A novel PSO algorithm based on an incremental-PID-controlled search strategy

Abstract

Many variants of Particle Swarm Optimization (PSO) algorithm have been proposed in literature in the past two decades. The two research major topics, namely particle movement trajectory analysis and parameter selection, have also been intensively explored. Focused on the two issues, this paper transforms the velocity updating strategy in PSO algorithm to an incremental PID controller and thus develops a new incremental PID controlled search strategy. To endow this new PID-controlled deterministic search strategy with a stochastic nature, a mutation operation is applied on the particles and their exemplars. Based on such learning concepts, a novel incremental PID-controlled PSO, called IPID-PSO in short, algorithm is proposed. The proposed IPID-PSO algorithm is applied to eight benchmark problems of function optimization widely considered in literature. Compare with other two PSO variants, the IPID-PSO algorithm outperforms them on most of the eight functions. This superiority shows the usefulness of the proposed search strategy. Finally, the IPID-PSO is employed to estimate the parameters of the Jiles–Atherton (J–A) model, which is usually adopted to describe the hysteresis loop of magnetic materials. The results demonstrate that the proposed IPID-PSO is also effective for the J–A model parameter estimation. Therefore it can be concluded that the IPID-PSO algorithm provides a promising and satisfactory solution to real-world applications.

Appendix 1: The eight benchmark functions examined

The following eight benchmark functions (Salomon 1996; Yao et al. 1999; Lee and Yao 2004), among which $f_1$ and $f_2$ are two uni-modal functions and $f_3$ to $f_8$ are six multi-modal functions of the independent $D$-dimensional variable x, are minimized by using the IPID-PSO algorithm in Sect. 4:

(1) Sphere function

$$\begin{aligned} f_1 (\mathbf {x})=\sum \limits _{i=1}^D {x_i ^2} \end{aligned}$$

(14)

(2) Rosenbrock’s function

$$\begin{aligned} f_2 (\mathbf {x})=\sum \limits _{i=1}^{D-1} {\left( 100(x_i ^2-x_{i+1} )^2+(x_i -1)^2\right) } \end{aligned}$$

(15)

(3) Ackley’s function

$$\begin{aligned} f_3 (\mathbf {x})&= -20\exp \left( -0.2\sqrt{\frac{1}{D}\sum \limits _{i=1}^D {x_i ^2} } \right) \nonumber \\&-\exp \left( \frac{1}{D}\sum \limits _{i=1}^D {\cos (2\pi x_i )} \right) +20+e \end{aligned}$$

(16)

(4) Griewanks’s function

$$\begin{aligned} f_4 (\mathbf {x})=\sum \limits _{i=1}^D {\frac{x_i ^2}{4,000}} -\prod \limits _{i=1}^D {\cos \left( \frac{x_i }{\sqrt{i} }\right) } +1 \end{aligned}$$

(17)

(5) Weierstrass function

$$\begin{aligned} \begin{array}{l} f_5 (\mathbf {x})=\sum \limits _{i=1}^D \left( \sum \limits _{k=0}^{k\max } {\left[ a^k\cos (2\pi b^k(x_i +0.5))\right] } \right) \\ \qquad \qquad -D\sum \limits _{k=0}^{k\max } {\left[ a^k\cos (2\pi b^k\cdot 0.5)\right] } \\ a=0.5,b=3,k\max =20 \\ \end{array} \end{aligned}$$

(18)

(6) Rastrigin’s function

$$\begin{aligned} f_6 (\mathbf {x})=\sum \limits _{i=1}^D {(x_i ^2-10\cos (2\pi x_i )+10)} \end{aligned}$$

(19)

(7) (Non-continuous) Rastrigin’s function

$$\begin{aligned} \begin{array}{l} f_7 (\mathbf {x})=\sum \limits _{i=1}^D {(y_i ^2-10\cos (2\pi y_i )+10)} \\ y_i =\left\{ {{\begin{array}{ll} {x_i } &{} {\left| {x_i } \right| <1/2} \\ {\frac{\mathrm{round}(2x_i )}{2}} &{} {\left| {x_i } \right| >=1/2} \\ \end{array} }} \right. \\ \end{array} \end{aligned}$$

(20)

(8) Schwefel’s function

$$\begin{aligned} f_8 ( \mathbf {x})=418.9829\times D-\sum \limits _{i=1}^D {x_i \cdot \sin (\left| {x_i } \right| ^{\frac{1}{2}})} \end{aligned}$$

(21)