Research ArticleA Simplified Output Regulator for a Class ofTakagi-Sugeno Fuzzy ModelsTonatiuh Hernndez-Corts, Jess A. Meda Campaa,Luis A. Pramo Carranza, and Julio C. Gmez MancillaInstituto Polit ecnico Nacional, SEPI-ESIME Zacatenco, Avenue IPN S/N, 07738 M exico, DF, MexicoCorrespondence should be addressed to Jes us A. Meda Campa na; jmedac@ipn.mxReceived 10 January 2015; Revised 1 April 2015; Accepted 1 April 2015Academic Editor: Qingling ZhangCopyright 2015 Tonatiuh Hern andez-Cort es et al. TisisanopenaccessarticledistributedundertheCreativeCommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.Tis paper is devoted to solve the regulation problemon the basis of local regulators, which are combined using new membershipfunctions. As a result, the exact tracking of references is achieved. Te design of linear local regulators is suggested in this paper,but now adequate membership functions are computed in order to ensure the proper combination of the local regulators inthe interpolation regions. Tese membership functions, which are given as mathematical expressions, solve the fuzzy regulationproblem in a relative simple way. Te form of the new membership functions is systematically derived for a class of Takagi-Sugeno(T-S) fuzzy systems. Some numerical examples are used to illustrate the viability of the proposed approach.1. IntroductionOne of the most important problems incontrol theoryconsists of fnding a controller capable of taking the outputs ofa plant towards the reference signals generated by an externalsystem, named exosystem.Tis problem has been studied by several authors due toits wide applicability in mechanical systems, aeronautics, andtelematics, just to name a few.Te works of Francis [1] and Francis and Wonham [2]haveshownthat thesolvabilityof amultivariablelinearregulator problem corresponds to the solvability of a systemof two linear matrix equations, called Francis Equations. Alsotheyhaveshownthat, inthecaseof errorfeedback, theregulator which solves the problem includes the exosystem.Tis property is commonly knowas Internal Model Principle.Later, Isidori, and Byrnes [3] showed that the result estab-lished by Francis could be extended to the nonlinear feld asa general case and that the equations of Francis represent aparticular case of a set of nonlinear equations. Tey showedthat the solvability for the nonlinear case depends on thesolution of a set of nonlinear partial diferential equations,called Francis-Isidori-Byrnes (FIB) equations. Unfortunately,such equations are too difcult to solve in a practical manner,in general.On the other hand, some techniques have been provento be an alternative to solve this problem by combining thetheoryof outputregulationandtheTakagi-Sugenofuzzymodeling [49].In [10], the authors propose an approach based on theweighted summation of local linear regulators in order tosynchronize chaotic systems by means of regulation theory.However, to ensure the exact output regulation, two condi-tions need to be fulflled: (1) the same input matrix for allsubsystems, that is, 1 = 2 = =

, where is the numberof rules in the fuzzy model, and (2) same zero error manifold(()) for every local subsystem.Later in [11], the exact output regulator is directlydesignedfortheoverall T-Sfuzzymodel. Althoughsucha controller achieves the exact output regulationwherethe weighted summation of linear local regulators fails, itsexpression may be very large.For that reason, in the present paper the simplicity ofthe fuzzy regulator obtained from the weighted summationof linear local regulators is exploited; the efectiveness of thecontroller given in this approach guarantees at least for a classHindawi Publishing CorporationMathematical Problems in EngineeringArticle ID 1481732 Mathematical Problems in Engineeringof T-S fuzzy models the exact output regulation. To this end,new membership functions will be systematically computedin order to adequately combine the linear local regulators,guaranteeing in this way the proper performance of the fuzzyregulator in the interpolation regions. A preliminary resulthas been given in [6], where the new membership functionsare approximated by sof computing techniques.Te mainidea comes fromthe fact that each local control-ler is valid, at least, for its corresponding local system, whilethe fuzzy interpolation regions require more attention at themoment of evaluating the performance of the overall fuzzycontroller. For that reason, the proposed approach consistsof fnding new membership functions capable of adequatelycombine adjacent local controllers in order to achieve thecontrol goal.So, the main contribution of the present work is to fnda control law for a class of Takagi-Sugeno fuzzy models, inorder to achieve exact output regulation on the basis of localregulatorsandcomputingof newmembershipfunctions,even if diferent input matrices appear in the linear localsubsystems. Consequently, one of the restrictions given in[10] is avoided, and there is no need of verifying the existenceof the fuzzy regulator for all 0 [11]. Besides, the newmembership functions, allowing the proper combination thelocal regulators, are given as a mathematical expressions.Te rest of the paper is organized as follows. In Section 2the nonlinear regulation problem formulation is given witha brief review of the Takagi-Sugeno models and the fuzzyregulationproblem. Te mainresult is developed inSection3.InSection4someexamplesarepresentedandfnally, inSection 5, some conclusions are drawn.2. The Output Regulation ProblemConsider a nonlinear system defned by () = (() , () , ()) , (1)() = (()) , (2) () = (()) , (3)

ref () = (()) , (4) () = (() , ()) , (5)where () R

is the state vector of the plant, () R

is the state vector of the exosystem, which generates thereference and/or the perturbation signals, and () R

isthe input signal. Equation (5) refers to diference betweenoutput system of the plant (() R

) and the referencesignal (ref() R

), that is, ((), ()) = () ref =(()) (()), taking into account that . Besides, itis assumed that (, , ), (, ) and () are

functions(for some large ) of their arguments and also that (0, 0, 0) =0, (0) = 0, and (0, 0) = 0 [12].Clearly, by linearizing (1)(5) around = 0 one gets () = () + () + () ,() = () , () = () ,

ref () = () , () = () () .(6)Tus, the Nonlinear Regulator Problem [3,13] consists offnding a controller () = ((), ()), such that the closed-loop system, () = ()+((), 0), has an asymptoticallystable equilibrium point, and the solution of the system (6)satisfes lim () = 0.So, bydefning (())asthesteady-statezeroerrormanifold and (()) as the steady-state input, the followingtheorem gives the conditions for the solution of nonlinearregulation problem.Teorem 1. Suppose that () = (()) is Poisson stable anda gain exists such that the matrix + is stable and thereexist mappings

() = (()) and

= (()) with (0) =0 and (0) = 0 satisfying(())() (()) = ((()) , () , (())) ,0 = ((()) , ()) .(7)Ten the signal control for the nonlinear regulation is given by () = (() (())) + (()) . (8)Proof. See [12, 13].Nonlinearpartial diferential equations (7)areknownasFrancis-Isidori-Byrnes(FIB)equationsandtheirlinearcounterparts are obtainedwhenthe mappings

() =(()) and

() = (()) become into

() = ()and

() = (), respectively. Tus, the linear problemis reduced to solve a set of linear matrix equations (Francisequations) [1]: = + + ,0 = .(9)2.1. Te Exact Output Fuzzy Regulation Problem. Takagi andSugeno proposed a fuzzy model composed by a set of linearsubsystem with IF-THEN rules capable of relating physicalknowledge, linguistic characteristics, and properties of thesystem. Suchamodel successfullyrepresentsanonlinearsystem at least in a predefned region of phase space [14].It is important to remark that in this work the exosystem ispurely linear, because the computation of new membershipfunctions for the general case is still an open problem.Te T-S model for system (1)(5) is given by [15].Model. Rule :IF 1() is

1 and and

() is

,Mathematical Problems in Engineering 3THEN () =

() +

() +

() , () = () , () =

() () , = 1, 2, . . . , ,(10)where is the number of rules in the model and

is thefuzzy sets defned on the basis of the knowledge of the system.Ten, the regulation problem defned by (1)(5) can berepresented through the T-S fuzzy model; that is, [10] () = =1

( ()) {

() +

() +

()} , () = () , () = =1

( ())

() () ,(11)where () R

is the state vector of the plant, () R

is the state vector of the exosystem, () R

is the inputsignal, and () R

and

(()) is the normalized weightof each rule, which depends on the membership function forthe premise variable

() in

, where

( ()) = =1

(

()) ,

( ()) =

( ())

=1

( ()),

=1

( ()) = 1,

( ()) 0,(12)with () = [1() 2()

()] as a function of (), =1, . . . , , and = 1, . . . , .Te Exact Fuzzy Regulator Problem consists of fndinga controller () = ((), ()), such that the closed-loopsystem, () = =1

( ()) {

() +

(() , 0)} , (13)has anasymptotically stable equilibriumpoint, andthesolution of system (11) satisfes lim () = 0.From [10, 12, 13] the desired overall fuzzy controller canbe represented as () = =1

( ())

[() (())] + (()) . (14)Considering that approximations for mappings (()) and(()) can be obtained by (()) = =1

( ())

() , (()) = =1

( ())

() ,(15)then, the solution of the fuzzy regulation problem requires toobtain

and

from the linear local regulators problemsincluded in (11) and defned by

=

+

+

,0 =

(16)for = 1, . . . , .Tus, the following controller is obtained: () = =1

( ())

[() =1

( ())

()]+ =1

( ())

() .(17)However, accordingto[10], theexact fuzzyregulationisachieved with this controller only when 1 = =

= ,and(1) input matrix is equal for all subsystems; that is, 1 = =

= , or(2) the steady-state zero error manifold is equal for allsubsystems; that is, 1 = =

= .Inthepresent work, restrictions1and2areavoidedby computing new membership functions for a class of T-Sfuzzy systems. On the other hand, in [11] the Exact OutputFuzzy Regulation Problem (EOFRP) was solved by fndingthe steady-state zero error manifold

() = (()) andthe steady-state input

() = (()) for the overall T-S fuzzy problem through

() = ()

() = () (()) (steady-state error equation), resulting in (()) =

()() and (()) =

()(), where

() and

() arecontinuous time-variant matrices and (()) =

()() isa 1 function [12].Within the next section, the exact output regulation, for aclass of T-S fuzzy systems, is obtained by proposing a solutionfor (()) on the basis of diferent membership functions inthe regulator.3. The Exact Output Fuzzy Regulation byLocal RegulatorsInthissection, aparticularclassof T-Sfuzzymodelsisconsidered to solve the exact output regulation on the basis oflinear local controllers. So, the main goal is to fnd an overallregulator based on the fuzzy summation of local regulatorsconsidering adequate membership functions. Clearly, such4 Mathematical Problems in Engineeringmembership functions are not necessarily the same includedin the fuzzy plant. Tus, the steady-state input (()) can bedefned as (()) = =1

(())

() , (18)where

(())arenewmembershipfunctions, suchthatthefuzzyoutputregulatorobtainedfromlocal regulatorscoincides with the controller given in [11], at least for theclass of fuzzy models considered. Tis novel approach onlyrequires the computation of the linear local controllers andthe computationof the newmembership functions whichwillbe presented below. Te regulator obtained in this way is validfor all 0, while in [11] this condition needs to be verifed.Consider the following matrices:

= [[[[[[[[0 1 0 00 0 1 0... ... ... d ...

1

2

3

]]]]]]]],

= [[[[[[[[00...

1]]]]]]]], = [[[[[[[0 1 0 00 0 1 0... ... ... d ...

1 2 3

]]]]]]],

= [1 0 0] , = [1 0 0] ,(19)where

represents the elements of the last row of

and

1represents the element of the last row of

with = 1, . . . , and = 1, . . . , . It is worth mentioning that

and arematrices of the same dimension, as shown in (19). Notice thatmatrices in the formof and can be used to generate a greatvariety of signals, ensuring, inthat way, the applicability of theapproach in a great number of cases. Terefore, from (14) thecontrol input can be defned by () = =1

(())

{() (())}+ =1

(())

() ,(20)because, as mentioned before, () is a function of () and insteady-state () = (()).On the other hand, from

, and from (6), it can beconcluded that 1

1() = 1

1() in steady-state, while from(19) the following exosystem can be easily derived: 1 = 2, 2 = 3,...

= 1

1 + 2

2 + +

,

= =1

.(21)Besides, each subsystem can be rewritten as follows: 1 = 2, 2 = 3,...

=

1

1 +

2

2 + +

+

1

,.(22)By equating the time derivative of the fuzzy output withthe time derivative of the reference signal, it results in 1 1 =

1 1. Furthermore, from(21) and(22), it is possibletoconclude that

1

= 1

, (23)where = 1, 2, . . . , . From this analysis, two results can beobtained. First = [[[[[[[[[[[

1

1 0 00 1

1 0... ... d ...0 0 1

1]]]]]]]]]]], (24)and due to the shape of (19) is constant and common for allsubsystems. Tus, (()) = (). Second, the local steady-state input is defned by

, = (1/1) 1

1 (1/1)

1

1

1+ (1/1) 2

2 (1/1)

2

2

1+ + (1/1)

(1/1)

1.(25)Mathematical Problems in Engineering 5Te previous expression can be rewritten in a compact formas follows:

, =

=1 (1/1) [

]

1 = 1, . . . , ,(26)and from (18) the fuzzy steady-state input is

() = =1

(()) ,= =1

(())

() , = 1, . . . , .(27)Fromthe previous analysis, each

can be computed from

= 1

1 [

1

1

1 2

2

1

1 ]. (28)Now, it is possible to rewrite the fuzzy model as [11] () = () () + () () + () () , (29) () = () , (30) () = () () () , (31)where() = =1

( ())

,() = =1

( ())

,() = =1

( ())

,() = =1

( ())

.(32)For the sake of simplicity, in the following analysis it isconsidered that

= 0, = 1, . . . , , that is, () = 0. However,taking similar steps as those used below, the result can beeasily extended to the case involving () = 0.Tus, from (19) and (29) it can be observed that the lastequation

in (29) is given by

= [1 (()) (

1

1 + +

)+ +

(()) (

1

1 + +

)]+ [1 (())

1 + +

(())

1]

() ,(33)where compact form is

= =1

(()) ( =1

) + ( =1

(())

1)

() ,(34)consideringthat insteady-state () = (), thetotalsteady-state input can be obtained from (23), resulting in

=1

(()) (

1

1 =1

)+ ( =1

(())

1)

() = 1

1 =1

.(35)Consequently, the steady-state input for the total fuzzy systemcan be constructed by

()= (1/1)

=1

(1/1)

=1

(()) (

=1

)

=1

(())

1.(36)Now, a new set of fuzzy membership functions

(()),satisfying

=1

(()) = 1 with

(()) 0, = 1, . . . , ,will be computed in order to obtain a similar steady-stateinput to (36) but formed from the fuzzy summation of locallinear regulators. At this point, the local regulators and totalregulator are defned as follows:

, = (()) = =1

(())

()= =1

(())

=1 (1/1) [

]

1,

= (())= (1/1)

=1

(1/1)

=1

(()) (

=1

)

=1

(())

1,(37)respectively. Tus, the local fuzzy regulators will be multipliedbythenewmembershipfunctionsandtheresultwill beequated to the global fuzzy regulator (36) as follows:(1/1)

=1

(1/1)

=1

(()) (

=1

)

=1

(())

1= =1

(())

=1 (1/1) [

]

1.(38)So, the adequate membership functions are

(()) =

1

(())

=1

(())

1, (39)with = 1, 2, . . . , . It is important to remark that the newmembership functions

(()) are given in terms of

(())for simplicity. However,

(()) can be removed from thenew membership functions (39), when

(()) is replacedby its corresponding expression. Besides,

(()) is directlyobtained from

(()) by considering that in steady-state() = ().6 Mathematical Problems in EngineeringNotice that (39) fulflls conditions

=1

(()) = 1,

(()) 0,(40)for all = 1, 2, . . . , .Remark 2. It can be observed that (39) is always valid whentheentriesofinputmatrix

1havethesamesign; ifthiscondition is not fulflled the denominators of the new fuzzymembership functions will present singularities. Neverthe-less, thesesingularitiesmayappearoutsidetheoperationregion, allowingmorefexibility, butsuchcasesrequireaparticular study of the system, which is beyond the scope ofthis work.Te following theorem provides the conditions for theexistence of the exact output fuzzy regulator for a class of T-Sfuzzy models.Teorem3. Te exact fuzzy output regulationwith fullinformation for T-S fuzzy systems in the form of (11), defnedbymatrices (19), is solvable if (a) the signof entries forcorresponding position, inside input matrices

for =1, . . . , , is the same, (b) there exists a fuzzy stabilizer () =

=1

(())

fortheT-Sfuzzysystem[16], and(c)theexosystem () = () is Poisson stable. Moreover, the ExactOutput Fuzzy Regulation Problem is solvable by the controller: () = =1

(())

{() ()}+ =1

(())

() ,(41)where

(()) can be readily obtained from (39).Proof. From the previous analysis, the existence of mappings(()) = ()and (()) =

=1

(())

()isguaranteed when the local subsystems are defned by (19);that is, the form of (()) corresponds to a diagonal matrixand will be the same for all subsystems, while (()) can bedirectly substitute.On the other hand, condition (a) avoids singularities inthe new membership functions

(()), while the inclusionof condition (b) has been thoroughly discussed in [12, 13, 1719], and it implies that the steady-state manifold can be madeasymptotically stable by the action of the fuzzy stabilizer.Finally, condition(c) isintroducedtoavoidthat thereference signal converges to zero, because if the referencetends tozero, thentheregulationproblemturns intoastabilizationone, whichcanbesolvedbymeans of thefuzzy stabilizer. In addition, as mentioned before, this kindof matrix canbeusedtogenerateagreat number ofreference signals. Te rest of the proof follows directly fromthe previous analysis.Remark 4. Although, the PDC approach [14] can be used tocompute the fuzzy stabilizer, more relaxed approaches can beconsidered to verify condition (b). For instance [16, 2025].4. Exact Output Fuzzy Regulation Problem fora Class of T-S Fuzzy Systems with MultipleInputs and Multiple Outputs (MIMO Case)Nowtheproblemwill beextendedtoT-SfuzzyMIMOmodels defned by matrices:

= [

] ,

= [

] , = [

] ,

= [

] ,

= [

] ,(42)for = 1, . . . , , with = [[[[[[[1 0 00 1 0... ... d ...0 0 1]]]]]]], = [[[[[[[0 0 00 0 0... ... d ...0 0 0]]]]]]],

= [[[[[[[[

(+1),1

(+1),2

(+1),3

(+1),

(+2),1

(+2),2

(+2),3

(+2),......... d ...

,1

,2

,3

,]]]]]]]],

= [[[[[[[[

+1 0 00

+2 0... ... d ...00

]]]]]]]], = [[[[[[[

(+1),1 (+1),2 (+1),3 (+1),

(+2),1 (+2),2 (+2),3 (+2),......... d ...

,1 ,2 ,3 ,]]]]]]],

=

= [[[[[[[0 0 00 0 0... ... d ...0 0 0]]]]]]],Mathematical Problems in Engineering 7

= [[[[[[[

1 0 00 2 0... ... d ...0 0

]]]]]]],

= [[[[[[[

1 0 00 2 0... ... d ...0 0

]]]]]]],(43)where = ,

R, R(),

R, R()(),

R,

R,

R,and R. As before, this proposal is given because agreat number of mechanical systems can be represented inthis form.Notice that this analysis cannot be omitted because thecomputation of the new membership functions is not thesame as that developed in the previous section. From(42) thefollowing exosystem can be easily derived: 1 = +1, 2 = +2,... +1 = (+1),1

1 + (+1),2

2 + + (+1),

= =1

(+1),

, +2 = (+2),1

1 + (+2),2

2 + + (+2),

= =1

(+2),

,...

= ,1

1 + ,2

2 + + ,

= =1

,

.(44)Moreover, each subsystem can be described by 1 = +1, 2 = +2,... +1 =

(+1),1

1 +

(+1),2

2 + + (+1),

+

+1

= =1

(+1),

+

+1

, +2 =

(+2),1

1 +

(+2),2

2 + + (+2),

+

+2

= =1

(+2),

+

1+2

,...

= ,1

1 + ,2

2 + + ,

+

= =1

,

+

.(45)As before, using expressions (44) and (45), considering 1

1 =

1

1, 2

2 = 2

2, . . . ,

=

, andperformingsuccessive substitutions of = with = 1, . . . , , thefollowing results are obtained: = [[[[[[[

1 0 00 2 0... ... d ...0 0

]]]]]]], (46)where R, and = 1, 2, . . . , ; also 1 =

1/1, 2 = 2/2, . . . ,

=

/

, +1 = 1/1, +2 =

2/2, . . . , + =

/

, and so forth. Notice that the valuesof repeat themselves every entry up to . Additionally, is constant and common for all subsystems while the localsteady-state inputs are defned by

=[[[[[[[[[[[[[[

1

(+1),1

1 1

(+1),1

1

1+1+ 1

(+1),1

2 2

(+1),1

2

1+1+ + 1

(+1),

(+1),

1+1

2

(+2),1

1 1

(+2),1

1

1+2+ 2

(+1),2

2 2

(+2),2

2

1+2+ + 2

(+2),

(+2),

1+2...

,1

1 1

,1

1

1

+

,2

2 2

,2

2

1

+ +

,

,

1

]]]]]]]]]]]]]], (47)8 Mathematical Problems in Engineeringfor = 1, . . . , . In compact form, (47) turns into

=[[[[[[[[[[[[[[[

=1

1

(+1),

(+1),

+1

=1

2

(+2),

(+2),

+2...

=1

,

,

]]]]]]]]]]]]]]](48)and from (27) the local matrices

have the form

=[[[[[[[[[[[[[[

1

(+1),1 1

(+1),1

+1

1

(+1),2 2

(+1),2

+1 1

,1

(+1),

+1

2

(+2),1 1

(+2),1

+2

2

(+2),2 2

(+2),2

+2 2

,1

(+2),

+2...... d...

,1 1

,1

,2 2

,2

,1

,

]]]]]]]]]]]]]]. (49)Now, again from the fuzzy plant given in (42) and theglobal fuzzy model (29), the expressions for +1, . . . ,

are +1 = =1

(()) ( =1

(+1),

)+ ( =1

(())

+1)

+2 = =1

(()) ( =1

(+2),

)+ ( =1

(())

+2)

...

= =1

(()) ( =1

,

)+ ( =1

(())

)

.(50)So, by applying successive substitutions as before with (44),(45) andknowingthat insteady-state

=

, thefollowing expressions are obtained:

=1

(()) ( =1

(+1),

)+ ( =1

(())

+1)

= 1 =1

(+1),

=1

(()) ( =1

(+2),

)+ ( =1

(())

+2)

= 2 =1

(+2),

... 1=1

(()) ( =1

,

)+ ( =1

(())

)

=

=1

,

.(51)Mathematical Problems in Engineering 9Terefore, the steady-state input for the overall fuzzy systemis

=[[[[[[[[[[[[[[[

1

=1 (+1),

=1

(()) (

=1

(+1),

)

=1

(())

+1

2

=1 (+2),

=1

(()) (

=1

(+2),

)

=1

(())

+2...

=1 ,

=1

(()) (

=1

,

)

=1

(())

]]]]]]]]]]]]]]].(52)But, the desired form of the steady-state is

() = (()) = =1

(())

() . (53)Consequently, the new membership functions,

(()),can be obtained afer equating (52) and (53), resulting in

(()) = [[[[[[[

,1,1 0 00 ,2,2 0...... d ...00 ,,]]]]]]], (54)where

,, =

+

(())

=1

(())

+(55)for all = 1, . . . , ; = = 1, . . . , . It can be readily observedthat the sum of the corresponding elements is equal to oneand ,,(()) 0.Tese new membership functions

(()) are organizedin a matrix form; for that reason

=1

(()) = (where is the identity matrix), the arrangement of (54) results as aconsequence of ensuring (20) and again this representationis valid when the values of the input matrices

have thesame sign at corresponding positions. Clearly, this conditionavoidssingularitiesin(55). Asbefore,

(())isdirectlyobtained from

(()) by considering that in steady-state() = (). At this point, the following theorem can bedefned.Teorem5. Te exact fuzzy output regulationwith fullinformationfor T-SfuzzyMIMOsystems inthe formof(11), defnedbymatrices(42), issolvableif (a)thesignofentries for corresponding position, inside input matrices

for = 1, . . . , , is the same, (b) there exists a fuzzy stabilizer() =

=1

(())

for the fuzzy system [16], and (c) theexosystem () = () is Poisson stable.Moreover, the Exact Output Fuzzy Regulation Problem forMIMO systems is solvable by the controller: () = =1

(())

{() ()}+ =1

(())

() ,(56)where

(()) can be readily obtained from (54) and (55).Proof. It follows directly fromTeorem3 andpreviousanalysis.Remark 6. Notice that

(()) is defned as a matrix andthis is not a typical membership function; however,

(())contains a set of membership functions related to each entryof

. Tis form allows us to design controller (20).5. Numerical Examples5.1. Simple Input Case. Consider the problem of balancingand swing of an inverted pendulum on a cart. Te motionequations for this system are [14] 1 () = 2 () , (57) 2 ()= sin(1 ()) 22 sin (21 ())/2 cos (1 ()) ()4/3 cos2 (1 ()),(58)where 1() denotes the angle (in radians) of the pendulumfromthevertical and 2()istheangularvelocity; =9.8 m/s2 is the gravity constant, is the mass of the pen-dulum, is the mass of the cart, 2 is the length of thependulum, and is the force applied to the cart (in newtons); = 1/( + ), with = 2.0 kg, = 8.0 kg, and 2 = 1.0 min the simulations. Te goal is to track the reference defnedby the exosystem within the range 1 (/2, /2). To thisend, a fuzzy model representing the nonlinear dynamics is asfollows.Rule 1.IF 1() is about 0,THEN () = 1() + 1().Rule 2.IF 1() is about /2(|1| < /2),THEN () = 2() + 2().10 Mathematical Problems in Engineeringh1h2x190 90(deg)100Figure 1: Membership functions of two-rule model.Membership functions for Rules 1 and 2 are showninFigure 1.In this example, the T-S fuzzy model of the plant is defned bymatrices:

1 = [ 0117.2941 0], 1 = [ 00.1765],

2 = [ 014.1585 0], 2 = [ 00.1500],

1 = 2 = [1 0] ,(59)while the exosystem can be constructed from the followingmatrices: = [ 0 11 0], = [1 0] , (60)which allows us to generate a great kind of sinusoidal refer-ences. Figure 2 shows the simulation results afer applying thecontroller: () = =1

(())

[() =1

(())

()]+ =1

(())

() ,(61)defnedin[10]. Asexpected, controller(61)isunabletoachieveexactregulation, althoughtrackingerrorremainsbounded.Now, using the method derived in this work, the localmatrices

can be readily obtained from (28), resulting in1 = [103.6667 0] , 2 = [34.3836 0] . (62)Besides, knowing that 1/1 = 1, matrix is obtained from(24): = [1 00 1], (63)while the newmembership functions computed from(39) are

1 (()) =(1 (())) (0.1765)1 (()) (0.1765) + 2 (()) (0.1500),

2 (()) =(2 (())) (0.1500)1 (()) (0.1765) + 2 (()) (0.1500).(64)On the other hand, the fuzzy stabilizer for this system isconstructed by means of the PDCapproach developed in[14],which is formed by

1 = [123.4898 6.9979] , 2 = [58.6405 7.7962] .(65)Figure 3 shows simulation with the controller defnedin(20). Observehowthenewmembershipfunctions arecapable of combining the local regulators in the proper wayto accomplish asymptotic tracking.Toverifytheefectivenessof thelatterregulator, thefuzzy controller is applied on the original system (57). Tesimulation results appear in Figure 4. Notice that the trackingerror depicted in Figure 4(b) is due to the approximationprovidedbytheT-Sfuzzymodel. Suchanerrorcanbereduced by considering a better approximation or an exactrepresentation of the nonlinear system, as the one given bythe nonlinear sector approach [14].5.2. MultipleInputs andMultipleOutputs (MIMO) Case.Now, consider a T-S fuzzy system of two rules, defned by thefollowing matrices:

1 = [[[[[[0 0 1 00 0 0 16 4 2 42 3 1 5]]]]]],

1 = [[[[[[0 00 07 00 2]]]]]],

2 = [[[[[[0 0 1 00 0 0 13 4 2 12 4 6 2]]]]]],

2 = [[[[[[0 00 03 00 5]]]]]],

1 = 2 = [1 0 0 00 1 0 0],(66)Mathematical Problems in Engineering 110 5 10 15 20 25 30 35 4040302010010203040TimeAmplitudex1 versus w1x1w1(a)0 5 10 15 20 25 30 35 400.40.30.20.100.10.20.3Tracking errorTimeAmplitudee(b)Figure 2: (a) Output versus reference and (b) tracking error. When the fuzzy controller is designed on the basis of linear regulators and thefuzzy memberships of the plant are considered in the construction the fuzzy regulator.0 5 10 15 20 25 30 35 40403020100102030TimeAmplitudex1 versus w1x1w1(a)0 5 10 15 20 25 30 35 400.60.50.40.30.20.100.10.20.3Tracking errorTimeAmplitudee(b)Figure 3: (a) Output reference and (b) tracking error. When the fuzzy controller is designed on local regulators and the adequate membershipfunctions for the fuzzy regulator are computed.0 5 10 15 20 25 30 35 4040302010010203040TimeAmplitudex1 versus w1x1w1(a)0 5 10 15 20 25 30 35 400.60.50.40.30.20.100.1Tracking errorTimeAmplitudee(b)Figure 4: (a) Output reference and (b) tracking error. When the fuzzy controller is designed on local regulators and the adequate membershipfunctions for the fuzzy regulator are computed.12 Mathematical Problems in Engineeringh1h2x16018000Figure 5: Membership functions of two-rule model.and an exosystem defned by = [[[[[[0 0 1 00 0 0 14 0 0 00 16 0 0]]]]]], = [1 0 0 00 1 0 0].(67)Membershipfunctions for Rules 1 and2 are showninFigure 5.Matrices defning the local regulators can be easily obtainedby using (49). For this example they are1 = [ 1.4286 0.5714 0.2857 0.57141.0000 9.5000 0.5000 2.5000],2 = [ 2.3333 1.3333 0.6667 0.33330.4000 4.0000 1.2000 0.4000].(68)On the other hand, 1/1 = 1 = 1, 2/2 = 2 = 1,

1/1 = 3 = 1, 2/2 = 4 = 1, and by (46), and matrix is = [[[[[[1 0 0 00 1 0 00 0 1 00 0 0 1]]]]]]. (69)Figures 6 and 7 showthe results obtained using linear regula-tors defned in (61).Now, by computing the new membership functions forthe fuzzy regulator through (54), one gets

1 (())= [[[[(1 (())) (7)1 (()) (3) + 2 (()) (3)00(1 (())) (2)1 (()) (2) + 2 (()) (5)]]]],

2 (())= [[[[(2 (())) (3)1 (()) (3) + 2 (()) (3)00(2 (())) (5)1 (()) (2) + 2 (()) (5)]]]].(70)As previously, the fuzzy stabilizer for this example is alsoconstructed by means of the PDCapproach developed in[14],resulting in

1 = [ 1.0286 0.5714 0.6000 0.57141.0000 2.2500 0.5000 3.7500],

2 = [ 1.4000 1.3333 1.4000 0.33330.4000 1.1000 1.2000 0.9000].(71)Tus, Figures 8 and 9 show the simulation results whenthe controller defned by (20) is applied.Besides, Figure 10 shows the shape of the new member-ship functions, which are clearly diferent from the originalones given in Figure 5.5.3. More General Problems. Consider the T-S fuzzy MIMOsystem with multiple inputs defned by the following matri-ces:

1 = [[[[[[[[[[[[0 0 1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 16 4 2 4 5 32 3 1 5 2 4]]]]]]]]]]]],

1 = [[[[[[[[[[[[0 00 00 00 05 00 2]]]]]]]]]]]],

2 = [[[[[[[[[[[[0 0 1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 16 1 2 5 4 60 2 7 4 5 2]]]]]]]]]]]],

2 = [[[[[[[[[[[[0 00 00 00 07 00 8]]]]]]]]]]]],

1 = 2 = [0 1 1 0 0 01 0 1 0 0 0].(72)Mathematical Problems in Engineering 130 5 10 15 20 25 301510505101520TimeAmplitudex1 versus w1x1w1(a)0 5 10 15 20 25 304202468TimeAmplitudeTracking error e1e1(b)Figure 6: (a) 1 versus 1 and (b) tracking error 1. When the fuzzy controller is designed on the basis of linear regulators and the fuzzymemberships of the plant are considered in the construction the fuzzy regulator.0 5 10 15 20 25 303020100102030TimeAmplitudex2 versus w2x2w2(a)0 5 10 15 20 25 30151050510TimeAmplitudeTracking error e2e2(b)Figure 7: (a) 2 versus 2 and (b) tracking error 2. When the fuzzy controller is designed on the basis of lineal regulators and the fuzzymemberships of the plant are considered in the construction the fuzzy regulator.0 5 10 15 20 25 301510505101520TimeAmplitudex1 versus w1x1w1(a)0 5 10 15 20 25 301012345678TimeAmplitudeTracking error e1e1(b)Figure 8: (a) 1 versus 1 and (b) tracking error 1. When the fuzzy controller is designed on the basis of local regulators and the adequatemembership functions for the fuzzy regulator are computed.Te reference signals will be generated by an exosystemdefned by = [ 0 11 0], = [1 00 1]. (73)Membership functions for Rules 1 and 2 are 1(()) =(80 1())/140 and 2(()) = 1 1(). Notice that for thiscase, = and

= . Tis implies that

and

mustbe obtained by solving Francis Equations for each subsystem.However, due to formof the fuzzy plant it is possible to verifythat 1 = 2.14 Mathematical Problems in Engineering0 5 10 15 20 25 30403020100102030TimeAmplitudex2 versus w2x2w2(a)0 5 10 15 20 25 301086420TimeAmplitudeTracking error e2e2(b)Figure 9: (a) 2 versus 2 and (b) tracking error 2. When the fuzzy controller is designed on the basis of local regulators and the adequatemembership functions for the fuzzy regulator are computed.60 40 20 0 20 40 60 8000.20.40.60.81Membership functionsAmplitude

1(1,1) (x1(t))

2(1,1) (x1(t))(a)60 40 20 0 20 40 60 8000.20.40.60.81Membership functionsAmplitude

1(2,2) (x1(t))

2(2,2) (x1(t))(b)Figure 10: (a) Membership functions corresponding to position (1,1) of 1(()) and 2(()). (b) Membership functions corresponding toposition (2,2) 1(()) and 2(()).So, by applying (9) for each local subsystem with = 1, 2,it results in1 = 2 = [[[[[[[[[[[[0.5000 0.50001.5000 0.50000.5000 0.50000.5000 1.50000.5000 0.50001.5000 0.5000]]]]]]]]]]]],1 = [ 0.7000 0.30002.0000 5.5000],2 = [ 2.3571 1.21430.4375 0.1875].(74)Again, the fuzzy stabilizer for this system is constructedby means of the PDC approach developed in [14], yielding

1 = [[[1.2229 0.0609 4.0284 0.1291 2.5666 0.40611.6919 8.3997 3.2934 12.7153 1.5732 6.1836]]],

2 = [[[0.8735 0.4721 2.8775 1.3780 1.6904 0.71870.6730 1.9749 1.5734 3.0538 0.4817 1.2959]]].(75)Te behavior obtainedbyapplyingcontroller (61) isdepictedinFigures 11 and12. Notice the exact outputregulation is not achieved.Mathematical Problems in Engineering 150 5 10 15 20 25 306040200204060TimeAmplitudex1 versus w1x1w1(a)0 5 10 15 20 25 30807060504030201001020TimeAmplitudeTracking error e1e1(b)Figure 11: (a) 1 versus 1 and (b) tracking error 1. When the fuzzy controller is designed on the basis of linear regulators and the fuzzymemberships of the plant are considered in the construction the fuzzy regulator.0 5 10 15 20 25 30806040200204060TimeAmplitudex2 versus w2x2w2(a)0 5 10 15 20 25 30151050510TimeAmplitudeTracking error e2e2(b)Figure 12: (a) 2 versus 2 and (b) tracking error 2. When the fuzzy controller is designed on the basis of linear regulators and the fuzzymemberships of the plant are considered in the construction the fuzzy regulator.Now, the adequate membership functions are computedfrom (54), resulting in

1 (())= [[[[(1 (())) (5)1 (()) (5) + 2 (()) (7)00(1 (())) (2)1 (()) (2) + 2 (()) (8)]]]],

2 (())= [[[[(2 (())) (7)1 (()) (5) + 2 (()) (7)00(2 (())) (8)1 (()) (2) + 2 (()) (8)]]]].(76)When

(()) is replaced by its corresponding expressions,one gets

1 (()) = [[[[1225

1 () + 140 5200 1 () 8031 () + 320]]]],

2 (()) = [[[[72 1225

1 () + 4100041 () + 24031 () + 320]]]].(77)Te results of applying control law(20) on the fuzzy plantare given in Figures 13 and 14.Tecomputed membershipfunctions for thefuzzyregulatoraregiveninFigure15. Again, itcanbereadilyobserved that these new membership functions are diferentfrom the original ones.Notice that, although this example does not fulfll theconditions of Teorem5, the membership functions obtainedfrom(39) allowthe exact tracking for the fuzzy problem. Tis16 Mathematical Problems in Engineering0 5 10 15 20 25 30806040200204060TimeAmplitudex1 versus w1x1w1(a)0 5 10 15 20 25 308070605040302010010TimeAmplitudeTracking error e1e1(b)Figure 13: (a) 1 versus 1 and (b) tracking error 1. When the fuzzy controller is designed on the basis of local regulators and the adequatemembership functions for the fuzzy regulator are computed.0 5 10 15 20 25 30806040200204060TimeAmplitudex2 versus w2x2w2(a)0 5 10 15 20 25 3086420246810TimeAmplitudeTracking error e2e2(b)Figure 14: (a) 2 versus 2 and (b) tracking error 2. When the fuzzy controller is designed on the basis of local regulators and the adequatemembership functions for the fuzzy regulator are computed.60 40 20 0 20 40 60 8000.20.40.60.81Membership functionsAmplitude

1(1,1) (x1(t))

2(1,1) (x1(t))(a)60 40 20 0 20 40 60 8000.20.40.60.81Membership functionsAmplitude

1(2,2) (x1(t))

2(2,2) (x1(t))(b)Figure 15: (a) Membership functions corresponding to position (1,1) of 1(()) and 2(()). (b) Membership functions corresponding toposition (2,2) 1(()), 2(()).Mathematical Problems in Engineering 17suggests that the approach presented in this work may beapplied on a bigger class of T-S models; however, this studyis not completed yet.6. ConclusionsA fuzzy regulator for continuous-time systems, based on thecombination of linear regulators adjusted by diferent mem-bership functions, has been presented. Te main advantageis that analytical expressions for the membership functions,which allowthe proper combination of the local regulators inorder to guarantee the exact regulation, can be easily applied.As a consequence, for a class of T-S fuzzy models, the pre-sented result allows the exact output regulation on the basisof local regulators. Besides, this class of T-S fuzzy modelscanbecommonlyfoundinmechanics, electromechanics,electronics, robotics, and so forth.Some numerical examples are usedto illustrate theapplicability of the proposed approach, even in cases wherethe dimensions of the plant and the exosystem are diferentbetweeneachother. Tis suggests that the approachpresentedin this work may be applied on a bigger class of T-S models.Finally, themethodproposedinthisworkavoidsthedisadvantage of constructing an exact fuzzy regulator basedonoverall T-Sfuzzysystemwhoseexpressionmayresultvery large. Instead, the given approach ofers a simple way todesign the complete regulator based on local regulators butwith membership functions which are not necessarily of thesame shape as those included in the fuzzy plant.Conflict of InterestsTeauthorsdeclarethat thereisnoconfict of interestsregarding the publication of this paper.AcknowledgmentsTis work was partially supported by CONACYT (ConsejoNacional de Ciencia y Tecnologa) through scholarship SNI(Sistema Nacional de Investigadores) and by IPN (InstitutoPolit ecnicoNacional)throughresearchProjects20140659and 20150487 and scholarships EDI (Estmulo al Desempe node los Investigadores), COFAA (Comisi on de Operaci on yFomento de Actividades Acad emicas), and BEIFI (Beca deEstmulo Institucional de Formaci on de Investigadores).References[1]B. A. Francis, Telinearmultivariableregulatorproblem,SIAM Journal on Control and Optimization, vol. 15, no. 3, pp.486505, 1977.[2]B. A. Francis and W. M. Wonham, Te internal model principleof control theory, Automatica, vol. 12, no. 5, pp. 457465, 1976.[3]A. Isidori and C. I. Byrnes, Output regulation of nonlinearsystems, IEEE Transactions on Automatic Control, vol. 35, no.2, pp. 131140, 1990.[4]B. Castillo-Toledo, S. Di Gennaro, and F. Jurado, Trajectorytracking for a quadrotor via fuzzy regulation, in Proceedings ofthe World Automation Congress (WAC 12), pp. 16, June 2012.[5]S. Chen, Output regulation of nonlinear singularly perturbedsystems based on T-S fuzzy model, Journal of Control Teoryand Applications, vol. 3, no. 4, pp. 399403, 2005.[6]R. Tapia-Herrera, J. A. Meda-Campa na, S. Alc&; ntara-Montes, T. Hern&; ndez-Cort&; s, and L. Salgado-Conrado,Tuning of a TS fuzzy output regulator using thesteepest descent approach and ANFIS, Mathematical Problemsin Engineering, vol. 2013, Article ID 873430, 14 pages, 2013.[7]K.-Y. Lian and J.-J. Liou, Output tracking control for fuzzysystems viaoutput feedbackdesign,IEEETransactions onFuzzy Systems, vol. 14, no. 5, pp. 628639, 2006.[8]X.-J. Ma andZ.-Q. Sun, Output tracking andregulationof nonlinear systembasedontakagi-sugenofuzzymodel,IEEE Transactions on Systems, Man, and Cybernetics, Part B:Cybernetics, vol. 30, no. 1, pp. 4759, 2000.[9]H. J. Lee, J. B. Park, andY. H. Joo, Commentsonoutputtracking and regulation of nonlinear system based on Takagi-Sugeno fuzzy model

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Xu, Robust controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctions, Information Sciences, vol. 174, no. 3-4, pp. 174197,2005.Submit your manuscripts athttp://www.hindawi.comHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014MathematicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Mathematical Problems in EngineeringHindawi Publishing Corporationhttp://www.hindawi.comDierential EquationsInternational Journal ofVolume 2014Applied MathematicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Journal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Mathematical PhysicsAdvances inComplex AnalysisJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014OptimizationJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014International Journal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Operations ResearchAdvances inJournal ofHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Function SpacesAbstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014International Journal of Mathematics and Mathematical SciencesHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014The Scientifc World JournalHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014AlgebraDiscrete Dynamics in Nature and SocietyHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Decision SciencesAdvances inDiscrete MathematicsJournal ofHindawi Publishing Corporationhttp://www.hindawi.comVolume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014Stochastic AnalysisInternational Journal of