chaos synchronization

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Research ArticleSynchronization of Discrete-Time Chaotic Fuzzy Systems by means of Fuzzy Output Regulation Using Genetic Algorithm

Tonatiuh Hernández Cortés,1 A. Verónica Curtidor López,1 Jorge Rodríguez Valdez,1

Jesús A. Meda Campaña,1 Ricardo Tapia Herrera,1 and José de Jesús Rubio2

Instituto Polit ́ecnico Nacional, SEPI-ESIME Zacatenco, Avenida IPN S/N, M ́exico, DF, MexicoInstituto Polit ́ecnico Nacional, SEPI-ESIME Azcapotzalco, Avenida de las Granjas No. ,

Azcapotzalco, M ́exico, DF, Mexico

Correspondence should be addressed to onatiuh Hernández Cortés; tona [email protected]

Received August ; Accepted November

Academic Editor: Rongwei Guo

Copyright © onatiuh Hernández Cortés et al. Tis 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.

Te synchronization o chaotic systems, described by discrete-time -S uzzy models, is treated by means o uzzy output regulationtheory. Te conditions or designing a discrete-time output regulator are given in this paper. Besides, when the system does notulll the conditions or exact tracking, a new regulator based on genetic algorithms is considered. Te genetic algorithms are

used to approximate the adequate membership unctions, which allow the adequate combination o local regulators. As a result,the tracking error is signicantly reduced. Both the Complete Synchronization and the Generalized Synchronization problem arestudied. Some numerical examples are used to illustrate the effectiveness o the proposed approach.

1. Introduction

A special nonlinear dynamical phenomenon, known aschaos, emerged in mid-s and reached applicable tech-nology in the late s and was considered as one o the three monumental discoveries o the twentieth century.On the other hand, uzzy logic, a set theory and thenan innite-valued logic, gets a wide applicability in many

industrial, commercial, and technical elds, ranging romcontrol, automation, and articial intelligence, just to namea ew. Fuzzy logic and chaos had been considered by many researches and engineers as undamental concepts and the-ories and their broad applicability in technology as well. Teinteraction between uzzy logic andchaoshas been developedor the last years leading to research topics as uzzy modeling o chaotic systems using akagi-Sugeno models,linguistic descriptions o chaotic systems, uzzy control o chaos, synchronization, and a combination o uzzy chaos orengineering applications [, ].

In the s, Rechenberg [] introduced “evolutionstrategies,” a method to optimize real-valued parameters or

devices such as airoils. Tis idea was urther developedby Schweel in []. Genetic algorithms (GAs) were initially developed by Bremermann [] in but popularized anddeveloped by Holland in the s. In contrast with evolutionstrategies and evolutionary programming, Holland’s ideawas not to design algorithms to solve specic problems butrather to ormally study the phenomenon o adaptation, as itoccurs in nature, and develop ways in which the mechanisms

o natural adaptation might be transerred into computersystems []. Te genetic algorithm is presented as abstractiono biological evolution and theoretical ramework or adapta-tion or moving rom one population o “chromosomes” (e.g.,strings o ones and zeros, or “bits”) to a new population by using a kind o “natural selection” together with the genetics-inspired operators o crossover, mutation, and inversion.Each chromosome consists o “genes” (e.g., bits); each geneis being an instance o a particular “allele” (e.g., or ).Te operator selection chooses those chromosomes in thepopulation that will be allowed to reproduce and thoseadjusted chromosomes produce more offspring than the lessones [].

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015, Article ID 198371, 18 pageshttp://dx.doi.org/10.1155/2015/198371


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Mathematical Problems in Engineering

According to Fogel and Aderson [], Bremermann wasthe rst to implement real-coded genetic algorithms as wellas providing a mathematic model o GA known as the one-max unction. In contrast to genetic algorithms, Evolutionary Strategies were initially developed or the purpose o param-eter optimization. Te idea was to imitate the principles o

organic evolution in experimental parameter optimizationor applications such as pipe bending or PID control or anonlinear system [].

Synchronization o chaotic systems is one o the moreexiting problems in control science and can be reerred atleast to Huygen’s observations []; it is understood as one o the trajectories o two autonomous chaotic systems, startingrom nearly initial conditions and converging to the other,andremains as → ∞; in [] it was reported that some kindo chaotic systems possesses a sel-synchronization property.However, not all chaotic systems can be decomposed intwo separate responses subsystems and be able to synchro-nize the drive system. Te ideas o these works have ledto improvement in many elds, such as communications[], encrypted systems, the complex inormation processingwithin the human brain, coupled biochemical reactors, andearthquake engineering [].

Synchronization can be classied as ollows: CompleteSynchronization: it is when two identical chaos oscillators aremutually coupled and one drives to the other; GeneralizedSynchronization: it differs rom the previous case by the actthat there are different chaos oscillators and the states o oneare completely dened by the other; Phase Synchronization: itoccurswhen the coupled oscillators arenot identical andhavedifferent amplitude that is still unsynchronized, while thephaseso oscillators evolve to be synchronized[]. Itis worthmentioning that studies in synchronization o nonlinearsystems have been reormulated based on the previous resultsrom classical control theory such as [–].

In this paper, the uzzy output regulation theory andakagi-Sugeno (-S) uzzy models are used to solve theComplete and Generalized Synchronization by using linearlocal regulators. Isidori and Byrnes [] showed that theoutput regulation established by Francis could be extendedor a nonlinear sector as a general case, resulting in a set o nonlinear partialdifferential equations calledFrancis-Isidori-Byrnes (FIB). Unortunately these equations in many casesare too difficult to solve in a practical manner. For this reasonin [] the approach based on the weighted summationo local linear regulators is presented and in [] the new

membership unctions in the regulator are approximated by sof computing techniques.

So, the main contribution o the present work is to nda control law or synchronizing o chaotic systems describedby discrete-time akagi-Sugeno uzzy models, rst when thesystem ullls the ollowing: (1) the input matrix or allsubsystems is the same and (2) the local regulators sharethe same zero error maniold (). In this way, the resultsgiven in [] are extended to the discrete-time domain. Onthe other hand, when the system master-slave does not ulllthe aorementioned conditions, new membership unctionsare computed in order to enhance the perormance o theuzzy regulator. Such proposed membership unctions are

different rom those given in the plant or exosystem and aretuned by using the GA. Te tuning o the new membershipunctions, which is as generalized bell-shaped unction, isgiven by optimization o the orm parameter.

Te rest o the paper is organized as ollows. In Section the discrete-time output regulation problem ormulation

is given with a brie review o the akagi-Sugeno modelsand the discrete-time uzzy regulation problem. In Section the tuning o membership unctions by means o GAs isthoroughly discussed. In Section Complete and General-ized Synchronization with some examples are presented andnally, in Section , some conclusions are drawn.

2. The Discrete-Time Fuzzy OutputRegulation Problem

Consider a nonlinear discrete-time system dened by

+1 = , , , () = , ()+1 = , ()

re , = , () = ℎ, , ()

where ∈ R is the state vector o the plant, ∈ ⊂R is the state vector o the exosystem, which generates the

reerence and/or the perturbation signals, and ∈ R isthe input signal. Equation () reers to difference betweenoutput system o the plant ( ∈ R) and the reerence signal(ref , ∈ R

); that is, ℎ(, ) = − ref , = () − ()and take into account that ≤ . Besides, it is assumed that(, , ), ℎ(, ), and () are analytical unctionsand also that (0, 0, 0) = 0, (0) = 0, and ℎ(0, 0) = 0 [].

Clearly, by linearizing ()–() around = 0, one gets+1 = + + , = ,

+1 = ,re , = ,

=

−

.

()

Tus, the Nonlinear Regulator Problem [, ] consists o nding a controller = (, ), such that the closed-loopsystem +1 = + (, 0) has an asymptotically stableequilibrium point, and the solution o system () satiseslim→∞ = 0.

So, by dening () as the steady-state zero errormaniold and () as the steady-state input, the ollowingtheorem gives the conditions or the solution o nonlinearregulation problem.

Teorem . Suppose that +1 = () is Poisson stable and there exists a gain

such that the matrix

+is stable and


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there exist mappings () = () and = () with(0) = 0 and (0) = 0 satisying = , , ,

0 = ℎ , . ()Ten the control signal or the nonlinear regulation is given by

= − + . ()Proo. See [–].

Te equation set () is known as Discrete-ime Francis-Isidori-Byrnes (DFIB) equations and linear counterpart isobtained when the mappings , = () and , = ()transorm into , = Π and , = Γ, respectively.Tus, the problem is reduced to solve linear matrix equations[] given by

Π = Π + Γ + ,0 = Π − . ().. Te Discrete-ime Output Fuzzy Regulation Problem.akagi and Sugeno proposed a uzzy model composed o a set o linear subsystems with IF-HEN rules capable o relating physical knowledge, linguistic characteristics, andproperties o the system. Such a model successully representsa nonlinear system at least in a predened region o phasespace []. Te -S model or the plant and exosystem can bedescribed as ollows []:

Plant Model

Rule :IF 1,1, is 1,1 and . . . and 1,1, is 1,1 ,

HEN+1 = + + , = ,

= 1,2 , . . . , 1,()

where 1 is the number o rules in the model o theplant and the sets1, are the uzzy sets dened basedon the previous dynamic knowledge o the system.

Exosystem Model

Rule :IF 2,1, is 2,1 and . . . and 2,2, is 2,2 ,

HEN+1 = ,ref , = Q, = 1,2 , . . . , 2, ()

where 2 is the number o rules in the model o theexosystem and

2, are the uzzy sets.

Ten, the regulation problem dened by ()–() can berepresented through the -S discrete-time uzzy model; thatis, []

+1 = 1=1

ℎ1, 1, + + ,

+1 = 2=1

ℎ2, 2, ,

= 1=1

ℎ1, 1, − 2=1

ℎ2, 2,,

()

where ∈ R is the state vector o the plant, ∈ R isthe state vector o the exosystem, ∈ R is the input signal, ∈ R, and ℎ∗,() is the normalized weight o each rule, 1or the plant and 2 or the exosystem, which depends on themembership unction or the premise variable ∗, in ∗,;that is,

∗, ∗, = ∗∏=1

∗, ∗,, ,

ℎ∗, ∗, = ∗, ∗,∑∗=1 ∗, ∗, ,∗=1

ℎ∗, ∗, = 1,ℎ∗, ∗, ≥ 0

()

with ∗, = [∗,1, ∗,2, ⋅ ⋅ ⋅ ∗,∗,] as a unction o and/or , = 1, . . . , ∗ and = 1, . . . ,.Te discrete-time uzzy output regulation problem consists

o nding a controller = (, ), such that the closed-loop system with no external signal

+1 = 1=1

ℎ1, 1, + ,0 ()has an asymptotically stable equilibrium point.

Te solution o system () satises

lim→∞

= 0. ()

In orderto achieve the synchronization o chaotic systemsdescribed by a -S discrete-time uzzy model it is necessary to ulll () [, ]. Ten

= 1

=1

ℎ1, 1, + + , ()

0 = 1=1

ℎ1, 1, − 2=1

ℎ2, 2,, ()


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where () is the zero error steady-state maniold whichbecomes invariant by the effect o the steady-state input().

Assuming the mappings () and () as

�̃�

=

1

=1ℎ1,

1,

2

=1ℎ2,

2,

Π,

, ()

̃ = 1=1

ℎ1, 1, 2=1

ℎ2, 2, Γ,, ()respectively, with Π, and Γ, as a solution o 1 ⋅2 lineal localproblems,

Π = Π + Γ + , ()0 = Π, − , ()

or all = 1, . . . , 1 and = 1, . . . , 2, the ollowing control law can be obtained [, , ]:

= 1ℎ=1

ℎ1, 1,

⋅ −

1=1

ℎ1, 1, 2=1

ℎ2, 2,Π+ 1

=1

ℎ1, 1, 2=1

ℎ2, 2, Γ.

()

However, by substitution o () and () in () and ()and considering

() the steady-state zero error maniold () = Π,that is, Π = Π,() the input matrices = and/or Γ = Γ,

or all = 1, . . . , 1 and = 1, . . . , 2, the ollowing controlsignal emerges:

= 1ℎ=1

ℎ1, 1, − Π

+ 1=1

ℎ1, 1, 2=1

ℎ2, 2, Γ.()

On the other hand, the existence o a uzzy stabilizer o theorm = ∑1=1 ℎ1,(1,), ensuring that the tracking errorconverges asymptotically to zero, can be obtained rom theParallel Distributed Compensator (PDC) [, ] or anotherstability analysis or -S uzzy models such as [].

Remark . Te control signal in () is given by the substitu-tion o () and () in () and (); the proposed controllerprovides the ollowing advantages:

() All parameters included in the controller are known;this includes the membership unctions o the plantandexosystem, which are well denedin the-S uzzy model.

On the other hand, Π and Γ come directly romsolving o 1 ∗ 2 local linear problems equivalentto solving the Francis equations; such problems canbe easily solved by using programs like Matlab orMathematica.

() In the case when the

1

∗ 2 local linear problems lead

to Π ≠ Π, then at least a bounded error is ensured.() It is clear that when Π = Π, the term∑1=1 ℎ1,(1,)∑2=1 ℎ2,(2,)Π changes to Πleading to controller dened in ().

() Te ollowing condition: the input matrices = and/or Γ = Γ avoids the crossed terms in thesolutions o () and (), allowing the exact uzzy output regulation.

() Te proposed controller can be seen as a simplesubstitution o the aorementioned elements.

On the other hand, the ollowing disadvantages canappear:

() I the condition is that the steady-state zero errormaniold () ≠ Π, that is, Π ≠ Π, then, it willbe necessary to adjust the local regulator by means o new membership unctions. Please reer to Section .

() As expected, the complexity o the controllerincreases according to the number o localsubsystems.

Te ollowing theorem provides the conditions or theexistence o the exact uzzy output regulator or a discrete-time -S uzzy models.

Teorem . Te exact uzzy output regulation with ull inormation o systems dened as () is solvable i (a) thereexists the same zero error steady-state maniold () = Π ;(b) there exist = ∑1=1 ℎ,(1,) or the uzzy system;(c) the exosystem +1 = () is Poisson stable, and the input matrices or all subsystems are equal. Moreover, the Exact Output Fuzzy Regulation Problem is solvable by the controller

= 1ℎ=1

ℎ1, 1, − Π

+ 1=1

ℎ1, 1, 2=1

ℎ2, 2, Γ.()

Proo. From the previous analysis, the existence o mappings() = Π and () = ∑1=1 ℎ1,(1,) ∑2=1 ℎ2,(2,)Γis guaranteed when the input matrices or all subsystems are equal, and the solution o 1 ⋅ 2 lineal local problems is

Π = Π + Γ + ,0 = Π, − , ()

leading to Π = Π or all = 1, . . . , 1 and = 1, . . . , 2.On the other hand, the inclusiono condition (b)has been

thoroughly discussed in [, , , , –], and it impliesthe existence o a uzzy stabilizer.


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Condition (c) ensures the nonexistence o crossed termsin the local Francis equations. Finally, condition (d) is intro-duced to avoid the act that the reerence signal converges tozero, which would turn the regulation problem into a simplestability problem. Te rest o the proo ollows directly romthe previous analysis.

3. The Output Regulator by means of Local Regulators and Tuning of New Membership Functions

In this section, a discrete-time -S uzzy model is consideredto solve the exact output regulation on the basis o linearlocal controllers. So, the main goal is to nd a completeregulator based on the uzzy summation o local regulatorsusing adequate membership unctions, such that the resultgiven in Teorem can be relaxed []. Tese membershipunctions are not necessarily the same included in the uzzy plant, as is described in (). Tus, the steady-state input

() can be dened as =

1=1

1, 1, 2=1

2, 2, Γ, ()where 1,(1,) and 2,(2,) are new membership unc-tions, such that the uzzy output regulator obtained romlocal regulators provides the exact uzzy output regulation.Tis approach requires the computation o the linear localcontrollers and the computation o the new membershipunctions. In this work, such unctions are represented by theollowing expression:

1, = 11 + − /2 , ∀ = 1, . . . , 1, ()

2, = 11 + − /2, ∀ = 1, . . . , 2. ()

1, and 2, are well known as generalized bell-shaped mem-bership unctions and the parameters , , , , , and determine the orm, center, and amplitude, respectively.Tereore, rom (), the input can be dened by

= 1

=1ℎ1, − + 1

=1

1, 2=1

2, Γ,()

because∗, is a unctiono and in steady-state = ().Ten, or tuning the membership unctions () and (),

the parameters and will be optimizedby means o geneticalgorithms, ensuring the correct interpolation between thelocal linear regulators. Te oregoing can be summarized inthe control scheme depicted in Figure .

o thisend,the ollowingalgorithm shouldbe carried out.

Algorithm . Main steps to take into account to solve thetuning membership unctions problem are as ollows []:

() Start with a randomly generated population o nl-bitchromosomes (candidate solutions to a problem).Tetraditional representation is binary as in Figure .

A binary stringis called“chromosome.” Each positiontherein is called “gene” and the value in this positionis named “allele.”

() Calculate the tness () o each chromosome inthe population.

() Repeat the ollowing steps until n offspring have beencreated:

(i) Select a pair o parent chromosomes rom thecurrent population, the probability o selectionbeing an increasing unction o tness. Selectionis done “with replacement,” meaning that thesame chromosome can be selected more than

once to become a parent.(ii) With probability (the “crossover probabil-

ity” or “crossover rate”), crossover the pair ata randomly chosen point (chosen with uni-orm probability) to orm two offspring. I nocrossover takes place, orm two offspring thatare exact copies o their respective parents.(Notethat here thecrossover rate is denedto bethe probability that two parents will cross overin a single point.)

(iii) Mutate the two offspring at each locus withprobability (the mutation probability ormutation rate), and place the resulting chromo-

somes in the new population.

() Replace the current population with the new popula-tion.

() Go to step ().

Tis process ends when the tness unction reaches a value less than or equal to a predened bound, or when themaximum number o iterations is reached. Each iteration o this process is called a generation.

In order to apply a genetic algorithm it requires theollowing ve basic components:

(i) A representation o the potential solutions to theproblem.

(ii) One way to create an initial population o possiblesolutions (typically a random process).

(iii) An evaluation unction to play the role o the environ-ment, classiying the solutionsin terms o its “tness.”

(iv) Genetic operators that alter the composition o thechromosomes that will be produced or generations.

(v) Values or the various parameters used by the geneticalgorithm (population size, crossover probability,mutation probability, maximum number o genera-tions, etc.).


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FuzzyexosystemFuzzy plant

Fuzzy controller

Regulator

Stabilizer

Bell-shaped mf’s

eGenetic algorithm

uk yk wk

xk

+

+

+

−

ai bi ci

aj bj cj

(ai, aj)

yref ,k

r1

∑i=1 h1,ixkki xk − wk

r1

∑i=1

1,iwkr2

∑j=1

2,jwkΓijwk1,iwk =

1

1 + xk − ci ai i2b

2,jwk = 1

1 + wk − cj aj 2bj

F : Control scheme or uzzy output regulation and genetic algorithms.

0 1 1 0 1 1 0 1 0 0 1 1 1 0 0 1

String 1 String 2 String 3 String 4

F : Binary string commonly used in genetic algorithms.

Te aorementioned can be summarized by the owchartdepicted in Figure .

4. Synchronization of T-S Discrete-TimeFuzzy Systems

Te stabilization and synchronization o chaotic systems aretwo o the most challenging and stimulating problems dueto their capabilities o describing a great variety o very interesting phenomena in physics, biology, chemistry, andengineering, to name a ew. In this section, the regulationtheory is used to synchronize chaotic systems described by -S discrete-time uzzy models. Both the drive system and theresponse system are modeled by the same attractor (R ̈ossler’sequation) with the difference that response system can beinuenced by an input signal. Tis type o synchronizationis known as Complete Synchronization (CS) [].

Considering two Rössler chaotic oscillators as �̇� = ()(drive system) and �̇� = (,, ) (response system), theordinary differential equations o these systems are

̇1 = −2 + 3 , ̇2 = 1 + 2,

̇3 = 1 − − 1 3,R ̈ossler attractor

Drive system

,

̇1 = −2 + 3 , ̇2 = 1 + 2,

̇3 = 1 − − 1 3 + ,R ̈ossler attractor

Response system.()

According to [], these systems can be exactly repre-sented by means o the ollowing two-rule -S uzzy modelswhen , and are constants and 1 ∈ [ − , + ] with > 0. Tus

̇ () = 2

=1ℎ1, 1 () { () + ̃ () , ̇ () = 2

=1

ℎ2, 1 () ̃ () ,

() = 2=1

ℎ1, 1 () ()

− 2=1

ℎ2, 1 () () ,

()


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Begin

Random initial

population

Individual performance

evaluation

Reproduction

Crossover

Mutation

Anothergeneration

Membership functions

performance

End

Yes

No

GA

Chromosome selection

for reproduction

e s

=

1 2 x

1 , k

− w

1 , k

2

O b j e c t i v e f u n c t i o n

F : Flowchart o genetic algorithms.

where

1 =

0 −1 −11 0 0 −

,

̃1 =

0 −1 −11 0

0 −

,

2 = 0 −1 −11 0 0

,

̃2 =

0 −1 −11 0 0

,

̃1 = ̃2 = 0 0 1 , = = 1 0 0 .

()

Besides, the membership unctions or such systems are

ℎ1,1 1 = 12 1 + − 1 ,

ℎ1,2

1

= 12 1 −

− 1 ,ℎ2,1 1 = 12 1 + − 1 ,

ℎ2,2 1 = 12 1 − − 1 .

()

Now, the continuous-time -S uzzy model can be convertedto the ollowing discrete counterpart by using []

= exp = + + 2 22! + ⋅ ⋅ ⋅ ,

= ∫

0exp i = − −1 .

()

Tereore, the discrete-time -S uzzy model is given by

+1 = 2=1

ℎ1, 1, + + ,

+1 = 2=1

ℎ2, 1, ,

= 2

=1ℎ1,

1,

− 2

=1ℎ2,

1,

,

()

where

1 =

1 − − + 1 0 0 1 −

,

1 =

1 − − + 1 0 0 1 −

,

2 =

1 − − + 1 0 0 + 1

,

2 =

1 − − + 1 0 0 + 1

,

1 = 2 = 0 0 , = = 1 0 0 .

()


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0

0

5

105

0

2

4

6

8

−5

−5

−2

−10−10

x1x2

x 3

Continuous-time fuzzy Rössler attractor

(a)

0

5

10

0

5

0

2

4

6

8

−2

x 1, kx 2 ,k

x 3

, k

−5

−5

−10

Discrete-time fuzzy Rössler attractor

(b)

F : (a) Continuous -S uzzy model or a R ̈ossler attractor and (b) discrete -S uzzy model or a R ̈ossler attractor.

Besides, the membership unctions or the -S discrete-time

uzzy model are

ℎ1,1 1, = 12 1 + − 1, ,

ℎ1,2 1, = 12 1 + − 1, ,

ℎ1,2 1, = 12 1 + − 1, ,

ℎ2,2 1, = 12 1 + − 1, ,

()

with

= 0.34,

= 0.4,

= 4.5,

= 10,

= 0.34,

= 0.4,

= 4.5, = 10, and as the sampling time.Figure (b) shows the trajectory o the discrete-time version o the continuous-time -S uzzy R ̈ossler model, with = 0.00357. It can be seen that the overall shape o thetrajectory is similar to that in Figure (a).

Ten, by using the approach derived in this work androm () and (), the zero error steady-state maniold() = Π is

Π =

1 0 00 1 0

0 0 1

. ()

Γ areΓ1,1 = 0 0 0 ,Γ1,2 = 0 0 20 ,Γ2,1 = 0 0 −20 ,Γ2,2 = 0 0 0 .

()

On the other hand, the uzzy stabilizer or this system iscomputed by means o Ackermann’s ormula, and by locatingthe eigenvalues at

() subsystem [0.9980+0.0062 0.9980−0.0062 0.9982],() subsystem [0.9980+0.0062 0.9980−0.0062 0.9982],

the ollowing gains are obtained:

1 = 3.1629 0.9257 8.0354 ,2 = 3.1629 0.9257 −11.9646 . ()

Remark . It is important to veriy that the uzzy eedback stabilizer is valid or the overall -S uzzy model, by checkingthat the eigenvalues o the interpolation regions are inside theunit circle also [].

Ten, by setting the initial conditions as1, = 5, 2, = 0,3, = 6, 1, = 1, 2, = 0, and 1, = 0 and by applyingthe controller (), the results depicted in Figures and areobtained.

Te tracking or the drive states and response states isdrawn in Figure .

.. Generalized Synchronization. Te discrete-time uzzy synchronization problem is solvable when the conditions o Teorem are ullled. However, when two chaotic systemsare different, ullling these conditions is not so common.Tis is because the local regulators have different zero error

steady-state maniolds, in general []. From the regulationpoint o view, the problem can be seen as nding, i it ispossible, a transormation : → regarding mappingthe trajectories o the drive system into the ones o theresponse systems; that is, = (); this is known asGeneralized Synchronization [, ] and satises

lim→∞

( (0)) − ( (0)) = 0, ()with (0) and (0) as initial conditions.

Now, consider the chaotic drive system as �̇� = () and

̇ = (,,) as the response system.


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0 5 10 15 20 25 30 35 40

0

5

10

Time

A m p l i t u d e

−10

−5

−15

Control uk

uk

F : Control signal or the Complete Synchronization o twodiscrete-time -S uzzy R ̈ossler systems.

0 5 10 15 20 25 30 35 40

0

1

2

3

4

Time

A m p

l i t u d e

−1

−2

−3

Tracking error e1,k = x1,k − w1,k

e1,k

F : racking error signal or the Complete Synchronizationo two discrete-time -S uzzy Rössler systems.

Te equations or the aorementioned systems aredescribed as ollows:

̇1 = 2 − 1 , ̇2 = 1 − 2 − 13, ̇3 = 12 − 3,Lorenz Attractor

Drive system

̇1 = 2 − 1 , ̇2 = ( − ) 1 − 13 + 2 + , ̇3 = 12 − 3,Chen’s attractor

Response system

()

As beore these systems can be exactly represented by means o the ollowing two-rule continuous-time -S uzzy models when 1 ∈ [min , max ] and 1 ∈ [1,2].Tereore,

̇ () = 2=1

ℎ1, 1 () { () + ̃ () , ̇ () = 2

=1

ℎ2, 1 () ̃ () ,

() = 2=1

ℎ1, 1 () ()− 2

=1

ℎ2, 1 () () ,()

where

1 =

− 0 − −min0 min −

,

̃1 =

− 0 −1 −10 1 −

,

2 =

− 0

− −max0 max − ,̃2 =

− 0 −1 −20 2 −

,

̃1 = ̃2 = 0 1 0 , = = 1 0 0 .

()

Ten, by using (), the discrete counterpart is obtained:

+1

= 2

=1ℎ1,

1,

+

+

,+1 = 2=1

ℎ2, 1, , =

2=1

ℎ1, 1, − 2=1

ℎ2, 1, ,()

where

1 =

1 − 0− ( − ) + 1 −min

0 min 1 − ,

1 = 1 − 0 1 − −1

0 1 1 − ,

2 =

1 − 0− ( − ) + 1 −max

0 max 1 + ,

2 =

1 − 0 1 − −2

0 2

1 −

,


10/19


0 5 10 15 20 25 30 35 40

0

2

4

6

8

A m p l i t u d e

−6−4

−2

x1,k versus w1,k

x1,kw1,k

(a)

0 5 10 15 20 25 30 35 40

0

2

4

A m p l i t u d e

−6

−4

−2

−8

x2,k versus w2,k

x2,kw2,k

(b)

0 5 10 15 20 25 30 35 40

0

2

4

6

8

A m p l i t u d e

−2

−4

x3,k versus w3,k

x3,kw3,k

(c)

F : States o drive and response tracking or a Complete Synchronization.

1 = 2 = 0 0 , = = 1 0 0 ,

()

with = 35, = 3, = 28, = 10, = 8/3, and = 28 and as the sampling time. Notice that, in thiscase, the Generalized Synchronization problem consists o the tracking o 1 by 1. Tis can be inerred by the ormo and . Te membership unctions or this system aredened as ollows:

Plant Membership Functions

ℎ1,1 1, = −1, + maxmax − min ,

ℎ1,2 1, = 1, − minmax − min .()

Exosystem Membership Functions

ℎ2,1 1, = −1, + 12 − 1 ,

ℎ2,2 1, = 1, − 22 − 1 .()

Tey are depicted in Figure .Figures (a) and (b) show the behavior o the two

discrete-time -S uzzy models, with

(0) = [1 1 1] and

(0) = [1 0 0]. Ten, by using the approach derived inthis work and rom () and (), the zero error steady-statemaniold or each subsystem is

Π1,1 =

1 0 00.7143 0.2857 03.0041 −0.0143 1.2861

,

Π1,2 =

1 0 00.7143 0.2857 03.0041 −0.0143 −1.2861

,

Π2,1 =

1 0 00.7143 0.2857 0−3.0041 0.0143 −1.2861

,

Π2,2 =

1 0 00.7143 0.2857 0−3.0041 0.0143 1.2861

.

()

Γ areΓ1,1 = −102.2648 −0.7142 −30.0121 ,Γ1,2 = −102.2648 −0.7142 30.0121 ,Γ2,1 = −102.2648 −0.7142 −30.0121 ,

Γ2,2

= −102.2648 −0.7142 30.0121 .

()


11/19


0

1

0

1

Xmin Xmax M1 M2

h1,1(x1,k) h1,2 (x1,k ) h2,1(w1,k h2,2(w1,k ))

w1,kx1,k

F : Membership unctions or Generalized Synchronization.

020

40 0

50

0

20

40

60

80

Discrete-time fuzzy Chen attractor

−50

−40−20

x1

x 3

x2

(a)

010200

10

20

30

40

50

Discrete-time fuzzy Lorenz attractor

0

50

−50

−20−10x1

x 3

x2

(b)

F : (a) Discrete-time -S uzzy model or a Chen attractor and (b) discrete -S uzzy model or a Lorenz attractor.

0 5 10 15 20 25 30 35 40

A m p l i t u d e

0

50

100

150

Time

−50

−100

−150

Control uk

uk

F : Control signal or the Generalized Synchronization o Chen-Lorenz discrete-time -S uzzy systems.

On the other hand, the uzzy stabilizer or this system iscomputed by means o Ackermann’s ormula, and by locatingthe eigenvalues at

() subsystem 0.9596 0.9955 0.82,() subsystem 0.9596 0.9955 0.82,

the ollowing gains are obtained:

1 = −14.3228 −214.9000 −19.6566 ,2 = −14.3228 −214.9000 19.6566 . ()

Notice that the stability is ensured in the uzzy interpolationregion by these gains, or all

≥ 0.

0 5 10 15 20 25 30 35 40

Time

012

34

A m p l i t u d e

−1−2−3

−4

−5


e1

F : racking error 1, signal or the Generalized Synchro-nization o Chen-Lorenz discrete-time -S uzzy systems.

As can be readily observed, conditions o Teorem arenot ullled because the zero error steady-state maniold isnot the same or the local regulators. However, the trackingerror can be bounded by using the controller dened in ().Figures – depicted the behavior o the controller ()

with (0) = [1 1 1] and (0) = [1 0 0] as the initialconditions.

.. Generalized Synchronization by Using Genetic Algorithms.Te main objective o integrating genetic algorithms is toobtain values o the parameters (, , and ) o the new membership unctions o the regulator and reducing, in this


12/19


0 5 10 15 20 25 30 35 40

05

101520

A m p l i t u d e

−5−10

−15−20

x1,kw1,k

x1,k versus w1,k

(a)

0 5 10 15 20 25 30 35 40

0

10

20

30

A m p l i t u d e

−10

−20

−30

x2,k versus w2,k

x2,kw2,k

(b)

0 5 10 15 20 25 30 35 4005

101520253035404550

A m p l i t u d e

x3,k versus w3,k

x3,kw3,k

(c)

F : States o drive and response system or a Generalized Synchronization.

way, the tracking error. o thisend, it is important to considerthe ollowing:

(i) Consider the representation o the potential solutionso the problem; the common representation is binary.

(ii) An initial population o possible solutions, in arandom process, is selected (population size affectsthe efficiency and perormance o GA). For this case,a population size equal to chromosomes is chosen.

(iii) A tness unction, which indicates how good or bada certain solution is, is dened. In this case the tnessunction is the mean square tracking error, given by the ollowing expression:

= 12 1, − 1,2

. ()For the representation o possible solutions it is necessary

to know the variables to optimize; or this case it has and parameters or = 1, 2 and = 1, 2 dened in () and ();besides, , , , and will be xed with a constant value;that is, 1 = 3, 1 = 30 or () considering 1,2 = 1 − 1,1;1 = 3, 1 = 20 or () also considering 2,2 = 1 − 2,1. Teabovementioned are represented in Figure .

It is also important to know the interval value in whichthe variable will be operating; or this case, such intervalis

1,

∈ [−30 30]. Tus, the size o each variable, in bits,

and the chromosome length can be computed by using theollowing expression []:

var = (int ↑) log2 limupper − limlower 10accurate , ()where var is the size o each variable in bits, the term (int ↑)is the decimal integer value, and limupper = 30 and limlower =−30 with an accuracy in . Tereore, the variable size is equalto bits, but there are two variables, 1 and 2; thus thechromosomes length is bits. Tereore, the chromosomesare represented in Figure .

Te initial population is random binary. o assess eachindividual in the objective unction is needed to decode, inthis case, a real number. o this end, the ollowing expressionis used:

Var = Dec val limupper − limlower2length(bit) − 1 + limlower . ()Var is a decimal value (phenotype) or each binary chromo-some (genotype), Dec val is the decimal value or each binary string, and length(bit) is equal to bits.

In order to compute the individual aptitude is necessary to introduce the value o each parameter in the bell-shapedmembership unctions1 and 2. Te tracking error unction() disregards the nonoptimal solution and allows theoptimal perormance by considering that 1, tracks 1,. Inable the individual aptitude is depicted.


13/19


: Individual aptitude.

Chromosome Binary string Decoded integer Fitness unction

(−., −.) . (., .) .

(., .) .

(−., −.) . (−., .) . (−., −.) . (−., .) . (., −.) . (., .) . (., −.) . (−., .) . (−., .) . (., .) . (., −.) .

(

−., .) .

(−., −.) . (., .) . (., −.) . (−., .) . (−., −.) .

0 10 20 30

0

0.2

0.4

0.6

0.8

1

Initial membership functions

−30 −20 −10

1,1(x1,k)

1,2 (x1,k)

(a)

0

0.2

0.4

0.6

0.8

1


0 10 20 30−30 −20 −10

2,1(w1,k)

2,2(w1,k)

(b)

F : Initial membership unctions or the proposed uzzy regulator tuned by GAs.

1 1 0 1 0 0 1 0 0 0 1 1 1 0 0 1 0 1 0 0 1 0 1 0 0 0 1 1 1 0 0 1

a1 a2

F : Individual binary representation.


14/19


Each tness unction value is converted into a set point ortness value. o this end, the ollowing expression is used:

tness val = (⋆) − min, ()where (⋆) is the tness unction value and min is theminimum tness unction value.

Te two chromosomes with better tness (elite chromo-somes) can live and produce offspring in the next generation.Tere are many methodsto select a pair o chromosomes; themost popular one is named proportional selection method :

(a) Calculate the total tness value

tness = 20=1

tness () . ()(b) Compute the probability or each chromosome

= tness ()tness . ()(c) Calculate the cumulative probability or each chro-

mosome

= =0

. ()Te total tness value tness is equal to ..Te probabilities and are shown in able .

(d) Generate a random number in the range [0, 1].(e) I − 1 < ≤ , then select the chromosome to be

the one o the parents.

() Repeat (d) and (e) to obtain the other parent.Apply crossover operation on the selected pair, i they

have been chosen or crossover (based on probability o crossover = 1.0). Te most applied crossover operationis single point crossover. Based on the probability o bitmutation = 0.01, ip the correspondent bit i selectedor mutation. At this point, the process o producing a pairo offspring rom two selected parents is nished.

Te elite chromosomes o the previous population are notsubject to mutation.

In the ollowing simulations the states 1,, controlsignal , and tracking error 1, depicted in Figures –,respectively, are obtained afer applying () and by replacing

the new membership unctions adjusted by GAs. Te nalmembership unctions afer tuning the orm parameter by means o GA are depicted in Figure . It can be readily observed that these new membership unctions are differentrom the original ones (see Figure ). Notice that, in thisexample, the tracking error (Figure ) is less than the errorobtained by using the approach discussed in Section .Tis is due to the new membership unctions which allow reducing considerably the tracking error. Tis suggests thatthe approach presented in this work may be improved by tuning the parameters o center and amplitude in the new membership unctions; however this study is not completedyet.

: Individual aptitude.

Chromosome . .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. . . .

. .

. .

. .

. .

.. Pseudouzzy Generalized Synchronization by Using GAs.In thissection the Generalized Synchronization by using GAsand linear local regulator design will be addressed. However,the crossed terms within the local regulators are arbitrarily

removed rom the design process.Tus, the steady-state input() can be dened as =

1=1

1, 1, Γ, ()where 1,(1,) are new membership unctions, such thatthe uzzy output regulator obtained rom local regulatorsprovides the exact uzzy output regulation. Tis approachrequires the computation o the linear local controllers andthe computation o the new membership unctions; suchunctions are represented by () and (). Tereore the nalcontrol system is dened by

= 1

=1ℎ1, − +

1

=11, Γ, ()

because∗, is a unctiono and in steady-state = ().For this case the initial membership unctions proposed

are shown in Figure . As it can be seen, these membershipunctions are notproperlyuzzy becausethey do notulll theconvex sum; that is, ∑1=1 () ≠ 0. Even so, these unctionswill be adjusted by GAs in order to ensure the outputregulation by using input (). Te ollowing simulationprovides a better behavior in the Lorenz-Chen discrete-timeuzzy system by using different membership unction andtuning by GAs. Such results are depicted in Figures –.


15/19


0

0.2

0.4

0.6

0.8

1

Final membership functions

0 10 20 30−30 −20 −10

1,1(x1,k)

1,2(x1,k)

(a)

0

0.2

0.4

0.6

0.8

1


0 10 20 30−30 −20 −10

2,1 (w1,k)

2,2(w1,k)

(b)

F : Final membership unctions or the proposed uzzy regulator tuned by GAs.

0 5 10 15 20 25 30 35 40

05

101520

Time

A m p l i t u d e

−5−10−15−20

x1,k versus w1,k

x1,kw1,k

F : States 1, and 1, or the Generalized Synchronizationo Chen-Lorenz discrete-time -S uzzy systems.

0 5 10 15 20 25 30 35 40

Time

A m p l i t u d e

050

100150200

−50−100

−150−200

Control uk

uk


Notice that the nal membership unction depicted inFigure is different rom the initial proposed in Figure and also notice that the sum o these new interpolation

A m p l i t u d e

0 5 10 15 20 25 30 35 40

Time

00.5

−0.5−1

−1.5−2

−2.5−3

−3.5−4

−4.5


e1,k

F : racking error 1, signal or the Generalized Synchro-nization o Chen-Lorenz discrete-time -S uzzy systems.

unctions is not equal to one. For that reason this approachis called Pseudouzzy Generalized Synchronization.

5. Conclusions

A uzzy output regulator or discrete-time systems, based on

the combination o linear regulators combined by differentmembership unctions, has been presented. Synchronizationo discrete-time chaos attractors can be possible; by meanso uzzy output regulation, sufficient conditions or thecontroller are given. However, when the conditions can notbe ullled, new membership unctions in the regulator areincluded; these ones are optimized by genetic algorithms.Te main advantage is that membership unctions, whichallow the proper combination o the local regulators, canbe easily obtained by means o GAs. As a consequence, thepresented result allows a very precise synchronization o chaotic systems described by -S discrete-time uzzy modelson the basis o local regulators.


16/19


0

0.2

0.4

0.6

0.8

1


0 10 20 30−30 −20 −10

1(w1,k)

2(w1,k)

F : Initial membership unctions or the pseudouzzy regu-lator tuned by GAs.

05

10152025

A m p l i t u d e

−5

−15−10

−200 5 10 15 20 25 30 35 40

Time

x1,k versus w1,k

x1,kw1,k

F : States 1, and 1, or the Generalized Synchronizationo Chen-Lorenz discrete-time -S uzzy systems.

050

100150200250300

A m p l i t u d e

−50−100−150−200

0 5 10 15 20 25 30 35 40Time

Control uk

uk


Complete and Generalized Synchronization are used toillustrate the applicability o the proposed approach. Besides,the method proposed in this work avoids the disadvantageo constructing an exact uzzy regulator based on overall -Suzzy system which may result to be very large. Instead, thegiven approach offers a simple way to design the complete

0

0.5

1

1.5

2

Time

A m p l i t u d e

−0.5

−1−1.5

0 5 10 15 20 25 30 35 40


e1,k

F : racking error 1, or the Generalized Synchronizationo Chen-Lorenz discrete-time -S uzzy systems.

0

0.2

0.4

0.6

0.8

1


0 10 20 30−30 −20 −10

1(w1,k)

2(w1,k)

F : Final interpolation unctions or the pseudouzzy regu-lator tuned by GAs.

regulator based on local regulators but with membershipunctions optimized by sof computing.

Conflict of Interests

Te authors declare that there is no conict o interestsregarding the publication o this paper.

Acknowledgments

Tis work is partially supported by Consejo Nacional deCiencia y ecnologı́a (CONACY) through Scholarship SNIand by Instituto Polit́ecnico Nacional (IPN) through researchProjects and and Scholarships COFAA,EDI, and BEIFI.

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