Parameters identification of chaotic system by chaotic gravitational search algorithm

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Abstract

In this paper, for the parameter identification problem of chaotic system, a chaotic gravitational search algorithm (CGSA) is proposed. At first, an iterative chaotic map with infinite collapses is introduced and chaotic local search (CLS) is designed, then CLS and basic gravitational search are combined in the procedure frame. The CGSA is composed of coarse gravitational search and fine chaotic local search, while chaotic search seeks the optimal solution further, based on the current best solution found by the coarse gravitational search. In order to show the effectiveness of CGSA, both offline and online parameter identifications of Lorenz system are conducted in comparative experiments, while the performances of CGSA are compared with GA, PSO and GSA. The results demonstrate the effectiveness and efficiency of CGSA in solving the problem of parameter identification of chaotic system, and the improvement to GSA has been verified.

Introduction

As an interesting phenomenon of nonlinear system, Chaos is a bounded unstable dynamic behavior generated by determined nonlinear equations and exhibiting sensitive dependence on initial state values. In scientific and engineering researches, many nonlinear systems have exhibited phenomenon of chaos, and control and synchronization of chaotic systems have been investigated intensely in various fields during recent years [1], [2], [3], [4], [5], [6]. Many methods have been developed to control and synchronize chaotic system, with a condition that parameters of the chaotic system are known in advance. This is the reason that parameter identification of chaotic systems has become an important issue in the past decade.

For the problem of parameter identification of chaotic system, some studies focused on synchronization-based methods. A feedback-based synchronization method and an adaptive control method were both introduced to estimate parameters for several chaotic systems [7], furthermore the approach proposed was also used to estimate one parameter of the transmitter for chaotic signal communication [8]. Wu et al. studied a method to identify parameters when the initial value was unknown by adding the chaotic orbits into a bi-search means [9]. In [10], several feedback control gains were introduced to synchronize the model system and the original physical system. In [11], an adaptive control-based synchronization method was presented for parameter identification for a modified chaotic oscillator.

Parameter identification of chaotic system is a multi-dimensional optimization problem, more and more studies tried to solve the problem by intelligent optimization methods. Typically, genetic algorithm (GA) was adopted to estimate parameters of Lorenz chaotic system [12]. Chaotic ant swarm algorithm was used to identify parameters of Logistic iteration system and Lorenz system [13]. Particle swarm optimization (PSO) [14] and differential evolution (DE) [15] were used for the parameters identification of Lorenz system respectively. In [16], Chang et al. introduced a evolutionary programming (EP) approach to solve this problem of chaotic system identification for Lorenz, Lü and Chen systems. Nelder–Mead simplex search and differential evolution algorithm (NMDE) was applied for parameter identification of Lorenz chaotic system [17].

As a different intelligent optimization algorithm compared with GA and PSO, gravitational search algorithm (GSA) is a newly developed heuristic optimization method based on the law of gravity and mass interactions [18]. GSA has been confirmed higher performance in solving various nonlinear functions, compared with some well-know search methods. In [19], GSA was introduced to apply in parameter identification of hydraulic turbine governing system. As the development of GSA, researchers tried to improve this algorithm by different ways. In [19], Li proposed an improved GSA by combing of the search strategy of particle swarm optimization and GSA. In [31], Sarafrazi has tried to improve the ability of GSA to further explore and exploit the search space by introducing the disruption operator. In this paper, we try to improve the GSA for parameter identification of chaotic system. Hybridization is nowadays recognized to be an essential aspect of high performing algorithms. Chaotic variables can go through every state in a certain area according to their own regularity without repetition. Due to the ergodic and dynamic properties of chaos variables, chaos search is more capable of hill-climbing and escaping from local optima than random search [20], and thus has been applied to the area of optimization computation. In the last two decades, various chaos-based optimization algorithms, for example, a chaos-based simulated annealing algorithm (CSA) [21], a hybrid chaotic ant swarm optimization [22], chaotic bee colony algorithms (CABC) [23] and chaotic particle swarm optimization algorithms (CPSO) [24], [25], have been proposed for solving complex optimization problems more effectively.

Motivated by the aforementioned researches, the goal of this paper is to present a chaotic gravitational search algorithm (CGSA), in which the chaotic search is embedded as a local search approach into the procedures of GSA. The architecture of the hybrid algorithm is emerged by switching the chaotic search and GSA to each other according to certain conditions. By doing so, the chaotic search will directly improve the current solutions found by GSA, leading a faster convergence speed, and further giving a higher probability to jump out of local optima. Compared with the improved GSA algorithm in [31], the strategy in this paper is totally different. In [31], the GSA was improved in the original procedure frame by introducing the disruption operator, which decided whether the positions of agents should be modified or not, thus adjusting the exploring and exploiting ability of GSA. However, in this paper, we introduce chaotic search to improve the searching ability after gravitational search has finished. The search process of CGSA composes of two phrases, gravitational search and chaotic search. The proposed CGSA is applied to parameter identification of chaotic system. The feasibility of this algorithm is demonstrated through identifying the parameters of Lorenz chaotic system. For offline identification, the performance of the proposed CGSA is compared with the GA and PSO in terms of parameter accuracy and computation time.

The paper is organized as follows. In Section 2, the problem of estimating chaotic system is discussed and the problem formulation is addressed. The GSA is briefly introduced and then the CGSA is proposed in Section 3. In Section 4, the simulation experiments are designed and the results are discussed. Finally, the conclusion is drawn in Section 5.

Section snippets

Problem formulation

Considering the following n-dimensional chaotic system:X˙=F(X,Θ)where X =  (x1, x2,  , xn)T  Rn denotes the state vector, Θ =  (θ1, θ2,  , θm)T  Rm is the unknown parameters vector. Generally many chaotic systems, such as Lorenz system, Lü system, Chen system, Rössler system and Chua’s circuit, etc., can be expressed by (1).

In order to estimate the unknown parameters in (1), a parameter identification system is defined below:X^˙=F(X^,Θ^)where X=(xˆ1,xˆ2,,xˆn)TRn and Θ^=(θˆ1,θˆ2,,θˆm)TRm are the

Brief introduction of gravitational search algorithm

GSA is a newly developed stochastic search algorithm based on the law of gravity and mass interactions. In GSA, the search agents are a collection of masses which interact with each other based on the Newtonian gravity and the laws of motion, completely different from other well-known population-based optimization method inspired by swarm behaviors in nature, such as particle swarm optimization, genetic algorithm. In GSA, agents are considered as objects and their performance are measured by

Chaotic system

In this section, typical chaotic system is taken as example to show the parameter identification performance of the proposed CGSA.

Lorenz system is described as:x˙=σ(y-x)y˙=ρx-xz-yz˙=xy-βzThe system parameters σ, ρ and β determine the behavior of chaotic system. The approximate range of parameter values that causes a chaotic behavior of the Lorenz oscillator is 4 < σ < 14, 24 < ρ <  90, and 1.5 < β < 4.5 [30]. when σ = 10, ρ = 28 and β = 8/3, the chaotic behavior of Lorenz system is shown in Fig. 2. The Lorenz

Conclusions

In this paper, a chaotic gravitational search algorithm (CGSA) is proposed and applied in the parameter identification of chaotic system. In CGSA, the search process is separated as two phrases: namely coarse gravitational search and fine chaotic search. In this algorithm, ICMIC map is introduced and chaotic local search is designed to search around the optimal solution based on the current best solution found by GSA. To show the effectiveness of the proposed CGSA, it has been applied to

Acknowledgments

This paper is supported by the Research Fund for the Doctoral Program of Higher Education of China (No. 20110142120020), the National Natural Science Foundation of China (Nos. 51109088 and 51079057), the Open Research Fund Program of Key Laboratory of Transients in Hydraulic Machinery, Ministry of Education (No. SLJX2011006) and the Fundamental Research Funds for the Central Universities, HUST (No. 2011QN066).

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