Parameters Estimation of Uncertain Fractional-Order Chaotic Systems via a Modified Artificial Bee Colony Algorithm

Parameter estimation for fractional-order chaotic systems has been an interesting and important issue in theory and various fields of application. In this paper, fractional orders, as well as systematic parameters of fractional-order chaotic systems are considered by treating fractional orders as additional parameters. The parameter estimation is transformed into a multidimensional optimization problem, and an effective modified artificial bee colony algorithm is proposed to solve this problem. Numerical simulations are conducted on two typical fractional-order chaotic systems to show the effectiveness of the proposed scheme.


Introduction
Recently, considerable attention has been give to making use of the great potential of fractional calculus in physics [1], electrical circuit theory [2] and control systems [3].In particular, a significant role is played in chaos control theory, and many control methods have been devised for fractional-order chaotic systems under the condition of known fractional orders and systematic parameters.However, in the real world, the fractional orders and systematic parameters of fractional-order chaotic systems cannot be exactly known.Therefore, to achieve the goal of the control and synchronization of fractional-order chaotic systems, the parameters should be estimated beforehand if they are unknown.
However, direct measurement of parameters for the fractional-order chaotic systems is very difficult, so estimating system parameters through an observed chaotic scalar time series based on parameter estimation methods is feasible and valuable.Recently, by formulating the problem as a multidimensional optimization problem and treating the fractional order as one of the parameters to be estimated, many optimization approaches have been applied to solve this problem, such as the particle swarm optimization (PSO) [4][5][6] and differential evolution (DE) [7][8][9].
The artificial bee colony (ABC) algorithm is one of the most recently introduced swarm-based optimization algorithms.It was developed by Karaboga in 2005 based on simulating the foraging behavior of a honeybee colony and has been shown to be competitive with other population-based algorithms for the global numerical optimization problem with the advantage of employing fewer control parameters [10][11][12][13][14][15].For instance, apart from the maximum evaluation number and population size, a standard GA has three more control parameters (crossover rate, mutation rate, generation gap) [16], a standard DE has at least two control parameters (crossover rate, scaling factor) [17] and a basic PSO has three control parameters (cognitive and social factors, inertia weight) [4].Besides, limit values for the velocities v max have a significant effect on the performance of PSO.The ABC algorithm has only one control parameter (limit) apart from colony size and maximum cycle number.Although it uses less control parameters, the performance of the ABC algorithm is better than or similar to that of these algorithms, and it can be efficiently used for solving multimodal and multidimensional optimization problems.
Similar to other population-based algorithms, the ABC algorithm can be further improved to enhance the exploration and exploitation abilities.Therefore, in this paper, a modified artificial bee colony (MABC) algorithm is put forward.Additionally, it is modified by employing two new searching equations.The MABC algorithm is further used to estimate the parameters of uncertain fractional-order chaotic systems via a functional extrema model.Numerical simulations are performed to estimate two well-known fractional-order chaotic systems and statistically compared with some typical existing methods.The simulation results demonstrate the good performance and the superiority of the MABC algorithm, and thus, the MABC algorithm proves to be a promising candidate for parameter estimation of uncertain fractional-order chaotic systems.
The rest of the paper is organized as follows.In Section 2, some preliminaries are given.In Section 3, a brief problem formulation is described.Section 4 describes the MABC algorithm in sufficient detail.The simulation results based on the MABC algorithm are given in Section 5, and the compared results based on PSO and DE are also presented.The paper ends with conclusions in Section 6.

Caputo Fractional-Order Derivative
There are several definitions of fractional-order derivatives.There are three best-known definitions: the Grunwald-Letnikov, Riemann-Liouville and Caputo definitions [18].In particular, the Caputo fractional-order derivative has the same initial conditions as integer-order derivatives, which is well understood in physical situations and more applicable to real-world problems.Thus, the Caputo fractional-order derivative is introduced in this paper.

The Standard Artificial Bee Colony (ABC) Algorithm
The standard artificial bee colony (ABC) algorithm proposed by Karaboga in 2005 is a competitive optimization technique that simulates the intelligent foraging behavior of a honeybee colony [10][11][12].In fact, the ABC algorithm is an iteration optimization technique similar to other population-based algorithms.In the ABC algorithm, the colony of artificial bees contains three groups of bees: employed bees, onlooker bees and scouts.The first half of the colony consists of the employed artificial bees, and the second half includes the onlookers.For every food source, there is only one employed bee.In other words, the number of food sources is equal to the number of employed bees.The employed bee of an abandoned food source becomes a scout.The search carried out by the artificial bees can be described as follows.

Initialization of the Population
The initial population of solutions is filled with SN number of n-dimensional real-valued vectors, which are generated randomly (i.e., food sources).Let ) represent the i-th food source in the population, and then, each food source is generated as follows: x min,j and x max,j are the lower and upper bounds for the dimension j, respectively.These food sources are randomly assigned to SN number of employed bees, and their fitness is evaluated accordingly.

The Employed Bee Phase
At this stage, for each position of the employed bee's food source X i , a new food source position V i is generated via the equation as follows: where k = 1, 2, • • • , SN and j = 1, 2, • • • , D. k and j are randomly generated, and k must be different from i. φ i,j is a random number in [−1, 1].The above explanation implies that the other components of V i , except for dimension j, are the same as the ones of X i .Then, a greedy selection is made between X i and V i .That is, once V i is obtained, it will be evaluated and compared with X i .If the fitness of V i is equal to or better than that of X i , consequently, V i will replace X i and correspondingly become a member of the population.Otherwise, X i is retained.

Calculating Probability Values Referring to the Probability Selection
After finishing the update process, employed bees share their nectar amount information, which is related to the food sources, with the onlooker bees in the nearby hive.An onlooker bee evaluates the nectar information taken from all employed bees and chooses a food source through a probability value p i , which is calculated by the following form: where f it i denotes the fitness value of solution X i .It is obvious that the higher the fitness value of solution X i , the higher the probability of selecting the i-th food source corresponding to solution X i is.
Besides, the fitness value f it i is defined as follows: where f (X i ) represents the objective function value of the decision vector X i .

The Onlooker Bee Phase
Based on the probability value p i calculated by Equation ( 4), each onlooker bee randomly chooses a food source corresponding to the solution X i with a probability value p i .Thereafter, it makes a modification (i.e., V i ) around the chosen food source by the Equation (3).Similarly to the phase of employed bees, the greedy selection mechanism is applied to select a better solution between solution X i and V i .

The Scout Bee Phase
In this phase, the number trial denotes the times that the food source position corresponding to the solution X i is not improved continuously in the honeybee's memory.Additionally, once the value of the number trial is more than the predetermined parameter limit employed in the ABC algorithm, the corresponding employed bee abandons the food source and becomes a scout bee.The scout bee produces a new food source randomly as Equation (2).

Framework of the Standard Artificial Bee Colony Algorithm
Based on the above introduction, the framework of the standard artificial bee colony algorithm can be given as below: Algorithm 1: Framework of the standard artificial bee colony algorithm.
Step (0) Predefine some parameters: SN (population size), D (the dimension of optimization problem), LOW ER (the lower bound of searching space), UP P ER (the upper bound of searching space), limit (control parameter), MCN (maximum cycle number).
Step (1) The population initialization phase: Step (1.1) Randomly generate 0.5 * SN points in the searching space to form an initial population via Equation (2).
Step (1.2) Evaluate the objective function values of population.

End While
Step (5) The scout bee's phase: If max(trial i ) > limit, replace X i with a new candidate solution generated via Equation (2).
Step (6) Set cycle = cycle + 1, and if cycle > MCN, then stop and output the best solution achieved so far, otherwise, go to Step 2.

Problem Formulation
Consider the following fractional-order chaotic system: where The corresponding estimated system can be written as: where T is a set of estimated systematic parameters and α = ( α1 , α2 , • • • , αn ) T is the estimated fractional orders.Besides, Systems ( 6) and ( 7) have the same initial conditions Y 0 .
In general, to identify the fractional-order chaotic System (6), it can be transformed into a functional extrema model as follows: where k = 1, 2, • • • , N is the sampling time point and N denotes the length of data used for parameter estimation.Y k and Ỹk respectively denote the state vector of the original System (6) and the estimated System (7) at time kh.h is the step size employed in the predictor-corrector approach for the numerical solutions of fractional differential equations [19].• is the Euclid norm.Ω is the searching area admitted for parameters θ, where fractional orders α are considered as special variables.The parameter identification of System (6) can be achieved by searching suitable θ and α in the searching space Ω, such that the objective function is minimized.In other words, the main task is to find the best combination of the independent variables of α and θ for the objective function.

A Modified Artificial Bee Colony Algorithm
In order to improve the convergence speed and accuracy of the standard ABC algorithm, some modifications in terms of the searching equations are proposed.Additionally, they are described in detail in the following subsections.

Two Modified Solution Searching Equations
To improve the performance of the standard ABC algorithm, one effective research trend is to investigate searching equations.So far, a variety of searching equations have been put forward [20][21][22][23][24].The most representative one is GABCproposed by Zhu and Kwong [21].In this method, in order to improve the exploitation and make full use of the information of the gbest solution to guide the search of candidate solution, a modified searching equation enlightened by PSO is suggested as follows: where the third term in the right-hand side of Equation ( 9) is a new added term called the gbest term, x best,j is the j-th element of the global best solution and ψ i,j is a uniform random number in [0, 1.5].
Nevertheless, based on the experiment results shown in [21], it can be found that the improvement of the algorithm is not notable.Thus, owing to the poor performance of the above Equation (3) and Equation ( 9), two new searching equations are presented to be used by the employed and onlooker bees, separately.The two equations are written as below: where x best,j is the j-th element of the global best solution.Similar to PSO algorithms, x i,j denotes the j-th element of the best previous solution of employed bee i (i.e., the local-best position or its experience).r1 and r2 are distinct integers randomly selected from {1, 2, • • • , SN} and are also different from i and best.j ∈ {1, 2, • • • , n} is a randomly chosen index.p i represents the current probability of the i-th employed bee.λ, µ are a random number in the range [0, 1], and ν is a random number in the range From Equation (10), due to the first term, the new candidate solution is generated around X r1 , which is a randomly selected individual from the population.The randomly selected individual X r1 can bring more information to the searching equation and avoid being trapped into the local optimum, which is good at exploration.Besides, despite the guidance of X best not being used in the first term, Equation (10) can also take full advantage of x i,j and X best in the latter two terms to drive the new candidate solution towards the current best solution, which can guarantee its convergence speed.Therefore, the last two terms of Equation (10) do well in the exploitation.Generally speaking, Equation (10) can keep the exploration and exploitation well balanced.However, because of the guidance of X r1 in the first term, Equation (10) pays more attention to the exploration.According to [10], in the employed bee phase, the searching process focuses on the exploration, which is consistent with the emphasis of Equation (10).Therefore, Equation ( 10) is chosen as the searching equation for the employed bees.
In Equation (11), owing to the first term, the new candidate solution is generated around X best , which stands for the current global best solution.The current probability is introduced in the second term to make full use of the information of the current population to further enhance the exploitation.Thus, the first two terms have effects on the exploitation ability.In addition, to keep balance corresponding to the first two terms in Equation (11), the third term is recommended to improve the diversity of the population.To sum up, Equation ( 11) may be beneficial to not only the convergence, but also the diversity of the population.However, due to the guidance of X best in the first term, Equation ( 11) places more emphasis on the exploitation.According to [10], the onlooker bee phase concentrates on the exploitation, which is inconsistent with the emphasis of Equation (11).Therefore, Equation ( 11) is selected as the searching equation for the onlooker bees.

The Proposed Method
In view of the above, a new algorithm is proposed.Two new searching equations are proposed to generate new candidate solutions on the employed bee's phase and onlooker bee's phase, respectively.The pseudo-code of the modified ABC algorithm (here, we call it MABC) is given below: Algorithm 2: Framework of the modified ABC algorithm.
Step (1) The population initialization phase: Step (1.1) Randomly generate 0.5 * SN points in the search space to form an initial population via Equation (2).
Step (1.2) Evaluate the objective function values of the population.

End While
Step (5) The scout bee's phase: If max(trial i ) > limit, replace X i with a new candidate solution generated via Equation (2).
Step (6) Set cycle = cycle + 1, and if cycle > MCN, then stop and output the best solution achieved so far; otherwise, go to Step 2.

Simulations
To verify the effectiveness of the MABC algorithm, two typical fractional-order chaotic systems are chosen to test the performance.The simulations were done using MATLAB (Version 7.1, MathWorks, Natick, MA, USA) on an Intel(R) Core(TM) i5-3470 CPU, 3.2 GHz with 4 GB of RAM.In order to calculate the objective function, the number of samples is set as 300 and the step size is 0.01.As is well known, the larger the population and the maximum cycle number, the larger the probability of finding the global optimum.However, a larger population and maximum cycle number implies a larger number of function evaluations.In the following simulations, for the MABC, ABC, PSO and DE algorithms, the population size (SN) and maximum cycle number (MCN) are all set as: SN = 100, MCN = 100.Besides, for the MABC and ABC algorithms, according to [10], to make a fair comparison and have good searching ability, the control parameter limit is chosen as 15.For the parameters of the DE algorithm, according to [7], the scaling factor F = 0.5 is usually a good choice.If the parameter F is smaller, the population may converge prematurely.On the contrary, the convergence speed decreases.Besides, regarding the crossover rate CR, a large CR often speeds up the convergence.However, from a certain value upwards, the convergence speed may decrease or the population may also converge prematurely.A good choice for the crossover rate is a value between CR = 0.3 and CR = 0.9.In the following experiments, the parameters of DE are selected as: F = 0.5, CR = 0.7.For PSO, empirical results have shown that an inertia weight w = 0.7298 and cognitive and social factors with c1 = c2 = 1.49618 provide good convergent behavior [4,25].Therefore, in the following simulation, the parameters for PSO are set as: w = 0.7298, c1 = c2 = 1.49618.To make a fair comparison, the searching spaces of the parameters are the same for all algorithms.
Example 1.The fractional-order economic system [26,27] is described as: when (a, b, c) = (1, 0.1, 1), (α 1 , α 2 , α 3 ) = (0.90, 0.85, 0.95) and the initial point is (2, −1, 1), System ( 12) is chaotic.To show the performance of the MABC algorithm clearly, the true values of fractional orders α 1 , α 2 , α 3 and systematic parameter a are assumed as unknown parameters, which need to be estimated.The searching spaces of the unknown parameters and orders are set as The statistical results of the best, the mean and the worst estimated parameters with the corresponding relative error values over 15 independent runs are shown in Table 1.From Table 1, it can be easily seen that the estimated value generated by the MABC algorithm is closer to the true parameter value, which means that it is more accurate than the standard ABC, PSO and DE algorithms.Besides, it can also be clearly found that the relative error values obtained by the MABC algorithm and marked with black are all smaller than those of the standard ABC, PSO and DE algorithms, which can further prove that the MABC algorithm has higher calculation accuracy.What is more, the best fitness value obtained by the MABC algorithm is better than those obtained by the standard ABC, PSO and DE algorithms.
The evolutionary curves of the parameters and fitness values estimated by the various algorithms are shown in Figures 1-3 in a single run.From Figures 1-3, it can be clearly seen that convergence processes of the parameters and fitness values of MABC algorithm are much better than other algorithms.The estimated parameters can be closer to the true values than the standard ABC, PSO and DE algorithms.Additionally, the relative error values and fitness values obtained by the MABC algorithm decline faster than the other algorithms.In other words, it can be concluded that the MABC algorithms can more efficiently identify a fractional-order economic system than the standard ABC algorithm, as well as the existing PSO and DE algorithms.Example 2. The fractional-order Rössler system [26,28] is described as: when (a, b, c) = (0.5, 0.2, 10), (α 1 , α 2 , α 3 ) = (0.90, 0.85, 0.95) and the initial point is (0.5, 1.5, 0.1), System (13) is chaotic.In this example, the fractional orders α 1 , α 2 , α 3 are selected as unknown parameters to be estimated.The searching spaces of the unknown parameters are set as To show the performance of MABC algorithm, the statistical results of the best, the mean and the worst estimated parameters by various algorithms over 15 independent runs are listed in Table 2. From Table 2, it can be seen that the MABC algorithm has more accurate results than those of the standard ABC, PSO and DE algorithms.Figures 4-6 depict the convergence profile of the evolutionary processes of the estimated parameters and the fitness values.From the figures, it can be seen that the MABC algorithm still can converge to the optimal solution more rapidly than the other algorithms.Based on the above two examples, the proposed scheme is significantly better than all of the other listed algorithms, and it has a faster convergence speed and higher calculation accuracy in estimating the unknown fractional orders and systematic parameters of fractional-order systems.

Conclusions
In this paper, the parameter estimation of fractional-order chaotic systems is converted to an optimization problem from the prospect of optimization.A modified artificial bee colony algorithm is proposed to solve the optimization problem via a functional extrema model.In simulations, the proposed method is employed to identify two typical fractional-order chaotic systems.It is shown that, for the given parameter configurations and maximum number of iterations, the modified ABC algorithm could estimate the unknown fractional orders and parameters of the uncertain fractional-order chaotic systems more rapidly, more accurately and more stably than standard the ABC, PSO and DE algorithms.Additionally, the proposed method can avoid the design of the parameter update law in the synchronization analysis of the fractional-order chaotic systems with unknown parameters.Furthermore, although this paper is mainly concentrated on the parameter estimation problem of the fractional-order chaotic systems, the proposed method is also a useful tool for the study of various numerical optimization problems in physics and other related areas.

Figure 1 .Figure 2 .Figure 3 .
Figure 1.Estimated parameter values with various algorithms on a fractional-order economic system.

Figure 4 .
Figure 4.Estimated parameter values with various algorithms on a fractional-order Rössler system.

Table 1 .
Simulation results of various algorithm for System (12) over 15 independent runs.MABC, modified ABC; DE, differential evolution.

Table 2 .
Simulation results of various algorithm for System (13) over 15 independent runs.