Citation: Rosi´c, M.; Sedak, M.; Simi´c,
M.; Pejovi´c, P. Chaos-Enhanced
Adaptive Hybrid Butterfly Particle
Swarm Optimization Algorithm for
Passive Target Localization. Sensors
2022, 22, 5739. https://doi.org/
10.3390/s22155739
Academic Editor: M. Osman Tokhi
Received: 24 June 2022
Accepted: 23 July 2022
Published: 31 July 2022
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Article
Chaos-Enhanced Adaptive Hybrid Butterfly Particle Swarm
Optimization Algorithm for Passive Target Localization
Maja Rosi´c
1,
* , Miloš Sedak
1
, Mirjana Simi´c
2
and Predrag Pejovi´c
2
1
Faculty of Mechanical Engineering, University of Belgrade, 11000 Belgrade, Serbia; msedak@mas.bg.ac.rs
2
School of Electrical Engineering, University of Belgrade, 11000 Belgrade, Serbia; mira@etf.rs (M.S.);
peja@etf.rs (P.P.)
* Correspondence: mrosic@mas.bg.ac.rs
Abstract:
This paper considers the problem of finding the position of a passive target using noisy
time difference of arrival (TDOA) measurements, obtained from multiple transmitters and a single
receiver. The maximum likelihood (ML) estimator’s objective function is extremely nonlinear and
non-convex, making it impossible to use traditional optimization techniques. In this regard, this
paper proposes the chaos-enhanced adaptive hybrid butterfly particle swarm optimization algo-
rithm, named CAHBPSO, as the hybridization of butterfly optimization (BOA) and particle swarm
optimization (PSO) algorithms, to estimate passive target position. In the proposed algorithm, an
adaptive strategy is employed to update the sensory fragrance of BOA algorithm, and chaos theory is
incorporated into the inertia weight of PSO algorithm. Furthermore, an adaptive switch probability
is employed to combine global and local search phases of BOA with the PSO algorithm. Additionally,
the semidefinite programming is employed to convert the considered problem into a convex one. The
statistical comparison on CEC2014 benchmark problems shows that the proposed algorithm provides
a better performance compared to well-known algorithms. The CAHBPSO method surpasses the
BOA, PSO and semidefinite programming (SDP) algorithms for a broad spectrum of noise, according
to simulation findings, and achieves the Cramer–Rao lower bound (CRLB).
Keywords:
localization; time difference of arrival; butterfly optimization algorithm; hybrid optimiza-
tion; particle swarm optimization; Cramer-Rao lower bound
1. Introduction
Determining the location of a passive target based on time difference of arrival (TDOA)
measurements from multiple transmitters and a single receiver is a key element in many
technologies, such as radar or sonar, telecommunications, mobile communications [
1
,
2
], etc.
In general, two groups of localization approaches, active and passive, may be distinguished.
The active localization approach takes into account the scenario when in the localization the
target is actively involved. However, in the second group, the target does not participate in
the localization process and merely serves to reflect the transmitter’s signals [
3
]. The global
positioning system (GPS) has been widely used to determine the position of an object in
outdoor environments [
4
]. However, this localization system cannot provide satisfactory
performance in indoor, underwater acoustics, and urban environments, where the satellite
signals are unavailable [
5
,
6
]. Therefore, passive target localization has become widely
used in various applications, as an effective alternative to the GPS and other conventional
localization systems.
Hence, the localization of a passive target is considered in this paper, where the noisy
TDOA measurements are employed. The range measurements are calculated from the
difference in the time it takes for a signal coming from a transmitter via the target to a
receiver and the time required for a signal coming directly from the transmitter to a receiver.
Therefore, the unknown position of a target becomes difficult to estimate since the TDOA
Sensors 2022, 22, 5739. https://doi.org/10.3390/s22155739 https://www.mdpi.com/journal/sensors
