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

Authors: Jun Wang; Bihua Zhou; Shudao Zhou
Journal: Computational Intelligence and Neuroscience
 (2016)

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## 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.


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## Body

## 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 [1–4]. 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) [5–8], genetic algorithm (GA) [9–12], 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 [14–16]. 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 Section2, 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 nonlinearn-dimension chaotic system:(1)X˙=FX,X0,θ,where  X=(x1,x2,…,xn)T∈Rn denotes the state vector of the chaotic system, X˙ is the derivative of X, X0=(x10,x20,…,xn0)T∈Rn denotes the initial state of system, and θ=(θ1,θ2,…,θd)T is a set of original parameters. Suppose the structure of the system (1) is known; then the estimated system can be written as (2)X~˙=FX~,X0,θ~,where X~=(x~1,x~2,…,x~n)T∈Rn 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:(3)Fθ~=1M∑i=1MX-X~2,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 pa∈[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:Basic cuckoo search algorithm.
Begin
Generate iteration timet=1
Initialized with random vector values and parameters
Evaluate the fitness of each individual (nest) and determine the best individual with the best fitness value
While (stopping criterion is not met or t < MaxGeneration)
Get a Cuckoo randomly by local random walk or Lévy Flights
Evaluate its fitnessFm
Choose a nest amongn (say, n) randomly
If (Fm<Fn)
replacen by the new nest m
End If
A fraction (pa) of worse nests are abandoned and new ones are built
Keep the best nests with quality solutions
Rank the solutions and find the current best
Update the generation numbert=t+1
End while
EndFurthermore, the algorithm used a balanced combination of a local random walk and the global explorative random walk, controlled by a switching parameterpa. The local random walk can be written as(4)xit+1=xit+αs⊗Hpa-ε⊗xjt-xkt,where xjt and xkt 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 [14–17]:(5)xit+1=xit+α⊕Lévys,λ.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:(6)Lévys,λ=λΓλsin⁡πλ/2π1s1+λ.


### 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 LM(QK), where Q is the prime number, M=QJ, and J is a positive integer satisfying K=(QJ-1)/(Q-1). The brief procedure of constructing the orthogonal array LM(QK)=[ai,j]M×K is described as shown in Procedure 1.Procedure 1:Procedure constructing the orthogonal array.
Step  1. Construct the basic columns
Fork=1 to J
j=Qk-1-1Q-1+1
Fori=1 to QJ
ai,j=i-1QJ-Kmod⁡Q
End for
End for
Step  2. Construct the non-basic columns
Fork=2 to J
j=Qk-1-1Q-1+1
Fors=1 to j-1
Fort=1 to Q-1
aj+(s-1)(Q-1)+1=as×t+ajmod⁡Q
End for
End for
End for
Step  3. Increment ai,j by one for 1≤i≤M, 1≤j≤NThe procedure of the orthogonal design algorithm is elaborated as shown in Algorithm2 and for more detailed information on the orthogonal design strategy, please refer to [17–19].Algorithm 2:Orthogonal design algorithm.
Begin
(1)  Construct the orthogonal array following the above steps
(2)  Randomly select two solutions from the population
(3)  Quantize the domain formed by the two solutions
(4)  Randomly generatek-1 integers p1⋯pk-1
(5)  UseLM(QK) to generate M potential offspring
(6)  Randomly select a solution from the population
(7)  Compare the solution with the best solution from the orthogonalM offspring and obtain the better solution
EndThe procedure of simulated annealing algorithm is simply stated as shown in Algorithm3 [20], and for more detailed information on the simulated annealing, please refer to [20–22].Algorithm 3:Simulated annealing algorithm.
Begin
(1)  Given a configuration of the elements of the system, randomly displace the elements on a time,
by a small amount and calculate the resulting change in the energy,ΔE
(2)If  ΔE<0
Then accept the displacement and use the resulting configuration as the starting point for the next iteration
Else
Accept the displacement with probabilityP(ΔE)=exp⁡(-ΔE/kbT) where T is the current temperature and
kb is Boltzmann’s constant
(3)  Repetition of this step continues until equilibrium is achieved
EndBased 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 Algorithm4.Algorithm 4:Improved cuckoo search algorithm.
Begin
(1)  Initial the parameter values of the algorithm, generate the random initial vector values and set the iteration numbert=1.
(2)  Evaluate fitness values of each individual and determine the current best individual with the best objective value.
Check whether the stopping criterion is met. If the stopping criterion is met, then output the best solution;
otherwise update the iteration numbert=t+1 and continue the iteration process until the stopping criterion is met.
(3)  Keep the best solution of the last iteration, and get a set of new solutionsXnew=[x1t+1,…,xi(t+1),…,xK(t+1)] by
Lévy flight, the Lévy flight is performed according to (4).
(4)  Evaluate the fitness valueFi(t+1) of the new solution xi(t+1), and compare Fi(t+1) with Fi(t) which represents the solution of
thetth iteration. If Fi(t+1) is better than Fi(t), then replace xi(t) by xi(t+1), otherwise, not abandoning the solution xi(t+1)
at once but accepting the solution with probabilityp=e(-|Δf|/(K⋅T)), where Δf is the change in the fitness value
Δf=Fi(t+1)-Fi(t). K is Boltzmann’s constant. T is the current temperature. Select a random variable ri∈[0,1], i∈[1,N],
ifp≥ri, then accept the new solution and use the solution as the starting point for the next iteration. Otherwise,
abandon the solution, then a set of solutionXnew′=[x11(t+1),x22(t+1),…,xii(t+1)] are obtained.
(5)  A fraction (pa) of worse nests are abandoned and new ones are built.
(6)  Implement the orthogonal design strategy procedures.
(7)  Go to step (2)
End


## 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 pa∈[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:Basic cuckoo search algorithm.
Begin
Generate iteration timet=1
Initialized with random vector values and parameters
Evaluate the fitness of each individual (nest) and determine the best individual with the best fitness value
While (stopping criterion is not met or t < MaxGeneration)
Get a Cuckoo randomly by local random walk or Lévy Flights
Evaluate its fitnessFm
Choose a nest amongn (say, n) randomly
If (Fm<Fn)
replacen by the new nest m
End If
A fraction (pa) of worse nests are abandoned and new ones are built
Keep the best nests with quality solutions
Rank the solutions and find the current best
Update the generation numbert=t+1
End while
EndFurthermore, the algorithm used a balanced combination of a local random walk and the global explorative random walk, controlled by a switching parameterpa. The local random walk can be written as(4)xit+1=xit+αs⊗Hpa-ε⊗xjt-xkt,where xjt and xkt 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 [14–17]:(5)xit+1=xit+α⊕Lévys,λ.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:(6)Lévys,λ=λΓλsin⁡πλ/2π1s1+λ.


## 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 LM(QK), where Q is the prime number, M=QJ, and J is a positive integer satisfying K=(QJ-1)/(Q-1). The brief procedure of constructing the orthogonal array LM(QK)=[ai,j]M×K is described as shown in Procedure 1.Procedure 1:Procedure constructing the orthogonal array.
Step  1. Construct the basic columns
Fork=1 to J
j=Qk-1-1Q-1+1
Fori=1 to QJ
ai,j=i-1QJ-Kmod⁡Q
End for
End for
Step  2. Construct the non-basic columns
Fork=2 to J
j=Qk-1-1Q-1+1
Fors=1 to j-1
Fort=1 to Q-1
aj+(s-1)(Q-1)+1=as×t+ajmod⁡Q
End for
End for
End for
Step  3. Increment ai,j by one for 1≤i≤M, 1≤j≤NThe procedure of the orthogonal design algorithm is elaborated as shown in Algorithm2 and for more detailed information on the orthogonal design strategy, please refer to [17–19].Algorithm 2:Orthogonal design algorithm.
Begin
(1)  Construct the orthogonal array following the above steps
(2)  Randomly select two solutions from the population
(3)  Quantize the domain formed by the two solutions
(4)  Randomly generatek-1 integers p1⋯pk-1
(5)  UseLM(QK) to generate M potential offspring
(6)  Randomly select a solution from the population
(7)  Compare the solution with the best solution from the orthogonalM offspring and obtain the better solution
EndThe procedure of simulated annealing algorithm is simply stated as shown in Algorithm3 [20], and for more detailed information on the simulated annealing, please refer to [20–22].Algorithm 3:Simulated annealing algorithm.
Begin
(1)  Given a configuration of the elements of the system, randomly displace the elements on a time,
by a small amount and calculate the resulting change in the energy,ΔE
(2)If  ΔE<0
Then accept the displacement and use the resulting configuration as the starting point for the next iteration
Else
Accept the displacement with probabilityP(ΔE)=exp⁡(-ΔE/kbT) where T is the current temperature and
kb is Boltzmann’s constant
(3)  Repetition of this step continues until equilibrium is achieved
EndBased 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 Algorithm4.Algorithm 4:Improved cuckoo search algorithm.
Begin
(1)  Initial the parameter values of the algorithm, generate the random initial vector values and set the iteration numbert=1.
(2)  Evaluate fitness values of each individual and determine the current best individual with the best objective value.
Check whether the stopping criterion is met. If the stopping criterion is met, then output the best solution;
otherwise update the iteration numbert=t+1 and continue the iteration process until the stopping criterion is met.
(3)  Keep the best solution of the last iteration, and get a set of new solutionsXnew=[x1t+1,…,xi(t+1),…,xK(t+1)] by
Lévy flight, the Lévy flight is performed according to (4).
(4)  Evaluate the fitness valueFi(t+1) of the new solution xi(t+1), and compare Fi(t+1) with Fi(t) which represents the solution of
thetth iteration. If Fi(t+1) is better than Fi(t), then replace xi(t) by xi(t+1), otherwise, not abandoning the solution xi(t+1)
at once but accepting the solution with probabilityp=e(-|Δf|/(K⋅T)), where Δf is the change in the fitness value
Δf=Fi(t+1)-Fi(t). K is Boltzmann’s constant. T is the current temperature. Select a random variable ri∈[0,1], i∈[1,N],
ifp≥ri, then accept the new solution and use the solution as the starting point for the next iteration. Otherwise,
abandon the solution, then a set of solutionXnew′=[x11(t+1),x22(t+1),…,xii(t+1)] are obtained.
(5)  A fraction (pa) of worse nests are abandoned and new ones are built.
(6)  Implement the orthogonal design strategy procedures.
(7)  Go to step (2)
End


## 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:(7)x˙=σ1y-x,y˙=σ2x-xz-y,z˙=xy-σ3z,where (x,y,z) is the state variables; σ1,σ2,σ3 are the unknown chaotic system parameters which need to be estimated. The real parameters of the system are σ1=10, σ2=28, and σ3=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 T0=100. Consider(8)Tn=T0ln⁡1+n.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:(9)H=1M∑n=1Mx~n-xn2+y~n-yn2+z~n-zn2.Figure1 shows the convergence process of the fitness values and three parameters (σ1,σ2,σ3) during the iterations in a single experiment.Figure 1
The convergence process of fitness function value and three parameters (σ1,σ2,σ3) 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 Table1. The results based on CS (the best parameter setting is pa=0.25, pa=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 cr=0.8, mu=0.1, where cr is the crossover rate and mu 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
0.113969It can be seen from Table1 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 (σ1,σ2,σ3) during the iterations in a single experiment under the noise condition.Figure 2
The convergence process of fitness function value and three parameters (σ1,σ2,σ3) 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 Table2.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
0.255996It can be seen from Table2 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:(10)x˙=σ1y-x,y˙=σ3-σ1x-σ3y-xz,z˙=xy-σ2z,where (x,y,z) is the state variables; σ1,σ2,σ3 are the unknown chaotic system parameters which need to be estimated. The real parameters of the system are σ1=35, σ2=3, and σ3=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 T0=100. The convergence process of the fitness values and three parameters (σ1,σ2,σ3) 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
The convergence process of fitness function value and three parameters (σ1,σ2,σ3) during the iterations in a single experiment under the noiseless condition.It can be seen from Table3 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
0.010562As 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 (σ1,σ2,σ3) during the iterations in a single experiment under the noise condition.Figure 4
The convergence process of fitness function value and three parameters (σ1,σ2,σ3) during the iterations in a single experiment under the noise condition.It can be seen from Table4 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


## 4.1. Lorenz Chaotic System

Lorenz chaotic system equation [23] is expressed as follows:(7)x˙=σ1y-x,y˙=σ2x-xz-y,z˙=xy-σ3z,where (x,y,z) is the state variables; σ1,σ2,σ3 are the unknown chaotic system parameters which need to be estimated. The real parameters of the system are σ1=10, σ2=28, and σ3=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 T0=100. Consider(8)Tn=T0ln⁡1+n.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:(9)H=1M∑n=1Mx~n-xn2+y~n-yn2+z~n-zn2.Figure1 shows the convergence process of the fitness values and three parameters (σ1,σ2,σ3) during the iterations in a single experiment.Figure 1
The convergence process of fitness function value and three parameters (σ1,σ2,σ3) 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 Table1. The results based on CS (the best parameter setting is pa=0.25, pa=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 cr=0.8, mu=0.1, where cr is the crossover rate and mu 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
0.113969It can be seen from Table1 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 (σ1,σ2,σ3) during the iterations in a single experiment under the noise condition.Figure 2
The convergence process of fitness function value and three parameters (σ1,σ2,σ3) 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 Table2.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
0.255996It can be seen from Table2 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:(10)x˙=σ1y-x,y˙=σ3-σ1x-σ3y-xz,z˙=xy-σ2z,where (x,y,z) is the state variables; σ1,σ2,σ3 are the unknown chaotic system parameters which need to be estimated. The real parameters of the system are σ1=35, σ2=3, and σ3=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 T0=100. The convergence process of the fitness values and three parameters (σ1,σ2,σ3) 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
The convergence process of fitness function value and three parameters (σ1,σ2,σ3) during the iterations in a single experiment under the noiseless condition.It can be seen from Table3 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
0.010562As 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 (σ1,σ2,σ3) during the iterations in a single experiment under the noise condition.Figure 4
The convergence process of fitness function value and three parameters (σ1,σ2,σ3) during the iterations in a single experiment under the noise condition.It can be seen from Table4 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.



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*Source: 2959370-2016-01-12.xml*