Particle Swarm Optimization

Author: CislunarSpace
Site: https://cislunarspace.cn

Definition

Particle Swarm Optimization (PSO) is a metaheuristic optimization algorithm based on swarm intelligence, simulating bird foraging behavior to find optimal solutions in multi-dimensional space through individual experience and group collaboration. PSO is simple, fast-converging, with few parameters, widely used in continuous space optimization.

Algorithm Origin

Proposed by Kennedy and Eberhart in 1995 based on social cognitive models:

Source	Correspondence
Bird flock foraging	Optimization search
Individual experience	Individual best pbest
Group collaboration	Global best gbest

Basic Principles

Particle Representation

Each particle represents a candidate solution:

\mathbf{x}_i = (x_{i1}, x_{i2}, ..., x_{iD})

Where $D$ is dimensionality.

Velocity Update

\mathbf{v}_i^{n+1} = w \mathbf{v}_i^n + c_1 r_1 (\mathbf{pbest}_i - \mathbf{x}_i^n) + c_2 r_2 (\mathbf{gbest} - \mathbf{x}_i^n)

Position Update

\mathbf{x}_i^{n+1} = \mathbf{x}_i^n + \mathbf{v}_i^{n+1}

Parameter Description

Parameter	Meaning	Typical Value
$w$	Inertia weight	0.4-0.9
$c_1$	Individual learning factor	2.0
$c_2$	Social learning factor	2.0
$r_1, r_2$	[0,1] random numbers	-
$v_{max}$	Maximum velocity	$0.1 \times x_{max}$

Applications in Stratospheric Airships

Trajectory Optimization

Optimization Variable	Dimension	Range
Heading angle sequence	$N \times 1$	$[0, 2\pi)$
Altitude profile	$N \times 1$	$[h_{min}, h_{max}]$
Time allocation	$N \times 1$	$[0, T_{max}]$

Objective Function

\min J = \int_0^T P_{prop}(\mathbf{x}, t) dt + \lambda_1 |h_{error}| + \lambda_2 |d_{station} - R_{target}|

References

Kennedy J, Eberhart R. Particle Swarm Optimization[C]. IEEE International Conference on Neural Networks, 1995.
Zhang Y, et al. PSO-based Trajectory Optimization for Stratospheric Airship[J]. AIAA Journal of Aerospace Systems, 2025.