Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 324738 13 pageshttpdxdoiorg1011552013324738

Research ArticleDelay-Dependent Stability Analysis of Uncertain Fuzzy Systemswith State and Input Delays under Imperfect Premise Matching

Zejian Zhang12 Xiao-Zhi Gao2 Kai Zenger2 and Xianlin Huang1

1 Center for Control Theory and Guidance Technology Harbin Institute of Technology Harbin 150001 China2Department of Automation and Systems Technology Aalto University School of Electrical Engineering 00076 Aalto Finland

Correspondence should be addressed to Zejian Zhang zzhanghitgmailcom

Received 1 December 2012 Accepted 4 February 2013

Academic Editor Peng Shi

Copyright copy 2013 Zejian Zhang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper discusses the stability and stabilization problem for uncertain T-S fuzzy systemswith time-varying state and input delaysA new augmented Lyapunov function with an additional triple-integral term and different membership functions of the fuzzymodels and fuzzy controllers are introduced to derive the stability criterion which is less conservative than the existing resultsMoreover a new flexibility design method is also provided Some numerical examples are given to demonstrate the effectivenessand less conservativeness of the proposed method

1 Introduction

Since the Takagi-Sugeno (T-S) [1] fuzzy systems were firstlyproposed in 1985 their stability analysis has received con-siderable research attention [2 3] However most of theexisting results are only for the T-S fuzzy systems free oftime delay Actually time delay often occurs inmany practicalsystems such as [4 5] It has been shown that the existenceof time delay usually becomes the source of instability anddeteriorated performance of systems Therefore study of thetime delay is important in both theory and practice [6] Thefirst stability analysis work on the T-S fuzzy systemswith timedelay is done in [7 8] by using the Lyapunov-Razumikhinfunctional approach If some uncertainties exist in a T-Sfuzzy time delay system they may also significantly affectthe system performances and even cause unstable systemoutputs Therefore the issue of the stability for the uncertainT-S fuzzy time delay systems has been widely explored [9ndash11]In the literature two basic approaches have been utilized thatis delay-independent approach [12] and delay-dependentapproach The latter makes use of the information on thelength of the delays and it is less conservative than the formeroneA lot of stability analysis results have been reported basedon the delay-dependent approach [13 14]

During the recent years some research work for differenttypes of the delays of the T-S fuzzy systems has beenpublished such as constant delay [15 16] bounded timedelay [17] time varying delay [18] and interval time varyingdelay [19 20] However all these results are only for the T-S fuzzy systems with state delays Thus they may be invalidwhen applied to the systems with input delays As we knowinput delays extensively exist in industrial processes andcan cause instability or serious performance deteriorationIn fact in modern industrial systems sensors controllersand plants are often connected together and the sampleddata and controller signals are transmitted through networksIn view of this the input delays should be taken intoconsideration for robust controller design Intensive resultson the stabilization for the T-S fuzzy systems with state andinput delays are reported in [21ndash28] For example the workin [21] is based on the linear systems with input delays In[22 24] the authors only consider the fuzzy systems withconstant input delays and their results are conservative In[23] the authors study the fuzzy systems with both the stateand input delays Unfortunately the results are obtainedwithout any uncertainty and the state delay is assumed tobe equal to the input delay In [26] the uncertainty has beenconsidered in the analysis but the state delay is also assumed

2 Mathematical Problems in Engineering

to be the same as the input delay Some interesting results forthe uncertain fuzzy systems with state and input delays havebeen obtained in [25 27 28] most of which introduce someLyapunov-Krasovskii functions containing integral termsfor example int119905

119905minus120591119909119879(119904)119876119909(119904)119889119904 and double-integral terms

int0

minus120591int119905

119905+120579119879(119904)119876(119904)119889119904 119889120579 Several triple-integral terms are

used in the Lyapunov function [29] to yield less conservativeresults for the fuzzy systems with state delay A large portionof robust controller design topics have been investigated onthe basis of the Parallel Distribution Compensation (PDC)design technique where the fuzzy controller shares the samepremise membership functions as those of the T-S fuzzy timedelay model [7ndash11 15ndash20 22ndash28] As a matter of fact if themembership functions in the premise of the fuzzy rules ofthe fuzzy controllers are allowed to be designed arbitrarily wecan even achieve better design flexibility For instance a fuzzycontroller not sharing the same premise rules as those of theT-S fuzzy model referred to as imperfect premise matchingis employed to control the nonlinear plants [30 31] and[12] extends the available results to the T-S fuzzy time-delaysystems with only the state delay

In this paper an augmented Lyapunov-Krasovskii func-tion that contains a triple-integral term is introduced to inves-tigate the stability and stabilization problem for uncertain T-S fuzzy systems with the state and input delays under theimperfect premise matching in which the fuzzy time delaymodel and fuzzy controller are with different premise Someless conservative delay-dependent stability and robust stabil-ity conditions are obtained by two integral inequalities anda parameterized model transformation method Moreoverdifferent from the general PDC design technique a newdesign approach of robust stable controllers is proposed Twosimulation examples are further given to illustrate that theproposed design methods are less conservative and moreflexible

2 System Description and Preliminaries

Let 119903 be the number of the fuzzy rules describing the timedelay nonlinear plant The 119894th rule can be represented asfollows

If 1198911(119909(119905)) is119872119894

1and and 119891

119901(119909(119905)) is119872119894

119901

(119905) = (1198601119894+ Δ1198601119894(119905)) 119909 (119905) + (119860

2119894+ Δ1198602119894(119905))

times 119909 (119905 minus 1198891(119905)) + (119861

119894+ Δ119861119894(119905)) 119906 (119905 minus 119889

2(119905))

119909 (119905) = 120593 (119905) 119905 isin [minusmax (ℎ1 ℎ2) 0]

(1)

where 119872119894120572is a fuzzy term of rule 119894 corresponding to the

function119891120572(119909(119905)) 120572 = 1 2 119901 119894 = 1 2 119903 119909(119905) isin 119877119899 is

the system state vector and 119906(119905) isin 119877119898 is the input vectorThematrices 119860

1119894 1198602119894 and 119861

119894 119894 = 1 2 119903 are of appropriate

dimensionsThe initial condition 120593(119905) is a continuous vector-valued function The delays 119889

1(119905) and 119889

2(119905) are time varying

and satisfy

0 le 1198891(119905) le ℎ

1 119889

1(119905) le 120583

1

0 le 1198892(119905) le ℎ

2 119889

2(119905) le 120583

2

(2)

where ℎ119894are constants representing the upper bound of the

delay 120583 is a positive constant Δ1198601119894 Δ1198602119894 and Δ119861

119894denote

the uncertainties in the system and they are the form of

[Δ1198601119894

Δ1198602119894

Δ119861119894] = 119863

119894119870119894(119905) [1198641119894

1198642119894

119864119887119894]

119894 = 1 2 119903(3)

where 119863119894 1198641119894 1198642119894 and 119864

119887119894are known constant matrices

of appropriate dimensions and 119870119894(119905) is unknown matrix

function satisfying 119870119894

119879(119905)119870119894(119905) le 119868 119868 is an appropriately

dimensioned identity matrix Hence the overall fuzzy modelcan be formulized as follows

(119905) =119903

sum119894=1

119908119894(119909 (119905)) [(119860

1119894+ Δ1198601119894(119905)) 119909 (119905) + (119860

2119894+ Δ1198602119894(119905))

times 119909 (119905 minus 1198891(119905)) + (119861

119894+ Δ119861119894(119905))

times119906 (119905 minus 1198892(119905))]

(4)

where119903

sum119894=1

119908119894(119909 (119905)) = 1 119908

119894(119909 (119905)) ge 0

119908119894(119909 (119905)) =

120583119894(119909 (119905))

sum119903119894=1120583119894(119909 (119905))

120583119894(119909 (119905)) =

119901

prod120572=1

120583119872119894

120572

(119891120572(119909 (119905)))

(5)

119908119894(119909(119905)) is the normalized grade ofmembership function that

is a nonlinear function of 119909(119905) 120583119872119894

120572

(119891120572(119909(119905))) is the grade of

the membership corresponding to the fuzzy term of119872119894120572

Different from the popular PDC design technique thefollowing fuzzy control law under imperfect premise match-ing is employed to deal with the problem of stabilization viastate feedback Under the imperfect premise matching the119895th rule of the fuzzy controller is defined as follows

If 1198921(119909(119905)) is119873119894

1and and 119892

119902(119909(119905)) is119873119894

119902

119906 (119905) = 119865119895119909 (119905) 119895 = 1 2 119903 (6)

where 119873119895120573denotes the fuzzy set 119902 is a positive integer and

119865119895isin 119877119898times119899 is the feedback gain of rule 119895 The state feedback

fuzzy control law is represented by

119906 (119905) =119903

sum119895=1

119898119895(119909 (119905)) 119865

119895119909 (119905) (7)

Mathematical Problems in Engineering 3

where119903

sum119895=1

119898119895(119909 (119905)) = 1 119898

119895(119909 (119905)) ge 0

119898119895(119909 (119905)) =

V119895(119909 (119905))

sum119903119895=1

V119895(119909 (119905))

V119895(119909 (119905)) =

119902

prod120573=1

V119873119895

120573

(119892120573(119909 (119905)))

(8)

119898119895(119909(119905)) is the normalized grade of membership function

that is a nonlinear function of 119909(119905) V119873119895

120573

(119892120573(119909(119905))) is the grade

of the membership corresponding to the fuzzy term of119873119895120573

As the time varying delay is included in the control inputwe have

119906 (119905 minus 1198892(119905)) =

119903

sum119895=1

119898119895(119909 (119905 minus 119889

2(119905))) 119865

119895119909 (119905 minus 119889

2(119905)) (9)

where119903

sum119895=1

119898119895(119909 (119905 minus 119889

2(119905))) = 1 119898

119895(119909 (119905 minus 119889

2(119905))) ge 0 (10)

Lemma 1 (see [32] (Schur complement)) Given constantmatrices Ω

1 Ω2 and Ω

3 where Ω

1= Ω1198791and Ω

2= Ω1198792 there

is Ω1+ Ω1198793Ωminus12Ω3lt 0 if and only if

[Ω1 Ω1198793

lowast minusΩ2

] lt 0 or [minusΩ2 Ω3

lowast minusΩ1

] lt 0 (11)

Lemma 2 (see [33]) Let Q = Q119879 D E and K(119905) satisfiesK119879(119905)K(119905) le I the following inequality holds

119876 + 119863119870 (119905) 119864 + 119864119879119870119879 (119905) 119863119879 lt 0 (12)

if and only if the following inequality holds for any smaller 120576 gt0

119876 + 120576minus1119863119863119879 + 120576119864119879119864 lt 0 (13)

Lemma 3 (see [34]) For any constant matrix Σ = Σ119879 gt 0and a scalar 120591 gt 0 such that the following integrations are welldefined we have

minus int119905

119905minus120591

119909119879 (119904) Σ119909 (119904) 119889119904

le minus1120591(int119905

119905minus120591

119909 (119904) 119889119904)119879

Σ(int119905

119905minus120591

119909 (119904) 119889119904)

minus int0

minus120591

int119905

119905+120579

119909119879 (119904) Σ119909 (119904) 119889119904 119889120579

le minus11205912(int0

minus120591

int119905

119905+120579

119909 (119904) 119889119904 119889120579)119879

Σ(int0

minus120591

int119905

119905+120579

119909 (119904) 119889119904 119889120579)

(14)

3 Main Results

In this section some new delay-dependent criteria for theT-S fuzzy systems with state and input delays are proposedby introducing a novel Lyapunov function with an additionaltriple-integral term under the imperfect premise matchingMoreover a robust stabilization criterion is also investigated

31 Stability of Nominal Fuzzy Systems Firstly we considerthe control design of a state feedback control law underthe imperfect premise matching that stabilizes the followingnominal fuzzy time varying delay system

(119905) =119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))

times [1198601119894119909 (119905) + 119860

2119894119909 (119905 minus 119889

1(119905)) + 119861

119894119865119895119909

times (119905 minus 1198892(119905))]

119909 (119905) = 120593 (119905) 119905 isin [minusmax (ℎ1 ℎ2) 0]

(15)

Theorem 4 Given scalars ℎ1

ge 0 ℎ2

ge 0 1205831 1205832 and

119905119894(119894 = 2 8) the closed-loop system (15) is asymptotically

stable for any 0 le 119889119894(119905) le ℎ

119894(119894 = 1 2) via the

imperfect premise matching controller design technique if themembership functions of the fuzzy model and fuzzy controllersatisfy 119898

119895(119909(119905 minus 119889

2(119905))) minus 120588

119895119908119895(119909(119905)) ge 0 for all 119895 119909(119905) and

119909(119905 minus 1198892(119905)) where 0 lt 120588

119895lt 1 and there exist matrices

11987511

gt 0 11987522

gt 0 11988211

gt 0 11988222

gt 0 11988511

gt 0 11988522

gt0 11987311gt 0 119873

22gt 0 119876

119894gt 0 119872

119894gt 0 119877

119894gt 0 (119894 = 1 2)

Λ119894= Λ119879119894isin 1198778119899times8119899 gt 0 (i = 1 2 119903) and

119875 = [11987511 11987512lowast 119875

22

] gt 0 119882 = [11988211 11988212lowast 119882

22

] gt 0

119885 = [11988511 11988512lowast 119885

22

] gt 0 119873 = [11987311 11987312lowast 119873

22

] gt 0

(16)

and real matrices 119884119895(119895 = 1 2 119903) 119875

12119882121198851211987312 and

119883 such that (30)ndash(32) hold In addition the stabilizing controllaw is given by

119906 (119905) =119903

sum119895=1

119898119895(119909 (119905 minus 119889

2(119905))) 119884

119895119883minus119879119909 (119905) (17)

Proof Choose a fuzzy weighting-dependent Lyapunov-Krasovskii functional candidate as

119881 (119909 (119905)) = 1198811(119909 (119905)) + 119881

2(119909 (119905)) + 119881

3(119909 (119905)) + 119881

4(119909 (119905))

+ 1198815(119909 (119905))

4 Mathematical Problems in Engineering

1198811(119909 (119905)) = 120585119879

1(119905) 119875120585

1(119905) + 120585119879

2(119905)119882120585

2(119905)

1198812(119909 (119905)) = int

0

minusℎ1

int119905

119905+120579

120585119879 (119904) 119885120585 (119904) 119889 119904119889120579

+ int0

minusℎ2

int119905

119905+120579

120585119879 (119904)119873120585 (119904) 119889119904 119889120579

1198813(119909 (119905)) = int

119905

119905minus1198891(119905)

119909119879 (119904) 1198761119909 (119904) 119889119904

+ int119905

119905minus1198892(119905)

119909119879 (119904) 1198762119909 (119904) 119889119904

1198814(119909 (119905)) = int

119905

119905minusℎ1

119909119879 (119904)1198721119909 (119904) 119889119904 + int

119905

119905minusℎ2

119909119879 (119904)1198722119909 (119904) 119889119904

1198815(119909 (119905)) = int

0

minusℎ1

int0

120579

int119905

119905+120582

119879 (119904) 1198771 (119904) 119889119904 119889120582 119889120579

+ int0

minusℎ2

int0

120579

int119905

119905+120582

119879 (119904) 1198772 (119904) 119889119904 119889120582 119889120579

(18)

where

1205851198791(119905) = [119909119879 (119905) (int

119905

119905minusℎ1

119909 (119904) 119889119904)119879

]

1205851198792(119905) = [119909119879 (119905) (int

119905

119905minusℎ2

119909 (119904) 119889119904)119879

]

120585119879 (119905) = [119909119879 (119905) 119879 (119905)]

119875 = [11987511 11987512lowast 11987522

] gt 0 119882 = [11988211 11988212lowast 11988222

] gt 0

119885 = [11988511 11988512lowast 11988522

] gt 0 119873 = [11987311 11987312lowast 11987322

] gt 0

(19)

and 119876119894 119872119894 and 119877

119894(119894 = 1 2) are the positive-definite

matrices to be determinedThe derivatives of119881(119909(119905)) along the trajectories of system

(15) are as follows

1(119909 (119905)) = 2120585119879

1(119905) 119875 1205851(119905) + 2120585119879

2(119905)119882 120585

2(119905)

= 2 [119909119879 (119905) (int119905

119905minusℎ1

119909 (119904) 119889119904)119879

] [11987511 11987512lowast 11987522

]

times [ (119905)119909 (119905) minus 119909 (119905 minus ℎ

1)]

+ 2 [119909119879 (119905) (int119905

119905minusℎ2

119909 (119904) 119889119904)119879

] [11988211 11988212lowast 11988222

]

times [ (119905)119909 (119905) minus 119909 (119905 minus ℎ

2)]

2(119909 (119905)) = ℎ

1[119909119879 (119905) 119879 (119905)] [11988511 11988512lowast 119885

22

] [119909 (119905) (119905)]

minusint119905

119905minusℎ1

[119909119879 (119904) 119879 (119904)] [11988511 11988512lowast 11988522

] [119909 (119904) (119904)] 119889119904

+ ℎ2[119909119879 (119905) 119879 (119905)] [11987311 11987312lowast 119873

22

] [119909 (119905) (119905)]

minusint119905

119905minusℎ2

[119909119879 (119904) 119879 (119904)] [11987311 11987312lowast 11987322

] [119909 (119904) (119904)] 119889119904

3(119909 (119905)) = 119909119879 (119905) 119876

1119909 (119905) minus (1 minus 119889

1(119905)) 119909119879 (119905 minus 119889

1(119905))

times 1198761119909 (119905 minus 119889

1(119905)) + 119909119879 (119905) 119876

2119909 (119905)

minus (1 minus 1198892(119905)) 119909119879 (119905 minus 119889

2(119905)) 119876

2119909 (119905 minus 119889

2(119905))

le 119909119879 (119905) 1198761119909 (119905) + 119909119879 (119905) 119876

2119909 (119905) minus (1 minus 120583

1)

times 119909119879 (119905 minus 1198891(119905)) 119876

1119909 (119905 minus 119889

1(119905)) minus (1 minus 120583

2)

times 119909119879 (119905 minus 1198892(119905)) 119876

2119909 (119905 minus 119889

2(119905))

4(119909 (119905)) = 119909119879 (119905)119872

1119909 (119905) minus 119909119879 (119905 minus ℎ

1)1198721119909 (119905 minus ℎ

1)

+ 119909119879 (119905)1198722119909 (119905) minus 119909119879 (119905 minus ℎ

2)1198722119909 (119905 minus ℎ

2)

5(119909 (119905)) =

12ℎ21119879 (119905) 119877

1 (119905) minus int

0

minusℎ

int119905

119905+120579

119879 (119904)

times 1198771 (119904) 119889119904 119889120579 +

12ℎ22119879 (119905) 119877

2 (119905)

minus int0

minusℎ

int119905

119905+120579

119879 (119904) 1198772 (119904) 119889119904 119889120579

(20)

From (15) the following equation can be obtained for anymatrices 119879

119894 119894 = 1 2 8

2119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))

times [119909119879 (119905) 1198791+ 119909119879 (119905 minus 119889

1(119905)) 1198792+119909119879 (119905 minus ℎ

1) 1198793

+ (int119905

119905minusℎ1

119909 (119904) 119889119904)119879

1198794+ 119909119879 (119905 minus 119889

2(119905)) 1198795+ 119909119879

times (119905 minus ℎ2) 1198796+ (int119905

119905minusℎ2

119909 (119904) 119889119904)119879

1198797+ 119879 (119905) 119879

8]

times [1198601119894119909 (119905) minus 119860

2119894119909 (119905 minus 119889

1(119905))

minus119861119894119865119895119909 (119905 minus 119889

2(119905)) minus (119905)] = 0

(21)

Mathematical Problems in Engineering 5

By Lemma 3 we have

minus int119905

119905minusℎ1

[119909119905 (119904) 119879 (119904)] [11988511 11988512lowast 11988522

] [119909 (119904) (119904)] 119889119904

le minus1ℎ1

[(int119905

119905minusℎ1

119909 (119904) 119889119904)119879

119909119879 (119905) minus 119909119879 (119905 minus ℎ1)]

times [11988511 11988512lowast 11988522

][

[

int119905

119905minusℎ1

119909 (119904) 119889119904

119909 (119905) minus 119909 (119905 minus ℎ1)]

]

minus int119905

119905minusℎ2

[119909119879 (119904) 119879 (119904)] [11987311 11987312lowast 11987322

] [119909 (119904) (119904)] 119889119904

le minus1ℎ2

[(int119905

119905minusℎ2

119909 (119904) 119889119904)119879

119909119879 (119905) minus 119909119879 (119905 minus ℎ1)]

times [11987311 11987312lowast 11987322

][

[

int119905

119905minusℎ2

119909 (119904) 119889119904

119909 (119905) minus 119909 (119905 minus ℎ2)]

]

minus int0

minusℎ119894

int119905

119905+120579

119879 (119904) 119877119894 (119904) 119889119904 119889120579

le minus2ℎ119894

2(int0

minusℎ119894

int119905

119905+120579

(119904) 119889119904 119889120579)119879

119877119894(int0

minusℎ119894

int119905

119905+120579

(119904) 119889119904 119889120579)

= minus2ℎ119894

2(ℎ119894119909 (119905) minus int

119905

119905minusℎ119894

119909 (119904) 119889119904)119879

times 119877119894(ℎ119894119909 (119905) minus int

119905

119905minusℎ119894

119909 (119904) 119889119904) 119894 = 1 2

(22)

Using the above inequalities andwith the zero quantities (21)we can obtain

(119909119905)

le 2 [119909119879 (119905) (int119905

119905minusℎ1

119909 (119904) 119889119904)119879

] [11987511 11987512lowast 11987522

]

times [ (119905)119909 (119905) minus 119909 (119905 minus ℎ

1)] + ℎ1 [119909

119879 (119905) 119879 (119905)]

times [11988511 11988512lowast 11988522

] [119909 (119905) (119905)] + ℎ2 [119909119879 (119905) 119879 (119905)] [11987311 11987312lowast 119873

22

]

times [119909 (119905) (119905)] minus1ℎ1

[(int119905

119905minusℎ1

119909 (119904) 119889119904)119879

119909119879 (119905) minus 119909119879 (119905 minus ℎ1)]

times [11988511 11988512lowast 11988522

][

[

int119905

119905minusℎ1

119909 (119904) 119889119904

119909 (119905) minus 119909 (119905 minus ℎ1)]

]

minus1ℎ2

[(int119905

119905minusℎ2

119909 (119904) 119889119904)119879

119909119879 (119905) minus 119909119879 (119905 minus ℎ2)]

times [11987311 11987312lowast 11987322

][

[

int119905

119905minusℎ2

119909 (119904) 119889119904

119909 (119905) minus 119909 (119905 minus ℎ2)]

]+ 119909119879 (119905) 119876

1119909 (119905)

+ 119909119879 (119905) 1198762119909 (119905) minus (1 minus 120583

1) 119909119879 (119905 minus 119889

1(119905)) 119876

1119909

times (119905 minus 1198891(119905)) minus (1 minus 120583

2) 119909119879 (119905 minus 119889

2(119905)) 119876

2119909

times (119905 minus 1198892(119905)) + 119909119879 (119905)119872

1119909 (119905) minus 119909119879 (119905 minus ℎ

1)1198721119909

times (119905 minus ℎ1) + 119909119879 (119905)119872

2119909 (119905) minus 119909119879 (119905 minus ℎ

2)1198722119909

times (119905 minus ℎ2) +

12ℎ21119879 (119905) 119877

1 (119905) minus

2ℎ1

2

times(ℎ1119909 (119905)minusint

119905

119905minusℎ1

119909 (119904) 119889119904)119879

1198771(ℎ1119909 (119905)minusint

119905

119905minusℎ1

119909 (119904) 119889119904)

+12ℎ2

2119879 (119905) 1198772 (119905)minus

2ℎ22

(ℎ2119909 (119905)minusint

119905

119905minusℎ2

119909 (119904) 119889119904)119879

times1198772(ℎ2119909 (119905) minus int

119905

119905minusℎ2

119909 (119904) 119889119904)

+ 2 [119909119879 (119905) (int119905

119905minusℎ2

119909 (119904) 119889119904)119879

]

times [11988211 11988212lowast 11988222

] [ (119905)119909 (119905) minus 119909 (119905 minus ℎ

2)]

+ 2119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))

times [119909119879 (119905) 1198791+ 119909119879 (119905 minus 119889

1(119905)) 1198792+ 119909119879 (119905 minus ℎ

1) 1198793

+ (int119905

119905minusℎ1

119909 (119904) 119889119904)119879

1198794+ 119909119879 (119905 minus 119889

2(119905)) 1198795+ 119909119879

times (119905 minus ℎ2) 1198796+(int119905

119905minusℎ2

119909 (119904) 119889119904)119879

1198797+ 119879 (119905) 119879

8]

times [1198601119894119909 (119905) minus 119860

2119894119909 (119905 minus 119889

1(119905))

minus119861119894119865119895119909 (119905 minus 119889

2(119905)) minus (119905)]

= 120577119879 (119905)(119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))Φ

119894119895)120577 (119905)

(23)

6 Mathematical Problems in Engineering

where

120577119879 (119905) = [119909119879 (119905) 119909119879 (119905 minus 1198891(119905)) 119909119879 (119905 minus ℎ

1) (int

119905

119905minusℎ1

119909 (119904) 119889119904)119879

119909119879 (119905 minus 1198892(119905)) 119909119879 (119905 minus ℎ

2) (int

119905

119905minusℎ2

119909 (119904) 119889119904)119879

119879 (119905)] (24)

Φ119894119895=

[[[[[[

[

Φ11119894 Φ12119894 Φ13 Φ14 Φ15119894119895 Φ16 Φ17 Φ18

lowast Φ22119894 119860119879

2119894119879119879

3119860119879

2119894119879119879

4Φ25119894119895 119860

119879

2119894119879119879

6119860119879

2119894119879119879

7Φ28119894

lowast lowast Φ33119894 minus11987522 1198793119861119894119865119895 0 0 minus1198793

lowast lowast lowast Φ44 1198794119861119894119865119895 0 0 Φ48

lowast lowast lowast lowast Φ55119894119895 119865119879

119895119861119879

119894119879119879

6119865119879

119895119861119879

119894119879119879

7Φ58119894

lowast lowast lowast lowast lowast Φ66 Φ67 minus1198796

lowast lowast lowast lowast lowast lowast Φ77 Φ78

lowast lowast lowast lowast lowast lowast lowast Φ88

]]]]]]

]

Φ11119894= 11987512+ 11987511987912+11988212+11988211987912+ ℎ111988511+ ℎ211987311

minus1ℎ1

11988522minus1ℎ2

11987322+ 1198761+ 1198762+1198721+1198722

minus 21198771minus 21198772+ 11987911198601119894+ 11986011987911198941198791198791

Φ12119894= 11987911198602119894+ 11986011987911198941198791198792

Φ13= minus11987512minus1ℎ1

11988522+ 11986011987911198941198791198793

Φ14= 11987522minus1ℎ2

11988511987912+1ℎ1

1198771+ 11986011987911198941198791198794

Φ15119894119895

= 1198791119861119894119865119895+ 11986011987911198941198791198795

Φ16= minus119882

12minus1ℎ2

11987322+ 11986011987911198941198791198796

Φ17= 11988222minus1ℎ2

11987311987912+2ℎ22

1198772+ 11986011987911198941198791198797

Φ18= 11987511+11988211+ ℎ111988512+ ℎ211987312minus 1198791+ 11986011987911198941198791198798

Φ22119894= minus (1 minus 120583

1) 1198761+ 11987921198602119894+ 11986011987921198941198791198792

Φ25119894119895

= 1198792119861119894119865119895+ 11986011987921198941198791198795

Φ28119894= minus1198792+ 11986011987921198941198791198798

Φ33= minus

1ℎ11988522minus1198721

Φ44= minus

1ℎ1

11988511minus2ℎ21

1198771

Φ48= 11987511987912minus 1198794

Φ55119894= minus (1 minus 120583

2) 1198762+ 1198795119861119894119865119895+ 1198651198791198951198611198791198941198791198795

Φ58119894= minus1198795+ 1198651198791198951198611198791198941198791198798

Φ66= minus

1ℎ2

11987322minus1198722

Φ67= minus

1ℎ2

11987311987912minus11988222

Φ77= minus

1ℎ2

11987311minus2ℎ22

1198772

Φ78= 11988211987912minus 1198797

Φ88= ℎ111988522+ ℎ211987322+12ℎ211198771+12ℎ221198772minus 1198798

(25)

From (25) it is obvious that if

119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))Φ

119894119895lt 0 (26)

(119909(119905)) lt 0From (26) we can discover that the feedback gains

are predefined The following proof presents the controllerdesign method We first pre- and postmultiply both thediag [119883 119883 119883 119883 119883 119883 119883 119883 ] and transpose to both sides of (25)and pre- and postmultiply the diag [119883 119883 ] transpose to bothsides of 119875 119882 119885 and 119873 and the 119883 and transpose to bothsides of 119876

119894119872119894 and 119877

119894 119894 = 1 2 Let 119879

119894= 1199051198941198791(119894 = 2 8)

and denote new variables 119883 = 119879minus11 1198771= 119883119877

1119883119879 119876

1=

1198831198761119883119879119872

1= 119883119872

1119883119879 119877

2= 119883119877

2119883119879 119876

2= 119883119876

2119883119879 119872

2=

1198831198722119883119879 Λ

119894= 119883Λ

119894119883119879 Λ

119895= 119883Λ

119895119883119879 119894 = 1 2 119903 119875

11=

11988311987511119883119879 119875

12= 119883119875

12119883119879 119875

22= 119883119875

22119883119879 119882

11= 119883119882

11119883119879

11988212= 119883119882

12119883119879 119882

22= 119883119882

22119883119879 119885

11= 119883119885

11119883119879 119885

12=

11988311988512119883119879 119885

22= 119883119885

22119883119879 119873

11= 119883119873

11119883119879 119873

12= 119883119873

12119883119879

and11987322= 119883119873

22119883119879 With 119865

119894= 119884119894119883minus119879 119894 = 1 2 119903we next

get

119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))Φ

119894119895lt 0 (27)

Mathematical Problems in Engineering 7

where

Φ119894119895=

[[[[[[[[

[

Φ11119894 Φ12119894 Φ13 Φ14 Φ15119894119895 Φ16 Φ17 Φ18

lowast Φ22119894 1199053119883119860119879

21198941199054119883119860119879

2119894Φ25119894119895 1199056119883119860

119879

21198941199057119883119860119879

2119894Φ28119894

lowast lowast Φ33119894 minus11987522 1199053119861119894119884119895 0 0 minus1199053119883119879

lowast lowast lowast Φ44 1199054119861119894119884119895 0 0 Φ48

lowast lowast lowast lowast Φ55119894119895 1199056119884119879

119895119861119879

1198941199057119884119879

119895119861119879

119894Φ58119894

lowast lowast lowast lowast lowast Φ66 Φ67 minus1199056119883119879

lowast lowast lowast lowast lowast lowast Φ77 Φ78

lowast lowast lowast lowast lowast lowast lowast Φ88

]]]]]]]]

]

Φ11119894= 11987512+ 119875119879

12+11988212+119882119879

12+ ℎ111988511

+ ℎ211987311minus1ℎ1

11988522minus1ℎ2

11987322+ 1198761+ 1198762

+1198721+1198722minus 21198771minus 21198772+ 1198601119894119883119879 + 119883119860119879

1119894

Φ12119894= 1198602119894119883119879 + 119905

21198831198601198791119894

Φ13= minus11987512minus1ℎ 111988522+ 11990531198831198601198791119894

Φ14= 11987522minus1ℎ1

119885119879

12+1ℎ1

1198771+ 11990541198831198601198791119894

Φ15119894119895

= 119861119894119884119895+ 11990551198831198601198791119894

Φ16= minus119882

12minus1ℎ2

11987322+ 11990561198831198601198791119894

Φ17= 11988222minus1ℎ2

119873119879

12+2ℎ22

1198772+ 11990571198831198601198791119894

Φ18= 11987511+11988211+ ℎ111988512+ ℎ211987312minus 119883119879 + 119905

81198831198601198791119894

Φ22119894= minus (1 minus 120583

1) 1198761+ 11990521198602119894119883119879 + 119905

21198831198601198792119894

Φ25119894119895

= 1199052119861119894119884119895+ 11990551198831198601198792119894

Φ28119894= minus1199052119883119879 + 119905

81198831198601198792119894

Φ33= minus

1ℎ1

11988522minus1198721

Φ44= minus

1ℎ1

11988511minus2ℎ21

1198771

Φ48= 119875119879

12minus 1199054119883119879

Φ55119894= minus (1 minus 120583

2) 1198762+ 1199055119861119894119884119895+ 1199055119884119879119895119861119879119894

Φ58119894= minus1199055119883119879 + 119905

8119884119879119895119861119879119894

Φ66= minus

1ℎ2

11987322minus1198722

Φ67= minus

1ℎ2

119873119879

12minus11988222

Φ77= minus

1ℎ2

11987311minus2ℎ22

1198772

Φ78= 119882119879

12minus 1199057119883119879

Φ88= ℎ111988522+ ℎ211987322+12ℎ211198771+12ℎ221198772minus 1199058119883119879

(28)

Here ldquolowastrdquo denotes the transpose elements in the symmetricpositions

If (27) holds then (119909(119905)) lt 0 Considersum119903119894=1sum119903119895=1

119908119894(119909(119905))(119908

119895(119909(119905)) minus 119898

119895(119909(119905 minus 119889

2(119905))))Λ

119894= 0

where Λ119894= Λ119879119894isin 1198778119899times8119899 gt 0 119894 = 1 2 119903 are arbitrary

matrices These terms are introduced to (27) to alleviate theconservativeness From (27) we also have

Φ =119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))Φ

119894119895

=119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905)) (119908

119895(119909 (119905)) minus 119898

119895(119909 (119905 minus 119889

2(119905)))

+120588119895119908119895(119909 (119905)) minus 120588

119895119908119895(119909 (119905))) Λ

119894

+119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))Φ

119894119895

=119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905)) 119908

119895(119909 (119905)) (120588

119895Φ119894119895minus 120588119895Λ119894+ Λ119894)

+119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905)) (119898

119895(119909 (119905 minus 119889

2(119905))) minus 120588

119895119908119895(119909 (119905)))

times (Φ119894119895minus Λ119894)

le119903

sum119894=1

1199082119894(120588119894Φ119894119894minus 120588119894Λ119894+ Λ119894)

+119903

sum119894=1

119903

sum119895=1

119908119894(119898119895minus 120588119895119908119895) (Φ119894119895minus Λ119894)

+119903

sum119894=1

sum119894lt119895

119908119894119908119895(120588119895Φ119894119895+ 120588119894Φ119895119894minus 120588119895Λ119894minus 120588119894Λ119895

+Λ119894+ Λ119895)

(29)

with119898119895(119909(119905 minus 119889

2(119905))) minus 120588

119895119908119895(119909(119905)) ge 0 for all 119895 119909(119905) and 119909(119905 minus

1198892(119905)) Let

Φ119894119895minus Λ119894lt 0 (30)

120588119894Φ119894119894minus 120588119894Λ119894+ Λ119894lt 0 (31)

120588119895Φ119894119895+ 120588119894Φ119895119894minus 120588119895Λ119894minus 120588119894Λ119895+ Λ119894+ Λ119895le 0 119894 lt 119895 (32)

for all 119894 119895 = 1 2 119903

8 Mathematical Problems in Engineering

Since (119909(119905)) lt 0 the fuzzy control system with the timevarying state and input delays (15) is asymptotically stablewith the state feedback control law (17)

32 Robust Stability of Uncertain Fuzzy Systems We alsoexamine the design of a robust stable controller for theuncertain system (4) under the imperfect premise matchingConsider the following uncertain fuzzy time varying delaycontrol system

(119905) =119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))

times [(1198601119894+ Δ1198601119894(119905)) 119909 (119905) + (119860

2119894+ Δ1198602119894(119905))

times 119909 (119905 minus 1198891(119905)) + (119861

119894+ Δ119861119894(119905))

times119906 (119905 minus 1198892(119905))]

119909 (119905) = 120593 (119905) 119905 isin [minusmax (ℎ1 ℎ2) 0]

(33)

Based on the above results of Theorem 4 a robust stabiliza-tion criterion for theT-S fuzzy systemswith time varying stateand input delays is investigated The following result can beobtained

Theorem 5 Given scalars ℎ1ge 0 ℎ

2ge 0 120583

1 1205832 and 119905

119894(119894 =

2 8) the uncertain fuzzy control systems with the timevarying state and input delays (33) is robustly stable for any0 le 119889

119894(119905) le ℎ

119894(119894 = 1 2) via the imperfect premise matching

controller design technique if the membership functions of thefuzzy model and fuzzy controller satisfy 119898

119895(119909(119905 minus 119889

2(119905))) minus

120588119895119908119895(119909(119905)) ge 0 for all 119895 119909(119905) and 119909(119905 minus 119889

2(119905)) where 0 lt

120588119895lt 1 and there exist common matrices 119875

11gt 0 119875

22gt 0

11988211gt 0 119882

22gt 0 119885

11gt 0 119885

22gt 0 119873

11gt 0 119873

22gt 0

119876119894gt 0 119872

119894gt 0 119877

119894gt 0 (119894 = 1 2) and Λ

119894= ΛT119894isin 1198778119899times8119899 gt

0 (i = 1 2 r) 119875 gt 0 119882 gt 0 119885 gt 0 119873 gt 0 and somematrices 119884

119895(119895 = 1 2 119903) 119883 and scalars 120576

1119894gt 0 120576

2119894gt 0

120576119887119894119895gt 0 satisfy the following LMIs

Ψ119894119895minus Λ119894lt 0 (34)

120588119894Ψ119894119894minus 120588119894Λ119894+ Λ119894lt 0 (35)

120588119895Ψ119894119895+ 120588119894Ψ119895119894minus 120588119895Λ119894minus 120588119894Λ119895+ Λ119894+ Λ119895 le 0 119894 lt 119895 (36)

where Ψ119894119895is defined as in (44) The state feedback gains can be

constructed as 119865119894= 119884119894119883minus119879

Proof If 1198601119894 1198602119894 and 119861

119894in Φ119894119895are replaced with 119860

1119894+

119863119894119870119894(119905)1198641119894 1198602119894+ 119863119894119870119894(119905)1198642119894 and 119861

119894+ 119863119894119870119894(119905)119864119887119894in (28) of

Theorem 4 respectively (27) for system (33) can be rewrittenas

119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889 (119905))) Ω

119894119895lt 0 (37)

where

Ω119894119895= Φ119894119895+ 1198631198791119894119870119894(119905) 119864119894+ 119864119879119894119870119879119894(119905) 1198631119894+ 1198631198792119894119870119894(119905) 1198642119894119895

+ 1198641198792119894119895119870119879119894(119905) 1198632119894+ 119863119879119887119894119870119894(119905) 119864119887119894119895+ 119864119879119887119894119895119870119879119894(119905) 119863119887119894

1198631119894= [1198631198791198940 0 0 0 0 0 0]

1198632119894= [0 119863119879

1198940 0 0 0 0 0]

119863119887119894= [0 0 0 0 119863119879

1198940 0 0]

(38)

1198641119894= [1198641119894119883119879 11990521198641119894119883119879 11990531198641119894119883119879 11990541198641119894119883119879 119905

51198641119894119883119879 11990561198641119894119883119879 11990571198641119894119883119879 11990581198641119894119883119879]

1198642119894= [1198642119894119883119879 11990521198642119894119883119879 11990531198642119894119883119879 11990541198642119894119883119879 11990551198642119894119883119879 11990561198642119894119883119879 11990571198642119894119883119879 11990581198642119894119883119879]

119864119887119894119895= [119864119887119894119884119895 1199052119864119887119894119884119895 1199053119864119887119894119884119895 1199054119864119887119894119884119895 1199055119864119887119894119884119895 1199056119864119887119894119884119895 1199057119864119887119894119884119895 1199058119864119887119894119884119895]

(39)

If Ω119894119895

lt 0 (37) holds According to Lemma 2 it isstraightforward to know that Ω

119894119895lt 0 is true if for each 119894

119895 there exists scalars 1205761119894gt 0 120576

2119894gt 0 and 120576

119887119894119895gt 0 such that

the following inequality holds

Φ119894119895+ 120576111989411986311987911198941198631119894+ 120576minus1111989411986411987911198941198641119894+ 120576211989411986311987921198941198632119894

+ 120576minus1211989411986411987921198941198642119894+ 120576119887119894119895119863119879119887119894119863119887119894+ 120576minus1119887119894119895119864119879119887119894119895119864119887119894119895lt 0

(40)

Based on Schur complement (40) is equivalent to thefollowing inequality

Ψ119894119895=[[[

[

Φ119894119895

1198641198791119894

1198641198792119894

119864119879119887119894119895

lowast minus1205761119894119868 0 0

lowast lowast minus1205762119894119868 0

lowast lowast lowast minus120576119887119894119895119868

]]]

]

lt 0 (41)

Mathematical Problems in Engineering 9

where

Φ119894119895=

[[[[[[[[

[

Φ11119894 Φ12119894 Φ13 Φ14 Φ15119894119895 Φ16 Φ17 Φ18

lowast Φ22119894 1199053119883119860119879

21198941199054119883119860119879

2119894Φ25119894119895 1199056119883119860

119879

21198941199057119883119860119879

2119894Φ28119894

lowast lowast Φ33119894 minus11987522 1199053119861119894119884119895 0 0 minus1199053119883119879

lowast lowast lowast Φ44 1199054119861119894119884119895 0 0 Φ48

lowast lowast lowast lowast Φ55119894119895 1199056119884119879

119895119861119879

1198941199057119884119879

119895119861119879

119894Φ58119894

lowast lowast lowast lowast lowast Φ66 Φ67 minus1199056119883119879

lowast lowast lowast lowast lowast lowast Φ77 Φ78

lowast lowast lowast lowast lowast lowast lowast Φ88

]]]]]]]]

]

lt 0

Φ11119894= Φ11119894+ 1205761119894119863119894119863119879119894

Φ22119894= Φ22119894+ 1205762119894119863119894119863119879119894

Φ55119894119895

= Φ55119894119895

+ 120576119887119894119863119894119863119879119894

(42)

and Φ are the same as in Theorem 4 Therefore (41) holds ifand only if the following inequality holds

119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889 (119905))) Ψ

119894119895lt 0 (43)

To alleviate the conservativeness of our robust stabilityanalysis similar to the analysis approach of Theorem 4 weintroduce sum119903

119894=1sum119903119895=1

119908119894(119908119895minus 119898119895)Λ119894= 0 to (43) and the

following proof is the same as in Theorem 4 We can obtainthe conditions and (34)ndash(36) Thus the fuzzy control systemwith the time varying state and input delays (34) is robuststable on the basis of the control law 119865

119894= 119884119894119883minus119879under these

conditions and (34)ndash(36) hold

Remark 6 If we set 119908119894(119909(119905)) = 119898

119895(119909(119905)) 119889

1(119905) = 119889

2(119905)

the closed-loop system in this paper has the same structureas that of paper [26] On the other hand let 119908

119894(119909(119905)) =

119898119895(119909(119905)) and 119889

1(119905) = 119889

2(119905) = 119889 we can obtain the system

in [16] Therefore the system studied in this paper is moregeneral Additionally the information of the membershipfunctions of the fuzzy time delay models and controllers areall considered in the stability analysis If the fuzzymodels andfuzzy controllers share the same fuzzymembership functionsthat is 119908

119894(119909(119905)) = 119898

119895(119909(119905)) the stability analysis can be

referred to [25 27 28] In other words the problem in [2527 28] is actually a special case of the one investigated in thispaper

Remark 7 The augmented Lyapunov function with an addi-tional triple-integral term is introduced in analyzing thestability problem for the T-S fuzzy systems with time varyingstate and input delays In addition two integral inequalitiesare used to derive Theorem 4 and less free-weighting matri-ces are introduced which lead to get less conservative resultsIt can be observed that some existing results are the specialcases of this paper For example the method in [25] is from(18) with 119875

12= 0 119875

22= 0 119885

12= 11988522= 0 119873

12= 11987322= 0

and 1198771= 1198772= 0 in Theorem 4 The results in [22] are based

on system (33) with 1198892(119905) = 0 and (18) with119872

2= 0 119885

12=

11988522= 0 119876

1= 1198762= 0 and 119877

1= 1198772= 0

Remark 8 Different from the general PDC technique thedesign method under the imperfect premise matching ismuch more flexible because the membership functions ofthe fuzzy controllers do not need to be chosen the same asthose of the fuzzy time delay models Instead they can bedesigned arbitrarilyThus the design flexibility is significantlyenhanced On the other hand some simple membershipfunctions of the fuzzy controllers might be employed whichcan reduce the implementation cost

Remark 9 In [7ndash11 15ndash20] the authors have considered therobust control for uncertain T-S fuzzy systems with statedelays Some delay-dependent conditions and fuzzy con-trollers are obtained with a traditional Lyapunov-Krasovskiifunctionalmethod andPDCmethod which all are the specialcases of Theorem 5 in this paper

4 Numerical Examples

In this section two numerical examples are given to illus-trate the conservativeness and effectiveness of the proposedmethods The first example compares our techniques withthe existing ones in the literature for stability analysis whichshows that Theorem 4 in this paper is less conservative thanthe other results The second example is used to illustrate theadvantage of the robust stability conditions ofTheorem 5 anddemonstrate how to design robust fuzzy controllers by usingour approach

Example 10 Consider the following fuzzy system with stateand input delays

(119905) =2

sum119894=1

119908119894(120579 (119905)) [119860

1119894119909 (119905) + 119860

2119894119909 (119905 minus 119889

1(119905))

+119861119894119906 (119905 minus 119889

2(119905))]

11986011= [0 06

0 1 ] 11986021= [05 09

0 2 ]

11986012= [1 0

1 0] 11986022= [09 0

1 16]

1198611= 1198612= [11]

(44)

The fuzzy membership functions are selected as [31]

1199081(1199091(119905)) = (1 minus

119888 (119905) sin (10038161003816100381610038161199091 (119905)1003816100381610038161003816minus4)5

1 + expminus1001199091(119905)3(1minus1199091(119905)))

timescos (119909

1(119905))2

1 + expminus251199091(119905)(3+(1199091(119905)042))

times 1199082(1199091(119905)) = 1 minus 119908

1(1199091(119905))

(45)

10 Mathematical Problems in Engineering

Table 1 The maximum allowable time delay and feedback gains(1205831= 0 120583

2= 0)

Paper ℎ1

ℎ2 Feedback gains

[26] 03120 031201198651= [10598 minus56598]

1198652= [minus13068 minus41167]

Theorem 4 033 0571198651= [minus00042 minus00449]1198652= [00048 00717]

where

1199091(119905) isin [minus

1205872

1205872]

119888 (119905) =sin (119909

1(119905)) + 140

isin [minus005 005] (46)

If we set 1198891(119905) = 119889

2(119905) in system (44) we can obtain the

system in [26] which implies that our system ismore generalEmploying the LMIs in [26] and those inTheorem 4yields themaximum state and input delays ℎ

1 and ℎ

2 which guarantee

the stability of system (44) and the feedback gains for 1205831= 0

and 1205832= 0 when 120588

1= 075 120588

2= 095 119905

2= 01 119905

3=

02 1199054= 07 119905

5= 01 119905

6= 04 119905

7= 01 and 119905

8= 13 as

given in Table 1 which clearly shows the superiority of theresults derived in this paper over those obtained from [26]

Example 11 Consider the well-known truck-trailer systemwhich can be described by the following T-S fuzzy system

(119905) =2

sum119894=1

119908119894(120579 (119905)) [(119860

1119894+ Δ1198601119894) 119909 (119905) + 119860

2119894119909 (119905 minus 119889

1(119905))

+ (119861119894+ Δ119861119894) 119906 (119905 minus 119889

2(119905))]

(47)

11986011=

[[[[[[[[[

[

minus119886V1199051198711199050

0 0

119886V1199051198711199050

0 0

119886V21199052

21198711199050

V1199051199050

0

]]]]]]]]]

]

11986021=

[[[[[[[[[

[

minus (1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)V21199052

21198711199050

0 0

]]]]]]]]]

]

1198611=[[[[

[

V1199051198971199050

0

0

]]]]

]

11986012=

[[[[[[[[[

[

minus119886V1199051198711199050

0 0

119886V1199051198711199050

0 0

119886119889V21199052

21198711199050

119889V1199051199050

0

]]]]]]]]]

]

11986022=

[[[[[[[[[

[

minus (1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)119889V21199052

21198711199050

0 0

]]]]]]]]]

]

(48)

1198612=[[[[

[

V1199051198971199050

0

0

]]]]

]

(49)

where 119897 = 28 119871 = 55 V = minus10 119905 = 20 1199050= 05

119889 = 101199050120587 119863

119894= [02555 02555 02555]119879 119864

1119894= 1198642119894=

[01 0 0]119879 119864119887119894

= [01 0 0]119879 (119894 = 1 2) and the fuzzymembership functions are selected the same as inExample 10

(1) Let 119886 = 1 system (47) is the same as the fuzzy systemof [22] which implies that our system ismore generalTable 2 gives themaximum input delay value of ℎ

2 for

which the stabilization is guaranteed byTheorem 5 aswell as the state feedback gains when 120588

1= 075 120588

2=

095 1199052= 01 119905

3= 02 119905

4= 07 119905

5= 01 119905

6=

04 1199057= 08 and 119905

8= 2 It is clearly visible that the

results derived in this paper are better than those from[22 28]

(2) In the case of 119886 = 1 the state and input delays of system(47) are all exist The approach proposed in [22] cannot be used to get the maximum of state delay ℎ

1

as the system in [22] only contains the input delayHowever the method in [28] and Theorem 5 can beused Here let 119886 = 07 120588

1= 075 120588

2= 095 119905

2=

01 1199053= 02 119905

4= 07 119905

5= 01 119905

6= 04 119905

7=

08 and 1199058= 3 and Table 3 provides the maximum

upper bounds of ℎ1and ℎ

2and the state feedback

gains for which the robust stabilization is guaranteedbyTheorem 5

By Theorem 5 we can conclude that the uncertain fuzzymodel (47) is robust stable based on the following fuzzycontrol law

119906 (119905 minus ℎ2) =2

sum119895=1

119898119895(1199091(119905 minus ℎ

2)) 119865119895119909 (119905 minus ℎ

2) (50)

With the PDC design technique it is required that thefuzzy controller must share the same fuzzy membershipfunctions as those of the fuzzy time delay model Under such

Mathematical Problems in Engineering 11

Table 2 The maximum allowable time delay and feedback gains(1205832= 0)

Paper ℎ2 Feedback gains

[22] 0751198651= [34227 minus03535 00045]

1198652= [35215 minus03617 00056]

[28] 0861198651= [33219 minus02406 00025]

1198652= [33272 minus02494 00026]

Theorem 5 121198651= [minus00212 00093 minus00024]1198652= [minus00250 00044 00014]

Table 3 The maximum allowable time delay and feedback gains(1205831= 1205832= 0)

Paper ℎ1

ℎ2 Feedback gains

[28] 01 0551198651= [33219 minus02406 00025]

1198652= [33272 minus02494 00026]

Theorem 5 04 171198651= [minus00128 00090 minus00017]1198652= [minus00194 00012 00009]

a condition the implementation cost of the fuzzy controlleris high by employing these complex membership functionsHowever fromTheorem 5 the membership functions do notneed to be chosen the same as119908

1(1199091(119905)) and119908

2(1199091(119905)) Thus

we can select some simple membership functions insteadsuch as

1198981(1199091(119905 minus ℎ

2)) = 075 exp

(minus1199091(119905 minus ℎ

2) minus 038)2

2 times 0382+ 005

1198982(1199091(119905 minus ℎ

2)) = 1 minus 119898

1(1199091(119905 minus ℎ

2))

(51)

We also assume that 1205881= 075 and 120588

2= 095 such that

119898119895(1199091(119905 minus ℎ

2)) minus 120588

119895119908119895(1199091(119905)) gt 0 for 119895 = 1 2 and 119909

1(119905 minus ℎ

2)

The fuzzy control law 119906(119905 minus ℎ2) = 119898

1[minus00128 00090 minus

00017]119909 + 1198982[minus00194 00012 00009]119909 can be employed

to stabilize the uncertain fuzzy control systems in (47) withthe state and input delays Figures 1 and 2 illustrate the stablestate response and control input under the initial conditionof 119909(0) = [4 minus 1 2]119879

Remark 12 In Example 11 our method offers less conser-vative results in the sense of allowing longer time delayand obtaining smaller feedback control gains Moreovercompared with [22ndash28] where the fuzzy controllermust havethe same fuzzy membership functions as those of the fuzzytime delay model the above membership functions of ourfuzzy controller are much simper which can considerablylower the implementation cost of the fuzzy controller andenhance the design flexibility

Remark 13 If the function of 119888(119905) in 1199081(1199091(119905)) is unknown

on the basis of the PDC design technique [7ndash11 15ndash20 22ndash28] the fuzzy controller cannot be implemented as 119888(119905) isnot available However using the proposed design method

0 5 10 15

0

2

4

6Closed-loop response

minus12

minus10

minus8

minus6

minus4

minus2

119905 (s)

119909(119905)

1199092

11990931199091

Figure 1The state responses of system (47) with ℎ1= 04 ℎ

2= 17

0 5 10 15

0

5

10

15

20

25

30

119905 (s)

minus20

minus15

minus10

minus5

119906(119905)

Figure 2 The control input of system (47) with ℎ1= 04 ℎ

2= 17

we can select some simper and well-known membershipfunctions that are different from those of the fuzzy time delaymodels for example119898

1(1199091(119905)) and119898

2(1199091(119905)) so as to realize

the fuzzy controller In other words the proposed designapproach can be applied even for the fuzzy time delay modelswith uncertain grades of membership

5 Conclusions

In this paper we study the robust stability and stabilizationproblems for general nonlinear fuzzy systems with timevarying state and input delays A less conservative delay-dependent robust stability criterion has been obtained byintroducing a novel augmented Lyapunov function withan additional triple-integral term Moreover a new design

12 Mathematical Problems in Engineering

approach different from the traditional PDC design tech-nique is proposed which can significantly improve the designflexibility and reduce the implementation cost of the fuzzycontroller by arbitrarily selecting simple fuzzy membershipfunctions Two numerical examples are used to illustrate theconservativeness and effectiveness of our proposed methods

Acknowledgments

This research work was funded by the NSFC under Grantno 61273095 and the Academy of Finland under Grants no135225 and no 127299

References

[1] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[2] W H Ho S H Chen I T Chen J H Chou and C C ShuldquoDesign of stable and quadratic-optimal static output feed-back controllers for T-S-fuzzy-model-based control systems anintegrative computational approachrdquo International Journal ofInnovative Computing Information and Control vol 8 no 1App 403ndash418 2012

[3] X J Su L G Wu P Shi and Y D Song ldquo119867infinmodel reduction

of T-S fuzzy stochastic systemsrdquo IEEE Transactions on SystemsMan and Cybermetics B vol 42 no 6 pp 1574ndash1585 2012

[4] R Yang P Shi G-P Liu andH Gao ldquoNetwork-based feedbackcontrol for systems with mixed delays based on quantizationand dropout compensationrdquo Automatica vol 47 no 12 pp2805ndash2809 2011

[5] XM Zhang Z J Zhang andG P Lu ldquoFault detection for state-delay fuzzy systems subject to random communication delayrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 4 pp 2439ndash2451 2012

[6] F B Li and X Zhang ldquoDelay-range-dependent robust 119867infin

filtering for singuar LPV systems with time variant delayrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 1 pp 339ndash353 2013

[7] Y Y Cao and P M Frank ldquoAnalysis and synthesis of nonlineartime-delay systems via fuzzy control approachrdquo IEEE Transac-tions on Fuzzy Systems vol 8 no 2 pp 200ndash211 2000

[8] Y-Y Cao and P M Frank ldquoStability analysis and synthesis ofnonlinear time-delay systems via linear Takagi-Sugeno fuzzymodelsrdquo Fuzzy Sets and Systems vol 124 no 2 pp 213ndash2292001

[9] O M Kwon and J H Park ldquoDelay-range-dependent stabiliza-tion of uncertain dynamic systems with interval time-varyingdelaysrdquo Applied Mathematics and Computation vol 208 no 1pp 58ndash68 2009

[10] E Tian D Yue and Y Zhang ldquoDelay-dependent robust 119867infin

control for T-S fuzzy system with interval time-varying delayrdquoFuzzy Sets and Systems vol 160 no 12 pp 1708ndash1719 2009

[11] J Y An T Li G LWen and R Li ldquoNew stability conditions foruncertain T-S fuzzy systems with interval time-varying delayrdquoInternational Journal of Control Automation and Systems vol10 no 3 pp 490ndash497 2012

[12] Z J Zhang X L Huang X J Ban and X Z Gao ldquoStabilityanalysis and controller design of T-S fuzzy systems with time-delay under imperfect premise matchingrdquo Journal of BeijingInstitute of Technology vol 21 no 3 pp 387ndash393 2012

[13] O M Kwon M J Park S M Lee and J H Park ldquoAugmentedLyapunov-Krasovskii functional approaches to robust stabilitycriteria for uncertain Takagi-Sugeno fuzzy systems with time-varying delaysrdquo Fuzzy Sets and Systems vol 201 no 16 pp 1ndash192012

[14] F Liu M Wu Y He and R Yokoyama ldquoNew delay-dependentstability criteria for T-S fuzzy systems with time-varying delayrdquoFuzzy Sets and Systems vol 161 no 15 pp 2033ndash2042 2010

[15] B Chen X Liu and S Tong ldquoNew delay-dependent stabiliza-tion conditions of T-S fuzzy systems with constant delayrdquo FuzzySets and Systems vol 158 no 20 pp 2209ndash2224 2007

[16] J Yoneyama ldquoRobust stability and stabilization for uncertainTakagi-Sugeno fuzzy time-delay systemsrdquo Fuzzy Sets and Sys-tems vol 158 no 2 pp 115ndash134 2007

[17] C Lin Q-G Wang and T H Lee ldquoDelay-dependent LMIconditions for stability and stabilization of T-S fuzzy systemswith bounded time-delayrdquo Fuzzy Sets and Systems vol 157 no9 pp 1229ndash1247 2006

[18] X J Su P Shi L G Wu and Y D Song ldquoA novel approachto filter design for T-S fuzzy discrete-time systems with time-varying delayrdquo IEEE Transactions on Fuzzy Systems vol 20 no6 pp 1114ndash1129 2012

[19] C Peng and Q-L Han ldquoDelay-range-dependent robust stabi-lization for uncertain T-S fuzzy control systems with intervaltime-varying delaysrdquo Information Sciences vol 181 no 19 pp4287ndash4299 2011

[20] S T Li Y W Jing and X M Liu ldquoRobust control for a class ofT-S fuzzy systemswith interval time-varying delayrdquo InternatinalJournal of Control Automation and Systems vol 10 no 4 pp737ndash743 2012

[21] X-M Zhang M Wu J-H She and Y He ldquoDelay-dependentstabilization of linear systems with time-varying state and inputdelaysrdquo Automatica vol 41 no 8 pp 1405ndash1412 2005

[22] B Chen X Liu and S Tong ldquoRobust fuzzy control of nonlinearsystems with input delayrdquo Chaos Solitons amp Fractals vol 37 no3 pp 894ndash901 2008

[23] C Lin Q-G Wang and T H Lee ldquoDelay-dependent LMIconditions for stability and stabilization of T-S fuzzy systemswith bounded time-delayrdquo Fuzzy Sets and Systems vol 157 no9 pp 1229ndash1247 2006

[24] H J Lee J B Park and Y H Joo ldquoRobust control for uncertainTakagi-Sugeno fuzzy systems with time-varying input delayrdquoJournal of Dynamic Systems Measurement and Control Trans-actions of the ASME vol 127 no 2 pp 302ndash306 2005

[25] C-H Lien and K-W Yu ldquoRobust control for Takagi-Sugenofuzzy systems with time-varying state and input delaysrdquo ChaosSolitons and Fractals vol 35 no 5 pp 1003ndash1008 2008

[26] Y K Su B Chen S Y Zhang and C Lin ldquoDelay-dependentstabilization of T-S fuzzy systems with time delayrdquo Control andDecision Kongzhi yu Juece vol 24 no 6 pp 921ndash927 2009

[27] B Chen X Liu C Lin and K Liu ldquoRobust 119867infin

control ofTakagi-Sugeno fuzzy systems with state and input time delaysrdquoFuzzy Sets and Systems vol 160 no 4 pp 403ndash422 2009

[28] L Li and X Liu ldquoNew results on delay-dependent robuststability criteria of uncertain fuzzy systems with state and inputdelaysrdquo Information Sciences vol 179 no 8 pp 1134ndash1148 2009

[29] Z J Zhang X J Ban X L Huang and X Z Gao ldquoNew delay-dependent robust stablity for T-S fuzzy systems with intervevaltime-varying delayrdquo in Proccedings of the 5th International JointConference on Computational Science and Optimization pp826ndash830 Harbin China 2012

Mathematical Problems in Engineering 13

[30] H K Lam and F H F Leung ldquoLMI-based stability andperformance design of fuzzy control systems fuzzy modelsand controllers with different premisesrdquo in Proceedings of theInternational Conference on Fuzzy Systerm pp 9599ndash9506Vancouver Canada 2006

[31] H K Lam andMNarimani ldquoStability analysis and perfomancedesign for fuzzy-model-based control system under imperfectpremise matchingrdquo IEEE Transactions on Fuzzy Systems vol 17no 4 pp 949ndash961 2009

[32] S Boyd L E Ghaoui and E Feron Linear Matrix Inequality inSystems and ControlTheory SIAM Philadelphia Pa USA 1994

[33] I R Petersen and C V Hollot ldquoA Riccati equation approach tothe stabilization of uncertain linear systemsrdquo Automatica vol22 no 4 pp 397ndash411 1986

[34] J Sun G P Liu and J Chen ldquoDelay-dependent stabilityand stabilization of neutral time-delay systemsrdquo InternationalJournal of Robust andNonlinear Control vol 19 no 12 pp 1364ndash1375 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

to be the same as the input delay Some interesting results forthe uncertain fuzzy systems with state and input delays havebeen obtained in [25 27 28] most of which introduce someLyapunov-Krasovskii functions containing integral termsfor example int119905

119905minus120591119909119879(119904)119876119909(119904)119889119904 and double-integral terms

int0

minus120591int119905

119905+120579119879(119904)119876(119904)119889119904 119889120579 Several triple-integral terms are

used in the Lyapunov function [29] to yield less conservativeresults for the fuzzy systems with state delay A large portionof robust controller design topics have been investigated onthe basis of the Parallel Distribution Compensation (PDC)design technique where the fuzzy controller shares the samepremise membership functions as those of the T-S fuzzy timedelay model [7ndash11 15ndash20 22ndash28] As a matter of fact if themembership functions in the premise of the fuzzy rules ofthe fuzzy controllers are allowed to be designed arbitrarily wecan even achieve better design flexibility For instance a fuzzycontroller not sharing the same premise rules as those of theT-S fuzzy model referred to as imperfect premise matchingis employed to control the nonlinear plants [30 31] and[12] extends the available results to the T-S fuzzy time-delaysystems with only the state delay

In this paper an augmented Lyapunov-Krasovskii func-tion that contains a triple-integral term is introduced to inves-tigate the stability and stabilization problem for uncertain T-S fuzzy systems with the state and input delays under theimperfect premise matching in which the fuzzy time delaymodel and fuzzy controller are with different premise Someless conservative delay-dependent stability and robust stabil-ity conditions are obtained by two integral inequalities anda parameterized model transformation method Moreoverdifferent from the general PDC design technique a newdesign approach of robust stable controllers is proposed Twosimulation examples are further given to illustrate that theproposed design methods are less conservative and moreflexible

2 System Description and Preliminaries

Let 119903 be the number of the fuzzy rules describing the timedelay nonlinear plant The 119894th rule can be represented asfollows

If 1198911(119909(119905)) is119872119894

1and and 119891

119901(119909(119905)) is119872119894

119901

(119905) = (1198601119894+ Δ1198601119894(119905)) 119909 (119905) + (119860

2119894+ Δ1198602119894(119905))

times 119909 (119905 minus 1198891(119905)) + (119861

119894+ Δ119861119894(119905)) 119906 (119905 minus 119889

2(119905))

119909 (119905) = 120593 (119905) 119905 isin [minusmax (ℎ1 ℎ2) 0]

(1)

where 119872119894120572is a fuzzy term of rule 119894 corresponding to the

function119891120572(119909(119905)) 120572 = 1 2 119901 119894 = 1 2 119903 119909(119905) isin 119877119899 is

the system state vector and 119906(119905) isin 119877119898 is the input vectorThematrices 119860

1119894 1198602119894 and 119861

119894 119894 = 1 2 119903 are of appropriate

dimensionsThe initial condition 120593(119905) is a continuous vector-valued function The delays 119889

1(119905) and 119889

2(119905) are time varying

and satisfy

0 le 1198891(119905) le ℎ

1 119889

1(119905) le 120583

1

0 le 1198892(119905) le ℎ

2 119889

2(119905) le 120583

2

(2)

where ℎ119894are constants representing the upper bound of the

delay 120583 is a positive constant Δ1198601119894 Δ1198602119894 and Δ119861

119894denote

the uncertainties in the system and they are the form of

[Δ1198601119894

Δ1198602119894

Δ119861119894] = 119863

119894119870119894(119905) [1198641119894

1198642119894

119864119887119894]

119894 = 1 2 119903(3)

where 119863119894 1198641119894 1198642119894 and 119864

119887119894are known constant matrices

of appropriate dimensions and 119870119894(119905) is unknown matrix

function satisfying 119870119894

119879(119905)119870119894(119905) le 119868 119868 is an appropriately

dimensioned identity matrix Hence the overall fuzzy modelcan be formulized as follows

(119905) =119903

sum119894=1

119908119894(119909 (119905)) [(119860

1119894+ Δ1198601119894(119905)) 119909 (119905) + (119860

2119894+ Δ1198602119894(119905))

times 119909 (119905 minus 1198891(119905)) + (119861

119894+ Δ119861119894(119905))

times119906 (119905 minus 1198892(119905))]

(4)

where119903

sum119894=1

119908119894(119909 (119905)) = 1 119908

119894(119909 (119905)) ge 0

119908119894(119909 (119905)) =

120583119894(119909 (119905))

sum119903119894=1120583119894(119909 (119905))

120583119894(119909 (119905)) =

119901

prod120572=1

120583119872119894

120572

(119891120572(119909 (119905)))

(5)

119908119894(119909(119905)) is the normalized grade ofmembership function that

is a nonlinear function of 119909(119905) 120583119872119894

120572

(119891120572(119909(119905))) is the grade of

the membership corresponding to the fuzzy term of119872119894120572

Different from the popular PDC design technique thefollowing fuzzy control law under imperfect premise match-ing is employed to deal with the problem of stabilization viastate feedback Under the imperfect premise matching the119895th rule of the fuzzy controller is defined as follows

If 1198921(119909(119905)) is119873119894

1and and 119892

119902(119909(119905)) is119873119894

119902

119906 (119905) = 119865119895119909 (119905) 119895 = 1 2 119903 (6)

where 119873119895120573denotes the fuzzy set 119902 is a positive integer and

119865119895isin 119877119898times119899 is the feedback gain of rule 119895 The state feedback

fuzzy control law is represented by

119906 (119905) =119903

sum119895=1

119898119895(119909 (119905)) 119865

119895119909 (119905) (7)

Mathematical Problems in Engineering 3

where119903

sum119895=1

119898119895(119909 (119905)) = 1 119898

119895(119909 (119905)) ge 0

119898119895(119909 (119905)) =

V119895(119909 (119905))

sum119903119895=1

V119895(119909 (119905))

V119895(119909 (119905)) =

119902

prod120573=1

V119873119895

120573

(119892120573(119909 (119905)))

(8)

119898119895(119909(119905)) is the normalized grade of membership function

that is a nonlinear function of 119909(119905) V119873119895

120573

(119892120573(119909(119905))) is the grade

of the membership corresponding to the fuzzy term of119873119895120573

As the time varying delay is included in the control inputwe have

119906 (119905 minus 1198892(119905)) =

119903

sum119895=1

119898119895(119909 (119905 minus 119889

2(119905))) 119865

119895119909 (119905 minus 119889

2(119905)) (9)

where119903

sum119895=1

119898119895(119909 (119905 minus 119889

2(119905))) = 1 119898

119895(119909 (119905 minus 119889

2(119905))) ge 0 (10)

Lemma 1 (see [32] (Schur complement)) Given constantmatrices Ω

1 Ω2 and Ω

3 where Ω

1= Ω1198791and Ω

2= Ω1198792 there

is Ω1+ Ω1198793Ωminus12Ω3lt 0 if and only if

[Ω1 Ω1198793

lowast minusΩ2

] lt 0 or [minusΩ2 Ω3

lowast minusΩ1

] lt 0 (11)

Lemma 2 (see [33]) Let Q = Q119879 D E and K(119905) satisfiesK119879(119905)K(119905) le I the following inequality holds

119876 + 119863119870 (119905) 119864 + 119864119879119870119879 (119905) 119863119879 lt 0 (12)

if and only if the following inequality holds for any smaller 120576 gt0

119876 + 120576minus1119863119863119879 + 120576119864119879119864 lt 0 (13)

Lemma 3 (see [34]) For any constant matrix Σ = Σ119879 gt 0and a scalar 120591 gt 0 such that the following integrations are welldefined we have

minus int119905

119905minus120591

119909119879 (119904) Σ119909 (119904) 119889119904

le minus1120591(int119905

119905minus120591

119909 (119904) 119889119904)119879

Σ(int119905

119905minus120591

119909 (119904) 119889119904)

minus int0

minus120591

int119905

119905+120579

119909119879 (119904) Σ119909 (119904) 119889119904 119889120579

le minus11205912(int0

minus120591

int119905

119905+120579

119909 (119904) 119889119904 119889120579)119879

Σ(int0

minus120591

int119905

119905+120579

119909 (119904) 119889119904 119889120579)

(14)

3 Main Results

In this section some new delay-dependent criteria for theT-S fuzzy systems with state and input delays are proposedby introducing a novel Lyapunov function with an additionaltriple-integral term under the imperfect premise matchingMoreover a robust stabilization criterion is also investigated

31 Stability of Nominal Fuzzy Systems Firstly we considerthe control design of a state feedback control law underthe imperfect premise matching that stabilizes the followingnominal fuzzy time varying delay system

(119905) =119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))

times [1198601119894119909 (119905) + 119860

2119894119909 (119905 minus 119889

1(119905)) + 119861

119894119865119895119909

times (119905 minus 1198892(119905))]

119909 (119905) = 120593 (119905) 119905 isin [minusmax (ℎ1 ℎ2) 0]

(15)

Theorem 4 Given scalars ℎ1

ge 0 ℎ2

ge 0 1205831 1205832 and

119905119894(119894 = 2 8) the closed-loop system (15) is asymptotically

stable for any 0 le 119889119894(119905) le ℎ

119894(119894 = 1 2) via the

imperfect premise matching controller design technique if themembership functions of the fuzzy model and fuzzy controllersatisfy 119898

119895(119909(119905 minus 119889

2(119905))) minus 120588

119895119908119895(119909(119905)) ge 0 for all 119895 119909(119905) and

119909(119905 minus 1198892(119905)) where 0 lt 120588

119895lt 1 and there exist matrices

11987511

gt 0 11987522

gt 0 11988211

gt 0 11988222

gt 0 11988511

gt 0 11988522

gt0 11987311gt 0 119873

22gt 0 119876

119894gt 0 119872

119894gt 0 119877

119894gt 0 (119894 = 1 2)

Λ119894= Λ119879119894isin 1198778119899times8119899 gt 0 (i = 1 2 119903) and

119875 = [11987511 11987512lowast 119875

22

] gt 0 119882 = [11988211 11988212lowast 119882

22

] gt 0

119885 = [11988511 11988512lowast 119885

22

] gt 0 119873 = [11987311 11987312lowast 119873

22

] gt 0

(16)

and real matrices 119884119895(119895 = 1 2 119903) 119875

12119882121198851211987312 and

119883 such that (30)ndash(32) hold In addition the stabilizing controllaw is given by

119906 (119905) =119903

sum119895=1

119898119895(119909 (119905 minus 119889

2(119905))) 119884

119895119883minus119879119909 (119905) (17)

Proof Choose a fuzzy weighting-dependent Lyapunov-Krasovskii functional candidate as

119881 (119909 (119905)) = 1198811(119909 (119905)) + 119881

2(119909 (119905)) + 119881

3(119909 (119905)) + 119881

4(119909 (119905))

+ 1198815(119909 (119905))

4 Mathematical Problems in Engineering

1198811(119909 (119905)) = 120585119879

1(119905) 119875120585

1(119905) + 120585119879

2(119905)119882120585

2(119905)

1198812(119909 (119905)) = int

0

minusℎ1

int119905

119905+120579

120585119879 (119904) 119885120585 (119904) 119889 119904119889120579

+ int0

minusℎ2

int119905

119905+120579

120585119879 (119904)119873120585 (119904) 119889119904 119889120579

1198813(119909 (119905)) = int

119905

119905minus1198891(119905)

119909119879 (119904) 1198761119909 (119904) 119889119904

+ int119905

119905minus1198892(119905)

119909119879 (119904) 1198762119909 (119904) 119889119904

1198814(119909 (119905)) = int

119905

119905minusℎ1

119909119879 (119904)1198721119909 (119904) 119889119904 + int

119905

119905minusℎ2

119909119879 (119904)1198722119909 (119904) 119889119904

1198815(119909 (119905)) = int

0

minusℎ1

int0

120579

int119905

119905+120582

119879 (119904) 1198771 (119904) 119889119904 119889120582 119889120579

+ int0

minusℎ2

int0

120579

int119905

119905+120582

119879 (119904) 1198772 (119904) 119889119904 119889120582 119889120579

(18)

where

1205851198791(119905) = [119909119879 (119905) (int

119905

119905minusℎ1

119909 (119904) 119889119904)119879

]

1205851198792(119905) = [119909119879 (119905) (int

119905

119905minusℎ2

119909 (119904) 119889119904)119879

]

120585119879 (119905) = [119909119879 (119905) 119879 (119905)]

119875 = [11987511 11987512lowast 11987522

] gt 0 119882 = [11988211 11988212lowast 11988222

] gt 0

119885 = [11988511 11988512lowast 11988522

] gt 0 119873 = [11987311 11987312lowast 11987322

] gt 0

(19)

and 119876119894 119872119894 and 119877

119894(119894 = 1 2) are the positive-definite

matrices to be determinedThe derivatives of119881(119909(119905)) along the trajectories of system

(15) are as follows

1(119909 (119905)) = 2120585119879

1(119905) 119875 1205851(119905) + 2120585119879

2(119905)119882 120585

2(119905)

= 2 [119909119879 (119905) (int119905

119905minusℎ1

119909 (119904) 119889119904)119879

] [11987511 11987512lowast 11987522

]

times [ (119905)119909 (119905) minus 119909 (119905 minus ℎ

1)]

+ 2 [119909119879 (119905) (int119905

119905minusℎ2

119909 (119904) 119889119904)119879

] [11988211 11988212lowast 11988222

]

times [ (119905)119909 (119905) minus 119909 (119905 minus ℎ

2)]

2(119909 (119905)) = ℎ

1[119909119879 (119905) 119879 (119905)] [11988511 11988512lowast 119885

22

] [119909 (119905) (119905)]

minusint119905

119905minusℎ1

[119909119879 (119904) 119879 (119904)] [11988511 11988512lowast 11988522

] [119909 (119904) (119904)] 119889119904

+ ℎ2[119909119879 (119905) 119879 (119905)] [11987311 11987312lowast 119873

22

] [119909 (119905) (119905)]

minusint119905

119905minusℎ2

[119909119879 (119904) 119879 (119904)] [11987311 11987312lowast 11987322

] [119909 (119904) (119904)] 119889119904

3(119909 (119905)) = 119909119879 (119905) 119876

1119909 (119905) minus (1 minus 119889

1(119905)) 119909119879 (119905 minus 119889

1(119905))

times 1198761119909 (119905 minus 119889

1(119905)) + 119909119879 (119905) 119876

2119909 (119905)

minus (1 minus 1198892(119905)) 119909119879 (119905 minus 119889

2(119905)) 119876

2119909 (119905 minus 119889

2(119905))

le 119909119879 (119905) 1198761119909 (119905) + 119909119879 (119905) 119876

2119909 (119905) minus (1 minus 120583

1)

times 119909119879 (119905 minus 1198891(119905)) 119876

1119909 (119905 minus 119889

1(119905)) minus (1 minus 120583

2)

times 119909119879 (119905 minus 1198892(119905)) 119876

2119909 (119905 minus 119889

2(119905))

4(119909 (119905)) = 119909119879 (119905)119872

1119909 (119905) minus 119909119879 (119905 minus ℎ

1)1198721119909 (119905 minus ℎ

1)

+ 119909119879 (119905)1198722119909 (119905) minus 119909119879 (119905 minus ℎ

2)1198722119909 (119905 minus ℎ

2)

5(119909 (119905)) =

12ℎ21119879 (119905) 119877

1 (119905) minus int

0

minusℎ

int119905

119905+120579

119879 (119904)

times 1198771 (119904) 119889119904 119889120579 +

12ℎ22119879 (119905) 119877

2 (119905)

minus int0

minusℎ

int119905

119905+120579

119879 (119904) 1198772 (119904) 119889119904 119889120579

(20)

From (15) the following equation can be obtained for anymatrices 119879

119894 119894 = 1 2 8

2119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))

times [119909119879 (119905) 1198791+ 119909119879 (119905 minus 119889

1(119905)) 1198792+119909119879 (119905 minus ℎ

1) 1198793

+ (int119905

119905minusℎ1

119909 (119904) 119889119904)119879

1198794+ 119909119879 (119905 minus 119889

2(119905)) 1198795+ 119909119879

times (119905 minus ℎ2) 1198796+ (int119905

119905minusℎ2

119909 (119904) 119889119904)119879

1198797+ 119879 (119905) 119879

8]

times [1198601119894119909 (119905) minus 119860

2119894119909 (119905 minus 119889

1(119905))

minus119861119894119865119895119909 (119905 minus 119889

2(119905)) minus (119905)] = 0

(21)

Mathematical Problems in Engineering 5

By Lemma 3 we have

minus int119905

119905minusℎ1

[119909119905 (119904) 119879 (119904)] [11988511 11988512lowast 11988522

] [119909 (119904) (119904)] 119889119904

le minus1ℎ1

[(int119905

119905minusℎ1

119909 (119904) 119889119904)119879

119909119879 (119905) minus 119909119879 (119905 minus ℎ1)]

times [11988511 11988512lowast 11988522

][

[

int119905

119905minusℎ1

119909 (119904) 119889119904

119909 (119905) minus 119909 (119905 minus ℎ1)]

]

minus int119905

119905minusℎ2

[119909119879 (119904) 119879 (119904)] [11987311 11987312lowast 11987322

] [119909 (119904) (119904)] 119889119904

le minus1ℎ2

[(int119905

119905minusℎ2

119909 (119904) 119889119904)119879

119909119879 (119905) minus 119909119879 (119905 minus ℎ1)]

times [11987311 11987312lowast 11987322

][

[

int119905

119905minusℎ2

119909 (119904) 119889119904

119909 (119905) minus 119909 (119905 minus ℎ2)]

]

minus int0

minusℎ119894

int119905

119905+120579

119879 (119904) 119877119894 (119904) 119889119904 119889120579

le minus2ℎ119894

2(int0

minusℎ119894

int119905

119905+120579

(119904) 119889119904 119889120579)119879

119877119894(int0

minusℎ119894

int119905

119905+120579

(119904) 119889119904 119889120579)

= minus2ℎ119894

2(ℎ119894119909 (119905) minus int

119905

119905minusℎ119894

119909 (119904) 119889119904)119879

times 119877119894(ℎ119894119909 (119905) minus int

119905

119905minusℎ119894

119909 (119904) 119889119904) 119894 = 1 2

(22)

Using the above inequalities andwith the zero quantities (21)we can obtain

(119909119905)

le 2 [119909119879 (119905) (int119905

119905minusℎ1

119909 (119904) 119889119904)119879

] [11987511 11987512lowast 11987522

]

times [ (119905)119909 (119905) minus 119909 (119905 minus ℎ

1)] + ℎ1 [119909

119879 (119905) 119879 (119905)]

times [11988511 11988512lowast 11988522

] [119909 (119905) (119905)] + ℎ2 [119909119879 (119905) 119879 (119905)] [11987311 11987312lowast 119873

22

]

times [119909 (119905) (119905)] minus1ℎ1

[(int119905

119905minusℎ1

119909 (119904) 119889119904)119879

119909119879 (119905) minus 119909119879 (119905 minus ℎ1)]

times [11988511 11988512lowast 11988522

][

[

int119905

119905minusℎ1

119909 (119904) 119889119904

119909 (119905) minus 119909 (119905 minus ℎ1)]

]

minus1ℎ2

[(int119905

119905minusℎ2

119909 (119904) 119889119904)119879

119909119879 (119905) minus 119909119879 (119905 minus ℎ2)]

times [11987311 11987312lowast 11987322

][

[

int119905

119905minusℎ2

119909 (119904) 119889119904

119909 (119905) minus 119909 (119905 minus ℎ2)]

]+ 119909119879 (119905) 119876

1119909 (119905)

+ 119909119879 (119905) 1198762119909 (119905) minus (1 minus 120583

1) 119909119879 (119905 minus 119889

1(119905)) 119876

1119909

times (119905 minus 1198891(119905)) minus (1 minus 120583

2) 119909119879 (119905 minus 119889

2(119905)) 119876

2119909

times (119905 minus 1198892(119905)) + 119909119879 (119905)119872

1119909 (119905) minus 119909119879 (119905 minus ℎ

1)1198721119909

times (119905 minus ℎ1) + 119909119879 (119905)119872

2119909 (119905) minus 119909119879 (119905 minus ℎ

2)1198722119909

times (119905 minus ℎ2) +

12ℎ21119879 (119905) 119877

1 (119905) minus

2ℎ1

2

times(ℎ1119909 (119905)minusint

119905

119905minusℎ1

119909 (119904) 119889119904)119879

1198771(ℎ1119909 (119905)minusint

119905

119905minusℎ1

119909 (119904) 119889119904)

+12ℎ2

2119879 (119905) 1198772 (119905)minus

2ℎ22

(ℎ2119909 (119905)minusint

119905

119905minusℎ2

119909 (119904) 119889119904)119879

times1198772(ℎ2119909 (119905) minus int

119905

119905minusℎ2

119909 (119904) 119889119904)

+ 2 [119909119879 (119905) (int119905

119905minusℎ2

119909 (119904) 119889119904)119879

]

times [11988211 11988212lowast 11988222

] [ (119905)119909 (119905) minus 119909 (119905 minus ℎ

2)]

+ 2119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))

times [119909119879 (119905) 1198791+ 119909119879 (119905 minus 119889

1(119905)) 1198792+ 119909119879 (119905 minus ℎ

1) 1198793

+ (int119905

119905minusℎ1

119909 (119904) 119889119904)119879

1198794+ 119909119879 (119905 minus 119889

2(119905)) 1198795+ 119909119879

times (119905 minus ℎ2) 1198796+(int119905

119905minusℎ2

119909 (119904) 119889119904)119879

1198797+ 119879 (119905) 119879

8]

times [1198601119894119909 (119905) minus 119860

2119894119909 (119905 minus 119889

1(119905))

minus119861119894119865119895119909 (119905 minus 119889

2(119905)) minus (119905)]

= 120577119879 (119905)(119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))Φ

119894119895)120577 (119905)

(23)

6 Mathematical Problems in Engineering

where

120577119879 (119905) = [119909119879 (119905) 119909119879 (119905 minus 1198891(119905)) 119909119879 (119905 minus ℎ

1) (int

119905

119905minusℎ1

119909 (119904) 119889119904)119879

119909119879 (119905 minus 1198892(119905)) 119909119879 (119905 minus ℎ

2) (int

119905

119905minusℎ2

119909 (119904) 119889119904)119879

119879 (119905)] (24)

Φ119894119895=

[[[[[[

[

Φ11119894 Φ12119894 Φ13 Φ14 Φ15119894119895 Φ16 Φ17 Φ18

lowast Φ22119894 119860119879

2119894119879119879

3119860119879

2119894119879119879

4Φ25119894119895 119860

119879

2119894119879119879

6119860119879

2119894119879119879

7Φ28119894

lowast lowast Φ33119894 minus11987522 1198793119861119894119865119895 0 0 minus1198793

lowast lowast lowast Φ44 1198794119861119894119865119895 0 0 Φ48

lowast lowast lowast lowast Φ55119894119895 119865119879

119895119861119879

119894119879119879

6119865119879

119895119861119879

119894119879119879

7Φ58119894

lowast lowast lowast lowast lowast Φ66 Φ67 minus1198796

lowast lowast lowast lowast lowast lowast Φ77 Φ78

lowast lowast lowast lowast lowast lowast lowast Φ88

]]]]]]

]

Φ11119894= 11987512+ 11987511987912+11988212+11988211987912+ ℎ111988511+ ℎ211987311

minus1ℎ1

11988522minus1ℎ2

11987322+ 1198761+ 1198762+1198721+1198722

minus 21198771minus 21198772+ 11987911198601119894+ 11986011987911198941198791198791

Φ12119894= 11987911198602119894+ 11986011987911198941198791198792

Φ13= minus11987512minus1ℎ1

11988522+ 11986011987911198941198791198793

Φ14= 11987522minus1ℎ2

11988511987912+1ℎ1

1198771+ 11986011987911198941198791198794

Φ15119894119895

= 1198791119861119894119865119895+ 11986011987911198941198791198795

Φ16= minus119882

12minus1ℎ2

11987322+ 11986011987911198941198791198796

Φ17= 11988222minus1ℎ2

11987311987912+2ℎ22

1198772+ 11986011987911198941198791198797

Φ18= 11987511+11988211+ ℎ111988512+ ℎ211987312minus 1198791+ 11986011987911198941198791198798

Φ22119894= minus (1 minus 120583

1) 1198761+ 11987921198602119894+ 11986011987921198941198791198792

Φ25119894119895

= 1198792119861119894119865119895+ 11986011987921198941198791198795

Φ28119894= minus1198792+ 11986011987921198941198791198798

Φ33= minus

1ℎ11988522minus1198721

Φ44= minus

1ℎ1

11988511minus2ℎ21

1198771

Φ48= 11987511987912minus 1198794

Φ55119894= minus (1 minus 120583

2) 1198762+ 1198795119861119894119865119895+ 1198651198791198951198611198791198941198791198795

Φ58119894= minus1198795+ 1198651198791198951198611198791198941198791198798

Φ66= minus

1ℎ2

11987322minus1198722

Φ67= minus

1ℎ2

11987311987912minus11988222

Φ77= minus

1ℎ2

11987311minus2ℎ22

1198772

Φ78= 11988211987912minus 1198797

Φ88= ℎ111988522+ ℎ211987322+12ℎ211198771+12ℎ221198772minus 1198798

(25)

From (25) it is obvious that if

119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))Φ

119894119895lt 0 (26)

(119909(119905)) lt 0From (26) we can discover that the feedback gains

are predefined The following proof presents the controllerdesign method We first pre- and postmultiply both thediag [119883 119883 119883 119883 119883 119883 119883 119883 ] and transpose to both sides of (25)and pre- and postmultiply the diag [119883 119883 ] transpose to bothsides of 119875 119882 119885 and 119873 and the 119883 and transpose to bothsides of 119876

119894119872119894 and 119877

119894 119894 = 1 2 Let 119879

119894= 1199051198941198791(119894 = 2 8)

and denote new variables 119883 = 119879minus11 1198771= 119883119877

1119883119879 119876

1=

1198831198761119883119879119872

1= 119883119872

1119883119879 119877

2= 119883119877

2119883119879 119876

2= 119883119876

2119883119879 119872

2=

1198831198722119883119879 Λ

119894= 119883Λ

119894119883119879 Λ

119895= 119883Λ

119895119883119879 119894 = 1 2 119903 119875

11=

11988311987511119883119879 119875

12= 119883119875

12119883119879 119875

22= 119883119875

22119883119879 119882

11= 119883119882

11119883119879

11988212= 119883119882

12119883119879 119882

22= 119883119882

22119883119879 119885

11= 119883119885

11119883119879 119885

12=

11988311988512119883119879 119885

22= 119883119885

22119883119879 119873

11= 119883119873

11119883119879 119873

12= 119883119873

12119883119879

and11987322= 119883119873

22119883119879 With 119865

119894= 119884119894119883minus119879 119894 = 1 2 119903we next

get

119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))Φ

119894119895lt 0 (27)

Mathematical Problems in Engineering 7

where

Φ119894119895=

[[[[[[[[

[

Φ11119894 Φ12119894 Φ13 Φ14 Φ15119894119895 Φ16 Φ17 Φ18

lowast Φ22119894 1199053119883119860119879

21198941199054119883119860119879

2119894Φ25119894119895 1199056119883119860

119879

21198941199057119883119860119879

2119894Φ28119894

lowast lowast Φ33119894 minus11987522 1199053119861119894119884119895 0 0 minus1199053119883119879

lowast lowast lowast Φ44 1199054119861119894119884119895 0 0 Φ48

lowast lowast lowast lowast Φ55119894119895 1199056119884119879

119895119861119879

1198941199057119884119879

119895119861119879

119894Φ58119894

lowast lowast lowast lowast lowast Φ66 Φ67 minus1199056119883119879

lowast lowast lowast lowast lowast lowast Φ77 Φ78

lowast lowast lowast lowast lowast lowast lowast Φ88

]]]]]]]]

]

Φ11119894= 11987512+ 119875119879

12+11988212+119882119879

12+ ℎ111988511

+ ℎ211987311minus1ℎ1

11988522minus1ℎ2

11987322+ 1198761+ 1198762

+1198721+1198722minus 21198771minus 21198772+ 1198601119894119883119879 + 119883119860119879

1119894

Φ12119894= 1198602119894119883119879 + 119905

21198831198601198791119894

Φ13= minus11987512minus1ℎ 111988522+ 11990531198831198601198791119894

Φ14= 11987522minus1ℎ1

119885119879

12+1ℎ1

1198771+ 11990541198831198601198791119894

Φ15119894119895

= 119861119894119884119895+ 11990551198831198601198791119894

Φ16= minus119882

12minus1ℎ2

11987322+ 11990561198831198601198791119894

Φ17= 11988222minus1ℎ2

119873119879

12+2ℎ22

1198772+ 11990571198831198601198791119894

Φ18= 11987511+11988211+ ℎ111988512+ ℎ211987312minus 119883119879 + 119905

81198831198601198791119894

Φ22119894= minus (1 minus 120583

1) 1198761+ 11990521198602119894119883119879 + 119905

21198831198601198792119894

Φ25119894119895

= 1199052119861119894119884119895+ 11990551198831198601198792119894

Φ28119894= minus1199052119883119879 + 119905

81198831198601198792119894

Φ33= minus

1ℎ1

11988522minus1198721

Φ44= minus

1ℎ1

11988511minus2ℎ21

1198771

Φ48= 119875119879

12minus 1199054119883119879

Φ55119894= minus (1 minus 120583

2) 1198762+ 1199055119861119894119884119895+ 1199055119884119879119895119861119879119894

Φ58119894= minus1199055119883119879 + 119905

8119884119879119895119861119879119894

Φ66= minus

1ℎ2

11987322minus1198722

Φ67= minus

1ℎ2

119873119879

12minus11988222

Φ77= minus

1ℎ2

11987311minus2ℎ22

1198772

Φ78= 119882119879

12minus 1199057119883119879

Φ88= ℎ111988522+ ℎ211987322+12ℎ211198771+12ℎ221198772minus 1199058119883119879

(28)

Here ldquolowastrdquo denotes the transpose elements in the symmetricpositions

If (27) holds then (119909(119905)) lt 0 Considersum119903119894=1sum119903119895=1

119908119894(119909(119905))(119908

119895(119909(119905)) minus 119898

119895(119909(119905 minus 119889

2(119905))))Λ

119894= 0

where Λ119894= Λ119879119894isin 1198778119899times8119899 gt 0 119894 = 1 2 119903 are arbitrary

matrices These terms are introduced to (27) to alleviate theconservativeness From (27) we also have

Φ =119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))Φ

119894119895

=119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905)) (119908

119895(119909 (119905)) minus 119898

119895(119909 (119905 minus 119889

2(119905)))

+120588119895119908119895(119909 (119905)) minus 120588

119895119908119895(119909 (119905))) Λ

119894

+119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))Φ

119894119895

=119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905)) 119908

119895(119909 (119905)) (120588

119895Φ119894119895minus 120588119895Λ119894+ Λ119894)

+119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905)) (119898

119895(119909 (119905 minus 119889

2(119905))) minus 120588

119895119908119895(119909 (119905)))

times (Φ119894119895minus Λ119894)

le119903

sum119894=1

1199082119894(120588119894Φ119894119894minus 120588119894Λ119894+ Λ119894)

+119903

sum119894=1

119903

sum119895=1

119908119894(119898119895minus 120588119895119908119895) (Φ119894119895minus Λ119894)

+119903

sum119894=1

sum119894lt119895

119908119894119908119895(120588119895Φ119894119895+ 120588119894Φ119895119894minus 120588119895Λ119894minus 120588119894Λ119895

+Λ119894+ Λ119895)

(29)

with119898119895(119909(119905 minus 119889

2(119905))) minus 120588

119895119908119895(119909(119905)) ge 0 for all 119895 119909(119905) and 119909(119905 minus

1198892(119905)) Let

Φ119894119895minus Λ119894lt 0 (30)

120588119894Φ119894119894minus 120588119894Λ119894+ Λ119894lt 0 (31)

120588119895Φ119894119895+ 120588119894Φ119895119894minus 120588119895Λ119894minus 120588119894Λ119895+ Λ119894+ Λ119895le 0 119894 lt 119895 (32)

for all 119894 119895 = 1 2 119903

8 Mathematical Problems in Engineering

Since (119909(119905)) lt 0 the fuzzy control system with the timevarying state and input delays (15) is asymptotically stablewith the state feedback control law (17)

32 Robust Stability of Uncertain Fuzzy Systems We alsoexamine the design of a robust stable controller for theuncertain system (4) under the imperfect premise matchingConsider the following uncertain fuzzy time varying delaycontrol system

(119905) =119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))

times [(1198601119894+ Δ1198601119894(119905)) 119909 (119905) + (119860

2119894+ Δ1198602119894(119905))

times 119909 (119905 minus 1198891(119905)) + (119861

119894+ Δ119861119894(119905))

times119906 (119905 minus 1198892(119905))]

119909 (119905) = 120593 (119905) 119905 isin [minusmax (ℎ1 ℎ2) 0]

(33)

Based on the above results of Theorem 4 a robust stabiliza-tion criterion for theT-S fuzzy systemswith time varying stateand input delays is investigated The following result can beobtained

Theorem 5 Given scalars ℎ1ge 0 ℎ

2ge 0 120583

1 1205832 and 119905

119894(119894 =

2 8) the uncertain fuzzy control systems with the timevarying state and input delays (33) is robustly stable for any0 le 119889

119894(119905) le ℎ

119894(119894 = 1 2) via the imperfect premise matching

controller design technique if the membership functions of thefuzzy model and fuzzy controller satisfy 119898

119895(119909(119905 minus 119889

2(119905))) minus

120588119895119908119895(119909(119905)) ge 0 for all 119895 119909(119905) and 119909(119905 minus 119889

2(119905)) where 0 lt

120588119895lt 1 and there exist common matrices 119875

11gt 0 119875

22gt 0

11988211gt 0 119882

22gt 0 119885

11gt 0 119885

22gt 0 119873

11gt 0 119873

22gt 0

119876119894gt 0 119872

119894gt 0 119877

119894gt 0 (119894 = 1 2) and Λ

119894= ΛT119894isin 1198778119899times8119899 gt

0 (i = 1 2 r) 119875 gt 0 119882 gt 0 119885 gt 0 119873 gt 0 and somematrices 119884

119895(119895 = 1 2 119903) 119883 and scalars 120576

1119894gt 0 120576

2119894gt 0

120576119887119894119895gt 0 satisfy the following LMIs

Ψ119894119895minus Λ119894lt 0 (34)

120588119894Ψ119894119894minus 120588119894Λ119894+ Λ119894lt 0 (35)

120588119895Ψ119894119895+ 120588119894Ψ119895119894minus 120588119895Λ119894minus 120588119894Λ119895+ Λ119894+ Λ119895 le 0 119894 lt 119895 (36)

where Ψ119894119895is defined as in (44) The state feedback gains can be

constructed as 119865119894= 119884119894119883minus119879

Proof If 1198601119894 1198602119894 and 119861

119894in Φ119894119895are replaced with 119860

1119894+

119863119894119870119894(119905)1198641119894 1198602119894+ 119863119894119870119894(119905)1198642119894 and 119861

119894+ 119863119894119870119894(119905)119864119887119894in (28) of

Theorem 4 respectively (27) for system (33) can be rewrittenas

119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889 (119905))) Ω

119894119895lt 0 (37)

where

Ω119894119895= Φ119894119895+ 1198631198791119894119870119894(119905) 119864119894+ 119864119879119894119870119879119894(119905) 1198631119894+ 1198631198792119894119870119894(119905) 1198642119894119895

+ 1198641198792119894119895119870119879119894(119905) 1198632119894+ 119863119879119887119894119870119894(119905) 119864119887119894119895+ 119864119879119887119894119895119870119879119894(119905) 119863119887119894

1198631119894= [1198631198791198940 0 0 0 0 0 0]

1198632119894= [0 119863119879

1198940 0 0 0 0 0]

119863119887119894= [0 0 0 0 119863119879

1198940 0 0]

(38)

1198641119894= [1198641119894119883119879 11990521198641119894119883119879 11990531198641119894119883119879 11990541198641119894119883119879 119905

51198641119894119883119879 11990561198641119894119883119879 11990571198641119894119883119879 11990581198641119894119883119879]

1198642119894= [1198642119894119883119879 11990521198642119894119883119879 11990531198642119894119883119879 11990541198642119894119883119879 11990551198642119894119883119879 11990561198642119894119883119879 11990571198642119894119883119879 11990581198642119894119883119879]

119864119887119894119895= [119864119887119894119884119895 1199052119864119887119894119884119895 1199053119864119887119894119884119895 1199054119864119887119894119884119895 1199055119864119887119894119884119895 1199056119864119887119894119884119895 1199057119864119887119894119884119895 1199058119864119887119894119884119895]

(39)

If Ω119894119895

lt 0 (37) holds According to Lemma 2 it isstraightforward to know that Ω

119894119895lt 0 is true if for each 119894

119895 there exists scalars 1205761119894gt 0 120576

2119894gt 0 and 120576

119887119894119895gt 0 such that

the following inequality holds

Φ119894119895+ 120576111989411986311987911198941198631119894+ 120576minus1111989411986411987911198941198641119894+ 120576211989411986311987921198941198632119894

+ 120576minus1211989411986411987921198941198642119894+ 120576119887119894119895119863119879119887119894119863119887119894+ 120576minus1119887119894119895119864119879119887119894119895119864119887119894119895lt 0

(40)

Based on Schur complement (40) is equivalent to thefollowing inequality

Ψ119894119895=[[[

[

Φ119894119895

1198641198791119894

1198641198792119894

119864119879119887119894119895

lowast minus1205761119894119868 0 0

lowast lowast minus1205762119894119868 0

lowast lowast lowast minus120576119887119894119895119868

]]]

]

lt 0 (41)

Mathematical Problems in Engineering 9

where

Φ119894119895=

[[[[[[[[

[

Φ11119894 Φ12119894 Φ13 Φ14 Φ15119894119895 Φ16 Φ17 Φ18

lowast Φ22119894 1199053119883119860119879

21198941199054119883119860119879

2119894Φ25119894119895 1199056119883119860

119879

21198941199057119883119860119879

2119894Φ28119894

lowast lowast Φ33119894 minus11987522 1199053119861119894119884119895 0 0 minus1199053119883119879

lowast lowast lowast Φ44 1199054119861119894119884119895 0 0 Φ48

lowast lowast lowast lowast Φ55119894119895 1199056119884119879

119895119861119879

1198941199057119884119879

119895119861119879

119894Φ58119894

lowast lowast lowast lowast lowast Φ66 Φ67 minus1199056119883119879

lowast lowast lowast lowast lowast lowast Φ77 Φ78

lowast lowast lowast lowast lowast lowast lowast Φ88

]]]]]]]]

]

lt 0

Φ11119894= Φ11119894+ 1205761119894119863119894119863119879119894

Φ22119894= Φ22119894+ 1205762119894119863119894119863119879119894

Φ55119894119895

= Φ55119894119895

+ 120576119887119894119863119894119863119879119894

(42)

and Φ are the same as in Theorem 4 Therefore (41) holds ifand only if the following inequality holds

119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889 (119905))) Ψ

119894119895lt 0 (43)

To alleviate the conservativeness of our robust stabilityanalysis similar to the analysis approach of Theorem 4 weintroduce sum119903

119894=1sum119903119895=1

119908119894(119908119895minus 119898119895)Λ119894= 0 to (43) and the

following proof is the same as in Theorem 4 We can obtainthe conditions and (34)ndash(36) Thus the fuzzy control systemwith the time varying state and input delays (34) is robuststable on the basis of the control law 119865

119894= 119884119894119883minus119879under these

conditions and (34)ndash(36) hold

Remark 6 If we set 119908119894(119909(119905)) = 119898

119895(119909(119905)) 119889

1(119905) = 119889

2(119905)

the closed-loop system in this paper has the same structureas that of paper [26] On the other hand let 119908

119894(119909(119905)) =

119898119895(119909(119905)) and 119889

1(119905) = 119889

2(119905) = 119889 we can obtain the system

in [16] Therefore the system studied in this paper is moregeneral Additionally the information of the membershipfunctions of the fuzzy time delay models and controllers areall considered in the stability analysis If the fuzzymodels andfuzzy controllers share the same fuzzymembership functionsthat is 119908

119894(119909(119905)) = 119898

119895(119909(119905)) the stability analysis can be

referred to [25 27 28] In other words the problem in [2527 28] is actually a special case of the one investigated in thispaper

Remark 7 The augmented Lyapunov function with an addi-tional triple-integral term is introduced in analyzing thestability problem for the T-S fuzzy systems with time varyingstate and input delays In addition two integral inequalitiesare used to derive Theorem 4 and less free-weighting matri-ces are introduced which lead to get less conservative resultsIt can be observed that some existing results are the specialcases of this paper For example the method in [25] is from(18) with 119875

12= 0 119875

22= 0 119885

12= 11988522= 0 119873

12= 11987322= 0

and 1198771= 1198772= 0 in Theorem 4 The results in [22] are based

on system (33) with 1198892(119905) = 0 and (18) with119872

2= 0 119885

12=

11988522= 0 119876

1= 1198762= 0 and 119877

1= 1198772= 0

Remark 8 Different from the general PDC technique thedesign method under the imperfect premise matching ismuch more flexible because the membership functions ofthe fuzzy controllers do not need to be chosen the same asthose of the fuzzy time delay models Instead they can bedesigned arbitrarilyThus the design flexibility is significantlyenhanced On the other hand some simple membershipfunctions of the fuzzy controllers might be employed whichcan reduce the implementation cost

Remark 9 In [7ndash11 15ndash20] the authors have considered therobust control for uncertain T-S fuzzy systems with statedelays Some delay-dependent conditions and fuzzy con-trollers are obtained with a traditional Lyapunov-Krasovskiifunctionalmethod andPDCmethod which all are the specialcases of Theorem 5 in this paper

4 Numerical Examples

In this section two numerical examples are given to illus-trate the conservativeness and effectiveness of the proposedmethods The first example compares our techniques withthe existing ones in the literature for stability analysis whichshows that Theorem 4 in this paper is less conservative thanthe other results The second example is used to illustrate theadvantage of the robust stability conditions ofTheorem 5 anddemonstrate how to design robust fuzzy controllers by usingour approach

Example 10 Consider the following fuzzy system with stateand input delays

(119905) =2

sum119894=1

119908119894(120579 (119905)) [119860

1119894119909 (119905) + 119860

2119894119909 (119905 minus 119889

1(119905))

+119861119894119906 (119905 minus 119889

2(119905))]

11986011= [0 06

0 1 ] 11986021= [05 09

0 2 ]

11986012= [1 0

1 0] 11986022= [09 0

1 16]

1198611= 1198612= [11]

(44)

The fuzzy membership functions are selected as [31]

1199081(1199091(119905)) = (1 minus

119888 (119905) sin (10038161003816100381610038161199091 (119905)1003816100381610038161003816minus4)5

1 + expminus1001199091(119905)3(1minus1199091(119905)))

timescos (119909

1(119905))2

1 + expminus251199091(119905)(3+(1199091(119905)042))

times 1199082(1199091(119905)) = 1 minus 119908

1(1199091(119905))

(45)

10 Mathematical Problems in Engineering

Table 1 The maximum allowable time delay and feedback gains(1205831= 0 120583

2= 0)

Paper ℎ1

ℎ2 Feedback gains

[26] 03120 031201198651= [10598 minus56598]

1198652= [minus13068 minus41167]

Theorem 4 033 0571198651= [minus00042 minus00449]1198652= [00048 00717]

where

1199091(119905) isin [minus

1205872

1205872]

119888 (119905) =sin (119909

1(119905)) + 140

isin [minus005 005] (46)

If we set 1198891(119905) = 119889

2(119905) in system (44) we can obtain the

system in [26] which implies that our system ismore generalEmploying the LMIs in [26] and those inTheorem 4yields themaximum state and input delays ℎ

1 and ℎ

2 which guarantee

the stability of system (44) and the feedback gains for 1205831= 0

and 1205832= 0 when 120588

1= 075 120588

2= 095 119905

2= 01 119905

3=

02 1199054= 07 119905

5= 01 119905

6= 04 119905

7= 01 and 119905

8= 13 as

given in Table 1 which clearly shows the superiority of theresults derived in this paper over those obtained from [26]

Example 11 Consider the well-known truck-trailer systemwhich can be described by the following T-S fuzzy system

(119905) =2

sum119894=1

119908119894(120579 (119905)) [(119860

1119894+ Δ1198601119894) 119909 (119905) + 119860

2119894119909 (119905 minus 119889

1(119905))

+ (119861119894+ Δ119861119894) 119906 (119905 minus 119889

2(119905))]

(47)

11986011=

[[[[[[[[[

[

minus119886V1199051198711199050

0 0

119886V1199051198711199050

0 0

119886V21199052

21198711199050

V1199051199050

0

]]]]]]]]]

]

11986021=

[[[[[[[[[

[

minus (1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)V21199052

21198711199050

0 0

]]]]]]]]]

]

1198611=[[[[

[

V1199051198971199050

0

0

]]]]

]

11986012=

[[[[[[[[[

[

minus119886V1199051198711199050

0 0

119886V1199051198711199050

0 0

119886119889V21199052

21198711199050

119889V1199051199050

0

]]]]]]]]]

]

11986022=

[[[[[[[[[

[

minus (1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)119889V21199052

21198711199050

0 0

]]]]]]]]]

]

(48)

1198612=[[[[

[

V1199051198971199050

0

0

]]]]

]

(49)

where 119897 = 28 119871 = 55 V = minus10 119905 = 20 1199050= 05

119889 = 101199050120587 119863

119894= [02555 02555 02555]119879 119864

1119894= 1198642119894=

[01 0 0]119879 119864119887119894

= [01 0 0]119879 (119894 = 1 2) and the fuzzymembership functions are selected the same as inExample 10

(1) Let 119886 = 1 system (47) is the same as the fuzzy systemof [22] which implies that our system ismore generalTable 2 gives themaximum input delay value of ℎ

2 for

which the stabilization is guaranteed byTheorem 5 aswell as the state feedback gains when 120588

1= 075 120588

2=

095 1199052= 01 119905

3= 02 119905

4= 07 119905

5= 01 119905

6=

04 1199057= 08 and 119905

8= 2 It is clearly visible that the

results derived in this paper are better than those from[22 28]

(2) In the case of 119886 = 1 the state and input delays of system(47) are all exist The approach proposed in [22] cannot be used to get the maximum of state delay ℎ

1

as the system in [22] only contains the input delayHowever the method in [28] and Theorem 5 can beused Here let 119886 = 07 120588

1= 075 120588

2= 095 119905

2=

01 1199053= 02 119905

4= 07 119905

5= 01 119905

6= 04 119905

7=

08 and 1199058= 3 and Table 3 provides the maximum

upper bounds of ℎ1and ℎ

2and the state feedback

gains for which the robust stabilization is guaranteedbyTheorem 5

By Theorem 5 we can conclude that the uncertain fuzzymodel (47) is robust stable based on the following fuzzycontrol law

119906 (119905 minus ℎ2) =2

sum119895=1

119898119895(1199091(119905 minus ℎ

2)) 119865119895119909 (119905 minus ℎ

2) (50)

With the PDC design technique it is required that thefuzzy controller must share the same fuzzy membershipfunctions as those of the fuzzy time delay model Under such

Mathematical Problems in Engineering 11

Table 2 The maximum allowable time delay and feedback gains(1205832= 0)

Paper ℎ2 Feedback gains

[22] 0751198651= [34227 minus03535 00045]

1198652= [35215 minus03617 00056]

[28] 0861198651= [33219 minus02406 00025]

1198652= [33272 minus02494 00026]

Theorem 5 121198651= [minus00212 00093 minus00024]1198652= [minus00250 00044 00014]

Table 3 The maximum allowable time delay and feedback gains(1205831= 1205832= 0)

Paper ℎ1

ℎ2 Feedback gains

[28] 01 0551198651= [33219 minus02406 00025]

1198652= [33272 minus02494 00026]

Theorem 5 04 171198651= [minus00128 00090 minus00017]1198652= [minus00194 00012 00009]

a condition the implementation cost of the fuzzy controlleris high by employing these complex membership functionsHowever fromTheorem 5 the membership functions do notneed to be chosen the same as119908

1(1199091(119905)) and119908

2(1199091(119905)) Thus

we can select some simple membership functions insteadsuch as

1198981(1199091(119905 minus ℎ

2)) = 075 exp

(minus1199091(119905 minus ℎ

2) minus 038)2

2 times 0382+ 005

1198982(1199091(119905 minus ℎ

2)) = 1 minus 119898

1(1199091(119905 minus ℎ

2))

(51)

We also assume that 1205881= 075 and 120588

2= 095 such that

119898119895(1199091(119905 minus ℎ

2)) minus 120588

119895119908119895(1199091(119905)) gt 0 for 119895 = 1 2 and 119909

1(119905 minus ℎ

2)

The fuzzy control law 119906(119905 minus ℎ2) = 119898

1[minus00128 00090 minus

00017]119909 + 1198982[minus00194 00012 00009]119909 can be employed

to stabilize the uncertain fuzzy control systems in (47) withthe state and input delays Figures 1 and 2 illustrate the stablestate response and control input under the initial conditionof 119909(0) = [4 minus 1 2]119879

Remark 12 In Example 11 our method offers less conser-vative results in the sense of allowing longer time delayand obtaining smaller feedback control gains Moreovercompared with [22ndash28] where the fuzzy controllermust havethe same fuzzy membership functions as those of the fuzzytime delay model the above membership functions of ourfuzzy controller are much simper which can considerablylower the implementation cost of the fuzzy controller andenhance the design flexibility

Remark 13 If the function of 119888(119905) in 1199081(1199091(119905)) is unknown

on the basis of the PDC design technique [7ndash11 15ndash20 22ndash28] the fuzzy controller cannot be implemented as 119888(119905) isnot available However using the proposed design method

0 5 10 15

0

2

4

6Closed-loop response

minus12

minus10

minus8

minus6

minus4

minus2

119905 (s)

119909(119905)

1199092

11990931199091

Figure 1The state responses of system (47) with ℎ1= 04 ℎ

2= 17

0 5 10 15

0

5

10

15

20

25

30

119905 (s)

minus20

minus15

minus10

minus5

119906(119905)

Figure 2 The control input of system (47) with ℎ1= 04 ℎ

2= 17

we can select some simper and well-known membershipfunctions that are different from those of the fuzzy time delaymodels for example119898

1(1199091(119905)) and119898

2(1199091(119905)) so as to realize

the fuzzy controller In other words the proposed designapproach can be applied even for the fuzzy time delay modelswith uncertain grades of membership

5 Conclusions

In this paper we study the robust stability and stabilizationproblems for general nonlinear fuzzy systems with timevarying state and input delays A less conservative delay-dependent robust stability criterion has been obtained byintroducing a novel augmented Lyapunov function withan additional triple-integral term Moreover a new design

12 Mathematical Problems in Engineering

approach different from the traditional PDC design tech-nique is proposed which can significantly improve the designflexibility and reduce the implementation cost of the fuzzycontroller by arbitrarily selecting simple fuzzy membershipfunctions Two numerical examples are used to illustrate theconservativeness and effectiveness of our proposed methods

Acknowledgments

This research work was funded by the NSFC under Grantno 61273095 and the Academy of Finland under Grants no135225 and no 127299

References

[1] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[2] W H Ho S H Chen I T Chen J H Chou and C C ShuldquoDesign of stable and quadratic-optimal static output feed-back controllers for T-S-fuzzy-model-based control systems anintegrative computational approachrdquo International Journal ofInnovative Computing Information and Control vol 8 no 1App 403ndash418 2012

[3] X J Su L G Wu P Shi and Y D Song ldquo119867infinmodel reduction

of T-S fuzzy stochastic systemsrdquo IEEE Transactions on SystemsMan and Cybermetics B vol 42 no 6 pp 1574ndash1585 2012

[4] R Yang P Shi G-P Liu andH Gao ldquoNetwork-based feedbackcontrol for systems with mixed delays based on quantizationand dropout compensationrdquo Automatica vol 47 no 12 pp2805ndash2809 2011

[5] XM Zhang Z J Zhang andG P Lu ldquoFault detection for state-delay fuzzy systems subject to random communication delayrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 4 pp 2439ndash2451 2012

[6] F B Li and X Zhang ldquoDelay-range-dependent robust 119867infin

filtering for singuar LPV systems with time variant delayrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 1 pp 339ndash353 2013

[7] Y Y Cao and P M Frank ldquoAnalysis and synthesis of nonlineartime-delay systems via fuzzy control approachrdquo IEEE Transac-tions on Fuzzy Systems vol 8 no 2 pp 200ndash211 2000

[8] Y-Y Cao and P M Frank ldquoStability analysis and synthesis ofnonlinear time-delay systems via linear Takagi-Sugeno fuzzymodelsrdquo Fuzzy Sets and Systems vol 124 no 2 pp 213ndash2292001

[9] O M Kwon and J H Park ldquoDelay-range-dependent stabiliza-tion of uncertain dynamic systems with interval time-varyingdelaysrdquo Applied Mathematics and Computation vol 208 no 1pp 58ndash68 2009

[10] E Tian D Yue and Y Zhang ldquoDelay-dependent robust 119867infin

control for T-S fuzzy system with interval time-varying delayrdquoFuzzy Sets and Systems vol 160 no 12 pp 1708ndash1719 2009

[11] J Y An T Li G LWen and R Li ldquoNew stability conditions foruncertain T-S fuzzy systems with interval time-varying delayrdquoInternational Journal of Control Automation and Systems vol10 no 3 pp 490ndash497 2012

[12] Z J Zhang X L Huang X J Ban and X Z Gao ldquoStabilityanalysis and controller design of T-S fuzzy systems with time-delay under imperfect premise matchingrdquo Journal of BeijingInstitute of Technology vol 21 no 3 pp 387ndash393 2012

[13] O M Kwon M J Park S M Lee and J H Park ldquoAugmentedLyapunov-Krasovskii functional approaches to robust stabilitycriteria for uncertain Takagi-Sugeno fuzzy systems with time-varying delaysrdquo Fuzzy Sets and Systems vol 201 no 16 pp 1ndash192012

[14] F Liu M Wu Y He and R Yokoyama ldquoNew delay-dependentstability criteria for T-S fuzzy systems with time-varying delayrdquoFuzzy Sets and Systems vol 161 no 15 pp 2033ndash2042 2010

[15] B Chen X Liu and S Tong ldquoNew delay-dependent stabiliza-tion conditions of T-S fuzzy systems with constant delayrdquo FuzzySets and Systems vol 158 no 20 pp 2209ndash2224 2007

[16] J Yoneyama ldquoRobust stability and stabilization for uncertainTakagi-Sugeno fuzzy time-delay systemsrdquo Fuzzy Sets and Sys-tems vol 158 no 2 pp 115ndash134 2007

[17] C Lin Q-G Wang and T H Lee ldquoDelay-dependent LMIconditions for stability and stabilization of T-S fuzzy systemswith bounded time-delayrdquo Fuzzy Sets and Systems vol 157 no9 pp 1229ndash1247 2006

[18] X J Su P Shi L G Wu and Y D Song ldquoA novel approachto filter design for T-S fuzzy discrete-time systems with time-varying delayrdquo IEEE Transactions on Fuzzy Systems vol 20 no6 pp 1114ndash1129 2012

[19] C Peng and Q-L Han ldquoDelay-range-dependent robust stabi-lization for uncertain T-S fuzzy control systems with intervaltime-varying delaysrdquo Information Sciences vol 181 no 19 pp4287ndash4299 2011

[20] S T Li Y W Jing and X M Liu ldquoRobust control for a class ofT-S fuzzy systemswith interval time-varying delayrdquo InternatinalJournal of Control Automation and Systems vol 10 no 4 pp737ndash743 2012

[21] X-M Zhang M Wu J-H She and Y He ldquoDelay-dependentstabilization of linear systems with time-varying state and inputdelaysrdquo Automatica vol 41 no 8 pp 1405ndash1412 2005

[22] B Chen X Liu and S Tong ldquoRobust fuzzy control of nonlinearsystems with input delayrdquo Chaos Solitons amp Fractals vol 37 no3 pp 894ndash901 2008

[23] C Lin Q-G Wang and T H Lee ldquoDelay-dependent LMIconditions for stability and stabilization of T-S fuzzy systemswith bounded time-delayrdquo Fuzzy Sets and Systems vol 157 no9 pp 1229ndash1247 2006

[24] H J Lee J B Park and Y H Joo ldquoRobust control for uncertainTakagi-Sugeno fuzzy systems with time-varying input delayrdquoJournal of Dynamic Systems Measurement and Control Trans-actions of the ASME vol 127 no 2 pp 302ndash306 2005

[25] C-H Lien and K-W Yu ldquoRobust control for Takagi-Sugenofuzzy systems with time-varying state and input delaysrdquo ChaosSolitons and Fractals vol 35 no 5 pp 1003ndash1008 2008

[26] Y K Su B Chen S Y Zhang and C Lin ldquoDelay-dependentstabilization of T-S fuzzy systems with time delayrdquo Control andDecision Kongzhi yu Juece vol 24 no 6 pp 921ndash927 2009

[27] B Chen X Liu C Lin and K Liu ldquoRobust 119867infin

control ofTakagi-Sugeno fuzzy systems with state and input time delaysrdquoFuzzy Sets and Systems vol 160 no 4 pp 403ndash422 2009

[28] L Li and X Liu ldquoNew results on delay-dependent robuststability criteria of uncertain fuzzy systems with state and inputdelaysrdquo Information Sciences vol 179 no 8 pp 1134ndash1148 2009

[29] Z J Zhang X J Ban X L Huang and X Z Gao ldquoNew delay-dependent robust stablity for T-S fuzzy systems with intervevaltime-varying delayrdquo in Proccedings of the 5th International JointConference on Computational Science and Optimization pp826ndash830 Harbin China 2012

Mathematical Problems in Engineering 13

[30] H K Lam and F H F Leung ldquoLMI-based stability andperformance design of fuzzy control systems fuzzy modelsand controllers with different premisesrdquo in Proceedings of theInternational Conference on Fuzzy Systerm pp 9599ndash9506Vancouver Canada 2006

[31] H K Lam andMNarimani ldquoStability analysis and perfomancedesign for fuzzy-model-based control system under imperfectpremise matchingrdquo IEEE Transactions on Fuzzy Systems vol 17no 4 pp 949ndash961 2009

[32] S Boyd L E Ghaoui and E Feron Linear Matrix Inequality inSystems and ControlTheory SIAM Philadelphia Pa USA 1994

[33] I R Petersen and C V Hollot ldquoA Riccati equation approach tothe stabilization of uncertain linear systemsrdquo Automatica vol22 no 4 pp 397ndash411 1986

[34] J Sun G P Liu and J Chen ldquoDelay-dependent stabilityand stabilization of neutral time-delay systemsrdquo InternationalJournal of Robust andNonlinear Control vol 19 no 12 pp 1364ndash1375 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

where119903

sum119895=1

119898119895(119909 (119905)) = 1 119898

119895(119909 (119905)) ge 0

119898119895(119909 (119905)) =

V119895(119909 (119905))

sum119903119895=1

V119895(119909 (119905))

V119895(119909 (119905)) =

119902

prod120573=1

V119873119895

120573

(119892120573(119909 (119905)))

(8)

119898119895(119909(119905)) is the normalized grade of membership function

that is a nonlinear function of 119909(119905) V119873119895

120573

(119892120573(119909(119905))) is the grade

of the membership corresponding to the fuzzy term of119873119895120573

As the time varying delay is included in the control inputwe have

119906 (119905 minus 1198892(119905)) =

119903

sum119895=1

119898119895(119909 (119905 minus 119889

2(119905))) 119865

119895119909 (119905 minus 119889

2(119905)) (9)

where119903

sum119895=1

119898119895(119909 (119905 minus 119889

2(119905))) = 1 119898

119895(119909 (119905 minus 119889

2(119905))) ge 0 (10)

Lemma 1 (see [32] (Schur complement)) Given constantmatrices Ω

1 Ω2 and Ω

3 where Ω

1= Ω1198791and Ω

2= Ω1198792 there

is Ω1+ Ω1198793Ωminus12Ω3lt 0 if and only if

[Ω1 Ω1198793

lowast minusΩ2

] lt 0 or [minusΩ2 Ω3

lowast minusΩ1

] lt 0 (11)

Lemma 2 (see [33]) Let Q = Q119879 D E and K(119905) satisfiesK119879(119905)K(119905) le I the following inequality holds

119876 + 119863119870 (119905) 119864 + 119864119879119870119879 (119905) 119863119879 lt 0 (12)

if and only if the following inequality holds for any smaller 120576 gt0

119876 + 120576minus1119863119863119879 + 120576119864119879119864 lt 0 (13)

Lemma 3 (see [34]) For any constant matrix Σ = Σ119879 gt 0and a scalar 120591 gt 0 such that the following integrations are welldefined we have

minus int119905

119905minus120591

119909119879 (119904) Σ119909 (119904) 119889119904

le minus1120591(int119905

119905minus120591

119909 (119904) 119889119904)119879

Σ(int119905

119905minus120591

119909 (119904) 119889119904)

minus int0

minus120591

int119905

119905+120579

119909119879 (119904) Σ119909 (119904) 119889119904 119889120579

le minus11205912(int0

minus120591

int119905

119905+120579

119909 (119904) 119889119904 119889120579)119879

Σ(int0

minus120591

int119905

119905+120579

119909 (119904) 119889119904 119889120579)

(14)

3 Main Results

In this section some new delay-dependent criteria for theT-S fuzzy systems with state and input delays are proposedby introducing a novel Lyapunov function with an additionaltriple-integral term under the imperfect premise matchingMoreover a robust stabilization criterion is also investigated

31 Stability of Nominal Fuzzy Systems Firstly we considerthe control design of a state feedback control law underthe imperfect premise matching that stabilizes the followingnominal fuzzy time varying delay system

(119905) =119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))

times [1198601119894119909 (119905) + 119860

2119894119909 (119905 minus 119889

1(119905)) + 119861

119894119865119895119909

times (119905 minus 1198892(119905))]

119909 (119905) = 120593 (119905) 119905 isin [minusmax (ℎ1 ℎ2) 0]

(15)

Theorem 4 Given scalars ℎ1

ge 0 ℎ2

ge 0 1205831 1205832 and

119905119894(119894 = 2 8) the closed-loop system (15) is asymptotically

stable for any 0 le 119889119894(119905) le ℎ

119894(119894 = 1 2) via the

imperfect premise matching controller design technique if themembership functions of the fuzzy model and fuzzy controllersatisfy 119898

119895(119909(119905 minus 119889

2(119905))) minus 120588

119895119908119895(119909(119905)) ge 0 for all 119895 119909(119905) and

119909(119905 minus 1198892(119905)) where 0 lt 120588

119895lt 1 and there exist matrices

11987511

gt 0 11987522

gt 0 11988211

gt 0 11988222

gt 0 11988511

gt 0 11988522

gt0 11987311gt 0 119873

22gt 0 119876

119894gt 0 119872

119894gt 0 119877

119894gt 0 (119894 = 1 2)

Λ119894= Λ119879119894isin 1198778119899times8119899 gt 0 (i = 1 2 119903) and

119875 = [11987511 11987512lowast 119875

22

] gt 0 119882 = [11988211 11988212lowast 119882

22

] gt 0

119885 = [11988511 11988512lowast 119885

22

] gt 0 119873 = [11987311 11987312lowast 119873

22

] gt 0

(16)

and real matrices 119884119895(119895 = 1 2 119903) 119875

12119882121198851211987312 and

119883 such that (30)ndash(32) hold In addition the stabilizing controllaw is given by

119906 (119905) =119903

sum119895=1

119898119895(119909 (119905 minus 119889

2(119905))) 119884

119895119883minus119879119909 (119905) (17)

Proof Choose a fuzzy weighting-dependent Lyapunov-Krasovskii functional candidate as

119881 (119909 (119905)) = 1198811(119909 (119905)) + 119881

2(119909 (119905)) + 119881

3(119909 (119905)) + 119881

4(119909 (119905))

+ 1198815(119909 (119905))

4 Mathematical Problems in Engineering

1198811(119909 (119905)) = 120585119879

1(119905) 119875120585

1(119905) + 120585119879

2(119905)119882120585

2(119905)

1198812(119909 (119905)) = int

0

minusℎ1

int119905

119905+120579

120585119879 (119904) 119885120585 (119904) 119889 119904119889120579

+ int0

minusℎ2

int119905

119905+120579

120585119879 (119904)119873120585 (119904) 119889119904 119889120579

1198813(119909 (119905)) = int

119905

119905minus1198891(119905)

119909119879 (119904) 1198761119909 (119904) 119889119904

+ int119905

119905minus1198892(119905)

119909119879 (119904) 1198762119909 (119904) 119889119904

1198814(119909 (119905)) = int

119905

119905minusℎ1

119909119879 (119904)1198721119909 (119904) 119889119904 + int

119905

119905minusℎ2

119909119879 (119904)1198722119909 (119904) 119889119904

1198815(119909 (119905)) = int

0

minusℎ1

int0

120579

int119905

119905+120582

119879 (119904) 1198771 (119904) 119889119904 119889120582 119889120579

+ int0

minusℎ2

int0

120579

int119905

119905+120582

119879 (119904) 1198772 (119904) 119889119904 119889120582 119889120579

(18)

where

1205851198791(119905) = [119909119879 (119905) (int

119905

119905minusℎ1

119909 (119904) 119889119904)119879

]

1205851198792(119905) = [119909119879 (119905) (int

119905

119905minusℎ2

119909 (119904) 119889119904)119879

]

120585119879 (119905) = [119909119879 (119905) 119879 (119905)]

119875 = [11987511 11987512lowast 11987522

] gt 0 119882 = [11988211 11988212lowast 11988222

] gt 0

119885 = [11988511 11988512lowast 11988522

] gt 0 119873 = [11987311 11987312lowast 11987322

] gt 0

(19)

and 119876119894 119872119894 and 119877

119894(119894 = 1 2) are the positive-definite

matrices to be determinedThe derivatives of119881(119909(119905)) along the trajectories of system

(15) are as follows

1(119909 (119905)) = 2120585119879

1(119905) 119875 1205851(119905) + 2120585119879

2(119905)119882 120585

2(119905)

= 2 [119909119879 (119905) (int119905

119905minusℎ1

119909 (119904) 119889119904)119879

] [11987511 11987512lowast 11987522

]

times [ (119905)119909 (119905) minus 119909 (119905 minus ℎ

1)]

+ 2 [119909119879 (119905) (int119905

119905minusℎ2

119909 (119904) 119889119904)119879

] [11988211 11988212lowast 11988222

]

times [ (119905)119909 (119905) minus 119909 (119905 minus ℎ

2)]

2(119909 (119905)) = ℎ

1[119909119879 (119905) 119879 (119905)] [11988511 11988512lowast 119885

22

] [119909 (119905) (119905)]

minusint119905

119905minusℎ1

[119909119879 (119904) 119879 (119904)] [11988511 11988512lowast 11988522

] [119909 (119904) (119904)] 119889119904

+ ℎ2[119909119879 (119905) 119879 (119905)] [11987311 11987312lowast 119873

22

] [119909 (119905) (119905)]

minusint119905

119905minusℎ2

[119909119879 (119904) 119879 (119904)] [11987311 11987312lowast 11987322

] [119909 (119904) (119904)] 119889119904

3(119909 (119905)) = 119909119879 (119905) 119876

1119909 (119905) minus (1 minus 119889

1(119905)) 119909119879 (119905 minus 119889

1(119905))

times 1198761119909 (119905 minus 119889

1(119905)) + 119909119879 (119905) 119876

2119909 (119905)

minus (1 minus 1198892(119905)) 119909119879 (119905 minus 119889

2(119905)) 119876

2119909 (119905 minus 119889

2(119905))

le 119909119879 (119905) 1198761119909 (119905) + 119909119879 (119905) 119876

2119909 (119905) minus (1 minus 120583

1)

times 119909119879 (119905 minus 1198891(119905)) 119876

1119909 (119905 minus 119889

1(119905)) minus (1 minus 120583

2)

times 119909119879 (119905 minus 1198892(119905)) 119876

2119909 (119905 minus 119889

2(119905))

4(119909 (119905)) = 119909119879 (119905)119872

1119909 (119905) minus 119909119879 (119905 minus ℎ

1)1198721119909 (119905 minus ℎ

1)

+ 119909119879 (119905)1198722119909 (119905) minus 119909119879 (119905 minus ℎ

2)1198722119909 (119905 minus ℎ

2)

5(119909 (119905)) =

12ℎ21119879 (119905) 119877

1 (119905) minus int

0

minusℎ

int119905

119905+120579

119879 (119904)

times 1198771 (119904) 119889119904 119889120579 +

12ℎ22119879 (119905) 119877

2 (119905)

minus int0

minusℎ

int119905

119905+120579

119879 (119904) 1198772 (119904) 119889119904 119889120579

(20)

From (15) the following equation can be obtained for anymatrices 119879

119894 119894 = 1 2 8

2119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))

times [119909119879 (119905) 1198791+ 119909119879 (119905 minus 119889

1(119905)) 1198792+119909119879 (119905 minus ℎ

1) 1198793

+ (int119905

119905minusℎ1

119909 (119904) 119889119904)119879

1198794+ 119909119879 (119905 minus 119889

2(119905)) 1198795+ 119909119879

times (119905 minus ℎ2) 1198796+ (int119905

119905minusℎ2

119909 (119904) 119889119904)119879

1198797+ 119879 (119905) 119879

8]

times [1198601119894119909 (119905) minus 119860

2119894119909 (119905 minus 119889

1(119905))

minus119861119894119865119895119909 (119905 minus 119889

2(119905)) minus (119905)] = 0

(21)

Mathematical Problems in Engineering 5

By Lemma 3 we have

minus int119905

119905minusℎ1

[119909119905 (119904) 119879 (119904)] [11988511 11988512lowast 11988522

] [119909 (119904) (119904)] 119889119904

le minus1ℎ1

[(int119905

119905minusℎ1

119909 (119904) 119889119904)119879

119909119879 (119905) minus 119909119879 (119905 minus ℎ1)]

times [11988511 11988512lowast 11988522

][

[

int119905

119905minusℎ1

119909 (119904) 119889119904

119909 (119905) minus 119909 (119905 minus ℎ1)]

]

minus int119905

119905minusℎ2

[119909119879 (119904) 119879 (119904)] [11987311 11987312lowast 11987322

] [119909 (119904) (119904)] 119889119904

le minus1ℎ2

[(int119905

119905minusℎ2

119909 (119904) 119889119904)119879

119909119879 (119905) minus 119909119879 (119905 minus ℎ1)]

times [11987311 11987312lowast 11987322

][

[

int119905

119905minusℎ2

119909 (119904) 119889119904

119909 (119905) minus 119909 (119905 minus ℎ2)]

]

minus int0

minusℎ119894

int119905

119905+120579

119879 (119904) 119877119894 (119904) 119889119904 119889120579

le minus2ℎ119894

2(int0

minusℎ119894

int119905

119905+120579

(119904) 119889119904 119889120579)119879

119877119894(int0

minusℎ119894

int119905

119905+120579

(119904) 119889119904 119889120579)

= minus2ℎ119894

2(ℎ119894119909 (119905) minus int

119905

119905minusℎ119894

119909 (119904) 119889119904)119879

times 119877119894(ℎ119894119909 (119905) minus int

119905

119905minusℎ119894

119909 (119904) 119889119904) 119894 = 1 2

(22)

Using the above inequalities andwith the zero quantities (21)we can obtain

(119909119905)

le 2 [119909119879 (119905) (int119905

119905minusℎ1

119909 (119904) 119889119904)119879

] [11987511 11987512lowast 11987522

]

times [ (119905)119909 (119905) minus 119909 (119905 minus ℎ

1)] + ℎ1 [119909

119879 (119905) 119879 (119905)]

times [11988511 11988512lowast 11988522

] [119909 (119905) (119905)] + ℎ2 [119909119879 (119905) 119879 (119905)] [11987311 11987312lowast 119873

22

]

times [119909 (119905) (119905)] minus1ℎ1

[(int119905

119905minusℎ1

119909 (119904) 119889119904)119879

119909119879 (119905) minus 119909119879 (119905 minus ℎ1)]

times [11988511 11988512lowast 11988522

][

[

int119905

119905minusℎ1

119909 (119904) 119889119904

119909 (119905) minus 119909 (119905 minus ℎ1)]

]

minus1ℎ2

[(int119905

119905minusℎ2

119909 (119904) 119889119904)119879

119909119879 (119905) minus 119909119879 (119905 minus ℎ2)]

times [11987311 11987312lowast 11987322

][

[

int119905

119905minusℎ2

119909 (119904) 119889119904

119909 (119905) minus 119909 (119905 minus ℎ2)]

]+ 119909119879 (119905) 119876

1119909 (119905)

+ 119909119879 (119905) 1198762119909 (119905) minus (1 minus 120583

1) 119909119879 (119905 minus 119889

1(119905)) 119876

1119909

times (119905 minus 1198891(119905)) minus (1 minus 120583

2) 119909119879 (119905 minus 119889

2(119905)) 119876

2119909

times (119905 minus 1198892(119905)) + 119909119879 (119905)119872

1119909 (119905) minus 119909119879 (119905 minus ℎ

1)1198721119909

times (119905 minus ℎ1) + 119909119879 (119905)119872

2119909 (119905) minus 119909119879 (119905 minus ℎ

2)1198722119909

times (119905 minus ℎ2) +

12ℎ21119879 (119905) 119877

1 (119905) minus

2ℎ1

2

times(ℎ1119909 (119905)minusint

119905

119905minusℎ1

119909 (119904) 119889119904)119879

1198771(ℎ1119909 (119905)minusint

119905

119905minusℎ1

119909 (119904) 119889119904)

+12ℎ2

2119879 (119905) 1198772 (119905)minus

2ℎ22

(ℎ2119909 (119905)minusint

119905

119905minusℎ2

119909 (119904) 119889119904)119879

times1198772(ℎ2119909 (119905) minus int

119905

119905minusℎ2

119909 (119904) 119889119904)

+ 2 [119909119879 (119905) (int119905

119905minusℎ2

119909 (119904) 119889119904)119879

]

times [11988211 11988212lowast 11988222

] [ (119905)119909 (119905) minus 119909 (119905 minus ℎ

2)]

+ 2119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))

times [119909119879 (119905) 1198791+ 119909119879 (119905 minus 119889

1(119905)) 1198792+ 119909119879 (119905 minus ℎ

1) 1198793

+ (int119905

119905minusℎ1

119909 (119904) 119889119904)119879

1198794+ 119909119879 (119905 minus 119889

2(119905)) 1198795+ 119909119879

times (119905 minus ℎ2) 1198796+(int119905

119905minusℎ2

119909 (119904) 119889119904)119879

1198797+ 119879 (119905) 119879

8]

times [1198601119894119909 (119905) minus 119860

2119894119909 (119905 minus 119889

1(119905))

minus119861119894119865119895119909 (119905 minus 119889

2(119905)) minus (119905)]

= 120577119879 (119905)(119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))Φ

119894119895)120577 (119905)

(23)

6 Mathematical Problems in Engineering

where

120577119879 (119905) = [119909119879 (119905) 119909119879 (119905 minus 1198891(119905)) 119909119879 (119905 minus ℎ

1) (int

119905

119905minusℎ1

119909 (119904) 119889119904)119879

119909119879 (119905 minus 1198892(119905)) 119909119879 (119905 minus ℎ

2) (int

119905

119905minusℎ2

119909 (119904) 119889119904)119879

119879 (119905)] (24)

Φ119894119895=

[[[[[[

[

Φ11119894 Φ12119894 Φ13 Φ14 Φ15119894119895 Φ16 Φ17 Φ18

lowast Φ22119894 119860119879

2119894119879119879

3119860119879

2119894119879119879

4Φ25119894119895 119860

119879

2119894119879119879

6119860119879

2119894119879119879

7Φ28119894

lowast lowast Φ33119894 minus11987522 1198793119861119894119865119895 0 0 minus1198793

lowast lowast lowast Φ44 1198794119861119894119865119895 0 0 Φ48

lowast lowast lowast lowast Φ55119894119895 119865119879

119895119861119879

119894119879119879

6119865119879

119895119861119879

119894119879119879

7Φ58119894

lowast lowast lowast lowast lowast Φ66 Φ67 minus1198796

lowast lowast lowast lowast lowast lowast Φ77 Φ78

lowast lowast lowast lowast lowast lowast lowast Φ88

]]]]]]

]

Φ11119894= 11987512+ 11987511987912+11988212+11988211987912+ ℎ111988511+ ℎ211987311

minus1ℎ1

11988522minus1ℎ2

11987322+ 1198761+ 1198762+1198721+1198722

minus 21198771minus 21198772+ 11987911198601119894+ 11986011987911198941198791198791

Φ12119894= 11987911198602119894+ 11986011987911198941198791198792

Φ13= minus11987512minus1ℎ1

11988522+ 11986011987911198941198791198793

Φ14= 11987522minus1ℎ2

11988511987912+1ℎ1

1198771+ 11986011987911198941198791198794

Φ15119894119895

= 1198791119861119894119865119895+ 11986011987911198941198791198795

Φ16= minus119882

12minus1ℎ2

11987322+ 11986011987911198941198791198796

Φ17= 11988222minus1ℎ2

11987311987912+2ℎ22

1198772+ 11986011987911198941198791198797

Φ18= 11987511+11988211+ ℎ111988512+ ℎ211987312minus 1198791+ 11986011987911198941198791198798

Φ22119894= minus (1 minus 120583

1) 1198761+ 11987921198602119894+ 11986011987921198941198791198792

Φ25119894119895

= 1198792119861119894119865119895+ 11986011987921198941198791198795

Φ28119894= minus1198792+ 11986011987921198941198791198798

Φ33= minus

1ℎ11988522minus1198721

Φ44= minus

1ℎ1

11988511minus2ℎ21

1198771

Φ48= 11987511987912minus 1198794

Φ55119894= minus (1 minus 120583

2) 1198762+ 1198795119861119894119865119895+ 1198651198791198951198611198791198941198791198795

Φ58119894= minus1198795+ 1198651198791198951198611198791198941198791198798

Φ66= minus

1ℎ2

11987322minus1198722

Φ67= minus

1ℎ2

11987311987912minus11988222

Φ77= minus

1ℎ2

11987311minus2ℎ22

1198772

Φ78= 11988211987912minus 1198797

Φ88= ℎ111988522+ ℎ211987322+12ℎ211198771+12ℎ221198772minus 1198798

(25)

From (25) it is obvious that if

119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))Φ

119894119895lt 0 (26)

(119909(119905)) lt 0From (26) we can discover that the feedback gains

are predefined The following proof presents the controllerdesign method We first pre- and postmultiply both thediag [119883 119883 119883 119883 119883 119883 119883 119883 ] and transpose to both sides of (25)and pre- and postmultiply the diag [119883 119883 ] transpose to bothsides of 119875 119882 119885 and 119873 and the 119883 and transpose to bothsides of 119876

119894119872119894 and 119877

119894 119894 = 1 2 Let 119879

119894= 1199051198941198791(119894 = 2 8)

and denote new variables 119883 = 119879minus11 1198771= 119883119877

1119883119879 119876

1=

1198831198761119883119879119872

1= 119883119872

1119883119879 119877

2= 119883119877

2119883119879 119876

2= 119883119876

2119883119879 119872

2=

1198831198722119883119879 Λ

119894= 119883Λ

119894119883119879 Λ

119895= 119883Λ

119895119883119879 119894 = 1 2 119903 119875

11=

11988311987511119883119879 119875

12= 119883119875

12119883119879 119875

22= 119883119875

22119883119879 119882

11= 119883119882

11119883119879

11988212= 119883119882

12119883119879 119882

22= 119883119882

22119883119879 119885

11= 119883119885

11119883119879 119885

12=

11988311988512119883119879 119885

22= 119883119885

22119883119879 119873

11= 119883119873

11119883119879 119873

12= 119883119873

12119883119879

and11987322= 119883119873

22119883119879 With 119865

119894= 119884119894119883minus119879 119894 = 1 2 119903we next

get

119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))Φ

119894119895lt 0 (27)

Mathematical Problems in Engineering 7

where

Φ119894119895=

[[[[[[[[

[

Φ11119894 Φ12119894 Φ13 Φ14 Φ15119894119895 Φ16 Φ17 Φ18

lowast Φ22119894 1199053119883119860119879

21198941199054119883119860119879

2119894Φ25119894119895 1199056119883119860

119879

21198941199057119883119860119879

2119894Φ28119894

lowast lowast Φ33119894 minus11987522 1199053119861119894119884119895 0 0 minus1199053119883119879

lowast lowast lowast Φ44 1199054119861119894119884119895 0 0 Φ48

lowast lowast lowast lowast Φ55119894119895 1199056119884119879

119895119861119879

1198941199057119884119879

119895119861119879

119894Φ58119894

lowast lowast lowast lowast lowast Φ66 Φ67 minus1199056119883119879

lowast lowast lowast lowast lowast lowast Φ77 Φ78

lowast lowast lowast lowast lowast lowast lowast Φ88

]]]]]]]]

]

Φ11119894= 11987512+ 119875119879

12+11988212+119882119879

12+ ℎ111988511

+ ℎ211987311minus1ℎ1

11988522minus1ℎ2

11987322+ 1198761+ 1198762

+1198721+1198722minus 21198771minus 21198772+ 1198601119894119883119879 + 119883119860119879

1119894

Φ12119894= 1198602119894119883119879 + 119905

21198831198601198791119894

Φ13= minus11987512minus1ℎ 111988522+ 11990531198831198601198791119894

Φ14= 11987522minus1ℎ1

119885119879

12+1ℎ1

1198771+ 11990541198831198601198791119894

Φ15119894119895

= 119861119894119884119895+ 11990551198831198601198791119894

Φ16= minus119882

12minus1ℎ2

11987322+ 11990561198831198601198791119894

Φ17= 11988222minus1ℎ2

119873119879

12+2ℎ22

1198772+ 11990571198831198601198791119894

Φ18= 11987511+11988211+ ℎ111988512+ ℎ211987312minus 119883119879 + 119905

81198831198601198791119894

Φ22119894= minus (1 minus 120583

1) 1198761+ 11990521198602119894119883119879 + 119905

21198831198601198792119894

Φ25119894119895

= 1199052119861119894119884119895+ 11990551198831198601198792119894

Φ28119894= minus1199052119883119879 + 119905

81198831198601198792119894

Φ33= minus

1ℎ1

11988522minus1198721

Φ44= minus

1ℎ1

11988511minus2ℎ21

1198771

Φ48= 119875119879

12minus 1199054119883119879

Φ55119894= minus (1 minus 120583

2) 1198762+ 1199055119861119894119884119895+ 1199055119884119879119895119861119879119894

Φ58119894= minus1199055119883119879 + 119905

8119884119879119895119861119879119894

Φ66= minus

1ℎ2

11987322minus1198722

Φ67= minus

1ℎ2

119873119879

12minus11988222

Φ77= minus

1ℎ2

11987311minus2ℎ22

1198772

Φ78= 119882119879

12minus 1199057119883119879

Φ88= ℎ111988522+ ℎ211987322+12ℎ211198771+12ℎ221198772minus 1199058119883119879

(28)

Here ldquolowastrdquo denotes the transpose elements in the symmetricpositions

If (27) holds then (119909(119905)) lt 0 Considersum119903119894=1sum119903119895=1

119908119894(119909(119905))(119908

119895(119909(119905)) minus 119898

119895(119909(119905 minus 119889

2(119905))))Λ

119894= 0

where Λ119894= Λ119879119894isin 1198778119899times8119899 gt 0 119894 = 1 2 119903 are arbitrary

matrices These terms are introduced to (27) to alleviate theconservativeness From (27) we also have

Φ =119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))Φ

119894119895

=119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905)) (119908

119895(119909 (119905)) minus 119898

119895(119909 (119905 minus 119889

2(119905)))

+120588119895119908119895(119909 (119905)) minus 120588

119895119908119895(119909 (119905))) Λ

119894

+119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))Φ

119894119895

=119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905)) 119908

119895(119909 (119905)) (120588

119895Φ119894119895minus 120588119895Λ119894+ Λ119894)

+119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905)) (119898

119895(119909 (119905 minus 119889

2(119905))) minus 120588

119895119908119895(119909 (119905)))

times (Φ119894119895minus Λ119894)

le119903

sum119894=1

1199082119894(120588119894Φ119894119894minus 120588119894Λ119894+ Λ119894)

+119903

sum119894=1

119903

sum119895=1

119908119894(119898119895minus 120588119895119908119895) (Φ119894119895minus Λ119894)

+119903

sum119894=1

sum119894lt119895

119908119894119908119895(120588119895Φ119894119895+ 120588119894Φ119895119894minus 120588119895Λ119894minus 120588119894Λ119895

+Λ119894+ Λ119895)

(29)

with119898119895(119909(119905 minus 119889

2(119905))) minus 120588

119895119908119895(119909(119905)) ge 0 for all 119895 119909(119905) and 119909(119905 minus

1198892(119905)) Let

Φ119894119895minus Λ119894lt 0 (30)

120588119894Φ119894119894minus 120588119894Λ119894+ Λ119894lt 0 (31)

120588119895Φ119894119895+ 120588119894Φ119895119894minus 120588119895Λ119894minus 120588119894Λ119895+ Λ119894+ Λ119895le 0 119894 lt 119895 (32)

for all 119894 119895 = 1 2 119903

8 Mathematical Problems in Engineering

Since (119909(119905)) lt 0 the fuzzy control system with the timevarying state and input delays (15) is asymptotically stablewith the state feedback control law (17)

32 Robust Stability of Uncertain Fuzzy Systems We alsoexamine the design of a robust stable controller for theuncertain system (4) under the imperfect premise matchingConsider the following uncertain fuzzy time varying delaycontrol system

(119905) =119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))

times [(1198601119894+ Δ1198601119894(119905)) 119909 (119905) + (119860

2119894+ Δ1198602119894(119905))

times 119909 (119905 minus 1198891(119905)) + (119861

119894+ Δ119861119894(119905))

times119906 (119905 minus 1198892(119905))]

119909 (119905) = 120593 (119905) 119905 isin [minusmax (ℎ1 ℎ2) 0]

(33)

Based on the above results of Theorem 4 a robust stabiliza-tion criterion for theT-S fuzzy systemswith time varying stateand input delays is investigated The following result can beobtained

Theorem 5 Given scalars ℎ1ge 0 ℎ

2ge 0 120583

1 1205832 and 119905

119894(119894 =

2 8) the uncertain fuzzy control systems with the timevarying state and input delays (33) is robustly stable for any0 le 119889

119894(119905) le ℎ

119894(119894 = 1 2) via the imperfect premise matching

controller design technique if the membership functions of thefuzzy model and fuzzy controller satisfy 119898

119895(119909(119905 minus 119889

2(119905))) minus

120588119895119908119895(119909(119905)) ge 0 for all 119895 119909(119905) and 119909(119905 minus 119889

2(119905)) where 0 lt

120588119895lt 1 and there exist common matrices 119875

11gt 0 119875

22gt 0

11988211gt 0 119882

22gt 0 119885

11gt 0 119885

22gt 0 119873

11gt 0 119873

22gt 0

119876119894gt 0 119872

119894gt 0 119877

119894gt 0 (119894 = 1 2) and Λ

119894= ΛT119894isin 1198778119899times8119899 gt

0 (i = 1 2 r) 119875 gt 0 119882 gt 0 119885 gt 0 119873 gt 0 and somematrices 119884

119895(119895 = 1 2 119903) 119883 and scalars 120576

1119894gt 0 120576

2119894gt 0

120576119887119894119895gt 0 satisfy the following LMIs

Ψ119894119895minus Λ119894lt 0 (34)

120588119894Ψ119894119894minus 120588119894Λ119894+ Λ119894lt 0 (35)

120588119895Ψ119894119895+ 120588119894Ψ119895119894minus 120588119895Λ119894minus 120588119894Λ119895+ Λ119894+ Λ119895 le 0 119894 lt 119895 (36)

where Ψ119894119895is defined as in (44) The state feedback gains can be

constructed as 119865119894= 119884119894119883minus119879

Proof If 1198601119894 1198602119894 and 119861

119894in Φ119894119895are replaced with 119860

1119894+

119863119894119870119894(119905)1198641119894 1198602119894+ 119863119894119870119894(119905)1198642119894 and 119861

119894+ 119863119894119870119894(119905)119864119887119894in (28) of

Theorem 4 respectively (27) for system (33) can be rewrittenas

119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889 (119905))) Ω

119894119895lt 0 (37)

where

Ω119894119895= Φ119894119895+ 1198631198791119894119870119894(119905) 119864119894+ 119864119879119894119870119879119894(119905) 1198631119894+ 1198631198792119894119870119894(119905) 1198642119894119895

+ 1198641198792119894119895119870119879119894(119905) 1198632119894+ 119863119879119887119894119870119894(119905) 119864119887119894119895+ 119864119879119887119894119895119870119879119894(119905) 119863119887119894

1198631119894= [1198631198791198940 0 0 0 0 0 0]

1198632119894= [0 119863119879

1198940 0 0 0 0 0]

119863119887119894= [0 0 0 0 119863119879

1198940 0 0]

(38)

1198641119894= [1198641119894119883119879 11990521198641119894119883119879 11990531198641119894119883119879 11990541198641119894119883119879 119905

51198641119894119883119879 11990561198641119894119883119879 11990571198641119894119883119879 11990581198641119894119883119879]

1198642119894= [1198642119894119883119879 11990521198642119894119883119879 11990531198642119894119883119879 11990541198642119894119883119879 11990551198642119894119883119879 11990561198642119894119883119879 11990571198642119894119883119879 11990581198642119894119883119879]

119864119887119894119895= [119864119887119894119884119895 1199052119864119887119894119884119895 1199053119864119887119894119884119895 1199054119864119887119894119884119895 1199055119864119887119894119884119895 1199056119864119887119894119884119895 1199057119864119887119894119884119895 1199058119864119887119894119884119895]

(39)

If Ω119894119895

lt 0 (37) holds According to Lemma 2 it isstraightforward to know that Ω

119894119895lt 0 is true if for each 119894

119895 there exists scalars 1205761119894gt 0 120576

2119894gt 0 and 120576

119887119894119895gt 0 such that

the following inequality holds

Φ119894119895+ 120576111989411986311987911198941198631119894+ 120576minus1111989411986411987911198941198641119894+ 120576211989411986311987921198941198632119894

+ 120576minus1211989411986411987921198941198642119894+ 120576119887119894119895119863119879119887119894119863119887119894+ 120576minus1119887119894119895119864119879119887119894119895119864119887119894119895lt 0

(40)

Based on Schur complement (40) is equivalent to thefollowing inequality

Ψ119894119895=[[[

[

Φ119894119895

1198641198791119894

1198641198792119894

119864119879119887119894119895

lowast minus1205761119894119868 0 0

lowast lowast minus1205762119894119868 0

lowast lowast lowast minus120576119887119894119895119868

]]]

]

lt 0 (41)

Mathematical Problems in Engineering 9

where

Φ119894119895=

[[[[[[[[

[

Φ11119894 Φ12119894 Φ13 Φ14 Φ15119894119895 Φ16 Φ17 Φ18

lowast Φ22119894 1199053119883119860119879

21198941199054119883119860119879

2119894Φ25119894119895 1199056119883119860

119879

21198941199057119883119860119879

2119894Φ28119894

lowast lowast Φ33119894 minus11987522 1199053119861119894119884119895 0 0 minus1199053119883119879

lowast lowast lowast Φ44 1199054119861119894119884119895 0 0 Φ48

lowast lowast lowast lowast Φ55119894119895 1199056119884119879

119895119861119879

1198941199057119884119879

119895119861119879

119894Φ58119894

lowast lowast lowast lowast lowast Φ66 Φ67 minus1199056119883119879

lowast lowast lowast lowast lowast lowast Φ77 Φ78

lowast lowast lowast lowast lowast lowast lowast Φ88

]]]]]]]]

]

lt 0

Φ11119894= Φ11119894+ 1205761119894119863119894119863119879119894

Φ22119894= Φ22119894+ 1205762119894119863119894119863119879119894

Φ55119894119895

= Φ55119894119895

+ 120576119887119894119863119894119863119879119894

(42)

and Φ are the same as in Theorem 4 Therefore (41) holds ifand only if the following inequality holds

119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889 (119905))) Ψ

119894119895lt 0 (43)

To alleviate the conservativeness of our robust stabilityanalysis similar to the analysis approach of Theorem 4 weintroduce sum119903

119894=1sum119903119895=1

119908119894(119908119895minus 119898119895)Λ119894= 0 to (43) and the

following proof is the same as in Theorem 4 We can obtainthe conditions and (34)ndash(36) Thus the fuzzy control systemwith the time varying state and input delays (34) is robuststable on the basis of the control law 119865

119894= 119884119894119883minus119879under these

conditions and (34)ndash(36) hold

Remark 6 If we set 119908119894(119909(119905)) = 119898

119895(119909(119905)) 119889

1(119905) = 119889

2(119905)

the closed-loop system in this paper has the same structureas that of paper [26] On the other hand let 119908

119894(119909(119905)) =

119898119895(119909(119905)) and 119889

1(119905) = 119889

2(119905) = 119889 we can obtain the system

in [16] Therefore the system studied in this paper is moregeneral Additionally the information of the membershipfunctions of the fuzzy time delay models and controllers areall considered in the stability analysis If the fuzzymodels andfuzzy controllers share the same fuzzymembership functionsthat is 119908

119894(119909(119905)) = 119898

119895(119909(119905)) the stability analysis can be

referred to [25 27 28] In other words the problem in [2527 28] is actually a special case of the one investigated in thispaper

Remark 7 The augmented Lyapunov function with an addi-tional triple-integral term is introduced in analyzing thestability problem for the T-S fuzzy systems with time varyingstate and input delays In addition two integral inequalitiesare used to derive Theorem 4 and less free-weighting matri-ces are introduced which lead to get less conservative resultsIt can be observed that some existing results are the specialcases of this paper For example the method in [25] is from(18) with 119875

12= 0 119875

22= 0 119885

12= 11988522= 0 119873

12= 11987322= 0

and 1198771= 1198772= 0 in Theorem 4 The results in [22] are based

on system (33) with 1198892(119905) = 0 and (18) with119872

2= 0 119885

12=

11988522= 0 119876

1= 1198762= 0 and 119877

1= 1198772= 0

Remark 8 Different from the general PDC technique thedesign method under the imperfect premise matching ismuch more flexible because the membership functions ofthe fuzzy controllers do not need to be chosen the same asthose of the fuzzy time delay models Instead they can bedesigned arbitrarilyThus the design flexibility is significantlyenhanced On the other hand some simple membershipfunctions of the fuzzy controllers might be employed whichcan reduce the implementation cost

Remark 9 In [7ndash11 15ndash20] the authors have considered therobust control for uncertain T-S fuzzy systems with statedelays Some delay-dependent conditions and fuzzy con-trollers are obtained with a traditional Lyapunov-Krasovskiifunctionalmethod andPDCmethod which all are the specialcases of Theorem 5 in this paper

4 Numerical Examples

In this section two numerical examples are given to illus-trate the conservativeness and effectiveness of the proposedmethods The first example compares our techniques withthe existing ones in the literature for stability analysis whichshows that Theorem 4 in this paper is less conservative thanthe other results The second example is used to illustrate theadvantage of the robust stability conditions ofTheorem 5 anddemonstrate how to design robust fuzzy controllers by usingour approach

Example 10 Consider the following fuzzy system with stateand input delays

(119905) =2

sum119894=1

119908119894(120579 (119905)) [119860

1119894119909 (119905) + 119860

2119894119909 (119905 minus 119889

1(119905))

+119861119894119906 (119905 minus 119889

2(119905))]

11986011= [0 06

0 1 ] 11986021= [05 09

0 2 ]

11986012= [1 0

1 0] 11986022= [09 0

1 16]

1198611= 1198612= [11]

(44)

The fuzzy membership functions are selected as [31]

1199081(1199091(119905)) = (1 minus

119888 (119905) sin (10038161003816100381610038161199091 (119905)1003816100381610038161003816minus4)5

1 + expminus1001199091(119905)3(1minus1199091(119905)))

timescos (119909

1(119905))2

1 + expminus251199091(119905)(3+(1199091(119905)042))

times 1199082(1199091(119905)) = 1 minus 119908

1(1199091(119905))

(45)

10 Mathematical Problems in Engineering

Table 1 The maximum allowable time delay and feedback gains(1205831= 0 120583

2= 0)

Paper ℎ1

ℎ2 Feedback gains

[26] 03120 031201198651= [10598 minus56598]

1198652= [minus13068 minus41167]

Theorem 4 033 0571198651= [minus00042 minus00449]1198652= [00048 00717]

where

1199091(119905) isin [minus

1205872

1205872]

119888 (119905) =sin (119909

1(119905)) + 140

isin [minus005 005] (46)

If we set 1198891(119905) = 119889

2(119905) in system (44) we can obtain the

system in [26] which implies that our system ismore generalEmploying the LMIs in [26] and those inTheorem 4yields themaximum state and input delays ℎ

1 and ℎ

2 which guarantee

the stability of system (44) and the feedback gains for 1205831= 0

and 1205832= 0 when 120588

1= 075 120588

2= 095 119905

2= 01 119905

3=

02 1199054= 07 119905

5= 01 119905

6= 04 119905

7= 01 and 119905

8= 13 as

given in Table 1 which clearly shows the superiority of theresults derived in this paper over those obtained from [26]

Example 11 Consider the well-known truck-trailer systemwhich can be described by the following T-S fuzzy system

(119905) =2

sum119894=1

119908119894(120579 (119905)) [(119860

1119894+ Δ1198601119894) 119909 (119905) + 119860

2119894119909 (119905 minus 119889

1(119905))

+ (119861119894+ Δ119861119894) 119906 (119905 minus 119889

2(119905))]

(47)

11986011=

[[[[[[[[[

[

minus119886V1199051198711199050

0 0

119886V1199051198711199050

0 0

119886V21199052

21198711199050

V1199051199050

0

]]]]]]]]]

]

11986021=

[[[[[[[[[

[

minus (1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)V21199052

21198711199050

0 0

]]]]]]]]]

]

1198611=[[[[

[

V1199051198971199050

0

0

]]]]

]

11986012=

[[[[[[[[[

[

minus119886V1199051198711199050

0 0

119886V1199051198711199050

0 0

119886119889V21199052

21198711199050

119889V1199051199050

0

]]]]]]]]]

]

11986022=

[[[[[[[[[

[

minus (1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)119889V21199052

21198711199050

0 0

]]]]]]]]]

]

(48)

1198612=[[[[

[

V1199051198971199050

0

0

]]]]

]

(49)

where 119897 = 28 119871 = 55 V = minus10 119905 = 20 1199050= 05

119889 = 101199050120587 119863

119894= [02555 02555 02555]119879 119864

1119894= 1198642119894=

[01 0 0]119879 119864119887119894

= [01 0 0]119879 (119894 = 1 2) and the fuzzymembership functions are selected the same as inExample 10

(1) Let 119886 = 1 system (47) is the same as the fuzzy systemof [22] which implies that our system ismore generalTable 2 gives themaximum input delay value of ℎ

2 for

which the stabilization is guaranteed byTheorem 5 aswell as the state feedback gains when 120588

1= 075 120588

2=

095 1199052= 01 119905

3= 02 119905

4= 07 119905

5= 01 119905

6=

04 1199057= 08 and 119905

8= 2 It is clearly visible that the

results derived in this paper are better than those from[22 28]

(2) In the case of 119886 = 1 the state and input delays of system(47) are all exist The approach proposed in [22] cannot be used to get the maximum of state delay ℎ

1

as the system in [22] only contains the input delayHowever the method in [28] and Theorem 5 can beused Here let 119886 = 07 120588

1= 075 120588

2= 095 119905

2=

01 1199053= 02 119905

4= 07 119905

5= 01 119905

6= 04 119905

7=

08 and 1199058= 3 and Table 3 provides the maximum

upper bounds of ℎ1and ℎ

2and the state feedback

gains for which the robust stabilization is guaranteedbyTheorem 5

By Theorem 5 we can conclude that the uncertain fuzzymodel (47) is robust stable based on the following fuzzycontrol law

119906 (119905 minus ℎ2) =2

sum119895=1

119898119895(1199091(119905 minus ℎ

2)) 119865119895119909 (119905 minus ℎ

2) (50)

With the PDC design technique it is required that thefuzzy controller must share the same fuzzy membershipfunctions as those of the fuzzy time delay model Under such

Mathematical Problems in Engineering 11

Table 2 The maximum allowable time delay and feedback gains(1205832= 0)

Paper ℎ2 Feedback gains

[22] 0751198651= [34227 minus03535 00045]

1198652= [35215 minus03617 00056]

[28] 0861198651= [33219 minus02406 00025]

1198652= [33272 minus02494 00026]

Theorem 5 121198651= [minus00212 00093 minus00024]1198652= [minus00250 00044 00014]

Table 3 The maximum allowable time delay and feedback gains(1205831= 1205832= 0)

Paper ℎ1

ℎ2 Feedback gains

[28] 01 0551198651= [33219 minus02406 00025]

1198652= [33272 minus02494 00026]

Theorem 5 04 171198651= [minus00128 00090 minus00017]1198652= [minus00194 00012 00009]

a condition the implementation cost of the fuzzy controlleris high by employing these complex membership functionsHowever fromTheorem 5 the membership functions do notneed to be chosen the same as119908

1(1199091(119905)) and119908

2(1199091(119905)) Thus

we can select some simple membership functions insteadsuch as

1198981(1199091(119905 minus ℎ

2)) = 075 exp

(minus1199091(119905 minus ℎ

2) minus 038)2

2 times 0382+ 005

1198982(1199091(119905 minus ℎ

2)) = 1 minus 119898

1(1199091(119905 minus ℎ

2))

(51)

We also assume that 1205881= 075 and 120588

2= 095 such that

119898119895(1199091(119905 minus ℎ

2)) minus 120588

119895119908119895(1199091(119905)) gt 0 for 119895 = 1 2 and 119909

1(119905 minus ℎ

2)

The fuzzy control law 119906(119905 minus ℎ2) = 119898

1[minus00128 00090 minus

00017]119909 + 1198982[minus00194 00012 00009]119909 can be employed

to stabilize the uncertain fuzzy control systems in (47) withthe state and input delays Figures 1 and 2 illustrate the stablestate response and control input under the initial conditionof 119909(0) = [4 minus 1 2]119879

Remark 12 In Example 11 our method offers less conser-vative results in the sense of allowing longer time delayand obtaining smaller feedback control gains Moreovercompared with [22ndash28] where the fuzzy controllermust havethe same fuzzy membership functions as those of the fuzzytime delay model the above membership functions of ourfuzzy controller are much simper which can considerablylower the implementation cost of the fuzzy controller andenhance the design flexibility

Remark 13 If the function of 119888(119905) in 1199081(1199091(119905)) is unknown

on the basis of the PDC design technique [7ndash11 15ndash20 22ndash28] the fuzzy controller cannot be implemented as 119888(119905) isnot available However using the proposed design method

0 5 10 15

0

2

4

6Closed-loop response

minus12

minus10

minus8

minus6

minus4

minus2

119905 (s)

119909(119905)

1199092

11990931199091

Figure 1The state responses of system (47) with ℎ1= 04 ℎ

2= 17

0 5 10 15

0

5

10

15

20

25

30

119905 (s)

minus20

minus15

minus10

minus5

119906(119905)

Figure 2 The control input of system (47) with ℎ1= 04 ℎ

2= 17

we can select some simper and well-known membershipfunctions that are different from those of the fuzzy time delaymodels for example119898

1(1199091(119905)) and119898

2(1199091(119905)) so as to realize

the fuzzy controller In other words the proposed designapproach can be applied even for the fuzzy time delay modelswith uncertain grades of membership

5 Conclusions

In this paper we study the robust stability and stabilizationproblems for general nonlinear fuzzy systems with timevarying state and input delays A less conservative delay-dependent robust stability criterion has been obtained byintroducing a novel augmented Lyapunov function withan additional triple-integral term Moreover a new design

12 Mathematical Problems in Engineering

approach different from the traditional PDC design tech-nique is proposed which can significantly improve the designflexibility and reduce the implementation cost of the fuzzycontroller by arbitrarily selecting simple fuzzy membershipfunctions Two numerical examples are used to illustrate theconservativeness and effectiveness of our proposed methods

Acknowledgments

This research work was funded by the NSFC under Grantno 61273095 and the Academy of Finland under Grants no135225 and no 127299

References

[1] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[2] W H Ho S H Chen I T Chen J H Chou and C C ShuldquoDesign of stable and quadratic-optimal static output feed-back controllers for T-S-fuzzy-model-based control systems anintegrative computational approachrdquo International Journal ofInnovative Computing Information and Control vol 8 no 1App 403ndash418 2012

[3] X J Su L G Wu P Shi and Y D Song ldquo119867infinmodel reduction

of T-S fuzzy stochastic systemsrdquo IEEE Transactions on SystemsMan and Cybermetics B vol 42 no 6 pp 1574ndash1585 2012

[4] R Yang P Shi G-P Liu andH Gao ldquoNetwork-based feedbackcontrol for systems with mixed delays based on quantizationand dropout compensationrdquo Automatica vol 47 no 12 pp2805ndash2809 2011

[5] XM Zhang Z J Zhang andG P Lu ldquoFault detection for state-delay fuzzy systems subject to random communication delayrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 4 pp 2439ndash2451 2012

[6] F B Li and X Zhang ldquoDelay-range-dependent robust 119867infin

filtering for singuar LPV systems with time variant delayrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 1 pp 339ndash353 2013

[7] Y Y Cao and P M Frank ldquoAnalysis and synthesis of nonlineartime-delay systems via fuzzy control approachrdquo IEEE Transac-tions on Fuzzy Systems vol 8 no 2 pp 200ndash211 2000

[8] Y-Y Cao and P M Frank ldquoStability analysis and synthesis ofnonlinear time-delay systems via linear Takagi-Sugeno fuzzymodelsrdquo Fuzzy Sets and Systems vol 124 no 2 pp 213ndash2292001

[9] O M Kwon and J H Park ldquoDelay-range-dependent stabiliza-tion of uncertain dynamic systems with interval time-varyingdelaysrdquo Applied Mathematics and Computation vol 208 no 1pp 58ndash68 2009

[10] E Tian D Yue and Y Zhang ldquoDelay-dependent robust 119867infin

control for T-S fuzzy system with interval time-varying delayrdquoFuzzy Sets and Systems vol 160 no 12 pp 1708ndash1719 2009

[11] J Y An T Li G LWen and R Li ldquoNew stability conditions foruncertain T-S fuzzy systems with interval time-varying delayrdquoInternational Journal of Control Automation and Systems vol10 no 3 pp 490ndash497 2012

[12] Z J Zhang X L Huang X J Ban and X Z Gao ldquoStabilityanalysis and controller design of T-S fuzzy systems with time-delay under imperfect premise matchingrdquo Journal of BeijingInstitute of Technology vol 21 no 3 pp 387ndash393 2012

[13] O M Kwon M J Park S M Lee and J H Park ldquoAugmentedLyapunov-Krasovskii functional approaches to robust stabilitycriteria for uncertain Takagi-Sugeno fuzzy systems with time-varying delaysrdquo Fuzzy Sets and Systems vol 201 no 16 pp 1ndash192012

[14] F Liu M Wu Y He and R Yokoyama ldquoNew delay-dependentstability criteria for T-S fuzzy systems with time-varying delayrdquoFuzzy Sets and Systems vol 161 no 15 pp 2033ndash2042 2010

[15] B Chen X Liu and S Tong ldquoNew delay-dependent stabiliza-tion conditions of T-S fuzzy systems with constant delayrdquo FuzzySets and Systems vol 158 no 20 pp 2209ndash2224 2007

[16] J Yoneyama ldquoRobust stability and stabilization for uncertainTakagi-Sugeno fuzzy time-delay systemsrdquo Fuzzy Sets and Sys-tems vol 158 no 2 pp 115ndash134 2007

[17] C Lin Q-G Wang and T H Lee ldquoDelay-dependent LMIconditions for stability and stabilization of T-S fuzzy systemswith bounded time-delayrdquo Fuzzy Sets and Systems vol 157 no9 pp 1229ndash1247 2006

[18] X J Su P Shi L G Wu and Y D Song ldquoA novel approachto filter design for T-S fuzzy discrete-time systems with time-varying delayrdquo IEEE Transactions on Fuzzy Systems vol 20 no6 pp 1114ndash1129 2012

[19] C Peng and Q-L Han ldquoDelay-range-dependent robust stabi-lization for uncertain T-S fuzzy control systems with intervaltime-varying delaysrdquo Information Sciences vol 181 no 19 pp4287ndash4299 2011

[20] S T Li Y W Jing and X M Liu ldquoRobust control for a class ofT-S fuzzy systemswith interval time-varying delayrdquo InternatinalJournal of Control Automation and Systems vol 10 no 4 pp737ndash743 2012

[21] X-M Zhang M Wu J-H She and Y He ldquoDelay-dependentstabilization of linear systems with time-varying state and inputdelaysrdquo Automatica vol 41 no 8 pp 1405ndash1412 2005

[22] B Chen X Liu and S Tong ldquoRobust fuzzy control of nonlinearsystems with input delayrdquo Chaos Solitons amp Fractals vol 37 no3 pp 894ndash901 2008

[23] C Lin Q-G Wang and T H Lee ldquoDelay-dependent LMIconditions for stability and stabilization of T-S fuzzy systemswith bounded time-delayrdquo Fuzzy Sets and Systems vol 157 no9 pp 1229ndash1247 2006

[24] H J Lee J B Park and Y H Joo ldquoRobust control for uncertainTakagi-Sugeno fuzzy systems with time-varying input delayrdquoJournal of Dynamic Systems Measurement and Control Trans-actions of the ASME vol 127 no 2 pp 302ndash306 2005

[25] C-H Lien and K-W Yu ldquoRobust control for Takagi-Sugenofuzzy systems with time-varying state and input delaysrdquo ChaosSolitons and Fractals vol 35 no 5 pp 1003ndash1008 2008

[26] Y K Su B Chen S Y Zhang and C Lin ldquoDelay-dependentstabilization of T-S fuzzy systems with time delayrdquo Control andDecision Kongzhi yu Juece vol 24 no 6 pp 921ndash927 2009

[27] B Chen X Liu C Lin and K Liu ldquoRobust 119867infin

control ofTakagi-Sugeno fuzzy systems with state and input time delaysrdquoFuzzy Sets and Systems vol 160 no 4 pp 403ndash422 2009

[28] L Li and X Liu ldquoNew results on delay-dependent robuststability criteria of uncertain fuzzy systems with state and inputdelaysrdquo Information Sciences vol 179 no 8 pp 1134ndash1148 2009

[29] Z J Zhang X J Ban X L Huang and X Z Gao ldquoNew delay-dependent robust stablity for T-S fuzzy systems with intervevaltime-varying delayrdquo in Proccedings of the 5th International JointConference on Computational Science and Optimization pp826ndash830 Harbin China 2012

Mathematical Problems in Engineering 13

[30] H K Lam and F H F Leung ldquoLMI-based stability andperformance design of fuzzy control systems fuzzy modelsand controllers with different premisesrdquo in Proceedings of theInternational Conference on Fuzzy Systerm pp 9599ndash9506Vancouver Canada 2006

[31] H K Lam andMNarimani ldquoStability analysis and perfomancedesign for fuzzy-model-based control system under imperfectpremise matchingrdquo IEEE Transactions on Fuzzy Systems vol 17no 4 pp 949ndash961 2009

[32] S Boyd L E Ghaoui and E Feron Linear Matrix Inequality inSystems and ControlTheory SIAM Philadelphia Pa USA 1994

[33] I R Petersen and C V Hollot ldquoA Riccati equation approach tothe stabilization of uncertain linear systemsrdquo Automatica vol22 no 4 pp 397ndash411 1986

[34] J Sun G P Liu and J Chen ldquoDelay-dependent stabilityand stabilization of neutral time-delay systemsrdquo InternationalJournal of Robust andNonlinear Control vol 19 no 12 pp 1364ndash1375 2009

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

1198811(119909 (119905)) = 120585119879

1(119905) 119875120585

1(119905) + 120585119879

2(119905)119882120585

2(119905)

1198812(119909 (119905)) = int

0

minusℎ1

int119905

119905+120579

120585119879 (119904) 119885120585 (119904) 119889 119904119889120579

+ int0

minusℎ2

int119905

119905+120579

120585119879 (119904)119873120585 (119904) 119889119904 119889120579

1198813(119909 (119905)) = int

119905

119905minus1198891(119905)

119909119879 (119904) 1198761119909 (119904) 119889119904

+ int119905

119905minus1198892(119905)

119909119879 (119904) 1198762119909 (119904) 119889119904

1198814(119909 (119905)) = int

119905

119905minusℎ1

119909119879 (119904)1198721119909 (119904) 119889119904 + int

119905

119905minusℎ2

119909119879 (119904)1198722119909 (119904) 119889119904

1198815(119909 (119905)) = int

0

minusℎ1

int0

120579

int119905

119905+120582

119879 (119904) 1198771 (119904) 119889119904 119889120582 119889120579

+ int0

minusℎ2

int0

120579

int119905

119905+120582

119879 (119904) 1198772 (119904) 119889119904 119889120582 119889120579

(18)

where

1205851198791(119905) = [119909119879 (119905) (int

119905

119905minusℎ1

119909 (119904) 119889119904)119879

]

1205851198792(119905) = [119909119879 (119905) (int

119905

119905minusℎ2

119909 (119904) 119889119904)119879

]

120585119879 (119905) = [119909119879 (119905) 119879 (119905)]

119875 = [11987511 11987512lowast 11987522

] gt 0 119882 = [11988211 11988212lowast 11988222

] gt 0

119885 = [11988511 11988512lowast 11988522

] gt 0 119873 = [11987311 11987312lowast 11987322

] gt 0

(19)

and 119876119894 119872119894 and 119877

119894(119894 = 1 2) are the positive-definite

matrices to be determinedThe derivatives of119881(119909(119905)) along the trajectories of system

(15) are as follows

1(119909 (119905)) = 2120585119879

1(119905) 119875 1205851(119905) + 2120585119879

2(119905)119882 120585

2(119905)

= 2 [119909119879 (119905) (int119905

119905minusℎ1

119909 (119904) 119889119904)119879

] [11987511 11987512lowast 11987522

]

times [ (119905)119909 (119905) minus 119909 (119905 minus ℎ

1)]

+ 2 [119909119879 (119905) (int119905

119905minusℎ2

119909 (119904) 119889119904)119879

] [11988211 11988212lowast 11988222

]

times [ (119905)119909 (119905) minus 119909 (119905 minus ℎ

2)]

2(119909 (119905)) = ℎ

1[119909119879 (119905) 119879 (119905)] [11988511 11988512lowast 119885

22

] [119909 (119905) (119905)]

minusint119905

119905minusℎ1

[119909119879 (119904) 119879 (119904)] [11988511 11988512lowast 11988522

] [119909 (119904) (119904)] 119889119904

+ ℎ2[119909119879 (119905) 119879 (119905)] [11987311 11987312lowast 119873

22

] [119909 (119905) (119905)]

minusint119905

119905minusℎ2

[119909119879 (119904) 119879 (119904)] [11987311 11987312lowast 11987322

] [119909 (119904) (119904)] 119889119904

3(119909 (119905)) = 119909119879 (119905) 119876

1119909 (119905) minus (1 minus 119889

1(119905)) 119909119879 (119905 minus 119889

1(119905))

times 1198761119909 (119905 minus 119889

1(119905)) + 119909119879 (119905) 119876

2119909 (119905)

minus (1 minus 1198892(119905)) 119909119879 (119905 minus 119889

2(119905)) 119876

2119909 (119905 minus 119889

2(119905))

le 119909119879 (119905) 1198761119909 (119905) + 119909119879 (119905) 119876

2119909 (119905) minus (1 minus 120583

1)

times 119909119879 (119905 minus 1198891(119905)) 119876

1119909 (119905 minus 119889

1(119905)) minus (1 minus 120583

2)

times 119909119879 (119905 minus 1198892(119905)) 119876

2119909 (119905 minus 119889

2(119905))

4(119909 (119905)) = 119909119879 (119905)119872

1119909 (119905) minus 119909119879 (119905 minus ℎ

1)1198721119909 (119905 minus ℎ

1)

+ 119909119879 (119905)1198722119909 (119905) minus 119909119879 (119905 minus ℎ

2)1198722119909 (119905 minus ℎ

2)

5(119909 (119905)) =

12ℎ21119879 (119905) 119877

1 (119905) minus int

0

minusℎ

int119905

119905+120579

119879 (119904)

times 1198771 (119904) 119889119904 119889120579 +

12ℎ22119879 (119905) 119877

2 (119905)

minus int0

minusℎ

int119905

119905+120579

119879 (119904) 1198772 (119904) 119889119904 119889120579

(20)

From (15) the following equation can be obtained for anymatrices 119879

119894 119894 = 1 2 8

2119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))

times [119909119879 (119905) 1198791+ 119909119879 (119905 minus 119889

1(119905)) 1198792+119909119879 (119905 minus ℎ

1) 1198793

+ (int119905

119905minusℎ1

119909 (119904) 119889119904)119879

1198794+ 119909119879 (119905 minus 119889

2(119905)) 1198795+ 119909119879

times (119905 minus ℎ2) 1198796+ (int119905

119905minusℎ2

119909 (119904) 119889119904)119879

1198797+ 119879 (119905) 119879

8]

times [1198601119894119909 (119905) minus 119860

2119894119909 (119905 minus 119889

1(119905))

minus119861119894119865119895119909 (119905 minus 119889

2(119905)) minus (119905)] = 0

(21)

Mathematical Problems in Engineering 5

By Lemma 3 we have

minus int119905

119905minusℎ1

[119909119905 (119904) 119879 (119904)] [11988511 11988512lowast 11988522

] [119909 (119904) (119904)] 119889119904

le minus1ℎ1

[(int119905

119905minusℎ1

119909 (119904) 119889119904)119879

119909119879 (119905) minus 119909119879 (119905 minus ℎ1)]

times [11988511 11988512lowast 11988522

][

[

int119905

119905minusℎ1

119909 (119904) 119889119904

119909 (119905) minus 119909 (119905 minus ℎ1)]

]

minus int119905

119905minusℎ2

[119909119879 (119904) 119879 (119904)] [11987311 11987312lowast 11987322

] [119909 (119904) (119904)] 119889119904

le minus1ℎ2

[(int119905

119905minusℎ2

119909 (119904) 119889119904)119879

119909119879 (119905) minus 119909119879 (119905 minus ℎ1)]

times [11987311 11987312lowast 11987322

][

[

int119905

119905minusℎ2

119909 (119904) 119889119904

119909 (119905) minus 119909 (119905 minus ℎ2)]

]

minus int0

minusℎ119894

int119905

119905+120579

119879 (119904) 119877119894 (119904) 119889119904 119889120579

le minus2ℎ119894

2(int0

minusℎ119894

int119905

119905+120579

(119904) 119889119904 119889120579)119879

119877119894(int0

minusℎ119894

int119905

119905+120579

(119904) 119889119904 119889120579)

= minus2ℎ119894

2(ℎ119894119909 (119905) minus int

119905

119905minusℎ119894

119909 (119904) 119889119904)119879

times 119877119894(ℎ119894119909 (119905) minus int

119905

119905minusℎ119894

119909 (119904) 119889119904) 119894 = 1 2

(22)

Using the above inequalities andwith the zero quantities (21)we can obtain

(119909119905)

le 2 [119909119879 (119905) (int119905

119905minusℎ1

119909 (119904) 119889119904)119879

] [11987511 11987512lowast 11987522

]

times [ (119905)119909 (119905) minus 119909 (119905 minus ℎ

1)] + ℎ1 [119909

119879 (119905) 119879 (119905)]

times [11988511 11988512lowast 11988522

] [119909 (119905) (119905)] + ℎ2 [119909119879 (119905) 119879 (119905)] [11987311 11987312lowast 119873

22

]

times [119909 (119905) (119905)] minus1ℎ1

[(int119905

119905minusℎ1

119909 (119904) 119889119904)119879

119909119879 (119905) minus 119909119879 (119905 minus ℎ1)]

times [11988511 11988512lowast 11988522

][

[

int119905

119905minusℎ1

119909 (119904) 119889119904

119909 (119905) minus 119909 (119905 minus ℎ1)]

]

minus1ℎ2

[(int119905

119905minusℎ2

119909 (119904) 119889119904)119879

119909119879 (119905) minus 119909119879 (119905 minus ℎ2)]

times [11987311 11987312lowast 11987322

][

[

int119905

119905minusℎ2

119909 (119904) 119889119904

119909 (119905) minus 119909 (119905 minus ℎ2)]

]+ 119909119879 (119905) 119876

1119909 (119905)

+ 119909119879 (119905) 1198762119909 (119905) minus (1 minus 120583

1) 119909119879 (119905 minus 119889

1(119905)) 119876

1119909

times (119905 minus 1198891(119905)) minus (1 minus 120583

2) 119909119879 (119905 minus 119889

2(119905)) 119876

2119909

times (119905 minus 1198892(119905)) + 119909119879 (119905)119872

1119909 (119905) minus 119909119879 (119905 minus ℎ

1)1198721119909

times (119905 minus ℎ1) + 119909119879 (119905)119872

2119909 (119905) minus 119909119879 (119905 minus ℎ

2)1198722119909

times (119905 minus ℎ2) +

12ℎ21119879 (119905) 119877

1 (119905) minus

2ℎ1

2

times(ℎ1119909 (119905)minusint

119905

119905minusℎ1

119909 (119904) 119889119904)119879

1198771(ℎ1119909 (119905)minusint

119905

119905minusℎ1

119909 (119904) 119889119904)

+12ℎ2

2119879 (119905) 1198772 (119905)minus

2ℎ22

(ℎ2119909 (119905)minusint

119905

119905minusℎ2

119909 (119904) 119889119904)119879

times1198772(ℎ2119909 (119905) minus int

119905

119905minusℎ2

119909 (119904) 119889119904)

+ 2 [119909119879 (119905) (int119905

119905minusℎ2

119909 (119904) 119889119904)119879

]

times [11988211 11988212lowast 11988222

] [ (119905)119909 (119905) minus 119909 (119905 minus ℎ

2)]

+ 2119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))

times [119909119879 (119905) 1198791+ 119909119879 (119905 minus 119889

1(119905)) 1198792+ 119909119879 (119905 minus ℎ

1) 1198793

+ (int119905

119905minusℎ1

119909 (119904) 119889119904)119879

1198794+ 119909119879 (119905 minus 119889

2(119905)) 1198795+ 119909119879

times (119905 minus ℎ2) 1198796+(int119905

119905minusℎ2

119909 (119904) 119889119904)119879

1198797+ 119879 (119905) 119879

8]

times [1198601119894119909 (119905) minus 119860

2119894119909 (119905 minus 119889

1(119905))

minus119861119894119865119895119909 (119905 minus 119889

2(119905)) minus (119905)]

= 120577119879 (119905)(119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))Φ

119894119895)120577 (119905)

(23)

6 Mathematical Problems in Engineering

where

120577119879 (119905) = [119909119879 (119905) 119909119879 (119905 minus 1198891(119905)) 119909119879 (119905 minus ℎ

1) (int

119905

119905minusℎ1

119909 (119904) 119889119904)119879

119909119879 (119905 minus 1198892(119905)) 119909119879 (119905 minus ℎ

2) (int

119905

119905minusℎ2

119909 (119904) 119889119904)119879

119879 (119905)] (24)

Φ119894119895=

[[[[[[

[

Φ11119894 Φ12119894 Φ13 Φ14 Φ15119894119895 Φ16 Φ17 Φ18

lowast Φ22119894 119860119879

2119894119879119879

3119860119879

2119894119879119879

4Φ25119894119895 119860

119879

2119894119879119879

6119860119879

2119894119879119879

7Φ28119894

lowast lowast Φ33119894 minus11987522 1198793119861119894119865119895 0 0 minus1198793

lowast lowast lowast Φ44 1198794119861119894119865119895 0 0 Φ48

lowast lowast lowast lowast Φ55119894119895 119865119879

119895119861119879

119894119879119879

6119865119879

119895119861119879

119894119879119879

7Φ58119894

lowast lowast lowast lowast lowast Φ66 Φ67 minus1198796

lowast lowast lowast lowast lowast lowast Φ77 Φ78

lowast lowast lowast lowast lowast lowast lowast Φ88

]]]]]]

]

Φ11119894= 11987512+ 11987511987912+11988212+11988211987912+ ℎ111988511+ ℎ211987311

minus1ℎ1

11988522minus1ℎ2

11987322+ 1198761+ 1198762+1198721+1198722

minus 21198771minus 21198772+ 11987911198601119894+ 11986011987911198941198791198791

Φ12119894= 11987911198602119894+ 11986011987911198941198791198792

Φ13= minus11987512minus1ℎ1

11988522+ 11986011987911198941198791198793

Φ14= 11987522minus1ℎ2

11988511987912+1ℎ1

1198771+ 11986011987911198941198791198794

Φ15119894119895

= 1198791119861119894119865119895+ 11986011987911198941198791198795

Φ16= minus119882

12minus1ℎ2

11987322+ 11986011987911198941198791198796

Φ17= 11988222minus1ℎ2

11987311987912+2ℎ22

1198772+ 11986011987911198941198791198797

Φ18= 11987511+11988211+ ℎ111988512+ ℎ211987312minus 1198791+ 11986011987911198941198791198798

Φ22119894= minus (1 minus 120583

1) 1198761+ 11987921198602119894+ 11986011987921198941198791198792

Φ25119894119895

= 1198792119861119894119865119895+ 11986011987921198941198791198795

Φ28119894= minus1198792+ 11986011987921198941198791198798

Φ33= minus

1ℎ11988522minus1198721

Φ44= minus

1ℎ1

11988511minus2ℎ21

1198771

Φ48= 11987511987912minus 1198794

Φ55119894= minus (1 minus 120583

2) 1198762+ 1198795119861119894119865119895+ 1198651198791198951198611198791198941198791198795

Φ58119894= minus1198795+ 1198651198791198951198611198791198941198791198798

Φ66= minus

1ℎ2

11987322minus1198722

Φ67= minus

1ℎ2

11987311987912minus11988222

Φ77= minus

1ℎ2

11987311minus2ℎ22

1198772

Φ78= 11988211987912minus 1198797

Φ88= ℎ111988522+ ℎ211987322+12ℎ211198771+12ℎ221198772minus 1198798

(25)

From (25) it is obvious that if

119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))Φ

119894119895lt 0 (26)

(119909(119905)) lt 0From (26) we can discover that the feedback gains

are predefined The following proof presents the controllerdesign method We first pre- and postmultiply both thediag [119883 119883 119883 119883 119883 119883 119883 119883 ] and transpose to both sides of (25)and pre- and postmultiply the diag [119883 119883 ] transpose to bothsides of 119875 119882 119885 and 119873 and the 119883 and transpose to bothsides of 119876

119894119872119894 and 119877

119894 119894 = 1 2 Let 119879

119894= 1199051198941198791(119894 = 2 8)

and denote new variables 119883 = 119879minus11 1198771= 119883119877

1119883119879 119876

1=

1198831198761119883119879119872

1= 119883119872

1119883119879 119877

2= 119883119877

2119883119879 119876

2= 119883119876

2119883119879 119872

2=

1198831198722119883119879 Λ

119894= 119883Λ

119894119883119879 Λ

119895= 119883Λ

119895119883119879 119894 = 1 2 119903 119875

11=

11988311987511119883119879 119875

12= 119883119875

12119883119879 119875

22= 119883119875

22119883119879 119882

11= 119883119882

11119883119879

11988212= 119883119882

12119883119879 119882

22= 119883119882

22119883119879 119885

11= 119883119885

11119883119879 119885

12=

11988311988512119883119879 119885

22= 119883119885

22119883119879 119873

11= 119883119873

11119883119879 119873

12= 119883119873

12119883119879

and11987322= 119883119873

22119883119879 With 119865

119894= 119884119894119883minus119879 119894 = 1 2 119903we next

get

119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))Φ

119894119895lt 0 (27)

Mathematical Problems in Engineering 7

where

Φ119894119895=

[[[[[[[[

[

Φ11119894 Φ12119894 Φ13 Φ14 Φ15119894119895 Φ16 Φ17 Φ18

lowast Φ22119894 1199053119883119860119879

21198941199054119883119860119879

2119894Φ25119894119895 1199056119883119860

119879

21198941199057119883119860119879

2119894Φ28119894

lowast lowast Φ33119894 minus11987522 1199053119861119894119884119895 0 0 minus1199053119883119879

lowast lowast lowast Φ44 1199054119861119894119884119895 0 0 Φ48

lowast lowast lowast lowast Φ55119894119895 1199056119884119879

119895119861119879

1198941199057119884119879

119895119861119879

119894Φ58119894

lowast lowast lowast lowast lowast Φ66 Φ67 minus1199056119883119879

lowast lowast lowast lowast lowast lowast Φ77 Φ78

lowast lowast lowast lowast lowast lowast lowast Φ88

]]]]]]]]

]

Φ11119894= 11987512+ 119875119879

12+11988212+119882119879

12+ ℎ111988511

+ ℎ211987311minus1ℎ1

11988522minus1ℎ2

11987322+ 1198761+ 1198762

+1198721+1198722minus 21198771minus 21198772+ 1198601119894119883119879 + 119883119860119879

1119894

Φ12119894= 1198602119894119883119879 + 119905

21198831198601198791119894

Φ13= minus11987512minus1ℎ 111988522+ 11990531198831198601198791119894

Φ14= 11987522minus1ℎ1

119885119879

12+1ℎ1

1198771+ 11990541198831198601198791119894

Φ15119894119895

= 119861119894119884119895+ 11990551198831198601198791119894

Φ16= minus119882

12minus1ℎ2

11987322+ 11990561198831198601198791119894

Φ17= 11988222minus1ℎ2

119873119879

12+2ℎ22

1198772+ 11990571198831198601198791119894

Φ18= 11987511+11988211+ ℎ111988512+ ℎ211987312minus 119883119879 + 119905

81198831198601198791119894

Φ22119894= minus (1 minus 120583

1) 1198761+ 11990521198602119894119883119879 + 119905

21198831198601198792119894

Φ25119894119895

= 1199052119861119894119884119895+ 11990551198831198601198792119894

Φ28119894= minus1199052119883119879 + 119905

81198831198601198792119894

Φ33= minus

1ℎ1

11988522minus1198721

Φ44= minus

1ℎ1

11988511minus2ℎ21

1198771

Φ48= 119875119879

12minus 1199054119883119879

Φ55119894= minus (1 minus 120583

2) 1198762+ 1199055119861119894119884119895+ 1199055119884119879119895119861119879119894

Φ58119894= minus1199055119883119879 + 119905

8119884119879119895119861119879119894

Φ66= minus

1ℎ2

11987322minus1198722

Φ67= minus

1ℎ2

119873119879

12minus11988222

Φ77= minus

1ℎ2

11987311minus2ℎ22

1198772

Φ78= 119882119879

12minus 1199057119883119879

Φ88= ℎ111988522+ ℎ211987322+12ℎ211198771+12ℎ221198772minus 1199058119883119879

(28)

Here ldquolowastrdquo denotes the transpose elements in the symmetricpositions

If (27) holds then (119909(119905)) lt 0 Considersum119903119894=1sum119903119895=1

119908119894(119909(119905))(119908

119895(119909(119905)) minus 119898

119895(119909(119905 minus 119889

2(119905))))Λ

119894= 0

where Λ119894= Λ119879119894isin 1198778119899times8119899 gt 0 119894 = 1 2 119903 are arbitrary

matrices These terms are introduced to (27) to alleviate theconservativeness From (27) we also have

Φ =119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))Φ

119894119895

=119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905)) (119908

119895(119909 (119905)) minus 119898

119895(119909 (119905 minus 119889

2(119905)))

+120588119895119908119895(119909 (119905)) minus 120588

119895119908119895(119909 (119905))) Λ

119894

+119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))Φ

119894119895

=119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905)) 119908

119895(119909 (119905)) (120588

119895Φ119894119895minus 120588119895Λ119894+ Λ119894)

+119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905)) (119898

119895(119909 (119905 minus 119889

2(119905))) minus 120588

119895119908119895(119909 (119905)))

times (Φ119894119895minus Λ119894)

le119903

sum119894=1

1199082119894(120588119894Φ119894119894minus 120588119894Λ119894+ Λ119894)

+119903

sum119894=1

119903

sum119895=1

119908119894(119898119895minus 120588119895119908119895) (Φ119894119895minus Λ119894)

+119903

sum119894=1

sum119894lt119895

119908119894119908119895(120588119895Φ119894119895+ 120588119894Φ119895119894minus 120588119895Λ119894minus 120588119894Λ119895

+Λ119894+ Λ119895)

(29)

with119898119895(119909(119905 minus 119889

2(119905))) minus 120588

119895119908119895(119909(119905)) ge 0 for all 119895 119909(119905) and 119909(119905 minus

1198892(119905)) Let

Φ119894119895minus Λ119894lt 0 (30)

120588119894Φ119894119894minus 120588119894Λ119894+ Λ119894lt 0 (31)

120588119895Φ119894119895+ 120588119894Φ119895119894minus 120588119895Λ119894minus 120588119894Λ119895+ Λ119894+ Λ119895le 0 119894 lt 119895 (32)

for all 119894 119895 = 1 2 119903

8 Mathematical Problems in Engineering

Since (119909(119905)) lt 0 the fuzzy control system with the timevarying state and input delays (15) is asymptotically stablewith the state feedback control law (17)

32 Robust Stability of Uncertain Fuzzy Systems We alsoexamine the design of a robust stable controller for theuncertain system (4) under the imperfect premise matchingConsider the following uncertain fuzzy time varying delaycontrol system

(119905) =119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))

times [(1198601119894+ Δ1198601119894(119905)) 119909 (119905) + (119860

2119894+ Δ1198602119894(119905))

times 119909 (119905 minus 1198891(119905)) + (119861

119894+ Δ119861119894(119905))

times119906 (119905 minus 1198892(119905))]

119909 (119905) = 120593 (119905) 119905 isin [minusmax (ℎ1 ℎ2) 0]

(33)

Based on the above results of Theorem 4 a robust stabiliza-tion criterion for theT-S fuzzy systemswith time varying stateand input delays is investigated The following result can beobtained

Theorem 5 Given scalars ℎ1ge 0 ℎ

2ge 0 120583

1 1205832 and 119905

119894(119894 =

2 8) the uncertain fuzzy control systems with the timevarying state and input delays (33) is robustly stable for any0 le 119889

119894(119905) le ℎ

119894(119894 = 1 2) via the imperfect premise matching

controller design technique if the membership functions of thefuzzy model and fuzzy controller satisfy 119898

119895(119909(119905 minus 119889

2(119905))) minus

120588119895119908119895(119909(119905)) ge 0 for all 119895 119909(119905) and 119909(119905 minus 119889

2(119905)) where 0 lt

120588119895lt 1 and there exist common matrices 119875

11gt 0 119875

22gt 0

11988211gt 0 119882

22gt 0 119885

11gt 0 119885

22gt 0 119873

11gt 0 119873

22gt 0

119876119894gt 0 119872

119894gt 0 119877

119894gt 0 (119894 = 1 2) and Λ

119894= ΛT119894isin 1198778119899times8119899 gt

0 (i = 1 2 r) 119875 gt 0 119882 gt 0 119885 gt 0 119873 gt 0 and somematrices 119884

119895(119895 = 1 2 119903) 119883 and scalars 120576

1119894gt 0 120576

2119894gt 0

120576119887119894119895gt 0 satisfy the following LMIs

Ψ119894119895minus Λ119894lt 0 (34)

120588119894Ψ119894119894minus 120588119894Λ119894+ Λ119894lt 0 (35)

120588119895Ψ119894119895+ 120588119894Ψ119895119894minus 120588119895Λ119894minus 120588119894Λ119895+ Λ119894+ Λ119895 le 0 119894 lt 119895 (36)

where Ψ119894119895is defined as in (44) The state feedback gains can be

constructed as 119865119894= 119884119894119883minus119879

Proof If 1198601119894 1198602119894 and 119861

119894in Φ119894119895are replaced with 119860

1119894+

119863119894119870119894(119905)1198641119894 1198602119894+ 119863119894119870119894(119905)1198642119894 and 119861

119894+ 119863119894119870119894(119905)119864119887119894in (28) of

Theorem 4 respectively (27) for system (33) can be rewrittenas

119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889 (119905))) Ω

119894119895lt 0 (37)

where

Ω119894119895= Φ119894119895+ 1198631198791119894119870119894(119905) 119864119894+ 119864119879119894119870119879119894(119905) 1198631119894+ 1198631198792119894119870119894(119905) 1198642119894119895

+ 1198641198792119894119895119870119879119894(119905) 1198632119894+ 119863119879119887119894119870119894(119905) 119864119887119894119895+ 119864119879119887119894119895119870119879119894(119905) 119863119887119894

1198631119894= [1198631198791198940 0 0 0 0 0 0]

1198632119894= [0 119863119879

1198940 0 0 0 0 0]

119863119887119894= [0 0 0 0 119863119879

1198940 0 0]

(38)

1198641119894= [1198641119894119883119879 11990521198641119894119883119879 11990531198641119894119883119879 11990541198641119894119883119879 119905

51198641119894119883119879 11990561198641119894119883119879 11990571198641119894119883119879 11990581198641119894119883119879]

1198642119894= [1198642119894119883119879 11990521198642119894119883119879 11990531198642119894119883119879 11990541198642119894119883119879 11990551198642119894119883119879 11990561198642119894119883119879 11990571198642119894119883119879 11990581198642119894119883119879]

119864119887119894119895= [119864119887119894119884119895 1199052119864119887119894119884119895 1199053119864119887119894119884119895 1199054119864119887119894119884119895 1199055119864119887119894119884119895 1199056119864119887119894119884119895 1199057119864119887119894119884119895 1199058119864119887119894119884119895]

(39)

If Ω119894119895

lt 0 (37) holds According to Lemma 2 it isstraightforward to know that Ω

119894119895lt 0 is true if for each 119894

119895 there exists scalars 1205761119894gt 0 120576

2119894gt 0 and 120576

119887119894119895gt 0 such that

the following inequality holds

Φ119894119895+ 120576111989411986311987911198941198631119894+ 120576minus1111989411986411987911198941198641119894+ 120576211989411986311987921198941198632119894

+ 120576minus1211989411986411987921198941198642119894+ 120576119887119894119895119863119879119887119894119863119887119894+ 120576minus1119887119894119895119864119879119887119894119895119864119887119894119895lt 0

(40)

Based on Schur complement (40) is equivalent to thefollowing inequality

Ψ119894119895=[[[

[

Φ119894119895

1198641198791119894

1198641198792119894

119864119879119887119894119895

lowast minus1205761119894119868 0 0

lowast lowast minus1205762119894119868 0

lowast lowast lowast minus120576119887119894119895119868

]]]

]

lt 0 (41)

Mathematical Problems in Engineering 9

where

Φ119894119895=

[[[[[[[[

[

Φ11119894 Φ12119894 Φ13 Φ14 Φ15119894119895 Φ16 Φ17 Φ18

lowast Φ22119894 1199053119883119860119879

21198941199054119883119860119879

2119894Φ25119894119895 1199056119883119860

119879

21198941199057119883119860119879

2119894Φ28119894

lowast lowast Φ33119894 minus11987522 1199053119861119894119884119895 0 0 minus1199053119883119879

lowast lowast lowast Φ44 1199054119861119894119884119895 0 0 Φ48

lowast lowast lowast lowast Φ55119894119895 1199056119884119879

119895119861119879

1198941199057119884119879

119895119861119879

119894Φ58119894

lowast lowast lowast lowast lowast Φ66 Φ67 minus1199056119883119879

lowast lowast lowast lowast lowast lowast Φ77 Φ78

lowast lowast lowast lowast lowast lowast lowast Φ88

]]]]]]]]

]

lt 0

Φ11119894= Φ11119894+ 1205761119894119863119894119863119879119894

Φ22119894= Φ22119894+ 1205762119894119863119894119863119879119894

Φ55119894119895

= Φ55119894119895

+ 120576119887119894119863119894119863119879119894

(42)

and Φ are the same as in Theorem 4 Therefore (41) holds ifand only if the following inequality holds

119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889 (119905))) Ψ

119894119895lt 0 (43)

To alleviate the conservativeness of our robust stabilityanalysis similar to the analysis approach of Theorem 4 weintroduce sum119903

119894=1sum119903119895=1

119908119894(119908119895minus 119898119895)Λ119894= 0 to (43) and the

following proof is the same as in Theorem 4 We can obtainthe conditions and (34)ndash(36) Thus the fuzzy control systemwith the time varying state and input delays (34) is robuststable on the basis of the control law 119865

119894= 119884119894119883minus119879under these

conditions and (34)ndash(36) hold

Remark 6 If we set 119908119894(119909(119905)) = 119898

119895(119909(119905)) 119889

1(119905) = 119889

2(119905)

the closed-loop system in this paper has the same structureas that of paper [26] On the other hand let 119908

119894(119909(119905)) =

119898119895(119909(119905)) and 119889

1(119905) = 119889

2(119905) = 119889 we can obtain the system

in [16] Therefore the system studied in this paper is moregeneral Additionally the information of the membershipfunctions of the fuzzy time delay models and controllers areall considered in the stability analysis If the fuzzymodels andfuzzy controllers share the same fuzzymembership functionsthat is 119908

119894(119909(119905)) = 119898

119895(119909(119905)) the stability analysis can be

referred to [25 27 28] In other words the problem in [2527 28] is actually a special case of the one investigated in thispaper

Remark 7 The augmented Lyapunov function with an addi-tional triple-integral term is introduced in analyzing thestability problem for the T-S fuzzy systems with time varyingstate and input delays In addition two integral inequalitiesare used to derive Theorem 4 and less free-weighting matri-ces are introduced which lead to get less conservative resultsIt can be observed that some existing results are the specialcases of this paper For example the method in [25] is from(18) with 119875

12= 0 119875

22= 0 119885

12= 11988522= 0 119873

12= 11987322= 0

and 1198771= 1198772= 0 in Theorem 4 The results in [22] are based

on system (33) with 1198892(119905) = 0 and (18) with119872

2= 0 119885

12=

11988522= 0 119876

1= 1198762= 0 and 119877

1= 1198772= 0

Remark 8 Different from the general PDC technique thedesign method under the imperfect premise matching ismuch more flexible because the membership functions ofthe fuzzy controllers do not need to be chosen the same asthose of the fuzzy time delay models Instead they can bedesigned arbitrarilyThus the design flexibility is significantlyenhanced On the other hand some simple membershipfunctions of the fuzzy controllers might be employed whichcan reduce the implementation cost

Remark 9 In [7ndash11 15ndash20] the authors have considered therobust control for uncertain T-S fuzzy systems with statedelays Some delay-dependent conditions and fuzzy con-trollers are obtained with a traditional Lyapunov-Krasovskiifunctionalmethod andPDCmethod which all are the specialcases of Theorem 5 in this paper

4 Numerical Examples

In this section two numerical examples are given to illus-trate the conservativeness and effectiveness of the proposedmethods The first example compares our techniques withthe existing ones in the literature for stability analysis whichshows that Theorem 4 in this paper is less conservative thanthe other results The second example is used to illustrate theadvantage of the robust stability conditions ofTheorem 5 anddemonstrate how to design robust fuzzy controllers by usingour approach

Example 10 Consider the following fuzzy system with stateand input delays

(119905) =2

sum119894=1

119908119894(120579 (119905)) [119860

1119894119909 (119905) + 119860

2119894119909 (119905 minus 119889

1(119905))

+119861119894119906 (119905 minus 119889

2(119905))]

11986011= [0 06

0 1 ] 11986021= [05 09

0 2 ]

11986012= [1 0

1 0] 11986022= [09 0

1 16]

1198611= 1198612= [11]

(44)

The fuzzy membership functions are selected as [31]

1199081(1199091(119905)) = (1 minus

119888 (119905) sin (10038161003816100381610038161199091 (119905)1003816100381610038161003816minus4)5

1 + expminus1001199091(119905)3(1minus1199091(119905)))

timescos (119909

1(119905))2

1 + expminus251199091(119905)(3+(1199091(119905)042))

times 1199082(1199091(119905)) = 1 minus 119908

1(1199091(119905))

(45)

10 Mathematical Problems in Engineering

Table 1 The maximum allowable time delay and feedback gains(1205831= 0 120583

2= 0)

Paper ℎ1

ℎ2 Feedback gains

[26] 03120 031201198651= [10598 minus56598]

1198652= [minus13068 minus41167]

Theorem 4 033 0571198651= [minus00042 minus00449]1198652= [00048 00717]

where

1199091(119905) isin [minus

1205872

1205872]

119888 (119905) =sin (119909

1(119905)) + 140

isin [minus005 005] (46)

If we set 1198891(119905) = 119889

2(119905) in system (44) we can obtain the

system in [26] which implies that our system ismore generalEmploying the LMIs in [26] and those inTheorem 4yields themaximum state and input delays ℎ

1 and ℎ

2 which guarantee

the stability of system (44) and the feedback gains for 1205831= 0

and 1205832= 0 when 120588

1= 075 120588

2= 095 119905

2= 01 119905

3=

02 1199054= 07 119905

5= 01 119905

6= 04 119905

7= 01 and 119905

8= 13 as

given in Table 1 which clearly shows the superiority of theresults derived in this paper over those obtained from [26]

Example 11 Consider the well-known truck-trailer systemwhich can be described by the following T-S fuzzy system

(119905) =2

sum119894=1

119908119894(120579 (119905)) [(119860

1119894+ Δ1198601119894) 119909 (119905) + 119860

2119894119909 (119905 minus 119889

1(119905))

+ (119861119894+ Δ119861119894) 119906 (119905 minus 119889

2(119905))]

(47)

11986011=

[[[[[[[[[

[

minus119886V1199051198711199050

0 0

119886V1199051198711199050

0 0

119886V21199052

21198711199050

V1199051199050

0

]]]]]]]]]

]

11986021=

[[[[[[[[[

[

minus (1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)V21199052

21198711199050

0 0

]]]]]]]]]

]

1198611=[[[[

[

V1199051198971199050

0

0

]]]]

]

11986012=

[[[[[[[[[

[

minus119886V1199051198711199050

0 0

119886V1199051198711199050

0 0

119886119889V21199052

21198711199050

119889V1199051199050

0

]]]]]]]]]

]

11986022=

[[[[[[[[[

[

minus (1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)119889V21199052

21198711199050

0 0

]]]]]]]]]

]

(48)

1198612=[[[[

[

V1199051198971199050

0

0

]]]]

]

(49)

where 119897 = 28 119871 = 55 V = minus10 119905 = 20 1199050= 05

119889 = 101199050120587 119863

119894= [02555 02555 02555]119879 119864

1119894= 1198642119894=

[01 0 0]119879 119864119887119894

= [01 0 0]119879 (119894 = 1 2) and the fuzzymembership functions are selected the same as inExample 10

(1) Let 119886 = 1 system (47) is the same as the fuzzy systemof [22] which implies that our system ismore generalTable 2 gives themaximum input delay value of ℎ

2 for

which the stabilization is guaranteed byTheorem 5 aswell as the state feedback gains when 120588

1= 075 120588

2=

095 1199052= 01 119905

3= 02 119905

4= 07 119905

5= 01 119905

6=

04 1199057= 08 and 119905

8= 2 It is clearly visible that the

results derived in this paper are better than those from[22 28]

(2) In the case of 119886 = 1 the state and input delays of system(47) are all exist The approach proposed in [22] cannot be used to get the maximum of state delay ℎ

1

as the system in [22] only contains the input delayHowever the method in [28] and Theorem 5 can beused Here let 119886 = 07 120588

1= 075 120588

2= 095 119905

2=

01 1199053= 02 119905

4= 07 119905

5= 01 119905

6= 04 119905

7=

08 and 1199058= 3 and Table 3 provides the maximum

upper bounds of ℎ1and ℎ

2and the state feedback

gains for which the robust stabilization is guaranteedbyTheorem 5

By Theorem 5 we can conclude that the uncertain fuzzymodel (47) is robust stable based on the following fuzzycontrol law

119906 (119905 minus ℎ2) =2

sum119895=1

119898119895(1199091(119905 minus ℎ

2)) 119865119895119909 (119905 minus ℎ

2) (50)

With the PDC design technique it is required that thefuzzy controller must share the same fuzzy membershipfunctions as those of the fuzzy time delay model Under such

Mathematical Problems in Engineering 11

Table 2 The maximum allowable time delay and feedback gains(1205832= 0)

Paper ℎ2 Feedback gains

[22] 0751198651= [34227 minus03535 00045]

1198652= [35215 minus03617 00056]

[28] 0861198651= [33219 minus02406 00025]

1198652= [33272 minus02494 00026]

Theorem 5 121198651= [minus00212 00093 minus00024]1198652= [minus00250 00044 00014]

Table 3 The maximum allowable time delay and feedback gains(1205831= 1205832= 0)

Paper ℎ1

ℎ2 Feedback gains

[28] 01 0551198651= [33219 minus02406 00025]

1198652= [33272 minus02494 00026]

Theorem 5 04 171198651= [minus00128 00090 minus00017]1198652= [minus00194 00012 00009]

a condition the implementation cost of the fuzzy controlleris high by employing these complex membership functionsHowever fromTheorem 5 the membership functions do notneed to be chosen the same as119908

1(1199091(119905)) and119908

2(1199091(119905)) Thus

we can select some simple membership functions insteadsuch as

1198981(1199091(119905 minus ℎ

2)) = 075 exp

(minus1199091(119905 minus ℎ

2) minus 038)2

2 times 0382+ 005

1198982(1199091(119905 minus ℎ

2)) = 1 minus 119898

1(1199091(119905 minus ℎ

2))

(51)

We also assume that 1205881= 075 and 120588

2= 095 such that

119898119895(1199091(119905 minus ℎ

2)) minus 120588

119895119908119895(1199091(119905)) gt 0 for 119895 = 1 2 and 119909

1(119905 minus ℎ

2)

The fuzzy control law 119906(119905 minus ℎ2) = 119898

1[minus00128 00090 minus

00017]119909 + 1198982[minus00194 00012 00009]119909 can be employed

to stabilize the uncertain fuzzy control systems in (47) withthe state and input delays Figures 1 and 2 illustrate the stablestate response and control input under the initial conditionof 119909(0) = [4 minus 1 2]119879

Remark 12 In Example 11 our method offers less conser-vative results in the sense of allowing longer time delayand obtaining smaller feedback control gains Moreovercompared with [22ndash28] where the fuzzy controllermust havethe same fuzzy membership functions as those of the fuzzytime delay model the above membership functions of ourfuzzy controller are much simper which can considerablylower the implementation cost of the fuzzy controller andenhance the design flexibility

Remark 13 If the function of 119888(119905) in 1199081(1199091(119905)) is unknown

on the basis of the PDC design technique [7ndash11 15ndash20 22ndash28] the fuzzy controller cannot be implemented as 119888(119905) isnot available However using the proposed design method

0 5 10 15

0

2

4

6Closed-loop response

minus12

minus10

minus8

minus6

minus4

minus2

119905 (s)

119909(119905)

1199092

11990931199091

Figure 1The state responses of system (47) with ℎ1= 04 ℎ

2= 17

0 5 10 15

0

5

10

15

20

25

30

119905 (s)

minus20

minus15

minus10

minus5

119906(119905)

Figure 2 The control input of system (47) with ℎ1= 04 ℎ

2= 17

we can select some simper and well-known membershipfunctions that are different from those of the fuzzy time delaymodels for example119898

1(1199091(119905)) and119898

2(1199091(119905)) so as to realize

the fuzzy controller In other words the proposed designapproach can be applied even for the fuzzy time delay modelswith uncertain grades of membership

5 Conclusions

In this paper we study the robust stability and stabilizationproblems for general nonlinear fuzzy systems with timevarying state and input delays A less conservative delay-dependent robust stability criterion has been obtained byintroducing a novel augmented Lyapunov function withan additional triple-integral term Moreover a new design

12 Mathematical Problems in Engineering

approach different from the traditional PDC design tech-nique is proposed which can significantly improve the designflexibility and reduce the implementation cost of the fuzzycontroller by arbitrarily selecting simple fuzzy membershipfunctions Two numerical examples are used to illustrate theconservativeness and effectiveness of our proposed methods

Acknowledgments

This research work was funded by the NSFC under Grantno 61273095 and the Academy of Finland under Grants no135225 and no 127299

References

[1] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[2] W H Ho S H Chen I T Chen J H Chou and C C ShuldquoDesign of stable and quadratic-optimal static output feed-back controllers for T-S-fuzzy-model-based control systems anintegrative computational approachrdquo International Journal ofInnovative Computing Information and Control vol 8 no 1App 403ndash418 2012

[3] X J Su L G Wu P Shi and Y D Song ldquo119867infinmodel reduction

of T-S fuzzy stochastic systemsrdquo IEEE Transactions on SystemsMan and Cybermetics B vol 42 no 6 pp 1574ndash1585 2012

[4] R Yang P Shi G-P Liu andH Gao ldquoNetwork-based feedbackcontrol for systems with mixed delays based on quantizationand dropout compensationrdquo Automatica vol 47 no 12 pp2805ndash2809 2011

[5] XM Zhang Z J Zhang andG P Lu ldquoFault detection for state-delay fuzzy systems subject to random communication delayrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 4 pp 2439ndash2451 2012

[6] F B Li and X Zhang ldquoDelay-range-dependent robust 119867infin

filtering for singuar LPV systems with time variant delayrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 1 pp 339ndash353 2013

[7] Y Y Cao and P M Frank ldquoAnalysis and synthesis of nonlineartime-delay systems via fuzzy control approachrdquo IEEE Transac-tions on Fuzzy Systems vol 8 no 2 pp 200ndash211 2000

[8] Y-Y Cao and P M Frank ldquoStability analysis and synthesis ofnonlinear time-delay systems via linear Takagi-Sugeno fuzzymodelsrdquo Fuzzy Sets and Systems vol 124 no 2 pp 213ndash2292001

[9] O M Kwon and J H Park ldquoDelay-range-dependent stabiliza-tion of uncertain dynamic systems with interval time-varyingdelaysrdquo Applied Mathematics and Computation vol 208 no 1pp 58ndash68 2009

[10] E Tian D Yue and Y Zhang ldquoDelay-dependent robust 119867infin

control for T-S fuzzy system with interval time-varying delayrdquoFuzzy Sets and Systems vol 160 no 12 pp 1708ndash1719 2009

[11] J Y An T Li G LWen and R Li ldquoNew stability conditions foruncertain T-S fuzzy systems with interval time-varying delayrdquoInternational Journal of Control Automation and Systems vol10 no 3 pp 490ndash497 2012

[12] Z J Zhang X L Huang X J Ban and X Z Gao ldquoStabilityanalysis and controller design of T-S fuzzy systems with time-delay under imperfect premise matchingrdquo Journal of BeijingInstitute of Technology vol 21 no 3 pp 387ndash393 2012

[13] O M Kwon M J Park S M Lee and J H Park ldquoAugmentedLyapunov-Krasovskii functional approaches to robust stabilitycriteria for uncertain Takagi-Sugeno fuzzy systems with time-varying delaysrdquo Fuzzy Sets and Systems vol 201 no 16 pp 1ndash192012

[14] F Liu M Wu Y He and R Yokoyama ldquoNew delay-dependentstability criteria for T-S fuzzy systems with time-varying delayrdquoFuzzy Sets and Systems vol 161 no 15 pp 2033ndash2042 2010

[15] B Chen X Liu and S Tong ldquoNew delay-dependent stabiliza-tion conditions of T-S fuzzy systems with constant delayrdquo FuzzySets and Systems vol 158 no 20 pp 2209ndash2224 2007

[16] J Yoneyama ldquoRobust stability and stabilization for uncertainTakagi-Sugeno fuzzy time-delay systemsrdquo Fuzzy Sets and Sys-tems vol 158 no 2 pp 115ndash134 2007

[17] C Lin Q-G Wang and T H Lee ldquoDelay-dependent LMIconditions for stability and stabilization of T-S fuzzy systemswith bounded time-delayrdquo Fuzzy Sets and Systems vol 157 no9 pp 1229ndash1247 2006

[18] X J Su P Shi L G Wu and Y D Song ldquoA novel approachto filter design for T-S fuzzy discrete-time systems with time-varying delayrdquo IEEE Transactions on Fuzzy Systems vol 20 no6 pp 1114ndash1129 2012

[19] C Peng and Q-L Han ldquoDelay-range-dependent robust stabi-lization for uncertain T-S fuzzy control systems with intervaltime-varying delaysrdquo Information Sciences vol 181 no 19 pp4287ndash4299 2011

[20] S T Li Y W Jing and X M Liu ldquoRobust control for a class ofT-S fuzzy systemswith interval time-varying delayrdquo InternatinalJournal of Control Automation and Systems vol 10 no 4 pp737ndash743 2012

[21] X-M Zhang M Wu J-H She and Y He ldquoDelay-dependentstabilization of linear systems with time-varying state and inputdelaysrdquo Automatica vol 41 no 8 pp 1405ndash1412 2005

[22] B Chen X Liu and S Tong ldquoRobust fuzzy control of nonlinearsystems with input delayrdquo Chaos Solitons amp Fractals vol 37 no3 pp 894ndash901 2008

[23] C Lin Q-G Wang and T H Lee ldquoDelay-dependent LMIconditions for stability and stabilization of T-S fuzzy systemswith bounded time-delayrdquo Fuzzy Sets and Systems vol 157 no9 pp 1229ndash1247 2006

[24] H J Lee J B Park and Y H Joo ldquoRobust control for uncertainTakagi-Sugeno fuzzy systems with time-varying input delayrdquoJournal of Dynamic Systems Measurement and Control Trans-actions of the ASME vol 127 no 2 pp 302ndash306 2005

[25] C-H Lien and K-W Yu ldquoRobust control for Takagi-Sugenofuzzy systems with time-varying state and input delaysrdquo ChaosSolitons and Fractals vol 35 no 5 pp 1003ndash1008 2008

[26] Y K Su B Chen S Y Zhang and C Lin ldquoDelay-dependentstabilization of T-S fuzzy systems with time delayrdquo Control andDecision Kongzhi yu Juece vol 24 no 6 pp 921ndash927 2009

[27] B Chen X Liu C Lin and K Liu ldquoRobust 119867infin

control ofTakagi-Sugeno fuzzy systems with state and input time delaysrdquoFuzzy Sets and Systems vol 160 no 4 pp 403ndash422 2009

[28] L Li and X Liu ldquoNew results on delay-dependent robuststability criteria of uncertain fuzzy systems with state and inputdelaysrdquo Information Sciences vol 179 no 8 pp 1134ndash1148 2009

[29] Z J Zhang X J Ban X L Huang and X Z Gao ldquoNew delay-dependent robust stablity for T-S fuzzy systems with intervevaltime-varying delayrdquo in Proccedings of the 5th International JointConference on Computational Science and Optimization pp826ndash830 Harbin China 2012

Mathematical Problems in Engineering 13

[30] H K Lam and F H F Leung ldquoLMI-based stability andperformance design of fuzzy control systems fuzzy modelsand controllers with different premisesrdquo in Proceedings of theInternational Conference on Fuzzy Systerm pp 9599ndash9506Vancouver Canada 2006

[31] H K Lam andMNarimani ldquoStability analysis and perfomancedesign for fuzzy-model-based control system under imperfectpremise matchingrdquo IEEE Transactions on Fuzzy Systems vol 17no 4 pp 949ndash961 2009

[32] S Boyd L E Ghaoui and E Feron Linear Matrix Inequality inSystems and ControlTheory SIAM Philadelphia Pa USA 1994

[33] I R Petersen and C V Hollot ldquoA Riccati equation approach tothe stabilization of uncertain linear systemsrdquo Automatica vol22 no 4 pp 397ndash411 1986

[34] J Sun G P Liu and J Chen ldquoDelay-dependent stabilityand stabilization of neutral time-delay systemsrdquo InternationalJournal of Robust andNonlinear Control vol 19 no 12 pp 1364ndash1375 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

By Lemma 3 we have

minus int119905

119905minusℎ1

[119909119905 (119904) 119879 (119904)] [11988511 11988512lowast 11988522

] [119909 (119904) (119904)] 119889119904

le minus1ℎ1

[(int119905

119905minusℎ1

119909 (119904) 119889119904)119879

119909119879 (119905) minus 119909119879 (119905 minus ℎ1)]

times [11988511 11988512lowast 11988522

][

[

int119905

119905minusℎ1

119909 (119904) 119889119904

119909 (119905) minus 119909 (119905 minus ℎ1)]

]

minus int119905

119905minusℎ2

[119909119879 (119904) 119879 (119904)] [11987311 11987312lowast 11987322

] [119909 (119904) (119904)] 119889119904

le minus1ℎ2

[(int119905

119905minusℎ2

119909 (119904) 119889119904)119879

119909119879 (119905) minus 119909119879 (119905 minus ℎ1)]

times [11987311 11987312lowast 11987322

][

[

int119905

119905minusℎ2

119909 (119904) 119889119904

119909 (119905) minus 119909 (119905 minus ℎ2)]

]

minus int0

minusℎ119894

int119905

119905+120579

119879 (119904) 119877119894 (119904) 119889119904 119889120579

le minus2ℎ119894

2(int0

minusℎ119894

int119905

119905+120579

(119904) 119889119904 119889120579)119879

119877119894(int0

minusℎ119894

int119905

119905+120579

(119904) 119889119904 119889120579)

= minus2ℎ119894

2(ℎ119894119909 (119905) minus int

119905

119905minusℎ119894

119909 (119904) 119889119904)119879

times 119877119894(ℎ119894119909 (119905) minus int

119905

119905minusℎ119894

119909 (119904) 119889119904) 119894 = 1 2

(22)

Using the above inequalities andwith the zero quantities (21)we can obtain

(119909119905)

le 2 [119909119879 (119905) (int119905

119905minusℎ1

119909 (119904) 119889119904)119879

] [11987511 11987512lowast 11987522

]

times [ (119905)119909 (119905) minus 119909 (119905 minus ℎ

1)] + ℎ1 [119909

119879 (119905) 119879 (119905)]

times [11988511 11988512lowast 11988522

] [119909 (119905) (119905)] + ℎ2 [119909119879 (119905) 119879 (119905)] [11987311 11987312lowast 119873

22

]

times [119909 (119905) (119905)] minus1ℎ1

[(int119905

119905minusℎ1

119909 (119904) 119889119904)119879

119909119879 (119905) minus 119909119879 (119905 minus ℎ1)]

times [11988511 11988512lowast 11988522

][

[

int119905

119905minusℎ1

119909 (119904) 119889119904

119909 (119905) minus 119909 (119905 minus ℎ1)]

]

minus1ℎ2

[(int119905

119905minusℎ2

119909 (119904) 119889119904)119879

119909119879 (119905) minus 119909119879 (119905 minus ℎ2)]

times [11987311 11987312lowast 11987322

][

[

int119905

119905minusℎ2

119909 (119904) 119889119904

119909 (119905) minus 119909 (119905 minus ℎ2)]

]+ 119909119879 (119905) 119876

1119909 (119905)

+ 119909119879 (119905) 1198762119909 (119905) minus (1 minus 120583

1) 119909119879 (119905 minus 119889

1(119905)) 119876

1119909

times (119905 minus 1198891(119905)) minus (1 minus 120583

2) 119909119879 (119905 minus 119889

2(119905)) 119876

2119909

times (119905 minus 1198892(119905)) + 119909119879 (119905)119872

1119909 (119905) minus 119909119879 (119905 minus ℎ

1)1198721119909

times (119905 minus ℎ1) + 119909119879 (119905)119872

2119909 (119905) minus 119909119879 (119905 minus ℎ

2)1198722119909

times (119905 minus ℎ2) +

12ℎ21119879 (119905) 119877

1 (119905) minus

2ℎ1

2

times(ℎ1119909 (119905)minusint

119905

119905minusℎ1

119909 (119904) 119889119904)119879

1198771(ℎ1119909 (119905)minusint

119905

119905minusℎ1

119909 (119904) 119889119904)

+12ℎ2

2119879 (119905) 1198772 (119905)minus

2ℎ22

(ℎ2119909 (119905)minusint

119905

119905minusℎ2

119909 (119904) 119889119904)119879

times1198772(ℎ2119909 (119905) minus int

119905

119905minusℎ2

119909 (119904) 119889119904)

+ 2 [119909119879 (119905) (int119905

119905minusℎ2

119909 (119904) 119889119904)119879

]

times [11988211 11988212lowast 11988222

] [ (119905)119909 (119905) minus 119909 (119905 minus ℎ

2)]

+ 2119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))

times [119909119879 (119905) 1198791+ 119909119879 (119905 minus 119889

1(119905)) 1198792+ 119909119879 (119905 minus ℎ

1) 1198793

+ (int119905

119905minusℎ1

119909 (119904) 119889119904)119879

1198794+ 119909119879 (119905 minus 119889

2(119905)) 1198795+ 119909119879

times (119905 minus ℎ2) 1198796+(int119905

119905minusℎ2

119909 (119904) 119889119904)119879

1198797+ 119879 (119905) 119879

8]

times [1198601119894119909 (119905) minus 119860

2119894119909 (119905 minus 119889

1(119905))

minus119861119894119865119895119909 (119905 minus 119889

2(119905)) minus (119905)]

= 120577119879 (119905)(119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))Φ

119894119895)120577 (119905)

(23)

6 Mathematical Problems in Engineering

where

120577119879 (119905) = [119909119879 (119905) 119909119879 (119905 minus 1198891(119905)) 119909119879 (119905 minus ℎ

1) (int

119905

119905minusℎ1

119909 (119904) 119889119904)119879

119909119879 (119905 minus 1198892(119905)) 119909119879 (119905 minus ℎ

2) (int

119905

119905minusℎ2

119909 (119904) 119889119904)119879

119879 (119905)] (24)

Φ119894119895=

[[[[[[

[

Φ11119894 Φ12119894 Φ13 Φ14 Φ15119894119895 Φ16 Φ17 Φ18

lowast Φ22119894 119860119879

2119894119879119879

3119860119879

2119894119879119879

4Φ25119894119895 119860

119879

2119894119879119879

6119860119879

2119894119879119879

7Φ28119894

lowast lowast Φ33119894 minus11987522 1198793119861119894119865119895 0 0 minus1198793

lowast lowast lowast Φ44 1198794119861119894119865119895 0 0 Φ48

lowast lowast lowast lowast Φ55119894119895 119865119879

119895119861119879

119894119879119879

6119865119879

119895119861119879

119894119879119879

7Φ58119894

lowast lowast lowast lowast lowast Φ66 Φ67 minus1198796

lowast lowast lowast lowast lowast lowast Φ77 Φ78

lowast lowast lowast lowast lowast lowast lowast Φ88

]]]]]]

]

Φ11119894= 11987512+ 11987511987912+11988212+11988211987912+ ℎ111988511+ ℎ211987311

minus1ℎ1

11988522minus1ℎ2

11987322+ 1198761+ 1198762+1198721+1198722

minus 21198771minus 21198772+ 11987911198601119894+ 11986011987911198941198791198791

Φ12119894= 11987911198602119894+ 11986011987911198941198791198792

Φ13= minus11987512minus1ℎ1

11988522+ 11986011987911198941198791198793

Φ14= 11987522minus1ℎ2

11988511987912+1ℎ1

1198771+ 11986011987911198941198791198794

Φ15119894119895

= 1198791119861119894119865119895+ 11986011987911198941198791198795

Φ16= minus119882

12minus1ℎ2

11987322+ 11986011987911198941198791198796

Φ17= 11988222minus1ℎ2

11987311987912+2ℎ22

1198772+ 11986011987911198941198791198797

Φ18= 11987511+11988211+ ℎ111988512+ ℎ211987312minus 1198791+ 11986011987911198941198791198798

Φ22119894= minus (1 minus 120583

1) 1198761+ 11987921198602119894+ 11986011987921198941198791198792

Φ25119894119895

= 1198792119861119894119865119895+ 11986011987921198941198791198795

Φ28119894= minus1198792+ 11986011987921198941198791198798

Φ33= minus

1ℎ11988522minus1198721

Φ44= minus

1ℎ1

11988511minus2ℎ21

1198771

Φ48= 11987511987912minus 1198794

Φ55119894= minus (1 minus 120583

2) 1198762+ 1198795119861119894119865119895+ 1198651198791198951198611198791198941198791198795

Φ58119894= minus1198795+ 1198651198791198951198611198791198941198791198798

Φ66= minus

1ℎ2

11987322minus1198722

Φ67= minus

1ℎ2

11987311987912minus11988222

Φ77= minus

1ℎ2

11987311minus2ℎ22

1198772

Φ78= 11988211987912minus 1198797

Φ88= ℎ111988522+ ℎ211987322+12ℎ211198771+12ℎ221198772minus 1198798

(25)

From (25) it is obvious that if

119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))Φ

119894119895lt 0 (26)

(119909(119905)) lt 0From (26) we can discover that the feedback gains

are predefined The following proof presents the controllerdesign method We first pre- and postmultiply both thediag [119883 119883 119883 119883 119883 119883 119883 119883 ] and transpose to both sides of (25)and pre- and postmultiply the diag [119883 119883 ] transpose to bothsides of 119875 119882 119885 and 119873 and the 119883 and transpose to bothsides of 119876

119894119872119894 and 119877

119894 119894 = 1 2 Let 119879

119894= 1199051198941198791(119894 = 2 8)

and denote new variables 119883 = 119879minus11 1198771= 119883119877

1119883119879 119876

1=

1198831198761119883119879119872

1= 119883119872

1119883119879 119877

2= 119883119877

2119883119879 119876

2= 119883119876

2119883119879 119872

2=

1198831198722119883119879 Λ

119894= 119883Λ

119894119883119879 Λ

119895= 119883Λ

119895119883119879 119894 = 1 2 119903 119875

11=

11988311987511119883119879 119875

12= 119883119875

12119883119879 119875

22= 119883119875

22119883119879 119882

11= 119883119882

11119883119879

11988212= 119883119882

12119883119879 119882

22= 119883119882

22119883119879 119885

11= 119883119885

11119883119879 119885

12=

11988311988512119883119879 119885

22= 119883119885

22119883119879 119873

11= 119883119873

11119883119879 119873

12= 119883119873

12119883119879

and11987322= 119883119873

22119883119879 With 119865

119894= 119884119894119883minus119879 119894 = 1 2 119903we next

get

119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))Φ

119894119895lt 0 (27)

Mathematical Problems in Engineering 7

where

Φ119894119895=

[[[[[[[[

[

Φ11119894 Φ12119894 Φ13 Φ14 Φ15119894119895 Φ16 Φ17 Φ18

lowast Φ22119894 1199053119883119860119879

21198941199054119883119860119879

2119894Φ25119894119895 1199056119883119860

119879

21198941199057119883119860119879

2119894Φ28119894

lowast lowast Φ33119894 minus11987522 1199053119861119894119884119895 0 0 minus1199053119883119879

lowast lowast lowast Φ44 1199054119861119894119884119895 0 0 Φ48

lowast lowast lowast lowast Φ55119894119895 1199056119884119879

119895119861119879

1198941199057119884119879

119895119861119879

119894Φ58119894

lowast lowast lowast lowast lowast Φ66 Φ67 minus1199056119883119879

lowast lowast lowast lowast lowast lowast Φ77 Φ78

lowast lowast lowast lowast lowast lowast lowast Φ88

]]]]]]]]

]

Φ11119894= 11987512+ 119875119879

12+11988212+119882119879

12+ ℎ111988511

+ ℎ211987311minus1ℎ1

11988522minus1ℎ2

11987322+ 1198761+ 1198762

+1198721+1198722minus 21198771minus 21198772+ 1198601119894119883119879 + 119883119860119879

1119894

Φ12119894= 1198602119894119883119879 + 119905

21198831198601198791119894

Φ13= minus11987512minus1ℎ 111988522+ 11990531198831198601198791119894

Φ14= 11987522minus1ℎ1

119885119879

12+1ℎ1

1198771+ 11990541198831198601198791119894

Φ15119894119895

= 119861119894119884119895+ 11990551198831198601198791119894

Φ16= minus119882

12minus1ℎ2

11987322+ 11990561198831198601198791119894

Φ17= 11988222minus1ℎ2

119873119879

12+2ℎ22

1198772+ 11990571198831198601198791119894

Φ18= 11987511+11988211+ ℎ111988512+ ℎ211987312minus 119883119879 + 119905

81198831198601198791119894

Φ22119894= minus (1 minus 120583

1) 1198761+ 11990521198602119894119883119879 + 119905

21198831198601198792119894

Φ25119894119895

= 1199052119861119894119884119895+ 11990551198831198601198792119894

Φ28119894= minus1199052119883119879 + 119905

81198831198601198792119894

Φ33= minus

1ℎ1

11988522minus1198721

Φ44= minus

1ℎ1

11988511minus2ℎ21

1198771

Φ48= 119875119879

12minus 1199054119883119879

Φ55119894= minus (1 minus 120583

2) 1198762+ 1199055119861119894119884119895+ 1199055119884119879119895119861119879119894

Φ58119894= minus1199055119883119879 + 119905

8119884119879119895119861119879119894

Φ66= minus

1ℎ2

11987322minus1198722

Φ67= minus

1ℎ2

119873119879

12minus11988222

Φ77= minus

1ℎ2

11987311minus2ℎ22

1198772

Φ78= 119882119879

12minus 1199057119883119879

Φ88= ℎ111988522+ ℎ211987322+12ℎ211198771+12ℎ221198772minus 1199058119883119879

(28)

Here ldquolowastrdquo denotes the transpose elements in the symmetricpositions

If (27) holds then (119909(119905)) lt 0 Considersum119903119894=1sum119903119895=1

119908119894(119909(119905))(119908

119895(119909(119905)) minus 119898

119895(119909(119905 minus 119889

2(119905))))Λ

119894= 0

where Λ119894= Λ119879119894isin 1198778119899times8119899 gt 0 119894 = 1 2 119903 are arbitrary

matrices These terms are introduced to (27) to alleviate theconservativeness From (27) we also have

Φ =119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))Φ

119894119895

=119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905)) (119908

119895(119909 (119905)) minus 119898

119895(119909 (119905 minus 119889

2(119905)))

+120588119895119908119895(119909 (119905)) minus 120588

119895119908119895(119909 (119905))) Λ

119894

+119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))Φ

119894119895

=119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905)) 119908

119895(119909 (119905)) (120588

119895Φ119894119895minus 120588119895Λ119894+ Λ119894)

+119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905)) (119898

119895(119909 (119905 minus 119889

2(119905))) minus 120588

119895119908119895(119909 (119905)))

times (Φ119894119895minus Λ119894)

le119903

sum119894=1

1199082119894(120588119894Φ119894119894minus 120588119894Λ119894+ Λ119894)

+119903

sum119894=1

119903

sum119895=1

119908119894(119898119895minus 120588119895119908119895) (Φ119894119895minus Λ119894)

+119903

sum119894=1

sum119894lt119895

119908119894119908119895(120588119895Φ119894119895+ 120588119894Φ119895119894minus 120588119895Λ119894minus 120588119894Λ119895

+Λ119894+ Λ119895)

(29)

with119898119895(119909(119905 minus 119889

2(119905))) minus 120588

119895119908119895(119909(119905)) ge 0 for all 119895 119909(119905) and 119909(119905 minus

1198892(119905)) Let

Φ119894119895minus Λ119894lt 0 (30)

120588119894Φ119894119894minus 120588119894Λ119894+ Λ119894lt 0 (31)

120588119895Φ119894119895+ 120588119894Φ119895119894minus 120588119895Λ119894minus 120588119894Λ119895+ Λ119894+ Λ119895le 0 119894 lt 119895 (32)

for all 119894 119895 = 1 2 119903

8 Mathematical Problems in Engineering

Since (119909(119905)) lt 0 the fuzzy control system with the timevarying state and input delays (15) is asymptotically stablewith the state feedback control law (17)

32 Robust Stability of Uncertain Fuzzy Systems We alsoexamine the design of a robust stable controller for theuncertain system (4) under the imperfect premise matchingConsider the following uncertain fuzzy time varying delaycontrol system

(119905) =119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))

times [(1198601119894+ Δ1198601119894(119905)) 119909 (119905) + (119860

2119894+ Δ1198602119894(119905))

times 119909 (119905 minus 1198891(119905)) + (119861

119894+ Δ119861119894(119905))

times119906 (119905 minus 1198892(119905))]

119909 (119905) = 120593 (119905) 119905 isin [minusmax (ℎ1 ℎ2) 0]

(33)

Based on the above results of Theorem 4 a robust stabiliza-tion criterion for theT-S fuzzy systemswith time varying stateand input delays is investigated The following result can beobtained

Theorem 5 Given scalars ℎ1ge 0 ℎ

2ge 0 120583

1 1205832 and 119905

119894(119894 =

2 8) the uncertain fuzzy control systems with the timevarying state and input delays (33) is robustly stable for any0 le 119889

119894(119905) le ℎ

119894(119894 = 1 2) via the imperfect premise matching

controller design technique if the membership functions of thefuzzy model and fuzzy controller satisfy 119898

119895(119909(119905 minus 119889

2(119905))) minus

120588119895119908119895(119909(119905)) ge 0 for all 119895 119909(119905) and 119909(119905 minus 119889

2(119905)) where 0 lt

120588119895lt 1 and there exist common matrices 119875

11gt 0 119875

22gt 0

11988211gt 0 119882

22gt 0 119885

11gt 0 119885

22gt 0 119873

11gt 0 119873

22gt 0

119876119894gt 0 119872

119894gt 0 119877

119894gt 0 (119894 = 1 2) and Λ

119894= ΛT119894isin 1198778119899times8119899 gt

0 (i = 1 2 r) 119875 gt 0 119882 gt 0 119885 gt 0 119873 gt 0 and somematrices 119884

119895(119895 = 1 2 119903) 119883 and scalars 120576

1119894gt 0 120576

2119894gt 0

120576119887119894119895gt 0 satisfy the following LMIs

Ψ119894119895minus Λ119894lt 0 (34)

120588119894Ψ119894119894minus 120588119894Λ119894+ Λ119894lt 0 (35)

120588119895Ψ119894119895+ 120588119894Ψ119895119894minus 120588119895Λ119894minus 120588119894Λ119895+ Λ119894+ Λ119895 le 0 119894 lt 119895 (36)

where Ψ119894119895is defined as in (44) The state feedback gains can be

constructed as 119865119894= 119884119894119883minus119879

Proof If 1198601119894 1198602119894 and 119861

119894in Φ119894119895are replaced with 119860

1119894+

119863119894119870119894(119905)1198641119894 1198602119894+ 119863119894119870119894(119905)1198642119894 and 119861

119894+ 119863119894119870119894(119905)119864119887119894in (28) of

Theorem 4 respectively (27) for system (33) can be rewrittenas

119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889 (119905))) Ω

119894119895lt 0 (37)

where

Ω119894119895= Φ119894119895+ 1198631198791119894119870119894(119905) 119864119894+ 119864119879119894119870119879119894(119905) 1198631119894+ 1198631198792119894119870119894(119905) 1198642119894119895

+ 1198641198792119894119895119870119879119894(119905) 1198632119894+ 119863119879119887119894119870119894(119905) 119864119887119894119895+ 119864119879119887119894119895119870119879119894(119905) 119863119887119894

1198631119894= [1198631198791198940 0 0 0 0 0 0]

1198632119894= [0 119863119879

1198940 0 0 0 0 0]

119863119887119894= [0 0 0 0 119863119879

1198940 0 0]

(38)

1198641119894= [1198641119894119883119879 11990521198641119894119883119879 11990531198641119894119883119879 11990541198641119894119883119879 119905

51198641119894119883119879 11990561198641119894119883119879 11990571198641119894119883119879 11990581198641119894119883119879]

1198642119894= [1198642119894119883119879 11990521198642119894119883119879 11990531198642119894119883119879 11990541198642119894119883119879 11990551198642119894119883119879 11990561198642119894119883119879 11990571198642119894119883119879 11990581198642119894119883119879]

119864119887119894119895= [119864119887119894119884119895 1199052119864119887119894119884119895 1199053119864119887119894119884119895 1199054119864119887119894119884119895 1199055119864119887119894119884119895 1199056119864119887119894119884119895 1199057119864119887119894119884119895 1199058119864119887119894119884119895]

(39)

If Ω119894119895

lt 0 (37) holds According to Lemma 2 it isstraightforward to know that Ω

119894119895lt 0 is true if for each 119894

119895 there exists scalars 1205761119894gt 0 120576

2119894gt 0 and 120576

119887119894119895gt 0 such that

the following inequality holds

Φ119894119895+ 120576111989411986311987911198941198631119894+ 120576minus1111989411986411987911198941198641119894+ 120576211989411986311987921198941198632119894

+ 120576minus1211989411986411987921198941198642119894+ 120576119887119894119895119863119879119887119894119863119887119894+ 120576minus1119887119894119895119864119879119887119894119895119864119887119894119895lt 0

(40)

Based on Schur complement (40) is equivalent to thefollowing inequality

Ψ119894119895=[[[

[

Φ119894119895

1198641198791119894

1198641198792119894

119864119879119887119894119895

lowast minus1205761119894119868 0 0

lowast lowast minus1205762119894119868 0

lowast lowast lowast minus120576119887119894119895119868

]]]

]

lt 0 (41)

Mathematical Problems in Engineering 9

where

Φ119894119895=

[[[[[[[[

[

Φ11119894 Φ12119894 Φ13 Φ14 Φ15119894119895 Φ16 Φ17 Φ18

lowast Φ22119894 1199053119883119860119879

21198941199054119883119860119879

2119894Φ25119894119895 1199056119883119860

119879

21198941199057119883119860119879

2119894Φ28119894

lowast lowast Φ33119894 minus11987522 1199053119861119894119884119895 0 0 minus1199053119883119879

lowast lowast lowast Φ44 1199054119861119894119884119895 0 0 Φ48

lowast lowast lowast lowast Φ55119894119895 1199056119884119879

119895119861119879

1198941199057119884119879

119895119861119879

119894Φ58119894

lowast lowast lowast lowast lowast Φ66 Φ67 minus1199056119883119879

lowast lowast lowast lowast lowast lowast Φ77 Φ78

lowast lowast lowast lowast lowast lowast lowast Φ88

]]]]]]]]

]

lt 0

Φ11119894= Φ11119894+ 1205761119894119863119894119863119879119894

Φ22119894= Φ22119894+ 1205762119894119863119894119863119879119894

Φ55119894119895

= Φ55119894119895

+ 120576119887119894119863119894119863119879119894

(42)

and Φ are the same as in Theorem 4 Therefore (41) holds ifand only if the following inequality holds

119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889 (119905))) Ψ

119894119895lt 0 (43)

To alleviate the conservativeness of our robust stabilityanalysis similar to the analysis approach of Theorem 4 weintroduce sum119903

119894=1sum119903119895=1

119908119894(119908119895minus 119898119895)Λ119894= 0 to (43) and the

following proof is the same as in Theorem 4 We can obtainthe conditions and (34)ndash(36) Thus the fuzzy control systemwith the time varying state and input delays (34) is robuststable on the basis of the control law 119865

119894= 119884119894119883minus119879under these

conditions and (34)ndash(36) hold

Remark 6 If we set 119908119894(119909(119905)) = 119898

119895(119909(119905)) 119889

1(119905) = 119889

2(119905)

the closed-loop system in this paper has the same structureas that of paper [26] On the other hand let 119908

119894(119909(119905)) =

119898119895(119909(119905)) and 119889

1(119905) = 119889

2(119905) = 119889 we can obtain the system

in [16] Therefore the system studied in this paper is moregeneral Additionally the information of the membershipfunctions of the fuzzy time delay models and controllers areall considered in the stability analysis If the fuzzymodels andfuzzy controllers share the same fuzzymembership functionsthat is 119908

119894(119909(119905)) = 119898

119895(119909(119905)) the stability analysis can be

referred to [25 27 28] In other words the problem in [2527 28] is actually a special case of the one investigated in thispaper

Remark 7 The augmented Lyapunov function with an addi-tional triple-integral term is introduced in analyzing thestability problem for the T-S fuzzy systems with time varyingstate and input delays In addition two integral inequalitiesare used to derive Theorem 4 and less free-weighting matri-ces are introduced which lead to get less conservative resultsIt can be observed that some existing results are the specialcases of this paper For example the method in [25] is from(18) with 119875

12= 0 119875

22= 0 119885

12= 11988522= 0 119873

12= 11987322= 0

and 1198771= 1198772= 0 in Theorem 4 The results in [22] are based

on system (33) with 1198892(119905) = 0 and (18) with119872

2= 0 119885

12=

11988522= 0 119876

1= 1198762= 0 and 119877

1= 1198772= 0

Remark 8 Different from the general PDC technique thedesign method under the imperfect premise matching ismuch more flexible because the membership functions ofthe fuzzy controllers do not need to be chosen the same asthose of the fuzzy time delay models Instead they can bedesigned arbitrarilyThus the design flexibility is significantlyenhanced On the other hand some simple membershipfunctions of the fuzzy controllers might be employed whichcan reduce the implementation cost

Remark 9 In [7ndash11 15ndash20] the authors have considered therobust control for uncertain T-S fuzzy systems with statedelays Some delay-dependent conditions and fuzzy con-trollers are obtained with a traditional Lyapunov-Krasovskiifunctionalmethod andPDCmethod which all are the specialcases of Theorem 5 in this paper

4 Numerical Examples

In this section two numerical examples are given to illus-trate the conservativeness and effectiveness of the proposedmethods The first example compares our techniques withthe existing ones in the literature for stability analysis whichshows that Theorem 4 in this paper is less conservative thanthe other results The second example is used to illustrate theadvantage of the robust stability conditions ofTheorem 5 anddemonstrate how to design robust fuzzy controllers by usingour approach

Example 10 Consider the following fuzzy system with stateand input delays

(119905) =2

sum119894=1

119908119894(120579 (119905)) [119860

1119894119909 (119905) + 119860

2119894119909 (119905 minus 119889

1(119905))

+119861119894119906 (119905 minus 119889

2(119905))]

11986011= [0 06

0 1 ] 11986021= [05 09

0 2 ]

11986012= [1 0

1 0] 11986022= [09 0

1 16]

1198611= 1198612= [11]

(44)

The fuzzy membership functions are selected as [31]

1199081(1199091(119905)) = (1 minus

119888 (119905) sin (10038161003816100381610038161199091 (119905)1003816100381610038161003816minus4)5

1 + expminus1001199091(119905)3(1minus1199091(119905)))

timescos (119909

1(119905))2

1 + expminus251199091(119905)(3+(1199091(119905)042))

times 1199082(1199091(119905)) = 1 minus 119908

1(1199091(119905))

(45)

10 Mathematical Problems in Engineering

Table 1 The maximum allowable time delay and feedback gains(1205831= 0 120583

2= 0)

Paper ℎ1

ℎ2 Feedback gains

[26] 03120 031201198651= [10598 minus56598]

1198652= [minus13068 minus41167]

Theorem 4 033 0571198651= [minus00042 minus00449]1198652= [00048 00717]

where

1199091(119905) isin [minus

1205872

1205872]

119888 (119905) =sin (119909

1(119905)) + 140

isin [minus005 005] (46)

If we set 1198891(119905) = 119889

2(119905) in system (44) we can obtain the

system in [26] which implies that our system ismore generalEmploying the LMIs in [26] and those inTheorem 4yields themaximum state and input delays ℎ

1 and ℎ

2 which guarantee

the stability of system (44) and the feedback gains for 1205831= 0

and 1205832= 0 when 120588

1= 075 120588

2= 095 119905

2= 01 119905

3=

02 1199054= 07 119905

5= 01 119905

6= 04 119905

7= 01 and 119905

8= 13 as

given in Table 1 which clearly shows the superiority of theresults derived in this paper over those obtained from [26]

Example 11 Consider the well-known truck-trailer systemwhich can be described by the following T-S fuzzy system

(119905) =2

sum119894=1

119908119894(120579 (119905)) [(119860

1119894+ Δ1198601119894) 119909 (119905) + 119860

2119894119909 (119905 minus 119889

1(119905))

+ (119861119894+ Δ119861119894) 119906 (119905 minus 119889

2(119905))]

(47)

11986011=

[[[[[[[[[

[

minus119886V1199051198711199050

0 0

119886V1199051198711199050

0 0

119886V21199052

21198711199050

V1199051199050

0

]]]]]]]]]

]

11986021=

[[[[[[[[[

[

minus (1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)V21199052

21198711199050

0 0

]]]]]]]]]

]

1198611=[[[[

[

V1199051198971199050

0

0

]]]]

]

11986012=

[[[[[[[[[

[

minus119886V1199051198711199050

0 0

119886V1199051198711199050

0 0

119886119889V21199052

21198711199050

119889V1199051199050

0

]]]]]]]]]

]

11986022=

[[[[[[[[[

[

minus (1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)119889V21199052

21198711199050

0 0

]]]]]]]]]

]

(48)

1198612=[[[[

[

V1199051198971199050

0

0

]]]]

]

(49)

where 119897 = 28 119871 = 55 V = minus10 119905 = 20 1199050= 05

119889 = 101199050120587 119863

119894= [02555 02555 02555]119879 119864

1119894= 1198642119894=

[01 0 0]119879 119864119887119894

= [01 0 0]119879 (119894 = 1 2) and the fuzzymembership functions are selected the same as inExample 10

(1) Let 119886 = 1 system (47) is the same as the fuzzy systemof [22] which implies that our system ismore generalTable 2 gives themaximum input delay value of ℎ

2 for

which the stabilization is guaranteed byTheorem 5 aswell as the state feedback gains when 120588

1= 075 120588

2=

095 1199052= 01 119905

3= 02 119905

4= 07 119905

5= 01 119905

6=

04 1199057= 08 and 119905

8= 2 It is clearly visible that the

results derived in this paper are better than those from[22 28]

(2) In the case of 119886 = 1 the state and input delays of system(47) are all exist The approach proposed in [22] cannot be used to get the maximum of state delay ℎ

1

as the system in [22] only contains the input delayHowever the method in [28] and Theorem 5 can beused Here let 119886 = 07 120588

1= 075 120588

2= 095 119905

2=

01 1199053= 02 119905

4= 07 119905

5= 01 119905

6= 04 119905

7=

08 and 1199058= 3 and Table 3 provides the maximum

upper bounds of ℎ1and ℎ

2and the state feedback

gains for which the robust stabilization is guaranteedbyTheorem 5

By Theorem 5 we can conclude that the uncertain fuzzymodel (47) is robust stable based on the following fuzzycontrol law

119906 (119905 minus ℎ2) =2

sum119895=1

119898119895(1199091(119905 minus ℎ

2)) 119865119895119909 (119905 minus ℎ

2) (50)

With the PDC design technique it is required that thefuzzy controller must share the same fuzzy membershipfunctions as those of the fuzzy time delay model Under such

Mathematical Problems in Engineering 11

Table 2 The maximum allowable time delay and feedback gains(1205832= 0)

Paper ℎ2 Feedback gains

[22] 0751198651= [34227 minus03535 00045]

1198652= [35215 minus03617 00056]

[28] 0861198651= [33219 minus02406 00025]

1198652= [33272 minus02494 00026]

Theorem 5 121198651= [minus00212 00093 minus00024]1198652= [minus00250 00044 00014]

Table 3 The maximum allowable time delay and feedback gains(1205831= 1205832= 0)

Paper ℎ1

ℎ2 Feedback gains

[28] 01 0551198651= [33219 minus02406 00025]

1198652= [33272 minus02494 00026]

Theorem 5 04 171198651= [minus00128 00090 minus00017]1198652= [minus00194 00012 00009]

a condition the implementation cost of the fuzzy controlleris high by employing these complex membership functionsHowever fromTheorem 5 the membership functions do notneed to be chosen the same as119908

1(1199091(119905)) and119908

2(1199091(119905)) Thus

we can select some simple membership functions insteadsuch as

1198981(1199091(119905 minus ℎ

2)) = 075 exp

(minus1199091(119905 minus ℎ

2) minus 038)2

2 times 0382+ 005

1198982(1199091(119905 minus ℎ

2)) = 1 minus 119898

1(1199091(119905 minus ℎ

2))

(51)

We also assume that 1205881= 075 and 120588

2= 095 such that

119898119895(1199091(119905 minus ℎ

2)) minus 120588

119895119908119895(1199091(119905)) gt 0 for 119895 = 1 2 and 119909

1(119905 minus ℎ

2)

The fuzzy control law 119906(119905 minus ℎ2) = 119898

1[minus00128 00090 minus

00017]119909 + 1198982[minus00194 00012 00009]119909 can be employed

to stabilize the uncertain fuzzy control systems in (47) withthe state and input delays Figures 1 and 2 illustrate the stablestate response and control input under the initial conditionof 119909(0) = [4 minus 1 2]119879

Remark 12 In Example 11 our method offers less conser-vative results in the sense of allowing longer time delayand obtaining smaller feedback control gains Moreovercompared with [22ndash28] where the fuzzy controllermust havethe same fuzzy membership functions as those of the fuzzytime delay model the above membership functions of ourfuzzy controller are much simper which can considerablylower the implementation cost of the fuzzy controller andenhance the design flexibility

Remark 13 If the function of 119888(119905) in 1199081(1199091(119905)) is unknown

on the basis of the PDC design technique [7ndash11 15ndash20 22ndash28] the fuzzy controller cannot be implemented as 119888(119905) isnot available However using the proposed design method

0 5 10 15

0

2

4

6Closed-loop response

minus12

minus10

minus8

minus6

minus4

minus2

119905 (s)

119909(119905)

1199092

11990931199091

Figure 1The state responses of system (47) with ℎ1= 04 ℎ

2= 17

0 5 10 15

0

5

10

15

20

25

30

119905 (s)

minus20

minus15

minus10

minus5

119906(119905)

Figure 2 The control input of system (47) with ℎ1= 04 ℎ

2= 17

we can select some simper and well-known membershipfunctions that are different from those of the fuzzy time delaymodels for example119898

1(1199091(119905)) and119898

2(1199091(119905)) so as to realize

the fuzzy controller In other words the proposed designapproach can be applied even for the fuzzy time delay modelswith uncertain grades of membership

5 Conclusions

In this paper we study the robust stability and stabilizationproblems for general nonlinear fuzzy systems with timevarying state and input delays A less conservative delay-dependent robust stability criterion has been obtained byintroducing a novel augmented Lyapunov function withan additional triple-integral term Moreover a new design

12 Mathematical Problems in Engineering

approach different from the traditional PDC design tech-nique is proposed which can significantly improve the designflexibility and reduce the implementation cost of the fuzzycontroller by arbitrarily selecting simple fuzzy membershipfunctions Two numerical examples are used to illustrate theconservativeness and effectiveness of our proposed methods

Acknowledgments

This research work was funded by the NSFC under Grantno 61273095 and the Academy of Finland under Grants no135225 and no 127299

References

[1] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[2] W H Ho S H Chen I T Chen J H Chou and C C ShuldquoDesign of stable and quadratic-optimal static output feed-back controllers for T-S-fuzzy-model-based control systems anintegrative computational approachrdquo International Journal ofInnovative Computing Information and Control vol 8 no 1App 403ndash418 2012

[3] X J Su L G Wu P Shi and Y D Song ldquo119867infinmodel reduction

of T-S fuzzy stochastic systemsrdquo IEEE Transactions on SystemsMan and Cybermetics B vol 42 no 6 pp 1574ndash1585 2012

[4] R Yang P Shi G-P Liu andH Gao ldquoNetwork-based feedbackcontrol for systems with mixed delays based on quantizationand dropout compensationrdquo Automatica vol 47 no 12 pp2805ndash2809 2011

[5] XM Zhang Z J Zhang andG P Lu ldquoFault detection for state-delay fuzzy systems subject to random communication delayrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 4 pp 2439ndash2451 2012

[6] F B Li and X Zhang ldquoDelay-range-dependent robust 119867infin

filtering for singuar LPV systems with time variant delayrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 1 pp 339ndash353 2013

[7] Y Y Cao and P M Frank ldquoAnalysis and synthesis of nonlineartime-delay systems via fuzzy control approachrdquo IEEE Transac-tions on Fuzzy Systems vol 8 no 2 pp 200ndash211 2000

[8] Y-Y Cao and P M Frank ldquoStability analysis and synthesis ofnonlinear time-delay systems via linear Takagi-Sugeno fuzzymodelsrdquo Fuzzy Sets and Systems vol 124 no 2 pp 213ndash2292001

[9] O M Kwon and J H Park ldquoDelay-range-dependent stabiliza-tion of uncertain dynamic systems with interval time-varyingdelaysrdquo Applied Mathematics and Computation vol 208 no 1pp 58ndash68 2009

[10] E Tian D Yue and Y Zhang ldquoDelay-dependent robust 119867infin

control for T-S fuzzy system with interval time-varying delayrdquoFuzzy Sets and Systems vol 160 no 12 pp 1708ndash1719 2009

[11] J Y An T Li G LWen and R Li ldquoNew stability conditions foruncertain T-S fuzzy systems with interval time-varying delayrdquoInternational Journal of Control Automation and Systems vol10 no 3 pp 490ndash497 2012

[12] Z J Zhang X L Huang X J Ban and X Z Gao ldquoStabilityanalysis and controller design of T-S fuzzy systems with time-delay under imperfect premise matchingrdquo Journal of BeijingInstitute of Technology vol 21 no 3 pp 387ndash393 2012

[13] O M Kwon M J Park S M Lee and J H Park ldquoAugmentedLyapunov-Krasovskii functional approaches to robust stabilitycriteria for uncertain Takagi-Sugeno fuzzy systems with time-varying delaysrdquo Fuzzy Sets and Systems vol 201 no 16 pp 1ndash192012

[14] F Liu M Wu Y He and R Yokoyama ldquoNew delay-dependentstability criteria for T-S fuzzy systems with time-varying delayrdquoFuzzy Sets and Systems vol 161 no 15 pp 2033ndash2042 2010

[15] B Chen X Liu and S Tong ldquoNew delay-dependent stabiliza-tion conditions of T-S fuzzy systems with constant delayrdquo FuzzySets and Systems vol 158 no 20 pp 2209ndash2224 2007

[16] J Yoneyama ldquoRobust stability and stabilization for uncertainTakagi-Sugeno fuzzy time-delay systemsrdquo Fuzzy Sets and Sys-tems vol 158 no 2 pp 115ndash134 2007

[17] C Lin Q-G Wang and T H Lee ldquoDelay-dependent LMIconditions for stability and stabilization of T-S fuzzy systemswith bounded time-delayrdquo Fuzzy Sets and Systems vol 157 no9 pp 1229ndash1247 2006

[18] X J Su P Shi L G Wu and Y D Song ldquoA novel approachto filter design for T-S fuzzy discrete-time systems with time-varying delayrdquo IEEE Transactions on Fuzzy Systems vol 20 no6 pp 1114ndash1129 2012

[19] C Peng and Q-L Han ldquoDelay-range-dependent robust stabi-lization for uncertain T-S fuzzy control systems with intervaltime-varying delaysrdquo Information Sciences vol 181 no 19 pp4287ndash4299 2011

[20] S T Li Y W Jing and X M Liu ldquoRobust control for a class ofT-S fuzzy systemswith interval time-varying delayrdquo InternatinalJournal of Control Automation and Systems vol 10 no 4 pp737ndash743 2012

[21] X-M Zhang M Wu J-H She and Y He ldquoDelay-dependentstabilization of linear systems with time-varying state and inputdelaysrdquo Automatica vol 41 no 8 pp 1405ndash1412 2005

[22] B Chen X Liu and S Tong ldquoRobust fuzzy control of nonlinearsystems with input delayrdquo Chaos Solitons amp Fractals vol 37 no3 pp 894ndash901 2008

[23] C Lin Q-G Wang and T H Lee ldquoDelay-dependent LMIconditions for stability and stabilization of T-S fuzzy systemswith bounded time-delayrdquo Fuzzy Sets and Systems vol 157 no9 pp 1229ndash1247 2006

[24] H J Lee J B Park and Y H Joo ldquoRobust control for uncertainTakagi-Sugeno fuzzy systems with time-varying input delayrdquoJournal of Dynamic Systems Measurement and Control Trans-actions of the ASME vol 127 no 2 pp 302ndash306 2005

[25] C-H Lien and K-W Yu ldquoRobust control for Takagi-Sugenofuzzy systems with time-varying state and input delaysrdquo ChaosSolitons and Fractals vol 35 no 5 pp 1003ndash1008 2008

[26] Y K Su B Chen S Y Zhang and C Lin ldquoDelay-dependentstabilization of T-S fuzzy systems with time delayrdquo Control andDecision Kongzhi yu Juece vol 24 no 6 pp 921ndash927 2009

[27] B Chen X Liu C Lin and K Liu ldquoRobust 119867infin

control ofTakagi-Sugeno fuzzy systems with state and input time delaysrdquoFuzzy Sets and Systems vol 160 no 4 pp 403ndash422 2009

[28] L Li and X Liu ldquoNew results on delay-dependent robuststability criteria of uncertain fuzzy systems with state and inputdelaysrdquo Information Sciences vol 179 no 8 pp 1134ndash1148 2009

[29] Z J Zhang X J Ban X L Huang and X Z Gao ldquoNew delay-dependent robust stablity for T-S fuzzy systems with intervevaltime-varying delayrdquo in Proccedings of the 5th International JointConference on Computational Science and Optimization pp826ndash830 Harbin China 2012

Mathematical Problems in Engineering 13

[30] H K Lam and F H F Leung ldquoLMI-based stability andperformance design of fuzzy control systems fuzzy modelsand controllers with different premisesrdquo in Proceedings of theInternational Conference on Fuzzy Systerm pp 9599ndash9506Vancouver Canada 2006

[31] H K Lam andMNarimani ldquoStability analysis and perfomancedesign for fuzzy-model-based control system under imperfectpremise matchingrdquo IEEE Transactions on Fuzzy Systems vol 17no 4 pp 949ndash961 2009

[32] S Boyd L E Ghaoui and E Feron Linear Matrix Inequality inSystems and ControlTheory SIAM Philadelphia Pa USA 1994

[33] I R Petersen and C V Hollot ldquoA Riccati equation approach tothe stabilization of uncertain linear systemsrdquo Automatica vol22 no 4 pp 397ndash411 1986

[34] J Sun G P Liu and J Chen ldquoDelay-dependent stabilityand stabilization of neutral time-delay systemsrdquo InternationalJournal of Robust andNonlinear Control vol 19 no 12 pp 1364ndash1375 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

where

120577119879 (119905) = [119909119879 (119905) 119909119879 (119905 minus 1198891(119905)) 119909119879 (119905 minus ℎ

1) (int

119905

119905minusℎ1

119909 (119904) 119889119904)119879

119909119879 (119905 minus 1198892(119905)) 119909119879 (119905 minus ℎ

2) (int

119905

119905minusℎ2

119909 (119904) 119889119904)119879

119879 (119905)] (24)

Φ119894119895=

[[[[[[

[

Φ11119894 Φ12119894 Φ13 Φ14 Φ15119894119895 Φ16 Φ17 Φ18

lowast Φ22119894 119860119879

2119894119879119879

3119860119879

2119894119879119879

4Φ25119894119895 119860

119879

2119894119879119879

6119860119879

2119894119879119879

7Φ28119894

lowast lowast Φ33119894 minus11987522 1198793119861119894119865119895 0 0 minus1198793

lowast lowast lowast Φ44 1198794119861119894119865119895 0 0 Φ48

lowast lowast lowast lowast Φ55119894119895 119865119879

119895119861119879

119894119879119879

6119865119879

119895119861119879

119894119879119879

7Φ58119894

lowast lowast lowast lowast lowast Φ66 Φ67 minus1198796

lowast lowast lowast lowast lowast lowast Φ77 Φ78

lowast lowast lowast lowast lowast lowast lowast Φ88

]]]]]]

]

Φ11119894= 11987512+ 11987511987912+11988212+11988211987912+ ℎ111988511+ ℎ211987311

minus1ℎ1

11988522minus1ℎ2

11987322+ 1198761+ 1198762+1198721+1198722

minus 21198771minus 21198772+ 11987911198601119894+ 11986011987911198941198791198791

Φ12119894= 11987911198602119894+ 11986011987911198941198791198792

Φ13= minus11987512minus1ℎ1

11988522+ 11986011987911198941198791198793

Φ14= 11987522minus1ℎ2

11988511987912+1ℎ1

1198771+ 11986011987911198941198791198794

Φ15119894119895

= 1198791119861119894119865119895+ 11986011987911198941198791198795

Φ16= minus119882

12minus1ℎ2

11987322+ 11986011987911198941198791198796

Φ17= 11988222minus1ℎ2

11987311987912+2ℎ22

1198772+ 11986011987911198941198791198797

Φ18= 11987511+11988211+ ℎ111988512+ ℎ211987312minus 1198791+ 11986011987911198941198791198798

Φ22119894= minus (1 minus 120583

1) 1198761+ 11987921198602119894+ 11986011987921198941198791198792

Φ25119894119895

= 1198792119861119894119865119895+ 11986011987921198941198791198795

Φ28119894= minus1198792+ 11986011987921198941198791198798

Φ33= minus

1ℎ11988522minus1198721

Φ44= minus

1ℎ1

11988511minus2ℎ21

1198771

Φ48= 11987511987912minus 1198794

Φ55119894= minus (1 minus 120583

2) 1198762+ 1198795119861119894119865119895+ 1198651198791198951198611198791198941198791198795

Φ58119894= minus1198795+ 1198651198791198951198611198791198941198791198798

Φ66= minus

1ℎ2

11987322minus1198722

Φ67= minus

1ℎ2

11987311987912minus11988222

Φ77= minus

1ℎ2

11987311minus2ℎ22

1198772

Φ78= 11988211987912minus 1198797

Φ88= ℎ111988522+ ℎ211987322+12ℎ211198771+12ℎ221198772minus 1198798

(25)

From (25) it is obvious that if

119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))Φ

119894119895lt 0 (26)

(119909(119905)) lt 0From (26) we can discover that the feedback gains

are predefined The following proof presents the controllerdesign method We first pre- and postmultiply both thediag [119883 119883 119883 119883 119883 119883 119883 119883 ] and transpose to both sides of (25)and pre- and postmultiply the diag [119883 119883 ] transpose to bothsides of 119875 119882 119885 and 119873 and the 119883 and transpose to bothsides of 119876

119894119872119894 and 119877

119894 119894 = 1 2 Let 119879

119894= 1199051198941198791(119894 = 2 8)

and denote new variables 119883 = 119879minus11 1198771= 119883119877

1119883119879 119876

1=

1198831198761119883119879119872

1= 119883119872

1119883119879 119877

2= 119883119877

2119883119879 119876

2= 119883119876

2119883119879 119872

2=

1198831198722119883119879 Λ

119894= 119883Λ

119894119883119879 Λ

119895= 119883Λ

119895119883119879 119894 = 1 2 119903 119875

11=

11988311987511119883119879 119875

12= 119883119875

12119883119879 119875

22= 119883119875

22119883119879 119882

11= 119883119882

11119883119879

11988212= 119883119882

12119883119879 119882

22= 119883119882

22119883119879 119885

11= 119883119885

11119883119879 119885

12=

11988311988512119883119879 119885

22= 119883119885

22119883119879 119873

11= 119883119873

11119883119879 119873

12= 119883119873

12119883119879

and11987322= 119883119873

22119883119879 With 119865

119894= 119884119894119883minus119879 119894 = 1 2 119903we next

get

119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))Φ

119894119895lt 0 (27)

Mathematical Problems in Engineering 7

where

Φ119894119895=

[[[[[[[[

[

Φ11119894 Φ12119894 Φ13 Φ14 Φ15119894119895 Φ16 Φ17 Φ18

lowast Φ22119894 1199053119883119860119879

21198941199054119883119860119879

2119894Φ25119894119895 1199056119883119860

119879

21198941199057119883119860119879

2119894Φ28119894

lowast lowast Φ33119894 minus11987522 1199053119861119894119884119895 0 0 minus1199053119883119879

lowast lowast lowast Φ44 1199054119861119894119884119895 0 0 Φ48

lowast lowast lowast lowast Φ55119894119895 1199056119884119879

119895119861119879

1198941199057119884119879

119895119861119879

119894Φ58119894

lowast lowast lowast lowast lowast Φ66 Φ67 minus1199056119883119879

lowast lowast lowast lowast lowast lowast Φ77 Φ78

lowast lowast lowast lowast lowast lowast lowast Φ88

]]]]]]]]

]

Φ11119894= 11987512+ 119875119879

12+11988212+119882119879

12+ ℎ111988511

+ ℎ211987311minus1ℎ1

11988522minus1ℎ2

11987322+ 1198761+ 1198762

+1198721+1198722minus 21198771minus 21198772+ 1198601119894119883119879 + 119883119860119879

1119894

Φ12119894= 1198602119894119883119879 + 119905

21198831198601198791119894

Φ13= minus11987512minus1ℎ 111988522+ 11990531198831198601198791119894

Φ14= 11987522minus1ℎ1

119885119879

12+1ℎ1

1198771+ 11990541198831198601198791119894

Φ15119894119895

= 119861119894119884119895+ 11990551198831198601198791119894

Φ16= minus119882

12minus1ℎ2

11987322+ 11990561198831198601198791119894

Φ17= 11988222minus1ℎ2

119873119879

12+2ℎ22

1198772+ 11990571198831198601198791119894

Φ18= 11987511+11988211+ ℎ111988512+ ℎ211987312minus 119883119879 + 119905

81198831198601198791119894

Φ22119894= minus (1 minus 120583

1) 1198761+ 11990521198602119894119883119879 + 119905

21198831198601198792119894

Φ25119894119895

= 1199052119861119894119884119895+ 11990551198831198601198792119894

Φ28119894= minus1199052119883119879 + 119905

81198831198601198792119894

Φ33= minus

1ℎ1

11988522minus1198721

Φ44= minus

1ℎ1

11988511minus2ℎ21

1198771

Φ48= 119875119879

12minus 1199054119883119879

Φ55119894= minus (1 minus 120583

2) 1198762+ 1199055119861119894119884119895+ 1199055119884119879119895119861119879119894

Φ58119894= minus1199055119883119879 + 119905

8119884119879119895119861119879119894

Φ66= minus

1ℎ2

11987322minus1198722

Φ67= minus

1ℎ2

119873119879

12minus11988222

Φ77= minus

1ℎ2

11987311minus2ℎ22

1198772

Φ78= 119882119879

12minus 1199057119883119879

Φ88= ℎ111988522+ ℎ211987322+12ℎ211198771+12ℎ221198772minus 1199058119883119879

(28)

Here ldquolowastrdquo denotes the transpose elements in the symmetricpositions

If (27) holds then (119909(119905)) lt 0 Considersum119903119894=1sum119903119895=1

119908119894(119909(119905))(119908

119895(119909(119905)) minus 119898

119895(119909(119905 minus 119889

2(119905))))Λ

119894= 0

where Λ119894= Λ119879119894isin 1198778119899times8119899 gt 0 119894 = 1 2 119903 are arbitrary

matrices These terms are introduced to (27) to alleviate theconservativeness From (27) we also have

Φ =119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))Φ

119894119895

=119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905)) (119908

119895(119909 (119905)) minus 119898

119895(119909 (119905 minus 119889

2(119905)))

+120588119895119908119895(119909 (119905)) minus 120588

119895119908119895(119909 (119905))) Λ

119894

+119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))Φ

119894119895

=119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905)) 119908

119895(119909 (119905)) (120588

119895Φ119894119895minus 120588119895Λ119894+ Λ119894)

+119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905)) (119898

119895(119909 (119905 minus 119889

2(119905))) minus 120588

119895119908119895(119909 (119905)))

times (Φ119894119895minus Λ119894)

le119903

sum119894=1

1199082119894(120588119894Φ119894119894minus 120588119894Λ119894+ Λ119894)

+119903

sum119894=1

119903

sum119895=1

119908119894(119898119895minus 120588119895119908119895) (Φ119894119895minus Λ119894)

+119903

sum119894=1

sum119894lt119895

119908119894119908119895(120588119895Φ119894119895+ 120588119894Φ119895119894minus 120588119895Λ119894minus 120588119894Λ119895

+Λ119894+ Λ119895)

(29)

with119898119895(119909(119905 minus 119889

2(119905))) minus 120588

119895119908119895(119909(119905)) ge 0 for all 119895 119909(119905) and 119909(119905 minus

1198892(119905)) Let

Φ119894119895minus Λ119894lt 0 (30)

120588119894Φ119894119894minus 120588119894Λ119894+ Λ119894lt 0 (31)

120588119895Φ119894119895+ 120588119894Φ119895119894minus 120588119895Λ119894minus 120588119894Λ119895+ Λ119894+ Λ119895le 0 119894 lt 119895 (32)

for all 119894 119895 = 1 2 119903

8 Mathematical Problems in Engineering

Since (119909(119905)) lt 0 the fuzzy control system with the timevarying state and input delays (15) is asymptotically stablewith the state feedback control law (17)

32 Robust Stability of Uncertain Fuzzy Systems We alsoexamine the design of a robust stable controller for theuncertain system (4) under the imperfect premise matchingConsider the following uncertain fuzzy time varying delaycontrol system

(119905) =119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))

times [(1198601119894+ Δ1198601119894(119905)) 119909 (119905) + (119860

2119894+ Δ1198602119894(119905))

times 119909 (119905 minus 1198891(119905)) + (119861

119894+ Δ119861119894(119905))

times119906 (119905 minus 1198892(119905))]

119909 (119905) = 120593 (119905) 119905 isin [minusmax (ℎ1 ℎ2) 0]

(33)

Based on the above results of Theorem 4 a robust stabiliza-tion criterion for theT-S fuzzy systemswith time varying stateand input delays is investigated The following result can beobtained

Theorem 5 Given scalars ℎ1ge 0 ℎ

2ge 0 120583

1 1205832 and 119905

119894(119894 =

2 8) the uncertain fuzzy control systems with the timevarying state and input delays (33) is robustly stable for any0 le 119889

119894(119905) le ℎ

119894(119894 = 1 2) via the imperfect premise matching

controller design technique if the membership functions of thefuzzy model and fuzzy controller satisfy 119898

119895(119909(119905 minus 119889

2(119905))) minus

120588119895119908119895(119909(119905)) ge 0 for all 119895 119909(119905) and 119909(119905 minus 119889

2(119905)) where 0 lt

120588119895lt 1 and there exist common matrices 119875

11gt 0 119875

22gt 0

11988211gt 0 119882

22gt 0 119885

11gt 0 119885

22gt 0 119873

11gt 0 119873

22gt 0

119876119894gt 0 119872

119894gt 0 119877

119894gt 0 (119894 = 1 2) and Λ

119894= ΛT119894isin 1198778119899times8119899 gt

0 (i = 1 2 r) 119875 gt 0 119882 gt 0 119885 gt 0 119873 gt 0 and somematrices 119884

119895(119895 = 1 2 119903) 119883 and scalars 120576

1119894gt 0 120576

2119894gt 0

120576119887119894119895gt 0 satisfy the following LMIs

Ψ119894119895minus Λ119894lt 0 (34)

120588119894Ψ119894119894minus 120588119894Λ119894+ Λ119894lt 0 (35)

120588119895Ψ119894119895+ 120588119894Ψ119895119894minus 120588119895Λ119894minus 120588119894Λ119895+ Λ119894+ Λ119895 le 0 119894 lt 119895 (36)

where Ψ119894119895is defined as in (44) The state feedback gains can be

constructed as 119865119894= 119884119894119883minus119879

Proof If 1198601119894 1198602119894 and 119861

119894in Φ119894119895are replaced with 119860

1119894+

119863119894119870119894(119905)1198641119894 1198602119894+ 119863119894119870119894(119905)1198642119894 and 119861

119894+ 119863119894119870119894(119905)119864119887119894in (28) of

Theorem 4 respectively (27) for system (33) can be rewrittenas

119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889 (119905))) Ω

119894119895lt 0 (37)

where

Ω119894119895= Φ119894119895+ 1198631198791119894119870119894(119905) 119864119894+ 119864119879119894119870119879119894(119905) 1198631119894+ 1198631198792119894119870119894(119905) 1198642119894119895

+ 1198641198792119894119895119870119879119894(119905) 1198632119894+ 119863119879119887119894119870119894(119905) 119864119887119894119895+ 119864119879119887119894119895119870119879119894(119905) 119863119887119894

1198631119894= [1198631198791198940 0 0 0 0 0 0]

1198632119894= [0 119863119879

1198940 0 0 0 0 0]

119863119887119894= [0 0 0 0 119863119879

1198940 0 0]

(38)

1198641119894= [1198641119894119883119879 11990521198641119894119883119879 11990531198641119894119883119879 11990541198641119894119883119879 119905

51198641119894119883119879 11990561198641119894119883119879 11990571198641119894119883119879 11990581198641119894119883119879]

1198642119894= [1198642119894119883119879 11990521198642119894119883119879 11990531198642119894119883119879 11990541198642119894119883119879 11990551198642119894119883119879 11990561198642119894119883119879 11990571198642119894119883119879 11990581198642119894119883119879]

119864119887119894119895= [119864119887119894119884119895 1199052119864119887119894119884119895 1199053119864119887119894119884119895 1199054119864119887119894119884119895 1199055119864119887119894119884119895 1199056119864119887119894119884119895 1199057119864119887119894119884119895 1199058119864119887119894119884119895]

(39)

If Ω119894119895

lt 0 (37) holds According to Lemma 2 it isstraightforward to know that Ω

119894119895lt 0 is true if for each 119894

119895 there exists scalars 1205761119894gt 0 120576

2119894gt 0 and 120576

119887119894119895gt 0 such that

the following inequality holds

Φ119894119895+ 120576111989411986311987911198941198631119894+ 120576minus1111989411986411987911198941198641119894+ 120576211989411986311987921198941198632119894

+ 120576minus1211989411986411987921198941198642119894+ 120576119887119894119895119863119879119887119894119863119887119894+ 120576minus1119887119894119895119864119879119887119894119895119864119887119894119895lt 0

(40)

Based on Schur complement (40) is equivalent to thefollowing inequality

Ψ119894119895=[[[

[

Φ119894119895

1198641198791119894

1198641198792119894

119864119879119887119894119895

lowast minus1205761119894119868 0 0

lowast lowast minus1205762119894119868 0

lowast lowast lowast minus120576119887119894119895119868

]]]

]

lt 0 (41)

Mathematical Problems in Engineering 9

where

Φ119894119895=

[[[[[[[[

[

Φ11119894 Φ12119894 Φ13 Φ14 Φ15119894119895 Φ16 Φ17 Φ18

lowast Φ22119894 1199053119883119860119879

21198941199054119883119860119879

2119894Φ25119894119895 1199056119883119860

119879

21198941199057119883119860119879

2119894Φ28119894

lowast lowast Φ33119894 minus11987522 1199053119861119894119884119895 0 0 minus1199053119883119879

lowast lowast lowast Φ44 1199054119861119894119884119895 0 0 Φ48

lowast lowast lowast lowast Φ55119894119895 1199056119884119879

119895119861119879

1198941199057119884119879

119895119861119879

119894Φ58119894

lowast lowast lowast lowast lowast Φ66 Φ67 minus1199056119883119879

lowast lowast lowast lowast lowast lowast Φ77 Φ78

lowast lowast lowast lowast lowast lowast lowast Φ88

]]]]]]]]

]

lt 0

Φ11119894= Φ11119894+ 1205761119894119863119894119863119879119894

Φ22119894= Φ22119894+ 1205762119894119863119894119863119879119894

Φ55119894119895

= Φ55119894119895

+ 120576119887119894119863119894119863119879119894

(42)

and Φ are the same as in Theorem 4 Therefore (41) holds ifand only if the following inequality holds

119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889 (119905))) Ψ

119894119895lt 0 (43)

To alleviate the conservativeness of our robust stabilityanalysis similar to the analysis approach of Theorem 4 weintroduce sum119903

119894=1sum119903119895=1

119908119894(119908119895minus 119898119895)Λ119894= 0 to (43) and the

following proof is the same as in Theorem 4 We can obtainthe conditions and (34)ndash(36) Thus the fuzzy control systemwith the time varying state and input delays (34) is robuststable on the basis of the control law 119865

119894= 119884119894119883minus119879under these

conditions and (34)ndash(36) hold

Remark 6 If we set 119908119894(119909(119905)) = 119898

119895(119909(119905)) 119889

1(119905) = 119889

2(119905)

the closed-loop system in this paper has the same structureas that of paper [26] On the other hand let 119908

119894(119909(119905)) =

119898119895(119909(119905)) and 119889

1(119905) = 119889

2(119905) = 119889 we can obtain the system

in [16] Therefore the system studied in this paper is moregeneral Additionally the information of the membershipfunctions of the fuzzy time delay models and controllers areall considered in the stability analysis If the fuzzymodels andfuzzy controllers share the same fuzzymembership functionsthat is 119908

119894(119909(119905)) = 119898

119895(119909(119905)) the stability analysis can be

referred to [25 27 28] In other words the problem in [2527 28] is actually a special case of the one investigated in thispaper

Remark 7 The augmented Lyapunov function with an addi-tional triple-integral term is introduced in analyzing thestability problem for the T-S fuzzy systems with time varyingstate and input delays In addition two integral inequalitiesare used to derive Theorem 4 and less free-weighting matri-ces are introduced which lead to get less conservative resultsIt can be observed that some existing results are the specialcases of this paper For example the method in [25] is from(18) with 119875

12= 0 119875

22= 0 119885

12= 11988522= 0 119873

12= 11987322= 0

and 1198771= 1198772= 0 in Theorem 4 The results in [22] are based

on system (33) with 1198892(119905) = 0 and (18) with119872

2= 0 119885

12=

11988522= 0 119876

1= 1198762= 0 and 119877

1= 1198772= 0

Remark 8 Different from the general PDC technique thedesign method under the imperfect premise matching ismuch more flexible because the membership functions ofthe fuzzy controllers do not need to be chosen the same asthose of the fuzzy time delay models Instead they can bedesigned arbitrarilyThus the design flexibility is significantlyenhanced On the other hand some simple membershipfunctions of the fuzzy controllers might be employed whichcan reduce the implementation cost

Remark 9 In [7ndash11 15ndash20] the authors have considered therobust control for uncertain T-S fuzzy systems with statedelays Some delay-dependent conditions and fuzzy con-trollers are obtained with a traditional Lyapunov-Krasovskiifunctionalmethod andPDCmethod which all are the specialcases of Theorem 5 in this paper

4 Numerical Examples

In this section two numerical examples are given to illus-trate the conservativeness and effectiveness of the proposedmethods The first example compares our techniques withthe existing ones in the literature for stability analysis whichshows that Theorem 4 in this paper is less conservative thanthe other results The second example is used to illustrate theadvantage of the robust stability conditions ofTheorem 5 anddemonstrate how to design robust fuzzy controllers by usingour approach

Example 10 Consider the following fuzzy system with stateand input delays

(119905) =2

sum119894=1

119908119894(120579 (119905)) [119860

1119894119909 (119905) + 119860

2119894119909 (119905 minus 119889

1(119905))

+119861119894119906 (119905 minus 119889

2(119905))]

11986011= [0 06

0 1 ] 11986021= [05 09

0 2 ]

11986012= [1 0

1 0] 11986022= [09 0

1 16]

1198611= 1198612= [11]

(44)

The fuzzy membership functions are selected as [31]

1199081(1199091(119905)) = (1 minus

119888 (119905) sin (10038161003816100381610038161199091 (119905)1003816100381610038161003816minus4)5

1 + expminus1001199091(119905)3(1minus1199091(119905)))

timescos (119909

1(119905))2

1 + expminus251199091(119905)(3+(1199091(119905)042))

times 1199082(1199091(119905)) = 1 minus 119908

1(1199091(119905))

(45)

10 Mathematical Problems in Engineering

Table 1 The maximum allowable time delay and feedback gains(1205831= 0 120583

2= 0)

Paper ℎ1

ℎ2 Feedback gains

[26] 03120 031201198651= [10598 minus56598]

1198652= [minus13068 minus41167]

Theorem 4 033 0571198651= [minus00042 minus00449]1198652= [00048 00717]

where

1199091(119905) isin [minus

1205872

1205872]

119888 (119905) =sin (119909

1(119905)) + 140

isin [minus005 005] (46)

If we set 1198891(119905) = 119889

2(119905) in system (44) we can obtain the

system in [26] which implies that our system ismore generalEmploying the LMIs in [26] and those inTheorem 4yields themaximum state and input delays ℎ

1 and ℎ

2 which guarantee

the stability of system (44) and the feedback gains for 1205831= 0

and 1205832= 0 when 120588

1= 075 120588

2= 095 119905

2= 01 119905

3=

02 1199054= 07 119905

5= 01 119905

6= 04 119905

7= 01 and 119905

8= 13 as

given in Table 1 which clearly shows the superiority of theresults derived in this paper over those obtained from [26]

Example 11 Consider the well-known truck-trailer systemwhich can be described by the following T-S fuzzy system

(119905) =2

sum119894=1

119908119894(120579 (119905)) [(119860

1119894+ Δ1198601119894) 119909 (119905) + 119860

2119894119909 (119905 minus 119889

1(119905))

+ (119861119894+ Δ119861119894) 119906 (119905 minus 119889

2(119905))]

(47)

11986011=

[[[[[[[[[

[

minus119886V1199051198711199050

0 0

119886V1199051198711199050

0 0

119886V21199052

21198711199050

V1199051199050

0

]]]]]]]]]

]

11986021=

[[[[[[[[[

[

minus (1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)V21199052

21198711199050

0 0

]]]]]]]]]

]

1198611=[[[[

[

V1199051198971199050

0

0

]]]]

]

11986012=

[[[[[[[[[

[

minus119886V1199051198711199050

0 0

119886V1199051198711199050

0 0

119886119889V21199052

21198711199050

119889V1199051199050

0

]]]]]]]]]

]

11986022=

[[[[[[[[[

[

minus (1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)119889V21199052

21198711199050

0 0

]]]]]]]]]

]

(48)

1198612=[[[[

[

V1199051198971199050

0

0

]]]]

]

(49)

where 119897 = 28 119871 = 55 V = minus10 119905 = 20 1199050= 05

119889 = 101199050120587 119863

119894= [02555 02555 02555]119879 119864

1119894= 1198642119894=

[01 0 0]119879 119864119887119894

= [01 0 0]119879 (119894 = 1 2) and the fuzzymembership functions are selected the same as inExample 10

(1) Let 119886 = 1 system (47) is the same as the fuzzy systemof [22] which implies that our system ismore generalTable 2 gives themaximum input delay value of ℎ

2 for

which the stabilization is guaranteed byTheorem 5 aswell as the state feedback gains when 120588

1= 075 120588

2=

095 1199052= 01 119905

3= 02 119905

4= 07 119905

5= 01 119905

6=

04 1199057= 08 and 119905

8= 2 It is clearly visible that the

results derived in this paper are better than those from[22 28]

(2) In the case of 119886 = 1 the state and input delays of system(47) are all exist The approach proposed in [22] cannot be used to get the maximum of state delay ℎ

1

as the system in [22] only contains the input delayHowever the method in [28] and Theorem 5 can beused Here let 119886 = 07 120588

1= 075 120588

2= 095 119905

2=

01 1199053= 02 119905

4= 07 119905

5= 01 119905

6= 04 119905

7=

08 and 1199058= 3 and Table 3 provides the maximum

upper bounds of ℎ1and ℎ

2and the state feedback

gains for which the robust stabilization is guaranteedbyTheorem 5

By Theorem 5 we can conclude that the uncertain fuzzymodel (47) is robust stable based on the following fuzzycontrol law

119906 (119905 minus ℎ2) =2

sum119895=1

119898119895(1199091(119905 minus ℎ

2)) 119865119895119909 (119905 minus ℎ

2) (50)

With the PDC design technique it is required that thefuzzy controller must share the same fuzzy membershipfunctions as those of the fuzzy time delay model Under such

Mathematical Problems in Engineering 11

Table 2 The maximum allowable time delay and feedback gains(1205832= 0)

Paper ℎ2 Feedback gains

[22] 0751198651= [34227 minus03535 00045]

1198652= [35215 minus03617 00056]

[28] 0861198651= [33219 minus02406 00025]

1198652= [33272 minus02494 00026]

Theorem 5 121198651= [minus00212 00093 minus00024]1198652= [minus00250 00044 00014]

Table 3 The maximum allowable time delay and feedback gains(1205831= 1205832= 0)

Paper ℎ1

ℎ2 Feedback gains

[28] 01 0551198651= [33219 minus02406 00025]

1198652= [33272 minus02494 00026]

Theorem 5 04 171198651= [minus00128 00090 minus00017]1198652= [minus00194 00012 00009]

a condition the implementation cost of the fuzzy controlleris high by employing these complex membership functionsHowever fromTheorem 5 the membership functions do notneed to be chosen the same as119908

1(1199091(119905)) and119908

2(1199091(119905)) Thus

we can select some simple membership functions insteadsuch as

1198981(1199091(119905 minus ℎ

2)) = 075 exp

(minus1199091(119905 minus ℎ

2) minus 038)2

2 times 0382+ 005

1198982(1199091(119905 minus ℎ

2)) = 1 minus 119898

1(1199091(119905 minus ℎ

2))

(51)

We also assume that 1205881= 075 and 120588

2= 095 such that

119898119895(1199091(119905 minus ℎ

2)) minus 120588

119895119908119895(1199091(119905)) gt 0 for 119895 = 1 2 and 119909

1(119905 minus ℎ

2)

The fuzzy control law 119906(119905 minus ℎ2) = 119898

1[minus00128 00090 minus

00017]119909 + 1198982[minus00194 00012 00009]119909 can be employed

to stabilize the uncertain fuzzy control systems in (47) withthe state and input delays Figures 1 and 2 illustrate the stablestate response and control input under the initial conditionof 119909(0) = [4 minus 1 2]119879

Remark 12 In Example 11 our method offers less conser-vative results in the sense of allowing longer time delayand obtaining smaller feedback control gains Moreovercompared with [22ndash28] where the fuzzy controllermust havethe same fuzzy membership functions as those of the fuzzytime delay model the above membership functions of ourfuzzy controller are much simper which can considerablylower the implementation cost of the fuzzy controller andenhance the design flexibility

Remark 13 If the function of 119888(119905) in 1199081(1199091(119905)) is unknown

on the basis of the PDC design technique [7ndash11 15ndash20 22ndash28] the fuzzy controller cannot be implemented as 119888(119905) isnot available However using the proposed design method

0 5 10 15

0

2

4

6Closed-loop response

minus12

minus10

minus8

minus6

minus4

minus2

119905 (s)

119909(119905)

1199092

11990931199091

Figure 1The state responses of system (47) with ℎ1= 04 ℎ

2= 17

0 5 10 15

0

5

10

15

20

25

30

119905 (s)

minus20

minus15

minus10

minus5

119906(119905)

Figure 2 The control input of system (47) with ℎ1= 04 ℎ

2= 17

we can select some simper and well-known membershipfunctions that are different from those of the fuzzy time delaymodels for example119898

1(1199091(119905)) and119898

2(1199091(119905)) so as to realize

the fuzzy controller In other words the proposed designapproach can be applied even for the fuzzy time delay modelswith uncertain grades of membership

5 Conclusions

In this paper we study the robust stability and stabilizationproblems for general nonlinear fuzzy systems with timevarying state and input delays A less conservative delay-dependent robust stability criterion has been obtained byintroducing a novel augmented Lyapunov function withan additional triple-integral term Moreover a new design

12 Mathematical Problems in Engineering

approach different from the traditional PDC design tech-nique is proposed which can significantly improve the designflexibility and reduce the implementation cost of the fuzzycontroller by arbitrarily selecting simple fuzzy membershipfunctions Two numerical examples are used to illustrate theconservativeness and effectiveness of our proposed methods

Acknowledgments

This research work was funded by the NSFC under Grantno 61273095 and the Academy of Finland under Grants no135225 and no 127299

References

[1] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[2] W H Ho S H Chen I T Chen J H Chou and C C ShuldquoDesign of stable and quadratic-optimal static output feed-back controllers for T-S-fuzzy-model-based control systems anintegrative computational approachrdquo International Journal ofInnovative Computing Information and Control vol 8 no 1App 403ndash418 2012

[3] X J Su L G Wu P Shi and Y D Song ldquo119867infinmodel reduction

of T-S fuzzy stochastic systemsrdquo IEEE Transactions on SystemsMan and Cybermetics B vol 42 no 6 pp 1574ndash1585 2012

[4] R Yang P Shi G-P Liu andH Gao ldquoNetwork-based feedbackcontrol for systems with mixed delays based on quantizationand dropout compensationrdquo Automatica vol 47 no 12 pp2805ndash2809 2011

[5] XM Zhang Z J Zhang andG P Lu ldquoFault detection for state-delay fuzzy systems subject to random communication delayrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 4 pp 2439ndash2451 2012

[6] F B Li and X Zhang ldquoDelay-range-dependent robust 119867infin

filtering for singuar LPV systems with time variant delayrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 1 pp 339ndash353 2013

[7] Y Y Cao and P M Frank ldquoAnalysis and synthesis of nonlineartime-delay systems via fuzzy control approachrdquo IEEE Transac-tions on Fuzzy Systems vol 8 no 2 pp 200ndash211 2000

[8] Y-Y Cao and P M Frank ldquoStability analysis and synthesis ofnonlinear time-delay systems via linear Takagi-Sugeno fuzzymodelsrdquo Fuzzy Sets and Systems vol 124 no 2 pp 213ndash2292001

[9] O M Kwon and J H Park ldquoDelay-range-dependent stabiliza-tion of uncertain dynamic systems with interval time-varyingdelaysrdquo Applied Mathematics and Computation vol 208 no 1pp 58ndash68 2009

[10] E Tian D Yue and Y Zhang ldquoDelay-dependent robust 119867infin

control for T-S fuzzy system with interval time-varying delayrdquoFuzzy Sets and Systems vol 160 no 12 pp 1708ndash1719 2009

[11] J Y An T Li G LWen and R Li ldquoNew stability conditions foruncertain T-S fuzzy systems with interval time-varying delayrdquoInternational Journal of Control Automation and Systems vol10 no 3 pp 490ndash497 2012

[12] Z J Zhang X L Huang X J Ban and X Z Gao ldquoStabilityanalysis and controller design of T-S fuzzy systems with time-delay under imperfect premise matchingrdquo Journal of BeijingInstitute of Technology vol 21 no 3 pp 387ndash393 2012

[13] O M Kwon M J Park S M Lee and J H Park ldquoAugmentedLyapunov-Krasovskii functional approaches to robust stabilitycriteria for uncertain Takagi-Sugeno fuzzy systems with time-varying delaysrdquo Fuzzy Sets and Systems vol 201 no 16 pp 1ndash192012

[14] F Liu M Wu Y He and R Yokoyama ldquoNew delay-dependentstability criteria for T-S fuzzy systems with time-varying delayrdquoFuzzy Sets and Systems vol 161 no 15 pp 2033ndash2042 2010

[15] B Chen X Liu and S Tong ldquoNew delay-dependent stabiliza-tion conditions of T-S fuzzy systems with constant delayrdquo FuzzySets and Systems vol 158 no 20 pp 2209ndash2224 2007

[16] J Yoneyama ldquoRobust stability and stabilization for uncertainTakagi-Sugeno fuzzy time-delay systemsrdquo Fuzzy Sets and Sys-tems vol 158 no 2 pp 115ndash134 2007

[17] C Lin Q-G Wang and T H Lee ldquoDelay-dependent LMIconditions for stability and stabilization of T-S fuzzy systemswith bounded time-delayrdquo Fuzzy Sets and Systems vol 157 no9 pp 1229ndash1247 2006

[18] X J Su P Shi L G Wu and Y D Song ldquoA novel approachto filter design for T-S fuzzy discrete-time systems with time-varying delayrdquo IEEE Transactions on Fuzzy Systems vol 20 no6 pp 1114ndash1129 2012

[19] C Peng and Q-L Han ldquoDelay-range-dependent robust stabi-lization for uncertain T-S fuzzy control systems with intervaltime-varying delaysrdquo Information Sciences vol 181 no 19 pp4287ndash4299 2011

[20] S T Li Y W Jing and X M Liu ldquoRobust control for a class ofT-S fuzzy systemswith interval time-varying delayrdquo InternatinalJournal of Control Automation and Systems vol 10 no 4 pp737ndash743 2012

[21] X-M Zhang M Wu J-H She and Y He ldquoDelay-dependentstabilization of linear systems with time-varying state and inputdelaysrdquo Automatica vol 41 no 8 pp 1405ndash1412 2005

[22] B Chen X Liu and S Tong ldquoRobust fuzzy control of nonlinearsystems with input delayrdquo Chaos Solitons amp Fractals vol 37 no3 pp 894ndash901 2008

[23] C Lin Q-G Wang and T H Lee ldquoDelay-dependent LMIconditions for stability and stabilization of T-S fuzzy systemswith bounded time-delayrdquo Fuzzy Sets and Systems vol 157 no9 pp 1229ndash1247 2006

[24] H J Lee J B Park and Y H Joo ldquoRobust control for uncertainTakagi-Sugeno fuzzy systems with time-varying input delayrdquoJournal of Dynamic Systems Measurement and Control Trans-actions of the ASME vol 127 no 2 pp 302ndash306 2005

[25] C-H Lien and K-W Yu ldquoRobust control for Takagi-Sugenofuzzy systems with time-varying state and input delaysrdquo ChaosSolitons and Fractals vol 35 no 5 pp 1003ndash1008 2008

[26] Y K Su B Chen S Y Zhang and C Lin ldquoDelay-dependentstabilization of T-S fuzzy systems with time delayrdquo Control andDecision Kongzhi yu Juece vol 24 no 6 pp 921ndash927 2009

[27] B Chen X Liu C Lin and K Liu ldquoRobust 119867infin

control ofTakagi-Sugeno fuzzy systems with state and input time delaysrdquoFuzzy Sets and Systems vol 160 no 4 pp 403ndash422 2009

[28] L Li and X Liu ldquoNew results on delay-dependent robuststability criteria of uncertain fuzzy systems with state and inputdelaysrdquo Information Sciences vol 179 no 8 pp 1134ndash1148 2009

[29] Z J Zhang X J Ban X L Huang and X Z Gao ldquoNew delay-dependent robust stablity for T-S fuzzy systems with intervevaltime-varying delayrdquo in Proccedings of the 5th International JointConference on Computational Science and Optimization pp826ndash830 Harbin China 2012

Mathematical Problems in Engineering 13

[30] H K Lam and F H F Leung ldquoLMI-based stability andperformance design of fuzzy control systems fuzzy modelsand controllers with different premisesrdquo in Proceedings of theInternational Conference on Fuzzy Systerm pp 9599ndash9506Vancouver Canada 2006

[31] H K Lam andMNarimani ldquoStability analysis and perfomancedesign for fuzzy-model-based control system under imperfectpremise matchingrdquo IEEE Transactions on Fuzzy Systems vol 17no 4 pp 949ndash961 2009

[32] S Boyd L E Ghaoui and E Feron Linear Matrix Inequality inSystems and ControlTheory SIAM Philadelphia Pa USA 1994

[33] I R Petersen and C V Hollot ldquoA Riccati equation approach tothe stabilization of uncertain linear systemsrdquo Automatica vol22 no 4 pp 397ndash411 1986

[34] J Sun G P Liu and J Chen ldquoDelay-dependent stabilityand stabilization of neutral time-delay systemsrdquo InternationalJournal of Robust andNonlinear Control vol 19 no 12 pp 1364ndash1375 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

where

Φ119894119895=

[[[[[[[[

[

Φ11119894 Φ12119894 Φ13 Φ14 Φ15119894119895 Φ16 Φ17 Φ18

lowast Φ22119894 1199053119883119860119879

21198941199054119883119860119879

2119894Φ25119894119895 1199056119883119860

119879

21198941199057119883119860119879

2119894Φ28119894

lowast lowast Φ33119894 minus11987522 1199053119861119894119884119895 0 0 minus1199053119883119879

lowast lowast lowast Φ44 1199054119861119894119884119895 0 0 Φ48

lowast lowast lowast lowast Φ55119894119895 1199056119884119879

119895119861119879

1198941199057119884119879

119895119861119879

119894Φ58119894

lowast lowast lowast lowast lowast Φ66 Φ67 minus1199056119883119879

lowast lowast lowast lowast lowast lowast Φ77 Φ78

lowast lowast lowast lowast lowast lowast lowast Φ88

]]]]]]]]

]

Φ11119894= 11987512+ 119875119879

12+11988212+119882119879

12+ ℎ111988511

+ ℎ211987311minus1ℎ1

11988522minus1ℎ2

11987322+ 1198761+ 1198762

+1198721+1198722minus 21198771minus 21198772+ 1198601119894119883119879 + 119883119860119879

1119894

Φ12119894= 1198602119894119883119879 + 119905

21198831198601198791119894

Φ13= minus11987512minus1ℎ 111988522+ 11990531198831198601198791119894

Φ14= 11987522minus1ℎ1

119885119879

12+1ℎ1

1198771+ 11990541198831198601198791119894

Φ15119894119895

= 119861119894119884119895+ 11990551198831198601198791119894

Φ16= minus119882

12minus1ℎ2

11987322+ 11990561198831198601198791119894

Φ17= 11988222minus1ℎ2

119873119879

12+2ℎ22

1198772+ 11990571198831198601198791119894

Φ18= 11987511+11988211+ ℎ111988512+ ℎ211987312minus 119883119879 + 119905

81198831198601198791119894

Φ22119894= minus (1 minus 120583

1) 1198761+ 11990521198602119894119883119879 + 119905

21198831198601198792119894

Φ25119894119895

= 1199052119861119894119884119895+ 11990551198831198601198792119894

Φ28119894= minus1199052119883119879 + 119905

81198831198601198792119894

Φ33= minus

1ℎ1

11988522minus1198721

Φ44= minus

1ℎ1

11988511minus2ℎ21

1198771

Φ48= 119875119879

12minus 1199054119883119879

Φ55119894= minus (1 minus 120583

2) 1198762+ 1199055119861119894119884119895+ 1199055119884119879119895119861119879119894

Φ58119894= minus1199055119883119879 + 119905

8119884119879119895119861119879119894

Φ66= minus

1ℎ2

11987322minus1198722

Φ67= minus

1ℎ2

119873119879

12minus11988222

Φ77= minus

1ℎ2

11987311minus2ℎ22

1198772

Φ78= 119882119879

12minus 1199057119883119879

Φ88= ℎ111988522+ ℎ211987322+12ℎ211198771+12ℎ221198772minus 1199058119883119879

(28)

Here ldquolowastrdquo denotes the transpose elements in the symmetricpositions

If (27) holds then (119909(119905)) lt 0 Considersum119903119894=1sum119903119895=1

119908119894(119909(119905))(119908

119895(119909(119905)) minus 119898

119895(119909(119905 minus 119889

2(119905))))Λ

119894= 0

where Λ119894= Λ119879119894isin 1198778119899times8119899 gt 0 119894 = 1 2 119903 are arbitrary

matrices These terms are introduced to (27) to alleviate theconservativeness From (27) we also have

Φ =119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))Φ

119894119895

=119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905)) (119908

119895(119909 (119905)) minus 119898

119895(119909 (119905 minus 119889

2(119905)))

+120588119895119908119895(119909 (119905)) minus 120588

119895119908119895(119909 (119905))) Λ

119894

+119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))Φ

119894119895

=119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905)) 119908

119895(119909 (119905)) (120588

119895Φ119894119895minus 120588119895Λ119894+ Λ119894)

+119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905)) (119898

119895(119909 (119905 minus 119889

2(119905))) minus 120588

119895119908119895(119909 (119905)))

times (Φ119894119895minus Λ119894)

le119903

sum119894=1

1199082119894(120588119894Φ119894119894minus 120588119894Λ119894+ Λ119894)

+119903

sum119894=1

119903

sum119895=1

119908119894(119898119895minus 120588119895119908119895) (Φ119894119895minus Λ119894)

+119903

sum119894=1

sum119894lt119895

119908119894119908119895(120588119895Φ119894119895+ 120588119894Φ119895119894minus 120588119895Λ119894minus 120588119894Λ119895

+Λ119894+ Λ119895)

(29)

with119898119895(119909(119905 minus 119889

2(119905))) minus 120588

119895119908119895(119909(119905)) ge 0 for all 119895 119909(119905) and 119909(119905 minus

1198892(119905)) Let

Φ119894119895minus Λ119894lt 0 (30)

120588119894Φ119894119894minus 120588119894Λ119894+ Λ119894lt 0 (31)

120588119895Φ119894119895+ 120588119894Φ119895119894minus 120588119895Λ119894minus 120588119894Λ119895+ Λ119894+ Λ119895le 0 119894 lt 119895 (32)

for all 119894 119895 = 1 2 119903

8 Mathematical Problems in Engineering

Since (119909(119905)) lt 0 the fuzzy control system with the timevarying state and input delays (15) is asymptotically stablewith the state feedback control law (17)

32 Robust Stability of Uncertain Fuzzy Systems We alsoexamine the design of a robust stable controller for theuncertain system (4) under the imperfect premise matchingConsider the following uncertain fuzzy time varying delaycontrol system

(119905) =119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))

times [(1198601119894+ Δ1198601119894(119905)) 119909 (119905) + (119860

2119894+ Δ1198602119894(119905))

times 119909 (119905 minus 1198891(119905)) + (119861

119894+ Δ119861119894(119905))

times119906 (119905 minus 1198892(119905))]

119909 (119905) = 120593 (119905) 119905 isin [minusmax (ℎ1 ℎ2) 0]

(33)

Based on the above results of Theorem 4 a robust stabiliza-tion criterion for theT-S fuzzy systemswith time varying stateand input delays is investigated The following result can beobtained

Theorem 5 Given scalars ℎ1ge 0 ℎ

2ge 0 120583

1 1205832 and 119905

119894(119894 =

2 8) the uncertain fuzzy control systems with the timevarying state and input delays (33) is robustly stable for any0 le 119889

119894(119905) le ℎ

119894(119894 = 1 2) via the imperfect premise matching

controller design technique if the membership functions of thefuzzy model and fuzzy controller satisfy 119898

119895(119909(119905 minus 119889

2(119905))) minus

120588119895119908119895(119909(119905)) ge 0 for all 119895 119909(119905) and 119909(119905 minus 119889

2(119905)) where 0 lt

120588119895lt 1 and there exist common matrices 119875

11gt 0 119875

22gt 0

11988211gt 0 119882

22gt 0 119885

11gt 0 119885

22gt 0 119873

11gt 0 119873

22gt 0

119876119894gt 0 119872

119894gt 0 119877

119894gt 0 (119894 = 1 2) and Λ

119894= ΛT119894isin 1198778119899times8119899 gt

0 (i = 1 2 r) 119875 gt 0 119882 gt 0 119885 gt 0 119873 gt 0 and somematrices 119884

119895(119895 = 1 2 119903) 119883 and scalars 120576

1119894gt 0 120576

2119894gt 0

120576119887119894119895gt 0 satisfy the following LMIs

Ψ119894119895minus Λ119894lt 0 (34)

120588119894Ψ119894119894minus 120588119894Λ119894+ Λ119894lt 0 (35)

120588119895Ψ119894119895+ 120588119894Ψ119895119894minus 120588119895Λ119894minus 120588119894Λ119895+ Λ119894+ Λ119895 le 0 119894 lt 119895 (36)

where Ψ119894119895is defined as in (44) The state feedback gains can be

constructed as 119865119894= 119884119894119883minus119879

Proof If 1198601119894 1198602119894 and 119861

119894in Φ119894119895are replaced with 119860

1119894+

119863119894119870119894(119905)1198641119894 1198602119894+ 119863119894119870119894(119905)1198642119894 and 119861

119894+ 119863119894119870119894(119905)119864119887119894in (28) of

Theorem 4 respectively (27) for system (33) can be rewrittenas

119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889 (119905))) Ω

119894119895lt 0 (37)

where

Ω119894119895= Φ119894119895+ 1198631198791119894119870119894(119905) 119864119894+ 119864119879119894119870119879119894(119905) 1198631119894+ 1198631198792119894119870119894(119905) 1198642119894119895

+ 1198641198792119894119895119870119879119894(119905) 1198632119894+ 119863119879119887119894119870119894(119905) 119864119887119894119895+ 119864119879119887119894119895119870119879119894(119905) 119863119887119894

1198631119894= [1198631198791198940 0 0 0 0 0 0]

1198632119894= [0 119863119879

1198940 0 0 0 0 0]

119863119887119894= [0 0 0 0 119863119879

1198940 0 0]

(38)

1198641119894= [1198641119894119883119879 11990521198641119894119883119879 11990531198641119894119883119879 11990541198641119894119883119879 119905

51198641119894119883119879 11990561198641119894119883119879 11990571198641119894119883119879 11990581198641119894119883119879]

1198642119894= [1198642119894119883119879 11990521198642119894119883119879 11990531198642119894119883119879 11990541198642119894119883119879 11990551198642119894119883119879 11990561198642119894119883119879 11990571198642119894119883119879 11990581198642119894119883119879]

119864119887119894119895= [119864119887119894119884119895 1199052119864119887119894119884119895 1199053119864119887119894119884119895 1199054119864119887119894119884119895 1199055119864119887119894119884119895 1199056119864119887119894119884119895 1199057119864119887119894119884119895 1199058119864119887119894119884119895]

(39)

If Ω119894119895

lt 0 (37) holds According to Lemma 2 it isstraightforward to know that Ω

119894119895lt 0 is true if for each 119894

119895 there exists scalars 1205761119894gt 0 120576

2119894gt 0 and 120576

119887119894119895gt 0 such that

the following inequality holds

Φ119894119895+ 120576111989411986311987911198941198631119894+ 120576minus1111989411986411987911198941198641119894+ 120576211989411986311987921198941198632119894

+ 120576minus1211989411986411987921198941198642119894+ 120576119887119894119895119863119879119887119894119863119887119894+ 120576minus1119887119894119895119864119879119887119894119895119864119887119894119895lt 0

(40)

Based on Schur complement (40) is equivalent to thefollowing inequality

Ψ119894119895=[[[

[

Φ119894119895

1198641198791119894

1198641198792119894

119864119879119887119894119895

lowast minus1205761119894119868 0 0

lowast lowast minus1205762119894119868 0

lowast lowast lowast minus120576119887119894119895119868

]]]

]

lt 0 (41)

Mathematical Problems in Engineering 9

where

Φ119894119895=

[[[[[[[[

[

Φ11119894 Φ12119894 Φ13 Φ14 Φ15119894119895 Φ16 Φ17 Φ18

lowast Φ22119894 1199053119883119860119879

21198941199054119883119860119879

2119894Φ25119894119895 1199056119883119860

119879

21198941199057119883119860119879

2119894Φ28119894

lowast lowast Φ33119894 minus11987522 1199053119861119894119884119895 0 0 minus1199053119883119879

lowast lowast lowast Φ44 1199054119861119894119884119895 0 0 Φ48

lowast lowast lowast lowast Φ55119894119895 1199056119884119879

119895119861119879

1198941199057119884119879

119895119861119879

119894Φ58119894

lowast lowast lowast lowast lowast Φ66 Φ67 minus1199056119883119879

lowast lowast lowast lowast lowast lowast Φ77 Φ78

lowast lowast lowast lowast lowast lowast lowast Φ88

]]]]]]]]

]

lt 0

Φ11119894= Φ11119894+ 1205761119894119863119894119863119879119894

Φ22119894= Φ22119894+ 1205762119894119863119894119863119879119894

Φ55119894119895

= Φ55119894119895

+ 120576119887119894119863119894119863119879119894

(42)

and Φ are the same as in Theorem 4 Therefore (41) holds ifand only if the following inequality holds

119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889 (119905))) Ψ

119894119895lt 0 (43)

To alleviate the conservativeness of our robust stabilityanalysis similar to the analysis approach of Theorem 4 weintroduce sum119903

119894=1sum119903119895=1

119908119894(119908119895minus 119898119895)Λ119894= 0 to (43) and the

following proof is the same as in Theorem 4 We can obtainthe conditions and (34)ndash(36) Thus the fuzzy control systemwith the time varying state and input delays (34) is robuststable on the basis of the control law 119865

119894= 119884119894119883minus119879under these

conditions and (34)ndash(36) hold

Remark 6 If we set 119908119894(119909(119905)) = 119898

119895(119909(119905)) 119889

1(119905) = 119889

2(119905)

the closed-loop system in this paper has the same structureas that of paper [26] On the other hand let 119908

119894(119909(119905)) =

119898119895(119909(119905)) and 119889

1(119905) = 119889

2(119905) = 119889 we can obtain the system

in [16] Therefore the system studied in this paper is moregeneral Additionally the information of the membershipfunctions of the fuzzy time delay models and controllers areall considered in the stability analysis If the fuzzymodels andfuzzy controllers share the same fuzzymembership functionsthat is 119908

119894(119909(119905)) = 119898

119895(119909(119905)) the stability analysis can be

referred to [25 27 28] In other words the problem in [2527 28] is actually a special case of the one investigated in thispaper

Remark 7 The augmented Lyapunov function with an addi-tional triple-integral term is introduced in analyzing thestability problem for the T-S fuzzy systems with time varyingstate and input delays In addition two integral inequalitiesare used to derive Theorem 4 and less free-weighting matri-ces are introduced which lead to get less conservative resultsIt can be observed that some existing results are the specialcases of this paper For example the method in [25] is from(18) with 119875

12= 0 119875

22= 0 119885

12= 11988522= 0 119873

12= 11987322= 0

and 1198771= 1198772= 0 in Theorem 4 The results in [22] are based

on system (33) with 1198892(119905) = 0 and (18) with119872

2= 0 119885

12=

11988522= 0 119876

1= 1198762= 0 and 119877

1= 1198772= 0

Remark 8 Different from the general PDC technique thedesign method under the imperfect premise matching ismuch more flexible because the membership functions ofthe fuzzy controllers do not need to be chosen the same asthose of the fuzzy time delay models Instead they can bedesigned arbitrarilyThus the design flexibility is significantlyenhanced On the other hand some simple membershipfunctions of the fuzzy controllers might be employed whichcan reduce the implementation cost

Remark 9 In [7ndash11 15ndash20] the authors have considered therobust control for uncertain T-S fuzzy systems with statedelays Some delay-dependent conditions and fuzzy con-trollers are obtained with a traditional Lyapunov-Krasovskiifunctionalmethod andPDCmethod which all are the specialcases of Theorem 5 in this paper

4 Numerical Examples

In this section two numerical examples are given to illus-trate the conservativeness and effectiveness of the proposedmethods The first example compares our techniques withthe existing ones in the literature for stability analysis whichshows that Theorem 4 in this paper is less conservative thanthe other results The second example is used to illustrate theadvantage of the robust stability conditions ofTheorem 5 anddemonstrate how to design robust fuzzy controllers by usingour approach

Example 10 Consider the following fuzzy system with stateand input delays

(119905) =2

sum119894=1

119908119894(120579 (119905)) [119860

1119894119909 (119905) + 119860

2119894119909 (119905 minus 119889

1(119905))

+119861119894119906 (119905 minus 119889

2(119905))]

11986011= [0 06

0 1 ] 11986021= [05 09

0 2 ]

11986012= [1 0

1 0] 11986022= [09 0

1 16]

1198611= 1198612= [11]

(44)

The fuzzy membership functions are selected as [31]

1199081(1199091(119905)) = (1 minus

119888 (119905) sin (10038161003816100381610038161199091 (119905)1003816100381610038161003816minus4)5

1 + expminus1001199091(119905)3(1minus1199091(119905)))

timescos (119909

1(119905))2

1 + expminus251199091(119905)(3+(1199091(119905)042))

times 1199082(1199091(119905)) = 1 minus 119908

1(1199091(119905))

(45)

10 Mathematical Problems in Engineering

Table 1 The maximum allowable time delay and feedback gains(1205831= 0 120583

2= 0)

Paper ℎ1

ℎ2 Feedback gains

[26] 03120 031201198651= [10598 minus56598]

1198652= [minus13068 minus41167]

Theorem 4 033 0571198651= [minus00042 minus00449]1198652= [00048 00717]

where

1199091(119905) isin [minus

1205872

1205872]

119888 (119905) =sin (119909

1(119905)) + 140

isin [minus005 005] (46)

If we set 1198891(119905) = 119889

2(119905) in system (44) we can obtain the

system in [26] which implies that our system ismore generalEmploying the LMIs in [26] and those inTheorem 4yields themaximum state and input delays ℎ

1 and ℎ

2 which guarantee

the stability of system (44) and the feedback gains for 1205831= 0

and 1205832= 0 when 120588

1= 075 120588

2= 095 119905

2= 01 119905

3=

02 1199054= 07 119905

5= 01 119905

6= 04 119905

7= 01 and 119905

8= 13 as

given in Table 1 which clearly shows the superiority of theresults derived in this paper over those obtained from [26]

Example 11 Consider the well-known truck-trailer systemwhich can be described by the following T-S fuzzy system

(119905) =2

sum119894=1

119908119894(120579 (119905)) [(119860

1119894+ Δ1198601119894) 119909 (119905) + 119860

2119894119909 (119905 minus 119889

1(119905))

+ (119861119894+ Δ119861119894) 119906 (119905 minus 119889

2(119905))]

(47)

11986011=

[[[[[[[[[

[

minus119886V1199051198711199050

0 0

119886V1199051198711199050

0 0

119886V21199052

21198711199050

V1199051199050

0

]]]]]]]]]

]

11986021=

[[[[[[[[[

[

minus (1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)V21199052

21198711199050

0 0

]]]]]]]]]

]

1198611=[[[[

[

V1199051198971199050

0

0

]]]]

]

11986012=

[[[[[[[[[

[

minus119886V1199051198711199050

0 0

119886V1199051198711199050

0 0

119886119889V21199052

21198711199050

119889V1199051199050

0

]]]]]]]]]

]

11986022=

[[[[[[[[[

[

minus (1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)119889V21199052

21198711199050

0 0

]]]]]]]]]

]

(48)

1198612=[[[[

[

V1199051198971199050

0

0

]]]]

]

(49)

where 119897 = 28 119871 = 55 V = minus10 119905 = 20 1199050= 05

119889 = 101199050120587 119863

119894= [02555 02555 02555]119879 119864

1119894= 1198642119894=

[01 0 0]119879 119864119887119894

= [01 0 0]119879 (119894 = 1 2) and the fuzzymembership functions are selected the same as inExample 10

(1) Let 119886 = 1 system (47) is the same as the fuzzy systemof [22] which implies that our system ismore generalTable 2 gives themaximum input delay value of ℎ

2 for

which the stabilization is guaranteed byTheorem 5 aswell as the state feedback gains when 120588

1= 075 120588

2=

095 1199052= 01 119905

3= 02 119905

4= 07 119905

5= 01 119905

6=

04 1199057= 08 and 119905

8= 2 It is clearly visible that the

results derived in this paper are better than those from[22 28]

(2) In the case of 119886 = 1 the state and input delays of system(47) are all exist The approach proposed in [22] cannot be used to get the maximum of state delay ℎ

1

as the system in [22] only contains the input delayHowever the method in [28] and Theorem 5 can beused Here let 119886 = 07 120588

1= 075 120588

2= 095 119905

2=

01 1199053= 02 119905

4= 07 119905

5= 01 119905

6= 04 119905

7=

08 and 1199058= 3 and Table 3 provides the maximum

upper bounds of ℎ1and ℎ

2and the state feedback

gains for which the robust stabilization is guaranteedbyTheorem 5

By Theorem 5 we can conclude that the uncertain fuzzymodel (47) is robust stable based on the following fuzzycontrol law

119906 (119905 minus ℎ2) =2

sum119895=1

119898119895(1199091(119905 minus ℎ

2)) 119865119895119909 (119905 minus ℎ

2) (50)

With the PDC design technique it is required that thefuzzy controller must share the same fuzzy membershipfunctions as those of the fuzzy time delay model Under such

Mathematical Problems in Engineering 11

Table 2 The maximum allowable time delay and feedback gains(1205832= 0)

Paper ℎ2 Feedback gains

[22] 0751198651= [34227 minus03535 00045]

1198652= [35215 minus03617 00056]

[28] 0861198651= [33219 minus02406 00025]

1198652= [33272 minus02494 00026]

Theorem 5 121198651= [minus00212 00093 minus00024]1198652= [minus00250 00044 00014]

Table 3 The maximum allowable time delay and feedback gains(1205831= 1205832= 0)

Paper ℎ1

ℎ2 Feedback gains

[28] 01 0551198651= [33219 minus02406 00025]

1198652= [33272 minus02494 00026]

Theorem 5 04 171198651= [minus00128 00090 minus00017]1198652= [minus00194 00012 00009]

a condition the implementation cost of the fuzzy controlleris high by employing these complex membership functionsHowever fromTheorem 5 the membership functions do notneed to be chosen the same as119908

1(1199091(119905)) and119908

2(1199091(119905)) Thus

we can select some simple membership functions insteadsuch as

1198981(1199091(119905 minus ℎ

2)) = 075 exp

(minus1199091(119905 minus ℎ

2) minus 038)2

2 times 0382+ 005

1198982(1199091(119905 minus ℎ

2)) = 1 minus 119898

1(1199091(119905 minus ℎ

2))

(51)

We also assume that 1205881= 075 and 120588

2= 095 such that

119898119895(1199091(119905 minus ℎ

2)) minus 120588

119895119908119895(1199091(119905)) gt 0 for 119895 = 1 2 and 119909

1(119905 minus ℎ

2)

The fuzzy control law 119906(119905 minus ℎ2) = 119898

1[minus00128 00090 minus

00017]119909 + 1198982[minus00194 00012 00009]119909 can be employed

to stabilize the uncertain fuzzy control systems in (47) withthe state and input delays Figures 1 and 2 illustrate the stablestate response and control input under the initial conditionof 119909(0) = [4 minus 1 2]119879

Remark 12 In Example 11 our method offers less conser-vative results in the sense of allowing longer time delayand obtaining smaller feedback control gains Moreovercompared with [22ndash28] where the fuzzy controllermust havethe same fuzzy membership functions as those of the fuzzytime delay model the above membership functions of ourfuzzy controller are much simper which can considerablylower the implementation cost of the fuzzy controller andenhance the design flexibility

Remark 13 If the function of 119888(119905) in 1199081(1199091(119905)) is unknown

on the basis of the PDC design technique [7ndash11 15ndash20 22ndash28] the fuzzy controller cannot be implemented as 119888(119905) isnot available However using the proposed design method

0 5 10 15

0

2

4

6Closed-loop response

minus12

minus10

minus8

minus6

minus4

minus2

119905 (s)

119909(119905)

1199092

11990931199091

Figure 1The state responses of system (47) with ℎ1= 04 ℎ

2= 17

0 5 10 15

0

5

10

15

20

25

30

119905 (s)

minus20

minus15

minus10

minus5

119906(119905)

Figure 2 The control input of system (47) with ℎ1= 04 ℎ

2= 17

we can select some simper and well-known membershipfunctions that are different from those of the fuzzy time delaymodels for example119898

1(1199091(119905)) and119898

2(1199091(119905)) so as to realize

the fuzzy controller In other words the proposed designapproach can be applied even for the fuzzy time delay modelswith uncertain grades of membership

5 Conclusions

In this paper we study the robust stability and stabilizationproblems for general nonlinear fuzzy systems with timevarying state and input delays A less conservative delay-dependent robust stability criterion has been obtained byintroducing a novel augmented Lyapunov function withan additional triple-integral term Moreover a new design

12 Mathematical Problems in Engineering

approach different from the traditional PDC design tech-nique is proposed which can significantly improve the designflexibility and reduce the implementation cost of the fuzzycontroller by arbitrarily selecting simple fuzzy membershipfunctions Two numerical examples are used to illustrate theconservativeness and effectiveness of our proposed methods

Acknowledgments

This research work was funded by the NSFC under Grantno 61273095 and the Academy of Finland under Grants no135225 and no 127299

References

[1] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[2] W H Ho S H Chen I T Chen J H Chou and C C ShuldquoDesign of stable and quadratic-optimal static output feed-back controllers for T-S-fuzzy-model-based control systems anintegrative computational approachrdquo International Journal ofInnovative Computing Information and Control vol 8 no 1App 403ndash418 2012

[3] X J Su L G Wu P Shi and Y D Song ldquo119867infinmodel reduction

of T-S fuzzy stochastic systemsrdquo IEEE Transactions on SystemsMan and Cybermetics B vol 42 no 6 pp 1574ndash1585 2012

[4] R Yang P Shi G-P Liu andH Gao ldquoNetwork-based feedbackcontrol for systems with mixed delays based on quantizationand dropout compensationrdquo Automatica vol 47 no 12 pp2805ndash2809 2011

[5] XM Zhang Z J Zhang andG P Lu ldquoFault detection for state-delay fuzzy systems subject to random communication delayrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 4 pp 2439ndash2451 2012

[6] F B Li and X Zhang ldquoDelay-range-dependent robust 119867infin

filtering for singuar LPV systems with time variant delayrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 1 pp 339ndash353 2013

[7] Y Y Cao and P M Frank ldquoAnalysis and synthesis of nonlineartime-delay systems via fuzzy control approachrdquo IEEE Transac-tions on Fuzzy Systems vol 8 no 2 pp 200ndash211 2000

[8] Y-Y Cao and P M Frank ldquoStability analysis and synthesis ofnonlinear time-delay systems via linear Takagi-Sugeno fuzzymodelsrdquo Fuzzy Sets and Systems vol 124 no 2 pp 213ndash2292001

[9] O M Kwon and J H Park ldquoDelay-range-dependent stabiliza-tion of uncertain dynamic systems with interval time-varyingdelaysrdquo Applied Mathematics and Computation vol 208 no 1pp 58ndash68 2009

[10] E Tian D Yue and Y Zhang ldquoDelay-dependent robust 119867infin

control for T-S fuzzy system with interval time-varying delayrdquoFuzzy Sets and Systems vol 160 no 12 pp 1708ndash1719 2009

[11] J Y An T Li G LWen and R Li ldquoNew stability conditions foruncertain T-S fuzzy systems with interval time-varying delayrdquoInternational Journal of Control Automation and Systems vol10 no 3 pp 490ndash497 2012

[12] Z J Zhang X L Huang X J Ban and X Z Gao ldquoStabilityanalysis and controller design of T-S fuzzy systems with time-delay under imperfect premise matchingrdquo Journal of BeijingInstitute of Technology vol 21 no 3 pp 387ndash393 2012

[13] O M Kwon M J Park S M Lee and J H Park ldquoAugmentedLyapunov-Krasovskii functional approaches to robust stabilitycriteria for uncertain Takagi-Sugeno fuzzy systems with time-varying delaysrdquo Fuzzy Sets and Systems vol 201 no 16 pp 1ndash192012

[14] F Liu M Wu Y He and R Yokoyama ldquoNew delay-dependentstability criteria for T-S fuzzy systems with time-varying delayrdquoFuzzy Sets and Systems vol 161 no 15 pp 2033ndash2042 2010

[15] B Chen X Liu and S Tong ldquoNew delay-dependent stabiliza-tion conditions of T-S fuzzy systems with constant delayrdquo FuzzySets and Systems vol 158 no 20 pp 2209ndash2224 2007

[16] J Yoneyama ldquoRobust stability and stabilization for uncertainTakagi-Sugeno fuzzy time-delay systemsrdquo Fuzzy Sets and Sys-tems vol 158 no 2 pp 115ndash134 2007

[17] C Lin Q-G Wang and T H Lee ldquoDelay-dependent LMIconditions for stability and stabilization of T-S fuzzy systemswith bounded time-delayrdquo Fuzzy Sets and Systems vol 157 no9 pp 1229ndash1247 2006

[18] X J Su P Shi L G Wu and Y D Song ldquoA novel approachto filter design for T-S fuzzy discrete-time systems with time-varying delayrdquo IEEE Transactions on Fuzzy Systems vol 20 no6 pp 1114ndash1129 2012

[19] C Peng and Q-L Han ldquoDelay-range-dependent robust stabi-lization for uncertain T-S fuzzy control systems with intervaltime-varying delaysrdquo Information Sciences vol 181 no 19 pp4287ndash4299 2011

[20] S T Li Y W Jing and X M Liu ldquoRobust control for a class ofT-S fuzzy systemswith interval time-varying delayrdquo InternatinalJournal of Control Automation and Systems vol 10 no 4 pp737ndash743 2012

[21] X-M Zhang M Wu J-H She and Y He ldquoDelay-dependentstabilization of linear systems with time-varying state and inputdelaysrdquo Automatica vol 41 no 8 pp 1405ndash1412 2005

[22] B Chen X Liu and S Tong ldquoRobust fuzzy control of nonlinearsystems with input delayrdquo Chaos Solitons amp Fractals vol 37 no3 pp 894ndash901 2008

[23] C Lin Q-G Wang and T H Lee ldquoDelay-dependent LMIconditions for stability and stabilization of T-S fuzzy systemswith bounded time-delayrdquo Fuzzy Sets and Systems vol 157 no9 pp 1229ndash1247 2006

[24] H J Lee J B Park and Y H Joo ldquoRobust control for uncertainTakagi-Sugeno fuzzy systems with time-varying input delayrdquoJournal of Dynamic Systems Measurement and Control Trans-actions of the ASME vol 127 no 2 pp 302ndash306 2005

[25] C-H Lien and K-W Yu ldquoRobust control for Takagi-Sugenofuzzy systems with time-varying state and input delaysrdquo ChaosSolitons and Fractals vol 35 no 5 pp 1003ndash1008 2008

[26] Y K Su B Chen S Y Zhang and C Lin ldquoDelay-dependentstabilization of T-S fuzzy systems with time delayrdquo Control andDecision Kongzhi yu Juece vol 24 no 6 pp 921ndash927 2009

[27] B Chen X Liu C Lin and K Liu ldquoRobust 119867infin

control ofTakagi-Sugeno fuzzy systems with state and input time delaysrdquoFuzzy Sets and Systems vol 160 no 4 pp 403ndash422 2009

[28] L Li and X Liu ldquoNew results on delay-dependent robuststability criteria of uncertain fuzzy systems with state and inputdelaysrdquo Information Sciences vol 179 no 8 pp 1134ndash1148 2009

[29] Z J Zhang X J Ban X L Huang and X Z Gao ldquoNew delay-dependent robust stablity for T-S fuzzy systems with intervevaltime-varying delayrdquo in Proccedings of the 5th International JointConference on Computational Science and Optimization pp826ndash830 Harbin China 2012

Mathematical Problems in Engineering 13

[30] H K Lam and F H F Leung ldquoLMI-based stability andperformance design of fuzzy control systems fuzzy modelsand controllers with different premisesrdquo in Proceedings of theInternational Conference on Fuzzy Systerm pp 9599ndash9506Vancouver Canada 2006

[31] H K Lam andMNarimani ldquoStability analysis and perfomancedesign for fuzzy-model-based control system under imperfectpremise matchingrdquo IEEE Transactions on Fuzzy Systems vol 17no 4 pp 949ndash961 2009

[32] S Boyd L E Ghaoui and E Feron Linear Matrix Inequality inSystems and ControlTheory SIAM Philadelphia Pa USA 1994

[33] I R Petersen and C V Hollot ldquoA Riccati equation approach tothe stabilization of uncertain linear systemsrdquo Automatica vol22 no 4 pp 397ndash411 1986

[34] J Sun G P Liu and J Chen ldquoDelay-dependent stabilityand stabilization of neutral time-delay systemsrdquo InternationalJournal of Robust andNonlinear Control vol 19 no 12 pp 1364ndash1375 2009

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Since (119909(119905)) lt 0 the fuzzy control system with the timevarying state and input delays (15) is asymptotically stablewith the state feedback control law (17)

32 Robust Stability of Uncertain Fuzzy Systems We alsoexamine the design of a robust stable controller for theuncertain system (4) under the imperfect premise matchingConsider the following uncertain fuzzy time varying delaycontrol system

(119905) =119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889

2(119905)))

times [(1198601119894+ Δ1198601119894(119905)) 119909 (119905) + (119860

2119894+ Δ1198602119894(119905))

times 119909 (119905 minus 1198891(119905)) + (119861

119894+ Δ119861119894(119905))

times119906 (119905 minus 1198892(119905))]

119909 (119905) = 120593 (119905) 119905 isin [minusmax (ℎ1 ℎ2) 0]

(33)

Based on the above results of Theorem 4 a robust stabiliza-tion criterion for theT-S fuzzy systemswith time varying stateand input delays is investigated The following result can beobtained

Theorem 5 Given scalars ℎ1ge 0 ℎ

2ge 0 120583

1 1205832 and 119905

119894(119894 =

2 8) the uncertain fuzzy control systems with the timevarying state and input delays (33) is robustly stable for any0 le 119889

119894(119905) le ℎ

119894(119894 = 1 2) via the imperfect premise matching

controller design technique if the membership functions of thefuzzy model and fuzzy controller satisfy 119898

119895(119909(119905 minus 119889

2(119905))) minus

120588119895119908119895(119909(119905)) ge 0 for all 119895 119909(119905) and 119909(119905 minus 119889

2(119905)) where 0 lt

120588119895lt 1 and there exist common matrices 119875

11gt 0 119875

22gt 0

11988211gt 0 119882

22gt 0 119885

11gt 0 119885

22gt 0 119873

11gt 0 119873

22gt 0

119876119894gt 0 119872

119894gt 0 119877

119894gt 0 (119894 = 1 2) and Λ

119894= ΛT119894isin 1198778119899times8119899 gt

0 (i = 1 2 r) 119875 gt 0 119882 gt 0 119885 gt 0 119873 gt 0 and somematrices 119884

119895(119895 = 1 2 119903) 119883 and scalars 120576

1119894gt 0 120576

2119894gt 0

120576119887119894119895gt 0 satisfy the following LMIs

Ψ119894119895minus Λ119894lt 0 (34)

120588119894Ψ119894119894minus 120588119894Λ119894+ Λ119894lt 0 (35)

120588119895Ψ119894119895+ 120588119894Ψ119895119894minus 120588119895Λ119894minus 120588119894Λ119895+ Λ119894+ Λ119895 le 0 119894 lt 119895 (36)

where Ψ119894119895is defined as in (44) The state feedback gains can be

constructed as 119865119894= 119884119894119883minus119879

Proof If 1198601119894 1198602119894 and 119861

119894in Φ119894119895are replaced with 119860

1119894+

119863119894119870119894(119905)1198641119894 1198602119894+ 119863119894119870119894(119905)1198642119894 and 119861

119894+ 119863119894119870119894(119905)119864119887119894in (28) of

Theorem 4 respectively (27) for system (33) can be rewrittenas

119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889 (119905))) Ω

119894119895lt 0 (37)

where

Ω119894119895= Φ119894119895+ 1198631198791119894119870119894(119905) 119864119894+ 119864119879119894119870119879119894(119905) 1198631119894+ 1198631198792119894119870119894(119905) 1198642119894119895

+ 1198641198792119894119895119870119879119894(119905) 1198632119894+ 119863119879119887119894119870119894(119905) 119864119887119894119895+ 119864119879119887119894119895119870119879119894(119905) 119863119887119894

1198631119894= [1198631198791198940 0 0 0 0 0 0]

1198632119894= [0 119863119879

1198940 0 0 0 0 0]

119863119887119894= [0 0 0 0 119863119879

1198940 0 0]

(38)

1198641119894= [1198641119894119883119879 11990521198641119894119883119879 11990531198641119894119883119879 11990541198641119894119883119879 119905

51198641119894119883119879 11990561198641119894119883119879 11990571198641119894119883119879 11990581198641119894119883119879]

1198642119894= [1198642119894119883119879 11990521198642119894119883119879 11990531198642119894119883119879 11990541198642119894119883119879 11990551198642119894119883119879 11990561198642119894119883119879 11990571198642119894119883119879 11990581198642119894119883119879]

119864119887119894119895= [119864119887119894119884119895 1199052119864119887119894119884119895 1199053119864119887119894119884119895 1199054119864119887119894119884119895 1199055119864119887119894119884119895 1199056119864119887119894119884119895 1199057119864119887119894119884119895 1199058119864119887119894119884119895]

(39)

If Ω119894119895

lt 0 (37) holds According to Lemma 2 it isstraightforward to know that Ω

119894119895lt 0 is true if for each 119894

119895 there exists scalars 1205761119894gt 0 120576

2119894gt 0 and 120576

119887119894119895gt 0 such that

the following inequality holds

Φ119894119895+ 120576111989411986311987911198941198631119894+ 120576minus1111989411986411987911198941198641119894+ 120576211989411986311987921198941198632119894

+ 120576minus1211989411986411987921198941198642119894+ 120576119887119894119895119863119879119887119894119863119887119894+ 120576minus1119887119894119895119864119879119887119894119895119864119887119894119895lt 0

(40)

Based on Schur complement (40) is equivalent to thefollowing inequality

Ψ119894119895=[[[

[

Φ119894119895

1198641198791119894

1198641198792119894

119864119879119887119894119895

lowast minus1205761119894119868 0 0

lowast lowast minus1205762119894119868 0

lowast lowast lowast minus120576119887119894119895119868

]]]

]

lt 0 (41)

Mathematical Problems in Engineering 9

where

Φ119894119895=

[[[[[[[[

[

Φ11119894 Φ12119894 Φ13 Φ14 Φ15119894119895 Φ16 Φ17 Φ18

lowast Φ22119894 1199053119883119860119879

21198941199054119883119860119879

2119894Φ25119894119895 1199056119883119860

119879

21198941199057119883119860119879

2119894Φ28119894

lowast lowast Φ33119894 minus11987522 1199053119861119894119884119895 0 0 minus1199053119883119879

lowast lowast lowast Φ44 1199054119861119894119884119895 0 0 Φ48

lowast lowast lowast lowast Φ55119894119895 1199056119884119879

119895119861119879

1198941199057119884119879

119895119861119879

119894Φ58119894

lowast lowast lowast lowast lowast Φ66 Φ67 minus1199056119883119879

lowast lowast lowast lowast lowast lowast Φ77 Φ78

lowast lowast lowast lowast lowast lowast lowast Φ88

]]]]]]]]

]

lt 0

Φ11119894= Φ11119894+ 1205761119894119863119894119863119879119894

Φ22119894= Φ22119894+ 1205762119894119863119894119863119879119894

Φ55119894119895

= Φ55119894119895

+ 120576119887119894119863119894119863119879119894

(42)

and Φ are the same as in Theorem 4 Therefore (41) holds ifand only if the following inequality holds

119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889 (119905))) Ψ

119894119895lt 0 (43)

To alleviate the conservativeness of our robust stabilityanalysis similar to the analysis approach of Theorem 4 weintroduce sum119903

119894=1sum119903119895=1

119908119894(119908119895minus 119898119895)Λ119894= 0 to (43) and the

following proof is the same as in Theorem 4 We can obtainthe conditions and (34)ndash(36) Thus the fuzzy control systemwith the time varying state and input delays (34) is robuststable on the basis of the control law 119865

119894= 119884119894119883minus119879under these

conditions and (34)ndash(36) hold

Remark 6 If we set 119908119894(119909(119905)) = 119898

119895(119909(119905)) 119889

1(119905) = 119889

2(119905)

the closed-loop system in this paper has the same structureas that of paper [26] On the other hand let 119908

119894(119909(119905)) =

119898119895(119909(119905)) and 119889

1(119905) = 119889

2(119905) = 119889 we can obtain the system

in [16] Therefore the system studied in this paper is moregeneral Additionally the information of the membershipfunctions of the fuzzy time delay models and controllers areall considered in the stability analysis If the fuzzymodels andfuzzy controllers share the same fuzzymembership functionsthat is 119908

119894(119909(119905)) = 119898

119895(119909(119905)) the stability analysis can be

referred to [25 27 28] In other words the problem in [2527 28] is actually a special case of the one investigated in thispaper

Remark 7 The augmented Lyapunov function with an addi-tional triple-integral term is introduced in analyzing thestability problem for the T-S fuzzy systems with time varyingstate and input delays In addition two integral inequalitiesare used to derive Theorem 4 and less free-weighting matri-ces are introduced which lead to get less conservative resultsIt can be observed that some existing results are the specialcases of this paper For example the method in [25] is from(18) with 119875

12= 0 119875

22= 0 119885

12= 11988522= 0 119873

12= 11987322= 0

and 1198771= 1198772= 0 in Theorem 4 The results in [22] are based

on system (33) with 1198892(119905) = 0 and (18) with119872

2= 0 119885

12=

11988522= 0 119876

1= 1198762= 0 and 119877

1= 1198772= 0

Remark 8 Different from the general PDC technique thedesign method under the imperfect premise matching ismuch more flexible because the membership functions ofthe fuzzy controllers do not need to be chosen the same asthose of the fuzzy time delay models Instead they can bedesigned arbitrarilyThus the design flexibility is significantlyenhanced On the other hand some simple membershipfunctions of the fuzzy controllers might be employed whichcan reduce the implementation cost

Remark 9 In [7ndash11 15ndash20] the authors have considered therobust control for uncertain T-S fuzzy systems with statedelays Some delay-dependent conditions and fuzzy con-trollers are obtained with a traditional Lyapunov-Krasovskiifunctionalmethod andPDCmethod which all are the specialcases of Theorem 5 in this paper

4 Numerical Examples

In this section two numerical examples are given to illus-trate the conservativeness and effectiveness of the proposedmethods The first example compares our techniques withthe existing ones in the literature for stability analysis whichshows that Theorem 4 in this paper is less conservative thanthe other results The second example is used to illustrate theadvantage of the robust stability conditions ofTheorem 5 anddemonstrate how to design robust fuzzy controllers by usingour approach

Example 10 Consider the following fuzzy system with stateand input delays

(119905) =2

sum119894=1

119908119894(120579 (119905)) [119860

1119894119909 (119905) + 119860

2119894119909 (119905 minus 119889

1(119905))

+119861119894119906 (119905 minus 119889

2(119905))]

11986011= [0 06

0 1 ] 11986021= [05 09

0 2 ]

11986012= [1 0

1 0] 11986022= [09 0

1 16]

1198611= 1198612= [11]

(44)

The fuzzy membership functions are selected as [31]

1199081(1199091(119905)) = (1 minus

119888 (119905) sin (10038161003816100381610038161199091 (119905)1003816100381610038161003816minus4)5

1 + expminus1001199091(119905)3(1minus1199091(119905)))

timescos (119909

1(119905))2

1 + expminus251199091(119905)(3+(1199091(119905)042))

times 1199082(1199091(119905)) = 1 minus 119908

1(1199091(119905))

(45)

10 Mathematical Problems in Engineering

Table 1 The maximum allowable time delay and feedback gains(1205831= 0 120583

2= 0)

Paper ℎ1

ℎ2 Feedback gains

[26] 03120 031201198651= [10598 minus56598]

1198652= [minus13068 minus41167]

Theorem 4 033 0571198651= [minus00042 minus00449]1198652= [00048 00717]

where

1199091(119905) isin [minus

1205872

1205872]

119888 (119905) =sin (119909

1(119905)) + 140

isin [minus005 005] (46)

If we set 1198891(119905) = 119889

2(119905) in system (44) we can obtain the

system in [26] which implies that our system ismore generalEmploying the LMIs in [26] and those inTheorem 4yields themaximum state and input delays ℎ

1 and ℎ

2 which guarantee

the stability of system (44) and the feedback gains for 1205831= 0

and 1205832= 0 when 120588

1= 075 120588

2= 095 119905

2= 01 119905

3=

02 1199054= 07 119905

5= 01 119905

6= 04 119905

7= 01 and 119905

8= 13 as

given in Table 1 which clearly shows the superiority of theresults derived in this paper over those obtained from [26]

Example 11 Consider the well-known truck-trailer systemwhich can be described by the following T-S fuzzy system

(119905) =2

sum119894=1

119908119894(120579 (119905)) [(119860

1119894+ Δ1198601119894) 119909 (119905) + 119860

2119894119909 (119905 minus 119889

1(119905))

+ (119861119894+ Δ119861119894) 119906 (119905 minus 119889

2(119905))]

(47)

11986011=

[[[[[[[[[

[

minus119886V1199051198711199050

0 0

119886V1199051198711199050

0 0

119886V21199052

21198711199050

V1199051199050

0

]]]]]]]]]

]

11986021=

[[[[[[[[[

[

minus (1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)V21199052

21198711199050

0 0

]]]]]]]]]

]

1198611=[[[[

[

V1199051198971199050

0

0

]]]]

]

11986012=

[[[[[[[[[

[

minus119886V1199051198711199050

0 0

119886V1199051198711199050

0 0

119886119889V21199052

21198711199050

119889V1199051199050

0

]]]]]]]]]

]

11986022=

[[[[[[[[[

[

minus (1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)119889V21199052

21198711199050

0 0

]]]]]]]]]

]

(48)

1198612=[[[[

[

V1199051198971199050

0

0

]]]]

]

(49)

where 119897 = 28 119871 = 55 V = minus10 119905 = 20 1199050= 05

119889 = 101199050120587 119863

119894= [02555 02555 02555]119879 119864

1119894= 1198642119894=

[01 0 0]119879 119864119887119894

= [01 0 0]119879 (119894 = 1 2) and the fuzzymembership functions are selected the same as inExample 10

(1) Let 119886 = 1 system (47) is the same as the fuzzy systemof [22] which implies that our system ismore generalTable 2 gives themaximum input delay value of ℎ

2 for

which the stabilization is guaranteed byTheorem 5 aswell as the state feedback gains when 120588

1= 075 120588

2=

095 1199052= 01 119905

3= 02 119905

4= 07 119905

5= 01 119905

6=

04 1199057= 08 and 119905

8= 2 It is clearly visible that the

results derived in this paper are better than those from[22 28]

(2) In the case of 119886 = 1 the state and input delays of system(47) are all exist The approach proposed in [22] cannot be used to get the maximum of state delay ℎ

1

as the system in [22] only contains the input delayHowever the method in [28] and Theorem 5 can beused Here let 119886 = 07 120588

1= 075 120588

2= 095 119905

2=

01 1199053= 02 119905

4= 07 119905

5= 01 119905

6= 04 119905

7=

08 and 1199058= 3 and Table 3 provides the maximum

upper bounds of ℎ1and ℎ

2and the state feedback

gains for which the robust stabilization is guaranteedbyTheorem 5

By Theorem 5 we can conclude that the uncertain fuzzymodel (47) is robust stable based on the following fuzzycontrol law

119906 (119905 minus ℎ2) =2

sum119895=1

119898119895(1199091(119905 minus ℎ

2)) 119865119895119909 (119905 minus ℎ

2) (50)

With the PDC design technique it is required that thefuzzy controller must share the same fuzzy membershipfunctions as those of the fuzzy time delay model Under such

Mathematical Problems in Engineering 11

Table 2 The maximum allowable time delay and feedback gains(1205832= 0)

Paper ℎ2 Feedback gains

[22] 0751198651= [34227 minus03535 00045]

1198652= [35215 minus03617 00056]

[28] 0861198651= [33219 minus02406 00025]

1198652= [33272 minus02494 00026]

Theorem 5 121198651= [minus00212 00093 minus00024]1198652= [minus00250 00044 00014]

Table 3 The maximum allowable time delay and feedback gains(1205831= 1205832= 0)

Paper ℎ1

ℎ2 Feedback gains

[28] 01 0551198651= [33219 minus02406 00025]

1198652= [33272 minus02494 00026]

Theorem 5 04 171198651= [minus00128 00090 minus00017]1198652= [minus00194 00012 00009]

a condition the implementation cost of the fuzzy controlleris high by employing these complex membership functionsHowever fromTheorem 5 the membership functions do notneed to be chosen the same as119908

1(1199091(119905)) and119908

2(1199091(119905)) Thus

we can select some simple membership functions insteadsuch as

1198981(1199091(119905 minus ℎ

2)) = 075 exp

(minus1199091(119905 minus ℎ

2) minus 038)2

2 times 0382+ 005

1198982(1199091(119905 minus ℎ

2)) = 1 minus 119898

1(1199091(119905 minus ℎ

2))

(51)

We also assume that 1205881= 075 and 120588

2= 095 such that

119898119895(1199091(119905 minus ℎ

2)) minus 120588

119895119908119895(1199091(119905)) gt 0 for 119895 = 1 2 and 119909

1(119905 minus ℎ

2)

The fuzzy control law 119906(119905 minus ℎ2) = 119898

1[minus00128 00090 minus

00017]119909 + 1198982[minus00194 00012 00009]119909 can be employed

to stabilize the uncertain fuzzy control systems in (47) withthe state and input delays Figures 1 and 2 illustrate the stablestate response and control input under the initial conditionof 119909(0) = [4 minus 1 2]119879

Remark 12 In Example 11 our method offers less conser-vative results in the sense of allowing longer time delayand obtaining smaller feedback control gains Moreovercompared with [22ndash28] where the fuzzy controllermust havethe same fuzzy membership functions as those of the fuzzytime delay model the above membership functions of ourfuzzy controller are much simper which can considerablylower the implementation cost of the fuzzy controller andenhance the design flexibility

Remark 13 If the function of 119888(119905) in 1199081(1199091(119905)) is unknown

on the basis of the PDC design technique [7ndash11 15ndash20 22ndash28] the fuzzy controller cannot be implemented as 119888(119905) isnot available However using the proposed design method

0 5 10 15

0

2

4

6Closed-loop response

minus12

minus10

minus8

minus6

minus4

minus2

119905 (s)

119909(119905)

1199092

11990931199091

Figure 1The state responses of system (47) with ℎ1= 04 ℎ

2= 17

0 5 10 15

0

5

10

15

20

25

30

119905 (s)

minus20

minus15

minus10

minus5

119906(119905)

Figure 2 The control input of system (47) with ℎ1= 04 ℎ

2= 17

we can select some simper and well-known membershipfunctions that are different from those of the fuzzy time delaymodels for example119898

1(1199091(119905)) and119898

2(1199091(119905)) so as to realize

the fuzzy controller In other words the proposed designapproach can be applied even for the fuzzy time delay modelswith uncertain grades of membership

5 Conclusions

In this paper we study the robust stability and stabilizationproblems for general nonlinear fuzzy systems with timevarying state and input delays A less conservative delay-dependent robust stability criterion has been obtained byintroducing a novel augmented Lyapunov function withan additional triple-integral term Moreover a new design

12 Mathematical Problems in Engineering

approach different from the traditional PDC design tech-nique is proposed which can significantly improve the designflexibility and reduce the implementation cost of the fuzzycontroller by arbitrarily selecting simple fuzzy membershipfunctions Two numerical examples are used to illustrate theconservativeness and effectiveness of our proposed methods

Acknowledgments

This research work was funded by the NSFC under Grantno 61273095 and the Academy of Finland under Grants no135225 and no 127299

References

[1] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[2] W H Ho S H Chen I T Chen J H Chou and C C ShuldquoDesign of stable and quadratic-optimal static output feed-back controllers for T-S-fuzzy-model-based control systems anintegrative computational approachrdquo International Journal ofInnovative Computing Information and Control vol 8 no 1App 403ndash418 2012

[3] X J Su L G Wu P Shi and Y D Song ldquo119867infinmodel reduction

of T-S fuzzy stochastic systemsrdquo IEEE Transactions on SystemsMan and Cybermetics B vol 42 no 6 pp 1574ndash1585 2012

[4] R Yang P Shi G-P Liu andH Gao ldquoNetwork-based feedbackcontrol for systems with mixed delays based on quantizationand dropout compensationrdquo Automatica vol 47 no 12 pp2805ndash2809 2011

[5] XM Zhang Z J Zhang andG P Lu ldquoFault detection for state-delay fuzzy systems subject to random communication delayrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 4 pp 2439ndash2451 2012

[6] F B Li and X Zhang ldquoDelay-range-dependent robust 119867infin

filtering for singuar LPV systems with time variant delayrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 1 pp 339ndash353 2013

[7] Y Y Cao and P M Frank ldquoAnalysis and synthesis of nonlineartime-delay systems via fuzzy control approachrdquo IEEE Transac-tions on Fuzzy Systems vol 8 no 2 pp 200ndash211 2000

[8] Y-Y Cao and P M Frank ldquoStability analysis and synthesis ofnonlinear time-delay systems via linear Takagi-Sugeno fuzzymodelsrdquo Fuzzy Sets and Systems vol 124 no 2 pp 213ndash2292001

[9] O M Kwon and J H Park ldquoDelay-range-dependent stabiliza-tion of uncertain dynamic systems with interval time-varyingdelaysrdquo Applied Mathematics and Computation vol 208 no 1pp 58ndash68 2009

[10] E Tian D Yue and Y Zhang ldquoDelay-dependent robust 119867infin

control for T-S fuzzy system with interval time-varying delayrdquoFuzzy Sets and Systems vol 160 no 12 pp 1708ndash1719 2009

[11] J Y An T Li G LWen and R Li ldquoNew stability conditions foruncertain T-S fuzzy systems with interval time-varying delayrdquoInternational Journal of Control Automation and Systems vol10 no 3 pp 490ndash497 2012

[12] Z J Zhang X L Huang X J Ban and X Z Gao ldquoStabilityanalysis and controller design of T-S fuzzy systems with time-delay under imperfect premise matchingrdquo Journal of BeijingInstitute of Technology vol 21 no 3 pp 387ndash393 2012

[13] O M Kwon M J Park S M Lee and J H Park ldquoAugmentedLyapunov-Krasovskii functional approaches to robust stabilitycriteria for uncertain Takagi-Sugeno fuzzy systems with time-varying delaysrdquo Fuzzy Sets and Systems vol 201 no 16 pp 1ndash192012

[14] F Liu M Wu Y He and R Yokoyama ldquoNew delay-dependentstability criteria for T-S fuzzy systems with time-varying delayrdquoFuzzy Sets and Systems vol 161 no 15 pp 2033ndash2042 2010

[15] B Chen X Liu and S Tong ldquoNew delay-dependent stabiliza-tion conditions of T-S fuzzy systems with constant delayrdquo FuzzySets and Systems vol 158 no 20 pp 2209ndash2224 2007

[16] J Yoneyama ldquoRobust stability and stabilization for uncertainTakagi-Sugeno fuzzy time-delay systemsrdquo Fuzzy Sets and Sys-tems vol 158 no 2 pp 115ndash134 2007

[17] C Lin Q-G Wang and T H Lee ldquoDelay-dependent LMIconditions for stability and stabilization of T-S fuzzy systemswith bounded time-delayrdquo Fuzzy Sets and Systems vol 157 no9 pp 1229ndash1247 2006

[18] X J Su P Shi L G Wu and Y D Song ldquoA novel approachto filter design for T-S fuzzy discrete-time systems with time-varying delayrdquo IEEE Transactions on Fuzzy Systems vol 20 no6 pp 1114ndash1129 2012

[19] C Peng and Q-L Han ldquoDelay-range-dependent robust stabi-lization for uncertain T-S fuzzy control systems with intervaltime-varying delaysrdquo Information Sciences vol 181 no 19 pp4287ndash4299 2011

[20] S T Li Y W Jing and X M Liu ldquoRobust control for a class ofT-S fuzzy systemswith interval time-varying delayrdquo InternatinalJournal of Control Automation and Systems vol 10 no 4 pp737ndash743 2012

[21] X-M Zhang M Wu J-H She and Y He ldquoDelay-dependentstabilization of linear systems with time-varying state and inputdelaysrdquo Automatica vol 41 no 8 pp 1405ndash1412 2005

[22] B Chen X Liu and S Tong ldquoRobust fuzzy control of nonlinearsystems with input delayrdquo Chaos Solitons amp Fractals vol 37 no3 pp 894ndash901 2008

[23] C Lin Q-G Wang and T H Lee ldquoDelay-dependent LMIconditions for stability and stabilization of T-S fuzzy systemswith bounded time-delayrdquo Fuzzy Sets and Systems vol 157 no9 pp 1229ndash1247 2006

[24] H J Lee J B Park and Y H Joo ldquoRobust control for uncertainTakagi-Sugeno fuzzy systems with time-varying input delayrdquoJournal of Dynamic Systems Measurement and Control Trans-actions of the ASME vol 127 no 2 pp 302ndash306 2005

[25] C-H Lien and K-W Yu ldquoRobust control for Takagi-Sugenofuzzy systems with time-varying state and input delaysrdquo ChaosSolitons and Fractals vol 35 no 5 pp 1003ndash1008 2008

[26] Y K Su B Chen S Y Zhang and C Lin ldquoDelay-dependentstabilization of T-S fuzzy systems with time delayrdquo Control andDecision Kongzhi yu Juece vol 24 no 6 pp 921ndash927 2009

[27] B Chen X Liu C Lin and K Liu ldquoRobust 119867infin

control ofTakagi-Sugeno fuzzy systems with state and input time delaysrdquoFuzzy Sets and Systems vol 160 no 4 pp 403ndash422 2009

[28] L Li and X Liu ldquoNew results on delay-dependent robuststability criteria of uncertain fuzzy systems with state and inputdelaysrdquo Information Sciences vol 179 no 8 pp 1134ndash1148 2009

[29] Z J Zhang X J Ban X L Huang and X Z Gao ldquoNew delay-dependent robust stablity for T-S fuzzy systems with intervevaltime-varying delayrdquo in Proccedings of the 5th International JointConference on Computational Science and Optimization pp826ndash830 Harbin China 2012

Mathematical Problems in Engineering 13

[30] H K Lam and F H F Leung ldquoLMI-based stability andperformance design of fuzzy control systems fuzzy modelsand controllers with different premisesrdquo in Proceedings of theInternational Conference on Fuzzy Systerm pp 9599ndash9506Vancouver Canada 2006

[31] H K Lam andMNarimani ldquoStability analysis and perfomancedesign for fuzzy-model-based control system under imperfectpremise matchingrdquo IEEE Transactions on Fuzzy Systems vol 17no 4 pp 949ndash961 2009

[32] S Boyd L E Ghaoui and E Feron Linear Matrix Inequality inSystems and ControlTheory SIAM Philadelphia Pa USA 1994

[33] I R Petersen and C V Hollot ldquoA Riccati equation approach tothe stabilization of uncertain linear systemsrdquo Automatica vol22 no 4 pp 397ndash411 1986

[34] J Sun G P Liu and J Chen ldquoDelay-dependent stabilityand stabilization of neutral time-delay systemsrdquo InternationalJournal of Robust andNonlinear Control vol 19 no 12 pp 1364ndash1375 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

where

Φ119894119895=

[[[[[[[[

[

Φ11119894 Φ12119894 Φ13 Φ14 Φ15119894119895 Φ16 Φ17 Φ18

lowast Φ22119894 1199053119883119860119879

21198941199054119883119860119879

2119894Φ25119894119895 1199056119883119860

119879

21198941199057119883119860119879

2119894Φ28119894

lowast lowast Φ33119894 minus11987522 1199053119861119894119884119895 0 0 minus1199053119883119879

lowast lowast lowast Φ44 1199054119861119894119884119895 0 0 Φ48

lowast lowast lowast lowast Φ55119894119895 1199056119884119879

119895119861119879

1198941199057119884119879

119895119861119879

119894Φ58119894

lowast lowast lowast lowast lowast Φ66 Φ67 minus1199056119883119879

lowast lowast lowast lowast lowast lowast Φ77 Φ78

lowast lowast lowast lowast lowast lowast lowast Φ88

]]]]]]]]

]

lt 0

Φ11119894= Φ11119894+ 1205761119894119863119894119863119879119894

Φ22119894= Φ22119894+ 1205762119894119863119894119863119879119894

Φ55119894119895

= Φ55119894119895

+ 120576119887119894119863119894119863119879119894

(42)

and Φ are the same as in Theorem 4 Therefore (41) holds ifand only if the following inequality holds

119903

sum119894=1

119903

sum119895=1

119908119894(119909 (119905))119898

119895(119909 (119905 minus 119889 (119905))) Ψ

119894119895lt 0 (43)

To alleviate the conservativeness of our robust stabilityanalysis similar to the analysis approach of Theorem 4 weintroduce sum119903

119894=1sum119903119895=1

119908119894(119908119895minus 119898119895)Λ119894= 0 to (43) and the

following proof is the same as in Theorem 4 We can obtainthe conditions and (34)ndash(36) Thus the fuzzy control systemwith the time varying state and input delays (34) is robuststable on the basis of the control law 119865

119894= 119884119894119883minus119879under these

conditions and (34)ndash(36) hold

Remark 6 If we set 119908119894(119909(119905)) = 119898

119895(119909(119905)) 119889

1(119905) = 119889

2(119905)

the closed-loop system in this paper has the same structureas that of paper [26] On the other hand let 119908

119894(119909(119905)) =

119898119895(119909(119905)) and 119889

1(119905) = 119889

2(119905) = 119889 we can obtain the system

in [16] Therefore the system studied in this paper is moregeneral Additionally the information of the membershipfunctions of the fuzzy time delay models and controllers areall considered in the stability analysis If the fuzzymodels andfuzzy controllers share the same fuzzymembership functionsthat is 119908

119894(119909(119905)) = 119898

119895(119909(119905)) the stability analysis can be

referred to [25 27 28] In other words the problem in [2527 28] is actually a special case of the one investigated in thispaper

Remark 7 The augmented Lyapunov function with an addi-tional triple-integral term is introduced in analyzing thestability problem for the T-S fuzzy systems with time varyingstate and input delays In addition two integral inequalitiesare used to derive Theorem 4 and less free-weighting matri-ces are introduced which lead to get less conservative resultsIt can be observed that some existing results are the specialcases of this paper For example the method in [25] is from(18) with 119875

12= 0 119875

22= 0 119885

12= 11988522= 0 119873

12= 11987322= 0

and 1198771= 1198772= 0 in Theorem 4 The results in [22] are based

on system (33) with 1198892(119905) = 0 and (18) with119872

2= 0 119885

12=

11988522= 0 119876

1= 1198762= 0 and 119877

1= 1198772= 0

Remark 8 Different from the general PDC technique thedesign method under the imperfect premise matching ismuch more flexible because the membership functions ofthe fuzzy controllers do not need to be chosen the same asthose of the fuzzy time delay models Instead they can bedesigned arbitrarilyThus the design flexibility is significantlyenhanced On the other hand some simple membershipfunctions of the fuzzy controllers might be employed whichcan reduce the implementation cost

Remark 9 In [7ndash11 15ndash20] the authors have considered therobust control for uncertain T-S fuzzy systems with statedelays Some delay-dependent conditions and fuzzy con-trollers are obtained with a traditional Lyapunov-Krasovskiifunctionalmethod andPDCmethod which all are the specialcases of Theorem 5 in this paper

4 Numerical Examples

In this section two numerical examples are given to illus-trate the conservativeness and effectiveness of the proposedmethods The first example compares our techniques withthe existing ones in the literature for stability analysis whichshows that Theorem 4 in this paper is less conservative thanthe other results The second example is used to illustrate theadvantage of the robust stability conditions ofTheorem 5 anddemonstrate how to design robust fuzzy controllers by usingour approach

Example 10 Consider the following fuzzy system with stateand input delays

(119905) =2

sum119894=1

119908119894(120579 (119905)) [119860

1119894119909 (119905) + 119860

2119894119909 (119905 minus 119889

1(119905))

+119861119894119906 (119905 minus 119889

2(119905))]

11986011= [0 06

0 1 ] 11986021= [05 09

0 2 ]

11986012= [1 0

1 0] 11986022= [09 0

1 16]

1198611= 1198612= [11]

(44)

The fuzzy membership functions are selected as [31]

1199081(1199091(119905)) = (1 minus

119888 (119905) sin (10038161003816100381610038161199091 (119905)1003816100381610038161003816minus4)5

1 + expminus1001199091(119905)3(1minus1199091(119905)))

timescos (119909

1(119905))2

1 + expminus251199091(119905)(3+(1199091(119905)042))

times 1199082(1199091(119905)) = 1 minus 119908

1(1199091(119905))

(45)

10 Mathematical Problems in Engineering

Table 1 The maximum allowable time delay and feedback gains(1205831= 0 120583

2= 0)

Paper ℎ1

ℎ2 Feedback gains

[26] 03120 031201198651= [10598 minus56598]

1198652= [minus13068 minus41167]

Theorem 4 033 0571198651= [minus00042 minus00449]1198652= [00048 00717]

where

1199091(119905) isin [minus

1205872

1205872]

119888 (119905) =sin (119909

1(119905)) + 140

isin [minus005 005] (46)

If we set 1198891(119905) = 119889

2(119905) in system (44) we can obtain the

system in [26] which implies that our system ismore generalEmploying the LMIs in [26] and those inTheorem 4yields themaximum state and input delays ℎ

1 and ℎ

2 which guarantee

the stability of system (44) and the feedback gains for 1205831= 0

and 1205832= 0 when 120588

1= 075 120588

2= 095 119905

2= 01 119905

3=

02 1199054= 07 119905

5= 01 119905

6= 04 119905

7= 01 and 119905

8= 13 as

given in Table 1 which clearly shows the superiority of theresults derived in this paper over those obtained from [26]

Example 11 Consider the well-known truck-trailer systemwhich can be described by the following T-S fuzzy system

(119905) =2

sum119894=1

119908119894(120579 (119905)) [(119860

1119894+ Δ1198601119894) 119909 (119905) + 119860

2119894119909 (119905 minus 119889

1(119905))

+ (119861119894+ Δ119861119894) 119906 (119905 minus 119889

2(119905))]

(47)

11986011=

[[[[[[[[[

[

minus119886V1199051198711199050

0 0

119886V1199051198711199050

0 0

119886V21199052

21198711199050

V1199051199050

0

]]]]]]]]]

]

11986021=

[[[[[[[[[

[

minus (1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)V21199052

21198711199050

0 0

]]]]]]]]]

]

1198611=[[[[

[

V1199051198971199050

0

0

]]]]

]

11986012=

[[[[[[[[[

[

minus119886V1199051198711199050

0 0

119886V1199051198711199050

0 0

119886119889V21199052

21198711199050

119889V1199051199050

0

]]]]]]]]]

]

11986022=

[[[[[[[[[

[

minus (1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)119889V21199052

21198711199050

0 0

]]]]]]]]]

]

(48)

1198612=[[[[

[

V1199051198971199050

0

0

]]]]

]

(49)

where 119897 = 28 119871 = 55 V = minus10 119905 = 20 1199050= 05

119889 = 101199050120587 119863

119894= [02555 02555 02555]119879 119864

1119894= 1198642119894=

[01 0 0]119879 119864119887119894

= [01 0 0]119879 (119894 = 1 2) and the fuzzymembership functions are selected the same as inExample 10

(1) Let 119886 = 1 system (47) is the same as the fuzzy systemof [22] which implies that our system ismore generalTable 2 gives themaximum input delay value of ℎ

2 for

which the stabilization is guaranteed byTheorem 5 aswell as the state feedback gains when 120588

1= 075 120588

2=

095 1199052= 01 119905

3= 02 119905

4= 07 119905

5= 01 119905

6=

04 1199057= 08 and 119905

8= 2 It is clearly visible that the

results derived in this paper are better than those from[22 28]

(2) In the case of 119886 = 1 the state and input delays of system(47) are all exist The approach proposed in [22] cannot be used to get the maximum of state delay ℎ

1

as the system in [22] only contains the input delayHowever the method in [28] and Theorem 5 can beused Here let 119886 = 07 120588

1= 075 120588

2= 095 119905

2=

01 1199053= 02 119905

4= 07 119905

5= 01 119905

6= 04 119905

7=

08 and 1199058= 3 and Table 3 provides the maximum

upper bounds of ℎ1and ℎ

2and the state feedback

gains for which the robust stabilization is guaranteedbyTheorem 5

By Theorem 5 we can conclude that the uncertain fuzzymodel (47) is robust stable based on the following fuzzycontrol law

119906 (119905 minus ℎ2) =2

sum119895=1

119898119895(1199091(119905 minus ℎ

2)) 119865119895119909 (119905 minus ℎ

2) (50)

With the PDC design technique it is required that thefuzzy controller must share the same fuzzy membershipfunctions as those of the fuzzy time delay model Under such

Mathematical Problems in Engineering 11

Table 2 The maximum allowable time delay and feedback gains(1205832= 0)

Paper ℎ2 Feedback gains

[22] 0751198651= [34227 minus03535 00045]

1198652= [35215 minus03617 00056]

[28] 0861198651= [33219 minus02406 00025]

1198652= [33272 minus02494 00026]

Theorem 5 121198651= [minus00212 00093 minus00024]1198652= [minus00250 00044 00014]

Table 3 The maximum allowable time delay and feedback gains(1205831= 1205832= 0)

Paper ℎ1

ℎ2 Feedback gains

[28] 01 0551198651= [33219 minus02406 00025]

1198652= [33272 minus02494 00026]

Theorem 5 04 171198651= [minus00128 00090 minus00017]1198652= [minus00194 00012 00009]

a condition the implementation cost of the fuzzy controlleris high by employing these complex membership functionsHowever fromTheorem 5 the membership functions do notneed to be chosen the same as119908

1(1199091(119905)) and119908

2(1199091(119905)) Thus

we can select some simple membership functions insteadsuch as

1198981(1199091(119905 minus ℎ

2)) = 075 exp

(minus1199091(119905 minus ℎ

2) minus 038)2

2 times 0382+ 005

1198982(1199091(119905 minus ℎ

2)) = 1 minus 119898

1(1199091(119905 minus ℎ

2))

(51)

We also assume that 1205881= 075 and 120588

2= 095 such that

119898119895(1199091(119905 minus ℎ

2)) minus 120588

119895119908119895(1199091(119905)) gt 0 for 119895 = 1 2 and 119909

1(119905 minus ℎ

2)

The fuzzy control law 119906(119905 minus ℎ2) = 119898

1[minus00128 00090 minus

00017]119909 + 1198982[minus00194 00012 00009]119909 can be employed

to stabilize the uncertain fuzzy control systems in (47) withthe state and input delays Figures 1 and 2 illustrate the stablestate response and control input under the initial conditionof 119909(0) = [4 minus 1 2]119879

Remark 12 In Example 11 our method offers less conser-vative results in the sense of allowing longer time delayand obtaining smaller feedback control gains Moreovercompared with [22ndash28] where the fuzzy controllermust havethe same fuzzy membership functions as those of the fuzzytime delay model the above membership functions of ourfuzzy controller are much simper which can considerablylower the implementation cost of the fuzzy controller andenhance the design flexibility

Remark 13 If the function of 119888(119905) in 1199081(1199091(119905)) is unknown

on the basis of the PDC design technique [7ndash11 15ndash20 22ndash28] the fuzzy controller cannot be implemented as 119888(119905) isnot available However using the proposed design method

0 5 10 15

0

2

4

6Closed-loop response

minus12

minus10

minus8

minus6

minus4

minus2

119905 (s)

119909(119905)

1199092

11990931199091

Figure 1The state responses of system (47) with ℎ1= 04 ℎ

2= 17

0 5 10 15

0

5

10

15

20

25

30

119905 (s)

minus20

minus15

minus10

minus5

119906(119905)

Figure 2 The control input of system (47) with ℎ1= 04 ℎ

2= 17

we can select some simper and well-known membershipfunctions that are different from those of the fuzzy time delaymodels for example119898

1(1199091(119905)) and119898

2(1199091(119905)) so as to realize

the fuzzy controller In other words the proposed designapproach can be applied even for the fuzzy time delay modelswith uncertain grades of membership

5 Conclusions

In this paper we study the robust stability and stabilizationproblems for general nonlinear fuzzy systems with timevarying state and input delays A less conservative delay-dependent robust stability criterion has been obtained byintroducing a novel augmented Lyapunov function withan additional triple-integral term Moreover a new design

12 Mathematical Problems in Engineering

approach different from the traditional PDC design tech-nique is proposed which can significantly improve the designflexibility and reduce the implementation cost of the fuzzycontroller by arbitrarily selecting simple fuzzy membershipfunctions Two numerical examples are used to illustrate theconservativeness and effectiveness of our proposed methods

Acknowledgments

This research work was funded by the NSFC under Grantno 61273095 and the Academy of Finland under Grants no135225 and no 127299

References

[1] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[2] W H Ho S H Chen I T Chen J H Chou and C C ShuldquoDesign of stable and quadratic-optimal static output feed-back controllers for T-S-fuzzy-model-based control systems anintegrative computational approachrdquo International Journal ofInnovative Computing Information and Control vol 8 no 1App 403ndash418 2012

[3] X J Su L G Wu P Shi and Y D Song ldquo119867infinmodel reduction

of T-S fuzzy stochastic systemsrdquo IEEE Transactions on SystemsMan and Cybermetics B vol 42 no 6 pp 1574ndash1585 2012

[4] R Yang P Shi G-P Liu andH Gao ldquoNetwork-based feedbackcontrol for systems with mixed delays based on quantizationand dropout compensationrdquo Automatica vol 47 no 12 pp2805ndash2809 2011

[5] XM Zhang Z J Zhang andG P Lu ldquoFault detection for state-delay fuzzy systems subject to random communication delayrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 4 pp 2439ndash2451 2012

[6] F B Li and X Zhang ldquoDelay-range-dependent robust 119867infin

filtering for singuar LPV systems with time variant delayrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 1 pp 339ndash353 2013

[7] Y Y Cao and P M Frank ldquoAnalysis and synthesis of nonlineartime-delay systems via fuzzy control approachrdquo IEEE Transac-tions on Fuzzy Systems vol 8 no 2 pp 200ndash211 2000

[8] Y-Y Cao and P M Frank ldquoStability analysis and synthesis ofnonlinear time-delay systems via linear Takagi-Sugeno fuzzymodelsrdquo Fuzzy Sets and Systems vol 124 no 2 pp 213ndash2292001

[9] O M Kwon and J H Park ldquoDelay-range-dependent stabiliza-tion of uncertain dynamic systems with interval time-varyingdelaysrdquo Applied Mathematics and Computation vol 208 no 1pp 58ndash68 2009

[10] E Tian D Yue and Y Zhang ldquoDelay-dependent robust 119867infin

control for T-S fuzzy system with interval time-varying delayrdquoFuzzy Sets and Systems vol 160 no 12 pp 1708ndash1719 2009

[11] J Y An T Li G LWen and R Li ldquoNew stability conditions foruncertain T-S fuzzy systems with interval time-varying delayrdquoInternational Journal of Control Automation and Systems vol10 no 3 pp 490ndash497 2012

[12] Z J Zhang X L Huang X J Ban and X Z Gao ldquoStabilityanalysis and controller design of T-S fuzzy systems with time-delay under imperfect premise matchingrdquo Journal of BeijingInstitute of Technology vol 21 no 3 pp 387ndash393 2012

[13] O M Kwon M J Park S M Lee and J H Park ldquoAugmentedLyapunov-Krasovskii functional approaches to robust stabilitycriteria for uncertain Takagi-Sugeno fuzzy systems with time-varying delaysrdquo Fuzzy Sets and Systems vol 201 no 16 pp 1ndash192012

[14] F Liu M Wu Y He and R Yokoyama ldquoNew delay-dependentstability criteria for T-S fuzzy systems with time-varying delayrdquoFuzzy Sets and Systems vol 161 no 15 pp 2033ndash2042 2010

[15] B Chen X Liu and S Tong ldquoNew delay-dependent stabiliza-tion conditions of T-S fuzzy systems with constant delayrdquo FuzzySets and Systems vol 158 no 20 pp 2209ndash2224 2007

[16] J Yoneyama ldquoRobust stability and stabilization for uncertainTakagi-Sugeno fuzzy time-delay systemsrdquo Fuzzy Sets and Sys-tems vol 158 no 2 pp 115ndash134 2007

[17] C Lin Q-G Wang and T H Lee ldquoDelay-dependent LMIconditions for stability and stabilization of T-S fuzzy systemswith bounded time-delayrdquo Fuzzy Sets and Systems vol 157 no9 pp 1229ndash1247 2006

[18] X J Su P Shi L G Wu and Y D Song ldquoA novel approachto filter design for T-S fuzzy discrete-time systems with time-varying delayrdquo IEEE Transactions on Fuzzy Systems vol 20 no6 pp 1114ndash1129 2012

[19] C Peng and Q-L Han ldquoDelay-range-dependent robust stabi-lization for uncertain T-S fuzzy control systems with intervaltime-varying delaysrdquo Information Sciences vol 181 no 19 pp4287ndash4299 2011

[20] S T Li Y W Jing and X M Liu ldquoRobust control for a class ofT-S fuzzy systemswith interval time-varying delayrdquo InternatinalJournal of Control Automation and Systems vol 10 no 4 pp737ndash743 2012

[21] X-M Zhang M Wu J-H She and Y He ldquoDelay-dependentstabilization of linear systems with time-varying state and inputdelaysrdquo Automatica vol 41 no 8 pp 1405ndash1412 2005

[22] B Chen X Liu and S Tong ldquoRobust fuzzy control of nonlinearsystems with input delayrdquo Chaos Solitons amp Fractals vol 37 no3 pp 894ndash901 2008

[23] C Lin Q-G Wang and T H Lee ldquoDelay-dependent LMIconditions for stability and stabilization of T-S fuzzy systemswith bounded time-delayrdquo Fuzzy Sets and Systems vol 157 no9 pp 1229ndash1247 2006

[24] H J Lee J B Park and Y H Joo ldquoRobust control for uncertainTakagi-Sugeno fuzzy systems with time-varying input delayrdquoJournal of Dynamic Systems Measurement and Control Trans-actions of the ASME vol 127 no 2 pp 302ndash306 2005

[25] C-H Lien and K-W Yu ldquoRobust control for Takagi-Sugenofuzzy systems with time-varying state and input delaysrdquo ChaosSolitons and Fractals vol 35 no 5 pp 1003ndash1008 2008

[26] Y K Su B Chen S Y Zhang and C Lin ldquoDelay-dependentstabilization of T-S fuzzy systems with time delayrdquo Control andDecision Kongzhi yu Juece vol 24 no 6 pp 921ndash927 2009

[27] B Chen X Liu C Lin and K Liu ldquoRobust 119867infin

control ofTakagi-Sugeno fuzzy systems with state and input time delaysrdquoFuzzy Sets and Systems vol 160 no 4 pp 403ndash422 2009

[28] L Li and X Liu ldquoNew results on delay-dependent robuststability criteria of uncertain fuzzy systems with state and inputdelaysrdquo Information Sciences vol 179 no 8 pp 1134ndash1148 2009

[29] Z J Zhang X J Ban X L Huang and X Z Gao ldquoNew delay-dependent robust stablity for T-S fuzzy systems with intervevaltime-varying delayrdquo in Proccedings of the 5th International JointConference on Computational Science and Optimization pp826ndash830 Harbin China 2012

Mathematical Problems in Engineering 13

[30] H K Lam and F H F Leung ldquoLMI-based stability andperformance design of fuzzy control systems fuzzy modelsand controllers with different premisesrdquo in Proceedings of theInternational Conference on Fuzzy Systerm pp 9599ndash9506Vancouver Canada 2006

[31] H K Lam andMNarimani ldquoStability analysis and perfomancedesign for fuzzy-model-based control system under imperfectpremise matchingrdquo IEEE Transactions on Fuzzy Systems vol 17no 4 pp 949ndash961 2009

[32] S Boyd L E Ghaoui and E Feron Linear Matrix Inequality inSystems and ControlTheory SIAM Philadelphia Pa USA 1994

[33] I R Petersen and C V Hollot ldquoA Riccati equation approach tothe stabilization of uncertain linear systemsrdquo Automatica vol22 no 4 pp 397ndash411 1986

[34] J Sun G P Liu and J Chen ldquoDelay-dependent stabilityand stabilization of neutral time-delay systemsrdquo InternationalJournal of Robust andNonlinear Control vol 19 no 12 pp 1364ndash1375 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

Table 1 The maximum allowable time delay and feedback gains(1205831= 0 120583

2= 0)

Paper ℎ1

ℎ2 Feedback gains

[26] 03120 031201198651= [10598 minus56598]

1198652= [minus13068 minus41167]

Theorem 4 033 0571198651= [minus00042 minus00449]1198652= [00048 00717]

where

1199091(119905) isin [minus

1205872

1205872]

119888 (119905) =sin (119909

1(119905)) + 140

isin [minus005 005] (46)

If we set 1198891(119905) = 119889

2(119905) in system (44) we can obtain the

system in [26] which implies that our system ismore generalEmploying the LMIs in [26] and those inTheorem 4yields themaximum state and input delays ℎ

1 and ℎ

2 which guarantee

the stability of system (44) and the feedback gains for 1205831= 0

and 1205832= 0 when 120588

1= 075 120588

2= 095 119905

2= 01 119905

3=

02 1199054= 07 119905

5= 01 119905

6= 04 119905

7= 01 and 119905

8= 13 as

given in Table 1 which clearly shows the superiority of theresults derived in this paper over those obtained from [26]

Example 11 Consider the well-known truck-trailer systemwhich can be described by the following T-S fuzzy system

(119905) =2

sum119894=1

119908119894(120579 (119905)) [(119860

1119894+ Δ1198601119894) 119909 (119905) + 119860

2119894119909 (119905 minus 119889

1(119905))

+ (119861119894+ Δ119861119894) 119906 (119905 minus 119889

2(119905))]

(47)

11986011=

[[[[[[[[[

[

minus119886V1199051198711199050

0 0

119886V1199051198711199050

0 0

119886V21199052

21198711199050

V1199051199050

0

]]]]]]]]]

]

11986021=

[[[[[[[[[

[

minus (1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)V21199052

21198711199050

0 0

]]]]]]]]]

]

1198611=[[[[

[

V1199051198971199050

0

0

]]]]

]

11986012=

[[[[[[[[[

[

minus119886V1199051198711199050

0 0

119886V1199051198711199050

0 0

119886119889V21199052

21198711199050

119889V1199051199050

0

]]]]]]]]]

]

11986022=

[[[[[[[[[

[

minus (1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)V1199051198711199050

0 0

(1 minus 119886)119889V21199052

21198711199050

0 0

]]]]]]]]]

]

(48)

1198612=[[[[

[

V1199051198971199050

0

0

]]]]

]

(49)

where 119897 = 28 119871 = 55 V = minus10 119905 = 20 1199050= 05

119889 = 101199050120587 119863

119894= [02555 02555 02555]119879 119864

1119894= 1198642119894=

[01 0 0]119879 119864119887119894

= [01 0 0]119879 (119894 = 1 2) and the fuzzymembership functions are selected the same as inExample 10

(1) Let 119886 = 1 system (47) is the same as the fuzzy systemof [22] which implies that our system ismore generalTable 2 gives themaximum input delay value of ℎ

2 for

which the stabilization is guaranteed byTheorem 5 aswell as the state feedback gains when 120588

1= 075 120588

2=

095 1199052= 01 119905

3= 02 119905

4= 07 119905

5= 01 119905

6=

04 1199057= 08 and 119905

8= 2 It is clearly visible that the

results derived in this paper are better than those from[22 28]

(2) In the case of 119886 = 1 the state and input delays of system(47) are all exist The approach proposed in [22] cannot be used to get the maximum of state delay ℎ

1

as the system in [22] only contains the input delayHowever the method in [28] and Theorem 5 can beused Here let 119886 = 07 120588

1= 075 120588

2= 095 119905

2=

01 1199053= 02 119905

4= 07 119905

5= 01 119905

6= 04 119905

7=

08 and 1199058= 3 and Table 3 provides the maximum

upper bounds of ℎ1and ℎ

2and the state feedback

gains for which the robust stabilization is guaranteedbyTheorem 5

By Theorem 5 we can conclude that the uncertain fuzzymodel (47) is robust stable based on the following fuzzycontrol law

119906 (119905 minus ℎ2) =2

sum119895=1

119898119895(1199091(119905 minus ℎ

2)) 119865119895119909 (119905 minus ℎ

2) (50)

With the PDC design technique it is required that thefuzzy controller must share the same fuzzy membershipfunctions as those of the fuzzy time delay model Under such

Mathematical Problems in Engineering 11

Table 2 The maximum allowable time delay and feedback gains(1205832= 0)

Paper ℎ2 Feedback gains

[22] 0751198651= [34227 minus03535 00045]

1198652= [35215 minus03617 00056]

[28] 0861198651= [33219 minus02406 00025]

1198652= [33272 minus02494 00026]

Theorem 5 121198651= [minus00212 00093 minus00024]1198652= [minus00250 00044 00014]

Table 3 The maximum allowable time delay and feedback gains(1205831= 1205832= 0)

Paper ℎ1

ℎ2 Feedback gains

[28] 01 0551198651= [33219 minus02406 00025]

1198652= [33272 minus02494 00026]

Theorem 5 04 171198651= [minus00128 00090 minus00017]1198652= [minus00194 00012 00009]

a condition the implementation cost of the fuzzy controlleris high by employing these complex membership functionsHowever fromTheorem 5 the membership functions do notneed to be chosen the same as119908

1(1199091(119905)) and119908

2(1199091(119905)) Thus

we can select some simple membership functions insteadsuch as

1198981(1199091(119905 minus ℎ

2)) = 075 exp

(minus1199091(119905 minus ℎ

2) minus 038)2

2 times 0382+ 005

1198982(1199091(119905 minus ℎ

2)) = 1 minus 119898

1(1199091(119905 minus ℎ

2))

(51)

We also assume that 1205881= 075 and 120588

2= 095 such that

119898119895(1199091(119905 minus ℎ

2)) minus 120588

119895119908119895(1199091(119905)) gt 0 for 119895 = 1 2 and 119909

1(119905 minus ℎ

2)

The fuzzy control law 119906(119905 minus ℎ2) = 119898

1[minus00128 00090 minus

00017]119909 + 1198982[minus00194 00012 00009]119909 can be employed

to stabilize the uncertain fuzzy control systems in (47) withthe state and input delays Figures 1 and 2 illustrate the stablestate response and control input under the initial conditionof 119909(0) = [4 minus 1 2]119879

Remark 12 In Example 11 our method offers less conser-vative results in the sense of allowing longer time delayand obtaining smaller feedback control gains Moreovercompared with [22ndash28] where the fuzzy controllermust havethe same fuzzy membership functions as those of the fuzzytime delay model the above membership functions of ourfuzzy controller are much simper which can considerablylower the implementation cost of the fuzzy controller andenhance the design flexibility

Remark 13 If the function of 119888(119905) in 1199081(1199091(119905)) is unknown

on the basis of the PDC design technique [7ndash11 15ndash20 22ndash28] the fuzzy controller cannot be implemented as 119888(119905) isnot available However using the proposed design method

0 5 10 15

0

2

4

6Closed-loop response

minus12

minus10

minus8

minus6

minus4

minus2

119905 (s)

119909(119905)

1199092

11990931199091

Figure 1The state responses of system (47) with ℎ1= 04 ℎ

2= 17

0 5 10 15

0

5

10

15

20

25

30

119905 (s)

minus20

minus15

minus10

minus5

119906(119905)

Figure 2 The control input of system (47) with ℎ1= 04 ℎ

2= 17

we can select some simper and well-known membershipfunctions that are different from those of the fuzzy time delaymodels for example119898

1(1199091(119905)) and119898

2(1199091(119905)) so as to realize

the fuzzy controller In other words the proposed designapproach can be applied even for the fuzzy time delay modelswith uncertain grades of membership

5 Conclusions

In this paper we study the robust stability and stabilizationproblems for general nonlinear fuzzy systems with timevarying state and input delays A less conservative delay-dependent robust stability criterion has been obtained byintroducing a novel augmented Lyapunov function withan additional triple-integral term Moreover a new design

12 Mathematical Problems in Engineering

approach different from the traditional PDC design tech-nique is proposed which can significantly improve the designflexibility and reduce the implementation cost of the fuzzycontroller by arbitrarily selecting simple fuzzy membershipfunctions Two numerical examples are used to illustrate theconservativeness and effectiveness of our proposed methods

Acknowledgments

This research work was funded by the NSFC under Grantno 61273095 and the Academy of Finland under Grants no135225 and no 127299

References

[1] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[2] W H Ho S H Chen I T Chen J H Chou and C C ShuldquoDesign of stable and quadratic-optimal static output feed-back controllers for T-S-fuzzy-model-based control systems anintegrative computational approachrdquo International Journal ofInnovative Computing Information and Control vol 8 no 1App 403ndash418 2012

[3] X J Su L G Wu P Shi and Y D Song ldquo119867infinmodel reduction

of T-S fuzzy stochastic systemsrdquo IEEE Transactions on SystemsMan and Cybermetics B vol 42 no 6 pp 1574ndash1585 2012

[4] R Yang P Shi G-P Liu andH Gao ldquoNetwork-based feedbackcontrol for systems with mixed delays based on quantizationand dropout compensationrdquo Automatica vol 47 no 12 pp2805ndash2809 2011

[5] XM Zhang Z J Zhang andG P Lu ldquoFault detection for state-delay fuzzy systems subject to random communication delayrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 4 pp 2439ndash2451 2012

[6] F B Li and X Zhang ldquoDelay-range-dependent robust 119867infin

filtering for singuar LPV systems with time variant delayrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 1 pp 339ndash353 2013

[7] Y Y Cao and P M Frank ldquoAnalysis and synthesis of nonlineartime-delay systems via fuzzy control approachrdquo IEEE Transac-tions on Fuzzy Systems vol 8 no 2 pp 200ndash211 2000

[8] Y-Y Cao and P M Frank ldquoStability analysis and synthesis ofnonlinear time-delay systems via linear Takagi-Sugeno fuzzymodelsrdquo Fuzzy Sets and Systems vol 124 no 2 pp 213ndash2292001

[9] O M Kwon and J H Park ldquoDelay-range-dependent stabiliza-tion of uncertain dynamic systems with interval time-varyingdelaysrdquo Applied Mathematics and Computation vol 208 no 1pp 58ndash68 2009

[10] E Tian D Yue and Y Zhang ldquoDelay-dependent robust 119867infin

control for T-S fuzzy system with interval time-varying delayrdquoFuzzy Sets and Systems vol 160 no 12 pp 1708ndash1719 2009

[11] J Y An T Li G LWen and R Li ldquoNew stability conditions foruncertain T-S fuzzy systems with interval time-varying delayrdquoInternational Journal of Control Automation and Systems vol10 no 3 pp 490ndash497 2012

[12] Z J Zhang X L Huang X J Ban and X Z Gao ldquoStabilityanalysis and controller design of T-S fuzzy systems with time-delay under imperfect premise matchingrdquo Journal of BeijingInstitute of Technology vol 21 no 3 pp 387ndash393 2012

[13] O M Kwon M J Park S M Lee and J H Park ldquoAugmentedLyapunov-Krasovskii functional approaches to robust stabilitycriteria for uncertain Takagi-Sugeno fuzzy systems with time-varying delaysrdquo Fuzzy Sets and Systems vol 201 no 16 pp 1ndash192012

[14] F Liu M Wu Y He and R Yokoyama ldquoNew delay-dependentstability criteria for T-S fuzzy systems with time-varying delayrdquoFuzzy Sets and Systems vol 161 no 15 pp 2033ndash2042 2010

[15] B Chen X Liu and S Tong ldquoNew delay-dependent stabiliza-tion conditions of T-S fuzzy systems with constant delayrdquo FuzzySets and Systems vol 158 no 20 pp 2209ndash2224 2007

[16] J Yoneyama ldquoRobust stability and stabilization for uncertainTakagi-Sugeno fuzzy time-delay systemsrdquo Fuzzy Sets and Sys-tems vol 158 no 2 pp 115ndash134 2007

[17] C Lin Q-G Wang and T H Lee ldquoDelay-dependent LMIconditions for stability and stabilization of T-S fuzzy systemswith bounded time-delayrdquo Fuzzy Sets and Systems vol 157 no9 pp 1229ndash1247 2006

[18] X J Su P Shi L G Wu and Y D Song ldquoA novel approachto filter design for T-S fuzzy discrete-time systems with time-varying delayrdquo IEEE Transactions on Fuzzy Systems vol 20 no6 pp 1114ndash1129 2012

[19] C Peng and Q-L Han ldquoDelay-range-dependent robust stabi-lization for uncertain T-S fuzzy control systems with intervaltime-varying delaysrdquo Information Sciences vol 181 no 19 pp4287ndash4299 2011

[20] S T Li Y W Jing and X M Liu ldquoRobust control for a class ofT-S fuzzy systemswith interval time-varying delayrdquo InternatinalJournal of Control Automation and Systems vol 10 no 4 pp737ndash743 2012

[21] X-M Zhang M Wu J-H She and Y He ldquoDelay-dependentstabilization of linear systems with time-varying state and inputdelaysrdquo Automatica vol 41 no 8 pp 1405ndash1412 2005

[22] B Chen X Liu and S Tong ldquoRobust fuzzy control of nonlinearsystems with input delayrdquo Chaos Solitons amp Fractals vol 37 no3 pp 894ndash901 2008

[23] C Lin Q-G Wang and T H Lee ldquoDelay-dependent LMIconditions for stability and stabilization of T-S fuzzy systemswith bounded time-delayrdquo Fuzzy Sets and Systems vol 157 no9 pp 1229ndash1247 2006

[24] H J Lee J B Park and Y H Joo ldquoRobust control for uncertainTakagi-Sugeno fuzzy systems with time-varying input delayrdquoJournal of Dynamic Systems Measurement and Control Trans-actions of the ASME vol 127 no 2 pp 302ndash306 2005

[25] C-H Lien and K-W Yu ldquoRobust control for Takagi-Sugenofuzzy systems with time-varying state and input delaysrdquo ChaosSolitons and Fractals vol 35 no 5 pp 1003ndash1008 2008

[26] Y K Su B Chen S Y Zhang and C Lin ldquoDelay-dependentstabilization of T-S fuzzy systems with time delayrdquo Control andDecision Kongzhi yu Juece vol 24 no 6 pp 921ndash927 2009

[27] B Chen X Liu C Lin and K Liu ldquoRobust 119867infin

control ofTakagi-Sugeno fuzzy systems with state and input time delaysrdquoFuzzy Sets and Systems vol 160 no 4 pp 403ndash422 2009

[28] L Li and X Liu ldquoNew results on delay-dependent robuststability criteria of uncertain fuzzy systems with state and inputdelaysrdquo Information Sciences vol 179 no 8 pp 1134ndash1148 2009

[29] Z J Zhang X J Ban X L Huang and X Z Gao ldquoNew delay-dependent robust stablity for T-S fuzzy systems with intervevaltime-varying delayrdquo in Proccedings of the 5th International JointConference on Computational Science and Optimization pp826ndash830 Harbin China 2012

Mathematical Problems in Engineering 13

[30] H K Lam and F H F Leung ldquoLMI-based stability andperformance design of fuzzy control systems fuzzy modelsand controllers with different premisesrdquo in Proceedings of theInternational Conference on Fuzzy Systerm pp 9599ndash9506Vancouver Canada 2006

[31] H K Lam andMNarimani ldquoStability analysis and perfomancedesign for fuzzy-model-based control system under imperfectpremise matchingrdquo IEEE Transactions on Fuzzy Systems vol 17no 4 pp 949ndash961 2009

[32] S Boyd L E Ghaoui and E Feron Linear Matrix Inequality inSystems and ControlTheory SIAM Philadelphia Pa USA 1994

[33] I R Petersen and C V Hollot ldquoA Riccati equation approach tothe stabilization of uncertain linear systemsrdquo Automatica vol22 no 4 pp 397ndash411 1986

[34] J Sun G P Liu and J Chen ldquoDelay-dependent stabilityand stabilization of neutral time-delay systemsrdquo InternationalJournal of Robust andNonlinear Control vol 19 no 12 pp 1364ndash1375 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

Table 2 The maximum allowable time delay and feedback gains(1205832= 0)

Paper ℎ2 Feedback gains

[22] 0751198651= [34227 minus03535 00045]

1198652= [35215 minus03617 00056]

[28] 0861198651= [33219 minus02406 00025]

1198652= [33272 minus02494 00026]

Theorem 5 121198651= [minus00212 00093 minus00024]1198652= [minus00250 00044 00014]

Table 3 The maximum allowable time delay and feedback gains(1205831= 1205832= 0)

Paper ℎ1

ℎ2 Feedback gains

[28] 01 0551198651= [33219 minus02406 00025]

1198652= [33272 minus02494 00026]

Theorem 5 04 171198651= [minus00128 00090 minus00017]1198652= [minus00194 00012 00009]

a condition the implementation cost of the fuzzy controlleris high by employing these complex membership functionsHowever fromTheorem 5 the membership functions do notneed to be chosen the same as119908

1(1199091(119905)) and119908

2(1199091(119905)) Thus

we can select some simple membership functions insteadsuch as

1198981(1199091(119905 minus ℎ

2)) = 075 exp

(minus1199091(119905 minus ℎ

2) minus 038)2

2 times 0382+ 005

1198982(1199091(119905 minus ℎ

2)) = 1 minus 119898

1(1199091(119905 minus ℎ

2))

(51)

We also assume that 1205881= 075 and 120588

2= 095 such that

119898119895(1199091(119905 minus ℎ

2)) minus 120588

119895119908119895(1199091(119905)) gt 0 for 119895 = 1 2 and 119909

1(119905 minus ℎ

2)

The fuzzy control law 119906(119905 minus ℎ2) = 119898

1[minus00128 00090 minus

00017]119909 + 1198982[minus00194 00012 00009]119909 can be employed

to stabilize the uncertain fuzzy control systems in (47) withthe state and input delays Figures 1 and 2 illustrate the stablestate response and control input under the initial conditionof 119909(0) = [4 minus 1 2]119879

Remark 12 In Example 11 our method offers less conser-vative results in the sense of allowing longer time delayand obtaining smaller feedback control gains Moreovercompared with [22ndash28] where the fuzzy controllermust havethe same fuzzy membership functions as those of the fuzzytime delay model the above membership functions of ourfuzzy controller are much simper which can considerablylower the implementation cost of the fuzzy controller andenhance the design flexibility

Remark 13 If the function of 119888(119905) in 1199081(1199091(119905)) is unknown

on the basis of the PDC design technique [7ndash11 15ndash20 22ndash28] the fuzzy controller cannot be implemented as 119888(119905) isnot available However using the proposed design method

0 5 10 15

0

2

4

6Closed-loop response

minus12

minus10

minus8

minus6

minus4

minus2

119905 (s)

119909(119905)

1199092

11990931199091

Figure 1The state responses of system (47) with ℎ1= 04 ℎ

2= 17

0 5 10 15

0

5

10

15

20

25

30

119905 (s)

minus20

minus15

minus10

minus5

119906(119905)

Figure 2 The control input of system (47) with ℎ1= 04 ℎ

2= 17

we can select some simper and well-known membershipfunctions that are different from those of the fuzzy time delaymodels for example119898

1(1199091(119905)) and119898

2(1199091(119905)) so as to realize

the fuzzy controller In other words the proposed designapproach can be applied even for the fuzzy time delay modelswith uncertain grades of membership

5 Conclusions

In this paper we study the robust stability and stabilizationproblems for general nonlinear fuzzy systems with timevarying state and input delays A less conservative delay-dependent robust stability criterion has been obtained byintroducing a novel augmented Lyapunov function withan additional triple-integral term Moreover a new design

12 Mathematical Problems in Engineering

approach different from the traditional PDC design tech-nique is proposed which can significantly improve the designflexibility and reduce the implementation cost of the fuzzycontroller by arbitrarily selecting simple fuzzy membershipfunctions Two numerical examples are used to illustrate theconservativeness and effectiveness of our proposed methods

Acknowledgments

This research work was funded by the NSFC under Grantno 61273095 and the Academy of Finland under Grants no135225 and no 127299

References

[1] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[2] W H Ho S H Chen I T Chen J H Chou and C C ShuldquoDesign of stable and quadratic-optimal static output feed-back controllers for T-S-fuzzy-model-based control systems anintegrative computational approachrdquo International Journal ofInnovative Computing Information and Control vol 8 no 1App 403ndash418 2012

[3] X J Su L G Wu P Shi and Y D Song ldquo119867infinmodel reduction

of T-S fuzzy stochastic systemsrdquo IEEE Transactions on SystemsMan and Cybermetics B vol 42 no 6 pp 1574ndash1585 2012

[4] R Yang P Shi G-P Liu andH Gao ldquoNetwork-based feedbackcontrol for systems with mixed delays based on quantizationand dropout compensationrdquo Automatica vol 47 no 12 pp2805ndash2809 2011

[5] XM Zhang Z J Zhang andG P Lu ldquoFault detection for state-delay fuzzy systems subject to random communication delayrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 4 pp 2439ndash2451 2012

[6] F B Li and X Zhang ldquoDelay-range-dependent robust 119867infin

filtering for singuar LPV systems with time variant delayrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 1 pp 339ndash353 2013

[7] Y Y Cao and P M Frank ldquoAnalysis and synthesis of nonlineartime-delay systems via fuzzy control approachrdquo IEEE Transac-tions on Fuzzy Systems vol 8 no 2 pp 200ndash211 2000

[8] Y-Y Cao and P M Frank ldquoStability analysis and synthesis ofnonlinear time-delay systems via linear Takagi-Sugeno fuzzymodelsrdquo Fuzzy Sets and Systems vol 124 no 2 pp 213ndash2292001

[9] O M Kwon and J H Park ldquoDelay-range-dependent stabiliza-tion of uncertain dynamic systems with interval time-varyingdelaysrdquo Applied Mathematics and Computation vol 208 no 1pp 58ndash68 2009

[10] E Tian D Yue and Y Zhang ldquoDelay-dependent robust 119867infin

control for T-S fuzzy system with interval time-varying delayrdquoFuzzy Sets and Systems vol 160 no 12 pp 1708ndash1719 2009

[11] J Y An T Li G LWen and R Li ldquoNew stability conditions foruncertain T-S fuzzy systems with interval time-varying delayrdquoInternational Journal of Control Automation and Systems vol10 no 3 pp 490ndash497 2012

[12] Z J Zhang X L Huang X J Ban and X Z Gao ldquoStabilityanalysis and controller design of T-S fuzzy systems with time-delay under imperfect premise matchingrdquo Journal of BeijingInstitute of Technology vol 21 no 3 pp 387ndash393 2012

[13] O M Kwon M J Park S M Lee and J H Park ldquoAugmentedLyapunov-Krasovskii functional approaches to robust stabilitycriteria for uncertain Takagi-Sugeno fuzzy systems with time-varying delaysrdquo Fuzzy Sets and Systems vol 201 no 16 pp 1ndash192012

[14] F Liu M Wu Y He and R Yokoyama ldquoNew delay-dependentstability criteria for T-S fuzzy systems with time-varying delayrdquoFuzzy Sets and Systems vol 161 no 15 pp 2033ndash2042 2010

[15] B Chen X Liu and S Tong ldquoNew delay-dependent stabiliza-tion conditions of T-S fuzzy systems with constant delayrdquo FuzzySets and Systems vol 158 no 20 pp 2209ndash2224 2007

[16] J Yoneyama ldquoRobust stability and stabilization for uncertainTakagi-Sugeno fuzzy time-delay systemsrdquo Fuzzy Sets and Sys-tems vol 158 no 2 pp 115ndash134 2007

[17] C Lin Q-G Wang and T H Lee ldquoDelay-dependent LMIconditions for stability and stabilization of T-S fuzzy systemswith bounded time-delayrdquo Fuzzy Sets and Systems vol 157 no9 pp 1229ndash1247 2006

[18] X J Su P Shi L G Wu and Y D Song ldquoA novel approachto filter design for T-S fuzzy discrete-time systems with time-varying delayrdquo IEEE Transactions on Fuzzy Systems vol 20 no6 pp 1114ndash1129 2012

[19] C Peng and Q-L Han ldquoDelay-range-dependent robust stabi-lization for uncertain T-S fuzzy control systems with intervaltime-varying delaysrdquo Information Sciences vol 181 no 19 pp4287ndash4299 2011

[20] S T Li Y W Jing and X M Liu ldquoRobust control for a class ofT-S fuzzy systemswith interval time-varying delayrdquo InternatinalJournal of Control Automation and Systems vol 10 no 4 pp737ndash743 2012

[21] X-M Zhang M Wu J-H She and Y He ldquoDelay-dependentstabilization of linear systems with time-varying state and inputdelaysrdquo Automatica vol 41 no 8 pp 1405ndash1412 2005

[22] B Chen X Liu and S Tong ldquoRobust fuzzy control of nonlinearsystems with input delayrdquo Chaos Solitons amp Fractals vol 37 no3 pp 894ndash901 2008

[23] C Lin Q-G Wang and T H Lee ldquoDelay-dependent LMIconditions for stability and stabilization of T-S fuzzy systemswith bounded time-delayrdquo Fuzzy Sets and Systems vol 157 no9 pp 1229ndash1247 2006

[24] H J Lee J B Park and Y H Joo ldquoRobust control for uncertainTakagi-Sugeno fuzzy systems with time-varying input delayrdquoJournal of Dynamic Systems Measurement and Control Trans-actions of the ASME vol 127 no 2 pp 302ndash306 2005

[25] C-H Lien and K-W Yu ldquoRobust control for Takagi-Sugenofuzzy systems with time-varying state and input delaysrdquo ChaosSolitons and Fractals vol 35 no 5 pp 1003ndash1008 2008

[26] Y K Su B Chen S Y Zhang and C Lin ldquoDelay-dependentstabilization of T-S fuzzy systems with time delayrdquo Control andDecision Kongzhi yu Juece vol 24 no 6 pp 921ndash927 2009

[27] B Chen X Liu C Lin and K Liu ldquoRobust 119867infin

control ofTakagi-Sugeno fuzzy systems with state and input time delaysrdquoFuzzy Sets and Systems vol 160 no 4 pp 403ndash422 2009

[28] L Li and X Liu ldquoNew results on delay-dependent robuststability criteria of uncertain fuzzy systems with state and inputdelaysrdquo Information Sciences vol 179 no 8 pp 1134ndash1148 2009

[29] Z J Zhang X J Ban X L Huang and X Z Gao ldquoNew delay-dependent robust stablity for T-S fuzzy systems with intervevaltime-varying delayrdquo in Proccedings of the 5th International JointConference on Computational Science and Optimization pp826ndash830 Harbin China 2012

Mathematical Problems in Engineering 13

[30] H K Lam and F H F Leung ldquoLMI-based stability andperformance design of fuzzy control systems fuzzy modelsand controllers with different premisesrdquo in Proceedings of theInternational Conference on Fuzzy Systerm pp 9599ndash9506Vancouver Canada 2006

[31] H K Lam andMNarimani ldquoStability analysis and perfomancedesign for fuzzy-model-based control system under imperfectpremise matchingrdquo IEEE Transactions on Fuzzy Systems vol 17no 4 pp 949ndash961 2009

[32] S Boyd L E Ghaoui and E Feron Linear Matrix Inequality inSystems and ControlTheory SIAM Philadelphia Pa USA 1994

[33] I R Petersen and C V Hollot ldquoA Riccati equation approach tothe stabilization of uncertain linear systemsrdquo Automatica vol22 no 4 pp 397ndash411 1986

[34] J Sun G P Liu and J Chen ldquoDelay-dependent stabilityand stabilization of neutral time-delay systemsrdquo InternationalJournal of Robust andNonlinear Control vol 19 no 12 pp 1364ndash1375 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

approach different from the traditional PDC design tech-nique is proposed which can significantly improve the designflexibility and reduce the implementation cost of the fuzzycontroller by arbitrarily selecting simple fuzzy membershipfunctions Two numerical examples are used to illustrate theconservativeness and effectiveness of our proposed methods

Acknowledgments

This research work was funded by the NSFC under Grantno 61273095 and the Academy of Finland under Grants no135225 and no 127299

References

[1] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[2] W H Ho S H Chen I T Chen J H Chou and C C ShuldquoDesign of stable and quadratic-optimal static output feed-back controllers for T-S-fuzzy-model-based control systems anintegrative computational approachrdquo International Journal ofInnovative Computing Information and Control vol 8 no 1App 403ndash418 2012

[3] X J Su L G Wu P Shi and Y D Song ldquo119867infinmodel reduction

of T-S fuzzy stochastic systemsrdquo IEEE Transactions on SystemsMan and Cybermetics B vol 42 no 6 pp 1574ndash1585 2012

[4] R Yang P Shi G-P Liu andH Gao ldquoNetwork-based feedbackcontrol for systems with mixed delays based on quantizationand dropout compensationrdquo Automatica vol 47 no 12 pp2805ndash2809 2011

[5] XM Zhang Z J Zhang andG P Lu ldquoFault detection for state-delay fuzzy systems subject to random communication delayrdquoInternational Journal of Innovative Computing Information andControl vol 8 no 4 pp 2439ndash2451 2012

[6] F B Li and X Zhang ldquoDelay-range-dependent robust 119867infin

filtering for singuar LPV systems with time variant delayrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 1 pp 339ndash353 2013

[7] Y Y Cao and P M Frank ldquoAnalysis and synthesis of nonlineartime-delay systems via fuzzy control approachrdquo IEEE Transac-tions on Fuzzy Systems vol 8 no 2 pp 200ndash211 2000

[8] Y-Y Cao and P M Frank ldquoStability analysis and synthesis ofnonlinear time-delay systems via linear Takagi-Sugeno fuzzymodelsrdquo Fuzzy Sets and Systems vol 124 no 2 pp 213ndash2292001

[9] O M Kwon and J H Park ldquoDelay-range-dependent stabiliza-tion of uncertain dynamic systems with interval time-varyingdelaysrdquo Applied Mathematics and Computation vol 208 no 1pp 58ndash68 2009

[10] E Tian D Yue and Y Zhang ldquoDelay-dependent robust 119867infin

control for T-S fuzzy system with interval time-varying delayrdquoFuzzy Sets and Systems vol 160 no 12 pp 1708ndash1719 2009

[11] J Y An T Li G LWen and R Li ldquoNew stability conditions foruncertain T-S fuzzy systems with interval time-varying delayrdquoInternational Journal of Control Automation and Systems vol10 no 3 pp 490ndash497 2012

[12] Z J Zhang X L Huang X J Ban and X Z Gao ldquoStabilityanalysis and controller design of T-S fuzzy systems with time-delay under imperfect premise matchingrdquo Journal of BeijingInstitute of Technology vol 21 no 3 pp 387ndash393 2012

[13] O M Kwon M J Park S M Lee and J H Park ldquoAugmentedLyapunov-Krasovskii functional approaches to robust stabilitycriteria for uncertain Takagi-Sugeno fuzzy systems with time-varying delaysrdquo Fuzzy Sets and Systems vol 201 no 16 pp 1ndash192012

[14] F Liu M Wu Y He and R Yokoyama ldquoNew delay-dependentstability criteria for T-S fuzzy systems with time-varying delayrdquoFuzzy Sets and Systems vol 161 no 15 pp 2033ndash2042 2010

[15] B Chen X Liu and S Tong ldquoNew delay-dependent stabiliza-tion conditions of T-S fuzzy systems with constant delayrdquo FuzzySets and Systems vol 158 no 20 pp 2209ndash2224 2007

[16] J Yoneyama ldquoRobust stability and stabilization for uncertainTakagi-Sugeno fuzzy time-delay systemsrdquo Fuzzy Sets and Sys-tems vol 158 no 2 pp 115ndash134 2007

[17] C Lin Q-G Wang and T H Lee ldquoDelay-dependent LMIconditions for stability and stabilization of T-S fuzzy systemswith bounded time-delayrdquo Fuzzy Sets and Systems vol 157 no9 pp 1229ndash1247 2006

[18] X J Su P Shi L G Wu and Y D Song ldquoA novel approachto filter design for T-S fuzzy discrete-time systems with time-varying delayrdquo IEEE Transactions on Fuzzy Systems vol 20 no6 pp 1114ndash1129 2012

[19] C Peng and Q-L Han ldquoDelay-range-dependent robust stabi-lization for uncertain T-S fuzzy control systems with intervaltime-varying delaysrdquo Information Sciences vol 181 no 19 pp4287ndash4299 2011

[20] S T Li Y W Jing and X M Liu ldquoRobust control for a class ofT-S fuzzy systemswith interval time-varying delayrdquo InternatinalJournal of Control Automation and Systems vol 10 no 4 pp737ndash743 2012

[21] X-M Zhang M Wu J-H She and Y He ldquoDelay-dependentstabilization of linear systems with time-varying state and inputdelaysrdquo Automatica vol 41 no 8 pp 1405ndash1412 2005

[22] B Chen X Liu and S Tong ldquoRobust fuzzy control of nonlinearsystems with input delayrdquo Chaos Solitons amp Fractals vol 37 no3 pp 894ndash901 2008

[23] C Lin Q-G Wang and T H Lee ldquoDelay-dependent LMIconditions for stability and stabilization of T-S fuzzy systemswith bounded time-delayrdquo Fuzzy Sets and Systems vol 157 no9 pp 1229ndash1247 2006

[24] H J Lee J B Park and Y H Joo ldquoRobust control for uncertainTakagi-Sugeno fuzzy systems with time-varying input delayrdquoJournal of Dynamic Systems Measurement and Control Trans-actions of the ASME vol 127 no 2 pp 302ndash306 2005

[25] C-H Lien and K-W Yu ldquoRobust control for Takagi-Sugenofuzzy systems with time-varying state and input delaysrdquo ChaosSolitons and Fractals vol 35 no 5 pp 1003ndash1008 2008

[26] Y K Su B Chen S Y Zhang and C Lin ldquoDelay-dependentstabilization of T-S fuzzy systems with time delayrdquo Control andDecision Kongzhi yu Juece vol 24 no 6 pp 921ndash927 2009

[27] B Chen X Liu C Lin and K Liu ldquoRobust 119867infin

control ofTakagi-Sugeno fuzzy systems with state and input time delaysrdquoFuzzy Sets and Systems vol 160 no 4 pp 403ndash422 2009

[28] L Li and X Liu ldquoNew results on delay-dependent robuststability criteria of uncertain fuzzy systems with state and inputdelaysrdquo Information Sciences vol 179 no 8 pp 1134ndash1148 2009

[29] Z J Zhang X J Ban X L Huang and X Z Gao ldquoNew delay-dependent robust stablity for T-S fuzzy systems with intervevaltime-varying delayrdquo in Proccedings of the 5th International JointConference on Computational Science and Optimization pp826ndash830 Harbin China 2012

Mathematical Problems in Engineering 13

[30] H K Lam and F H F Leung ldquoLMI-based stability andperformance design of fuzzy control systems fuzzy modelsand controllers with different premisesrdquo in Proceedings of theInternational Conference on Fuzzy Systerm pp 9599ndash9506Vancouver Canada 2006

[31] H K Lam andMNarimani ldquoStability analysis and perfomancedesign for fuzzy-model-based control system under imperfectpremise matchingrdquo IEEE Transactions on Fuzzy Systems vol 17no 4 pp 949ndash961 2009

[32] S Boyd L E Ghaoui and E Feron Linear Matrix Inequality inSystems and ControlTheory SIAM Philadelphia Pa USA 1994

[33] I R Petersen and C V Hollot ldquoA Riccati equation approach tothe stabilization of uncertain linear systemsrdquo Automatica vol22 no 4 pp 397ndash411 1986

[34] J Sun G P Liu and J Chen ldquoDelay-dependent stabilityand stabilization of neutral time-delay systemsrdquo InternationalJournal of Robust andNonlinear Control vol 19 no 12 pp 1364ndash1375 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 13

[30] H K Lam and F H F Leung ldquoLMI-based stability andperformance design of fuzzy control systems fuzzy modelsand controllers with different premisesrdquo in Proceedings of theInternational Conference on Fuzzy Systerm pp 9599ndash9506Vancouver Canada 2006

[31] H K Lam andMNarimani ldquoStability analysis and perfomancedesign for fuzzy-model-based control system under imperfectpremise matchingrdquo IEEE Transactions on Fuzzy Systems vol 17no 4 pp 949ndash961 2009

[32] S Boyd L E Ghaoui and E Feron Linear Matrix Inequality inSystems and ControlTheory SIAM Philadelphia Pa USA 1994

[33] I R Petersen and C V Hollot ldquoA Riccati equation approach tothe stabilization of uncertain linear systemsrdquo Automatica vol22 no 4 pp 397ndash411 1986

[34] J Sun G P Liu and J Chen ldquoDelay-dependent stabilityand stabilization of neutral time-delay systemsrdquo InternationalJournal of Robust andNonlinear Control vol 19 no 12 pp 1364ndash1375 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of