Research ArticleAdaptive Stabilization Control for a Class of Complex NonlinearSystems Based on T-S Fuzzy Bilinear Model

Jinsheng Xing and Naizheng Shi

School of Mathematics amp Computer Science Shanxi Normal University Linfen 041004 China

Correspondence should be addressed to Jinsheng Xing xjs6448126com

Received 24 July 2014 Accepted 15 December 2014

Academic Editor Lakshmanan Shanmugam

Copyright copy 2015 J Xing and N Shi 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 proposes a stable adaptive fuzzy control scheme for a class of nonlinear systems with multiple inputs The multipleinputs T-S fuzzy bilinear model is established to represent the unknown complex systems A parallel distributed compensation(PDC) method is utilized to design the fuzzy controller without considering the error due to fuzzy modelling and the sufficientconditions of the closed-loop system stability with respect to decay rate 120572 are derived by linear matrix inequalities (LMIs)Then theerrors caused by fuzzy modelling are considered and the method of adaptive control is used to reduce the effect of the modellingerrors and dynamic performance of the closed-loop system is improved By Lyapunov stability criterion the resulting closed-loopsystem is proved to be asymptotically stable The main contribution is to deal with the differences between the T-S fuzzy bilinearmodel and the real system a global asymptotically stable adaptive control scheme is presented for real complex systems Finallyillustrative examples are provided to demonstrate the effectiveness of the results proposed in this paper

1 Introduction

Takagi-Sugeno (T-S) model-based fuzzy control is an effec-tive and flexible tool for control of nonlinear systems It hasattracted wide attention [1ndash6] The robust fuzzy control andadaptive fuzzy control approaches based on T-S fuzzy linearmodel with parameters uncertainties have been extensivelystudied in [2 7ndash9] and the references therein The adaptivefuzzy control based on T-S fuzzy model is divided into adap-tive control based on static T-S fuzzy model (or fuzzy logicsystems) and adaptive control based on dynamic T-S fuzzymodel Up to nowmany results for the adaptive control basedon fuzzy logic system have appeared Compared with theadaptive control based on fuzzy logic system the develop-ment of the adaptive control based on dynamic T-S fuzzymodel is relatively slow until recent years the latter hasbecome a focus issue in the control community Novelresearch results have emerged in [10ndash19] A novel direct T-Sfuzzy neural online modelling and control method for a classof nonlinear systems with parametric uncertainties has beenproposed which utilized T-S fuzzy neural model to approxi-mate the virtual linear system and designed the online identi-fication algorithm and robust adaptive tracking controller in

[10 11] respectivelyThen the stability of the resulting closed-loop systems was proved by Lyapunov stability criterion In[13] the adaptive output tracking control problem of param-eter strict-feedback system was discussed and the concept ofvirtual variable also was proposed based on LMI In [14] theproblem of simultaneously estimating the state and unknowninputs was considered in T-S fuzzy systems sufficient con-ditions were given for the stability of the observer Thecomputation of the observer gains was based on solving aseries of LMIs In [17] a T-S fuzzymodel is proposed to repre-sent the nonlinear model of microelectromechanical systemsgyroscope a robust adaptive slidingmode control with onlineidentification for the upper bounds of external disturbancesand an adaptive estimator for the model uncertainty parame-ters are proposed in the Lyapunov framework In [18] a newnormal form of a global noncanonical form T-S fuzzy modelwas derived and a new solution framework was developedfor adaptive control of general discrete-time state-space T-Sfuzzy systems with a relative degree In [19] a novel designschemeof stable adaptive fuzzy control for a class of nonlinearsystems was proposedThe parallel distributed compensation(PDC) method was utilized to design the fuzzy controllerwithout the fuzzy modelling error The sufficient conditions

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 659521 11 pageshttpdxdoiorg1011552015659521

2 Mathematical Problems in Engineering

with respect to decay rate 120572 were derived by linear matrixinequalities (LMIs) and by small-gain theorem The adaptivecompensation term was adopted to reduce the effect of themodelling However the consequence parts of the above T-Sfuzzy models [10ndash19] were linear dynamic model or approxi-mate linear dynamicmodel so thesemethods have inevitabledefects for some nonlinear systems

It is known that bilinear models can describe many phys-ical systems and dynamical processes in engineering fields[20 21]There are twomain advantages of the bilinear systemOne is that it provides a better approximation to a nonlinearsystem than a linear one and another is that many realphysical processes may be appropriately modelled as bilinearsystems when the linear models are inadequate Consideringthe advantages of bilinear systems and T-S fuzzy control thefuzzy control based on the T-S fuzzy model with bilinear ruleconsequence attracted the interest of researchers [22ndash26]TheT-S fuzzy bilinear model may be suitable for some classesof nonlinear plants The robust stabilization for continuous-time fuzzy system with local bilinear model was studied in[22] and then the result was extended to the fuzzy systemwith time-delay only in the state [23] The problem of robuststabilization for discrete-time fuzzy systemwith local bilinearmodel was investigated in [24] Reference [25] focuses on theproblem of nonfragile guaranteed cost control for a class of T-S discrete-time fuzzy bilinear systems Based on the paralleldistributed compensation approach the sufficient conditionswere derived such that the closed-loop system was asymptot-ically stable and the cost function value was no more than acertain upper bound in the presence of the additive controllergain perturbations In [26] an observer-based fuzzy controldesign was given for discrete-time T-S fuzzy bilinear systemsIn [27 28] authors proposed robust stability conditions forstochastic fuzzy impulsive recurrent neural networks withtime-varying delays and uncertain stochastic fuzzy recurrentneural networks with mixed time-varying delays

However when there are differences between T-S fuzzybilinear model and reality systems these results will not beapplied

Considering the differences of the fuzzy model and thereality systems in the paper a stable adaptive fuzzy controlfor complex nonlinear systems is presented based onmultipleinputs T-S fuzzy bilinear system with parameters uncertain-ties In consideration of themodelling error an adaptive fuzzycontrol is proposed to compensate for the issues At first theconcept of the so-called PDC and LMI approach is employedto design the state feedback controller without consideringthe error caused by fuzzymodellingThe sufficient conditionswith respect to decay rate 120572 are derived in the sense ofLyapunov asymptotic stabilityThen the error caused by fuzzymodelling is considered an adaptive compensation term isdesigned to reduce the effect of the modelling error Thecontributions of this paper are as follows (i) the differencesbetween T-S fuzzy bilinear model and the real system areconsidered in themodelling and analysis (ii) a global asymp-totical stable adaptive control scheme is presented for realsystems (iii) a sufficient condition of the closed-loop systemsis given Finally theoretical analysis verifies that the state

converges to zero and all signals of the closed-loop systemsare bounded

2 Problem Statement and Basic Assumptions

Consider the nonlinear system in the following form

119894= 119909119894+1

119894 = 1 119899 minus 1

119899= 119891 (119909) + 119892

119879(119909) 119906

(1)

where 119909 = (1199091 1199092 119909

119899)119879

isin 119877119899 and 119906 isin 119877

119898 are the vectorsof state and control input respectively 119891(119909) is the unknowncontinuous function 119892(119909) is the vector of unknown continu-ous control gain function which satisfies 119892

119895(119909) ge 119892

119895min gt 0119895 = 1 2 119898

Definition 1 (see [19]) System (1) under the input being zerois globally asymptotically stable with decay rate 120572 if thereexists a scalar 120572 gt 0 such that

(119909 (119905)) le minus2120572119881 (119909 (119905)) (2)

where 119881(119909(119905)) = 119909119879(119905)119875119909(119905) is the Lyapunov function candi-

date and 119875 gt 0

Lemma 2 (see [25]) Given two matrices119860 and 119861 with appro-priate dimensions one has 119860119879119861 + 119861

119879119860le 119860119879119860

+ 119861119879119861

In this paper our objective is to design an adaptive fuzzycontroller so that the closed-loop systems are asymptoticallystable that is the states of the closed-loop system converge tozero and all signals of the closed-loop systems are bounded

System (1) can be expressed in terms of the T-S fuzzymodel as followsPlant rule 119894

IF 1199111 (

119905) is 119872119894

1 1199112 (

119905) is 119872119894

2 119911

119892 (119905) is 119872

119894

119892

THEN (119905) = 119860119894119909 (119905) + 119861

119894119906 (119905) + 119862

119894119906 (119905) 119909 (119905) + 119876Δ119891

119894 (119909 119906)

119894 = 1 2 119903

(3)

where 119911(119905) = [1199111(119905) 1199112(119905) 119911

119892(119905)] isin 119877

119892 are known premisevariables that may be functions of the state variables 119872119894

119895is

the fuzzy set 119903 is the number of the rules 119876 = [0 0 1]119879

119862119894119906(119905) = sum

119898

119895=1119862119894119895119906119895 119862119894119895

isin 119877119899times119899 is constant matrices and

Δ119891119894(119909 119906) = (Δ119886

119894119909 + Δ119887

119894119906 + Δ119862

119894119906119909) denotes the model error

of the 119894th bilinear model in the 119894th fuzzy space (also called the119894th fuzzy rule)

The modelling error terms are defined as follows

Δ119886119894= (Δ119886

1198941 Δ1198861198942 Δ119886

119894119899) isin 1198771times119899

Δ119887119894= (Δ119887

1198941 Δ1198871198942 Δ119887

119894119898) isin 1198771times119898

Δ119862119894119906 (119905) =

119898

sum

119895=1

Δ119862119894119895119906119895

Δ119862119894119895

= [Δ1198881198941198951

Δ1198881198941198952

sdot sdot sdot Δ119888119894119895119899

] isin 1198771times119899

(4)

Mathematical Problems in Engineering 3

119860119894isin 119877119899times119899 119861

119894isin 119877119899times119898 and 119862

119894119895are constant matrices which

have of the following forms

119860119894=

[

[

[

[

[

[

[

[

[

0 1 0 sdot sdot sdot 0 0

0 0 1 sdot sdot sdot 0 0

0 0 0 sdot sdot sdot 1 0

0 0 0 sdot sdot sdot 0 1

1198861198941

1198861198942

1198861198943

sdot sdot sdot 119886119894(119899minus1)

119886119894119899

]

]

]

]

]

]

]

]

]

119861119894=

[

[

[

[

[

[

[

[

[

0 0 0 sdot sdot sdot 0 0

0 0 0 sdot sdot sdot 0 0

0 0 0 sdot sdot sdot 0 0

0 0 0 sdot sdot sdot 0 0

1198871198941

1198871198942

1198871198943

sdot sdot sdot 119887119894(119898minus1)

119887119894119898

]

]

]

]

]

]

]

]

]

119862119894119895

=

[

[

[

[

[

[

[

[

[

0 0 0 sdot sdot sdot 0 0

0 0 0 sdot sdot sdot 0 0

0 0 0 sdot sdot sdot 0 0

0 0 0 sdot sdot sdot 0 0

1198881198941198951

1198881198941198952

1198881198941198953

sdot sdot sdot 119888119894119895(119899minus1)

119888119894119895119899

]

]

]

]

]

]

]

]

]

(5)

By using the fuzzy inference method with a singletonfuzzification product inference and centre average defuzzi-fication the overall fuzzy model is of the following form

(119905) =

119903

sum

119894=1

ℎ119894 (119911 (119905)) [119860 119894

119909 (119905) + 119861119894119906 (119905) + 119862

119894119906 (119905) 119909 (119905)

+ 119876Δ119891119894 (119909 119906)]

(6)

where

ℎ119894 (119911 (119905)) =

119908119894 (119911 (119905))

sum119903

119895=1119908119895 (

119911 (119905))

119908119894 (119911 (119905)) =

119892

prod

119895=1

119872119894

119895(119911119895 (

119905))

(7)

We assume 119908119894(119911(119905)) ge 0 sum119903

119894=1119908119894(119911(119905)) gt 0 for all 119905

Therefore we have

ℎ119894 (119911 (119905)) ge 0

119903

sum

119894=1

ℎ119894 (119911 (119905)) = 1 119894 = 1 2 119903 (8)

By comparison with (1) and (6) it is easy to see that

119891 (119909) =

119903

sum

119894=1

ℎ119894 (119911 (119905)) (119886119894

+ Δ119886119894) 119909 (119905)

119892119895 (

119909) =

119903

sum

119894=1

ℎ119894 (119911 (119905)) (119887119894119895

+ Δ119887119894119895

+ 119888119894119895119909 (119905) + Δ119862

119894119895119909 (119905))

(9)

where 119886119894= (1198861198941 1198861198942 119886

119894119899) isin 119877

1times119899 119887119894= (1198871198941 1198871198942 119887

119894119898) isin

1198771times119898 and 119888

119894119895= (1198881198941198951

1198881198941198952

119888119894119895119899

)

Remark 3 From now on unless confusion arises argumentssuch as 119911(119905) in ℎ

119894(119911(119905)) will be omitted just for notational

convenience

3 Control Design and Stability Analysis

System (6) can be represented by following the T-S fuzzymodel without considering the modelling error that isΔ119891119894(119909 119906) equiv 0 119894 = 1 2 119903 Consider

Plant rule 119894

IF 1199111 (

119905) is 119872119894

1 1199112 (

119905) is 119872119894

2 119911

119892 (119905) is 119872

119894

119892

THEN (119905) = 119860119894119909 (119905) + 119861

119894119906 (119905) + 119862

119894119906 (119905) 119909 (119905)

119894 = 1 2 119903

(10)

By using the fuzzy inference method with a singletonfuzzification product inference and centre average defuzzi-fication the overall fuzzy model is of the following form

(119905) =

119903

sum

119894=1

ℎ119894[119860119894119909 (119905) + 119861

119894119906 (119905) + 119862

119894119906 (119905) 119909 (119905)] (11)

Based on the idea of PDC the 119895th state-feedback con-troller is designed as follows

Plant rule 119894

IF 1199111 (

119905) is 119872119894

1 1199112 (

119905) is 119872119894

2 119911

119892 (119905) is 119872

119894

119892

THEN 119906119902119895

=

120588119895119865119894119895119909 (119905)

radic1 + 119909119879(119905) 119865119879

119894119895119865119894119895119909 (119905)

119894 = 1 2 119903

(12)

where 119865119894119895

isin 1198771times119899 is a vector to be determined and 120588

119895gt 0 is a

scalar to be assignedThe overall fuzzy control law can be represented by

119906119902119895

=

119903

sum

119894=1

ℎ119894

120588119895119865119894119895119909 (119905)

radic1 + 119909119879(119905) 119865119879

119894119895119865119894119895119909 (119905)

=

119903

sum

119894=1

ℎ119894120588119895sin 120579119894119895

=

119903

sum

119894=1

ℎ119894120588119895119865119894119895cos 120579119894119895119909 (119905)

(13)

where

sin 120579119894119895

=

119865119894119895119909 (119905)

radic1 + 119909119879(119905) 119865119879

119894119895119865119894119895119909 (119905)

cos 120579119894119895

=

1

radic1 + 119909119879(119905) 119865119879

119894119895119865119894119895119909 (119905)

120579119894119895

isin [minus

120587

2

120587

2

]

1 le 119894 le 119903 1 le 119895 le 119898

(14)

4 Mathematical Problems in Engineering

Substituting (13) into (11) one can get the closed-loopsystem

(119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895(119860119894+

119898

sum

119896=1

119861119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

119862119894119896120588119896sin 120579119895119896

)119909 (119905)

(15)

where 119861119894119896denotes the 119896th column of the 119861

119894

Theorem 4 Given positive scalars 120588119896(1 le 119896 le 119898) if there

exist a symmetric positive definite matrix119880 and some constantmatrices 119882

119894119895 such that LMIs (16) and (17) hold

[

[

[

119860119894119880 + 119880119860

119879

119894+ 120588 + 2120572119880 lowast lowast

119861119894119882119894

minus119868

119862119894119880 minus119868

]

]

]

lt 0 1 le 119894 le 119903 (16)

[

[

[

[

[

[

[

[

119872119894

lowast lowast lowast lowast

119861119894119882119895

minus119868

119861119895119882119894

minus119868

119862119894119880 minus119868

119862119895119880 minus119868

]

]

]

]

]

]

]

]

lt 0 1 le 119894 lt 119895 le 119903 (17)

where 120588 = sum119898

119896=11205882

119896 119872119894= 119860119894119880+119880119860

119879

119894+119860119895119880+119880119860

119879

119895+2120588+2120572119880

119861119894119882119894=

[

[

[

11986111989411198821198941

119861119894119898

119882119894119898

]

]

]

119862119894=

[

[

[

1198621198941

119862119894119898

]

]

]

119868 =[

[

119868

d119868

]

]

(18)

Then the FBS (15) is globally asymptotically stable with decayrate 120572 via the fuzzy feedback controller (13) and the gains canbe determined by 119865

119894119895= 119882119894119895119880minus1

Proof Consider the Lyapunov function candidate as follows

1198811 (

119909 (119905)) = 119909119879(119905) 119875119909 (119905) (19)

where 119875 = 119880minus1

Applying Schur complement lemma inequality (16) canbe written as follows

119880119860119879

119894+ 119860119894119880 +

119898

sum

119896=1

1205882

119896+

119898

sum

119896=1

(119861119894119896119882119894119896)119879119861119894119896119882119894119896

+

119898

sum

119896=1

(119862119894119896119880)119879(119862119894119896119880) + 2120572119880 lt 0

(20)

Premultiplying and postmultiplying (20) by 119875 respectivelywe have

119860119879

119894119875 + 119875119860

119894+

119898

sum

119896=1

1205882

119896119875119875 +

119898

sum

119896=1

(119861119894119896119865119894119896)119879119861119894119896119865119894119896

+

119898

sum

119896=1

119862119879

119894119896119862119894119896

+ 2120572119875 lt 0

(21)

Applying a similar procedure to inequality (17) we can obtain

119860119879

119894119875 + 119875119860

119894+ 119860119879

119895119875 + 119875119860

119895+ 2

119898

sum

119896=1

1205882

119896119875119875

+

119898

sum

119896=1

(119861119894119896119865119895119896

)

119879

(119861119894119896119865119895119896

) +

119898

sum

119896=1

119862119879

119894119896119862119894119896

+

119898

sum

119896=1

(119861119895119896

119865119894119896)

119879

(119861119895119896

119865119894119896) +

119898

sum

119896=1

119862119879

119895119896119862119895119896

+ 2120572119875 lt 0

(22)

The time derivative of 1198811is

1 (

119909 (119905)) = 119879119875119909 + 119909

119879119875 (23)

By substituting (15) into (23) we can get

1 (

119909 (119905)) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895119909119879

[119860119894+

119898

sum

119896=1

119861119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

119862119894119896120588119896sin 120579119895119896

]

119879

119875

+ 119875[119860119894+

119898

sum

119896=1

119861119894119896120588119896

times 119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

119862119894119896120588119896sin 120579119895119896

]

119909

=

119903

sum

119894=1

ℎ2

119894119909119879Λ119894119894119909 +

119903

sum

119894lt119895

ℎ119894ℎ119895119909119879Λ119894119895119909

(24)

Mathematical Problems in Engineering 5

where

Λ119894119894

= 119860119879

119894119875 + 119875119860

119894+

119898

sum

119896=1

120588119896sin 120579119894119896

(119862119879

119894119896119875 + 119875119862

119894119896)

+

119898

sum

119896=1

120588119896cos 120579119894119896

[(119861119894119896119865119894119896)119879119875 + 119875119861

119894119896119865119894119896]

Λ119894119895

= 119860119879

119894119875 + 119875119860

119894+

119898

sum

119896=1

120588119896sin 120579119894119896

(119862119879

119894119896119875 + 119875119862

119894119896)

+

119898

sum

119896=1

120588119896cos 120579119894119896

[(119861119894119896119865119895119896

)

119879

119875 + 119875119861119894119896119865119895119896

]

+ 119860119879

119895119875 + 119875119860

119895+

119898

sum

119896=1

120588119896sin 120579119895119896

(119862119879

119895119896119875 + 119875119862

119895119896)

+

119898

sum

119896=1

120588119896cos 120579119895119896

[(119861119895119896

119865119894119896)

119879

119875 + 119875119861119895119896

119865119894119896]

(25)

First by premultiplying and postmultiplyingΛ119894119894by119880 we can

obtain

119880Λ119894119894119880 = 119880119860

119879

119894+ 119860119894119880 +

119898

sum

119896=1

120588119896sin 120579119894119896

(119880119862119879

119894119896+ 119862119894119896119880)

+

119898

sum

119896=1

120588119896cos 120579119894119896

[119880 (119861119894119896119865119894119896)119879+ 119861119894119896119865119894119896119880]

(26)

According to Lemma 2 we can get the following

[120588119896119861119894119896119865119894119896119880 + 120588119896119880 (119861119894119896119865119894119896)119879] cos 120579

119894119896

le 1205882

119896cos2120579119894119896

+ (119861119894119896119882119894119896)119879119861119894119896119882119894119896

119880119862119879

119894119896120588119896sin 120579119894119896

+ 119862119894119896119880120588119896sin 120579119894119896

le 1205882

119896sin2120579119894119896

+ (119862119894119896119880)119879(119862119894119896119880)

(27)

From (26) and (27) we can obtain

119880Λ119894119894119880 le 119880119860

119879

119894+ 119860119894119880 +

119898

sum

119896=1

1205882

119896+

119898

sum

119896=1

(119862119894119896119880)119879119862119894119896119880

+

119898

sum

119896=1

(119861119894119896119882119894119896)119879119861119894119896119882119894119896

(28)

Applying similar procedures (26)ndash(28) to Λ119894119895 we can

obtain

119880Λ119894119895119880 le 119880119860

119879

119894+ 119860119894119880 + 119880119860

119879

119895+ 119860119895119880 + 2

119898

sum

119896=1

1205882

119896

+

119898

sum

119896=1

(119861119894119896119882119895119896

)

119879

(119861119894119896119882119895119896

) +

119898

sum

119896=1

(119862119894119896119880)119879119862119894119896119880

+

119898

sum

119896=1

(119861119895119896

119882119894119896)

119879

(119861119895119896

119882119894119896) +

119898

sum

119896=1

(119862119895119896

119880)

119879

119862119895119896

119880

(29)

Substituting (28) and (29) into (24) we obtain

1 (

119909 (119905)) le

119903

sum

119894=1

ℎ2

119894119909119879119875[119880119860

119879

119894+ 119860119894119880 +

119898

sum

119896=1

1205882

119896

+

119898

sum

119896=1

(119861119894119896119882119894119896)119879119861119894119896119882119894119896

+

119898

sum

119896=1

(119862119894119896119880)119879(119862119894119896119880)]119875119909

+

119903

sum

119894lt119895

ℎ119894ℎ119895119909119879119875[119880119860

119879

119894+ 119860119894119880 + 119880119860

119879

119895

+ 119860119895119880 + 2

119898

sum

119896=1

1205882

119896

+

119898

sum

119896=1

(119861119894119896119882119895119896

)

119879

(119861119894119896119882119895119896

)

+

119898

sum

119896=1

(119862119894119896119880)119879119862119894119896119880

+

119898

sum

119896=1

(119861119895119896

119882119894119896)

119879

(119861119895119896

119882119894119896)

+

119898

sum

119896=1

(119862119895119896

119880)

119879

119862119895119896

119880]119875119909

(30)By substituting (21) and (22) into (30) we can obtain

1 (

119909 (119905)) le minus2120572119875119909 = minus21205721198811 (

119909 (119905)) (31)Then by Definition 1 the closed-loop fuzzy system (15)

is globally asymptotically stable with decay rate 120572 Thiscompletes the proof of Theorem 4

Next the modelling error in (6) is considered and anadaptive compensation term is adopted to reduce the effectsof the modelling error

Adopt the fuzzy controller in the following form119906 = 119906119902119895

+ 119906119904119895

119895 = 1 2 119898 (32)where compensator 119906

119904119895will be designed later

Substituting (32) into (6) yields

(119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895[(119860119894+

119898

sum

119896=1

119861119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

119862119894119896120588119896sin 120579119895119896

)119909 (119905)

+ 119876(Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588119896sin 120579119895119896

)119909 (119905)

6 Mathematical Problems in Engineering

+

119898

sum

119896=1

(119861119894119896

+ 119876Δ119887119894119896

+ 119862119894119896119909 (119905)

+ 119876Δ119862119894119896119909 (119905)) 119906119904119895

]

(33)

Suppose that there exists an unknown constant 120582 such that

120582 ge

119903

sum

119894=1

119903

sum

119895=1

1003817100381710038171003817100381710038171003817100381710038171003817

Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588119896sin 120579119895119896

1003817100381710038171003817100381710038171003817100381710038171003817

(34)

Then from 0 le ℎ119894(119911(119905)) le 1 and

10038171003817100381710038171003817100381710038171003817100381710038171003817

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895(Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588 sin 120579

119895119896)

10038171003817100381710038171003817100381710038171003817100381710038171003817

le

119903

sum

119894=1

119903

sum

119895=1

1003817100381710038171003817100381710038171003817100381710038171003817

Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588 sin 120579

119895119896

1003817100381710038171003817100381710038171003817100381710038171003817

(35)

we have

10038171003817100381710038171003817100381710038171003817100381710038171003817

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895(Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588 sin 120579

119895119896)

1003817100381710038171003817100381710038171003817100381710038171003817

le 120582

(36)

It is easy to see that we can choose a function vector119867120582such

that

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895(Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588119896sin 120579119895119896

) = 120582119867120582

(37)

and 119867120582 le 1

Remark 5 Here assumption (34) is reasonable in many realsystems due to its boundedness such as chaotic system [19]for example Example 1 in the paper satisfies the assumptionOn the other hand the uncertain terms of the consideredsystems in the existing literature [2 18 22ndash24] satisfy thecondition of (34)

Denote

120596 (119905) = 119867120582119909 (119905) (38)

Substituting (37) and (38) into (33) we can obtain thefollowing feedback system

(119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895[(119860119894+

119898

sum

119896=1

119861119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

119862119894119896120588119896sin 120579119895119896

)119909 (119905)

+

119898

sum

119896=1

(119861119894119896

+ 119876Δ119887119894119896

+ 119862119894119896119909 (119905)

+ 119876Δ119862119894119896119909 (119905)) 119906119904119895

] + 119876120582120596 (119905)

(39)

Choose the adaptive compensator as follows

119906119904119895

= minus

1

2119898119892119895min1205742119876119879119875119888119909 (119905) 119895 = 1 2 119898 (40)

where 119888 = 1205822 119888 is the parameter estimation of 119888 and 120574 gt 0 is

a gain constantChoose the adaptive law as follows

=

1

21205781205742119909119879(119905) 119875119876119876

119879119875119909 (119905) (41)

where 120578 gt 0 is a gain constant which determines the rate ofadaptation

Substituting (40) into (39) yields

(119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895[(119860119894+

119898

sum

119896=1

119861119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

119862119894119896120588119896sin 120579119895119896

)119909 (119905)

+

119898

sum

119896=1

(119861119894119896

+ 119876Δ119887119894119896

+ 119862119894119896119909 (119905)

+ 119876Δ119862119894119896119909 (119905))

times (minus

119876119879119875119888119909 (119905)

2119898119892119895min1205742)] + 119876120582120596 (119905)

(42)

Theorem 6 Consider the uncertain nonlinear system (1) withcontrol law defined by (32) (13) and (40) and the parameterupdated by the adaptive law (41) If there exist a symmetricpositive definite matrix 119875 and some matrices 119865

119894119895(1 le 119894 119895 le 119903)

satisfying the LMIs (16) and (17) and the design parameter ischosen as

0 lt 120574 lt radic2120572120582min (119875) (43)

then the closed-loop system (42) is asymptotically stable and allsignals of the closed-loop system (42) are bounded

Mathematical Problems in Engineering 7

Proof Consider the Lyapunov function candidate

119881 = 119909119879(119905) 119875119909 (119905) + 120578 (119888 minus 119888)

2 (44)

where 120578 gt 0Let

119866119894119895

= 119860119894+

119898

sum

119896=1

119861119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

119862119894119896120588119896sin 120579119895119896

(45)

The time derivative of 119881 is

= 119879119875119909 + 119909

119879119875 + 2120578 (119888 minus 119888)

(46)

Substituting (39) into (46) we can obtain

=

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895[119909119879(119905)

times 119866119894119895

119879119875 + 119875119866

119894119895+ 2119875119876

times (Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588119896sin 120579119895119896

)119909 (119905)

+ 2119909119879(119905) 119875

119898

sum

119896=1

(119861119894119896

+ 119876Δ119887119894119896

+ 119862119894119896119909 (119905)

+ 119876Δ119862119894119896119909 (119905)) 119906119904119896

]

+ 2120578 (119888 minus 119888)

=

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895119909119879(119905) (119866119894119895

119879119875 + 119875119866

119894119895) 119909 (119905)

+ 2119909119879(119905) 119875119876120582120596 (119905)

+ 2119909119879(119905) 119875119876

119898

sum

119895=1

119892119895 (

119909) 119906119904119895+ 2120578 (119888 minus 119888)

(47)

From the proof of Theorem 4 we get

le minus2120572119909119879(119905) 119875119909 (119905) + 2119909

119879(119905) 119875119876120582120596 (119905)

+ 2119909119879(119905) 119875119876

119898

sum

119895=1

119892119895 (

119909) 119906119904119895+ 2120578 (119888 minus 119888)

(48)

It is easy to see that

2119909119879119875119876120582120596 minus 120574

21205962+ 12057421205962

= minus120574210038171003817100381710038171003817100381710038171003817

120596 minus

1

1205742119909119879119875119876120582

10038171003817100381710038171003817100381710038171003817

2

+

1

12057421199091198791198751198761205822119876119879119875119909 + 120574

21205962

le

1205822

1205742119909119879119875119876119876119879119875119909 + 120574

21205962

(49)

Substituting (49) into (48) yields

le minus2120572119909119879(119905) 119875119909 (119905) +

1205822

1205742119909119879119875119876119876119879119875119909 + 120574

21205962

+ 2119909119879(119905) 119875119876

119898

sum

119895=1

119892119895 (

119909) 119906119904119895+ 2120578 (119888 minus 119888)

(50)

Substituting (40) and (41) into (50) we obtain

le minus2120572119909119879(119905) 119875119909 (119905) + (minus119888 + 119888)

1

1205742119909119879119875119876119876119879119875119909

+ 2120578 (119888 minus 119888)

+ 12057421205962

= minus2120572119909119879(119905) 119875119909 (119905) + 120574

21205962

le minus2120572120582min (119875) 119909 (119905)2+ 12057421205962

(51)

where 1205962= 120596119879120596 = 119909

119879119867120582

119879119867120582119909 le 119867

120582

119879119867120582 sdot 119909

2le 119909

2By choosing 0 lt 120574 lt radic2120572120582min(119875) we can get lt 0

then we have that the states119909(119905) rarr 0 as 119905 approaches infinityvia LaSalle invariance principle and 119881(119905) is bounded From(44) we can obtain that states 119909 and 119888 are bounded thereforethe boundedness of 119906

119904119895is ensured from (40) Similarly from

(13) we can obtain that 119906119902119895

is boundedThen it can be provedthat (1) the closed-loop system (42) is asymptotically stableand (2) all signals of the closed-loop system (42) are bounded

4 Simulations

In this section we will give two examples to show theefficiency of the proposed approach The first example isan unknown chaotic system and the second example is aparameter uncertain T-S fuzzy bilinear system with multipleinputs

Example 1 Consider the following chaotic system with con-trol input

1= 1199092

2= minus01119909

2minus 1199091

3+ 12 cos 119905 + 119906

(52)

When 119906(119905) = 0 and the initial states are chosen as 119909(0) =

(2 2)119879 the states phase portrait of system (52) is shown in

Figure 1

8 Mathematical Problems in Engineering

0 1 2 3 4

0

2

4

6

8

10

minus10

minus8

minus6

minus4

minus4 minus3

minus2

minus2 minus1

x2

x1

Figure 1 The phase portrait of the chaotic system

System (52) can be modelled as the following T-S fuzzybilinear model

Rule 1 IF 1199091 (

119905) is about 0

THEN (119905) = (1198601+ Δ1198601) 119909 (119905) + (119861

1+ Δ1198611) 119906 (119905)

+ (1198621+ Δ1198621) 119909 (119905) 119906 (119905)

Rule 2 IF 1199091 (

119905) is about plusmn 2

THEN (119905) = (1198602+ Δ1198602) 119909 (119905) + (119861

2+ Δ1198612) 119906 (119905)

+ (1198622+ Δ1198622) 119909 (119905) 119906 (119905)

Rule 3 IF 1199091 (

119905) is about plusmn 4

THEN (119905) = (1198603+ Δ1198603) 119909 (119905) + (119861

3+ Δ1198613) 119906 (119905)

+ (1198623+ Δ1198623) 119909 (119905) 119906 (119905)

(53)

where 119909(119905) = (1199091(119905) 1199092(119905))119879 1198601= 1198602= 1198603= [0 1

minus1 minus1] 1198611=

1198612= 1198613= [0

minus1] and 119862

1= 1198622= 1198623= [0 1

1 1] Choose 120572 = 03

120588 = 009 1198651= [0 minus1] 119865

2= [minus1 minus1] and 119865

3= [minus1 minus1] By

solving LMIs (16)-(17) one can obtain

119875 = [

93637 46993

46993 106246] 120582min (119875) = 52528 (54)

Utilize the controllers (32) (13) and (40) and the parameterupdated law (41) to control system (52) The design parame-ters are chosen as 120578 = 2 120574 = 2 lt radic120582min(119875) the initial condi-tions are chosen as 119909(0) = (2 minus2)

119879 119888(0) = 0 and the relation-ship functions are selected as shown in Figure 2 The simula-tion results are shown in Figures 3 4 5 and 6 In Figures 3ndash6the curves of states control input and adaptive updatedparameter for the T-S fuzzy bilinear system are drawn by solidlines respectively while the curves of states control inputand adaptive updated parameter for T-S fuzzy linear systemare depicted by dotted lines respectively By comparison

Rule 3 Rule 2 Rule 1 Rule 2 Rule 3

minus4 minus3 minus2 minus1 0 1 2 3 4

x1

M(x

1)

0

02

04

06

08

1

Figure 2 The relationship functions

0 1 2 3 4 5 6 7 8 9 10Time (s)

minus05

0

05

1

15

2

x1

T-S FBST-S FLS

Figure 3 The state 1199091response curves

the convergence rates of the states of two systems are almostthe same though the state and control amplitudes of T-S fuzzybilinear system (FBS) are smaller thanT-S fuzzy linear system(FLS) Thus the proposed method has some advantages ofperformance over the existing approach [16]

Example 2 Consider the following parametric uncertainmultiple inputs bilinear fuzzy system

Rule 1 IF 1199091 (

119905) is 1198711

THEN (119905) = 1198601119909 (119905) + 119861

1119906 (119905) + 119862

1119906 (119905) 119909 (119905)

+ 119876 (Δ1198861119909 (119905) + Δ119887

1119906 (119905)

+ Δ1198621119906 (119905) 119909 (119905))

Rule 2 IF 1199092 (

119905) is 1198712

THEN (119905) = 1198602119909 (119905) + 119861

2119906 (119905) + 119862

2119906 (119905) 119909 (119905)

+ 119876 (Δ1198862119909 (119905) + Δ119887

2119906 (119905)

+ Δ1198622119906 (119905) 119909 (119905))

(55)

Mathematical Problems in Engineering 9

0 1 2 3 4 5 6 7 8 9 10Time (s)

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

x2

T-S FBST-S FLS

Figure 4 The state 1199092response curves

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

12

Time (s)

Con

trol u

minus4

minus2

T-S FBST-S FLS

Figure 5 The control curves

where 119909(119905) = (1199091(119905) 1199092(119905))119879 1198601= 1198602= [0 1

minus1 minus1] 11986111

= [0

1]

11986112

= [0

minus1] 11986121

= 11986122

= [0

1] 11986211

= [0 0

minus1 1] 11986212

= 11986221

=

11986222

= [0 0

minus1 minus1] 120572 = 03 120588

1= 005 120588

2= 004 119865

11=

[minus12 minus18] 11986512

= [minus08 minus09] 11986521

= [minus05 minus12] and11986522

= [minus1 minus1] using LMI technique to solve (16)-(17) wecan get a feasible solution as

119875 = [

168099 85379

85379 180100] 120582min (119875) = 88510 (56)

Apply the controllers (32) (13) and (40) and the parametersupdated law (41) to system (55) The design parameters arechosen as 120578 = 2 120574 = 2 lt radic2120572120582min(119875) The initial conditions

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Time (s)

Para

met

er c

T-S FBST-S FLS

Figure 6 The curves of adaptive updated parameters

0 5 10 15 20 25 30Time (s)

minus05

0

05

1

15

2

Stat

esx1

T-S FBST-S FLS

Figure 7 Responses of system state 1199091(T-S FBS solid line T-S FLS

dotted line)

are 119909(0) = (2 minus08)119879 119888(0) = 0 The simulation results are

shown in Figures 7 8 9 10 and 11Through the comparison between T-S fuzzy linear model

and bilinear one we can see that the settling time of thesystems is almost the same under the same initial conditionsalthough responses of T-S fuzzy bilinear system (FBS) stateamplitudes are smaller than T-S fuzzy linear system (FLS)and the demand of the control input of the system is lowThus the proposedmethod has better dynamic performancesthan the existing ones based on T-S fuzzy linear model

10 Mathematical Problems in Engineering

0 5 10 15 20 25 30Time (s)

minus1

minus08

minus06

minus04

minus02

0

02

04

Stat

esx2

T-S FBST-S FLS

Figure 8 Responses of system state 1199092(T-S FBS solid line T-S FLS

dotted line)

0 5 10 15 20 25 30Time (s)

minus15

minus1

minus05

0

05

1

15

T-S FBST-S FLS

Con

trol i

nput

u1

Figure 9 Control input1199061(T-S FBS solid line T-S FLS dotted line)

5 Conclusion

This paper proposes a new modelling method based onthe multiple inputs T-S fuzzy bilinear model which is usedto approximate nonlinear system the parallel distributedcompensation (PDC) method is utilized to design the fuzzycontroller without considering the error caused by fuzzymodellingThe sufficient conditionswith respect to decay rate120572 are derived by linear matrix inequalities (LMIs) The errorcaused by fuzzy modelling is considered and the method ofadaptive control is used to reduce the effect of the modelling

0 5 10 15 20 25 30Time (s)

minus08

minus06

minus04

minus02

0

02

04

06

Con

trol i

nput

u2

T-S FBST-S FLS

Figure 10 Control input 1199062(T-S FBS solid line T-S FLS dotted

line)

0 5 10 15 20 25 30Time (s)

0

01

02

03

04

05

06

07

08

09

Adap

tive p

aram

eter

c

T-S FBST-S FLS

Figure 11 Adaptive parameter 119888 (T-S FBS solid line T-S FLS dottedline)

error By Lyapunov stability criterion the resulting closed-loop system is proved to be asymptotically stable Finally twoillustrative examples are provided to show that the approachbased T-S fuzzy bilinear systems have some advantages ofperformance over the existing methods based on T-S fuzzylinear system The future research work is to extend theapproach to general system such as discrete-time systemsstochastic systems and time-delay systems

Mathematical Problems in Engineering 11

Conflict of Interests

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

Acknowledgment

This project was supported by the Soft Science Foundation ofShanxi Province (2011041033-3)

References

[1] K Tanaka and H O Wang Fuzzy Control Systems Design andAnalysis A Linear Matrix Inequality Approach Wiley-Intersci-ence 2001

[2] G Feng ldquoA survey on analysis and design of model-based fuzzycontrol systemsrdquo IEEE Transactions on Fuzzy Systems vol 14no 5 pp 676ndash697 2006

[3] K Tanaka T Ikeda and H O Wang ldquoFuzzy regulators andfuzzy observers relaxed stability conditions and LMI-baseddesignsrdquo IEEE Transactions on Fuzzy Systems vol 6 no 2 pp250ndash265 1998

[4] E Kim and H Lee ldquoNew approaches to relaxed quadraticstability condition of fuzzy control systemsrdquo IEEE Transactionson Fuzzy Systems vol 8 no 5 pp 523ndash534 2000

[5] K Tanaka andM Sugeno ldquoStability analysis and design of fuzzycontrol systemsrdquo Fuzzy Sets and Systems vol 45 no 2 pp 135ndash156 1992

[6] H-N Wu and H-X Li ldquoNew approach to delay-dependentstability analysis and stabilization for continuous-time fuzzysystems with time-varying delayrdquo IEEE Transactions on FuzzySystems vol 15 no 3 pp 482ndash493 2007

[7] F Zheng Q-G Wang and T H Lee ldquoAdaptive and robustcontroller design for uncertain nonlinear systems via fuzzymodeling approachrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 34 no 1 pp 166ndash178 2004

[8] C L Hwang ldquoA novel Takagi-Sugeno-based robust adaptivefuzzy sliding-mode controllerrdquo IEEE Transactions on FuzzySystems vol 12 no 5 pp 676ndash687 2004

[9] S Dong Adaptive Fuzzy Control of Nonlinear System Scienceand Technology Publishing House Beijing China 2006

[10] W-Y Wang Y-H Chien Y-G Leu and T-T Lee ldquoAdaptiveT-S fuzzy-neural modeling and control for general MIMOunknown non-affine nonlinear systems using projection updatelawsrdquo Automatica vol 46 no 5 pp 852ndash863 2010

[11] Y-H Chien W-Y Wang Y-G Leu and T-T Lee ldquoRobustadaptive controller design for a class of uncertain nonlinear sys-tems using online T-S fuzzy-neural modeling approachrdquo IEEETransactions on Systems Man and Cybernetics Part B Cyber-netics vol 41 no 2 pp 542ndash552 2011

[12] S P Moustakidis G A Rovithakis and J B Theocharis ldquoAnadaptive neuro-fuzzy tracking control formulti-input nonlineardynamic systemsrdquo Automatica vol 44 no 5 pp 1418ndash14252008

[13] K-Y Lian and H-W Tu ldquoLMI-Based adaptive tracking controlfor parametric strict-feedback systemsrdquo IEEE Transactions onFuzzy Systems vol 16 no 5 pp 1245ndash1258 2008

[14] Z Lendek J Lauber T M Guerra R Babuka and B De Schut-ter ldquoAdaptive observers for TS fuzzy systems with unknownpolynomial inputsrdquo Fuzzy Sets and Systems vol 161 no 15 pp2043ndash2065 2010

[15] C-H Hyun C-W Park and S Kim ldquoTakagi-Sugeno fuzzymodel based indirect adaptive fuzzy observer and controllerdesignrdquo Information Sciences vol 180 no 11 pp 2314ndash23272010

[16] Y-H Chang W-S Chan and C-W Chang ldquoT-S fuzzy model-based adaptive dynamic surface control for ball and beamsystemrdquo IEEE Transactions on Industrial Electronics vol 60 no6 pp 2251ndash2263 2013

[17] S TWang and J T Fei ldquoRobust adaptive slidingmode control ofMEMS gyroscope using T-S fuzzymodelrdquoNonlinear Dynamicsvol 77 no 1-2 pp 361ndash371 2014

[18] R Qi G Tao C Tan and X Yao ldquoAdaptive control of discrete-time state-space T-S fuzzy systems with general relative degreerdquoFuzzy Sets and Systems vol 217 pp 22ndash40 2013

[19] H B Jiang J J Yu and C G Zhou ldquoStable adaptive fuzzycontrol of nonlinear systems using small-gain theorem and LMIapproachrdquo Journal of ControlTheory andApplications vol 8 no4 pp 527ndash532 2010

[20] R R Mohler Bilinear Control Processes Academic Press NewYork NY USA 1973

[21] D L Elliott Bilinear Systems in Encyclopedia of Electrical Engi-neering Wiley New York NY USA 2001

[22] T-H S Li and S-H Tsai ldquoT-S fuzzy bilinear model and fuzzycontroller design for a class of nonlinear systemsrdquo IEEE Trans-actions on Fuzzy Systems vol 15 no 3 pp 494ndash506 2007

[23] S H Tsai and TH S Li ldquoRobust fuzzy control of a class of fuzzybilinear systems with time-delayrdquo Chaos Solitons and Fractalsvol 39 no 5 pp 2028ndash2040 2009

[24] T-H S Li S-H Tsai J-Z Lee M-Y Hsiao and C-H ChaoldquoRobust 119867

infinfuzzy control for a class of uncertain discrete

fuzzy bilinear systemsrdquo IEEE Transactions on SystemsMan andCybernetics Part B Cybernetics vol 38 no 2 pp 510ndash527 2008

[25] G Zhang J-M Li and Y-W Ge ldquoNonfragile guaranteed costcontrol of discrete-time fuzzy bilinear system with time-delayrdquoJournal of Dynamic Systems Measurement and Control Trans-actions of the ASME vol 136 no 4 Article ID 044502 2014

[26] J R Li J M Li and Z L Xia ldquoObserver-based fuzzy controldesign for discrete-time T-S fuzzy bilinear systemsrdquo Interna-tional Journal of Uncertainty Fuzziness and Knowledge-BasedSystems vol 21 no 3 pp 435ndash454 2013

[27] M S Ali ldquoRobust stability of stochastic fuzzy impulsive recur-rent neural networks with time varying delaysrdquo Iranian Journalof Fuzzy Systems vol 11 no 4 pp 1ndash13 2014

[28] M Syed Ali ldquoRobust stability analysis of Takagi-Sugeno uncer-tain stochastic fuzzy recurrent neural networks with mixedtime-varying delaysrdquo Chinese Physics B vol 20 no 8 ArticleID 080201 2011

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

with respect to decay rate 120572 were derived by linear matrixinequalities (LMIs) and by small-gain theorem The adaptivecompensation term was adopted to reduce the effect of themodelling However the consequence parts of the above T-Sfuzzy models [10ndash19] were linear dynamic model or approxi-mate linear dynamicmodel so thesemethods have inevitabledefects for some nonlinear systems

It is known that bilinear models can describe many phys-ical systems and dynamical processes in engineering fields[20 21]There are twomain advantages of the bilinear systemOne is that it provides a better approximation to a nonlinearsystem than a linear one and another is that many realphysical processes may be appropriately modelled as bilinearsystems when the linear models are inadequate Consideringthe advantages of bilinear systems and T-S fuzzy control thefuzzy control based on the T-S fuzzy model with bilinear ruleconsequence attracted the interest of researchers [22ndash26]TheT-S fuzzy bilinear model may be suitable for some classesof nonlinear plants The robust stabilization for continuous-time fuzzy system with local bilinear model was studied in[22] and then the result was extended to the fuzzy systemwith time-delay only in the state [23] The problem of robuststabilization for discrete-time fuzzy systemwith local bilinearmodel was investigated in [24] Reference [25] focuses on theproblem of nonfragile guaranteed cost control for a class of T-S discrete-time fuzzy bilinear systems Based on the paralleldistributed compensation approach the sufficient conditionswere derived such that the closed-loop system was asymptot-ically stable and the cost function value was no more than acertain upper bound in the presence of the additive controllergain perturbations In [26] an observer-based fuzzy controldesign was given for discrete-time T-S fuzzy bilinear systemsIn [27 28] authors proposed robust stability conditions forstochastic fuzzy impulsive recurrent neural networks withtime-varying delays and uncertain stochastic fuzzy recurrentneural networks with mixed time-varying delays

However when there are differences between T-S fuzzybilinear model and reality systems these results will not beapplied

Considering the differences of the fuzzy model and thereality systems in the paper a stable adaptive fuzzy controlfor complex nonlinear systems is presented based onmultipleinputs T-S fuzzy bilinear system with parameters uncertain-ties In consideration of themodelling error an adaptive fuzzycontrol is proposed to compensate for the issues At first theconcept of the so-called PDC and LMI approach is employedto design the state feedback controller without consideringthe error caused by fuzzymodellingThe sufficient conditionswith respect to decay rate 120572 are derived in the sense ofLyapunov asymptotic stabilityThen the error caused by fuzzymodelling is considered an adaptive compensation term isdesigned to reduce the effect of the modelling error Thecontributions of this paper are as follows (i) the differencesbetween T-S fuzzy bilinear model and the real system areconsidered in themodelling and analysis (ii) a global asymp-totical stable adaptive control scheme is presented for realsystems (iii) a sufficient condition of the closed-loop systemsis given Finally theoretical analysis verifies that the state

converges to zero and all signals of the closed-loop systemsare bounded

2 Problem Statement and Basic Assumptions

Consider the nonlinear system in the following form

119894= 119909119894+1

119894 = 1 119899 minus 1

119899= 119891 (119909) + 119892

119879(119909) 119906

(1)

where 119909 = (1199091 1199092 119909

119899)119879

isin 119877119899 and 119906 isin 119877

119898 are the vectorsof state and control input respectively 119891(119909) is the unknowncontinuous function 119892(119909) is the vector of unknown continu-ous control gain function which satisfies 119892

119895(119909) ge 119892

119895min gt 0119895 = 1 2 119898

Definition 1 (see [19]) System (1) under the input being zerois globally asymptotically stable with decay rate 120572 if thereexists a scalar 120572 gt 0 such that

(119909 (119905)) le minus2120572119881 (119909 (119905)) (2)

where 119881(119909(119905)) = 119909119879(119905)119875119909(119905) is the Lyapunov function candi-

date and 119875 gt 0

Lemma 2 (see [25]) Given two matrices119860 and 119861 with appro-priate dimensions one has 119860119879119861 + 119861

119879119860le 119860119879119860

+ 119861119879119861

In this paper our objective is to design an adaptive fuzzycontroller so that the closed-loop systems are asymptoticallystable that is the states of the closed-loop system converge tozero and all signals of the closed-loop systems are bounded

System (1) can be expressed in terms of the T-S fuzzymodel as followsPlant rule 119894

IF 1199111 (

119905) is 119872119894

1 1199112 (

119905) is 119872119894

2 119911

119892 (119905) is 119872

119894

119892

THEN (119905) = 119860119894119909 (119905) + 119861

119894119906 (119905) + 119862

119894119906 (119905) 119909 (119905) + 119876Δ119891

119894 (119909 119906)

119894 = 1 2 119903

(3)

where 119911(119905) = [1199111(119905) 1199112(119905) 119911

119892(119905)] isin 119877

119892 are known premisevariables that may be functions of the state variables 119872119894

119895is

the fuzzy set 119903 is the number of the rules 119876 = [0 0 1]119879

119862119894119906(119905) = sum

119898

119895=1119862119894119895119906119895 119862119894119895

isin 119877119899times119899 is constant matrices and

Δ119891119894(119909 119906) = (Δ119886

119894119909 + Δ119887

119894119906 + Δ119862

119894119906119909) denotes the model error

of the 119894th bilinear model in the 119894th fuzzy space (also called the119894th fuzzy rule)

The modelling error terms are defined as follows

Δ119886119894= (Δ119886

1198941 Δ1198861198942 Δ119886

119894119899) isin 1198771times119899

Δ119887119894= (Δ119887

1198941 Δ1198871198942 Δ119887

119894119898) isin 1198771times119898

Δ119862119894119906 (119905) =

119898

sum

119895=1

Δ119862119894119895119906119895

Δ119862119894119895

= [Δ1198881198941198951

Δ1198881198941198952

sdot sdot sdot Δ119888119894119895119899

] isin 1198771times119899

(4)

Mathematical Problems in Engineering 3

119860119894isin 119877119899times119899 119861

119894isin 119877119899times119898 and 119862

119894119895are constant matrices which

have of the following forms

119860119894=

[

[

[

[

[

[

[

[

[

0 1 0 sdot sdot sdot 0 0

0 0 1 sdot sdot sdot 0 0

0 0 0 sdot sdot sdot 1 0

0 0 0 sdot sdot sdot 0 1

1198861198941

1198861198942

1198861198943

sdot sdot sdot 119886119894(119899minus1)

119886119894119899

]

]

]

]

]

]

]

]

]

119861119894=

[

[

[

[

[

[

[

[

[

0 0 0 sdot sdot sdot 0 0

0 0 0 sdot sdot sdot 0 0

0 0 0 sdot sdot sdot 0 0

0 0 0 sdot sdot sdot 0 0

1198871198941

1198871198942

1198871198943

sdot sdot sdot 119887119894(119898minus1)

119887119894119898

]

]

]

]

]

]

]

]

]

119862119894119895

=

[

[

[

[

[

[

[

[

[

0 0 0 sdot sdot sdot 0 0

0 0 0 sdot sdot sdot 0 0

0 0 0 sdot sdot sdot 0 0

0 0 0 sdot sdot sdot 0 0

1198881198941198951

1198881198941198952

1198881198941198953

sdot sdot sdot 119888119894119895(119899minus1)

119888119894119895119899

]

]

]

]

]

]

]

]

]

(5)

By using the fuzzy inference method with a singletonfuzzification product inference and centre average defuzzi-fication the overall fuzzy model is of the following form

(119905) =

119903

sum

119894=1

ℎ119894 (119911 (119905)) [119860 119894

119909 (119905) + 119861119894119906 (119905) + 119862

119894119906 (119905) 119909 (119905)

+ 119876Δ119891119894 (119909 119906)]

(6)

where

ℎ119894 (119911 (119905)) =

119908119894 (119911 (119905))

sum119903

119895=1119908119895 (

119911 (119905))

119908119894 (119911 (119905)) =

119892

prod

119895=1

119872119894

119895(119911119895 (

119905))

(7)

We assume 119908119894(119911(119905)) ge 0 sum119903

119894=1119908119894(119911(119905)) gt 0 for all 119905

Therefore we have

ℎ119894 (119911 (119905)) ge 0

119903

sum

119894=1

ℎ119894 (119911 (119905)) = 1 119894 = 1 2 119903 (8)

By comparison with (1) and (6) it is easy to see that

119891 (119909) =

119903

sum

119894=1

ℎ119894 (119911 (119905)) (119886119894

+ Δ119886119894) 119909 (119905)

119892119895 (

119909) =

119903

sum

119894=1

ℎ119894 (119911 (119905)) (119887119894119895

+ Δ119887119894119895

+ 119888119894119895119909 (119905) + Δ119862

119894119895119909 (119905))

(9)

where 119886119894= (1198861198941 1198861198942 119886

119894119899) isin 119877

1times119899 119887119894= (1198871198941 1198871198942 119887

119894119898) isin

1198771times119898 and 119888

119894119895= (1198881198941198951

1198881198941198952

119888119894119895119899

)

Remark 3 From now on unless confusion arises argumentssuch as 119911(119905) in ℎ

119894(119911(119905)) will be omitted just for notational

convenience

3 Control Design and Stability Analysis

System (6) can be represented by following the T-S fuzzymodel without considering the modelling error that isΔ119891119894(119909 119906) equiv 0 119894 = 1 2 119903 Consider

Plant rule 119894

IF 1199111 (

119905) is 119872119894

1 1199112 (

119905) is 119872119894

2 119911

119892 (119905) is 119872

119894

119892

THEN (119905) = 119860119894119909 (119905) + 119861

119894119906 (119905) + 119862

119894119906 (119905) 119909 (119905)

119894 = 1 2 119903

(10)

By using the fuzzy inference method with a singletonfuzzification product inference and centre average defuzzi-fication the overall fuzzy model is of the following form

(119905) =

119903

sum

119894=1

ℎ119894[119860119894119909 (119905) + 119861

119894119906 (119905) + 119862

119894119906 (119905) 119909 (119905)] (11)

Based on the idea of PDC the 119895th state-feedback con-troller is designed as follows

Plant rule 119894

IF 1199111 (

119905) is 119872119894

1 1199112 (

119905) is 119872119894

2 119911

119892 (119905) is 119872

119894

119892

THEN 119906119902119895

=

120588119895119865119894119895119909 (119905)

radic1 + 119909119879(119905) 119865119879

119894119895119865119894119895119909 (119905)

119894 = 1 2 119903

(12)

where 119865119894119895

isin 1198771times119899 is a vector to be determined and 120588

119895gt 0 is a

scalar to be assignedThe overall fuzzy control law can be represented by

119906119902119895

=

119903

sum

119894=1

ℎ119894

120588119895119865119894119895119909 (119905)

radic1 + 119909119879(119905) 119865119879

119894119895119865119894119895119909 (119905)

=

119903

sum

119894=1

ℎ119894120588119895sin 120579119894119895

=

119903

sum

119894=1

ℎ119894120588119895119865119894119895cos 120579119894119895119909 (119905)

(13)

where

sin 120579119894119895

=

119865119894119895119909 (119905)

radic1 + 119909119879(119905) 119865119879

119894119895119865119894119895119909 (119905)

cos 120579119894119895

=

1

radic1 + 119909119879(119905) 119865119879

119894119895119865119894119895119909 (119905)

120579119894119895

isin [minus

120587

2

120587

2

]

1 le 119894 le 119903 1 le 119895 le 119898

(14)

4 Mathematical Problems in Engineering

Substituting (13) into (11) one can get the closed-loopsystem

(119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895(119860119894+

119898

sum

119896=1

119861119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

119862119894119896120588119896sin 120579119895119896

)119909 (119905)

(15)

where 119861119894119896denotes the 119896th column of the 119861

119894

Theorem 4 Given positive scalars 120588119896(1 le 119896 le 119898) if there

exist a symmetric positive definite matrix119880 and some constantmatrices 119882

119894119895 such that LMIs (16) and (17) hold

[

[

[

119860119894119880 + 119880119860

119879

119894+ 120588 + 2120572119880 lowast lowast

119861119894119882119894

minus119868

119862119894119880 minus119868

]

]

]

lt 0 1 le 119894 le 119903 (16)

[

[

[

[

[

[

[

[

119872119894

lowast lowast lowast lowast

119861119894119882119895

minus119868

119861119895119882119894

minus119868

119862119894119880 minus119868

119862119895119880 minus119868

]

]

]

]

]

]

]

]

lt 0 1 le 119894 lt 119895 le 119903 (17)

where 120588 = sum119898

119896=11205882

119896 119872119894= 119860119894119880+119880119860

119879

119894+119860119895119880+119880119860

119879

119895+2120588+2120572119880

119861119894119882119894=

[

[

[

11986111989411198821198941

119861119894119898

119882119894119898

]

]

]

119862119894=

[

[

[

1198621198941

119862119894119898

]

]

]

119868 =[

[

119868

d119868

]

]

(18)

Then the FBS (15) is globally asymptotically stable with decayrate 120572 via the fuzzy feedback controller (13) and the gains canbe determined by 119865

119894119895= 119882119894119895119880minus1

Proof Consider the Lyapunov function candidate as follows

1198811 (

119909 (119905)) = 119909119879(119905) 119875119909 (119905) (19)

where 119875 = 119880minus1

Applying Schur complement lemma inequality (16) canbe written as follows

119880119860119879

119894+ 119860119894119880 +

119898

sum

119896=1

1205882

119896+

119898

sum

119896=1

(119861119894119896119882119894119896)119879119861119894119896119882119894119896

+

119898

sum

119896=1

(119862119894119896119880)119879(119862119894119896119880) + 2120572119880 lt 0

(20)

Premultiplying and postmultiplying (20) by 119875 respectivelywe have

119860119879

119894119875 + 119875119860

119894+

119898

sum

119896=1

1205882

119896119875119875 +

119898

sum

119896=1

(119861119894119896119865119894119896)119879119861119894119896119865119894119896

+

119898

sum

119896=1

119862119879

119894119896119862119894119896

+ 2120572119875 lt 0

(21)

Applying a similar procedure to inequality (17) we can obtain

119860119879

119894119875 + 119875119860

119894+ 119860119879

119895119875 + 119875119860

119895+ 2

119898

sum

119896=1

1205882

119896119875119875

+

119898

sum

119896=1

(119861119894119896119865119895119896

)

119879

(119861119894119896119865119895119896

) +

119898

sum

119896=1

119862119879

119894119896119862119894119896

+

119898

sum

119896=1

(119861119895119896

119865119894119896)

119879

(119861119895119896

119865119894119896) +

119898

sum

119896=1

119862119879

119895119896119862119895119896

+ 2120572119875 lt 0

(22)

The time derivative of 1198811is

1 (

119909 (119905)) = 119879119875119909 + 119909

119879119875 (23)

By substituting (15) into (23) we can get

1 (

119909 (119905)) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895119909119879

[119860119894+

119898

sum

119896=1

119861119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

119862119894119896120588119896sin 120579119895119896

]

119879

119875

+ 119875[119860119894+

119898

sum

119896=1

119861119894119896120588119896

times 119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

119862119894119896120588119896sin 120579119895119896

]

119909

=

119903

sum

119894=1

ℎ2

119894119909119879Λ119894119894119909 +

119903

sum

119894lt119895

ℎ119894ℎ119895119909119879Λ119894119895119909

(24)

Mathematical Problems in Engineering 5

where

Λ119894119894

= 119860119879

119894119875 + 119875119860

119894+

119898

sum

119896=1

120588119896sin 120579119894119896

(119862119879

119894119896119875 + 119875119862

119894119896)

+

119898

sum

119896=1

120588119896cos 120579119894119896

[(119861119894119896119865119894119896)119879119875 + 119875119861

119894119896119865119894119896]

Λ119894119895

= 119860119879

119894119875 + 119875119860

119894+

119898

sum

119896=1

120588119896sin 120579119894119896

(119862119879

119894119896119875 + 119875119862

119894119896)

+

119898

sum

119896=1

120588119896cos 120579119894119896

[(119861119894119896119865119895119896

)

119879

119875 + 119875119861119894119896119865119895119896

]

+ 119860119879

119895119875 + 119875119860

119895+

119898

sum

119896=1

120588119896sin 120579119895119896

(119862119879

119895119896119875 + 119875119862

119895119896)

+

119898

sum

119896=1

120588119896cos 120579119895119896

[(119861119895119896

119865119894119896)

119879

119875 + 119875119861119895119896

119865119894119896]

(25)

First by premultiplying and postmultiplyingΛ119894119894by119880 we can

obtain

119880Λ119894119894119880 = 119880119860

119879

119894+ 119860119894119880 +

119898

sum

119896=1

120588119896sin 120579119894119896

(119880119862119879

119894119896+ 119862119894119896119880)

+

119898

sum

119896=1

120588119896cos 120579119894119896

[119880 (119861119894119896119865119894119896)119879+ 119861119894119896119865119894119896119880]

(26)

According to Lemma 2 we can get the following

[120588119896119861119894119896119865119894119896119880 + 120588119896119880 (119861119894119896119865119894119896)119879] cos 120579

119894119896

le 1205882

119896cos2120579119894119896

+ (119861119894119896119882119894119896)119879119861119894119896119882119894119896

119880119862119879

119894119896120588119896sin 120579119894119896

+ 119862119894119896119880120588119896sin 120579119894119896

le 1205882

119896sin2120579119894119896

+ (119862119894119896119880)119879(119862119894119896119880)

(27)

From (26) and (27) we can obtain

119880Λ119894119894119880 le 119880119860

119879

119894+ 119860119894119880 +

119898

sum

119896=1

1205882

119896+

119898

sum

119896=1

(119862119894119896119880)119879119862119894119896119880

+

119898

sum

119896=1

(119861119894119896119882119894119896)119879119861119894119896119882119894119896

(28)

Applying similar procedures (26)ndash(28) to Λ119894119895 we can

obtain

119880Λ119894119895119880 le 119880119860

119879

119894+ 119860119894119880 + 119880119860

119879

119895+ 119860119895119880 + 2

119898

sum

119896=1

1205882

119896

+

119898

sum

119896=1

(119861119894119896119882119895119896

)

119879

(119861119894119896119882119895119896

) +

119898

sum

119896=1

(119862119894119896119880)119879119862119894119896119880

+

119898

sum

119896=1

(119861119895119896

119882119894119896)

119879

(119861119895119896

119882119894119896) +

119898

sum

119896=1

(119862119895119896

119880)

119879

119862119895119896

119880

(29)

Substituting (28) and (29) into (24) we obtain

1 (

119909 (119905)) le

119903

sum

119894=1

ℎ2

119894119909119879119875[119880119860

119879

119894+ 119860119894119880 +

119898

sum

119896=1

1205882

119896

+

119898

sum

119896=1

(119861119894119896119882119894119896)119879119861119894119896119882119894119896

+

119898

sum

119896=1

(119862119894119896119880)119879(119862119894119896119880)]119875119909

+

119903

sum

119894lt119895

ℎ119894ℎ119895119909119879119875[119880119860

119879

119894+ 119860119894119880 + 119880119860

119879

119895

+ 119860119895119880 + 2

119898

sum

119896=1

1205882

119896

+

119898

sum

119896=1

(119861119894119896119882119895119896

)

119879

(119861119894119896119882119895119896

)

+

119898

sum

119896=1

(119862119894119896119880)119879119862119894119896119880

+

119898

sum

119896=1

(119861119895119896

119882119894119896)

119879

(119861119895119896

119882119894119896)

+

119898

sum

119896=1

(119862119895119896

119880)

119879

119862119895119896

119880]119875119909

(30)By substituting (21) and (22) into (30) we can obtain

1 (

119909 (119905)) le minus2120572119875119909 = minus21205721198811 (

119909 (119905)) (31)Then by Definition 1 the closed-loop fuzzy system (15)

is globally asymptotically stable with decay rate 120572 Thiscompletes the proof of Theorem 4

Next the modelling error in (6) is considered and anadaptive compensation term is adopted to reduce the effectsof the modelling error

Adopt the fuzzy controller in the following form119906 = 119906119902119895

+ 119906119904119895

119895 = 1 2 119898 (32)where compensator 119906

119904119895will be designed later

Substituting (32) into (6) yields

(119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895[(119860119894+

119898

sum

119896=1

119861119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

119862119894119896120588119896sin 120579119895119896

)119909 (119905)

+ 119876(Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588119896sin 120579119895119896

)119909 (119905)

6 Mathematical Problems in Engineering

+

119898

sum

119896=1

(119861119894119896

+ 119876Δ119887119894119896

+ 119862119894119896119909 (119905)

+ 119876Δ119862119894119896119909 (119905)) 119906119904119895

]

(33)

Suppose that there exists an unknown constant 120582 such that

120582 ge

119903

sum

119894=1

119903

sum

119895=1

1003817100381710038171003817100381710038171003817100381710038171003817

Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588119896sin 120579119895119896

1003817100381710038171003817100381710038171003817100381710038171003817

(34)

Then from 0 le ℎ119894(119911(119905)) le 1 and

10038171003817100381710038171003817100381710038171003817100381710038171003817

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895(Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588 sin 120579

119895119896)

10038171003817100381710038171003817100381710038171003817100381710038171003817

le

119903

sum

119894=1

119903

sum

119895=1

1003817100381710038171003817100381710038171003817100381710038171003817

Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588 sin 120579

119895119896

1003817100381710038171003817100381710038171003817100381710038171003817

(35)

we have

10038171003817100381710038171003817100381710038171003817100381710038171003817

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895(Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588 sin 120579

119895119896)

1003817100381710038171003817100381710038171003817100381710038171003817

le 120582

(36)

It is easy to see that we can choose a function vector119867120582such

that

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895(Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588119896sin 120579119895119896

) = 120582119867120582

(37)

and 119867120582 le 1

Remark 5 Here assumption (34) is reasonable in many realsystems due to its boundedness such as chaotic system [19]for example Example 1 in the paper satisfies the assumptionOn the other hand the uncertain terms of the consideredsystems in the existing literature [2 18 22ndash24] satisfy thecondition of (34)

Denote

120596 (119905) = 119867120582119909 (119905) (38)

Substituting (37) and (38) into (33) we can obtain thefollowing feedback system

(119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895[(119860119894+

119898

sum

119896=1

119861119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

119862119894119896120588119896sin 120579119895119896

)119909 (119905)

+

119898

sum

119896=1

(119861119894119896

+ 119876Δ119887119894119896

+ 119862119894119896119909 (119905)

+ 119876Δ119862119894119896119909 (119905)) 119906119904119895

] + 119876120582120596 (119905)

(39)

Choose the adaptive compensator as follows

119906119904119895

= minus

1

2119898119892119895min1205742119876119879119875119888119909 (119905) 119895 = 1 2 119898 (40)

where 119888 = 1205822 119888 is the parameter estimation of 119888 and 120574 gt 0 is

a gain constantChoose the adaptive law as follows

=

1

21205781205742119909119879(119905) 119875119876119876

119879119875119909 (119905) (41)

where 120578 gt 0 is a gain constant which determines the rate ofadaptation

Substituting (40) into (39) yields

(119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895[(119860119894+

119898

sum

119896=1

119861119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

119862119894119896120588119896sin 120579119895119896

)119909 (119905)

+

119898

sum

119896=1

(119861119894119896

+ 119876Δ119887119894119896

+ 119862119894119896119909 (119905)

+ 119876Δ119862119894119896119909 (119905))

times (minus

119876119879119875119888119909 (119905)

2119898119892119895min1205742)] + 119876120582120596 (119905)

(42)

Theorem 6 Consider the uncertain nonlinear system (1) withcontrol law defined by (32) (13) and (40) and the parameterupdated by the adaptive law (41) If there exist a symmetricpositive definite matrix 119875 and some matrices 119865

119894119895(1 le 119894 119895 le 119903)

satisfying the LMIs (16) and (17) and the design parameter ischosen as

0 lt 120574 lt radic2120572120582min (119875) (43)

then the closed-loop system (42) is asymptotically stable and allsignals of the closed-loop system (42) are bounded

Mathematical Problems in Engineering 7

Proof Consider the Lyapunov function candidate

119881 = 119909119879(119905) 119875119909 (119905) + 120578 (119888 minus 119888)

2 (44)

where 120578 gt 0Let

119866119894119895

= 119860119894+

119898

sum

119896=1

119861119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

119862119894119896120588119896sin 120579119895119896

(45)

The time derivative of 119881 is

= 119879119875119909 + 119909

119879119875 + 2120578 (119888 minus 119888)

(46)

Substituting (39) into (46) we can obtain

=

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895[119909119879(119905)

times 119866119894119895

119879119875 + 119875119866

119894119895+ 2119875119876

times (Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588119896sin 120579119895119896

)119909 (119905)

+ 2119909119879(119905) 119875

119898

sum

119896=1

(119861119894119896

+ 119876Δ119887119894119896

+ 119862119894119896119909 (119905)

+ 119876Δ119862119894119896119909 (119905)) 119906119904119896

]

+ 2120578 (119888 minus 119888)

=

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895119909119879(119905) (119866119894119895

119879119875 + 119875119866

119894119895) 119909 (119905)

+ 2119909119879(119905) 119875119876120582120596 (119905)

+ 2119909119879(119905) 119875119876

119898

sum

119895=1

119892119895 (

119909) 119906119904119895+ 2120578 (119888 minus 119888)

(47)

From the proof of Theorem 4 we get

le minus2120572119909119879(119905) 119875119909 (119905) + 2119909

119879(119905) 119875119876120582120596 (119905)

+ 2119909119879(119905) 119875119876

119898

sum

119895=1

119892119895 (

119909) 119906119904119895+ 2120578 (119888 minus 119888)

(48)

It is easy to see that

2119909119879119875119876120582120596 minus 120574

21205962+ 12057421205962

= minus120574210038171003817100381710038171003817100381710038171003817

120596 minus

1

1205742119909119879119875119876120582

10038171003817100381710038171003817100381710038171003817

2

+

1

12057421199091198791198751198761205822119876119879119875119909 + 120574

21205962

le

1205822

1205742119909119879119875119876119876119879119875119909 + 120574

21205962

(49)

Substituting (49) into (48) yields

le minus2120572119909119879(119905) 119875119909 (119905) +

1205822

1205742119909119879119875119876119876119879119875119909 + 120574

21205962

+ 2119909119879(119905) 119875119876

119898

sum

119895=1

119892119895 (

119909) 119906119904119895+ 2120578 (119888 minus 119888)

(50)

Substituting (40) and (41) into (50) we obtain

le minus2120572119909119879(119905) 119875119909 (119905) + (minus119888 + 119888)

1

1205742119909119879119875119876119876119879119875119909

+ 2120578 (119888 minus 119888)

+ 12057421205962

= minus2120572119909119879(119905) 119875119909 (119905) + 120574

21205962

le minus2120572120582min (119875) 119909 (119905)2+ 12057421205962

(51)

where 1205962= 120596119879120596 = 119909

119879119867120582

119879119867120582119909 le 119867

120582

119879119867120582 sdot 119909

2le 119909

2By choosing 0 lt 120574 lt radic2120572120582min(119875) we can get lt 0

then we have that the states119909(119905) rarr 0 as 119905 approaches infinityvia LaSalle invariance principle and 119881(119905) is bounded From(44) we can obtain that states 119909 and 119888 are bounded thereforethe boundedness of 119906

119904119895is ensured from (40) Similarly from

(13) we can obtain that 119906119902119895

is boundedThen it can be provedthat (1) the closed-loop system (42) is asymptotically stableand (2) all signals of the closed-loop system (42) are bounded

4 Simulations

In this section we will give two examples to show theefficiency of the proposed approach The first example isan unknown chaotic system and the second example is aparameter uncertain T-S fuzzy bilinear system with multipleinputs

Example 1 Consider the following chaotic system with con-trol input

1= 1199092

2= minus01119909

2minus 1199091

3+ 12 cos 119905 + 119906

(52)

When 119906(119905) = 0 and the initial states are chosen as 119909(0) =

(2 2)119879 the states phase portrait of system (52) is shown in

Figure 1

8 Mathematical Problems in Engineering

0 1 2 3 4

0

2

4

6

8

10

minus10

minus8

minus6

minus4

minus4 minus3

minus2

minus2 minus1

x2

x1

Figure 1 The phase portrait of the chaotic system

System (52) can be modelled as the following T-S fuzzybilinear model

Rule 1 IF 1199091 (

119905) is about 0

THEN (119905) = (1198601+ Δ1198601) 119909 (119905) + (119861

1+ Δ1198611) 119906 (119905)

+ (1198621+ Δ1198621) 119909 (119905) 119906 (119905)

Rule 2 IF 1199091 (

119905) is about plusmn 2

THEN (119905) = (1198602+ Δ1198602) 119909 (119905) + (119861

2+ Δ1198612) 119906 (119905)

+ (1198622+ Δ1198622) 119909 (119905) 119906 (119905)

Rule 3 IF 1199091 (

119905) is about plusmn 4

THEN (119905) = (1198603+ Δ1198603) 119909 (119905) + (119861

3+ Δ1198613) 119906 (119905)

+ (1198623+ Δ1198623) 119909 (119905) 119906 (119905)

(53)

where 119909(119905) = (1199091(119905) 1199092(119905))119879 1198601= 1198602= 1198603= [0 1

minus1 minus1] 1198611=

1198612= 1198613= [0

minus1] and 119862

1= 1198622= 1198623= [0 1

1 1] Choose 120572 = 03

120588 = 009 1198651= [0 minus1] 119865

2= [minus1 minus1] and 119865

3= [minus1 minus1] By

solving LMIs (16)-(17) one can obtain

119875 = [

93637 46993

46993 106246] 120582min (119875) = 52528 (54)

Utilize the controllers (32) (13) and (40) and the parameterupdated law (41) to control system (52) The design parame-ters are chosen as 120578 = 2 120574 = 2 lt radic120582min(119875) the initial condi-tions are chosen as 119909(0) = (2 minus2)

119879 119888(0) = 0 and the relation-ship functions are selected as shown in Figure 2 The simula-tion results are shown in Figures 3 4 5 and 6 In Figures 3ndash6the curves of states control input and adaptive updatedparameter for the T-S fuzzy bilinear system are drawn by solidlines respectively while the curves of states control inputand adaptive updated parameter for T-S fuzzy linear systemare depicted by dotted lines respectively By comparison

Rule 3 Rule 2 Rule 1 Rule 2 Rule 3

minus4 minus3 minus2 minus1 0 1 2 3 4

x1

M(x

1)

0

02

04

06

08

1

Figure 2 The relationship functions

0 1 2 3 4 5 6 7 8 9 10Time (s)

minus05

0

05

1

15

2

x1

T-S FBST-S FLS

Figure 3 The state 1199091response curves

the convergence rates of the states of two systems are almostthe same though the state and control amplitudes of T-S fuzzybilinear system (FBS) are smaller thanT-S fuzzy linear system(FLS) Thus the proposed method has some advantages ofperformance over the existing approach [16]

Example 2 Consider the following parametric uncertainmultiple inputs bilinear fuzzy system

Rule 1 IF 1199091 (

119905) is 1198711

THEN (119905) = 1198601119909 (119905) + 119861

1119906 (119905) + 119862

1119906 (119905) 119909 (119905)

+ 119876 (Δ1198861119909 (119905) + Δ119887

1119906 (119905)

+ Δ1198621119906 (119905) 119909 (119905))

Rule 2 IF 1199092 (

119905) is 1198712

THEN (119905) = 1198602119909 (119905) + 119861

2119906 (119905) + 119862

2119906 (119905) 119909 (119905)

+ 119876 (Δ1198862119909 (119905) + Δ119887

2119906 (119905)

+ Δ1198622119906 (119905) 119909 (119905))

(55)

Mathematical Problems in Engineering 9

0 1 2 3 4 5 6 7 8 9 10Time (s)

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

x2

T-S FBST-S FLS

Figure 4 The state 1199092response curves

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

12

Time (s)

Con

trol u

minus4

minus2

T-S FBST-S FLS

Figure 5 The control curves

where 119909(119905) = (1199091(119905) 1199092(119905))119879 1198601= 1198602= [0 1

minus1 minus1] 11986111

= [0

1]

11986112

= [0

minus1] 11986121

= 11986122

= [0

1] 11986211

= [0 0

minus1 1] 11986212

= 11986221

=

11986222

= [0 0

minus1 minus1] 120572 = 03 120588

1= 005 120588

2= 004 119865

11=

[minus12 minus18] 11986512

= [minus08 minus09] 11986521

= [minus05 minus12] and11986522

= [minus1 minus1] using LMI technique to solve (16)-(17) wecan get a feasible solution as

119875 = [

168099 85379

85379 180100] 120582min (119875) = 88510 (56)

Apply the controllers (32) (13) and (40) and the parametersupdated law (41) to system (55) The design parameters arechosen as 120578 = 2 120574 = 2 lt radic2120572120582min(119875) The initial conditions

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Time (s)

Para

met

er c

T-S FBST-S FLS

Figure 6 The curves of adaptive updated parameters

0 5 10 15 20 25 30Time (s)

minus05

0

05

1

15

2

Stat

esx1

T-S FBST-S FLS

Figure 7 Responses of system state 1199091(T-S FBS solid line T-S FLS

dotted line)

are 119909(0) = (2 minus08)119879 119888(0) = 0 The simulation results are

shown in Figures 7 8 9 10 and 11Through the comparison between T-S fuzzy linear model

and bilinear one we can see that the settling time of thesystems is almost the same under the same initial conditionsalthough responses of T-S fuzzy bilinear system (FBS) stateamplitudes are smaller than T-S fuzzy linear system (FLS)and the demand of the control input of the system is lowThus the proposedmethod has better dynamic performancesthan the existing ones based on T-S fuzzy linear model

10 Mathematical Problems in Engineering

0 5 10 15 20 25 30Time (s)

minus1

minus08

minus06

minus04

minus02

0

02

04

Stat

esx2

T-S FBST-S FLS

Figure 8 Responses of system state 1199092(T-S FBS solid line T-S FLS

dotted line)

0 5 10 15 20 25 30Time (s)

minus15

minus1

minus05

0

05

1

15

T-S FBST-S FLS

Con

trol i

nput

u1

Figure 9 Control input1199061(T-S FBS solid line T-S FLS dotted line)

5 Conclusion

This paper proposes a new modelling method based onthe multiple inputs T-S fuzzy bilinear model which is usedto approximate nonlinear system the parallel distributedcompensation (PDC) method is utilized to design the fuzzycontroller without considering the error caused by fuzzymodellingThe sufficient conditionswith respect to decay rate120572 are derived by linear matrix inequalities (LMIs) The errorcaused by fuzzy modelling is considered and the method ofadaptive control is used to reduce the effect of the modelling

0 5 10 15 20 25 30Time (s)

minus08

minus06

minus04

minus02

0

02

04

06

Con

trol i

nput

u2

T-S FBST-S FLS

Figure 10 Control input 1199062(T-S FBS solid line T-S FLS dotted

line)

0 5 10 15 20 25 30Time (s)

0

01

02

03

04

05

06

07

08

09

Adap

tive p

aram

eter

c

T-S FBST-S FLS

Figure 11 Adaptive parameter 119888 (T-S FBS solid line T-S FLS dottedline)

error By Lyapunov stability criterion the resulting closed-loop system is proved to be asymptotically stable Finally twoillustrative examples are provided to show that the approachbased T-S fuzzy bilinear systems have some advantages ofperformance over the existing methods based on T-S fuzzylinear system The future research work is to extend theapproach to general system such as discrete-time systemsstochastic systems and time-delay systems

Mathematical Problems in Engineering 11

Conflict of Interests

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

Acknowledgment

This project was supported by the Soft Science Foundation ofShanxi Province (2011041033-3)

References

[1] K Tanaka and H O Wang Fuzzy Control Systems Design andAnalysis A Linear Matrix Inequality Approach Wiley-Intersci-ence 2001

[2] G Feng ldquoA survey on analysis and design of model-based fuzzycontrol systemsrdquo IEEE Transactions on Fuzzy Systems vol 14no 5 pp 676ndash697 2006

[3] K Tanaka T Ikeda and H O Wang ldquoFuzzy regulators andfuzzy observers relaxed stability conditions and LMI-baseddesignsrdquo IEEE Transactions on Fuzzy Systems vol 6 no 2 pp250ndash265 1998

[4] E Kim and H Lee ldquoNew approaches to relaxed quadraticstability condition of fuzzy control systemsrdquo IEEE Transactionson Fuzzy Systems vol 8 no 5 pp 523ndash534 2000

[5] K Tanaka andM Sugeno ldquoStability analysis and design of fuzzycontrol systemsrdquo Fuzzy Sets and Systems vol 45 no 2 pp 135ndash156 1992

[6] H-N Wu and H-X Li ldquoNew approach to delay-dependentstability analysis and stabilization for continuous-time fuzzysystems with time-varying delayrdquo IEEE Transactions on FuzzySystems vol 15 no 3 pp 482ndash493 2007

[7] F Zheng Q-G Wang and T H Lee ldquoAdaptive and robustcontroller design for uncertain nonlinear systems via fuzzymodeling approachrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 34 no 1 pp 166ndash178 2004

[8] C L Hwang ldquoA novel Takagi-Sugeno-based robust adaptivefuzzy sliding-mode controllerrdquo IEEE Transactions on FuzzySystems vol 12 no 5 pp 676ndash687 2004

[9] S Dong Adaptive Fuzzy Control of Nonlinear System Scienceand Technology Publishing House Beijing China 2006

[10] W-Y Wang Y-H Chien Y-G Leu and T-T Lee ldquoAdaptiveT-S fuzzy-neural modeling and control for general MIMOunknown non-affine nonlinear systems using projection updatelawsrdquo Automatica vol 46 no 5 pp 852ndash863 2010

[11] Y-H Chien W-Y Wang Y-G Leu and T-T Lee ldquoRobustadaptive controller design for a class of uncertain nonlinear sys-tems using online T-S fuzzy-neural modeling approachrdquo IEEETransactions on Systems Man and Cybernetics Part B Cyber-netics vol 41 no 2 pp 542ndash552 2011

[12] S P Moustakidis G A Rovithakis and J B Theocharis ldquoAnadaptive neuro-fuzzy tracking control formulti-input nonlineardynamic systemsrdquo Automatica vol 44 no 5 pp 1418ndash14252008

[13] K-Y Lian and H-W Tu ldquoLMI-Based adaptive tracking controlfor parametric strict-feedback systemsrdquo IEEE Transactions onFuzzy Systems vol 16 no 5 pp 1245ndash1258 2008

[14] Z Lendek J Lauber T M Guerra R Babuka and B De Schut-ter ldquoAdaptive observers for TS fuzzy systems with unknownpolynomial inputsrdquo Fuzzy Sets and Systems vol 161 no 15 pp2043ndash2065 2010

[15] C-H Hyun C-W Park and S Kim ldquoTakagi-Sugeno fuzzymodel based indirect adaptive fuzzy observer and controllerdesignrdquo Information Sciences vol 180 no 11 pp 2314ndash23272010

[16] Y-H Chang W-S Chan and C-W Chang ldquoT-S fuzzy model-based adaptive dynamic surface control for ball and beamsystemrdquo IEEE Transactions on Industrial Electronics vol 60 no6 pp 2251ndash2263 2013

[17] S TWang and J T Fei ldquoRobust adaptive slidingmode control ofMEMS gyroscope using T-S fuzzymodelrdquoNonlinear Dynamicsvol 77 no 1-2 pp 361ndash371 2014

[18] R Qi G Tao C Tan and X Yao ldquoAdaptive control of discrete-time state-space T-S fuzzy systems with general relative degreerdquoFuzzy Sets and Systems vol 217 pp 22ndash40 2013

[19] H B Jiang J J Yu and C G Zhou ldquoStable adaptive fuzzycontrol of nonlinear systems using small-gain theorem and LMIapproachrdquo Journal of ControlTheory andApplications vol 8 no4 pp 527ndash532 2010

[20] R R Mohler Bilinear Control Processes Academic Press NewYork NY USA 1973

[21] D L Elliott Bilinear Systems in Encyclopedia of Electrical Engi-neering Wiley New York NY USA 2001

[22] T-H S Li and S-H Tsai ldquoT-S fuzzy bilinear model and fuzzycontroller design for a class of nonlinear systemsrdquo IEEE Trans-actions on Fuzzy Systems vol 15 no 3 pp 494ndash506 2007

[23] S H Tsai and TH S Li ldquoRobust fuzzy control of a class of fuzzybilinear systems with time-delayrdquo Chaos Solitons and Fractalsvol 39 no 5 pp 2028ndash2040 2009

[24] T-H S Li S-H Tsai J-Z Lee M-Y Hsiao and C-H ChaoldquoRobust 119867

infinfuzzy control for a class of uncertain discrete

fuzzy bilinear systemsrdquo IEEE Transactions on SystemsMan andCybernetics Part B Cybernetics vol 38 no 2 pp 510ndash527 2008

[25] G Zhang J-M Li and Y-W Ge ldquoNonfragile guaranteed costcontrol of discrete-time fuzzy bilinear system with time-delayrdquoJournal of Dynamic Systems Measurement and Control Trans-actions of the ASME vol 136 no 4 Article ID 044502 2014

[26] J R Li J M Li and Z L Xia ldquoObserver-based fuzzy controldesign for discrete-time T-S fuzzy bilinear systemsrdquo Interna-tional Journal of Uncertainty Fuzziness and Knowledge-BasedSystems vol 21 no 3 pp 435ndash454 2013

[27] M S Ali ldquoRobust stability of stochastic fuzzy impulsive recur-rent neural networks with time varying delaysrdquo Iranian Journalof Fuzzy Systems vol 11 no 4 pp 1ndash13 2014

[28] M Syed Ali ldquoRobust stability analysis of Takagi-Sugeno uncer-tain stochastic fuzzy recurrent neural networks with mixedtime-varying delaysrdquo Chinese Physics B vol 20 no 8 ArticleID 080201 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

119860119894isin 119877119899times119899 119861

119894isin 119877119899times119898 and 119862

119894119895are constant matrices which

have of the following forms

119860119894=

[

[

[

[

[

[

[

[

[

0 1 0 sdot sdot sdot 0 0

0 0 1 sdot sdot sdot 0 0

0 0 0 sdot sdot sdot 1 0

0 0 0 sdot sdot sdot 0 1

1198861198941

1198861198942

1198861198943

sdot sdot sdot 119886119894(119899minus1)

119886119894119899

]

]

]

]

]

]

]

]

]

119861119894=

[

[

[

[

[

[

[

[

[

0 0 0 sdot sdot sdot 0 0

0 0 0 sdot sdot sdot 0 0

0 0 0 sdot sdot sdot 0 0

0 0 0 sdot sdot sdot 0 0

1198871198941

1198871198942

1198871198943

sdot sdot sdot 119887119894(119898minus1)

119887119894119898

]

]

]

]

]

]

]

]

]

119862119894119895

=

[

[

[

[

[

[

[

[

[

0 0 0 sdot sdot sdot 0 0

0 0 0 sdot sdot sdot 0 0

0 0 0 sdot sdot sdot 0 0

0 0 0 sdot sdot sdot 0 0

1198881198941198951

1198881198941198952

1198881198941198953

sdot sdot sdot 119888119894119895(119899minus1)

119888119894119895119899

]

]

]

]

]

]

]

]

]

(5)

By using the fuzzy inference method with a singletonfuzzification product inference and centre average defuzzi-fication the overall fuzzy model is of the following form

(119905) =

119903

sum

119894=1

ℎ119894 (119911 (119905)) [119860 119894

119909 (119905) + 119861119894119906 (119905) + 119862

119894119906 (119905) 119909 (119905)

+ 119876Δ119891119894 (119909 119906)]

(6)

where

ℎ119894 (119911 (119905)) =

119908119894 (119911 (119905))

sum119903

119895=1119908119895 (

119911 (119905))

119908119894 (119911 (119905)) =

119892

prod

119895=1

119872119894

119895(119911119895 (

119905))

(7)

We assume 119908119894(119911(119905)) ge 0 sum119903

119894=1119908119894(119911(119905)) gt 0 for all 119905

Therefore we have

ℎ119894 (119911 (119905)) ge 0

119903

sum

119894=1

ℎ119894 (119911 (119905)) = 1 119894 = 1 2 119903 (8)

By comparison with (1) and (6) it is easy to see that

119891 (119909) =

119903

sum

119894=1

ℎ119894 (119911 (119905)) (119886119894

+ Δ119886119894) 119909 (119905)

119892119895 (

119909) =

119903

sum

119894=1

ℎ119894 (119911 (119905)) (119887119894119895

+ Δ119887119894119895

+ 119888119894119895119909 (119905) + Δ119862

119894119895119909 (119905))

(9)

where 119886119894= (1198861198941 1198861198942 119886

119894119899) isin 119877

1times119899 119887119894= (1198871198941 1198871198942 119887

119894119898) isin

1198771times119898 and 119888

119894119895= (1198881198941198951

1198881198941198952

119888119894119895119899

)

Remark 3 From now on unless confusion arises argumentssuch as 119911(119905) in ℎ

119894(119911(119905)) will be omitted just for notational

convenience

3 Control Design and Stability Analysis

System (6) can be represented by following the T-S fuzzymodel without considering the modelling error that isΔ119891119894(119909 119906) equiv 0 119894 = 1 2 119903 Consider

Plant rule 119894

IF 1199111 (

119905) is 119872119894

1 1199112 (

119905) is 119872119894

2 119911

119892 (119905) is 119872

119894

119892

THEN (119905) = 119860119894119909 (119905) + 119861

119894119906 (119905) + 119862

119894119906 (119905) 119909 (119905)

119894 = 1 2 119903

(10)

By using the fuzzy inference method with a singletonfuzzification product inference and centre average defuzzi-fication the overall fuzzy model is of the following form

(119905) =

119903

sum

119894=1

ℎ119894[119860119894119909 (119905) + 119861

119894119906 (119905) + 119862

119894119906 (119905) 119909 (119905)] (11)

Based on the idea of PDC the 119895th state-feedback con-troller is designed as follows

Plant rule 119894

IF 1199111 (

119905) is 119872119894

1 1199112 (

119905) is 119872119894

2 119911

119892 (119905) is 119872

119894

119892

THEN 119906119902119895

=

120588119895119865119894119895119909 (119905)

radic1 + 119909119879(119905) 119865119879

119894119895119865119894119895119909 (119905)

119894 = 1 2 119903

(12)

where 119865119894119895

isin 1198771times119899 is a vector to be determined and 120588

119895gt 0 is a

scalar to be assignedThe overall fuzzy control law can be represented by

119906119902119895

=

119903

sum

119894=1

ℎ119894

120588119895119865119894119895119909 (119905)

radic1 + 119909119879(119905) 119865119879

119894119895119865119894119895119909 (119905)

=

119903

sum

119894=1

ℎ119894120588119895sin 120579119894119895

=

119903

sum

119894=1

ℎ119894120588119895119865119894119895cos 120579119894119895119909 (119905)

(13)

where

sin 120579119894119895

=

119865119894119895119909 (119905)

radic1 + 119909119879(119905) 119865119879

119894119895119865119894119895119909 (119905)

cos 120579119894119895

=

1

radic1 + 119909119879(119905) 119865119879

119894119895119865119894119895119909 (119905)

120579119894119895

isin [minus

120587

2

120587

2

]

1 le 119894 le 119903 1 le 119895 le 119898

(14)

4 Mathematical Problems in Engineering

Substituting (13) into (11) one can get the closed-loopsystem

(119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895(119860119894+

119898

sum

119896=1

119861119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

119862119894119896120588119896sin 120579119895119896

)119909 (119905)

(15)

where 119861119894119896denotes the 119896th column of the 119861

119894

Theorem 4 Given positive scalars 120588119896(1 le 119896 le 119898) if there

exist a symmetric positive definite matrix119880 and some constantmatrices 119882

119894119895 such that LMIs (16) and (17) hold

[

[

[

119860119894119880 + 119880119860

119879

119894+ 120588 + 2120572119880 lowast lowast

119861119894119882119894

minus119868

119862119894119880 minus119868

]

]

]

lt 0 1 le 119894 le 119903 (16)

[

[

[

[

[

[

[

[

119872119894

lowast lowast lowast lowast

119861119894119882119895

minus119868

119861119895119882119894

minus119868

119862119894119880 minus119868

119862119895119880 minus119868

]

]

]

]

]

]

]

]

lt 0 1 le 119894 lt 119895 le 119903 (17)

where 120588 = sum119898

119896=11205882

119896 119872119894= 119860119894119880+119880119860

119879

119894+119860119895119880+119880119860

119879

119895+2120588+2120572119880

119861119894119882119894=

[

[

[

11986111989411198821198941

119861119894119898

119882119894119898

]

]

]

119862119894=

[

[

[

1198621198941

119862119894119898

]

]

]

119868 =[

[

119868

d119868

]

]

(18)

Then the FBS (15) is globally asymptotically stable with decayrate 120572 via the fuzzy feedback controller (13) and the gains canbe determined by 119865

119894119895= 119882119894119895119880minus1

Proof Consider the Lyapunov function candidate as follows

1198811 (

119909 (119905)) = 119909119879(119905) 119875119909 (119905) (19)

where 119875 = 119880minus1

Applying Schur complement lemma inequality (16) canbe written as follows

119880119860119879

119894+ 119860119894119880 +

119898

sum

119896=1

1205882

119896+

119898

sum

119896=1

(119861119894119896119882119894119896)119879119861119894119896119882119894119896

+

119898

sum

119896=1

(119862119894119896119880)119879(119862119894119896119880) + 2120572119880 lt 0

(20)

Premultiplying and postmultiplying (20) by 119875 respectivelywe have

119860119879

119894119875 + 119875119860

119894+

119898

sum

119896=1

1205882

119896119875119875 +

119898

sum

119896=1

(119861119894119896119865119894119896)119879119861119894119896119865119894119896

+

119898

sum

119896=1

119862119879

119894119896119862119894119896

+ 2120572119875 lt 0

(21)

Applying a similar procedure to inequality (17) we can obtain

119860119879

119894119875 + 119875119860

119894+ 119860119879

119895119875 + 119875119860

119895+ 2

119898

sum

119896=1

1205882

119896119875119875

+

119898

sum

119896=1

(119861119894119896119865119895119896

)

119879

(119861119894119896119865119895119896

) +

119898

sum

119896=1

119862119879

119894119896119862119894119896

+

119898

sum

119896=1

(119861119895119896

119865119894119896)

119879

(119861119895119896

119865119894119896) +

119898

sum

119896=1

119862119879

119895119896119862119895119896

+ 2120572119875 lt 0

(22)

The time derivative of 1198811is

1 (

119909 (119905)) = 119879119875119909 + 119909

119879119875 (23)

By substituting (15) into (23) we can get

1 (

119909 (119905)) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895119909119879

[119860119894+

119898

sum

119896=1

119861119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

119862119894119896120588119896sin 120579119895119896

]

119879

119875

+ 119875[119860119894+

119898

sum

119896=1

119861119894119896120588119896

times 119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

119862119894119896120588119896sin 120579119895119896

]

119909

=

119903

sum

119894=1

ℎ2

119894119909119879Λ119894119894119909 +

119903

sum

119894lt119895

ℎ119894ℎ119895119909119879Λ119894119895119909

(24)

Mathematical Problems in Engineering 5

where

Λ119894119894

= 119860119879

119894119875 + 119875119860

119894+

119898

sum

119896=1

120588119896sin 120579119894119896

(119862119879

119894119896119875 + 119875119862

119894119896)

+

119898

sum

119896=1

120588119896cos 120579119894119896

[(119861119894119896119865119894119896)119879119875 + 119875119861

119894119896119865119894119896]

Λ119894119895

= 119860119879

119894119875 + 119875119860

119894+

119898

sum

119896=1

120588119896sin 120579119894119896

(119862119879

119894119896119875 + 119875119862

119894119896)

+

119898

sum

119896=1

120588119896cos 120579119894119896

[(119861119894119896119865119895119896

)

119879

119875 + 119875119861119894119896119865119895119896

]

+ 119860119879

119895119875 + 119875119860

119895+

119898

sum

119896=1

120588119896sin 120579119895119896

(119862119879

119895119896119875 + 119875119862

119895119896)

+

119898

sum

119896=1

120588119896cos 120579119895119896

[(119861119895119896

119865119894119896)

119879

119875 + 119875119861119895119896

119865119894119896]

(25)

First by premultiplying and postmultiplyingΛ119894119894by119880 we can

obtain

119880Λ119894119894119880 = 119880119860

119879

119894+ 119860119894119880 +

119898

sum

119896=1

120588119896sin 120579119894119896

(119880119862119879

119894119896+ 119862119894119896119880)

+

119898

sum

119896=1

120588119896cos 120579119894119896

[119880 (119861119894119896119865119894119896)119879+ 119861119894119896119865119894119896119880]

(26)

According to Lemma 2 we can get the following

[120588119896119861119894119896119865119894119896119880 + 120588119896119880 (119861119894119896119865119894119896)119879] cos 120579

119894119896

le 1205882

119896cos2120579119894119896

+ (119861119894119896119882119894119896)119879119861119894119896119882119894119896

119880119862119879

119894119896120588119896sin 120579119894119896

+ 119862119894119896119880120588119896sin 120579119894119896

le 1205882

119896sin2120579119894119896

+ (119862119894119896119880)119879(119862119894119896119880)

(27)

From (26) and (27) we can obtain

119880Λ119894119894119880 le 119880119860

119879

119894+ 119860119894119880 +

119898

sum

119896=1

1205882

119896+

119898

sum

119896=1

(119862119894119896119880)119879119862119894119896119880

+

119898

sum

119896=1

(119861119894119896119882119894119896)119879119861119894119896119882119894119896

(28)

Applying similar procedures (26)ndash(28) to Λ119894119895 we can

obtain

119880Λ119894119895119880 le 119880119860

119879

119894+ 119860119894119880 + 119880119860

119879

119895+ 119860119895119880 + 2

119898

sum

119896=1

1205882

119896

+

119898

sum

119896=1

(119861119894119896119882119895119896

)

119879

(119861119894119896119882119895119896

) +

119898

sum

119896=1

(119862119894119896119880)119879119862119894119896119880

+

119898

sum

119896=1

(119861119895119896

119882119894119896)

119879

(119861119895119896

119882119894119896) +

119898

sum

119896=1

(119862119895119896

119880)

119879

119862119895119896

119880

(29)

Substituting (28) and (29) into (24) we obtain

1 (

119909 (119905)) le

119903

sum

119894=1

ℎ2

119894119909119879119875[119880119860

119879

119894+ 119860119894119880 +

119898

sum

119896=1

1205882

119896

+

119898

sum

119896=1

(119861119894119896119882119894119896)119879119861119894119896119882119894119896

+

119898

sum

119896=1

(119862119894119896119880)119879(119862119894119896119880)]119875119909

+

119903

sum

119894lt119895

ℎ119894ℎ119895119909119879119875[119880119860

119879

119894+ 119860119894119880 + 119880119860

119879

119895

+ 119860119895119880 + 2

119898

sum

119896=1

1205882

119896

+

119898

sum

119896=1

(119861119894119896119882119895119896

)

119879

(119861119894119896119882119895119896

)

+

119898

sum

119896=1

(119862119894119896119880)119879119862119894119896119880

+

119898

sum

119896=1

(119861119895119896

119882119894119896)

119879

(119861119895119896

119882119894119896)

+

119898

sum

119896=1

(119862119895119896

119880)

119879

119862119895119896

119880]119875119909

(30)By substituting (21) and (22) into (30) we can obtain

1 (

119909 (119905)) le minus2120572119875119909 = minus21205721198811 (

119909 (119905)) (31)Then by Definition 1 the closed-loop fuzzy system (15)

is globally asymptotically stable with decay rate 120572 Thiscompletes the proof of Theorem 4

Next the modelling error in (6) is considered and anadaptive compensation term is adopted to reduce the effectsof the modelling error

Adopt the fuzzy controller in the following form119906 = 119906119902119895

+ 119906119904119895

119895 = 1 2 119898 (32)where compensator 119906

119904119895will be designed later

Substituting (32) into (6) yields

(119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895[(119860119894+

119898

sum

119896=1

119861119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

119862119894119896120588119896sin 120579119895119896

)119909 (119905)

+ 119876(Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588119896sin 120579119895119896

)119909 (119905)

6 Mathematical Problems in Engineering

+

119898

sum

119896=1

(119861119894119896

+ 119876Δ119887119894119896

+ 119862119894119896119909 (119905)

+ 119876Δ119862119894119896119909 (119905)) 119906119904119895

]

(33)

Suppose that there exists an unknown constant 120582 such that

120582 ge

119903

sum

119894=1

119903

sum

119895=1

1003817100381710038171003817100381710038171003817100381710038171003817

Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588119896sin 120579119895119896

1003817100381710038171003817100381710038171003817100381710038171003817

(34)

Then from 0 le ℎ119894(119911(119905)) le 1 and

10038171003817100381710038171003817100381710038171003817100381710038171003817

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895(Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588 sin 120579

119895119896)

10038171003817100381710038171003817100381710038171003817100381710038171003817

le

119903

sum

119894=1

119903

sum

119895=1

1003817100381710038171003817100381710038171003817100381710038171003817

Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588 sin 120579

119895119896

1003817100381710038171003817100381710038171003817100381710038171003817

(35)

we have

10038171003817100381710038171003817100381710038171003817100381710038171003817

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895(Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588 sin 120579

119895119896)

1003817100381710038171003817100381710038171003817100381710038171003817

le 120582

(36)

It is easy to see that we can choose a function vector119867120582such

that

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895(Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588119896sin 120579119895119896

) = 120582119867120582

(37)

and 119867120582 le 1

Remark 5 Here assumption (34) is reasonable in many realsystems due to its boundedness such as chaotic system [19]for example Example 1 in the paper satisfies the assumptionOn the other hand the uncertain terms of the consideredsystems in the existing literature [2 18 22ndash24] satisfy thecondition of (34)

Denote

120596 (119905) = 119867120582119909 (119905) (38)

Substituting (37) and (38) into (33) we can obtain thefollowing feedback system

(119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895[(119860119894+

119898

sum

119896=1

119861119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

119862119894119896120588119896sin 120579119895119896

)119909 (119905)

+

119898

sum

119896=1

(119861119894119896

+ 119876Δ119887119894119896

+ 119862119894119896119909 (119905)

+ 119876Δ119862119894119896119909 (119905)) 119906119904119895

] + 119876120582120596 (119905)

(39)

Choose the adaptive compensator as follows

119906119904119895

= minus

1

2119898119892119895min1205742119876119879119875119888119909 (119905) 119895 = 1 2 119898 (40)

where 119888 = 1205822 119888 is the parameter estimation of 119888 and 120574 gt 0 is

a gain constantChoose the adaptive law as follows

=

1

21205781205742119909119879(119905) 119875119876119876

119879119875119909 (119905) (41)

where 120578 gt 0 is a gain constant which determines the rate ofadaptation

Substituting (40) into (39) yields

(119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895[(119860119894+

119898

sum

119896=1

119861119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

119862119894119896120588119896sin 120579119895119896

)119909 (119905)

+

119898

sum

119896=1

(119861119894119896

+ 119876Δ119887119894119896

+ 119862119894119896119909 (119905)

+ 119876Δ119862119894119896119909 (119905))

times (minus

119876119879119875119888119909 (119905)

2119898119892119895min1205742)] + 119876120582120596 (119905)

(42)

Theorem 6 Consider the uncertain nonlinear system (1) withcontrol law defined by (32) (13) and (40) and the parameterupdated by the adaptive law (41) If there exist a symmetricpositive definite matrix 119875 and some matrices 119865

119894119895(1 le 119894 119895 le 119903)

satisfying the LMIs (16) and (17) and the design parameter ischosen as

0 lt 120574 lt radic2120572120582min (119875) (43)

then the closed-loop system (42) is asymptotically stable and allsignals of the closed-loop system (42) are bounded

Mathematical Problems in Engineering 7

Proof Consider the Lyapunov function candidate

119881 = 119909119879(119905) 119875119909 (119905) + 120578 (119888 minus 119888)

2 (44)

where 120578 gt 0Let

119866119894119895

= 119860119894+

119898

sum

119896=1

119861119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

119862119894119896120588119896sin 120579119895119896

(45)

The time derivative of 119881 is

= 119879119875119909 + 119909

119879119875 + 2120578 (119888 minus 119888)

(46)

Substituting (39) into (46) we can obtain

=

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895[119909119879(119905)

times 119866119894119895

119879119875 + 119875119866

119894119895+ 2119875119876

times (Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588119896sin 120579119895119896

)119909 (119905)

+ 2119909119879(119905) 119875

119898

sum

119896=1

(119861119894119896

+ 119876Δ119887119894119896

+ 119862119894119896119909 (119905)

+ 119876Δ119862119894119896119909 (119905)) 119906119904119896

]

+ 2120578 (119888 minus 119888)

=

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895119909119879(119905) (119866119894119895

119879119875 + 119875119866

119894119895) 119909 (119905)

+ 2119909119879(119905) 119875119876120582120596 (119905)

+ 2119909119879(119905) 119875119876

119898

sum

119895=1

119892119895 (

119909) 119906119904119895+ 2120578 (119888 minus 119888)

(47)

From the proof of Theorem 4 we get

le minus2120572119909119879(119905) 119875119909 (119905) + 2119909

119879(119905) 119875119876120582120596 (119905)

+ 2119909119879(119905) 119875119876

119898

sum

119895=1

119892119895 (

119909) 119906119904119895+ 2120578 (119888 minus 119888)

(48)

It is easy to see that

2119909119879119875119876120582120596 minus 120574

21205962+ 12057421205962

= minus120574210038171003817100381710038171003817100381710038171003817

120596 minus

1

1205742119909119879119875119876120582

10038171003817100381710038171003817100381710038171003817

2

+

1

12057421199091198791198751198761205822119876119879119875119909 + 120574

21205962

le

1205822

1205742119909119879119875119876119876119879119875119909 + 120574

21205962

(49)

Substituting (49) into (48) yields

le minus2120572119909119879(119905) 119875119909 (119905) +

1205822

1205742119909119879119875119876119876119879119875119909 + 120574

21205962

+ 2119909119879(119905) 119875119876

119898

sum

119895=1

119892119895 (

119909) 119906119904119895+ 2120578 (119888 minus 119888)

(50)

Substituting (40) and (41) into (50) we obtain

le minus2120572119909119879(119905) 119875119909 (119905) + (minus119888 + 119888)

1

1205742119909119879119875119876119876119879119875119909

+ 2120578 (119888 minus 119888)

+ 12057421205962

= minus2120572119909119879(119905) 119875119909 (119905) + 120574

21205962

le minus2120572120582min (119875) 119909 (119905)2+ 12057421205962

(51)

where 1205962= 120596119879120596 = 119909

119879119867120582

119879119867120582119909 le 119867

120582

119879119867120582 sdot 119909

2le 119909

2By choosing 0 lt 120574 lt radic2120572120582min(119875) we can get lt 0

then we have that the states119909(119905) rarr 0 as 119905 approaches infinityvia LaSalle invariance principle and 119881(119905) is bounded From(44) we can obtain that states 119909 and 119888 are bounded thereforethe boundedness of 119906

119904119895is ensured from (40) Similarly from

(13) we can obtain that 119906119902119895

is boundedThen it can be provedthat (1) the closed-loop system (42) is asymptotically stableand (2) all signals of the closed-loop system (42) are bounded

4 Simulations

In this section we will give two examples to show theefficiency of the proposed approach The first example isan unknown chaotic system and the second example is aparameter uncertain T-S fuzzy bilinear system with multipleinputs

Example 1 Consider the following chaotic system with con-trol input

1= 1199092

2= minus01119909

2minus 1199091

3+ 12 cos 119905 + 119906

(52)

When 119906(119905) = 0 and the initial states are chosen as 119909(0) =

(2 2)119879 the states phase portrait of system (52) is shown in

Figure 1

8 Mathematical Problems in Engineering

0 1 2 3 4

0

2

4

6

8

10

minus10

minus8

minus6

minus4

minus4 minus3

minus2

minus2 minus1

x2

x1

Figure 1 The phase portrait of the chaotic system

System (52) can be modelled as the following T-S fuzzybilinear model

Rule 1 IF 1199091 (

119905) is about 0

THEN (119905) = (1198601+ Δ1198601) 119909 (119905) + (119861

1+ Δ1198611) 119906 (119905)

+ (1198621+ Δ1198621) 119909 (119905) 119906 (119905)

Rule 2 IF 1199091 (

119905) is about plusmn 2

THEN (119905) = (1198602+ Δ1198602) 119909 (119905) + (119861

2+ Δ1198612) 119906 (119905)

+ (1198622+ Δ1198622) 119909 (119905) 119906 (119905)

Rule 3 IF 1199091 (

119905) is about plusmn 4

THEN (119905) = (1198603+ Δ1198603) 119909 (119905) + (119861

3+ Δ1198613) 119906 (119905)

+ (1198623+ Δ1198623) 119909 (119905) 119906 (119905)

(53)

where 119909(119905) = (1199091(119905) 1199092(119905))119879 1198601= 1198602= 1198603= [0 1

minus1 minus1] 1198611=

1198612= 1198613= [0

minus1] and 119862

1= 1198622= 1198623= [0 1

1 1] Choose 120572 = 03

120588 = 009 1198651= [0 minus1] 119865

2= [minus1 minus1] and 119865

3= [minus1 minus1] By

solving LMIs (16)-(17) one can obtain

119875 = [

93637 46993

46993 106246] 120582min (119875) = 52528 (54)

Utilize the controllers (32) (13) and (40) and the parameterupdated law (41) to control system (52) The design parame-ters are chosen as 120578 = 2 120574 = 2 lt radic120582min(119875) the initial condi-tions are chosen as 119909(0) = (2 minus2)

119879 119888(0) = 0 and the relation-ship functions are selected as shown in Figure 2 The simula-tion results are shown in Figures 3 4 5 and 6 In Figures 3ndash6the curves of states control input and adaptive updatedparameter for the T-S fuzzy bilinear system are drawn by solidlines respectively while the curves of states control inputand adaptive updated parameter for T-S fuzzy linear systemare depicted by dotted lines respectively By comparison

Rule 3 Rule 2 Rule 1 Rule 2 Rule 3

minus4 minus3 minus2 minus1 0 1 2 3 4

x1

M(x

1)

0

02

04

06

08

1

Figure 2 The relationship functions

0 1 2 3 4 5 6 7 8 9 10Time (s)

minus05

0

05

1

15

2

x1

T-S FBST-S FLS

Figure 3 The state 1199091response curves

the convergence rates of the states of two systems are almostthe same though the state and control amplitudes of T-S fuzzybilinear system (FBS) are smaller thanT-S fuzzy linear system(FLS) Thus the proposed method has some advantages ofperformance over the existing approach [16]

Example 2 Consider the following parametric uncertainmultiple inputs bilinear fuzzy system

Rule 1 IF 1199091 (

119905) is 1198711

THEN (119905) = 1198601119909 (119905) + 119861

1119906 (119905) + 119862

1119906 (119905) 119909 (119905)

+ 119876 (Δ1198861119909 (119905) + Δ119887

1119906 (119905)

+ Δ1198621119906 (119905) 119909 (119905))

Rule 2 IF 1199092 (

119905) is 1198712

THEN (119905) = 1198602119909 (119905) + 119861

2119906 (119905) + 119862

2119906 (119905) 119909 (119905)

+ 119876 (Δ1198862119909 (119905) + Δ119887

2119906 (119905)

+ Δ1198622119906 (119905) 119909 (119905))

(55)

Mathematical Problems in Engineering 9

0 1 2 3 4 5 6 7 8 9 10Time (s)

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

x2

T-S FBST-S FLS

Figure 4 The state 1199092response curves

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

12

Time (s)

Con

trol u

minus4

minus2

T-S FBST-S FLS

Figure 5 The control curves

where 119909(119905) = (1199091(119905) 1199092(119905))119879 1198601= 1198602= [0 1

minus1 minus1] 11986111

= [0

1]

11986112

= [0

minus1] 11986121

= 11986122

= [0

1] 11986211

= [0 0

minus1 1] 11986212

= 11986221

=

11986222

= [0 0

minus1 minus1] 120572 = 03 120588

1= 005 120588

2= 004 119865

11=

[minus12 minus18] 11986512

= [minus08 minus09] 11986521

= [minus05 minus12] and11986522

= [minus1 minus1] using LMI technique to solve (16)-(17) wecan get a feasible solution as

119875 = [

168099 85379

85379 180100] 120582min (119875) = 88510 (56)

Apply the controllers (32) (13) and (40) and the parametersupdated law (41) to system (55) The design parameters arechosen as 120578 = 2 120574 = 2 lt radic2120572120582min(119875) The initial conditions

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Time (s)

Para

met

er c

T-S FBST-S FLS

Figure 6 The curves of adaptive updated parameters

0 5 10 15 20 25 30Time (s)

minus05

0

05

1

15

2

Stat

esx1

T-S FBST-S FLS

Figure 7 Responses of system state 1199091(T-S FBS solid line T-S FLS

dotted line)

are 119909(0) = (2 minus08)119879 119888(0) = 0 The simulation results are

shown in Figures 7 8 9 10 and 11Through the comparison between T-S fuzzy linear model

and bilinear one we can see that the settling time of thesystems is almost the same under the same initial conditionsalthough responses of T-S fuzzy bilinear system (FBS) stateamplitudes are smaller than T-S fuzzy linear system (FLS)and the demand of the control input of the system is lowThus the proposedmethod has better dynamic performancesthan the existing ones based on T-S fuzzy linear model

10 Mathematical Problems in Engineering

0 5 10 15 20 25 30Time (s)

minus1

minus08

minus06

minus04

minus02

0

02

04

Stat

esx2

T-S FBST-S FLS

Figure 8 Responses of system state 1199092(T-S FBS solid line T-S FLS

dotted line)

0 5 10 15 20 25 30Time (s)

minus15

minus1

minus05

0

05

1

15

T-S FBST-S FLS

Con

trol i

nput

u1

Figure 9 Control input1199061(T-S FBS solid line T-S FLS dotted line)

5 Conclusion

This paper proposes a new modelling method based onthe multiple inputs T-S fuzzy bilinear model which is usedto approximate nonlinear system the parallel distributedcompensation (PDC) method is utilized to design the fuzzycontroller without considering the error caused by fuzzymodellingThe sufficient conditionswith respect to decay rate120572 are derived by linear matrix inequalities (LMIs) The errorcaused by fuzzy modelling is considered and the method ofadaptive control is used to reduce the effect of the modelling

0 5 10 15 20 25 30Time (s)

minus08

minus06

minus04

minus02

0

02

04

06

Con

trol i

nput

u2

T-S FBST-S FLS

Figure 10 Control input 1199062(T-S FBS solid line T-S FLS dotted

line)

0 5 10 15 20 25 30Time (s)

0

01

02

03

04

05

06

07

08

09

Adap

tive p

aram

eter

c

T-S FBST-S FLS

Figure 11 Adaptive parameter 119888 (T-S FBS solid line T-S FLS dottedline)

error By Lyapunov stability criterion the resulting closed-loop system is proved to be asymptotically stable Finally twoillustrative examples are provided to show that the approachbased T-S fuzzy bilinear systems have some advantages ofperformance over the existing methods based on T-S fuzzylinear system The future research work is to extend theapproach to general system such as discrete-time systemsstochastic systems and time-delay systems

Mathematical Problems in Engineering 11

Conflict of Interests

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

Acknowledgment

This project was supported by the Soft Science Foundation ofShanxi Province (2011041033-3)

References

[1] K Tanaka and H O Wang Fuzzy Control Systems Design andAnalysis A Linear Matrix Inequality Approach Wiley-Intersci-ence 2001

[2] G Feng ldquoA survey on analysis and design of model-based fuzzycontrol systemsrdquo IEEE Transactions on Fuzzy Systems vol 14no 5 pp 676ndash697 2006

[3] K Tanaka T Ikeda and H O Wang ldquoFuzzy regulators andfuzzy observers relaxed stability conditions and LMI-baseddesignsrdquo IEEE Transactions on Fuzzy Systems vol 6 no 2 pp250ndash265 1998

[4] E Kim and H Lee ldquoNew approaches to relaxed quadraticstability condition of fuzzy control systemsrdquo IEEE Transactionson Fuzzy Systems vol 8 no 5 pp 523ndash534 2000

[5] K Tanaka andM Sugeno ldquoStability analysis and design of fuzzycontrol systemsrdquo Fuzzy Sets and Systems vol 45 no 2 pp 135ndash156 1992

[6] H-N Wu and H-X Li ldquoNew approach to delay-dependentstability analysis and stabilization for continuous-time fuzzysystems with time-varying delayrdquo IEEE Transactions on FuzzySystems vol 15 no 3 pp 482ndash493 2007

[7] F Zheng Q-G Wang and T H Lee ldquoAdaptive and robustcontroller design for uncertain nonlinear systems via fuzzymodeling approachrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 34 no 1 pp 166ndash178 2004

[8] C L Hwang ldquoA novel Takagi-Sugeno-based robust adaptivefuzzy sliding-mode controllerrdquo IEEE Transactions on FuzzySystems vol 12 no 5 pp 676ndash687 2004

[9] S Dong Adaptive Fuzzy Control of Nonlinear System Scienceand Technology Publishing House Beijing China 2006

[10] W-Y Wang Y-H Chien Y-G Leu and T-T Lee ldquoAdaptiveT-S fuzzy-neural modeling and control for general MIMOunknown non-affine nonlinear systems using projection updatelawsrdquo Automatica vol 46 no 5 pp 852ndash863 2010

[11] Y-H Chien W-Y Wang Y-G Leu and T-T Lee ldquoRobustadaptive controller design for a class of uncertain nonlinear sys-tems using online T-S fuzzy-neural modeling approachrdquo IEEETransactions on Systems Man and Cybernetics Part B Cyber-netics vol 41 no 2 pp 542ndash552 2011

[12] S P Moustakidis G A Rovithakis and J B Theocharis ldquoAnadaptive neuro-fuzzy tracking control formulti-input nonlineardynamic systemsrdquo Automatica vol 44 no 5 pp 1418ndash14252008

[13] K-Y Lian and H-W Tu ldquoLMI-Based adaptive tracking controlfor parametric strict-feedback systemsrdquo IEEE Transactions onFuzzy Systems vol 16 no 5 pp 1245ndash1258 2008

[14] Z Lendek J Lauber T M Guerra R Babuka and B De Schut-ter ldquoAdaptive observers for TS fuzzy systems with unknownpolynomial inputsrdquo Fuzzy Sets and Systems vol 161 no 15 pp2043ndash2065 2010

[15] C-H Hyun C-W Park and S Kim ldquoTakagi-Sugeno fuzzymodel based indirect adaptive fuzzy observer and controllerdesignrdquo Information Sciences vol 180 no 11 pp 2314ndash23272010

[16] Y-H Chang W-S Chan and C-W Chang ldquoT-S fuzzy model-based adaptive dynamic surface control for ball and beamsystemrdquo IEEE Transactions on Industrial Electronics vol 60 no6 pp 2251ndash2263 2013

[17] S TWang and J T Fei ldquoRobust adaptive slidingmode control ofMEMS gyroscope using T-S fuzzymodelrdquoNonlinear Dynamicsvol 77 no 1-2 pp 361ndash371 2014

[18] R Qi G Tao C Tan and X Yao ldquoAdaptive control of discrete-time state-space T-S fuzzy systems with general relative degreerdquoFuzzy Sets and Systems vol 217 pp 22ndash40 2013

[19] H B Jiang J J Yu and C G Zhou ldquoStable adaptive fuzzycontrol of nonlinear systems using small-gain theorem and LMIapproachrdquo Journal of ControlTheory andApplications vol 8 no4 pp 527ndash532 2010

[20] R R Mohler Bilinear Control Processes Academic Press NewYork NY USA 1973

[21] D L Elliott Bilinear Systems in Encyclopedia of Electrical Engi-neering Wiley New York NY USA 2001

[22] T-H S Li and S-H Tsai ldquoT-S fuzzy bilinear model and fuzzycontroller design for a class of nonlinear systemsrdquo IEEE Trans-actions on Fuzzy Systems vol 15 no 3 pp 494ndash506 2007

[23] S H Tsai and TH S Li ldquoRobust fuzzy control of a class of fuzzybilinear systems with time-delayrdquo Chaos Solitons and Fractalsvol 39 no 5 pp 2028ndash2040 2009

[24] T-H S Li S-H Tsai J-Z Lee M-Y Hsiao and C-H ChaoldquoRobust 119867

infinfuzzy control for a class of uncertain discrete

fuzzy bilinear systemsrdquo IEEE Transactions on SystemsMan andCybernetics Part B Cybernetics vol 38 no 2 pp 510ndash527 2008

[25] G Zhang J-M Li and Y-W Ge ldquoNonfragile guaranteed costcontrol of discrete-time fuzzy bilinear system with time-delayrdquoJournal of Dynamic Systems Measurement and Control Trans-actions of the ASME vol 136 no 4 Article ID 044502 2014

[26] J R Li J M Li and Z L Xia ldquoObserver-based fuzzy controldesign for discrete-time T-S fuzzy bilinear systemsrdquo Interna-tional Journal of Uncertainty Fuzziness and Knowledge-BasedSystems vol 21 no 3 pp 435ndash454 2013

[27] M S Ali ldquoRobust stability of stochastic fuzzy impulsive recur-rent neural networks with time varying delaysrdquo Iranian Journalof Fuzzy Systems vol 11 no 4 pp 1ndash13 2014

[28] M Syed Ali ldquoRobust stability analysis of Takagi-Sugeno uncer-tain stochastic fuzzy recurrent neural networks with mixedtime-varying delaysrdquo Chinese Physics B vol 20 no 8 ArticleID 080201 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

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4 Mathematical Problems in Engineering

Substituting (13) into (11) one can get the closed-loopsystem

(119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895(119860119894+

119898

sum

119896=1

119861119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

119862119894119896120588119896sin 120579119895119896

)119909 (119905)

(15)

where 119861119894119896denotes the 119896th column of the 119861

119894

Theorem 4 Given positive scalars 120588119896(1 le 119896 le 119898) if there

exist a symmetric positive definite matrix119880 and some constantmatrices 119882

119894119895 such that LMIs (16) and (17) hold

[

[

[

119860119894119880 + 119880119860

119879

119894+ 120588 + 2120572119880 lowast lowast

119861119894119882119894

minus119868

119862119894119880 minus119868

]

]

]

lt 0 1 le 119894 le 119903 (16)

[

[

[

[

[

[

[

[

119872119894

lowast lowast lowast lowast

119861119894119882119895

minus119868

119861119895119882119894

minus119868

119862119894119880 minus119868

119862119895119880 minus119868

]

]

]

]

]

]

]

]

lt 0 1 le 119894 lt 119895 le 119903 (17)

where 120588 = sum119898

119896=11205882

119896 119872119894= 119860119894119880+119880119860

119879

119894+119860119895119880+119880119860

119879

119895+2120588+2120572119880

119861119894119882119894=

[

[

[

11986111989411198821198941

119861119894119898

119882119894119898

]

]

]

119862119894=

[

[

[

1198621198941

119862119894119898

]

]

]

119868 =[

[

119868

d119868

]

]

(18)

Then the FBS (15) is globally asymptotically stable with decayrate 120572 via the fuzzy feedback controller (13) and the gains canbe determined by 119865

119894119895= 119882119894119895119880minus1

Proof Consider the Lyapunov function candidate as follows

1198811 (

119909 (119905)) = 119909119879(119905) 119875119909 (119905) (19)

where 119875 = 119880minus1

Applying Schur complement lemma inequality (16) canbe written as follows

119880119860119879

119894+ 119860119894119880 +

119898

sum

119896=1

1205882

119896+

119898

sum

119896=1

(119861119894119896119882119894119896)119879119861119894119896119882119894119896

+

119898

sum

119896=1

(119862119894119896119880)119879(119862119894119896119880) + 2120572119880 lt 0

(20)

Premultiplying and postmultiplying (20) by 119875 respectivelywe have

119860119879

119894119875 + 119875119860

119894+

119898

sum

119896=1

1205882

119896119875119875 +

119898

sum

119896=1

(119861119894119896119865119894119896)119879119861119894119896119865119894119896

+

119898

sum

119896=1

119862119879

119894119896119862119894119896

+ 2120572119875 lt 0

(21)

Applying a similar procedure to inequality (17) we can obtain

119860119879

119894119875 + 119875119860

119894+ 119860119879

119895119875 + 119875119860

119895+ 2

119898

sum

119896=1

1205882

119896119875119875

+

119898

sum

119896=1

(119861119894119896119865119895119896

)

119879

(119861119894119896119865119895119896

) +

119898

sum

119896=1

119862119879

119894119896119862119894119896

+

119898

sum

119896=1

(119861119895119896

119865119894119896)

119879

(119861119895119896

119865119894119896) +

119898

sum

119896=1

119862119879

119895119896119862119895119896

+ 2120572119875 lt 0

(22)

The time derivative of 1198811is

1 (

119909 (119905)) = 119879119875119909 + 119909

119879119875 (23)

By substituting (15) into (23) we can get

1 (

119909 (119905)) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895119909119879

[119860119894+

119898

sum

119896=1

119861119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

119862119894119896120588119896sin 120579119895119896

]

119879

119875

+ 119875[119860119894+

119898

sum

119896=1

119861119894119896120588119896

times 119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

119862119894119896120588119896sin 120579119895119896

]

119909

=

119903

sum

119894=1

ℎ2

119894119909119879Λ119894119894119909 +

119903

sum

119894lt119895

ℎ119894ℎ119895119909119879Λ119894119895119909

(24)

Mathematical Problems in Engineering 5

where

Λ119894119894

= 119860119879

119894119875 + 119875119860

119894+

119898

sum

119896=1

120588119896sin 120579119894119896

(119862119879

119894119896119875 + 119875119862

119894119896)

+

119898

sum

119896=1

120588119896cos 120579119894119896

[(119861119894119896119865119894119896)119879119875 + 119875119861

119894119896119865119894119896]

Λ119894119895

= 119860119879

119894119875 + 119875119860

119894+

119898

sum

119896=1

120588119896sin 120579119894119896

(119862119879

119894119896119875 + 119875119862

119894119896)

+

119898

sum

119896=1

120588119896cos 120579119894119896

[(119861119894119896119865119895119896

)

119879

119875 + 119875119861119894119896119865119895119896

]

+ 119860119879

119895119875 + 119875119860

119895+

119898

sum

119896=1

120588119896sin 120579119895119896

(119862119879

119895119896119875 + 119875119862

119895119896)

+

119898

sum

119896=1

120588119896cos 120579119895119896

[(119861119895119896

119865119894119896)

119879

119875 + 119875119861119895119896

119865119894119896]

(25)

First by premultiplying and postmultiplyingΛ119894119894by119880 we can

obtain

119880Λ119894119894119880 = 119880119860

119879

119894+ 119860119894119880 +

119898

sum

119896=1

120588119896sin 120579119894119896

(119880119862119879

119894119896+ 119862119894119896119880)

+

119898

sum

119896=1

120588119896cos 120579119894119896

[119880 (119861119894119896119865119894119896)119879+ 119861119894119896119865119894119896119880]

(26)

According to Lemma 2 we can get the following

[120588119896119861119894119896119865119894119896119880 + 120588119896119880 (119861119894119896119865119894119896)119879] cos 120579

119894119896

le 1205882

119896cos2120579119894119896

+ (119861119894119896119882119894119896)119879119861119894119896119882119894119896

119880119862119879

119894119896120588119896sin 120579119894119896

+ 119862119894119896119880120588119896sin 120579119894119896

le 1205882

119896sin2120579119894119896

+ (119862119894119896119880)119879(119862119894119896119880)

(27)

From (26) and (27) we can obtain

119880Λ119894119894119880 le 119880119860

119879

119894+ 119860119894119880 +

119898

sum

119896=1

1205882

119896+

119898

sum

119896=1

(119862119894119896119880)119879119862119894119896119880

+

119898

sum

119896=1

(119861119894119896119882119894119896)119879119861119894119896119882119894119896

(28)

Applying similar procedures (26)ndash(28) to Λ119894119895 we can

obtain

119880Λ119894119895119880 le 119880119860

119879

119894+ 119860119894119880 + 119880119860

119879

119895+ 119860119895119880 + 2

119898

sum

119896=1

1205882

119896

+

119898

sum

119896=1

(119861119894119896119882119895119896

)

119879

(119861119894119896119882119895119896

) +

119898

sum

119896=1

(119862119894119896119880)119879119862119894119896119880

+

119898

sum

119896=1

(119861119895119896

119882119894119896)

119879

(119861119895119896

119882119894119896) +

119898

sum

119896=1

(119862119895119896

119880)

119879

119862119895119896

119880

(29)

Substituting (28) and (29) into (24) we obtain

1 (

119909 (119905)) le

119903

sum

119894=1

ℎ2

119894119909119879119875[119880119860

119879

119894+ 119860119894119880 +

119898

sum

119896=1

1205882

119896

+

119898

sum

119896=1

(119861119894119896119882119894119896)119879119861119894119896119882119894119896

+

119898

sum

119896=1

(119862119894119896119880)119879(119862119894119896119880)]119875119909

+

119903

sum

119894lt119895

ℎ119894ℎ119895119909119879119875[119880119860

119879

119894+ 119860119894119880 + 119880119860

119879

119895

+ 119860119895119880 + 2

119898

sum

119896=1

1205882

119896

+

119898

sum

119896=1

(119861119894119896119882119895119896

)

119879

(119861119894119896119882119895119896

)

+

119898

sum

119896=1

(119862119894119896119880)119879119862119894119896119880

+

119898

sum

119896=1

(119861119895119896

119882119894119896)

119879

(119861119895119896

119882119894119896)

+

119898

sum

119896=1

(119862119895119896

119880)

119879

119862119895119896

119880]119875119909

(30)By substituting (21) and (22) into (30) we can obtain

1 (

119909 (119905)) le minus2120572119875119909 = minus21205721198811 (

119909 (119905)) (31)Then by Definition 1 the closed-loop fuzzy system (15)

is globally asymptotically stable with decay rate 120572 Thiscompletes the proof of Theorem 4

Next the modelling error in (6) is considered and anadaptive compensation term is adopted to reduce the effectsof the modelling error

Adopt the fuzzy controller in the following form119906 = 119906119902119895

+ 119906119904119895

119895 = 1 2 119898 (32)where compensator 119906

119904119895will be designed later

Substituting (32) into (6) yields

(119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895[(119860119894+

119898

sum

119896=1

119861119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

119862119894119896120588119896sin 120579119895119896

)119909 (119905)

+ 119876(Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588119896sin 120579119895119896

)119909 (119905)

6 Mathematical Problems in Engineering

+

119898

sum

119896=1

(119861119894119896

+ 119876Δ119887119894119896

+ 119862119894119896119909 (119905)

+ 119876Δ119862119894119896119909 (119905)) 119906119904119895

]

(33)

Suppose that there exists an unknown constant 120582 such that

120582 ge

119903

sum

119894=1

119903

sum

119895=1

1003817100381710038171003817100381710038171003817100381710038171003817

Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588119896sin 120579119895119896

1003817100381710038171003817100381710038171003817100381710038171003817

(34)

Then from 0 le ℎ119894(119911(119905)) le 1 and

10038171003817100381710038171003817100381710038171003817100381710038171003817

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895(Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588 sin 120579

119895119896)

10038171003817100381710038171003817100381710038171003817100381710038171003817

le

119903

sum

119894=1

119903

sum

119895=1

1003817100381710038171003817100381710038171003817100381710038171003817

Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588 sin 120579

119895119896

1003817100381710038171003817100381710038171003817100381710038171003817

(35)

we have

10038171003817100381710038171003817100381710038171003817100381710038171003817

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895(Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588 sin 120579

119895119896)

1003817100381710038171003817100381710038171003817100381710038171003817

le 120582

(36)

It is easy to see that we can choose a function vector119867120582such

that

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895(Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588119896sin 120579119895119896

) = 120582119867120582

(37)

and 119867120582 le 1

Remark 5 Here assumption (34) is reasonable in many realsystems due to its boundedness such as chaotic system [19]for example Example 1 in the paper satisfies the assumptionOn the other hand the uncertain terms of the consideredsystems in the existing literature [2 18 22ndash24] satisfy thecondition of (34)

Denote

120596 (119905) = 119867120582119909 (119905) (38)

Substituting (37) and (38) into (33) we can obtain thefollowing feedback system

(119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895[(119860119894+

119898

sum

119896=1

119861119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

119862119894119896120588119896sin 120579119895119896

)119909 (119905)

+

119898

sum

119896=1

(119861119894119896

+ 119876Δ119887119894119896

+ 119862119894119896119909 (119905)

+ 119876Δ119862119894119896119909 (119905)) 119906119904119895

] + 119876120582120596 (119905)

(39)

Choose the adaptive compensator as follows

119906119904119895

= minus

1

2119898119892119895min1205742119876119879119875119888119909 (119905) 119895 = 1 2 119898 (40)

where 119888 = 1205822 119888 is the parameter estimation of 119888 and 120574 gt 0 is

a gain constantChoose the adaptive law as follows

=

1

21205781205742119909119879(119905) 119875119876119876

119879119875119909 (119905) (41)

where 120578 gt 0 is a gain constant which determines the rate ofadaptation

Substituting (40) into (39) yields

(119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895[(119860119894+

119898

sum

119896=1

119861119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

119862119894119896120588119896sin 120579119895119896

)119909 (119905)

+

119898

sum

119896=1

(119861119894119896

+ 119876Δ119887119894119896

+ 119862119894119896119909 (119905)

+ 119876Δ119862119894119896119909 (119905))

times (minus

119876119879119875119888119909 (119905)

2119898119892119895min1205742)] + 119876120582120596 (119905)

(42)

Theorem 6 Consider the uncertain nonlinear system (1) withcontrol law defined by (32) (13) and (40) and the parameterupdated by the adaptive law (41) If there exist a symmetricpositive definite matrix 119875 and some matrices 119865

119894119895(1 le 119894 119895 le 119903)

satisfying the LMIs (16) and (17) and the design parameter ischosen as

0 lt 120574 lt radic2120572120582min (119875) (43)

then the closed-loop system (42) is asymptotically stable and allsignals of the closed-loop system (42) are bounded

Mathematical Problems in Engineering 7

Proof Consider the Lyapunov function candidate

119881 = 119909119879(119905) 119875119909 (119905) + 120578 (119888 minus 119888)

2 (44)

where 120578 gt 0Let

119866119894119895

= 119860119894+

119898

sum

119896=1

119861119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

119862119894119896120588119896sin 120579119895119896

(45)

The time derivative of 119881 is

= 119879119875119909 + 119909

119879119875 + 2120578 (119888 minus 119888)

(46)

Substituting (39) into (46) we can obtain

=

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895[119909119879(119905)

times 119866119894119895

119879119875 + 119875119866

119894119895+ 2119875119876

times (Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588119896sin 120579119895119896

)119909 (119905)

+ 2119909119879(119905) 119875

119898

sum

119896=1

(119861119894119896

+ 119876Δ119887119894119896

+ 119862119894119896119909 (119905)

+ 119876Δ119862119894119896119909 (119905)) 119906119904119896

]

+ 2120578 (119888 minus 119888)

=

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895119909119879(119905) (119866119894119895

119879119875 + 119875119866

119894119895) 119909 (119905)

+ 2119909119879(119905) 119875119876120582120596 (119905)

+ 2119909119879(119905) 119875119876

119898

sum

119895=1

119892119895 (

119909) 119906119904119895+ 2120578 (119888 minus 119888)

(47)

From the proof of Theorem 4 we get

le minus2120572119909119879(119905) 119875119909 (119905) + 2119909

119879(119905) 119875119876120582120596 (119905)

+ 2119909119879(119905) 119875119876

119898

sum

119895=1

119892119895 (

119909) 119906119904119895+ 2120578 (119888 minus 119888)

(48)

It is easy to see that

2119909119879119875119876120582120596 minus 120574

21205962+ 12057421205962

= minus120574210038171003817100381710038171003817100381710038171003817

120596 minus

1

1205742119909119879119875119876120582

10038171003817100381710038171003817100381710038171003817

2

+

1

12057421199091198791198751198761205822119876119879119875119909 + 120574

21205962

le

1205822

1205742119909119879119875119876119876119879119875119909 + 120574

21205962

(49)

Substituting (49) into (48) yields

le minus2120572119909119879(119905) 119875119909 (119905) +

1205822

1205742119909119879119875119876119876119879119875119909 + 120574

21205962

+ 2119909119879(119905) 119875119876

119898

sum

119895=1

119892119895 (

119909) 119906119904119895+ 2120578 (119888 minus 119888)

(50)

Substituting (40) and (41) into (50) we obtain

le minus2120572119909119879(119905) 119875119909 (119905) + (minus119888 + 119888)

1

1205742119909119879119875119876119876119879119875119909

+ 2120578 (119888 minus 119888)

+ 12057421205962

= minus2120572119909119879(119905) 119875119909 (119905) + 120574

21205962

le minus2120572120582min (119875) 119909 (119905)2+ 12057421205962

(51)

where 1205962= 120596119879120596 = 119909

119879119867120582

119879119867120582119909 le 119867

120582

119879119867120582 sdot 119909

2le 119909

2By choosing 0 lt 120574 lt radic2120572120582min(119875) we can get lt 0

then we have that the states119909(119905) rarr 0 as 119905 approaches infinityvia LaSalle invariance principle and 119881(119905) is bounded From(44) we can obtain that states 119909 and 119888 are bounded thereforethe boundedness of 119906

119904119895is ensured from (40) Similarly from

(13) we can obtain that 119906119902119895

is boundedThen it can be provedthat (1) the closed-loop system (42) is asymptotically stableand (2) all signals of the closed-loop system (42) are bounded

4 Simulations

In this section we will give two examples to show theefficiency of the proposed approach The first example isan unknown chaotic system and the second example is aparameter uncertain T-S fuzzy bilinear system with multipleinputs

Example 1 Consider the following chaotic system with con-trol input

1= 1199092

2= minus01119909

2minus 1199091

3+ 12 cos 119905 + 119906

(52)

When 119906(119905) = 0 and the initial states are chosen as 119909(0) =

(2 2)119879 the states phase portrait of system (52) is shown in

Figure 1

8 Mathematical Problems in Engineering

0 1 2 3 4

0

2

4

6

8

10

minus10

minus8

minus6

minus4

minus4 minus3

minus2

minus2 minus1

x2

x1

Figure 1 The phase portrait of the chaotic system

System (52) can be modelled as the following T-S fuzzybilinear model

Rule 1 IF 1199091 (

119905) is about 0

THEN (119905) = (1198601+ Δ1198601) 119909 (119905) + (119861

1+ Δ1198611) 119906 (119905)

+ (1198621+ Δ1198621) 119909 (119905) 119906 (119905)

Rule 2 IF 1199091 (

119905) is about plusmn 2

THEN (119905) = (1198602+ Δ1198602) 119909 (119905) + (119861

2+ Δ1198612) 119906 (119905)

+ (1198622+ Δ1198622) 119909 (119905) 119906 (119905)

Rule 3 IF 1199091 (

119905) is about plusmn 4

THEN (119905) = (1198603+ Δ1198603) 119909 (119905) + (119861

3+ Δ1198613) 119906 (119905)

+ (1198623+ Δ1198623) 119909 (119905) 119906 (119905)

(53)

where 119909(119905) = (1199091(119905) 1199092(119905))119879 1198601= 1198602= 1198603= [0 1

minus1 minus1] 1198611=

1198612= 1198613= [0

minus1] and 119862

1= 1198622= 1198623= [0 1

1 1] Choose 120572 = 03

120588 = 009 1198651= [0 minus1] 119865

2= [minus1 minus1] and 119865

3= [minus1 minus1] By

solving LMIs (16)-(17) one can obtain

119875 = [

93637 46993

46993 106246] 120582min (119875) = 52528 (54)

Utilize the controllers (32) (13) and (40) and the parameterupdated law (41) to control system (52) The design parame-ters are chosen as 120578 = 2 120574 = 2 lt radic120582min(119875) the initial condi-tions are chosen as 119909(0) = (2 minus2)

119879 119888(0) = 0 and the relation-ship functions are selected as shown in Figure 2 The simula-tion results are shown in Figures 3 4 5 and 6 In Figures 3ndash6the curves of states control input and adaptive updatedparameter for the T-S fuzzy bilinear system are drawn by solidlines respectively while the curves of states control inputand adaptive updated parameter for T-S fuzzy linear systemare depicted by dotted lines respectively By comparison

Rule 3 Rule 2 Rule 1 Rule 2 Rule 3

minus4 minus3 minus2 minus1 0 1 2 3 4

x1

M(x

1)

0

02

04

06

08

1

Figure 2 The relationship functions

0 1 2 3 4 5 6 7 8 9 10Time (s)

minus05

0

05

1

15

2

x1

T-S FBST-S FLS

Figure 3 The state 1199091response curves

the convergence rates of the states of two systems are almostthe same though the state and control amplitudes of T-S fuzzybilinear system (FBS) are smaller thanT-S fuzzy linear system(FLS) Thus the proposed method has some advantages ofperformance over the existing approach [16]

Example 2 Consider the following parametric uncertainmultiple inputs bilinear fuzzy system

Rule 1 IF 1199091 (

119905) is 1198711

THEN (119905) = 1198601119909 (119905) + 119861

1119906 (119905) + 119862

1119906 (119905) 119909 (119905)

+ 119876 (Δ1198861119909 (119905) + Δ119887

1119906 (119905)

+ Δ1198621119906 (119905) 119909 (119905))

Rule 2 IF 1199092 (

119905) is 1198712

THEN (119905) = 1198602119909 (119905) + 119861

2119906 (119905) + 119862

2119906 (119905) 119909 (119905)

+ 119876 (Δ1198862119909 (119905) + Δ119887

2119906 (119905)

+ Δ1198622119906 (119905) 119909 (119905))

(55)

Mathematical Problems in Engineering 9

0 1 2 3 4 5 6 7 8 9 10Time (s)

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

x2

T-S FBST-S FLS

Figure 4 The state 1199092response curves

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

12

Time (s)

Con

trol u

minus4

minus2

T-S FBST-S FLS

Figure 5 The control curves

where 119909(119905) = (1199091(119905) 1199092(119905))119879 1198601= 1198602= [0 1

minus1 minus1] 11986111

= [0

1]

11986112

= [0

minus1] 11986121

= 11986122

= [0

1] 11986211

= [0 0

minus1 1] 11986212

= 11986221

=

11986222

= [0 0

minus1 minus1] 120572 = 03 120588

1= 005 120588

2= 004 119865

11=

[minus12 minus18] 11986512

= [minus08 minus09] 11986521

= [minus05 minus12] and11986522

= [minus1 minus1] using LMI technique to solve (16)-(17) wecan get a feasible solution as

119875 = [

168099 85379

85379 180100] 120582min (119875) = 88510 (56)

Apply the controllers (32) (13) and (40) and the parametersupdated law (41) to system (55) The design parameters arechosen as 120578 = 2 120574 = 2 lt radic2120572120582min(119875) The initial conditions

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Time (s)

Para

met

er c

T-S FBST-S FLS

Figure 6 The curves of adaptive updated parameters

0 5 10 15 20 25 30Time (s)

minus05

0

05

1

15

2

Stat

esx1

T-S FBST-S FLS

Figure 7 Responses of system state 1199091(T-S FBS solid line T-S FLS

dotted line)

are 119909(0) = (2 minus08)119879 119888(0) = 0 The simulation results are

shown in Figures 7 8 9 10 and 11Through the comparison between T-S fuzzy linear model

and bilinear one we can see that the settling time of thesystems is almost the same under the same initial conditionsalthough responses of T-S fuzzy bilinear system (FBS) stateamplitudes are smaller than T-S fuzzy linear system (FLS)and the demand of the control input of the system is lowThus the proposedmethod has better dynamic performancesthan the existing ones based on T-S fuzzy linear model

10 Mathematical Problems in Engineering

0 5 10 15 20 25 30Time (s)

minus1

minus08

minus06

minus04

minus02

0

02

04

Stat

esx2

T-S FBST-S FLS

Figure 8 Responses of system state 1199092(T-S FBS solid line T-S FLS

dotted line)

0 5 10 15 20 25 30Time (s)

minus15

minus1

minus05

0

05

1

15

T-S FBST-S FLS

Con

trol i

nput

u1

Figure 9 Control input1199061(T-S FBS solid line T-S FLS dotted line)

5 Conclusion

This paper proposes a new modelling method based onthe multiple inputs T-S fuzzy bilinear model which is usedto approximate nonlinear system the parallel distributedcompensation (PDC) method is utilized to design the fuzzycontroller without considering the error caused by fuzzymodellingThe sufficient conditionswith respect to decay rate120572 are derived by linear matrix inequalities (LMIs) The errorcaused by fuzzy modelling is considered and the method ofadaptive control is used to reduce the effect of the modelling

0 5 10 15 20 25 30Time (s)

minus08

minus06

minus04

minus02

0

02

04

06

Con

trol i

nput

u2

T-S FBST-S FLS

Figure 10 Control input 1199062(T-S FBS solid line T-S FLS dotted

line)

0 5 10 15 20 25 30Time (s)

0

01

02

03

04

05

06

07

08

09

Adap

tive p

aram

eter

c

T-S FBST-S FLS

Figure 11 Adaptive parameter 119888 (T-S FBS solid line T-S FLS dottedline)

error By Lyapunov stability criterion the resulting closed-loop system is proved to be asymptotically stable Finally twoillustrative examples are provided to show that the approachbased T-S fuzzy bilinear systems have some advantages ofperformance over the existing methods based on T-S fuzzylinear system The future research work is to extend theapproach to general system such as discrete-time systemsstochastic systems and time-delay systems

Mathematical Problems in Engineering 11

Conflict of Interests

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

Acknowledgment

This project was supported by the Soft Science Foundation ofShanxi Province (2011041033-3)

References

[1] K Tanaka and H O Wang Fuzzy Control Systems Design andAnalysis A Linear Matrix Inequality Approach Wiley-Intersci-ence 2001

[2] G Feng ldquoA survey on analysis and design of model-based fuzzycontrol systemsrdquo IEEE Transactions on Fuzzy Systems vol 14no 5 pp 676ndash697 2006

[3] K Tanaka T Ikeda and H O Wang ldquoFuzzy regulators andfuzzy observers relaxed stability conditions and LMI-baseddesignsrdquo IEEE Transactions on Fuzzy Systems vol 6 no 2 pp250ndash265 1998

[4] E Kim and H Lee ldquoNew approaches to relaxed quadraticstability condition of fuzzy control systemsrdquo IEEE Transactionson Fuzzy Systems vol 8 no 5 pp 523ndash534 2000

[5] K Tanaka andM Sugeno ldquoStability analysis and design of fuzzycontrol systemsrdquo Fuzzy Sets and Systems vol 45 no 2 pp 135ndash156 1992

[6] H-N Wu and H-X Li ldquoNew approach to delay-dependentstability analysis and stabilization for continuous-time fuzzysystems with time-varying delayrdquo IEEE Transactions on FuzzySystems vol 15 no 3 pp 482ndash493 2007

[7] F Zheng Q-G Wang and T H Lee ldquoAdaptive and robustcontroller design for uncertain nonlinear systems via fuzzymodeling approachrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 34 no 1 pp 166ndash178 2004

[8] C L Hwang ldquoA novel Takagi-Sugeno-based robust adaptivefuzzy sliding-mode controllerrdquo IEEE Transactions on FuzzySystems vol 12 no 5 pp 676ndash687 2004

[9] S Dong Adaptive Fuzzy Control of Nonlinear System Scienceand Technology Publishing House Beijing China 2006

[10] W-Y Wang Y-H Chien Y-G Leu and T-T Lee ldquoAdaptiveT-S fuzzy-neural modeling and control for general MIMOunknown non-affine nonlinear systems using projection updatelawsrdquo Automatica vol 46 no 5 pp 852ndash863 2010

[11] Y-H Chien W-Y Wang Y-G Leu and T-T Lee ldquoRobustadaptive controller design for a class of uncertain nonlinear sys-tems using online T-S fuzzy-neural modeling approachrdquo IEEETransactions on Systems Man and Cybernetics Part B Cyber-netics vol 41 no 2 pp 542ndash552 2011

[12] S P Moustakidis G A Rovithakis and J B Theocharis ldquoAnadaptive neuro-fuzzy tracking control formulti-input nonlineardynamic systemsrdquo Automatica vol 44 no 5 pp 1418ndash14252008

[13] K-Y Lian and H-W Tu ldquoLMI-Based adaptive tracking controlfor parametric strict-feedback systemsrdquo IEEE Transactions onFuzzy Systems vol 16 no 5 pp 1245ndash1258 2008

[14] Z Lendek J Lauber T M Guerra R Babuka and B De Schut-ter ldquoAdaptive observers for TS fuzzy systems with unknownpolynomial inputsrdquo Fuzzy Sets and Systems vol 161 no 15 pp2043ndash2065 2010

[15] C-H Hyun C-W Park and S Kim ldquoTakagi-Sugeno fuzzymodel based indirect adaptive fuzzy observer and controllerdesignrdquo Information Sciences vol 180 no 11 pp 2314ndash23272010

[16] Y-H Chang W-S Chan and C-W Chang ldquoT-S fuzzy model-based adaptive dynamic surface control for ball and beamsystemrdquo IEEE Transactions on Industrial Electronics vol 60 no6 pp 2251ndash2263 2013

[17] S TWang and J T Fei ldquoRobust adaptive slidingmode control ofMEMS gyroscope using T-S fuzzymodelrdquoNonlinear Dynamicsvol 77 no 1-2 pp 361ndash371 2014

[18] R Qi G Tao C Tan and X Yao ldquoAdaptive control of discrete-time state-space T-S fuzzy systems with general relative degreerdquoFuzzy Sets and Systems vol 217 pp 22ndash40 2013

[19] H B Jiang J J Yu and C G Zhou ldquoStable adaptive fuzzycontrol of nonlinear systems using small-gain theorem and LMIapproachrdquo Journal of ControlTheory andApplications vol 8 no4 pp 527ndash532 2010

[20] R R Mohler Bilinear Control Processes Academic Press NewYork NY USA 1973

[21] D L Elliott Bilinear Systems in Encyclopedia of Electrical Engi-neering Wiley New York NY USA 2001

[22] T-H S Li and S-H Tsai ldquoT-S fuzzy bilinear model and fuzzycontroller design for a class of nonlinear systemsrdquo IEEE Trans-actions on Fuzzy Systems vol 15 no 3 pp 494ndash506 2007

[23] S H Tsai and TH S Li ldquoRobust fuzzy control of a class of fuzzybilinear systems with time-delayrdquo Chaos Solitons and Fractalsvol 39 no 5 pp 2028ndash2040 2009

[24] T-H S Li S-H Tsai J-Z Lee M-Y Hsiao and C-H ChaoldquoRobust 119867

infinfuzzy control for a class of uncertain discrete

fuzzy bilinear systemsrdquo IEEE Transactions on SystemsMan andCybernetics Part B Cybernetics vol 38 no 2 pp 510ndash527 2008

[25] G Zhang J-M Li and Y-W Ge ldquoNonfragile guaranteed costcontrol of discrete-time fuzzy bilinear system with time-delayrdquoJournal of Dynamic Systems Measurement and Control Trans-actions of the ASME vol 136 no 4 Article ID 044502 2014

[26] J R Li J M Li and Z L Xia ldquoObserver-based fuzzy controldesign for discrete-time T-S fuzzy bilinear systemsrdquo Interna-tional Journal of Uncertainty Fuzziness and Knowledge-BasedSystems vol 21 no 3 pp 435ndash454 2013

[27] M S Ali ldquoRobust stability of stochastic fuzzy impulsive recur-rent neural networks with time varying delaysrdquo Iranian Journalof Fuzzy Systems vol 11 no 4 pp 1ndash13 2014

[28] M Syed Ali ldquoRobust stability analysis of Takagi-Sugeno uncer-tain stochastic fuzzy recurrent neural networks with mixedtime-varying delaysrdquo Chinese Physics B vol 20 no 8 ArticleID 080201 2011

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 5

where

Λ119894119894

= 119860119879

119894119875 + 119875119860

119894+

119898

sum

119896=1

120588119896sin 120579119894119896

(119862119879

119894119896119875 + 119875119862

119894119896)

+

119898

sum

119896=1

120588119896cos 120579119894119896

[(119861119894119896119865119894119896)119879119875 + 119875119861

119894119896119865119894119896]

Λ119894119895

= 119860119879

119894119875 + 119875119860

119894+

119898

sum

119896=1

120588119896sin 120579119894119896

(119862119879

119894119896119875 + 119875119862

119894119896)

+

119898

sum

119896=1

120588119896cos 120579119894119896

[(119861119894119896119865119895119896

)

119879

119875 + 119875119861119894119896119865119895119896

]

+ 119860119879

119895119875 + 119875119860

119895+

119898

sum

119896=1

120588119896sin 120579119895119896

(119862119879

119895119896119875 + 119875119862

119895119896)

+

119898

sum

119896=1

120588119896cos 120579119895119896

[(119861119895119896

119865119894119896)

119879

119875 + 119875119861119895119896

119865119894119896]

(25)

First by premultiplying and postmultiplyingΛ119894119894by119880 we can

obtain

119880Λ119894119894119880 = 119880119860

119879

119894+ 119860119894119880 +

119898

sum

119896=1

120588119896sin 120579119894119896

(119880119862119879

119894119896+ 119862119894119896119880)

+

119898

sum

119896=1

120588119896cos 120579119894119896

[119880 (119861119894119896119865119894119896)119879+ 119861119894119896119865119894119896119880]

(26)

According to Lemma 2 we can get the following

[120588119896119861119894119896119865119894119896119880 + 120588119896119880 (119861119894119896119865119894119896)119879] cos 120579

119894119896

le 1205882

119896cos2120579119894119896

+ (119861119894119896119882119894119896)119879119861119894119896119882119894119896

119880119862119879

119894119896120588119896sin 120579119894119896

+ 119862119894119896119880120588119896sin 120579119894119896

le 1205882

119896sin2120579119894119896

+ (119862119894119896119880)119879(119862119894119896119880)

(27)

From (26) and (27) we can obtain

119880Λ119894119894119880 le 119880119860

119879

119894+ 119860119894119880 +

119898

sum

119896=1

1205882

119896+

119898

sum

119896=1

(119862119894119896119880)119879119862119894119896119880

+

119898

sum

119896=1

(119861119894119896119882119894119896)119879119861119894119896119882119894119896

(28)

Applying similar procedures (26)ndash(28) to Λ119894119895 we can

obtain

119880Λ119894119895119880 le 119880119860

119879

119894+ 119860119894119880 + 119880119860

119879

119895+ 119860119895119880 + 2

119898

sum

119896=1

1205882

119896

+

119898

sum

119896=1

(119861119894119896119882119895119896

)

119879

(119861119894119896119882119895119896

) +

119898

sum

119896=1

(119862119894119896119880)119879119862119894119896119880

+

119898

sum

119896=1

(119861119895119896

119882119894119896)

119879

(119861119895119896

119882119894119896) +

119898

sum

119896=1

(119862119895119896

119880)

119879

119862119895119896

119880

(29)

Substituting (28) and (29) into (24) we obtain

1 (

119909 (119905)) le

119903

sum

119894=1

ℎ2

119894119909119879119875[119880119860

119879

119894+ 119860119894119880 +

119898

sum

119896=1

1205882

119896

+

119898

sum

119896=1

(119861119894119896119882119894119896)119879119861119894119896119882119894119896

+

119898

sum

119896=1

(119862119894119896119880)119879(119862119894119896119880)]119875119909

+

119903

sum

119894lt119895

ℎ119894ℎ119895119909119879119875[119880119860

119879

119894+ 119860119894119880 + 119880119860

119879

119895

+ 119860119895119880 + 2

119898

sum

119896=1

1205882

119896

+

119898

sum

119896=1

(119861119894119896119882119895119896

)

119879

(119861119894119896119882119895119896

)

+

119898

sum

119896=1

(119862119894119896119880)119879119862119894119896119880

+

119898

sum

119896=1

(119861119895119896

119882119894119896)

119879

(119861119895119896

119882119894119896)

+

119898

sum

119896=1

(119862119895119896

119880)

119879

119862119895119896

119880]119875119909

(30)By substituting (21) and (22) into (30) we can obtain

1 (

119909 (119905)) le minus2120572119875119909 = minus21205721198811 (

119909 (119905)) (31)Then by Definition 1 the closed-loop fuzzy system (15)

is globally asymptotically stable with decay rate 120572 Thiscompletes the proof of Theorem 4

Next the modelling error in (6) is considered and anadaptive compensation term is adopted to reduce the effectsof the modelling error

Adopt the fuzzy controller in the following form119906 = 119906119902119895

+ 119906119904119895

119895 = 1 2 119898 (32)where compensator 119906

119904119895will be designed later

Substituting (32) into (6) yields

(119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895[(119860119894+

119898

sum

119896=1

119861119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

119862119894119896120588119896sin 120579119895119896

)119909 (119905)

+ 119876(Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588119896sin 120579119895119896

)119909 (119905)

6 Mathematical Problems in Engineering

+

119898

sum

119896=1

(119861119894119896

+ 119876Δ119887119894119896

+ 119862119894119896119909 (119905)

+ 119876Δ119862119894119896119909 (119905)) 119906119904119895

]

(33)

Suppose that there exists an unknown constant 120582 such that

120582 ge

119903

sum

119894=1

119903

sum

119895=1

1003817100381710038171003817100381710038171003817100381710038171003817

Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588119896sin 120579119895119896

1003817100381710038171003817100381710038171003817100381710038171003817

(34)

Then from 0 le ℎ119894(119911(119905)) le 1 and

10038171003817100381710038171003817100381710038171003817100381710038171003817

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895(Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588 sin 120579

119895119896)

10038171003817100381710038171003817100381710038171003817100381710038171003817

le

119903

sum

119894=1

119903

sum

119895=1

1003817100381710038171003817100381710038171003817100381710038171003817

Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588 sin 120579

119895119896

1003817100381710038171003817100381710038171003817100381710038171003817

(35)

we have

10038171003817100381710038171003817100381710038171003817100381710038171003817

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895(Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588 sin 120579

119895119896)

1003817100381710038171003817100381710038171003817100381710038171003817

le 120582

(36)

It is easy to see that we can choose a function vector119867120582such

that

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895(Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588119896sin 120579119895119896

) = 120582119867120582

(37)

and 119867120582 le 1

Remark 5 Here assumption (34) is reasonable in many realsystems due to its boundedness such as chaotic system [19]for example Example 1 in the paper satisfies the assumptionOn the other hand the uncertain terms of the consideredsystems in the existing literature [2 18 22ndash24] satisfy thecondition of (34)

Denote

120596 (119905) = 119867120582119909 (119905) (38)

Substituting (37) and (38) into (33) we can obtain thefollowing feedback system

(119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895[(119860119894+

119898

sum

119896=1

119861119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

119862119894119896120588119896sin 120579119895119896

)119909 (119905)

+

119898

sum

119896=1

(119861119894119896

+ 119876Δ119887119894119896

+ 119862119894119896119909 (119905)

+ 119876Δ119862119894119896119909 (119905)) 119906119904119895

] + 119876120582120596 (119905)

(39)

Choose the adaptive compensator as follows

119906119904119895

= minus

1

2119898119892119895min1205742119876119879119875119888119909 (119905) 119895 = 1 2 119898 (40)

where 119888 = 1205822 119888 is the parameter estimation of 119888 and 120574 gt 0 is

a gain constantChoose the adaptive law as follows

=

1

21205781205742119909119879(119905) 119875119876119876

119879119875119909 (119905) (41)

where 120578 gt 0 is a gain constant which determines the rate ofadaptation

Substituting (40) into (39) yields

(119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895[(119860119894+

119898

sum

119896=1

119861119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

119862119894119896120588119896sin 120579119895119896

)119909 (119905)

+

119898

sum

119896=1

(119861119894119896

+ 119876Δ119887119894119896

+ 119862119894119896119909 (119905)

+ 119876Δ119862119894119896119909 (119905))

times (minus

119876119879119875119888119909 (119905)

2119898119892119895min1205742)] + 119876120582120596 (119905)

(42)

Theorem 6 Consider the uncertain nonlinear system (1) withcontrol law defined by (32) (13) and (40) and the parameterupdated by the adaptive law (41) If there exist a symmetricpositive definite matrix 119875 and some matrices 119865

119894119895(1 le 119894 119895 le 119903)

satisfying the LMIs (16) and (17) and the design parameter ischosen as

0 lt 120574 lt radic2120572120582min (119875) (43)

then the closed-loop system (42) is asymptotically stable and allsignals of the closed-loop system (42) are bounded

Mathematical Problems in Engineering 7

Proof Consider the Lyapunov function candidate

119881 = 119909119879(119905) 119875119909 (119905) + 120578 (119888 minus 119888)

2 (44)

where 120578 gt 0Let

119866119894119895

= 119860119894+

119898

sum

119896=1

119861119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

119862119894119896120588119896sin 120579119895119896

(45)

The time derivative of 119881 is

= 119879119875119909 + 119909

119879119875 + 2120578 (119888 minus 119888)

(46)

Substituting (39) into (46) we can obtain

=

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895[119909119879(119905)

times 119866119894119895

119879119875 + 119875119866

119894119895+ 2119875119876

times (Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588119896sin 120579119895119896

)119909 (119905)

+ 2119909119879(119905) 119875

119898

sum

119896=1

(119861119894119896

+ 119876Δ119887119894119896

+ 119862119894119896119909 (119905)

+ 119876Δ119862119894119896119909 (119905)) 119906119904119896

]

+ 2120578 (119888 minus 119888)

=

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895119909119879(119905) (119866119894119895

119879119875 + 119875119866

119894119895) 119909 (119905)

+ 2119909119879(119905) 119875119876120582120596 (119905)

+ 2119909119879(119905) 119875119876

119898

sum

119895=1

119892119895 (

119909) 119906119904119895+ 2120578 (119888 minus 119888)

(47)

From the proof of Theorem 4 we get

le minus2120572119909119879(119905) 119875119909 (119905) + 2119909

119879(119905) 119875119876120582120596 (119905)

+ 2119909119879(119905) 119875119876

119898

sum

119895=1

119892119895 (

119909) 119906119904119895+ 2120578 (119888 minus 119888)

(48)

It is easy to see that

2119909119879119875119876120582120596 minus 120574

21205962+ 12057421205962

= minus120574210038171003817100381710038171003817100381710038171003817

120596 minus

1

1205742119909119879119875119876120582

10038171003817100381710038171003817100381710038171003817

2

+

1

12057421199091198791198751198761205822119876119879119875119909 + 120574

21205962

le

1205822

1205742119909119879119875119876119876119879119875119909 + 120574

21205962

(49)

Substituting (49) into (48) yields

le minus2120572119909119879(119905) 119875119909 (119905) +

1205822

1205742119909119879119875119876119876119879119875119909 + 120574

21205962

+ 2119909119879(119905) 119875119876

119898

sum

119895=1

119892119895 (

119909) 119906119904119895+ 2120578 (119888 minus 119888)

(50)

Substituting (40) and (41) into (50) we obtain

le minus2120572119909119879(119905) 119875119909 (119905) + (minus119888 + 119888)

1

1205742119909119879119875119876119876119879119875119909

+ 2120578 (119888 minus 119888)

+ 12057421205962

= minus2120572119909119879(119905) 119875119909 (119905) + 120574

21205962

le minus2120572120582min (119875) 119909 (119905)2+ 12057421205962

(51)

where 1205962= 120596119879120596 = 119909

119879119867120582

119879119867120582119909 le 119867

120582

119879119867120582 sdot 119909

2le 119909

2By choosing 0 lt 120574 lt radic2120572120582min(119875) we can get lt 0

then we have that the states119909(119905) rarr 0 as 119905 approaches infinityvia LaSalle invariance principle and 119881(119905) is bounded From(44) we can obtain that states 119909 and 119888 are bounded thereforethe boundedness of 119906

119904119895is ensured from (40) Similarly from

(13) we can obtain that 119906119902119895

is boundedThen it can be provedthat (1) the closed-loop system (42) is asymptotically stableand (2) all signals of the closed-loop system (42) are bounded

4 Simulations

In this section we will give two examples to show theefficiency of the proposed approach The first example isan unknown chaotic system and the second example is aparameter uncertain T-S fuzzy bilinear system with multipleinputs

Example 1 Consider the following chaotic system with con-trol input

1= 1199092

2= minus01119909

2minus 1199091

3+ 12 cos 119905 + 119906

(52)

When 119906(119905) = 0 and the initial states are chosen as 119909(0) =

(2 2)119879 the states phase portrait of system (52) is shown in

Figure 1

8 Mathematical Problems in Engineering

0 1 2 3 4

0

2

4

6

8

10

minus10

minus8

minus6

minus4

minus4 minus3

minus2

minus2 minus1

x2

x1

Figure 1 The phase portrait of the chaotic system

System (52) can be modelled as the following T-S fuzzybilinear model

Rule 1 IF 1199091 (

119905) is about 0

THEN (119905) = (1198601+ Δ1198601) 119909 (119905) + (119861

1+ Δ1198611) 119906 (119905)

+ (1198621+ Δ1198621) 119909 (119905) 119906 (119905)

Rule 2 IF 1199091 (

119905) is about plusmn 2

THEN (119905) = (1198602+ Δ1198602) 119909 (119905) + (119861

2+ Δ1198612) 119906 (119905)

+ (1198622+ Δ1198622) 119909 (119905) 119906 (119905)

Rule 3 IF 1199091 (

119905) is about plusmn 4

THEN (119905) = (1198603+ Δ1198603) 119909 (119905) + (119861

3+ Δ1198613) 119906 (119905)

+ (1198623+ Δ1198623) 119909 (119905) 119906 (119905)

(53)

where 119909(119905) = (1199091(119905) 1199092(119905))119879 1198601= 1198602= 1198603= [0 1

minus1 minus1] 1198611=

1198612= 1198613= [0

minus1] and 119862

1= 1198622= 1198623= [0 1

1 1] Choose 120572 = 03

120588 = 009 1198651= [0 minus1] 119865

2= [minus1 minus1] and 119865

3= [minus1 minus1] By

solving LMIs (16)-(17) one can obtain

119875 = [

93637 46993

46993 106246] 120582min (119875) = 52528 (54)

Utilize the controllers (32) (13) and (40) and the parameterupdated law (41) to control system (52) The design parame-ters are chosen as 120578 = 2 120574 = 2 lt radic120582min(119875) the initial condi-tions are chosen as 119909(0) = (2 minus2)

119879 119888(0) = 0 and the relation-ship functions are selected as shown in Figure 2 The simula-tion results are shown in Figures 3 4 5 and 6 In Figures 3ndash6the curves of states control input and adaptive updatedparameter for the T-S fuzzy bilinear system are drawn by solidlines respectively while the curves of states control inputand adaptive updated parameter for T-S fuzzy linear systemare depicted by dotted lines respectively By comparison

Rule 3 Rule 2 Rule 1 Rule 2 Rule 3

minus4 minus3 minus2 minus1 0 1 2 3 4

x1

M(x

1)

0

02

04

06

08

1

Figure 2 The relationship functions

0 1 2 3 4 5 6 7 8 9 10Time (s)

minus05

0

05

1

15

2

x1

T-S FBST-S FLS

Figure 3 The state 1199091response curves

the convergence rates of the states of two systems are almostthe same though the state and control amplitudes of T-S fuzzybilinear system (FBS) are smaller thanT-S fuzzy linear system(FLS) Thus the proposed method has some advantages ofperformance over the existing approach [16]

Example 2 Consider the following parametric uncertainmultiple inputs bilinear fuzzy system

Rule 1 IF 1199091 (

119905) is 1198711

THEN (119905) = 1198601119909 (119905) + 119861

1119906 (119905) + 119862

1119906 (119905) 119909 (119905)

+ 119876 (Δ1198861119909 (119905) + Δ119887

1119906 (119905)

+ Δ1198621119906 (119905) 119909 (119905))

Rule 2 IF 1199092 (

119905) is 1198712

THEN (119905) = 1198602119909 (119905) + 119861

2119906 (119905) + 119862

2119906 (119905) 119909 (119905)

+ 119876 (Δ1198862119909 (119905) + Δ119887

2119906 (119905)

+ Δ1198622119906 (119905) 119909 (119905))

(55)

Mathematical Problems in Engineering 9

0 1 2 3 4 5 6 7 8 9 10Time (s)

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

x2

T-S FBST-S FLS

Figure 4 The state 1199092response curves

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

12

Time (s)

Con

trol u

minus4

minus2

T-S FBST-S FLS

Figure 5 The control curves

where 119909(119905) = (1199091(119905) 1199092(119905))119879 1198601= 1198602= [0 1

minus1 minus1] 11986111

= [0

1]

11986112

= [0

minus1] 11986121

= 11986122

= [0

1] 11986211

= [0 0

minus1 1] 11986212

= 11986221

=

11986222

= [0 0

minus1 minus1] 120572 = 03 120588

1= 005 120588

2= 004 119865

11=

[minus12 minus18] 11986512

= [minus08 minus09] 11986521

= [minus05 minus12] and11986522

= [minus1 minus1] using LMI technique to solve (16)-(17) wecan get a feasible solution as

119875 = [

168099 85379

85379 180100] 120582min (119875) = 88510 (56)

Apply the controllers (32) (13) and (40) and the parametersupdated law (41) to system (55) The design parameters arechosen as 120578 = 2 120574 = 2 lt radic2120572120582min(119875) The initial conditions

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Time (s)

Para

met

er c

T-S FBST-S FLS

Figure 6 The curves of adaptive updated parameters

0 5 10 15 20 25 30Time (s)

minus05

0

05

1

15

2

Stat

esx1

T-S FBST-S FLS

Figure 7 Responses of system state 1199091(T-S FBS solid line T-S FLS

dotted line)

are 119909(0) = (2 minus08)119879 119888(0) = 0 The simulation results are

shown in Figures 7 8 9 10 and 11Through the comparison between T-S fuzzy linear model

and bilinear one we can see that the settling time of thesystems is almost the same under the same initial conditionsalthough responses of T-S fuzzy bilinear system (FBS) stateamplitudes are smaller than T-S fuzzy linear system (FLS)and the demand of the control input of the system is lowThus the proposedmethod has better dynamic performancesthan the existing ones based on T-S fuzzy linear model

10 Mathematical Problems in Engineering

0 5 10 15 20 25 30Time (s)

minus1

minus08

minus06

minus04

minus02

0

02

04

Stat

esx2

T-S FBST-S FLS

Figure 8 Responses of system state 1199092(T-S FBS solid line T-S FLS

dotted line)

0 5 10 15 20 25 30Time (s)

minus15

minus1

minus05

0

05

1

15

T-S FBST-S FLS

Con

trol i

nput

u1

Figure 9 Control input1199061(T-S FBS solid line T-S FLS dotted line)

5 Conclusion

This paper proposes a new modelling method based onthe multiple inputs T-S fuzzy bilinear model which is usedto approximate nonlinear system the parallel distributedcompensation (PDC) method is utilized to design the fuzzycontroller without considering the error caused by fuzzymodellingThe sufficient conditionswith respect to decay rate120572 are derived by linear matrix inequalities (LMIs) The errorcaused by fuzzy modelling is considered and the method ofadaptive control is used to reduce the effect of the modelling

0 5 10 15 20 25 30Time (s)

minus08

minus06

minus04

minus02

0

02

04

06

Con

trol i

nput

u2

T-S FBST-S FLS

Figure 10 Control input 1199062(T-S FBS solid line T-S FLS dotted

line)

0 5 10 15 20 25 30Time (s)

0

01

02

03

04

05

06

07

08

09

Adap

tive p

aram

eter

c

T-S FBST-S FLS

Figure 11 Adaptive parameter 119888 (T-S FBS solid line T-S FLS dottedline)

error By Lyapunov stability criterion the resulting closed-loop system is proved to be asymptotically stable Finally twoillustrative examples are provided to show that the approachbased T-S fuzzy bilinear systems have some advantages ofperformance over the existing methods based on T-S fuzzylinear system The future research work is to extend theapproach to general system such as discrete-time systemsstochastic systems and time-delay systems

Mathematical Problems in Engineering 11

Conflict of Interests

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

Acknowledgment

This project was supported by the Soft Science Foundation ofShanxi Province (2011041033-3)

References

[1] K Tanaka and H O Wang Fuzzy Control Systems Design andAnalysis A Linear Matrix Inequality Approach Wiley-Intersci-ence 2001

[2] G Feng ldquoA survey on analysis and design of model-based fuzzycontrol systemsrdquo IEEE Transactions on Fuzzy Systems vol 14no 5 pp 676ndash697 2006

[3] K Tanaka T Ikeda and H O Wang ldquoFuzzy regulators andfuzzy observers relaxed stability conditions and LMI-baseddesignsrdquo IEEE Transactions on Fuzzy Systems vol 6 no 2 pp250ndash265 1998

[4] E Kim and H Lee ldquoNew approaches to relaxed quadraticstability condition of fuzzy control systemsrdquo IEEE Transactionson Fuzzy Systems vol 8 no 5 pp 523ndash534 2000

[5] K Tanaka andM Sugeno ldquoStability analysis and design of fuzzycontrol systemsrdquo Fuzzy Sets and Systems vol 45 no 2 pp 135ndash156 1992

[6] H-N Wu and H-X Li ldquoNew approach to delay-dependentstability analysis and stabilization for continuous-time fuzzysystems with time-varying delayrdquo IEEE Transactions on FuzzySystems vol 15 no 3 pp 482ndash493 2007

[7] F Zheng Q-G Wang and T H Lee ldquoAdaptive and robustcontroller design for uncertain nonlinear systems via fuzzymodeling approachrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 34 no 1 pp 166ndash178 2004

[8] C L Hwang ldquoA novel Takagi-Sugeno-based robust adaptivefuzzy sliding-mode controllerrdquo IEEE Transactions on FuzzySystems vol 12 no 5 pp 676ndash687 2004

[9] S Dong Adaptive Fuzzy Control of Nonlinear System Scienceand Technology Publishing House Beijing China 2006

[10] W-Y Wang Y-H Chien Y-G Leu and T-T Lee ldquoAdaptiveT-S fuzzy-neural modeling and control for general MIMOunknown non-affine nonlinear systems using projection updatelawsrdquo Automatica vol 46 no 5 pp 852ndash863 2010

[11] Y-H Chien W-Y Wang Y-G Leu and T-T Lee ldquoRobustadaptive controller design for a class of uncertain nonlinear sys-tems using online T-S fuzzy-neural modeling approachrdquo IEEETransactions on Systems Man and Cybernetics Part B Cyber-netics vol 41 no 2 pp 542ndash552 2011

[12] S P Moustakidis G A Rovithakis and J B Theocharis ldquoAnadaptive neuro-fuzzy tracking control formulti-input nonlineardynamic systemsrdquo Automatica vol 44 no 5 pp 1418ndash14252008

[13] K-Y Lian and H-W Tu ldquoLMI-Based adaptive tracking controlfor parametric strict-feedback systemsrdquo IEEE Transactions onFuzzy Systems vol 16 no 5 pp 1245ndash1258 2008

[14] Z Lendek J Lauber T M Guerra R Babuka and B De Schut-ter ldquoAdaptive observers for TS fuzzy systems with unknownpolynomial inputsrdquo Fuzzy Sets and Systems vol 161 no 15 pp2043ndash2065 2010

[15] C-H Hyun C-W Park and S Kim ldquoTakagi-Sugeno fuzzymodel based indirect adaptive fuzzy observer and controllerdesignrdquo Information Sciences vol 180 no 11 pp 2314ndash23272010

[16] Y-H Chang W-S Chan and C-W Chang ldquoT-S fuzzy model-based adaptive dynamic surface control for ball and beamsystemrdquo IEEE Transactions on Industrial Electronics vol 60 no6 pp 2251ndash2263 2013

[17] S TWang and J T Fei ldquoRobust adaptive slidingmode control ofMEMS gyroscope using T-S fuzzymodelrdquoNonlinear Dynamicsvol 77 no 1-2 pp 361ndash371 2014

[18] R Qi G Tao C Tan and X Yao ldquoAdaptive control of discrete-time state-space T-S fuzzy systems with general relative degreerdquoFuzzy Sets and Systems vol 217 pp 22ndash40 2013

[19] H B Jiang J J Yu and C G Zhou ldquoStable adaptive fuzzycontrol of nonlinear systems using small-gain theorem and LMIapproachrdquo Journal of ControlTheory andApplications vol 8 no4 pp 527ndash532 2010

[20] R R Mohler Bilinear Control Processes Academic Press NewYork NY USA 1973

[21] D L Elliott Bilinear Systems in Encyclopedia of Electrical Engi-neering Wiley New York NY USA 2001

[22] T-H S Li and S-H Tsai ldquoT-S fuzzy bilinear model and fuzzycontroller design for a class of nonlinear systemsrdquo IEEE Trans-actions on Fuzzy Systems vol 15 no 3 pp 494ndash506 2007

[23] S H Tsai and TH S Li ldquoRobust fuzzy control of a class of fuzzybilinear systems with time-delayrdquo Chaos Solitons and Fractalsvol 39 no 5 pp 2028ndash2040 2009

[24] T-H S Li S-H Tsai J-Z Lee M-Y Hsiao and C-H ChaoldquoRobust 119867

infinfuzzy control for a class of uncertain discrete

fuzzy bilinear systemsrdquo IEEE Transactions on SystemsMan andCybernetics Part B Cybernetics vol 38 no 2 pp 510ndash527 2008

[25] G Zhang J-M Li and Y-W Ge ldquoNonfragile guaranteed costcontrol of discrete-time fuzzy bilinear system with time-delayrdquoJournal of Dynamic Systems Measurement and Control Trans-actions of the ASME vol 136 no 4 Article ID 044502 2014

[26] J R Li J M Li and Z L Xia ldquoObserver-based fuzzy controldesign for discrete-time T-S fuzzy bilinear systemsrdquo Interna-tional Journal of Uncertainty Fuzziness and Knowledge-BasedSystems vol 21 no 3 pp 435ndash454 2013

[27] M S Ali ldquoRobust stability of stochastic fuzzy impulsive recur-rent neural networks with time varying delaysrdquo Iranian Journalof Fuzzy Systems vol 11 no 4 pp 1ndash13 2014

[28] M Syed Ali ldquoRobust stability analysis of Takagi-Sugeno uncer-tain stochastic fuzzy recurrent neural networks with mixedtime-varying delaysrdquo Chinese Physics B vol 20 no 8 ArticleID 080201 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

6 Mathematical Problems in Engineering

+

119898

sum

119896=1

(119861119894119896

+ 119876Δ119887119894119896

+ 119862119894119896119909 (119905)

+ 119876Δ119862119894119896119909 (119905)) 119906119904119895

]

(33)

Suppose that there exists an unknown constant 120582 such that

120582 ge

119903

sum

119894=1

119903

sum

119895=1

1003817100381710038171003817100381710038171003817100381710038171003817

Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588119896sin 120579119895119896

1003817100381710038171003817100381710038171003817100381710038171003817

(34)

Then from 0 le ℎ119894(119911(119905)) le 1 and

10038171003817100381710038171003817100381710038171003817100381710038171003817

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895(Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588 sin 120579

119895119896)

10038171003817100381710038171003817100381710038171003817100381710038171003817

le

119903

sum

119894=1

119903

sum

119895=1

1003817100381710038171003817100381710038171003817100381710038171003817

Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588 sin 120579

119895119896

1003817100381710038171003817100381710038171003817100381710038171003817

(35)

we have

10038171003817100381710038171003817100381710038171003817100381710038171003817

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895(Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588 sin 120579

119895119896)

1003817100381710038171003817100381710038171003817100381710038171003817

le 120582

(36)

It is easy to see that we can choose a function vector119867120582such

that

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895(Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588119896sin 120579119895119896

) = 120582119867120582

(37)

and 119867120582 le 1

Remark 5 Here assumption (34) is reasonable in many realsystems due to its boundedness such as chaotic system [19]for example Example 1 in the paper satisfies the assumptionOn the other hand the uncertain terms of the consideredsystems in the existing literature [2 18 22ndash24] satisfy thecondition of (34)

Denote

120596 (119905) = 119867120582119909 (119905) (38)

Substituting (37) and (38) into (33) we can obtain thefollowing feedback system

(119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895[(119860119894+

119898

sum

119896=1

119861119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

119862119894119896120588119896sin 120579119895119896

)119909 (119905)

+

119898

sum

119896=1

(119861119894119896

+ 119876Δ119887119894119896

+ 119862119894119896119909 (119905)

+ 119876Δ119862119894119896119909 (119905)) 119906119904119895

] + 119876120582120596 (119905)

(39)

Choose the adaptive compensator as follows

119906119904119895

= minus

1

2119898119892119895min1205742119876119879119875119888119909 (119905) 119895 = 1 2 119898 (40)

where 119888 = 1205822 119888 is the parameter estimation of 119888 and 120574 gt 0 is

a gain constantChoose the adaptive law as follows

=

1

21205781205742119909119879(119905) 119875119876119876

119879119875119909 (119905) (41)

where 120578 gt 0 is a gain constant which determines the rate ofadaptation

Substituting (40) into (39) yields

(119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895[(119860119894+

119898

sum

119896=1

119861119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

119862119894119896120588119896sin 120579119895119896

)119909 (119905)

+

119898

sum

119896=1

(119861119894119896

+ 119876Δ119887119894119896

+ 119862119894119896119909 (119905)

+ 119876Δ119862119894119896119909 (119905))

times (minus

119876119879119875119888119909 (119905)

2119898119892119895min1205742)] + 119876120582120596 (119905)

(42)

Theorem 6 Consider the uncertain nonlinear system (1) withcontrol law defined by (32) (13) and (40) and the parameterupdated by the adaptive law (41) If there exist a symmetricpositive definite matrix 119875 and some matrices 119865

119894119895(1 le 119894 119895 le 119903)

satisfying the LMIs (16) and (17) and the design parameter ischosen as

0 lt 120574 lt radic2120572120582min (119875) (43)

then the closed-loop system (42) is asymptotically stable and allsignals of the closed-loop system (42) are bounded

Mathematical Problems in Engineering 7

Proof Consider the Lyapunov function candidate

119881 = 119909119879(119905) 119875119909 (119905) + 120578 (119888 minus 119888)

2 (44)

where 120578 gt 0Let

119866119894119895

= 119860119894+

119898

sum

119896=1

119861119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

119862119894119896120588119896sin 120579119895119896

(45)

The time derivative of 119881 is

= 119879119875119909 + 119909

119879119875 + 2120578 (119888 minus 119888)

(46)

Substituting (39) into (46) we can obtain

=

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895[119909119879(119905)

times 119866119894119895

119879119875 + 119875119866

119894119895+ 2119875119876

times (Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588119896sin 120579119895119896

)119909 (119905)

+ 2119909119879(119905) 119875

119898

sum

119896=1

(119861119894119896

+ 119876Δ119887119894119896

+ 119862119894119896119909 (119905)

+ 119876Δ119862119894119896119909 (119905)) 119906119904119896

]

+ 2120578 (119888 minus 119888)

=

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895119909119879(119905) (119866119894119895

119879119875 + 119875119866

119894119895) 119909 (119905)

+ 2119909119879(119905) 119875119876120582120596 (119905)

+ 2119909119879(119905) 119875119876

119898

sum

119895=1

119892119895 (

119909) 119906119904119895+ 2120578 (119888 minus 119888)

(47)

From the proof of Theorem 4 we get

le minus2120572119909119879(119905) 119875119909 (119905) + 2119909

119879(119905) 119875119876120582120596 (119905)

+ 2119909119879(119905) 119875119876

119898

sum

119895=1

119892119895 (

119909) 119906119904119895+ 2120578 (119888 minus 119888)

(48)

It is easy to see that

2119909119879119875119876120582120596 minus 120574

21205962+ 12057421205962

= minus120574210038171003817100381710038171003817100381710038171003817

120596 minus

1

1205742119909119879119875119876120582

10038171003817100381710038171003817100381710038171003817

2

+

1

12057421199091198791198751198761205822119876119879119875119909 + 120574

21205962

le

1205822

1205742119909119879119875119876119876119879119875119909 + 120574

21205962

(49)

Substituting (49) into (48) yields

le minus2120572119909119879(119905) 119875119909 (119905) +

1205822

1205742119909119879119875119876119876119879119875119909 + 120574

21205962

+ 2119909119879(119905) 119875119876

119898

sum

119895=1

119892119895 (

119909) 119906119904119895+ 2120578 (119888 minus 119888)

(50)

Substituting (40) and (41) into (50) we obtain

le minus2120572119909119879(119905) 119875119909 (119905) + (minus119888 + 119888)

1

1205742119909119879119875119876119876119879119875119909

+ 2120578 (119888 minus 119888)

+ 12057421205962

= minus2120572119909119879(119905) 119875119909 (119905) + 120574

21205962

le minus2120572120582min (119875) 119909 (119905)2+ 12057421205962

(51)

where 1205962= 120596119879120596 = 119909

119879119867120582

119879119867120582119909 le 119867

120582

119879119867120582 sdot 119909

2le 119909

2By choosing 0 lt 120574 lt radic2120572120582min(119875) we can get lt 0

then we have that the states119909(119905) rarr 0 as 119905 approaches infinityvia LaSalle invariance principle and 119881(119905) is bounded From(44) we can obtain that states 119909 and 119888 are bounded thereforethe boundedness of 119906

119904119895is ensured from (40) Similarly from

(13) we can obtain that 119906119902119895

is boundedThen it can be provedthat (1) the closed-loop system (42) is asymptotically stableand (2) all signals of the closed-loop system (42) are bounded

4 Simulations

In this section we will give two examples to show theefficiency of the proposed approach The first example isan unknown chaotic system and the second example is aparameter uncertain T-S fuzzy bilinear system with multipleinputs

Example 1 Consider the following chaotic system with con-trol input

1= 1199092

2= minus01119909

2minus 1199091

3+ 12 cos 119905 + 119906

(52)

When 119906(119905) = 0 and the initial states are chosen as 119909(0) =

(2 2)119879 the states phase portrait of system (52) is shown in

Figure 1

8 Mathematical Problems in Engineering

0 1 2 3 4

0

2

4

6

8

10

minus10

minus8

minus6

minus4

minus4 minus3

minus2

minus2 minus1

x2

x1

Figure 1 The phase portrait of the chaotic system

System (52) can be modelled as the following T-S fuzzybilinear model

Rule 1 IF 1199091 (

119905) is about 0

THEN (119905) = (1198601+ Δ1198601) 119909 (119905) + (119861

1+ Δ1198611) 119906 (119905)

+ (1198621+ Δ1198621) 119909 (119905) 119906 (119905)

Rule 2 IF 1199091 (

119905) is about plusmn 2

THEN (119905) = (1198602+ Δ1198602) 119909 (119905) + (119861

2+ Δ1198612) 119906 (119905)

+ (1198622+ Δ1198622) 119909 (119905) 119906 (119905)

Rule 3 IF 1199091 (

119905) is about plusmn 4

THEN (119905) = (1198603+ Δ1198603) 119909 (119905) + (119861

3+ Δ1198613) 119906 (119905)

+ (1198623+ Δ1198623) 119909 (119905) 119906 (119905)

(53)

where 119909(119905) = (1199091(119905) 1199092(119905))119879 1198601= 1198602= 1198603= [0 1

minus1 minus1] 1198611=

1198612= 1198613= [0

minus1] and 119862

1= 1198622= 1198623= [0 1

1 1] Choose 120572 = 03

120588 = 009 1198651= [0 minus1] 119865

2= [minus1 minus1] and 119865

3= [minus1 minus1] By

solving LMIs (16)-(17) one can obtain

119875 = [

93637 46993

46993 106246] 120582min (119875) = 52528 (54)

Utilize the controllers (32) (13) and (40) and the parameterupdated law (41) to control system (52) The design parame-ters are chosen as 120578 = 2 120574 = 2 lt radic120582min(119875) the initial condi-tions are chosen as 119909(0) = (2 minus2)

119879 119888(0) = 0 and the relation-ship functions are selected as shown in Figure 2 The simula-tion results are shown in Figures 3 4 5 and 6 In Figures 3ndash6the curves of states control input and adaptive updatedparameter for the T-S fuzzy bilinear system are drawn by solidlines respectively while the curves of states control inputand adaptive updated parameter for T-S fuzzy linear systemare depicted by dotted lines respectively By comparison

Rule 3 Rule 2 Rule 1 Rule 2 Rule 3

minus4 minus3 minus2 minus1 0 1 2 3 4

x1

M(x

1)

0

02

04

06

08

1

Figure 2 The relationship functions

0 1 2 3 4 5 6 7 8 9 10Time (s)

minus05

0

05

1

15

2

x1

T-S FBST-S FLS

Figure 3 The state 1199091response curves

the convergence rates of the states of two systems are almostthe same though the state and control amplitudes of T-S fuzzybilinear system (FBS) are smaller thanT-S fuzzy linear system(FLS) Thus the proposed method has some advantages ofperformance over the existing approach [16]

Example 2 Consider the following parametric uncertainmultiple inputs bilinear fuzzy system

Rule 1 IF 1199091 (

119905) is 1198711

THEN (119905) = 1198601119909 (119905) + 119861

1119906 (119905) + 119862

1119906 (119905) 119909 (119905)

+ 119876 (Δ1198861119909 (119905) + Δ119887

1119906 (119905)

+ Δ1198621119906 (119905) 119909 (119905))

Rule 2 IF 1199092 (

119905) is 1198712

THEN (119905) = 1198602119909 (119905) + 119861

2119906 (119905) + 119862

2119906 (119905) 119909 (119905)

+ 119876 (Δ1198862119909 (119905) + Δ119887

2119906 (119905)

+ Δ1198622119906 (119905) 119909 (119905))

(55)

Mathematical Problems in Engineering 9

0 1 2 3 4 5 6 7 8 9 10Time (s)

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

x2

T-S FBST-S FLS

Figure 4 The state 1199092response curves

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

12

Time (s)

Con

trol u

minus4

minus2

T-S FBST-S FLS

Figure 5 The control curves

where 119909(119905) = (1199091(119905) 1199092(119905))119879 1198601= 1198602= [0 1

minus1 minus1] 11986111

= [0

1]

11986112

= [0

minus1] 11986121

= 11986122

= [0

1] 11986211

= [0 0

minus1 1] 11986212

= 11986221

=

11986222

= [0 0

minus1 minus1] 120572 = 03 120588

1= 005 120588

2= 004 119865

11=

[minus12 minus18] 11986512

= [minus08 minus09] 11986521

= [minus05 minus12] and11986522

= [minus1 minus1] using LMI technique to solve (16)-(17) wecan get a feasible solution as

119875 = [

168099 85379

85379 180100] 120582min (119875) = 88510 (56)

Apply the controllers (32) (13) and (40) and the parametersupdated law (41) to system (55) The design parameters arechosen as 120578 = 2 120574 = 2 lt radic2120572120582min(119875) The initial conditions

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Time (s)

Para

met

er c

T-S FBST-S FLS

Figure 6 The curves of adaptive updated parameters

0 5 10 15 20 25 30Time (s)

minus05

0

05

1

15

2

Stat

esx1

T-S FBST-S FLS

Figure 7 Responses of system state 1199091(T-S FBS solid line T-S FLS

dotted line)

are 119909(0) = (2 minus08)119879 119888(0) = 0 The simulation results are

shown in Figures 7 8 9 10 and 11Through the comparison between T-S fuzzy linear model

and bilinear one we can see that the settling time of thesystems is almost the same under the same initial conditionsalthough responses of T-S fuzzy bilinear system (FBS) stateamplitudes are smaller than T-S fuzzy linear system (FLS)and the demand of the control input of the system is lowThus the proposedmethod has better dynamic performancesthan the existing ones based on T-S fuzzy linear model

10 Mathematical Problems in Engineering

0 5 10 15 20 25 30Time (s)

minus1

minus08

minus06

minus04

minus02

0

02

04

Stat

esx2

T-S FBST-S FLS

Figure 8 Responses of system state 1199092(T-S FBS solid line T-S FLS

dotted line)

0 5 10 15 20 25 30Time (s)

minus15

minus1

minus05

0

05

1

15

T-S FBST-S FLS

Con

trol i

nput

u1

Figure 9 Control input1199061(T-S FBS solid line T-S FLS dotted line)

5 Conclusion

This paper proposes a new modelling method based onthe multiple inputs T-S fuzzy bilinear model which is usedto approximate nonlinear system the parallel distributedcompensation (PDC) method is utilized to design the fuzzycontroller without considering the error caused by fuzzymodellingThe sufficient conditionswith respect to decay rate120572 are derived by linear matrix inequalities (LMIs) The errorcaused by fuzzy modelling is considered and the method ofadaptive control is used to reduce the effect of the modelling

0 5 10 15 20 25 30Time (s)

minus08

minus06

minus04

minus02

0

02

04

06

Con

trol i

nput

u2

T-S FBST-S FLS

Figure 10 Control input 1199062(T-S FBS solid line T-S FLS dotted

line)

0 5 10 15 20 25 30Time (s)

0

01

02

03

04

05

06

07

08

09

Adap

tive p

aram

eter

c

T-S FBST-S FLS

Figure 11 Adaptive parameter 119888 (T-S FBS solid line T-S FLS dottedline)

error By Lyapunov stability criterion the resulting closed-loop system is proved to be asymptotically stable Finally twoillustrative examples are provided to show that the approachbased T-S fuzzy bilinear systems have some advantages ofperformance over the existing methods based on T-S fuzzylinear system The future research work is to extend theapproach to general system such as discrete-time systemsstochastic systems and time-delay systems

Mathematical Problems in Engineering 11

Conflict of Interests

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

Acknowledgment

This project was supported by the Soft Science Foundation ofShanxi Province (2011041033-3)

References

[1] K Tanaka and H O Wang Fuzzy Control Systems Design andAnalysis A Linear Matrix Inequality Approach Wiley-Intersci-ence 2001

[2] G Feng ldquoA survey on analysis and design of model-based fuzzycontrol systemsrdquo IEEE Transactions on Fuzzy Systems vol 14no 5 pp 676ndash697 2006

[3] K Tanaka T Ikeda and H O Wang ldquoFuzzy regulators andfuzzy observers relaxed stability conditions and LMI-baseddesignsrdquo IEEE Transactions on Fuzzy Systems vol 6 no 2 pp250ndash265 1998

[4] E Kim and H Lee ldquoNew approaches to relaxed quadraticstability condition of fuzzy control systemsrdquo IEEE Transactionson Fuzzy Systems vol 8 no 5 pp 523ndash534 2000

[5] K Tanaka andM Sugeno ldquoStability analysis and design of fuzzycontrol systemsrdquo Fuzzy Sets and Systems vol 45 no 2 pp 135ndash156 1992

[6] H-N Wu and H-X Li ldquoNew approach to delay-dependentstability analysis and stabilization for continuous-time fuzzysystems with time-varying delayrdquo IEEE Transactions on FuzzySystems vol 15 no 3 pp 482ndash493 2007

[7] F Zheng Q-G Wang and T H Lee ldquoAdaptive and robustcontroller design for uncertain nonlinear systems via fuzzymodeling approachrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 34 no 1 pp 166ndash178 2004

[8] C L Hwang ldquoA novel Takagi-Sugeno-based robust adaptivefuzzy sliding-mode controllerrdquo IEEE Transactions on FuzzySystems vol 12 no 5 pp 676ndash687 2004

[9] S Dong Adaptive Fuzzy Control of Nonlinear System Scienceand Technology Publishing House Beijing China 2006

[10] W-Y Wang Y-H Chien Y-G Leu and T-T Lee ldquoAdaptiveT-S fuzzy-neural modeling and control for general MIMOunknown non-affine nonlinear systems using projection updatelawsrdquo Automatica vol 46 no 5 pp 852ndash863 2010

[11] Y-H Chien W-Y Wang Y-G Leu and T-T Lee ldquoRobustadaptive controller design for a class of uncertain nonlinear sys-tems using online T-S fuzzy-neural modeling approachrdquo IEEETransactions on Systems Man and Cybernetics Part B Cyber-netics vol 41 no 2 pp 542ndash552 2011

[12] S P Moustakidis G A Rovithakis and J B Theocharis ldquoAnadaptive neuro-fuzzy tracking control formulti-input nonlineardynamic systemsrdquo Automatica vol 44 no 5 pp 1418ndash14252008

[13] K-Y Lian and H-W Tu ldquoLMI-Based adaptive tracking controlfor parametric strict-feedback systemsrdquo IEEE Transactions onFuzzy Systems vol 16 no 5 pp 1245ndash1258 2008

[14] Z Lendek J Lauber T M Guerra R Babuka and B De Schut-ter ldquoAdaptive observers for TS fuzzy systems with unknownpolynomial inputsrdquo Fuzzy Sets and Systems vol 161 no 15 pp2043ndash2065 2010

[15] C-H Hyun C-W Park and S Kim ldquoTakagi-Sugeno fuzzymodel based indirect adaptive fuzzy observer and controllerdesignrdquo Information Sciences vol 180 no 11 pp 2314ndash23272010

[16] Y-H Chang W-S Chan and C-W Chang ldquoT-S fuzzy model-based adaptive dynamic surface control for ball and beamsystemrdquo IEEE Transactions on Industrial Electronics vol 60 no6 pp 2251ndash2263 2013

[17] S TWang and J T Fei ldquoRobust adaptive slidingmode control ofMEMS gyroscope using T-S fuzzymodelrdquoNonlinear Dynamicsvol 77 no 1-2 pp 361ndash371 2014

[18] R Qi G Tao C Tan and X Yao ldquoAdaptive control of discrete-time state-space T-S fuzzy systems with general relative degreerdquoFuzzy Sets and Systems vol 217 pp 22ndash40 2013

[19] H B Jiang J J Yu and C G Zhou ldquoStable adaptive fuzzycontrol of nonlinear systems using small-gain theorem and LMIapproachrdquo Journal of ControlTheory andApplications vol 8 no4 pp 527ndash532 2010

[20] R R Mohler Bilinear Control Processes Academic Press NewYork NY USA 1973

[21] D L Elliott Bilinear Systems in Encyclopedia of Electrical Engi-neering Wiley New York NY USA 2001

[22] T-H S Li and S-H Tsai ldquoT-S fuzzy bilinear model and fuzzycontroller design for a class of nonlinear systemsrdquo IEEE Trans-actions on Fuzzy Systems vol 15 no 3 pp 494ndash506 2007

[23] S H Tsai and TH S Li ldquoRobust fuzzy control of a class of fuzzybilinear systems with time-delayrdquo Chaos Solitons and Fractalsvol 39 no 5 pp 2028ndash2040 2009

[24] T-H S Li S-H Tsai J-Z Lee M-Y Hsiao and C-H ChaoldquoRobust 119867

infinfuzzy control for a class of uncertain discrete

fuzzy bilinear systemsrdquo IEEE Transactions on SystemsMan andCybernetics Part B Cybernetics vol 38 no 2 pp 510ndash527 2008

[25] G Zhang J-M Li and Y-W Ge ldquoNonfragile guaranteed costcontrol of discrete-time fuzzy bilinear system with time-delayrdquoJournal of Dynamic Systems Measurement and Control Trans-actions of the ASME vol 136 no 4 Article ID 044502 2014

[26] J R Li J M Li and Z L Xia ldquoObserver-based fuzzy controldesign for discrete-time T-S fuzzy bilinear systemsrdquo Interna-tional Journal of Uncertainty Fuzziness and Knowledge-BasedSystems vol 21 no 3 pp 435ndash454 2013

[27] M S Ali ldquoRobust stability of stochastic fuzzy impulsive recur-rent neural networks with time varying delaysrdquo Iranian Journalof Fuzzy Systems vol 11 no 4 pp 1ndash13 2014

[28] M Syed Ali ldquoRobust stability analysis of Takagi-Sugeno uncer-tain stochastic fuzzy recurrent neural networks with mixedtime-varying delaysrdquo Chinese Physics B vol 20 no 8 ArticleID 080201 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

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

Proof Consider the Lyapunov function candidate

119881 = 119909119879(119905) 119875119909 (119905) + 120578 (119888 minus 119888)

2 (44)

where 120578 gt 0Let

119866119894119895

= 119860119894+

119898

sum

119896=1

119861119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

119862119894119896120588119896sin 120579119895119896

(45)

The time derivative of 119881 is

= 119879119875119909 + 119909

119879119875 + 2120578 (119888 minus 119888)

(46)

Substituting (39) into (46) we can obtain

=

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895[119909119879(119905)

times 119866119894119895

119879119875 + 119875119866

119894119895+ 2119875119876

times (Δ119886119894+

119898

sum

119896=1

Δ119887119894119896120588119896119865119895119896

cos 120579119895119896

+

119898

sum

119896=1

Δ119862119894119896120588119896sin 120579119895119896

)119909 (119905)

+ 2119909119879(119905) 119875

119898

sum

119896=1

(119861119894119896

+ 119876Δ119887119894119896

+ 119862119894119896119909 (119905)

+ 119876Δ119862119894119896119909 (119905)) 119906119904119896

]

+ 2120578 (119888 minus 119888)

=

119903

sum

119894=1

119903

sum

119895=1

ℎ119894ℎ119895119909119879(119905) (119866119894119895

119879119875 + 119875119866

119894119895) 119909 (119905)

+ 2119909119879(119905) 119875119876120582120596 (119905)

+ 2119909119879(119905) 119875119876

119898

sum

119895=1

119892119895 (

119909) 119906119904119895+ 2120578 (119888 minus 119888)

(47)

From the proof of Theorem 4 we get

le minus2120572119909119879(119905) 119875119909 (119905) + 2119909

119879(119905) 119875119876120582120596 (119905)

+ 2119909119879(119905) 119875119876

119898

sum

119895=1

119892119895 (

119909) 119906119904119895+ 2120578 (119888 minus 119888)

(48)

It is easy to see that

2119909119879119875119876120582120596 minus 120574

21205962+ 12057421205962

= minus120574210038171003817100381710038171003817100381710038171003817

120596 minus

1

1205742119909119879119875119876120582

10038171003817100381710038171003817100381710038171003817

2

+

1

12057421199091198791198751198761205822119876119879119875119909 + 120574

21205962

le

1205822

1205742119909119879119875119876119876119879119875119909 + 120574

21205962

(49)

Substituting (49) into (48) yields

le minus2120572119909119879(119905) 119875119909 (119905) +

1205822

1205742119909119879119875119876119876119879119875119909 + 120574

21205962

+ 2119909119879(119905) 119875119876

119898

sum

119895=1

119892119895 (

119909) 119906119904119895+ 2120578 (119888 minus 119888)

(50)

Substituting (40) and (41) into (50) we obtain

le minus2120572119909119879(119905) 119875119909 (119905) + (minus119888 + 119888)

1

1205742119909119879119875119876119876119879119875119909

+ 2120578 (119888 minus 119888)

+ 12057421205962

= minus2120572119909119879(119905) 119875119909 (119905) + 120574

21205962

le minus2120572120582min (119875) 119909 (119905)2+ 12057421205962

(51)

where 1205962= 120596119879120596 = 119909

119879119867120582

119879119867120582119909 le 119867

120582

119879119867120582 sdot 119909

2le 119909

2By choosing 0 lt 120574 lt radic2120572120582min(119875) we can get lt 0

then we have that the states119909(119905) rarr 0 as 119905 approaches infinityvia LaSalle invariance principle and 119881(119905) is bounded From(44) we can obtain that states 119909 and 119888 are bounded thereforethe boundedness of 119906

119904119895is ensured from (40) Similarly from

(13) we can obtain that 119906119902119895

is boundedThen it can be provedthat (1) the closed-loop system (42) is asymptotically stableand (2) all signals of the closed-loop system (42) are bounded

4 Simulations

In this section we will give two examples to show theefficiency of the proposed approach The first example isan unknown chaotic system and the second example is aparameter uncertain T-S fuzzy bilinear system with multipleinputs

Example 1 Consider the following chaotic system with con-trol input

1= 1199092

2= minus01119909

2minus 1199091

3+ 12 cos 119905 + 119906

(52)

When 119906(119905) = 0 and the initial states are chosen as 119909(0) =

(2 2)119879 the states phase portrait of system (52) is shown in

Figure 1

8 Mathematical Problems in Engineering

0 1 2 3 4

0

2

4

6

8

10

minus10

minus8

minus6

minus4

minus4 minus3

minus2

minus2 minus1

x2

x1

Figure 1 The phase portrait of the chaotic system

System (52) can be modelled as the following T-S fuzzybilinear model

Rule 1 IF 1199091 (

119905) is about 0

THEN (119905) = (1198601+ Δ1198601) 119909 (119905) + (119861

1+ Δ1198611) 119906 (119905)

+ (1198621+ Δ1198621) 119909 (119905) 119906 (119905)

Rule 2 IF 1199091 (

119905) is about plusmn 2

THEN (119905) = (1198602+ Δ1198602) 119909 (119905) + (119861

2+ Δ1198612) 119906 (119905)

+ (1198622+ Δ1198622) 119909 (119905) 119906 (119905)

Rule 3 IF 1199091 (

119905) is about plusmn 4

THEN (119905) = (1198603+ Δ1198603) 119909 (119905) + (119861

3+ Δ1198613) 119906 (119905)

+ (1198623+ Δ1198623) 119909 (119905) 119906 (119905)

(53)

where 119909(119905) = (1199091(119905) 1199092(119905))119879 1198601= 1198602= 1198603= [0 1

minus1 minus1] 1198611=

1198612= 1198613= [0

minus1] and 119862

1= 1198622= 1198623= [0 1

1 1] Choose 120572 = 03

120588 = 009 1198651= [0 minus1] 119865

2= [minus1 minus1] and 119865

3= [minus1 minus1] By

solving LMIs (16)-(17) one can obtain

119875 = [

93637 46993

46993 106246] 120582min (119875) = 52528 (54)

Utilize the controllers (32) (13) and (40) and the parameterupdated law (41) to control system (52) The design parame-ters are chosen as 120578 = 2 120574 = 2 lt radic120582min(119875) the initial condi-tions are chosen as 119909(0) = (2 minus2)

119879 119888(0) = 0 and the relation-ship functions are selected as shown in Figure 2 The simula-tion results are shown in Figures 3 4 5 and 6 In Figures 3ndash6the curves of states control input and adaptive updatedparameter for the T-S fuzzy bilinear system are drawn by solidlines respectively while the curves of states control inputand adaptive updated parameter for T-S fuzzy linear systemare depicted by dotted lines respectively By comparison

Rule 3 Rule 2 Rule 1 Rule 2 Rule 3

minus4 minus3 minus2 minus1 0 1 2 3 4

x1

M(x

1)

0

02

04

06

08

1

Figure 2 The relationship functions

0 1 2 3 4 5 6 7 8 9 10Time (s)

minus05

0

05

1

15

2

x1

T-S FBST-S FLS

Figure 3 The state 1199091response curves

the convergence rates of the states of two systems are almostthe same though the state and control amplitudes of T-S fuzzybilinear system (FBS) are smaller thanT-S fuzzy linear system(FLS) Thus the proposed method has some advantages ofperformance over the existing approach [16]

Example 2 Consider the following parametric uncertainmultiple inputs bilinear fuzzy system

Rule 1 IF 1199091 (

119905) is 1198711

THEN (119905) = 1198601119909 (119905) + 119861

1119906 (119905) + 119862

1119906 (119905) 119909 (119905)

+ 119876 (Δ1198861119909 (119905) + Δ119887

1119906 (119905)

+ Δ1198621119906 (119905) 119909 (119905))

Rule 2 IF 1199092 (

119905) is 1198712

THEN (119905) = 1198602119909 (119905) + 119861

2119906 (119905) + 119862

2119906 (119905) 119909 (119905)

+ 119876 (Δ1198862119909 (119905) + Δ119887

2119906 (119905)

+ Δ1198622119906 (119905) 119909 (119905))

(55)

Mathematical Problems in Engineering 9

0 1 2 3 4 5 6 7 8 9 10Time (s)

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

x2

T-S FBST-S FLS

Figure 4 The state 1199092response curves

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

12

Time (s)

Con

trol u

minus4

minus2

T-S FBST-S FLS

Figure 5 The control curves

where 119909(119905) = (1199091(119905) 1199092(119905))119879 1198601= 1198602= [0 1

minus1 minus1] 11986111

= [0

1]

11986112

= [0

minus1] 11986121

= 11986122

= [0

1] 11986211

= [0 0

minus1 1] 11986212

= 11986221

=

11986222

= [0 0

minus1 minus1] 120572 = 03 120588

1= 005 120588

2= 004 119865

11=

[minus12 minus18] 11986512

= [minus08 minus09] 11986521

= [minus05 minus12] and11986522

= [minus1 minus1] using LMI technique to solve (16)-(17) wecan get a feasible solution as

119875 = [

168099 85379

85379 180100] 120582min (119875) = 88510 (56)

Apply the controllers (32) (13) and (40) and the parametersupdated law (41) to system (55) The design parameters arechosen as 120578 = 2 120574 = 2 lt radic2120572120582min(119875) The initial conditions

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Time (s)

Para

met

er c

T-S FBST-S FLS

Figure 6 The curves of adaptive updated parameters

0 5 10 15 20 25 30Time (s)

minus05

0

05

1

15

2

Stat

esx1

T-S FBST-S FLS

Figure 7 Responses of system state 1199091(T-S FBS solid line T-S FLS

dotted line)

are 119909(0) = (2 minus08)119879 119888(0) = 0 The simulation results are

shown in Figures 7 8 9 10 and 11Through the comparison between T-S fuzzy linear model

and bilinear one we can see that the settling time of thesystems is almost the same under the same initial conditionsalthough responses of T-S fuzzy bilinear system (FBS) stateamplitudes are smaller than T-S fuzzy linear system (FLS)and the demand of the control input of the system is lowThus the proposedmethod has better dynamic performancesthan the existing ones based on T-S fuzzy linear model

10 Mathematical Problems in Engineering

0 5 10 15 20 25 30Time (s)

minus1

minus08

minus06

minus04

minus02

0

02

04

Stat

esx2

T-S FBST-S FLS

Figure 8 Responses of system state 1199092(T-S FBS solid line T-S FLS

dotted line)

0 5 10 15 20 25 30Time (s)

minus15

minus1

minus05

0

05

1

15

T-S FBST-S FLS

Con

trol i

nput

u1

Figure 9 Control input1199061(T-S FBS solid line T-S FLS dotted line)

5 Conclusion

This paper proposes a new modelling method based onthe multiple inputs T-S fuzzy bilinear model which is usedto approximate nonlinear system the parallel distributedcompensation (PDC) method is utilized to design the fuzzycontroller without considering the error caused by fuzzymodellingThe sufficient conditionswith respect to decay rate120572 are derived by linear matrix inequalities (LMIs) The errorcaused by fuzzy modelling is considered and the method ofadaptive control is used to reduce the effect of the modelling

0 5 10 15 20 25 30Time (s)

minus08

minus06

minus04

minus02

0

02

04

06

Con

trol i

nput

u2

T-S FBST-S FLS

Figure 10 Control input 1199062(T-S FBS solid line T-S FLS dotted

line)

0 5 10 15 20 25 30Time (s)

0

01

02

03

04

05

06

07

08

09

Adap

tive p

aram

eter

c

T-S FBST-S FLS

Figure 11 Adaptive parameter 119888 (T-S FBS solid line T-S FLS dottedline)

error By Lyapunov stability criterion the resulting closed-loop system is proved to be asymptotically stable Finally twoillustrative examples are provided to show that the approachbased T-S fuzzy bilinear systems have some advantages ofperformance over the existing methods based on T-S fuzzylinear system The future research work is to extend theapproach to general system such as discrete-time systemsstochastic systems and time-delay systems

Mathematical Problems in Engineering 11

Conflict of Interests

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

Acknowledgment

This project was supported by the Soft Science Foundation ofShanxi Province (2011041033-3)

References

[1] K Tanaka and H O Wang Fuzzy Control Systems Design andAnalysis A Linear Matrix Inequality Approach Wiley-Intersci-ence 2001

[2] G Feng ldquoA survey on analysis and design of model-based fuzzycontrol systemsrdquo IEEE Transactions on Fuzzy Systems vol 14no 5 pp 676ndash697 2006

[3] K Tanaka T Ikeda and H O Wang ldquoFuzzy regulators andfuzzy observers relaxed stability conditions and LMI-baseddesignsrdquo IEEE Transactions on Fuzzy Systems vol 6 no 2 pp250ndash265 1998

[4] E Kim and H Lee ldquoNew approaches to relaxed quadraticstability condition of fuzzy control systemsrdquo IEEE Transactionson Fuzzy Systems vol 8 no 5 pp 523ndash534 2000

[5] K Tanaka andM Sugeno ldquoStability analysis and design of fuzzycontrol systemsrdquo Fuzzy Sets and Systems vol 45 no 2 pp 135ndash156 1992

[6] H-N Wu and H-X Li ldquoNew approach to delay-dependentstability analysis and stabilization for continuous-time fuzzysystems with time-varying delayrdquo IEEE Transactions on FuzzySystems vol 15 no 3 pp 482ndash493 2007

[7] F Zheng Q-G Wang and T H Lee ldquoAdaptive and robustcontroller design for uncertain nonlinear systems via fuzzymodeling approachrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 34 no 1 pp 166ndash178 2004

[8] C L Hwang ldquoA novel Takagi-Sugeno-based robust adaptivefuzzy sliding-mode controllerrdquo IEEE Transactions on FuzzySystems vol 12 no 5 pp 676ndash687 2004

[9] S Dong Adaptive Fuzzy Control of Nonlinear System Scienceand Technology Publishing House Beijing China 2006

[10] W-Y Wang Y-H Chien Y-G Leu and T-T Lee ldquoAdaptiveT-S fuzzy-neural modeling and control for general MIMOunknown non-affine nonlinear systems using projection updatelawsrdquo Automatica vol 46 no 5 pp 852ndash863 2010

[11] Y-H Chien W-Y Wang Y-G Leu and T-T Lee ldquoRobustadaptive controller design for a class of uncertain nonlinear sys-tems using online T-S fuzzy-neural modeling approachrdquo IEEETransactions on Systems Man and Cybernetics Part B Cyber-netics vol 41 no 2 pp 542ndash552 2011

[12] S P Moustakidis G A Rovithakis and J B Theocharis ldquoAnadaptive neuro-fuzzy tracking control formulti-input nonlineardynamic systemsrdquo Automatica vol 44 no 5 pp 1418ndash14252008

[13] K-Y Lian and H-W Tu ldquoLMI-Based adaptive tracking controlfor parametric strict-feedback systemsrdquo IEEE Transactions onFuzzy Systems vol 16 no 5 pp 1245ndash1258 2008

[14] Z Lendek J Lauber T M Guerra R Babuka and B De Schut-ter ldquoAdaptive observers for TS fuzzy systems with unknownpolynomial inputsrdquo Fuzzy Sets and Systems vol 161 no 15 pp2043ndash2065 2010

[15] C-H Hyun C-W Park and S Kim ldquoTakagi-Sugeno fuzzymodel based indirect adaptive fuzzy observer and controllerdesignrdquo Information Sciences vol 180 no 11 pp 2314ndash23272010

[16] Y-H Chang W-S Chan and C-W Chang ldquoT-S fuzzy model-based adaptive dynamic surface control for ball and beamsystemrdquo IEEE Transactions on Industrial Electronics vol 60 no6 pp 2251ndash2263 2013

[17] S TWang and J T Fei ldquoRobust adaptive slidingmode control ofMEMS gyroscope using T-S fuzzymodelrdquoNonlinear Dynamicsvol 77 no 1-2 pp 361ndash371 2014

[18] R Qi G Tao C Tan and X Yao ldquoAdaptive control of discrete-time state-space T-S fuzzy systems with general relative degreerdquoFuzzy Sets and Systems vol 217 pp 22ndash40 2013

[19] H B Jiang J J Yu and C G Zhou ldquoStable adaptive fuzzycontrol of nonlinear systems using small-gain theorem and LMIapproachrdquo Journal of ControlTheory andApplications vol 8 no4 pp 527ndash532 2010

[20] R R Mohler Bilinear Control Processes Academic Press NewYork NY USA 1973

[21] D L Elliott Bilinear Systems in Encyclopedia of Electrical Engi-neering Wiley New York NY USA 2001

[22] T-H S Li and S-H Tsai ldquoT-S fuzzy bilinear model and fuzzycontroller design for a class of nonlinear systemsrdquo IEEE Trans-actions on Fuzzy Systems vol 15 no 3 pp 494ndash506 2007

[23] S H Tsai and TH S Li ldquoRobust fuzzy control of a class of fuzzybilinear systems with time-delayrdquo Chaos Solitons and Fractalsvol 39 no 5 pp 2028ndash2040 2009

[24] T-H S Li S-H Tsai J-Z Lee M-Y Hsiao and C-H ChaoldquoRobust 119867

infinfuzzy control for a class of uncertain discrete

fuzzy bilinear systemsrdquo IEEE Transactions on SystemsMan andCybernetics Part B Cybernetics vol 38 no 2 pp 510ndash527 2008

[25] G Zhang J-M Li and Y-W Ge ldquoNonfragile guaranteed costcontrol of discrete-time fuzzy bilinear system with time-delayrdquoJournal of Dynamic Systems Measurement and Control Trans-actions of the ASME vol 136 no 4 Article ID 044502 2014

[26] J R Li J M Li and Z L Xia ldquoObserver-based fuzzy controldesign for discrete-time T-S fuzzy bilinear systemsrdquo Interna-tional Journal of Uncertainty Fuzziness and Knowledge-BasedSystems vol 21 no 3 pp 435ndash454 2013

[27] M S Ali ldquoRobust stability of stochastic fuzzy impulsive recur-rent neural networks with time varying delaysrdquo Iranian Journalof Fuzzy Systems vol 11 no 4 pp 1ndash13 2014

[28] M Syed Ali ldquoRobust stability analysis of Takagi-Sugeno uncer-tain stochastic fuzzy recurrent neural networks with mixedtime-varying delaysrdquo Chinese Physics B vol 20 no 8 ArticleID 080201 2011

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

8 Mathematical Problems in Engineering

0 1 2 3 4

0

2

4

6

8

10

minus10

minus8

minus6

minus4

minus4 minus3

minus2

minus2 minus1

x2

x1

Figure 1 The phase portrait of the chaotic system

System (52) can be modelled as the following T-S fuzzybilinear model

Rule 1 IF 1199091 (

119905) is about 0

THEN (119905) = (1198601+ Δ1198601) 119909 (119905) + (119861

1+ Δ1198611) 119906 (119905)

+ (1198621+ Δ1198621) 119909 (119905) 119906 (119905)

Rule 2 IF 1199091 (

119905) is about plusmn 2

THEN (119905) = (1198602+ Δ1198602) 119909 (119905) + (119861

2+ Δ1198612) 119906 (119905)

+ (1198622+ Δ1198622) 119909 (119905) 119906 (119905)

Rule 3 IF 1199091 (

119905) is about plusmn 4

THEN (119905) = (1198603+ Δ1198603) 119909 (119905) + (119861

3+ Δ1198613) 119906 (119905)

+ (1198623+ Δ1198623) 119909 (119905) 119906 (119905)

(53)

where 119909(119905) = (1199091(119905) 1199092(119905))119879 1198601= 1198602= 1198603= [0 1

minus1 minus1] 1198611=

1198612= 1198613= [0

minus1] and 119862

1= 1198622= 1198623= [0 1

1 1] Choose 120572 = 03

120588 = 009 1198651= [0 minus1] 119865

2= [minus1 minus1] and 119865

3= [minus1 minus1] By

solving LMIs (16)-(17) one can obtain

119875 = [

93637 46993

46993 106246] 120582min (119875) = 52528 (54)

Utilize the controllers (32) (13) and (40) and the parameterupdated law (41) to control system (52) The design parame-ters are chosen as 120578 = 2 120574 = 2 lt radic120582min(119875) the initial condi-tions are chosen as 119909(0) = (2 minus2)

119879 119888(0) = 0 and the relation-ship functions are selected as shown in Figure 2 The simula-tion results are shown in Figures 3 4 5 and 6 In Figures 3ndash6the curves of states control input and adaptive updatedparameter for the T-S fuzzy bilinear system are drawn by solidlines respectively while the curves of states control inputand adaptive updated parameter for T-S fuzzy linear systemare depicted by dotted lines respectively By comparison

Rule 3 Rule 2 Rule 1 Rule 2 Rule 3

minus4 minus3 minus2 minus1 0 1 2 3 4

x1

M(x

1)

0

02

04

06

08

1

Figure 2 The relationship functions

0 1 2 3 4 5 6 7 8 9 10Time (s)

minus05

0

05

1

15

2

x1

T-S FBST-S FLS

Figure 3 The state 1199091response curves

the convergence rates of the states of two systems are almostthe same though the state and control amplitudes of T-S fuzzybilinear system (FBS) are smaller thanT-S fuzzy linear system(FLS) Thus the proposed method has some advantages ofperformance over the existing approach [16]

Example 2 Consider the following parametric uncertainmultiple inputs bilinear fuzzy system

Rule 1 IF 1199091 (

119905) is 1198711

THEN (119905) = 1198601119909 (119905) + 119861

1119906 (119905) + 119862

1119906 (119905) 119909 (119905)

+ 119876 (Δ1198861119909 (119905) + Δ119887

1119906 (119905)

+ Δ1198621119906 (119905) 119909 (119905))

Rule 2 IF 1199092 (

119905) is 1198712

THEN (119905) = 1198602119909 (119905) + 119861

2119906 (119905) + 119862

2119906 (119905) 119909 (119905)

+ 119876 (Δ1198862119909 (119905) + Δ119887

2119906 (119905)

+ Δ1198622119906 (119905) 119909 (119905))

(55)

Mathematical Problems in Engineering 9

0 1 2 3 4 5 6 7 8 9 10Time (s)

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

x2

T-S FBST-S FLS

Figure 4 The state 1199092response curves

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

12

Time (s)

Con

trol u

minus4

minus2

T-S FBST-S FLS

Figure 5 The control curves

where 119909(119905) = (1199091(119905) 1199092(119905))119879 1198601= 1198602= [0 1

minus1 minus1] 11986111

= [0

1]

11986112

= [0

minus1] 11986121

= 11986122

= [0

1] 11986211

= [0 0

minus1 1] 11986212

= 11986221

=

11986222

= [0 0

minus1 minus1] 120572 = 03 120588

1= 005 120588

2= 004 119865

11=

[minus12 minus18] 11986512

= [minus08 minus09] 11986521

= [minus05 minus12] and11986522

= [minus1 minus1] using LMI technique to solve (16)-(17) wecan get a feasible solution as

119875 = [

168099 85379

85379 180100] 120582min (119875) = 88510 (56)

Apply the controllers (32) (13) and (40) and the parametersupdated law (41) to system (55) The design parameters arechosen as 120578 = 2 120574 = 2 lt radic2120572120582min(119875) The initial conditions

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Time (s)

Para

met

er c

T-S FBST-S FLS

Figure 6 The curves of adaptive updated parameters

0 5 10 15 20 25 30Time (s)

minus05

0

05

1

15

2

Stat

esx1

T-S FBST-S FLS

Figure 7 Responses of system state 1199091(T-S FBS solid line T-S FLS

dotted line)

are 119909(0) = (2 minus08)119879 119888(0) = 0 The simulation results are

shown in Figures 7 8 9 10 and 11Through the comparison between T-S fuzzy linear model

and bilinear one we can see that the settling time of thesystems is almost the same under the same initial conditionsalthough responses of T-S fuzzy bilinear system (FBS) stateamplitudes are smaller than T-S fuzzy linear system (FLS)and the demand of the control input of the system is lowThus the proposedmethod has better dynamic performancesthan the existing ones based on T-S fuzzy linear model

10 Mathematical Problems in Engineering

0 5 10 15 20 25 30Time (s)

minus1

minus08

minus06

minus04

minus02

0

02

04

Stat

esx2

T-S FBST-S FLS

Figure 8 Responses of system state 1199092(T-S FBS solid line T-S FLS

dotted line)

0 5 10 15 20 25 30Time (s)

minus15

minus1

minus05

0

05

1

15

T-S FBST-S FLS

Con

trol i

nput

u1

Figure 9 Control input1199061(T-S FBS solid line T-S FLS dotted line)

5 Conclusion

This paper proposes a new modelling method based onthe multiple inputs T-S fuzzy bilinear model which is usedto approximate nonlinear system the parallel distributedcompensation (PDC) method is utilized to design the fuzzycontroller without considering the error caused by fuzzymodellingThe sufficient conditionswith respect to decay rate120572 are derived by linear matrix inequalities (LMIs) The errorcaused by fuzzy modelling is considered and the method ofadaptive control is used to reduce the effect of the modelling

0 5 10 15 20 25 30Time (s)

minus08

minus06

minus04

minus02

0

02

04

06

Con

trol i

nput

u2

T-S FBST-S FLS

Figure 10 Control input 1199062(T-S FBS solid line T-S FLS dotted

line)

0 5 10 15 20 25 30Time (s)

0

01

02

03

04

05

06

07

08

09

Adap

tive p

aram

eter

c

T-S FBST-S FLS

Figure 11 Adaptive parameter 119888 (T-S FBS solid line T-S FLS dottedline)

error By Lyapunov stability criterion the resulting closed-loop system is proved to be asymptotically stable Finally twoillustrative examples are provided to show that the approachbased T-S fuzzy bilinear systems have some advantages ofperformance over the existing methods based on T-S fuzzylinear system The future research work is to extend theapproach to general system such as discrete-time systemsstochastic systems and time-delay systems

Mathematical Problems in Engineering 11

Conflict of Interests

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

Acknowledgment

This project was supported by the Soft Science Foundation ofShanxi Province (2011041033-3)

References

[1] K Tanaka and H O Wang Fuzzy Control Systems Design andAnalysis A Linear Matrix Inequality Approach Wiley-Intersci-ence 2001

[2] G Feng ldquoA survey on analysis and design of model-based fuzzycontrol systemsrdquo IEEE Transactions on Fuzzy Systems vol 14no 5 pp 676ndash697 2006

[3] K Tanaka T Ikeda and H O Wang ldquoFuzzy regulators andfuzzy observers relaxed stability conditions and LMI-baseddesignsrdquo IEEE Transactions on Fuzzy Systems vol 6 no 2 pp250ndash265 1998

[4] E Kim and H Lee ldquoNew approaches to relaxed quadraticstability condition of fuzzy control systemsrdquo IEEE Transactionson Fuzzy Systems vol 8 no 5 pp 523ndash534 2000

[5] K Tanaka andM Sugeno ldquoStability analysis and design of fuzzycontrol systemsrdquo Fuzzy Sets and Systems vol 45 no 2 pp 135ndash156 1992

[6] H-N Wu and H-X Li ldquoNew approach to delay-dependentstability analysis and stabilization for continuous-time fuzzysystems with time-varying delayrdquo IEEE Transactions on FuzzySystems vol 15 no 3 pp 482ndash493 2007

[7] F Zheng Q-G Wang and T H Lee ldquoAdaptive and robustcontroller design for uncertain nonlinear systems via fuzzymodeling approachrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 34 no 1 pp 166ndash178 2004

[8] C L Hwang ldquoA novel Takagi-Sugeno-based robust adaptivefuzzy sliding-mode controllerrdquo IEEE Transactions on FuzzySystems vol 12 no 5 pp 676ndash687 2004

[9] S Dong Adaptive Fuzzy Control of Nonlinear System Scienceand Technology Publishing House Beijing China 2006

[10] W-Y Wang Y-H Chien Y-G Leu and T-T Lee ldquoAdaptiveT-S fuzzy-neural modeling and control for general MIMOunknown non-affine nonlinear systems using projection updatelawsrdquo Automatica vol 46 no 5 pp 852ndash863 2010

[11] Y-H Chien W-Y Wang Y-G Leu and T-T Lee ldquoRobustadaptive controller design for a class of uncertain nonlinear sys-tems using online T-S fuzzy-neural modeling approachrdquo IEEETransactions on Systems Man and Cybernetics Part B Cyber-netics vol 41 no 2 pp 542ndash552 2011

[12] S P Moustakidis G A Rovithakis and J B Theocharis ldquoAnadaptive neuro-fuzzy tracking control formulti-input nonlineardynamic systemsrdquo Automatica vol 44 no 5 pp 1418ndash14252008

[13] K-Y Lian and H-W Tu ldquoLMI-Based adaptive tracking controlfor parametric strict-feedback systemsrdquo IEEE Transactions onFuzzy Systems vol 16 no 5 pp 1245ndash1258 2008

[14] Z Lendek J Lauber T M Guerra R Babuka and B De Schut-ter ldquoAdaptive observers for TS fuzzy systems with unknownpolynomial inputsrdquo Fuzzy Sets and Systems vol 161 no 15 pp2043ndash2065 2010

[15] C-H Hyun C-W Park and S Kim ldquoTakagi-Sugeno fuzzymodel based indirect adaptive fuzzy observer and controllerdesignrdquo Information Sciences vol 180 no 11 pp 2314ndash23272010

[16] Y-H Chang W-S Chan and C-W Chang ldquoT-S fuzzy model-based adaptive dynamic surface control for ball and beamsystemrdquo IEEE Transactions on Industrial Electronics vol 60 no6 pp 2251ndash2263 2013

[17] S TWang and J T Fei ldquoRobust adaptive slidingmode control ofMEMS gyroscope using T-S fuzzymodelrdquoNonlinear Dynamicsvol 77 no 1-2 pp 361ndash371 2014

[18] R Qi G Tao C Tan and X Yao ldquoAdaptive control of discrete-time state-space T-S fuzzy systems with general relative degreerdquoFuzzy Sets and Systems vol 217 pp 22ndash40 2013

[19] H B Jiang J J Yu and C G Zhou ldquoStable adaptive fuzzycontrol of nonlinear systems using small-gain theorem and LMIapproachrdquo Journal of ControlTheory andApplications vol 8 no4 pp 527ndash532 2010

[20] R R Mohler Bilinear Control Processes Academic Press NewYork NY USA 1973

[21] D L Elliott Bilinear Systems in Encyclopedia of Electrical Engi-neering Wiley New York NY USA 2001

[22] T-H S Li and S-H Tsai ldquoT-S fuzzy bilinear model and fuzzycontroller design for a class of nonlinear systemsrdquo IEEE Trans-actions on Fuzzy Systems vol 15 no 3 pp 494ndash506 2007

[23] S H Tsai and TH S Li ldquoRobust fuzzy control of a class of fuzzybilinear systems with time-delayrdquo Chaos Solitons and Fractalsvol 39 no 5 pp 2028ndash2040 2009

[24] T-H S Li S-H Tsai J-Z Lee M-Y Hsiao and C-H ChaoldquoRobust 119867

infinfuzzy control for a class of uncertain discrete

fuzzy bilinear systemsrdquo IEEE Transactions on SystemsMan andCybernetics Part B Cybernetics vol 38 no 2 pp 510ndash527 2008

[25] G Zhang J-M Li and Y-W Ge ldquoNonfragile guaranteed costcontrol of discrete-time fuzzy bilinear system with time-delayrdquoJournal of Dynamic Systems Measurement and Control Trans-actions of the ASME vol 136 no 4 Article ID 044502 2014

[26] J R Li J M Li and Z L Xia ldquoObserver-based fuzzy controldesign for discrete-time T-S fuzzy bilinear systemsrdquo Interna-tional Journal of Uncertainty Fuzziness and Knowledge-BasedSystems vol 21 no 3 pp 435ndash454 2013

[27] M S Ali ldquoRobust stability of stochastic fuzzy impulsive recur-rent neural networks with time varying delaysrdquo Iranian Journalof Fuzzy Systems vol 11 no 4 pp 1ndash13 2014

[28] M Syed Ali ldquoRobust stability analysis of Takagi-Sugeno uncer-tain stochastic fuzzy recurrent neural networks with mixedtime-varying delaysrdquo Chinese Physics B vol 20 no 8 ArticleID 080201 2011

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

0 1 2 3 4 5 6 7 8 9 10Time (s)

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

x2

T-S FBST-S FLS

Figure 4 The state 1199092response curves

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

12

Time (s)

Con

trol u

minus4

minus2

T-S FBST-S FLS

Figure 5 The control curves

where 119909(119905) = (1199091(119905) 1199092(119905))119879 1198601= 1198602= [0 1

minus1 minus1] 11986111

= [0

1]

11986112

= [0

minus1] 11986121

= 11986122

= [0

1] 11986211

= [0 0

minus1 1] 11986212

= 11986221

=

11986222

= [0 0

minus1 minus1] 120572 = 03 120588

1= 005 120588

2= 004 119865

11=

[minus12 minus18] 11986512

= [minus08 minus09] 11986521

= [minus05 minus12] and11986522

= [minus1 minus1] using LMI technique to solve (16)-(17) wecan get a feasible solution as

119875 = [

168099 85379

85379 180100] 120582min (119875) = 88510 (56)

Apply the controllers (32) (13) and (40) and the parametersupdated law (41) to system (55) The design parameters arechosen as 120578 = 2 120574 = 2 lt radic2120572120582min(119875) The initial conditions

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Time (s)

Para

met

er c

T-S FBST-S FLS

Figure 6 The curves of adaptive updated parameters

0 5 10 15 20 25 30Time (s)

minus05

0

05

1

15

2

Stat

esx1

T-S FBST-S FLS

Figure 7 Responses of system state 1199091(T-S FBS solid line T-S FLS

dotted line)

are 119909(0) = (2 minus08)119879 119888(0) = 0 The simulation results are

shown in Figures 7 8 9 10 and 11Through the comparison between T-S fuzzy linear model

and bilinear one we can see that the settling time of thesystems is almost the same under the same initial conditionsalthough responses of T-S fuzzy bilinear system (FBS) stateamplitudes are smaller than T-S fuzzy linear system (FLS)and the demand of the control input of the system is lowThus the proposedmethod has better dynamic performancesthan the existing ones based on T-S fuzzy linear model

10 Mathematical Problems in Engineering

0 5 10 15 20 25 30Time (s)

minus1

minus08

minus06

minus04

minus02

0

02

04

Stat

esx2

T-S FBST-S FLS

Figure 8 Responses of system state 1199092(T-S FBS solid line T-S FLS

dotted line)

0 5 10 15 20 25 30Time (s)

minus15

minus1

minus05

0

05

1

15

T-S FBST-S FLS

Con

trol i

nput

u1

Figure 9 Control input1199061(T-S FBS solid line T-S FLS dotted line)

5 Conclusion

This paper proposes a new modelling method based onthe multiple inputs T-S fuzzy bilinear model which is usedto approximate nonlinear system the parallel distributedcompensation (PDC) method is utilized to design the fuzzycontroller without considering the error caused by fuzzymodellingThe sufficient conditionswith respect to decay rate120572 are derived by linear matrix inequalities (LMIs) The errorcaused by fuzzy modelling is considered and the method ofadaptive control is used to reduce the effect of the modelling

0 5 10 15 20 25 30Time (s)

minus08

minus06

minus04

minus02

0

02

04

06

Con

trol i

nput

u2

T-S FBST-S FLS

Figure 10 Control input 1199062(T-S FBS solid line T-S FLS dotted

line)

0 5 10 15 20 25 30Time (s)

0

01

02

03

04

05

06

07

08

09

Adap

tive p

aram

eter

c

T-S FBST-S FLS

Figure 11 Adaptive parameter 119888 (T-S FBS solid line T-S FLS dottedline)

error By Lyapunov stability criterion the resulting closed-loop system is proved to be asymptotically stable Finally twoillustrative examples are provided to show that the approachbased T-S fuzzy bilinear systems have some advantages ofperformance over the existing methods based on T-S fuzzylinear system The future research work is to extend theapproach to general system such as discrete-time systemsstochastic systems and time-delay systems

Mathematical Problems in Engineering 11

Conflict of Interests

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

Acknowledgment

This project was supported by the Soft Science Foundation ofShanxi Province (2011041033-3)

References

[1] K Tanaka and H O Wang Fuzzy Control Systems Design andAnalysis A Linear Matrix Inequality Approach Wiley-Intersci-ence 2001

[2] G Feng ldquoA survey on analysis and design of model-based fuzzycontrol systemsrdquo IEEE Transactions on Fuzzy Systems vol 14no 5 pp 676ndash697 2006

[3] K Tanaka T Ikeda and H O Wang ldquoFuzzy regulators andfuzzy observers relaxed stability conditions and LMI-baseddesignsrdquo IEEE Transactions on Fuzzy Systems vol 6 no 2 pp250ndash265 1998

[4] E Kim and H Lee ldquoNew approaches to relaxed quadraticstability condition of fuzzy control systemsrdquo IEEE Transactionson Fuzzy Systems vol 8 no 5 pp 523ndash534 2000

[5] K Tanaka andM Sugeno ldquoStability analysis and design of fuzzycontrol systemsrdquo Fuzzy Sets and Systems vol 45 no 2 pp 135ndash156 1992

[6] H-N Wu and H-X Li ldquoNew approach to delay-dependentstability analysis and stabilization for continuous-time fuzzysystems with time-varying delayrdquo IEEE Transactions on FuzzySystems vol 15 no 3 pp 482ndash493 2007

[7] F Zheng Q-G Wang and T H Lee ldquoAdaptive and robustcontroller design for uncertain nonlinear systems via fuzzymodeling approachrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 34 no 1 pp 166ndash178 2004

[8] C L Hwang ldquoA novel Takagi-Sugeno-based robust adaptivefuzzy sliding-mode controllerrdquo IEEE Transactions on FuzzySystems vol 12 no 5 pp 676ndash687 2004

[9] S Dong Adaptive Fuzzy Control of Nonlinear System Scienceand Technology Publishing House Beijing China 2006

[10] W-Y Wang Y-H Chien Y-G Leu and T-T Lee ldquoAdaptiveT-S fuzzy-neural modeling and control for general MIMOunknown non-affine nonlinear systems using projection updatelawsrdquo Automatica vol 46 no 5 pp 852ndash863 2010

[11] Y-H Chien W-Y Wang Y-G Leu and T-T Lee ldquoRobustadaptive controller design for a class of uncertain nonlinear sys-tems using online T-S fuzzy-neural modeling approachrdquo IEEETransactions on Systems Man and Cybernetics Part B Cyber-netics vol 41 no 2 pp 542ndash552 2011

[12] S P Moustakidis G A Rovithakis and J B Theocharis ldquoAnadaptive neuro-fuzzy tracking control formulti-input nonlineardynamic systemsrdquo Automatica vol 44 no 5 pp 1418ndash14252008

[13] K-Y Lian and H-W Tu ldquoLMI-Based adaptive tracking controlfor parametric strict-feedback systemsrdquo IEEE Transactions onFuzzy Systems vol 16 no 5 pp 1245ndash1258 2008

[14] Z Lendek J Lauber T M Guerra R Babuka and B De Schut-ter ldquoAdaptive observers for TS fuzzy systems with unknownpolynomial inputsrdquo Fuzzy Sets and Systems vol 161 no 15 pp2043ndash2065 2010

[15] C-H Hyun C-W Park and S Kim ldquoTakagi-Sugeno fuzzymodel based indirect adaptive fuzzy observer and controllerdesignrdquo Information Sciences vol 180 no 11 pp 2314ndash23272010

[16] Y-H Chang W-S Chan and C-W Chang ldquoT-S fuzzy model-based adaptive dynamic surface control for ball and beamsystemrdquo IEEE Transactions on Industrial Electronics vol 60 no6 pp 2251ndash2263 2013

[17] S TWang and J T Fei ldquoRobust adaptive slidingmode control ofMEMS gyroscope using T-S fuzzymodelrdquoNonlinear Dynamicsvol 77 no 1-2 pp 361ndash371 2014

[18] R Qi G Tao C Tan and X Yao ldquoAdaptive control of discrete-time state-space T-S fuzzy systems with general relative degreerdquoFuzzy Sets and Systems vol 217 pp 22ndash40 2013

[19] H B Jiang J J Yu and C G Zhou ldquoStable adaptive fuzzycontrol of nonlinear systems using small-gain theorem and LMIapproachrdquo Journal of ControlTheory andApplications vol 8 no4 pp 527ndash532 2010

[20] R R Mohler Bilinear Control Processes Academic Press NewYork NY USA 1973

[21] D L Elliott Bilinear Systems in Encyclopedia of Electrical Engi-neering Wiley New York NY USA 2001

[22] T-H S Li and S-H Tsai ldquoT-S fuzzy bilinear model and fuzzycontroller design for a class of nonlinear systemsrdquo IEEE Trans-actions on Fuzzy Systems vol 15 no 3 pp 494ndash506 2007

[23] S H Tsai and TH S Li ldquoRobust fuzzy control of a class of fuzzybilinear systems with time-delayrdquo Chaos Solitons and Fractalsvol 39 no 5 pp 2028ndash2040 2009

[24] T-H S Li S-H Tsai J-Z Lee M-Y Hsiao and C-H ChaoldquoRobust 119867

infinfuzzy control for a class of uncertain discrete

fuzzy bilinear systemsrdquo IEEE Transactions on SystemsMan andCybernetics Part B Cybernetics vol 38 no 2 pp 510ndash527 2008

[25] G Zhang J-M Li and Y-W Ge ldquoNonfragile guaranteed costcontrol of discrete-time fuzzy bilinear system with time-delayrdquoJournal of Dynamic Systems Measurement and Control Trans-actions of the ASME vol 136 no 4 Article ID 044502 2014

[26] J R Li J M Li and Z L Xia ldquoObserver-based fuzzy controldesign for discrete-time T-S fuzzy bilinear systemsrdquo Interna-tional Journal of Uncertainty Fuzziness and Knowledge-BasedSystems vol 21 no 3 pp 435ndash454 2013

[27] M S Ali ldquoRobust stability of stochastic fuzzy impulsive recur-rent neural networks with time varying delaysrdquo Iranian Journalof Fuzzy Systems vol 11 no 4 pp 1ndash13 2014

[28] M Syed Ali ldquoRobust stability analysis of Takagi-Sugeno uncer-tain stochastic fuzzy recurrent neural networks with mixedtime-varying delaysrdquo Chinese Physics B vol 20 no 8 ArticleID 080201 2011

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

0 5 10 15 20 25 30Time (s)

minus1

minus08

minus06

minus04

minus02

0

02

04

Stat

esx2

T-S FBST-S FLS

Figure 8 Responses of system state 1199092(T-S FBS solid line T-S FLS

dotted line)

0 5 10 15 20 25 30Time (s)

minus15

minus1

minus05

0

05

1

15

T-S FBST-S FLS

Con

trol i

nput

u1

Figure 9 Control input1199061(T-S FBS solid line T-S FLS dotted line)

5 Conclusion

This paper proposes a new modelling method based onthe multiple inputs T-S fuzzy bilinear model which is usedto approximate nonlinear system the parallel distributedcompensation (PDC) method is utilized to design the fuzzycontroller without considering the error caused by fuzzymodellingThe sufficient conditionswith respect to decay rate120572 are derived by linear matrix inequalities (LMIs) The errorcaused by fuzzy modelling is considered and the method ofadaptive control is used to reduce the effect of the modelling

0 5 10 15 20 25 30Time (s)

minus08

minus06

minus04

minus02

0

02

04

06

Con

trol i

nput

u2

T-S FBST-S FLS

Figure 10 Control input 1199062(T-S FBS solid line T-S FLS dotted

line)

0 5 10 15 20 25 30Time (s)

0

01

02

03

04

05

06

07

08

09

Adap

tive p

aram

eter

c

T-S FBST-S FLS

Figure 11 Adaptive parameter 119888 (T-S FBS solid line T-S FLS dottedline)

error By Lyapunov stability criterion the resulting closed-loop system is proved to be asymptotically stable Finally twoillustrative examples are provided to show that the approachbased T-S fuzzy bilinear systems have some advantages ofperformance over the existing methods based on T-S fuzzylinear system The future research work is to extend theapproach to general system such as discrete-time systemsstochastic systems and time-delay systems

Mathematical Problems in Engineering 11

Conflict of Interests

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

Acknowledgment

This project was supported by the Soft Science Foundation ofShanxi Province (2011041033-3)

References

[1] K Tanaka and H O Wang Fuzzy Control Systems Design andAnalysis A Linear Matrix Inequality Approach Wiley-Intersci-ence 2001

[2] G Feng ldquoA survey on analysis and design of model-based fuzzycontrol systemsrdquo IEEE Transactions on Fuzzy Systems vol 14no 5 pp 676ndash697 2006

[3] K Tanaka T Ikeda and H O Wang ldquoFuzzy regulators andfuzzy observers relaxed stability conditions and LMI-baseddesignsrdquo IEEE Transactions on Fuzzy Systems vol 6 no 2 pp250ndash265 1998

[4] E Kim and H Lee ldquoNew approaches to relaxed quadraticstability condition of fuzzy control systemsrdquo IEEE Transactionson Fuzzy Systems vol 8 no 5 pp 523ndash534 2000

[5] K Tanaka andM Sugeno ldquoStability analysis and design of fuzzycontrol systemsrdquo Fuzzy Sets and Systems vol 45 no 2 pp 135ndash156 1992

[6] H-N Wu and H-X Li ldquoNew approach to delay-dependentstability analysis and stabilization for continuous-time fuzzysystems with time-varying delayrdquo IEEE Transactions on FuzzySystems vol 15 no 3 pp 482ndash493 2007

[7] F Zheng Q-G Wang and T H Lee ldquoAdaptive and robustcontroller design for uncertain nonlinear systems via fuzzymodeling approachrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 34 no 1 pp 166ndash178 2004

[8] C L Hwang ldquoA novel Takagi-Sugeno-based robust adaptivefuzzy sliding-mode controllerrdquo IEEE Transactions on FuzzySystems vol 12 no 5 pp 676ndash687 2004

[9] S Dong Adaptive Fuzzy Control of Nonlinear System Scienceand Technology Publishing House Beijing China 2006

[10] W-Y Wang Y-H Chien Y-G Leu and T-T Lee ldquoAdaptiveT-S fuzzy-neural modeling and control for general MIMOunknown non-affine nonlinear systems using projection updatelawsrdquo Automatica vol 46 no 5 pp 852ndash863 2010

[11] Y-H Chien W-Y Wang Y-G Leu and T-T Lee ldquoRobustadaptive controller design for a class of uncertain nonlinear sys-tems using online T-S fuzzy-neural modeling approachrdquo IEEETransactions on Systems Man and Cybernetics Part B Cyber-netics vol 41 no 2 pp 542ndash552 2011

[12] S P Moustakidis G A Rovithakis and J B Theocharis ldquoAnadaptive neuro-fuzzy tracking control formulti-input nonlineardynamic systemsrdquo Automatica vol 44 no 5 pp 1418ndash14252008

[13] K-Y Lian and H-W Tu ldquoLMI-Based adaptive tracking controlfor parametric strict-feedback systemsrdquo IEEE Transactions onFuzzy Systems vol 16 no 5 pp 1245ndash1258 2008

[14] Z Lendek J Lauber T M Guerra R Babuka and B De Schut-ter ldquoAdaptive observers for TS fuzzy systems with unknownpolynomial inputsrdquo Fuzzy Sets and Systems vol 161 no 15 pp2043ndash2065 2010

[15] C-H Hyun C-W Park and S Kim ldquoTakagi-Sugeno fuzzymodel based indirect adaptive fuzzy observer and controllerdesignrdquo Information Sciences vol 180 no 11 pp 2314ndash23272010

[16] Y-H Chang W-S Chan and C-W Chang ldquoT-S fuzzy model-based adaptive dynamic surface control for ball and beamsystemrdquo IEEE Transactions on Industrial Electronics vol 60 no6 pp 2251ndash2263 2013

[17] S TWang and J T Fei ldquoRobust adaptive slidingmode control ofMEMS gyroscope using T-S fuzzymodelrdquoNonlinear Dynamicsvol 77 no 1-2 pp 361ndash371 2014

[18] R Qi G Tao C Tan and X Yao ldquoAdaptive control of discrete-time state-space T-S fuzzy systems with general relative degreerdquoFuzzy Sets and Systems vol 217 pp 22ndash40 2013

[19] H B Jiang J J Yu and C G Zhou ldquoStable adaptive fuzzycontrol of nonlinear systems using small-gain theorem and LMIapproachrdquo Journal of ControlTheory andApplications vol 8 no4 pp 527ndash532 2010

[20] R R Mohler Bilinear Control Processes Academic Press NewYork NY USA 1973

[21] D L Elliott Bilinear Systems in Encyclopedia of Electrical Engi-neering Wiley New York NY USA 2001

[22] T-H S Li and S-H Tsai ldquoT-S fuzzy bilinear model and fuzzycontroller design for a class of nonlinear systemsrdquo IEEE Trans-actions on Fuzzy Systems vol 15 no 3 pp 494ndash506 2007

[23] S H Tsai and TH S Li ldquoRobust fuzzy control of a class of fuzzybilinear systems with time-delayrdquo Chaos Solitons and Fractalsvol 39 no 5 pp 2028ndash2040 2009

[24] T-H S Li S-H Tsai J-Z Lee M-Y Hsiao and C-H ChaoldquoRobust 119867

infinfuzzy control for a class of uncertain discrete

fuzzy bilinear systemsrdquo IEEE Transactions on SystemsMan andCybernetics Part B Cybernetics vol 38 no 2 pp 510ndash527 2008

[25] G Zhang J-M Li and Y-W Ge ldquoNonfragile guaranteed costcontrol of discrete-time fuzzy bilinear system with time-delayrdquoJournal of Dynamic Systems Measurement and Control Trans-actions of the ASME vol 136 no 4 Article ID 044502 2014

[26] J R Li J M Li and Z L Xia ldquoObserver-based fuzzy controldesign for discrete-time T-S fuzzy bilinear systemsrdquo Interna-tional Journal of Uncertainty Fuzziness and Knowledge-BasedSystems vol 21 no 3 pp 435ndash454 2013

[27] M S Ali ldquoRobust stability of stochastic fuzzy impulsive recur-rent neural networks with time varying delaysrdquo Iranian Journalof Fuzzy Systems vol 11 no 4 pp 1ndash13 2014

[28] M Syed Ali ldquoRobust stability analysis of Takagi-Sugeno uncer-tain stochastic fuzzy recurrent neural networks with mixedtime-varying delaysrdquo Chinese Physics B vol 20 no 8 ArticleID 080201 2011

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

Conflict of Interests

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

Acknowledgment

This project was supported by the Soft Science Foundation ofShanxi Province (2011041033-3)

References

[1] K Tanaka and H O Wang Fuzzy Control Systems Design andAnalysis A Linear Matrix Inequality Approach Wiley-Intersci-ence 2001

[2] G Feng ldquoA survey on analysis and design of model-based fuzzycontrol systemsrdquo IEEE Transactions on Fuzzy Systems vol 14no 5 pp 676ndash697 2006

[3] K Tanaka T Ikeda and H O Wang ldquoFuzzy regulators andfuzzy observers relaxed stability conditions and LMI-baseddesignsrdquo IEEE Transactions on Fuzzy Systems vol 6 no 2 pp250ndash265 1998

[4] E Kim and H Lee ldquoNew approaches to relaxed quadraticstability condition of fuzzy control systemsrdquo IEEE Transactionson Fuzzy Systems vol 8 no 5 pp 523ndash534 2000

[5] K Tanaka andM Sugeno ldquoStability analysis and design of fuzzycontrol systemsrdquo Fuzzy Sets and Systems vol 45 no 2 pp 135ndash156 1992

[6] H-N Wu and H-X Li ldquoNew approach to delay-dependentstability analysis and stabilization for continuous-time fuzzysystems with time-varying delayrdquo IEEE Transactions on FuzzySystems vol 15 no 3 pp 482ndash493 2007

[7] F Zheng Q-G Wang and T H Lee ldquoAdaptive and robustcontroller design for uncertain nonlinear systems via fuzzymodeling approachrdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 34 no 1 pp 166ndash178 2004

[8] C L Hwang ldquoA novel Takagi-Sugeno-based robust adaptivefuzzy sliding-mode controllerrdquo IEEE Transactions on FuzzySystems vol 12 no 5 pp 676ndash687 2004

[9] S Dong Adaptive Fuzzy Control of Nonlinear System Scienceand Technology Publishing House Beijing China 2006

[10] W-Y Wang Y-H Chien Y-G Leu and T-T Lee ldquoAdaptiveT-S fuzzy-neural modeling and control for general MIMOunknown non-affine nonlinear systems using projection updatelawsrdquo Automatica vol 46 no 5 pp 852ndash863 2010

[11] Y-H Chien W-Y Wang Y-G Leu and T-T Lee ldquoRobustadaptive controller design for a class of uncertain nonlinear sys-tems using online T-S fuzzy-neural modeling approachrdquo IEEETransactions on Systems Man and Cybernetics Part B Cyber-netics vol 41 no 2 pp 542ndash552 2011

[12] S P Moustakidis G A Rovithakis and J B Theocharis ldquoAnadaptive neuro-fuzzy tracking control formulti-input nonlineardynamic systemsrdquo Automatica vol 44 no 5 pp 1418ndash14252008

[13] K-Y Lian and H-W Tu ldquoLMI-Based adaptive tracking controlfor parametric strict-feedback systemsrdquo IEEE Transactions onFuzzy Systems vol 16 no 5 pp 1245ndash1258 2008

[14] Z Lendek J Lauber T M Guerra R Babuka and B De Schut-ter ldquoAdaptive observers for TS fuzzy systems with unknownpolynomial inputsrdquo Fuzzy Sets and Systems vol 161 no 15 pp2043ndash2065 2010

[15] C-H Hyun C-W Park and S Kim ldquoTakagi-Sugeno fuzzymodel based indirect adaptive fuzzy observer and controllerdesignrdquo Information Sciences vol 180 no 11 pp 2314ndash23272010

[16] Y-H Chang W-S Chan and C-W Chang ldquoT-S fuzzy model-based adaptive dynamic surface control for ball and beamsystemrdquo IEEE Transactions on Industrial Electronics vol 60 no6 pp 2251ndash2263 2013

[17] S TWang and J T Fei ldquoRobust adaptive slidingmode control ofMEMS gyroscope using T-S fuzzymodelrdquoNonlinear Dynamicsvol 77 no 1-2 pp 361ndash371 2014

[18] R Qi G Tao C Tan and X Yao ldquoAdaptive control of discrete-time state-space T-S fuzzy systems with general relative degreerdquoFuzzy Sets and Systems vol 217 pp 22ndash40 2013

[19] H B Jiang J J Yu and C G Zhou ldquoStable adaptive fuzzycontrol of nonlinear systems using small-gain theorem and LMIapproachrdquo Journal of ControlTheory andApplications vol 8 no4 pp 527ndash532 2010

[20] R R Mohler Bilinear Control Processes Academic Press NewYork NY USA 1973

[21] D L Elliott Bilinear Systems in Encyclopedia of Electrical Engi-neering Wiley New York NY USA 2001

[22] T-H S Li and S-H Tsai ldquoT-S fuzzy bilinear model and fuzzycontroller design for a class of nonlinear systemsrdquo IEEE Trans-actions on Fuzzy Systems vol 15 no 3 pp 494ndash506 2007

[23] S H Tsai and TH S Li ldquoRobust fuzzy control of a class of fuzzybilinear systems with time-delayrdquo Chaos Solitons and Fractalsvol 39 no 5 pp 2028ndash2040 2009

[24] T-H S Li S-H Tsai J-Z Lee M-Y Hsiao and C-H ChaoldquoRobust 119867

infinfuzzy control for a class of uncertain discrete

fuzzy bilinear systemsrdquo IEEE Transactions on SystemsMan andCybernetics Part B Cybernetics vol 38 no 2 pp 510ndash527 2008

[25] G Zhang J-M Li and Y-W Ge ldquoNonfragile guaranteed costcontrol of discrete-time fuzzy bilinear system with time-delayrdquoJournal of Dynamic Systems Measurement and Control Trans-actions of the ASME vol 136 no 4 Article ID 044502 2014

[26] J R Li J M Li and Z L Xia ldquoObserver-based fuzzy controldesign for discrete-time T-S fuzzy bilinear systemsrdquo Interna-tional Journal of Uncertainty Fuzziness and Knowledge-BasedSystems vol 21 no 3 pp 435ndash454 2013

[27] M S Ali ldquoRobust stability of stochastic fuzzy impulsive recur-rent neural networks with time varying delaysrdquo Iranian Journalof Fuzzy Systems vol 11 no 4 pp 1ndash13 2014

[28] M Syed Ali ldquoRobust stability analysis of Takagi-Sugeno uncer-tain stochastic fuzzy recurrent neural networks with mixedtime-varying delaysrdquo Chinese Physics B vol 20 no 8 ArticleID 080201 2011

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