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. 2016 Jan 12;2016:2959370. doi: 10.1155/2016/2959370

An Improved Cuckoo Search Optimization Algorithm for the Problem of Chaotic Systems Parameter Estimation

Jun Wang 1,*, Bihua Zhou 1, Shudao Zhou 2
PMCID: PMC4737452  PMID: 26880874

Abstract

This paper proposes an improved cuckoo search (ICS) algorithm to establish the parameters of chaotic systems. In order to improve the optimization capability of the basic cuckoo search (CS) algorithm, the orthogonal design and simulated annealing operation are incorporated in the CS algorithm to enhance the exploitation search ability. Then the proposed algorithm is used to establish parameters of the Lorenz chaotic system and Chen chaotic system under the noiseless and noise condition, respectively. The numerical results demonstrate that the algorithm can estimate parameters with high accuracy and reliability. Finally, the results are compared with the CS algorithm, genetic algorithm, and particle swarm optimization algorithm, and the compared results demonstrate the method is energy-efficient and superior.

1. Introduction

Chaos is a universal complex dynamical phenomenon, lurking in many nonlinear systems, such as communication systems and meteorological systems. The control and synchronization of chaos has been widely studied [14]. Parameter estimation is a prerequisite to accomplish the control and synchronization of chaos. During recent years many parameter estimation methods have been proposed, such as particle swarm optimization (PSO) [58], genetic algorithm (GA) [912], and mathematical methods of multiple shooting [13]. However, the GA and PSO algorithms are easily trapped into local-best solution that affects the quality of solutions; the precisions of PSO, GA, and multiple shooting are not high enough. Recently, a novel and robust metaheuristic based method called cuckoo search algorithm was proposed by Yang and Deb [1416]. The algorithm proved to be very promising and could outperform existing algorithms such as GA and PSO [14]. However, the relatively poor ability of local searching is a drawback, and it is necessary to further improve the performance of CS algorithm to obtain a higher-quality solution. The basic principle of the ICS algorithm is to integrate the orthogonal design and simulated annealing operation to enhance the exploitation optimization capacity.

The remaining sections of this paper are organized as follows. In Section 2, a brief formulation of chaotic system parameters estimation is described. Section 3 elaborates the ICS algorithm, and the results established upon the proposed algorithm and some compared algorithms are given in Section 4. The paper ends with conclusions in Section 5.

2. Problem Formulation

A problem of parameter estimation can be converted into a problem of multidimensional optimization by constructing the proper fitness function.

Let the following equation be a continuous nonlinear n-dimension chaotic system:

X˙=FX,X0,θ, (1)

where  X = (x 1, x 2,…, x n)TR n denotes the state vector of the chaotic system, X˙ is the derivative of X, X 0 = (x 10, x 20,…, x n 0)TR n denotes the initial state of system, and θ = (θ 1, θ 2,…, θ d)T is a set of original parameters. Suppose the structure of the system (1) is known; then the estimated system can be written as

X~˙=FX~,X0,θ~, (2)

where X~=(x~1,x~2,,x~n)TRn denotes the state vector of the estimated system; θ~=(θ~1,θ~2,,θ~d)T is a set of estimated parameters. In order to convert the parameter estimation problem into optimization problem, the following objective fitness function is defined:

Fθ~=1Mi=1MXX~2, (3)

where i = 1,2,…, M is the sampling time point and M denotes the length of data used for parameter estimation. The parameter estimation of system (1) can be achieved by searching the most proper values of θ~ such that the objective function (3) is globally minimized.

It can be found that (3) is a multidimensional nonlinear function with multiple local search optima; it is easily trapped into local optimal solution and the computation amount is great, so it is not easy to search the globally optimal solution effectively and accurately using traditional general methods. In the paper an improved CS algorithm is proposed to solve the complex optimization problem.

3. Improved CS Algorithm

3.1. Basic CS Algorithm

The basic CS algorithm is based on the brood parasitism of some cuckoo species by laying their eggs in the nests of other host birds. For simplicity in describing the basic CS, the following three ideal rules are used [14]: (1) Each cuckoo lays one egg at a time, and dumps it in a randomly chosen set; (2) the best nests with high-quality eggs will be carried over to the next generations; (3) the number of available host nests is fixed, and the egg laid by a cuckoo is discovered by the host bird with a probability p a ∈ [0,1]. In this case, the host bird can either get rid of the egg away or simply abandon the nest and build a complex new nest. Based on the above rules, the basic CS algorithm is described as shown in Algorithm 1 [14].

Algorithm 1.

Algorithm 1

Basic cuckoo search algorithm.

Furthermore, the algorithm used a balanced combination of a local random walk and the global explorative random walk, controlled by a switching parameter p a. The local random walk can be written as

xit+1=xit+αsHpaεxjtxkt, (4)

where x j t and x k t are two different solutions selected randomly by random permutation, H is a Heaviside function, ε is a random number drawn from a uniform distribution, and s is the step size.

On the other hand, the global random walk is carried out by using Lévy flights [1417]:

xit+1=xit+αLévys,λ. (5)

Here, α > 0 is the step size scaling factor; Lévy(s, λ) is the step-lengths that are distributed according to the following probability distribution shown in (6) which has an infinite variance with an infinite mean:

Lévys,λ=λΓλsinπλ/2π1s1+λ. (6)

3.2. ICS Algorithm

In order to further improve searching ability of the algorithm, the orthogonal design and simulated annealing operation are integrated into the CS algorithm. The basic idea of the orthogonal design is to utilize the properties of the fractional experiment to efficiently determine the best combination of levels [17]. An orthogonal array of K factors with Q levels and M combinations is denoted as L M(Q K), where Q is the prime number, M = Q J, and J is a positive integer satisfying K = (Q J − 1)/(Q − 1). The brief procedure of constructing the orthogonal array L M(Q K) = [a i,j]M×K is described as shown in Procedure 1.

Procedure 1: Procedure constructing the orthogonal array.

  • Step  1. Construct the basic columns

  •  For k = 1 to J

  •    j=Qk-1-1Q-1+1

  •   For i = 1 to Q J

  •     ai,j=i-1QJ-Kmod⁡Q

  •   End for

  •  End for

  • Step  2. Construct the non-basic columns

  •  For k = 2 to J

  •    j=Qk-1-1Q-1+1

  •   For s = 1 to j − 1

  •    For t = 1 to Q − 1

  • a j+(s−1)(Q−1)+1 = (a s × t + a j)mod⁡Q

  •    End for

  •   End for

  •  End for

  • Step  3. Increment a i,j by one for 1 ≤ iM, 1 ≤ jN

The procedure of the orthogonal design algorithm is elaborated as shown in Algorithm 2 and for more detailed information on the orthogonal design strategy, please refer to [1719].

Algorithm 2.

Algorithm 2

Orthogonal design algorithm.

The procedure of simulated annealing algorithm is simply stated as shown in Algorithm 3 [20], and for more detailed information on the simulated annealing, please refer to [2022].

Algorithm 3.

Algorithm 3

Simulated annealing algorithm.

Based on the above description of the orthogonal design strategy and simulated annealing operation, the detailed procedures for parameter estimation with the ICS algorithm can be summarized as shown in Algorithm 4.

Algorithm 4.

Algorithm 4

Improved cuckoo search algorithm.

4. Simulation Results

To demonstrate the effectiveness of the improved algorithm, the algorithm is used to estimate parameters of Lorenz chaotic system [23] and Chen chaotic system [24].

4.1. Lorenz Chaotic System

Lorenz chaotic system equation [23] is expressed as follows:

x˙=σ1yx,y˙=σ2xxzy,z˙=xyσ3z, (7)

where (x, y, z) is the state variables; σ 1, σ 2, σ 3 are the unknown chaotic system parameters which need to be estimated. The real parameters of the system are σ 1 = 10, σ 2 = 28, and σ 3 = 8/3 which ensure a chaotic behavior, in order to obtain the values of some state variables, the fourth-order Runge-Kutta algorithm is used to solve (7), and the integral step is h = 0.01. Then a series of state variables values are obtained and 100 state variables of different times ({(x(n), y(n), z(n)),   n = 1, 2, …, 100}) are chosen to be the sample data. The parameters of the algorithm are set as follows: the max iteration number is N = 200, the sample size is M = 100, the annealing mode is shown in (8) where n is the iteration number, and the initial temperature is T 0 = 100. Consider

Tn=T0ln1+n. (8)

The objective (fitness) function H is shown in (9), where (x(n), y(n), z(n)) is the nth state variable that corresponds to the true system parameters and  (x~(n),y~(n),z~(n)) is the nth state variable that corresponds to the estimated system parameters:

H=1Mn=1Mx~nxn2+y~nyn2+z~nzn2. (9)

Figure 1 shows the convergence process of the fitness values and three parameters (σ 1, σ 2, σ 3) during the iterations in a single experiment.

Figure 1.

Figure 1

The convergence process of fitness function value and three parameters (σ 1, σ 2, σ 3) during the iterations in a single experiment under the noiseless condition.

In order to eliminate the difference of each experiment, the algorithm is also executed 50 times; then the mean value of the 50 experiments is taken as the final estimated value; the mean value and best value of the 50 experiments are listed in Table 1. The results based on CS (the best parameter setting is p a = 0.25, p a = 0.01), PSO (the best parameter setting is w = 0.8, c = 1.5, where w is the inertia weight and c is acceleration factor), and GA (the best parameter setting is c r = 0.8, m u = 0.1, where c r is the crossover rate and m u is the mutation rate) are also listed in Table 1.

Table 1.

The statistical results based on different methods in the noiseless condition.

Mean value Best value
ICS CS PSO GA ICS CS PSO GA
σ 1 10.000000 9.998736 9.985012 10.082051 10.000000 9.999927 9.995510 10.026911
σ 2 28.000000 28.000005 28.014411 27.881034 28.000000 28.000002 28.001304 28.004702
σ 3 2.666667 2.666661 2.668102 2.681882 2.666667 2.666665 2.666802 2.669018
H 1.1822e − 010 3.7614e − 004 0.069517 0.331901 1.2933e − 011 2.9556e − 005 0.011377 0.113969

It can be seen from Table 1 that the best fitness values obtained by ICS algorithm are quite better than the other algorithms. The mean values of the established parameters are also with higher precision than others. The estimated values are close to the true values infinitely. It can be concluded in general that the ICS algorithm contributes to superior performance, CS performs nest-best, PSO is better than GA, and GA performs worst.

As the actual chaotic systems always associate with noise, in order to test the performance of parameter estimation in the noise condition, the noise sequences are added to the original sample data. The white noise is added to the state variables {(x(n), y(n), z(n)),   n = 1, 2, …, 100}; the range of the noise sequences is from −0.1 to 0.1. Figure 2 shows the convergence process of the fitness values and three parameters (σ 1, σ 2, σ 3) during the iterations in a single experiment under the noise condition.

Figure 2.

Figure 2

The convergence process of fitness function value and three parameters (σ 1, σ 2, σ 3) during the iterations in a single experiment under the noise condition.

In order to eliminate the difference of each experiment, the algorithm is executed 50 times, then the mean value of the 50 experiments is taken as the final estimated value, and the corresponding results are listed in Table 2.

Table 2.

The statistical results by different algorithms in the noise condition.

Mean value Best value
ICS CS PSO GA ICS CS PSO GA
σ 1 9.996110 10.080014 9.844606 10.217998 9.998941 10.001565 9.881002 10.044011
σ 2 28.002272 27.980212 27.860013 27.661201 28.000099 28.001995 28.022441 27.900189
σ 3 2.666590 2.658890 2.700198 2.659880 2.666675 2.667704 2.675596 2.670228
H 0.008402 0.0390389 0.221401 0.500227 0.000908 0.001228 0.050931 0.255996

It can be seen from Table 2 that the four algorithms all have a certain capability of identification of parameters, but the performance of ICS is much better than the other algorithms; it supplies more robust and precise results; although the precision of the estimated parameters is declined compared with the results in the noiseless condition, the precision is still satisfactory. Then it can be concluded that the ICS algorithm possesses a powerful capability for parameters identification in the noise condition.

4.2. Chen Chaotic System

Chen chaotic system equation [24] is expressed as follows:

x˙=σ1yx,y˙=σ3σ1xσ3yxz,z˙=xyσ2z, (10)

where (x, y, z) is the state variables; σ 1, σ 2, σ 3 are the unknown chaotic system parameters which need to be estimated. The real parameters of the system are σ 1 = 35, σ 2 = 3, and σ 3 = 28 which ensure a chaotic behavior, the fourth-order Runge-Kutta algorithm is used to solve (10), and the integral step is h = 0.01. Then a series of state variables values are obtained and 100 state variables of different times ({(x(n), y(n), z(n)),   n = 1, 2, …, 100}) are chosen to be the sample data. The parameters of the algorithm are set as follows: the max iteration number is N = 200, the sample size is M = 100, the annealing mode is shown in (8) where n is the iteration number, and the initial temperature is T 0 = 100. The convergence process of the fitness values and three parameters (σ 1, σ 2, σ 3) during the iterations in a single experiment is shown in Figure 3. In order to eliminate the difference of each experiment, the algorithm is executed 50 times, then the mean value of the 50 experiments is taken as the final estimated value, and the corresponding results are listed in Table 2.

Figure 3.

Figure 3

The convergence process of fitness function value and three parameters (σ 1, σ 2, σ 3) during the iterations in a single experiment under the noiseless condition.

It can be seen from Table 3 that the best fitness values obtained by ICS algorithm are quite better than the other algorithms. The mean values of the established parameters are also with higher precision than others. The estimated values are close to the true values asymptotically. It can be concluded in general that the ICS algorithm contributes to superior performance, CS performs nest-best, PSO is better than GA, and GA performs worst.

Table 3.

The statistical results by different algorithms in the noiseless condition.

Mean value Best value
ICS CS PSO GA ICS CS PSO GA
σ 1 34.999438 35.089675 34.844278 33.535396 34.999945 34.996661 34.782290 35.102699
σ 2 2.999951 2.999081 3.012977 3.005031 2.999999 2.999907 2.998694 2.991955
σ 3 27.999757 28.043810 27.917648 27.291109 27.999974 27.998427 27.895888 28.053975
H 4.6438e − 007 4.6628e − 004 0.027312 0.115259 2.9658e − 010 3.2051e − 006 0.003312 0.010562

As the actual chaotic systems always come along with noise, in order to test the performance of parameter estimation in the noise condition, the noise sequences are added to the original sample data. The white noise is added to the state variables {(x(n), y(n), z(n)),   n = 1, 2, …, 100}; the range of the noise sequences is from −0.1 to 0.1. Figure 4 shows the convergence process of the fitness values and three parameters (σ 1, σ 2, σ 3) during the iterations in a single experiment under the noise condition.

Figure 4.

Figure 4

The convergence process of fitness function value and three parameters (σ 1, σ 2, σ 3) during the iterations in a single experiment under the noise condition.

It can be seen from Table 4 that the four algorithms all have a certain capability of identification of parameters, but the performance of ICS is much better than the other algorithms; it supplies more robust and precise results; although the precision of the estimated parameters is declined compared with the results in the noiseless condition, the precision is still satisfactory. Then it can be concluded that the ICS algorithm possesses a powerful capability for parameters identification in the noise condition.

Table 4.

The statistical results by different algorithms in the noise condition.

Mean value Best value
ICS CS PSO GA ICS CS PSO GA
σ 1 35.421272 35.859540 34.108398 35.962547 34.970096 35.112205 35.313896 10.044011
σ 2 2.996796 2.993281 2.970277 3.052984 2.999311 2.997309 2.980455 27.900189
σ 3 28.204815 28.418425 27.588450 28.434651 27.985940 28.055617 28.162608 2.670228
H 0.009334 0.038112 0.244276 0.591957 1.5986e − 004 0.001495 0.062901 0.255996

5. Conclusion

In this paper, an energy-efficient and superior ICS algorithm is proposed to estimate chaotic system parameters. The estimated results demonstrate the strong capabilities and effectiveness of the proposed algorithm, compared with the CS, PSO, and GA algorithms; the ICS algorithm supplies more robust and precise results. Besides, the algorithm also has a more powerful capability of noise immunity. In general, the proposed ICS algorithm is a feasible, energy-efficient, and promising method for parameters estimation of chaotic systems.

Acknowledgment

This work is supported by National Natural Science Foundation of China, 61271106.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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