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

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Yang, C, Yu, Z, Wang, P, Yu, Z, Karimi, H and Feng, Zhiguang (2013) Robust l2-l∞ Filtering for Discrete-Time Delay Systems. Mathematical Problems in Engineering, 2013. ISSN 1024-123X

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Volume 2013, Article ID 408941,10pages http://dx.doi.org/10.1155/2013/408941

Research Article

Robust 𝑙 ₂ - 𝑙 _∞ Filtering for Discrete-Time Delay Systems

Chengming Yang,

¹

Zhandong Yu,

¹

Pinchao Wang,

²

Zhen Yu,

¹

Hamid Reza Karimi,

³

and Zhiguang Feng

⁴

1College of Engineering, Bohai University, Jinzhou, Liaoning 121013, China

2School of Astronautics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China

3Department of Engineering, Faculty of Engineering and Science, University of Agder, 4898 Grimstad, Norway

4College of Information Science and Technology, Bohai University, Jinzhou, Liaoning 121013, China

Correspondence should be addressed to Zhiguang Feng; oncharge@hku.hk Received 11 September 2013; Accepted 8 October 2013

Academic Editor: Baoyong Zhang

Copyright © 2013 Chengming Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 the𝑙₂-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, sufficient𝑙₂-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 robust𝑙₂-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 esti- mation 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 robust𝐿₂-𝐿_∞filtering for stochastic systems and the exponential 𝐿₂-𝐿_∞ filtering for Markovian jump sys- tems 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_∞fil- tering 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 the𝑙₂-𝑙_∞performance anal- ysis 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 con- servative result is obtained. Moreover, the desired filter for nominal systems with time-varying delay is obtained by solv- ing 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 iden- tity 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_𝑘√𝑥^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 − 𝑑 (𝑘)) + 𝐵 (𝜎) 𝑤 (𝑘) , 𝑦 (𝑘) = 𝐶 (𝜎) 𝑥 (𝑘) + 𝐶_𝑑(𝜎) 𝑥 (𝑘 − 𝑑 (𝑘)) + 𝐷 (𝜎) 𝑤 (𝑘) ,

𝑧 (𝑘) = 𝐿 (𝜎) 𝑥 (𝑘) + 𝐿_𝑑(𝜎) 𝑥 (𝑘 − 𝑑 (𝑘)) + 𝐺 (𝜎) 𝑤 (𝑘) , 𝑥 (𝑘) = 𝜙 (𝑘) , 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 to𝑙₂and𝜙(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), and𝐺(𝜎) are system matrices and satisfy

𝐴 (𝜎) = 𝐴 + Δ𝐴 (𝜎) , 𝐴_𝑑(𝜎) = 𝐴_𝑑+ Δ𝐴_𝑑(𝜎) , 𝐵 (𝜎) = 𝐵 + Δ𝐵 (𝜎) ,

𝐶 (𝜎) = 𝐶 + Δ𝐶 (𝜎) , 𝐶_𝑑(𝜎) = 𝐶_𝑑+ Δ𝐶_𝑑(𝜎) , 𝐷 (𝜎) = 𝐷 + Δ𝐷 (𝜎) ,

𝐿 (𝜎) = 𝐿 + Δ𝐿 (𝜎) , 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_𝐴 𝑁_𝐴𝑑 𝑁_𝐵] , Δ^𝑇₁(𝜎) Δ₁(𝜎) ≤ 𝐼, [Δ𝐶 (𝜎) Δ𝐶_𝑑(𝜎) Δ𝐷 (𝜎)] = 𝑀₂Δ₂(𝜎) [𝑁_𝐶 𝑁_𝐶𝑑 𝑁_𝐷] , Δ^𝑇₂(𝜎) Δ₂(𝜎) ≤ 𝐼, [Δ𝐿 (𝜎) Δ𝐿_𝑑(𝜎) Δ𝐺 (𝜎)] = 𝑀₃Δ₃(𝜎) [𝑁_𝐿 𝑁_𝐿𝑑 𝑁_𝐺] ,

Δ^𝑇₃(𝜎) Δ₃(𝜎) ≤ 𝐼, (4) where𝜎 ∈ ΘandΘis a compact set inR. The system in (1) is assumed to be asymptotically stable. Our purpose is to design a full order linear filter for the estimate of𝑧(𝑘):

̂𝑥 (𝑘 + 1) = 𝐴_𝑓̂𝑥 (𝑘) + 𝐵_𝑓𝑦 (𝑘) , ̂𝑥 (0) = 0,

̂𝑧 (𝑘) = 𝐶_𝑓̂𝑥 (𝑘) + 𝐷_𝑓𝑦 (𝑘) , (5) where𝐴_𝑓,𝐵_𝑓,𝐶_𝑓, and𝐷_𝑓are 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) Φ̃𝑥 (𝑘 − 𝑑 (𝑘)) + ̃𝐵 (𝜎) 𝑤 (𝑘)

̃𝑧 (𝑘) = ̃𝐿 (𝜎) ̃𝑥 (𝑘) + ̃𝐿_𝑑(𝜎) Φ̃𝑥 (𝑘 − 𝑑 (𝑘)) + ̃𝐺 (𝜎) 𝑤 (𝑘)

̃𝑥 (𝑘) = [𝜙^𝑇 (𝑘) 0]^𝑇, 𝑘 = −𝑑₂, −𝑑₂+ 1, . . . , 0, (6) whereΦ = [𝐼 0]and

𝐴 (𝜎) = [ 𝐴 (𝜎)̃ 0

𝐵_𝑓𝐶 (𝜎) 𝐴_𝑓] , 𝐴̃_𝑑(𝜎) = [ 𝐴^𝑑(𝜎) 𝐵_𝑓𝐶_𝑑(𝜎)] ,

̃𝐵 = [ 𝐵 (𝜎)𝐵_𝑓𝐷 (𝜎)] ,

̃𝐿 (𝑘) = [𝐿 (𝜎) − 𝐷𝑓𝐶 (𝜎) −𝐶_𝑓] ,

̃𝐿_𝑑(𝜎) = 𝐿_𝑑(𝜎) − 𝐷_𝑓𝐶_𝑑(𝜎) , 𝐺 (𝜎) = 𝐺 (𝜎) − 𝐷̃ _𝑓𝐷 (𝜎) . (7)

The nominal system of (6) is system (6) without uncer- tainty; 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 and𝑉 > 0, the following inequality holds:

𝑈𝑉⁻¹𝑈^𝑇≥ 𝑈 + 𝑈^𝑇− 𝑉. (8) Lemma 2 (see [19]). Let𝑓₁, 𝑓₂, . . . , 𝑓_𝑁:R^𝑚 → Rhave pos- itive values in a subset𝐷ofR^𝑚. Then, the reciprocally convex combination of𝑓_𝑖over𝐷satisfies

{𝛼𝑖|𝛼𝑖>0,∑min𝑖𝛼𝑖=1}∑

𝑖

1

𝛼_𝑖𝑓_𝑖(𝑘) = ∑

𝑖

𝑓_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 matrix𝑀 > 0, integers𝑎 ≤ 𝑏, vector function𝑤:{𝑎, 𝑎 + 1, . . . , 𝑏} → R^𝑛, then

− (𝑏 − 𝑎 + 1)∑^𝑏

𝑖=𝑎

𝑤^𝑇(𝑖) 𝑀𝑤 (𝑖) ≤ −(∑^𝑏

𝑖=𝑎

𝑤(𝑖))

𝑇

𝑀 (∑^𝑏

𝑖=𝑎𝑤 (𝑖)) . (11) Lemma 4. Given a symmetric matrix𝑄and matrices𝐻,𝐸 with appropriate dimensions, then

𝑄 + sym(𝐻𝐹𝐸) < 0, (12) for all𝐹^𝑇𝐹 ≤ 𝐼, if and only if there exists a scalar𝜀 > 0such that

𝑄 + 𝜀𝐸^𝑇𝐸 + 𝜀⁻¹𝐻𝐻^𝑇< 0. (13) Definition 5. Given a scalar𝛾 > 0, the filtering error̃𝑧(𝑘)in (6) is said to satisfy the𝑙₂-𝑙_∞disturbance attenuation level𝛾 under zero initial state, and the following condition is sat- isfied:

‖̃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 the𝑙₂-𝑙_∞performance defined inDefinition 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 an𝑙₂-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 scalar𝛾 > 0, the nominal system of (6) is asymptotically stable with an𝑙₂-𝑙_∞performance if there exist matrices𝑃 > 0, 𝑄₃ > 0, 𝑄_𝑖> 0, 𝑖 = 1, 2,𝑆_𝑗> 0, 𝑗 = 1, 2, and𝑀such that the following LMIs hold:

[𝑆² 𝑀

⋆ 𝑆₂] ≥ 0, (15)

[[ [[ [ [

𝑃 0 0 ̃𝐿^𝑇

⋆ 𝑄₃ 0 ̃𝐿^𝑇_𝑑

⋆ ⋆ 𝐼 ̃𝐺^𝑇

⋆ ⋆ ⋆ 𝛾²𝐼 ]] ]] ] ]

> 0, (16)

Π =̃ [[ [[ [[ [[ [[ [[ [

Π̃₁₁ Φ^𝑇𝑆₁ 0 0 0 ̃Π₁₆𝑆₁ ̃Π₁₇𝑆₂ 𝐴̃^𝑇𝑃

⋆ Π̃₂₂ Π̃₂₃ 𝑀^𝑇 0 0 0 0

⋆ ⋆ Π̃₃₃ Π̃₃₄ 0 𝑑₁𝐴^𝑇_𝑑𝑆₁ 𝑑𝐴̃ ^𝑇_𝑑𝑆₂ 𝐴̃^𝑇_𝑑𝑃

⋆ ⋆ ⋆ ̃Π₄₄ 0 0 0 0

⋆ ⋆ ⋆ ⋆ −𝐼 𝑑₁𝐵^𝑇𝑆₁ 𝑑𝐵̃ ^𝑇𝑆₂ ̃𝐵^𝑇𝑃

⋆ ⋆ ⋆ ⋆ ⋆ −𝑆₁ 0 0

⋆ ⋆ ⋆ ⋆ ⋆ ⋆ −𝑆₂ 0

⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ −𝑃

]] ]] ]] ]] ]] ]] ]

< 0,

(17) where

̃Π₁₁= −𝑃 + 𝑄₁+ 𝑄₂+ ( ̃𝑑 + 1) Φ^𝑇𝑄₃Φ − Φ^𝑇𝑆₁Φ, Π̃₂₂= −𝑆₂− 𝑄₁− 𝑆₁, Π̃₂₃= 𝑆₂− 𝑀^𝑇,

̃Π₃₃= −𝑄₃− 2𝑆₂+ sym(𝑀) , Π̃₃₄= 𝑆₂− 𝑀^𝑇, Π̃₄₄= −𝑄₂− 𝑆₂,

Π̃₁₆= 𝑑₁Φ^𝑇(𝐴 − 𝐼)^𝑇, Π̃₁₇= ̃𝑑Φ^𝑇(𝐴 − 𝐼)^𝑇, 𝑄_𝑖=diag{𝑄_𝑖, 𝑄_𝑖} , 𝑖 = 1, 2, ̃𝑑 = 𝑑₂− 𝑑₁.

(18)

Proof. First, the asymptotic stability of the nominal system of (6) is proved. We denotẽ𝜂(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) 𝑃̃𝑥 (𝑘) , 𝑉₂(𝑘) =∑²

𝑗=1 𝑘−1∑

𝑖=𝑘−𝑑𝑗

̃𝑥^𝑇(𝑖) 𝑄_𝑖̃𝑥 (𝑖) ,

𝑉₃(𝑘) = ^𝑘−1∑

𝑖=𝑘−𝑑(𝑘)

̃𝑥^𝑇(𝑖) Φ^𝑇𝑄₃Φ̃𝑥 (𝑖)

+ ^−𝑑∑¹

𝑗=−𝑑2+1 𝑘−1∑

𝑖=𝑘+𝑗̃𝑥^𝑇(𝑖) Φ^𝑇𝑄₃Φ̃𝑥 (𝑖) , 𝑉₄(𝑘) = ∑⁻¹

𝑗=−𝑑1

𝑘−1∑

𝑖=𝑘+𝑗

𝑑₁̃𝜂^𝑇(𝑖) Φ^𝑇𝑆₁Φ̃𝜂 (𝑖) ,

𝑉₅(𝑘) =^−𝑑∑¹⁻¹

𝑗=−𝑑2

𝑘−1∑

𝑖=𝑘+𝑗

𝑑̃𝜂̃ ^𝑇(𝑖) Φ^𝑇𝑆₂Φ̃𝜂 (𝑖) .

(20) Calculating the forward difference of𝑉(𝑘)along the trajecto- ries of filtering error system (6) with𝑤(𝑘) = 0yields

Δ𝑉₁(𝑘) = ̃𝑥^𝑇(𝑘 + 1) 𝑃̃𝑥 (𝑘 + 1) − ̃𝑥^𝑇(𝑘) 𝑃̃𝑥 (𝑘)

= ( ̃𝐴̃𝑥(𝑘) + ̃𝐴_𝑑Φ̃𝑥(𝑘 − 𝑑(𝑘)))^𝑇

× 𝑃 ( ̃𝐴̃𝑥 (𝑘) + ̃𝐴_𝑑Φ̃𝑥 (𝑘 − 𝑑 (𝑘)))

− ̃𝑥^𝑇(𝑘) 𝑃̃𝑥 (𝑘) ,

(21)

Δ𝑉₂(𝑘) =∑²

𝑗=1

̃𝑥^𝑇(𝑘) 𝑄_𝑗̃𝑥 (𝑘)

−∑²

𝑗=1

̃𝑥 (𝑘 − 𝑑_𝑗) 𝑄_𝑗̃𝑥 (𝑘 − 𝑑_𝑗)

≤∑²

𝑗=1̃𝑥^𝑇(𝑘) 𝑄_𝑗̃𝑥 (𝑘)

−∑²

𝑗=1̃𝑥 (𝑘 − 𝑑_𝑗) Φ^𝑇𝑄_𝑗Φ̃𝑥 (𝑘 − 𝑑_𝑗) ,

(22)

Δ𝑉₃(𝑘) = ( ̃𝑑 + 1) ̃𝑥^𝑇(𝑘) Φ^𝑇𝑄₃Φ̃𝑥 (𝑘)

+ ^𝑘−1∑

𝑖=𝑘+1−𝑑(𝑘+1)

̃𝑥^𝑇(𝑖) Φ^𝑇𝑄₃Φ̃𝑥 (𝑖)

− ^𝑘−1∑

𝑖=𝑘+1−𝑑(𝑘)

̃𝑥^𝑇(𝑖) Φ^𝑇𝑄₃Φ̃𝑥 (𝑖)

− ̃𝑥^𝑇(𝑘 − 𝑑 (𝑘)) Φ^𝑇𝑄₃Φ̃𝑥 (𝑘 − 𝑑 (𝑘))

− ^𝑘−𝑑∑¹

𝑖=𝑘−𝑑2+1

̃𝑥^𝑇(𝑖) Φ^𝑇𝑄₃Φ̃𝑥 (𝑖)

= ( ̃𝑑 + 1) ̃𝑥^𝑇(𝑘) Φ^𝑇𝑄₃Φ̃𝑥 (𝑘) + ^𝑘−1∑

𝑖=𝑘+1−𝑑1

̃𝑥^𝑇(𝑖) Φ^𝑇𝑄₃Φ̃𝑥 (𝑖)

+ ^𝑘−𝑑∑¹

𝑖=𝑘+1−𝑑(𝑘+1)̃𝑥^𝑇(𝑖) Φ^𝑇𝑄₃Φ̃𝑥 (𝑖)

− ̃𝑥^𝑇(𝑘 − 𝑑 (𝑘)) Φ^𝑇𝑄₃Φ̃𝑥 (𝑘 − 𝑑 (𝑘))

− ^𝑘−1∑

𝑖=𝑘+1−𝑑(𝑘)

̃𝑥^𝑇(𝑖) Φ^𝑇𝑄₃Φ̃𝑥 (𝑖)

− ^𝑘−𝑑∑¹

𝑖=𝑘−𝑑₂+1̃𝑥^𝑇(𝑖) Φ^𝑇𝑄₃Φ̃𝑥 (𝑖)

≤ ( ̃𝑑 + 1) ̃𝑥^𝑇(𝑘) Φ^𝑇𝑄₃Φ̃𝑥 (𝑘)

− ̃𝑥^𝑇(𝑘 − 𝑑 (𝑘)) Φ^𝑇𝑄₃Φ̃𝑥 (𝑘 − 𝑑 (𝑘)) .

(23) By usingLemma 3, we have

Δ𝑉₄(𝑘) = 𝑑²₁̃𝜂^𝑇(𝑘) Φ^𝑇𝑆₁Φ̃𝜂 (𝑘)

− 𝑑₁ ^𝑘−1∑

𝑖=𝑘−𝑑₁̃𝜂^𝑇(𝑖) Φ^𝑇𝑆₁Φ̃𝜂 (𝑖)

≤ 𝑑²₁((𝐴 − 𝐼) Φ̃𝑥 (𝑘) + 𝐴_𝑑Φ̃𝑥 (𝑘 − 𝑑 (𝑘)))^𝑇

× 𝑆₁((𝐴 − 𝐼) Φ̃𝑥 (𝑘) + 𝐴𝑑Φ̃𝑥 (𝑘 − 𝑑 (𝑘)))

− (Φ̃𝑥 (𝑘) − Φ̃𝑥 (𝑘 − 𝑑₁))^𝑇

× 𝑆₁(Φ̃𝑥 (𝑘) − Φ̃𝑥 (𝑘 − 𝑑₁)) .

(24)

Since[^𝑆_{⋆ 𝑆}^2𝑀₂] ≥ 0, the following inequality holds:

[[ [[ [

√𝛼₁

𝛼₂(𝑥 (𝑘 − 𝑑 (𝑘)) − 𝑥 (𝑘 − 𝑑2))

−√𝛼₂

𝛼₁(𝑥 (𝑘 − 𝑑₁) − 𝑥 (𝑘 − 𝑑 (𝑘))) ]] ]] ]

𝑇

× [𝑆² 𝑀

⋆ 𝑆₂]

×[[[[ [

√𝛼₁

𝛼₂(𝑥 (𝑘 − 𝑑 (𝑘)) − 𝑥 (𝑘 − 𝑑₂))

−√𝛼₂

𝛼₁ (𝑥 (𝑘 − 𝑑₁) − 𝑥 (𝑘 − 𝑑 (𝑘))) ]] ]] ]

≥ 0,

(25)

where𝛼₁ = (𝑑₂ − 𝑑(𝑘))/ ̃𝑑and 𝛼₂ = (𝑑(𝑘) − 𝑑₁)/ ̃𝑑. Then employingLemma 2, for𝑑₁< 𝑑(𝑘) < 𝑑₂, we have

Δ𝑉₅(𝑘) = ̃𝑑²̃𝜂^𝑇(𝑘) Φ^𝑇𝑆₂Φ̃𝜂 (𝑘)

− ̃𝑑^{𝑘−𝑑(𝑘)−1}∑

𝑖=𝑘−𝑑2

̃𝜂^𝑇(𝑖) Φ^𝑇𝑆₂Φ̃𝜂 (𝑖)

− ̃𝑑^𝑘−𝑑∑¹⁻¹

𝑖=𝑘−𝑑(𝑘)

̃𝜂^𝑇(𝑖) Φ^𝑇𝑆₂Φ̃𝜂 (𝑖)

≤ ̃𝑑²̃𝜂^𝑇(𝑘) Φ^𝑇𝑆₂Φ̃𝜂 (𝑘)

− 𝑑̃

𝑑₂− 𝑑 (𝑘)(^{𝑘−𝑑(𝑘)−1}∑

𝑖=𝑘−𝑑2

Φ̃𝜂(𝑖))

𝑇

𝑆₂(^{𝑘−𝑑(𝑘)−1}∑

𝑖=𝑘−𝑑2

Φ̃𝜂 (𝑖))

− 𝑑̃

𝑑 (𝑘) − 𝑑₁(^𝑘−𝑑∑¹⁻¹

𝑖=𝑘−𝑑(𝑘)

Φ̃𝜂(𝑖))

𝑇

𝑆₂(^𝑘−𝑑∑¹⁻¹

𝑖=𝑘−𝑑(𝑘)Φ̃𝜂 (𝑖))

≤ ̃𝑑²((𝐴 − 𝐼)Φ̃𝑥(𝑘) + 𝐴_𝑑Φ̃𝑥(𝑘 − 𝑑(𝑘)))^𝑇

× 𝑆₂((𝐴 − 𝐼) Φ̃𝑥 (𝑘) + 𝐴𝑑Φ̃𝑥 (𝑘 − 𝑑 (𝑘)))

− [𝑥(𝑘 − 𝑑(𝑘)) − 𝑥(𝑘 − 𝑑²) 𝑥(𝑘 − 𝑑₁) − 𝑥(𝑘 − 𝑑(𝑘))]

𝑇[𝑆² 𝑀

⋆ 𝑆₂]

× [𝑥 (𝑘 − 𝑑 (𝑘)) − 𝑥 (𝑘 − 𝑑²) 𝑥 (𝑘 − 𝑑₁) − 𝑥 (𝑘 − 𝑑 (𝑘))] .

(26) Note that when𝑑(𝑘) = 𝑑₁or𝑑(𝑘) = 𝑑₂, it yields𝑥(𝑘 − 𝑑₁) − 𝑥(𝑘 − 𝑑(𝑘)) = 0or𝑥(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^𝑇(𝑘 − 𝑑1) Φ^𝑇 ̃𝑥^𝑇(𝑘 − 𝑑 (𝑘)) Φ^𝑇 ̃𝑥^𝑇(𝑘 − 𝑑2) Φ^𝑇]^𝑇,

Π_𝑠=[[[ [

̃Π₁₁ Φ^𝑇𝑆₁ 0 0

⋆ Π̃₂₂ Π̃₂₃ 𝑀^𝑇

⋆ ⋆ Π̃₃₃ Π̃₃₄

⋆ ⋆ ⋆ ̃Π₄₄ ]] ] ]

+[[[ [

̃Π₁₆ 0 𝑑₁𝐴^𝑇_𝑑

0 ]] ] ]

𝑆₁[[[ [

Π̃₁₆ 0 𝑑₁𝐴^𝑇_𝑑

0 ]] ] ]

𝑇

+[[[ [

Π̃₁₇

̃0 𝑑𝐴^𝑇_𝑑

0 ]] ] ]

𝑆₂[[[ [

̃Π₁₇

̃0 𝑑𝐴^𝑇_𝑑

0 ]] ] ]

𝑇

+[[[ [

𝐴̃^𝑇

̃0 𝐴^𝑇_𝑑

0 ]] ] ]

𝑃[[[ [

𝐴̃^𝑇

̃0 𝐴^𝑇_𝑑

0 ]] ] ]

𝑇

.

(28)

On the other hand, the following inequality can be obtained from (17):

Π_𝑠1= [[ [[ [[ [[ [ [

Π̃₁₁ Φ^𝑇𝑆₁ 0 0 ̃Π₁₆𝑆₁ Π̃₁₇𝑆₂ 𝐴̃^𝑇𝑃

⋆ ̃Π₂₂ Π̃₂₃ 𝑀^𝑇 0 0 0

⋆ ⋆ Π̃₃₃ ̃Π₃₄ 𝑑₁𝐴^𝑇_𝑑𝑆₁ 𝑑𝐴̃ ^𝑇_𝑑𝑆₂ 𝐴̃^𝑇_𝑑𝑃

⋆ ⋆ ⋆ ̃Π₄₄ 0 0 0

⋆ ⋆ ⋆ ⋆ −𝑆₁ 0 0

⋆ ⋆ ⋆ ⋆ ⋆ −𝑆₂ 0

⋆ ⋆ ⋆ ⋆ ⋆ ⋆ −𝑃

]] ]] ]] ]] ] ]

< 0

(29) which is equivalent toΠ_𝑠 < 0. Hence,Δ𝑉(𝑘) < 0 which implies that the filtering error system in (6) with𝑤(𝑘) = 0 is asymptotically stable.

Next, we show the𝑙₂-𝑙_∞ performance of system (6). To this end, we define

𝐽 (𝑘) = 𝑉 (𝑘) −^𝑘−1∑

𝑖=0𝑤^𝑇(𝑖) 𝑤 (𝑖) . (30) 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=0[Δ𝑉 (𝑖) − 𝑤^𝑇(𝑖) 𝑤 (𝑖)]

=^𝑘−1∑

𝑖=0

𝜉^𝑇(𝑖) (Π + Π^𝑇₁𝑃Π₁+ 𝑑₁²Π^𝑇₂𝑆₁Π₂ +𝑑²₁₂Π^𝑇₂𝑆₂Π₂) 𝜉 (𝑖) ,

(31)

where 𝜉 (𝑖)

= [̃𝑥^𝑇(𝑖) ̃𝑥^𝑇(𝑖 − 𝑑₁) Φ^𝑇 ̃𝑥^𝑇(𝑖 − 𝑑 (𝑖)) Φ^𝑇 ̃𝑥^𝑇(𝑖 − 𝑑₂) Φ^𝑇 𝑤 (𝑖)]^𝑇,

Π = [[ [[ [[ [

̃Π₁₁ Φ^𝑇𝑆₁ 0 0 Π̃₁₅

⋆ Π̃₂₂ Π̃₂₃ 𝑀^𝑇 0

⋆ ⋆ Π̃₃₃ Π̃₃₄ −̃𝐿^𝑇_𝑑𝑆

⋆ ⋆ ⋆ ̃Π₄₄ 0

⋆ ⋆ ⋆ ⋆ Π̃₅₅

]] ]] ]] ] ,

Π₁= [ ̃𝐴 0 ̃𝐴_𝑑 0 ̃𝐵] , Π₂= [(𝐴 − 𝐼) Φ 0 𝐴_𝑑 0 𝐵] ,

Π₃= [̃𝐿 0 ̃𝐿_𝑑 0 ̃𝐺] .

(32) By using Schur complement equivalence, the inequality in (17) is equivalent toΠ+Π^𝑇₁𝑃Π₁+𝑑₁²Π^𝑇₂𝑆₁Π₂+𝑑²₁₂Π^𝑇₂𝑆₂Π₂< 0.

Then we have𝐽(𝑘) < 0; that is, 𝑉 (𝑘) <^𝑘−1∑

𝑖=0

𝑤^𝑇(𝑖) 𝑤 (𝑖) . (33)

On the other hand, it yields from (16) and (33) that

̃𝑧^𝑇(𝑘) ̃𝑧 (𝑘) = 𝜂^𝑇(𝑘) [̃𝐿 ̃𝐿_𝑑 𝐺]̃^𝑇[̃𝐿 ̃𝐿_𝑑 𝐺] 𝜂 (𝑘)̃

≤ 𝛾²𝜂^𝑇(𝑘) [ [

𝑃 0 0

⋆ 𝑄₃ 0

⋆ ⋆ 𝐼 ] ]

𝜂 (𝑘)

≤ 𝛾²(𝑉 (𝑘) + 𝑤^𝑇(𝑘) 𝑤 (𝑘))

≤ 𝛾²∑^𝑘

𝑖=0

𝑤^𝑇(𝑖) 𝑤 (𝑖) ≤ 𝛾²∑^∞

𝑖=0

𝑤^𝑇(𝑖) 𝑤 (𝑖) , (34)

where

𝜂 (𝑘) = [ [

̃𝑥 (𝑘) Φ̃𝑥 (𝑘 − 𝑑 (𝑘))

𝑤 (𝑘) ] ]

. (35)

Then, we have‖̃𝑧‖_∞ < 𝛾‖𝑤‖₂by taking the supremum over time𝑘 > 0. ByDefinition 5, the filtering error̃𝑧(𝑘)satisfies a given𝑙₂-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 inte- gral term

−^𝑘−𝑑∑¹⁻¹

𝑖=𝑘−𝑑₂

(𝑑₂− 𝑑₁) 𝜂^𝑇(𝑖) 𝑆𝜂 (𝑖)

= −^{𝑘−𝑑(𝑘)−1}∑

𝑖=𝑘−𝑑2

(𝑑₂− 𝑑₁) 𝜂^𝑇(𝑖) 𝑆𝜂 (𝑖)

− ^𝑘−𝑑∑¹⁻¹

𝑖=𝑘−𝑑(𝑘)

(𝑑₂− 𝑑₁) 𝜂^𝑇(𝑖) 𝑆𝜂 (𝑖)

(36)

with𝑑₁≤ 𝑑(𝑘) ≤ 𝑑₂, 𝜂(𝑖) = 𝑥(𝑖 + 1) − 𝑥(𝑖)by the term

− [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)]

(37)

which is a special case of the following term with𝑀 = 0

− [𝑥(𝑘 − 𝑑(𝑘)) − 𝑥(𝑘 − 𝑑¹) 𝑥(𝑘 − 𝑑(𝑘)) − 𝑥(𝑘 − 𝑑₂)]

𝑇

× [𝑆 𝑀⋆ 𝑆 ] [𝑥 (𝑘 − 𝑑 (𝑘)) − 𝑥 (𝑘 − 𝑑₁) 𝑥 (𝑘 − 𝑑 (𝑡)) − 𝑥 (𝑘 − 𝑑₂)]

(38)

with[_{⋆ 𝑆}^{𝑆 𝑀}] ≥ 0, which is one of the advantages of recip- rocally 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 matrices𝐻₁,𝐻₂, and𝑇, which is presented in the following theorem.

Theorem 8. Given a scalar𝛾 > 0, the nominal system of (6) is asymptotically stable with an𝑙₂-𝑙_∞performance if there exist matrices𝑃 > 0, 𝑄_𝑖 > 0,𝑖 = 1, 2, 3,𝑆_𝑗 > 0,𝐻_𝑗,𝑗 = 1, 2,𝑇, and𝑀, such that the following LMIs hold:

[𝑆² 𝑀

⋆ 𝑆₂] ≥ 0, (39)

[[ [[ [ [

𝑃 0 0 ̃𝐿^𝑇

⋆ 𝑄₃ 0 ̃𝐿^𝑇_𝑑

⋆ ⋆ 𝐼 ̃𝐺^𝑇

⋆ ⋆ ⋆ 𝛾²𝐼 ]] ]] ] ]

> 0, (40)

Ω =̃ [[ [[ [[ [[ [[ [[ [

Π̃₁₁ Φ^𝑇𝑆₁ 0 0 0 Π̃₁₆𝐻₁^𝑇 ̃Π₁₇𝐻₂^𝑇 𝐴̃^𝑇𝑇^𝑇

⋆ ̃Π₂₂ ̃Π₂₃ 𝑀^𝑇 0 0 0 0

⋆ ⋆ ̃Π₃₃ ̃Π₃₄ 0 𝑑₁𝐴^𝑇_𝑑𝐻₁^𝑇 𝑑𝐴̃ ^𝑇_𝑑𝐻₂^𝑇 𝐴̃^𝑇_𝑑𝑇^𝑇

⋆ ⋆ ⋆ ̃Π₄₄ 0 0 0 0

⋆ ⋆ ⋆ ⋆ −𝐼 𝑑₁𝐵^𝑇𝐻₁^𝑇 𝑑𝐵̃ ^𝑇𝐻^𝑇₂ ̃𝐵^𝑇𝑇^𝑇

⋆ ⋆ ⋆ ⋆ ⋆ 𝑆₁− 𝐻₁^𝑇− 𝐻₁ 0 0

⋆ ⋆ ⋆ ⋆ ⋆ ⋆ 𝑆₂− 𝐻₂^𝑇− 𝐻₂ 0

⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ 𝑃 − 𝑇^𝑇− 𝑇

]] ]] ]] ]] ]] ]] ]

< 0 , (41)

whereΠ̃_𝑖𝑖,𝑖 = 1, . . . , 4,̃Π₁₆,Π̃₁₇,̃Π₂₃, and̃Π₃₄are defined in (17).

Proof. On one hand, if (17) holds, then there exist𝐻_𝑗= 𝐻_𝑗^𝑇= 𝑆_𝑗, 𝑗 = 1, 2, and𝑇 = 𝑇^𝑇 = 𝑃such that (41) holds. On the other hand, if (41) holds, we have the following inequality based onLemma 1:

[[ [[ [[ [[ [

Π̃₁₁ Φ^𝑇𝑆₁ 0 0 0 Π̃₁₆𝐻₁^𝑇 Π̃₁₇𝐻₂^𝑇 𝐴̃^𝑇𝑇^𝑇

⋆ ̃Π₂₂ ̃Π₂₃ 𝑀^𝑇 0 0 0 0

⋆ ⋆ ̃Π₃₃ Π̃₃₄ 0 𝑑₁𝐴^𝑇_𝑑𝐻₁^𝑇 𝑑𝐴̃ ^𝑇_𝑑𝐻₂^𝑇 𝐴̃^𝑇_𝑑𝑇^𝑇

⋆ ⋆ ⋆ ̃Π₄₄ 0 0 0 0

⋆ ⋆ ⋆ ⋆ −𝐼 𝑑₁𝐵^𝑇𝐻^𝑇₁ 𝑑𝐵̃^𝑇𝐻₂^𝑇 ̃𝐵^𝑇𝑇^𝑇

⋆ ⋆ ⋆ ⋆ ⋆ −𝐻₁𝑆⁻¹₁ 𝐻₁^𝑇 0 0

⋆ ⋆ ⋆ ⋆ ⋆ ⋆ −𝐻₂𝑆⁻¹₂ 𝐻₂^𝑇 0

⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ −𝑇𝑃⁻¹𝑇^𝑇

]] ]] ]] ]] ]

< 0.

(42)

In addition, matrices𝐻_𝑗,𝑗 = 1, 2, and𝑇are nonsingular due to𝑆_𝑗−𝐻_𝑗^𝑇−𝐻_𝑗 < 0,𝑗 = 1, 2, and𝑃−𝑇^𝑇−𝑇 < 0. Then, pre- and promultiplying (42) by diag{𝐼, 𝐼, 𝐼, 𝐼, 𝐼, 𝑆₁𝐻₁⁻¹, 𝑆₂𝐻₂⁻¹, 𝑃𝑇⁻¹} and its transpose yields (17). Therefore, the equivalence between (41) 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 ofTheorem 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 an𝑙₂-𝑙_∞performance if there exist matrices[^𝑃_{⋆ 𝑃}^1𝑃2₃] > 0,𝑄₃ > 0,𝑄̃_𝑙 > 0,𝑄_𝑙> 0,𝑙 = 1, 2,𝑆_𝑗 > 0, 𝐻_𝑗,𝐹_𝑗,𝑗 = 1, 2, diagonal matrix𝑁 > 0,𝑇₁, and𝑀such that the following set of LMIs hold:

[𝑆² 𝑀

⋆ 𝑆₂] ≥ 0 (43)

Ω = [Ξ Γ⋆ Λ] < 0 (44)

Υ = [[ [[ [[ [[ [[ [

𝑃₁ 𝑃₂ 0 0 (𝐿 − 𝐷_𝑓𝐶)^𝑇

⋆ 𝑃₃ 0 0 −𝐶^𝑇_𝑓

⋆ ⋆ 𝑄₃ 0 (𝐿_𝑑− 𝐷_𝑓𝐶_𝑑)^𝑇

⋆ ⋆ ⋆ 𝐼 (𝐺 − 𝐷_𝑓𝐶_𝑑)^𝑇

⋆ ⋆ ⋆ ⋆ 𝛾²𝐼 ]] ]] ]] ]] ]] ]

> 0, (45)

where

Ξ = [[ [[ [[ [ [

Ξ₁₁ −𝑃₂ 𝑆₁ 𝐶^𝑇𝑁𝐶_𝑑 0 0

⋆ Ξ₂₂ 0 0 0 0

⋆ ⋆ Ξ₃₃ 𝑆₂− 𝑀^𝑇 𝑀^𝑇 0

⋆ ⋆ ⋆ Ξ₄₄ 𝑆₂− 𝑀^𝑇 0

⋆ ⋆ ⋆ ⋆ Ξ₅₅ 0

⋆ ⋆ ⋆ ⋆ ⋆ −𝐼

]] ]] ]] ] ] ,

Γ=

[[ [[ [[ [[ [[ [ [

Γ₁₁ Γ₁₂ 𝐴^𝑇𝑇₁^𝑇+ 𝐶^𝑇A_𝑠0𝐵^𝑇_𝑓 𝐴^𝑇𝐹₁^𝑇+ 𝐶^𝑇A_𝑠0𝐵^𝑇_𝑓

0 0 𝐴^𝑇_𝑓 𝐴^𝑇_𝑓

0 0 0 0

𝑑₁𝐴^𝑇_𝑑𝐻₁^𝑇 𝑑𝐴̃ ^𝑇_𝑑𝐻^𝑇₂ 𝐴^𝑇_𝑑𝑇₁^𝑇+ 𝐶^𝑇_𝑑A_𝑠0𝐵^𝑇_𝑓 𝐴^𝑇_𝑑𝐹₁^𝑇+ 𝐶^𝑇_𝑑𝑖A_𝑠0𝐵^𝑇_𝑓

0 0 0 0

𝑑₁𝐵^𝑇𝐻₁^𝑇 𝑑𝐵̃ ^𝑇𝐻₂^𝑇 𝐵^𝑇𝑇₁^𝑇+ 𝐷^𝑇A_𝑠0𝐵^𝑇_𝑓 𝐵^𝑇𝐹₁^𝑇+ 𝐷^𝑇A_𝑠0𝐵^𝑇_𝑓 ]] ]] ]] ]] ]] ] ]

,

Λ=[[[ [

𝑆₁− 𝐻₁− 𝐻₁^𝑇 0 0 0

⋆ 𝑆₂− 𝐻₂− 𝐻^𝑇₂ 0 0

⋆ ⋆ 𝑃₁− 𝑇₁− 𝑇₁^𝑇 𝑃₂− 𝐹₂− 𝐹₁^𝑇

⋆ ⋆ ⋆ 𝑃₃− 𝐹₂− 𝐹₂^𝑇

]] ] ] ,

Ξ₁₁ = − (𝑃₁+ 𝑆₁) + 𝑄₁+ 𝑄₂+ ( ̃𝑑 + 1) 𝑄₃, Ξ₂₂= −𝑃₃+ ̃𝑄₁+ ̃𝑄₂ ,

Ξ₃₃= −𝑆₂− 𝑄₁− 𝑆₁, Ξ₄₄= −𝑄₃+ sym(𝑀_𝑖− 𝑆_2𝑖) ,

Ξ₅₅= −𝑄₂− 𝑆₂,

Γ₁₁= 𝑑₁(𝐴 − 𝐼)^𝑇𝐻₁^𝑇, Γ₁₂= ̃𝑑(𝐴 − 𝐼)^𝑇𝐻₂^𝑇. (46) Moreover, a suitable𝑙₂-𝑙_∞filter is given by

𝐴_𝑓= 𝐴_𝑓𝐹₂⁻¹, 𝐵_𝑓= 𝐵_𝑓, 𝐶_𝑓= 𝐶_𝑓𝐹₂⁻¹,

𝐷_𝑓= 𝐷_𝑓. (47)

Proof. Firstly, we introduce four matrices𝑇₁,𝑇₂,𝑇₃, and𝑇₄ with𝑇₄invertible and define

𝐽₁= [𝐼 0

0 𝑇₂𝑇₄⁻¹] , 𝐹₁= 𝑇₂𝑇₄⁻¹𝑇₃, 𝐹₂= 𝑇₂𝑇₄^−𝑇𝑇₂^𝑇, 𝑄̃_𝑙= 𝑇₂𝑇₄⁻¹𝑄_𝑙𝑇₄^−𝑇𝑇₂^𝑇, 𝐽 =diag{𝐽₁, 𝐼, 𝐼, 𝐼, 𝐼, 𝐼, 𝐼, 𝐼, 𝐽₁} ,

[𝑃¹ 𝑃₂

⋆ 𝑃₃] = 𝐽₁𝑃𝐽₁^𝑇, 𝑇 = [𝑇¹ 𝑇₂ 𝑇₃ 𝑇₄] , 𝐽₂=diag{𝐽₁, 𝐼, 𝐼, 𝐼} ,

[𝐴_𝑓 𝐵_𝑓

𝐶_𝑓 𝐷_𝑓] = [𝑇² 0

0 𝐼] [𝐴_𝑓 𝐵_𝑓

𝐶_𝑓 𝐷_𝑓] [𝑇^4−𝑇𝑇₂^𝑇 0 0 𝐼] .

(48) From (44), we have𝐹₂ + 𝐹₂^𝑇 = 𝑇₂𝑇₄^−𝑇𝑇₂^𝑇+ 𝑇₂𝑇₄⁻¹𝑇₂^𝑇 > 0 which implies that 𝑇₂ is nonsingular. Hence, 𝐽and 𝐽₂ are

nonsingular. The inequality in (45) can be obtained by pre- an promultiplying (33) with𝐽₂ and𝐽₂^𝑇, respectively. Noting that

Ω = 𝐽̃Ω𝐽^𝑇 (49)

we haveΩ < 0. On the other hand, becausẽ 𝑇₂and𝑇₄cannot be obtained from (44), we cannot determine the filters from (48). However, we can construct an equivalent filter transfer function from𝑦(𝑘)tõ𝑧(𝑘):

𝑇_̃𝑧𝑦= 𝐶_𝑓(𝑧𝐼 − 𝐴_𝑓)⁻¹𝐵_𝑓+ 𝐷_𝑓

= 𝐶_𝑓𝑇₂^−𝑇𝑇₄^𝑇(𝑧𝐼 − 𝑇₂⁻¹𝐴_𝑓𝑇₂^−𝑇𝑇₄^𝑇)⁻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.

(50)

Therefore, the desired filter can be obtained from (47). 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 an𝑙₂-𝑙_∞performance if there exist matrices [^𝑃_{⋆ 𝑃}^1𝑃2₃] > 0, 𝑄₃ > 0, 𝑄̃_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 scalars𝜀_𝑖 > 0,𝑖 = 1, . . . , 4such that the following set of LMIs hold:

[𝑆² 𝑀

⋆ 𝑆₂] ≥ 0, (51)

[ [

Ω + 𝜀₃Ω^𝑇₁Ω₁+ 𝜀₄Ω^𝑇₂Ω₂ Ω₃ Ω₄

⋆ −𝜀₃𝐼 0

⋆ ⋆ −𝜀₄𝐼

] ]

< 0, (52)

[[ [[ [[ [[ [[ [[ [[ [ [

Υ₁₁ 𝑃₂ Υ₁₃ Υ₁₄ (𝐿 − 𝐷_𝑓𝐶)^𝑇 0 0

⋆ 𝑃₃ 0 0 −𝐶^𝑇_𝑓 0 0

⋆ ⋆ Υ₃₃ Υ₃₄ (𝐿_𝑑− 𝐷_𝑓𝐶_𝑑)^𝑇 0 0

⋆ ⋆ ⋆ Υ₄₄ (𝐺 − 𝐷_𝑓𝐶_𝑑)^𝑇 0 0

⋆ ⋆ ⋆ ⋆ 𝛾²𝐼 𝑀₃ 0

⋆ ⋆ ⋆ ⋆ ⋆ 𝜀₁𝐼 𝐷_𝑓𝑀₂

⋆ ⋆ ⋆ ⋆ ⋆ ⋆ 𝜀₂𝐼

]] ]] ]] ]] ]] ]] ]] ] ]

> 0,

(53) whereΩis defined in(44)and

Ω₁= [𝑁_𝐴 0 0 𝑁_𝐴𝑑 0 𝑁_𝐵 0 0 0 0] ,

Ω₂= [𝑁_𝐶 0 0 𝑁_𝐶𝑑 0 𝑁_𝐷 0 0 0 0] , Ω₃

= [0 0 0 0 0 0 𝑑₁𝑀₁^𝑇𝐻₁^𝑇 𝑑𝑀̃ ^𝑇₁𝐻₂^𝑇 𝑀₁^𝑇𝑇₁^𝑇 𝑀₁^𝑇𝐹₁^𝑇]^𝑇, Ω₄= [0 0 0 0 0 0 0 0 𝑀₂^𝑇̃𝐵^𝑇_𝑓 𝑀₂^𝑇̃𝐵^𝑇_𝑓]^𝑇,

Υ₁₁= 𝑃₁− 𝜀₁𝑁_𝐿^𝑇𝑁_𝐿− 𝜀₂𝑁_𝐶^𝑇𝑁_𝐶, Υ₁₃= −𝜀₁𝑁_𝐿^𝑇𝑁_𝐿𝑑− 𝜀₂𝑁_𝐶^𝑇𝑁_𝐶𝑑,

Υ₁₄= −𝜀₁𝑁_𝐿^𝑇𝑁_𝐺− 𝜀₂𝑁_𝐶^𝑇𝑁_𝐷, Υ₃₃= 𝑄₃− 𝜀₁𝑁_𝐿𝑑^𝑇𝑁_𝐿𝑑− 𝜀₂𝑁_𝐶𝑑^𝑇𝑁_𝐶𝑑,

Υ₃₄= −𝜀₁𝑁_𝐿𝑑^𝑇𝑁_𝐺− 𝜀₂𝑁_𝐶𝑑^𝑇 𝑁_𝐷, Υ₄₄= 𝐼 − 𝜀₁𝑁_𝐺^𝑇𝑁_𝐺− 𝜀₂𝑁_𝐷^𝑇𝑁_𝐷.

(54) Moreover, a suitable𝑙₂-𝑙_∞filter is given by

𝐴_𝑓= 𝐴_𝑓𝐹₂⁻¹, 𝐵_𝑓= 𝐵_𝑓, 𝐶_𝑓= 𝐶_𝑓𝐹₂⁻¹, 𝐷_𝑓= 𝐷_𝑓.

(55)

Proof. Firstly, replace matrices𝐴,𝐴_𝑑,𝐵,𝐶,𝐶_𝑑, and𝐷in (44) with𝐴+Δ𝐴,𝐴_𝑑+Δ𝐴_𝑑,𝐵+Δ𝐵, 𝐶+Δ𝐶,𝐶_𝑑+Δ𝐶_𝑑, and𝐷+Δ𝐷, respectively, and the following inequality is obtained:

Ω +sym(Ω^𝑇₁Δ^𝑇₁Ω^𝑇₃) +sym(Ω^𝑇₂Δ^𝑇₂Ω^𝑇₄) < 0, (56) where Ω_𝑖, 𝑖 = 1, . . . , 4are defined in (52). 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. (57) Then by using Schur complement equivalence, the inequality in (57) is equivalent to (44). Substituting𝐿,𝐿_𝑑, and𝐺in (45) with𝐿 + Δ𝐿,𝐿_𝑑+ Δ𝐿_𝑑, and𝐺 + Δ𝐺, respectively, we can get Υ +sym(Υ₁^𝑇Δ^𝑇₃Υ₃^𝑇) +sym(Υ₂^𝑇Δ^𝑇₂Υ₄^𝑇) > 0, (58) where

Υ₁= [𝑁_𝐿 𝑁_𝐿𝑑 𝑁_𝐺 0] , Υ₂= [𝑁_𝐶 𝑁_𝐶𝑑 𝑁_𝐷 0]

Υ₃= [0 0 0 𝑀^𝑇₃] , Υ₄= [0 0 0 𝑀₂^𝑇𝐷^𝑇_𝑓] . (59)