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Research ArticleObservers of Fuzzy Descriptor Systems with Time-Delays

Hongbiao Fan, Min Meng, and Jun-e Feng

School of Mathematics, Shandong University, Jinan 250100, China

Correspondence should be addressed to Jun-e Feng; [email protected]

Received 25 March 2014; Revised 11 June 2014; Accepted 13 June 2014; Published 6 July 2014

Academic Editor: Elena Braverman

Copyright © 2014 Hongbiao Fan 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.

For discrete fuzzy descriptor systems with time-delays, the problem of designing fuzzy observers is investigated in this paper. Basedon an equivalent transformation, discrete fuzzy descriptor systems with time-delays are converted into standard discrete systemswith time-delays. Then, via linear matrix inequality (LMI) approach, both delay-dependent and delay-independent conditions forthe existence of fuzzy state observers are obtained. Finally, two numerical examples are provided to illustrate the proposed method.

1. Introduction

For many practical engineering systems, increased pro-ductivity has led to new operating conditions, which aremore challenging. Such conditions would affect system’sperformance. To improve efficiency, a number of methodshave been proposed, such as fault-tolerant control [1], faultdetection, and isolation [2]. As far as we know, most practicalsystems are nonlinear, and it has been proved that any smoothnonlinear system can be accurately presented by Takagi-Sugeno (T-S) models, which were firstly presented by Takagiand Sugeno in 1985 [3]. Therefore, a lot of attention has beenattracted by them and some important results have beenobtained [4, 5]. On the other hand, in many real systems,a nature phenomenon is after-effect. Because of the con-tinuously expanded physical settings and capabilities, thecommon point-to-point communication form does not workwell in modern industry any longer. So it is necessary toseek a new platform. Due to comprehensive diagnosis, lowcost, and so on, online communication was introduced, andmore and more networks are considered in control loops.However, since the communication media possesses time-sharing, time-delays can not be avoided in the control loops.In addition, time-delays often cause instability and affectsystem’s performance. Therefore, many researchers haveinvestigated time-delayed systems [6–8]. However, there arefew papers referring to discrete-time fuzzy systems simulta-neously including singularity and time-delays.

Interconnected systems, which are described as differentnames in [9], have been concerned in many researches.Increasing attention is paid to both theory and practical use ofinterconnected systems. So its fundamental theory and appli-cations have involved a wide field during recent years. In realworld, there are also many interconnected systems, such aslarge electric networks, electric power systems, and differenttypes of societal systems. One such system consists of a seriesof independent subsystems, but all subsystems are interre-lated with some goals. Recently, for the stability and stabiliza-tion of interconnected systems, a number of methods havebeen used [10, 11]. For a class of uncertain nonlinear descrip-tor systems with input saturation, [12] considered robuststabilization. Furthermore, with mode-dependent time-varying delays, uncertain discrete-time switched systemswere considered to design 𝑙

2− 𝑙∞

filter in [13]. Indeed, T-S fuzzy model is also a kind of interconnected systems, andthis paper focuses on studying T-S fuzzy models. When theparameters and structure of a system are not known, a fuzzymodel can be used. T-S model, which is usually applied todescribe a nonlinear complex system, is one of themost com-mon types of fuzzymodels. A T-S fuzzymodel includesmanyfuzzy rules and the consequent part of each rule is in facta local model. A large number of problems about T-S fuzzymodel have been considered.

Utilizing the approach of linear matrix inequality (LMI),[14] investigated 𝐷-stability and nonfragile control for T-Sfuzzy discrete-time descriptor systems with multiple delays.

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014, Article ID 714518, 9 pageshttp://dx.doi.org/10.1155/2014/714518

2 Abstract and Applied Analysis

Subject to stochastic perturbation and time-varying delay, thepassivity and passification problems for T-S fuzzy descriptorsystems were studied in [15]. There are also some advancedmethods to deal with T-S fuzzy time-delayed systems. Forexample, a linear lower dimensional model was used toapproximate the original discrete-time fuzzy system withtime-delays in [16], and the model approximation was castedinto a sequential minimization problem with LMI con-straints. Distributed fuzzy filters were designed for a classof sensor networks, which were described by discrete-timeT-S fuzzy systems with time-varying delays and multipleprobabilistic packet losses, in [17]. About stability analysisand stabilization for a class of discrete-time T-S fuzzy systemswith time-varying state delay, a novel delay-partitioningmethodwas developed in [18] and a stability condition, whichis much less conservative than most existing results, wasderived by the new idea. Taking advantage of similar delay-partitioning approach, [19] analyzed dissipativity of T-S fuzzytime-delayed descriptor systems. Also for a class of discrete-time T-S fuzzy time-delay systems, the problem of reliablefilter design with strict dissipativity was considered in [20]and a sufficient condition of reliable dissipativity analysis wasproposed. Additionally, filter matrices can be obtained bysolving a convex optimization problem. This paper focuseson designing observers for discrete-time fuzzy descriptorsystems with time-delays via LMI approach, which is oftenused. Furthermore, we try to investigate the correspondingproblems with delay-partitioning method in our next work.

For a system, the control is often based on state feedback.However, all states of a system are not always available. Atthis time, it seems very important to estimate system states.So many scholars begin to focus on the problem of designingobservers and filters. For example, a proportional multiple-integral observer was investigated for fuzzy chaotic modelwith unknown input in [21]. As for descriptor time-delayedsystems with Markovian jump, [22] discussed designinglinear memoryless observers. And a method of designingdelay-dependent 𝐻

∞filters for singular systems with time-

delayswas considered in [23]. For fuzzy systems, considerableattention should be paid to fuzzy observers and some achieve-ments have been earned. In [24], a T-S fuzzy systemwas firstlytransformed into a standard form and then a sliding modelfuzzy observer could be constructed. With the approach in[24], it is effective to deal with matched and unmatcheduncertainties in fuzzymodels. In order to estimate the systemstate and output disturbance at the same time, Gao and hiscooperators tried to design a novel fuzzy observer [25], whichconsidered two cases for the output matrices. When eachsubsystem has the same output matrices, an augmented fuzzydescriptor system could be constructed. When the outputmatrices of all subsystems are different, a standard T-S fuzzysystem was studied. For each case, T-S fuzzy state-spaceobservers were designed, respectively.Moreover, the observertechniques proposed were applied to the fault estimation.In 2007, Marx et al. gave a method of designing decouplingobservers for T-S descriptor systems with unknown inputs[26], where the proposed observer could be used to performfault diagnosis. It is worth noting that only continuoussystems are concerned in most existing works. This paper

mainly discusses observer design for discrete-time-delayeddescriptor systems with T-S fuzzy model.

The paper is organized as follows. Section 2 introducesthe problem to be investigated. In Section 3, both delay-independent and delay-dependent sufficient conditions forthe existence of fuzzy state observer are given. For eachsufficient condition, two numerical examples are shown inSection 4. At last a short conclusion is included in Section 5.

2. Problem Formulation

In this section, we will briefly describe the problem to bestudied. Through this paper, the following discrete-time-delayed descriptor system is considered.

Plant form is as follows.Rule 𝑗: IF 𝜃

1(𝑘) is 𝑀

𝑗1, 𝜃2(𝑘) is 𝑀

𝑗2,. . ., and 𝜃

𝑝(𝑘) is 𝑀

𝑗𝑝,

THEN

𝐸𝑥 (𝑘 + 1) = 𝐴𝑗𝑥 (𝑘) + 𝐴

𝑑𝑗𝑥 (𝑘 − 𝑑) + 𝐵

𝑗𝑢 (𝑘)

𝑦𝑗(𝑘) = 𝐶

𝑗𝑥 (𝑘)

𝑥 (𝑘) = 𝜙 (𝑘) , 𝑘 = −𝑑, −𝑑 + 1, . . . , 0,

(1)

where 𝑀𝑗𝑠, 𝜃𝑠(𝑘) (1 ≤ 𝑠 ≤ 𝑝) are fuzzy sets and premise

variables, respectively. 𝑥(𝑘) ∈ R𝑛 is the system state, themeasurable output is 𝑦(𝑘) ∈ R𝑞, and 𝑢(𝑘) ∈ R𝑝 representsthe control input.The initial condition is denoted by𝜙(𝑘). It isassumed that𝐸 ∈ R𝑛×𝑛 is singular; that is, rank𝐸 = 𝑟 < 𝑛.Theremaining matrices, 𝐴

𝑗, 𝐴𝑑𝑗, 𝐵𝑗, and 𝐶

𝑗, are known. Besides,

1 ≤ 𝑗 ≤ 𝐿, where 𝐿 is the number of IF-THEN rules. 𝑑, 𝐿, 𝑝are positive integer numbers.

We make the following assumption.

Assumption 1. Consider

rank [𝐸

𝐶𝑗

] = 𝑛, 1 ≤ 𝑗 ≤ 𝐿, (2)

so there exists a matrix pair [𝑇𝑗𝑁𝑗] for each 𝑗 ∈ {1, 2, . . . , 𝐿}

such that

𝑇𝑗𝐸 + 𝑁

𝑗𝐶𝑗= 𝐼𝑛, 1 ≤ 𝑗 ≤ 𝐿. (3)

Remark 2. Because of the singularity of matrix 𝐸 in fuzzysystem (1), we give Assumption 1. Condition (2) is the sameas that in [27], where another condition is also needed. In thispaper, only one is enough.

Now we try to transform system (1) into a form as astandard system. Denote by𝑋

+ the pseudoinverse of amatrix𝑋. Then the general solution to (3), with condition (2), is

[𝑇𝑗 𝑁𝑗] = [

𝐸

𝐶𝑗

]

+

+ 𝑍(𝐼𝑛+𝑞

− [𝐸

𝐶𝑗

] [𝐸

𝐶𝑗

]

+

) , (4)

where matrix 𝑇𝑗(1 ≤ 𝑗 ≤ 𝐿) is nonsingular, which

can be achieved via designing the arbitrary matrix 𝑍 withappropriate dimension.

Abstract and Applied Analysis 3

According to equality (3), the first equation of fuzzysystem (1) is changed into

𝑥 (𝑘 + 1) = 𝑇𝑗𝐴𝑗𝑥 (𝑘) + 𝑇

𝑗𝐴𝑑𝑗𝑥 (𝑘 − 𝑑) + 𝑇

𝑗𝐵𝑗𝑢 (𝑘)

+ 𝑁𝑗𝑦𝑗(𝑘 + 1) ,

(5)

where 1 ≤ 𝑗 ≤ 𝐿.Let

ℎ𝑗(𝜃 (𝑘)) =

∏𝑝

𝑠=1𝑀𝑗𝑠

(𝜃𝑗(𝑘))

∑𝐿

𝑗=1∏𝑝

𝑠=1𝑀𝑗𝑠

(𝜃𝑗(𝑘))

, 1 ≤ 𝑗 ≤ 𝐿, (6)

with 𝜃(𝑘) = [𝜃1(𝑘)𝜃2(𝑘) ⋅ ⋅ ⋅ 𝜃

𝑝(𝑘)]. 𝑀

𝑗𝑠(𝜃𝑗(𝑘)) means the

grade of membership of 𝜃𝑗(𝑘) in 𝑀

𝑗𝑠. Obviously, 0 ≤

𝑀𝑗𝑠(𝜃𝑗(𝑘)) ≤ 1. Therefore, ℎ

𝑗(𝜃(𝑘)) ≥ 0 (1 ≤ 𝑗 ≤ 𝐿) and

∑𝐿

𝑗=1ℎ𝑗(𝜃(𝑘)) = 1 for all 𝑘. For fuzzy system (1), the final state

and output are given as follows:

𝑥 (𝑘 + 1) =

𝐿

∑

𝑗=1

ℎ𝑗(𝜃 (𝑘)) [𝑇

𝑗𝐴𝑗𝑥 (𝑘) + 𝑇

𝑗𝐴𝑑𝑗𝑥 (𝑘 − 𝑑)

+𝑇𝑗𝐵𝑗𝑢 (𝑘) + 𝑁

𝑗𝑦𝑗(𝑘 + 1)] ,

𝑦 (𝑘) =

𝐿

∑

𝑗=1

ℎ𝑗(𝜃 (𝑘)) 𝐶

𝑗𝑥 (𝑘) ,

𝑥 (𝑘) = 𝜙 (𝑘) , 𝑘 = −𝑑, −𝑑 + 1, . . . , 0.

(7)

Define

[𝐴 (𝜃) 𝐴𝑑(𝜃) 𝐵 (𝜃) 𝐶 (𝜃)]

≜

𝐿

∑

𝑗=1

ℎ𝑗(𝜃 (𝑘)) [𝐴𝑗 𝐴

𝑑𝑗𝐵𝑗

𝐶𝑗] ,

(8)

where [𝐴𝑗

𝐴𝑑𝑗

𝐵𝑗] = 𝑇𝑗[𝐴𝑗 𝐴

𝑑𝑗𝐵𝑗].Then system (7) can

be written as

𝑥 (𝑘 + 1) = 𝐴 (𝜃) 𝑥 (𝑘) + 𝐴𝑑(𝜃) 𝑥 (𝑘 − 𝑑) + 𝐵 (𝜃) 𝑢 (𝑘)

+

𝐿

∑

𝑗=1

ℎ𝑗(𝜃 (𝑘))𝑁

𝑗𝑦𝑗(𝑘 + 1) ,

𝑦 (𝑘) = 𝐶 (𝜃) 𝑥 (𝑘) ,

𝑥 (𝑘) = 𝜙 (𝑘) , 𝑘 = −𝑑, −𝑑 + 1, . . . , 0.

(9)

Through the above analysis, singular fuzzy system (1) istransformed into the ordinary linear system (9), for which theobserver will be designed. In this paper, the state observer asin the following form is considered.

Observer rule 𝑗: IF 𝜃1(𝑘) is𝑀

𝑗1, 𝜃2(𝑘) is𝑀

𝑗2,. . ., and 𝜃

𝑝(𝑘)

is 𝑀𝑗𝑝, THEN

𝑥 (𝑘 + 1) = 𝑇𝑗𝐴𝑗𝑥 (𝑘) + 𝑇

𝑗𝐴𝑑𝑗𝑥 (𝑘 − 𝑑) + 𝑇

𝑗𝐵𝑗𝑢 (𝑘)

+ 𝐺𝑗(𝑦 (𝑘) − 𝑦 (𝑘)) + 𝑁

𝑗𝑦𝑗(𝑘 + 1) ,

𝑦𝑗(𝑘) = 𝐶

𝑗𝑥 (𝑘) ,

(10)

where 1 ≤ 𝑗 ≤ 𝐿, 𝑥(𝑘) ∈ R𝑛 is the estimate of state 𝑥(𝑘), andmatrices 𝐺

𝑗(𝑗 = 1, 2, . . . , 𝐿) are observation error matrices.

Denote by 𝑦(𝑘) and 𝑦(𝑘) the final output of fuzzy system andfuzzy observer, respectively. With the same weight ℎ

𝑗(𝜃(𝑘))

and notations in system (7) the final estimated state andoutput of fuzzy observer (10) can be described as

𝑥 (𝑘 + 1) = 𝐴 (𝜃) 𝑥 (𝑘) + 𝐴𝑑(𝜃) 𝑥 (𝑘 − 𝑑) + 𝐵 (𝜃) 𝑢 (𝑘)

+ 𝐺 (𝜃) (𝑦 (𝑘) − 𝑦 (𝑘)) + 𝑦 (𝑘 + 1) ,

𝑦 (𝑘) = 𝐶 (𝜃) 𝑥 (𝑘) ,

(11)

where 𝑦(𝑘) = ∑𝐿

𝑗=1ℎ𝑗(𝜃(𝑘))𝑁

𝑗𝑦𝑗(𝑘) and 𝐺(𝜃) ≜

∑𝐿

𝑗=1ℎ𝑗(𝜃(𝑘))𝐺

𝑗.

Thus the problem of this paper focuses on finding matri-ces 𝐺𝑗(𝑗 = 1, 2, . . . , 𝐿) such that model (11) is an observer of

system (9).

3. Main Results

This section discusses how to design fuzzy state observersfor discrete descriptor systems with time-delays. And forexistence of fuzzy observer, two different sufficient conditionsare derived. We give the Schur complement Lemma first.

Lemma 3 (see [28]). The LMI

[𝑀 𝑆

𝑆𝑇

𝑅] > 0, (12)

where 𝑀 = 𝑀𝑇, 𝑅 = 𝑅

𝑇 are equivalent to

𝑅 > 0, 𝑀 − 𝑆𝑅−1

𝑆𝑇

> 0. (13)

Now a sufficient condition about existence of consideredobserver, which does not rely on time-delay, is presented bythe following theorem.

Theorem 4. Model (10) is a state observer of fuzzy system (1),if there exist common matrices 𝑃 > 0 and 𝑄 > 0 and matrices𝑊𝑗(𝑗 = 1, 2, . . . , 𝐿) such that

Λ𝑖𝑖< 0, 𝑖 = 1, 2, . . . , 𝐿,

Λ𝑖𝑗+ Λ𝑗𝑖

< 0, 1 ≤ 𝑖 < 𝑗 ≤ 𝐿,

(14)

where

Λ𝑖𝑗

= [

[

𝑄 − 𝑃 0 𝐴𝑇

𝑖𝑃 + 𝐶

𝑇

𝑗𝑊𝑖

∗ −𝑄 𝐴𝑇

𝑑𝑖𝑃

∗ ∗ −𝑃

]

]

, (15)


and ∗ denotes matrix entries implied by the symmetry of amatrix through this paper. Furthermore, the observer gainmatrices can be obtained as

𝐺𝑖= 𝑃−1

𝑊𝑇

𝑖, 1 ≤ 𝑖 ≤ 𝐿. (16)

Proof. Define

𝑒 (𝑘) = 𝑥 (𝑘) − 𝑥 (𝑘) . (17)

From system (9) and observer (11), the error dynamic equa-tion of estimation error 𝑒(𝑘) can be derived as

𝑒 (𝑘 + 1) = [𝐴 (𝜃) + 𝐺 (𝜃) 𝐶 (𝜃)] 𝑒 (𝑘) + 𝐴𝑑(𝜃) 𝑒 (𝑘 − 𝑑) .

(18)

Take a Lyapunov function as

𝑉 (𝑘) = 𝑒𝑇(𝑘) 𝑃𝑒 (𝑘) +

𝑑

∑

𝑖=1

𝑒𝑇(𝑘 − 𝑖) 𝑄𝑒 (𝑘 − 𝑖) , (19)

where 𝑃,𝑄 > 0; then

Δ𝑉 (𝑘) = 𝑉 (𝑘 + 1) − 𝑉 (𝑘)

= 𝑒𝑇(𝑘 + 1) 𝑃𝑒 (𝑘 + 1) − 𝑒

𝑇(𝑘) 𝑃𝑒 (𝑘)

+

𝑑

∑

𝑖=1

𝑒𝑇(𝑘 + 1 − 𝑖) 𝑄𝑒 (𝑘 + 1 − 𝑖)

−

𝑑

∑

𝑖=1

𝑒𝑇(𝑘 − 𝑖) 𝑄𝑒 (𝑘 − 𝑖)

= 𝑒𝑇(𝑘 + 1) 𝑃𝑒 (𝑘 + 1) − 𝑒


+ 𝑒𝑇(𝑘) 𝑄𝑒 (𝑘) − 𝑒

𝑇(𝑘 − 𝑑)𝑄𝑒 (𝑘 − 𝑑)

= {[𝐴 (𝜃) + 𝐺 (𝜃) 𝐶 (𝜃)] 𝑒 (𝑘) + 𝐴𝑑(𝜃) 𝑒 (𝑘 − 𝑑)}

𝑇

× 𝑃 {[𝐴 (𝜃) + 𝐺 (𝜃) 𝐶 (𝜃)] 𝑒 (𝑘) + 𝐴𝑑(𝜃) 𝑒 (𝑘 − 𝑑)}

− 𝑒𝑇(𝑘) 𝑃𝑒 (𝑘) + 𝑒

𝑇(𝑘) 𝑄𝑒 (𝑘)

− 𝑒𝑇(𝑘 − 𝑑)𝑄𝑒 (𝑘 − 𝑑) .

(20)

Let 𝜂(𝑘) = [𝑒𝑇(𝑘) 𝑒𝑇(𝑘 − 𝑑)]

𝑇; thus

Δ𝑉 (𝑘) = 𝜂𝑇(𝑘) Λ𝜂 (𝑘) , (21)

where

Λ = [

[

𝜆1(𝜃) 𝜆

2(𝜃)

𝜆𝑇

2(𝜃) 𝜆

3(𝜃)

]

]

, (22)

while 𝜆1(𝜃) = [𝐴(𝜃) + 𝐺(𝜃)𝐶(𝜃)]

𝑇𝑃[𝐴(𝜃) + 𝐺(𝜃)𝐶(𝜃)] +

𝑄 − 𝑃, 𝜆2(𝜃) = [𝐴(𝜃) + 𝐺(𝜃)𝐶(𝜃)]

𝑇𝑃𝐴𝑑(𝜃), and 𝜆

3(𝜃) =

𝐴𝑇

𝑑(𝜃)𝑃𝐴

𝑑(𝜃) − 𝑄.

The inequalityΔ𝑉(𝑘) < 0 holds for all 𝜂(𝑘) ̸= 0 if and onlyif

Λ < 0. (23)

From the Lyapunov stable theory, if inequality (23) holds, theerror system (18) would be asymptotically stable.

On the other hand, according to Lemma 3, inequality (23)is equivalent to

[[

[

𝑄 − 𝑃 0 [𝐴 (𝜃) + 𝐺 (𝜃) 𝐶 (𝜃)]𝑇

𝑃


𝑑(𝜃) 𝑃

∗ ∗ −𝑃

]]

]

< 0. (24)

With the definition of coefficientmatrices in system (9), it canbe obtained from inequality (24) that

𝐿

∑

𝑖=1

𝐿

∑

𝑗=1

ℎ𝑖(𝜃 (𝑘)) ℎ

𝑗(𝜃 (𝑘))

[[

[

𝑄 − 𝑃 0 (𝐴𝑖+ 𝐺𝑖𝐶𝑗)𝑇

𝑃


𝑑𝑖𝑃

∗ ∗ −𝑃

]]

]

< 0,

(25)

which is equivalent to

𝐿

∑

𝑖=1

𝐿

∑

𝑗=1


𝑗(𝜃 (𝑘)) Λ

𝑖𝑗< 0. (26)

And inequality (26) can be rewritten as

𝐿

∑

𝑖=1

ℎ𝑖(𝜃 (𝑘)) Λ

𝑖𝑖+

𝐿−1

∑

𝑖=1

𝐿

∑

𝑗=𝑖+1


𝑗(𝜃 (𝑘)) (Λ

𝑖𝑗+ Λ𝑗𝑖) < 0.

(27)

Taking𝑊𝑖= 𝐺𝑇

𝑖𝑃, it follows that conditions (14) are sufficient

to guarantee (26) is correct. Inequality (23) holds and theerror system (18) is asymptotically stable.

Obviously, conditions in Theorem 4 are not relevant todelay 𝑑. Next, we will give another result which depends ontime-delay.

Theorem5. For fuzzy system (1), there is a fuzzy state observerin form of (10) if there exist common matrices 𝑃 > 0, 𝑈 > 0,and 𝑅 > 0 and matrices 𝑊

𝑗(𝑗 = 1, 2, . . . , 𝐿) such that

Ψ𝑖𝑖< 0, 𝑖 = 1, 2, . . . , 𝐿,

Ψ𝑖𝑗+ Ψ𝑗𝑖

< 0, 1 ≤ 𝑖 < 𝑗 ≤ 𝐿,(28)

where

Ψ𝑖𝑗

=

[[[[[

[

𝑈 − 𝑃 0 𝐴𝑇

𝑖𝑃 + 𝐶

𝑇

𝑗𝑊𝑖

𝐴𝑇

𝑖𝑃 + 𝐶

𝑇

𝑗𝑊𝑖− 𝑃

∗ −𝑈 𝐴𝑇

𝑑𝑖𝑃 𝐴

𝑇

𝑑𝑖𝑃

∗ ∗ −𝑃 0

∗ ∗ ∗ −1

𝑑𝑅

]]]]]

]

.

(29)


Moreover, the observer gain matrices can be calculated as

𝐺𝑖= 𝑃−1

𝑊𝑇

𝑖, 1 ≤ 𝑖 ≤ 𝐿. (30)

Proof. According to proof of Theorem 4, the dynamic equa-tion of error system is (18). Construct a Lyapunov functionas

𝑉 (𝑘)

= 𝑒𝑇(𝑘) 𝑃𝑒 (𝑘) +

𝑘−1

∑

𝑙=𝑘−𝑑

𝑒𝑇(𝑙) 𝑈𝑒 (𝑙)

+

0

∑

𝑠=−𝑑+1

𝑘−1

∑

𝑙=𝑘−1+𝑠

[𝑒 (𝑙 + 1) − 𝑒 (𝑙)]𝑇𝑍 [𝑒 (𝑙 + 1) − 𝑒 (𝑙)] ,

(31)

where 𝑃,𝑈, 𝑍 > 0. Then

Δ𝑉 (𝑘)

= 𝑉 (𝑘 + 1) − 𝑉 (𝑘)

= 𝑒𝑇(𝑘 + 1) 𝑃𝑒 (𝑘 + 1) − 𝑒


+

𝑘

∑

𝑙=𝑘−𝑑+1

𝑒𝑇(𝑙) 𝑈𝑒 (𝑙) −

𝑘−1

∑

𝑙=𝑘−𝑑

𝑒𝑇(𝑙) 𝑈𝑒 (𝑙)

+

0

∑

𝑠=−𝑑+1

{

𝑘

∑

𝑙=𝑘+𝑠

[𝑒 (𝑙 + 1) − 𝑒 (𝑙)]𝑇𝑍 [𝑒 (𝑙 + 1) − 𝑒 (𝑙)]

−

𝑘−1

∑

𝑙=𝑘−1+𝑠

[𝑒 (𝑙 + 1) − 𝑒 (𝑙)]𝑇

×𝑍 [𝑒 (𝑙 + 1) − 𝑒 (𝑙)] }

= 𝑒𝑇(𝑘 + 1) 𝑃𝑒 (𝑘 + 1) − 𝑒

𝑇(𝑘) 𝑃𝑒 (𝑘) + 𝑒

𝑇(𝑘) 𝑈𝑒 (𝑘)

− 𝑒𝑇(𝑘 − 𝑑)𝑈𝑒 (𝑘 − 𝑑)

−

0

∑

𝑠=−𝑑+1

{[𝑒 (𝑘 + 𝑠) − 𝑒 (𝑘 − 1 + 𝑠)]𝑇

×𝑍 [𝑒 (𝑘 + 𝑠) − 𝑒 (𝑘 − 1 + 𝑠)]}

+ 𝑑[𝑒 (𝑘 + 1) − 𝑒 (𝑘)]𝑇𝑍 [𝑒 (𝑘 + 1) − 𝑒 (𝑘)]

(18)

= 𝑒𝑇(𝑘) 𝜔1(𝜃) 𝑒 (𝑘) + 𝑒

𝑇(𝑘 − 𝑑) 𝜔

2(𝜃) 𝑒 (𝑘 − 𝑑)

+ 𝑒𝑇(𝑘) 𝜔3(𝜃) 𝑒 (𝑘 − 𝑑) + 𝑒

𝑇(𝑘 − 𝑑) 𝜔

𝑇

3(𝜃) 𝑒 (𝑘)

−

0

∑

𝑠=−𝑑+1

[𝑒 (𝑘 + 𝑠) − 𝑒 (𝑘 − 1 + 𝑠)]𝑇

× 𝑍 [𝑒 (𝑘 + 𝑠) − 𝑒 (𝑘 − 1 + 𝑠)] ,

(32)

where

𝜔1(𝜃) = [𝐴 (𝜃) + 𝐺 (𝜃) 𝐶 (𝜃)]

𝑇

𝑃 [𝐴 (𝜃) + 𝐺 (𝜃) 𝐶 (𝜃)]

− 𝑃 + 𝑈 + 𝜔4(𝜃) ,

𝜔2(𝜃) = 𝐴

𝑇

𝑑(𝜃) 𝑃𝐴

𝑑(𝜃) − 𝑈 + 𝑑𝐴

𝑇

𝑑(𝜃) 𝑍𝐴

𝑑(𝜃) ,

𝜔3(𝜃) = [𝐴 (𝜃) + 𝐺 (𝜃) 𝐶 (𝜃)]

𝑇

𝑃𝐴𝑑(𝜃)

+ 𝑑[𝐴 (𝜃) + 𝐺 (𝜃) 𝐶 (𝜃) − 𝐼]𝑇

𝑍𝐴𝑑(𝜃) ,

𝜔4(𝜃) = 𝑑[𝐴 (𝜃) + 𝐺 (𝜃) 𝐶 (𝜃) − 𝐼]

𝑇

× 𝑍 [𝐴 (𝜃) + 𝐺 (𝜃) 𝐶 (𝜃) − 𝐼] .

(33)

Define 𝜂(𝑘) = [𝑒𝑇(𝑘) 𝑒𝑇(𝑘 − 𝑑)]

𝑇; then

Δ𝑉 (𝑘) = 𝜂𝑇(𝑘) Ψ𝜂 (𝑘)

−

0

∑

𝑠=−𝑑+1

[𝑒 (𝑘 + 𝑠) − 𝑒 (𝑘 − 1 + 𝑠)]𝑇

× 𝑍 [𝑒 (𝑘 + 𝑠) − 𝑒 (𝑘 − 1 + 𝑠)]

≤ 𝜂𝑇(𝑘) Ψ𝜂 (𝑘) ,

(34)

where

Ψ = [

[

𝜔1(𝜃) 𝜔

3(𝜃)

𝜔𝑇

3(𝜃) 𝜔

2(𝜃)

]

]

. (35)

If

Ψ < 0, (36)

then Δ𝑉(𝑘) < 0 and the error system (18) is asymptoticallystable.

By Lemma 3, inequality (36) is equivalent to


[[[[[

[

𝑈 − 𝑃 0 [𝐴 (𝜃) + 𝐺 (𝜃) 𝐶 (𝜃)]𝑇

𝑃 [𝐴 (𝜃) + 𝐺 (𝜃) 𝐶 (𝜃) − 𝐼]𝑇


𝑑(𝜃) 𝑃 𝐴

𝑇

𝑑(𝜃)

∗ ∗ −𝑃 0

∗ ∗ ∗ −1

𝑑𝑍−1

]]]]]

]

< 0. (37)

Define 𝐽 ≜ diag[𝐼, 𝐼, 𝐼, 𝑃]. Pre- and postmultiplying inequal-ity (37) by matrix 𝐽, we can obtain

[[[[[

[

𝑈 − 𝑃 0 [𝐴 (𝜃) + 𝐺 (𝜃) 𝐶 (𝜃)]𝑇

𝑃 [𝐴 (𝜃) + 𝐺 (𝜃) 𝐶 (𝜃) − 𝐼]𝑇

𝑃


𝑑(𝜃) 𝑃 𝐴

𝑇

𝑑(𝜃) 𝑃

∗ ∗ −𝑃 0

∗ ∗ ∗ −1

𝑑𝑃𝑍−1

𝑃

]]]]]

]

< 0. (38)

Taking 𝑅 = 𝑃𝑍−1

𝑃 and 𝑊𝑖

= 𝐺𝑇

𝑖𝑃, it is derived that

inequality (38) is equivalent to

𝐿

∑

𝑖=1

𝐿

∑

𝑗=1


𝑗(𝜃 (𝑘)) Ψ

𝑖𝑗< 0. (39)

That is𝐿

∑

𝑖=1

ℎ𝑖(𝜃 (𝑘)) Ψ

𝑖𝑖+

𝐿−1

∑

𝑖=1

𝐿

∑

𝑗=𝑖+1


𝑗(𝜃 (𝑘)) (Ψ

𝑖𝑗+ Ψ𝑗𝑖) < 0.

(40)

Combining with inequalities (28), we can know thatinequality (39) holds and condition (36) is correct. Thus theerror system (18) is asymptotically stable.

Remark 6. It is easy to see thatTheorem 5 impliesTheorem 4.In fact, with 𝜑

𝑖𝑗≜ [𝑃𝐴

𝑖+ 𝑊𝑇

𝑖𝐶𝑗

− 𝑃 𝑃𝐴𝑑𝑖

0]𝑇

(𝑖, 𝑗 =

1, 2, . . . , 𝐿), we have

Ψ𝑖𝑗

= [

[

Λ𝑖𝑗

𝜑𝑖𝑗

∗ −1

𝑑𝑅]

]

, (41)

where matrices Ψ𝑖𝑗and Λ

𝑖𝑗are defined in Theorems 5 and

4, respectively. So, according to Lemma 3, inequalities (28)are equivalent to Λ

𝑖𝑖+ 𝑑𝜑𝑖𝑖𝑅−1

𝜑𝑇

𝑖𝑖< 0 (𝑖 = 1, 2, . . . , 𝐿) and

Λ𝑖𝑗+Λ𝑗𝑖+ (𝑑/2)(𝜑

𝑖𝑗+𝜑𝑗𝑖)𝑅−1

(𝜑𝑖𝑗+𝜑𝑗𝑖)𝑇

< 0 (1 ≤ 𝑖 < 𝑗 ≤ 𝐿),respectively. Since𝑅 > 0, from inequalities (28) we can obtainthat Λ

𝑖𝑖< 0 (𝑖 = 1, 2, . . . , 𝐿) and Λ

𝑖𝑗+ Λ𝑗𝑖

< 0 (1 ≤ 𝑖 <

𝑗 ≤ 𝐿). Therefore, the conditions of Theorem 5 are strongerthan those of Theorem 4. On the other hand, if there donot exist compatible solutions to inequalities (28) for somefuzzy descriptor system, we may design its observer based onTheorem 4.

Remark 7. For fuzzy state-space systems, the results of thispaper can also be applied. Choosing 𝐸 = 𝐼

𝑛, a discrete fuzzy

state-space system can be derived from fuzzy system (1) with-out time-delays. Let 𝑇

𝑗= 𝐼𝑛, 𝑁𝑗= 0, and 𝐴

𝑑𝑗= 0 (1 ≤ 𝑗 ≤

𝐿). Then, the presented observer in [4] can also be obtainedfrom model (10).

At the end, we give an algorithm to design observer forfuzzy descriptor system (1).

Algorithm 8. The following steps are introduced to determineobserver (10) for fuzzy descriptor system (1) according toTheorem 4.

(1) Verify condition (2). If it holds, then go to step 2.Otherwise, stop.

(2) According to formulation (4), solve (3) and guaranteethat matrix 𝑇

𝑗(1 ≤ 𝑗 ≤ 𝐿) is nonsingular. Then

transform fuzzy system (1) into system (9) withsolutions to (3).

(3) Solve LMIs (14). If they are solvable, go to step 4.Otherwise, stop.

(4) Give fuzzy observer (10), with 𝐺𝑖= 𝑃−1

𝑊𝑇

𝑖(1 ≤ 𝑖 ≤

𝐿).

The steps of designing observer for fuzzy descriptorsystem (1) based onTheorem 5 are similar to Algorithm 8 andomitted here.

Remark 9. In this paper, we firstly consider observer designfor fuzzy discrete systems with singularity and time-delayssimultaneously. Comparedwith existingwork [27], the singu-larity of fuzzy system can be easily eliminated under only onesimple assumption. Finally, both delay-dependent and delay-independent conditions are derived for obtaining observergain matrices.


4. Examples

In this section, two numerical examples are provided. Thefirst one is to design a fuzzy observer according toTheorem 4and the second one illustrates Theorem 5.

Example 10. Consider a discrete fuzzy descriptor systemwithtime-delays as follows.

Plant Rule 1: IF 𝑥1(𝑘) is 𝑀

1(𝑥1(𝑘)), THEN

𝐸𝑥 (𝑘 + 1) = 𝐴1𝑥 (𝑘) + 𝐴

𝑑1𝑥 (𝑘 − 𝑑) + 𝐵

1𝑢 (𝑘) ,

𝑦1(𝑘) = 𝐶

1𝑥 (𝑘) ,

𝑥 (𝑘) = 𝜙 (𝑘) , 𝑘 = −𝑑, −𝑑 + 1, . . . , 0.

(42)

Plant Rule 2: IF 𝑥1(𝑘) is 𝑀

2(𝑥1(𝑘)), THEN

𝐸𝑥 (𝑘 + 1) = 𝐴2𝑥 (𝑘) + 𝐴

𝑑2𝑥 (𝑘 − 𝑑) + 𝐵

2𝑢 (𝑘) ,

𝑦2(𝑘) = 𝐶

2𝑥 (𝑘) ,

𝑥 (𝑘) = 𝜙 (𝑘) , 𝑘 = −𝑑, −𝑑 + 1, . . . , 0,

(43)

where

𝐸 = [1 0

0 0] , 𝐴

1= [

0 1

−1 1] , 𝐴

2= [

0 −1

−1 1] ,

𝐴𝑑1

= [−0.1 0

0.2 0.1] , 𝐴

𝑑2= [

0 0.1

−0.2 0] ,

𝐵1= [

0

1] , 𝐵

2= [

0

−1] , 𝐶

1= [

0

1]

𝑇

,

𝐶2= [

1

1]

𝑇

,

(44)

and membership functions are

𝑀1(𝑥1(𝑘)) ≜ 1 −

𝑥2

1(𝑘)

2.25,

𝑀2(𝑥1(𝑘)) ≜

𝑥2

1(𝑘)

2.25.

(45)

Orbits of membership functions for Rules 1 and 2 are shownin Figure 1.

Coefficient matrices of fuzzy descriptor system are given,and it can be easily verified that matrices 𝐸 and 𝐶

𝑗(𝑗 = 1, 2)

satisfy condition (2). Based on formulation (4), a solution to(3) is shown as

𝑇1= [

1 0

0 −1] , 𝑁

1= [

0

1] ,

𝑇2= [

1 0

−1 −1] , 𝑁

2= [

0

1] .

(46)

0 0.5 1 1.50

0.10.20.30.40.50.60.70.80.9

1

Mem

bers

hip

Rule 1Rule 2

−1.5

x1(k)

−1 −0.5

Figure 1: Membership functions.

Then fuzzy descriptor system is converted into the follow-ing ordinary system:

𝑥 (𝑘 + 1) = 𝐴 (𝑘) 𝑥 (𝑘) + 𝐴𝑑(𝑘) 𝑥 (𝑘 − 𝑑) + 𝐵𝑢 (𝑘)

+

2

∑

𝑗=1

𝑀𝑗(𝑥1(𝑘))𝑁

𝑗𝑦𝑗(𝑘 + 1) ,

𝑦 (𝑘) = 𝐶 (𝑘) 𝑥 (𝑘) ,

𝑥 (𝑘) = 𝜙 (𝑘) , 𝑘 = −𝑑, −𝑑 + 1, . . . , 0,

(47)

where

[𝐴 (𝑘) 𝐴𝑑(𝑘) 𝐵 (𝑘) 𝐶 (𝑘)]

=

2

∑

𝑗=1

𝑀𝑗(𝑥1(𝑘)) [𝑇𝑗𝐴𝑗 𝑇

𝑗𝐴𝑑𝑗

𝑇𝑗𝐵𝑗

𝐶𝑗] .

(48)

By using MATLAB LMI Toolbox and solving LMIs inTheorem 4, we obtain

𝑃 = [227.2273 −25.4600

−25.4600 61.9384] , 𝑊

1= [

−199.6323

57.1439]

𝑇

,

𝑄 = [39.4769 3.5449

3.5449 27.3497] , 𝑊

2= [

181.1576

−32.4363]

𝑇

,

(49)

so

𝐺1= [

−0.8126

0.5886] , 𝐺

2= [

0.7742

−0.2054] . (50)

The state trajectories of discrete-time-delay fuzzy descriptorsystem are shown in Figure 2 and their estimations aredepicted by Figure 3. On the other hand, from Figure 4 weknow that the error system is asymptotically stable.

Example 11. Consider the fuzzy system in Example 10.Assume that time-delay 𝑑 = 2, and take the same solution to


0 5 10 15 20

00.5

11.5

k

x

−1.5

−1

−0.5

x1

x2

Figure 2: State of the fuzzy descriptor system.

0 5 10 15 20k

00.5

11.5

x

−1.5

−1

−0.5

Estimate of x1Estimate of x2

Figure 3: State of the fuzzy observer.

0 5 10 15 20k

e 00.5

11.5

−1.5

−1

−0.5

e1

e2

Figure 4: Error system.

0 5 10 15 20k

x 00.5

11.5

−1.5

−1

−0.5

Estimate of x1Estimate of x2

Figure 5: State of the fuzzy observer.

0 5 10 15 20k

e 00.5

11.5

−1.5

−1

−0.5

e1

e2

Figure 6: Error system.

(3) as that in Example 10. According to Theorem 5 and usingMATLAB LMI Toolbox, we obtain

𝑃 = [0.8184 −0.0655

−0.0655 0.1411] , 𝑊

1= [

−0.7240

0.1516]

𝑇

,

𝑈 = [0.0658 0.0075

0.0075 0.0657] , 𝑊

2= [

0.7072

−0.0666]

𝑇

,

𝑅 = [14.4443 −2.7589

−2.7589 5.1602] .

(51)

Based on matrices 𝑃 and 𝑊𝑗(𝑗 = 1, 2), we can derive

𝐺1= [

−0.8295

0.6896] , 𝐺

2= [

0.8582

−0.0735] . (52)

State trajectories of fuzzy system are the same as thosein Figure 2, and Figure 5 is state response of fuzzy observer.Furthermore, from the error system depicted in Figure 6, weknow that error converges to zero asymptotically.

Remark 12. From these two examples above, it is obvious thaterror in Figure 6 converges faster than that in Figure 4. Insummary, fuzzy observer designed according to Theorem 5is better; that is, the convergence time of error is shorter.However, if there is no appropriate observer for some discretefuzzy descriptor system based on Theorem 5, we may designits observer depending onTheorem 4.

5. Conclusion

This paper has discussed how to design fuzzy observers fordiscrete descriptor systems with time-delays. According toa simple transformation, the singularity of considered fuzzysystems has been eliminated. Then two sufficient conditionsfor the existence of fuzzy observer have been derived.Notably, one condition depends on time-delays and theother does not. Finally, different fuzzy observers have beendesigned for the same discrete-time-delayed descriptor sys-tem. By comparison, the fuzzy observer depending on delaycondition is better. Additionally, our future work will dealwith fuzzy time-delayed descriptor systems by using delay-partitioning method.


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

This work was partially supported by NNSF of China(61174141, 61374025), Research Awards Young and Middle-Aged Scientists of Shandong Province (BS2011SF009,BS2011DX019), and Excellent Youth Foundation of ShandongProvince (JQ201219).

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