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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 systemâsperformance. 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 affectsystemâs performance. Therefore, many researchers haveinvestigated time-delayed systems [6â8]. 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 đ
2â đâ
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 đ·-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 đ»
âfilters 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 đ: IF đ
1(đ) is đ
đ1, đ2(đ) is đ
đ2,. . ., and đ
đ(đ) is đ
đđ,
THEN
đžđ„ (đ + 1) = đŽđđ„ (đ) + đŽ
đđđ„ (đ â đ) + đ”
đđą (đ)
đŠđ(đ) = đ¶
đđ„ (đ)
đ„ (đ) = đ (đ) , đ = âđ, âđ + 1, . . . , 0,
(1)
where đđđ , đđ (đ) (1 †đ †đ) are fuzzy sets and premise
variables, respectively. đ„(đ) â Rđ is the system state, themeasurable output is đŠ(đ) â Rđ, and đą(đ) â Rđ representsthe control input.The initial condition is denoted byđ(đ). It isassumed thatđž â RđĂđ is singular; that is, rankđž = đ < đ.Theremaining matrices, đŽ
đ, đŽđđ, đ”đ, and đ¶
đ, are known. Besides,
1 †đ †đż, where đż is the number of IF-THEN rules. đ, đż, đare positive integer numbers.
We make the following assumption.
Assumption 1. Consider
rank [đž
đ¶đ
] = đ, 1 †đ †đż, (2)
so there exists a matrix pair [đđđđ] for each đ â {1, 2, . . . , đż}
such that
đđđž + đ
đđ¶đ= đŒđ, 1 †đ †đż. (3)
Remark 2. Because of the singularity of matrix đž 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 byđ
+ the pseudoinverse of amatrixđ. Then the general solution to (3), with condition (2), is
[đđ đđ] = [
đž
đ¶đ
]
+
+ đ(đŒđ+đ
â [đž
đ¶đ
] [đž
đ¶đ
]
+
) , (4)
where matrix đđ(1 †đ †đż) is nonsingular, which
can be achieved via designing the arbitrary matrix đ withappropriate dimension.
Abstract and Applied Analysis 3
According to equality (3), the first equation of fuzzysystem (1) is changed into
đ„ (đ + 1) = đđđŽđđ„ (đ) + đ
đđŽđđđ„ (đ â đ) + đ
đđ”đđą (đ)
+ đđđŠđ(đ + 1) ,
(5)
where 1 †đ †đż.Let
âđ(đ (đ)) =
âđ
đ =1đđđ
(đđ(đ))
âđż
đ=1âđ
đ =1đđđ
(đđ(đ))
, 1 †đ †đż, (6)
with đ(đ) = [đ1(đ)đ2(đ) â â â đ
đ(đ)]. đ
đđ (đđ(đ)) means the
grade of membership of đđ(đ) in đ
đđ . Obviously, 0 â€
đđđ (đđ(đ)) †1. Therefore, â
đ(đ(đ)) â„ 0 (1 †đ †đż) and
âđż
đ=1âđ(đ(đ)) = 1 for all đ. For fuzzy system (1), the final state
and output are given as follows:
đ„ (đ + 1) =
đż
â
đ=1
âđ(đ (đ)) [đ
đđŽđđ„ (đ) + đ
đđŽđđđ„ (đ â đ)
+đđđ”đđą (đ) + đ
đđŠđ(đ + 1)] ,
đŠ (đ) =
đż
â
đ=1
âđ(đ (đ)) đ¶
đđ„ (đ) ,
đ„ (đ) = đ (đ) , đ = âđ, âđ + 1, . . . , 0.
(7)
Define
[đŽ (đ) đŽđ(đ) đ” (đ) đ¶ (đ)]
â
đż
â
đ=1
âđ(đ (đ)) [đŽđ đŽ
đđđ”đ
đ¶đ] ,
(8)
where [đŽđ
đŽđđ
đ”đ] = đđ[đŽđ đŽ
đđđ”đ].Then system (7) can
be written as
đ„ (đ + 1) = đŽ (đ) đ„ (đ) + đŽđ(đ) đ„ (đ â đ) + đ” (đ) đą (đ)
+
đż
â
đ=1
âđ(đ (đ))đ
đđŠđ(đ + 1) ,
đŠ (đ) = đ¶ (đ) đ„ (đ) ,
đ„ (đ) = đ (đ) , đ = âđ, âđ + 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 đ: IF đ1(đ) isđ
đ1, đ2(đ) isđ
đ2,. . ., and đ
đ(đ)
is đđđ, THEN
đ„ (đ + 1) = đđđŽđđ„ (đ) + đ
đđŽđđđ„ (đ â đ) + đ
đđ”đđą (đ)
+ đșđ(đŠ (đ) â đŠ (đ)) + đ
đđŠđ(đ + 1) ,
đŠđ(đ) = đ¶
đđ„ (đ) ,
(10)
where 1 †đ †đż, đ„(đ) â Rđ is the estimate of state đ„(đ), andmatrices đș
đ(đ = 1, 2, . . . , đż) are observation error matrices.
Denote by đŠ(đ) and đŠ(đ) the final output of fuzzy system andfuzzy observer, respectively. With the same weight â
đ(đ(đ))
and notations in system (7) the final estimated state andoutput of fuzzy observer (10) can be described as
đ„ (đ + 1) = đŽ (đ) đ„ (đ) + đŽđ(đ) đ„ (đ â đ) + đ” (đ) đą (đ)
+ đș (đ) (đŠ (đ) â đŠ (đ)) + đŠ (đ + 1) ,
đŠ (đ) = đ¶ (đ) đ„ (đ) ,
(11)
where đŠ(đ) = âđż
đ=1âđ(đ(đ))đ
đđŠđ(đ) and đș(đ) â
âđż
đ=1âđ(đ(đ))đș
đ.
Thus the problem of this paper focuses on finding matri-ces đșđ(đ = 1, 2, . . . , đż) 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
[đ đ
đđ
đ ] > 0, (12)
where đ = đđ, đ = đ
đ are equivalent to
đ > 0, đ â đđ â1
đđ
> 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 đ > 0 and đ > 0 and matricesđđ(đ = 1, 2, . . . , đż) such that
Îđđ< 0, đ = 1, 2, . . . , đż,
Îđđ+ Îđđ
< 0, 1 †đ < đ †đż,
(14)
where
Îđđ
= [
[
đ â đ 0 đŽđ
đđ + đ¶
đ
đđđ
â âđ đŽđ
đđđ
â â âđ
]
]
, (15)
4 Abstract and Applied Analysis
and â denotes matrix entries implied by the symmetry of amatrix through this paper. Furthermore, the observer gainmatrices can be obtained as
đșđ= đâ1
đđ
đ, 1 †đ †đż. (16)
Proof. Define
đ (đ) = đ„ (đ) â đ„ (đ) . (17)
From system (9) and observer (11), the error dynamic equa-tion of estimation error đ(đ) can be derived as
đ (đ + 1) = [đŽ (đ) + đș (đ) đ¶ (đ)] đ (đ) + đŽđ(đ) đ (đ â đ) .
(18)
Take a Lyapunov function as
đ (đ) = đđ(đ) đđ (đ) +
đ
â
đ=1
đđ(đ â đ) đđ (đ â đ) , (19)
where đ,đ > 0; then
Îđ (đ) = đ (đ + 1) â đ (đ)
= đđ(đ + 1) đđ (đ + 1) â đ
đ(đ) đđ (đ)
+
đ
â
đ=1
đđ(đ + 1 â đ) đđ (đ + 1 â đ)
â
đ
â
đ=1
đđ(đ â đ) đđ (đ â đ)
= đđ(đ + 1) đđ (đ + 1) â đ
đ(đ) đđ (đ)
+ đđ(đ) đđ (đ) â đ
đ(đ â đ)đđ (đ â đ)
= {[đŽ (đ) + đș (đ) đ¶ (đ)] đ (đ) + đŽđ(đ) đ (đ â đ)}
đ
Ă đ {[đŽ (đ) + đș (đ) đ¶ (đ)] đ (đ) + đŽđ(đ) đ (đ â đ)}
â đđ(đ) đđ (đ) + đ
đ(đ) đđ (đ)
â đđ(đ â đ)đđ (đ â đ) .
(20)
Let đ(đ) = [đđ(đ) đđ(đ â đ)]
đ; thus
Îđ (đ) = đđ(đ) Îđ (đ) , (21)
where
Î = [
[
đ1(đ) đ
2(đ)
đđ
2(đ) đ
3(đ)
]
]
, (22)
while đ1(đ) = [đŽ(đ) + đș(đ)đ¶(đ)]
đđ[đŽ(đ) + đș(đ)đ¶(đ)] +
đ â đ, đ2(đ) = [đŽ(đ) + đș(đ)đ¶(đ)]
đđđŽđ(đ), and đ
3(đ) =
đŽđ
đ(đ)đđŽ
đ(đ) â đ.
The inequalityÎđ(đ) < 0 holds for all đ(đ) Ìž= 0 if and onlyif
Î < 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
[[
[
đ â đ 0 [đŽ (đ) + đș (đ) đ¶ (đ)]đ
đ
â âđ đŽđ
đ(đ) đ
â â âđ
]]
]
< 0. (24)
With the definition of coefficientmatrices in system (9), it canbe obtained from inequality (24) that
đż
â
đ=1
đż
â
đ=1
âđ(đ (đ)) â
đ(đ (đ))
[[
[
đ â đ 0 (đŽđ+ đșđđ¶đ)đ
đ
â âđ đŽđ
đđđ
â â âđ
]]
]
< 0,
(25)
which is equivalent to
đż
â
đ=1
đż
â
đ=1
âđ(đ (đ)) â
đ(đ (đ)) Î
đđ< 0. (26)
And inequality (26) can be rewritten as
đż
â
đ=1
âđ(đ (đ)) Î
đđ+
đżâ1
â
đ=1
đż
â
đ=đ+1
âđ(đ (đ)) â
đ(đ (đ)) (Î
đđ+ Îđđ) < 0.
(27)
Takingđđ= đșđ
đđ, 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 đ. 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 đ > 0, đ > 0,and đ > 0 and matrices đ
đ(đ = 1, 2, . . . , đż) such that
Κđđ< 0, đ = 1, 2, . . . , đż,
Κđđ+ Κđđ
< 0, 1 †đ < đ †đż,(28)
where
Κđđ
=
[[[[[
[
đ â đ 0 đŽđ
đđ + đ¶
đ
đđđ
đŽđ
đđ + đ¶
đ
đđđâ đ
â âđ đŽđ
đđđ đŽ
đ
đđđ
â â âđ 0
â â â â1
đđ
]]]]]
]
.
(29)
Abstract and Applied Analysis 5
Moreover, the observer gain matrices can be calculated as
đșđ= đâ1
đđ
đ, 1 †đ †đż. (30)
Proof. According to proof of Theorem 4, the dynamic equa-tion of error system is (18). Construct a Lyapunov functionas
đ (đ)
= đđ(đ) đđ (đ) +
đâ1
â
đ=đâđ
đđ(đ) đđ (đ)
+
0
â
đ =âđ+1
đâ1
â
đ=đâ1+đ
[đ (đ + 1) â đ (đ)]đđ [đ (đ + 1) â đ (đ)] ,
(31)
where đ,đ, đ > 0. Then
Îđ (đ)
= đ (đ + 1) â đ (đ)
= đđ(đ + 1) đđ (đ + 1) â đ
đ(đ) đđ (đ)
+
đ
â
đ=đâđ+1
đđ(đ) đđ (đ) â
đâ1
â
đ=đâđ
đđ(đ) đđ (đ)
+
0
â
đ =âđ+1
{
đ
â
đ=đ+đ
[đ (đ + 1) â đ (đ)]đđ [đ (đ + 1) â đ (đ)]
â
đâ1
â
đ=đâ1+đ
[đ (đ + 1) â đ (đ)]đ
Ăđ [đ (đ + 1) â đ (đ)] }
= đđ(đ + 1) đđ (đ + 1) â đ
đ(đ) đđ (đ) + đ
đ(đ) đđ (đ)
â đđ(đ â đ)đđ (đ â đ)
â
0
â
đ =âđ+1
{[đ (đ + đ ) â đ (đ â 1 + đ )]đ
Ăđ [đ (đ + đ ) â đ (đ â 1 + đ )]}
+ đ[đ (đ + 1) â đ (đ)]đđ [đ (đ + 1) â đ (đ)]
(18)
= đđ(đ) đ1(đ) đ (đ) + đ
đ(đ â đ) đ
2(đ) đ (đ â đ)
+ đđ(đ) đ3(đ) đ (đ â đ) + đ
đ(đ â đ) đ
đ
3(đ) đ (đ)
â
0
â
đ =âđ+1
[đ (đ + đ ) â đ (đ â 1 + đ )]đ
Ă đ [đ (đ + đ ) â đ (đ â 1 + đ )] ,
(32)
where
đ1(đ) = [đŽ (đ) + đș (đ) đ¶ (đ)]
đ
đ [đŽ (đ) + đș (đ) đ¶ (đ)]
â đ + đ + đ4(đ) ,
đ2(đ) = đŽ
đ
đ(đ) đđŽ
đ(đ) â đ + đđŽ
đ
đ(đ) đđŽ
đ(đ) ,
đ3(đ) = [đŽ (đ) + đș (đ) đ¶ (đ)]
đ
đđŽđ(đ)
+ đ[đŽ (đ) + đș (đ) đ¶ (đ) â đŒ]đ
đđŽđ(đ) ,
đ4(đ) = đ[đŽ (đ) + đș (đ) đ¶ (đ) â đŒ]
đ
Ă đ [đŽ (đ) + đș (đ) đ¶ (đ) â đŒ] .
(33)
Define đ(đ) = [đđ(đ) đđ(đ â đ)]
đ; then
Îđ (đ) = đđ(đ) Κđ (đ)
â
0
â
đ =âđ+1
[đ (đ + đ ) â đ (đ â 1 + đ )]đ
Ă đ [đ (đ + đ ) â đ (đ â 1 + đ )]
†đđ(đ) Κđ (đ) ,
(34)
where
Κ = [
[
đ1(đ) đ
3(đ)
đđ
3(đ) đ
2(đ)
]
]
. (35)
If
Κ < 0, (36)
then Îđ(đ) < 0 and the error system (18) is asymptoticallystable.
By Lemma 3, inequality (36) is equivalent to
6 Abstract and Applied Analysis
[[[[[
[
đ â đ 0 [đŽ (đ) + đș (đ) đ¶ (đ)]đ
đ [đŽ (đ) + đș (đ) đ¶ (đ) â đŒ]đ
â âđ đŽđ
đ(đ) đ đŽ
đ
đ(đ)
â â âđ 0
â â â â1
đđâ1
]]]]]
]
< 0. (37)
Define đœ â diag[đŒ, đŒ, đŒ, đ]. Pre- and postmultiplying inequal-ity (37) by matrix đœ, we can obtain
[[[[[
[
đ â đ 0 [đŽ (đ) + đș (đ) đ¶ (đ)]đ
đ [đŽ (đ) + đș (đ) đ¶ (đ) â đŒ]đ
đ
â âđ đŽđ
đ(đ) đ đŽ
đ
đ(đ) đ
â â âđ 0
â â â â1
đđđâ1
đ
]]]]]
]
< 0. (38)
Taking đ = đđâ1
đ and đđ
= đșđ
đđ, it is derived that
inequality (38) is equivalent to
đż
â
đ=1
đż
â
đ=1
âđ(đ (đ)) â
đ(đ (đ)) Κ
đđ< 0. (39)
That isđż
â
đ=1
âđ(đ (đ)) Κ
đđ+
đżâ1
â
đ=1
đż
â
đ=đ+1
âđ(đ (đ)) â
đ(đ (đ)) (Κ
đđ+ Κđđ) < 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 đ
đđâ [đđŽ
đ+ đđ
đđ¶đ
â đ đđŽđđ
0]đ
(đ, đ =
1, 2, . . . , đż), we have
Κđđ
= [
[
Îđđ
đđđ
â â1
đđ ]
]
, (41)
where matrices Κđđand Î
đđare defined in Theorems 5 and
4, respectively. So, according to Lemma 3, inequalities (28)are equivalent to Î
đđ+ đđđđđ â1
đđ
đđ< 0 (đ = 1, 2, . . . , đż) and
Îđđ+Îđđ+ (đ/2)(đ
đđ+đđđ)đ â1
(đđđ+đđđ)đ
< 0 (1 †đ < đ †đż),respectively. Sinceđ > 0, from inequalities (28) we can obtainthat Î
đđ< 0 (đ = 1, 2, . . . , đż) and Î
đđ+ Îđđ
< 0 (1 †đ <
đ †đż). 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 đž = đŒ
đ, a discrete fuzzy
state-space system can be derived from fuzzy system (1) with-out time-delays. Let đ
đ= đŒđ, đđ= 0, and đŽ
đđ= 0 (1 †đ â€
đż). 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 đ
đ(1 †đ †đż) 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 đșđ= đâ1
đđ
đ(1 †đ â€
đż).
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.
Abstract and Applied Analysis 7
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 đ„1(đ) is đ
1(đ„1(đ)), THEN
đžđ„ (đ + 1) = đŽ1đ„ (đ) + đŽ
đ1đ„ (đ â đ) + đ”
1đą (đ) ,
đŠ1(đ) = đ¶
1đ„ (đ) ,
đ„ (đ) = đ (đ) , đ = âđ, âđ + 1, . . . , 0.
(42)
Plant Rule 2: IF đ„1(đ) is đ
2(đ„1(đ)), THEN
đžđ„ (đ + 1) = đŽ2đ„ (đ) + đŽ
đ2đ„ (đ â đ) + đ”
2đą (đ) ,
đŠ2(đ) = đ¶
2đ„ (đ) ,
đ„ (đ) = đ (đ) , đ = âđ, âđ + 1, . . . , 0,
(43)
where
đž = [1 0
0 0] , đŽ
1= [
0 1
â1 1] , đŽ
2= [
0 â1
â1 1] ,
đŽđ1
= [â0.1 0
0.2 0.1] , đŽ
đ2= [
0 0.1
â0.2 0] ,
đ”1= [
0
1] , đ”
2= [
0
â1] , đ¶
1= [
0
1]
đ
,
đ¶2= [
1
1]
đ
,
(44)
and membership functions are
đ1(đ„1(đ)) â 1 â
đ„2
1(đ)
2.25,
đ2(đ„1(đ)) â
đ„2
1(đ)
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 đž and đ¶
đ(đ = 1, 2)
satisfy condition (2). Based on formulation (4), a solution to(3) is shown as
đ1= [
1 0
0 â1] , đ
1= [
0
1] ,
đ2= [
1 0
â1 â1] , đ
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
â1.5
x1(k)
â1 â0.5
Figure 1: Membership functions.
Then fuzzy descriptor system is converted into the follow-ing ordinary system:
đ„ (đ + 1) = đŽ (đ) đ„ (đ) + đŽđ(đ) đ„ (đ â đ) + đ”đą (đ)
+
2
â
đ=1
đđ(đ„1(đ))đ
đđŠđ(đ + 1) ,
đŠ (đ) = đ¶ (đ) đ„ (đ) ,
đ„ (đ) = đ (đ) , đ = âđ, âđ + 1, . . . , 0,
(47)
where
[đŽ (đ) đŽđ(đ) đ” (đ) đ¶ (đ)]
=
2
â
đ=1
đđ(đ„1(đ)) [đđđŽđ đ
đđŽđđ
đđđ”đ
đ¶đ] .
(48)
By using MATLAB LMI Toolbox and solving LMIs inTheorem 4, we obtain
đ = [227.2273 â25.4600
â25.4600 61.9384] , đ
1= [
â199.6323
57.1439]
đ
,
đ = [39.4769 3.5449
3.5449 27.3497] , đ
2= [
181.1576
â32.4363]
đ
,
(49)
so
đș1= [
â0.8126
0.5886] , đș
2= [
0.7742
â0.2054] . (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 đ = 2, and take the same solution to
8 Abstract and Applied Analysis
0 5 10 15 20
00.5
11.5
k
x
â1.5
â1
â0.5
x1
x2
Figure 2: State of the fuzzy descriptor system.
0 5 10 15 20k
00.5
11.5
x
â1.5
â1
â0.5
Estimate of x1Estimate of x2
Figure 3: State of the fuzzy observer.
0 5 10 15 20k
e 00.5
11.5
â1.5
â1
â0.5
e1
e2
Figure 4: Error system.
0 5 10 15 20k
x 00.5
11.5
â1.5
â1
â0.5
Estimate of x1Estimate of x2
Figure 5: State of the fuzzy observer.
0 5 10 15 20k
e 00.5
11.5
â1.5
â1
â0.5
e1
e2
Figure 6: Error system.
(3) as that in Example 10. According to Theorem 5 and usingMATLAB LMI Toolbox, we obtain
đ = [0.8184 â0.0655
â0.0655 0.1411] , đ
1= [
â0.7240
0.1516]
đ
,
đ = [0.0658 0.0075
0.0075 0.0657] , đ
2= [
0.7072
â0.0666]
đ
,
đ = [14.4443 â2.7589
â2.7589 5.1602] .
(51)
Based on matrices đ and đđ(đ = 1, 2), we can derive
đș1= [
â0.8295
0.6896] , đș
2= [
0.8582
â0.0735] . (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.
Abstract and Applied Analysis 9
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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