Journal of Control Science and Engineering

Volume 2019, Article ID 7478498, 11 pages

https://doi.org/10.1155/2019/7478498

## An Improved Convergence Particle Swarm Optimization Algorithm with Random Sampling of Control Parameters

^{1}School of Electrical Engineering, Henan University of Technology, Zhengzhou, Henan 450001, China^{2}Zhengzhou Key Laboratory of Machine Perception and Intelligence Systems, Zhengzhou, Henan 450001, China

Correspondence should be addressed to Tianfei Chen; moc.361@iefnait_nehc

Received 2 January 2019; Revised 2 April 2019; Accepted 7 May 2019; Published 2 June 2019

Academic Editor: Ai-Guo Wu

Copyright © 2019 Lijun Sun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Although particle swarm optimization (PSO) has been widely used to address various complicated engineering problems, it still needs to overcome the several shortcomings of PSO, e.g., premature convergence and low accuracy. Its final optimization result is related to the control parameters selection; therefore, an improved convergence particle swarm optimization algorithm with random sampling of control parameters is proposed. For the proposed algorithm, the random sampling strategy of control parameters is designed, which can promote the flexibility of algorithm parameters and simultaneously enhance the updating randomness for both particle velocity and position. According to the convergence analysis of PSO, the sampling range for inertial weight is determined after both the acceleration factors have already been sampled in their respective value interval, to ensure convergence for every evolution step of algorithm. Besides that, in order to make full use of dimension information of some better particles, the stochastic correction approach on each dimension for the population optimum value has been adopted. The final experiments results demonstrate that the proposed algorithm further improves the convergence rate while maintaining higher convergence accuracy, compared with basic particle swarm optimization and other variants.

#### 1. Introduction

PSO, proposed by Kennedy and Eberhart [1], is an evolutionary algorithm based on swarm intelligence which simulates birds or fish predation, and it has already attracted a lot of interest from scholars and researchers for the reason that PSO has simple structure, strong maneuverability, easy realization, and other characteristics. Up to now, PSO has been successfully applied in many areas [2–6], and meanwhile some improved versions of PSO have also been studied accordingly [7–11].

Inertial weight, which is the relatively important control parameter of PSO, is first introduced by Shi [12] into the basic evolution equations of algorithm, and from then on, some research about the influence of inertial weight on optimization performance has been carried out. In [13], Alfi proposes an adaptive particle swarm optimization algorithm in which a dynamic inertia weight is used. Based on the Bayesian theory, an adaptive adjustment strategy for inertia weight is designed by Zhang [14], and simultaneously the historical positions of particles are fully used. Although the convergence precision of this improved PSO is higher, the convergence speed becomes slow. After Han [15] compared several common updating ways for inertial weight, it can be concluded that the convergence performances of the simulated annealing updating method and the linear decreasing method are relatively better, but the acceleration coefficient, when it changed, has big influence on the convergence performance. Besides that, a linear updating way for acceleration coefficient* c*_{1} and* c*_{2} is put forward by Ratnaweera [16] through the corresponding parameter analysis, and Yamaguchi [17] proposed an adaptive adjustment strategy for acceleration coefficient according to the updated positions of particles. Furthermore, Liang [18] proposed a comprehensive learning particle swarm optimization algorithm using a novel learning method to improve the convergence performance. However, all the above literatures improve the convergence performance by modifying control parameters of algorithm, and they are only demonstrated through numerical simulation experiments, but because of lack of the corresponding theoretical convergence analysis, perhaps some part of the actual evolutionary process is divergent and unstable.

The convergence of PSO algorithms should be based on the framework of random search algorithm [19], and it has been proved by Van den Bergh [20] that PSO is not a global optimization algorithm and also cannot be guaranteed to converge to a local optimum. On this basis, Trelea [21] utilized the linear dynamic system theory with constant coefficient to analyze the stability of basic PSO, and Clerc [22] established a constraint model of PSO described by only five parameters; then the convergence and trajectories of particles in phase plane were analyzed. Starting with the Markov chain formed by particles states, Ren [23] has pointed out that this Markov chain does not have conditions for stationary processes, and then it was proved that PSO is not globally convergent in the view of transition probability. On the basis of stochastic system theory, Jin [24] analyzed the mean square convergence of PSO and provided a sufficient condition of convergence. Although some these literatures for convergence analysis supply sufficient conditions, it is still not given how to adjust control parameters of algorithm in evolution process in order to get better convergence performance.

In view of the problems mentioned above, this paper proposes an improved convergence particle swarm optimization algorithm with random sampling of control parameters (SC-PSO), and the main contribution of the present work is delineated as follows.

The random sampling strategy is designed to improve the flexibility of control parameters, which also can strengthen the position updating randomness of particles to enhance the exploration ability of PSO and help to jump out of local optimum.

In order to ensure the convergence of the algorithm, the inertia weight is selected around the center part of the convergence interval, and the phenomenon of “oscillation” and “two steps forward, one step back” can be prevented.

Due to the weakness of exploitation caused by the random sampling strategy, the intermediate particle updating strategy is devised to update the optimal position of swarm population in every evolution step. In addition, the optimal value is updated dimensionally, and the dimensionality information of different particles is used to randomly select the value, so as to find the better position in the dimension.

This paper is organized as follows. Section 2 introduces the basic PSO and gives its theoretical analysis of convergence. Section 3 describes the proposed algorithm in detail. Section 4 presents the test functions, the parameters setting of each algorithm, the results, and discussions. Conclusions are given in Section 5.

#### 2. PSO Algorithm

##### 2.1. Basic PSO

While PSO is running, each particle is regarded as a feasible solution to the optimization problem in search space and the flight behavior of particles can be treated as the search process of all individuals; then the velocity of particles is dynamically updated according to the historical optimal position of particle and the optimal position of swarm population. It is assumed that the swarm population is composed of particles in* D* dimensional space, and the historical optimal position of the* i*_{th} particle is represented by , , and the optimal position of swarm population is denoted as . In every evolution step, the velocity and position for each particle are updated by dynamically tracking its corresponding historical optimal position and the optimal position of swarm population. The detailed equations are expressed as follows: where* t* shows the iteration number and indicates dimension; thus* x*_{i,d}(*t*) is the* d*_{th} dimension variable of the* i*_{th} particle in the* t*_{th} iteration, and variables* v*_{i,d}(*t*),* p*_{g,d}(*t*), and* p*_{i,d}(*t*) have the similar meanings in turn;* ω* is inertial weight, and

*c*

_{1}and

*c*

_{2}denote acceleration coefficients;

*r*

_{1}and

*r*

_{2}are random numbers uniformly distributed in interval .

According to the detailed optimization problem to be settled, the objective function should be set, and the objective function values of each particle are corresponding fitness values. The fitness value can be used to not only measure the position of particles but also update the historical optimal position of particles and the optimal position of swarm population.

##### 2.2. Convergence Analysis

The convergence of particles trajectories is determined by control parameters of algorithm, and in order to facilitate analysis and generality, the case of a single particle system having only one dimension is taken as an example. After that, the basic evolution equations (1) and (2) can be transformed into the dynamic equation form.where we define , , and . If the historical optimal position of particle finally converges to the optimal position of swarm population , the dynamic equation (3) can be arranged as follows:

For (4), [24] has given a sufficient condition of mean square convergence through theoretical analysis, and this condition is expressed aswhere

According to formula (5) and (6), it is easy to get the relationship between inertial weight and acceleration coefficients* c*_{1} and* c*_{2} at the situation of convergence.

#### 3. Our Proposal: SC-PSO Algorithm

##### 3.1. Random Sampling Strategy for Control Parameters

For basic PSO, control parameters have great impact on the performance of algorithm. If they are assigned inappropriately, the trajectories of particles cannot converge and may even be unstable, which will cause that the optimal solution of optimization problems cannot be found. At present, the control parameters are usually chosen according to the experience or experiments from engineers, so it is not flexible and the exploration ability of PSO has also been greatly restricted.

Random sampling strategy is designed to improve the flexibility of control parameters and enhance the exploration ability of PSO to help to jump out of local optimum. On the basis of the conclusion from [24], the convergence of PSO should be considered when the parameters are randomly selected. First of all, acceleration coefficients* c*_{1} and* c*_{2} are, respectively, uniformly sampled in their corresponding value interval, and the parameters* μ* and

*can be computed using formula (6). According to the condition of mean square convergence from formula (5), the sampling interval for inertial weight*

*σ**can be solved.where and , respectively, denote the lower bound and the upper bound.*

*ω*Finally, the inertial weight ought to be sampled in the above computed interval. However, in order to avoid the phenomenon of “oscillation” and “two steps forward, one step back,” the inertia weight is selected around the center part of the convergence interval of* ω*. According to formula (8) and (9), Figure 1 shows the relationship between

*and*

*μ**for mean square convergence of PSO. For example, when the acceleration coefficients have been already sampled,*

*ω**c*

_{1}=

*c*

_{2}=2, and then parameters

*and*

*μ*

*σ*^{2}can be computed,

*=2,*

*μ*

*σ*^{2}=2/3. The convergence interval of inertial weight is . In practice, the inertial weight is uniformly selected around the center part of the above interval to avoid the phenomenon of “oscillation” and “two steps forward, one step back.”