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

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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

1_{College of Engineering, Bohai University, Jinzhou, Liaoning 121013, China}

2_{School of Astronautics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China}

3_{Department of Engineering, Faculty of Engineering and Science, University of Agder, 4898 Grimstad, Norway} 4_{College of Information Science and Technology, Bohai University, Jinzhou, Liaoning 121013, China}

Correspondence should be addressed to Zhiguang Feng; [email protected]

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

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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. 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 iden-tity 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‖_∞ = sup_𝑘√𝑥𝑇(𝑘)𝑥(𝑘) and

‖𝑥‖₂ = √∑∞_𝑘=0𝑥𝑇_{(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) + 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,

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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 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) .

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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) [𝑁𝐴 𝑁𝐴𝑑 𝑁𝐵] ,

Δ𝑇₁(𝜎) Δ1(𝜎) ≤ 𝐼,

[Δ𝐶 (𝜎) Δ𝐶𝑑(𝜎) Δ𝐷 (𝜎)] = 𝑀2Δ2(𝜎) [𝑁𝐶 𝑁𝐶𝑑 𝑁𝐷] ,

Δ𝑇₂(𝜎) Δ2(𝜎) ≤ 𝐼,

[Δ𝐿 (𝜎) Δ𝐿𝑑(𝜎) Δ𝐺 (𝜎)] = 𝑀3Δ3(𝜎) [𝑁𝐿 𝑁𝐿𝑑 𝑁𝐺] ,

Δ𝑇₃(𝜎) Δ3(𝜎) ≤ 𝐼,

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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) = 𝐴𝑓̂𝑥 (𝑘) + 𝐵𝑓𝑦 (𝑘) , ̂𝑥 (0) = 0,

̂𝑧 (𝑘) = 𝐶𝑓̂𝑥 (𝑘) + 𝐷𝑓𝑦 (𝑘) ,

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where𝐴_𝑓,𝐵_𝑓,𝐶_𝑓, and𝐷_𝑓are filter gains to be determined.

Let the augmented state vector ̃𝑥(𝑘) = [𝑥𝑇(𝑘) ̂𝑥𝑇(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) + ̃𝐿𝑑(𝜎) Φ̃𝑥 (𝑘 − 𝑑 (𝑘)) + ̃𝐺 (𝜎) 𝑤 (𝑘)

̃𝑥 (𝑘) = [𝜙𝑇 _{(𝑘) 0]}𝑇_{, 𝑘 = −𝑑}

2, −𝑑2+ 1, . . . , 0,

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whereΦ = [𝐼 0]and

̃

𝐴 (𝜎) = [ 𝐴 (𝜎)_𝐵 0

𝑓𝐶 (𝜎) 𝐴𝑓] , 𝐴̃𝑑(𝜎) = [ 𝐴

𝑑(𝜎)

𝐵_𝑓𝐶_𝑑(𝜎)] ,

̃𝐵 = [ 𝐵 (𝜎)_𝐵

𝑓𝐷 (𝜎)] ,

̃𝐿 (𝑘) = [𝐿 (𝜎) − 𝐷𝑓𝐶 (𝜎) −𝐶𝑓] ,

̃𝐿_𝑑(𝜎) = 𝐿_𝑑(𝜎) − 𝐷_𝑓𝐶_𝑑(𝜎) , 𝐺 (𝜎) = 𝐺 (𝜎) − 𝐷̃ 𝑓𝐷 (𝜎) .

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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−1𝑈𝑇≥ 𝑈 + 𝑈𝑇− 𝑉. (8)

Lemma 2 (see [19]). Let𝑓₁, 𝑓₂, . . . , 𝑓_𝑁:R𝑚 → Rhave

pos-itive values in a subset𝐷ofR𝑚. Then, the reciprocally convex

combination of𝑓_𝑖over𝐷satisfies

min

{𝛼𝑖|𝛼𝑖>0,∑𝑖𝛼𝑖=1}∑_𝑖

1

𝛼_𝑖𝑓𝑖(𝑘) = ∑_𝑖 𝑓𝑖(𝑘) +max𝑔𝑖,𝑗(𝑘)_{𝑖 ̸= 𝑗}∑𝑔𝑖,𝑗(𝑘) (9)

subject to

{𝑔_𝑖,𝑗:R𝑚 󳨀→ R, 𝑔_𝑗,𝑖(𝑘) = 𝑔_𝑖,𝑗(𝑘) , [𝑓𝑖(𝑘) 𝑔𝑖,𝑗(𝑘)

𝑔_𝑗,𝑖(𝑘) 𝑓𝑗(𝑘) ] ≥ 0} .

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Lemma 3. For any constant matrix𝑀 > 0, integers𝑎 ≤ 𝑏,

vector function𝑤:{𝑎, 𝑎 + 1, . . . , 𝑏} → R𝑛, then

− (𝑏 − 𝑎 + 1)∑𝑏

𝑖=𝑎

𝑤𝑇(𝑖) 𝑀𝑤 (𝑖) ≤ −(∑𝑏

𝑖=𝑎 𝑤(𝑖))

𝑇

𝑀 (∑𝑏

𝑖=𝑎𝑤 (𝑖)) .

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Lemma 4. Given a symmetric matrix𝑄and matrices𝐻,𝐸

with appropriate dimensions, then

𝑄 + sym(𝐻𝐹𝐸) < 0, (12)

for all𝐹𝑇𝐹 ≤ 𝐼, if and only if there exists a scalar𝜀 > 0such

that

𝑄 + 𝜀𝐸𝑇𝐸 + 𝜀−1𝐻𝐻𝑇< 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 ̃𝐿𝑇

⋆ 𝑄₃ 0 ̃𝐿𝑇

𝑑

⋆ ⋆ 𝐼 ̃𝐺𝑇

⋆ ⋆ ⋆ 𝛾2_𝐼

] ] ] ] ] ]

> 0, (16)

̃ Π =

[ [ [ [ [ [ [ [ [ [ [ [ [ ̃

Π11 Φ𝑇𝑆1 0 0 0 ̃Π16𝑆1 ̃Π17𝑆2 𝐴̃𝑇𝑃

⋆ Π̃₂₂ Π̃₂₃ 𝑀𝑇 ₀ ₀ ₀ ₀

⋆ ⋆ Π̃₃₃ Π̃₃₄ 0 𝑑₁𝐴𝑇_𝑑𝑆₁ 𝑑𝐴̃ 𝑇_𝑑𝑆₂ 𝐴̃𝑇_𝑑𝑃

⋆ ⋆ ⋆ ̃Π₄₄ 0 0 0 0

⋆ ⋆ ⋆ ⋆ −𝐼 𝑑₁𝐵𝑇_𝑆

1 𝑑𝐵̃ 𝑇𝑆2 ̃𝐵𝑇𝑃

⋆ ⋆ ⋆ ⋆ ⋆ −𝑆₁ 0 0

⋆ ⋆ ⋆ ⋆ ⋆ ⋆ −𝑆₂ 0

⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ −𝑃

] ] ] ] ] ] ] ] ] ] ] ] ] < 0,

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where

̃

Π₁₁= −𝑃 + 𝑄₁+ 𝑄₂+ ( ̃𝑑 + 1) Φ𝑇𝑄₃Φ − Φ𝑇𝑆₁Φ,

̃

Π₂₂= −𝑆₂− 𝑄₁− 𝑆₁, Π̃₂₃= 𝑆₂− 𝑀𝑇,

̃

Π₃₃= −𝑄₃− 2𝑆₂+ sym(𝑀) , Π̃₃₄= 𝑆₂− 𝑀𝑇,

̃

Π₄₄= −𝑄₂− 𝑆₂,

̃

Π₁₆= 𝑑₁Φ𝑇(𝐴 − 𝐼)𝑇, Π̃₁₇= ̃𝑑Φ𝑇(𝐴 − 𝐼)𝑇,

𝑄_𝑖=diag{𝑄_𝑖, 𝑄_𝑖} , 𝑖 = 1, 2, ̃𝑑 = 𝑑₂− 𝑑₁.

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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) = ̃𝑥𝑇(𝑘) 𝑃̃𝑥 (𝑘) ,

𝑉₂(𝑘) =∑2

𝑗=1 𝑘−1 ∑ 𝑖=𝑘−𝑑𝑗

̃𝑥𝑇_{(𝑖) 𝑄}

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𝑉₃(𝑘) = 𝑘−1∑ 𝑖=𝑘−𝑑(𝑘)

̃𝑥𝑇(𝑖) Φ𝑇𝑄₃Φ̃𝑥 (𝑖)

+ −𝑑1∑

𝑗=−𝑑2+1 𝑘−1 ∑

𝑖=𝑘+𝑗̃𝑥

𝑇_{(𝑖) Φ}𝑇_𝑄

3Φ̃𝑥 (𝑖) ,

𝑉₄(𝑘) = ∑−1

𝑗=−𝑑1 𝑘−1 ∑ 𝑖=𝑘+𝑗

𝑑₁̃𝜂𝑇(𝑖) Φ𝑇𝑆₁Φ̃𝜂 (𝑖) ,

𝑉5(𝑘) =

−𝑑1−1 ∑ 𝑗=−𝑑2

𝑘−1 ∑ 𝑖=𝑘+𝑗

̃

𝑑̃𝜂𝑇(𝑖) Φ𝑇𝑆2Φ̃𝜂 (𝑖) .

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

Δ𝑉₁(𝑘) = ̃𝑥𝑇(𝑘 + 1) 𝑃̃𝑥 (𝑘 + 1) − ̃𝑥𝑇(𝑘) 𝑃̃𝑥 (𝑘)

= ( ̃𝐴̃𝑥(𝑘) + ̃𝐴_𝑑Φ̃𝑥(𝑘 − 𝑑(𝑘)))𝑇

× 𝑃 ( ̃𝐴̃𝑥 (𝑘) + ̃𝐴_𝑑Φ̃𝑥 (𝑘 − 𝑑 (𝑘)))

− ̃𝑥𝑇_{(𝑘) 𝑃̃𝑥 (𝑘) ,}

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Δ𝑉₂(𝑘) =∑2

𝑗=1

̃𝑥𝑇_{(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 − 𝑑𝑗

) Φ𝑇_𝑄

𝑗Φ̃𝑥 (𝑘 − 𝑑𝑗) ,

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Δ𝑉₃(𝑘) = ( ̃𝑑 + 1) ̃𝑥𝑇(𝑘) Φ𝑇𝑄₃Φ̃𝑥 (𝑘)

+ 𝑘−1∑

𝑖=𝑘+1−𝑑(𝑘+1)

− 𝑘−1∑

𝑖=𝑘+1−𝑑(𝑘)

̃𝑥𝑇_{(𝑖) Φ}𝑇_𝑄_{3Φ̃𝑥 (𝑖)}

− ̃𝑥𝑇(𝑘 − 𝑑 (𝑘)) Φ𝑇𝑄₃Φ̃𝑥 (𝑘 − 𝑑 (𝑘))

− 𝑘−𝑑1∑

𝑖=𝑘−𝑑2+1

= ( ̃𝑑 + 1) ̃𝑥𝑇(𝑘) Φ𝑇𝑄₃Φ̃𝑥 (𝑘)

+ 𝑘−1∑

𝑖=𝑘+1−𝑑1

̃𝑥𝑇_{(𝑖) Φ}𝑇_𝑄

3Φ̃𝑥 (𝑖)

+ 𝑘−𝑑1∑

𝑖=𝑘+1−𝑑(𝑘+1)̃𝑥

𝑇_{(𝑖) Φ}𝑇_𝑄

3Φ̃𝑥 (𝑖)

− ̃𝑥𝑇(𝑘 − 𝑑 (𝑘)) Φ𝑇𝑄3Φ̃𝑥 (𝑘 − 𝑑 (𝑘))

− 𝑘−1∑

𝑖=𝑘+1−𝑑(𝑘)

̃𝑥𝑇_{(𝑖) Φ}𝑇_𝑄_{3Φ̃𝑥 (𝑖)}

− 𝑘−𝑑∑1

𝑖=𝑘−𝑑2+1

≤ ( ̃𝑑 + 1) ̃𝑥𝑇(𝑘) Φ𝑇𝑄3Φ̃𝑥 (𝑘)

− ̃𝑥𝑇(𝑘 − 𝑑 (𝑘)) Φ𝑇𝑄₃Φ̃𝑥 (𝑘 − 𝑑 (𝑘)) .

(23)

By usingLemma 3, we have

Δ𝑉₄(𝑘) = 𝑑2₁̃𝜂𝑇(𝑘) Φ𝑇𝑆₁Φ̃𝜂 (𝑘)

− 𝑑₁ 𝑘−1∑

𝑖=𝑘−𝑑1

̃𝜂𝑇(𝑖) Φ𝑇𝑆₁Φ̃𝜂 (𝑖)

≤ 𝑑2

1((𝐴 − 𝐼) Φ̃𝑥 (𝑘) + 𝐴𝑑Φ̃𝑥 (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[𝑆2_{⋆ 𝑆2}𝑀] ≥ 0, the following inequality holds:

[ [ [ [ [

√𝛼_𝛼1

2(𝑥 (𝑘 − 𝑑 (𝑘)) − 𝑥 (𝑘 − 𝑑2))

−√𝛼2

𝛼₁(𝑥 (𝑘 − 𝑑1) − 𝑥 (𝑘 − 𝑑 (𝑘)))

] ] ] ] ] 𝑇

× [𝑆2 𝑀

⋆ 𝑆2]

×[[[_[

[

√𝛼_𝛼1

2(𝑥 (𝑘 − 𝑑 (𝑘)) − 𝑥 (𝑘 − 𝑑2))

−√𝛼2

𝛼₁ (𝑥 (𝑘 − 𝑑1) − 𝑥 (𝑘 − 𝑑 (𝑘)))

] ] ] ] ]

≥ 0,

(5)

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

Δ𝑉₅(𝑘) = ̃𝑑2_̃𝜂𝑇_{(𝑘) Φ}𝑇_𝑆

2Φ̃𝜂 (𝑘)

− ̃𝑑𝑘−𝑑(𝑘)−1∑

𝑖=𝑘−𝑑2

̃𝜂𝑇(𝑖) Φ𝑇𝑆₂Φ̃𝜂 (𝑖)

− ̃𝑑𝑘−𝑑∑1−1

𝑖=𝑘−𝑑(𝑘)

̃𝜂𝑇(𝑖) Φ𝑇𝑆₂Φ̃𝜂 (𝑖)

≤ ̃𝑑2̃𝜂𝑇(𝑘) Φ𝑇𝑆₂Φ̃𝜂 (𝑘)

−_𝑑 𝑑̃

2− 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−𝑑1−1 ∑ 𝑖=𝑘−𝑑(𝑘)

Φ̃𝜂(𝑖)) 𝑇

𝑆2(

𝑘−𝑑1−1 ∑

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) = 𝜁𝑇(𝑘) Π_𝑠𝜁 (𝑘) , (27)

where

𝜁 (𝑘)

= [̃𝑥𝑇_{(𝑘) ̃𝑥}𝑇_{(𝑘 − 𝑑1) Φ}𝑇 _̃𝑥𝑇_{(𝑘 − 𝑑 (𝑘)) Φ}𝑇 _̃𝑥𝑇_{(𝑘 − 𝑑2) Φ}𝑇_]𝑇_,

Π_𝑠=[[_[

[ ̃

Π₁₁ Φ𝑇_𝑆

1 0 0

⋆ Π̃₂₂ Π̃₂₃ 𝑀𝑇

⋆ ⋆ Π̃₃₃ Π̃₃₄

⋆ ⋆ ⋆ ̃Π₄₄

] ] ] ] +[[_[ [ ̃ Π₁₆ 0

𝑑₁𝐴𝑇_𝑑

0 ] ] ] ]

𝑆₁[[_[

[ ̃

Π₁₆

0

𝑑₁𝐴𝑇_𝑑

0 ] ] ] ] 𝑇 +[[_[ [ ̃ Π₁₇ 0 ̃ 𝑑𝐴𝑇 𝑑 0 ] ] ] ]

𝑆₂[[_[

[ ̃ Π₁₇ 0 ̃ 𝑑𝐴𝑇 𝑑 0 ] ] ] ] 𝑇 +[[_[ [ ̃ 𝐴𝑇 0 ̃ 𝐴𝑇 𝑑 0 ] ] ] ] 𝑃[[_[ [ ̃ 𝐴𝑇 0 ̃ 𝐴𝑇 𝑑 0 ] ] ] ] 𝑇 . (28)

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

Π_𝑠1=

[ [ [ [ [ [ [ [ [ [ ̃

1 0 0 ̃Π16𝑆1 Π̃17𝑆2 𝐴̃𝑇𝑃

⋆ ̃Π₂₂ Π̃₂₃ 𝑀𝑇 0 0 0

⋆ ⋆ Π̃₃₃ ̃Π₃₄ 𝑑₁𝐴𝑇

𝑑𝑆1 𝑑𝐴̃ 𝑇𝑑𝑆2 𝐴̃𝑇𝑑𝑃

⋆ ⋆ ⋆ ̃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

𝜉𝑇(𝑖) (Π + Π𝑇₁𝑃Π₁+ 𝑑₁2Π𝑇₂𝑆₁Π₂

+𝑑2₁₂Π𝑇₂𝑆₂Π₂) 𝜉 (𝑖) ,

(31)

where

𝜉 (𝑖)

= [̃𝑥𝑇_{(𝑖) ̃𝑥}𝑇_{(𝑖 − 𝑑}

1) Φ𝑇 ̃𝑥𝑇(𝑖 − 𝑑 (𝑖)) Φ𝑇 ̃𝑥𝑇(𝑖 − 𝑑2) Φ𝑇 𝑤 (𝑖)]𝑇,

Π = [ [ [ [ [ [ [ ̃

1 0 0 Π̃15

⋆ Π̃₂₂ Π̃₂₃ 𝑀𝑇 0

⋆ ⋆ Π̃₃₃ Π̃₃₄ −̃𝐿𝑇

𝑑𝑆

⋆ ⋆ ⋆ ̃Π₄₄ 0

⋆ ⋆ ⋆ ⋆ Π̃₅₅

] ] ] ] ] ] ] ,

Π₁= [ ̃𝐴 0 ̃𝐴𝑑 0 ̃𝐵] ,

Π₂= [(𝐴 − 𝐼) Φ 0 𝐴𝑑 0 𝐵] ,

Π₃= [̃𝐿 0 ̃𝐿𝑑 0 ̃𝐺] .

(32)

By using Schur complement equivalence, the inequality in (17) is equivalent toΠ+Π𝑇₁𝑃Π₁+𝑑₁2Π𝑇₂𝑆₁Π₂+𝑑2₁₂Π𝑇₂𝑆₂Π₂< 0. Then we have𝐽(𝑘) < 0; that is,

𝑉 (𝑘) <𝑘−1∑

𝑖=0

(6)

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

̃𝑧𝑇(𝑘) ̃𝑧 (𝑘) = 𝜂𝑇(𝑘) [̃𝐿 ̃𝐿𝑑 𝐺]̃𝑇[̃𝐿 ̃𝐿𝑑 𝐺] 𝜂 (𝑘)̃

≤ 𝛾2𝜂𝑇(𝑘) [

[

𝑃 0 0

⋆ 𝑄₃ 0

⋆ ⋆ 𝐼 ] ]

𝜂 (𝑘)

≤ 𝛾2(𝑉 (𝑘) + 𝑤𝑇(𝑘) 𝑤 (𝑘))

≤ 𝛾2∑𝑘

𝑖=0

𝑤𝑇(𝑖) 𝑤 (𝑖) ≤ 𝛾2∑∞

𝑖=0

𝑤𝑇(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) 𝜂𝑇(𝑖) 𝑆𝜂 (𝑖)

= −𝑘−𝑑(𝑘)−1∑

𝑖=𝑘−𝑑2

(𝑑2− 𝑑1) 𝜂𝑇(𝑖) 𝑆𝜂 (𝑖)

− 𝑘−𝑑1−1∑

𝑖=𝑘−𝑑(𝑘)

(𝑑₂− 𝑑₁) 𝜂𝑇(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.

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 ̃𝐿𝑇

⋆ 𝑄₃ 0 ̃𝐿𝑇

𝑑

⋆ ⋆ 𝐼 ̃𝐺𝑇

⋆ ⋆ ⋆ 𝛾2_𝐼

] ] ] ] ] ]

> 0, (40)

̃ Ω =

[ [ [ [ [ [ [ [ [ [ [ [ [ ̃

1 0 0 0 Π̃16𝐻1𝑇 ̃Π17𝐻2𝑇 𝐴̃𝑇𝑇𝑇

⋆ ̃Π₂₂ ̃Π₂₃ 𝑀𝑇 0 0 0 0

⋆ ⋆ ̃Π₃₃ ̃Π₃₄ 0 𝑑₁𝐴𝑇

𝑑𝐻1𝑇 𝑑𝐴̃ 𝑇𝑑𝐻2𝑇 𝐴̃𝑇𝑑𝑇𝑇

⋆ ⋆ ⋆ ̃Π₄₄ 0 0 0 0

⋆ ⋆ ⋆ ⋆ −𝐼 𝑑₁𝐵𝑇_𝐻𝑇

1 𝑑𝐵̃ 𝑇𝐻𝑇2 ̃𝐵𝑇𝑇𝑇

⋆ ⋆ ⋆ ⋆ ⋆ 𝑆₁− 𝐻𝑇

1 − 𝐻1 0 0

⋆ ⋆ ⋆ ⋆ ⋆ ⋆ 𝑆₂− 𝐻₂𝑇− 𝐻₂ 0

⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ 𝑃 − 𝑇𝑇_{− 𝑇}

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

(7)

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

(17).

Proof. On one hand, if (17) holds, then there exist𝐻_𝑗= 𝐻_𝑗𝑇=

𝑆𝑗, 𝑗 = 1, 2, and𝑇 = 𝑇𝑇 = 𝑃such that (41) holds. On the

other hand, if (41) holds, we have the following inequality based onLemma 1:

[ [ [ [ [ [ [ [ [ ̃

Π11 Φ𝑇𝑆1 0 0 0 Π̃16𝐻1𝑇 Π̃17𝐻2𝑇 𝐴̃𝑇𝑇𝑇

⋆ ̃Π22 ̃Π23 𝑀𝑇 0 0 0 0

⋆ ⋆ ̃Π33 Π̃34 0 𝑑1𝐴𝑇𝑑𝐻1𝑇 𝑑𝐴̃ 𝑇𝑑𝐻2𝑇 𝐴̃𝑇𝑑𝑇𝑇

⋆ ⋆ ⋆ ̃Π44 0 0 0 0

⋆ ⋆ ⋆ ⋆ −𝐼 𝑑1𝐵𝑇𝐻𝑇1 𝑑𝐵̃𝑇𝐻2𝑇 ̃𝐵𝑇𝑇𝑇

⋆ ⋆ ⋆ ⋆ ⋆ −𝐻1𝑆−11 𝐻1𝑇 0 0

⋆ ⋆ ⋆ ⋆ ⋆ ⋆ −𝐻2𝑆−12 𝐻2𝑇 0

⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ −𝑇𝑃−1_𝑇𝑇

] ] ] ] ] ] ] ] ] < 0. (42)

In addition, matrices𝐻_𝑗,𝑗 = 1, 2, and𝑇are nonsingular due

to𝑆_𝑗−𝐻_𝑗𝑇−𝐻_𝑗 < 0,𝑗 = 1, 2, and𝑃−𝑇𝑇−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.

matrices[𝑃1_{⋆ 𝑃}𝑃2

3] > 0,𝑄3 > 0,𝑄̃𝑙 > 0,𝑄𝑙> 0,𝑙 = 1, 2,𝑆𝑗 > 0,

𝐻𝑗,𝐹𝑗,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 − 𝐷𝑓𝐶)𝑇

⋆ 𝑃₃ 0 0 −𝐶𝑇_𝑓

⋆ ⋆ 𝑄₃ 0 (𝐿_𝑑− 𝐷_𝑓𝐶_𝑑)𝑇

⋆ ⋆ ⋆ 𝐼 (𝐺 − 𝐷_𝑓𝐶_𝑑)𝑇

⋆ ⋆ ⋆ ⋆ 𝛾2𝐼

] ] ] ] ] ] ] ] ] ] ]

> 0, (45)

where Ξ = [ [ [ [ [ [ [ [

Ξ₁₁ −𝑃₂ 𝑆₁ 𝐶𝑇_𝑁𝐶

𝑑 0 0

⋆ Ξ₂₂ 0 0 0 0

⋆ ⋆ Ξ₃₃ 𝑆₂− 𝑀𝑇 _𝑀𝑇 ₀

⋆ ⋆ ⋆ Ξ44 𝑆2− 𝑀𝑇 0

⋆ ⋆ ⋆ ⋆ Ξ₅₅ 0

⋆ ⋆ ⋆ ⋆ ⋆ −𝐼 ] ] ] ] ] ] ] ] , Γ= [ [ [ [ [ [ [ [ [ [ [ [

Γ11 Γ12 𝐴𝑇𝑇1𝑇+ 𝐶𝑇A𝑠0𝐵𝑇𝑓 𝐴𝑇𝐹1𝑇+ 𝐶𝑇A𝑠0𝐵𝑇𝑓

0 0 𝐴𝑇𝑓 𝐴𝑇𝑓

0 0 0 0

𝑑1𝐴𝑇𝑑𝐻1𝑇 𝑑𝐴̃ 𝑇𝑑𝐻𝑇2 𝐴𝑇𝑑𝑇1𝑇+ 𝐶𝑇𝑑A𝑠0𝐵𝑇𝑓 𝐴𝑇𝑑𝐹1𝑇+ 𝐶𝑇𝑑𝑖A𝑠0𝐵𝑇𝑓

0 0 0 0

𝑑1𝐵𝑇𝐻1𝑇 𝑑𝐵̃ 𝑇𝐻2𝑇 𝐵𝑇𝑇1𝑇+ 𝐷𝑇A𝑠0𝐵𝑇𝑓 𝐵𝑇𝐹1𝑇+ 𝐷𝑇A𝑠0𝐵𝑇𝑓

] ] ] ] ] ] ] ] ] ] ] ] , Λ=[[_[ [

𝑆1− 𝐻1− 𝐻1𝑇 0 0 0

⋆ 𝑆2− 𝐻2− 𝐻𝑇2 0 0

⋆ ⋆ 𝑃1− 𝑇1− 𝑇1𝑇 𝑃2− 𝐹2− 𝐹1𝑇

⋆ ⋆ ⋆ 𝑃3− 𝐹2− 𝐹2𝑇

] ] ] ] ,

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)𝑇_𝐻𝑇

1, Γ12= ̃𝑑(𝐴 − 𝐼)𝑇𝐻2𝑇.

(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 𝑇}0

2𝑇4−1] , 𝐹1= 𝑇2𝑇

−1

4 𝑇3, 𝐹2= 𝑇2𝑇4−𝑇𝑇2𝑇,

̃

𝑄𝑙= 𝑇2𝑇4−1𝑄𝑙𝑇4−𝑇𝑇2𝑇, 𝐽 =diag{𝐽1, 𝐼, 𝐼, 𝐼, 𝐼, 𝐼, 𝐼, 𝐼, 𝐽1} ,

[𝑃1 𝑃2

⋆ 𝑃3] = 𝐽1𝑃𝐽

𝑇

1, 𝑇 = [𝑇_𝑇1₃ 𝑇_𝑇2₄] ,

𝐽₂=diag{𝐽₁, 𝐼, 𝐼, 𝐼} ,

[𝐴𝑓 𝐵𝑓

𝐶_𝑓 𝐷_𝑓] = [𝑇0 𝐼] [2 0 𝐴𝐶𝑓𝑓 𝐷𝐵𝑓𝑓] [𝑇

−𝑇

4 𝑇2𝑇 0

0 𝐼] .

(48)

(8)

nonsingular. The inequality in (45) can be obtained by pre-an promultiplying (33) with𝐽₂ and𝐽₂𝑇, respectively. Noting that

Ω = 𝐽̃Ω𝐽𝑇 (49)

we haveΩ < 0̃ . On the other hand, because𝑇₂and𝑇₄cannot be obtained from (44), we cannot determine the filters from (48). However, we can construct an equivalent filter transfer function from𝑦(𝑘)tõ𝑧(𝑘):

𝑇̃𝑧𝑦= 𝐶𝑓(𝑧𝐼 − 𝐴𝑓)−1𝐵𝑓+ 𝐷𝑓

= 𝐶𝑓𝑇2−𝑇𝑇4𝑇(𝑧𝐼 − 𝑇2−1𝐴𝑓𝑇2−𝑇𝑇4𝑇)

−1

𝑇₂−1𝐵𝑓+ 𝐷𝑓

= 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 [𝑃1𝑃2

⋆ 𝑃3] > 0, 𝑄3 > 0, 𝑄̃𝑙 > 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 + 𝜀₃Ω𝑇

1Ω1+ 𝜀4Ω𝑇2Ω2 Ω3 Ω4

⋆ −𝜀₃𝐼 0

⋆ ⋆ −𝜀₄𝐼

] ]

< 0, (52)

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

Υ₁₁ 𝑃₂ Υ₁₃ Υ₁₄ (𝐿 − 𝐷_𝑓𝐶)𝑇 0 0

⋆ 𝑃₃ 0 0 −𝐶𝑇_𝑓 0 0

⋆ ⋆ Υ₃₃ Υ₃₄ (𝐿_𝑑− 𝐷_𝑓𝐶_𝑑)𝑇 0 0

⋆ ⋆ ⋆ Υ44 (𝐺 − 𝐷𝑓𝐶𝑑)𝑇 0 0

⋆ ⋆ ⋆ ⋆ 𝛾2_𝐼 _𝑀

3 0

⋆ ⋆ ⋆ ⋆ ⋆ 𝜀1𝐼 𝐷𝑓𝑀2

⋆ ⋆ ⋆ ⋆ ⋆ ⋆ 𝜀₂𝐼

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

> 0,

(53)

whereΩis defined in(44)and

Ω₁= [𝑁_𝐴 0 0 𝑁_𝐴𝑑 0 𝑁_𝐵 0 0 0 0] ,

Ω₂= [𝑁𝐶 0 0 𝑁𝐶𝑑 0 𝑁𝐷 0 0 0 0] ,

Ω₃

= [0 0 0 0 0 0 𝑑₁𝑀𝑇

1𝐻1𝑇 𝑑𝑀̃ 𝑇1𝐻2𝑇 𝑀1𝑇𝑇1𝑇 𝑀1𝑇𝐹1𝑇]

𝑇 ,

Ω₄= [0 0 0 0 0 0 0 0 𝑀𝑇

2 ̃𝐵𝑇𝑓 𝑀2𝑇̃𝐵𝑇𝑓]𝑇,

Υ₁₁= 𝑃₁− 𝜀₁𝑁_𝐿𝑇𝑁_𝐿− 𝜀₂𝑁_𝐶𝑇𝑁_𝐶,

Υ₁₃= −𝜀₁𝑁_𝐿𝑇𝑁_𝐿𝑑− 𝜀₂𝑁_𝐶𝑇𝑁_𝐶𝑑,

Υ₁₄= −𝜀₁𝑁_𝐿𝑇𝑁_𝐺− 𝜀₂𝑁_𝐶𝑇𝑁_𝐷,

Υ₃₃= 𝑄₃− 𝜀₁𝑁_𝐿𝑑𝑇𝑁_𝐿𝑑− 𝜀₂𝑁_𝐶𝑑𝑇𝑁_𝐶𝑑,

Υ34= −𝜀1𝑁𝐿𝑑𝑇𝑁𝐺− 𝜀2𝑁𝐶𝑑𝑇 𝑁𝐷,

Υ₄₄= 𝐼 − 𝜀₁𝑁_𝐺𝑇𝑁_𝐺− 𝜀₂𝑁_𝐷𝑇𝑁_𝐷.

(54)

Moreover, a suitable𝑙₂-𝑙_∞filter is given by

𝐴_𝑓= 𝐴_𝑓𝐹₂−1, 𝐵_𝑓= 𝐵_𝑓, 𝐶_𝑓= 𝐶_𝑓𝐹₂−1,

𝐷𝑓= 𝐷𝑓.

(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(Ω𝑇₁Δ𝑇₁Ω𝑇₃) +sym(Ω𝑇₂Δ𝑇₂Ω𝑇₄) < 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 + 𝜀₃Ω𝑇₁Ω₁+ 𝜀₃−1Ω₃Ω𝑇₃+ 𝜀₄Ω𝑇₂Ω₂+ 𝜀₄−1Ω₄Ω𝑇₄ < 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(Υ₁𝑇Δ𝑇₃Υ₃𝑇) +sym(Υ₂𝑇Δ𝑇₂Υ₄𝑇) > 0, (58)

where

Υ₁= [𝑁𝐿 𝑁𝐿𝑑 𝑁𝐺 0] , Υ2= [𝑁𝐶 𝑁𝐶𝑑 𝑁𝐷 0]

Υ₃= [0 0 0 𝑀𝑇

3] , Υ4= [0 0 0 𝑀2𝑇𝐷𝑇𝑓] .

(9)

By following the similar line, the equivalence between (58) and (53) 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 −0.5_{0.2 0.5 ] ,} 𝐴𝑑= [0.1 0_{0 0.2] ,}

𝐵 = [−1_{1 ] ,} 𝐶 = [−0.1 −1] , 𝐶_𝑑= [−0.1 0.6] ,

𝐿 = [1 0.2] , 𝐿_𝑑= [0.5 0.6], 𝐷 = 0.1,

𝐺 = −0.5.

(60)

For different delay cases, the different minima of𝛾 can be calculated by solving the LMIs inTheorem 9. When the upper bound of the time-varying delay is 5, that is, 𝑑₂ = 5, the minima of𝛾for a given𝑑₁are listed inTable 1.

Moreover, when𝑑₁ = 2,𝑑₂ = 5, the corresponding𝑙₂-𝑙_∞ filter is given as follows:

𝐴_𝑓= [2.6553 −1.9535_{2.2783 −1.5631] ,} 𝐵_𝑓= [−1.5405_{−1.9024] ,}

𝐶_𝑓= [−1.3230 1.0479] , 𝐷_𝑓= 0.2378.

(61)

When uncertainty appears in the system,Theorem 10will be used for the desired filter design. The following uncertainty parameters are considered:

𝑀₁= [0.35 0_{−0.2 0.1] ,} 𝑁_𝐴= [0.2 0.4_{0 0.5] ,}

𝑁_𝐴𝑑 = [0.1 0_{0 0.2] ,} 𝑁_𝐵= [0.3_{0.1] ,}

𝑀₂= 0.2, 𝑁_𝐶= [0.15 −0.22] ,

𝑁𝐶𝑑= [−0.3 0.2] , 𝑁𝐷= −0.5,

𝑀₃= −0.4, 𝑁_𝐿= [−0.25 −0.2] ,

𝑁_𝐿𝑑= [0.13 0.32] , 𝑁_𝐷= 0.2.

(62)

Similarly, the allowed minimal values of𝛾 can be obtained by solving the LMIs inTheorem 10. For𝑑₂ = 5, the different minimum allowed𝛾 are listed inTable 2 for the uncertain system with different𝑑₁.

Moreover, when𝑑₁= 2, 𝑑₂= 5, the desired filter is given as follows:

𝐴_𝑓= [−0.1261 0.2147_{−0.3333 0.5671] ,} 𝐵_𝑓= [−0.1725_{−0.2622] ,}

𝐶_𝑓= [−0.0076 0.0125] , 𝐷_𝑓= 0.1106.

(63)

Table 1: Minimum allowed𝛾for𝑑₂= 5.

Methods 𝑑1

1 2 3 4

Theorem 9 1.8371 1.5652 1.3585 1.1861

Table 2: Minimum allowed𝛾for𝑑₂= 5.

Methods 𝑑1

1 2 3 4

Theorem 10 3.4157 2.6514 2.1568 1.8405

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 sufficient𝑙₂-𝑙_∞ performance analysis conditions in terms of LMIs have been proposed to render the filtering error systems asymptotically stable with an𝑙₂-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).

References

[1] H. Gao and C. Wang, “A delay-dependent approach to robust

𝐻_∞filtering for uncertain discrete-time state-delayed systems,”

IEEE Transactions on Signal Processing, vol. 52, no. 6, pp. 1631– 1640, 2004.

[2] J. Lam, Z. Shu, S. Xu, and E. K. Boukas, “Robust𝐻_∞control of descriptor discrete-time Markovian jump systems,” Interna-tional Journal of Control, vol. 80, no. 3, pp. 374–385, 2007. [3] S. Ma, C. Zhang, and Z. Cheng, “Delay-dependent robust𝐻_∞

control for uncertain discrete-time singular systems with time-delays,”Journal of Computational and Applied Mathematics, vol. 217, no. 1, pp. 194–211, 2008.

[4] L. Wu, P. Shi, H. Gao, and C. Wang, “A new approach to robust𝐻_∞filtering for uncertain systems with both discrete and distributed delays,”Circuits, Systems, and Signal Processing, vol. 26, no. 2, pp. 229–247, 2007.

(10)

Transactions on Industrial Electronics, vol. 61, no. 1, pp. 436–446, 2014.

[6] T. Li, W. X. Zheng, and C. Lin, “Delay-slope-dependent stability results of recurrent neural networks,”IEEE Transactions on Neural Networks, vol. 22, no. 12, pp. 2138–2143, 2011.

[7] Q. Zhou, P. Shi, S. Xu, and H. Li, “Observer-based adaptive neu-ral network control for nonlinear stochastic systems with time-delay,”IEEE Transactions on Neural Networks and Learning Systems, vol. 24, no. 1, pp. 71–80, 2013.

[8] X. Zhu and G. Yang, “Jensen inequality approach to stability analysis of discrete-time systems with time-varying delay,” in

Proceedings of the American Control Conference (ACC ’08), pp. 1644–1649, Seattle, Wash, USA, June 2008.

[9] X. Meng, J. Lam, H. Gao, and B. Du, “A delay-partitioning approach to the stability analysis of discrete-time systems,”

Automatica, vol. 46, no. 3, pp. 610–614, 2010.

[10] H. Shao and Q. L. Han, “New stability criteria for linear discrete-time systems with interval-like time-varying delays,”

IEEE Transactions on Automatic Control, vol. 56, no. 3, pp. 619– 625, 2011.

[11] X. Meng, H. Gao, and S. Mou, “A new parameter-dependent approach to robust energy-to-peak filter design,”Circuits, Sys-tems, and Signal Processing, vol. 26, no. 4, pp. 451–471, 2007. [12] H. Zhang, A. S. Mehr, and Y. Shi, “Improved robust

energy-to-peak filtering for uncertain linear systems,”Signal Processing, vol. 90, no. 9, pp. 2667–2675, 2010.

[13] B. Zhang and S. Xu, “Robust𝐿₂−𝐿_∞filtering for uncertain non-linearly parameterized stochastic systems with time-varying delays,”Circuits, Systems, and Signal Processing, vol. 26, no. 5, pp. 751–772, 2007.

[14] B. Zhang and Y. Li, “Exponential𝐿₂−𝐿_∞filtering for distributed delay systems with markovian jumping parameters,” Signal Processing, vol. 93, no. 1, pp. 206–216, 2013.

[15] H. Gao and T. Chen, “New results on stability of discrete-time systems with time-varying state delay,”IEEE Transactions on Automatic Control, vol. 52, no. 2, pp. 328–334, 2007.

[16] H. Liu, F. Sun, and Z. Sun, “Reduced-order filtering with energy-to-peak performance for discrete-time Markovian jumping sys-tems,”IMA Journal of Mathematical Control and Information, vol. 21, no. 2, pp. 143–158, 2004.

[17] H. Zhang, Y. Shi, and A. S. Mehr, “Robust energy-to-peak filter-ing for networked systems with time-varyfilter-ing delays and ran-domly missing data,”IET Control Theory & Applications, vol. 4, no. 12, pp. 2921–2936, 2010.

[18] J. Qiu, G. Feng, and J. Yang, “New results on robust energy-to-peak filtering for discrete-time switched polytopic linear systems with time-varying delay,”IET Control Theory & Appli-cations, vol. 2, no. 9, pp. 795–806, 2008.

[19] P. G. Park, J. W. Ko, and C. Jeong, “Reciprocally convex approach to stability of systems with time-varying delays,”Automatica, vol. 47, no. 1, pp. 235–238, 2011.

[20] E. K. Boukas,Control of Singular Systems with Random Abrupt Changes, Communications and Control Engineering Series, Springer, Berlin, Germany, 2008.

[21] Z. Feng, J. Lam, and H. Gao, “𝛼-dissipativity analysis of singular time-delay systems,”Automatica, vol. 47, no. 11, pp. 2548–2552, 2011.

[22] S. Mou, H. Gao, and T. Chen, “New delay-range-dependent sta-bility condition for linear system,” inProceedings of the 7th World Congress on Intelligent Control and Automation (WCICA ’08), pp. 313–316, Chongqing, China, June 2008.

[23] S. Yin, S. Ding, A. Haghani, H. Hao, and P. Zhang, “A compar-ison study of basic data-driven fault diagnosis and process monitoring methods on the benchmark Tennessee Eastman process,”Journal of Process Control, vol. 22, no. 9, pp. 1567–1581, 2012.

[24] S. Yin, X. Yang, and H. R. Karimi, “Data-driven adaptive observer for fault diagnosis,”Mathematical Problems in Engi-neering, vol. 2012, Article ID 832836, 21 pages, 2012.

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