Research ArticleA Simplified Output Regulator for a Class ofTakagi-Sugeno Fuzzy Models

Tonatiuh Hernaacutendez-Corteacutes Jesuacutes A Meda CampantildeaLuis A Paacuteramo Carranza and Julio C Goacutemez Mancilla

Instituto Politecnico Nacional SEPI-ESIME Zacatenco Avenue IPN SN 07738 Mexico DF Mexico

Correspondence should be addressed to Jesus A Meda Campana jmedacipnmx

Received 10 January 2015 Revised 1 April 2015 Accepted 1 April 2015

Academic Editor Qingling Zhang

Copyright copy 2015 Tonatiuh Hernandez-Cortes et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

This paper is devoted to solve the regulation problem on the basis of local regulators which are combined using ldquonewrdquomembershipfunctions As a result the exact tracking of references is achieved The design of linear local regulators is suggested in this paperbut now adequate membership functions are computed in order to ensure the proper combination of the local regulators inthe interpolation regions These membership functions which are given as mathematical expressions solve the fuzzy regulationproblem in a relative simple way The 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 Introduction

One of the most important problems in control theoryconsists of finding a controller capable of taking the outputs ofa plant towards the reference signals generated by an externalsystem named exosystem

This problem has been studied by several authors due toits wide applicability in mechanical systems aeronautics andtelematics just to name a few

The works of Francis [1] and Francis and Wonham [2]have shown that the solvability of a multivariable linearregulator problem corresponds to the solvability of a systemof two linearmatrix equations called Francis Equations Alsothey have shown that in the case of error feedback theregulator which solves the problem includes the exosystemThis property is commonly know as InternalModel Principle

Later Isidori and Byrnes [3] showed that the result estab-lished by Francis could be extended to the nonlinear field asa general case and that the equations of Francis represent aparticular case of a set of nonlinear equations They showedthat the solvability for the nonlinear case depends on thesolution of a set of nonlinear partial differential equationscalled Francis-Isidori-Byrnes (FIB) equations Unfortunately

such equations are too difficult to solve in a practical mannerin general

On the other hand some techniques have been provento be an alternative to solve this problem by combining thetheory of output regulation and the Takagi-Sugeno fuzzymodeling [4ndash9]

In [10] the authors propose an approach based on theweighted summation of local linear regulators in order tosynchronize chaotic systems by means of regulation theoryHowever to ensure the exact output regulation two condi-tions need to be fulfilled (1) the same input matrix for allsubsystems that is119861

1= 1198612= sdot sdot sdot = 119861

119903 where 119903 is the number

of rules in the fuzzy model and (2) same zero error manifold120587(119908(119905)) for every local subsystem

Later in [11] the exact output regulator is directlydesigned for the overall T-S fuzzy model Although sucha controller achieves the exact output regulation wherethe 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 effectiveness of thecontroller given in this approach guarantees at least for a class

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 148173 18 pageshttpdxdoiorg1011552015148173

2 Mathematical Problems in Engineering

of T-S fuzzy models the exact output regulation To this endnew membership functions will be systematically computedin order to adequately combine the linear local regulatorsguaranteeing 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 soft computing techniques

Themain idea comes from the 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 finding 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 finda control law for a class of Takagi-Sugeno fuzzy models inorder to achieve exact output regulation on the basis of localregulators and computing of new membership functionseven if different 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 119905 ge 0 [11] Besides the newmembership functions allowing the proper combination thelocal regulators are given as a mathematical expressions

The 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 fuzzyregulation problemThemain result is developed in Section 3In Section 4 some examples are presented and finally inSection 5 some conclusions are drawn

2 The Output Regulation Problem

Consider a nonlinear system defined by

(119905) = 119891 (119909 (119905) 120596 (119905) 119906 (119905)) (1)

119910 (119905) = 119888 (119909 (119905)) (2)

(119905) = 119904 (120596 (119905)) (3)

119910ref (119905) = 119902 (120596 (119905)) (4)

119890 (119905) = ℎ (119909 (119905) 120596 (119905)) (5)

where 119909(119905) isin R119899 is the state vector of the plant 119908(119905) isin 119882 sub

R119904 is the state vector of the exosystem which generates thereference andor the perturbation signals and 119906(119905) isin R119898 isthe input signal Equation (5) refers to difference betweenoutput system of the plant (119910(119905) isin R119898) and the referencesignal (119910ref(119905) isin R119898) that is ℎ(119909(119905) 120596(119905)) = 119910(119905) minus 119910ref =119888(119909(119905)) minus 119902(119909(119905)) taking into account that 119898 le 119899 Besides itis assumed that 119891(119909 119906 119908) ℎ(119909 119908) and 119908(119905) are 119862119896 functions(for some large 119896) of their arguments and also that119891(0 0 0) =0 119904(0) = 0 and ℎ(0 0) = 0 [12]

Clearly by linearizing (1)ndash(5) around 119909 = 0 one gets

(119905) = 119860119909 (119905) + 119861119906 (119905) + 119875119908 (119905)

119910 (119905) = 119862119909 (119905)

(119905) = 119878119908 (119905)

119910ref (119905) = 119876119908 (119905)

119890 (119905) = 119862119909 (119905) minus 119876119908 (119905)

(6)

Thus the Nonlinear Regulator Problem [3 13] consists offinding a controller 119906(119905) = 120572(119909(119905) 119908(119905)) such that the closed-loop system (119905) = 119860119909(119905)+119861120572(119909(119905) 0) has an asymptoticallystable equilibrium point and the solution of the system (6)satisfies lim

119905rarrinfin119890(119905) = 0

So by defining 120587(119908(119905)) as the steady-state zero errormanifold and 120574(119908(119905)) as the steady-state input the followingtheorem gives the conditions for the solution of nonlinearregulation problem

Theorem 1 Suppose that (119905) = 119904(119908(119905)) is Poisson stable anda gain119870 exists such that the matrix 119860+119861119870 is stable and thereexist mappings 119909

119904119904(119905) = 120587(119908(119905)) and 119906

119904119904= 120574(119908(119905))with 120587(0) =

0 and 120574(0) = 0 satisfying

120597120587 (119908 (119905))

120597119908 (119905)119904 (119908 (119905)) = 119891 (120587 (119908 (119905)) 119908 (119905) 120574 (119908 (119905)))

0 = ℎ (120587 (119908 (119905)) 119908 (119905))

(7)

Then the signal control for the nonlinear regulation is given by

119906 (119905) = 119870 (119909 (119905) minus 120587 (119908 (119905))) + 120574 (119908 (119905)) (8)

Proof See [12 13]

Nonlinear partial differential equations (7) are knownas Francis-Isidori-Byrnes (FIB) equations and their linearcounterparts are obtained when the mappings 119909

119904119904(119905) =

120587(119908(119905)) and 119906119904119904(119905) = 120574(119908(119905)) become into 119909

119904119904(119905) = Π119908(119905)

and 119906119904119904(119905) = Γ119908(119905) respectively Thus the linear problem

is reduced to solve a set of linear matrix equations (Francisequations) [1]

Π119878 = 119860Π + 119861Γ + 119875

0 = 119862Π minus 119876

(9)

21 The 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 Such a model successfully represents a nonlinearsystem at least in a predefined region of phase space [14]It is important to remark that in this work the exosystem ispurely ldquolinearrdquo because the computation of new membershipfunctions for the general case is still an open problem

The T-S model for system (1)ndash(5) is given by [15]

Model Rule 119894

IF 1199111(119905) is119872119894

1and sdot sdot sdot and 119911

119901(119905) is119872119894

119901

Mathematical Problems in Engineering 3

THEN

(119905) = 119860119894119909 (119905) + 119861

119894119906 (119905) + 119875

119894119908 (119905)

(119905) = 119878119908 (119905)

119890 (119905) = 119862119894119909 (119905) minus 119876119908 (119905)

119894 = 1 2 119903

(10)

where 119903 is the number of rules in the model and 119872119894

119895is the

fuzzy sets defined on the basis of the knowledge of the systemThen the regulation problem defined by (1)ndash(5) can be

represented through the T-S fuzzy model that is [10]

(119905) =

119903

sum

119894=1

ℎ119894(119911 (119905)) 119860

119894119909 (119905) + 119861

119894119906 (119905) + 119875

119894119908 (119905)

(119905) = 119878119908 (119905)

119890 (119905) =

119903

sum

119896=1

ℎ119894(119911 (119905)) 119862

119894119909 (119905) minus 119876119908 (119905)

(11)

where 119909(119905) isin R119899 is the state vector of the plant 119908(119905) isin R119904

is the state vector of the exosystem 119906(119905) isin R119898 is the inputsignal and 119890(119905) isin R119898 and ℎ

119894(119911(119905)) is the normalized weight

of each rule which depends on the membership function forthe premise variable 119911

119895(119905) in119872119894

119895 where

120596119894(119911 (119905)) =

119901

prod

119895=1

119872119894

119895(119911119895(119905))

ℎ119894(119911 (119905)) =

120596119894(119911 (119905))

sum119903

119894=1120596119894(119911 (119905))

119903

sum

119894=1

ℎ119894(119911 (119905)) = 1

ℎ119894(119911 (119905)) ge 0

(12)

with 119911(119905) = [1199111(119905) 1199112(119905) sdot sdot sdot 119911

119901(119905)] as a function of 119909(119905) 119894 =

1 119903 and 119895 = 1 119901The Exact Fuzzy Regulator Problem consists of finding

a controller 119906(119905) = 120572(119909(119905) 119908(119905)) such that the closed-loopsystem

(119905) =

119903

sum

119894=1

ℎ119894(119911 (119905)) 119860

119894119909 (119905) + 119861

119894120572 (119909 (119905) 0) (13)

has an asymptotically stable equilibrium point and thesolution of system (11) satisfies lim

119905rarrinfin119890(119905) = 0

From [10 12 13] the desired overall fuzzy controller canbe represented as

119906 (119905) =

119903

sum

ℎ=1

ℎ119894(119911 (119905)) 119870

119894 [119909 (119905) minus 120587 (119908 (119905))] + 120574 (119908 (119905)) (14)

Considering that approximations for mappings 120587(119908(119905)) and120574(119908(119905)) can be obtained by

(119908 (119905)) =

119903

sum

119894=1

ℎ119894(119911 (119905)) Π

119894119908 (119905)

120574 (119908 (119905)) =

119903

sum

119894=1

ℎ119894(119911 (119905)) Γ

119894119908 (119905)

(15)

then the solution of the fuzzy regulation problem requires toobtain Π

119894and Γ119894from the 119903 linear local regulators problems

included in (11) and defined by

Π119894119878 = 119860

119894Π119894+ 119861119894Γ119894+ 119875119894

0 = 119862119894Π119894minus 119876

(16)

for 119894 = 1 119903Thus the following controller is obtained

119906 (119905) =

119903

sum

119894=1

ℎ119894(119911 (119905)) 119870

119894[119909 (119905) minus

119903

sum

119894=1

ℎ119894(119911 (119905)) Π

119894119908 (119905)]

+

119903

sum

119894=1

ℎ119894(119911 (119905)) Γ

119894119908 (119905)

(17)

However according to [10] the exact fuzzy regulation isachieved with this controller only when Π

1= sdot sdot sdot = Π

119903= Π

and

(1) input matrix is equal for all subsystems that is 1198611=

sdot sdot sdot = 119861119903= 119861 or

(2) the steady-state zero error manifold is equal for allsubsystems that is Γ

1= sdot sdot sdot = Γ

119903= Γ

In the present work restrictions 1 and 2 are avoidedby 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 findingthe steady-state zero error manifold 119909

119904119904(119905) = 120587(119908(119905)) and

the steady-state input 119906119904119904(119905) = 120574(119908(119905)) for the overall T-

S fuzzy problem through 119890119904119904(119905) = 119909(119905) minus 119909

119904119904(119905) = 119909(119905) minus

120587(119908(119905)) (steady-state error equation) resulting in 120587(119908(119905)) =Π119891(119905)119908(119905) and 120574(119908(119905)) = Γ

119891(119905)119908(119905) whereΠ

119891(119905) and Γ

119891(119905) are

continuous time-variantmatrices and 120587(119908(119905)) = Π119891(119905)119908(119905) is

a 119862119896minus1 function [12]Within the next section the exact output regulation for a

class of T-S fuzzy systems is obtained by proposing a solutionfor 120574(119908(119905)) on the basis of different membership functions inthe regulator

3 The Exact Output Fuzzy Regulation byLocal Regulators

In this section a particular class of T-S fuzzy models isconsidered to solve the exact output regulation on the basis oflinear local controllers So the main goal is to find an overallregulator based on the fuzzy summation of local regulatorsconsidering adequate membership functions Clearly such

4 Mathematical Problems in Engineering

membership functions are not necessarily the same includedin the fuzzy plant Thus the steady-state input 120574(119908(119905)) can bedefined as

120574 (119908 (119905)) =

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905) (18)

where 120583119894(119908(119905)) are new membership functions such that

the fuzzy output regulator obtained from local regulatorscoincides with the controller given in [11] at least for theclass of fuzzy models considered This novel approach onlyrequires the computation of the linear local controllers andthe computation of the newmembership functionswhichwillbe presented belowThe regulator obtained in this way is validfor all 119905 ge 0 while in [11] this condition needs to be verified

Consider the following matrices

119860119894=

[[[[[[[

[

0 1 0 sdot sdot sdot 0

0 0 1 sdot sdot sdot 0

d

119886119894

1119886119894

2119886119894

3sdot sdot sdot 119886119894

119899

]]]]]]]

]

119861119894=

[[[[[[[

[

0

0

119887119894

1

]]]]]]]

]

119878 =

[[[[[[

[

0 1 0 sdot sdot sdot 0

0 0 1 sdot sdot sdot 0

d

119904111990421199043sdot sdot sdot 119904119899

]]]]]]

]

119862119894= [1198881 0 sdot sdot sdot 0]

119876 = [1199021 0 sdot sdot sdot 0]

(19)

where 119886119894119895represents the elements of the last row of 119860

119894and 1198871198941

represents the element of the last row of 119861119894with 119895 = 1 119899

and 119894 = 1 119903 It is worth mentioning that 119860119894and 119878 are

matrices of the same dimension as shown in (19) Notice thatmatrices in the formof 119878 and119876 can be used to generate a greatvariety of signals ensuring in thatway the applicability of theapproach in a great number of cases Therefore from (14) thecontrol input can be defined by

119906 (119905) =

119903

sum

119894=1

ℎ119894(119909 (119905)) 119896

119894119909 (119905) minus 120587 (119908 (119905))

+

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905)

(20)

because as mentioned before 119911(119905) is a function of 119909(119905) and insteady-state 119909(119905) = 120587(119908(119905))

On the other hand from 119862119894 119876 and from (6) it can be

concluded that 11988811199091(119905) = 119902

11199081(119905) in steady-state while from

(19) the following exosystem can be easily derived

1= 1199082

2= 1199083

119899= 11990411199081+ 11990421199082+ sdot sdot sdot + 119904

119899119908119899

119899=

119899

sum

119894=1

119904119894119908119894

(21)

Besides each subsystem can be rewritten as follows

1= 1199092

2= 1199093

119899= 119886119894

11199091+ 119886119894

21199092+ sdot sdot sdot + 119886

119894

119899119909119899+ 119887119894

1119906119904119904119894

(22)

By equating the time derivative of the fuzzy output withthe time derivative of the reference signal it results in 119888

11=

11990211 Furthermore from (21) and (22) it is possible to

conclude that

1198881119894= 1199021119894 (23)

where 119894 = 1 2 119899 From this analysis two results can beobtained First

Π =

[[[[[[[[[[

[

1199021

1198881

0 sdot sdot sdot 0

01199021

1198881

sdot sdot sdot 0

d

0 0 sdot sdot sdot1199021

1198881

]]]]]]]]]]

]

(24)

and due to the shape of (19)Π is constant and common for allsubsystemsThus 120587(119908(119905)) = Π119908(119905) Second the local steady-state input is defined by

119906119904119904119894

=(11990211198881) 11990411199081minus (11990211198881) 119886119894

11199081

119887119894

1

+(11990211198881) 11990421199082minus (11990211198881) 119886119894

21199082

119887119894

1

+ sdot sdot sdot +(11990211198881) 119904119899119908119899minus (11990211198881) 119886119894

119899119908119899

119887119894

1

(25)

Mathematical Problems in Engineering 5

The previous expression can be rewritten in a compact formas follows

119906119904119904119894

=

sum119899

119895=1(11990211198881) [119904119895119908119895minus 119886119894

119895119908119895]

119887119894

1

forall119894 = 1 119903 (26)

and from (18) the fuzzy steady-state input is

119906119904119904(119905) =

119903

sum

119894=1

120583119894(119908 (119905)) 119906

119904119904119894

=

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905) forall119894 = 1 119903

(27)

From the previous analysis each Γ119894can be computed from

Γ119894=1199021

1198881

[1199041minus 119886119894

1

119887119894

1

1199042minus 119886119894

2

119887119894

1

sdot sdot sdot119904119899minus 119886119894

119899

119887119894

1

] (28)

Now it is possible to rewrite the fuzzy model as [11]

(119905) = 119860 (119905) 119909 (119905) + 119861 (119905) 119906 (119905) + (119905) 119908 (119905) (29)

(119905) = 119878119908 (119905) (30)

119890 (119905) = 119862 (119905) 119909 (119905) minus 119876119908 (119905) (31)

where

119860 (119905) =

119903

sum

119894=1

ℎ119894(119911 (119905)) 119860

119894

119861 (119905) =

119903

sum

119894=1

ℎ119894(119911 (119905)) 119861

119894

(119905) =

119903

sum

119894=1

ℎ119894(119911 (119905)) 119875

119894

119862 (119905) =

119903

sum

119894=1

ℎ119894(119911 (119905)) 119862

119894

(32)

For the sake of simplicity in the following analysis it isconsidered that119875

119894= 0 119894 = 1 119903 that is (119905) = 0 However

taking similar steps as those used below the result can beeasily extended to the case involving (119905) = 0

Thus from (19) and (29) it can be observed that the lastequation

119899in (29) is given by

119899= [ℎ1(119909 (119905)) (119886

119894

11199091+ sdot sdot sdot + 119886

119894

119899119909119899)

+ sdot sdot sdot + ℎ119903(119909 (119905)) (119886

119894

11199091+ sdot sdot sdot + 119886

119894

119899119909119899)]

+ [ℎ1(119909 (119905)) 119887

119894

1+ sdot sdot sdot + ℎ

119903(119909 (119905)) 119887

119903

1] 119906119904119904(119905)

(33)

where compact form is

119899=

119903

sum

119894=1

ℎ119894(119909 (119905))(

119899

sum

119895=1

119886119894

119895119909119895) + (

119903

sum

119894=1

ℎ119894(119909 (119905)) 119887

119894

1)119906119904119904(119905)

(34)

considering that in steady-state 119909(119905) = Π119908(119905) the totalsteady-state input can be obtained from (23) resulting in

119903

sum

119894=1

ℎ119894(119908 (119905))(

1199021

1198881

119899

sum

119895=1

119886119894

119895119908119895)

+ (

119903

sum

119894=1

ℎ119894(119908 (119905)) 119887

119894

1)119906119904119904(119905) =

1199021

1198881

119899

sum

119894=1

119904119894119908119894

(35)

Consequently the steady-state input for the total fuzzy systemcan be constructed by

119906119904119904(119905)

=

(11990211198881)sum119899

119894=1119904119894119908119894minus (11990211198881)sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

1

(36)

Now a new set of fuzzy membership functions 120583119894(119908(119905))

satisfying sum119903119894=1

120583119894(119908(119905)) = 1 with 120583

119894(119908(119905)) ge 0 119894 = 1 119903

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 defined as follows

119906119904119904119894

= 120574 (119908 (119905)) =

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905)

=

119903

sum

119894=1

120583119894(119908 (119905))

sum119899

119895=1(11990211198881) [119904119895119908119895minus 119886119894

119895119908119895]

119887119894

1

119906119904119904= 120574 (119908 (119905))

=

(11990211198881)sum119899

119894=1119904119894119908119894minus (11990211198881)sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

1

(37)

respectivelyThus the local fuzzy regulatorswill bemultipliedby the new membership functions and the result will beequated to the global fuzzy regulator (36) as follows

(11990211198881)sum119899

119894=1119904119894119908119894minus (11990211198881)sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

1

=

119903

sum

119894=1

120583119894(119908 (119905))

sum119899

119895=1(11990211198881) [119904119895119908119895minus 119886119894

119895119908119895]

119887119894

1

(38)

So the adequate membership functions are

120583119894(119908 (119905)) =

119887119894

1ℎ119894(119908 (119905))

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

1

(39)

with 119894 = 1 2 119903 It is important to remark that the newmembership functions 120583

119894(119908(119905)) are given in terms of ℎ

119894(119908(119905))

for simplicity However ℎ119894(119908(119905)) can be removed from the

new membership functions (39) when ℎ119894(119908(119905)) is replaced

by its corresponding expression Besides ℎ119894(119908(119905)) is directly

obtained from ℎ119894(119909(119905)) by considering that in steady-state

119909(119905) = Π119908(119905)

6 Mathematical Problems in Engineering

Notice that (39) fulfills conditions119903

sum

119894=1

120583119894(119908 (119905)) = 1

120583119894(119908 (119905)) ge 0

(40)

for all 119894 = 1 2 119903

Remark 2 It can be observed that (39) is always valid whenthe entries of input matrix 119887

119894

1have the same sign if this

condition is not fulfilled the denominators of the new fuzzymembership functions will present singularities Neverthe-less these singularities may appear outside the operationregion allowing more flexibility but such cases require aparticular study of the system which is beyond the scope ofthis work

The following theorem provides the conditions for theexistence of the exact output fuzzy regulator for a class of T-Sfuzzy models

Theorem 3 The exact fuzzy output regulation with fullinformation for T-S fuzzy systems in the form of (11) definedby matrices (19) is solvable if (a) the sign of entries forcorresponding position inside input matrices 119861

119894for 119894 =

1 119903 is the same (b) there exists a fuzzy stabilizer 119906(119905) =sum119903

119894=1ℎ119894(119911(119905))119870

119894119909 for the T-S fuzzy system [16] and (c) the

exosystem (119905) = 119878119908(119905) is Poisson stable Moreover the ExactOutput Fuzzy Regulation Problem is solvable by the controller

119906 (119905) =

119903

sum

119894=1

ℎ119894(119909 (119905)) 119896

119894119909 (119905) minus Π119908 (119905)

+

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905)

(41)

where 120583119894(119908(119905)) can be readily obtained from (39)

Proof From the previous analysis the existence of mappings120587(119908(119905)) = Π119908(119905) and 120574(119908(119905)) = sum

119903

119894=1120583119894(119908(119905))Γ

119894119908(119905) is

guaranteed when the local subsystems are defined by (19)that is the form of 120587(119908(119905)) corresponds to a diagonal matrixand will be the same for all subsystems while 120574(119908(119905)) can bedirectly substitute

On the other hand condition (a) avoids singularities inthe new membership functions 120583

119894(119908(119905)) while the inclusion

of condition (b) has been thoroughly discussed in [12 13 17ndash19] and it implies that the steady-state manifold can be madeasymptotically stable by the action of the fuzzy stabilizer

Finally condition (c) is introduced to avoid that thereference signal converges to zero because if the referencetends to zero then the regulation problem turns into astabilization one which can be solved by means of thefuzzy stabilizer In addition as mentioned before this kindof matrix 119878 can be used to generate a great number ofreference signals The 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 20ndash25]

4 Exact Output Fuzzy Regulation Problem fora Class of T-S Fuzzy Systems with MultipleInputs and Multiple Outputs (MIMO Case)

Now the problem will be extended to T-S fuzzy MIMOmodels defined by matrices

119860119894= [

119874 119868

119863119894

] 119861119894= [

119874

119865119894

]

119878 = [119874 119868

119866] 119862

119894= [ 119867119862 119874

119862 ]

119876119894= [ 119867119876 119874

119876 ]

(42)

for 119894 = 1 119903 with

119868 =

[[[[[[

[

1 0 sdot sdot sdot 0

0 1 sdot sdot sdot 0

d

0 0 sdot sdot sdot 1

]]]]]]

]

119874 =

[[[[[[

[

0 0 sdot sdot sdot 0

0 0 sdot sdot sdot 0

d

0 0 sdot sdot sdot 0

]]]]]]

]

119863119894=

[[[[[[[

[

119886119894

(119896+1)1119886119894

(119896+1)2119886119894

(119896+1)3sdot sdot sdot 119886119894

(119896+1)119899

119886119894

(119896+2)1119886119894

(119896+2)2119886119894

(119896+2)3sdot sdot sdot 119886119894

(119896+2)119899

d

119886119894

1198991119886119894

1198992119886119894

1198993sdot sdot sdot 119886

119894

119899119899

]]]]]]]

]

119865119894=

[[[[[[[

[

119887119894

119896+10 sdot sdot sdot 0

0 119887119894

119896+2sdot sdot sdot 0

d

0 0 sdot sdot sdot 119887119894

119899

]]]]]]]

]

119866 =

[[[[[[

[

119904(119896+1)1

119904(119896+1)2

119904(119896+1)3

sdot sdot sdot 119904(119896+1)119899

119904(119896+2)1

119904(119896+2)2

119904(119896+2)3

sdot sdot sdot 119904(119896+2)119899

d

1199041198991

1199041198992

1199041198993

sdot sdot sdot 119904119899119899

]]]]]]

]

119874119862= 119874119876=

[[[[[[

[

0 0 sdot sdot sdot 0

0 0 sdot sdot sdot 0

d

0 0 sdot sdot sdot 0

]]]]]]

]

Mathematical Problems in Engineering 7

119867119862=

[[[[[[

[

11988810 sdot sdot sdot 0

0 1198882sdot sdot sdot 0

d

0 0 sdot sdot sdot 119888119898

]]]]]]

]

119867119876=

[[[[[[

[

1199021

0 sdot sdot sdot 0

0 1199022sdot sdot sdot 0

d

0 0 sdot sdot sdot 119902119898

]]]]]]

]

(43)

where 119896 = 119899 minus 119898 119865119894isin R119898times119898 119874 isin R(119899minus119898)times119898 119874

119862isin R119898times119898

119868 isin R(119899minus119898)times(119899minus119898) 119863119894isin R119898times119899 119867

119862isin R119898times119898 119867

119876isin R119898times119898

and 119866 isin R119898times119899 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= 119908119898+1

2= 119908119898+2

119896+1

= 119904(119896+1)1

1199081+ 119904(119896+1)2

1199082+ sdot sdot sdot + 119904

(119896+1)119899119908119899

=

119899

sum

119895=1

119904(119896+1)119895

119908119895

119896+2

= 119904(119896+2)1

1199081+ 119904(119896+2)2

1199082+ sdot sdot sdot + 119904

(119896+2)119899119908119899

=

119899

sum

119895=1

119904(119896+2)119895

119908119895

119899= 11990411989911199081+ 11990411989921199082+ sdot sdot sdot + 119904

119899119899119908119899=

119899

sum

119895=1

119904119899119895119908119895

(44)

Moreover each subsystem can be described by

1= 119909119898+1

2= 119909119898+2

119896+1

= 119886119894

(119896+1)11199091+ 119886119894

(119896+1)21199092+ sdot sdot sdot + 119886

(119896+1)119899119909119899

+ 119887119894

119896+1119906119894

119904119904

=

119899

sum

119895=1

119886119894

(119896+1)119895119909119895+ 119887119894

119896+1119906119894

119904119904

119896+2

= 119886119894

(119896+2)11199091+ 119886119894

(119896+2)21199092+ sdot sdot sdot + 119886

(119896+2)119899119909119899

+ 119887119894

119896+2119906119894

119904119904

=

119899

sum

119895=1

119886119894

(119896+2)119895119908119895+ 119887119894

1+2119906119894

119904119904

119899= 11990411989911199081+ 11990411989921199082+ sdot sdot sdot + 119904

119899119899119908119899+ 119887119894

119899119906119894

119904119904

=

119899

sum

119895=1

119904119899119895119908119895+ 119887119894

119899119906119894

119904119904

(45)

As before using expressions (44) and (45) considering 11988811199091=

11990211199081 11988821199092

= 11990221199082 119888

119898119909119898

= 119902119898119908119898 and performing

successive substitutions of ℓ= 120587ℓℓwith ℓ = 1 119898 the

following results are obtained

Π =

[[[[[[

[

1205871

0 sdot sdot sdot 0

0 1205872sdot sdot sdot 0

d

0 0 sdot sdot sdot 120587119899

]]]]]]

]

(46)

where Π isin R119899times119899 and 119894 = 1 2 119903 also 1205871

=

11990211198881 1205872= 11990221198882 120587

119898= 119902119898119888119898 120587119898+1

= 11990211198881 120587119898+2

=

11990221198882 120587

119898+119898= 119902119898119888119898 and so forth Notice that the values

of 120587lowastrepeat themselves every 119898 entry up to 119899 Additionally

Π is constant and common for all subsystems while the localsteady-state inputs are defined by

119906119894

119904119904=

[[[[[[[[[[[[[

[

1205871119904(119896+1)1

1199081minus 1205871119886119894

(119896+1)11199081

1198871

119896+1

+

1205871119904(119896+1)1

1199082minus 1205872119886119894

(119896+1)11199082

1198871

119896+1

+ sdot sdot sdot +

1205871119904(119896+1)119899

119908119899minus 120587119899119886119894

(119896+1)119899119908119899

1198871

119896+1

1205872119904(119896+2)1

1199081minus 1205871119886119894

(119896+2)11199081

1198871

119896+2

+

1205872119904(119896+1)2

1199082minus 1205872119886119894

(119896+2)21199082

1198871

119896+2

+ sdot sdot sdot +

1205872119904(119896+2)119899

119908119899minus 120587119899119886119894

(119896+2)119899119908119899

1198871

119896+2

12058711989911990411989911199081minus 1205871119886119894

11989911199081

1198871119899

+12058711989911990411989921199082minus 1205872119886119894

11989921199082

1198871119899

+ sdot sdot sdot +120587119899119904119899119899119908119899minus 120587119899119886119894

119899119899119908119899

1198871119899

]]]]]]]]]]]]]

]

(47)

8 Mathematical Problems in Engineering

for 119894 = 1 119903 In compact form (47) turns into

119906119894

119904119904=

[[[[[[[[[[[[[[

[

119899

sum

119895=1

1205871119904(119896+1)119895

119908119895minus 120587119895119886(119896+1)119895

119908119895

119887119894

119896+1

119899

sum

119895=1

1205872119904(119896+2)119895

119908119895minus 120587119895119886(119896+2)119895

119908119895

119887119894

119896+2

119899

sum

119895=1

120587119899119904119899119895119908119895minus 120587119895119886119899119895119908119895

119887119894119899

]]]]]]]]]]]]]]

]

(48)

and from (27) the local matrices Γ119894have the form

Γ119894=

[[[[[[[[[[[[[

[

1205871119904(119896+1)1

minus 1205871119886119894

(119896+1)1

119887119894

119896+1

1205871119904(119896+1)2

minus 1205872119886119894

(119896+1)2

119887119894

119896+1

sdot sdot sdot

12058711199041198991

minus 120587119899119886119894

(119896+1)119899

119887119894

119896+1

1205872119904(119896+2)1

minus 1205871119886119894

(119896+2)1

119887119894

119896+2

1205872119904(119896+2)2

minus 1205872119886119894

(119896+2)2

119887119894

119896+2

sdot sdot sdot

12058721199041198991

minus 120587119899119886119894

(119896+2)119899

119887119894

119896+2

d

1205871198991199041198991

minus 1205871119886119894

1198991

119887119894119899

1205871198991199041198992

minus 1205872119886119894

1198992

119887119894119899

sdot sdot sdot1205871198991199041198991

minus 120587119899119886119894

119899119899

119887119894119899

]]]]]]]]]]]]]

]

(49)

Now again from the fuzzy plant given in (42) and theglobal fuzzy model (29) the expressions for

119896+1

119899are

119896+1

=

119903

sum

119894=1

ℎ119894(119909 (119905))(

119899

sum

119895=1

119886119894

(119896+1)119895119909119895)

+ (

119903

sum

119894=1

ℎ119894(119909 (119905)) 119887

119894

119896+1)119906119904119904

119896+2

=

119903

sum

119894=1

ℎ119894(119909 (119905))(

119899

sum

119895=1

119886119894

(119896+2)119895119909119895)

+ (

119903

sum

119894=1

ℎ119894(119909 (119905)) 119887

119894

119896+2)119906119904119904

119899=

119903

sum

119894=1

ℎ119894(119909 (119905))(

119899

sum

119895=1

119886119894

119899119895119909119895)

+ (

119903

sum

119894=1

ℎ119894(119909 (119905)) 119887

119894

119899)119906119904119904

(50)

So by applying successive substitutions as before with (44)(45) and knowing that in steady-state 119909

119895= 120587119895119908119895 the

following expressions are obtained

119903

sum

119894=1

ℎ119894(119908 (119905))(

119899

sum

119895=1

119886119894

(119896+1)119895120587119895119908119895)

+ (

119903

sum

119894=1

ℎ119894(119908 (119905)) 119887

119894

119896+1)119906119904119904= 1205871

119899

sum

119894=1

119904(119896+1)119894

119908119894

119903

sum

119894=1

ℎ119894(119908 (119905))(

119899

sum

119895=1

119886119894

(119896+2)119895120587119895119908119895)

+ (

119903

sum

119894=1

ℎ119894(119908 (119905)) 119887

119894

119896+2)119906119904119904= 1205872

119899

sum

119894=1

119904(119896+2)

119894119908119894

1199031

sum

119894=1

ℎ119894(119908 (119905))(

119899

sum

119895=1

119886119894

119899119895120587119895119908119895)

+ (

119903

sum

119894=1

ℎ119894(119908 (119905)) 119887

119894

119899)119906119904119904= 120587119899

119899

sum

119894=1

119904119899119894119908119894

(51)

Mathematical Problems in Engineering 9

Therefore the steady-state input for the overall fuzzy systemis

119906119904119904

=

[[[[[[[[[[[[[[

[

1205871sum119899

119894=1119904(119896+1)119894

119908119894minus sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

(119896+1)119895120587119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

119896+1

1205872sum119899

119894=1119904(119896+2)119894

119908119894minus sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

(119896+2)119895120587119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

119896+2

120587119899sum119899

119894=1119904119899119895119908119894minus sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

119899119895120587119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

119899

]]]]]]]]]]]]]]

]

(52)

But the desired form of the steady-state is

119906119904119904(119905) = 120574 (119908 (119905)) =

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905) (53)

Consequently the new membership functions 120583119894(119908(119905))

can be obtained after equating (52) and (53) resulting in

120583119894(119908 (119905)) =

[[[[[[

[

12058311989411

0 sdot sdot sdot 0

0 12058311989422

sdot sdot sdot 0

d

0 0 sdot sdot sdot 120583119894119898119898

]]]]]]

]

(54)

where

120583119894119895119897

=

119887119894

119896+119895ℎ119894(119908 (119905))

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

119896+119895

(55)

for all 119894 = 1 119903 119895 = 119897 = 1 119898 It can be readily observedthat the sum of the corresponding elements is equal to oneand 120583

119894119895119897(119908(119905)) ge 0

These new membership functions 120583119894(119908(119905)) are organized

in a matrix form for that reason sum119903119894=1

120583119894(119908(119905)) = 119868 (where 119868

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 119861

119894have the

same sign at corresponding positions Clearly this conditionavoids singularities in (55) As before ℎ

119894(119908(119905)) is directly

obtained from ℎ119894(119909(119905)) by considering that in steady-state

119909(119905) = Π119908(119905) At this point the following theorem can bedefined

Theorem 5 The exact fuzzy output regulation with fullinformation for T-S fuzzy MIMO systems in the form of(11) defined by matrices (42) is solvable if (a) the sign ofentries for corresponding position inside input matrices 119861

119894for

119894 = 1 119903 is the same (b) there exists a fuzzy stabilizer119906(119905) = sum

119903

119894=1ℎ119894(119911(119905))119870

119894119909 for the fuzzy system [16] and (c) the

exosystem (119905) = 119878119908(119905) is Poisson stable

Moreover the Exact Output Fuzzy Regulation Problem forMIMO systems is solvable by the controller

119906 (119905) =

119903

sum

119894=1

ℎ119894(119909 (119905)) 119896

119894119909 (119905) minus Π119908 (119905)

+

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905)

(56)

where 120583119894(119908(119905)) can be readily obtained from (54) and (55)

Proof It follows directly from Theorem 3 and previousanalysis

Remark 6 Notice that 120583119894(119908(119905)) is defined as a matrix and

this is not a typical membership function however 120583119894(119908(119905))

contains a set of membership functions related to each entryof 119861119894 This form allows us to design controller (20)

5 Numerical Examples

51 Simple Input Case Consider the problem of balancingand swing of an inverted pendulum on a cart The motionequations for this system are [14]

1(119905) = 119909

2(119905) (57)

2(119905)

=119892 sin(119909

1(119905))minus 119886119898119897119909

2

2sin (2119909

1(119905))2 minus119886 cos (119909

1(119905)) 119906 (119905)

41198973 minus 119886119898119897 cos2 (1199091(119905))

(58)

where 1199091(119905) denotes the angle (in radians) of the pendulum

from the vertical and 1199092(119905) is the angular velocity 119892 =

98ms2 is the gravity constant 119898 is the mass of the pen-dulum 119872 is the mass of the cart 2119897 is the length of thependulum and 119906 is the force applied to the cart (in newtons)119886 = 1(119898 +119872) with119898 = 20 kg119872 = 80 kg and 2119897 = 10min the simulations The goal is to track the reference definedby the exosystem within the range 119909

1isin (minus1205872 1205872) To this

end a fuzzy model representing the nonlinear dynamics is asfollows

Rule 1

IF 1199091(119905) is about 0

THEN (119905) = 1198601119909(119905) + 119861

1119906(119905)

Rule 2

IF 1199091(119905) is about plusmn1205872(|119909

1| lt 1205872)

THEN (119905) = 1198602119909(119905) + 119861

2119906(119905)

10 Mathematical Problems in Engineering

h1

h2

x1minus90 90

(deg)

1

00

Figure 1 Membership functions of two-rule model

Membership functions for Rules 1 and 2 are shown in Figure 1In this example the T-S fuzzymodel of the plant is defined bymatrices

1198601= [

0 1

172941 0] 119861

1= [

0

minus01765]

1198602= [

0 1

41585 0] 119861

2= [

0

minus01500]

1198621= 1198622= [1 0]

(59)

while the exosystem can be constructed from the followingmatrices

119878 = [

0 1

minus1 0] 119876 = [1 0] (60)

which allows us to generate a great kind of sinusoidal refer-ences Figure 2 shows the simulation results after applying thecontroller

119906 (119905) =

119903

sum

119894=1

ℎ119894(119909 (119905)) 119870

119894[119909 (119905) minus

119903

sum

119894=1

ℎ119894(119908 (119905)) Π

119894119908 (119905)]

+

119903

sum

119894=1

ℎ119894(119908 (119905)) Γ

119894119908 (119905)

(61)

defined in [10] As expected controller (61) is unable toachieve exact regulation although tracking error remainsbounded

Now using the method derived in this work the localmatrices Γ

119894can be readily obtained from (28) resulting in

Γ1= [1036667 0] Γ

2= [343836 0] (62)

Besides knowing that 11990211198881= 1 matrix Π is obtained from

(24)

Π = [

1 0

0 1] (63)

while the newmembership functions computed from (39) are

1205831(119908 (119905)) =

(ℎ1(119908 (119905))) (minus01765)

ℎ1(119908 (119905)) (minus01765) + ℎ

2(119908 (119905)) (minus01500)

1205832(119908 (119905)) =

(ℎ2(119908 (119905))) (minus01500)

ℎ1(119908 (119905)) (minus01765) + ℎ

2(119908 (119905)) (minus01500)

(64)

On the other hand the fuzzy stabilizer for this system isconstructed bymeans of the PDC approach developed in [14]which is formed by

1198701= [1234898 69979] 119870

2= [586405 77962]

(65)

Figure 3 shows simulation with the controller definedin (20) Observe how the new membership functions arecapable of combining the local regulators in the proper wayto accomplish asymptotic tracking

To verify the effectiveness of the latter regulator thefuzzy controller is applied on the original system (57) Thesimulation results appear in Figure 4 Notice that the trackingerror depicted in Figure 4(b) is due to the approximationprovided by the T-S fuzzy model Such an error can bereduced by considering a better approximation or an exactrepresentation of the nonlinear system as the one given bythe nonlinear sector approach [14]

52 Multiple Inputs and Multiple Outputs (MIMO) CaseNow consider a T-S fuzzy system of two rules defined by thefollowing matrices

1198601=

[[[[[

[

0 0 1 0

0 0 0 1

6 4 2 4

2 3 1 5

]]]]]

]

1198611=

[[[[[

[

0 0

0 0

minus7 0

0 2

]]]]]

]

1198602=

[[[[[

[

0 0 1 0

0 0 0 1

3 4 2 1

2 4 6 2

]]]]]

]

1198612=

[[[[[

[

0 0

0 0

minus3 0

0 5

]]]]]

]

1198621= 1198622= [

1 0 0 0

0 1 0 0]

(66)

Mathematical Problems in Engineering 11

0 5 10 15 20 25 30 35 40minus40minus30minus20minus10

010203040

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30 35 40minus04minus03minus02minus01

0010203

Tracking error

Time

Am

plitu

de

e

(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 40minus40minus30minus20minus10

0102030

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30 35 40minus06minus05minus04minus03minus02minus01

0010203

Tracking error

Time

Am

plitu

de

e

(b)

Figure 3 (a) Output reference and (b) tracking errorWhen the fuzzy controller is designed on local regulators and the adequatemembershipfunctions for the fuzzy regulator are computed

0 5 10 15 20 25 30 35 40minus40minus30minus20minus10

010203040

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30 35 40minus06minus05minus04minus03minus02minus01

001

Tracking error

Time

Am

plitu

de

e

(b)

Figure 4 (a) Output reference and (b) tracking errorWhen the fuzzy controller is designed on local regulators and the adequatemembershipfunctions for the fuzzy regulator are computed

12 Mathematical Problems in Engineering

h1 h2

x1minus60

1

80

0

0

Figure 5 Membership functions of two-rule model

and an exosystem defined by

119878 =

[[[[[

[

0 0 1 0

0 0 0 1

minus4 0 0 0

0 minus16 0 0

]]]]]

]

119876 = [

1 0 0 0

0 1 0 0]

(67)

Membership functions forRules 1 and 2 are shown in Figure 5Matrices defining the local regulators can be easily obtainedby using (49) For this example they are

Γ1= [

14286 05714 02857 05714

minus10000 minus95000 minus05000 minus25000]

Γ2= [

23333 13333 06667 03333

minus04000 minus40000 minus12000 minus04000]

(68)

On the other hand 11990211198881= 1205871= 1 119902

21198882= 1205872= 1

11990211198881= 1205873= 1 119902

21198882= 1205874= 1 and by (46) and matrix Π is

Π =

[[[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

]]]]]

]

(69)

Figures 6 and 7 show the results obtained using linear regula-tors defined in (61)

Now by computing the new membership functions forthe fuzzy regulator through (54) one gets1205831(119908 (119905))

=

[[[

[

(ℎ1(119908 (119905))) (minus7)

ℎ1(119908 (119905)) (3) + ℎ

2(119908 (119905)) (3)

0

0(ℎ1(119908 (119905))) (2)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (5)

]]]

]

1205832(119908 (119905))

=

[[[

[

(ℎ2(119908 (119905))) (minus3)

ℎ1(119908 (119905)) (3) + ℎ

2(119908 (119905)) (3)

0

0(ℎ2(119908 (119905))) (5)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (5)

]]]

]

(70)

As previously the fuzzy stabilizer for this example is alsoconstructed bymeans of the PDC approach developed in [14]resulting in

1198701= [

10286 05714 06000 05714

minus10000 minus22500 minus05000 minus37500]

1198702= [

14000 13333 14000 03333

minus04000 minus11000 minus12000 minus09000]

(71)

Thus Figures 8 and 9 show the simulation results whenthe controller defined by (20) is applied

Besides Figure 10 shows the shape of the new member-ship functions which are clearly different from the originalones given in Figure 5

53 More General Problems Consider the T-S fuzzy MIMOsystem with multiple inputs defined by the following matri-ces

1198601=

[[[[[[[[[[[

[

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

minus6 4 2 4 5 3

2 3 minus1 5 minus2 4

]]]]]]]]]]]

]

1198611=

[[[[[[[[[[[

[

0 0

0 0

0 0

0 0

5 0

0 2

]]]]]]]]]]]

]

1198602=

[[[[[[[[[[[

[

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

minus6 1 2 minus5 4 6

0 2 minus7 4 5 2

]]]]]]]]]]]

]

1198612=

[[[[[[[[[[[

[

0 0

0 0

0 0

0 0

7 0

0 8

]]]]]]]]]]]

]

1198621= 1198622= [

0 1 1 0 0 0

1 0 1 0 0 0]

(72)

Mathematical Problems in Engineering 13

0 5 10 15 20 25 30minus15minus10

minus505

101520

Time

Am

plitu

dex1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus4minus2

02468

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 6 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus30minus20minus10

0102030

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus15

minus10

minus5

0

5

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 7 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of lineal regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus15minus10

minus505

101520

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus1

012345678

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 8 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

The reference signals will be generated by an exosystemdefined by

119878 = [

0 1

minus1 0] 119876 = [

1 0

0 1] (73)

Membership functions for Rules 1 and 2 are ℎ1(119909(119905)) =

(80 minus 1199091(119905))140 and ℎ

2(119909(119905)) = 1 minus ℎ

1(119905) Notice that for this

case 119899 = 119904 and 119862119894

= 119876 This implies that Π119894and Γ

119894must

be obtained by solving Francis Equations for each subsystemHowever due to form of the fuzzy plant it is possible to verifythat Π

1= Π2

14 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus40minus30minus20minus10

0102030

Time

Am

plitu

dex2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus10

minus8minus6minus4minus2

0

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 9 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(11) (x1(t))1205832(11) (x1(t))

(a)

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(22) (x1(t))1205832(22) (x1(t))

(b)

Figure 10 (a) Membership functions corresponding to position (11) of 1205831(119908(119905)) and 120583

2(119908(119905)) (b) Membership functions corresponding to

position (22) 1205831(119908(119905)) and 120583

2(119908(119905))

So by applying (9) for each local subsystem with 119894 = 1 2it results in

Π1= Π2=

[[[[[[[[[[[

[

05000 05000

15000 minus05000

minus05000 05000

05000 15000

minus05000 minus05000

minus15000 05000

]]]]]]]]]]]

]

Γ1= [

07000 minus03000

minus20000 minus55000]

Γ2= [

23571 12143

minus04375 minus01875]

(74)

Again the fuzzy stabilizer for this system is constructedby means of the PDC approach developed in [14] yielding

1198701=[[

[

minus12229 00609 minus40284 01291 minus25666 minus04061

16919 minus83997 32934 minus127153 15732 minus61836

]]

]

1198702=[[

[

minus08735 04721 minus28775 13780 minus16904 minus07187

06730 minus19749 15734 minus30538 minus04817 minus12959

]]

]

(75)

The behavior obtained by applying controller (61) isdepicted in Figures 11 and 12 Notice the exact outputregulation is not achieved

Mathematical Problems in Engineering 15

0 5 10 15 20 25 30minus60minus40minus20

0204060

Time

Am

plitu

dex1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus80minus70minus60minus50minus40minus30minus20minus10

01020

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 11 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus15

minus10

minus5

0

5

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 12 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

Now the adequate membership functions are computedfrom (54) resulting in

1205831(119908 (119905))

=

[[[

[

(ℎ1(119908 (119905))) (5)

ℎ1(119908 (119905)) (5) + ℎ

2(119908 (119905)) (7)

0

0(ℎ1(119908 (119905))) (2)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (8)

]]]

]

1205832(119908 (119905))

=

[[[

[

(ℎ2(119908 (119905))) (7)

ℎ1(119908 (119905)) (5) + ℎ

2(119908 (119905)) (7)

0

0(ℎ2(119908 (119905))) (8)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (8)

]]]

]

(76)

When ℎ119894(119908(119905)) is replaced by its corresponding expressions

one gets

1205831(119908 (119905)) =

[[[

[

1225

1199081(119905) + 140

minus5

20

0 minus1199081(119905) minus 80

31199081(119905) + 320

]]]

]

1205832(119908 (119905)) =

[[[

[

7

2minus

1225

1199081(119905) + 410

0

041199081(119905) + 240

31199081(119905) + 320

]]]

]

(77)

The results of applying control law (20) on the fuzzy plantare given in Figures 13 and 14

The ldquocomputedrdquo membership functions for the fuzzyregulator are given in Figure 15 Again it can be readilyobserved that these new membership functions are differentfrom the original ones

Notice that although this example does not fulfill theconditions ofTheorem 5 themembership functions obtainedfrom (39) allow the exact tracking for the fuzzy problemThis

16 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus80minus70minus60minus50minus40minus30minus20minus10

010

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 13 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus8minus6minus4minus2

02468

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 14 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(11) (x1(t))1205832(11) (x1(t))

(a)

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(22) (x1(t))1205832(22) (x1(t))

(b)

Figure 15 (a) Membership functions corresponding to position (11) of 1205831(119908(119905)) and 120583

2(119908(119905)) (b) Membership functions corresponding to

position (22) 1205831(119908(119905)) 120583

2(119908(119905))

Mathematical Problems in Engineering 17

suggests that the approach presented in this work may beapplied on a bigger class of T-S models however this studyis not completed yet

6 Conclusions

A fuzzy regulator for continuous-time systems based on thecombination of linear regulators adjusted by different mem-bership functions has been presented The main advantageis that analytical expressions for the membership functionswhich allow the proper combination of the local regulators inorder to guarantee the exact regulation can be easily appliedAs 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 modelscan be commonly found in mechanics electromechanicselectronics robotics and so forth

Some numerical examples are used to illustrate theapplicability of the proposed approach even in cases wherethe dimensions of the plant and the exosystem are differentbetween each otherThis suggests that the approach presentedin this work may be applied on a bigger class of T-S models

Finally the method proposed in this work avoids thedisadvantage of constructing an exact fuzzy regulator basedon overall T-S fuzzy system whose expression may resultvery large Instead the given approach offers 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 Interests

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

Acknowledgments

This work was partially supported by CONACYT (ConsejoNacional de Ciencia y Tecnologıa) through scholarship SNI(Sistema Nacional de Investigadores) and by IPN (InstitutoPolitecnico Nacional) through research Projects 20140659and 20150487 and scholarships EDI (Estımulo al Desempenode los Investigadores) COFAA (Comision de Operacion yFomento de Actividades Academicas) and BEIFI (Beca deEstımulo Institucional de Formacion de Investigadores)

References

[1] B A Francis ldquoThe linear multivariable regulator problemrdquoSIAM Journal on Control and Optimization vol 15 no 3 pp486ndash505 1977

[2] B A Francis andWMWonham ldquoThe internalmodel principleof control theoryrdquo Automatica vol 12 no 5 pp 457ndash465 1976

[3] A Isidori and C I Byrnes ldquoOutput regulation of nonlinearsystemsrdquo IEEE Transactions on Automatic Control vol 35 no2 pp 131ndash140 1990

[4] B Castillo-Toledo S Di Gennaro and F Jurado ldquoTrajectorytracking for a quadrotor via fuzzy regulationrdquo in Proceedings ofthe World Automation Congress (WAC rsquo12) pp 1ndash6 June 2012

[5] S Chen ldquoOutput regulation of nonlinear singularly perturbedsystems based on T-S fuzzy modelrdquo Journal of Control Theoryand Applications vol 3 no 4 pp 399ndash403 2005

[6] R Tapia-Herrera J A Meda-Campana S Alcamp119886119901119900119904 119886ntara-Montes T Hernamp119886119901119900119904 119886ndez-Cortamp119886119901119900119904 119890s and L Salgado-Conrado ldquoTuning of a TS fuzzy output regulator using thesteepest descent approach and ANFISrdquoMathematical Problemsin Engineering vol 2013 Article ID 873430 14 pages 2013

[7] K-Y Lian and J-J Liou ldquoOutput tracking control for fuzzysystems via output feedback designrdquo IEEE Transactions onFuzzy Systems vol 14 no 5 pp 628ndash639 2006

[8] X-J Ma and Z-Q Sun ldquoOutput tracking and regulationof nonlinear system based on takagi-sugeno fuzzy modelrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 30 no 1 pp 47ndash59 2000

[9] H J Lee J B Park and Y H Joo ldquoComments on lsquooutputtracking and regulation of nonlinear system based on Takagi-Sugeno fuzzy model1015840rdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 33 no 3 pp 521ndash523 2003

[10] J A Meda-Campana B Castillo-Toledo and G Chen ldquoSyn-chronization of chaotic systems from a fuzzy regulationapproachrdquo Fuzzy Sets and Systems vol 160 no 19 pp 2860ndash2875 2009

[11] J A Meda-Campana J C Gamp119886119901119900119904 119900mez-Mancilla and BCastillo-Toledo ldquoExact output regulation for nonlinear systemsdescribed by Takagi-Sugeno fuzzy modelsrdquo IEEE Transactionson Fuzzy Systems vol 20 no 2 pp 235ndash247 2012

[12] C Byrnes F Priscoli and A Isidori Output Regulation ofUncertain Nonlinear Systems Birkhauser 1997

[13] A Isidori Nonlinear Control Systems Communications andControl Engineering Series Springer Berlin Germany 3rdedition 1995

[14] K Tanaka and H O Wang Fuzzy Control Systems Designand Analysis A Linear Matrix Inequality Approach John Wileyamp119886119898119901 Sons New York NY USA 2001

[15] L-X Wang A Course in Fuzzy Systems and Control PrenticeHall Upper Saddle River NJ USA 1997

[16] I Abdelmalek N Golamp119886119901119900119904 119890a and M L Hadjili ldquoA newfuzzy Lyapunov approach to non-quadratic stabilization ofTakagi-Sugeno fuzzy modelsrdquo International Journal of AppliedMathematics andComputer Science vol 17 no 1 pp 39ndash51 2007

[17] J A Meda-Campana and B Castillo-Toledo ldquoOn the outputregulation for TS fuzzymodels using slidingmodesrdquo inProceed-ings of the American Control Conference (ACC rsquo05) pp 4062ndash4067 Portland Ore USA June 2005

[18] B Castillo-Toledo and J A Meda-Campana ldquoThe fuzzydiscrete-time robust regulation problem a LMI approachrdquo inProceedings of the 41st IEEE Conference on Decision and Controlvol 2 pp 2159ndash2164 IEEE Las Vegas Nev USA December2002

[19] B Castillo-Toledo J A Meda-Campana and A Titli ldquoAfuzzy output regulator for Takagi-Sugeno fuzzy modelsrdquo inProceedings of the IEEE International Symposium on IntelligentControl pp 310ndash315 Houston Tex USA October 2003

[20] B-C Ding H-X Sun and Y-E Qiao ldquoStability analysis of T-Sfuzzy control systems based on parameter-dependent Lyapunovfunctionrdquo Acta Automatica Sinica vol 31 no 4 pp 651ndash6542005

[21] M Narimani H K Lam R Dilmaghani and C Wolfe ldquoLMI-based stability analysis of fuzzy-model-based control systemsusing approximated polynomial membership functionsrdquo IEEE

18 Mathematical Problems in Engineering

Transactions on Systems Man and Cybernetics Part B Cyber-netics vol 41 no 3 pp 713ndash724 2011

[22] H K Lam ldquoPolynomial fuzzy-model-based control systemsstability analysis via piecewise-linear membership functionsrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 588ndash5932011

[23] K Tanaka T Hori and H O Wang ldquoA fuzzy Lyapunovapproach to fuzzy control system designrdquo in Proceedings of theAmerican Control Conference vol 6 pp 4790ndash4795 ArlingtonVa USA June 2001

[24] K Tanaka T Hori and H O Wang ldquoA multiple Lyapunovfunction approach to stabilization of fuzzy control systemsrdquoIEEE Transactions on Fuzzy Systems vol 11 no 4 pp 582ndash5892003

[25] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 174ndash1972005

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

of T-S fuzzy models the exact output regulation To this endnew membership functions will be systematically computedin order to adequately combine the linear local regulatorsguaranteeing 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 soft computing techniques

Themain idea comes from the 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 finding 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 finda control law for a class of Takagi-Sugeno fuzzy models inorder to achieve exact output regulation on the basis of localregulators and computing of new membership functionseven if different 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 119905 ge 0 [11] Besides the newmembership functions allowing the proper combination thelocal regulators are given as a mathematical expressions

The 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 fuzzyregulation problemThemain result is developed in Section 3In Section 4 some examples are presented and finally inSection 5 some conclusions are drawn

2 The Output Regulation Problem

Consider a nonlinear system defined by

(119905) = 119891 (119909 (119905) 120596 (119905) 119906 (119905)) (1)

119910 (119905) = 119888 (119909 (119905)) (2)

(119905) = 119904 (120596 (119905)) (3)

119910ref (119905) = 119902 (120596 (119905)) (4)

119890 (119905) = ℎ (119909 (119905) 120596 (119905)) (5)

where 119909(119905) isin R119899 is the state vector of the plant 119908(119905) isin 119882 sub

R119904 is the state vector of the exosystem which generates thereference andor the perturbation signals and 119906(119905) isin R119898 isthe input signal Equation (5) refers to difference betweenoutput system of the plant (119910(119905) isin R119898) and the referencesignal (119910ref(119905) isin R119898) that is ℎ(119909(119905) 120596(119905)) = 119910(119905) minus 119910ref =119888(119909(119905)) minus 119902(119909(119905)) taking into account that 119898 le 119899 Besides itis assumed that 119891(119909 119906 119908) ℎ(119909 119908) and 119908(119905) are 119862119896 functions(for some large 119896) of their arguments and also that119891(0 0 0) =0 119904(0) = 0 and ℎ(0 0) = 0 [12]

Clearly by linearizing (1)ndash(5) around 119909 = 0 one gets

(119905) = 119860119909 (119905) + 119861119906 (119905) + 119875119908 (119905)

119910 (119905) = 119862119909 (119905)

(119905) = 119878119908 (119905)

119910ref (119905) = 119876119908 (119905)

119890 (119905) = 119862119909 (119905) minus 119876119908 (119905)

(6)

Thus the Nonlinear Regulator Problem [3 13] consists offinding a controller 119906(119905) = 120572(119909(119905) 119908(119905)) such that the closed-loop system (119905) = 119860119909(119905)+119861120572(119909(119905) 0) has an asymptoticallystable equilibrium point and the solution of the system (6)satisfies lim

119905rarrinfin119890(119905) = 0

So by defining 120587(119908(119905)) as the steady-state zero errormanifold and 120574(119908(119905)) as the steady-state input the followingtheorem gives the conditions for the solution of nonlinearregulation problem

Theorem 1 Suppose that (119905) = 119904(119908(119905)) is Poisson stable anda gain119870 exists such that the matrix 119860+119861119870 is stable and thereexist mappings 119909

119904119904(119905) = 120587(119908(119905)) and 119906

119904119904= 120574(119908(119905))with 120587(0) =

0 and 120574(0) = 0 satisfying

120597120587 (119908 (119905))

120597119908 (119905)119904 (119908 (119905)) = 119891 (120587 (119908 (119905)) 119908 (119905) 120574 (119908 (119905)))

0 = ℎ (120587 (119908 (119905)) 119908 (119905))

(7)

Then the signal control for the nonlinear regulation is given by

119906 (119905) = 119870 (119909 (119905) minus 120587 (119908 (119905))) + 120574 (119908 (119905)) (8)

Proof See [12 13]

Nonlinear partial differential equations (7) are knownas Francis-Isidori-Byrnes (FIB) equations and their linearcounterparts are obtained when the mappings 119909

119904119904(119905) =

120587(119908(119905)) and 119906119904119904(119905) = 120574(119908(119905)) become into 119909

119904119904(119905) = Π119908(119905)

and 119906119904119904(119905) = Γ119908(119905) respectively Thus the linear problem

is reduced to solve a set of linear matrix equations (Francisequations) [1]

Π119878 = 119860Π + 119861Γ + 119875

0 = 119862Π minus 119876

(9)

21 The 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 Such a model successfully represents a nonlinearsystem at least in a predefined region of phase space [14]It is important to remark that in this work the exosystem ispurely ldquolinearrdquo because the computation of new membershipfunctions for the general case is still an open problem

The T-S model for system (1)ndash(5) is given by [15]

Model Rule 119894

IF 1199111(119905) is119872119894

1and sdot sdot sdot and 119911

119901(119905) is119872119894

119901

Mathematical Problems in Engineering 3

THEN

(119905) = 119860119894119909 (119905) + 119861

119894119906 (119905) + 119875

119894119908 (119905)

(119905) = 119878119908 (119905)

119890 (119905) = 119862119894119909 (119905) minus 119876119908 (119905)

119894 = 1 2 119903

(10)

where 119903 is the number of rules in the model and 119872119894

119895is the

fuzzy sets defined on the basis of the knowledge of the systemThen the regulation problem defined by (1)ndash(5) can be

represented through the T-S fuzzy model that is [10]

(119905) =

119903

sum

119894=1

ℎ119894(119911 (119905)) 119860

119894119909 (119905) + 119861

119894119906 (119905) + 119875

119894119908 (119905)

(119905) = 119878119908 (119905)

119890 (119905) =

119903

sum

119896=1

ℎ119894(119911 (119905)) 119862

119894119909 (119905) minus 119876119908 (119905)

(11)

where 119909(119905) isin R119899 is the state vector of the plant 119908(119905) isin R119904

is the state vector of the exosystem 119906(119905) isin R119898 is the inputsignal and 119890(119905) isin R119898 and ℎ

119894(119911(119905)) is the normalized weight

of each rule which depends on the membership function forthe premise variable 119911

119895(119905) in119872119894

119895 where

120596119894(119911 (119905)) =

119901

prod

119895=1

119872119894

119895(119911119895(119905))

ℎ119894(119911 (119905)) =

120596119894(119911 (119905))

sum119903

119894=1120596119894(119911 (119905))

119903

sum

119894=1

ℎ119894(119911 (119905)) = 1

ℎ119894(119911 (119905)) ge 0

(12)

with 119911(119905) = [1199111(119905) 1199112(119905) sdot sdot sdot 119911

119901(119905)] as a function of 119909(119905) 119894 =

1 119903 and 119895 = 1 119901The Exact Fuzzy Regulator Problem consists of finding

a controller 119906(119905) = 120572(119909(119905) 119908(119905)) such that the closed-loopsystem

(119905) =

119903

sum

119894=1

ℎ119894(119911 (119905)) 119860

119894119909 (119905) + 119861

119894120572 (119909 (119905) 0) (13)

has an asymptotically stable equilibrium point and thesolution of system (11) satisfies lim

119905rarrinfin119890(119905) = 0

From [10 12 13] the desired overall fuzzy controller canbe represented as

119906 (119905) =

119903

sum

ℎ=1

ℎ119894(119911 (119905)) 119870

119894 [119909 (119905) minus 120587 (119908 (119905))] + 120574 (119908 (119905)) (14)

Considering that approximations for mappings 120587(119908(119905)) and120574(119908(119905)) can be obtained by

(119908 (119905)) =

119903

sum

119894=1

ℎ119894(119911 (119905)) Π

119894119908 (119905)

120574 (119908 (119905)) =

119903

sum

119894=1

ℎ119894(119911 (119905)) Γ

119894119908 (119905)

(15)

then the solution of the fuzzy regulation problem requires toobtain Π

119894and Γ119894from the 119903 linear local regulators problems

included in (11) and defined by

Π119894119878 = 119860

119894Π119894+ 119861119894Γ119894+ 119875119894

0 = 119862119894Π119894minus 119876

(16)

for 119894 = 1 119903Thus the following controller is obtained

119906 (119905) =

119903

sum

119894=1

ℎ119894(119911 (119905)) 119870

119894[119909 (119905) minus

119903

sum

119894=1

ℎ119894(119911 (119905)) Π

119894119908 (119905)]

+

119903

sum

119894=1

ℎ119894(119911 (119905)) Γ

119894119908 (119905)

(17)

However according to [10] the exact fuzzy regulation isachieved with this controller only when Π

1= sdot sdot sdot = Π

119903= Π

and

(1) input matrix is equal for all subsystems that is 1198611=

sdot sdot sdot = 119861119903= 119861 or

(2) the steady-state zero error manifold is equal for allsubsystems that is Γ

1= sdot sdot sdot = Γ

119903= Γ

In the present work restrictions 1 and 2 are avoidedby 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 findingthe steady-state zero error manifold 119909

119904119904(119905) = 120587(119908(119905)) and

the steady-state input 119906119904119904(119905) = 120574(119908(119905)) for the overall T-

S fuzzy problem through 119890119904119904(119905) = 119909(119905) minus 119909

119904119904(119905) = 119909(119905) minus

120587(119908(119905)) (steady-state error equation) resulting in 120587(119908(119905)) =Π119891(119905)119908(119905) and 120574(119908(119905)) = Γ

119891(119905)119908(119905) whereΠ

119891(119905) and Γ

119891(119905) are

continuous time-variantmatrices and 120587(119908(119905)) = Π119891(119905)119908(119905) is

a 119862119896minus1 function [12]Within the next section the exact output regulation for a

class of T-S fuzzy systems is obtained by proposing a solutionfor 120574(119908(119905)) on the basis of different membership functions inthe regulator

3 The Exact Output Fuzzy Regulation byLocal Regulators

In this section a particular class of T-S fuzzy models isconsidered to solve the exact output regulation on the basis oflinear local controllers So the main goal is to find an overallregulator based on the fuzzy summation of local regulatorsconsidering adequate membership functions Clearly such

4 Mathematical Problems in Engineering

membership functions are not necessarily the same includedin the fuzzy plant Thus the steady-state input 120574(119908(119905)) can bedefined as

120574 (119908 (119905)) =

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905) (18)

where 120583119894(119908(119905)) are new membership functions such that

the fuzzy output regulator obtained from local regulatorscoincides with the controller given in [11] at least for theclass of fuzzy models considered This novel approach onlyrequires the computation of the linear local controllers andthe computation of the newmembership functionswhichwillbe presented belowThe regulator obtained in this way is validfor all 119905 ge 0 while in [11] this condition needs to be verified

Consider the following matrices

119860119894=

[[[[[[[

[

0 1 0 sdot sdot sdot 0

0 0 1 sdot sdot sdot 0

d

119886119894

1119886119894

2119886119894

3sdot sdot sdot 119886119894

119899

]]]]]]]

]

119861119894=

[[[[[[[

[

0

0

119887119894

1

]]]]]]]

]

119878 =

[[[[[[

[

0 1 0 sdot sdot sdot 0

0 0 1 sdot sdot sdot 0

d

119904111990421199043sdot sdot sdot 119904119899

]]]]]]

]

119862119894= [1198881 0 sdot sdot sdot 0]

119876 = [1199021 0 sdot sdot sdot 0]

(19)

where 119886119894119895represents the elements of the last row of 119860

119894and 1198871198941

represents the element of the last row of 119861119894with 119895 = 1 119899

and 119894 = 1 119903 It is worth mentioning that 119860119894and 119878 are

matrices of the same dimension as shown in (19) Notice thatmatrices in the formof 119878 and119876 can be used to generate a greatvariety of signals ensuring in thatway the applicability of theapproach in a great number of cases Therefore from (14) thecontrol input can be defined by

119906 (119905) =

119903

sum

119894=1

ℎ119894(119909 (119905)) 119896

119894119909 (119905) minus 120587 (119908 (119905))

+

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905)

(20)

because as mentioned before 119911(119905) is a function of 119909(119905) and insteady-state 119909(119905) = 120587(119908(119905))

On the other hand from 119862119894 119876 and from (6) it can be

concluded that 11988811199091(119905) = 119902

11199081(119905) in steady-state while from

(19) the following exosystem can be easily derived

1= 1199082

2= 1199083

119899= 11990411199081+ 11990421199082+ sdot sdot sdot + 119904

119899119908119899

119899=

119899

sum

119894=1

119904119894119908119894

(21)

Besides each subsystem can be rewritten as follows

1= 1199092

2= 1199093

119899= 119886119894

11199091+ 119886119894

21199092+ sdot sdot sdot + 119886

119894

119899119909119899+ 119887119894

1119906119904119904119894

(22)

By equating the time derivative of the fuzzy output withthe time derivative of the reference signal it results in 119888

11=

11990211 Furthermore from (21) and (22) it is possible to

conclude that

1198881119894= 1199021119894 (23)

where 119894 = 1 2 119899 From this analysis two results can beobtained First

Π =

[[[[[[[[[[

[

1199021

1198881

0 sdot sdot sdot 0

01199021

1198881

sdot sdot sdot 0

d

0 0 sdot sdot sdot1199021

1198881

]]]]]]]]]]

]

(24)

and due to the shape of (19)Π is constant and common for allsubsystemsThus 120587(119908(119905)) = Π119908(119905) Second the local steady-state input is defined by

119906119904119904119894

=(11990211198881) 11990411199081minus (11990211198881) 119886119894

11199081

119887119894

1

+(11990211198881) 11990421199082minus (11990211198881) 119886119894

21199082

119887119894

1

+ sdot sdot sdot +(11990211198881) 119904119899119908119899minus (11990211198881) 119886119894

119899119908119899

119887119894

1

(25)

Mathematical Problems in Engineering 5

The previous expression can be rewritten in a compact formas follows

119906119904119904119894

=

sum119899

119895=1(11990211198881) [119904119895119908119895minus 119886119894

119895119908119895]

119887119894

1

forall119894 = 1 119903 (26)

and from (18) the fuzzy steady-state input is

119906119904119904(119905) =

119903

sum

119894=1

120583119894(119908 (119905)) 119906

119904119904119894

=

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905) forall119894 = 1 119903

(27)

From the previous analysis each Γ119894can be computed from

Γ119894=1199021

1198881

[1199041minus 119886119894

1

119887119894

1

1199042minus 119886119894

2

119887119894

1

sdot sdot sdot119904119899minus 119886119894

119899

119887119894

1

] (28)

Now it is possible to rewrite the fuzzy model as [11]

(119905) = 119860 (119905) 119909 (119905) + 119861 (119905) 119906 (119905) + (119905) 119908 (119905) (29)

(119905) = 119878119908 (119905) (30)

119890 (119905) = 119862 (119905) 119909 (119905) minus 119876119908 (119905) (31)

where

119860 (119905) =

119903

sum

119894=1

ℎ119894(119911 (119905)) 119860

119894

119861 (119905) =

119903

sum

119894=1

ℎ119894(119911 (119905)) 119861

119894

(119905) =

119903

sum

119894=1

ℎ119894(119911 (119905)) 119875

119894

119862 (119905) =

119903

sum

119894=1

ℎ119894(119911 (119905)) 119862

119894

(32)

For the sake of simplicity in the following analysis it isconsidered that119875

119894= 0 119894 = 1 119903 that is (119905) = 0 However

taking similar steps as those used below the result can beeasily extended to the case involving (119905) = 0

Thus from (19) and (29) it can be observed that the lastequation

119899in (29) is given by

119899= [ℎ1(119909 (119905)) (119886

119894

11199091+ sdot sdot sdot + 119886

119894

119899119909119899)

+ sdot sdot sdot + ℎ119903(119909 (119905)) (119886

119894

11199091+ sdot sdot sdot + 119886

119894

119899119909119899)]

+ [ℎ1(119909 (119905)) 119887

119894

1+ sdot sdot sdot + ℎ

119903(119909 (119905)) 119887

119903

1] 119906119904119904(119905)

(33)

where compact form is

119899=

119903

sum

119894=1

ℎ119894(119909 (119905))(

119899

sum

119895=1

119886119894

119895119909119895) + (

119903

sum

119894=1

ℎ119894(119909 (119905)) 119887

119894

1)119906119904119904(119905)

(34)

considering that in steady-state 119909(119905) = Π119908(119905) the totalsteady-state input can be obtained from (23) resulting in

119903

sum

119894=1

ℎ119894(119908 (119905))(

1199021

1198881

119899

sum

119895=1

119886119894

119895119908119895)

+ (

119903

sum

119894=1

ℎ119894(119908 (119905)) 119887

119894

1)119906119904119904(119905) =

1199021

1198881

119899

sum

119894=1

119904119894119908119894

(35)

Consequently the steady-state input for the total fuzzy systemcan be constructed by

119906119904119904(119905)

=

(11990211198881)sum119899

119894=1119904119894119908119894minus (11990211198881)sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

1

(36)

Now a new set of fuzzy membership functions 120583119894(119908(119905))

satisfying sum119903119894=1

120583119894(119908(119905)) = 1 with 120583

119894(119908(119905)) ge 0 119894 = 1 119903

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 defined as follows

119906119904119904119894

= 120574 (119908 (119905)) =

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905)

=

119903

sum

119894=1

120583119894(119908 (119905))

sum119899

119895=1(11990211198881) [119904119895119908119895minus 119886119894

119895119908119895]

119887119894

1

119906119904119904= 120574 (119908 (119905))

=

(11990211198881)sum119899

119894=1119904119894119908119894minus (11990211198881)sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

1

(37)

respectivelyThus the local fuzzy regulatorswill bemultipliedby the new membership functions and the result will beequated to the global fuzzy regulator (36) as follows

(11990211198881)sum119899

119894=1119904119894119908119894minus (11990211198881)sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

1

=

119903

sum

119894=1

120583119894(119908 (119905))

sum119899

119895=1(11990211198881) [119904119895119908119895minus 119886119894

119895119908119895]

119887119894

1

(38)

So the adequate membership functions are

120583119894(119908 (119905)) =

119887119894

1ℎ119894(119908 (119905))

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

1

(39)

with 119894 = 1 2 119903 It is important to remark that the newmembership functions 120583

119894(119908(119905)) are given in terms of ℎ

119894(119908(119905))

for simplicity However ℎ119894(119908(119905)) can be removed from the

new membership functions (39) when ℎ119894(119908(119905)) is replaced

by its corresponding expression Besides ℎ119894(119908(119905)) is directly

obtained from ℎ119894(119909(119905)) by considering that in steady-state

119909(119905) = Π119908(119905)

6 Mathematical Problems in Engineering

Notice that (39) fulfills conditions119903

sum

119894=1

120583119894(119908 (119905)) = 1

120583119894(119908 (119905)) ge 0

(40)

for all 119894 = 1 2 119903

Remark 2 It can be observed that (39) is always valid whenthe entries of input matrix 119887

119894

1have the same sign if this

condition is not fulfilled the denominators of the new fuzzymembership functions will present singularities Neverthe-less these singularities may appear outside the operationregion allowing more flexibility but such cases require aparticular study of the system which is beyond the scope ofthis work

The following theorem provides the conditions for theexistence of the exact output fuzzy regulator for a class of T-Sfuzzy models

Theorem 3 The exact fuzzy output regulation with fullinformation for T-S fuzzy systems in the form of (11) definedby matrices (19) is solvable if (a) the sign of entries forcorresponding position inside input matrices 119861

119894for 119894 =

1 119903 is the same (b) there exists a fuzzy stabilizer 119906(119905) =sum119903

119894=1ℎ119894(119911(119905))119870

119894119909 for the T-S fuzzy system [16] and (c) the

exosystem (119905) = 119878119908(119905) is Poisson stable Moreover the ExactOutput Fuzzy Regulation Problem is solvable by the controller

119906 (119905) =

119903

sum

119894=1

ℎ119894(119909 (119905)) 119896

119894119909 (119905) minus Π119908 (119905)

+

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905)

(41)

where 120583119894(119908(119905)) can be readily obtained from (39)

Proof From the previous analysis the existence of mappings120587(119908(119905)) = Π119908(119905) and 120574(119908(119905)) = sum

119903

119894=1120583119894(119908(119905))Γ

119894119908(119905) is

guaranteed when the local subsystems are defined by (19)that is the form of 120587(119908(119905)) corresponds to a diagonal matrixand will be the same for all subsystems while 120574(119908(119905)) can bedirectly substitute

On the other hand condition (a) avoids singularities inthe new membership functions 120583

119894(119908(119905)) while the inclusion

of condition (b) has been thoroughly discussed in [12 13 17ndash19] and it implies that the steady-state manifold can be madeasymptotically stable by the action of the fuzzy stabilizer

Finally condition (c) is introduced to avoid that thereference signal converges to zero because if the referencetends to zero then the regulation problem turns into astabilization one which can be solved by means of thefuzzy stabilizer In addition as mentioned before this kindof matrix 119878 can be used to generate a great number ofreference signals The 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 20ndash25]

4 Exact Output Fuzzy Regulation Problem fora Class of T-S Fuzzy Systems with MultipleInputs and Multiple Outputs (MIMO Case)

Now the problem will be extended to T-S fuzzy MIMOmodels defined by matrices

119860119894= [

119874 119868

119863119894

] 119861119894= [

119874

119865119894

]

119878 = [119874 119868

119866] 119862

119894= [ 119867119862 119874

119862 ]

119876119894= [ 119867119876 119874

119876 ]

(42)

for 119894 = 1 119903 with

119868 =

[[[[[[

[

1 0 sdot sdot sdot 0

0 1 sdot sdot sdot 0

d

0 0 sdot sdot sdot 1

]]]]]]

]

119874 =

[[[[[[

[

0 0 sdot sdot sdot 0

0 0 sdot sdot sdot 0

d

0 0 sdot sdot sdot 0

]]]]]]

]

119863119894=

[[[[[[[

[

119886119894

(119896+1)1119886119894

(119896+1)2119886119894

(119896+1)3sdot sdot sdot 119886119894

(119896+1)119899

119886119894

(119896+2)1119886119894

(119896+2)2119886119894

(119896+2)3sdot sdot sdot 119886119894

(119896+2)119899

d

119886119894

1198991119886119894

1198992119886119894

1198993sdot sdot sdot 119886

119894

119899119899

]]]]]]]

]

119865119894=

[[[[[[[

[

119887119894

119896+10 sdot sdot sdot 0

0 119887119894

119896+2sdot sdot sdot 0

d

0 0 sdot sdot sdot 119887119894

119899

]]]]]]]

]

119866 =

[[[[[[

[

119904(119896+1)1

119904(119896+1)2

119904(119896+1)3

sdot sdot sdot 119904(119896+1)119899

119904(119896+2)1

119904(119896+2)2

119904(119896+2)3

sdot sdot sdot 119904(119896+2)119899

d

1199041198991

1199041198992

1199041198993

sdot sdot sdot 119904119899119899

]]]]]]

]

119874119862= 119874119876=

[[[[[[

[

0 0 sdot sdot sdot 0

0 0 sdot sdot sdot 0

d

0 0 sdot sdot sdot 0

]]]]]]

]

Mathematical Problems in Engineering 7

119867119862=

[[[[[[

[

11988810 sdot sdot sdot 0

0 1198882sdot sdot sdot 0

d

0 0 sdot sdot sdot 119888119898

]]]]]]

]

119867119876=

[[[[[[

[

1199021

0 sdot sdot sdot 0

0 1199022sdot sdot sdot 0

d

0 0 sdot sdot sdot 119902119898

]]]]]]

]

(43)

where 119896 = 119899 minus 119898 119865119894isin R119898times119898 119874 isin R(119899minus119898)times119898 119874

119862isin R119898times119898

119868 isin R(119899minus119898)times(119899minus119898) 119863119894isin R119898times119899 119867

119862isin R119898times119898 119867

119876isin R119898times119898

and 119866 isin R119898times119899 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= 119908119898+1

2= 119908119898+2

119896+1

= 119904(119896+1)1

1199081+ 119904(119896+1)2

1199082+ sdot sdot sdot + 119904

(119896+1)119899119908119899

=

119899

sum

119895=1

119904(119896+1)119895

119908119895

119896+2

= 119904(119896+2)1

1199081+ 119904(119896+2)2

1199082+ sdot sdot sdot + 119904

(119896+2)119899119908119899

=

119899

sum

119895=1

119904(119896+2)119895

119908119895

119899= 11990411989911199081+ 11990411989921199082+ sdot sdot sdot + 119904

119899119899119908119899=

119899

sum

119895=1

119904119899119895119908119895

(44)

Moreover each subsystem can be described by

1= 119909119898+1

2= 119909119898+2

119896+1

= 119886119894

(119896+1)11199091+ 119886119894

(119896+1)21199092+ sdot sdot sdot + 119886

(119896+1)119899119909119899

+ 119887119894

119896+1119906119894

119904119904

=

119899

sum

119895=1

119886119894

(119896+1)119895119909119895+ 119887119894

119896+1119906119894

119904119904

119896+2

= 119886119894

(119896+2)11199091+ 119886119894

(119896+2)21199092+ sdot sdot sdot + 119886

(119896+2)119899119909119899

+ 119887119894

119896+2119906119894

119904119904

=

119899

sum

119895=1

119886119894

(119896+2)119895119908119895+ 119887119894

1+2119906119894

119904119904

119899= 11990411989911199081+ 11990411989921199082+ sdot sdot sdot + 119904

119899119899119908119899+ 119887119894

119899119906119894

119904119904

=

119899

sum

119895=1

119904119899119895119908119895+ 119887119894

119899119906119894

119904119904

(45)

As before using expressions (44) and (45) considering 11988811199091=

11990211199081 11988821199092

= 11990221199082 119888

119898119909119898

= 119902119898119908119898 and performing

successive substitutions of ℓ= 120587ℓℓwith ℓ = 1 119898 the

following results are obtained

Π =

[[[[[[

[

1205871

0 sdot sdot sdot 0

0 1205872sdot sdot sdot 0

d

0 0 sdot sdot sdot 120587119899

]]]]]]

]

(46)

where Π isin R119899times119899 and 119894 = 1 2 119903 also 1205871

=

11990211198881 1205872= 11990221198882 120587

119898= 119902119898119888119898 120587119898+1

= 11990211198881 120587119898+2

=

11990221198882 120587

119898+119898= 119902119898119888119898 and so forth Notice that the values

of 120587lowastrepeat themselves every 119898 entry up to 119899 Additionally

Π is constant and common for all subsystems while the localsteady-state inputs are defined by

119906119894

119904119904=

[[[[[[[[[[[[[

[

1205871119904(119896+1)1

1199081minus 1205871119886119894

(119896+1)11199081

1198871

119896+1

+

1205871119904(119896+1)1

1199082minus 1205872119886119894

(119896+1)11199082

1198871

119896+1

+ sdot sdot sdot +

1205871119904(119896+1)119899

119908119899minus 120587119899119886119894

(119896+1)119899119908119899

1198871

119896+1

1205872119904(119896+2)1

1199081minus 1205871119886119894

(119896+2)11199081

1198871

119896+2

+

1205872119904(119896+1)2

1199082minus 1205872119886119894

(119896+2)21199082

1198871

119896+2

+ sdot sdot sdot +

1205872119904(119896+2)119899

119908119899minus 120587119899119886119894

(119896+2)119899119908119899

1198871

119896+2

12058711989911990411989911199081minus 1205871119886119894

11989911199081

1198871119899

+12058711989911990411989921199082minus 1205872119886119894

11989921199082

1198871119899

+ sdot sdot sdot +120587119899119904119899119899119908119899minus 120587119899119886119894

119899119899119908119899

1198871119899

]]]]]]]]]]]]]

]

(47)

8 Mathematical Problems in Engineering

for 119894 = 1 119903 In compact form (47) turns into

119906119894

119904119904=

[[[[[[[[[[[[[[

[

119899

sum

119895=1

1205871119904(119896+1)119895

119908119895minus 120587119895119886(119896+1)119895

119908119895

119887119894

119896+1

119899

sum

119895=1

1205872119904(119896+2)119895

119908119895minus 120587119895119886(119896+2)119895

119908119895

119887119894

119896+2

119899

sum

119895=1

120587119899119904119899119895119908119895minus 120587119895119886119899119895119908119895

119887119894119899

]]]]]]]]]]]]]]

]

(48)

and from (27) the local matrices Γ119894have the form

Γ119894=

[[[[[[[[[[[[[

[

1205871119904(119896+1)1

minus 1205871119886119894

(119896+1)1

119887119894

119896+1

1205871119904(119896+1)2

minus 1205872119886119894

(119896+1)2

119887119894

119896+1

sdot sdot sdot

12058711199041198991

minus 120587119899119886119894

(119896+1)119899

119887119894

119896+1

1205872119904(119896+2)1

minus 1205871119886119894

(119896+2)1

119887119894

119896+2

1205872119904(119896+2)2

minus 1205872119886119894

(119896+2)2

119887119894

119896+2

sdot sdot sdot

12058721199041198991

minus 120587119899119886119894

(119896+2)119899

119887119894

119896+2

d

1205871198991199041198991

minus 1205871119886119894

1198991

119887119894119899

1205871198991199041198992

minus 1205872119886119894

1198992

119887119894119899

sdot sdot sdot1205871198991199041198991

minus 120587119899119886119894

119899119899

119887119894119899

]]]]]]]]]]]]]

]

(49)

Now again from the fuzzy plant given in (42) and theglobal fuzzy model (29) the expressions for

119896+1

119899are

119896+1

=

119903

sum

119894=1

ℎ119894(119909 (119905))(

119899

sum

119895=1

119886119894

(119896+1)119895119909119895)

+ (

119903

sum

119894=1

ℎ119894(119909 (119905)) 119887

119894

119896+1)119906119904119904

119896+2

=

119903

sum

119894=1

ℎ119894(119909 (119905))(

119899

sum

119895=1

119886119894

(119896+2)119895119909119895)

+ (

119903

sum

119894=1

ℎ119894(119909 (119905)) 119887

119894

119896+2)119906119904119904

119899=

119903

sum

119894=1

ℎ119894(119909 (119905))(

119899

sum

119895=1

119886119894

119899119895119909119895)

+ (

119903

sum

119894=1

ℎ119894(119909 (119905)) 119887

119894

119899)119906119904119904

(50)

So by applying successive substitutions as before with (44)(45) and knowing that in steady-state 119909

119895= 120587119895119908119895 the

following expressions are obtained

119903

sum

119894=1

ℎ119894(119908 (119905))(

119899

sum

119895=1

119886119894

(119896+1)119895120587119895119908119895)

+ (

119903

sum

119894=1

ℎ119894(119908 (119905)) 119887

119894

119896+1)119906119904119904= 1205871

119899

sum

119894=1

119904(119896+1)119894

119908119894

119903

sum

119894=1

ℎ119894(119908 (119905))(

119899

sum

119895=1

119886119894

(119896+2)119895120587119895119908119895)

+ (

119903

sum

119894=1

ℎ119894(119908 (119905)) 119887

119894

119896+2)119906119904119904= 1205872

119899

sum

119894=1

119904(119896+2)

119894119908119894

1199031

sum

119894=1

ℎ119894(119908 (119905))(

119899

sum

119895=1

119886119894

119899119895120587119895119908119895)

+ (

119903

sum

119894=1

ℎ119894(119908 (119905)) 119887

119894

119899)119906119904119904= 120587119899

119899

sum

119894=1

119904119899119894119908119894

(51)

Mathematical Problems in Engineering 9

Therefore the steady-state input for the overall fuzzy systemis

119906119904119904

=

[[[[[[[[[[[[[[

[

1205871sum119899

119894=1119904(119896+1)119894

119908119894minus sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

(119896+1)119895120587119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

119896+1

1205872sum119899

119894=1119904(119896+2)119894

119908119894minus sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

(119896+2)119895120587119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

119896+2

120587119899sum119899

119894=1119904119899119895119908119894minus sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

119899119895120587119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

119899

]]]]]]]]]]]]]]

]

(52)

But the desired form of the steady-state is

119906119904119904(119905) = 120574 (119908 (119905)) =

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905) (53)

Consequently the new membership functions 120583119894(119908(119905))

can be obtained after equating (52) and (53) resulting in

120583119894(119908 (119905)) =

[[[[[[

[

12058311989411

0 sdot sdot sdot 0

0 12058311989422

sdot sdot sdot 0

d

0 0 sdot sdot sdot 120583119894119898119898

]]]]]]

]

(54)

where

120583119894119895119897

=

119887119894

119896+119895ℎ119894(119908 (119905))

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

119896+119895

(55)

for all 119894 = 1 119903 119895 = 119897 = 1 119898 It can be readily observedthat the sum of the corresponding elements is equal to oneand 120583

119894119895119897(119908(119905)) ge 0

These new membership functions 120583119894(119908(119905)) are organized

in a matrix form for that reason sum119903119894=1

120583119894(119908(119905)) = 119868 (where 119868

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 119861

119894have the

same sign at corresponding positions Clearly this conditionavoids singularities in (55) As before ℎ

119894(119908(119905)) is directly

obtained from ℎ119894(119909(119905)) by considering that in steady-state

119909(119905) = Π119908(119905) At this point the following theorem can bedefined

Theorem 5 The exact fuzzy output regulation with fullinformation for T-S fuzzy MIMO systems in the form of(11) defined by matrices (42) is solvable if (a) the sign ofentries for corresponding position inside input matrices 119861

119894for

119894 = 1 119903 is the same (b) there exists a fuzzy stabilizer119906(119905) = sum

119903

119894=1ℎ119894(119911(119905))119870

119894119909 for the fuzzy system [16] and (c) the

exosystem (119905) = 119878119908(119905) is Poisson stable

Moreover the Exact Output Fuzzy Regulation Problem forMIMO systems is solvable by the controller

119906 (119905) =

119903

sum

119894=1

ℎ119894(119909 (119905)) 119896

119894119909 (119905) minus Π119908 (119905)

+

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905)

(56)

where 120583119894(119908(119905)) can be readily obtained from (54) and (55)

Proof It follows directly from Theorem 3 and previousanalysis

Remark 6 Notice that 120583119894(119908(119905)) is defined as a matrix and

this is not a typical membership function however 120583119894(119908(119905))

contains a set of membership functions related to each entryof 119861119894 This form allows us to design controller (20)

5 Numerical Examples

51 Simple Input Case Consider the problem of balancingand swing of an inverted pendulum on a cart The motionequations for this system are [14]

1(119905) = 119909

2(119905) (57)

2(119905)

=119892 sin(119909

1(119905))minus 119886119898119897119909

2

2sin (2119909

1(119905))2 minus119886 cos (119909

1(119905)) 119906 (119905)

41198973 minus 119886119898119897 cos2 (1199091(119905))

(58)

where 1199091(119905) denotes the angle (in radians) of the pendulum

from the vertical and 1199092(119905) is the angular velocity 119892 =

98ms2 is the gravity constant 119898 is the mass of the pen-dulum 119872 is the mass of the cart 2119897 is the length of thependulum and 119906 is the force applied to the cart (in newtons)119886 = 1(119898 +119872) with119898 = 20 kg119872 = 80 kg and 2119897 = 10min the simulations The goal is to track the reference definedby the exosystem within the range 119909

1isin (minus1205872 1205872) To this

end a fuzzy model representing the nonlinear dynamics is asfollows

Rule 1

IF 1199091(119905) is about 0

THEN (119905) = 1198601119909(119905) + 119861

1119906(119905)

Rule 2

IF 1199091(119905) is about plusmn1205872(|119909

1| lt 1205872)

THEN (119905) = 1198602119909(119905) + 119861

2119906(119905)

10 Mathematical Problems in Engineering

h1

h2

x1minus90 90

(deg)

1

00

Figure 1 Membership functions of two-rule model

Membership functions for Rules 1 and 2 are shown in Figure 1In this example the T-S fuzzymodel of the plant is defined bymatrices

1198601= [

0 1

172941 0] 119861

1= [

0

minus01765]

1198602= [

0 1

41585 0] 119861

2= [

0

minus01500]

1198621= 1198622= [1 0]

(59)

while the exosystem can be constructed from the followingmatrices

119878 = [

0 1

minus1 0] 119876 = [1 0] (60)

which allows us to generate a great kind of sinusoidal refer-ences Figure 2 shows the simulation results after applying thecontroller

119906 (119905) =

119903

sum

119894=1

ℎ119894(119909 (119905)) 119870

119894[119909 (119905) minus

119903

sum

119894=1

ℎ119894(119908 (119905)) Π

119894119908 (119905)]

+

119903

sum

119894=1

ℎ119894(119908 (119905)) Γ

119894119908 (119905)

(61)

defined in [10] As expected controller (61) is unable toachieve exact regulation although tracking error remainsbounded

Now using the method derived in this work the localmatrices Γ

119894can be readily obtained from (28) resulting in

Γ1= [1036667 0] Γ

2= [343836 0] (62)

Besides knowing that 11990211198881= 1 matrix Π is obtained from

(24)

Π = [

1 0

0 1] (63)

while the newmembership functions computed from (39) are

1205831(119908 (119905)) =

(ℎ1(119908 (119905))) (minus01765)

ℎ1(119908 (119905)) (minus01765) + ℎ

2(119908 (119905)) (minus01500)

1205832(119908 (119905)) =

(ℎ2(119908 (119905))) (minus01500)

ℎ1(119908 (119905)) (minus01765) + ℎ

2(119908 (119905)) (minus01500)

(64)

On the other hand the fuzzy stabilizer for this system isconstructed bymeans of the PDC approach developed in [14]which is formed by

1198701= [1234898 69979] 119870

2= [586405 77962]

(65)

Figure 3 shows simulation with the controller definedin (20) Observe how the new membership functions arecapable of combining the local regulators in the proper wayto accomplish asymptotic tracking

To verify the effectiveness of the latter regulator thefuzzy controller is applied on the original system (57) Thesimulation results appear in Figure 4 Notice that the trackingerror depicted in Figure 4(b) is due to the approximationprovided by the T-S fuzzy model Such an error can bereduced by considering a better approximation or an exactrepresentation of the nonlinear system as the one given bythe nonlinear sector approach [14]

52 Multiple Inputs and Multiple Outputs (MIMO) CaseNow consider a T-S fuzzy system of two rules defined by thefollowing matrices

1198601=

[[[[[

[

0 0 1 0

0 0 0 1

6 4 2 4

2 3 1 5

]]]]]

]

1198611=

[[[[[

[

0 0

0 0

minus7 0

0 2

]]]]]

]

1198602=

[[[[[

[

0 0 1 0

0 0 0 1

3 4 2 1

2 4 6 2

]]]]]

]

1198612=

[[[[[

[

0 0

0 0

minus3 0

0 5

]]]]]

]

1198621= 1198622= [

1 0 0 0

0 1 0 0]

(66)

Mathematical Problems in Engineering 11

0 5 10 15 20 25 30 35 40minus40minus30minus20minus10

010203040

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30 35 40minus04minus03minus02minus01

0010203

Tracking error

Time

Am

plitu

de

e

(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 40minus40minus30minus20minus10

0102030

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30 35 40minus06minus05minus04minus03minus02minus01

0010203

Tracking error

Time

Am

plitu

de

e

(b)

Figure 3 (a) Output reference and (b) tracking errorWhen the fuzzy controller is designed on local regulators and the adequatemembershipfunctions for the fuzzy regulator are computed

0 5 10 15 20 25 30 35 40minus40minus30minus20minus10

010203040

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30 35 40minus06minus05minus04minus03minus02minus01

001

Tracking error

Time

Am

plitu

de

e

(b)

Figure 4 (a) Output reference and (b) tracking errorWhen the fuzzy controller is designed on local regulators and the adequatemembershipfunctions for the fuzzy regulator are computed

12 Mathematical Problems in Engineering

h1 h2

x1minus60

1

80

0

0

Figure 5 Membership functions of two-rule model

and an exosystem defined by

119878 =

[[[[[

[

0 0 1 0

0 0 0 1

minus4 0 0 0

0 minus16 0 0

]]]]]

]

119876 = [

1 0 0 0

0 1 0 0]

(67)

Membership functions forRules 1 and 2 are shown in Figure 5Matrices defining the local regulators can be easily obtainedby using (49) For this example they are

Γ1= [

14286 05714 02857 05714

minus10000 minus95000 minus05000 minus25000]

Γ2= [

23333 13333 06667 03333

minus04000 minus40000 minus12000 minus04000]

(68)

On the other hand 11990211198881= 1205871= 1 119902

21198882= 1205872= 1

11990211198881= 1205873= 1 119902

21198882= 1205874= 1 and by (46) and matrix Π is

Π =

[[[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

]]]]]

]

(69)

Figures 6 and 7 show the results obtained using linear regula-tors defined in (61)

Now by computing the new membership functions forthe fuzzy regulator through (54) one gets1205831(119908 (119905))

=

[[[

[

(ℎ1(119908 (119905))) (minus7)

ℎ1(119908 (119905)) (3) + ℎ

2(119908 (119905)) (3)

0

0(ℎ1(119908 (119905))) (2)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (5)

]]]

]

1205832(119908 (119905))

=

[[[

[

(ℎ2(119908 (119905))) (minus3)

ℎ1(119908 (119905)) (3) + ℎ

2(119908 (119905)) (3)

0

0(ℎ2(119908 (119905))) (5)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (5)

]]]

]

(70)

As previously the fuzzy stabilizer for this example is alsoconstructed bymeans of the PDC approach developed in [14]resulting in

1198701= [

10286 05714 06000 05714

minus10000 minus22500 minus05000 minus37500]

1198702= [

14000 13333 14000 03333

minus04000 minus11000 minus12000 minus09000]

(71)

Thus Figures 8 and 9 show the simulation results whenthe controller defined by (20) is applied

Besides Figure 10 shows the shape of the new member-ship functions which are clearly different from the originalones given in Figure 5

53 More General Problems Consider the T-S fuzzy MIMOsystem with multiple inputs defined by the following matri-ces

1198601=

[[[[[[[[[[[

[

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

minus6 4 2 4 5 3

2 3 minus1 5 minus2 4

]]]]]]]]]]]

]

1198611=

[[[[[[[[[[[

[

0 0

0 0

0 0

0 0

5 0

0 2

]]]]]]]]]]]

]

1198602=

[[[[[[[[[[[

[

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

minus6 1 2 minus5 4 6

0 2 minus7 4 5 2

]]]]]]]]]]]

]

1198612=

[[[[[[[[[[[

[

0 0

0 0

0 0

0 0

7 0

0 8

]]]]]]]]]]]

]

1198621= 1198622= [

0 1 1 0 0 0

1 0 1 0 0 0]

(72)

Mathematical Problems in Engineering 13

0 5 10 15 20 25 30minus15minus10

minus505

101520

Time

Am

plitu

dex1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus4minus2

02468

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 6 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus30minus20minus10

0102030

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus15

minus10

minus5

0

5

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 7 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of lineal regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus15minus10

minus505

101520

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus1

012345678

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 8 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

The reference signals will be generated by an exosystemdefined by

119878 = [

0 1

minus1 0] 119876 = [

1 0

0 1] (73)

Membership functions for Rules 1 and 2 are ℎ1(119909(119905)) =

(80 minus 1199091(119905))140 and ℎ

2(119909(119905)) = 1 minus ℎ

1(119905) Notice that for this

case 119899 = 119904 and 119862119894

= 119876 This implies that Π119894and Γ

119894must

be obtained by solving Francis Equations for each subsystemHowever due to form of the fuzzy plant it is possible to verifythat Π

1= Π2

14 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus40minus30minus20minus10

0102030

Time

Am

plitu

dex2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus10

minus8minus6minus4minus2

0

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 9 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(11) (x1(t))1205832(11) (x1(t))

(a)

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(22) (x1(t))1205832(22) (x1(t))

(b)

Figure 10 (a) Membership functions corresponding to position (11) of 1205831(119908(119905)) and 120583

2(119908(119905)) (b) Membership functions corresponding to

position (22) 1205831(119908(119905)) and 120583

2(119908(119905))

So by applying (9) for each local subsystem with 119894 = 1 2it results in

Π1= Π2=

[[[[[[[[[[[

[

05000 05000

15000 minus05000

minus05000 05000

05000 15000

minus05000 minus05000

minus15000 05000

]]]]]]]]]]]

]

Γ1= [

07000 minus03000

minus20000 minus55000]

Γ2= [

23571 12143

minus04375 minus01875]

(74)

Again the fuzzy stabilizer for this system is constructedby means of the PDC approach developed in [14] yielding

1198701=[[

[

minus12229 00609 minus40284 01291 minus25666 minus04061

16919 minus83997 32934 minus127153 15732 minus61836

]]

]

1198702=[[

[

minus08735 04721 minus28775 13780 minus16904 minus07187

06730 minus19749 15734 minus30538 minus04817 minus12959

]]

]

(75)

The behavior obtained by applying controller (61) isdepicted in Figures 11 and 12 Notice the exact outputregulation is not achieved

Mathematical Problems in Engineering 15

0 5 10 15 20 25 30minus60minus40minus20

0204060

Time

Am

plitu

dex1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus80minus70minus60minus50minus40minus30minus20minus10

01020

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 11 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus15

minus10

minus5

0

5

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 12 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

Now the adequate membership functions are computedfrom (54) resulting in

1205831(119908 (119905))

=

[[[

[

(ℎ1(119908 (119905))) (5)

ℎ1(119908 (119905)) (5) + ℎ

2(119908 (119905)) (7)

0

0(ℎ1(119908 (119905))) (2)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (8)

]]]

]

1205832(119908 (119905))

=

[[[

[

(ℎ2(119908 (119905))) (7)

ℎ1(119908 (119905)) (5) + ℎ

2(119908 (119905)) (7)

0

0(ℎ2(119908 (119905))) (8)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (8)

]]]

]

(76)

When ℎ119894(119908(119905)) is replaced by its corresponding expressions

one gets

1205831(119908 (119905)) =

[[[

[

1225

1199081(119905) + 140

minus5

20

0 minus1199081(119905) minus 80

31199081(119905) + 320

]]]

]

1205832(119908 (119905)) =

[[[

[

7

2minus

1225

1199081(119905) + 410

0

041199081(119905) + 240

31199081(119905) + 320

]]]

]

(77)

The results of applying control law (20) on the fuzzy plantare given in Figures 13 and 14

The ldquocomputedrdquo membership functions for the fuzzyregulator are given in Figure 15 Again it can be readilyobserved that these new membership functions are differentfrom the original ones

Notice that although this example does not fulfill theconditions ofTheorem 5 themembership functions obtainedfrom (39) allow the exact tracking for the fuzzy problemThis

16 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus80minus70minus60minus50minus40minus30minus20minus10

010

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 13 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus8minus6minus4minus2

02468

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 14 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(11) (x1(t))1205832(11) (x1(t))

(a)

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(22) (x1(t))1205832(22) (x1(t))

(b)

Figure 15 (a) Membership functions corresponding to position (11) of 1205831(119908(119905)) and 120583

2(119908(119905)) (b) Membership functions corresponding to

position (22) 1205831(119908(119905)) 120583

2(119908(119905))

Mathematical Problems in Engineering 17

suggests that the approach presented in this work may beapplied on a bigger class of T-S models however this studyis not completed yet

6 Conclusions

A fuzzy regulator for continuous-time systems based on thecombination of linear regulators adjusted by different mem-bership functions has been presented The main advantageis that analytical expressions for the membership functionswhich allow the proper combination of the local regulators inorder to guarantee the exact regulation can be easily appliedAs 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 modelscan be commonly found in mechanics electromechanicselectronics robotics and so forth

Some numerical examples are used to illustrate theapplicability of the proposed approach even in cases wherethe dimensions of the plant and the exosystem are differentbetween each otherThis suggests that the approach presentedin this work may be applied on a bigger class of T-S models

Finally the method proposed in this work avoids thedisadvantage of constructing an exact fuzzy regulator basedon overall T-S fuzzy system whose expression may resultvery large Instead the given approach offers 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 Interests

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

Acknowledgments

This work was partially supported by CONACYT (ConsejoNacional de Ciencia y Tecnologıa) through scholarship SNI(Sistema Nacional de Investigadores) and by IPN (InstitutoPolitecnico Nacional) through research Projects 20140659and 20150487 and scholarships EDI (Estımulo al Desempenode los Investigadores) COFAA (Comision de Operacion yFomento de Actividades Academicas) and BEIFI (Beca deEstımulo Institucional de Formacion de Investigadores)

References

[1] B A Francis ldquoThe linear multivariable regulator problemrdquoSIAM Journal on Control and Optimization vol 15 no 3 pp486ndash505 1977

[2] B A Francis andWMWonham ldquoThe internalmodel principleof control theoryrdquo Automatica vol 12 no 5 pp 457ndash465 1976

[3] A Isidori and C I Byrnes ldquoOutput regulation of nonlinearsystemsrdquo IEEE Transactions on Automatic Control vol 35 no2 pp 131ndash140 1990

[4] B Castillo-Toledo S Di Gennaro and F Jurado ldquoTrajectorytracking for a quadrotor via fuzzy regulationrdquo in Proceedings ofthe World Automation Congress (WAC rsquo12) pp 1ndash6 June 2012

[5] S Chen ldquoOutput regulation of nonlinear singularly perturbedsystems based on T-S fuzzy modelrdquo Journal of Control Theoryand Applications vol 3 no 4 pp 399ndash403 2005

[6] R Tapia-Herrera J A Meda-Campana S Alcamp119886119901119900119904 119886ntara-Montes T Hernamp119886119901119900119904 119886ndez-Cortamp119886119901119900119904 119890s and L Salgado-Conrado ldquoTuning of a TS fuzzy output regulator using thesteepest descent approach and ANFISrdquoMathematical Problemsin Engineering vol 2013 Article ID 873430 14 pages 2013

[7] K-Y Lian and J-J Liou ldquoOutput tracking control for fuzzysystems via output feedback designrdquo IEEE Transactions onFuzzy Systems vol 14 no 5 pp 628ndash639 2006

[8] X-J Ma and Z-Q Sun ldquoOutput tracking and regulationof nonlinear system based on takagi-sugeno fuzzy modelrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 30 no 1 pp 47ndash59 2000

[9] H J Lee J B Park and Y H Joo ldquoComments on lsquooutputtracking and regulation of nonlinear system based on Takagi-Sugeno fuzzy model1015840rdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 33 no 3 pp 521ndash523 2003

[10] J A Meda-Campana B Castillo-Toledo and G Chen ldquoSyn-chronization of chaotic systems from a fuzzy regulationapproachrdquo Fuzzy Sets and Systems vol 160 no 19 pp 2860ndash2875 2009

[11] J A Meda-Campana J C Gamp119886119901119900119904 119900mez-Mancilla and BCastillo-Toledo ldquoExact output regulation for nonlinear systemsdescribed by Takagi-Sugeno fuzzy modelsrdquo IEEE Transactionson Fuzzy Systems vol 20 no 2 pp 235ndash247 2012

[12] C Byrnes F Priscoli and A Isidori Output Regulation ofUncertain Nonlinear Systems Birkhauser 1997

[13] A Isidori Nonlinear Control Systems Communications andControl Engineering Series Springer Berlin Germany 3rdedition 1995

[14] K Tanaka and H O Wang Fuzzy Control Systems Designand Analysis A Linear Matrix Inequality Approach John Wileyamp119886119898119901 Sons New York NY USA 2001

[15] L-X Wang A Course in Fuzzy Systems and Control PrenticeHall Upper Saddle River NJ USA 1997

[16] I Abdelmalek N Golamp119886119901119900119904 119890a and M L Hadjili ldquoA newfuzzy Lyapunov approach to non-quadratic stabilization ofTakagi-Sugeno fuzzy modelsrdquo International Journal of AppliedMathematics andComputer Science vol 17 no 1 pp 39ndash51 2007

[17] J A Meda-Campana and B Castillo-Toledo ldquoOn the outputregulation for TS fuzzymodels using slidingmodesrdquo inProceed-ings of the American Control Conference (ACC rsquo05) pp 4062ndash4067 Portland Ore USA June 2005

[18] B Castillo-Toledo and J A Meda-Campana ldquoThe fuzzydiscrete-time robust regulation problem a LMI approachrdquo inProceedings of the 41st IEEE Conference on Decision and Controlvol 2 pp 2159ndash2164 IEEE Las Vegas Nev USA December2002

[19] B Castillo-Toledo J A Meda-Campana and A Titli ldquoAfuzzy output regulator for Takagi-Sugeno fuzzy modelsrdquo inProceedings of the IEEE International Symposium on IntelligentControl pp 310ndash315 Houston Tex USA October 2003

[20] B-C Ding H-X Sun and Y-E Qiao ldquoStability analysis of T-Sfuzzy control systems based on parameter-dependent Lyapunovfunctionrdquo Acta Automatica Sinica vol 31 no 4 pp 651ndash6542005

[21] M Narimani H K Lam R Dilmaghani and C Wolfe ldquoLMI-based stability analysis of fuzzy-model-based control systemsusing approximated polynomial membership functionsrdquo IEEE

18 Mathematical Problems in Engineering

Transactions on Systems Man and Cybernetics Part B Cyber-netics vol 41 no 3 pp 713ndash724 2011

[22] H K Lam ldquoPolynomial fuzzy-model-based control systemsstability analysis via piecewise-linear membership functionsrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 588ndash5932011

[23] K Tanaka T Hori and H O Wang ldquoA fuzzy Lyapunovapproach to fuzzy control system designrdquo in Proceedings of theAmerican Control Conference vol 6 pp 4790ndash4795 ArlingtonVa USA June 2001

[24] K Tanaka T Hori and H O Wang ldquoA multiple Lyapunovfunction approach to stabilization of fuzzy control systemsrdquoIEEE Transactions on Fuzzy Systems vol 11 no 4 pp 582ndash5892003

[25] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 174ndash1972005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

THEN

(119905) = 119860119894119909 (119905) + 119861

119894119906 (119905) + 119875

119894119908 (119905)

(119905) = 119878119908 (119905)

119890 (119905) = 119862119894119909 (119905) minus 119876119908 (119905)

119894 = 1 2 119903

(10)

where 119903 is the number of rules in the model and 119872119894

119895is the

fuzzy sets defined on the basis of the knowledge of the systemThen the regulation problem defined by (1)ndash(5) can be

represented through the T-S fuzzy model that is [10]

(119905) =

119903

sum

119894=1

ℎ119894(119911 (119905)) 119860

119894119909 (119905) + 119861

119894119906 (119905) + 119875

119894119908 (119905)

(119905) = 119878119908 (119905)

119890 (119905) =

119903

sum

119896=1

ℎ119894(119911 (119905)) 119862

119894119909 (119905) minus 119876119908 (119905)

(11)

where 119909(119905) isin R119899 is the state vector of the plant 119908(119905) isin R119904

is the state vector of the exosystem 119906(119905) isin R119898 is the inputsignal and 119890(119905) isin R119898 and ℎ

119894(119911(119905)) is the normalized weight

of each rule which depends on the membership function forthe premise variable 119911

119895(119905) in119872119894

119895 where

120596119894(119911 (119905)) =

119901

prod

119895=1

119872119894

119895(119911119895(119905))

ℎ119894(119911 (119905)) =

120596119894(119911 (119905))

sum119903

119894=1120596119894(119911 (119905))

119903

sum

119894=1

ℎ119894(119911 (119905)) = 1

ℎ119894(119911 (119905)) ge 0

(12)

with 119911(119905) = [1199111(119905) 1199112(119905) sdot sdot sdot 119911

119901(119905)] as a function of 119909(119905) 119894 =

1 119903 and 119895 = 1 119901The Exact Fuzzy Regulator Problem consists of finding

a controller 119906(119905) = 120572(119909(119905) 119908(119905)) such that the closed-loopsystem

(119905) =

119903

sum

119894=1

ℎ119894(119911 (119905)) 119860

119894119909 (119905) + 119861

119894120572 (119909 (119905) 0) (13)

has an asymptotically stable equilibrium point and thesolution of system (11) satisfies lim

119905rarrinfin119890(119905) = 0

From [10 12 13] the desired overall fuzzy controller canbe represented as

119906 (119905) =

119903

sum

ℎ=1

ℎ119894(119911 (119905)) 119870

119894 [119909 (119905) minus 120587 (119908 (119905))] + 120574 (119908 (119905)) (14)

Considering that approximations for mappings 120587(119908(119905)) and120574(119908(119905)) can be obtained by

(119908 (119905)) =

119903

sum

119894=1

ℎ119894(119911 (119905)) Π

119894119908 (119905)

120574 (119908 (119905)) =

119903

sum

119894=1

ℎ119894(119911 (119905)) Γ

119894119908 (119905)

(15)

then the solution of the fuzzy regulation problem requires toobtain Π

119894and Γ119894from the 119903 linear local regulators problems

included in (11) and defined by

Π119894119878 = 119860

119894Π119894+ 119861119894Γ119894+ 119875119894

0 = 119862119894Π119894minus 119876

(16)

for 119894 = 1 119903Thus the following controller is obtained

119906 (119905) =

119903

sum

119894=1

ℎ119894(119911 (119905)) 119870

119894[119909 (119905) minus

119903

sum

119894=1

ℎ119894(119911 (119905)) Π

119894119908 (119905)]

+

119903

sum

119894=1

ℎ119894(119911 (119905)) Γ

119894119908 (119905)

(17)

However according to [10] the exact fuzzy regulation isachieved with this controller only when Π

1= sdot sdot sdot = Π

119903= Π

and

(1) input matrix is equal for all subsystems that is 1198611=

sdot sdot sdot = 119861119903= 119861 or

(2) the steady-state zero error manifold is equal for allsubsystems that is Γ

1= sdot sdot sdot = Γ

119903= Γ

In the present work restrictions 1 and 2 are avoidedby 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 findingthe steady-state zero error manifold 119909

119904119904(119905) = 120587(119908(119905)) and

the steady-state input 119906119904119904(119905) = 120574(119908(119905)) for the overall T-

S fuzzy problem through 119890119904119904(119905) = 119909(119905) minus 119909

119904119904(119905) = 119909(119905) minus

120587(119908(119905)) (steady-state error equation) resulting in 120587(119908(119905)) =Π119891(119905)119908(119905) and 120574(119908(119905)) = Γ

119891(119905)119908(119905) whereΠ

119891(119905) and Γ

119891(119905) are

continuous time-variantmatrices and 120587(119908(119905)) = Π119891(119905)119908(119905) is

a 119862119896minus1 function [12]Within the next section the exact output regulation for a

class of T-S fuzzy systems is obtained by proposing a solutionfor 120574(119908(119905)) on the basis of different membership functions inthe regulator

3 The Exact Output Fuzzy Regulation byLocal Regulators

In this section a particular class of T-S fuzzy models isconsidered to solve the exact output regulation on the basis oflinear local controllers So the main goal is to find an overallregulator based on the fuzzy summation of local regulatorsconsidering adequate membership functions Clearly such

4 Mathematical Problems in Engineering

membership functions are not necessarily the same includedin the fuzzy plant Thus the steady-state input 120574(119908(119905)) can bedefined as

120574 (119908 (119905)) =

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905) (18)

where 120583119894(119908(119905)) are new membership functions such that

the fuzzy output regulator obtained from local regulatorscoincides with the controller given in [11] at least for theclass of fuzzy models considered This novel approach onlyrequires the computation of the linear local controllers andthe computation of the newmembership functionswhichwillbe presented belowThe regulator obtained in this way is validfor all 119905 ge 0 while in [11] this condition needs to be verified

Consider the following matrices

119860119894=

[[[[[[[

[

0 1 0 sdot sdot sdot 0

0 0 1 sdot sdot sdot 0

d

119886119894

1119886119894

2119886119894

3sdot sdot sdot 119886119894

119899

]]]]]]]

]

119861119894=

[[[[[[[

[

0

0

119887119894

1

]]]]]]]

]

119878 =

[[[[[[

[

0 1 0 sdot sdot sdot 0

0 0 1 sdot sdot sdot 0

d

119904111990421199043sdot sdot sdot 119904119899

]]]]]]

]

119862119894= [1198881 0 sdot sdot sdot 0]

119876 = [1199021 0 sdot sdot sdot 0]

(19)

where 119886119894119895represents the elements of the last row of 119860

119894and 1198871198941

represents the element of the last row of 119861119894with 119895 = 1 119899

and 119894 = 1 119903 It is worth mentioning that 119860119894and 119878 are

matrices of the same dimension as shown in (19) Notice thatmatrices in the formof 119878 and119876 can be used to generate a greatvariety of signals ensuring in thatway the applicability of theapproach in a great number of cases Therefore from (14) thecontrol input can be defined by

119906 (119905) =

119903

sum

119894=1

ℎ119894(119909 (119905)) 119896

119894119909 (119905) minus 120587 (119908 (119905))

+

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905)

(20)

because as mentioned before 119911(119905) is a function of 119909(119905) and insteady-state 119909(119905) = 120587(119908(119905))

On the other hand from 119862119894 119876 and from (6) it can be

concluded that 11988811199091(119905) = 119902

11199081(119905) in steady-state while from

(19) the following exosystem can be easily derived

1= 1199082

2= 1199083

119899= 11990411199081+ 11990421199082+ sdot sdot sdot + 119904

119899119908119899

119899=

119899

sum

119894=1

119904119894119908119894

(21)

Besides each subsystem can be rewritten as follows

1= 1199092

2= 1199093

119899= 119886119894

11199091+ 119886119894

21199092+ sdot sdot sdot + 119886

119894

119899119909119899+ 119887119894

1119906119904119904119894

(22)

By equating the time derivative of the fuzzy output withthe time derivative of the reference signal it results in 119888

11=

11990211 Furthermore from (21) and (22) it is possible to

conclude that

1198881119894= 1199021119894 (23)

where 119894 = 1 2 119899 From this analysis two results can beobtained First

Π =

[[[[[[[[[[

[

1199021

1198881

0 sdot sdot sdot 0

01199021

1198881

sdot sdot sdot 0

d

0 0 sdot sdot sdot1199021

1198881

]]]]]]]]]]

]

(24)

and due to the shape of (19)Π is constant and common for allsubsystemsThus 120587(119908(119905)) = Π119908(119905) Second the local steady-state input is defined by

119906119904119904119894

=(11990211198881) 11990411199081minus (11990211198881) 119886119894

11199081

119887119894

1

+(11990211198881) 11990421199082minus (11990211198881) 119886119894

21199082

119887119894

1

+ sdot sdot sdot +(11990211198881) 119904119899119908119899minus (11990211198881) 119886119894

119899119908119899

119887119894

1

(25)

Mathematical Problems in Engineering 5

The previous expression can be rewritten in a compact formas follows

119906119904119904119894

=

sum119899

119895=1(11990211198881) [119904119895119908119895minus 119886119894

119895119908119895]

119887119894

1

forall119894 = 1 119903 (26)

and from (18) the fuzzy steady-state input is

119906119904119904(119905) =

119903

sum

119894=1

120583119894(119908 (119905)) 119906

119904119904119894

=

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905) forall119894 = 1 119903

(27)

From the previous analysis each Γ119894can be computed from

Γ119894=1199021

1198881

[1199041minus 119886119894

1

119887119894

1

1199042minus 119886119894

2

119887119894

1

sdot sdot sdot119904119899minus 119886119894

119899

119887119894

1

] (28)

Now it is possible to rewrite the fuzzy model as [11]

(119905) = 119860 (119905) 119909 (119905) + 119861 (119905) 119906 (119905) + (119905) 119908 (119905) (29)

(119905) = 119878119908 (119905) (30)

119890 (119905) = 119862 (119905) 119909 (119905) minus 119876119908 (119905) (31)

where

119860 (119905) =

119903

sum

119894=1

ℎ119894(119911 (119905)) 119860

119894

119861 (119905) =

119903

sum

119894=1

ℎ119894(119911 (119905)) 119861

119894

(119905) =

119903

sum

119894=1

ℎ119894(119911 (119905)) 119875

119894

119862 (119905) =

119903

sum

119894=1

ℎ119894(119911 (119905)) 119862

119894

(32)

For the sake of simplicity in the following analysis it isconsidered that119875

119894= 0 119894 = 1 119903 that is (119905) = 0 However

taking similar steps as those used below the result can beeasily extended to the case involving (119905) = 0

Thus from (19) and (29) it can be observed that the lastequation

119899in (29) is given by

119899= [ℎ1(119909 (119905)) (119886

119894

11199091+ sdot sdot sdot + 119886

119894

119899119909119899)

+ sdot sdot sdot + ℎ119903(119909 (119905)) (119886

119894

11199091+ sdot sdot sdot + 119886

119894

119899119909119899)]

+ [ℎ1(119909 (119905)) 119887

119894

1+ sdot sdot sdot + ℎ

119903(119909 (119905)) 119887

119903

1] 119906119904119904(119905)

(33)

where compact form is

119899=

119903

sum

119894=1

ℎ119894(119909 (119905))(

119899

sum

119895=1

119886119894

119895119909119895) + (

119903

sum

119894=1

ℎ119894(119909 (119905)) 119887

119894

1)119906119904119904(119905)

(34)

considering that in steady-state 119909(119905) = Π119908(119905) the totalsteady-state input can be obtained from (23) resulting in

119903

sum

119894=1

ℎ119894(119908 (119905))(

1199021

1198881

119899

sum

119895=1

119886119894

119895119908119895)

+ (

119903

sum

119894=1

ℎ119894(119908 (119905)) 119887

119894

1)119906119904119904(119905) =

1199021

1198881

119899

sum

119894=1

119904119894119908119894

(35)

Consequently the steady-state input for the total fuzzy systemcan be constructed by

119906119904119904(119905)

=

(11990211198881)sum119899

119894=1119904119894119908119894minus (11990211198881)sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

1

(36)

Now a new set of fuzzy membership functions 120583119894(119908(119905))

satisfying sum119903119894=1

120583119894(119908(119905)) = 1 with 120583

119894(119908(119905)) ge 0 119894 = 1 119903

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 defined as follows

119906119904119904119894

= 120574 (119908 (119905)) =

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905)

=

119903

sum

119894=1

120583119894(119908 (119905))

sum119899

119895=1(11990211198881) [119904119895119908119895minus 119886119894

119895119908119895]

119887119894

1

119906119904119904= 120574 (119908 (119905))

=

(11990211198881)sum119899

119894=1119904119894119908119894minus (11990211198881)sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

1

(37)

respectivelyThus the local fuzzy regulatorswill bemultipliedby the new membership functions and the result will beequated to the global fuzzy regulator (36) as follows

(11990211198881)sum119899

119894=1119904119894119908119894minus (11990211198881)sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

1

=

119903

sum

119894=1

120583119894(119908 (119905))

sum119899

119895=1(11990211198881) [119904119895119908119895minus 119886119894

119895119908119895]

119887119894

1

(38)

So the adequate membership functions are

120583119894(119908 (119905)) =

119887119894

1ℎ119894(119908 (119905))

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

1

(39)

with 119894 = 1 2 119903 It is important to remark that the newmembership functions 120583

119894(119908(119905)) are given in terms of ℎ

119894(119908(119905))

for simplicity However ℎ119894(119908(119905)) can be removed from the

new membership functions (39) when ℎ119894(119908(119905)) is replaced

by its corresponding expression Besides ℎ119894(119908(119905)) is directly

obtained from ℎ119894(119909(119905)) by considering that in steady-state

119909(119905) = Π119908(119905)

6 Mathematical Problems in Engineering

Notice that (39) fulfills conditions119903

sum

119894=1

120583119894(119908 (119905)) = 1

120583119894(119908 (119905)) ge 0

(40)

for all 119894 = 1 2 119903

Remark 2 It can be observed that (39) is always valid whenthe entries of input matrix 119887

119894

1have the same sign if this

condition is not fulfilled the denominators of the new fuzzymembership functions will present singularities Neverthe-less these singularities may appear outside the operationregion allowing more flexibility but such cases require aparticular study of the system which is beyond the scope ofthis work

The following theorem provides the conditions for theexistence of the exact output fuzzy regulator for a class of T-Sfuzzy models

Theorem 3 The exact fuzzy output regulation with fullinformation for T-S fuzzy systems in the form of (11) definedby matrices (19) is solvable if (a) the sign of entries forcorresponding position inside input matrices 119861

119894for 119894 =

1 119903 is the same (b) there exists a fuzzy stabilizer 119906(119905) =sum119903

119894=1ℎ119894(119911(119905))119870

119894119909 for the T-S fuzzy system [16] and (c) the

exosystem (119905) = 119878119908(119905) is Poisson stable Moreover the ExactOutput Fuzzy Regulation Problem is solvable by the controller

119906 (119905) =

119903

sum

119894=1

ℎ119894(119909 (119905)) 119896

119894119909 (119905) minus Π119908 (119905)

+

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905)

(41)

where 120583119894(119908(119905)) can be readily obtained from (39)

Proof From the previous analysis the existence of mappings120587(119908(119905)) = Π119908(119905) and 120574(119908(119905)) = sum

119903

119894=1120583119894(119908(119905))Γ

119894119908(119905) is

guaranteed when the local subsystems are defined by (19)that is the form of 120587(119908(119905)) corresponds to a diagonal matrixand will be the same for all subsystems while 120574(119908(119905)) can bedirectly substitute

On the other hand condition (a) avoids singularities inthe new membership functions 120583

119894(119908(119905)) while the inclusion

of condition (b) has been thoroughly discussed in [12 13 17ndash19] and it implies that the steady-state manifold can be madeasymptotically stable by the action of the fuzzy stabilizer

Finally condition (c) is introduced to avoid that thereference signal converges to zero because if the referencetends to zero then the regulation problem turns into astabilization one which can be solved by means of thefuzzy stabilizer In addition as mentioned before this kindof matrix 119878 can be used to generate a great number ofreference signals The 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 20ndash25]

4 Exact Output Fuzzy Regulation Problem fora Class of T-S Fuzzy Systems with MultipleInputs and Multiple Outputs (MIMO Case)

Now the problem will be extended to T-S fuzzy MIMOmodels defined by matrices

119860119894= [

119874 119868

119863119894

] 119861119894= [

119874

119865119894

]

119878 = [119874 119868

119866] 119862

119894= [ 119867119862 119874

119862 ]

119876119894= [ 119867119876 119874

119876 ]

(42)

for 119894 = 1 119903 with

119868 =

[[[[[[

[

1 0 sdot sdot sdot 0

0 1 sdot sdot sdot 0

d

0 0 sdot sdot sdot 1

]]]]]]

]

119874 =

[[[[[[

[

0 0 sdot sdot sdot 0

0 0 sdot sdot sdot 0

d

0 0 sdot sdot sdot 0

]]]]]]

]

119863119894=

[[[[[[[

[

119886119894

(119896+1)1119886119894

(119896+1)2119886119894

(119896+1)3sdot sdot sdot 119886119894

(119896+1)119899

119886119894

(119896+2)1119886119894

(119896+2)2119886119894

(119896+2)3sdot sdot sdot 119886119894

(119896+2)119899

d

119886119894

1198991119886119894

1198992119886119894

1198993sdot sdot sdot 119886

119894

119899119899

]]]]]]]

]

119865119894=

[[[[[[[

[

119887119894

119896+10 sdot sdot sdot 0

0 119887119894

119896+2sdot sdot sdot 0

d

0 0 sdot sdot sdot 119887119894

119899

]]]]]]]

]

119866 =

[[[[[[

[

119904(119896+1)1

119904(119896+1)2

119904(119896+1)3

sdot sdot sdot 119904(119896+1)119899

119904(119896+2)1

119904(119896+2)2

119904(119896+2)3

sdot sdot sdot 119904(119896+2)119899

d

1199041198991

1199041198992

1199041198993

sdot sdot sdot 119904119899119899

]]]]]]

]

119874119862= 119874119876=

[[[[[[

[

0 0 sdot sdot sdot 0

0 0 sdot sdot sdot 0

d

0 0 sdot sdot sdot 0

]]]]]]

]

Mathematical Problems in Engineering 7

119867119862=

[[[[[[

[

11988810 sdot sdot sdot 0

0 1198882sdot sdot sdot 0

d

0 0 sdot sdot sdot 119888119898

]]]]]]

]

119867119876=

[[[[[[

[

1199021

0 sdot sdot sdot 0

0 1199022sdot sdot sdot 0

d

0 0 sdot sdot sdot 119902119898

]]]]]]

]

(43)

where 119896 = 119899 minus 119898 119865119894isin R119898times119898 119874 isin R(119899minus119898)times119898 119874

119862isin R119898times119898

119868 isin R(119899minus119898)times(119899minus119898) 119863119894isin R119898times119899 119867

119862isin R119898times119898 119867

119876isin R119898times119898

and 119866 isin R119898times119899 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= 119908119898+1

2= 119908119898+2

119896+1

= 119904(119896+1)1

1199081+ 119904(119896+1)2

1199082+ sdot sdot sdot + 119904

(119896+1)119899119908119899

=

119899

sum

119895=1

119904(119896+1)119895

119908119895

119896+2

= 119904(119896+2)1

1199081+ 119904(119896+2)2

1199082+ sdot sdot sdot + 119904

(119896+2)119899119908119899

=

119899

sum

119895=1

119904(119896+2)119895

119908119895

119899= 11990411989911199081+ 11990411989921199082+ sdot sdot sdot + 119904

119899119899119908119899=

119899

sum

119895=1

119904119899119895119908119895

(44)

Moreover each subsystem can be described by

1= 119909119898+1

2= 119909119898+2

119896+1

= 119886119894

(119896+1)11199091+ 119886119894

(119896+1)21199092+ sdot sdot sdot + 119886

(119896+1)119899119909119899

+ 119887119894

119896+1119906119894

119904119904

=

119899

sum

119895=1

119886119894

(119896+1)119895119909119895+ 119887119894

119896+1119906119894

119904119904

119896+2

= 119886119894

(119896+2)11199091+ 119886119894

(119896+2)21199092+ sdot sdot sdot + 119886

(119896+2)119899119909119899

+ 119887119894

119896+2119906119894

119904119904

=

119899

sum

119895=1

119886119894

(119896+2)119895119908119895+ 119887119894

1+2119906119894

119904119904

119899= 11990411989911199081+ 11990411989921199082+ sdot sdot sdot + 119904

119899119899119908119899+ 119887119894

119899119906119894

119904119904

=

119899

sum

119895=1

119904119899119895119908119895+ 119887119894

119899119906119894

119904119904

(45)

As before using expressions (44) and (45) considering 11988811199091=

11990211199081 11988821199092

= 11990221199082 119888

119898119909119898

= 119902119898119908119898 and performing

successive substitutions of ℓ= 120587ℓℓwith ℓ = 1 119898 the

following results are obtained

Π =

[[[[[[

[

1205871

0 sdot sdot sdot 0

0 1205872sdot sdot sdot 0

d

0 0 sdot sdot sdot 120587119899

]]]]]]

]

(46)

where Π isin R119899times119899 and 119894 = 1 2 119903 also 1205871

=

11990211198881 1205872= 11990221198882 120587

119898= 119902119898119888119898 120587119898+1

= 11990211198881 120587119898+2

=

11990221198882 120587

119898+119898= 119902119898119888119898 and so forth Notice that the values

of 120587lowastrepeat themselves every 119898 entry up to 119899 Additionally

Π is constant and common for all subsystems while the localsteady-state inputs are defined by

119906119894

119904119904=

[[[[[[[[[[[[[

[

1205871119904(119896+1)1

1199081minus 1205871119886119894

(119896+1)11199081

1198871

119896+1

+

1205871119904(119896+1)1

1199082minus 1205872119886119894

(119896+1)11199082

1198871

119896+1

+ sdot sdot sdot +

1205871119904(119896+1)119899

119908119899minus 120587119899119886119894

(119896+1)119899119908119899

1198871

119896+1

1205872119904(119896+2)1

1199081minus 1205871119886119894

(119896+2)11199081

1198871

119896+2

+

1205872119904(119896+1)2

1199082minus 1205872119886119894

(119896+2)21199082

1198871

119896+2

+ sdot sdot sdot +

1205872119904(119896+2)119899

119908119899minus 120587119899119886119894

(119896+2)119899119908119899

1198871

119896+2

12058711989911990411989911199081minus 1205871119886119894

11989911199081

1198871119899

+12058711989911990411989921199082minus 1205872119886119894

11989921199082

1198871119899

+ sdot sdot sdot +120587119899119904119899119899119908119899minus 120587119899119886119894

119899119899119908119899

1198871119899

]]]]]]]]]]]]]

]

(47)

8 Mathematical Problems in Engineering

for 119894 = 1 119903 In compact form (47) turns into

119906119894

119904119904=

[[[[[[[[[[[[[[

[

119899

sum

119895=1

1205871119904(119896+1)119895

119908119895minus 120587119895119886(119896+1)119895

119908119895

119887119894

119896+1

119899

sum

119895=1

1205872119904(119896+2)119895

119908119895minus 120587119895119886(119896+2)119895

119908119895

119887119894

119896+2

119899

sum

119895=1

120587119899119904119899119895119908119895minus 120587119895119886119899119895119908119895

119887119894119899

]]]]]]]]]]]]]]

]

(48)

and from (27) the local matrices Γ119894have the form

Γ119894=

[[[[[[[[[[[[[

[

1205871119904(119896+1)1

minus 1205871119886119894

(119896+1)1

119887119894

119896+1

1205871119904(119896+1)2

minus 1205872119886119894

(119896+1)2

119887119894

119896+1

sdot sdot sdot

12058711199041198991

minus 120587119899119886119894

(119896+1)119899

119887119894

119896+1

1205872119904(119896+2)1

minus 1205871119886119894

(119896+2)1

119887119894

119896+2

1205872119904(119896+2)2

minus 1205872119886119894

(119896+2)2

119887119894

119896+2

sdot sdot sdot

12058721199041198991

minus 120587119899119886119894

(119896+2)119899

119887119894

119896+2

d

1205871198991199041198991

minus 1205871119886119894

1198991

119887119894119899

1205871198991199041198992

minus 1205872119886119894

1198992

119887119894119899

sdot sdot sdot1205871198991199041198991

minus 120587119899119886119894

119899119899

119887119894119899

]]]]]]]]]]]]]

]

(49)

Now again from the fuzzy plant given in (42) and theglobal fuzzy model (29) the expressions for

119896+1

119899are

119896+1

=

119903

sum

119894=1

ℎ119894(119909 (119905))(

119899

sum

119895=1

119886119894

(119896+1)119895119909119895)

+ (

119903

sum

119894=1

ℎ119894(119909 (119905)) 119887

119894

119896+1)119906119904119904

119896+2

=

119903

sum

119894=1

ℎ119894(119909 (119905))(

119899

sum

119895=1

119886119894

(119896+2)119895119909119895)

+ (

119903

sum

119894=1

ℎ119894(119909 (119905)) 119887

119894

119896+2)119906119904119904

119899=

119903

sum

119894=1

ℎ119894(119909 (119905))(

119899

sum

119895=1

119886119894

119899119895119909119895)

+ (

119903

sum

119894=1

ℎ119894(119909 (119905)) 119887

119894

119899)119906119904119904

(50)

So by applying successive substitutions as before with (44)(45) and knowing that in steady-state 119909

119895= 120587119895119908119895 the

following expressions are obtained

119903

sum

119894=1

ℎ119894(119908 (119905))(

119899

sum

119895=1

119886119894

(119896+1)119895120587119895119908119895)

+ (

119903

sum

119894=1

ℎ119894(119908 (119905)) 119887

119894

119896+1)119906119904119904= 1205871

119899

sum

119894=1

119904(119896+1)119894

119908119894

119903

sum

119894=1

ℎ119894(119908 (119905))(

119899

sum

119895=1

119886119894

(119896+2)119895120587119895119908119895)

+ (

119903

sum

119894=1

ℎ119894(119908 (119905)) 119887

119894

119896+2)119906119904119904= 1205872

119899

sum

119894=1

119904(119896+2)

119894119908119894

1199031

sum

119894=1

ℎ119894(119908 (119905))(

119899

sum

119895=1

119886119894

119899119895120587119895119908119895)

+ (

119903

sum

119894=1

ℎ119894(119908 (119905)) 119887

119894

119899)119906119904119904= 120587119899

119899

sum

119894=1

119904119899119894119908119894

(51)

Mathematical Problems in Engineering 9

Therefore the steady-state input for the overall fuzzy systemis

119906119904119904

=

[[[[[[[[[[[[[[

[

1205871sum119899

119894=1119904(119896+1)119894

119908119894minus sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

(119896+1)119895120587119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

119896+1

1205872sum119899

119894=1119904(119896+2)119894

119908119894minus sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

(119896+2)119895120587119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

119896+2

120587119899sum119899

119894=1119904119899119895119908119894minus sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

119899119895120587119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

119899

]]]]]]]]]]]]]]

]

(52)

But the desired form of the steady-state is

119906119904119904(119905) = 120574 (119908 (119905)) =

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905) (53)

Consequently the new membership functions 120583119894(119908(119905))

can be obtained after equating (52) and (53) resulting in

120583119894(119908 (119905)) =

[[[[[[

[

12058311989411

0 sdot sdot sdot 0

0 12058311989422

sdot sdot sdot 0

d

0 0 sdot sdot sdot 120583119894119898119898

]]]]]]

]

(54)

where

120583119894119895119897

=

119887119894

119896+119895ℎ119894(119908 (119905))

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

119896+119895

(55)

for all 119894 = 1 119903 119895 = 119897 = 1 119898 It can be readily observedthat the sum of the corresponding elements is equal to oneand 120583

119894119895119897(119908(119905)) ge 0

These new membership functions 120583119894(119908(119905)) are organized

in a matrix form for that reason sum119903119894=1

120583119894(119908(119905)) = 119868 (where 119868

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 119861

119894have the

same sign at corresponding positions Clearly this conditionavoids singularities in (55) As before ℎ

119894(119908(119905)) is directly

obtained from ℎ119894(119909(119905)) by considering that in steady-state

119909(119905) = Π119908(119905) At this point the following theorem can bedefined

Theorem 5 The exact fuzzy output regulation with fullinformation for T-S fuzzy MIMO systems in the form of(11) defined by matrices (42) is solvable if (a) the sign ofentries for corresponding position inside input matrices 119861

119894for

119894 = 1 119903 is the same (b) there exists a fuzzy stabilizer119906(119905) = sum

119903

119894=1ℎ119894(119911(119905))119870

119894119909 for the fuzzy system [16] and (c) the

exosystem (119905) = 119878119908(119905) is Poisson stable

Moreover the Exact Output Fuzzy Regulation Problem forMIMO systems is solvable by the controller

119906 (119905) =

119903

sum

119894=1

ℎ119894(119909 (119905)) 119896

119894119909 (119905) minus Π119908 (119905)

+

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905)

(56)

where 120583119894(119908(119905)) can be readily obtained from (54) and (55)

Proof It follows directly from Theorem 3 and previousanalysis

Remark 6 Notice that 120583119894(119908(119905)) is defined as a matrix and

this is not a typical membership function however 120583119894(119908(119905))

contains a set of membership functions related to each entryof 119861119894 This form allows us to design controller (20)

5 Numerical Examples

51 Simple Input Case Consider the problem of balancingand swing of an inverted pendulum on a cart The motionequations for this system are [14]

1(119905) = 119909

2(119905) (57)

2(119905)

=119892 sin(119909

1(119905))minus 119886119898119897119909

2

2sin (2119909

1(119905))2 minus119886 cos (119909

1(119905)) 119906 (119905)

41198973 minus 119886119898119897 cos2 (1199091(119905))

(58)

where 1199091(119905) denotes the angle (in radians) of the pendulum

from the vertical and 1199092(119905) is the angular velocity 119892 =

98ms2 is the gravity constant 119898 is the mass of the pen-dulum 119872 is the mass of the cart 2119897 is the length of thependulum and 119906 is the force applied to the cart (in newtons)119886 = 1(119898 +119872) with119898 = 20 kg119872 = 80 kg and 2119897 = 10min the simulations The goal is to track the reference definedby the exosystem within the range 119909

1isin (minus1205872 1205872) To this

end a fuzzy model representing the nonlinear dynamics is asfollows

Rule 1

IF 1199091(119905) is about 0

THEN (119905) = 1198601119909(119905) + 119861

1119906(119905)

Rule 2

IF 1199091(119905) is about plusmn1205872(|119909

1| lt 1205872)

THEN (119905) = 1198602119909(119905) + 119861

2119906(119905)

10 Mathematical Problems in Engineering

h1

h2

x1minus90 90

(deg)

1

00

Figure 1 Membership functions of two-rule model

Membership functions for Rules 1 and 2 are shown in Figure 1In this example the T-S fuzzymodel of the plant is defined bymatrices

1198601= [

0 1

172941 0] 119861

1= [

0

minus01765]

1198602= [

0 1

41585 0] 119861

2= [

0

minus01500]

1198621= 1198622= [1 0]

(59)

while the exosystem can be constructed from the followingmatrices

119878 = [

0 1

minus1 0] 119876 = [1 0] (60)

which allows us to generate a great kind of sinusoidal refer-ences Figure 2 shows the simulation results after applying thecontroller

119906 (119905) =

119903

sum

119894=1

ℎ119894(119909 (119905)) 119870

119894[119909 (119905) minus

119903

sum

119894=1

ℎ119894(119908 (119905)) Π

119894119908 (119905)]

+

119903

sum

119894=1

ℎ119894(119908 (119905)) Γ

119894119908 (119905)

(61)

defined in [10] As expected controller (61) is unable toachieve exact regulation although tracking error remainsbounded

Now using the method derived in this work the localmatrices Γ

119894can be readily obtained from (28) resulting in

Γ1= [1036667 0] Γ

2= [343836 0] (62)

Besides knowing that 11990211198881= 1 matrix Π is obtained from

(24)

Π = [

1 0

0 1] (63)

while the newmembership functions computed from (39) are

1205831(119908 (119905)) =

(ℎ1(119908 (119905))) (minus01765)

ℎ1(119908 (119905)) (minus01765) + ℎ

2(119908 (119905)) (minus01500)

1205832(119908 (119905)) =

(ℎ2(119908 (119905))) (minus01500)

ℎ1(119908 (119905)) (minus01765) + ℎ

2(119908 (119905)) (minus01500)

(64)

On the other hand the fuzzy stabilizer for this system isconstructed bymeans of the PDC approach developed in [14]which is formed by

1198701= [1234898 69979] 119870

2= [586405 77962]

(65)

Figure 3 shows simulation with the controller definedin (20) Observe how the new membership functions arecapable of combining the local regulators in the proper wayto accomplish asymptotic tracking

To verify the effectiveness of the latter regulator thefuzzy controller is applied on the original system (57) Thesimulation results appear in Figure 4 Notice that the trackingerror depicted in Figure 4(b) is due to the approximationprovided by the T-S fuzzy model Such an error can bereduced by considering a better approximation or an exactrepresentation of the nonlinear system as the one given bythe nonlinear sector approach [14]

52 Multiple Inputs and Multiple Outputs (MIMO) CaseNow consider a T-S fuzzy system of two rules defined by thefollowing matrices

1198601=

[[[[[

[

0 0 1 0

0 0 0 1

6 4 2 4

2 3 1 5

]]]]]

]

1198611=

[[[[[

[

0 0

0 0

minus7 0

0 2

]]]]]

]

1198602=

[[[[[

[

0 0 1 0

0 0 0 1

3 4 2 1

2 4 6 2

]]]]]

]

1198612=

[[[[[

[

0 0

0 0

minus3 0

0 5

]]]]]

]

1198621= 1198622= [

1 0 0 0

0 1 0 0]

(66)

Mathematical Problems in Engineering 11

0 5 10 15 20 25 30 35 40minus40minus30minus20minus10

010203040

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30 35 40minus04minus03minus02minus01

0010203

Tracking error

Time

Am

plitu

de

e

(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 40minus40minus30minus20minus10

0102030

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30 35 40minus06minus05minus04minus03minus02minus01

0010203

Tracking error

Time

Am

plitu

de

e

(b)

Figure 3 (a) Output reference and (b) tracking errorWhen the fuzzy controller is designed on local regulators and the adequatemembershipfunctions for the fuzzy regulator are computed

0 5 10 15 20 25 30 35 40minus40minus30minus20minus10

010203040

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30 35 40minus06minus05minus04minus03minus02minus01

001

Tracking error

Time

Am

plitu

de

e

(b)

Figure 4 (a) Output reference and (b) tracking errorWhen the fuzzy controller is designed on local regulators and the adequatemembershipfunctions for the fuzzy regulator are computed

12 Mathematical Problems in Engineering

h1 h2

x1minus60

1

80

0

0

Figure 5 Membership functions of two-rule model

and an exosystem defined by

119878 =

[[[[[

[

0 0 1 0

0 0 0 1

minus4 0 0 0

0 minus16 0 0

]]]]]

]

119876 = [

1 0 0 0

0 1 0 0]

(67)

Membership functions forRules 1 and 2 are shown in Figure 5Matrices defining the local regulators can be easily obtainedby using (49) For this example they are

Γ1= [

14286 05714 02857 05714

minus10000 minus95000 minus05000 minus25000]

Γ2= [

23333 13333 06667 03333

minus04000 minus40000 minus12000 minus04000]

(68)

On the other hand 11990211198881= 1205871= 1 119902

21198882= 1205872= 1

11990211198881= 1205873= 1 119902

21198882= 1205874= 1 and by (46) and matrix Π is

Π =

[[[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

]]]]]

]

(69)

Figures 6 and 7 show the results obtained using linear regula-tors defined in (61)

Now by computing the new membership functions forthe fuzzy regulator through (54) one gets1205831(119908 (119905))

=

[[[

[

(ℎ1(119908 (119905))) (minus7)

ℎ1(119908 (119905)) (3) + ℎ

2(119908 (119905)) (3)

0

0(ℎ1(119908 (119905))) (2)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (5)

]]]

]

1205832(119908 (119905))

=

[[[

[

(ℎ2(119908 (119905))) (minus3)

ℎ1(119908 (119905)) (3) + ℎ

2(119908 (119905)) (3)

0

0(ℎ2(119908 (119905))) (5)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (5)

]]]

]

(70)

As previously the fuzzy stabilizer for this example is alsoconstructed bymeans of the PDC approach developed in [14]resulting in

1198701= [

10286 05714 06000 05714

minus10000 minus22500 minus05000 minus37500]

1198702= [

14000 13333 14000 03333

minus04000 minus11000 minus12000 minus09000]

(71)

Thus Figures 8 and 9 show the simulation results whenthe controller defined by (20) is applied

Besides Figure 10 shows the shape of the new member-ship functions which are clearly different from the originalones given in Figure 5

53 More General Problems Consider the T-S fuzzy MIMOsystem with multiple inputs defined by the following matri-ces

1198601=

[[[[[[[[[[[

[

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

minus6 4 2 4 5 3

2 3 minus1 5 minus2 4

]]]]]]]]]]]

]

1198611=

[[[[[[[[[[[

[

0 0

0 0

0 0

0 0

5 0

0 2

]]]]]]]]]]]

]

1198602=

[[[[[[[[[[[

[

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

minus6 1 2 minus5 4 6

0 2 minus7 4 5 2

]]]]]]]]]]]

]

1198612=

[[[[[[[[[[[

[

0 0

0 0

0 0

0 0

7 0

0 8

]]]]]]]]]]]

]

1198621= 1198622= [

0 1 1 0 0 0

1 0 1 0 0 0]

(72)

Mathematical Problems in Engineering 13

0 5 10 15 20 25 30minus15minus10

minus505

101520

Time

Am

plitu

dex1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus4minus2

02468

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 6 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus30minus20minus10

0102030

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus15

minus10

minus5

0

5

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 7 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of lineal regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus15minus10

minus505

101520

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus1

012345678

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 8 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

The reference signals will be generated by an exosystemdefined by

119878 = [

0 1

minus1 0] 119876 = [

1 0

0 1] (73)

Membership functions for Rules 1 and 2 are ℎ1(119909(119905)) =

(80 minus 1199091(119905))140 and ℎ

2(119909(119905)) = 1 minus ℎ

1(119905) Notice that for this

case 119899 = 119904 and 119862119894

= 119876 This implies that Π119894and Γ

119894must

be obtained by solving Francis Equations for each subsystemHowever due to form of the fuzzy plant it is possible to verifythat Π

1= Π2

14 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus40minus30minus20minus10

0102030

Time

Am

plitu

dex2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus10

minus8minus6minus4minus2

0

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 9 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(11) (x1(t))1205832(11) (x1(t))

(a)

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(22) (x1(t))1205832(22) (x1(t))

(b)

Figure 10 (a) Membership functions corresponding to position (11) of 1205831(119908(119905)) and 120583

2(119908(119905)) (b) Membership functions corresponding to

position (22) 1205831(119908(119905)) and 120583

2(119908(119905))

So by applying (9) for each local subsystem with 119894 = 1 2it results in

Π1= Π2=

[[[[[[[[[[[

[

05000 05000

15000 minus05000

minus05000 05000

05000 15000

minus05000 minus05000

minus15000 05000

]]]]]]]]]]]

]

Γ1= [

07000 minus03000

minus20000 minus55000]

Γ2= [

23571 12143

minus04375 minus01875]

(74)

Again the fuzzy stabilizer for this system is constructedby means of the PDC approach developed in [14] yielding

1198701=[[

[

minus12229 00609 minus40284 01291 minus25666 minus04061

16919 minus83997 32934 minus127153 15732 minus61836

]]

]

1198702=[[

[

minus08735 04721 minus28775 13780 minus16904 minus07187

06730 minus19749 15734 minus30538 minus04817 minus12959

]]

]

(75)

The behavior obtained by applying controller (61) isdepicted in Figures 11 and 12 Notice the exact outputregulation is not achieved

Mathematical Problems in Engineering 15

0 5 10 15 20 25 30minus60minus40minus20

0204060

Time

Am

plitu

dex1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus80minus70minus60minus50minus40minus30minus20minus10

01020

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 11 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus15

minus10

minus5

0

5

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 12 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

Now the adequate membership functions are computedfrom (54) resulting in

1205831(119908 (119905))

=

[[[

[

(ℎ1(119908 (119905))) (5)

ℎ1(119908 (119905)) (5) + ℎ

2(119908 (119905)) (7)

0

0(ℎ1(119908 (119905))) (2)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (8)

]]]

]

1205832(119908 (119905))

=

[[[

[

(ℎ2(119908 (119905))) (7)

ℎ1(119908 (119905)) (5) + ℎ

2(119908 (119905)) (7)

0

0(ℎ2(119908 (119905))) (8)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (8)

]]]

]

(76)

When ℎ119894(119908(119905)) is replaced by its corresponding expressions

one gets

1205831(119908 (119905)) =

[[[

[

1225

1199081(119905) + 140

minus5

20

0 minus1199081(119905) minus 80

31199081(119905) + 320

]]]

]

1205832(119908 (119905)) =

[[[

[

7

2minus

1225

1199081(119905) + 410

0

041199081(119905) + 240

31199081(119905) + 320

]]]

]

(77)

The results of applying control law (20) on the fuzzy plantare given in Figures 13 and 14

The ldquocomputedrdquo membership functions for the fuzzyregulator are given in Figure 15 Again it can be readilyobserved that these new membership functions are differentfrom the original ones

Notice that although this example does not fulfill theconditions ofTheorem 5 themembership functions obtainedfrom (39) allow the exact tracking for the fuzzy problemThis

16 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus80minus70minus60minus50minus40minus30minus20minus10

010

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 13 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus8minus6minus4minus2

02468

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 14 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(11) (x1(t))1205832(11) (x1(t))

(a)

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(22) (x1(t))1205832(22) (x1(t))

(b)

Figure 15 (a) Membership functions corresponding to position (11) of 1205831(119908(119905)) and 120583

2(119908(119905)) (b) Membership functions corresponding to

position (22) 1205831(119908(119905)) 120583

2(119908(119905))

Mathematical Problems in Engineering 17

suggests that the approach presented in this work may beapplied on a bigger class of T-S models however this studyis not completed yet

6 Conclusions

A fuzzy regulator for continuous-time systems based on thecombination of linear regulators adjusted by different mem-bership functions has been presented The main advantageis that analytical expressions for the membership functionswhich allow the proper combination of the local regulators inorder to guarantee the exact regulation can be easily appliedAs 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 modelscan be commonly found in mechanics electromechanicselectronics robotics and so forth

Some numerical examples are used to illustrate theapplicability of the proposed approach even in cases wherethe dimensions of the plant and the exosystem are differentbetween each otherThis suggests that the approach presentedin this work may be applied on a bigger class of T-S models

Finally the method proposed in this work avoids thedisadvantage of constructing an exact fuzzy regulator basedon overall T-S fuzzy system whose expression may resultvery large Instead the given approach offers 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 Interests

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

Acknowledgments

This work was partially supported by CONACYT (ConsejoNacional de Ciencia y Tecnologıa) through scholarship SNI(Sistema Nacional de Investigadores) and by IPN (InstitutoPolitecnico Nacional) through research Projects 20140659and 20150487 and scholarships EDI (Estımulo al Desempenode los Investigadores) COFAA (Comision de Operacion yFomento de Actividades Academicas) and BEIFI (Beca deEstımulo Institucional de Formacion de Investigadores)

References

[1] B A Francis ldquoThe linear multivariable regulator problemrdquoSIAM Journal on Control and Optimization vol 15 no 3 pp486ndash505 1977

[2] B A Francis andWMWonham ldquoThe internalmodel principleof control theoryrdquo Automatica vol 12 no 5 pp 457ndash465 1976

[3] A Isidori and C I Byrnes ldquoOutput regulation of nonlinearsystemsrdquo IEEE Transactions on Automatic Control vol 35 no2 pp 131ndash140 1990

[4] B Castillo-Toledo S Di Gennaro and F Jurado ldquoTrajectorytracking for a quadrotor via fuzzy regulationrdquo in Proceedings ofthe World Automation Congress (WAC rsquo12) pp 1ndash6 June 2012

[5] S Chen ldquoOutput regulation of nonlinear singularly perturbedsystems based on T-S fuzzy modelrdquo Journal of Control Theoryand Applications vol 3 no 4 pp 399ndash403 2005

[6] R Tapia-Herrera J A Meda-Campana S Alcamp119886119901119900119904 119886ntara-Montes T Hernamp119886119901119900119904 119886ndez-Cortamp119886119901119900119904 119890s and L Salgado-Conrado ldquoTuning of a TS fuzzy output regulator using thesteepest descent approach and ANFISrdquoMathematical Problemsin Engineering vol 2013 Article ID 873430 14 pages 2013

[7] K-Y Lian and J-J Liou ldquoOutput tracking control for fuzzysystems via output feedback designrdquo IEEE Transactions onFuzzy Systems vol 14 no 5 pp 628ndash639 2006

[8] X-J Ma and Z-Q Sun ldquoOutput tracking and regulationof nonlinear system based on takagi-sugeno fuzzy modelrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 30 no 1 pp 47ndash59 2000

[9] H J Lee J B Park and Y H Joo ldquoComments on lsquooutputtracking and regulation of nonlinear system based on Takagi-Sugeno fuzzy model1015840rdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 33 no 3 pp 521ndash523 2003

[10] J A Meda-Campana B Castillo-Toledo and G Chen ldquoSyn-chronization of chaotic systems from a fuzzy regulationapproachrdquo Fuzzy Sets and Systems vol 160 no 19 pp 2860ndash2875 2009

[11] J A Meda-Campana J C Gamp119886119901119900119904 119900mez-Mancilla and BCastillo-Toledo ldquoExact output regulation for nonlinear systemsdescribed by Takagi-Sugeno fuzzy modelsrdquo IEEE Transactionson Fuzzy Systems vol 20 no 2 pp 235ndash247 2012

[12] C Byrnes F Priscoli and A Isidori Output Regulation ofUncertain Nonlinear Systems Birkhauser 1997

[13] A Isidori Nonlinear Control Systems Communications andControl Engineering Series Springer Berlin Germany 3rdedition 1995

[14] K Tanaka and H O Wang Fuzzy Control Systems Designand Analysis A Linear Matrix Inequality Approach John Wileyamp119886119898119901 Sons New York NY USA 2001

[15] L-X Wang A Course in Fuzzy Systems and Control PrenticeHall Upper Saddle River NJ USA 1997

[16] I Abdelmalek N Golamp119886119901119900119904 119890a and M L Hadjili ldquoA newfuzzy Lyapunov approach to non-quadratic stabilization ofTakagi-Sugeno fuzzy modelsrdquo International Journal of AppliedMathematics andComputer Science vol 17 no 1 pp 39ndash51 2007

[17] J A Meda-Campana and B Castillo-Toledo ldquoOn the outputregulation for TS fuzzymodels using slidingmodesrdquo inProceed-ings of the American Control Conference (ACC rsquo05) pp 4062ndash4067 Portland Ore USA June 2005

[18] B Castillo-Toledo and J A Meda-Campana ldquoThe fuzzydiscrete-time robust regulation problem a LMI approachrdquo inProceedings of the 41st IEEE Conference on Decision and Controlvol 2 pp 2159ndash2164 IEEE Las Vegas Nev USA December2002

[19] B Castillo-Toledo J A Meda-Campana and A Titli ldquoAfuzzy output regulator for Takagi-Sugeno fuzzy modelsrdquo inProceedings of the IEEE International Symposium on IntelligentControl pp 310ndash315 Houston Tex USA October 2003

[20] B-C Ding H-X Sun and Y-E Qiao ldquoStability analysis of T-Sfuzzy control systems based on parameter-dependent Lyapunovfunctionrdquo Acta Automatica Sinica vol 31 no 4 pp 651ndash6542005

[21] M Narimani H K Lam R Dilmaghani and C Wolfe ldquoLMI-based stability analysis of fuzzy-model-based control systemsusing approximated polynomial membership functionsrdquo IEEE

18 Mathematical Problems in Engineering

Transactions on Systems Man and Cybernetics Part B Cyber-netics vol 41 no 3 pp 713ndash724 2011

[22] H K Lam ldquoPolynomial fuzzy-model-based control systemsstability analysis via piecewise-linear membership functionsrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 588ndash5932011

[23] K Tanaka T Hori and H O Wang ldquoA fuzzy Lyapunovapproach to fuzzy control system designrdquo in Proceedings of theAmerican Control Conference vol 6 pp 4790ndash4795 ArlingtonVa USA June 2001

[24] K Tanaka T Hori and H O Wang ldquoA multiple Lyapunovfunction approach to stabilization of fuzzy control systemsrdquoIEEE Transactions on Fuzzy Systems vol 11 no 4 pp 582ndash5892003

[25] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 174ndash1972005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

membership functions are not necessarily the same includedin the fuzzy plant Thus the steady-state input 120574(119908(119905)) can bedefined as

120574 (119908 (119905)) =

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905) (18)

where 120583119894(119908(119905)) are new membership functions such that

the fuzzy output regulator obtained from local regulatorscoincides with the controller given in [11] at least for theclass of fuzzy models considered This novel approach onlyrequires the computation of the linear local controllers andthe computation of the newmembership functionswhichwillbe presented belowThe regulator obtained in this way is validfor all 119905 ge 0 while in [11] this condition needs to be verified

Consider the following matrices

119860119894=

[[[[[[[

[

0 1 0 sdot sdot sdot 0

0 0 1 sdot sdot sdot 0

d

119886119894

1119886119894

2119886119894

3sdot sdot sdot 119886119894

119899

]]]]]]]

]

119861119894=

[[[[[[[

[

0

0

119887119894

1

]]]]]]]

]

119878 =

[[[[[[

[

0 1 0 sdot sdot sdot 0

0 0 1 sdot sdot sdot 0

d

119904111990421199043sdot sdot sdot 119904119899

]]]]]]

]

119862119894= [1198881 0 sdot sdot sdot 0]

119876 = [1199021 0 sdot sdot sdot 0]

(19)

where 119886119894119895represents the elements of the last row of 119860

119894and 1198871198941

represents the element of the last row of 119861119894with 119895 = 1 119899

and 119894 = 1 119903 It is worth mentioning that 119860119894and 119878 are

matrices of the same dimension as shown in (19) Notice thatmatrices in the formof 119878 and119876 can be used to generate a greatvariety of signals ensuring in thatway the applicability of theapproach in a great number of cases Therefore from (14) thecontrol input can be defined by

119906 (119905) =

119903

sum

119894=1

ℎ119894(119909 (119905)) 119896

119894119909 (119905) minus 120587 (119908 (119905))

+

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905)

(20)

because as mentioned before 119911(119905) is a function of 119909(119905) and insteady-state 119909(119905) = 120587(119908(119905))

On the other hand from 119862119894 119876 and from (6) it can be

concluded that 11988811199091(119905) = 119902

11199081(119905) in steady-state while from

(19) the following exosystem can be easily derived

1= 1199082

2= 1199083

119899= 11990411199081+ 11990421199082+ sdot sdot sdot + 119904

119899119908119899

119899=

119899

sum

119894=1

119904119894119908119894

(21)

Besides each subsystem can be rewritten as follows

1= 1199092

2= 1199093

119899= 119886119894

11199091+ 119886119894

21199092+ sdot sdot sdot + 119886

119894

119899119909119899+ 119887119894

1119906119904119904119894

(22)

By equating the time derivative of the fuzzy output withthe time derivative of the reference signal it results in 119888

11=

11990211 Furthermore from (21) and (22) it is possible to

conclude that

1198881119894= 1199021119894 (23)

where 119894 = 1 2 119899 From this analysis two results can beobtained First

Π =

[[[[[[[[[[

[

1199021

1198881

0 sdot sdot sdot 0

01199021

1198881

sdot sdot sdot 0

d

0 0 sdot sdot sdot1199021

1198881

]]]]]]]]]]

]

(24)

and due to the shape of (19)Π is constant and common for allsubsystemsThus 120587(119908(119905)) = Π119908(119905) Second the local steady-state input is defined by

119906119904119904119894

=(11990211198881) 11990411199081minus (11990211198881) 119886119894

11199081

119887119894

1

+(11990211198881) 11990421199082minus (11990211198881) 119886119894

21199082

119887119894

1

+ sdot sdot sdot +(11990211198881) 119904119899119908119899minus (11990211198881) 119886119894

119899119908119899

119887119894

1

(25)

Mathematical Problems in Engineering 5

The previous expression can be rewritten in a compact formas follows

119906119904119904119894

=

sum119899

119895=1(11990211198881) [119904119895119908119895minus 119886119894

119895119908119895]

119887119894

1

forall119894 = 1 119903 (26)

and from (18) the fuzzy steady-state input is

119906119904119904(119905) =

119903

sum

119894=1

120583119894(119908 (119905)) 119906

119904119904119894

=

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905) forall119894 = 1 119903

(27)

From the previous analysis each Γ119894can be computed from

Γ119894=1199021

1198881

[1199041minus 119886119894

1

119887119894

1

1199042minus 119886119894

2

119887119894

1

sdot sdot sdot119904119899minus 119886119894

119899

119887119894

1

] (28)

Now it is possible to rewrite the fuzzy model as [11]

(119905) = 119860 (119905) 119909 (119905) + 119861 (119905) 119906 (119905) + (119905) 119908 (119905) (29)

(119905) = 119878119908 (119905) (30)

119890 (119905) = 119862 (119905) 119909 (119905) minus 119876119908 (119905) (31)

where

119860 (119905) =

119903

sum

119894=1

ℎ119894(119911 (119905)) 119860

119894

119861 (119905) =

119903

sum

119894=1

ℎ119894(119911 (119905)) 119861

119894

(119905) =

119903

sum

119894=1

ℎ119894(119911 (119905)) 119875

119894

119862 (119905) =

119903

sum

119894=1

ℎ119894(119911 (119905)) 119862

119894

(32)

For the sake of simplicity in the following analysis it isconsidered that119875

119894= 0 119894 = 1 119903 that is (119905) = 0 However

taking similar steps as those used below the result can beeasily extended to the case involving (119905) = 0

Thus from (19) and (29) it can be observed that the lastequation

119899in (29) is given by

119899= [ℎ1(119909 (119905)) (119886

119894

11199091+ sdot sdot sdot + 119886

119894

119899119909119899)

+ sdot sdot sdot + ℎ119903(119909 (119905)) (119886

119894

11199091+ sdot sdot sdot + 119886

119894

119899119909119899)]

+ [ℎ1(119909 (119905)) 119887

119894

1+ sdot sdot sdot + ℎ

119903(119909 (119905)) 119887

119903

1] 119906119904119904(119905)

(33)

where compact form is

119899=

119903

sum

119894=1

ℎ119894(119909 (119905))(

119899

sum

119895=1

119886119894

119895119909119895) + (

119903

sum

119894=1

ℎ119894(119909 (119905)) 119887

119894

1)119906119904119904(119905)

(34)

considering that in steady-state 119909(119905) = Π119908(119905) the totalsteady-state input can be obtained from (23) resulting in

119903

sum

119894=1

ℎ119894(119908 (119905))(

1199021

1198881

119899

sum

119895=1

119886119894

119895119908119895)

+ (

119903

sum

119894=1

ℎ119894(119908 (119905)) 119887

119894

1)119906119904119904(119905) =

1199021

1198881

119899

sum

119894=1

119904119894119908119894

(35)

Consequently the steady-state input for the total fuzzy systemcan be constructed by

119906119904119904(119905)

=

(11990211198881)sum119899

119894=1119904119894119908119894minus (11990211198881)sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

1

(36)

Now a new set of fuzzy membership functions 120583119894(119908(119905))

satisfying sum119903119894=1

120583119894(119908(119905)) = 1 with 120583

119894(119908(119905)) ge 0 119894 = 1 119903

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 defined as follows

119906119904119904119894

= 120574 (119908 (119905)) =

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905)

=

119903

sum

119894=1

120583119894(119908 (119905))

sum119899

119895=1(11990211198881) [119904119895119908119895minus 119886119894

119895119908119895]

119887119894

1

119906119904119904= 120574 (119908 (119905))

=

(11990211198881)sum119899

119894=1119904119894119908119894minus (11990211198881)sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

1

(37)

respectivelyThus the local fuzzy regulatorswill bemultipliedby the new membership functions and the result will beequated to the global fuzzy regulator (36) as follows

(11990211198881)sum119899

119894=1119904119894119908119894minus (11990211198881)sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

1

=

119903

sum

119894=1

120583119894(119908 (119905))

sum119899

119895=1(11990211198881) [119904119895119908119895minus 119886119894

119895119908119895]

119887119894

1

(38)

So the adequate membership functions are

120583119894(119908 (119905)) =

119887119894

1ℎ119894(119908 (119905))

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

1

(39)

with 119894 = 1 2 119903 It is important to remark that the newmembership functions 120583

119894(119908(119905)) are given in terms of ℎ

119894(119908(119905))

for simplicity However ℎ119894(119908(119905)) can be removed from the

new membership functions (39) when ℎ119894(119908(119905)) is replaced

by its corresponding expression Besides ℎ119894(119908(119905)) is directly

obtained from ℎ119894(119909(119905)) by considering that in steady-state

119909(119905) = Π119908(119905)

6 Mathematical Problems in Engineering

Notice that (39) fulfills conditions119903

sum

119894=1

120583119894(119908 (119905)) = 1

120583119894(119908 (119905)) ge 0

(40)

for all 119894 = 1 2 119903

Remark 2 It can be observed that (39) is always valid whenthe entries of input matrix 119887

119894

1have the same sign if this

condition is not fulfilled the denominators of the new fuzzymembership functions will present singularities Neverthe-less these singularities may appear outside the operationregion allowing more flexibility but such cases require aparticular study of the system which is beyond the scope ofthis work

The following theorem provides the conditions for theexistence of the exact output fuzzy regulator for a class of T-Sfuzzy models

Theorem 3 The exact fuzzy output regulation with fullinformation for T-S fuzzy systems in the form of (11) definedby matrices (19) is solvable if (a) the sign of entries forcorresponding position inside input matrices 119861

119894for 119894 =

1 119903 is the same (b) there exists a fuzzy stabilizer 119906(119905) =sum119903

119894=1ℎ119894(119911(119905))119870

119894119909 for the T-S fuzzy system [16] and (c) the

exosystem (119905) = 119878119908(119905) is Poisson stable Moreover the ExactOutput Fuzzy Regulation Problem is solvable by the controller

119906 (119905) =

119903

sum

119894=1

ℎ119894(119909 (119905)) 119896

119894119909 (119905) minus Π119908 (119905)

+

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905)

(41)

where 120583119894(119908(119905)) can be readily obtained from (39)

Proof From the previous analysis the existence of mappings120587(119908(119905)) = Π119908(119905) and 120574(119908(119905)) = sum

119903

119894=1120583119894(119908(119905))Γ

119894119908(119905) is

guaranteed when the local subsystems are defined by (19)that is the form of 120587(119908(119905)) corresponds to a diagonal matrixand will be the same for all subsystems while 120574(119908(119905)) can bedirectly substitute

On the other hand condition (a) avoids singularities inthe new membership functions 120583

119894(119908(119905)) while the inclusion

of condition (b) has been thoroughly discussed in [12 13 17ndash19] and it implies that the steady-state manifold can be madeasymptotically stable by the action of the fuzzy stabilizer

Finally condition (c) is introduced to avoid that thereference signal converges to zero because if the referencetends to zero then the regulation problem turns into astabilization one which can be solved by means of thefuzzy stabilizer In addition as mentioned before this kindof matrix 119878 can be used to generate a great number ofreference signals The 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 20ndash25]

4 Exact Output Fuzzy Regulation Problem fora Class of T-S Fuzzy Systems with MultipleInputs and Multiple Outputs (MIMO Case)

Now the problem will be extended to T-S fuzzy MIMOmodels defined by matrices

119860119894= [

119874 119868

119863119894

] 119861119894= [

119874

119865119894

]

119878 = [119874 119868

119866] 119862

119894= [ 119867119862 119874

119862 ]

119876119894= [ 119867119876 119874

119876 ]

(42)

for 119894 = 1 119903 with

119868 =

[[[[[[

[

1 0 sdot sdot sdot 0

0 1 sdot sdot sdot 0

d

0 0 sdot sdot sdot 1

]]]]]]

]

119874 =

[[[[[[

[

0 0 sdot sdot sdot 0

0 0 sdot sdot sdot 0

d

0 0 sdot sdot sdot 0

]]]]]]

]

119863119894=

[[[[[[[

[

119886119894

(119896+1)1119886119894

(119896+1)2119886119894

(119896+1)3sdot sdot sdot 119886119894

(119896+1)119899

119886119894

(119896+2)1119886119894

(119896+2)2119886119894

(119896+2)3sdot sdot sdot 119886119894

(119896+2)119899

d

119886119894

1198991119886119894

1198992119886119894

1198993sdot sdot sdot 119886

119894

119899119899

]]]]]]]

]

119865119894=

[[[[[[[

[

119887119894

119896+10 sdot sdot sdot 0

0 119887119894

119896+2sdot sdot sdot 0

d

0 0 sdot sdot sdot 119887119894

119899

]]]]]]]

]

119866 =

[[[[[[

[

119904(119896+1)1

119904(119896+1)2

119904(119896+1)3

sdot sdot sdot 119904(119896+1)119899

119904(119896+2)1

119904(119896+2)2

119904(119896+2)3

sdot sdot sdot 119904(119896+2)119899

d

1199041198991

1199041198992

1199041198993

sdot sdot sdot 119904119899119899

]]]]]]

]

119874119862= 119874119876=

[[[[[[

[

0 0 sdot sdot sdot 0

0 0 sdot sdot sdot 0

d

0 0 sdot sdot sdot 0

]]]]]]

]

Mathematical Problems in Engineering 7

119867119862=

[[[[[[

[

11988810 sdot sdot sdot 0

0 1198882sdot sdot sdot 0

d

0 0 sdot sdot sdot 119888119898

]]]]]]

]

119867119876=

[[[[[[

[

1199021

0 sdot sdot sdot 0

0 1199022sdot sdot sdot 0

d

0 0 sdot sdot sdot 119902119898

]]]]]]

]

(43)

where 119896 = 119899 minus 119898 119865119894isin R119898times119898 119874 isin R(119899minus119898)times119898 119874

119862isin R119898times119898

119868 isin R(119899minus119898)times(119899minus119898) 119863119894isin R119898times119899 119867

119862isin R119898times119898 119867

119876isin R119898times119898

and 119866 isin R119898times119899 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= 119908119898+1

2= 119908119898+2

119896+1

= 119904(119896+1)1

1199081+ 119904(119896+1)2

1199082+ sdot sdot sdot + 119904

(119896+1)119899119908119899

=

119899

sum

119895=1

119904(119896+1)119895

119908119895

119896+2

= 119904(119896+2)1

1199081+ 119904(119896+2)2

1199082+ sdot sdot sdot + 119904

(119896+2)119899119908119899

=

119899

sum

119895=1

119904(119896+2)119895

119908119895

119899= 11990411989911199081+ 11990411989921199082+ sdot sdot sdot + 119904

119899119899119908119899=

119899

sum

119895=1

119904119899119895119908119895

(44)

Moreover each subsystem can be described by

1= 119909119898+1

2= 119909119898+2

119896+1

= 119886119894

(119896+1)11199091+ 119886119894

(119896+1)21199092+ sdot sdot sdot + 119886

(119896+1)119899119909119899

+ 119887119894

119896+1119906119894

119904119904

=

119899

sum

119895=1

119886119894

(119896+1)119895119909119895+ 119887119894

119896+1119906119894

119904119904

119896+2

= 119886119894

(119896+2)11199091+ 119886119894

(119896+2)21199092+ sdot sdot sdot + 119886

(119896+2)119899119909119899

+ 119887119894

119896+2119906119894

119904119904

=

119899

sum

119895=1

119886119894

(119896+2)119895119908119895+ 119887119894

1+2119906119894

119904119904

119899= 11990411989911199081+ 11990411989921199082+ sdot sdot sdot + 119904

119899119899119908119899+ 119887119894

119899119906119894

119904119904

=

119899

sum

119895=1

119904119899119895119908119895+ 119887119894

119899119906119894

119904119904

(45)

As before using expressions (44) and (45) considering 11988811199091=

11990211199081 11988821199092

= 11990221199082 119888

119898119909119898

= 119902119898119908119898 and performing

successive substitutions of ℓ= 120587ℓℓwith ℓ = 1 119898 the

following results are obtained

Π =

[[[[[[

[

1205871

0 sdot sdot sdot 0

0 1205872sdot sdot sdot 0

d

0 0 sdot sdot sdot 120587119899

]]]]]]

]

(46)

where Π isin R119899times119899 and 119894 = 1 2 119903 also 1205871

=

11990211198881 1205872= 11990221198882 120587

119898= 119902119898119888119898 120587119898+1

= 11990211198881 120587119898+2

=

11990221198882 120587

119898+119898= 119902119898119888119898 and so forth Notice that the values

of 120587lowastrepeat themselves every 119898 entry up to 119899 Additionally

Π is constant and common for all subsystems while the localsteady-state inputs are defined by

119906119894

119904119904=

[[[[[[[[[[[[[

[

1205871119904(119896+1)1

1199081minus 1205871119886119894

(119896+1)11199081

1198871

119896+1

+

1205871119904(119896+1)1

1199082minus 1205872119886119894

(119896+1)11199082

1198871

119896+1

+ sdot sdot sdot +

1205871119904(119896+1)119899

119908119899minus 120587119899119886119894

(119896+1)119899119908119899

1198871

119896+1

1205872119904(119896+2)1

1199081minus 1205871119886119894

(119896+2)11199081

1198871

119896+2

+

1205872119904(119896+1)2

1199082minus 1205872119886119894

(119896+2)21199082

1198871

119896+2

+ sdot sdot sdot +

1205872119904(119896+2)119899

119908119899minus 120587119899119886119894

(119896+2)119899119908119899

1198871

119896+2

12058711989911990411989911199081minus 1205871119886119894

11989911199081

1198871119899

+12058711989911990411989921199082minus 1205872119886119894

11989921199082

1198871119899

+ sdot sdot sdot +120587119899119904119899119899119908119899minus 120587119899119886119894

119899119899119908119899

1198871119899

]]]]]]]]]]]]]

]

(47)

8 Mathematical Problems in Engineering

for 119894 = 1 119903 In compact form (47) turns into

119906119894

119904119904=

[[[[[[[[[[[[[[

[

119899

sum

119895=1

1205871119904(119896+1)119895

119908119895minus 120587119895119886(119896+1)119895

119908119895

119887119894

119896+1

119899

sum

119895=1

1205872119904(119896+2)119895

119908119895minus 120587119895119886(119896+2)119895

119908119895

119887119894

119896+2

119899

sum

119895=1

120587119899119904119899119895119908119895minus 120587119895119886119899119895119908119895

119887119894119899

]]]]]]]]]]]]]]

]

(48)

and from (27) the local matrices Γ119894have the form

Γ119894=

[[[[[[[[[[[[[

[

1205871119904(119896+1)1

minus 1205871119886119894

(119896+1)1

119887119894

119896+1

1205871119904(119896+1)2

minus 1205872119886119894

(119896+1)2

119887119894

119896+1

sdot sdot sdot

12058711199041198991

minus 120587119899119886119894

(119896+1)119899

119887119894

119896+1

1205872119904(119896+2)1

minus 1205871119886119894

(119896+2)1

119887119894

119896+2

1205872119904(119896+2)2

minus 1205872119886119894

(119896+2)2

119887119894

119896+2

sdot sdot sdot

12058721199041198991

minus 120587119899119886119894

(119896+2)119899

119887119894

119896+2

d

1205871198991199041198991

minus 1205871119886119894

1198991

119887119894119899

1205871198991199041198992

minus 1205872119886119894

1198992

119887119894119899

sdot sdot sdot1205871198991199041198991

minus 120587119899119886119894

119899119899

119887119894119899

]]]]]]]]]]]]]

]

(49)

Now again from the fuzzy plant given in (42) and theglobal fuzzy model (29) the expressions for

119896+1

119899are

119896+1

=

119903

sum

119894=1

ℎ119894(119909 (119905))(

119899

sum

119895=1

119886119894

(119896+1)119895119909119895)

+ (

119903

sum

119894=1

ℎ119894(119909 (119905)) 119887

119894

119896+1)119906119904119904

119896+2

=

119903

sum

119894=1

ℎ119894(119909 (119905))(

119899

sum

119895=1

119886119894

(119896+2)119895119909119895)

+ (

119903

sum

119894=1

ℎ119894(119909 (119905)) 119887

119894

119896+2)119906119904119904

119899=

119903

sum

119894=1

ℎ119894(119909 (119905))(

119899

sum

119895=1

119886119894

119899119895119909119895)

+ (

119903

sum

119894=1

ℎ119894(119909 (119905)) 119887

119894

119899)119906119904119904

(50)

So by applying successive substitutions as before with (44)(45) and knowing that in steady-state 119909

119895= 120587119895119908119895 the

following expressions are obtained

119903

sum

119894=1

ℎ119894(119908 (119905))(

119899

sum

119895=1

119886119894

(119896+1)119895120587119895119908119895)

+ (

119903

sum

119894=1

ℎ119894(119908 (119905)) 119887

119894

119896+1)119906119904119904= 1205871

119899

sum

119894=1

119904(119896+1)119894

119908119894

119903

sum

119894=1

ℎ119894(119908 (119905))(

119899

sum

119895=1

119886119894

(119896+2)119895120587119895119908119895)

+ (

119903

sum

119894=1

ℎ119894(119908 (119905)) 119887

119894

119896+2)119906119904119904= 1205872

119899

sum

119894=1

119904(119896+2)

119894119908119894

1199031

sum

119894=1

ℎ119894(119908 (119905))(

119899

sum

119895=1

119886119894

119899119895120587119895119908119895)

+ (

119903

sum

119894=1

ℎ119894(119908 (119905)) 119887

119894

119899)119906119904119904= 120587119899

119899

sum

119894=1

119904119899119894119908119894

(51)

Mathematical Problems in Engineering 9

Therefore the steady-state input for the overall fuzzy systemis

119906119904119904

=

[[[[[[[[[[[[[[

[

1205871sum119899

119894=1119904(119896+1)119894

119908119894minus sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

(119896+1)119895120587119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

119896+1

1205872sum119899

119894=1119904(119896+2)119894

119908119894minus sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

(119896+2)119895120587119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

119896+2

120587119899sum119899

119894=1119904119899119895119908119894minus sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

119899119895120587119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

119899

]]]]]]]]]]]]]]

]

(52)

But the desired form of the steady-state is

119906119904119904(119905) = 120574 (119908 (119905)) =

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905) (53)

Consequently the new membership functions 120583119894(119908(119905))

can be obtained after equating (52) and (53) resulting in

120583119894(119908 (119905)) =

[[[[[[

[

12058311989411

0 sdot sdot sdot 0

0 12058311989422

sdot sdot sdot 0

d

0 0 sdot sdot sdot 120583119894119898119898

]]]]]]

]

(54)

where

120583119894119895119897

=

119887119894

119896+119895ℎ119894(119908 (119905))

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

119896+119895

(55)

for all 119894 = 1 119903 119895 = 119897 = 1 119898 It can be readily observedthat the sum of the corresponding elements is equal to oneand 120583

119894119895119897(119908(119905)) ge 0

These new membership functions 120583119894(119908(119905)) are organized

in a matrix form for that reason sum119903119894=1

120583119894(119908(119905)) = 119868 (where 119868

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 119861

119894have the

same sign at corresponding positions Clearly this conditionavoids singularities in (55) As before ℎ

119894(119908(119905)) is directly

obtained from ℎ119894(119909(119905)) by considering that in steady-state

119909(119905) = Π119908(119905) At this point the following theorem can bedefined

Theorem 5 The exact fuzzy output regulation with fullinformation for T-S fuzzy MIMO systems in the form of(11) defined by matrices (42) is solvable if (a) the sign ofentries for corresponding position inside input matrices 119861

119894for

119894 = 1 119903 is the same (b) there exists a fuzzy stabilizer119906(119905) = sum

119903

119894=1ℎ119894(119911(119905))119870

119894119909 for the fuzzy system [16] and (c) the

exosystem (119905) = 119878119908(119905) is Poisson stable

Moreover the Exact Output Fuzzy Regulation Problem forMIMO systems is solvable by the controller

119906 (119905) =

119903

sum

119894=1

ℎ119894(119909 (119905)) 119896

119894119909 (119905) minus Π119908 (119905)

+

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905)

(56)

where 120583119894(119908(119905)) can be readily obtained from (54) and (55)

Proof It follows directly from Theorem 3 and previousanalysis

Remark 6 Notice that 120583119894(119908(119905)) is defined as a matrix and

this is not a typical membership function however 120583119894(119908(119905))

contains a set of membership functions related to each entryof 119861119894 This form allows us to design controller (20)

5 Numerical Examples

51 Simple Input Case Consider the problem of balancingand swing of an inverted pendulum on a cart The motionequations for this system are [14]

1(119905) = 119909

2(119905) (57)

2(119905)

=119892 sin(119909

1(119905))minus 119886119898119897119909

2

2sin (2119909

1(119905))2 minus119886 cos (119909

1(119905)) 119906 (119905)

41198973 minus 119886119898119897 cos2 (1199091(119905))

(58)

where 1199091(119905) denotes the angle (in radians) of the pendulum

from the vertical and 1199092(119905) is the angular velocity 119892 =

98ms2 is the gravity constant 119898 is the mass of the pen-dulum 119872 is the mass of the cart 2119897 is the length of thependulum and 119906 is the force applied to the cart (in newtons)119886 = 1(119898 +119872) with119898 = 20 kg119872 = 80 kg and 2119897 = 10min the simulations The goal is to track the reference definedby the exosystem within the range 119909

1isin (minus1205872 1205872) To this

end a fuzzy model representing the nonlinear dynamics is asfollows

Rule 1

IF 1199091(119905) is about 0

THEN (119905) = 1198601119909(119905) + 119861

1119906(119905)

Rule 2

IF 1199091(119905) is about plusmn1205872(|119909

1| lt 1205872)

THEN (119905) = 1198602119909(119905) + 119861

2119906(119905)

10 Mathematical Problems in Engineering

h1

h2

x1minus90 90

(deg)

1

00

Figure 1 Membership functions of two-rule model

Membership functions for Rules 1 and 2 are shown in Figure 1In this example the T-S fuzzymodel of the plant is defined bymatrices

1198601= [

0 1

172941 0] 119861

1= [

0

minus01765]

1198602= [

0 1

41585 0] 119861

2= [

0

minus01500]

1198621= 1198622= [1 0]

(59)

while the exosystem can be constructed from the followingmatrices

119878 = [

0 1

minus1 0] 119876 = [1 0] (60)

which allows us to generate a great kind of sinusoidal refer-ences Figure 2 shows the simulation results after applying thecontroller

119906 (119905) =

119903

sum

119894=1

ℎ119894(119909 (119905)) 119870

119894[119909 (119905) minus

119903

sum

119894=1

ℎ119894(119908 (119905)) Π

119894119908 (119905)]

+

119903

sum

119894=1

ℎ119894(119908 (119905)) Γ

119894119908 (119905)

(61)

defined in [10] As expected controller (61) is unable toachieve exact regulation although tracking error remainsbounded

Now using the method derived in this work the localmatrices Γ

119894can be readily obtained from (28) resulting in

Γ1= [1036667 0] Γ

2= [343836 0] (62)

Besides knowing that 11990211198881= 1 matrix Π is obtained from

(24)

Π = [

1 0

0 1] (63)

while the newmembership functions computed from (39) are

1205831(119908 (119905)) =

(ℎ1(119908 (119905))) (minus01765)

ℎ1(119908 (119905)) (minus01765) + ℎ

2(119908 (119905)) (minus01500)

1205832(119908 (119905)) =

(ℎ2(119908 (119905))) (minus01500)

ℎ1(119908 (119905)) (minus01765) + ℎ

2(119908 (119905)) (minus01500)

(64)

On the other hand the fuzzy stabilizer for this system isconstructed bymeans of the PDC approach developed in [14]which is formed by

1198701= [1234898 69979] 119870

2= [586405 77962]

(65)

Figure 3 shows simulation with the controller definedin (20) Observe how the new membership functions arecapable of combining the local regulators in the proper wayto accomplish asymptotic tracking

To verify the effectiveness of the latter regulator thefuzzy controller is applied on the original system (57) Thesimulation results appear in Figure 4 Notice that the trackingerror depicted in Figure 4(b) is due to the approximationprovided by the T-S fuzzy model Such an error can bereduced by considering a better approximation or an exactrepresentation of the nonlinear system as the one given bythe nonlinear sector approach [14]

52 Multiple Inputs and Multiple Outputs (MIMO) CaseNow consider a T-S fuzzy system of two rules defined by thefollowing matrices

1198601=

[[[[[

[

0 0 1 0

0 0 0 1

6 4 2 4

2 3 1 5

]]]]]

]

1198611=

[[[[[

[

0 0

0 0

minus7 0

0 2

]]]]]

]

1198602=

[[[[[

[

0 0 1 0

0 0 0 1

3 4 2 1

2 4 6 2

]]]]]

]

1198612=

[[[[[

[

0 0

0 0

minus3 0

0 5

]]]]]

]

1198621= 1198622= [

1 0 0 0

0 1 0 0]

(66)

Mathematical Problems in Engineering 11

0 5 10 15 20 25 30 35 40minus40minus30minus20minus10

010203040

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30 35 40minus04minus03minus02minus01

0010203

Tracking error

Time

Am

plitu

de

e

(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 40minus40minus30minus20minus10

0102030

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30 35 40minus06minus05minus04minus03minus02minus01

0010203

Tracking error

Time

Am

plitu

de

e

(b)

Figure 3 (a) Output reference and (b) tracking errorWhen the fuzzy controller is designed on local regulators and the adequatemembershipfunctions for the fuzzy regulator are computed

0 5 10 15 20 25 30 35 40minus40minus30minus20minus10

010203040

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30 35 40minus06minus05minus04minus03minus02minus01

001

Tracking error

Time

Am

plitu

de

e

(b)

Figure 4 (a) Output reference and (b) tracking errorWhen the fuzzy controller is designed on local regulators and the adequatemembershipfunctions for the fuzzy regulator are computed

12 Mathematical Problems in Engineering

h1 h2

x1minus60

1

80

0

0

Figure 5 Membership functions of two-rule model

and an exosystem defined by

119878 =

[[[[[

[

0 0 1 0

0 0 0 1

minus4 0 0 0

0 minus16 0 0

]]]]]

]

119876 = [

1 0 0 0

0 1 0 0]

(67)

Membership functions forRules 1 and 2 are shown in Figure 5Matrices defining the local regulators can be easily obtainedby using (49) For this example they are

Γ1= [

14286 05714 02857 05714

minus10000 minus95000 minus05000 minus25000]

Γ2= [

23333 13333 06667 03333

minus04000 minus40000 minus12000 minus04000]

(68)

On the other hand 11990211198881= 1205871= 1 119902

21198882= 1205872= 1

11990211198881= 1205873= 1 119902

21198882= 1205874= 1 and by (46) and matrix Π is

Π =

[[[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

]]]]]

]

(69)

Figures 6 and 7 show the results obtained using linear regula-tors defined in (61)

Now by computing the new membership functions forthe fuzzy regulator through (54) one gets1205831(119908 (119905))

=

[[[

[

(ℎ1(119908 (119905))) (minus7)

ℎ1(119908 (119905)) (3) + ℎ

2(119908 (119905)) (3)

0

0(ℎ1(119908 (119905))) (2)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (5)

]]]

]

1205832(119908 (119905))

=

[[[

[

(ℎ2(119908 (119905))) (minus3)

ℎ1(119908 (119905)) (3) + ℎ

2(119908 (119905)) (3)

0

0(ℎ2(119908 (119905))) (5)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (5)

]]]

]

(70)

As previously the fuzzy stabilizer for this example is alsoconstructed bymeans of the PDC approach developed in [14]resulting in

1198701= [

10286 05714 06000 05714

minus10000 minus22500 minus05000 minus37500]

1198702= [

14000 13333 14000 03333

minus04000 minus11000 minus12000 minus09000]

(71)

Thus Figures 8 and 9 show the simulation results whenthe controller defined by (20) is applied

Besides Figure 10 shows the shape of the new member-ship functions which are clearly different from the originalones given in Figure 5

53 More General Problems Consider the T-S fuzzy MIMOsystem with multiple inputs defined by the following matri-ces

1198601=

[[[[[[[[[[[

[

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

minus6 4 2 4 5 3

2 3 minus1 5 minus2 4

]]]]]]]]]]]

]

1198611=

[[[[[[[[[[[

[

0 0

0 0

0 0

0 0

5 0

0 2

]]]]]]]]]]]

]

1198602=

[[[[[[[[[[[

[

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

minus6 1 2 minus5 4 6

0 2 minus7 4 5 2

]]]]]]]]]]]

]

1198612=

[[[[[[[[[[[

[

0 0

0 0

0 0

0 0

7 0

0 8

]]]]]]]]]]]

]

1198621= 1198622= [

0 1 1 0 0 0

1 0 1 0 0 0]

(72)

Mathematical Problems in Engineering 13

0 5 10 15 20 25 30minus15minus10

minus505

101520

Time

Am

plitu

dex1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus4minus2

02468

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 6 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus30minus20minus10

0102030

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus15

minus10

minus5

0

5

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 7 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of lineal regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus15minus10

minus505

101520

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus1

012345678

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 8 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

The reference signals will be generated by an exosystemdefined by

119878 = [

0 1

minus1 0] 119876 = [

1 0

0 1] (73)

Membership functions for Rules 1 and 2 are ℎ1(119909(119905)) =

(80 minus 1199091(119905))140 and ℎ

2(119909(119905)) = 1 minus ℎ

1(119905) Notice that for this

case 119899 = 119904 and 119862119894

= 119876 This implies that Π119894and Γ

119894must

be obtained by solving Francis Equations for each subsystemHowever due to form of the fuzzy plant it is possible to verifythat Π

1= Π2

14 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus40minus30minus20minus10

0102030

Time

Am

plitu

dex2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus10

minus8minus6minus4minus2

0

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 9 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(11) (x1(t))1205832(11) (x1(t))

(a)

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(22) (x1(t))1205832(22) (x1(t))

(b)

Figure 10 (a) Membership functions corresponding to position (11) of 1205831(119908(119905)) and 120583

2(119908(119905)) (b) Membership functions corresponding to

position (22) 1205831(119908(119905)) and 120583

2(119908(119905))

So by applying (9) for each local subsystem with 119894 = 1 2it results in

Π1= Π2=

[[[[[[[[[[[

[

05000 05000

15000 minus05000

minus05000 05000

05000 15000

minus05000 minus05000

minus15000 05000

]]]]]]]]]]]

]

Γ1= [

07000 minus03000

minus20000 minus55000]

Γ2= [

23571 12143

minus04375 minus01875]

(74)

Again the fuzzy stabilizer for this system is constructedby means of the PDC approach developed in [14] yielding

1198701=[[

[

minus12229 00609 minus40284 01291 minus25666 minus04061

16919 minus83997 32934 minus127153 15732 minus61836

]]

]

1198702=[[

[

minus08735 04721 minus28775 13780 minus16904 minus07187

06730 minus19749 15734 minus30538 minus04817 minus12959

]]

]

(75)

The behavior obtained by applying controller (61) isdepicted in Figures 11 and 12 Notice the exact outputregulation is not achieved

Mathematical Problems in Engineering 15

0 5 10 15 20 25 30minus60minus40minus20

0204060

Time

Am

plitu

dex1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus80minus70minus60minus50minus40minus30minus20minus10

01020

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 11 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus15

minus10

minus5

0

5

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 12 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

Now the adequate membership functions are computedfrom (54) resulting in

1205831(119908 (119905))

=

[[[

[

(ℎ1(119908 (119905))) (5)

ℎ1(119908 (119905)) (5) + ℎ

2(119908 (119905)) (7)

0

0(ℎ1(119908 (119905))) (2)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (8)

]]]

]

1205832(119908 (119905))

=

[[[

[

(ℎ2(119908 (119905))) (7)

ℎ1(119908 (119905)) (5) + ℎ

2(119908 (119905)) (7)

0

0(ℎ2(119908 (119905))) (8)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (8)

]]]

]

(76)

When ℎ119894(119908(119905)) is replaced by its corresponding expressions

one gets

1205831(119908 (119905)) =

[[[

[

1225

1199081(119905) + 140

minus5

20

0 minus1199081(119905) minus 80

31199081(119905) + 320

]]]

]

1205832(119908 (119905)) =

[[[

[

7

2minus

1225

1199081(119905) + 410

0

041199081(119905) + 240

31199081(119905) + 320

]]]

]

(77)

The results of applying control law (20) on the fuzzy plantare given in Figures 13 and 14

The ldquocomputedrdquo membership functions for the fuzzyregulator are given in Figure 15 Again it can be readilyobserved that these new membership functions are differentfrom the original ones

Notice that although this example does not fulfill theconditions ofTheorem 5 themembership functions obtainedfrom (39) allow the exact tracking for the fuzzy problemThis

16 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus80minus70minus60minus50minus40minus30minus20minus10

010

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 13 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus8minus6minus4minus2

02468

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 14 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(11) (x1(t))1205832(11) (x1(t))

(a)

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(22) (x1(t))1205832(22) (x1(t))

(b)

Figure 15 (a) Membership functions corresponding to position (11) of 1205831(119908(119905)) and 120583

2(119908(119905)) (b) Membership functions corresponding to

position (22) 1205831(119908(119905)) 120583

2(119908(119905))

Mathematical Problems in Engineering 17

suggests that the approach presented in this work may beapplied on a bigger class of T-S models however this studyis not completed yet

6 Conclusions

A fuzzy regulator for continuous-time systems based on thecombination of linear regulators adjusted by different mem-bership functions has been presented The main advantageis that analytical expressions for the membership functionswhich allow the proper combination of the local regulators inorder to guarantee the exact regulation can be easily appliedAs 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 modelscan be commonly found in mechanics electromechanicselectronics robotics and so forth

Some numerical examples are used to illustrate theapplicability of the proposed approach even in cases wherethe dimensions of the plant and the exosystem are differentbetween each otherThis suggests that the approach presentedin this work may be applied on a bigger class of T-S models

Finally the method proposed in this work avoids thedisadvantage of constructing an exact fuzzy regulator basedon overall T-S fuzzy system whose expression may resultvery large Instead the given approach offers 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 Interests

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

Acknowledgments

This work was partially supported by CONACYT (ConsejoNacional de Ciencia y Tecnologıa) through scholarship SNI(Sistema Nacional de Investigadores) and by IPN (InstitutoPolitecnico Nacional) through research Projects 20140659and 20150487 and scholarships EDI (Estımulo al Desempenode los Investigadores) COFAA (Comision de Operacion yFomento de Actividades Academicas) and BEIFI (Beca deEstımulo Institucional de Formacion de Investigadores)

References

[1] B A Francis ldquoThe linear multivariable regulator problemrdquoSIAM Journal on Control and Optimization vol 15 no 3 pp486ndash505 1977

[2] B A Francis andWMWonham ldquoThe internalmodel principleof control theoryrdquo Automatica vol 12 no 5 pp 457ndash465 1976

[3] A Isidori and C I Byrnes ldquoOutput regulation of nonlinearsystemsrdquo IEEE Transactions on Automatic Control vol 35 no2 pp 131ndash140 1990

[4] B Castillo-Toledo S Di Gennaro and F Jurado ldquoTrajectorytracking for a quadrotor via fuzzy regulationrdquo in Proceedings ofthe World Automation Congress (WAC rsquo12) pp 1ndash6 June 2012

[5] S Chen ldquoOutput regulation of nonlinear singularly perturbedsystems based on T-S fuzzy modelrdquo Journal of Control Theoryand Applications vol 3 no 4 pp 399ndash403 2005

[6] R Tapia-Herrera J A Meda-Campana S Alcamp119886119901119900119904 119886ntara-Montes T Hernamp119886119901119900119904 119886ndez-Cortamp119886119901119900119904 119890s and L Salgado-Conrado ldquoTuning of a TS fuzzy output regulator using thesteepest descent approach and ANFISrdquoMathematical Problemsin Engineering vol 2013 Article ID 873430 14 pages 2013

[7] K-Y Lian and J-J Liou ldquoOutput tracking control for fuzzysystems via output feedback designrdquo IEEE Transactions onFuzzy Systems vol 14 no 5 pp 628ndash639 2006

[8] X-J Ma and Z-Q Sun ldquoOutput tracking and regulationof nonlinear system based on takagi-sugeno fuzzy modelrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 30 no 1 pp 47ndash59 2000

[9] H J Lee J B Park and Y H Joo ldquoComments on lsquooutputtracking and regulation of nonlinear system based on Takagi-Sugeno fuzzy model1015840rdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 33 no 3 pp 521ndash523 2003

[10] J A Meda-Campana B Castillo-Toledo and G Chen ldquoSyn-chronization of chaotic systems from a fuzzy regulationapproachrdquo Fuzzy Sets and Systems vol 160 no 19 pp 2860ndash2875 2009

[11] J A Meda-Campana J C Gamp119886119901119900119904 119900mez-Mancilla and BCastillo-Toledo ldquoExact output regulation for nonlinear systemsdescribed by Takagi-Sugeno fuzzy modelsrdquo IEEE Transactionson Fuzzy Systems vol 20 no 2 pp 235ndash247 2012

[12] C Byrnes F Priscoli and A Isidori Output Regulation ofUncertain Nonlinear Systems Birkhauser 1997

[13] A Isidori Nonlinear Control Systems Communications andControl Engineering Series Springer Berlin Germany 3rdedition 1995

[14] K Tanaka and H O Wang Fuzzy Control Systems Designand Analysis A Linear Matrix Inequality Approach John Wileyamp119886119898119901 Sons New York NY USA 2001

[15] L-X Wang A Course in Fuzzy Systems and Control PrenticeHall Upper Saddle River NJ USA 1997

[16] I Abdelmalek N Golamp119886119901119900119904 119890a and M L Hadjili ldquoA newfuzzy Lyapunov approach to non-quadratic stabilization ofTakagi-Sugeno fuzzy modelsrdquo International Journal of AppliedMathematics andComputer Science vol 17 no 1 pp 39ndash51 2007

[17] J A Meda-Campana and B Castillo-Toledo ldquoOn the outputregulation for TS fuzzymodels using slidingmodesrdquo inProceed-ings of the American Control Conference (ACC rsquo05) pp 4062ndash4067 Portland Ore USA June 2005

[18] B Castillo-Toledo and J A Meda-Campana ldquoThe fuzzydiscrete-time robust regulation problem a LMI approachrdquo inProceedings of the 41st IEEE Conference on Decision and Controlvol 2 pp 2159ndash2164 IEEE Las Vegas Nev USA December2002

[19] B Castillo-Toledo J A Meda-Campana and A Titli ldquoAfuzzy output regulator for Takagi-Sugeno fuzzy modelsrdquo inProceedings of the IEEE International Symposium on IntelligentControl pp 310ndash315 Houston Tex USA October 2003

[20] B-C Ding H-X Sun and Y-E Qiao ldquoStability analysis of T-Sfuzzy control systems based on parameter-dependent Lyapunovfunctionrdquo Acta Automatica Sinica vol 31 no 4 pp 651ndash6542005

[21] M Narimani H K Lam R Dilmaghani and C Wolfe ldquoLMI-based stability analysis of fuzzy-model-based control systemsusing approximated polynomial membership functionsrdquo IEEE

18 Mathematical Problems in Engineering

Transactions on Systems Man and Cybernetics Part B Cyber-netics vol 41 no 3 pp 713ndash724 2011

[22] H K Lam ldquoPolynomial fuzzy-model-based control systemsstability analysis via piecewise-linear membership functionsrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 588ndash5932011

[23] K Tanaka T Hori and H O Wang ldquoA fuzzy Lyapunovapproach to fuzzy control system designrdquo in Proceedings of theAmerican Control Conference vol 6 pp 4790ndash4795 ArlingtonVa USA June 2001

[24] K Tanaka T Hori and H O Wang ldquoA multiple Lyapunovfunction approach to stabilization of fuzzy control systemsrdquoIEEE Transactions on Fuzzy Systems vol 11 no 4 pp 582ndash5892003

[25] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 174ndash1972005

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

The previous expression can be rewritten in a compact formas follows

119906119904119904119894

=

sum119899

119895=1(11990211198881) [119904119895119908119895minus 119886119894

119895119908119895]

119887119894

1

forall119894 = 1 119903 (26)

and from (18) the fuzzy steady-state input is

119906119904119904(119905) =

119903

sum

119894=1

120583119894(119908 (119905)) 119906

119904119904119894

=

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905) forall119894 = 1 119903

(27)

From the previous analysis each Γ119894can be computed from

Γ119894=1199021

1198881

[1199041minus 119886119894

1

119887119894

1

1199042minus 119886119894

2

119887119894

1

sdot sdot sdot119904119899minus 119886119894

119899

119887119894

1

] (28)

Now it is possible to rewrite the fuzzy model as [11]

(119905) = 119860 (119905) 119909 (119905) + 119861 (119905) 119906 (119905) + (119905) 119908 (119905) (29)

(119905) = 119878119908 (119905) (30)

119890 (119905) = 119862 (119905) 119909 (119905) minus 119876119908 (119905) (31)

where

119860 (119905) =

119903

sum

119894=1

ℎ119894(119911 (119905)) 119860

119894

119861 (119905) =

119903

sum

119894=1

ℎ119894(119911 (119905)) 119861

119894

(119905) =

119903

sum

119894=1

ℎ119894(119911 (119905)) 119875

119894

119862 (119905) =

119903

sum

119894=1

ℎ119894(119911 (119905)) 119862

119894

(32)

For the sake of simplicity in the following analysis it isconsidered that119875

119894= 0 119894 = 1 119903 that is (119905) = 0 However

taking similar steps as those used below the result can beeasily extended to the case involving (119905) = 0

Thus from (19) and (29) it can be observed that the lastequation

119899in (29) is given by

119899= [ℎ1(119909 (119905)) (119886

119894

11199091+ sdot sdot sdot + 119886

119894

119899119909119899)

+ sdot sdot sdot + ℎ119903(119909 (119905)) (119886

119894

11199091+ sdot sdot sdot + 119886

119894

119899119909119899)]

+ [ℎ1(119909 (119905)) 119887

119894

1+ sdot sdot sdot + ℎ

119903(119909 (119905)) 119887

119903

1] 119906119904119904(119905)

(33)

where compact form is

119899=

119903

sum

119894=1

ℎ119894(119909 (119905))(

119899

sum

119895=1

119886119894

119895119909119895) + (

119903

sum

119894=1

ℎ119894(119909 (119905)) 119887

119894

1)119906119904119904(119905)

(34)

considering that in steady-state 119909(119905) = Π119908(119905) the totalsteady-state input can be obtained from (23) resulting in

119903

sum

119894=1

ℎ119894(119908 (119905))(

1199021

1198881

119899

sum

119895=1

119886119894

119895119908119895)

+ (

119903

sum

119894=1

ℎ119894(119908 (119905)) 119887

119894

1)119906119904119904(119905) =

1199021

1198881

119899

sum

119894=1

119904119894119908119894

(35)

Consequently the steady-state input for the total fuzzy systemcan be constructed by

119906119904119904(119905)

=

(11990211198881)sum119899

119894=1119904119894119908119894minus (11990211198881)sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

1

(36)

Now a new set of fuzzy membership functions 120583119894(119908(119905))

satisfying sum119903119894=1

120583119894(119908(119905)) = 1 with 120583

119894(119908(119905)) ge 0 119894 = 1 119903

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 defined as follows

119906119904119904119894

= 120574 (119908 (119905)) =

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905)

=

119903

sum

119894=1

120583119894(119908 (119905))

sum119899

119895=1(11990211198881) [119904119895119908119895minus 119886119894

119895119908119895]

119887119894

1

119906119904119904= 120574 (119908 (119905))

=

(11990211198881)sum119899

119894=1119904119894119908119894minus (11990211198881)sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

1

(37)

respectivelyThus the local fuzzy regulatorswill bemultipliedby the new membership functions and the result will beequated to the global fuzzy regulator (36) as follows

(11990211198881)sum119899

119894=1119904119894119908119894minus (11990211198881)sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

1

=

119903

sum

119894=1

120583119894(119908 (119905))

sum119899

119895=1(11990211198881) [119904119895119908119895minus 119886119894

119895119908119895]

119887119894

1

(38)

So the adequate membership functions are

120583119894(119908 (119905)) =

119887119894

1ℎ119894(119908 (119905))

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

1

(39)

with 119894 = 1 2 119903 It is important to remark that the newmembership functions 120583

119894(119908(119905)) are given in terms of ℎ

119894(119908(119905))

for simplicity However ℎ119894(119908(119905)) can be removed from the

new membership functions (39) when ℎ119894(119908(119905)) is replaced

by its corresponding expression Besides ℎ119894(119908(119905)) is directly

obtained from ℎ119894(119909(119905)) by considering that in steady-state

119909(119905) = Π119908(119905)

6 Mathematical Problems in Engineering

Notice that (39) fulfills conditions119903

sum

119894=1

120583119894(119908 (119905)) = 1

120583119894(119908 (119905)) ge 0

(40)

for all 119894 = 1 2 119903

Remark 2 It can be observed that (39) is always valid whenthe entries of input matrix 119887

119894

1have the same sign if this

condition is not fulfilled the denominators of the new fuzzymembership functions will present singularities Neverthe-less these singularities may appear outside the operationregion allowing more flexibility but such cases require aparticular study of the system which is beyond the scope ofthis work

The following theorem provides the conditions for theexistence of the exact output fuzzy regulator for a class of T-Sfuzzy models

Theorem 3 The exact fuzzy output regulation with fullinformation for T-S fuzzy systems in the form of (11) definedby matrices (19) is solvable if (a) the sign of entries forcorresponding position inside input matrices 119861

119894for 119894 =

1 119903 is the same (b) there exists a fuzzy stabilizer 119906(119905) =sum119903

119894=1ℎ119894(119911(119905))119870

119894119909 for the T-S fuzzy system [16] and (c) the

exosystem (119905) = 119878119908(119905) is Poisson stable Moreover the ExactOutput Fuzzy Regulation Problem is solvable by the controller

119906 (119905) =

119903

sum

119894=1

ℎ119894(119909 (119905)) 119896

119894119909 (119905) minus Π119908 (119905)

+

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905)

(41)

where 120583119894(119908(119905)) can be readily obtained from (39)

Proof From the previous analysis the existence of mappings120587(119908(119905)) = Π119908(119905) and 120574(119908(119905)) = sum

119903

119894=1120583119894(119908(119905))Γ

119894119908(119905) is

guaranteed when the local subsystems are defined by (19)that is the form of 120587(119908(119905)) corresponds to a diagonal matrixand will be the same for all subsystems while 120574(119908(119905)) can bedirectly substitute

On the other hand condition (a) avoids singularities inthe new membership functions 120583

119894(119908(119905)) while the inclusion

of condition (b) has been thoroughly discussed in [12 13 17ndash19] and it implies that the steady-state manifold can be madeasymptotically stable by the action of the fuzzy stabilizer

Finally condition (c) is introduced to avoid that thereference signal converges to zero because if the referencetends to zero then the regulation problem turns into astabilization one which can be solved by means of thefuzzy stabilizer In addition as mentioned before this kindof matrix 119878 can be used to generate a great number ofreference signals The 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 20ndash25]

4 Exact Output Fuzzy Regulation Problem fora Class of T-S Fuzzy Systems with MultipleInputs and Multiple Outputs (MIMO Case)

Now the problem will be extended to T-S fuzzy MIMOmodels defined by matrices

119860119894= [

119874 119868

119863119894

] 119861119894= [

119874

119865119894

]

119878 = [119874 119868

119866] 119862

119894= [ 119867119862 119874

119862 ]

119876119894= [ 119867119876 119874

119876 ]

(42)

for 119894 = 1 119903 with

119868 =

[[[[[[

[

1 0 sdot sdot sdot 0

0 1 sdot sdot sdot 0

d

0 0 sdot sdot sdot 1

]]]]]]

]

119874 =

[[[[[[

[

0 0 sdot sdot sdot 0

0 0 sdot sdot sdot 0

d

0 0 sdot sdot sdot 0

]]]]]]

]

119863119894=

[[[[[[[

[

119886119894

(119896+1)1119886119894

(119896+1)2119886119894

(119896+1)3sdot sdot sdot 119886119894

(119896+1)119899

119886119894

(119896+2)1119886119894

(119896+2)2119886119894

(119896+2)3sdot sdot sdot 119886119894

(119896+2)119899

d

119886119894

1198991119886119894

1198992119886119894

1198993sdot sdot sdot 119886

119894

119899119899

]]]]]]]

]

119865119894=

[[[[[[[

[

119887119894

119896+10 sdot sdot sdot 0

0 119887119894

119896+2sdot sdot sdot 0

d

0 0 sdot sdot sdot 119887119894

119899

]]]]]]]

]

119866 =

[[[[[[

[

119904(119896+1)1

119904(119896+1)2

119904(119896+1)3

sdot sdot sdot 119904(119896+1)119899

119904(119896+2)1

119904(119896+2)2

119904(119896+2)3

sdot sdot sdot 119904(119896+2)119899

d

1199041198991

1199041198992

1199041198993

sdot sdot sdot 119904119899119899

]]]]]]

]

119874119862= 119874119876=

[[[[[[

[

0 0 sdot sdot sdot 0

0 0 sdot sdot sdot 0

d

0 0 sdot sdot sdot 0

]]]]]]

]

Mathematical Problems in Engineering 7

119867119862=

[[[[[[

[

11988810 sdot sdot sdot 0

0 1198882sdot sdot sdot 0

d

0 0 sdot sdot sdot 119888119898

]]]]]]

]

119867119876=

[[[[[[

[

1199021

0 sdot sdot sdot 0

0 1199022sdot sdot sdot 0

d

0 0 sdot sdot sdot 119902119898

]]]]]]

]

(43)

where 119896 = 119899 minus 119898 119865119894isin R119898times119898 119874 isin R(119899minus119898)times119898 119874

119862isin R119898times119898

119868 isin R(119899minus119898)times(119899minus119898) 119863119894isin R119898times119899 119867

119862isin R119898times119898 119867

119876isin R119898times119898

and 119866 isin R119898times119899 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= 119908119898+1

2= 119908119898+2

119896+1

= 119904(119896+1)1

1199081+ 119904(119896+1)2

1199082+ sdot sdot sdot + 119904

(119896+1)119899119908119899

=

119899

sum

119895=1

119904(119896+1)119895

119908119895

119896+2

= 119904(119896+2)1

1199081+ 119904(119896+2)2

1199082+ sdot sdot sdot + 119904

(119896+2)119899119908119899

=

119899

sum

119895=1

119904(119896+2)119895

119908119895

119899= 11990411989911199081+ 11990411989921199082+ sdot sdot sdot + 119904

119899119899119908119899=

119899

sum

119895=1

119904119899119895119908119895

(44)

Moreover each subsystem can be described by

1= 119909119898+1

2= 119909119898+2

119896+1

= 119886119894

(119896+1)11199091+ 119886119894

(119896+1)21199092+ sdot sdot sdot + 119886

(119896+1)119899119909119899

+ 119887119894

119896+1119906119894

119904119904

=

119899

sum

119895=1

119886119894

(119896+1)119895119909119895+ 119887119894

119896+1119906119894

119904119904

119896+2

= 119886119894

(119896+2)11199091+ 119886119894

(119896+2)21199092+ sdot sdot sdot + 119886

(119896+2)119899119909119899

+ 119887119894

119896+2119906119894

119904119904

=

119899

sum

119895=1

119886119894

(119896+2)119895119908119895+ 119887119894

1+2119906119894

119904119904

119899= 11990411989911199081+ 11990411989921199082+ sdot sdot sdot + 119904

119899119899119908119899+ 119887119894

119899119906119894

119904119904

=

119899

sum

119895=1

119904119899119895119908119895+ 119887119894

119899119906119894

119904119904

(45)

As before using expressions (44) and (45) considering 11988811199091=

11990211199081 11988821199092

= 11990221199082 119888

119898119909119898

= 119902119898119908119898 and performing

successive substitutions of ℓ= 120587ℓℓwith ℓ = 1 119898 the

following results are obtained

Π =

[[[[[[

[

1205871

0 sdot sdot sdot 0

0 1205872sdot sdot sdot 0

d

0 0 sdot sdot sdot 120587119899

]]]]]]

]

(46)

where Π isin R119899times119899 and 119894 = 1 2 119903 also 1205871

=

11990211198881 1205872= 11990221198882 120587

119898= 119902119898119888119898 120587119898+1

= 11990211198881 120587119898+2

=

11990221198882 120587

119898+119898= 119902119898119888119898 and so forth Notice that the values

of 120587lowastrepeat themselves every 119898 entry up to 119899 Additionally

Π is constant and common for all subsystems while the localsteady-state inputs are defined by

119906119894

119904119904=

[[[[[[[[[[[[[

[

1205871119904(119896+1)1

1199081minus 1205871119886119894

(119896+1)11199081

1198871

119896+1

+

1205871119904(119896+1)1

1199082minus 1205872119886119894

(119896+1)11199082

1198871

119896+1

+ sdot sdot sdot +

1205871119904(119896+1)119899

119908119899minus 120587119899119886119894

(119896+1)119899119908119899

1198871

119896+1

1205872119904(119896+2)1

1199081minus 1205871119886119894

(119896+2)11199081

1198871

119896+2

+

1205872119904(119896+1)2

1199082minus 1205872119886119894

(119896+2)21199082

1198871

119896+2

+ sdot sdot sdot +

1205872119904(119896+2)119899

119908119899minus 120587119899119886119894

(119896+2)119899119908119899

1198871

119896+2

12058711989911990411989911199081minus 1205871119886119894

11989911199081

1198871119899

+12058711989911990411989921199082minus 1205872119886119894

11989921199082

1198871119899

+ sdot sdot sdot +120587119899119904119899119899119908119899minus 120587119899119886119894

119899119899119908119899

1198871119899

]]]]]]]]]]]]]

]

(47)

8 Mathematical Problems in Engineering

for 119894 = 1 119903 In compact form (47) turns into

119906119894

119904119904=

[[[[[[[[[[[[[[

[

119899

sum

119895=1

1205871119904(119896+1)119895

119908119895minus 120587119895119886(119896+1)119895

119908119895

119887119894

119896+1

119899

sum

119895=1

1205872119904(119896+2)119895

119908119895minus 120587119895119886(119896+2)119895

119908119895

119887119894

119896+2

119899

sum

119895=1

120587119899119904119899119895119908119895minus 120587119895119886119899119895119908119895

119887119894119899

]]]]]]]]]]]]]]

]

(48)

and from (27) the local matrices Γ119894have the form

Γ119894=

[[[[[[[[[[[[[

[

1205871119904(119896+1)1

minus 1205871119886119894

(119896+1)1

119887119894

119896+1

1205871119904(119896+1)2

minus 1205872119886119894

(119896+1)2

119887119894

119896+1

sdot sdot sdot

12058711199041198991

minus 120587119899119886119894

(119896+1)119899

119887119894

119896+1

1205872119904(119896+2)1

minus 1205871119886119894

(119896+2)1

119887119894

119896+2

1205872119904(119896+2)2

minus 1205872119886119894

(119896+2)2

119887119894

119896+2

sdot sdot sdot

12058721199041198991

minus 120587119899119886119894

(119896+2)119899

119887119894

119896+2

d

1205871198991199041198991

minus 1205871119886119894

1198991

119887119894119899

1205871198991199041198992

minus 1205872119886119894

1198992

119887119894119899

sdot sdot sdot1205871198991199041198991

minus 120587119899119886119894

119899119899

119887119894119899

]]]]]]]]]]]]]

]

(49)

Now again from the fuzzy plant given in (42) and theglobal fuzzy model (29) the expressions for

119896+1

119899are

119896+1

=

119903

sum

119894=1

ℎ119894(119909 (119905))(

119899

sum

119895=1

119886119894

(119896+1)119895119909119895)

+ (

119903

sum

119894=1

ℎ119894(119909 (119905)) 119887

119894

119896+1)119906119904119904

119896+2

=

119903

sum

119894=1

ℎ119894(119909 (119905))(

119899

sum

119895=1

119886119894

(119896+2)119895119909119895)

+ (

119903

sum

119894=1

ℎ119894(119909 (119905)) 119887

119894

119896+2)119906119904119904

119899=

119903

sum

119894=1

ℎ119894(119909 (119905))(

119899

sum

119895=1

119886119894

119899119895119909119895)

+ (

119903

sum

119894=1

ℎ119894(119909 (119905)) 119887

119894

119899)119906119904119904

(50)

So by applying successive substitutions as before with (44)(45) and knowing that in steady-state 119909

119895= 120587119895119908119895 the

following expressions are obtained

119903

sum

119894=1

ℎ119894(119908 (119905))(

119899

sum

119895=1

119886119894

(119896+1)119895120587119895119908119895)

+ (

119903

sum

119894=1

ℎ119894(119908 (119905)) 119887

119894

119896+1)119906119904119904= 1205871

119899

sum

119894=1

119904(119896+1)119894

119908119894

119903

sum

119894=1

ℎ119894(119908 (119905))(

119899

sum

119895=1

119886119894

(119896+2)119895120587119895119908119895)

+ (

119903

sum

119894=1

ℎ119894(119908 (119905)) 119887

119894

119896+2)119906119904119904= 1205872

119899

sum

119894=1

119904(119896+2)

119894119908119894

1199031

sum

119894=1

ℎ119894(119908 (119905))(

119899

sum

119895=1

119886119894

119899119895120587119895119908119895)

+ (

119903

sum

119894=1

ℎ119894(119908 (119905)) 119887

119894

119899)119906119904119904= 120587119899

119899

sum

119894=1

119904119899119894119908119894

(51)

Mathematical Problems in Engineering 9

Therefore the steady-state input for the overall fuzzy systemis

119906119904119904

=

[[[[[[[[[[[[[[

[

1205871sum119899

119894=1119904(119896+1)119894

119908119894minus sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

(119896+1)119895120587119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

119896+1

1205872sum119899

119894=1119904(119896+2)119894

119908119894minus sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

(119896+2)119895120587119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

119896+2

120587119899sum119899

119894=1119904119899119895119908119894minus sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

119899119895120587119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

119899

]]]]]]]]]]]]]]

]

(52)

But the desired form of the steady-state is

119906119904119904(119905) = 120574 (119908 (119905)) =

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905) (53)

Consequently the new membership functions 120583119894(119908(119905))

can be obtained after equating (52) and (53) resulting in

120583119894(119908 (119905)) =

[[[[[[

[

12058311989411

0 sdot sdot sdot 0

0 12058311989422

sdot sdot sdot 0

d

0 0 sdot sdot sdot 120583119894119898119898

]]]]]]

]

(54)

where

120583119894119895119897

=

119887119894

119896+119895ℎ119894(119908 (119905))

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

119896+119895

(55)

for all 119894 = 1 119903 119895 = 119897 = 1 119898 It can be readily observedthat the sum of the corresponding elements is equal to oneand 120583

119894119895119897(119908(119905)) ge 0

These new membership functions 120583119894(119908(119905)) are organized

in a matrix form for that reason sum119903119894=1

120583119894(119908(119905)) = 119868 (where 119868

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 119861

119894have the

same sign at corresponding positions Clearly this conditionavoids singularities in (55) As before ℎ

119894(119908(119905)) is directly

obtained from ℎ119894(119909(119905)) by considering that in steady-state

119909(119905) = Π119908(119905) At this point the following theorem can bedefined

Theorem 5 The exact fuzzy output regulation with fullinformation for T-S fuzzy MIMO systems in the form of(11) defined by matrices (42) is solvable if (a) the sign ofentries for corresponding position inside input matrices 119861

119894for

119894 = 1 119903 is the same (b) there exists a fuzzy stabilizer119906(119905) = sum

119903

119894=1ℎ119894(119911(119905))119870

119894119909 for the fuzzy system [16] and (c) the

exosystem (119905) = 119878119908(119905) is Poisson stable

Moreover the Exact Output Fuzzy Regulation Problem forMIMO systems is solvable by the controller

119906 (119905) =

119903

sum

119894=1

ℎ119894(119909 (119905)) 119896

119894119909 (119905) minus Π119908 (119905)

+

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905)

(56)

where 120583119894(119908(119905)) can be readily obtained from (54) and (55)

Proof It follows directly from Theorem 3 and previousanalysis

Remark 6 Notice that 120583119894(119908(119905)) is defined as a matrix and

this is not a typical membership function however 120583119894(119908(119905))

contains a set of membership functions related to each entryof 119861119894 This form allows us to design controller (20)

5 Numerical Examples

51 Simple Input Case Consider the problem of balancingand swing of an inverted pendulum on a cart The motionequations for this system are [14]

1(119905) = 119909

2(119905) (57)

2(119905)

=119892 sin(119909

1(119905))minus 119886119898119897119909

2

2sin (2119909

1(119905))2 minus119886 cos (119909

1(119905)) 119906 (119905)

41198973 minus 119886119898119897 cos2 (1199091(119905))

(58)

where 1199091(119905) denotes the angle (in radians) of the pendulum

from the vertical and 1199092(119905) is the angular velocity 119892 =

98ms2 is the gravity constant 119898 is the mass of the pen-dulum 119872 is the mass of the cart 2119897 is the length of thependulum and 119906 is the force applied to the cart (in newtons)119886 = 1(119898 +119872) with119898 = 20 kg119872 = 80 kg and 2119897 = 10min the simulations The goal is to track the reference definedby the exosystem within the range 119909

1isin (minus1205872 1205872) To this

end a fuzzy model representing the nonlinear dynamics is asfollows

Rule 1

IF 1199091(119905) is about 0

THEN (119905) = 1198601119909(119905) + 119861

1119906(119905)

Rule 2

IF 1199091(119905) is about plusmn1205872(|119909

1| lt 1205872)

THEN (119905) = 1198602119909(119905) + 119861

2119906(119905)

10 Mathematical Problems in Engineering

h1

h2

x1minus90 90

(deg)

1

00

Figure 1 Membership functions of two-rule model

Membership functions for Rules 1 and 2 are shown in Figure 1In this example the T-S fuzzymodel of the plant is defined bymatrices

1198601= [

0 1

172941 0] 119861

1= [

0

minus01765]

1198602= [

0 1

41585 0] 119861

2= [

0

minus01500]

1198621= 1198622= [1 0]

(59)

while the exosystem can be constructed from the followingmatrices

119878 = [

0 1

minus1 0] 119876 = [1 0] (60)

which allows us to generate a great kind of sinusoidal refer-ences Figure 2 shows the simulation results after applying thecontroller

119906 (119905) =

119903

sum

119894=1

ℎ119894(119909 (119905)) 119870

119894[119909 (119905) minus

119903

sum

119894=1

ℎ119894(119908 (119905)) Π

119894119908 (119905)]

+

119903

sum

119894=1

ℎ119894(119908 (119905)) Γ

119894119908 (119905)

(61)

defined in [10] As expected controller (61) is unable toachieve exact regulation although tracking error remainsbounded

Now using the method derived in this work the localmatrices Γ

119894can be readily obtained from (28) resulting in

Γ1= [1036667 0] Γ

2= [343836 0] (62)

Besides knowing that 11990211198881= 1 matrix Π is obtained from

(24)

Π = [

1 0

0 1] (63)

while the newmembership functions computed from (39) are

1205831(119908 (119905)) =

(ℎ1(119908 (119905))) (minus01765)

ℎ1(119908 (119905)) (minus01765) + ℎ

2(119908 (119905)) (minus01500)

1205832(119908 (119905)) =

(ℎ2(119908 (119905))) (minus01500)

ℎ1(119908 (119905)) (minus01765) + ℎ

2(119908 (119905)) (minus01500)

(64)

On the other hand the fuzzy stabilizer for this system isconstructed bymeans of the PDC approach developed in [14]which is formed by

1198701= [1234898 69979] 119870

2= [586405 77962]

(65)

Figure 3 shows simulation with the controller definedin (20) Observe how the new membership functions arecapable of combining the local regulators in the proper wayto accomplish asymptotic tracking

To verify the effectiveness of the latter regulator thefuzzy controller is applied on the original system (57) Thesimulation results appear in Figure 4 Notice that the trackingerror depicted in Figure 4(b) is due to the approximationprovided by the T-S fuzzy model Such an error can bereduced by considering a better approximation or an exactrepresentation of the nonlinear system as the one given bythe nonlinear sector approach [14]

52 Multiple Inputs and Multiple Outputs (MIMO) CaseNow consider a T-S fuzzy system of two rules defined by thefollowing matrices

1198601=

[[[[[

[

0 0 1 0

0 0 0 1

6 4 2 4

2 3 1 5

]]]]]

]

1198611=

[[[[[

[

0 0

0 0

minus7 0

0 2

]]]]]

]

1198602=

[[[[[

[

0 0 1 0

0 0 0 1

3 4 2 1

2 4 6 2

]]]]]

]

1198612=

[[[[[

[

0 0

0 0

minus3 0

0 5

]]]]]

]

1198621= 1198622= [

1 0 0 0

0 1 0 0]

(66)

Mathematical Problems in Engineering 11

0 5 10 15 20 25 30 35 40minus40minus30minus20minus10

010203040

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30 35 40minus04minus03minus02minus01

0010203

Tracking error

Time

Am

plitu

de

e

(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 40minus40minus30minus20minus10

0102030

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30 35 40minus06minus05minus04minus03minus02minus01

0010203

Tracking error

Time

Am

plitu

de

e

(b)

Figure 3 (a) Output reference and (b) tracking errorWhen the fuzzy controller is designed on local regulators and the adequatemembershipfunctions for the fuzzy regulator are computed

0 5 10 15 20 25 30 35 40minus40minus30minus20minus10

010203040

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30 35 40minus06minus05minus04minus03minus02minus01

001

Tracking error

Time

Am

plitu

de

e

(b)

Figure 4 (a) Output reference and (b) tracking errorWhen the fuzzy controller is designed on local regulators and the adequatemembershipfunctions for the fuzzy regulator are computed

12 Mathematical Problems in Engineering

h1 h2

x1minus60

1

80

0

0

Figure 5 Membership functions of two-rule model

and an exosystem defined by

119878 =

[[[[[

[

0 0 1 0

0 0 0 1

minus4 0 0 0

0 minus16 0 0

]]]]]

]

119876 = [

1 0 0 0

0 1 0 0]

(67)

Membership functions forRules 1 and 2 are shown in Figure 5Matrices defining the local regulators can be easily obtainedby using (49) For this example they are

Γ1= [

14286 05714 02857 05714

minus10000 minus95000 minus05000 minus25000]

Γ2= [

23333 13333 06667 03333

minus04000 minus40000 minus12000 minus04000]

(68)

On the other hand 11990211198881= 1205871= 1 119902

21198882= 1205872= 1

11990211198881= 1205873= 1 119902

21198882= 1205874= 1 and by (46) and matrix Π is

Π =

[[[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

]]]]]

]

(69)

Figures 6 and 7 show the results obtained using linear regula-tors defined in (61)

Now by computing the new membership functions forthe fuzzy regulator through (54) one gets1205831(119908 (119905))

=

[[[

[

(ℎ1(119908 (119905))) (minus7)

ℎ1(119908 (119905)) (3) + ℎ

2(119908 (119905)) (3)

0

0(ℎ1(119908 (119905))) (2)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (5)

]]]

]

1205832(119908 (119905))

=

[[[

[

(ℎ2(119908 (119905))) (minus3)

ℎ1(119908 (119905)) (3) + ℎ

2(119908 (119905)) (3)

0

0(ℎ2(119908 (119905))) (5)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (5)

]]]

]

(70)

As previously the fuzzy stabilizer for this example is alsoconstructed bymeans of the PDC approach developed in [14]resulting in

1198701= [

10286 05714 06000 05714

minus10000 minus22500 minus05000 minus37500]

1198702= [

14000 13333 14000 03333

minus04000 minus11000 minus12000 minus09000]

(71)

Thus Figures 8 and 9 show the simulation results whenthe controller defined by (20) is applied

Besides Figure 10 shows the shape of the new member-ship functions which are clearly different from the originalones given in Figure 5

53 More General Problems Consider the T-S fuzzy MIMOsystem with multiple inputs defined by the following matri-ces

1198601=

[[[[[[[[[[[

[

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

minus6 4 2 4 5 3

2 3 minus1 5 minus2 4

]]]]]]]]]]]

]

1198611=

[[[[[[[[[[[

[

0 0

0 0

0 0

0 0

5 0

0 2

]]]]]]]]]]]

]

1198602=

[[[[[[[[[[[

[

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

minus6 1 2 minus5 4 6

0 2 minus7 4 5 2

]]]]]]]]]]]

]

1198612=

[[[[[[[[[[[

[

0 0

0 0

0 0

0 0

7 0

0 8

]]]]]]]]]]]

]

1198621= 1198622= [

0 1 1 0 0 0

1 0 1 0 0 0]

(72)

Mathematical Problems in Engineering 13

0 5 10 15 20 25 30minus15minus10

minus505

101520

Time

Am

plitu

dex1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus4minus2

02468

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 6 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus30minus20minus10

0102030

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus15

minus10

minus5

0

5

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 7 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of lineal regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus15minus10

minus505

101520

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus1

012345678

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 8 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

The reference signals will be generated by an exosystemdefined by

119878 = [

0 1

minus1 0] 119876 = [

1 0

0 1] (73)

Membership functions for Rules 1 and 2 are ℎ1(119909(119905)) =

(80 minus 1199091(119905))140 and ℎ

2(119909(119905)) = 1 minus ℎ

1(119905) Notice that for this

case 119899 = 119904 and 119862119894

= 119876 This implies that Π119894and Γ

119894must

be obtained by solving Francis Equations for each subsystemHowever due to form of the fuzzy plant it is possible to verifythat Π

1= Π2

14 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus40minus30minus20minus10

0102030

Time

Am

plitu

dex2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus10

minus8minus6minus4minus2

0

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 9 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(11) (x1(t))1205832(11) (x1(t))

(a)

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(22) (x1(t))1205832(22) (x1(t))

(b)

Figure 10 (a) Membership functions corresponding to position (11) of 1205831(119908(119905)) and 120583

2(119908(119905)) (b) Membership functions corresponding to

position (22) 1205831(119908(119905)) and 120583

2(119908(119905))

So by applying (9) for each local subsystem with 119894 = 1 2it results in

Π1= Π2=

[[[[[[[[[[[

[

05000 05000

15000 minus05000

minus05000 05000

05000 15000

minus05000 minus05000

minus15000 05000

]]]]]]]]]]]

]

Γ1= [

07000 minus03000

minus20000 minus55000]

Γ2= [

23571 12143

minus04375 minus01875]

(74)

Again the fuzzy stabilizer for this system is constructedby means of the PDC approach developed in [14] yielding

1198701=[[

[

minus12229 00609 minus40284 01291 minus25666 minus04061

16919 minus83997 32934 minus127153 15732 minus61836

]]

]

1198702=[[

[

minus08735 04721 minus28775 13780 minus16904 minus07187

06730 minus19749 15734 minus30538 minus04817 minus12959

]]

]

(75)

The behavior obtained by applying controller (61) isdepicted in Figures 11 and 12 Notice the exact outputregulation is not achieved

Mathematical Problems in Engineering 15

0 5 10 15 20 25 30minus60minus40minus20

0204060

Time

Am

plitu

dex1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus80minus70minus60minus50minus40minus30minus20minus10

01020

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 11 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus15

minus10

minus5

0

5

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 12 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

Now the adequate membership functions are computedfrom (54) resulting in

1205831(119908 (119905))

=

[[[

[

(ℎ1(119908 (119905))) (5)

ℎ1(119908 (119905)) (5) + ℎ

2(119908 (119905)) (7)

0

0(ℎ1(119908 (119905))) (2)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (8)

]]]

]

1205832(119908 (119905))

=

[[[

[

(ℎ2(119908 (119905))) (7)

ℎ1(119908 (119905)) (5) + ℎ

2(119908 (119905)) (7)

0

0(ℎ2(119908 (119905))) (8)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (8)

]]]

]

(76)

When ℎ119894(119908(119905)) is replaced by its corresponding expressions

one gets

1205831(119908 (119905)) =

[[[

[

1225

1199081(119905) + 140

minus5

20

0 minus1199081(119905) minus 80

31199081(119905) + 320

]]]

]

1205832(119908 (119905)) =

[[[

[

7

2minus

1225

1199081(119905) + 410

0

041199081(119905) + 240

31199081(119905) + 320

]]]

]

(77)

The results of applying control law (20) on the fuzzy plantare given in Figures 13 and 14

The ldquocomputedrdquo membership functions for the fuzzyregulator are given in Figure 15 Again it can be readilyobserved that these new membership functions are differentfrom the original ones

Notice that although this example does not fulfill theconditions ofTheorem 5 themembership functions obtainedfrom (39) allow the exact tracking for the fuzzy problemThis

16 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus80minus70minus60minus50minus40minus30minus20minus10

010

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 13 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus8minus6minus4minus2

02468

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 14 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(11) (x1(t))1205832(11) (x1(t))

(a)

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(22) (x1(t))1205832(22) (x1(t))

(b)

Figure 15 (a) Membership functions corresponding to position (11) of 1205831(119908(119905)) and 120583

2(119908(119905)) (b) Membership functions corresponding to

position (22) 1205831(119908(119905)) 120583

2(119908(119905))

Mathematical Problems in Engineering 17

suggests that the approach presented in this work may beapplied on a bigger class of T-S models however this studyis not completed yet

6 Conclusions

A fuzzy regulator for continuous-time systems based on thecombination of linear regulators adjusted by different mem-bership functions has been presented The main advantageis that analytical expressions for the membership functionswhich allow the proper combination of the local regulators inorder to guarantee the exact regulation can be easily appliedAs 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 modelscan be commonly found in mechanics electromechanicselectronics robotics and so forth

Some numerical examples are used to illustrate theapplicability of the proposed approach even in cases wherethe dimensions of the plant and the exosystem are differentbetween each otherThis suggests that the approach presentedin this work may be applied on a bigger class of T-S models

Finally the method proposed in this work avoids thedisadvantage of constructing an exact fuzzy regulator basedon overall T-S fuzzy system whose expression may resultvery large Instead the given approach offers 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 Interests

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

Acknowledgments

This work was partially supported by CONACYT (ConsejoNacional de Ciencia y Tecnologıa) through scholarship SNI(Sistema Nacional de Investigadores) and by IPN (InstitutoPolitecnico Nacional) through research Projects 20140659and 20150487 and scholarships EDI (Estımulo al Desempenode los Investigadores) COFAA (Comision de Operacion yFomento de Actividades Academicas) and BEIFI (Beca deEstımulo Institucional de Formacion de Investigadores)

References

[1] B A Francis ldquoThe linear multivariable regulator problemrdquoSIAM Journal on Control and Optimization vol 15 no 3 pp486ndash505 1977

[2] B A Francis andWMWonham ldquoThe internalmodel principleof control theoryrdquo Automatica vol 12 no 5 pp 457ndash465 1976

[3] A Isidori and C I Byrnes ldquoOutput regulation of nonlinearsystemsrdquo IEEE Transactions on Automatic Control vol 35 no2 pp 131ndash140 1990

[4] B Castillo-Toledo S Di Gennaro and F Jurado ldquoTrajectorytracking for a quadrotor via fuzzy regulationrdquo in Proceedings ofthe World Automation Congress (WAC rsquo12) pp 1ndash6 June 2012

[5] S Chen ldquoOutput regulation of nonlinear singularly perturbedsystems based on T-S fuzzy modelrdquo Journal of Control Theoryand Applications vol 3 no 4 pp 399ndash403 2005

[6] R Tapia-Herrera J A Meda-Campana S Alcamp119886119901119900119904 119886ntara-Montes T Hernamp119886119901119900119904 119886ndez-Cortamp119886119901119900119904 119890s and L Salgado-Conrado ldquoTuning of a TS fuzzy output regulator using thesteepest descent approach and ANFISrdquoMathematical Problemsin Engineering vol 2013 Article ID 873430 14 pages 2013

[7] K-Y Lian and J-J Liou ldquoOutput tracking control for fuzzysystems via output feedback designrdquo IEEE Transactions onFuzzy Systems vol 14 no 5 pp 628ndash639 2006

[8] X-J Ma and Z-Q Sun ldquoOutput tracking and regulationof nonlinear system based on takagi-sugeno fuzzy modelrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 30 no 1 pp 47ndash59 2000

[9] H J Lee J B Park and Y H Joo ldquoComments on lsquooutputtracking and regulation of nonlinear system based on Takagi-Sugeno fuzzy model1015840rdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 33 no 3 pp 521ndash523 2003

[10] J A Meda-Campana B Castillo-Toledo and G Chen ldquoSyn-chronization of chaotic systems from a fuzzy regulationapproachrdquo Fuzzy Sets and Systems vol 160 no 19 pp 2860ndash2875 2009

[11] J A Meda-Campana J C Gamp119886119901119900119904 119900mez-Mancilla and BCastillo-Toledo ldquoExact output regulation for nonlinear systemsdescribed by Takagi-Sugeno fuzzy modelsrdquo IEEE Transactionson Fuzzy Systems vol 20 no 2 pp 235ndash247 2012

[12] C Byrnes F Priscoli and A Isidori Output Regulation ofUncertain Nonlinear Systems Birkhauser 1997

[13] A Isidori Nonlinear Control Systems Communications andControl Engineering Series Springer Berlin Germany 3rdedition 1995

[14] K Tanaka and H O Wang Fuzzy Control Systems Designand Analysis A Linear Matrix Inequality Approach John Wileyamp119886119898119901 Sons New York NY USA 2001

[15] L-X Wang A Course in Fuzzy Systems and Control PrenticeHall Upper Saddle River NJ USA 1997

[16] I Abdelmalek N Golamp119886119901119900119904 119890a and M L Hadjili ldquoA newfuzzy Lyapunov approach to non-quadratic stabilization ofTakagi-Sugeno fuzzy modelsrdquo International Journal of AppliedMathematics andComputer Science vol 17 no 1 pp 39ndash51 2007

[17] J A Meda-Campana and B Castillo-Toledo ldquoOn the outputregulation for TS fuzzymodels using slidingmodesrdquo inProceed-ings of the American Control Conference (ACC rsquo05) pp 4062ndash4067 Portland Ore USA June 2005

[18] B Castillo-Toledo and J A Meda-Campana ldquoThe fuzzydiscrete-time robust regulation problem a LMI approachrdquo inProceedings of the 41st IEEE Conference on Decision and Controlvol 2 pp 2159ndash2164 IEEE Las Vegas Nev USA December2002

[19] B Castillo-Toledo J A Meda-Campana and A Titli ldquoAfuzzy output regulator for Takagi-Sugeno fuzzy modelsrdquo inProceedings of the IEEE International Symposium on IntelligentControl pp 310ndash315 Houston Tex USA October 2003

[20] B-C Ding H-X Sun and Y-E Qiao ldquoStability analysis of T-Sfuzzy control systems based on parameter-dependent Lyapunovfunctionrdquo Acta Automatica Sinica vol 31 no 4 pp 651ndash6542005

[21] M Narimani H K Lam R Dilmaghani and C Wolfe ldquoLMI-based stability analysis of fuzzy-model-based control systemsusing approximated polynomial membership functionsrdquo IEEE

18 Mathematical Problems in Engineering

Transactions on Systems Man and Cybernetics Part B Cyber-netics vol 41 no 3 pp 713ndash724 2011

[22] H K Lam ldquoPolynomial fuzzy-model-based control systemsstability analysis via piecewise-linear membership functionsrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 588ndash5932011

[23] K Tanaka T Hori and H O Wang ldquoA fuzzy Lyapunovapproach to fuzzy control system designrdquo in Proceedings of theAmerican Control Conference vol 6 pp 4790ndash4795 ArlingtonVa USA June 2001

[24] K Tanaka T Hori and H O Wang ldquoA multiple Lyapunovfunction approach to stabilization of fuzzy control systemsrdquoIEEE Transactions on Fuzzy Systems vol 11 no 4 pp 582ndash5892003

[25] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 174ndash1972005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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

6 Mathematical Problems in Engineering

Notice that (39) fulfills conditions119903

sum

119894=1

120583119894(119908 (119905)) = 1

120583119894(119908 (119905)) ge 0

(40)

for all 119894 = 1 2 119903

Remark 2 It can be observed that (39) is always valid whenthe entries of input matrix 119887

119894

1have the same sign if this

condition is not fulfilled the denominators of the new fuzzymembership functions will present singularities Neverthe-less these singularities may appear outside the operationregion allowing more flexibility but such cases require aparticular study of the system which is beyond the scope ofthis work

The following theorem provides the conditions for theexistence of the exact output fuzzy regulator for a class of T-Sfuzzy models

Theorem 3 The exact fuzzy output regulation with fullinformation for T-S fuzzy systems in the form of (11) definedby matrices (19) is solvable if (a) the sign of entries forcorresponding position inside input matrices 119861

119894for 119894 =

1 119903 is the same (b) there exists a fuzzy stabilizer 119906(119905) =sum119903

119894=1ℎ119894(119911(119905))119870

119894119909 for the T-S fuzzy system [16] and (c) the

exosystem (119905) = 119878119908(119905) is Poisson stable Moreover the ExactOutput Fuzzy Regulation Problem is solvable by the controller

119906 (119905) =

119903

sum

119894=1

ℎ119894(119909 (119905)) 119896

119894119909 (119905) minus Π119908 (119905)

+

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905)

(41)

where 120583119894(119908(119905)) can be readily obtained from (39)

Proof From the previous analysis the existence of mappings120587(119908(119905)) = Π119908(119905) and 120574(119908(119905)) = sum

119903

119894=1120583119894(119908(119905))Γ

119894119908(119905) is

guaranteed when the local subsystems are defined by (19)that is the form of 120587(119908(119905)) corresponds to a diagonal matrixand will be the same for all subsystems while 120574(119908(119905)) can bedirectly substitute

On the other hand condition (a) avoids singularities inthe new membership functions 120583

119894(119908(119905)) while the inclusion

of condition (b) has been thoroughly discussed in [12 13 17ndash19] and it implies that the steady-state manifold can be madeasymptotically stable by the action of the fuzzy stabilizer

Finally condition (c) is introduced to avoid that thereference signal converges to zero because if the referencetends to zero then the regulation problem turns into astabilization one which can be solved by means of thefuzzy stabilizer In addition as mentioned before this kindof matrix 119878 can be used to generate a great number ofreference signals The 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 20ndash25]

4 Exact Output Fuzzy Regulation Problem fora Class of T-S Fuzzy Systems with MultipleInputs and Multiple Outputs (MIMO Case)

Now the problem will be extended to T-S fuzzy MIMOmodels defined by matrices

119860119894= [

119874 119868

119863119894

] 119861119894= [

119874

119865119894

]

119878 = [119874 119868

119866] 119862

119894= [ 119867119862 119874

119862 ]

119876119894= [ 119867119876 119874

119876 ]

(42)

for 119894 = 1 119903 with

119868 =

[[[[[[

[

1 0 sdot sdot sdot 0

0 1 sdot sdot sdot 0

d

0 0 sdot sdot sdot 1

]]]]]]

]

119874 =

[[[[[[

[

0 0 sdot sdot sdot 0

0 0 sdot sdot sdot 0

d

0 0 sdot sdot sdot 0

]]]]]]

]

119863119894=

[[[[[[[

[

119886119894

(119896+1)1119886119894

(119896+1)2119886119894

(119896+1)3sdot sdot sdot 119886119894

(119896+1)119899

119886119894

(119896+2)1119886119894

(119896+2)2119886119894

(119896+2)3sdot sdot sdot 119886119894

(119896+2)119899

d

119886119894

1198991119886119894

1198992119886119894

1198993sdot sdot sdot 119886

119894

119899119899

]]]]]]]

]

119865119894=

[[[[[[[

[

119887119894

119896+10 sdot sdot sdot 0

0 119887119894

119896+2sdot sdot sdot 0

d

0 0 sdot sdot sdot 119887119894

119899

]]]]]]]

]

119866 =

[[[[[[

[

119904(119896+1)1

119904(119896+1)2

119904(119896+1)3

sdot sdot sdot 119904(119896+1)119899

119904(119896+2)1

119904(119896+2)2

119904(119896+2)3

sdot sdot sdot 119904(119896+2)119899

d

1199041198991

1199041198992

1199041198993

sdot sdot sdot 119904119899119899

]]]]]]

]

119874119862= 119874119876=

[[[[[[

[

0 0 sdot sdot sdot 0

0 0 sdot sdot sdot 0

d

0 0 sdot sdot sdot 0

]]]]]]

]

Mathematical Problems in Engineering 7

119867119862=

[[[[[[

[

11988810 sdot sdot sdot 0

0 1198882sdot sdot sdot 0

d

0 0 sdot sdot sdot 119888119898

]]]]]]

]

119867119876=

[[[[[[

[

1199021

0 sdot sdot sdot 0

0 1199022sdot sdot sdot 0

d

0 0 sdot sdot sdot 119902119898

]]]]]]

]

(43)

where 119896 = 119899 minus 119898 119865119894isin R119898times119898 119874 isin R(119899minus119898)times119898 119874

119862isin R119898times119898

119868 isin R(119899minus119898)times(119899minus119898) 119863119894isin R119898times119899 119867

119862isin R119898times119898 119867

119876isin R119898times119898

and 119866 isin R119898times119899 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= 119908119898+1

2= 119908119898+2

119896+1

= 119904(119896+1)1

1199081+ 119904(119896+1)2

1199082+ sdot sdot sdot + 119904

(119896+1)119899119908119899

=

119899

sum

119895=1

119904(119896+1)119895

119908119895

119896+2

= 119904(119896+2)1

1199081+ 119904(119896+2)2

1199082+ sdot sdot sdot + 119904

(119896+2)119899119908119899

=

119899

sum

119895=1

119904(119896+2)119895

119908119895

119899= 11990411989911199081+ 11990411989921199082+ sdot sdot sdot + 119904

119899119899119908119899=

119899

sum

119895=1

119904119899119895119908119895

(44)

Moreover each subsystem can be described by

1= 119909119898+1

2= 119909119898+2

119896+1

= 119886119894

(119896+1)11199091+ 119886119894

(119896+1)21199092+ sdot sdot sdot + 119886

(119896+1)119899119909119899

+ 119887119894

119896+1119906119894

119904119904

=

119899

sum

119895=1

119886119894

(119896+1)119895119909119895+ 119887119894

119896+1119906119894

119904119904

119896+2

= 119886119894

(119896+2)11199091+ 119886119894

(119896+2)21199092+ sdot sdot sdot + 119886

(119896+2)119899119909119899

+ 119887119894

119896+2119906119894

119904119904

=

119899

sum

119895=1

119886119894

(119896+2)119895119908119895+ 119887119894

1+2119906119894

119904119904

119899= 11990411989911199081+ 11990411989921199082+ sdot sdot sdot + 119904

119899119899119908119899+ 119887119894

119899119906119894

119904119904

=

119899

sum

119895=1

119904119899119895119908119895+ 119887119894

119899119906119894

119904119904

(45)

As before using expressions (44) and (45) considering 11988811199091=

11990211199081 11988821199092

= 11990221199082 119888

119898119909119898

= 119902119898119908119898 and performing

successive substitutions of ℓ= 120587ℓℓwith ℓ = 1 119898 the

following results are obtained

Π =

[[[[[[

[

1205871

0 sdot sdot sdot 0

0 1205872sdot sdot sdot 0

d

0 0 sdot sdot sdot 120587119899

]]]]]]

]

(46)

where Π isin R119899times119899 and 119894 = 1 2 119903 also 1205871

=

11990211198881 1205872= 11990221198882 120587

119898= 119902119898119888119898 120587119898+1

= 11990211198881 120587119898+2

=

11990221198882 120587

119898+119898= 119902119898119888119898 and so forth Notice that the values

of 120587lowastrepeat themselves every 119898 entry up to 119899 Additionally

Π is constant and common for all subsystems while the localsteady-state inputs are defined by

119906119894

119904119904=

[[[[[[[[[[[[[

[

1205871119904(119896+1)1

1199081minus 1205871119886119894

(119896+1)11199081

1198871

119896+1

+

1205871119904(119896+1)1

1199082minus 1205872119886119894

(119896+1)11199082

1198871

119896+1

+ sdot sdot sdot +

1205871119904(119896+1)119899

119908119899minus 120587119899119886119894

(119896+1)119899119908119899

1198871

119896+1

1205872119904(119896+2)1

1199081minus 1205871119886119894

(119896+2)11199081

1198871

119896+2

+

1205872119904(119896+1)2

1199082minus 1205872119886119894

(119896+2)21199082

1198871

119896+2

+ sdot sdot sdot +

1205872119904(119896+2)119899

119908119899minus 120587119899119886119894

(119896+2)119899119908119899

1198871

119896+2

12058711989911990411989911199081minus 1205871119886119894

11989911199081

1198871119899

+12058711989911990411989921199082minus 1205872119886119894

11989921199082

1198871119899

+ sdot sdot sdot +120587119899119904119899119899119908119899minus 120587119899119886119894

119899119899119908119899

1198871119899

]]]]]]]]]]]]]

]

(47)

8 Mathematical Problems in Engineering

for 119894 = 1 119903 In compact form (47) turns into

119906119894

119904119904=

[[[[[[[[[[[[[[

[

119899

sum

119895=1

1205871119904(119896+1)119895

119908119895minus 120587119895119886(119896+1)119895

119908119895

119887119894

119896+1

119899

sum

119895=1

1205872119904(119896+2)119895

119908119895minus 120587119895119886(119896+2)119895

119908119895

119887119894

119896+2

119899

sum

119895=1

120587119899119904119899119895119908119895minus 120587119895119886119899119895119908119895

119887119894119899

]]]]]]]]]]]]]]

]

(48)

and from (27) the local matrices Γ119894have the form

Γ119894=

[[[[[[[[[[[[[

[

1205871119904(119896+1)1

minus 1205871119886119894

(119896+1)1

119887119894

119896+1

1205871119904(119896+1)2

minus 1205872119886119894

(119896+1)2

119887119894

119896+1

sdot sdot sdot

12058711199041198991

minus 120587119899119886119894

(119896+1)119899

119887119894

119896+1

1205872119904(119896+2)1

minus 1205871119886119894

(119896+2)1

119887119894

119896+2

1205872119904(119896+2)2

minus 1205872119886119894

(119896+2)2

119887119894

119896+2

sdot sdot sdot

12058721199041198991

minus 120587119899119886119894

(119896+2)119899

119887119894

119896+2

d

1205871198991199041198991

minus 1205871119886119894

1198991

119887119894119899

1205871198991199041198992

minus 1205872119886119894

1198992

119887119894119899

sdot sdot sdot1205871198991199041198991

minus 120587119899119886119894

119899119899

119887119894119899

]]]]]]]]]]]]]

]

(49)

Now again from the fuzzy plant given in (42) and theglobal fuzzy model (29) the expressions for

119896+1

119899are

119896+1

=

119903

sum

119894=1

ℎ119894(119909 (119905))(

119899

sum

119895=1

119886119894

(119896+1)119895119909119895)

+ (

119903

sum

119894=1

ℎ119894(119909 (119905)) 119887

119894

119896+1)119906119904119904

119896+2

=

119903

sum

119894=1

ℎ119894(119909 (119905))(

119899

sum

119895=1

119886119894

(119896+2)119895119909119895)

+ (

119903

sum

119894=1

ℎ119894(119909 (119905)) 119887

119894

119896+2)119906119904119904

119899=

119903

sum

119894=1

ℎ119894(119909 (119905))(

119899

sum

119895=1

119886119894

119899119895119909119895)

+ (

119903

sum

119894=1

ℎ119894(119909 (119905)) 119887

119894

119899)119906119904119904

(50)

So by applying successive substitutions as before with (44)(45) and knowing that in steady-state 119909

119895= 120587119895119908119895 the

following expressions are obtained

119903

sum

119894=1

ℎ119894(119908 (119905))(

119899

sum

119895=1

119886119894

(119896+1)119895120587119895119908119895)

+ (

119903

sum

119894=1

ℎ119894(119908 (119905)) 119887

119894

119896+1)119906119904119904= 1205871

119899

sum

119894=1

119904(119896+1)119894

119908119894

119903

sum

119894=1

ℎ119894(119908 (119905))(

119899

sum

119895=1

119886119894

(119896+2)119895120587119895119908119895)

+ (

119903

sum

119894=1

ℎ119894(119908 (119905)) 119887

119894

119896+2)119906119904119904= 1205872

119899

sum

119894=1

119904(119896+2)

119894119908119894

1199031

sum

119894=1

ℎ119894(119908 (119905))(

119899

sum

119895=1

119886119894

119899119895120587119895119908119895)

+ (

119903

sum

119894=1

ℎ119894(119908 (119905)) 119887

119894

119899)119906119904119904= 120587119899

119899

sum

119894=1

119904119899119894119908119894

(51)

Mathematical Problems in Engineering 9

Therefore the steady-state input for the overall fuzzy systemis

119906119904119904

=

[[[[[[[[[[[[[[

[

1205871sum119899

119894=1119904(119896+1)119894

119908119894minus sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

(119896+1)119895120587119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

119896+1

1205872sum119899

119894=1119904(119896+2)119894

119908119894minus sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

(119896+2)119895120587119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

119896+2

120587119899sum119899

119894=1119904119899119895119908119894minus sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

119899119895120587119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

119899

]]]]]]]]]]]]]]

]

(52)

But the desired form of the steady-state is

119906119904119904(119905) = 120574 (119908 (119905)) =

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905) (53)

Consequently the new membership functions 120583119894(119908(119905))

can be obtained after equating (52) and (53) resulting in

120583119894(119908 (119905)) =

[[[[[[

[

12058311989411

0 sdot sdot sdot 0

0 12058311989422

sdot sdot sdot 0

d

0 0 sdot sdot sdot 120583119894119898119898

]]]]]]

]

(54)

where

120583119894119895119897

=

119887119894

119896+119895ℎ119894(119908 (119905))

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

119896+119895

(55)

for all 119894 = 1 119903 119895 = 119897 = 1 119898 It can be readily observedthat the sum of the corresponding elements is equal to oneand 120583

119894119895119897(119908(119905)) ge 0

These new membership functions 120583119894(119908(119905)) are organized

in a matrix form for that reason sum119903119894=1

120583119894(119908(119905)) = 119868 (where 119868

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 119861

119894have the

same sign at corresponding positions Clearly this conditionavoids singularities in (55) As before ℎ

119894(119908(119905)) is directly

obtained from ℎ119894(119909(119905)) by considering that in steady-state

119909(119905) = Π119908(119905) At this point the following theorem can bedefined

Theorem 5 The exact fuzzy output regulation with fullinformation for T-S fuzzy MIMO systems in the form of(11) defined by matrices (42) is solvable if (a) the sign ofentries for corresponding position inside input matrices 119861

119894for

119894 = 1 119903 is the same (b) there exists a fuzzy stabilizer119906(119905) = sum

119903

119894=1ℎ119894(119911(119905))119870

119894119909 for the fuzzy system [16] and (c) the

exosystem (119905) = 119878119908(119905) is Poisson stable

Moreover the Exact Output Fuzzy Regulation Problem forMIMO systems is solvable by the controller

119906 (119905) =

119903

sum

119894=1

ℎ119894(119909 (119905)) 119896

119894119909 (119905) minus Π119908 (119905)

+

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905)

(56)

where 120583119894(119908(119905)) can be readily obtained from (54) and (55)

Proof It follows directly from Theorem 3 and previousanalysis

Remark 6 Notice that 120583119894(119908(119905)) is defined as a matrix and

this is not a typical membership function however 120583119894(119908(119905))

contains a set of membership functions related to each entryof 119861119894 This form allows us to design controller (20)

5 Numerical Examples

51 Simple Input Case Consider the problem of balancingand swing of an inverted pendulum on a cart The motionequations for this system are [14]

1(119905) = 119909

2(119905) (57)

2(119905)

=119892 sin(119909

1(119905))minus 119886119898119897119909

2

2sin (2119909

1(119905))2 minus119886 cos (119909

1(119905)) 119906 (119905)

41198973 minus 119886119898119897 cos2 (1199091(119905))

(58)

where 1199091(119905) denotes the angle (in radians) of the pendulum

from the vertical and 1199092(119905) is the angular velocity 119892 =

98ms2 is the gravity constant 119898 is the mass of the pen-dulum 119872 is the mass of the cart 2119897 is the length of thependulum and 119906 is the force applied to the cart (in newtons)119886 = 1(119898 +119872) with119898 = 20 kg119872 = 80 kg and 2119897 = 10min the simulations The goal is to track the reference definedby the exosystem within the range 119909

1isin (minus1205872 1205872) To this

end a fuzzy model representing the nonlinear dynamics is asfollows

Rule 1

IF 1199091(119905) is about 0

THEN (119905) = 1198601119909(119905) + 119861

1119906(119905)

Rule 2

IF 1199091(119905) is about plusmn1205872(|119909

1| lt 1205872)

THEN (119905) = 1198602119909(119905) + 119861

2119906(119905)

10 Mathematical Problems in Engineering

h1

h2

x1minus90 90

(deg)

1

00

Figure 1 Membership functions of two-rule model

Membership functions for Rules 1 and 2 are shown in Figure 1In this example the T-S fuzzymodel of the plant is defined bymatrices

1198601= [

0 1

172941 0] 119861

1= [

0

minus01765]

1198602= [

0 1

41585 0] 119861

2= [

0

minus01500]

1198621= 1198622= [1 0]

(59)

while the exosystem can be constructed from the followingmatrices

119878 = [

0 1

minus1 0] 119876 = [1 0] (60)

which allows us to generate a great kind of sinusoidal refer-ences Figure 2 shows the simulation results after applying thecontroller

119906 (119905) =

119903

sum

119894=1

ℎ119894(119909 (119905)) 119870

119894[119909 (119905) minus

119903

sum

119894=1

ℎ119894(119908 (119905)) Π

119894119908 (119905)]

+

119903

sum

119894=1

ℎ119894(119908 (119905)) Γ

119894119908 (119905)

(61)

defined in [10] As expected controller (61) is unable toachieve exact regulation although tracking error remainsbounded

Now using the method derived in this work the localmatrices Γ

119894can be readily obtained from (28) resulting in

Γ1= [1036667 0] Γ

2= [343836 0] (62)

Besides knowing that 11990211198881= 1 matrix Π is obtained from

(24)

Π = [

1 0

0 1] (63)

while the newmembership functions computed from (39) are

1205831(119908 (119905)) =

(ℎ1(119908 (119905))) (minus01765)

ℎ1(119908 (119905)) (minus01765) + ℎ

2(119908 (119905)) (minus01500)

1205832(119908 (119905)) =

(ℎ2(119908 (119905))) (minus01500)

ℎ1(119908 (119905)) (minus01765) + ℎ

2(119908 (119905)) (minus01500)

(64)

On the other hand the fuzzy stabilizer for this system isconstructed bymeans of the PDC approach developed in [14]which is formed by

1198701= [1234898 69979] 119870

2= [586405 77962]

(65)

Figure 3 shows simulation with the controller definedin (20) Observe how the new membership functions arecapable of combining the local regulators in the proper wayto accomplish asymptotic tracking

To verify the effectiveness of the latter regulator thefuzzy controller is applied on the original system (57) Thesimulation results appear in Figure 4 Notice that the trackingerror depicted in Figure 4(b) is due to the approximationprovided by the T-S fuzzy model Such an error can bereduced by considering a better approximation or an exactrepresentation of the nonlinear system as the one given bythe nonlinear sector approach [14]

52 Multiple Inputs and Multiple Outputs (MIMO) CaseNow consider a T-S fuzzy system of two rules defined by thefollowing matrices

1198601=

[[[[[

[

0 0 1 0

0 0 0 1

6 4 2 4

2 3 1 5

]]]]]

]

1198611=

[[[[[

[

0 0

0 0

minus7 0

0 2

]]]]]

]

1198602=

[[[[[

[

0 0 1 0

0 0 0 1

3 4 2 1

2 4 6 2

]]]]]

]

1198612=

[[[[[

[

0 0

0 0

minus3 0

0 5

]]]]]

]

1198621= 1198622= [

1 0 0 0

0 1 0 0]

(66)

Mathematical Problems in Engineering 11

0 5 10 15 20 25 30 35 40minus40minus30minus20minus10

010203040

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30 35 40minus04minus03minus02minus01

0010203

Tracking error

Time

Am

plitu

de

e

(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 40minus40minus30minus20minus10

0102030

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30 35 40minus06minus05minus04minus03minus02minus01

0010203

Tracking error

Time

Am

plitu

de

e

(b)

Figure 3 (a) Output reference and (b) tracking errorWhen the fuzzy controller is designed on local regulators and the adequatemembershipfunctions for the fuzzy regulator are computed

0 5 10 15 20 25 30 35 40minus40minus30minus20minus10

010203040

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30 35 40minus06minus05minus04minus03minus02minus01

001

Tracking error

Time

Am

plitu

de

e

(b)

Figure 4 (a) Output reference and (b) tracking errorWhen the fuzzy controller is designed on local regulators and the adequatemembershipfunctions for the fuzzy regulator are computed

12 Mathematical Problems in Engineering

h1 h2

x1minus60

1

80

0

0

Figure 5 Membership functions of two-rule model

and an exosystem defined by

119878 =

[[[[[

[

0 0 1 0

0 0 0 1

minus4 0 0 0

0 minus16 0 0

]]]]]

]

119876 = [

1 0 0 0

0 1 0 0]

(67)

Membership functions forRules 1 and 2 are shown in Figure 5Matrices defining the local regulators can be easily obtainedby using (49) For this example they are

Γ1= [

14286 05714 02857 05714

minus10000 minus95000 minus05000 minus25000]

Γ2= [

23333 13333 06667 03333

minus04000 minus40000 minus12000 minus04000]

(68)

On the other hand 11990211198881= 1205871= 1 119902

21198882= 1205872= 1

11990211198881= 1205873= 1 119902

21198882= 1205874= 1 and by (46) and matrix Π is

Π =

[[[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

]]]]]

]

(69)

Figures 6 and 7 show the results obtained using linear regula-tors defined in (61)

Now by computing the new membership functions forthe fuzzy regulator through (54) one gets1205831(119908 (119905))

=

[[[

[

(ℎ1(119908 (119905))) (minus7)

ℎ1(119908 (119905)) (3) + ℎ

2(119908 (119905)) (3)

0

0(ℎ1(119908 (119905))) (2)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (5)

]]]

]

1205832(119908 (119905))

=

[[[

[

(ℎ2(119908 (119905))) (minus3)

ℎ1(119908 (119905)) (3) + ℎ

2(119908 (119905)) (3)

0

0(ℎ2(119908 (119905))) (5)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (5)

]]]

]

(70)

As previously the fuzzy stabilizer for this example is alsoconstructed bymeans of the PDC approach developed in [14]resulting in

1198701= [

10286 05714 06000 05714

minus10000 minus22500 minus05000 minus37500]

1198702= [

14000 13333 14000 03333

minus04000 minus11000 minus12000 minus09000]

(71)

Thus Figures 8 and 9 show the simulation results whenthe controller defined by (20) is applied

Besides Figure 10 shows the shape of the new member-ship functions which are clearly different from the originalones given in Figure 5

53 More General Problems Consider the T-S fuzzy MIMOsystem with multiple inputs defined by the following matri-ces

1198601=

[[[[[[[[[[[

[

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

minus6 4 2 4 5 3

2 3 minus1 5 minus2 4

]]]]]]]]]]]

]

1198611=

[[[[[[[[[[[

[

0 0

0 0

0 0

0 0

5 0

0 2

]]]]]]]]]]]

]

1198602=

[[[[[[[[[[[

[

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

minus6 1 2 minus5 4 6

0 2 minus7 4 5 2

]]]]]]]]]]]

]

1198612=

[[[[[[[[[[[

[

0 0

0 0

0 0

0 0

7 0

0 8

]]]]]]]]]]]

]

1198621= 1198622= [

0 1 1 0 0 0

1 0 1 0 0 0]

(72)

Mathematical Problems in Engineering 13

0 5 10 15 20 25 30minus15minus10

minus505

101520

Time

Am

plitu

dex1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus4minus2

02468

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 6 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus30minus20minus10

0102030

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus15

minus10

minus5

0

5

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 7 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of lineal regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus15minus10

minus505

101520

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus1

012345678

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 8 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

The reference signals will be generated by an exosystemdefined by

119878 = [

0 1

minus1 0] 119876 = [

1 0

0 1] (73)

Membership functions for Rules 1 and 2 are ℎ1(119909(119905)) =

(80 minus 1199091(119905))140 and ℎ

2(119909(119905)) = 1 minus ℎ

1(119905) Notice that for this

case 119899 = 119904 and 119862119894

= 119876 This implies that Π119894and Γ

119894must

be obtained by solving Francis Equations for each subsystemHowever due to form of the fuzzy plant it is possible to verifythat Π

1= Π2

14 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus40minus30minus20minus10

0102030

Time

Am

plitu

dex2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus10

minus8minus6minus4minus2

0

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 9 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(11) (x1(t))1205832(11) (x1(t))

(a)

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(22) (x1(t))1205832(22) (x1(t))

(b)

Figure 10 (a) Membership functions corresponding to position (11) of 1205831(119908(119905)) and 120583

2(119908(119905)) (b) Membership functions corresponding to

position (22) 1205831(119908(119905)) and 120583

2(119908(119905))

So by applying (9) for each local subsystem with 119894 = 1 2it results in

Π1= Π2=

[[[[[[[[[[[

[

05000 05000

15000 minus05000

minus05000 05000

05000 15000

minus05000 minus05000

minus15000 05000

]]]]]]]]]]]

]

Γ1= [

07000 minus03000

minus20000 minus55000]

Γ2= [

23571 12143

minus04375 minus01875]

(74)

Again the fuzzy stabilizer for this system is constructedby means of the PDC approach developed in [14] yielding

1198701=[[

[

minus12229 00609 minus40284 01291 minus25666 minus04061

16919 minus83997 32934 minus127153 15732 minus61836

]]

]

1198702=[[

[

minus08735 04721 minus28775 13780 minus16904 minus07187

06730 minus19749 15734 minus30538 minus04817 minus12959

]]

]

(75)

The behavior obtained by applying controller (61) isdepicted in Figures 11 and 12 Notice the exact outputregulation is not achieved

Mathematical Problems in Engineering 15

0 5 10 15 20 25 30minus60minus40minus20

0204060

Time

Am

plitu

dex1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus80minus70minus60minus50minus40minus30minus20minus10

01020

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 11 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus15

minus10

minus5

0

5

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 12 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

Now the adequate membership functions are computedfrom (54) resulting in

1205831(119908 (119905))

=

[[[

[

(ℎ1(119908 (119905))) (5)

ℎ1(119908 (119905)) (5) + ℎ

2(119908 (119905)) (7)

0

0(ℎ1(119908 (119905))) (2)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (8)

]]]

]

1205832(119908 (119905))

=

[[[

[

(ℎ2(119908 (119905))) (7)

ℎ1(119908 (119905)) (5) + ℎ

2(119908 (119905)) (7)

0

0(ℎ2(119908 (119905))) (8)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (8)

]]]

]

(76)

When ℎ119894(119908(119905)) is replaced by its corresponding expressions

one gets

1205831(119908 (119905)) =

[[[

[

1225

1199081(119905) + 140

minus5

20

0 minus1199081(119905) minus 80

31199081(119905) + 320

]]]

]

1205832(119908 (119905)) =

[[[

[

7

2minus

1225

1199081(119905) + 410

0

041199081(119905) + 240

31199081(119905) + 320

]]]

]

(77)

The results of applying control law (20) on the fuzzy plantare given in Figures 13 and 14

The ldquocomputedrdquo membership functions for the fuzzyregulator are given in Figure 15 Again it can be readilyobserved that these new membership functions are differentfrom the original ones

Notice that although this example does not fulfill theconditions ofTheorem 5 themembership functions obtainedfrom (39) allow the exact tracking for the fuzzy problemThis

16 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus80minus70minus60minus50minus40minus30minus20minus10

010

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 13 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus8minus6minus4minus2

02468

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 14 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(11) (x1(t))1205832(11) (x1(t))

(a)

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(22) (x1(t))1205832(22) (x1(t))

(b)

Figure 15 (a) Membership functions corresponding to position (11) of 1205831(119908(119905)) and 120583

2(119908(119905)) (b) Membership functions corresponding to

position (22) 1205831(119908(119905)) 120583

2(119908(119905))

Mathematical Problems in Engineering 17

suggests that the approach presented in this work may beapplied on a bigger class of T-S models however this studyis not completed yet

6 Conclusions

A fuzzy regulator for continuous-time systems based on thecombination of linear regulators adjusted by different mem-bership functions has been presented The main advantageis that analytical expressions for the membership functionswhich allow the proper combination of the local regulators inorder to guarantee the exact regulation can be easily appliedAs 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 modelscan be commonly found in mechanics electromechanicselectronics robotics and so forth

Some numerical examples are used to illustrate theapplicability of the proposed approach even in cases wherethe dimensions of the plant and the exosystem are differentbetween each otherThis suggests that the approach presentedin this work may be applied on a bigger class of T-S models

Finally the method proposed in this work avoids thedisadvantage of constructing an exact fuzzy regulator basedon overall T-S fuzzy system whose expression may resultvery large Instead the given approach offers 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 Interests

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

Acknowledgments

This work was partially supported by CONACYT (ConsejoNacional de Ciencia y Tecnologıa) through scholarship SNI(Sistema Nacional de Investigadores) and by IPN (InstitutoPolitecnico Nacional) through research Projects 20140659and 20150487 and scholarships EDI (Estımulo al Desempenode los Investigadores) COFAA (Comision de Operacion yFomento de Actividades Academicas) and BEIFI (Beca deEstımulo Institucional de Formacion de Investigadores)

References

[1] B A Francis ldquoThe linear multivariable regulator problemrdquoSIAM Journal on Control and Optimization vol 15 no 3 pp486ndash505 1977

[2] B A Francis andWMWonham ldquoThe internalmodel principleof control theoryrdquo Automatica vol 12 no 5 pp 457ndash465 1976

[3] A Isidori and C I Byrnes ldquoOutput regulation of nonlinearsystemsrdquo IEEE Transactions on Automatic Control vol 35 no2 pp 131ndash140 1990

[4] B Castillo-Toledo S Di Gennaro and F Jurado ldquoTrajectorytracking for a quadrotor via fuzzy regulationrdquo in Proceedings ofthe World Automation Congress (WAC rsquo12) pp 1ndash6 June 2012

[5] S Chen ldquoOutput regulation of nonlinear singularly perturbedsystems based on T-S fuzzy modelrdquo Journal of Control Theoryand Applications vol 3 no 4 pp 399ndash403 2005

[6] R Tapia-Herrera J A Meda-Campana S Alcamp119886119901119900119904 119886ntara-Montes T Hernamp119886119901119900119904 119886ndez-Cortamp119886119901119900119904 119890s and L Salgado-Conrado ldquoTuning of a TS fuzzy output regulator using thesteepest descent approach and ANFISrdquoMathematical Problemsin Engineering vol 2013 Article ID 873430 14 pages 2013

[7] K-Y Lian and J-J Liou ldquoOutput tracking control for fuzzysystems via output feedback designrdquo IEEE Transactions onFuzzy Systems vol 14 no 5 pp 628ndash639 2006

[8] X-J Ma and Z-Q Sun ldquoOutput tracking and regulationof nonlinear system based on takagi-sugeno fuzzy modelrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 30 no 1 pp 47ndash59 2000

[9] H J Lee J B Park and Y H Joo ldquoComments on lsquooutputtracking and regulation of nonlinear system based on Takagi-Sugeno fuzzy model1015840rdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 33 no 3 pp 521ndash523 2003

[10] J A Meda-Campana B Castillo-Toledo and G Chen ldquoSyn-chronization of chaotic systems from a fuzzy regulationapproachrdquo Fuzzy Sets and Systems vol 160 no 19 pp 2860ndash2875 2009

[11] J A Meda-Campana J C Gamp119886119901119900119904 119900mez-Mancilla and BCastillo-Toledo ldquoExact output regulation for nonlinear systemsdescribed by Takagi-Sugeno fuzzy modelsrdquo IEEE Transactionson Fuzzy Systems vol 20 no 2 pp 235ndash247 2012

[12] C Byrnes F Priscoli and A Isidori Output Regulation ofUncertain Nonlinear Systems Birkhauser 1997

[13] A Isidori Nonlinear Control Systems Communications andControl Engineering Series Springer Berlin Germany 3rdedition 1995

[14] K Tanaka and H O Wang Fuzzy Control Systems Designand Analysis A Linear Matrix Inequality Approach John Wileyamp119886119898119901 Sons New York NY USA 2001

[15] L-X Wang A Course in Fuzzy Systems and Control PrenticeHall Upper Saddle River NJ USA 1997

[16] I Abdelmalek N Golamp119886119901119900119904 119890a and M L Hadjili ldquoA newfuzzy Lyapunov approach to non-quadratic stabilization ofTakagi-Sugeno fuzzy modelsrdquo International Journal of AppliedMathematics andComputer Science vol 17 no 1 pp 39ndash51 2007

[17] J A Meda-Campana and B Castillo-Toledo ldquoOn the outputregulation for TS fuzzymodels using slidingmodesrdquo inProceed-ings of the American Control Conference (ACC rsquo05) pp 4062ndash4067 Portland Ore USA June 2005

[18] B Castillo-Toledo and J A Meda-Campana ldquoThe fuzzydiscrete-time robust regulation problem a LMI approachrdquo inProceedings of the 41st IEEE Conference on Decision and Controlvol 2 pp 2159ndash2164 IEEE Las Vegas Nev USA December2002

[19] B Castillo-Toledo J A Meda-Campana and A Titli ldquoAfuzzy output regulator for Takagi-Sugeno fuzzy modelsrdquo inProceedings of the IEEE International Symposium on IntelligentControl pp 310ndash315 Houston Tex USA October 2003

[20] B-C Ding H-X Sun and Y-E Qiao ldquoStability analysis of T-Sfuzzy control systems based on parameter-dependent Lyapunovfunctionrdquo Acta Automatica Sinica vol 31 no 4 pp 651ndash6542005

[21] M Narimani H K Lam R Dilmaghani and C Wolfe ldquoLMI-based stability analysis of fuzzy-model-based control systemsusing approximated polynomial membership functionsrdquo IEEE

18 Mathematical Problems in Engineering

Transactions on Systems Man and Cybernetics Part B Cyber-netics vol 41 no 3 pp 713ndash724 2011

[22] H K Lam ldquoPolynomial fuzzy-model-based control systemsstability analysis via piecewise-linear membership functionsrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 588ndash5932011

[23] K Tanaka T Hori and H O Wang ldquoA fuzzy Lyapunovapproach to fuzzy control system designrdquo in Proceedings of theAmerican Control Conference vol 6 pp 4790ndash4795 ArlingtonVa USA June 2001

[24] K Tanaka T Hori and H O Wang ldquoA multiple Lyapunovfunction approach to stabilization of fuzzy control systemsrdquoIEEE Transactions on Fuzzy Systems vol 11 no 4 pp 582ndash5892003

[25] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 174ndash1972005

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 7

119867119862=

[[[[[[

[

11988810 sdot sdot sdot 0

0 1198882sdot sdot sdot 0

d

0 0 sdot sdot sdot 119888119898

]]]]]]

]

119867119876=

[[[[[[

[

1199021

0 sdot sdot sdot 0

0 1199022sdot sdot sdot 0

d

0 0 sdot sdot sdot 119902119898

]]]]]]

]

(43)

where 119896 = 119899 minus 119898 119865119894isin R119898times119898 119874 isin R(119899minus119898)times119898 119874

119862isin R119898times119898

119868 isin R(119899minus119898)times(119899minus119898) 119863119894isin R119898times119899 119867

119862isin R119898times119898 119867

119876isin R119898times119898

and 119866 isin R119898times119899 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= 119908119898+1

2= 119908119898+2

119896+1

= 119904(119896+1)1

1199081+ 119904(119896+1)2

1199082+ sdot sdot sdot + 119904

(119896+1)119899119908119899

=

119899

sum

119895=1

119904(119896+1)119895

119908119895

119896+2

= 119904(119896+2)1

1199081+ 119904(119896+2)2

1199082+ sdot sdot sdot + 119904

(119896+2)119899119908119899

=

119899

sum

119895=1

119904(119896+2)119895

119908119895

119899= 11990411989911199081+ 11990411989921199082+ sdot sdot sdot + 119904

119899119899119908119899=

119899

sum

119895=1

119904119899119895119908119895

(44)

Moreover each subsystem can be described by

1= 119909119898+1

2= 119909119898+2

119896+1

= 119886119894

(119896+1)11199091+ 119886119894

(119896+1)21199092+ sdot sdot sdot + 119886

(119896+1)119899119909119899

+ 119887119894

119896+1119906119894

119904119904

=

119899

sum

119895=1

119886119894

(119896+1)119895119909119895+ 119887119894

119896+1119906119894

119904119904

119896+2

= 119886119894

(119896+2)11199091+ 119886119894

(119896+2)21199092+ sdot sdot sdot + 119886

(119896+2)119899119909119899

+ 119887119894

119896+2119906119894

119904119904

=

119899

sum

119895=1

119886119894

(119896+2)119895119908119895+ 119887119894

1+2119906119894

119904119904

119899= 11990411989911199081+ 11990411989921199082+ sdot sdot sdot + 119904

119899119899119908119899+ 119887119894

119899119906119894

119904119904

=

119899

sum

119895=1

119904119899119895119908119895+ 119887119894

119899119906119894

119904119904

(45)

As before using expressions (44) and (45) considering 11988811199091=

11990211199081 11988821199092

= 11990221199082 119888

119898119909119898

= 119902119898119908119898 and performing

successive substitutions of ℓ= 120587ℓℓwith ℓ = 1 119898 the

following results are obtained

Π =

[[[[[[

[

1205871

0 sdot sdot sdot 0

0 1205872sdot sdot sdot 0

d

0 0 sdot sdot sdot 120587119899

]]]]]]

]

(46)

where Π isin R119899times119899 and 119894 = 1 2 119903 also 1205871

=

11990211198881 1205872= 11990221198882 120587

119898= 119902119898119888119898 120587119898+1

= 11990211198881 120587119898+2

=

11990221198882 120587

119898+119898= 119902119898119888119898 and so forth Notice that the values

of 120587lowastrepeat themselves every 119898 entry up to 119899 Additionally

Π is constant and common for all subsystems while the localsteady-state inputs are defined by

119906119894

119904119904=

[[[[[[[[[[[[[

[

1205871119904(119896+1)1

1199081minus 1205871119886119894

(119896+1)11199081

1198871

119896+1

+

1205871119904(119896+1)1

1199082minus 1205872119886119894

(119896+1)11199082

1198871

119896+1

+ sdot sdot sdot +

1205871119904(119896+1)119899

119908119899minus 120587119899119886119894

(119896+1)119899119908119899

1198871

119896+1

1205872119904(119896+2)1

1199081minus 1205871119886119894

(119896+2)11199081

1198871

119896+2

+

1205872119904(119896+1)2

1199082minus 1205872119886119894

(119896+2)21199082

1198871

119896+2

+ sdot sdot sdot +

1205872119904(119896+2)119899

119908119899minus 120587119899119886119894

(119896+2)119899119908119899

1198871

119896+2

12058711989911990411989911199081minus 1205871119886119894

11989911199081

1198871119899

+12058711989911990411989921199082minus 1205872119886119894

11989921199082

1198871119899

+ sdot sdot sdot +120587119899119904119899119899119908119899minus 120587119899119886119894

119899119899119908119899

1198871119899

]]]]]]]]]]]]]

]

(47)

8 Mathematical Problems in Engineering

for 119894 = 1 119903 In compact form (47) turns into

119906119894

119904119904=

[[[[[[[[[[[[[[

[

119899

sum

119895=1

1205871119904(119896+1)119895

119908119895minus 120587119895119886(119896+1)119895

119908119895

119887119894

119896+1

119899

sum

119895=1

1205872119904(119896+2)119895

119908119895minus 120587119895119886(119896+2)119895

119908119895

119887119894

119896+2

119899

sum

119895=1

120587119899119904119899119895119908119895minus 120587119895119886119899119895119908119895

119887119894119899

]]]]]]]]]]]]]]

]

(48)

and from (27) the local matrices Γ119894have the form

Γ119894=

[[[[[[[[[[[[[

[

1205871119904(119896+1)1

minus 1205871119886119894

(119896+1)1

119887119894

119896+1

1205871119904(119896+1)2

minus 1205872119886119894

(119896+1)2

119887119894

119896+1

sdot sdot sdot

12058711199041198991

minus 120587119899119886119894

(119896+1)119899

119887119894

119896+1

1205872119904(119896+2)1

minus 1205871119886119894

(119896+2)1

119887119894

119896+2

1205872119904(119896+2)2

minus 1205872119886119894

(119896+2)2

119887119894

119896+2

sdot sdot sdot

12058721199041198991

minus 120587119899119886119894

(119896+2)119899

119887119894

119896+2

d

1205871198991199041198991

minus 1205871119886119894

1198991

119887119894119899

1205871198991199041198992

minus 1205872119886119894

1198992

119887119894119899

sdot sdot sdot1205871198991199041198991

minus 120587119899119886119894

119899119899

119887119894119899

]]]]]]]]]]]]]

]

(49)

Now again from the fuzzy plant given in (42) and theglobal fuzzy model (29) the expressions for

119896+1

119899are

119896+1

=

119903

sum

119894=1

ℎ119894(119909 (119905))(

119899

sum

119895=1

119886119894

(119896+1)119895119909119895)

+ (

119903

sum

119894=1

ℎ119894(119909 (119905)) 119887

119894

119896+1)119906119904119904

119896+2

=

119903

sum

119894=1

ℎ119894(119909 (119905))(

119899

sum

119895=1

119886119894

(119896+2)119895119909119895)

+ (

119903

sum

119894=1

ℎ119894(119909 (119905)) 119887

119894

119896+2)119906119904119904

119899=

119903

sum

119894=1

ℎ119894(119909 (119905))(

119899

sum

119895=1

119886119894

119899119895119909119895)

+ (

119903

sum

119894=1

ℎ119894(119909 (119905)) 119887

119894

119899)119906119904119904

(50)

So by applying successive substitutions as before with (44)(45) and knowing that in steady-state 119909

119895= 120587119895119908119895 the

following expressions are obtained

119903

sum

119894=1

ℎ119894(119908 (119905))(

119899

sum

119895=1

119886119894

(119896+1)119895120587119895119908119895)

+ (

119903

sum

119894=1

ℎ119894(119908 (119905)) 119887

119894

119896+1)119906119904119904= 1205871

119899

sum

119894=1

119904(119896+1)119894

119908119894

119903

sum

119894=1

ℎ119894(119908 (119905))(

119899

sum

119895=1

119886119894

(119896+2)119895120587119895119908119895)

+ (

119903

sum

119894=1

ℎ119894(119908 (119905)) 119887

119894

119896+2)119906119904119904= 1205872

119899

sum

119894=1

119904(119896+2)

119894119908119894

1199031

sum

119894=1

ℎ119894(119908 (119905))(

119899

sum

119895=1

119886119894

119899119895120587119895119908119895)

+ (

119903

sum

119894=1

ℎ119894(119908 (119905)) 119887

119894

119899)119906119904119904= 120587119899

119899

sum

119894=1

119904119899119894119908119894

(51)

Mathematical Problems in Engineering 9

Therefore the steady-state input for the overall fuzzy systemis

119906119904119904

=

[[[[[[[[[[[[[[

[

1205871sum119899

119894=1119904(119896+1)119894

119908119894minus sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

(119896+1)119895120587119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

119896+1

1205872sum119899

119894=1119904(119896+2)119894

119908119894minus sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

(119896+2)119895120587119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

119896+2

120587119899sum119899

119894=1119904119899119895119908119894minus sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

119899119895120587119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

119899

]]]]]]]]]]]]]]

]

(52)

But the desired form of the steady-state is

119906119904119904(119905) = 120574 (119908 (119905)) =

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905) (53)

Consequently the new membership functions 120583119894(119908(119905))

can be obtained after equating (52) and (53) resulting in

120583119894(119908 (119905)) =

[[[[[[

[

12058311989411

0 sdot sdot sdot 0

0 12058311989422

sdot sdot sdot 0

d

0 0 sdot sdot sdot 120583119894119898119898

]]]]]]

]

(54)

where

120583119894119895119897

=

119887119894

119896+119895ℎ119894(119908 (119905))

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

119896+119895

(55)

for all 119894 = 1 119903 119895 = 119897 = 1 119898 It can be readily observedthat the sum of the corresponding elements is equal to oneand 120583

119894119895119897(119908(119905)) ge 0

These new membership functions 120583119894(119908(119905)) are organized

in a matrix form for that reason sum119903119894=1

120583119894(119908(119905)) = 119868 (where 119868

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 119861

119894have the

same sign at corresponding positions Clearly this conditionavoids singularities in (55) As before ℎ

119894(119908(119905)) is directly

obtained from ℎ119894(119909(119905)) by considering that in steady-state

119909(119905) = Π119908(119905) At this point the following theorem can bedefined

Theorem 5 The exact fuzzy output regulation with fullinformation for T-S fuzzy MIMO systems in the form of(11) defined by matrices (42) is solvable if (a) the sign ofentries for corresponding position inside input matrices 119861

119894for

119894 = 1 119903 is the same (b) there exists a fuzzy stabilizer119906(119905) = sum

119903

119894=1ℎ119894(119911(119905))119870

119894119909 for the fuzzy system [16] and (c) the

exosystem (119905) = 119878119908(119905) is Poisson stable

Moreover the Exact Output Fuzzy Regulation Problem forMIMO systems is solvable by the controller

119906 (119905) =

119903

sum

119894=1

ℎ119894(119909 (119905)) 119896

119894119909 (119905) minus Π119908 (119905)

+

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905)

(56)

where 120583119894(119908(119905)) can be readily obtained from (54) and (55)

Proof It follows directly from Theorem 3 and previousanalysis

Remark 6 Notice that 120583119894(119908(119905)) is defined as a matrix and

this is not a typical membership function however 120583119894(119908(119905))

contains a set of membership functions related to each entryof 119861119894 This form allows us to design controller (20)

5 Numerical Examples

51 Simple Input Case Consider the problem of balancingand swing of an inverted pendulum on a cart The motionequations for this system are [14]

1(119905) = 119909

2(119905) (57)

2(119905)

=119892 sin(119909

1(119905))minus 119886119898119897119909

2

2sin (2119909

1(119905))2 minus119886 cos (119909

1(119905)) 119906 (119905)

41198973 minus 119886119898119897 cos2 (1199091(119905))

(58)

where 1199091(119905) denotes the angle (in radians) of the pendulum

from the vertical and 1199092(119905) is the angular velocity 119892 =

98ms2 is the gravity constant 119898 is the mass of the pen-dulum 119872 is the mass of the cart 2119897 is the length of thependulum and 119906 is the force applied to the cart (in newtons)119886 = 1(119898 +119872) with119898 = 20 kg119872 = 80 kg and 2119897 = 10min the simulations The goal is to track the reference definedby the exosystem within the range 119909

1isin (minus1205872 1205872) To this

end a fuzzy model representing the nonlinear dynamics is asfollows

Rule 1

IF 1199091(119905) is about 0

THEN (119905) = 1198601119909(119905) + 119861

1119906(119905)

Rule 2

IF 1199091(119905) is about plusmn1205872(|119909

1| lt 1205872)

THEN (119905) = 1198602119909(119905) + 119861

2119906(119905)

10 Mathematical Problems in Engineering

h1

h2

x1minus90 90

(deg)

1

00

Figure 1 Membership functions of two-rule model

Membership functions for Rules 1 and 2 are shown in Figure 1In this example the T-S fuzzymodel of the plant is defined bymatrices

1198601= [

0 1

172941 0] 119861

1= [

0

minus01765]

1198602= [

0 1

41585 0] 119861

2= [

0

minus01500]

1198621= 1198622= [1 0]

(59)

while the exosystem can be constructed from the followingmatrices

119878 = [

0 1

minus1 0] 119876 = [1 0] (60)

which allows us to generate a great kind of sinusoidal refer-ences Figure 2 shows the simulation results after applying thecontroller

119906 (119905) =

119903

sum

119894=1

ℎ119894(119909 (119905)) 119870

119894[119909 (119905) minus

119903

sum

119894=1

ℎ119894(119908 (119905)) Π

119894119908 (119905)]

+

119903

sum

119894=1

ℎ119894(119908 (119905)) Γ

119894119908 (119905)

(61)

defined in [10] As expected controller (61) is unable toachieve exact regulation although tracking error remainsbounded

Now using the method derived in this work the localmatrices Γ

119894can be readily obtained from (28) resulting in

Γ1= [1036667 0] Γ

2= [343836 0] (62)

Besides knowing that 11990211198881= 1 matrix Π is obtained from

(24)

Π = [

1 0

0 1] (63)

while the newmembership functions computed from (39) are

1205831(119908 (119905)) =

(ℎ1(119908 (119905))) (minus01765)

ℎ1(119908 (119905)) (minus01765) + ℎ

2(119908 (119905)) (minus01500)

1205832(119908 (119905)) =

(ℎ2(119908 (119905))) (minus01500)

ℎ1(119908 (119905)) (minus01765) + ℎ

2(119908 (119905)) (minus01500)

(64)

On the other hand the fuzzy stabilizer for this system isconstructed bymeans of the PDC approach developed in [14]which is formed by

1198701= [1234898 69979] 119870

2= [586405 77962]

(65)

Figure 3 shows simulation with the controller definedin (20) Observe how the new membership functions arecapable of combining the local regulators in the proper wayto accomplish asymptotic tracking

To verify the effectiveness of the latter regulator thefuzzy controller is applied on the original system (57) Thesimulation results appear in Figure 4 Notice that the trackingerror depicted in Figure 4(b) is due to the approximationprovided by the T-S fuzzy model Such an error can bereduced by considering a better approximation or an exactrepresentation of the nonlinear system as the one given bythe nonlinear sector approach [14]

52 Multiple Inputs and Multiple Outputs (MIMO) CaseNow consider a T-S fuzzy system of two rules defined by thefollowing matrices

1198601=

[[[[[

[

0 0 1 0

0 0 0 1

6 4 2 4

2 3 1 5

]]]]]

]

1198611=

[[[[[

[

0 0

0 0

minus7 0

0 2

]]]]]

]

1198602=

[[[[[

[

0 0 1 0

0 0 0 1

3 4 2 1

2 4 6 2

]]]]]

]

1198612=

[[[[[

[

0 0

0 0

minus3 0

0 5

]]]]]

]

1198621= 1198622= [

1 0 0 0

0 1 0 0]

(66)

Mathematical Problems in Engineering 11

0 5 10 15 20 25 30 35 40minus40minus30minus20minus10

010203040

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30 35 40minus04minus03minus02minus01

0010203

Tracking error

Time

Am

plitu

de

e

(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 40minus40minus30minus20minus10

0102030

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30 35 40minus06minus05minus04minus03minus02minus01

0010203

Tracking error

Time

Am

plitu

de

e

(b)

Figure 3 (a) Output reference and (b) tracking errorWhen the fuzzy controller is designed on local regulators and the adequatemembershipfunctions for the fuzzy regulator are computed

0 5 10 15 20 25 30 35 40minus40minus30minus20minus10

010203040

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30 35 40minus06minus05minus04minus03minus02minus01

001

Tracking error

Time

Am

plitu

de

e

(b)

Figure 4 (a) Output reference and (b) tracking errorWhen the fuzzy controller is designed on local regulators and the adequatemembershipfunctions for the fuzzy regulator are computed

12 Mathematical Problems in Engineering

h1 h2

x1minus60

1

80

0

0

Figure 5 Membership functions of two-rule model

and an exosystem defined by

119878 =

[[[[[

[

0 0 1 0

0 0 0 1

minus4 0 0 0

0 minus16 0 0

]]]]]

]

119876 = [

1 0 0 0

0 1 0 0]

(67)

Membership functions forRules 1 and 2 are shown in Figure 5Matrices defining the local regulators can be easily obtainedby using (49) For this example they are

Γ1= [

14286 05714 02857 05714

minus10000 minus95000 minus05000 minus25000]

Γ2= [

23333 13333 06667 03333

minus04000 minus40000 minus12000 minus04000]

(68)

On the other hand 11990211198881= 1205871= 1 119902

21198882= 1205872= 1

11990211198881= 1205873= 1 119902

21198882= 1205874= 1 and by (46) and matrix Π is

Π =

[[[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

]]]]]

]

(69)

Figures 6 and 7 show the results obtained using linear regula-tors defined in (61)

Now by computing the new membership functions forthe fuzzy regulator through (54) one gets1205831(119908 (119905))

=

[[[

[

(ℎ1(119908 (119905))) (minus7)

ℎ1(119908 (119905)) (3) + ℎ

2(119908 (119905)) (3)

0

0(ℎ1(119908 (119905))) (2)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (5)

]]]

]

1205832(119908 (119905))

=

[[[

[

(ℎ2(119908 (119905))) (minus3)

ℎ1(119908 (119905)) (3) + ℎ

2(119908 (119905)) (3)

0

0(ℎ2(119908 (119905))) (5)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (5)

]]]

]

(70)

As previously the fuzzy stabilizer for this example is alsoconstructed bymeans of the PDC approach developed in [14]resulting in

1198701= [

10286 05714 06000 05714

minus10000 minus22500 minus05000 minus37500]

1198702= [

14000 13333 14000 03333

minus04000 minus11000 minus12000 minus09000]

(71)

Thus Figures 8 and 9 show the simulation results whenthe controller defined by (20) is applied

Besides Figure 10 shows the shape of the new member-ship functions which are clearly different from the originalones given in Figure 5

53 More General Problems Consider the T-S fuzzy MIMOsystem with multiple inputs defined by the following matri-ces

1198601=

[[[[[[[[[[[

[

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

minus6 4 2 4 5 3

2 3 minus1 5 minus2 4

]]]]]]]]]]]

]

1198611=

[[[[[[[[[[[

[

0 0

0 0

0 0

0 0

5 0

0 2

]]]]]]]]]]]

]

1198602=

[[[[[[[[[[[

[

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

minus6 1 2 minus5 4 6

0 2 minus7 4 5 2

]]]]]]]]]]]

]

1198612=

[[[[[[[[[[[

[

0 0

0 0

0 0

0 0

7 0

0 8

]]]]]]]]]]]

]

1198621= 1198622= [

0 1 1 0 0 0

1 0 1 0 0 0]

(72)

Mathematical Problems in Engineering 13

0 5 10 15 20 25 30minus15minus10

minus505

101520

Time

Am

plitu

dex1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus4minus2

02468

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 6 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus30minus20minus10

0102030

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus15

minus10

minus5

0

5

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 7 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of lineal regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus15minus10

minus505

101520

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus1

012345678

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 8 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

The reference signals will be generated by an exosystemdefined by

119878 = [

0 1

minus1 0] 119876 = [

1 0

0 1] (73)

Membership functions for Rules 1 and 2 are ℎ1(119909(119905)) =

(80 minus 1199091(119905))140 and ℎ

2(119909(119905)) = 1 minus ℎ

1(119905) Notice that for this

case 119899 = 119904 and 119862119894

= 119876 This implies that Π119894and Γ

119894must

be obtained by solving Francis Equations for each subsystemHowever due to form of the fuzzy plant it is possible to verifythat Π

1= Π2

14 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus40minus30minus20minus10

0102030

Time

Am

plitu

dex2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus10

minus8minus6minus4minus2

0

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 9 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(11) (x1(t))1205832(11) (x1(t))

(a)

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(22) (x1(t))1205832(22) (x1(t))

(b)

Figure 10 (a) Membership functions corresponding to position (11) of 1205831(119908(119905)) and 120583

2(119908(119905)) (b) Membership functions corresponding to

position (22) 1205831(119908(119905)) and 120583

2(119908(119905))

So by applying (9) for each local subsystem with 119894 = 1 2it results in

Π1= Π2=

[[[[[[[[[[[

[

05000 05000

15000 minus05000

minus05000 05000

05000 15000

minus05000 minus05000

minus15000 05000

]]]]]]]]]]]

]

Γ1= [

07000 minus03000

minus20000 minus55000]

Γ2= [

23571 12143

minus04375 minus01875]

(74)

Again the fuzzy stabilizer for this system is constructedby means of the PDC approach developed in [14] yielding

1198701=[[

[

minus12229 00609 minus40284 01291 minus25666 minus04061

16919 minus83997 32934 minus127153 15732 minus61836

]]

]

1198702=[[

[

minus08735 04721 minus28775 13780 minus16904 minus07187

06730 minus19749 15734 minus30538 minus04817 minus12959

]]

]

(75)

The behavior obtained by applying controller (61) isdepicted in Figures 11 and 12 Notice the exact outputregulation is not achieved

Mathematical Problems in Engineering 15

0 5 10 15 20 25 30minus60minus40minus20

0204060

Time

Am

plitu

dex1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus80minus70minus60minus50minus40minus30minus20minus10

01020

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 11 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus15

minus10

minus5

0

5

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 12 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

Now the adequate membership functions are computedfrom (54) resulting in

1205831(119908 (119905))

=

[[[

[

(ℎ1(119908 (119905))) (5)

ℎ1(119908 (119905)) (5) + ℎ

2(119908 (119905)) (7)

0

0(ℎ1(119908 (119905))) (2)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (8)

]]]

]

1205832(119908 (119905))

=

[[[

[

(ℎ2(119908 (119905))) (7)

ℎ1(119908 (119905)) (5) + ℎ

2(119908 (119905)) (7)

0

0(ℎ2(119908 (119905))) (8)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (8)

]]]

]

(76)

When ℎ119894(119908(119905)) is replaced by its corresponding expressions

one gets

1205831(119908 (119905)) =

[[[

[

1225

1199081(119905) + 140

minus5

20

0 minus1199081(119905) minus 80

31199081(119905) + 320

]]]

]

1205832(119908 (119905)) =

[[[

[

7

2minus

1225

1199081(119905) + 410

0

041199081(119905) + 240

31199081(119905) + 320

]]]

]

(77)

The results of applying control law (20) on the fuzzy plantare given in Figures 13 and 14

The ldquocomputedrdquo membership functions for the fuzzyregulator are given in Figure 15 Again it can be readilyobserved that these new membership functions are differentfrom the original ones

Notice that although this example does not fulfill theconditions ofTheorem 5 themembership functions obtainedfrom (39) allow the exact tracking for the fuzzy problemThis

16 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus80minus70minus60minus50minus40minus30minus20minus10

010

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 13 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus8minus6minus4minus2

02468

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 14 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(11) (x1(t))1205832(11) (x1(t))

(a)

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(22) (x1(t))1205832(22) (x1(t))

(b)

Figure 15 (a) Membership functions corresponding to position (11) of 1205831(119908(119905)) and 120583

2(119908(119905)) (b) Membership functions corresponding to

position (22) 1205831(119908(119905)) 120583

2(119908(119905))

Mathematical Problems in Engineering 17

suggests that the approach presented in this work may beapplied on a bigger class of T-S models however this studyis not completed yet

6 Conclusions

A fuzzy regulator for continuous-time systems based on thecombination of linear regulators adjusted by different mem-bership functions has been presented The main advantageis that analytical expressions for the membership functionswhich allow the proper combination of the local regulators inorder to guarantee the exact regulation can be easily appliedAs 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 modelscan be commonly found in mechanics electromechanicselectronics robotics and so forth

Some numerical examples are used to illustrate theapplicability of the proposed approach even in cases wherethe dimensions of the plant and the exosystem are differentbetween each otherThis suggests that the approach presentedin this work may be applied on a bigger class of T-S models

Finally the method proposed in this work avoids thedisadvantage of constructing an exact fuzzy regulator basedon overall T-S fuzzy system whose expression may resultvery large Instead the given approach offers 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 Interests

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

Acknowledgments

This work was partially supported by CONACYT (ConsejoNacional de Ciencia y Tecnologıa) through scholarship SNI(Sistema Nacional de Investigadores) and by IPN (InstitutoPolitecnico Nacional) through research Projects 20140659and 20150487 and scholarships EDI (Estımulo al Desempenode los Investigadores) COFAA (Comision de Operacion yFomento de Actividades Academicas) and BEIFI (Beca deEstımulo Institucional de Formacion de Investigadores)

References

[1] B A Francis ldquoThe linear multivariable regulator problemrdquoSIAM Journal on Control and Optimization vol 15 no 3 pp486ndash505 1977

[2] B A Francis andWMWonham ldquoThe internalmodel principleof control theoryrdquo Automatica vol 12 no 5 pp 457ndash465 1976

[3] A Isidori and C I Byrnes ldquoOutput regulation of nonlinearsystemsrdquo IEEE Transactions on Automatic Control vol 35 no2 pp 131ndash140 1990

[4] B Castillo-Toledo S Di Gennaro and F Jurado ldquoTrajectorytracking for a quadrotor via fuzzy regulationrdquo in Proceedings ofthe World Automation Congress (WAC rsquo12) pp 1ndash6 June 2012

[5] S Chen ldquoOutput regulation of nonlinear singularly perturbedsystems based on T-S fuzzy modelrdquo Journal of Control Theoryand Applications vol 3 no 4 pp 399ndash403 2005

[6] R Tapia-Herrera J A Meda-Campana S Alcamp119886119901119900119904 119886ntara-Montes T Hernamp119886119901119900119904 119886ndez-Cortamp119886119901119900119904 119890s and L Salgado-Conrado ldquoTuning of a TS fuzzy output regulator using thesteepest descent approach and ANFISrdquoMathematical Problemsin Engineering vol 2013 Article ID 873430 14 pages 2013

[7] K-Y Lian and J-J Liou ldquoOutput tracking control for fuzzysystems via output feedback designrdquo IEEE Transactions onFuzzy Systems vol 14 no 5 pp 628ndash639 2006

[8] X-J Ma and Z-Q Sun ldquoOutput tracking and regulationof nonlinear system based on takagi-sugeno fuzzy modelrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 30 no 1 pp 47ndash59 2000

[9] H J Lee J B Park and Y H Joo ldquoComments on lsquooutputtracking and regulation of nonlinear system based on Takagi-Sugeno fuzzy model1015840rdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 33 no 3 pp 521ndash523 2003

[10] J A Meda-Campana B Castillo-Toledo and G Chen ldquoSyn-chronization of chaotic systems from a fuzzy regulationapproachrdquo Fuzzy Sets and Systems vol 160 no 19 pp 2860ndash2875 2009

[11] J A Meda-Campana J C Gamp119886119901119900119904 119900mez-Mancilla and BCastillo-Toledo ldquoExact output regulation for nonlinear systemsdescribed by Takagi-Sugeno fuzzy modelsrdquo IEEE Transactionson Fuzzy Systems vol 20 no 2 pp 235ndash247 2012

[12] C Byrnes F Priscoli and A Isidori Output Regulation ofUncertain Nonlinear Systems Birkhauser 1997

[13] A Isidori Nonlinear Control Systems Communications andControl Engineering Series Springer Berlin Germany 3rdedition 1995

[14] K Tanaka and H O Wang Fuzzy Control Systems Designand Analysis A Linear Matrix Inequality Approach John Wileyamp119886119898119901 Sons New York NY USA 2001

[15] L-X Wang A Course in Fuzzy Systems and Control PrenticeHall Upper Saddle River NJ USA 1997

[16] I Abdelmalek N Golamp119886119901119900119904 119890a and M L Hadjili ldquoA newfuzzy Lyapunov approach to non-quadratic stabilization ofTakagi-Sugeno fuzzy modelsrdquo International Journal of AppliedMathematics andComputer Science vol 17 no 1 pp 39ndash51 2007

[17] J A Meda-Campana and B Castillo-Toledo ldquoOn the outputregulation for TS fuzzymodels using slidingmodesrdquo inProceed-ings of the American Control Conference (ACC rsquo05) pp 4062ndash4067 Portland Ore USA June 2005

[18] B Castillo-Toledo and J A Meda-Campana ldquoThe fuzzydiscrete-time robust regulation problem a LMI approachrdquo inProceedings of the 41st IEEE Conference on Decision and Controlvol 2 pp 2159ndash2164 IEEE Las Vegas Nev USA December2002

[19] B Castillo-Toledo J A Meda-Campana and A Titli ldquoAfuzzy output regulator for Takagi-Sugeno fuzzy modelsrdquo inProceedings of the IEEE International Symposium on IntelligentControl pp 310ndash315 Houston Tex USA October 2003

[20] B-C Ding H-X Sun and Y-E Qiao ldquoStability analysis of T-Sfuzzy control systems based on parameter-dependent Lyapunovfunctionrdquo Acta Automatica Sinica vol 31 no 4 pp 651ndash6542005

[21] M Narimani H K Lam R Dilmaghani and C Wolfe ldquoLMI-based stability analysis of fuzzy-model-based control systemsusing approximated polynomial membership functionsrdquo IEEE

18 Mathematical Problems in Engineering

Transactions on Systems Man and Cybernetics Part B Cyber-netics vol 41 no 3 pp 713ndash724 2011

[22] H K Lam ldquoPolynomial fuzzy-model-based control systemsstability analysis via piecewise-linear membership functionsrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 588ndash5932011

[23] K Tanaka T Hori and H O Wang ldquoA fuzzy Lyapunovapproach to fuzzy control system designrdquo in Proceedings of theAmerican Control Conference vol 6 pp 4790ndash4795 ArlingtonVa USA June 2001

[24] K Tanaka T Hori and H O Wang ldquoA multiple Lyapunovfunction approach to stabilization of fuzzy control systemsrdquoIEEE Transactions on Fuzzy Systems vol 11 no 4 pp 582ndash5892003

[25] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 174ndash1972005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

for 119894 = 1 119903 In compact form (47) turns into

119906119894

119904119904=

[[[[[[[[[[[[[[

[

119899

sum

119895=1

1205871119904(119896+1)119895

119908119895minus 120587119895119886(119896+1)119895

119908119895

119887119894

119896+1

119899

sum

119895=1

1205872119904(119896+2)119895

119908119895minus 120587119895119886(119896+2)119895

119908119895

119887119894

119896+2

119899

sum

119895=1

120587119899119904119899119895119908119895minus 120587119895119886119899119895119908119895

119887119894119899

]]]]]]]]]]]]]]

]

(48)

and from (27) the local matrices Γ119894have the form

Γ119894=

[[[[[[[[[[[[[

[

1205871119904(119896+1)1

minus 1205871119886119894

(119896+1)1

119887119894

119896+1

1205871119904(119896+1)2

minus 1205872119886119894

(119896+1)2

119887119894

119896+1

sdot sdot sdot

12058711199041198991

minus 120587119899119886119894

(119896+1)119899

119887119894

119896+1

1205872119904(119896+2)1

minus 1205871119886119894

(119896+2)1

119887119894

119896+2

1205872119904(119896+2)2

minus 1205872119886119894

(119896+2)2

119887119894

119896+2

sdot sdot sdot

12058721199041198991

minus 120587119899119886119894

(119896+2)119899

119887119894

119896+2

d

1205871198991199041198991

minus 1205871119886119894

1198991

119887119894119899

1205871198991199041198992

minus 1205872119886119894

1198992

119887119894119899

sdot sdot sdot1205871198991199041198991

minus 120587119899119886119894

119899119899

119887119894119899

]]]]]]]]]]]]]

]

(49)

Now again from the fuzzy plant given in (42) and theglobal fuzzy model (29) the expressions for

119896+1

119899are

119896+1

=

119903

sum

119894=1

ℎ119894(119909 (119905))(

119899

sum

119895=1

119886119894

(119896+1)119895119909119895)

+ (

119903

sum

119894=1

ℎ119894(119909 (119905)) 119887

119894

119896+1)119906119904119904

119896+2

=

119903

sum

119894=1

ℎ119894(119909 (119905))(

119899

sum

119895=1

119886119894

(119896+2)119895119909119895)

+ (

119903

sum

119894=1

ℎ119894(119909 (119905)) 119887

119894

119896+2)119906119904119904

119899=

119903

sum

119894=1

ℎ119894(119909 (119905))(

119899

sum

119895=1

119886119894

119899119895119909119895)

+ (

119903

sum

119894=1

ℎ119894(119909 (119905)) 119887

119894

119899)119906119904119904

(50)

So by applying successive substitutions as before with (44)(45) and knowing that in steady-state 119909

119895= 120587119895119908119895 the

following expressions are obtained

119903

sum

119894=1

ℎ119894(119908 (119905))(

119899

sum

119895=1

119886119894

(119896+1)119895120587119895119908119895)

+ (

119903

sum

119894=1

ℎ119894(119908 (119905)) 119887

119894

119896+1)119906119904119904= 1205871

119899

sum

119894=1

119904(119896+1)119894

119908119894

119903

sum

119894=1

ℎ119894(119908 (119905))(

119899

sum

119895=1

119886119894

(119896+2)119895120587119895119908119895)

+ (

119903

sum

119894=1

ℎ119894(119908 (119905)) 119887

119894

119896+2)119906119904119904= 1205872

119899

sum

119894=1

119904(119896+2)

119894119908119894

1199031

sum

119894=1

ℎ119894(119908 (119905))(

119899

sum

119895=1

119886119894

119899119895120587119895119908119895)

+ (

119903

sum

119894=1

ℎ119894(119908 (119905)) 119887

119894

119899)119906119904119904= 120587119899

119899

sum

119894=1

119904119899119894119908119894

(51)

Mathematical Problems in Engineering 9

Therefore the steady-state input for the overall fuzzy systemis

119906119904119904

=

[[[[[[[[[[[[[[

[

1205871sum119899

119894=1119904(119896+1)119894

119908119894minus sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

(119896+1)119895120587119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

119896+1

1205872sum119899

119894=1119904(119896+2)119894

119908119894minus sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

(119896+2)119895120587119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

119896+2

120587119899sum119899

119894=1119904119899119895119908119894minus sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

119899119895120587119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

119899

]]]]]]]]]]]]]]

]

(52)

But the desired form of the steady-state is

119906119904119904(119905) = 120574 (119908 (119905)) =

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905) (53)

Consequently the new membership functions 120583119894(119908(119905))

can be obtained after equating (52) and (53) resulting in

120583119894(119908 (119905)) =

[[[[[[

[

12058311989411

0 sdot sdot sdot 0

0 12058311989422

sdot sdot sdot 0

d

0 0 sdot sdot sdot 120583119894119898119898

]]]]]]

]

(54)

where

120583119894119895119897

=

119887119894

119896+119895ℎ119894(119908 (119905))

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

119896+119895

(55)

for all 119894 = 1 119903 119895 = 119897 = 1 119898 It can be readily observedthat the sum of the corresponding elements is equal to oneand 120583

119894119895119897(119908(119905)) ge 0

These new membership functions 120583119894(119908(119905)) are organized

in a matrix form for that reason sum119903119894=1

120583119894(119908(119905)) = 119868 (where 119868

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 119861

119894have the

same sign at corresponding positions Clearly this conditionavoids singularities in (55) As before ℎ

119894(119908(119905)) is directly

obtained from ℎ119894(119909(119905)) by considering that in steady-state

119909(119905) = Π119908(119905) At this point the following theorem can bedefined

Theorem 5 The exact fuzzy output regulation with fullinformation for T-S fuzzy MIMO systems in the form of(11) defined by matrices (42) is solvable if (a) the sign ofentries for corresponding position inside input matrices 119861

119894for

119894 = 1 119903 is the same (b) there exists a fuzzy stabilizer119906(119905) = sum

119903

119894=1ℎ119894(119911(119905))119870

119894119909 for the fuzzy system [16] and (c) the

exosystem (119905) = 119878119908(119905) is Poisson stable

Moreover the Exact Output Fuzzy Regulation Problem forMIMO systems is solvable by the controller

119906 (119905) =

119903

sum

119894=1

ℎ119894(119909 (119905)) 119896

119894119909 (119905) minus Π119908 (119905)

+

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905)

(56)

where 120583119894(119908(119905)) can be readily obtained from (54) and (55)

Proof It follows directly from Theorem 3 and previousanalysis

Remark 6 Notice that 120583119894(119908(119905)) is defined as a matrix and

this is not a typical membership function however 120583119894(119908(119905))

contains a set of membership functions related to each entryof 119861119894 This form allows us to design controller (20)

5 Numerical Examples

51 Simple Input Case Consider the problem of balancingand swing of an inverted pendulum on a cart The motionequations for this system are [14]

1(119905) = 119909

2(119905) (57)

2(119905)

=119892 sin(119909

1(119905))minus 119886119898119897119909

2

2sin (2119909

1(119905))2 minus119886 cos (119909

1(119905)) 119906 (119905)

41198973 minus 119886119898119897 cos2 (1199091(119905))

(58)

where 1199091(119905) denotes the angle (in radians) of the pendulum

from the vertical and 1199092(119905) is the angular velocity 119892 =

98ms2 is the gravity constant 119898 is the mass of the pen-dulum 119872 is the mass of the cart 2119897 is the length of thependulum and 119906 is the force applied to the cart (in newtons)119886 = 1(119898 +119872) with119898 = 20 kg119872 = 80 kg and 2119897 = 10min the simulations The goal is to track the reference definedby the exosystem within the range 119909

1isin (minus1205872 1205872) To this

end a fuzzy model representing the nonlinear dynamics is asfollows

Rule 1

IF 1199091(119905) is about 0

THEN (119905) = 1198601119909(119905) + 119861

1119906(119905)

Rule 2

IF 1199091(119905) is about plusmn1205872(|119909

1| lt 1205872)

THEN (119905) = 1198602119909(119905) + 119861

2119906(119905)

10 Mathematical Problems in Engineering

h1

h2

x1minus90 90

(deg)

1

00

Figure 1 Membership functions of two-rule model

Membership functions for Rules 1 and 2 are shown in Figure 1In this example the T-S fuzzymodel of the plant is defined bymatrices

1198601= [

0 1

172941 0] 119861

1= [

0

minus01765]

1198602= [

0 1

41585 0] 119861

2= [

0

minus01500]

1198621= 1198622= [1 0]

(59)

while the exosystem can be constructed from the followingmatrices

119878 = [

0 1

minus1 0] 119876 = [1 0] (60)

which allows us to generate a great kind of sinusoidal refer-ences Figure 2 shows the simulation results after applying thecontroller

119906 (119905) =

119903

sum

119894=1

ℎ119894(119909 (119905)) 119870

119894[119909 (119905) minus

119903

sum

119894=1

ℎ119894(119908 (119905)) Π

119894119908 (119905)]

+

119903

sum

119894=1

ℎ119894(119908 (119905)) Γ

119894119908 (119905)

(61)

defined in [10] As expected controller (61) is unable toachieve exact regulation although tracking error remainsbounded

Now using the method derived in this work the localmatrices Γ

119894can be readily obtained from (28) resulting in

Γ1= [1036667 0] Γ

2= [343836 0] (62)

Besides knowing that 11990211198881= 1 matrix Π is obtained from

(24)

Π = [

1 0

0 1] (63)

while the newmembership functions computed from (39) are

1205831(119908 (119905)) =

(ℎ1(119908 (119905))) (minus01765)

ℎ1(119908 (119905)) (minus01765) + ℎ

2(119908 (119905)) (minus01500)

1205832(119908 (119905)) =

(ℎ2(119908 (119905))) (minus01500)

ℎ1(119908 (119905)) (minus01765) + ℎ

2(119908 (119905)) (minus01500)

(64)

On the other hand the fuzzy stabilizer for this system isconstructed bymeans of the PDC approach developed in [14]which is formed by

1198701= [1234898 69979] 119870

2= [586405 77962]

(65)

Figure 3 shows simulation with the controller definedin (20) Observe how the new membership functions arecapable of combining the local regulators in the proper wayto accomplish asymptotic tracking

To verify the effectiveness of the latter regulator thefuzzy controller is applied on the original system (57) Thesimulation results appear in Figure 4 Notice that the trackingerror depicted in Figure 4(b) is due to the approximationprovided by the T-S fuzzy model Such an error can bereduced by considering a better approximation or an exactrepresentation of the nonlinear system as the one given bythe nonlinear sector approach [14]

52 Multiple Inputs and Multiple Outputs (MIMO) CaseNow consider a T-S fuzzy system of two rules defined by thefollowing matrices

1198601=

[[[[[

[

0 0 1 0

0 0 0 1

6 4 2 4

2 3 1 5

]]]]]

]

1198611=

[[[[[

[

0 0

0 0

minus7 0

0 2

]]]]]

]

1198602=

[[[[[

[

0 0 1 0

0 0 0 1

3 4 2 1

2 4 6 2

]]]]]

]

1198612=

[[[[[

[

0 0

0 0

minus3 0

0 5

]]]]]

]

1198621= 1198622= [

1 0 0 0

0 1 0 0]

(66)

Mathematical Problems in Engineering 11

0 5 10 15 20 25 30 35 40minus40minus30minus20minus10

010203040

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30 35 40minus04minus03minus02minus01

0010203

Tracking error

Time

Am

plitu

de

e

(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 40minus40minus30minus20minus10

0102030

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30 35 40minus06minus05minus04minus03minus02minus01

0010203

Tracking error

Time

Am

plitu

de

e

(b)

Figure 3 (a) Output reference and (b) tracking errorWhen the fuzzy controller is designed on local regulators and the adequatemembershipfunctions for the fuzzy regulator are computed

0 5 10 15 20 25 30 35 40minus40minus30minus20minus10

010203040

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30 35 40minus06minus05minus04minus03minus02minus01

001

Tracking error

Time

Am

plitu

de

e

(b)

Figure 4 (a) Output reference and (b) tracking errorWhen the fuzzy controller is designed on local regulators and the adequatemembershipfunctions for the fuzzy regulator are computed

12 Mathematical Problems in Engineering

h1 h2

x1minus60

1

80

0

0

Figure 5 Membership functions of two-rule model

and an exosystem defined by

119878 =

[[[[[

[

0 0 1 0

0 0 0 1

minus4 0 0 0

0 minus16 0 0

]]]]]

]

119876 = [

1 0 0 0

0 1 0 0]

(67)

Membership functions forRules 1 and 2 are shown in Figure 5Matrices defining the local regulators can be easily obtainedby using (49) For this example they are

Γ1= [

14286 05714 02857 05714

minus10000 minus95000 minus05000 minus25000]

Γ2= [

23333 13333 06667 03333

minus04000 minus40000 minus12000 minus04000]

(68)

On the other hand 11990211198881= 1205871= 1 119902

21198882= 1205872= 1

11990211198881= 1205873= 1 119902

21198882= 1205874= 1 and by (46) and matrix Π is

Π =

[[[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

]]]]]

]

(69)

Figures 6 and 7 show the results obtained using linear regula-tors defined in (61)

Now by computing the new membership functions forthe fuzzy regulator through (54) one gets1205831(119908 (119905))

=

[[[

[

(ℎ1(119908 (119905))) (minus7)

ℎ1(119908 (119905)) (3) + ℎ

2(119908 (119905)) (3)

0

0(ℎ1(119908 (119905))) (2)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (5)

]]]

]

1205832(119908 (119905))

=

[[[

[

(ℎ2(119908 (119905))) (minus3)

ℎ1(119908 (119905)) (3) + ℎ

2(119908 (119905)) (3)

0

0(ℎ2(119908 (119905))) (5)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (5)

]]]

]

(70)

As previously the fuzzy stabilizer for this example is alsoconstructed bymeans of the PDC approach developed in [14]resulting in

1198701= [

10286 05714 06000 05714

minus10000 minus22500 minus05000 minus37500]

1198702= [

14000 13333 14000 03333

minus04000 minus11000 minus12000 minus09000]

(71)

Thus Figures 8 and 9 show the simulation results whenthe controller defined by (20) is applied

Besides Figure 10 shows the shape of the new member-ship functions which are clearly different from the originalones given in Figure 5

53 More General Problems Consider the T-S fuzzy MIMOsystem with multiple inputs defined by the following matri-ces

1198601=

[[[[[[[[[[[

[

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

minus6 4 2 4 5 3

2 3 minus1 5 minus2 4

]]]]]]]]]]]

]

1198611=

[[[[[[[[[[[

[

0 0

0 0

0 0

0 0

5 0

0 2

]]]]]]]]]]]

]

1198602=

[[[[[[[[[[[

[

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

minus6 1 2 minus5 4 6

0 2 minus7 4 5 2

]]]]]]]]]]]

]

1198612=

[[[[[[[[[[[

[

0 0

0 0

0 0

0 0

7 0

0 8

]]]]]]]]]]]

]

1198621= 1198622= [

0 1 1 0 0 0

1 0 1 0 0 0]

(72)

Mathematical Problems in Engineering 13

0 5 10 15 20 25 30minus15minus10

minus505

101520

Time

Am

plitu

dex1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus4minus2

02468

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 6 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus30minus20minus10

0102030

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus15

minus10

minus5

0

5

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 7 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of lineal regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus15minus10

minus505

101520

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus1

012345678

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 8 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

The reference signals will be generated by an exosystemdefined by

119878 = [

0 1

minus1 0] 119876 = [

1 0

0 1] (73)

Membership functions for Rules 1 and 2 are ℎ1(119909(119905)) =

(80 minus 1199091(119905))140 and ℎ

2(119909(119905)) = 1 minus ℎ

1(119905) Notice that for this

case 119899 = 119904 and 119862119894

= 119876 This implies that Π119894and Γ

119894must

be obtained by solving Francis Equations for each subsystemHowever due to form of the fuzzy plant it is possible to verifythat Π

1= Π2

14 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus40minus30minus20minus10

0102030

Time

Am

plitu

dex2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus10

minus8minus6minus4minus2

0

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 9 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(11) (x1(t))1205832(11) (x1(t))

(a)

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(22) (x1(t))1205832(22) (x1(t))

(b)

Figure 10 (a) Membership functions corresponding to position (11) of 1205831(119908(119905)) and 120583

2(119908(119905)) (b) Membership functions corresponding to

position (22) 1205831(119908(119905)) and 120583

2(119908(119905))

So by applying (9) for each local subsystem with 119894 = 1 2it results in

Π1= Π2=

[[[[[[[[[[[

[

05000 05000

15000 minus05000

minus05000 05000

05000 15000

minus05000 minus05000

minus15000 05000

]]]]]]]]]]]

]

Γ1= [

07000 minus03000

minus20000 minus55000]

Γ2= [

23571 12143

minus04375 minus01875]

(74)

Again the fuzzy stabilizer for this system is constructedby means of the PDC approach developed in [14] yielding

1198701=[[

[

minus12229 00609 minus40284 01291 minus25666 minus04061

16919 minus83997 32934 minus127153 15732 minus61836

]]

]

1198702=[[

[

minus08735 04721 minus28775 13780 minus16904 minus07187

06730 minus19749 15734 minus30538 minus04817 minus12959

]]

]

(75)

The behavior obtained by applying controller (61) isdepicted in Figures 11 and 12 Notice the exact outputregulation is not achieved

Mathematical Problems in Engineering 15

0 5 10 15 20 25 30minus60minus40minus20

0204060

Time

Am

plitu

dex1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus80minus70minus60minus50minus40minus30minus20minus10

01020

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 11 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus15

minus10

minus5

0

5

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 12 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

Now the adequate membership functions are computedfrom (54) resulting in

1205831(119908 (119905))

=

[[[

[

(ℎ1(119908 (119905))) (5)

ℎ1(119908 (119905)) (5) + ℎ

2(119908 (119905)) (7)

0

0(ℎ1(119908 (119905))) (2)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (8)

]]]

]

1205832(119908 (119905))

=

[[[

[

(ℎ2(119908 (119905))) (7)

ℎ1(119908 (119905)) (5) + ℎ

2(119908 (119905)) (7)

0

0(ℎ2(119908 (119905))) (8)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (8)

]]]

]

(76)

When ℎ119894(119908(119905)) is replaced by its corresponding expressions

one gets

1205831(119908 (119905)) =

[[[

[

1225

1199081(119905) + 140

minus5

20

0 minus1199081(119905) minus 80

31199081(119905) + 320

]]]

]

1205832(119908 (119905)) =

[[[

[

7

2minus

1225

1199081(119905) + 410

0

041199081(119905) + 240

31199081(119905) + 320

]]]

]

(77)

The results of applying control law (20) on the fuzzy plantare given in Figures 13 and 14

The ldquocomputedrdquo membership functions for the fuzzyregulator are given in Figure 15 Again it can be readilyobserved that these new membership functions are differentfrom the original ones

Notice that although this example does not fulfill theconditions ofTheorem 5 themembership functions obtainedfrom (39) allow the exact tracking for the fuzzy problemThis

16 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus80minus70minus60minus50minus40minus30minus20minus10

010

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 13 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus8minus6minus4minus2

02468

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 14 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(11) (x1(t))1205832(11) (x1(t))

(a)

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(22) (x1(t))1205832(22) (x1(t))

(b)

Figure 15 (a) Membership functions corresponding to position (11) of 1205831(119908(119905)) and 120583

2(119908(119905)) (b) Membership functions corresponding to

position (22) 1205831(119908(119905)) 120583

2(119908(119905))

Mathematical Problems in Engineering 17

suggests that the approach presented in this work may beapplied on a bigger class of T-S models however this studyis not completed yet

6 Conclusions

A fuzzy regulator for continuous-time systems based on thecombination of linear regulators adjusted by different mem-bership functions has been presented The main advantageis that analytical expressions for the membership functionswhich allow the proper combination of the local regulators inorder to guarantee the exact regulation can be easily appliedAs 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 modelscan be commonly found in mechanics electromechanicselectronics robotics and so forth

Some numerical examples are used to illustrate theapplicability of the proposed approach even in cases wherethe dimensions of the plant and the exosystem are differentbetween each otherThis suggests that the approach presentedin this work may be applied on a bigger class of T-S models

Finally the method proposed in this work avoids thedisadvantage of constructing an exact fuzzy regulator basedon overall T-S fuzzy system whose expression may resultvery large Instead the given approach offers 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 Interests

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

Acknowledgments

This work was partially supported by CONACYT (ConsejoNacional de Ciencia y Tecnologıa) through scholarship SNI(Sistema Nacional de Investigadores) and by IPN (InstitutoPolitecnico Nacional) through research Projects 20140659and 20150487 and scholarships EDI (Estımulo al Desempenode los Investigadores) COFAA (Comision de Operacion yFomento de Actividades Academicas) and BEIFI (Beca deEstımulo Institucional de Formacion de Investigadores)

References

[1] B A Francis ldquoThe linear multivariable regulator problemrdquoSIAM Journal on Control and Optimization vol 15 no 3 pp486ndash505 1977

[2] B A Francis andWMWonham ldquoThe internalmodel principleof control theoryrdquo Automatica vol 12 no 5 pp 457ndash465 1976

[3] A Isidori and C I Byrnes ldquoOutput regulation of nonlinearsystemsrdquo IEEE Transactions on Automatic Control vol 35 no2 pp 131ndash140 1990

[4] B Castillo-Toledo S Di Gennaro and F Jurado ldquoTrajectorytracking for a quadrotor via fuzzy regulationrdquo in Proceedings ofthe World Automation Congress (WAC rsquo12) pp 1ndash6 June 2012

[5] S Chen ldquoOutput regulation of nonlinear singularly perturbedsystems based on T-S fuzzy modelrdquo Journal of Control Theoryand Applications vol 3 no 4 pp 399ndash403 2005

[6] R Tapia-Herrera J A Meda-Campana S Alcamp119886119901119900119904 119886ntara-Montes T Hernamp119886119901119900119904 119886ndez-Cortamp119886119901119900119904 119890s and L Salgado-Conrado ldquoTuning of a TS fuzzy output regulator using thesteepest descent approach and ANFISrdquoMathematical Problemsin Engineering vol 2013 Article ID 873430 14 pages 2013

[7] K-Y Lian and J-J Liou ldquoOutput tracking control for fuzzysystems via output feedback designrdquo IEEE Transactions onFuzzy Systems vol 14 no 5 pp 628ndash639 2006

[8] X-J Ma and Z-Q Sun ldquoOutput tracking and regulationof nonlinear system based on takagi-sugeno fuzzy modelrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 30 no 1 pp 47ndash59 2000

[9] H J Lee J B Park and Y H Joo ldquoComments on lsquooutputtracking and regulation of nonlinear system based on Takagi-Sugeno fuzzy model1015840rdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 33 no 3 pp 521ndash523 2003

[10] J A Meda-Campana B Castillo-Toledo and G Chen ldquoSyn-chronization of chaotic systems from a fuzzy regulationapproachrdquo Fuzzy Sets and Systems vol 160 no 19 pp 2860ndash2875 2009

[11] J A Meda-Campana J C Gamp119886119901119900119904 119900mez-Mancilla and BCastillo-Toledo ldquoExact output regulation for nonlinear systemsdescribed by Takagi-Sugeno fuzzy modelsrdquo IEEE Transactionson Fuzzy Systems vol 20 no 2 pp 235ndash247 2012

[12] C Byrnes F Priscoli and A Isidori Output Regulation ofUncertain Nonlinear Systems Birkhauser 1997

[13] A Isidori Nonlinear Control Systems Communications andControl Engineering Series Springer Berlin Germany 3rdedition 1995

[14] K Tanaka and H O Wang Fuzzy Control Systems Designand Analysis A Linear Matrix Inequality Approach John Wileyamp119886119898119901 Sons New York NY USA 2001

[15] L-X Wang A Course in Fuzzy Systems and Control PrenticeHall Upper Saddle River NJ USA 1997

[16] I Abdelmalek N Golamp119886119901119900119904 119890a and M L Hadjili ldquoA newfuzzy Lyapunov approach to non-quadratic stabilization ofTakagi-Sugeno fuzzy modelsrdquo International Journal of AppliedMathematics andComputer Science vol 17 no 1 pp 39ndash51 2007

[17] J A Meda-Campana and B Castillo-Toledo ldquoOn the outputregulation for TS fuzzymodels using slidingmodesrdquo inProceed-ings of the American Control Conference (ACC rsquo05) pp 4062ndash4067 Portland Ore USA June 2005

[18] B Castillo-Toledo and J A Meda-Campana ldquoThe fuzzydiscrete-time robust regulation problem a LMI approachrdquo inProceedings of the 41st IEEE Conference on Decision and Controlvol 2 pp 2159ndash2164 IEEE Las Vegas Nev USA December2002

[19] B Castillo-Toledo J A Meda-Campana and A Titli ldquoAfuzzy output regulator for Takagi-Sugeno fuzzy modelsrdquo inProceedings of the IEEE International Symposium on IntelligentControl pp 310ndash315 Houston Tex USA October 2003

[20] B-C Ding H-X Sun and Y-E Qiao ldquoStability analysis of T-Sfuzzy control systems based on parameter-dependent Lyapunovfunctionrdquo Acta Automatica Sinica vol 31 no 4 pp 651ndash6542005

[21] M Narimani H K Lam R Dilmaghani and C Wolfe ldquoLMI-based stability analysis of fuzzy-model-based control systemsusing approximated polynomial membership functionsrdquo IEEE

18 Mathematical Problems in Engineering

Transactions on Systems Man and Cybernetics Part B Cyber-netics vol 41 no 3 pp 713ndash724 2011

[22] H K Lam ldquoPolynomial fuzzy-model-based control systemsstability analysis via piecewise-linear membership functionsrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 588ndash5932011

[23] K Tanaka T Hori and H O Wang ldquoA fuzzy Lyapunovapproach to fuzzy control system designrdquo in Proceedings of theAmerican Control Conference vol 6 pp 4790ndash4795 ArlingtonVa USA June 2001

[24] K Tanaka T Hori and H O Wang ldquoA multiple Lyapunovfunction approach to stabilization of fuzzy control systemsrdquoIEEE Transactions on Fuzzy Systems vol 11 no 4 pp 582ndash5892003

[25] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 174ndash1972005

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

Therefore the steady-state input for the overall fuzzy systemis

119906119904119904

=

[[[[[[[[[[[[[[

[

1205871sum119899

119894=1119904(119896+1)119894

119908119894minus sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

(119896+1)119895120587119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

119896+1

1205872sum119899

119894=1119904(119896+2)119894

119908119894minus sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

(119896+2)119895120587119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

119896+2

120587119899sum119899

119894=1119904119899119895119908119894minus sum119903

119894=1ℎ119894(119908 (119905)) (sum

119899

119895=1119886119894

119899119895120587119895119908119895)

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

119899

]]]]]]]]]]]]]]

]

(52)

But the desired form of the steady-state is

119906119904119904(119905) = 120574 (119908 (119905)) =

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905) (53)

Consequently the new membership functions 120583119894(119908(119905))

can be obtained after equating (52) and (53) resulting in

120583119894(119908 (119905)) =

[[[[[[

[

12058311989411

0 sdot sdot sdot 0

0 12058311989422

sdot sdot sdot 0

d

0 0 sdot sdot sdot 120583119894119898119898

]]]]]]

]

(54)

where

120583119894119895119897

=

119887119894

119896+119895ℎ119894(119908 (119905))

sum119903

119894=1ℎ119894(119908 (119905)) 119887

119894

119896+119895

(55)

for all 119894 = 1 119903 119895 = 119897 = 1 119898 It can be readily observedthat the sum of the corresponding elements is equal to oneand 120583

119894119895119897(119908(119905)) ge 0

These new membership functions 120583119894(119908(119905)) are organized

in a matrix form for that reason sum119903119894=1

120583119894(119908(119905)) = 119868 (where 119868

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 119861

119894have the

same sign at corresponding positions Clearly this conditionavoids singularities in (55) As before ℎ

119894(119908(119905)) is directly

obtained from ℎ119894(119909(119905)) by considering that in steady-state

119909(119905) = Π119908(119905) At this point the following theorem can bedefined

Theorem 5 The exact fuzzy output regulation with fullinformation for T-S fuzzy MIMO systems in the form of(11) defined by matrices (42) is solvable if (a) the sign ofentries for corresponding position inside input matrices 119861

119894for

119894 = 1 119903 is the same (b) there exists a fuzzy stabilizer119906(119905) = sum

119903

119894=1ℎ119894(119911(119905))119870

119894119909 for the fuzzy system [16] and (c) the

exosystem (119905) = 119878119908(119905) is Poisson stable

Moreover the Exact Output Fuzzy Regulation Problem forMIMO systems is solvable by the controller

119906 (119905) =

119903

sum

119894=1

ℎ119894(119909 (119905)) 119896

119894119909 (119905) minus Π119908 (119905)

+

119903

sum

119894=1

120583119894(119908 (119905)) Γ

119894119908 (119905)

(56)

where 120583119894(119908(119905)) can be readily obtained from (54) and (55)

Proof It follows directly from Theorem 3 and previousanalysis

Remark 6 Notice that 120583119894(119908(119905)) is defined as a matrix and

this is not a typical membership function however 120583119894(119908(119905))

contains a set of membership functions related to each entryof 119861119894 This form allows us to design controller (20)

5 Numerical Examples

51 Simple Input Case Consider the problem of balancingand swing of an inverted pendulum on a cart The motionequations for this system are [14]

1(119905) = 119909

2(119905) (57)

2(119905)

=119892 sin(119909

1(119905))minus 119886119898119897119909

2

2sin (2119909

1(119905))2 minus119886 cos (119909

1(119905)) 119906 (119905)

41198973 minus 119886119898119897 cos2 (1199091(119905))

(58)

where 1199091(119905) denotes the angle (in radians) of the pendulum

from the vertical and 1199092(119905) is the angular velocity 119892 =

98ms2 is the gravity constant 119898 is the mass of the pen-dulum 119872 is the mass of the cart 2119897 is the length of thependulum and 119906 is the force applied to the cart (in newtons)119886 = 1(119898 +119872) with119898 = 20 kg119872 = 80 kg and 2119897 = 10min the simulations The goal is to track the reference definedby the exosystem within the range 119909

1isin (minus1205872 1205872) To this

end a fuzzy model representing the nonlinear dynamics is asfollows

Rule 1

IF 1199091(119905) is about 0

THEN (119905) = 1198601119909(119905) + 119861

1119906(119905)

Rule 2

IF 1199091(119905) is about plusmn1205872(|119909

1| lt 1205872)

THEN (119905) = 1198602119909(119905) + 119861

2119906(119905)

10 Mathematical Problems in Engineering

h1

h2

x1minus90 90

(deg)

1

00

Figure 1 Membership functions of two-rule model

Membership functions for Rules 1 and 2 are shown in Figure 1In this example the T-S fuzzymodel of the plant is defined bymatrices

1198601= [

0 1

172941 0] 119861

1= [

0

minus01765]

1198602= [

0 1

41585 0] 119861

2= [

0

minus01500]

1198621= 1198622= [1 0]

(59)

while the exosystem can be constructed from the followingmatrices

119878 = [

0 1

minus1 0] 119876 = [1 0] (60)

which allows us to generate a great kind of sinusoidal refer-ences Figure 2 shows the simulation results after applying thecontroller

119906 (119905) =

119903

sum

119894=1

ℎ119894(119909 (119905)) 119870

119894[119909 (119905) minus

119903

sum

119894=1

ℎ119894(119908 (119905)) Π

119894119908 (119905)]

+

119903

sum

119894=1

ℎ119894(119908 (119905)) Γ

119894119908 (119905)

(61)

defined in [10] As expected controller (61) is unable toachieve exact regulation although tracking error remainsbounded

Now using the method derived in this work the localmatrices Γ

119894can be readily obtained from (28) resulting in

Γ1= [1036667 0] Γ

2= [343836 0] (62)

Besides knowing that 11990211198881= 1 matrix Π is obtained from

(24)

Π = [

1 0

0 1] (63)

while the newmembership functions computed from (39) are

1205831(119908 (119905)) =

(ℎ1(119908 (119905))) (minus01765)

ℎ1(119908 (119905)) (minus01765) + ℎ

2(119908 (119905)) (minus01500)

1205832(119908 (119905)) =

(ℎ2(119908 (119905))) (minus01500)

ℎ1(119908 (119905)) (minus01765) + ℎ

2(119908 (119905)) (minus01500)

(64)

On the other hand the fuzzy stabilizer for this system isconstructed bymeans of the PDC approach developed in [14]which is formed by

1198701= [1234898 69979] 119870

2= [586405 77962]

(65)

Figure 3 shows simulation with the controller definedin (20) Observe how the new membership functions arecapable of combining the local regulators in the proper wayto accomplish asymptotic tracking

To verify the effectiveness of the latter regulator thefuzzy controller is applied on the original system (57) Thesimulation results appear in Figure 4 Notice that the trackingerror depicted in Figure 4(b) is due to the approximationprovided by the T-S fuzzy model Such an error can bereduced by considering a better approximation or an exactrepresentation of the nonlinear system as the one given bythe nonlinear sector approach [14]

52 Multiple Inputs and Multiple Outputs (MIMO) CaseNow consider a T-S fuzzy system of two rules defined by thefollowing matrices

1198601=

[[[[[

[

0 0 1 0

0 0 0 1

6 4 2 4

2 3 1 5

]]]]]

]

1198611=

[[[[[

[

0 0

0 0

minus7 0

0 2

]]]]]

]

1198602=

[[[[[

[

0 0 1 0

0 0 0 1

3 4 2 1

2 4 6 2

]]]]]

]

1198612=

[[[[[

[

0 0

0 0

minus3 0

0 5

]]]]]

]

1198621= 1198622= [

1 0 0 0

0 1 0 0]

(66)

Mathematical Problems in Engineering 11

0 5 10 15 20 25 30 35 40minus40minus30minus20minus10

010203040

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30 35 40minus04minus03minus02minus01

0010203

Tracking error

Time

Am

plitu

de

e

(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 40minus40minus30minus20minus10

0102030

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30 35 40minus06minus05minus04minus03minus02minus01

0010203

Tracking error

Time

Am

plitu

de

e

(b)

Figure 3 (a) Output reference and (b) tracking errorWhen the fuzzy controller is designed on local regulators and the adequatemembershipfunctions for the fuzzy regulator are computed

0 5 10 15 20 25 30 35 40minus40minus30minus20minus10

010203040

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30 35 40minus06minus05minus04minus03minus02minus01

001

Tracking error

Time

Am

plitu

de

e

(b)

Figure 4 (a) Output reference and (b) tracking errorWhen the fuzzy controller is designed on local regulators and the adequatemembershipfunctions for the fuzzy regulator are computed

12 Mathematical Problems in Engineering

h1 h2

x1minus60

1

80

0

0

Figure 5 Membership functions of two-rule model

and an exosystem defined by

119878 =

[[[[[

[

0 0 1 0

0 0 0 1

minus4 0 0 0

0 minus16 0 0

]]]]]

]

119876 = [

1 0 0 0

0 1 0 0]

(67)

Membership functions forRules 1 and 2 are shown in Figure 5Matrices defining the local regulators can be easily obtainedby using (49) For this example they are

Γ1= [

14286 05714 02857 05714

minus10000 minus95000 minus05000 minus25000]

Γ2= [

23333 13333 06667 03333

minus04000 minus40000 minus12000 minus04000]

(68)

On the other hand 11990211198881= 1205871= 1 119902

21198882= 1205872= 1

11990211198881= 1205873= 1 119902

21198882= 1205874= 1 and by (46) and matrix Π is

Π =

[[[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

]]]]]

]

(69)

Figures 6 and 7 show the results obtained using linear regula-tors defined in (61)

Now by computing the new membership functions forthe fuzzy regulator through (54) one gets1205831(119908 (119905))

=

[[[

[

(ℎ1(119908 (119905))) (minus7)

ℎ1(119908 (119905)) (3) + ℎ

2(119908 (119905)) (3)

0

0(ℎ1(119908 (119905))) (2)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (5)

]]]

]

1205832(119908 (119905))

=

[[[

[

(ℎ2(119908 (119905))) (minus3)

ℎ1(119908 (119905)) (3) + ℎ

2(119908 (119905)) (3)

0

0(ℎ2(119908 (119905))) (5)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (5)

]]]

]

(70)

As previously the fuzzy stabilizer for this example is alsoconstructed bymeans of the PDC approach developed in [14]resulting in

1198701= [

10286 05714 06000 05714

minus10000 minus22500 minus05000 minus37500]

1198702= [

14000 13333 14000 03333

minus04000 minus11000 minus12000 minus09000]

(71)

Thus Figures 8 and 9 show the simulation results whenthe controller defined by (20) is applied

Besides Figure 10 shows the shape of the new member-ship functions which are clearly different from the originalones given in Figure 5

53 More General Problems Consider the T-S fuzzy MIMOsystem with multiple inputs defined by the following matri-ces

1198601=

[[[[[[[[[[[

[

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

minus6 4 2 4 5 3

2 3 minus1 5 minus2 4

]]]]]]]]]]]

]

1198611=

[[[[[[[[[[[

[

0 0

0 0

0 0

0 0

5 0

0 2

]]]]]]]]]]]

]

1198602=

[[[[[[[[[[[

[

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

minus6 1 2 minus5 4 6

0 2 minus7 4 5 2

]]]]]]]]]]]

]

1198612=

[[[[[[[[[[[

[

0 0

0 0

0 0

0 0

7 0

0 8

]]]]]]]]]]]

]

1198621= 1198622= [

0 1 1 0 0 0

1 0 1 0 0 0]

(72)

Mathematical Problems in Engineering 13

0 5 10 15 20 25 30minus15minus10

minus505

101520

Time

Am

plitu

dex1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus4minus2

02468

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 6 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus30minus20minus10

0102030

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus15

minus10

minus5

0

5

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 7 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of lineal regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus15minus10

minus505

101520

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus1

012345678

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 8 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

The reference signals will be generated by an exosystemdefined by

119878 = [

0 1

minus1 0] 119876 = [

1 0

0 1] (73)

Membership functions for Rules 1 and 2 are ℎ1(119909(119905)) =

(80 minus 1199091(119905))140 and ℎ

2(119909(119905)) = 1 minus ℎ

1(119905) Notice that for this

case 119899 = 119904 and 119862119894

= 119876 This implies that Π119894and Γ

119894must

be obtained by solving Francis Equations for each subsystemHowever due to form of the fuzzy plant it is possible to verifythat Π

1= Π2

14 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus40minus30minus20minus10

0102030

Time

Am

plitu

dex2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus10

minus8minus6minus4minus2

0

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 9 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(11) (x1(t))1205832(11) (x1(t))

(a)

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(22) (x1(t))1205832(22) (x1(t))

(b)

Figure 10 (a) Membership functions corresponding to position (11) of 1205831(119908(119905)) and 120583

2(119908(119905)) (b) Membership functions corresponding to

position (22) 1205831(119908(119905)) and 120583

2(119908(119905))

So by applying (9) for each local subsystem with 119894 = 1 2it results in

Π1= Π2=

[[[[[[[[[[[

[

05000 05000

15000 minus05000

minus05000 05000

05000 15000

minus05000 minus05000

minus15000 05000

]]]]]]]]]]]

]

Γ1= [

07000 minus03000

minus20000 minus55000]

Γ2= [

23571 12143

minus04375 minus01875]

(74)

Again the fuzzy stabilizer for this system is constructedby means of the PDC approach developed in [14] yielding

1198701=[[

[

minus12229 00609 minus40284 01291 minus25666 minus04061

16919 minus83997 32934 minus127153 15732 minus61836

]]

]

1198702=[[

[

minus08735 04721 minus28775 13780 minus16904 minus07187

06730 minus19749 15734 minus30538 minus04817 minus12959

]]

]

(75)

The behavior obtained by applying controller (61) isdepicted in Figures 11 and 12 Notice the exact outputregulation is not achieved

Mathematical Problems in Engineering 15

0 5 10 15 20 25 30minus60minus40minus20

0204060

Time

Am

plitu

dex1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus80minus70minus60minus50minus40minus30minus20minus10

01020

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 11 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus15

minus10

minus5

0

5

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 12 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

Now the adequate membership functions are computedfrom (54) resulting in

1205831(119908 (119905))

=

[[[

[

(ℎ1(119908 (119905))) (5)

ℎ1(119908 (119905)) (5) + ℎ

2(119908 (119905)) (7)

0

0(ℎ1(119908 (119905))) (2)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (8)

]]]

]

1205832(119908 (119905))

=

[[[

[

(ℎ2(119908 (119905))) (7)

ℎ1(119908 (119905)) (5) + ℎ

2(119908 (119905)) (7)

0

0(ℎ2(119908 (119905))) (8)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (8)

]]]

]

(76)

When ℎ119894(119908(119905)) is replaced by its corresponding expressions

one gets

1205831(119908 (119905)) =

[[[

[

1225

1199081(119905) + 140

minus5

20

0 minus1199081(119905) minus 80

31199081(119905) + 320

]]]

]

1205832(119908 (119905)) =

[[[

[

7

2minus

1225

1199081(119905) + 410

0

041199081(119905) + 240

31199081(119905) + 320

]]]

]

(77)

The results of applying control law (20) on the fuzzy plantare given in Figures 13 and 14

The ldquocomputedrdquo membership functions for the fuzzyregulator are given in Figure 15 Again it can be readilyobserved that these new membership functions are differentfrom the original ones

Notice that although this example does not fulfill theconditions ofTheorem 5 themembership functions obtainedfrom (39) allow the exact tracking for the fuzzy problemThis

16 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus80minus70minus60minus50minus40minus30minus20minus10

010

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 13 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus8minus6minus4minus2

02468

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 14 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(11) (x1(t))1205832(11) (x1(t))

(a)

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(22) (x1(t))1205832(22) (x1(t))

(b)

Figure 15 (a) Membership functions corresponding to position (11) of 1205831(119908(119905)) and 120583

2(119908(119905)) (b) Membership functions corresponding to

position (22) 1205831(119908(119905)) 120583

2(119908(119905))

Mathematical Problems in Engineering 17

suggests that the approach presented in this work may beapplied on a bigger class of T-S models however this studyis not completed yet

6 Conclusions

A fuzzy regulator for continuous-time systems based on thecombination of linear regulators adjusted by different mem-bership functions has been presented The main advantageis that analytical expressions for the membership functionswhich allow the proper combination of the local regulators inorder to guarantee the exact regulation can be easily appliedAs 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 modelscan be commonly found in mechanics electromechanicselectronics robotics and so forth

Some numerical examples are used to illustrate theapplicability of the proposed approach even in cases wherethe dimensions of the plant and the exosystem are differentbetween each otherThis suggests that the approach presentedin this work may be applied on a bigger class of T-S models

Finally the method proposed in this work avoids thedisadvantage of constructing an exact fuzzy regulator basedon overall T-S fuzzy system whose expression may resultvery large Instead the given approach offers 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 Interests

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

Acknowledgments

This work was partially supported by CONACYT (ConsejoNacional de Ciencia y Tecnologıa) through scholarship SNI(Sistema Nacional de Investigadores) and by IPN (InstitutoPolitecnico Nacional) through research Projects 20140659and 20150487 and scholarships EDI (Estımulo al Desempenode los Investigadores) COFAA (Comision de Operacion yFomento de Actividades Academicas) and BEIFI (Beca deEstımulo Institucional de Formacion de Investigadores)

References

[1] B A Francis ldquoThe linear multivariable regulator problemrdquoSIAM Journal on Control and Optimization vol 15 no 3 pp486ndash505 1977

[2] B A Francis andWMWonham ldquoThe internalmodel principleof control theoryrdquo Automatica vol 12 no 5 pp 457ndash465 1976

[3] A Isidori and C I Byrnes ldquoOutput regulation of nonlinearsystemsrdquo IEEE Transactions on Automatic Control vol 35 no2 pp 131ndash140 1990

[4] B Castillo-Toledo S Di Gennaro and F Jurado ldquoTrajectorytracking for a quadrotor via fuzzy regulationrdquo in Proceedings ofthe World Automation Congress (WAC rsquo12) pp 1ndash6 June 2012

[5] S Chen ldquoOutput regulation of nonlinear singularly perturbedsystems based on T-S fuzzy modelrdquo Journal of Control Theoryand Applications vol 3 no 4 pp 399ndash403 2005

[6] R Tapia-Herrera J A Meda-Campana S Alcamp119886119901119900119904 119886ntara-Montes T Hernamp119886119901119900119904 119886ndez-Cortamp119886119901119900119904 119890s and L Salgado-Conrado ldquoTuning of a TS fuzzy output regulator using thesteepest descent approach and ANFISrdquoMathematical Problemsin Engineering vol 2013 Article ID 873430 14 pages 2013

[7] K-Y Lian and J-J Liou ldquoOutput tracking control for fuzzysystems via output feedback designrdquo IEEE Transactions onFuzzy Systems vol 14 no 5 pp 628ndash639 2006

[8] X-J Ma and Z-Q Sun ldquoOutput tracking and regulationof nonlinear system based on takagi-sugeno fuzzy modelrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 30 no 1 pp 47ndash59 2000

[9] H J Lee J B Park and Y H Joo ldquoComments on lsquooutputtracking and regulation of nonlinear system based on Takagi-Sugeno fuzzy model1015840rdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 33 no 3 pp 521ndash523 2003

[10] J A Meda-Campana B Castillo-Toledo and G Chen ldquoSyn-chronization of chaotic systems from a fuzzy regulationapproachrdquo Fuzzy Sets and Systems vol 160 no 19 pp 2860ndash2875 2009

[11] J A Meda-Campana J C Gamp119886119901119900119904 119900mez-Mancilla and BCastillo-Toledo ldquoExact output regulation for nonlinear systemsdescribed by Takagi-Sugeno fuzzy modelsrdquo IEEE Transactionson Fuzzy Systems vol 20 no 2 pp 235ndash247 2012

[12] C Byrnes F Priscoli and A Isidori Output Regulation ofUncertain Nonlinear Systems Birkhauser 1997

[13] A Isidori Nonlinear Control Systems Communications andControl Engineering Series Springer Berlin Germany 3rdedition 1995

[14] K Tanaka and H O Wang Fuzzy Control Systems Designand Analysis A Linear Matrix Inequality Approach John Wileyamp119886119898119901 Sons New York NY USA 2001

[15] L-X Wang A Course in Fuzzy Systems and Control PrenticeHall Upper Saddle River NJ USA 1997

[16] I Abdelmalek N Golamp119886119901119900119904 119890a and M L Hadjili ldquoA newfuzzy Lyapunov approach to non-quadratic stabilization ofTakagi-Sugeno fuzzy modelsrdquo International Journal of AppliedMathematics andComputer Science vol 17 no 1 pp 39ndash51 2007

[17] J A Meda-Campana and B Castillo-Toledo ldquoOn the outputregulation for TS fuzzymodels using slidingmodesrdquo inProceed-ings of the American Control Conference (ACC rsquo05) pp 4062ndash4067 Portland Ore USA June 2005

[18] B Castillo-Toledo and J A Meda-Campana ldquoThe fuzzydiscrete-time robust regulation problem a LMI approachrdquo inProceedings of the 41st IEEE Conference on Decision and Controlvol 2 pp 2159ndash2164 IEEE Las Vegas Nev USA December2002

[19] B Castillo-Toledo J A Meda-Campana and A Titli ldquoAfuzzy output regulator for Takagi-Sugeno fuzzy modelsrdquo inProceedings of the IEEE International Symposium on IntelligentControl pp 310ndash315 Houston Tex USA October 2003

[20] B-C Ding H-X Sun and Y-E Qiao ldquoStability analysis of T-Sfuzzy control systems based on parameter-dependent Lyapunovfunctionrdquo Acta Automatica Sinica vol 31 no 4 pp 651ndash6542005

[21] M Narimani H K Lam R Dilmaghani and C Wolfe ldquoLMI-based stability analysis of fuzzy-model-based control systemsusing approximated polynomial membership functionsrdquo IEEE

18 Mathematical Problems in Engineering

Transactions on Systems Man and Cybernetics Part B Cyber-netics vol 41 no 3 pp 713ndash724 2011

[22] H K Lam ldquoPolynomial fuzzy-model-based control systemsstability analysis via piecewise-linear membership functionsrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 588ndash5932011

[23] K Tanaka T Hori and H O Wang ldquoA fuzzy Lyapunovapproach to fuzzy control system designrdquo in Proceedings of theAmerican Control Conference vol 6 pp 4790ndash4795 ArlingtonVa USA June 2001

[24] K Tanaka T Hori and H O Wang ldquoA multiple Lyapunovfunction approach to stabilization of fuzzy control systemsrdquoIEEE Transactions on Fuzzy Systems vol 11 no 4 pp 582ndash5892003

[25] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 174ndash1972005

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

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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

h1

h2

x1minus90 90

(deg)

1

00

Figure 1 Membership functions of two-rule model

Membership functions for Rules 1 and 2 are shown in Figure 1In this example the T-S fuzzymodel of the plant is defined bymatrices

1198601= [

0 1

172941 0] 119861

1= [

0

minus01765]

1198602= [

0 1

41585 0] 119861

2= [

0

minus01500]

1198621= 1198622= [1 0]

(59)

while the exosystem can be constructed from the followingmatrices

119878 = [

0 1

minus1 0] 119876 = [1 0] (60)

which allows us to generate a great kind of sinusoidal refer-ences Figure 2 shows the simulation results after applying thecontroller

119906 (119905) =

119903

sum

119894=1

ℎ119894(119909 (119905)) 119870

119894[119909 (119905) minus

119903

sum

119894=1

ℎ119894(119908 (119905)) Π

119894119908 (119905)]

+

119903

sum

119894=1

ℎ119894(119908 (119905)) Γ

119894119908 (119905)

(61)

defined in [10] As expected controller (61) is unable toachieve exact regulation although tracking error remainsbounded

Now using the method derived in this work the localmatrices Γ

119894can be readily obtained from (28) resulting in

Γ1= [1036667 0] Γ

2= [343836 0] (62)

Besides knowing that 11990211198881= 1 matrix Π is obtained from

(24)

Π = [

1 0

0 1] (63)

while the newmembership functions computed from (39) are

1205831(119908 (119905)) =

(ℎ1(119908 (119905))) (minus01765)

ℎ1(119908 (119905)) (minus01765) + ℎ

2(119908 (119905)) (minus01500)

1205832(119908 (119905)) =

(ℎ2(119908 (119905))) (minus01500)

ℎ1(119908 (119905)) (minus01765) + ℎ

2(119908 (119905)) (minus01500)

(64)

On the other hand the fuzzy stabilizer for this system isconstructed bymeans of the PDC approach developed in [14]which is formed by

1198701= [1234898 69979] 119870

2= [586405 77962]

(65)

Figure 3 shows simulation with the controller definedin (20) Observe how the new membership functions arecapable of combining the local regulators in the proper wayto accomplish asymptotic tracking

To verify the effectiveness of the latter regulator thefuzzy controller is applied on the original system (57) Thesimulation results appear in Figure 4 Notice that the trackingerror depicted in Figure 4(b) is due to the approximationprovided by the T-S fuzzy model Such an error can bereduced by considering a better approximation or an exactrepresentation of the nonlinear system as the one given bythe nonlinear sector approach [14]

52 Multiple Inputs and Multiple Outputs (MIMO) CaseNow consider a T-S fuzzy system of two rules defined by thefollowing matrices

1198601=

[[[[[

[

0 0 1 0

0 0 0 1

6 4 2 4

2 3 1 5

]]]]]

]

1198611=

[[[[[

[

0 0

0 0

minus7 0

0 2

]]]]]

]

1198602=

[[[[[

[

0 0 1 0

0 0 0 1

3 4 2 1

2 4 6 2

]]]]]

]

1198612=

[[[[[

[

0 0

0 0

minus3 0

0 5

]]]]]

]

1198621= 1198622= [

1 0 0 0

0 1 0 0]

(66)

Mathematical Problems in Engineering 11

0 5 10 15 20 25 30 35 40minus40minus30minus20minus10

010203040

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30 35 40minus04minus03minus02minus01

0010203

Tracking error

Time

Am

plitu

de

e

(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 40minus40minus30minus20minus10

0102030

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30 35 40minus06minus05minus04minus03minus02minus01

0010203

Tracking error

Time

Am

plitu

de

e

(b)

Figure 3 (a) Output reference and (b) tracking errorWhen the fuzzy controller is designed on local regulators and the adequatemembershipfunctions for the fuzzy regulator are computed

0 5 10 15 20 25 30 35 40minus40minus30minus20minus10

010203040

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30 35 40minus06minus05minus04minus03minus02minus01

001

Tracking error

Time

Am

plitu

de

e

(b)

Figure 4 (a) Output reference and (b) tracking errorWhen the fuzzy controller is designed on local regulators and the adequatemembershipfunctions for the fuzzy regulator are computed

12 Mathematical Problems in Engineering

h1 h2

x1minus60

1

80

0

0

Figure 5 Membership functions of two-rule model

and an exosystem defined by

119878 =

[[[[[

[

0 0 1 0

0 0 0 1

minus4 0 0 0

0 minus16 0 0

]]]]]

]

119876 = [

1 0 0 0

0 1 0 0]

(67)

Membership functions forRules 1 and 2 are shown in Figure 5Matrices defining the local regulators can be easily obtainedby using (49) For this example they are

Γ1= [

14286 05714 02857 05714

minus10000 minus95000 minus05000 minus25000]

Γ2= [

23333 13333 06667 03333

minus04000 minus40000 minus12000 minus04000]

(68)

On the other hand 11990211198881= 1205871= 1 119902

21198882= 1205872= 1

11990211198881= 1205873= 1 119902

21198882= 1205874= 1 and by (46) and matrix Π is

Π =

[[[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

]]]]]

]

(69)

Figures 6 and 7 show the results obtained using linear regula-tors defined in (61)

Now by computing the new membership functions forthe fuzzy regulator through (54) one gets1205831(119908 (119905))

=

[[[

[

(ℎ1(119908 (119905))) (minus7)

ℎ1(119908 (119905)) (3) + ℎ

2(119908 (119905)) (3)

0

0(ℎ1(119908 (119905))) (2)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (5)

]]]

]

1205832(119908 (119905))

=

[[[

[

(ℎ2(119908 (119905))) (minus3)

ℎ1(119908 (119905)) (3) + ℎ

2(119908 (119905)) (3)

0

0(ℎ2(119908 (119905))) (5)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (5)

]]]

]

(70)

As previously the fuzzy stabilizer for this example is alsoconstructed bymeans of the PDC approach developed in [14]resulting in

1198701= [

10286 05714 06000 05714

minus10000 minus22500 minus05000 minus37500]

1198702= [

14000 13333 14000 03333

minus04000 minus11000 minus12000 minus09000]

(71)

Thus Figures 8 and 9 show the simulation results whenthe controller defined by (20) is applied

Besides Figure 10 shows the shape of the new member-ship functions which are clearly different from the originalones given in Figure 5

53 More General Problems Consider the T-S fuzzy MIMOsystem with multiple inputs defined by the following matri-ces

1198601=

[[[[[[[[[[[

[

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

minus6 4 2 4 5 3

2 3 minus1 5 minus2 4

]]]]]]]]]]]

]

1198611=

[[[[[[[[[[[

[

0 0

0 0

0 0

0 0

5 0

0 2

]]]]]]]]]]]

]

1198602=

[[[[[[[[[[[

[

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

minus6 1 2 minus5 4 6

0 2 minus7 4 5 2

]]]]]]]]]]]

]

1198612=

[[[[[[[[[[[

[

0 0

0 0

0 0

0 0

7 0

0 8

]]]]]]]]]]]

]

1198621= 1198622= [

0 1 1 0 0 0

1 0 1 0 0 0]

(72)

Mathematical Problems in Engineering 13

0 5 10 15 20 25 30minus15minus10

minus505

101520

Time

Am

plitu

dex1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus4minus2

02468

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 6 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus30minus20minus10

0102030

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus15

minus10

minus5

0

5

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 7 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of lineal regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus15minus10

minus505

101520

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus1

012345678

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 8 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

The reference signals will be generated by an exosystemdefined by

119878 = [

0 1

minus1 0] 119876 = [

1 0

0 1] (73)

Membership functions for Rules 1 and 2 are ℎ1(119909(119905)) =

(80 minus 1199091(119905))140 and ℎ

2(119909(119905)) = 1 minus ℎ

1(119905) Notice that for this

case 119899 = 119904 and 119862119894

= 119876 This implies that Π119894and Γ

119894must

be obtained by solving Francis Equations for each subsystemHowever due to form of the fuzzy plant it is possible to verifythat Π

1= Π2

14 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus40minus30minus20minus10

0102030

Time

Am

plitu

dex2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus10

minus8minus6minus4minus2

0

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 9 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(11) (x1(t))1205832(11) (x1(t))

(a)

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(22) (x1(t))1205832(22) (x1(t))

(b)

Figure 10 (a) Membership functions corresponding to position (11) of 1205831(119908(119905)) and 120583

2(119908(119905)) (b) Membership functions corresponding to

position (22) 1205831(119908(119905)) and 120583

2(119908(119905))

So by applying (9) for each local subsystem with 119894 = 1 2it results in

Π1= Π2=

[[[[[[[[[[[

[

05000 05000

15000 minus05000

minus05000 05000

05000 15000

minus05000 minus05000

minus15000 05000

]]]]]]]]]]]

]

Γ1= [

07000 minus03000

minus20000 minus55000]

Γ2= [

23571 12143

minus04375 minus01875]

(74)

Again the fuzzy stabilizer for this system is constructedby means of the PDC approach developed in [14] yielding

1198701=[[

[

minus12229 00609 minus40284 01291 minus25666 minus04061

16919 minus83997 32934 minus127153 15732 minus61836

]]

]

1198702=[[

[

minus08735 04721 minus28775 13780 minus16904 minus07187

06730 minus19749 15734 minus30538 minus04817 minus12959

]]

]

(75)

The behavior obtained by applying controller (61) isdepicted in Figures 11 and 12 Notice the exact outputregulation is not achieved

Mathematical Problems in Engineering 15

0 5 10 15 20 25 30minus60minus40minus20

0204060

Time

Am

plitu

dex1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus80minus70minus60minus50minus40minus30minus20minus10

01020

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 11 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus15

minus10

minus5

0

5

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 12 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

Now the adequate membership functions are computedfrom (54) resulting in

1205831(119908 (119905))

=

[[[

[

(ℎ1(119908 (119905))) (5)

ℎ1(119908 (119905)) (5) + ℎ

2(119908 (119905)) (7)

0

0(ℎ1(119908 (119905))) (2)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (8)

]]]

]

1205832(119908 (119905))

=

[[[

[

(ℎ2(119908 (119905))) (7)

ℎ1(119908 (119905)) (5) + ℎ

2(119908 (119905)) (7)

0

0(ℎ2(119908 (119905))) (8)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (8)

]]]

]

(76)

When ℎ119894(119908(119905)) is replaced by its corresponding expressions

one gets

1205831(119908 (119905)) =

[[[

[

1225

1199081(119905) + 140

minus5

20

0 minus1199081(119905) minus 80

31199081(119905) + 320

]]]

]

1205832(119908 (119905)) =

[[[

[

7

2minus

1225

1199081(119905) + 410

0

041199081(119905) + 240

31199081(119905) + 320

]]]

]

(77)

The results of applying control law (20) on the fuzzy plantare given in Figures 13 and 14

The ldquocomputedrdquo membership functions for the fuzzyregulator are given in Figure 15 Again it can be readilyobserved that these new membership functions are differentfrom the original ones

Notice that although this example does not fulfill theconditions ofTheorem 5 themembership functions obtainedfrom (39) allow the exact tracking for the fuzzy problemThis

16 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus80minus70minus60minus50minus40minus30minus20minus10

010

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 13 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus8minus6minus4minus2

02468

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 14 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(11) (x1(t))1205832(11) (x1(t))

(a)

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(22) (x1(t))1205832(22) (x1(t))

(b)

Figure 15 (a) Membership functions corresponding to position (11) of 1205831(119908(119905)) and 120583

2(119908(119905)) (b) Membership functions corresponding to

position (22) 1205831(119908(119905)) 120583

2(119908(119905))

Mathematical Problems in Engineering 17

suggests that the approach presented in this work may beapplied on a bigger class of T-S models however this studyis not completed yet

6 Conclusions

A fuzzy regulator for continuous-time systems based on thecombination of linear regulators adjusted by different mem-bership functions has been presented The main advantageis that analytical expressions for the membership functionswhich allow the proper combination of the local regulators inorder to guarantee the exact regulation can be easily appliedAs 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 modelscan be commonly found in mechanics electromechanicselectronics robotics and so forth

Some numerical examples are used to illustrate theapplicability of the proposed approach even in cases wherethe dimensions of the plant and the exosystem are differentbetween each otherThis suggests that the approach presentedin this work may be applied on a bigger class of T-S models

Finally the method proposed in this work avoids thedisadvantage of constructing an exact fuzzy regulator basedon overall T-S fuzzy system whose expression may resultvery large Instead the given approach offers 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 Interests

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

Acknowledgments

This work was partially supported by CONACYT (ConsejoNacional de Ciencia y Tecnologıa) through scholarship SNI(Sistema Nacional de Investigadores) and by IPN (InstitutoPolitecnico Nacional) through research Projects 20140659and 20150487 and scholarships EDI (Estımulo al Desempenode los Investigadores) COFAA (Comision de Operacion yFomento de Actividades Academicas) and BEIFI (Beca deEstımulo Institucional de Formacion de Investigadores)

References

[1] B A Francis ldquoThe linear multivariable regulator problemrdquoSIAM Journal on Control and Optimization vol 15 no 3 pp486ndash505 1977

[2] B A Francis andWMWonham ldquoThe internalmodel principleof control theoryrdquo Automatica vol 12 no 5 pp 457ndash465 1976

[3] A Isidori and C I Byrnes ldquoOutput regulation of nonlinearsystemsrdquo IEEE Transactions on Automatic Control vol 35 no2 pp 131ndash140 1990

[4] B Castillo-Toledo S Di Gennaro and F Jurado ldquoTrajectorytracking for a quadrotor via fuzzy regulationrdquo in Proceedings ofthe World Automation Congress (WAC rsquo12) pp 1ndash6 June 2012

[5] S Chen ldquoOutput regulation of nonlinear singularly perturbedsystems based on T-S fuzzy modelrdquo Journal of Control Theoryand Applications vol 3 no 4 pp 399ndash403 2005

[6] R Tapia-Herrera J A Meda-Campana S Alcamp119886119901119900119904 119886ntara-Montes T Hernamp119886119901119900119904 119886ndez-Cortamp119886119901119900119904 119890s and L Salgado-Conrado ldquoTuning of a TS fuzzy output regulator using thesteepest descent approach and ANFISrdquoMathematical Problemsin Engineering vol 2013 Article ID 873430 14 pages 2013

[7] K-Y Lian and J-J Liou ldquoOutput tracking control for fuzzysystems via output feedback designrdquo IEEE Transactions onFuzzy Systems vol 14 no 5 pp 628ndash639 2006

[8] X-J Ma and Z-Q Sun ldquoOutput tracking and regulationof nonlinear system based on takagi-sugeno fuzzy modelrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 30 no 1 pp 47ndash59 2000

[9] H J Lee J B Park and Y H Joo ldquoComments on lsquooutputtracking and regulation of nonlinear system based on Takagi-Sugeno fuzzy model1015840rdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 33 no 3 pp 521ndash523 2003

[10] J A Meda-Campana B Castillo-Toledo and G Chen ldquoSyn-chronization of chaotic systems from a fuzzy regulationapproachrdquo Fuzzy Sets and Systems vol 160 no 19 pp 2860ndash2875 2009

[11] J A Meda-Campana J C Gamp119886119901119900119904 119900mez-Mancilla and BCastillo-Toledo ldquoExact output regulation for nonlinear systemsdescribed by Takagi-Sugeno fuzzy modelsrdquo IEEE Transactionson Fuzzy Systems vol 20 no 2 pp 235ndash247 2012

[12] C Byrnes F Priscoli and A Isidori Output Regulation ofUncertain Nonlinear Systems Birkhauser 1997

[13] A Isidori Nonlinear Control Systems Communications andControl Engineering Series Springer Berlin Germany 3rdedition 1995

[14] K Tanaka and H O Wang Fuzzy Control Systems Designand Analysis A Linear Matrix Inequality Approach John Wileyamp119886119898119901 Sons New York NY USA 2001

[15] L-X Wang A Course in Fuzzy Systems and Control PrenticeHall Upper Saddle River NJ USA 1997

[16] I Abdelmalek N Golamp119886119901119900119904 119890a and M L Hadjili ldquoA newfuzzy Lyapunov approach to non-quadratic stabilization ofTakagi-Sugeno fuzzy modelsrdquo International Journal of AppliedMathematics andComputer Science vol 17 no 1 pp 39ndash51 2007

[17] J A Meda-Campana and B Castillo-Toledo ldquoOn the outputregulation for TS fuzzymodels using slidingmodesrdquo inProceed-ings of the American Control Conference (ACC rsquo05) pp 4062ndash4067 Portland Ore USA June 2005

[18] B Castillo-Toledo and J A Meda-Campana ldquoThe fuzzydiscrete-time robust regulation problem a LMI approachrdquo inProceedings of the 41st IEEE Conference on Decision and Controlvol 2 pp 2159ndash2164 IEEE Las Vegas Nev USA December2002

[19] B Castillo-Toledo J A Meda-Campana and A Titli ldquoAfuzzy output regulator for Takagi-Sugeno fuzzy modelsrdquo inProceedings of the IEEE International Symposium on IntelligentControl pp 310ndash315 Houston Tex USA October 2003

[20] B-C Ding H-X Sun and Y-E Qiao ldquoStability analysis of T-Sfuzzy control systems based on parameter-dependent Lyapunovfunctionrdquo Acta Automatica Sinica vol 31 no 4 pp 651ndash6542005

[21] M Narimani H K Lam R Dilmaghani and C Wolfe ldquoLMI-based stability analysis of fuzzy-model-based control systemsusing approximated polynomial membership functionsrdquo IEEE

18 Mathematical Problems in Engineering

Transactions on Systems Man and Cybernetics Part B Cyber-netics vol 41 no 3 pp 713ndash724 2011

[22] H K Lam ldquoPolynomial fuzzy-model-based control systemsstability analysis via piecewise-linear membership functionsrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 588ndash5932011

[23] K Tanaka T Hori and H O Wang ldquoA fuzzy Lyapunovapproach to fuzzy control system designrdquo in Proceedings of theAmerican Control Conference vol 6 pp 4790ndash4795 ArlingtonVa USA June 2001

[24] K Tanaka T Hori and H O Wang ldquoA multiple Lyapunovfunction approach to stabilization of fuzzy control systemsrdquoIEEE Transactions on Fuzzy Systems vol 11 no 4 pp 582ndash5892003

[25] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 174ndash1972005

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

0 5 10 15 20 25 30 35 40minus40minus30minus20minus10

010203040

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30 35 40minus04minus03minus02minus01

0010203

Tracking error

Time

Am

plitu

de

e

(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 40minus40minus30minus20minus10

0102030

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30 35 40minus06minus05minus04minus03minus02minus01

0010203

Tracking error

Time

Am

plitu

de

e

(b)

Figure 3 (a) Output reference and (b) tracking errorWhen the fuzzy controller is designed on local regulators and the adequatemembershipfunctions for the fuzzy regulator are computed

0 5 10 15 20 25 30 35 40minus40minus30minus20minus10

010203040

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30 35 40minus06minus05minus04minus03minus02minus01

001

Tracking error

Time

Am

plitu

de

e

(b)

Figure 4 (a) Output reference and (b) tracking errorWhen the fuzzy controller is designed on local regulators and the adequatemembershipfunctions for the fuzzy regulator are computed

12 Mathematical Problems in Engineering

h1 h2

x1minus60

1

80

0

0

Figure 5 Membership functions of two-rule model

and an exosystem defined by

119878 =

[[[[[

[

0 0 1 0

0 0 0 1

minus4 0 0 0

0 minus16 0 0

]]]]]

]

119876 = [

1 0 0 0

0 1 0 0]

(67)

Membership functions forRules 1 and 2 are shown in Figure 5Matrices defining the local regulators can be easily obtainedby using (49) For this example they are

Γ1= [

14286 05714 02857 05714

minus10000 minus95000 minus05000 minus25000]

Γ2= [

23333 13333 06667 03333

minus04000 minus40000 minus12000 minus04000]

(68)

On the other hand 11990211198881= 1205871= 1 119902

21198882= 1205872= 1

11990211198881= 1205873= 1 119902

21198882= 1205874= 1 and by (46) and matrix Π is

Π =

[[[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

]]]]]

]

(69)

Figures 6 and 7 show the results obtained using linear regula-tors defined in (61)

Now by computing the new membership functions forthe fuzzy regulator through (54) one gets1205831(119908 (119905))

=

[[[

[

(ℎ1(119908 (119905))) (minus7)

ℎ1(119908 (119905)) (3) + ℎ

2(119908 (119905)) (3)

0

0(ℎ1(119908 (119905))) (2)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (5)

]]]

]

1205832(119908 (119905))

=

[[[

[

(ℎ2(119908 (119905))) (minus3)

ℎ1(119908 (119905)) (3) + ℎ

2(119908 (119905)) (3)

0

0(ℎ2(119908 (119905))) (5)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (5)

]]]

]

(70)

As previously the fuzzy stabilizer for this example is alsoconstructed bymeans of the PDC approach developed in [14]resulting in

1198701= [

10286 05714 06000 05714

minus10000 minus22500 minus05000 minus37500]

1198702= [

14000 13333 14000 03333

minus04000 minus11000 minus12000 minus09000]

(71)

Thus Figures 8 and 9 show the simulation results whenthe controller defined by (20) is applied

Besides Figure 10 shows the shape of the new member-ship functions which are clearly different from the originalones given in Figure 5

53 More General Problems Consider the T-S fuzzy MIMOsystem with multiple inputs defined by the following matri-ces

1198601=

[[[[[[[[[[[

[

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

minus6 4 2 4 5 3

2 3 minus1 5 minus2 4

]]]]]]]]]]]

]

1198611=

[[[[[[[[[[[

[

0 0

0 0

0 0

0 0

5 0

0 2

]]]]]]]]]]]

]

1198602=

[[[[[[[[[[[

[

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

minus6 1 2 minus5 4 6

0 2 minus7 4 5 2

]]]]]]]]]]]

]

1198612=

[[[[[[[[[[[

[

0 0

0 0

0 0

0 0

7 0

0 8

]]]]]]]]]]]

]

1198621= 1198622= [

0 1 1 0 0 0

1 0 1 0 0 0]

(72)

Mathematical Problems in Engineering 13

0 5 10 15 20 25 30minus15minus10

minus505

101520

Time

Am

plitu

dex1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus4minus2

02468

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 6 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus30minus20minus10

0102030

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus15

minus10

minus5

0

5

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 7 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of lineal regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus15minus10

minus505

101520

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus1

012345678

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 8 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

The reference signals will be generated by an exosystemdefined by

119878 = [

0 1

minus1 0] 119876 = [

1 0

0 1] (73)

Membership functions for Rules 1 and 2 are ℎ1(119909(119905)) =

(80 minus 1199091(119905))140 and ℎ

2(119909(119905)) = 1 minus ℎ

1(119905) Notice that for this

case 119899 = 119904 and 119862119894

= 119876 This implies that Π119894and Γ

119894must

be obtained by solving Francis Equations for each subsystemHowever due to form of the fuzzy plant it is possible to verifythat Π

1= Π2

14 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus40minus30minus20minus10

0102030

Time

Am

plitu

dex2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus10

minus8minus6minus4minus2

0

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 9 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(11) (x1(t))1205832(11) (x1(t))

(a)

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(22) (x1(t))1205832(22) (x1(t))

(b)

Figure 10 (a) Membership functions corresponding to position (11) of 1205831(119908(119905)) and 120583

2(119908(119905)) (b) Membership functions corresponding to

position (22) 1205831(119908(119905)) and 120583

2(119908(119905))

So by applying (9) for each local subsystem with 119894 = 1 2it results in

Π1= Π2=

[[[[[[[[[[[

[

05000 05000

15000 minus05000

minus05000 05000

05000 15000

minus05000 minus05000

minus15000 05000

]]]]]]]]]]]

]

Γ1= [

07000 minus03000

minus20000 minus55000]

Γ2= [

23571 12143

minus04375 minus01875]

(74)

Again the fuzzy stabilizer for this system is constructedby means of the PDC approach developed in [14] yielding

1198701=[[

[

minus12229 00609 minus40284 01291 minus25666 minus04061

16919 minus83997 32934 minus127153 15732 minus61836

]]

]

1198702=[[

[

minus08735 04721 minus28775 13780 minus16904 minus07187

06730 minus19749 15734 minus30538 minus04817 minus12959

]]

]

(75)

The behavior obtained by applying controller (61) isdepicted in Figures 11 and 12 Notice the exact outputregulation is not achieved

Mathematical Problems in Engineering 15

0 5 10 15 20 25 30minus60minus40minus20

0204060

Time

Am

plitu

dex1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus80minus70minus60minus50minus40minus30minus20minus10

01020

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 11 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus15

minus10

minus5

0

5

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 12 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

Now the adequate membership functions are computedfrom (54) resulting in

1205831(119908 (119905))

=

[[[

[

(ℎ1(119908 (119905))) (5)

ℎ1(119908 (119905)) (5) + ℎ

2(119908 (119905)) (7)

0

0(ℎ1(119908 (119905))) (2)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (8)

]]]

]

1205832(119908 (119905))

=

[[[

[

(ℎ2(119908 (119905))) (7)

ℎ1(119908 (119905)) (5) + ℎ

2(119908 (119905)) (7)

0

0(ℎ2(119908 (119905))) (8)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (8)

]]]

]

(76)

When ℎ119894(119908(119905)) is replaced by its corresponding expressions

one gets

1205831(119908 (119905)) =

[[[

[

1225

1199081(119905) + 140

minus5

20

0 minus1199081(119905) minus 80

31199081(119905) + 320

]]]

]

1205832(119908 (119905)) =

[[[

[

7

2minus

1225

1199081(119905) + 410

0

041199081(119905) + 240

31199081(119905) + 320

]]]

]

(77)

The results of applying control law (20) on the fuzzy plantare given in Figures 13 and 14

The ldquocomputedrdquo membership functions for the fuzzyregulator are given in Figure 15 Again it can be readilyobserved that these new membership functions are differentfrom the original ones

Notice that although this example does not fulfill theconditions ofTheorem 5 themembership functions obtainedfrom (39) allow the exact tracking for the fuzzy problemThis

16 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus80minus70minus60minus50minus40minus30minus20minus10

010

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 13 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus8minus6minus4minus2

02468

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 14 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(11) (x1(t))1205832(11) (x1(t))

(a)

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(22) (x1(t))1205832(22) (x1(t))

(b)

Figure 15 (a) Membership functions corresponding to position (11) of 1205831(119908(119905)) and 120583

2(119908(119905)) (b) Membership functions corresponding to

position (22) 1205831(119908(119905)) 120583

2(119908(119905))

Mathematical Problems in Engineering 17

suggests that the approach presented in this work may beapplied on a bigger class of T-S models however this studyis not completed yet

6 Conclusions

A fuzzy regulator for continuous-time systems based on thecombination of linear regulators adjusted by different mem-bership functions has been presented The main advantageis that analytical expressions for the membership functionswhich allow the proper combination of the local regulators inorder to guarantee the exact regulation can be easily appliedAs 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 modelscan be commonly found in mechanics electromechanicselectronics robotics and so forth

Some numerical examples are used to illustrate theapplicability of the proposed approach even in cases wherethe dimensions of the plant and the exosystem are differentbetween each otherThis suggests that the approach presentedin this work may be applied on a bigger class of T-S models

Finally the method proposed in this work avoids thedisadvantage of constructing an exact fuzzy regulator basedon overall T-S fuzzy system whose expression may resultvery large Instead the given approach offers 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 Interests

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

Acknowledgments

This work was partially supported by CONACYT (ConsejoNacional de Ciencia y Tecnologıa) through scholarship SNI(Sistema Nacional de Investigadores) and by IPN (InstitutoPolitecnico Nacional) through research Projects 20140659and 20150487 and scholarships EDI (Estımulo al Desempenode los Investigadores) COFAA (Comision de Operacion yFomento de Actividades Academicas) and BEIFI (Beca deEstımulo Institucional de Formacion de Investigadores)

References

[1] B A Francis ldquoThe linear multivariable regulator problemrdquoSIAM Journal on Control and Optimization vol 15 no 3 pp486ndash505 1977

[2] B A Francis andWMWonham ldquoThe internalmodel principleof control theoryrdquo Automatica vol 12 no 5 pp 457ndash465 1976

[3] A Isidori and C I Byrnes ldquoOutput regulation of nonlinearsystemsrdquo IEEE Transactions on Automatic Control vol 35 no2 pp 131ndash140 1990

[4] B Castillo-Toledo S Di Gennaro and F Jurado ldquoTrajectorytracking for a quadrotor via fuzzy regulationrdquo in Proceedings ofthe World Automation Congress (WAC rsquo12) pp 1ndash6 June 2012

[5] S Chen ldquoOutput regulation of nonlinear singularly perturbedsystems based on T-S fuzzy modelrdquo Journal of Control Theoryand Applications vol 3 no 4 pp 399ndash403 2005

[6] R Tapia-Herrera J A Meda-Campana S Alcamp119886119901119900119904 119886ntara-Montes T Hernamp119886119901119900119904 119886ndez-Cortamp119886119901119900119904 119890s and L Salgado-Conrado ldquoTuning of a TS fuzzy output regulator using thesteepest descent approach and ANFISrdquoMathematical Problemsin Engineering vol 2013 Article ID 873430 14 pages 2013

[7] K-Y Lian and J-J Liou ldquoOutput tracking control for fuzzysystems via output feedback designrdquo IEEE Transactions onFuzzy Systems vol 14 no 5 pp 628ndash639 2006

[8] X-J Ma and Z-Q Sun ldquoOutput tracking and regulationof nonlinear system based on takagi-sugeno fuzzy modelrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 30 no 1 pp 47ndash59 2000

[9] H J Lee J B Park and Y H Joo ldquoComments on lsquooutputtracking and regulation of nonlinear system based on Takagi-Sugeno fuzzy model1015840rdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 33 no 3 pp 521ndash523 2003

[10] J A Meda-Campana B Castillo-Toledo and G Chen ldquoSyn-chronization of chaotic systems from a fuzzy regulationapproachrdquo Fuzzy Sets and Systems vol 160 no 19 pp 2860ndash2875 2009

[11] J A Meda-Campana J C Gamp119886119901119900119904 119900mez-Mancilla and BCastillo-Toledo ldquoExact output regulation for nonlinear systemsdescribed by Takagi-Sugeno fuzzy modelsrdquo IEEE Transactionson Fuzzy Systems vol 20 no 2 pp 235ndash247 2012

[12] C Byrnes F Priscoli and A Isidori Output Regulation ofUncertain Nonlinear Systems Birkhauser 1997

[13] A Isidori Nonlinear Control Systems Communications andControl Engineering Series Springer Berlin Germany 3rdedition 1995

[14] K Tanaka and H O Wang Fuzzy Control Systems Designand Analysis A Linear Matrix Inequality Approach John Wileyamp119886119898119901 Sons New York NY USA 2001

[15] L-X Wang A Course in Fuzzy Systems and Control PrenticeHall Upper Saddle River NJ USA 1997

[16] I Abdelmalek N Golamp119886119901119900119904 119890a and M L Hadjili ldquoA newfuzzy Lyapunov approach to non-quadratic stabilization ofTakagi-Sugeno fuzzy modelsrdquo International Journal of AppliedMathematics andComputer Science vol 17 no 1 pp 39ndash51 2007

[17] J A Meda-Campana and B Castillo-Toledo ldquoOn the outputregulation for TS fuzzymodels using slidingmodesrdquo inProceed-ings of the American Control Conference (ACC rsquo05) pp 4062ndash4067 Portland Ore USA June 2005

[18] B Castillo-Toledo and J A Meda-Campana ldquoThe fuzzydiscrete-time robust regulation problem a LMI approachrdquo inProceedings of the 41st IEEE Conference on Decision and Controlvol 2 pp 2159ndash2164 IEEE Las Vegas Nev USA December2002

[19] B Castillo-Toledo J A Meda-Campana and A Titli ldquoAfuzzy output regulator for Takagi-Sugeno fuzzy modelsrdquo inProceedings of the IEEE International Symposium on IntelligentControl pp 310ndash315 Houston Tex USA October 2003

[20] B-C Ding H-X Sun and Y-E Qiao ldquoStability analysis of T-Sfuzzy control systems based on parameter-dependent Lyapunovfunctionrdquo Acta Automatica Sinica vol 31 no 4 pp 651ndash6542005

[21] M Narimani H K Lam R Dilmaghani and C Wolfe ldquoLMI-based stability analysis of fuzzy-model-based control systemsusing approximated polynomial membership functionsrdquo IEEE

18 Mathematical Problems in Engineering

Transactions on Systems Man and Cybernetics Part B Cyber-netics vol 41 no 3 pp 713ndash724 2011

[22] H K Lam ldquoPolynomial fuzzy-model-based control systemsstability analysis via piecewise-linear membership functionsrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 588ndash5932011

[23] K Tanaka T Hori and H O Wang ldquoA fuzzy Lyapunovapproach to fuzzy control system designrdquo in Proceedings of theAmerican Control Conference vol 6 pp 4790ndash4795 ArlingtonVa USA June 2001

[24] K Tanaka T Hori and H O Wang ldquoA multiple Lyapunovfunction approach to stabilization of fuzzy control systemsrdquoIEEE Transactions on Fuzzy Systems vol 11 no 4 pp 582ndash5892003

[25] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 174ndash1972005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

h1 h2

x1minus60

1

80

0

0

Figure 5 Membership functions of two-rule model

and an exosystem defined by

119878 =

[[[[[

[

0 0 1 0

0 0 0 1

minus4 0 0 0

0 minus16 0 0

]]]]]

]

119876 = [

1 0 0 0

0 1 0 0]

(67)

Membership functions forRules 1 and 2 are shown in Figure 5Matrices defining the local regulators can be easily obtainedby using (49) For this example they are

Γ1= [

14286 05714 02857 05714

minus10000 minus95000 minus05000 minus25000]

Γ2= [

23333 13333 06667 03333

minus04000 minus40000 minus12000 minus04000]

(68)

On the other hand 11990211198881= 1205871= 1 119902

21198882= 1205872= 1

11990211198881= 1205873= 1 119902

21198882= 1205874= 1 and by (46) and matrix Π is

Π =

[[[[[

[

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

]]]]]

]

(69)

Figures 6 and 7 show the results obtained using linear regula-tors defined in (61)

Now by computing the new membership functions forthe fuzzy regulator through (54) one gets1205831(119908 (119905))

=

[[[

[

(ℎ1(119908 (119905))) (minus7)

ℎ1(119908 (119905)) (3) + ℎ

2(119908 (119905)) (3)

0

0(ℎ1(119908 (119905))) (2)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (5)

]]]

]

1205832(119908 (119905))

=

[[[

[

(ℎ2(119908 (119905))) (minus3)

ℎ1(119908 (119905)) (3) + ℎ

2(119908 (119905)) (3)

0

0(ℎ2(119908 (119905))) (5)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (5)

]]]

]

(70)

As previously the fuzzy stabilizer for this example is alsoconstructed bymeans of the PDC approach developed in [14]resulting in

1198701= [

10286 05714 06000 05714

minus10000 minus22500 minus05000 minus37500]

1198702= [

14000 13333 14000 03333

minus04000 minus11000 minus12000 minus09000]

(71)

Thus Figures 8 and 9 show the simulation results whenthe controller defined by (20) is applied

Besides Figure 10 shows the shape of the new member-ship functions which are clearly different from the originalones given in Figure 5

53 More General Problems Consider the T-S fuzzy MIMOsystem with multiple inputs defined by the following matri-ces

1198601=

[[[[[[[[[[[

[

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

minus6 4 2 4 5 3

2 3 minus1 5 minus2 4

]]]]]]]]]]]

]

1198611=

[[[[[[[[[[[

[

0 0

0 0

0 0

0 0

5 0

0 2

]]]]]]]]]]]

]

1198602=

[[[[[[[[[[[

[

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

minus6 1 2 minus5 4 6

0 2 minus7 4 5 2

]]]]]]]]]]]

]

1198612=

[[[[[[[[[[[

[

0 0

0 0

0 0

0 0

7 0

0 8

]]]]]]]]]]]

]

1198621= 1198622= [

0 1 1 0 0 0

1 0 1 0 0 0]

(72)

Mathematical Problems in Engineering 13

0 5 10 15 20 25 30minus15minus10

minus505

101520

Time

Am

plitu

dex1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus4minus2

02468

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 6 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus30minus20minus10

0102030

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus15

minus10

minus5

0

5

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 7 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of lineal regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus15minus10

minus505

101520

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus1

012345678

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 8 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

The reference signals will be generated by an exosystemdefined by

119878 = [

0 1

minus1 0] 119876 = [

1 0

0 1] (73)

Membership functions for Rules 1 and 2 are ℎ1(119909(119905)) =

(80 minus 1199091(119905))140 and ℎ

2(119909(119905)) = 1 minus ℎ

1(119905) Notice that for this

case 119899 = 119904 and 119862119894

= 119876 This implies that Π119894and Γ

119894must

be obtained by solving Francis Equations for each subsystemHowever due to form of the fuzzy plant it is possible to verifythat Π

1= Π2

14 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus40minus30minus20minus10

0102030

Time

Am

plitu

dex2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus10

minus8minus6minus4minus2

0

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 9 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(11) (x1(t))1205832(11) (x1(t))

(a)

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(22) (x1(t))1205832(22) (x1(t))

(b)

Figure 10 (a) Membership functions corresponding to position (11) of 1205831(119908(119905)) and 120583

2(119908(119905)) (b) Membership functions corresponding to

position (22) 1205831(119908(119905)) and 120583

2(119908(119905))

So by applying (9) for each local subsystem with 119894 = 1 2it results in

Π1= Π2=

[[[[[[[[[[[

[

05000 05000

15000 minus05000

minus05000 05000

05000 15000

minus05000 minus05000

minus15000 05000

]]]]]]]]]]]

]

Γ1= [

07000 minus03000

minus20000 minus55000]

Γ2= [

23571 12143

minus04375 minus01875]

(74)

Again the fuzzy stabilizer for this system is constructedby means of the PDC approach developed in [14] yielding

1198701=[[

[

minus12229 00609 minus40284 01291 minus25666 minus04061

16919 minus83997 32934 minus127153 15732 minus61836

]]

]

1198702=[[

[

minus08735 04721 minus28775 13780 minus16904 minus07187

06730 minus19749 15734 minus30538 minus04817 minus12959

]]

]

(75)

The behavior obtained by applying controller (61) isdepicted in Figures 11 and 12 Notice the exact outputregulation is not achieved

Mathematical Problems in Engineering 15

0 5 10 15 20 25 30minus60minus40minus20

0204060

Time

Am

plitu

dex1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus80minus70minus60minus50minus40minus30minus20minus10

01020

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 11 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus15

minus10

minus5

0

5

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 12 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

Now the adequate membership functions are computedfrom (54) resulting in

1205831(119908 (119905))

=

[[[

[

(ℎ1(119908 (119905))) (5)

ℎ1(119908 (119905)) (5) + ℎ

2(119908 (119905)) (7)

0

0(ℎ1(119908 (119905))) (2)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (8)

]]]

]

1205832(119908 (119905))

=

[[[

[

(ℎ2(119908 (119905))) (7)

ℎ1(119908 (119905)) (5) + ℎ

2(119908 (119905)) (7)

0

0(ℎ2(119908 (119905))) (8)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (8)

]]]

]

(76)

When ℎ119894(119908(119905)) is replaced by its corresponding expressions

one gets

1205831(119908 (119905)) =

[[[

[

1225

1199081(119905) + 140

minus5

20

0 minus1199081(119905) minus 80

31199081(119905) + 320

]]]

]

1205832(119908 (119905)) =

[[[

[

7

2minus

1225

1199081(119905) + 410

0

041199081(119905) + 240

31199081(119905) + 320

]]]

]

(77)

The results of applying control law (20) on the fuzzy plantare given in Figures 13 and 14

The ldquocomputedrdquo membership functions for the fuzzyregulator are given in Figure 15 Again it can be readilyobserved that these new membership functions are differentfrom the original ones

Notice that although this example does not fulfill theconditions ofTheorem 5 themembership functions obtainedfrom (39) allow the exact tracking for the fuzzy problemThis

16 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus80minus70minus60minus50minus40minus30minus20minus10

010

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 13 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus8minus6minus4minus2

02468

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 14 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(11) (x1(t))1205832(11) (x1(t))

(a)

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(22) (x1(t))1205832(22) (x1(t))

(b)

Figure 15 (a) Membership functions corresponding to position (11) of 1205831(119908(119905)) and 120583

2(119908(119905)) (b) Membership functions corresponding to

position (22) 1205831(119908(119905)) 120583

2(119908(119905))

Mathematical Problems in Engineering 17

suggests that the approach presented in this work may beapplied on a bigger class of T-S models however this studyis not completed yet

6 Conclusions

A fuzzy regulator for continuous-time systems based on thecombination of linear regulators adjusted by different mem-bership functions has been presented The main advantageis that analytical expressions for the membership functionswhich allow the proper combination of the local regulators inorder to guarantee the exact regulation can be easily appliedAs 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 modelscan be commonly found in mechanics electromechanicselectronics robotics and so forth

Some numerical examples are used to illustrate theapplicability of the proposed approach even in cases wherethe dimensions of the plant and the exosystem are differentbetween each otherThis suggests that the approach presentedin this work may be applied on a bigger class of T-S models

Finally the method proposed in this work avoids thedisadvantage of constructing an exact fuzzy regulator basedon overall T-S fuzzy system whose expression may resultvery large Instead the given approach offers 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 Interests

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

Acknowledgments

This work was partially supported by CONACYT (ConsejoNacional de Ciencia y Tecnologıa) through scholarship SNI(Sistema Nacional de Investigadores) and by IPN (InstitutoPolitecnico Nacional) through research Projects 20140659and 20150487 and scholarships EDI (Estımulo al Desempenode los Investigadores) COFAA (Comision de Operacion yFomento de Actividades Academicas) and BEIFI (Beca deEstımulo Institucional de Formacion de Investigadores)

References

[1] B A Francis ldquoThe linear multivariable regulator problemrdquoSIAM Journal on Control and Optimization vol 15 no 3 pp486ndash505 1977

[2] B A Francis andWMWonham ldquoThe internalmodel principleof control theoryrdquo Automatica vol 12 no 5 pp 457ndash465 1976

[3] A Isidori and C I Byrnes ldquoOutput regulation of nonlinearsystemsrdquo IEEE Transactions on Automatic Control vol 35 no2 pp 131ndash140 1990

[4] B Castillo-Toledo S Di Gennaro and F Jurado ldquoTrajectorytracking for a quadrotor via fuzzy regulationrdquo in Proceedings ofthe World Automation Congress (WAC rsquo12) pp 1ndash6 June 2012

[5] S Chen ldquoOutput regulation of nonlinear singularly perturbedsystems based on T-S fuzzy modelrdquo Journal of Control Theoryand Applications vol 3 no 4 pp 399ndash403 2005

[6] R Tapia-Herrera J A Meda-Campana S Alcamp119886119901119900119904 119886ntara-Montes T Hernamp119886119901119900119904 119886ndez-Cortamp119886119901119900119904 119890s and L Salgado-Conrado ldquoTuning of a TS fuzzy output regulator using thesteepest descent approach and ANFISrdquoMathematical Problemsin Engineering vol 2013 Article ID 873430 14 pages 2013

[7] K-Y Lian and J-J Liou ldquoOutput tracking control for fuzzysystems via output feedback designrdquo IEEE Transactions onFuzzy Systems vol 14 no 5 pp 628ndash639 2006

[8] X-J Ma and Z-Q Sun ldquoOutput tracking and regulationof nonlinear system based on takagi-sugeno fuzzy modelrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 30 no 1 pp 47ndash59 2000

[9] H J Lee J B Park and Y H Joo ldquoComments on lsquooutputtracking and regulation of nonlinear system based on Takagi-Sugeno fuzzy model1015840rdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 33 no 3 pp 521ndash523 2003

[10] J A Meda-Campana B Castillo-Toledo and G Chen ldquoSyn-chronization of chaotic systems from a fuzzy regulationapproachrdquo Fuzzy Sets and Systems vol 160 no 19 pp 2860ndash2875 2009

[11] J A Meda-Campana J C Gamp119886119901119900119904 119900mez-Mancilla and BCastillo-Toledo ldquoExact output regulation for nonlinear systemsdescribed by Takagi-Sugeno fuzzy modelsrdquo IEEE Transactionson Fuzzy Systems vol 20 no 2 pp 235ndash247 2012

[12] C Byrnes F Priscoli and A Isidori Output Regulation ofUncertain Nonlinear Systems Birkhauser 1997

[13] A Isidori Nonlinear Control Systems Communications andControl Engineering Series Springer Berlin Germany 3rdedition 1995

[14] K Tanaka and H O Wang Fuzzy Control Systems Designand Analysis A Linear Matrix Inequality Approach John Wileyamp119886119898119901 Sons New York NY USA 2001

[15] L-X Wang A Course in Fuzzy Systems and Control PrenticeHall Upper Saddle River NJ USA 1997

[16] I Abdelmalek N Golamp119886119901119900119904 119890a and M L Hadjili ldquoA newfuzzy Lyapunov approach to non-quadratic stabilization ofTakagi-Sugeno fuzzy modelsrdquo International Journal of AppliedMathematics andComputer Science vol 17 no 1 pp 39ndash51 2007

[17] J A Meda-Campana and B Castillo-Toledo ldquoOn the outputregulation for TS fuzzymodels using slidingmodesrdquo inProceed-ings of the American Control Conference (ACC rsquo05) pp 4062ndash4067 Portland Ore USA June 2005

[18] B Castillo-Toledo and J A Meda-Campana ldquoThe fuzzydiscrete-time robust regulation problem a LMI approachrdquo inProceedings of the 41st IEEE Conference on Decision and Controlvol 2 pp 2159ndash2164 IEEE Las Vegas Nev USA December2002

[19] B Castillo-Toledo J A Meda-Campana and A Titli ldquoAfuzzy output regulator for Takagi-Sugeno fuzzy modelsrdquo inProceedings of the IEEE International Symposium on IntelligentControl pp 310ndash315 Houston Tex USA October 2003

[20] B-C Ding H-X Sun and Y-E Qiao ldquoStability analysis of T-Sfuzzy control systems based on parameter-dependent Lyapunovfunctionrdquo Acta Automatica Sinica vol 31 no 4 pp 651ndash6542005

[21] M Narimani H K Lam R Dilmaghani and C Wolfe ldquoLMI-based stability analysis of fuzzy-model-based control systemsusing approximated polynomial membership functionsrdquo IEEE

18 Mathematical Problems in Engineering

Transactions on Systems Man and Cybernetics Part B Cyber-netics vol 41 no 3 pp 713ndash724 2011

[22] H K Lam ldquoPolynomial fuzzy-model-based control systemsstability analysis via piecewise-linear membership functionsrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 588ndash5932011

[23] K Tanaka T Hori and H O Wang ldquoA fuzzy Lyapunovapproach to fuzzy control system designrdquo in Proceedings of theAmerican Control Conference vol 6 pp 4790ndash4795 ArlingtonVa USA June 2001

[24] K Tanaka T Hori and H O Wang ldquoA multiple Lyapunovfunction approach to stabilization of fuzzy control systemsrdquoIEEE Transactions on Fuzzy Systems vol 11 no 4 pp 582ndash5892003

[25] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 174ndash1972005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 13

0 5 10 15 20 25 30minus15minus10

minus505

101520

Time

Am

plitu

dex1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus4minus2

02468

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 6 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus30minus20minus10

0102030

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus15

minus10

minus5

0

5

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 7 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of lineal regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus15minus10

minus505

101520

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus1

012345678

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 8 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

The reference signals will be generated by an exosystemdefined by

119878 = [

0 1

minus1 0] 119876 = [

1 0

0 1] (73)

Membership functions for Rules 1 and 2 are ℎ1(119909(119905)) =

(80 minus 1199091(119905))140 and ℎ

2(119909(119905)) = 1 minus ℎ

1(119905) Notice that for this

case 119899 = 119904 and 119862119894

= 119876 This implies that Π119894and Γ

119894must

be obtained by solving Francis Equations for each subsystemHowever due to form of the fuzzy plant it is possible to verifythat Π

1= Π2

14 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus40minus30minus20minus10

0102030

Time

Am

plitu

dex2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus10

minus8minus6minus4minus2

0

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 9 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(11) (x1(t))1205832(11) (x1(t))

(a)

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(22) (x1(t))1205832(22) (x1(t))

(b)

Figure 10 (a) Membership functions corresponding to position (11) of 1205831(119908(119905)) and 120583

2(119908(119905)) (b) Membership functions corresponding to

position (22) 1205831(119908(119905)) and 120583

2(119908(119905))

So by applying (9) for each local subsystem with 119894 = 1 2it results in

Π1= Π2=

[[[[[[[[[[[

[

05000 05000

15000 minus05000

minus05000 05000

05000 15000

minus05000 minus05000

minus15000 05000

]]]]]]]]]]]

]

Γ1= [

07000 minus03000

minus20000 minus55000]

Γ2= [

23571 12143

minus04375 minus01875]

(74)

Again the fuzzy stabilizer for this system is constructedby means of the PDC approach developed in [14] yielding

1198701=[[

[

minus12229 00609 minus40284 01291 minus25666 minus04061

16919 minus83997 32934 minus127153 15732 minus61836

]]

]

1198702=[[

[

minus08735 04721 minus28775 13780 minus16904 minus07187

06730 minus19749 15734 minus30538 minus04817 minus12959

]]

]

(75)

The behavior obtained by applying controller (61) isdepicted in Figures 11 and 12 Notice the exact outputregulation is not achieved

Mathematical Problems in Engineering 15

0 5 10 15 20 25 30minus60minus40minus20

0204060

Time

Am

plitu

dex1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus80minus70minus60minus50minus40minus30minus20minus10

01020

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 11 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus15

minus10

minus5

0

5

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 12 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

Now the adequate membership functions are computedfrom (54) resulting in

1205831(119908 (119905))

=

[[[

[

(ℎ1(119908 (119905))) (5)

ℎ1(119908 (119905)) (5) + ℎ

2(119908 (119905)) (7)

0

0(ℎ1(119908 (119905))) (2)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (8)

]]]

]

1205832(119908 (119905))

=

[[[

[

(ℎ2(119908 (119905))) (7)

ℎ1(119908 (119905)) (5) + ℎ

2(119908 (119905)) (7)

0

0(ℎ2(119908 (119905))) (8)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (8)

]]]

]

(76)

When ℎ119894(119908(119905)) is replaced by its corresponding expressions

one gets

1205831(119908 (119905)) =

[[[

[

1225

1199081(119905) + 140

minus5

20

0 minus1199081(119905) minus 80

31199081(119905) + 320

]]]

]

1205832(119908 (119905)) =

[[[

[

7

2minus

1225

1199081(119905) + 410

0

041199081(119905) + 240

31199081(119905) + 320

]]]

]

(77)

The results of applying control law (20) on the fuzzy plantare given in Figures 13 and 14

The ldquocomputedrdquo membership functions for the fuzzyregulator are given in Figure 15 Again it can be readilyobserved that these new membership functions are differentfrom the original ones

Notice that although this example does not fulfill theconditions ofTheorem 5 themembership functions obtainedfrom (39) allow the exact tracking for the fuzzy problemThis

16 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus80minus70minus60minus50minus40minus30minus20minus10

010

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 13 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus8minus6minus4minus2

02468

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 14 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(11) (x1(t))1205832(11) (x1(t))

(a)

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(22) (x1(t))1205832(22) (x1(t))

(b)

Figure 15 (a) Membership functions corresponding to position (11) of 1205831(119908(119905)) and 120583

2(119908(119905)) (b) Membership functions corresponding to

position (22) 1205831(119908(119905)) 120583

2(119908(119905))

Mathematical Problems in Engineering 17

suggests that the approach presented in this work may beapplied on a bigger class of T-S models however this studyis not completed yet

6 Conclusions

A fuzzy regulator for continuous-time systems based on thecombination of linear regulators adjusted by different mem-bership functions has been presented The main advantageis that analytical expressions for the membership functionswhich allow the proper combination of the local regulators inorder to guarantee the exact regulation can be easily appliedAs 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 modelscan be commonly found in mechanics electromechanicselectronics robotics and so forth

Some numerical examples are used to illustrate theapplicability of the proposed approach even in cases wherethe dimensions of the plant and the exosystem are differentbetween each otherThis suggests that the approach presentedin this work may be applied on a bigger class of T-S models

Finally the method proposed in this work avoids thedisadvantage of constructing an exact fuzzy regulator basedon overall T-S fuzzy system whose expression may resultvery large Instead the given approach offers 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 Interests

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

Acknowledgments

This work was partially supported by CONACYT (ConsejoNacional de Ciencia y Tecnologıa) through scholarship SNI(Sistema Nacional de Investigadores) and by IPN (InstitutoPolitecnico Nacional) through research Projects 20140659and 20150487 and scholarships EDI (Estımulo al Desempenode los Investigadores) COFAA (Comision de Operacion yFomento de Actividades Academicas) and BEIFI (Beca deEstımulo Institucional de Formacion de Investigadores)

References

[1] B A Francis ldquoThe linear multivariable regulator problemrdquoSIAM Journal on Control and Optimization vol 15 no 3 pp486ndash505 1977

[2] B A Francis andWMWonham ldquoThe internalmodel principleof control theoryrdquo Automatica vol 12 no 5 pp 457ndash465 1976

[3] A Isidori and C I Byrnes ldquoOutput regulation of nonlinearsystemsrdquo IEEE Transactions on Automatic Control vol 35 no2 pp 131ndash140 1990

[4] B Castillo-Toledo S Di Gennaro and F Jurado ldquoTrajectorytracking for a quadrotor via fuzzy regulationrdquo in Proceedings ofthe World Automation Congress (WAC rsquo12) pp 1ndash6 June 2012

[5] S Chen ldquoOutput regulation of nonlinear singularly perturbedsystems based on T-S fuzzy modelrdquo Journal of Control Theoryand Applications vol 3 no 4 pp 399ndash403 2005

[6] R Tapia-Herrera J A Meda-Campana S Alcamp119886119901119900119904 119886ntara-Montes T Hernamp119886119901119900119904 119886ndez-Cortamp119886119901119900119904 119890s and L Salgado-Conrado ldquoTuning of a TS fuzzy output regulator using thesteepest descent approach and ANFISrdquoMathematical Problemsin Engineering vol 2013 Article ID 873430 14 pages 2013

[7] K-Y Lian and J-J Liou ldquoOutput tracking control for fuzzysystems via output feedback designrdquo IEEE Transactions onFuzzy Systems vol 14 no 5 pp 628ndash639 2006

[8] X-J Ma and Z-Q Sun ldquoOutput tracking and regulationof nonlinear system based on takagi-sugeno fuzzy modelrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 30 no 1 pp 47ndash59 2000

[9] H J Lee J B Park and Y H Joo ldquoComments on lsquooutputtracking and regulation of nonlinear system based on Takagi-Sugeno fuzzy model1015840rdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 33 no 3 pp 521ndash523 2003

[10] J A Meda-Campana B Castillo-Toledo and G Chen ldquoSyn-chronization of chaotic systems from a fuzzy regulationapproachrdquo Fuzzy Sets and Systems vol 160 no 19 pp 2860ndash2875 2009

[11] J A Meda-Campana J C Gamp119886119901119900119904 119900mez-Mancilla and BCastillo-Toledo ldquoExact output regulation for nonlinear systemsdescribed by Takagi-Sugeno fuzzy modelsrdquo IEEE Transactionson Fuzzy Systems vol 20 no 2 pp 235ndash247 2012

[12] C Byrnes F Priscoli and A Isidori Output Regulation ofUncertain Nonlinear Systems Birkhauser 1997

[13] A Isidori Nonlinear Control Systems Communications andControl Engineering Series Springer Berlin Germany 3rdedition 1995

[14] K Tanaka and H O Wang Fuzzy Control Systems Designand Analysis A Linear Matrix Inequality Approach John Wileyamp119886119898119901 Sons New York NY USA 2001

[15] L-X Wang A Course in Fuzzy Systems and Control PrenticeHall Upper Saddle River NJ USA 1997

[16] I Abdelmalek N Golamp119886119901119900119904 119890a and M L Hadjili ldquoA newfuzzy Lyapunov approach to non-quadratic stabilization ofTakagi-Sugeno fuzzy modelsrdquo International Journal of AppliedMathematics andComputer Science vol 17 no 1 pp 39ndash51 2007

[17] J A Meda-Campana and B Castillo-Toledo ldquoOn the outputregulation for TS fuzzymodels using slidingmodesrdquo inProceed-ings of the American Control Conference (ACC rsquo05) pp 4062ndash4067 Portland Ore USA June 2005

[18] B Castillo-Toledo and J A Meda-Campana ldquoThe fuzzydiscrete-time robust regulation problem a LMI approachrdquo inProceedings of the 41st IEEE Conference on Decision and Controlvol 2 pp 2159ndash2164 IEEE Las Vegas Nev USA December2002

[19] B Castillo-Toledo J A Meda-Campana and A Titli ldquoAfuzzy output regulator for Takagi-Sugeno fuzzy modelsrdquo inProceedings of the IEEE International Symposium on IntelligentControl pp 310ndash315 Houston Tex USA October 2003

[20] B-C Ding H-X Sun and Y-E Qiao ldquoStability analysis of T-Sfuzzy control systems based on parameter-dependent Lyapunovfunctionrdquo Acta Automatica Sinica vol 31 no 4 pp 651ndash6542005

[21] M Narimani H K Lam R Dilmaghani and C Wolfe ldquoLMI-based stability analysis of fuzzy-model-based control systemsusing approximated polynomial membership functionsrdquo IEEE

18 Mathematical Problems in Engineering

Transactions on Systems Man and Cybernetics Part B Cyber-netics vol 41 no 3 pp 713ndash724 2011

[22] H K Lam ldquoPolynomial fuzzy-model-based control systemsstability analysis via piecewise-linear membership functionsrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 588ndash5932011

[23] K Tanaka T Hori and H O Wang ldquoA fuzzy Lyapunovapproach to fuzzy control system designrdquo in Proceedings of theAmerican Control Conference vol 6 pp 4790ndash4795 ArlingtonVa USA June 2001

[24] K Tanaka T Hori and H O Wang ldquoA multiple Lyapunovfunction approach to stabilization of fuzzy control systemsrdquoIEEE Transactions on Fuzzy Systems vol 11 no 4 pp 582ndash5892003

[25] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 174ndash1972005

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

14 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus40minus30minus20minus10

0102030

Time

Am

plitu

dex2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus10

minus8minus6minus4minus2

0

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 9 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(11) (x1(t))1205832(11) (x1(t))

(a)

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(22) (x1(t))1205832(22) (x1(t))

(b)

Figure 10 (a) Membership functions corresponding to position (11) of 1205831(119908(119905)) and 120583

2(119908(119905)) (b) Membership functions corresponding to

position (22) 1205831(119908(119905)) and 120583

2(119908(119905))

So by applying (9) for each local subsystem with 119894 = 1 2it results in

Π1= Π2=

[[[[[[[[[[[

[

05000 05000

15000 minus05000

minus05000 05000

05000 15000

minus05000 minus05000

minus15000 05000

]]]]]]]]]]]

]

Γ1= [

07000 minus03000

minus20000 minus55000]

Γ2= [

23571 12143

minus04375 minus01875]

(74)

Again the fuzzy stabilizer for this system is constructedby means of the PDC approach developed in [14] yielding

1198701=[[

[

minus12229 00609 minus40284 01291 minus25666 minus04061

16919 minus83997 32934 minus127153 15732 minus61836

]]

]

1198702=[[

[

minus08735 04721 minus28775 13780 minus16904 minus07187

06730 minus19749 15734 minus30538 minus04817 minus12959

]]

]

(75)

The behavior obtained by applying controller (61) isdepicted in Figures 11 and 12 Notice the exact outputregulation is not achieved

Mathematical Problems in Engineering 15

0 5 10 15 20 25 30minus60minus40minus20

0204060

Time

Am

plitu

dex1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus80minus70minus60minus50minus40minus30minus20minus10

01020

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 11 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus15

minus10

minus5

0

5

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 12 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

Now the adequate membership functions are computedfrom (54) resulting in

1205831(119908 (119905))

=

[[[

[

(ℎ1(119908 (119905))) (5)

ℎ1(119908 (119905)) (5) + ℎ

2(119908 (119905)) (7)

0

0(ℎ1(119908 (119905))) (2)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (8)

]]]

]

1205832(119908 (119905))

=

[[[

[

(ℎ2(119908 (119905))) (7)

ℎ1(119908 (119905)) (5) + ℎ

2(119908 (119905)) (7)

0

0(ℎ2(119908 (119905))) (8)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (8)

]]]

]

(76)

When ℎ119894(119908(119905)) is replaced by its corresponding expressions

one gets

1205831(119908 (119905)) =

[[[

[

1225

1199081(119905) + 140

minus5

20

0 minus1199081(119905) minus 80

31199081(119905) + 320

]]]

]

1205832(119908 (119905)) =

[[[

[

7

2minus

1225

1199081(119905) + 410

0

041199081(119905) + 240

31199081(119905) + 320

]]]

]

(77)

The results of applying control law (20) on the fuzzy plantare given in Figures 13 and 14

The ldquocomputedrdquo membership functions for the fuzzyregulator are given in Figure 15 Again it can be readilyobserved that these new membership functions are differentfrom the original ones

Notice that although this example does not fulfill theconditions ofTheorem 5 themembership functions obtainedfrom (39) allow the exact tracking for the fuzzy problemThis

16 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus80minus70minus60minus50minus40minus30minus20minus10

010

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 13 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus8minus6minus4minus2

02468

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 14 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(11) (x1(t))1205832(11) (x1(t))

(a)

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(22) (x1(t))1205832(22) (x1(t))

(b)

Figure 15 (a) Membership functions corresponding to position (11) of 1205831(119908(119905)) and 120583

2(119908(119905)) (b) Membership functions corresponding to

position (22) 1205831(119908(119905)) 120583

2(119908(119905))

Mathematical Problems in Engineering 17

suggests that the approach presented in this work may beapplied on a bigger class of T-S models however this studyis not completed yet

6 Conclusions

A fuzzy regulator for continuous-time systems based on thecombination of linear regulators adjusted by different mem-bership functions has been presented The main advantageis that analytical expressions for the membership functionswhich allow the proper combination of the local regulators inorder to guarantee the exact regulation can be easily appliedAs 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 modelscan be commonly found in mechanics electromechanicselectronics robotics and so forth

Some numerical examples are used to illustrate theapplicability of the proposed approach even in cases wherethe dimensions of the plant and the exosystem are differentbetween each otherThis suggests that the approach presentedin this work may be applied on a bigger class of T-S models

Finally the method proposed in this work avoids thedisadvantage of constructing an exact fuzzy regulator basedon overall T-S fuzzy system whose expression may resultvery large Instead the given approach offers 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 Interests

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

Acknowledgments

This work was partially supported by CONACYT (ConsejoNacional de Ciencia y Tecnologıa) through scholarship SNI(Sistema Nacional de Investigadores) and by IPN (InstitutoPolitecnico Nacional) through research Projects 20140659and 20150487 and scholarships EDI (Estımulo al Desempenode los Investigadores) COFAA (Comision de Operacion yFomento de Actividades Academicas) and BEIFI (Beca deEstımulo Institucional de Formacion de Investigadores)

References

[1] B A Francis ldquoThe linear multivariable regulator problemrdquoSIAM Journal on Control and Optimization vol 15 no 3 pp486ndash505 1977

[2] B A Francis andWMWonham ldquoThe internalmodel principleof control theoryrdquo Automatica vol 12 no 5 pp 457ndash465 1976

[3] A Isidori and C I Byrnes ldquoOutput regulation of nonlinearsystemsrdquo IEEE Transactions on Automatic Control vol 35 no2 pp 131ndash140 1990

[4] B Castillo-Toledo S Di Gennaro and F Jurado ldquoTrajectorytracking for a quadrotor via fuzzy regulationrdquo in Proceedings ofthe World Automation Congress (WAC rsquo12) pp 1ndash6 June 2012

[5] S Chen ldquoOutput regulation of nonlinear singularly perturbedsystems based on T-S fuzzy modelrdquo Journal of Control Theoryand Applications vol 3 no 4 pp 399ndash403 2005

[6] R Tapia-Herrera J A Meda-Campana S Alcamp119886119901119900119904 119886ntara-Montes T Hernamp119886119901119900119904 119886ndez-Cortamp119886119901119900119904 119890s and L Salgado-Conrado ldquoTuning of a TS fuzzy output regulator using thesteepest descent approach and ANFISrdquoMathematical Problemsin Engineering vol 2013 Article ID 873430 14 pages 2013

[7] K-Y Lian and J-J Liou ldquoOutput tracking control for fuzzysystems via output feedback designrdquo IEEE Transactions onFuzzy Systems vol 14 no 5 pp 628ndash639 2006

[8] X-J Ma and Z-Q Sun ldquoOutput tracking and regulationof nonlinear system based on takagi-sugeno fuzzy modelrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 30 no 1 pp 47ndash59 2000

[9] H J Lee J B Park and Y H Joo ldquoComments on lsquooutputtracking and regulation of nonlinear system based on Takagi-Sugeno fuzzy model1015840rdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 33 no 3 pp 521ndash523 2003

[10] J A Meda-Campana B Castillo-Toledo and G Chen ldquoSyn-chronization of chaotic systems from a fuzzy regulationapproachrdquo Fuzzy Sets and Systems vol 160 no 19 pp 2860ndash2875 2009

[11] J A Meda-Campana J C Gamp119886119901119900119904 119900mez-Mancilla and BCastillo-Toledo ldquoExact output regulation for nonlinear systemsdescribed by Takagi-Sugeno fuzzy modelsrdquo IEEE Transactionson Fuzzy Systems vol 20 no 2 pp 235ndash247 2012

[12] C Byrnes F Priscoli and A Isidori Output Regulation ofUncertain Nonlinear Systems Birkhauser 1997

[13] A Isidori Nonlinear Control Systems Communications andControl Engineering Series Springer Berlin Germany 3rdedition 1995

[14] K Tanaka and H O Wang Fuzzy Control Systems Designand Analysis A Linear Matrix Inequality Approach John Wileyamp119886119898119901 Sons New York NY USA 2001

[15] L-X Wang A Course in Fuzzy Systems and Control PrenticeHall Upper Saddle River NJ USA 1997

[16] I Abdelmalek N Golamp119886119901119900119904 119890a and M L Hadjili ldquoA newfuzzy Lyapunov approach to non-quadratic stabilization ofTakagi-Sugeno fuzzy modelsrdquo International Journal of AppliedMathematics andComputer Science vol 17 no 1 pp 39ndash51 2007

[17] J A Meda-Campana and B Castillo-Toledo ldquoOn the outputregulation for TS fuzzymodels using slidingmodesrdquo inProceed-ings of the American Control Conference (ACC rsquo05) pp 4062ndash4067 Portland Ore USA June 2005

[18] B Castillo-Toledo and J A Meda-Campana ldquoThe fuzzydiscrete-time robust regulation problem a LMI approachrdquo inProceedings of the 41st IEEE Conference on Decision and Controlvol 2 pp 2159ndash2164 IEEE Las Vegas Nev USA December2002

[19] B Castillo-Toledo J A Meda-Campana and A Titli ldquoAfuzzy output regulator for Takagi-Sugeno fuzzy modelsrdquo inProceedings of the IEEE International Symposium on IntelligentControl pp 310ndash315 Houston Tex USA October 2003

[20] B-C Ding H-X Sun and Y-E Qiao ldquoStability analysis of T-Sfuzzy control systems based on parameter-dependent Lyapunovfunctionrdquo Acta Automatica Sinica vol 31 no 4 pp 651ndash6542005

[21] M Narimani H K Lam R Dilmaghani and C Wolfe ldquoLMI-based stability analysis of fuzzy-model-based control systemsusing approximated polynomial membership functionsrdquo IEEE

18 Mathematical Problems in Engineering

Transactions on Systems Man and Cybernetics Part B Cyber-netics vol 41 no 3 pp 713ndash724 2011

[22] H K Lam ldquoPolynomial fuzzy-model-based control systemsstability analysis via piecewise-linear membership functionsrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 588ndash5932011

[23] K Tanaka T Hori and H O Wang ldquoA fuzzy Lyapunovapproach to fuzzy control system designrdquo in Proceedings of theAmerican Control Conference vol 6 pp 4790ndash4795 ArlingtonVa USA June 2001

[24] K Tanaka T Hori and H O Wang ldquoA multiple Lyapunovfunction approach to stabilization of fuzzy control systemsrdquoIEEE Transactions on Fuzzy Systems vol 11 no 4 pp 582ndash5892003

[25] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 174ndash1972005

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 15

0 5 10 15 20 25 30minus60minus40minus20

0204060

Time

Am

plitu

dex1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus80minus70minus60minus50minus40minus30minus20minus10

01020

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 11 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus15

minus10

minus5

0

5

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 12 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of linear regulators and the fuzzy

memberships of the plant are considered in the construction the fuzzy regulator

Now the adequate membership functions are computedfrom (54) resulting in

1205831(119908 (119905))

=

[[[

[

(ℎ1(119908 (119905))) (5)

ℎ1(119908 (119905)) (5) + ℎ

2(119908 (119905)) (7)

0

0(ℎ1(119908 (119905))) (2)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (8)

]]]

]

1205832(119908 (119905))

=

[[[

[

(ℎ2(119908 (119905))) (7)

ℎ1(119908 (119905)) (5) + ℎ

2(119908 (119905)) (7)

0

0(ℎ2(119908 (119905))) (8)

ℎ1(119908 (119905)) (2) + ℎ

2(119908 (119905)) (8)

]]]

]

(76)

When ℎ119894(119908(119905)) is replaced by its corresponding expressions

one gets

1205831(119908 (119905)) =

[[[

[

1225

1199081(119905) + 140

minus5

20

0 minus1199081(119905) minus 80

31199081(119905) + 320

]]]

]

1205832(119908 (119905)) =

[[[

[

7

2minus

1225

1199081(119905) + 410

0

041199081(119905) + 240

31199081(119905) + 320

]]]

]

(77)

The results of applying control law (20) on the fuzzy plantare given in Figures 13 and 14

The ldquocomputedrdquo membership functions for the fuzzyregulator are given in Figure 15 Again it can be readilyobserved that these new membership functions are differentfrom the original ones

Notice that although this example does not fulfill theconditions ofTheorem 5 themembership functions obtainedfrom (39) allow the exact tracking for the fuzzy problemThis

16 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus80minus70minus60minus50minus40minus30minus20minus10

010

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 13 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus8minus6minus4minus2

02468

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 14 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(11) (x1(t))1205832(11) (x1(t))

(a)

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(22) (x1(t))1205832(22) (x1(t))

(b)

Figure 15 (a) Membership functions corresponding to position (11) of 1205831(119908(119905)) and 120583

2(119908(119905)) (b) Membership functions corresponding to

position (22) 1205831(119908(119905)) 120583

2(119908(119905))

Mathematical Problems in Engineering 17

suggests that the approach presented in this work may beapplied on a bigger class of T-S models however this studyis not completed yet

6 Conclusions

A fuzzy regulator for continuous-time systems based on thecombination of linear regulators adjusted by different mem-bership functions has been presented The main advantageis that analytical expressions for the membership functionswhich allow the proper combination of the local regulators inorder to guarantee the exact regulation can be easily appliedAs 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 modelscan be commonly found in mechanics electromechanicselectronics robotics and so forth

Some numerical examples are used to illustrate theapplicability of the proposed approach even in cases wherethe dimensions of the plant and the exosystem are differentbetween each otherThis suggests that the approach presentedin this work may be applied on a bigger class of T-S models

Finally the method proposed in this work avoids thedisadvantage of constructing an exact fuzzy regulator basedon overall T-S fuzzy system whose expression may resultvery large Instead the given approach offers 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 Interests

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

Acknowledgments

This work was partially supported by CONACYT (ConsejoNacional de Ciencia y Tecnologıa) through scholarship SNI(Sistema Nacional de Investigadores) and by IPN (InstitutoPolitecnico Nacional) through research Projects 20140659and 20150487 and scholarships EDI (Estımulo al Desempenode los Investigadores) COFAA (Comision de Operacion yFomento de Actividades Academicas) and BEIFI (Beca deEstımulo Institucional de Formacion de Investigadores)

References

[1] B A Francis ldquoThe linear multivariable regulator problemrdquoSIAM Journal on Control and Optimization vol 15 no 3 pp486ndash505 1977

[2] B A Francis andWMWonham ldquoThe internalmodel principleof control theoryrdquo Automatica vol 12 no 5 pp 457ndash465 1976

[3] A Isidori and C I Byrnes ldquoOutput regulation of nonlinearsystemsrdquo IEEE Transactions on Automatic Control vol 35 no2 pp 131ndash140 1990

[4] B Castillo-Toledo S Di Gennaro and F Jurado ldquoTrajectorytracking for a quadrotor via fuzzy regulationrdquo in Proceedings ofthe World Automation Congress (WAC rsquo12) pp 1ndash6 June 2012

[5] S Chen ldquoOutput regulation of nonlinear singularly perturbedsystems based on T-S fuzzy modelrdquo Journal of Control Theoryand Applications vol 3 no 4 pp 399ndash403 2005

[6] R Tapia-Herrera J A Meda-Campana S Alcamp119886119901119900119904 119886ntara-Montes T Hernamp119886119901119900119904 119886ndez-Cortamp119886119901119900119904 119890s and L Salgado-Conrado ldquoTuning of a TS fuzzy output regulator using thesteepest descent approach and ANFISrdquoMathematical Problemsin Engineering vol 2013 Article ID 873430 14 pages 2013

[7] K-Y Lian and J-J Liou ldquoOutput tracking control for fuzzysystems via output feedback designrdquo IEEE Transactions onFuzzy Systems vol 14 no 5 pp 628ndash639 2006

[8] X-J Ma and Z-Q Sun ldquoOutput tracking and regulationof nonlinear system based on takagi-sugeno fuzzy modelrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 30 no 1 pp 47ndash59 2000

[9] H J Lee J B Park and Y H Joo ldquoComments on lsquooutputtracking and regulation of nonlinear system based on Takagi-Sugeno fuzzy model1015840rdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 33 no 3 pp 521ndash523 2003

[10] J A Meda-Campana B Castillo-Toledo and G Chen ldquoSyn-chronization of chaotic systems from a fuzzy regulationapproachrdquo Fuzzy Sets and Systems vol 160 no 19 pp 2860ndash2875 2009

[11] J A Meda-Campana J C Gamp119886119901119900119904 119900mez-Mancilla and BCastillo-Toledo ldquoExact output regulation for nonlinear systemsdescribed by Takagi-Sugeno fuzzy modelsrdquo IEEE Transactionson Fuzzy Systems vol 20 no 2 pp 235ndash247 2012

[12] C Byrnes F Priscoli and A Isidori Output Regulation ofUncertain Nonlinear Systems Birkhauser 1997

[13] A Isidori Nonlinear Control Systems Communications andControl Engineering Series Springer Berlin Germany 3rdedition 1995

[14] K Tanaka and H O Wang Fuzzy Control Systems Designand Analysis A Linear Matrix Inequality Approach John Wileyamp119886119898119901 Sons New York NY USA 2001

[15] L-X Wang A Course in Fuzzy Systems and Control PrenticeHall Upper Saddle River NJ USA 1997

[16] I Abdelmalek N Golamp119886119901119900119904 119890a and M L Hadjili ldquoA newfuzzy Lyapunov approach to non-quadratic stabilization ofTakagi-Sugeno fuzzy modelsrdquo International Journal of AppliedMathematics andComputer Science vol 17 no 1 pp 39ndash51 2007

[17] J A Meda-Campana and B Castillo-Toledo ldquoOn the outputregulation for TS fuzzymodels using slidingmodesrdquo inProceed-ings of the American Control Conference (ACC rsquo05) pp 4062ndash4067 Portland Ore USA June 2005

[18] B Castillo-Toledo and J A Meda-Campana ldquoThe fuzzydiscrete-time robust regulation problem a LMI approachrdquo inProceedings of the 41st IEEE Conference on Decision and Controlvol 2 pp 2159ndash2164 IEEE Las Vegas Nev USA December2002

[19] B Castillo-Toledo J A Meda-Campana and A Titli ldquoAfuzzy output regulator for Takagi-Sugeno fuzzy modelsrdquo inProceedings of the IEEE International Symposium on IntelligentControl pp 310ndash315 Houston Tex USA October 2003

[20] B-C Ding H-X Sun and Y-E Qiao ldquoStability analysis of T-Sfuzzy control systems based on parameter-dependent Lyapunovfunctionrdquo Acta Automatica Sinica vol 31 no 4 pp 651ndash6542005

[21] M Narimani H K Lam R Dilmaghani and C Wolfe ldquoLMI-based stability analysis of fuzzy-model-based control systemsusing approximated polynomial membership functionsrdquo IEEE

18 Mathematical Problems in Engineering

Transactions on Systems Man and Cybernetics Part B Cyber-netics vol 41 no 3 pp 713ndash724 2011

[22] H K Lam ldquoPolynomial fuzzy-model-based control systemsstability analysis via piecewise-linear membership functionsrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 588ndash5932011

[23] K Tanaka T Hori and H O Wang ldquoA fuzzy Lyapunovapproach to fuzzy control system designrdquo in Proceedings of theAmerican Control Conference vol 6 pp 4790ndash4795 ArlingtonVa USA June 2001

[24] K Tanaka T Hori and H O Wang ldquoA multiple Lyapunovfunction approach to stabilization of fuzzy control systemsrdquoIEEE Transactions on Fuzzy Systems vol 11 no 4 pp 582ndash5892003

[25] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 174ndash1972005

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

16 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x1 versus w1

x1w1

(a)

0 5 10 15 20 25 30minus80minus70minus60minus50minus40minus30minus20minus10

010

Time

Am

plitu

de

Tracking error e1

e1

(b)

Figure 13 (a) 1199091versus 119908

1and (b) tracking error 119890

1 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

0 5 10 15 20 25 30minus80minus60minus40minus20

0204060

Time

Am

plitu

de

x2 versus w2

x2w2

(a)

0 5 10 15 20 25 30minus8minus6minus4minus2

02468

10

Time

Am

plitu

de

Tracking error e2

e2

(b)

Figure 14 (a) 1199092versus 119908

2and (b) tracking error 119890

2 When the fuzzy controller is designed on the basis of local regulators and the adequate

membership functions for the fuzzy regulator are computed

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(11) (x1(t))1205832(11) (x1(t))

(a)

minus60 minus40 minus20 0 20 40 60 80

0

02

04

06

08

1

Membership functions

Am

plitu

de

1205831(22) (x1(t))1205832(22) (x1(t))

(b)

Figure 15 (a) Membership functions corresponding to position (11) of 1205831(119908(119905)) and 120583

2(119908(119905)) (b) Membership functions corresponding to

position (22) 1205831(119908(119905)) 120583

2(119908(119905))

Mathematical Problems in Engineering 17

suggests that the approach presented in this work may beapplied on a bigger class of T-S models however this studyis not completed yet

6 Conclusions

A fuzzy regulator for continuous-time systems based on thecombination of linear regulators adjusted by different mem-bership functions has been presented The main advantageis that analytical expressions for the membership functionswhich allow the proper combination of the local regulators inorder to guarantee the exact regulation can be easily appliedAs 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 modelscan be commonly found in mechanics electromechanicselectronics robotics and so forth

Some numerical examples are used to illustrate theapplicability of the proposed approach even in cases wherethe dimensions of the plant and the exosystem are differentbetween each otherThis suggests that the approach presentedin this work may be applied on a bigger class of T-S models

Finally the method proposed in this work avoids thedisadvantage of constructing an exact fuzzy regulator basedon overall T-S fuzzy system whose expression may resultvery large Instead the given approach offers 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 Interests

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

Acknowledgments

This work was partially supported by CONACYT (ConsejoNacional de Ciencia y Tecnologıa) through scholarship SNI(Sistema Nacional de Investigadores) and by IPN (InstitutoPolitecnico Nacional) through research Projects 20140659and 20150487 and scholarships EDI (Estımulo al Desempenode los Investigadores) COFAA (Comision de Operacion yFomento de Actividades Academicas) and BEIFI (Beca deEstımulo Institucional de Formacion de Investigadores)

References

[1] B A Francis ldquoThe linear multivariable regulator problemrdquoSIAM Journal on Control and Optimization vol 15 no 3 pp486ndash505 1977

[2] B A Francis andWMWonham ldquoThe internalmodel principleof control theoryrdquo Automatica vol 12 no 5 pp 457ndash465 1976

[3] A Isidori and C I Byrnes ldquoOutput regulation of nonlinearsystemsrdquo IEEE Transactions on Automatic Control vol 35 no2 pp 131ndash140 1990

[4] B Castillo-Toledo S Di Gennaro and F Jurado ldquoTrajectorytracking for a quadrotor via fuzzy regulationrdquo in Proceedings ofthe World Automation Congress (WAC rsquo12) pp 1ndash6 June 2012

[5] S Chen ldquoOutput regulation of nonlinear singularly perturbedsystems based on T-S fuzzy modelrdquo Journal of Control Theoryand Applications vol 3 no 4 pp 399ndash403 2005

[6] R Tapia-Herrera J A Meda-Campana S Alcamp119886119901119900119904 119886ntara-Montes T Hernamp119886119901119900119904 119886ndez-Cortamp119886119901119900119904 119890s and L Salgado-Conrado ldquoTuning of a TS fuzzy output regulator using thesteepest descent approach and ANFISrdquoMathematical Problemsin Engineering vol 2013 Article ID 873430 14 pages 2013

[7] K-Y Lian and J-J Liou ldquoOutput tracking control for fuzzysystems via output feedback designrdquo IEEE Transactions onFuzzy Systems vol 14 no 5 pp 628ndash639 2006

[8] X-J Ma and Z-Q Sun ldquoOutput tracking and regulationof nonlinear system based on takagi-sugeno fuzzy modelrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 30 no 1 pp 47ndash59 2000

[9] H J Lee J B Park and Y H Joo ldquoComments on lsquooutputtracking and regulation of nonlinear system based on Takagi-Sugeno fuzzy model1015840rdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 33 no 3 pp 521ndash523 2003

[10] J A Meda-Campana B Castillo-Toledo and G Chen ldquoSyn-chronization of chaotic systems from a fuzzy regulationapproachrdquo Fuzzy Sets and Systems vol 160 no 19 pp 2860ndash2875 2009

[11] J A Meda-Campana J C Gamp119886119901119900119904 119900mez-Mancilla and BCastillo-Toledo ldquoExact output regulation for nonlinear systemsdescribed by Takagi-Sugeno fuzzy modelsrdquo IEEE Transactionson Fuzzy Systems vol 20 no 2 pp 235ndash247 2012

[12] C Byrnes F Priscoli and A Isidori Output Regulation ofUncertain Nonlinear Systems Birkhauser 1997

[13] A Isidori Nonlinear Control Systems Communications andControl Engineering Series Springer Berlin Germany 3rdedition 1995

[14] K Tanaka and H O Wang Fuzzy Control Systems Designand Analysis A Linear Matrix Inequality Approach John Wileyamp119886119898119901 Sons New York NY USA 2001

[15] L-X Wang A Course in Fuzzy Systems and Control PrenticeHall Upper Saddle River NJ USA 1997

[16] I Abdelmalek N Golamp119886119901119900119904 119890a and M L Hadjili ldquoA newfuzzy Lyapunov approach to non-quadratic stabilization ofTakagi-Sugeno fuzzy modelsrdquo International Journal of AppliedMathematics andComputer Science vol 17 no 1 pp 39ndash51 2007

[17] J A Meda-Campana and B Castillo-Toledo ldquoOn the outputregulation for TS fuzzymodels using slidingmodesrdquo inProceed-ings of the American Control Conference (ACC rsquo05) pp 4062ndash4067 Portland Ore USA June 2005

[18] B Castillo-Toledo and J A Meda-Campana ldquoThe fuzzydiscrete-time robust regulation problem a LMI approachrdquo inProceedings of the 41st IEEE Conference on Decision and Controlvol 2 pp 2159ndash2164 IEEE Las Vegas Nev USA December2002

[19] B Castillo-Toledo J A Meda-Campana and A Titli ldquoAfuzzy output regulator for Takagi-Sugeno fuzzy modelsrdquo inProceedings of the IEEE International Symposium on IntelligentControl pp 310ndash315 Houston Tex USA October 2003

[20] B-C Ding H-X Sun and Y-E Qiao ldquoStability analysis of T-Sfuzzy control systems based on parameter-dependent Lyapunovfunctionrdquo Acta Automatica Sinica vol 31 no 4 pp 651ndash6542005

[21] M Narimani H K Lam R Dilmaghani and C Wolfe ldquoLMI-based stability analysis of fuzzy-model-based control systemsusing approximated polynomial membership functionsrdquo IEEE

18 Mathematical Problems in Engineering

Transactions on Systems Man and Cybernetics Part B Cyber-netics vol 41 no 3 pp 713ndash724 2011

[22] H K Lam ldquoPolynomial fuzzy-model-based control systemsstability analysis via piecewise-linear membership functionsrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 588ndash5932011

[23] K Tanaka T Hori and H O Wang ldquoA fuzzy Lyapunovapproach to fuzzy control system designrdquo in Proceedings of theAmerican Control Conference vol 6 pp 4790ndash4795 ArlingtonVa USA June 2001

[24] K Tanaka T Hori and H O Wang ldquoA multiple Lyapunovfunction approach to stabilization of fuzzy control systemsrdquoIEEE Transactions on Fuzzy Systems vol 11 no 4 pp 582ndash5892003

[25] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 174ndash1972005

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 17

suggests that the approach presented in this work may beapplied on a bigger class of T-S models however this studyis not completed yet

6 Conclusions

A fuzzy regulator for continuous-time systems based on thecombination of linear regulators adjusted by different mem-bership functions has been presented The main advantageis that analytical expressions for the membership functionswhich allow the proper combination of the local regulators inorder to guarantee the exact regulation can be easily appliedAs 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 modelscan be commonly found in mechanics electromechanicselectronics robotics and so forth

Some numerical examples are used to illustrate theapplicability of the proposed approach even in cases wherethe dimensions of the plant and the exosystem are differentbetween each otherThis suggests that the approach presentedin this work may be applied on a bigger class of T-S models

Finally the method proposed in this work avoids thedisadvantage of constructing an exact fuzzy regulator basedon overall T-S fuzzy system whose expression may resultvery large Instead the given approach offers 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 Interests

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

Acknowledgments

This work was partially supported by CONACYT (ConsejoNacional de Ciencia y Tecnologıa) through scholarship SNI(Sistema Nacional de Investigadores) and by IPN (InstitutoPolitecnico Nacional) through research Projects 20140659and 20150487 and scholarships EDI (Estımulo al Desempenode los Investigadores) COFAA (Comision de Operacion yFomento de Actividades Academicas) and BEIFI (Beca deEstımulo Institucional de Formacion de Investigadores)

References

[1] B A Francis ldquoThe linear multivariable regulator problemrdquoSIAM Journal on Control and Optimization vol 15 no 3 pp486ndash505 1977

[2] B A Francis andWMWonham ldquoThe internalmodel principleof control theoryrdquo Automatica vol 12 no 5 pp 457ndash465 1976

[3] A Isidori and C I Byrnes ldquoOutput regulation of nonlinearsystemsrdquo IEEE Transactions on Automatic Control vol 35 no2 pp 131ndash140 1990

[4] B Castillo-Toledo S Di Gennaro and F Jurado ldquoTrajectorytracking for a quadrotor via fuzzy regulationrdquo in Proceedings ofthe World Automation Congress (WAC rsquo12) pp 1ndash6 June 2012

[5] S Chen ldquoOutput regulation of nonlinear singularly perturbedsystems based on T-S fuzzy modelrdquo Journal of Control Theoryand Applications vol 3 no 4 pp 399ndash403 2005

[6] R Tapia-Herrera J A Meda-Campana S Alcamp119886119901119900119904 119886ntara-Montes T Hernamp119886119901119900119904 119886ndez-Cortamp119886119901119900119904 119890s and L Salgado-Conrado ldquoTuning of a TS fuzzy output regulator using thesteepest descent approach and ANFISrdquoMathematical Problemsin Engineering vol 2013 Article ID 873430 14 pages 2013

[7] K-Y Lian and J-J Liou ldquoOutput tracking control for fuzzysystems via output feedback designrdquo IEEE Transactions onFuzzy Systems vol 14 no 5 pp 628ndash639 2006

[8] X-J Ma and Z-Q Sun ldquoOutput tracking and regulationof nonlinear system based on takagi-sugeno fuzzy modelrdquoIEEE Transactions on Systems Man and Cybernetics Part BCybernetics vol 30 no 1 pp 47ndash59 2000

[9] H J Lee J B Park and Y H Joo ldquoComments on lsquooutputtracking and regulation of nonlinear system based on Takagi-Sugeno fuzzy model1015840rdquo IEEE Transactions on Systems Man andCybernetics Part B Cybernetics vol 33 no 3 pp 521ndash523 2003

[10] J A Meda-Campana B Castillo-Toledo and G Chen ldquoSyn-chronization of chaotic systems from a fuzzy regulationapproachrdquo Fuzzy Sets and Systems vol 160 no 19 pp 2860ndash2875 2009

[11] J A Meda-Campana J C Gamp119886119901119900119904 119900mez-Mancilla and BCastillo-Toledo ldquoExact output regulation for nonlinear systemsdescribed by Takagi-Sugeno fuzzy modelsrdquo IEEE Transactionson Fuzzy Systems vol 20 no 2 pp 235ndash247 2012

[12] C Byrnes F Priscoli and A Isidori Output Regulation ofUncertain Nonlinear Systems Birkhauser 1997

[13] A Isidori Nonlinear Control Systems Communications andControl Engineering Series Springer Berlin Germany 3rdedition 1995

[14] K Tanaka and H O Wang Fuzzy Control Systems Designand Analysis A Linear Matrix Inequality Approach John Wileyamp119886119898119901 Sons New York NY USA 2001

[15] L-X Wang A Course in Fuzzy Systems and Control PrenticeHall Upper Saddle River NJ USA 1997

[16] I Abdelmalek N Golamp119886119901119900119904 119890a and M L Hadjili ldquoA newfuzzy Lyapunov approach to non-quadratic stabilization ofTakagi-Sugeno fuzzy modelsrdquo International Journal of AppliedMathematics andComputer Science vol 17 no 1 pp 39ndash51 2007

[17] J A Meda-Campana and B Castillo-Toledo ldquoOn the outputregulation for TS fuzzymodels using slidingmodesrdquo inProceed-ings of the American Control Conference (ACC rsquo05) pp 4062ndash4067 Portland Ore USA June 2005

[18] B Castillo-Toledo and J A Meda-Campana ldquoThe fuzzydiscrete-time robust regulation problem a LMI approachrdquo inProceedings of the 41st IEEE Conference on Decision and Controlvol 2 pp 2159ndash2164 IEEE Las Vegas Nev USA December2002

[19] B Castillo-Toledo J A Meda-Campana and A Titli ldquoAfuzzy output regulator for Takagi-Sugeno fuzzy modelsrdquo inProceedings of the IEEE International Symposium on IntelligentControl pp 310ndash315 Houston Tex USA October 2003

[20] B-C Ding H-X Sun and Y-E Qiao ldquoStability analysis of T-Sfuzzy control systems based on parameter-dependent Lyapunovfunctionrdquo Acta Automatica Sinica vol 31 no 4 pp 651ndash6542005

[21] M Narimani H K Lam R Dilmaghani and C Wolfe ldquoLMI-based stability analysis of fuzzy-model-based control systemsusing approximated polynomial membership functionsrdquo IEEE

18 Mathematical Problems in Engineering

Transactions on Systems Man and Cybernetics Part B Cyber-netics vol 41 no 3 pp 713ndash724 2011

[22] H K Lam ldquoPolynomial fuzzy-model-based control systemsstability analysis via piecewise-linear membership functionsrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 588ndash5932011

[23] K Tanaka T Hori and H O Wang ldquoA fuzzy Lyapunovapproach to fuzzy control system designrdquo in Proceedings of theAmerican Control Conference vol 6 pp 4790ndash4795 ArlingtonVa USA June 2001

[24] K Tanaka T Hori and H O Wang ldquoA multiple Lyapunovfunction approach to stabilization of fuzzy control systemsrdquoIEEE Transactions on Fuzzy Systems vol 11 no 4 pp 582ndash5892003

[25] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 174ndash1972005

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

18 Mathematical Problems in Engineering

Transactions on Systems Man and Cybernetics Part B Cyber-netics vol 41 no 3 pp 713ndash724 2011

[22] H K Lam ldquoPolynomial fuzzy-model-based control systemsstability analysis via piecewise-linear membership functionsrdquoIEEE Transactions on Fuzzy Systems vol 19 no 3 pp 588ndash5932011

[23] K Tanaka T Hori and H O Wang ldquoA fuzzy Lyapunovapproach to fuzzy control system designrdquo in Proceedings of theAmerican Control Conference vol 6 pp 4790ndash4795 ArlingtonVa USA June 2001

[24] K Tanaka T Hori and H O Wang ldquoA multiple Lyapunovfunction approach to stabilization of fuzzy control systemsrdquoIEEE Transactions on Fuzzy Systems vol 11 no 4 pp 582ndash5892003

[25] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 174ndash1972005

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