A variant with a time varying PID controller of particle swarm optimizers

Abstract

This paper presents a novel variant of particle swarm optimizers (PSOs) that we call the proportional integral derivative (PID) controller inspired particle swarm optimizer (PidSO), which uses a novel evolutionary strategy whereby a specified PID controller is used to improve particles’ local and global best positions information. This strategy enables PidSO to improve the diversity of swarm in a bid to discourage premature convergence and to perform a global search over the entire search space more efficiently. Empirical experiments were conducted on both analytically unimodal and multimodal test functions. The experimental results demonstrate that PidSO enhances the diversity of the swarm and features better search effectiveness and efficiency in solving most multimodal optimization problems when compared with other recent variants of PSOs and evolutionary optimization algorithms such as integral-controlled PSO (ICPSO), PID-controlled PSO (PIDCPSO), comprehensive learning PSO (CLPSO), I-population covariance matrix adaptation evolution strategy (IPOP-CMA-ES), and a multi algorithm genetically adaptive method for single objective optimization (AMALGAM-SO), and so on. Additionally, it has been observed that PidSO is able to achieve comparatively better success rates and success performances though it is more complex, and that the performance of PidSO is promoted by selecting proper law based controllers. Consequently, PidSO offers a new solution to real engineering optimization designs of industrial systems.

Introduction

Particle swarm optimization is a stochastic population-based algorithm which is modeled on the behaviors of insects swarming, animals herding, birds flocking, and fish schooling where these swarms search for food in a collaborative manner, and it was originally introduced by Kennedy and Eberhart in 1995 [7], [11]. It is usually used for the optimization of continuous nonlinear systems. Since the particle swarm optimization uses a simple swarm emulating mechanism to guide the particles to search for globally optimal solutions and implements easily, it has succeed in solving many real-world optimization problems.

Similar to other evolutionary computation algorithms, the particle swarm optimization also shares a population-based iterative evolution technique. Hence, it can computationally be inefficient as measured by the number of function evaluations (FEs) required. Moreover, it may easily get trapped in the local optimum when solving complex multimodal problems. In order to improve the performance of PSOs and achieve the specific goals of accelerating convergence speed and avoiding local optima, a number of variants of PSOs have been proposed so far in spite of being difficult to simultaneously demonstrate these expectations.

In this paper, we present a novel variant of PSOs which we name PidSO, where a specific PID controller is used to evolve particles’ local and global best positions. As we know, the PID controller, which has been widely used in the industry because of its simple structure and robust performance in a wide range of operating conditions, is the best known one among numerous process control methods. Our proposed objective is to elevate the standard PSO (SPSO) performance by a PID controller, namely, to achieve proper response from the proportional term, eliminate the steady-state errors that occur in particles’ movement from the integral term, and improve particles’ evolutionary dynamics from the derivative term. Empirical experiments were conducted on both analytically unimodal and multimodal functions. As a result, PidSO demonstrates better performances for most multimodal problems than other recent variants of PSOs and evolutionary optimization algorithms such as comprehensive learning PSO (CLPSO) [12], integral-controlled PSO (ICPSO) [4], PID-controlled PSO (PIDCPSO) [5], I-population covariance matrix adaptation evolution strategy (IPOP-CMA-ES) [2], and CMA-GA-DE version of a multi algorithm genetically adaptive method for single objective optimization (AMALGAM-SO) [18], and so on.

The remainder of the paper is outlined as follows. Section 2 reviews related work about the combination study of PID controllers and PSOs. Section 3 gives an in-depth and detailed description about SPSO, and then presents the design of an effective PID controller and its introduction into SPSO, so our proposed PidSO is completely created and utilized. Section 4 depicts the experiments on both given unimodal and multimodal test functions, and further discusses the experimental results. Finally, Section 5 details our conclusions and gives suggestions on future research work.