Chaos, Solitons and Fractals 41 (2009) 2526–2532

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Chaos, Solitons and Fractals

journal homepage: www.elsevier .com/locate /chaos

An improved harmony search algorithm for synchronizationof discrete-time chaotic systems

Leandro dos Santos Coelho *, Diego Luis de Andrade BernertIndustrial and Systems Engineering Graduate Program, LAS/PPGEPS, Pontifical Catholic University of Paraná, PUCPR,Imaculada Conceição, 1155, 80215-901 Curitiba, Paraná, Brazil

a r t i c l e i n f o

Article history:Accepted 18 September 2008

0960-0779/$ - see front matter � 2008 Elsevier Ltddoi:10.1016/j.chaos.2008.09.028

* Corresponding author.E-mail addresses: leandro.coelho@pucpr.br (L.d.

a b s t r a c t

The harmony search (HS) algorithm is a recently developed meta-heuristic algorithm, andhas been very successful in a wide variety of optimization problems. HS was conceptual-ized using an analogy with music improvisation process where music players improvisethe pitches of their instruments to obtain better harmony. The HS algorithm does notrequire initial values and uses a random search instead of a gradient search, so derivativeinformation is unnecessary. Furthermore, the HS algorithm is simple in concept, few inparameters, easy in implementation, imposes fewer mathematical requirements, and doesnot require initial value settings of the decision variables. In recent years, the investigationof synchronization and control problem for discrete chaotic systems has attracted muchattention, and many possible applications. The tuning of a proportional–integral–deriva-tive (PID) controller based on an improved HS (IHS) algorithm for synchronization oftwo identical discrete chaotic systems subject the different initial conditions is investi-gated in this paper. Simulation results of the IHS to determine the PID parameters to syn-chronization of two Hénon chaotic systems are compared with other HS approachesincluding classical HS and global-best HS. Numerical results reveal that the proposed IHSmethod is a powerful search and controller design optimization tool for synchronizationof chaotic systems.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Many nonlinear dynamical systems have been found to show a kind of behavior known as chaos. A chaotic dynamicalsystem has complex dynamical behaviors that possess some special features such as being extremely sensitive to tiny vari-ations of initial conditions, broad spectra of Fourier transforms, and fractal properties of the motion in phase space.

The study of synchronization phenomena in the dynamical systems started at Huggen’s experiments in 17th century.Meantime, during the last two decades, synchronization in chaotic dynamic systems has received a great deal of interestamong scientists from various research fields [1–15] since Pecora and Carroll [16] introduced a method to synchronizetwo identical chaotic systems with different initial conditions and the method was realized in electronic circuits.

A basic configuration for chaos synchronization is the master–slave (drive-response) pattern, where the response chaoticsystem must track the drive chaotic trajectory. Among others, potential applications of chaos synchronization include phys-ical, chemical, and ecological processes as well as secure communications.

Most recently, many types of design and optimization methods have been suggested for the synchronization of chaoticsystems via proportional–integral–derivative (PID) control such as linear matrix inequality [6], observers [1], evolutionaryprogramming [4,17], and particle swarm optimization [2].

. All rights reserved.

Santos Coelho), dbernert@gmail.com (D.L. de Andrade Bernert).

L.d. Santos Coelho, D.L. de Andrade Bernert / Chaos, Solitons and Fractals 41 (2009) 2526–2532 2527

In this paper, the tuning of a PID controller based on an improved harmony search (IHS) algorithm for synchronization oftwo identical discrete chaotic systems subject the different initial conditions is proposed. Harmony search (HS) algorithmproposed in [18] has been recently developed in an analogy with music improvisation process where musicians in an ensem-ble continue to polish their pitches in order to obtain better harmony. Jazz improvisation seeks to find musically pleasingharmony similar to the optimum design process which seeks to find optimum solution. The pitch of each musical instrumentdetermines the aesthetic quality, just as the objective function value is determined by the set of values assigned to each deci-sion variable [19]. In addition, HS uses a stochastic random search instead of a gradient search so that derivative informationis unnecessary.

Simulation results of the IHS to determine the PID parameters to synchronization of two Hénon chaotic systems are com-pared with other HS methods including the classical HS and global-best HS.

The outline of the remainder of this paper can be summarized as follows: Section 2 describes the problem, while Section 3explains the concepts of HS and IHS approach. Section 4 presents the simulation results of PID’s tuning and chaotic synchro-nization. Finally, Section 5 outlines a brief conclusion about this study.

2. Problem description

2.1. PID controller

PID control is a term usually used for denoting the control with the Proportional, Integral, and Derivative actions that hasbeen, for decades, the most common control technique used in process control [20,21]. The PID control has a simple structurethat is easily understood by field engineers and is robust against disturbance and system uncertainty.

As modelled in this paper, the transfer function of PID controller is described by the following equation in the continuouss-domain (Laplace operator)

GPIDðsÞ ¼ P þ I þ D ¼ UðsÞEðsÞ ¼ Kp þ

Ki

sþ Kd � s ð1Þ

or GPIDðsÞ ¼ Kp � 1þ 1Ti � s

þ Td � s� �

; ð2Þ

where U(s) and E(s) are the control (controller output) and tracking error signals in s-domain, respectively; Kp is the propor-tional gain, Ki is the integration gain, and Kd is the derivative gain. Ti is the integral action time or reset time and Td is re-ferred to as the derivation action time or rate time.

In this context, the output of the PID controller in time domain is given by

uðtÞ ¼ Kp � eðtÞ þ Ki

Z t

0eðsÞdsþ Kd �

deðtÞdt

; ð3Þ

where u(t) and e(t) are the control and tracking error signals in time domain, respectively. Using trapezoidal approximationsfor Eq. (3) to obtain the discrete control law, we have

uðkÞ ¼ uðk� 1Þ þ Kp � ½eðkÞ � eðk� 1Þ� þ Ki �Ts

2� ½eðkÞ � eðk� 1Þ� þ Kd �

Ts

2� ½eðkÞ � 2eðk� 1Þ þ eðk� 2Þ�; ð4Þ

where Ts is the sampling period. The proportional part of the PID controller reduces error responses to disturbances. Theintegral term of the error eliminates steady state error and the derivative term of error dampens the dynamic responseand thereby improves stability of the system. How to solve these three gains to meet the required performance is thekey-challenge in the PID control system. However, it is difficult to find the optimal set of PID gains for nonlinear dynamicalsystems.

2.2. Nonlinear discrete chaotic system

In this study, two identical delayed discrete chaotic systems are considered to be synchronized using the proposed PIDcontrol. The master system is given by a typical discrete chaotic system, Hénon map [22] is employed as an example in thispaper. Hénon map can be described as follows [23]:

x1ðkþ 1Þ ¼ �p � x21ðkÞ þ xðkÞ þ 1; ð5Þ

xðkþ 1Þ ¼ q � x1ðkÞ; ð6Þ

where p = 1.4, q = 0.3, and x is the master state. On the other hand, the corresponding slave system is described by

y1ðkþ 1Þ ¼ �p � y21ðkÞ þ yðkÞ þ 1; ð7Þ

yðkþ 1Þ ¼ q � y1ðkÞ þ uðkÞ; ð8Þ

2528 L.d. Santos Coelho, D.L. de Andrade Bernert / Chaos, Solitons and Fractals 41 (2009) 2526–2532

where y is the slave state and u is the external control force that adopts the PID control of Eq. (4). For two identical delayeddiscrete chaotic systems (6) and (8) without control u, the state trajectories of these two chaotic systems will quickly sep-arate each other if their initial conditions are not the same. However, the state trajectories can approach synchronization forany initial conditions if an appropriate controller is utilized [2]. Hence the purpose of this paper is to apply the IHS algorithmto find out the optimal PID control gains such that chaos synchronization for two identical delayed discrete chaotic systemsis achieved.

In the time domain, specifications for a control system design involve certain requirements associated with the time re-sponse of the system. The requirements are often expressed in terms of the standard quantities on the rise time, settlingtime, overshoot, peak time, and steady state error of a step response. For simplicity, the objective function F used in thisstudy is defined as

F ¼XM

k¼1

jxðkÞ � yðkÞj ¼XM

k¼1

jeðkÞj; ð9Þ

where e(k) is the error signal between the master and slave states and M is the total number of sampling points. The opti-mization problem involves finding g� ¼ ½K�p;K

�i ;K

�d� such that the F performance index of the system is minimized.

3. Fundamentals of harmony search algorithm

This section describes the proposed IHS algorithm. First, a brief overview of the HS is provided, and finally the optimiza-tion procedure of the proposed IHS algorithm is presented.

3.1. Harmony search algorithm

Recently, Geem et al. [18] proposed a new HS meta-heuristic algorithm that was inspired by musical process of searchingfor a perfect state of harmony. The harmony in music is analogous to the optimization solution vector, and the musician’simprovisations are analogous to local and global search schemes in optimization techniques [24]. The HS algorithm doesnot require initial values for the decision variables. Furthermore, instead of a gradient search, the HS algorithm uses a sto-chastic random search that is based on the harmony memory considering rate and the pitch adjusting rate so that derivativeinformation is unnecessary. Compared to earlier meta-heuristic optimization algorithms, the HS algorithm imposes fewermathematical requirements and can be easily adopted for various types of engineering optimization problems [19].

In the HS algorithm, musical performances seek a perfect state of harmony determined by aesthetic estimation, as theoptimization algorithms seek a best state (i.e., global optimum) determined by objective function value. It has been success-fully applied to various optimization problems in computation and engineering fields including economic dispatch of elec-trical energy [25], multicast routing [26], clustering [27], optimum design [19], traveling salesman problem [28], parameteroptimization of river flood model [29], design of pipeline network [30,31], and design of truss structures [32].

The optimization procedure of the HS algorithm consists of steps 1–5, as follows:

Step 1. Initialize the optimization problem and algorithm parameters.Step 2. Initialize the harmony memory (HM).Step 3. Improvise a new harmony from the HM.Step 4. Update the HM.Step 5. Repeat Steps 3 and 4 until the termination criterion is satisfied.

The detailed explanation of these steps can be found in [18,19,24] which are summarized in the following:

Step 1. Initialize the optimization problem and HS algorithm parameters. First, the optimization problem is specified asfollows:

Minimize F subject to xi 2 Xi; i ¼ 1; . . . ;N

where F is the objective function, x is the set of each decision variable ðxiÞ; Xi is the set of the possible range of valuesfor each design variable (continuous design variables), that is, xi;lower 6 Xi 6 xi;upper , where xi;lower and xi;upper are thelower and upper bounds for each decision variable; and N is the number of design variables. In this context, theHS algorithm parameters that are required to solve the optimization problem are also specified in this step. The num-ber of solution vectors in harmony memory (HMS) that is the size of the harmony memory matrix, harmony memoryconsidering rate (HMCR), pitch adjusting rate (PAR), and the maximum number of searches (stopping criterion) areselected in this step. Here, HMCR and PAR are parameters that are used to improve the solution vector. Both are de-fined in Step 3.

Step 2. Initialize the harmony memory (HM). The harmony memory (HM) is a memory location where all the solution vectors(sets of decision variables) are stored. In Step 2, the HM matrix, shown in Eq. (10), is filled with randomly generatedsolution vectors using uniform distribution, where

L.d. Santos Coelho, D.L. de Andrade Bernert / Chaos, Solitons and Fractals 41 (2009) 2526–2532 2529

HM ¼

x11 x1

2 � � � x1N�1 x1

N

x21 x2

2 � � � x2N�1 x2

N

..

. ... ..

. ... ..

.

xHMS�11 xHMS�1

2 � � � xHMS�1N�1 xHMS�1

N

xHMS1 xHMS

2 � � � xHMSN�1 xHMS

N

266666664

377777775: ð10Þ

Step 3. Improvise a new harmony from the HM. A new harmony vector, x0 ¼ ðx01; x02; . . . ; x0NÞ, is generated based on three rules:(i) memory consideration, (ii) pitch adjustment, and (iii) random selection. Generating a new harmony is called‘improvisation’. In the memory consideration, the value of the first decision variable ðx01Þ for the new vector is chosenfrom any value in the specified HM range ðx01 � xHMS

1 Þ. Values of the other decision variables ðx02; . . . ; x0NÞ are chosen inthe same manner. The HMCR, which varies between 0 and 1, is the rate of choosing one value from the historicalvalues stored in the HM, while (1 � HMCR) is the rate of randomly selecting one value from the possible range ofvalues.

x0i x0i 2 fx1

i ; x2i ; . . . ; xHMS

i g with probability HMCRx0i 2 Xi with probabilityð1-HMCRÞ:

(ð11Þ

After, every component obtained by the memory consideration is examined to determine whether it should be pitch-adjusted. This operation uses the PAR parameter, which is the rate of pitch adjustment as follows:

Pitch adjusting decision for x0i Yes with probability PARNo with probabilityð1-PARÞ:

�ð12Þ

The value of (1 � PAR) sets the rate of doing nothing. If the pitch adjustment decision for x0i is Yes, x0i is replaced asfollows:

x0i x0i � r � bw; ð13Þ

where bw is an arbitrary distance bandwidth, r is a random number generated using uniform distribution between 0and 1.In Step 3, HM consideration, pitch adjustment or random selection is applied to each variable of the new har-mony vector in turn.

Step 4. Update the HM. If the new harmony vector, x0 ¼ ðx01; x02; . . . ; x0NÞ is better than the worst harmony in the HM, judged interms of the objective function value, F, the new harmony is included in the HM and the existing worst harmony isexcluded from the HM.

Step 5. Repeat Steps 3 and 4 until the termination criterion is satisfied.

3.2. Proposed improved HS algorithm

To improve the performance of the HS algorithm and eliminate the drawbacks lie with fixed values of HMCR and PAR,Mahdavi et al. [33] proposed an improved harmony search algorithm that uses variable PAR and bw in improvisation step.Also, Omran and Mahdavi [34] proposed a new variant of harmony search, called the global-best harmony search (GHS), inwhich concepts from swarm intelligence are borrowed to enhance the performance of HSA such that the new harmony canmimic the best harmony in the HM.

The IHS proposed in this work has exactly the same steps of classical HS with exception that Step 3, where the IHS dynam-ically updates PAR in which concepts from dispersed particle swarm optimization [35] are adopted. In this case, PAR is up-date as follows:

PARðtÞ PARmin þ ðPARmax � PARminÞ � grade; ð14Þ

where PAR(t) is the pitch adjusting rate for generation t, PARmin is the minimum adjusting rate, PARmax is the minimumadjusting rate, and t is the generation number. The grade is updated according to the following expression:

grade ðFmaxðtÞ �meanðFÞÞðFmaxðtÞ � FminðtÞÞ

; ð15Þ

where FmaxðtÞ and FminðtÞ are the maximum and minimum function objective values in generation t, respectively; mean(F) ismean of objective function value of harmony memory.

4. Results of numerical simulation

In this section, we will illustrate the synchronization PID controller design for the above two systems given by Eqs. (6) and(8) with different initial value conditions x1ð0Þ ¼ 0:2; xð0Þ ¼ 0:3; y1ð0Þ ¼ 0:8, and y(0) = 0.5. We solved the optimization prob-lem with M = 30 and Ts ¼ 1.

Table 1Convergence results for synchronization of discrete chaotic systems in 30 runs

Optimization method Minimum F Mean F Maximum F Standard deviation of F

HS 1.9014 4.0649 13.6037 2.1525GHS 1.8566 3.6435 5.8747 0.6000IHS 1.8068 3.4277 4.9283 0.7430

Table 2Best results of PID controller gains and performance data using optimization method in 30 runs

Parameter HS GHS IHS

Kp 2.2204 � 10�16 2.2208 � 10�16 5.6967 � 10�7

Ki 7.5375 � 10�3 7.1739 � 10�3 3.2812 � 10�3

Kd 0.9427 0.9441 0.9442Mean of error signal 0.0484 0.0469 0.0453Variance of control signal 0.0223 0.0222 0.0224F 1.9014 1.8566 1.8068

0 5 10 15 20 25 30-1.5

-1

-0.5

0

0.5

1

1.5

k

stat

e re

spon

se

x(k)y(k)

0 5 10 15 20 25 30-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

k

erro

r sig

nal,

e

a b

Fig. 1. Best result using classical HS. (a) States responses; (b) error signal.

2530 L.d. Santos Coelho, D.L. de Andrade Bernert / Chaos, Solitons and Fractals 41 (2009) 2526–2532

In this section, to verify and demonstrate the effectiveness of the proposed method, we discuss the simulation results ofsynchronization between the master and slave states for different optimization approaches including IHS, classical HS [18]and global-best HS (GHS) [34].

Each optimization method was implemented in Matlab (MathWorks). All the programs were run on a 3.2 GHz Pentium IVprocessor with 2 GB of random access memory. In each case study, 30 independent runs were made for each of the optimi-zation methods involving 30 different initial trial solutions for each optimization method. In this paper, the optimization ap-proaches are adopted using 2000 cost function evaluations in each run. The lower and upper bounds of the search space usedin optimization methods were ðKp;Ki;KdÞ 2 ½0;1�. In HS approaches, the HM was initially structured with randomly gener-ated solution vectors within the bounds prescribed for this example (i.e., 0 to 1).

The total number of solution vectors in classical HS, i.e., the HMS, was 10, and the HMCR and the PAR were 0.85 and 0.45,respectively. In GHS and HIS, the setup were HMS = 10, HMCR = 0.85, PARmin = 0.01, and PARmax = 0.99. The dynamical updat-ing of bw in GHS had the same procedure and parameters adopted in [34].

Convergence results obtained by applying the different HS approaches for synchronization of two Hénon maps are sum-marized in Table 1. It can be seen from Table 1 that with the same maximum number of generations, IHS obtained bettermean and minimum F values than classical HS and GHS methods.

Table 2 presents the best results of the PID controller gains and performance data obtained using optimization ap-proaches. As shown in Table 2, results confirm that IHS was still superior in terms of efficiency than the classical HS andGHS methods for PID’s controller tuning.

Figs. 1–3 shows the state responses of the master and slave systems using the resulting PID controller gains obtained byclassical HS, GHS, and IHS approaches. It is observed that the PID controller design based on HS and IHS approaches pre-sented similar performance for synchronization of two chaotic systems. Moreover, it can be observed in Figs. 1 and 3 thatthe slave signal tracked the master signal as expected without any problem.

0 5 10 15 20 25 30-1.5

-1

-0.5

0

0.5

1

1.5

k

stat

e re

spon

se

x(k)y(k)

0 5 10 15 20 25 30-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

k

erro

r sig

nal,

e

a b

Fig. 2. Best result using GHS. (a) States responses; (b) error signal.

0 5 10 15 20 25 30-1.5

-1

-0.5

0

0.5

1

1.5

k

stat

e re

spon

se

x(k)y(k)

0 5 10 15 20 25 30-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

k

erro

r sig

nal,

ea b

Fig. 3. Best result using IHS. (a) States responses; (b) error signal.

L.d. Santos Coelho, D.L. de Andrade Bernert / Chaos, Solitons and Fractals 41 (2009) 2526–2532 2531

5. Conclusion

In this paper, an IHS approach to tune PID controller gains was proposed in a synchronization application of two chaoticsystems. The well-known Hénon mapping with different initial conditions for the master and slave is chosen as example toillustrate the proposed scheme, and simulations are also given to verify the effectiveness of the proposed IHS approach.According to our results the proposed method based on PID control and IHS tuning can be successfully applied to controland synchronization problems of Hénon mapping. Furthermore, the IHS was compared against classical HS and GHS withencouraging results.

As an extension of this paper, we may also incorporate the dynamic parameters updating techniques into the new IHSmethod to make it more promising in applications of parameter identification and advanced controllers tuning.

Acknowledgements

This work was supported by the National Council of Scientific and Technologic Development of Brazil – CNPq – underGrant 309646/2006-5/PQ.

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