂𝑁^𝑇_𝐷𝑁_𝐷. (55) Moreover, a suitable 𝑙₂-𝑙_∞ filter is given by 𝐴_𝑓=𝐴_𝑓𝐹₂⁻¹,𝐵_𝑓=𝐵_𝑓,𝐶_𝑓=𝐶_𝑓𝐹₂⁻¹,𝐷_𝑓=𝐷_𝑓. (56)

Proof. Firstly, replace matrices 𝐴, 𝐴_𝑑, 𝐵, 𝐶, 𝐶_𝑑, and 𝐷 in (45) with 𝐴+Δ𝐴, 𝐴_𝑑+Δ𝐴_𝑑, 𝐵+Δ𝐵,? 𝐶+Δ𝐶, 𝐶_𝑑+Δ𝐶_𝑑, and 𝐷+Δ𝐷, respectively, and the following inequality is obtained: Ω+symΩ^𝑇₁Δ^𝑇₁Ω^𝑇₃+symΩ^𝑇₂Δ^𝑇₂Ω^𝑇₄<0, (57) where Ω_𝑖, 𝑖=1,…,4 are defined in (53). Then by using Lemma 4, the above inequality holds if and only if Ω+𝜀₃Ω^𝑇₁Ω₁+𝜀₃⁻¹Ω₃Ω^𝑇₃+𝜀₄Ω^𝑇₂Ω₂+𝜀₄⁻¹Ω₄Ω^𝑇₄<0. (58) Then by using Schur complement equivalence, the inequality in (58) is equivalent to (45). Substituting 𝐿, 𝐿_𝑑, and 𝐺 in (46) with 𝐿+Δ𝐿,??𝐿_𝑑+Δ𝐿_𝑑, and 𝐺+Δ𝐺, respectively, we can get Υ+symΥ^𝑇₁Δ^𝑇₃Υ^𝑇₃+symΥ^𝑇₂Δ^𝑇₂Υ^𝑇₄>0, (59) where Υ₁=𝑁_𝐿𝑁_𝐿𝑑𝑁_𝐺0,Υ₂=𝑁_𝐶𝑁_𝐶𝑑𝑁_𝐷0Υ₃=000𝑀^𝑇₃,Υ₄=000𝑀^𝑇₂𝐷^𝑇_𝑓. (60) 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: 𝐴=0.1−0.50.20.5,𝐴_𝑑=,10.1000.2𝐵=−1,𝐶=−0.1−1,𝐶_𝑑=−0.10.6,𝐿=10.2,𝐿_𝑑=0.50.6,𝐷=0.1,𝐺=−0.5. (61) 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 allowed 𝛾 for 𝑑₂=5.

Moreover, when 𝑑₁=2, 𝑑₂=5, the corresponding 𝑙₂-𝑙_∞ filter is given as follows: 𝐴_𝑓=2.6553−1.95352.2783−1.5631,𝐵_𝑓=,𝐶−1.5405−1.9024_𝑓=−1.32301.0479,𝐷_𝑓=0.2378. (62) When uncertainty appears in the system, Theorem 10 will be used for the desired filter design. The following uncertainty parameters are considered: 𝑀₁=0.350−0.20.1,𝑁_𝐴=,𝑁0.20.400.5_𝐴𝑑=0.1000.2,𝑁_𝐵=,𝑀0.30.1₂=0.2,𝑁_𝐶=,𝑁0.15−0.22_𝐶𝑑=−0.30.2,𝑁_𝐷𝑀=−0.5,₃=−0.4,𝑁_𝐿=,𝑁−0.25−0.2_𝐿𝑑=0.130.32,𝑁_𝐷=0.2. (63) 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 allowed 𝛾 for 𝑑₂=5.

Moreover, when 𝑑₁=2, ??𝑑₂=5, the desired filter is given as follows: 𝐴_𝑓=−0.12610.2147−0.33330.5671,𝐵_𝑓=,𝐶−0.1725−0.2622_𝑓=−0.00760.0125,𝐷_𝑓=0.1106. (64)

5. Conclusions

The robust 𝑙₂-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].

Conflict of Interests

None of the authors of the paper has declared any conflict of interests.

Acknowledgment

This work was partially supported by Liaoning Provincial Natural Science Foundation of China (2013020227).