The IEEE International Conference on Industrial Informatics (INDIN 2008) DCC, Daejeon, Korea July 13-16, 2008

A New Approach in Discrete Chaos System Control by Differential Evolution Algorithm Fei Gao

Ju-Jang Lee

Department of Electrical Engineering and Computer Science Korea Advanced Institute of Science and Technology Daejeon, 305–701, Republic of Korea Email: [email protected]

Department of Electrical Engineering and Computer Science Korea Advanced Institute of Science and Technology Daejeon, 305–701, Republic of Korea Email: [email protected]

Abstract—In this paper, a novel chaos control approach by differential evolution (DE) Algorithm is proposed to control the inherent chaotic phenomenon in genetic algorithms. Firstly the new approach detects the dynamical behaviors of a new discrete chaos system with rational fraction in GA such as the extreme values and unstable periodic points. Secondly it directs the chaotic system to its unstable fix point from any initial point by global controlling factors {Uk } solved self–adaptively by DE. Thirdly it proposes a multi–model solution for chaos control through DE to stabilize the system on its unstable fix point. Simulation results are further presented to show the effectiveness and performance of the put method.

I. I NTRODUCTION Fractals, chaos, complexity and nonlinear science have established contacts with each other closely along with science developments, and the fields of society, economy, nature, engineering and technology are taking on more and more obviously not fabricative but intrinsic chaotic phenomena and fractal characters[1–3]. Simple iterative systems can generate non-linear dynamic system intricate completely and have been applied broadly, for example, the famous Logistic map is applied into spatio-temporal chaos, communications security etc [4,5]. Evolutionary algorithm (EA) is an umbrella term used to describe computer-based problem solving systems which use computational models of some of the known mechanisms of EVOLUTION as key elements in their design and implementation. Although simplistic from a biologist’s viewpoint, these algorithms are sufficiently complex to provide robust and powerful adaptive search mechanisms [6, 7]. Recently a lot of iterative systems are proposed basing on Logistic map in the studies of most well-know evolutionary computation method genetic algorithm (GA). For instance, a new discrete chaotic system with rational fraction is proposed to describe the inherent chaotic phenomenon in GA[4, 5]. Differential evolution (DE) algorithm is a novel minimization method of EAs, capable of handling non-differentiable, nonlinear and multi-modal objective functions, with few, easily chosen, control parameters [8, 9]. DE turned out to be the best genetic type of algorithm for solving the real-valued test function suite of the 1st international conference on evolutionary computation [9]. The crucial idea behind DE is a 978-1-4244-2171-8/08/$25.00 ⓒ2008 IEEE

scheme for generating trial parameter vectors. In this way no separate probability distribution has to be used which makes the scheme completely self-organizing [8, 9]. In this paper, a novel chaos control approach with differential evolution Algorithm (CCDE) is proposed to study the chaotic system introduced above. It combines following processes as a whole: detecting dynamical behaviors, directing the system to the fix point from any initial and stabilizing it on this fix points. The rest of this paper is organized as follows. In Section II, the main concepts of new chaotic system with rational fraction are described. In Section III the main processes of DE and a new technique to progress DE is introduced. A novel chaos control strategy CCDE is proposed in Section IV, and experimental results are reported and analyzed. The paper concludes with Section V. II. A NEW CHAOTIC SYSTEM WITH RATIONAL FRACTION A deterministic model in GA is required in the problems of multi-model function’s optimization and multi-objective optimization for it can control the frequency of evolution, reconcile peak values and create complexity and diversity. And chaos sequence can improve the performance of GA much more comparatively with random sequence as the analysis of GA’s convergence and numerical simulations show [2]. Recently a serial of non-linear model are selected in Ref [5] to describe the stochastic phenomena in the evolution process in GA through a lot of simulations and comparison. System (1) is one of them, indissolubly linking with GA [10]. xn+1 = f (xn ) =

1 − q · xn x2n + 0.1

(1)

where q is in [-q0 , q0 ], x is in [-10.0025 , 10.0025], and   1 1 · 10.0025 − ≈ 0.9990 (2) q0 = 10.0025 10.00252 + 0.1 System (1) is a discrete iterative system with rational fraction with different parameter q to describe GA’s random evolution process. It is much more complicated than Logistic map for it has more than one peak value because it can be seen as approximate combination of three parabolic maps for it has

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three concave-convex domains with some special parameter q. And it will have an impact upon fields of GA with studies going on [5]. III. T HE MAIN CONCEPT OF DIFFERENTIAL EVOLUTION ALGORITHM

DE algorithm grew out of Price’s attempts to solve the Chebychev Polynomial fitting Problem that had been posed to him by Storn [9]. It utilizes M n–dimensional vectors,xi = (xi1 , · · · , xin ) ∈ S, i = 1, · · · , M as a population for each iteration, called a generation, of the algorithm. At each generation, two operators, namely mutation and crossover (recombination), are applied on each individual, thus producing the new population. Then, a selection phase takes place, where each individual of the new population is compared to the corresponding individual of the old population, and the best between them is selected as a member of the population in the next generation [8, 10]. The details of the DE are given as below [8]: Algorithm 1. Differential Evolution Algorithm Step 1. Initialization. Random generateM individuals in feasible region S, G=0, crossover constant CR > 0, mutation constant CF = 0.5, G, define a fitness function f (x), value the population and label the best individual in current population as Q. Step 2. DE Evolution. g = g + 1, for each xi = (xi 1 , xi 2 , · · · , xi n ) (i =1,2,. . . , M ) : 1) Mutation. Random choose four mutually different individuals xa, xb, xc, xd in the current population to get a vector Dabcd =(xa − xb )+(xc − xd ), use it to generate new vector ξi = (ξi 1 , . . . , ξi n ) as below: ξi = Q + CF × Dabcd

(3)

2) Crossover. To get a new testing vector Ui = (ui 1 , . . . , ui n )with ξ i :  ξij , if (randb(j) < CR)|(j = rnbr(i)); uij = xij , if (randb(j) ≥ CR)&(j = rnbr(i)). (4) where rnbr(i) is random integer in {1, 2, · · · , n}, randb(j)is j-th random real in [0,1], j =1,2,. . . ,n. 3) Replacement. Remain the better one between xi and Ui :  Ui , if f (Ui ) < f (xi ); (5) xi = xi , if f (Ui ) ≥ f (xi ). Step 3. Updating. Find the current best Q and remain the better between Q and Q as the new Q. Step 4. Termination. If g > G , then export the Q , else go back to Step2. Like all the other kinds of EA, DE converge to local optimum easily. Mostly this is caused by the imbalance of exploitation and exploration of the evolutionary operators. To avoid exploitation excessively in redundant space and searching efficiency in the whole feasible space, make EA with relative fewer generations as a step, we put a novel technique through a technique space contraction simulating

the idea of sequential number theoretic optimization (SNTO) [11] as below. If a local optimum Qg is found, we define a new searching space D(t+1) = [a(t+1) , b(t+1) ] centering Qg from the current searching space D(t) = [a(t) , b(t) ] as below:  (t+1) (t) (t) (t) = max(xi − γci , ai ), ai (i = 1, 2, · · · , s) (t+1) (t) (t) (t) = min(xi + γci , bi ), bi (6) where γ in (0,1) is pre-given contraction ratio. Then we use EA to search in the new space to get a new optimum Qg , save the better one in Qg and Qg . When this space contraction technique is adopted to DE, a novel scheme SCDE comes out. DE in each contraction will have much fewer generations than a single DE. If K1 is the times of space contraction, G is the evolution generation of DE in each contraction, M is the size of the population, then the total evaluations of the fitness function is C = M ×(1+CR)×K × K1 . And if K1 =0, then SCDE will be the normal DE. In this way, DE is the special case of the new scheme SCDE. IV. A NOVEL CHAOS CONTROL STRATEGY CCDE Now we propose a novel chaos control strategy through DE (CCDE) to control the chaos system (1). CCDE consists of three aspects. Firstly it detects the dynamical behaviors of the system (1) such as the domain and unstable periodic points through DE. Secondly it directs system (1) to its unstable fix point from any initial point by global controlling factors {Uk } solved self-adaptively by DE. Thirdly it proposes a multi-model solution for chaos control through DE to stabilize system (1) on its unstable fix point. Ref [5] reports the unstable points of system (1) as below: q = −0.2972, the fix point xf = 1.0951; 2 period points are x2,1 = 1.0793, x2,2 = 1.1113. q = 0.06, 4 period points are x4,1 = 0.0412, x4,2 = 9.8305, x4,3 = −0.5795, x4,4 = 2.3293. A. Dynamical behaviors simulations by DE Firstly CCDE detects the dynamical behaviors of discrete chaotic such as its maximum and minimum and unstable periodic points of system (1). 1) Maximum and minimum: We can conclude the following theory easily. 1 − q · x, thenf (x, q) ≡ Theory 1: Letf (x, q) ≡ x2 +0.1 1 − q · x = f (−x, −q). 2 x +0.1 1 Theory 2[5] : Letf (x, q) ≡ x2 +0.1 − q · x, q is in [-q0 , q0 ], x is in [-10.0025 , 10.0025], then 10.00252 2 +0.1 − 10.0025 ≤ f (x, q) ≤ 10.0025. As DE is an effective way to solve nonlinear optimization, and the f (x, q) is only a two dimensional function, so we choose DE to find its value field in [-10.0025, 10.0025]×[-q0 ,q0 ]. For DE, population size M =20, CF=0.5, CR=0.1, K=30, the fitness function is functionf (x, q), and DE will evaluate the fitness function C = M ×(1+CR)×K=660 times.Let DE run 100 times independently.

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DE achieves f (x, q)’s max 10.0025 in probability 83% at (x, q)=(10.0025, -0.999001747378838) or (-10.0025, 0.999001747378838). And with the same parameters, DE achieves f (x, q)’s min -9.98252995631366 in probability 99% with correspond (x, q)= (10.0025, 0.999001747378838) or (-10.0025, 0.999001747378838). Fig.1 shows one of the successful searching processes by DE, where Aver. denotes average fitness value and Best denotes the best fitness value in the evolution. Fig.2 shows the system (1) figure with different parameter q. From Fig.1 and data detected above we know DE is effective in finding f (x, q)’s domain.

10

f

5

Aver. Min Fitness Best. Min Fitness Aver. Max Fitness Best. Max Fitness

0 −5 −10 0

10 Fig. 1.

15

f (x ,q)

10 5 0

Iterations

20

30

Process of DE to find f (x, q)’s field

q=0.1 q=0.5 q=0.9 q= − 0.1 q= − 0.5 q= − 0.9

−5 −10 −15

−10 Fig. 2.

−5

x

0

5

B. Directing chaos system and stabilize it through DE

10

System (1) with different q

2) Unstable periodic points: Let Φ=(Φ1 ,Φ2 ,. . . ,Φn )T : R → Rn (Φi : Rn → R, i=1,2,. . . , n) is a nonlinear system, we define a new function (Eq.7) below to get its different unstable period orbits.     (7) F (X) = Φ(p) (X) − X  n

2

Then X ∗ s.t. F (X)=0 is also the Φ’s pperiod points. When X ∗ is achieved, {X, Φ(1) (X), Φ(2) (X),. . . , Φ(P ) (X)} is the Φ’s p period orbit. We can judge the X ∗ ’s stability by the algorithm[12] as below: Algorithm 2. Judging the X ∗ ’s stability.   Step 0. LetX (0) , m=1, J = ∂Φ(x) ; ∂x  x=X (0)   ∂Φ(x) ,J = Step 1. X (m) = Φ(X (m−1) ), Jm = ∂x  x=X (m) Jm × J, m = m + 1; Step 2. Until m = P , otherwise go back to Step1; Step 3. Compute eigenvalue λ1 of J, if Im(λ1 ) = 0, and

λ1 ≈ 1, then it’s stable; Otherwise it’s unstable. The optimization of function (7) is difficult, so we choose CCDE to seek the system (1)’s unstable period points. With the times of space contraction is K1 =18, the evolution generation of DE in each contraction G =50, the size of the population M =80 in [-10.0025, 10.0025], CF=0.5, CR=0.05 then the total evaluations of the fitness function is C = M × (1+CR) ×K × K1 =75600. And when x−f (p) (x) s.t. ≤ 10−12 , we consider CCDE is successful. If two period points x1, x2 s.t. x1 − x2 ≤ 10−12 , we consider they are the same. Let CCDE run 50 times independently, the results are the following: When q = −0.0444, CCDE finds its unstable fix point xf = 0.982433140632848 in probability 94%, and two period unstable points x2,1 = 5.60653982323182, x2,2 = 0.280642886521397 in probability 90%. When q = −0.2972, CCDE finds its unstable fix point xf = 1.0951213808242 in probability 92%, and two period unstable points x2,1 = 1.11130797961875, x2,2 = 1.07934234385035 in probability 98%. When q = 0.06, CCDE finds its unstable fix point xf = 0.946792933569802 in probability 92%, two period unstable points x2,1 = 5.42337795697132, x2,2 = −0.291519325535561 in probability 94% and 4 period unstable points x4,1 = −0.579494728556509, x4,2 = 0.0412110138749085, x4,3 = 2.32932621198794, x4,4 = 9.83052881797208 in probability 86%. When q = 0.5, CCDE finds its unstable fix point xf = 0.835448639019916 in probability 88%. All these results are much more superior to the results reported in Ref. [5] in sense of x − f (p) (x)

Having the unstable period points above, Secondly CCDE directs system (1) to its unstable fix point from any initial point by global controlling factors {Uk } solved self-adaptively by DE. Thirdly CCDE proposes a multi-model solution for chaos control through DE to stabilize system (1) on its unstable fix point. Now CCDE directs the chaos system (1) into these points from any initial points. With the concept of global control [13], DE self adaptively find a chaotic orbit converging to an

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unstable period point and force system (1) from any initials to this period. We choose chaos system (1)’s unstable period point achieved in Subsection IV-A, that is with q = 0.5, its unstable fix point is xf = 0.835448639019916. The control objective is to make the system (8) below

Un

(8)

from any initial x1 on the mercy of control factors [U1 ,U2 ,. . . ,Um ] become Xm = xm+1 in U(xf , ε1 ) (ε1 =10−6 ) after m iterations. And we choose DE to realize this process for it’s only a problem of m-dimensional function optimization which can be resolve by DE. Let m = 8, CR = 0.005, CF = 0.5, K = 1000 and generate 50 points in [-2, 2]m , the termination condition is f1 ≤ε. The process of directing chaos is then translated into minimizing the function f1 : f1 ([U1 , U2 , . . ., U8 ]) = Xm − xf

0 −2 −4 0

40

n

60

80

100

CSDE control chaos

−3

3

When system is in U(xf , ε1 ), we put a multi-model solution for chaos control through DE to stabilize the system on U(xf , ε2 ) (ε2 =0.0011). The objective function is f2 :

1 Un

2

(10)

And we select CR=0.005, CF=0.5, K=300 is selected and random choose 50 points in [-0.01, 0.01] as the initial population. Then a novel chaos control strategy through DE as the third step of CCDE come into being as below: Algorithm 3. Chaos control strategy through DE Step 1. Choose SCDE to seek the unstable fix point xf of system (1) with q = 0.5; Step 2. Judge the stability of xf by Algorithm 2. If it is unstable, go into Step 3, otherwise go back to Step1; Step 3. For a given initial X0 = 0.92, use DE to find global control factors [U1 ,U2 ,. . . ,Um ], until f1 ≤ε, Xm = x8 , n = 8; Step 4. stabilize xn in U(xf ,ε2 ) with Uk detected by DE with the objective f2 , compute xn+1 by system (8); Step 5. n = n + 1, if n ≤ 100, go back Step 4; Otherwise output {xn } = {X1 , X2 , ..., X8 , x9 , ..., x100 }. For the chaos system we discussed, let Algorithm 3. run 50 times independently, the efficiency is 86%. We choose one of the success control process as Fig.3 shows, where xn is the point controlled,Un is the control factors, error=|Xn − xf |, Parameters denotes xn ,Un , error, n is the iteration. For Un from n=10is small enough, Fig. 4 shows {Un }. From the simulations above, we can conclude that the put chaos control strategy CCDE is efficient and robust for the new chaos system (1) in three aspects: finding value domain, detecting unstable period orbits, directing and stabilizing the chaos system.

20 Fig. 3.

(9)

f2 (Uk ) = xk+1 − xf

error

2

Parameters

xn+1 = f (xn ) + Uk

xn

4

x 10

Un

0 −1 −2 −3 −4

20

40 Fig. 4.

x

60

80

100

Control factors Un

V. C ONCLUSIONS In this paper we propose a novel chaos control scheme CCDE combining with the concept of evolutionary technique DE and some established control techniques to study the chaotic phenomena in GA. And the experiments done show the proposed scheme is robust and effective. Though simulations of CCDE are done to chaos system (1) from studies of GA, we can easily derive it into the other chaos systems[1,13]. ACKNOWLEDGMENTS We thank the anonymous reviewers for their constructive remarks and comments. The work is Supported by Brain Korea(BK)21 project from Korean government and NNSFC No. 10647141 from China. R EFERENCES [1] Rui J.P. de Figueiredo, and G. Chen, , Nonlinear Feedback Control Systems An Operator Theory Approach. Academic Press, New York, 1993.

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[2] R. Caponetto, L. Fortuna, Fazzino S. et.al, ”Chaotic sequences to improve the performance of evolutionary algorithms.” IEEE Trans Evolution Computation, vol. 7(3), pp. 289–304, 2003. [3] J. Q. Fang, Control Chaos and Develop High Technique(in Chinese), Atomic Energy Press, Beijing, 2002. [4] T. Kapitaniak, ”Stochastic resonance in chaotically forced systems”, Chaos, Solitons & Fractals , vol. 3, pp. 405–410, 1993. [5] J. A. Lu, X.Q. Wu, ”A new discrete chaotic system with rational fraction and its dynamical behaviors”, Chaos, Solitons & Fractals , vol. 22, pp. 311–319, 2004. [6] D. Whitley, ”An overview of evolutionary algorithms: Practical issues and common pitfalls”, Information and Software Technology, vol. 43(14), pp. 817–831, 2001. [7] R. Storn, K. Price, Differential evolution-a simple and efficient adaptive scheme for global optimization over continuous spaces. Journal of Global Optimization, vol. 11, pp. 341–359, 1997. [8] David Corne, Marco Dorigo, Fred Glover, New Ideas in Optimization (Advanced Topics in Computer Science). McGraw-Hill, 2004. [9] K. Price, R. Storn, J. A. Lampinen, Differential Evolution: A Practical Approach to Global Optimization (Natural Computing Series). Springer, 2005. [10] F. Gao, H. Q. Tong, ”UEAS: A Novel United Evolutionary Algorithm Scheme”, Lecture Notes in Computer Science, vol. 4234, pp. 772-780, 2006. [11] L. K. Hua, Y. Wang , 1981, Applications of Number theory to Numerical analysis. Springer -Verlag & Science Press, Berlin & Beijing. [12] K. Parsopoulos, M. Vrahatis, ”Computing periodic orbits of nonlinear mappings through particle swarm optimization”, Proc. of the 4th GRACM Congress on Computational Mechanics , Patras, Greece, 2002. [13] G. Chen, J. L¨u., Dynamics of the Lorenz System Family: Analysis, Control and Synchronization. Science Press, Beijing, 2003.

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