# Universe-inspired algorithms for Control Engineering: A review

Rodrigo M. C. Bernardo<EMAIL_ADDRESS>Delfim F. M. Torres
<EMAIL_ADDRESS>Carlos A. R. Herdeiro<EMAIL_ADDRESS>Marco P. Soares dos Santos
<EMAIL_ADDRESS>Center for Mechanical Technology & Automation (TEMA),
Department of Mechanical Engineering,
University of Aveiro, 3810-193 Aveiro, Portugal  Center for Research and
Development in Mathematics and Applications (CIDMA), Department of
Mathematics,
University of Aveiro, 3810-193 Aveiro, Portugal  Intelligent Systems Associate
Laboratory (LASI), Portugal

###### Abstract

Control algorithms have been proposed based on knowledge related to nature-
inspired mechanisms, including those based on the behavior of living beings.
This paper presents a review focused on major breakthroughs carried out in the
scope of applied control inspired by the gravitational attraction between
bodies. A control approach focused on Artificial Potential Fields was
identified, as well as four optimization metaheuristics: Gravitational Search
Algorithm, Black-Hole algorithm, Multi-Verse Optimizer, and Galactic Swarm
Optimization. A thorough analysis of ninety-one relevant papers was carried
out to highlight their performance and to identify the gravitational and
attraction foundations, as well as the universe laws supporting them. Included
are their standard formulations, as well as their improved, modified, hybrid,
cascade, fuzzy, chaotic and adaptive versions. Moreover, this review also
deeply delves into the impact of universe-inspired algorithms on control
problems of dynamic systems, providing an extensive list of control-related
applications, and their inherent advantages and limitations. Strong evidence
suggests that gravitation-inspired and black-hole dynamic-driven algorithms
can outperform other well-known algorithms in control engineering, even though
they have not been designed according to realistic astrophysical phenomena and
formulated according to astrophysics laws. Even so, they support future
research directions towards the development of high-sophisticated control laws
inspired by Newtonian/Einsteinian physics, such that effective control-
astrophysics bridges can be established and applied in a wide range of
applications.

###### keywords:

Nature-inspired control , Gravitational Search Algorithm , Black hole ,
Artificial Potential Field , Multi-Verse Optimizer , Galactic Swarm
Optimization , Optimization , Nonlinear control

††journal: This is a preprint of a paper that is published open access in
’Heliyon’ [https://doi.org/10.1016/j.heliyon.2024.e31771].

first-style = long-short, list/display = used, list/template = tabular, GSA
short = GSA, long = Gravitational Search Algorithm, tag = abbrev PI short =
PI, long = Proportional Integral Controller, tag = abbrev FLC short = FLC,
long = Fuzzy Logic Controller, tag = abbrev PID short = PID, long =
Proportional Integral Derivative Controller, tag = abbrev UPFC short = UPFC,
long = Unified Power Flow Controller, tag = abbrev THD short = THD, long =
Total Harmonic Distortion, tag = abbrev PIDF short = PIDF, long = PID
Controller with Derivative Filter, tag = abbrev ACO short = ACO, long = Ant
Colony Optimization, tag = abbrev BA short = BA, long = Bat Algorithm, tag =
abbrev SSA short = SSA, long = Salp Swarm Algorithm, tag = abbrev WOA short =
WOA, long = Whale Optimization Algorithm, tag = abbrev FOPID short = FOPID,
long = Fractional Order Proportional Derivative Proportional Integral
controller, tag = abbrev SMC short = SMC, long = Sliding Mode Controller, tag
= abbrev MPC short = MPC, long = Model Predictive Control, tag = abbrev RL
short = RL, long = Reinforcement Learning, tag = abbrev PS short = PS, long =
Pattern Search algorithm, tag = abbrev GA short = GA, long = Genetic
Algorithm, tag = abbrev BH short = BH, long = Black Hole algorithm, tag =
abbrev PSO short = PSO, long = Particle Swarm Optimization, tag = abbrev PIMR
short = PIMR, long = Proportional-Integral Multiresonant controller, tag =
abbrev MVO short = MVO, long = Multi-Verse Optimizer, tag = abbrev PID+DD
short = PID+DD, long = PID plus double-derivative controller, tag = abbrev
FPDPI short = FPDPI, long = Fractional Order Proportional Derivative
Proportional Integral controller, tag = abbrev NN short = NN, long = Neural
Network, tag = abbrev DE short = DE, long = Differential Evolution, tag =
abbrev ABC short = ABC, long = Artificial Bee Colony algorithm, tag = abbrev
GWO short = GWO, long = Grey Wolf Optimizer, tag = abbrev APF short = APF,
long = Artificial Potential Field, tag = abbrev GSO short = GSO, long =
Galactic Swarm Optimization, tag = abbrev [table]capposition=top
[figure]style=plain, subcapbesideposition=top

## Abbreviations

[include=abbrev, heading=None]

##  1 Introduction

In recent decades, control of non-linear systems has been one of the most
important topics in control theory [2]. Despite the massive use of non-linear
models for accurate prediction of physical systems, it is still difficult to
ensure high stability margins and desired performances in non-linear systems,
mainly if uncertainties must be overcome [3]. Researchers have been observing
nature seeking inspiration to solve complex real-world control-related
problems, since it is a clear example of a time-dependent process in a state
of optimization, according to evolutionary mechanisms. One can find many
natural processes in which a state of equilibrium and adaptation is reached,
which can be investigated for nature-inspired high-performance optimization
and control. Steer et al. [4] stated that the term nature refer ”to any part
of the physical universe which is not a product of intentional human design”.
These authors also distinguish between ‘strong’ inspiration and ‘weak’
inspiration, where the first one involves ”the investigation of some existing
problem-solving mechanism, the extraction of some qualitative process
description, and the application to some alternative purpose”, while the
second is the ”less formal role of some phenomenon in the creative stage of
solution formulation”.

Well known control methods do not consider the dynamics occurring in natural
phenomena (non-inspired control) or only consider some dynamics occurring in
biological structures (bio-inspired control). Many non-inspired control
methods were already proposed, such as the Proportional, Integral and
Derivative (PID) control, predictive control, optimal control, and sliding
mode control [5, 6, 7]. These are non-nature-inspired and employ ”artificial”
control approaches, often neglecting the rationality and effectiveness
inherent in natural systems. In the case of sliding mode control, it does
exhibit an attraction-like behavior, as the system state appears to be drawn
towards the sliding surface. However, this attraction to the sliding surface
is achieved through an artificial mechanism using a variable switching
structure. Nonetheless, these controllers are formulated using a non-natural
attraction, therefore they are not rooted in the natural behavior of celestial
bodies in the universe. Intelligent control and bio-inspired control,
including the design of Artificial Neural Networks and FLC [8, 9, 10], have
also been extensively applied. However, their usage often demands a non-
negligible degree of intuition and lacks interpretability [11, 12].

The main goal of this paper is to provide a literature review of the most
relevant studies that highlight major scientific achievements in the domain of
nature-inspired universe-conveyed control, to highlight their ability for
future applications in multiple areas. Regarding their application in control
systems engineering, scientific efforts have been centered on optimization and
development of metaheuristics, despite the excess of metaphorical heuristics
already reported [13]. No control methods have been found with mathematical
and physical formulations of gravitational attraction or black holes dynamics
directly in their composition, thus evidencing a literature gap to be
explored, where promising control methods may be designed using astrophysical
dynamics, as they may provide mechanisms of stability and robustness (e.g. the
strong gravitation field occurring in black holes). Moreover, the use of
spacetime curvatures may hold great potential to engineer high performance
trajectory tracking systems, as such phenomena ensure the shortest natural
path between two points/states. Indeed, this review perform a thoroughly
analysis to both optimization algorithms, already applied in the field of
Control Engineering, and the control algorithms themselves, as long as their
formulations are deeply related to gravitational attraction phenomena. This
goal was achieved by providing an extensive analysis to the main concepts from
which the original algorithms and related variants were developed, including
their performance, characteristics and applications.

After conducting an initial structured search, no actual control methods
inspired by gravitational attraction were found. The closest approach involves
the use of APF to introduce attraction or repulsion behavior into systems.
However, concerning optimization applied to control (e.g. the optimization of
controller parameters), four algorithms were identified: GSA, BH, MVO, and
GSO. To our knowledge, no literature reviews were already focused on the use
of gravitational phenomena in Control Engineering. Indeed, several review
papers were already published in the scope of GSA, BH, MVO, but they are
mainly focused in data clustering, classification or general optimization
problems [14, 15]. Besides, bio-inspired control methods (including those
inspired in swarm intelligence or evolutionary phenomena) are currently much
more explored than non-biological nature inspired control methods, even though
astrophysical phenomena hold potential to be used for developing high-
sophisticated control systems, due to their inherent gravitational attraction.

The contributions of this paper can be summarized as follows: (i)
Identification and exploration of control methods inspired by gravitational
attraction or black-hole attraction derived dynamics; (ii) Critical analysis
to the optimization algorithms already applied to control problems. Included
are the GSA, BH, MVO, and GSO, as well as their variants and modifications;
(iii) Critical analysis to the APF already applied to control problems; (iv)
Discussion on the potential advancements and limitations related to the use of
gravitational attraction and universe-inspired algorithms in control systems.
The ultimate goal is to contribute towards the development of high-
sophisticated control systems inspired by realistic astrophysical phenomena
and authentically formulated by Newtonian/Einsteinian physics.

##  2 Methods

###  2.1 Selection Criteria

In this paper we present a rigorous analysis of controllers and optimization
algorithms applied in control systems inspired in a specific natural
phenomenon: gravity, and related attraction between bodies. The Scopus
database was searched in the time interval between 2000 and 2023 by seeking
for the terms “gravit* AND control*”, “attrac* AND control*”, “black-hole AND
control*”, and “galactic AND control*”, in the title, abstract and keywords. A
search using the term “universe AND control” was also conducted; however, only
studies outside the scope were obtained. Control inspired in black-holes was
included as they are currently considered an extreme phenomenon where extreme
gravity and related extreme attraction conditions occur. The searching results
were limited to: (i) document type: journals; (ii) subject: engineering; (iii)
language: English. The compilation was further refined to remove documents
outside the scope of this review, which as carried out according to the
following rules:

  1. 1.

All the papers in control field obtained by searching the word ”attraction”,
but referring to multiple meanings of the term not related to control science
were removed (e.g. interest, liking, and tempting).

  2. 2.

All the papers which contain the term ”control” but do not refer to field of
control systems (e.g. attraction of ants by pheromones).

  3. 3.

All the papers that refer the terms ”attraction”, ”gravity” and ”black-holes”
but whose controllers were not inspired in the gravity phenomenon (e.g.,
control in zero-gravity, micro-gravity environment).

  4. 4.

All the papers in the third and fourth quartiles, according to the Clarivate
ranking, were removed, as we found they do not provide relevant content.

The search was completed in March 2024. Ninety-one relevant papers were
selected according to these criteria.

###  2.2 Literature Search Strategy

The following data were extracted and analyzed from the selected collection of
papers: (1) inspired control law (concept, architecture, and analytical
formulation); (2) inspired optimization algorithm (concept, type of
optimization, and analytical formulation); (3) differences in the main
concepts and main analytical formulations found in modified or hybrid versions
when compared to the original proposed versions; (4) application of the
proposed methods in the field of control systems; (5) relevant performance
indicators.

###  2.3 Terminology

In the last decades, a large number of different nature-inspired algorithms
and variants (e.g. modifications and hybridization) were proposed to overcome
relevant limitations mainly related to entrapment in local optima, premature
convergence, parameter tuning, and exploration and exploitation imbalance
[16]. Nevertheless, the adopted terminology to describe the different
algorithms has not been widely consensual among researchers. Hence, concerning
the conceptual differences between the original algorithm and its variants,
the following classification was established:

  * 1.

Standard: The algorithm is used in its original formulation without any
changes.

  * 2.

Improved: The algorithm was upgraded aiming to achieve superior performances,
but without affecting the original conceptualization, (no artificial
mechanisms were introduced).

  * 3.

Modified: The algorithm was modified aiming to achieve superior performances,
but in such a way that partially or totally loses the affinity to its original
conceptualization.

  * 4.

Hybrid: Merging of two algorithms aiming to achieve better performances in
comparison with their individual performance. The algorithms must be truly
combined in their formulation, i.e., they must not be formulated as an
individual sequencing.

  * 5.

Cascade: Two algorithms individually sequenced.

  * 6.

Fuzzy: Algorithm that include fuzzy logic in their conceptualization (e.g.,
for fine-tune parameterization).

  * 7.

Chaotic: Algorithm that include chaotic behaviors aiming to improve its
performance.

  * 8.

Adaptive: Algorithm that include time-dependent modifications to the original
algorithm throughout iterations (e.g., parametric modification).

##  3 Attraction inspired optimization algorithms applied to control of
dynamic systems

###  3.1 Gravitational Search algorithm

####  3.1.1 Overview

The GSA was firstly proposed by Rashedi et al. [17], who developed a heuristic
optimization method based on the Newton’s law of gravity from classical
physics. In this optimization algorithm, the search agents are represented by
bodies whose mass depends on their fitness [17]. The optimized solutions are
obtained by body attraction phenomena, since bodies are modelled by larger
masses to produce large attraction forces. Through this mechanism (Fig. 1)
inspired by the gravitational force, the agents converge towards the best
solution, which is represented by the body with the highest mass [17]. The
baseline for the GSA development was Newton’s law of universal gravitation
[18],

$\displaystyle\vec{\mathbf{F}}=-G\frac{M_{1}M_{2}}{r^{2}}\hat{\mathbf{r}},$

where $G$ is the gravitational constant, $M_{1}$ and $M_{2}$ are the bodies
mass that attract each other, and $r$ is the distance between the two bodies.
Although the GSA behaves as an artificial isolated system of masses with
dynamics defined by the laws of gravitation and motion, these laws may be
artificially modified from classic Newton law formulations, such that improved
results can be achieved.

According to the original conceptualization GSA [17], heavier bodies, which
correspond to good solutions, move slowly than the lighter ones, which ensures
the exploitation step of the algorithm. The method require to implement the
GSA as expressed in Fig. 3.

[] []
[]

Figure 1: (a) Forces due to gravitational attraction on a three-body system.
$F_{12}$ is the force that $M_{2}$ applies on $M_{1}$, $F_{13}$ is the force
that $M_{3}$ applies on $M_{1}$, $F_{R1}$ is the resultant force applied on
$M_{1}$ and $a_{1}$ is the acceleration due to $F_{R1}$; (b) Black Hole
structure. The Schwarzschild $(R_{S})$ radius is calculated by
$R_{S}=\frac{2GM}{c^{2}}$ where $M$ is the black hole mass, $G$ is the
gravitational constant and $c$ is the light speed. According to the Black Hole
theories [19], all objects that enter into the event horizon can not escape
due to the massive gravitational attraction force. (c) Illustration of white-
hole, black-hole and wormhole, respectively from left to right. Reproduced
with permission from Ref.[20]

The optimization problem is modelled as a system with $N$ mass-defined agents.
The position of the $i$th agent is defined by

$X_{i}=(x_{i}^{1},\ldots,x_{i}^{d},\ldots,x_{i}^{n})\;\;\;\;\;\textrm{for}\;i=1,2,\ldots,N$
(1)

where $x_{i}^{d}$ is the position of the $i$th agent in the $d$th dimension.
The gravitational constant $G$ at time $t$ is computed by

$G(t)=G_{0}\textup{exp}\left(-\beta\frac{t}{t_{max}}\right)\;,\;\beta<1,$ (2)

where $G_{0}$ is initial value of $G$. The Large Number hypothesis [21], which
was the first hypothesis proposing a time varying gravitational constant,
supported the paradigm stating that physical quantities should acquire
dynamically their current values. Indeed, the GSA was established by defining
the force acting on mass $i$ at time $(t)$ due to the presence of mass $j$ as
follows:

$F_{ij}^{d}(t)=G(t)\frac{M_{i}(t)\;M_{j}(t)}{R_{ij}(t)+\varepsilon}\left(x_{j}^{d}(t)-x_{i}^{d}(t)\right)$
(3)

where $\varepsilon$ is a small constant and $R_{ij}(t)$ is the Euclidean
distance between the two agents $i$ and $j$. The total force acting on agent
$i$ in the dimension $d$ is a randomly weighted sum of $d$th components of the
forces due to other agents:

$F_{i}^{d}(t)=\sum_{j=1,j\neq i}^{N}rand_{j}F_{ij}^{d}(t),$ (4)

where $rand_{j}$ is a random number in the interval $[0,1]$. The position of
the agents at the end of each iteration is calculated by:

$x_{i}^{d}(t+1)=x_{i}^{d}(t)+v_{i}^{d}(t+1),$ (5)
$v_{i}^{d}(t+1)=rand_{i}\;v_{i}^{d}(t)+a_{i}^{d}(t),$ (6)
$a_{i}^{d}(t)=\frac{F_{i}^{d}(t)}{M_{i}(t)},$

where $rand_{i}$ is a random value in the interval $[0,1]$; $M_{i}(t)$ is the
mass of $i$th agent at time $t$, and it is defined by

$M_{i}(t)=\frac{m_{i}(t)}{\sum_{j=1}^{N}m_{j}(t)},$
$m_{i}(t)=\frac{fit_{i}(t)-worst(t)}{best(t)-worst(t)},$

with $fit_{i}(t)$ the fitness value of the $i$th agent at time $t$, which
depends on the defined objective function; $best(t)$ and $worst(t)$ are
defined respectively by

$best(t)=\left\\{\begin{array}[]{cl}\min\limits_{j\in\left\\{1,\dots,N\right\\}}fit_{j}(t)&\textrm{,
if minimizing}\\\
\max\limits_{j\in\left\\{1,\dots,N\right\\}}fit_{j}(t)&\textrm{, if
maximizing}\end{array}\right.$ (7)

and

$worst(t)=\left\\{\begin{array}[]{cl}\max\limits_{j\in\left\\{1,\dots,N\right\\}}fit_{j}(t)&\textrm{,
if minimizing}\\\
\min\limits_{j\in\left\\{1,\dots,N\right\\}}fit_{j}(t)&\textrm{, if
maximizing}\end{array}\right.$ (8)

Functions (7) and (8) are problem-dependent, i.e, minimization problems
require a different formulation from maximization problems.

####  3.1.2 GSA and related variations applied in control

Twenty-seven control applications were found related to use of the standard
version of GSA, and thirty-six related to its variations
(LABEL:table:originalGSA and LABEL:table:variationsGSA). Applications of GSA
in control are mainly focused on optimal tuning of controllers gains,
searching of the best control parameters, and finding the best control
settings of complex systems. The main application field was electric energy
generation (62%), although they were already applied in the control of servo
systems (10%), as well as in applications with multiple constrains and
requiring optimization of multiple parameters, and also in applications in
which control problems are transformed in optimization problems. The PID (Fig.
2a), FLC, and UPFC, whose parameters were optimized by some version of GSA,
represent the majority of the study cases (29%, 16%, and 5%, respectively).

Concerning GSA variants , the most used algorithm in control applications was
the hybrid GSA-PSO, followed by chaotic mechanisms and improved versions of
GSA. Significant advantages have been found by using GSA optimization
algorithms. On the one hand, GSA provides [22, 23, 24]: (i) a good global
exploration capacity (good ability to search for new results); (ii) faster
convergence in comparison to other methods (e.g., PSO, GA ; (iii) high
computational efficiency; and (iv) higher accuracy in comparison to other
methods (e.g., GA, ACO , On the other hand, limitations of GSA are related to
[25, 26, 27]: (a) diversity loss of new solutions in the final search steps;
(b) possibility of getting stuck in local optima; (c) parametrisation of the
algorithm itself is required: their parameters have a significant influence in
the effectiveness of the algorithm. Three modifications to GSA were proposed
so far to improve their effectiveness, by complementing the advantages of
original GSA with the advantages inherent to mechanisms of other searching or
optimization methods: In order to find the optimal controller parameters of a
hydraulic turbine governing system, an increasing $\beta$ value and a
diversity based mutation were proposed [28]. The change performed on $\beta$
affects Eq. (2), allowing to obtain a better control in the balance between
exploration and exploitation. The second mechanism, triggered when the
population diversity is lower than a dynamic threshold, ensures that the
probability of agent mutation increases, such that the trap on local optima
solutions is avoided. On the other two modified variants, changes were
performed on Eq. (6). To adjust the balance between global exploration and
local exploitation, a simple mechanism based on a linear increasing $\gamma$
was introduced in Eq. (6) to divide the equation in two terms. Concerning the
problem of finding optimal UPFC settings, an improvement of 2% was achieved
with less iterations in comparison with original GSA [29]. Lu et al. [30]
suggested a more complex modification to the velocity update equation (6) ,
including the transmission of information between agents to allow that all
agents are updated based on the best ones, and adding memory to ensure that
the best individual position is stored and used to compute (6). This concept
is similar to the one used in PSO [31], despite it is differently formulated.

Some improved methods using non-complex concepts were found to conduct to more
effective results. To find the best thyristor controlled series compensator
location to control a power system, Mahapatra et al. [32] proposed a mechanism
to limit the maximum value of the velocity update (6), with a decreasing
maximum velocity, ensuring that the algorithm exploits the local search space
in the final search phase. A similar approach was tested to optimize the
thresholds and weights of a NN model [33]. Li et al. [34] proposed to perform
a mutation based on Gaussian and Cauchy distributions to enhance the
exploitation and exploration capabilities of GSA, respectively This method was
tested by optimizing the controller gains of a pump turbine governing system,
where the optimization capabilities of the improved GSA were highlighted in
contrast to the Ziegler-Nichols tuning approach (Fig. 2b).

Opposition-based optimization is a technique already tested with many other
optimization algorithms [35, 36, 37, 38, 39, 40]. Opposition optimization was
used with GSA and applied to control systems in order to find the optimal
control parameters of power systems [41, 42]. The main concept of opposition-
based optimization is to check the opposite solution $\breve{x}_{i}$, defined
as $\breve{x}_{i}=L+U-x_{i}$, where $L$ and $U$ are the lower and upper bounds
of the search space, respectively. If the opposite candidate is fitter than
the initial one, the opposite one is saved for the next iteration [43]. Such
optimization was also used with GSA, and applied to control systems to find
the optimal control parameters of power systems [41, 42].

Some processes, such as the GSA tuning, are hard to determine objectively.
However, Fuzzy Logic is a practical method of tuning the GSA parameters as it
can emulate the human reasoning in the use of imprecise information [44, 45,
46]. Aghaie et al. [44] proposed a fuzzy system to set the $\beta$ value in
Eq. (2). Such proposed fuzzy system output new $\beta$ values according to
four inputs: (1) the current iteration; (2) the progression level; (3) the
diversification; and (4) the previous $\beta$ value. The diversification of
population is given by

$div=\frac{r_{ave}-r_{min}}{r_{max}-r_{min}}~{},$

where $r$ is the euclidean distance between two agents and $r_{ave}$,
$r_{max}$ and $r_{min}$, are the average, maximum and minimum distances
between agents, respectively. The level of progression is defined by

$prog=\frac{fit_{ave}(t)-fit_{ave}(t-1)}{fit_{ave}(t)}~{}.$

The proposed set of rules is shown in Table 1.

Rule | Inputs |  |  |  | Output
---|---|---|---|---|---
| Iteration | $prog$ | $div$ | $\beta(t-1)$ | $\beta(t)$
1 | low | low | low | medium | low
2 | medium | low | low | high | medium
3 | high | low | high | medium | high
Table 1: Fuzzy rules used by Aghaie et al. [44] to determine the $\beta$
values.

Adaptation over iterations is other mechanism that has been employed by
researchers to enhance the GSA abilities. Two main conceptualizations were
found using adaptive GSA applied to control: by adapting $G$ and $\varepsilon$
values over time [47, 48], and by performing a mutation with an adaptive
probability, which is determined based on the success rate of the previous
mutations [49]. Applications of deterministic chaos can be observed in control
theory, computer science and physics; recently, chaotic-embedded GSA has also
been investigated as another mechanism to improve the GSA performance [50].
The use of chaotic maps allows to comprise additional layers of randomness to
the algorithm, enhancing the local search capabilities [51, 52]. By including
neural behavior, Vikas and Parhi [53] recently proposed a Modified Chaotic
Neural Oscillator-based Hyperbolic GSA (MHGSA) applied to humanoid robot path
planning. They reported the ability of this adaptive GSA to achieve short
paths in relation to original GSA and avoid obstacles.

The most common hybrid algorithms applied to control is the hybrid GSA-PSO,
due to the high similarities between GSA and PSO algorithms, which allows an
easy merging of their analytical formalization. Two versions were already
proposed: (1) a simplified one only considering the propagation of the best
solution through agents [54, 55, 56, 57, 58, 59, 60, 61, 62, 63]; and (2) a
more complex one that saves the personal best of each agent, adding memory to
the algorithm [64, 65]. Other versions may arise through the combination of
the various mechanisms mentioned above. Included is a chaotic hybrid GSA-PSO
designed to optimize the parameters of a robust controller aiming to solve the
load frequency problem of a micro grid, showing relevant results (improvement
up to $\sim 83\%$), as shown in Fig. 2c.

Figure 2: (a) Control of transient voltage on a hybrid energy system using a PID controller tuned by GSA. Adapted with permission from Ref. [66]. (b) Frequency control of a pump turbine governing system using a PID tuned by Ziegler–Nichols (ZN) method and by the proposed improved GSA (CGGSA). Adapted with permission from Ref. [34]. (c) Comparison of different robust controller settings applied to control micro grid output frequency deviation, where the proposed $H_{2}/H_{inf}$ was optimized by hybrid particle swarm optimization and gravitational search algorithm with chaotic map algorithm (CPSOGSA). The proposed method was faster in retrieve the reference frequency with significantly less overshoot. Adapted with permission from Ref. [67]. (d) Comparison of performance between Real Coded Genetic algorithm (RGA), PSO, GSA, and hybrid Real Coded Genetic - Pattern Search algorithm (RGA-PS). Adapted with permission from Ref. [68]. Table 2: Applications of original GSA in control systems found in literature from 2000 to 2023. NA - Not Applicable Controller |  Application Description | Reference
---|---|---
NA |  Find the optimal settings (e.g. generator terminal voltages, transformer settings, output of compensating devices) for the reactive power dispatch problem that minimize the active power loss and enhance voltage stability of power system. | [69]
UPFC |  Search of optimal gains of UPFC that exhibit greater robustness in the power system control. | [70]
NA |  Define the optimal switching angles of an inverter to minimize the THD. | [68]
State of charge feedback controller |  Optimize the controller parameters to smooth the impact of photovoltaic sources in the power grid. | [71]
PIDF |  Optimize the PIDF gains of an Automatic Generation Control to minimize the generator frequency deviations and the tie-line power error of interconnected power systems. | [72]
UPFC |  Search of optimal gains and location of multiple UPFC that minimize the power loss and the dispatch cost of the power system. | [73]
FLC |  Find the optimal membership functions parameters of FLC. The controller is applied to drive the speed of an induction motor. | [74]
NA |  Parametric optimization of ultrasonic machining processes | [75]
PID |  Tuning of PID gains to control a field-sensed magnetic suspension system | [76]
UPFC |  Find optimal settings of UPFC during the post-fault period | [77]
Type II/ Type III compensators |  Find the optimal gains, zeros and poles location of the compensators to control a DC-DC boost converter | [78]
PI |  Optimize the PI gains of an Automatic Generation Control to minimize the generator frequency deviations and the tie-line power error of interconnected power systems | [79]
Fuzzy PID |  Optimize the controller parameters for Automatic Generation Control of a multi-area multi-source power system | [80]
FLC |  Optimize the rules and membership functions of FLC to control the traffic flow | [81]
NA |  Optimize the switching angles of a reactive power compensator | [82]
Backstepping Control |  Optimize the controller parameters for the trajectory tracking control of autonomous quadrotor helicopter | [83]
NA |  Find the optimal settings to control the electric power generation system | [84]
NA |  Find the optimal electric vehicles controller settings that minimizes the voltage fluctuations and the degradation of batteries | [85]
PID |  Optimize the PID gains to control an inverted pendulum system | [86]
NA |  Find the optimal settings of a congestion management system in a power system under deregulated regime | [87]
FOPID |  Optimize the FOPID parameters to optimal control a micro grid system with various components | [88]
SMC |  Optimize the SMC parameters to control a dual-motor driving system | [89]
PIDF |  Optimize the PIDF gains to control a hybrid power system | [66]
NA |  Find optimal settings of a battery energy storage system | [90]
MPC |  Optimize the MPC parameters to determine online the optimal control sequence. Applied to a quadrotor | [91]
PI |  Determine the optimal parameters of a PI to control the voltage and frequency of a micro grid | [92]
RL-based control |  Find optimal initial weights and biases of the Neural Network controller to avoid instability. The controller was tested in a linear position servo system | [93]
Table 3: Applications of GSA variations in control systems found in literature from 2000 to 2023. NA - Not Applicable Variation |  Modified Equations |  Controller |  Application Description | Reference
---|---|---|---|---
Modified |  In (2) $\beta$ is updated as: $\beta(t)=\gamma\textup{sinh}\left(\eta\left(\frac{t}{t_{max}}-0.5\right)\right)+\lambda$
Perform the mutation: for each $x_{i}^{d}$, $x_{i}^{d}=z_{i}^{d}~{}\textup{if}~{}rand_{1}<P_{c}~{}\textup{where}~{}Z_{i}=X_{r_{1}}+rand_{2}(X_{r_{2}}-X_{r_{3}})$. The mutation occurs if $div<\varepsilon(t)~{}\textup{where}~{}\varepsilon(t)=\varepsilon_{0}\textup{exp}\left(-\mu t/t_{max}\right)$ |  PID |  Optimal parameter identification of a hydraulic turbine governing system | [28]
Substitute (6) by: $v_{i}^{d}(t+1)=\gamma\times gv_{i}^{d}(t+1)+(1-\gamma)\times lv_{i}^{d}(t+1)$ where $gv_{i}^{d}(t+1)=rand_{i}\times gv_{i}^{d}(t)+a_{i}^{d}(t)$ and $lv_{i}^{d}(t+1)=rand_{i}\times lv_{i}^{d}(t)+a_{i}^{d}(t)$ |  UPFC |  Find optimal UPFC settings that minimize the power losses | [29]
Substitute (6) by: $v_{i}^{d}(t+1)=rand_{0}v_{i}^{d}(t)+F\times a_{i}^{d}(t)+pbest_{i}^{d}(t)\left(1-e^{-c_{1}rand_{1}\times t}\right)+gbest_{i}^{d}(t)\left(1-e^{-c_{2}rand_{2}\times t}\right)$ where $F=\frac{1}{\sqrt{a}}e^{-\left(\phi/a\right)^{2}}\textup{cos}\left(5\phi/a\right)$ |  Neural Network controller |  Find neural network optimal parameters. The controller was applied on the integration of offshore wind and wave energy systems | [30]
Improved |  (3) is substituted by:
$F_{ij}^{d}(t)=G(t)\frac{M_{i}(t)\;M_{j}(t)}{R_{ij}(t)+\varepsilon x_{j}^{d}(t)}\left(x_{j}^{d}(t)-x_{i}^{d}(t)\right)$ |  FLC |  Optimize the FLC parameters. The controller was applied on a DC servo system | [94]
Perform a velocity limitation in (6): $-V_{max}\leq v_{i}^{d}\leq V_{max}$ where $V_{max}=V_{max0}\times\left[1-(t/t_{max})^{h}\right]$ with $V_{max0}=\alpha(x_{max}-x_{min}),\alpha\in]0,1]$ |  NA |  Find the optimal thyristor controlled series compensator location in a power system | [32]
Perform the following mutation to $X_{i}$:
$X_{i}^{new}=X_{i}\times(1+\alpha\times(\eta N(0,1)+(1-\eta)C(0,1)))$ where $N(0,1)$ and $C(0,1)$ are random numbers from the Gaussian and Cauchy distributions respectively. Then, a new vector is obtained as $X_{all}=[X_{new}X]$ and only the best $N$ solutions are selected. |  PID |  Optimize the PID gains applied to a pump turbine governing system | [34]
Substitute (6) by: $v_{i}^{d}(t+1)=rand_{i}w(t)v_{i}^{d}(t)+a_{i}^{d}(t)$, where $w(t)=w_{max}-\frac{w_{max}-w_{min}}{t_{max}}\times t$ |  NA |  Optimise the the neural network thresholds and weights. The neural network is used to filter the speed error that is used in the design of the servo system controller | [33]
Opposition based |  With a certain probability named jumping rate, $J_{r}$, after (5) the opposite solutions in relation to the actual population are verified: $Ox_{i}^{d}=\textup{min}^{d}+\textup{max}^{d}-x_{i}^{d}$. Then, the $N$ fittest agents from set $\\{X,OX\\}$ are selected. |  FLC |  Search the optimal control parameters of an autonomous power system. The goal is to enhance the transient response, minimize the overshoot and oscillations, and improve the damping factor | [41]
NA |  Search the optimal control parameters for the problem of optimal reactive power dispatch of power systems | [42]
Fuzzy based |  The $\beta$ parameter in (2) is defined by fuzzy system. |  NA |  Search the optimal design parameters of core patterns for nuclear reactors to solve the loading pattern optimization problem | [44]
The gravitational constant $G(t)$ and the parameter $\varepsilon$ in (3) are adapted using a fuzzy logic mechanism. |  PI |  Search the optimal design parameters of a PI controller for position control a servo system | [45]
Adaptive |  On the first 15% of iterations, (2) is given by: $G(t)=$ $G_{0}\left(\frac{1-\beta t}{t_{max}}\right)$. During next 45% iterations (2) is defined by $G(t)=G_{0}\textup{exp}\left(-\beta\frac{t}{t_{max}}\right)$ and $\varepsilon$ in (3) is given by $\varepsilon(t)=$ $\varepsilon_{0}-\varepsilon_{0}\frac{(t-0.15t_{max})}{0.85t_{max}}$. The remaining 40% of iterations $G(t)$ is set as a constant. |  FLC |  Find optimal parameters of a FLC for position control of a servo system
| [47]
PI |  Find optimal parameters of a PI for position control of a servo system
| [48]
Adaptive |  Two mutation mechanisms are considered: $X_{i,1}=X_{r_{1}}+rand_{1}(X_{r_{2}}-X_{r_{3}})+rand_{2}(X_{r_{4}}-X_{r_{5}})$ and $X_{i,2}=X_{r_{1}}+rand_{1}(X_{best}-X_{worst})$. On each iteration, the probability $P_{a}$ of occurring the mutation $X_{i,a}$ with $a=1,2$ is given by $P_{a}=\frac{\textup{sr}_{a}}{\textup{sr}_{1}+\textup{sr}_{2}}$, where $\textup{sr}_{a}$ is the success rate of the mutation mechanism on past iterations. |  NA |  Applied for optimal reactive power dispatch and voltage control in power system operation | [49]
Chaotic |  After (5), with a given probability, perform the chaotic search: $z_{i}(t)=\frac{x_{\textup{max}i}+x_{\textup{min}i}}{2}+\frac{x_{\textup{max}i}-x_{\textup{min}i}}{2}cx_{i}(t)$, where , $x_{\textup{max}i}=$ $x_{best}+\delta$ , $x_{\textup{min}i}=x_{best}-\delta$ and $cx_{i}(t)$ is a chaotic map $\in[-1,1]\setminus 0$. If $z_{i}(t)$ is fitter than $x_{best}$ then $z_{i}(t)$ is the new solution. |  NA |  Find the optimal parameters of a hydraulic turbine governing system fuzzy model
| [51]
Model-free controller |  The optimal model-free controller was designed according to the quadratic performance index and applied to a vibro-impact system
| [52]
(4) is replaced by: $F_{i}^{d}(t)=\sum_{j=1,j\neq i}^{N}C(t)F_{ij}^{d}(t)$, where $C(t)$ is a normalized chaotic map. |  Robust controller |  Optimal load frequency control settings applied to a micro grid | [67]
Robust FLC |  Find optimal controller parameters with application to the hydraulic turbine governing system | [95]
Chaotic neural oscillators |  (2) is replaced by $G(t)=G_{0}\times Z(t)$, with $Z(t)=0.5(1-L(t))$, where $L(t)=(V(t)-U(t))e^{-kt^{2}}+Y(t)$. Here, $Y(t)=\textup{tansig}(J(t))$. $U(t)$ and $V(t)$ are updated over iterations as following: $U(t+1)=\textup{tansig}\left(p_{1}L(t)+p_{2}U(t)-p_{3}V(t)+p_{4}(J(t)-\phi_{u})\right)$ and $V(t+1)=\textup{tansig}\left(q_{1}L(t)-q_{2}U(t)-q_{3}V(t)+q_{4}(J(t)-\phi_{v})\right)$. |  NA |  Find the optimal path for humanoid robot to avoid dynamic obstacles | [53]
Cascade |  After performing a global search using GSA, the Gradient Descent Method is used to perform a refined local search (see [96] additional details). |  Lead-lag phase compensator |  Find the optimal gains and lead-lag parameters of a power system stabilizer installed on synchronous generator | [97]
After performing a global search using GSA, the PS is used to perform a refined local search (see [98] for additional details). |  PID |  Find optimal automatic generation controller parameters to minimize frequency deviation of a multi-area electric power system | [99]
Hybrid GSA-FA |  (5) is replaced by: $x_{i}^{d}(t+1)=x_{i}^{d}(t)+\alpha(x_{j}-x_{i})+v_{i}^{d}(t+1)$, where $\alpha=\alpha_{0}e^{-\gamma r^{2}}$ |  PI |  Find optimal controller gains for the load frequency control problem of a power system | [100]
Hybrid GSA-GA |  In each main iteration, $K\%$ of population is selected to evolve by using GA [101], and remaining population evolves using GSA. Then, solutions of both methods are combined. This process is repeated until the maximum iteration is achieved. |  UPFC |  Find optimal controller parameters to minimize system oscillations of a power system
| [102]
FLC |  Find optimal controller parameters applied to speed control of a permanent magnet synchronous motor
| [103]
Hybrid GSA-PSO |  (6) is replaced by: $v_{i}^{d}(t+1)=w\times v_{i}^{d}(t)+k_{1}rand_{1}a_{i}^{d}(t)+k_{2}rand_{2}(x_{best}^{d}-x_{i}^{d}(t))$ |  PI |  Find optimal speed controller parameters to minimize the ripple of a switched reluctance motor | [54]
FLC/PID |  Find optimal automatic generation controller parameters to minimize the frequency deviation of a electric power system | [55]
NA |  Find optimal settings of a multi-valve steam turbines system for power generation | [56]
Hybrid GSA-PSO |  (6) is replaced by: $v_{i}^{d}(t+1)=w\times v_{i}^{d}(t)+k_{1}rand_{1}a_{i}^{d}(t)+k_{2}rand_{2}(x_{best}^{d}-x_{i}^{d}(t))$ |  NA |  Find optimal controller parameters of a power system stabilizer, applied to a multi-machine power system | [57]
Neural Network MPC |  Find optimal neural network parameters of a non-linear continuous stirred tank reactor model used in the MPC | [58]
PID |  Find optimal gains of PID to control the interconnection of two area power system | [59]
NA |  Find optimal settings for the optimal reactive power dispatch problem | [60]
NA |  The state estimation of a three-phase unbalanced distribution system is formulated as a nonlinear optimization problem which is solved by the proposed method | [61]
NA |  Applied to state-of-charge optimization (charging control) in the electric vehicles charging | [62]
Neural PID |  Find optimal initial settings of the controller applied as automatic load frequency controller of interconnected hybrid power system | [63]
(6) is replaced by: $v_{i}^{d}(t+1)=w\times v_{i}^{d}(t)+k_{1}rand_{1}(x_{best,i}^{d}-x_{i}^{d}(t))$ $+k_{2}rand_{2}\left(\frac{a_{i}^{d}(t)+x_{best}}{2}-x_{i}^{d}(t)\right)$ |  Fuzzy SMC |  Find optimal controller parameters to control a generator-based wind turbine, ensuring power extraction maximization and regulation of reactive power according to grid requirements | [64]
PID |  Optimal PID for load frequency control of multi-source deregulated power system | [65]

###  3.2 Black-Hole algorithm

####  3.2.1 Overview

A novel heuristic optimization method, motivated by the behavior of stars
around a black hole (Fig. 1), was proposed by Hatamlou et al.[104] in 2013:
the BH. Its search agents are represented by stars; the one with the highest
fitness value is established as the black hole-agent, which attracts all the
others aiming to mimic the behavior of a real black hole. A star-agent is
absorbed when it crosses the so-called Schwarzschild radius, which results in
the remotion of the agent from the search space. To maintain a balance in the
number of agents, a new star-agent is added at a random position of the search
space. Throughout the iterations, if any star-agent becomes fitter than the
black hole-agent, then the role of black hole-agent will be performed in the
next iteration. After a predefined stopping criterion, the optimal solution is
obtained by the the black hole-agent in the last iteration [104]. Even though
the BH is inspired by the behavior of the black hole phenomenon, it uses a
conceptual form not supported by (Newtonian or Relativistic) physical laws
already theorized to describe the dynamics of black holes.

Similarly to other population-based algorithms, the first step consists in
generating an initial population of candidate solutions randomly distributed
over the search space. Due to the small number of equations formulating this
algorithm, its implementation is not complex, even though its high efficiency
has been reported [105]. The BH algorithm is summarized in Fig. 3. The
original algorithm was established as follows. Let us consider a system with
$N$ agents, in which the position of the $i$th agent is defined by Eq. (1).
After the initialization, the fitness values of the agents are evaluated, and
the best candidate is selected as the black hole-agent. As natural black holes
absorb the stars surrounding them, the $i$th agent is dragged towards the
black hole. The positioning of each agent is defined by

$x_{i}(t+1)=x_{i}(t)+rand\;\left(x_{BH}-x_{i}(t)\right),$ (9)

where $rand$ is a random value in the interval $[0,1]$. While mimicking the
motion towards the black hole, if a star-agent becomes fitter than the black
hole-agent in this new position, both this star-agent and the black hole-agent
switch their positions. If during its movement, a star-agent crosses the event
horizon of the black hole-agent, this star-agent “dies” and a new one is
created randomly in the search space, to ensure a constant number of agents.
The radius that defines the event horizon is given by

$R=\frac{f_{BH}}{\sum_{i=1}^{N}f_{i}},$ (10)

where $f_{BH}$ is the fitness value of the black hole and $f_{i}$ is the
fitness value of the $i$th star (agent).

####  3.2.2 BH and variations applied in control

Few applications using the BH and its variations were reported, as summarized
in LABEL:table:BH. The BH was mostly applied in the optimization of parameters
and gains of controllers (the exception was the modified BH). By using the
original version of the BH, the operation strategy (focused on finding the set
point parameters) for a combined cooling, heating, and power system was
optimized, such that the energy consumption, the system cost and the carbon
dioxide emissions can be minimized [106]. Only a comparison with a PSO
algorithm was conducted: while the PSO algorithm achieved optimal results with
an objective function value minimized up to 0.595, the BH obtained was
minimized up to 0.58, which represents the slight improvement of 2.5%. The
original algorithm was also applied to enhance the power quality of an AC
micro grid by searching the optimal PIMR gains that minimize the Total
Harmonic Distortion (THD) [107]. The comparison was also carried out with PSO,
but higher performances were observed, namely 33% improved objective function
value and a faster convergence.

An improved version of the BH was found aiming to improve the motion of agents
by introducing a new concept that prevents the dispersion of solutions:
instead of a randomly generation, the obtained data of existing agents is used
to generate new members(LABEL:table:BH) [108]. This version was able to
achieve good results when used to optimize the parameters of membership
functions of a FLC. By computing the minimization of the carbon emissions,
this method was able to provide improvements of 17% and 14%, over the GA and
PSO, respectively [108]. A similar formulation was used for urban traffic
network control, leading to improvements of 29% compared to an already well-
established approach [109]. This improved version was also applied to optimize
the parameters of a Model-free SMC for a Frequency Load Controller, designed
to regulate a micro grid [110]. The proposed approach provided the best
regulation under load changes.

[] []
[] []

Figure 3: (a) Standard GSA algorithm. (b) Standard BH algorithm. (c) Standard
MOV algorithm. (d) Standard GSO algorithm.

[]
[]

Figure 4: (a) Error of grinding granularity using the proposed soft-sensor
model with different optimization algorithms. Note that the Golden Sine Levy-
flight BH (GSLBH-ELM) achieved better results than without Levy-flight
operator (GSBH-ELM) and than the base algorithm (BH-ELM). Adapted with
permission from Ref. [111]. (b) Displacement of a structure during an
earthquake with and without structure control (MVO-based Optimized FLC).
Reproduced with permission from Ref. [20].

A modified version was proposed to optimize an extreme learning machine soft-
sensor model to predict the grinding granularity [111]. Comparing to the
original BH algorithm, this modified BH was upgraded by applying two well-
known operators to the movement of the agents, namely the Golden Sine operator
[112, 113, 114, 115], and the Levy flight operator [116, 117, 118]. These
operators have already been used to modify other optimization algorithms,
namely BA, SSA, GWO and WOA [113, 115, 119, 120], and its ability to improve
the original BH algorithm has been recently demonstrated. Lower prediction
errors were achieved in comparison to other methods, namely the original BH
and the Golden Sine BH (without the Levy flight operator) (Fig. 4).

Table 4: Applications of BH and variations of it in control systems found in literature from 2000 to 2023. NA - Not Applicable Variation |  Modified Equations |  Controller |  Application Description | Reference
---|---|---|---|---
Standard |  |  |  |
NA |  NA |  Optimization of the set points of a combined cooling, heating, and power-ground source heat pump system. | [106]
NA |  PIMR |  Optimization of the controller gains. The controller was applied to improve the power quality components of an AC microgrid. | [107]
Modified |  |  |  |
After (9) perform the Golden Sine and Levy flight operators:
$x_{i}(t+1)=x_{i}(t)+a\cdot\textup{sign}(rand-0.5)\otimes s$
$x_{i}(t+1)=\left|\textup{sin}(r_{1})\right|x_{i}(t)-r_{2}\textup{sin}(r_{1})\left|m_{1}x_{BH}-m_{2}x_{i}(t)\right|$ where $a$, $m_{1}$ and $m_{2}$ are coefficients, $s$ is the step size vector for Levy flight, $r_{1}$ is a random number in $[0,2\pi]$ and $r_{2}$ is a random number in $[0,\pi]$ |  NA |  Optimization of extreme learning machine (ELM) soft-sensor model on grinding process | [111]
Improved |  |  |  |
(9) is changed for: $x_{i}(t+1)=x_{i}(t)+rand_{1}(x_{BH}-x_{i}(t))$ $+rand_{2}(x_{r}(t)-x_{i}(t))$ where $r\in[1,N]$ (10) is changed for: $R=\frac{\sum_{i=1}^{N}\left\|x_{i}-x_{mean}\right\|}{N}$ New candidate solutions are created by: $x_{new}=x_{BH}+$ $(2rand(1,N)-1)\frac{\textup{max}(\left\|x_{i}-x_{BH}\right\|)}{N}$ |  FLC |  Applied to FLC membership functions parameters optimization. The controller was applied in a multi-objective dynamic optimal power flow framework. | [108]
Model-Free SMC |  Applied to optimize the parameters of Model-Free SMC. The controller is then applied as Frequency Load Controller in a microgrid. | [110]
FLC |  Applied in control of traffic signal scheduling and phase succession to ensure smooth traffic flow with the objective of minimize the waiting time and average queue length. | [109]

###  3.3 Multiverse algorithm

####  3.3.1 Overview

The MVO is a recent population-based optimization algorithm inspired in the
multi-verse theory, focused on the interaction between universes, from which
white-holes, black-holes and wormholes emerge [121]. While white-holes present
similarities with universes under expansion, black-holes attract everything
with their extreme gravitational force. Wormholes are responsible for
connecting different parts of a universe, acting like space-time traveling
tunnels. These three cosmic objects of MVO are illustrated in Fig. 1.

Each solution of the MVO is analogous to a “universe”, and each solution
dimension is an object that can be transmitted through “white-holes”, “black-
holes” and “wormholes”. The objects are transferred from “white-holes” of a
source-universe-solution to “black-holes” of a destination-universe-solution.
Therefore, the population, corresponding to the set of universes-solutions, is
described as

$U=\begin{bmatrix}x_{1}^{1}&x_{1}^{2}&\cdots&x_{1}^{d}\\\
x_{2}^{1}&x_{2}^{2}&\cdots&x_{2}^{d}\\\ \vdots&\vdots&\ddots&\vdots\\\
x_{n}^{1}&x_{n}^{2}&\cdots&x_{n}^{d}\end{bmatrix}$

where $d$ is the dimension of search space, and $n$ is the number of candidate
solutions. For each object $x_{i}^{j}$, which denotes the $j$th variable of
$i$th universe-solution, the following comparison is performed:

$x_{i}^{j}=\left\\{\begin{matrix}x_{k}^{j}&r_{1}<NI(U_{i})\\\
x_{i}^{j}&r_{1}\geq NI(U_{i})\end{matrix}\right.$

where $NI(U_{i})$ is the normalized inflation rate of the $i$th universe,
$r_{1}$ is a random number in $[0,1]$, and $k$ is a universe-solution selected
by a roulette wheel selection mechanism [121]. The inflation rate of a
universe-solution is a value proportional to the fitness of the corresponding
solution. This mechanism performs the exchange of objects between universes-
solutions; in order to provide local changes, wormhole tunnels are established
between a specific universe-solution and the best universe-solution emerged at
time $t$. The formulation of such mechanism is the one that follows:

$x_{i}^{j}=\left\\{\begin{matrix}\left\\{\begin{matrix}X_{j}+\textup{TDR}\cdot(r_{4}\cdot(ub_{j}-lb_{j})+lb_{j})&r_{3}<0.5\\\
X_{j}-\textup{TDR}\cdot(r_{4}\cdot(ub_{j}-lb_{j})+lb_{j})&r_{3}\geq
0.5\end{matrix}\right.&r_{2}<\textup{WEP}\\\
x_{i}^{j}&r_{2}\geq\textup{WEP}\end{matrix}\right.$ (11)

where $X_{j}$ is the $j$th variable of the best solution, TDR (traveling
distance rate) and WEP (wormhole existence probability) are coefficients,
$lb_{j}$ and $ub_{j}$ are respectively the lower and upper bounds of the $j$th
variable, and $r_{2}$, $r_{3}$ and $r_{4}$ are random numbers defined in
$[0,1]$. The MVO algorithm is illustrated in Fig. 3.

####  3.3.2 MVO applied in control

Only this standard MVO was proposed so far, and its application is reduced to
optimization of controller parameters, namely PID-derived and FLC controllers
(Table 5). The first application in control systems of this method dates back
to 2017, which was engineered to search for the optimal parameters of a PID+DD
to operate as a Load Frequency Controller on a power system [122]. Recently,
the same problem was revisited using the same algorithm to optimize a FPDPI.
MVO was also applied in the active structural control of a building (vibration
control) by searching the optimal set of parameters for the membership
functions of a FLC [20], which reduced the vibration of a structure during
tests with Kobe earthquake data, as shown in Fig. 4. The performance of the
MVO was compared with other optimization metaheuristics (namely GA, PSO and
GWO) using test functions. However, in all the aforementioned papers, no
comparisons were found extending to other optimization methods considering
real case problems.

Variation |  Controller |  Application Description | Reference
---|---|---|---
Standard |  PID+DD |  Optimization of controller parameters. The controller was applied as a load frequency controller used to control the flow of steam to the turbines of a generator. | [122]
FLC |  Optimization of the membership functions. The FLC was applied in active control of structures in civil engineering. | [20]
FPDPI |  Optimization of the controller parameters. Controller applied in a multi area power system consisting in hydro, thermal, and gas power plants. | [123]
Table 5: Applications of MOV in control systems found in literature from 2000
to 2023.

###  3.4 Galactic Swarm Optimization algorithm

####  3.4.1 Overview

To enhance the equilibrium between exploration and exploitation, the GSO was
introduced in 2016 by Muthiah-Nakarajan and Noel [124]. This algorithm draws
inspiration from the movement of galaxies and the stars within them. Stars are
not uniformly distributed throughout the cosmos; rather, they cluster into
galaxies, which are not evenly distributed.

While GSO has been conceptualized based on PSO, the authors emphasize that
this choice was made primarily due to the simplicity of implementing PSO. They
assert that GSO could be implemented using any population optimization
heuristic [124]. Hence, the base method is not delineated to maintain
generality.

To implement GSO, $M$ galaxies are created, each containing $N$ different
stars. During each iteration of the algorithm, the core heuristic is executed
for each galaxy to determine the optimal solution within each galaxy. If a
superior solution to the current global one is discovered during this process,
the global solution is updated. At the conclusion of this phase, each galaxy
is represented by its best local solution. In the subsequent phase, the core
heuristic is applied, with the galaxies acting as the search agents, moving
towards the best solution. This iterative process continues until a stopping
criterion is met.

In summary, GSO entails the application of a fundamental algorithm to
ascertain the best local solution for each galaxy, followed by applying the
algorithm at a broader level to determine the best global solution.
Analogously, it can be conceptualized that during the initial phase, stars
converge within each galaxy toward the star with the greatest mass, whereas in
the subsequent phase, galaxies (clusters of stars) converge toward the galaxy
with the greatest mass. The GSO algorithm is illustrated in Fig. 3.

####  3.4.2 GSO applied in control

The utilization of GSO in system control remains relatively limited and under
explored: only five distinct applications of this algorithm were identified.
Among these cases, only one notably study employed the WOA algorithm as its
foundation, while the remaining cases utilized PSO, as outlined in [124].
These applications encompass the optimization of FLC membership functions
[125, 126], the refinement of micro grid parameters [127, 128], and the
determination of optimal control parameters for an IoT network [129]. A
summary of the research findings is presented in Table 6.

Embedded algorithm |  Controller |  Application Description | Reference
---|---|---|---
PSO |  FLC |  Optimization of the membership functions of the controller. The FLC was applied to an autonomous mobile robot for trajectory tracking under noise effects. | [125, 126]
PSO |  FLC |  Optimization of the membership functions. The FLC was applied to control the water level in a water tank. | [130]
PSO |  NA |  Optimization of the settings parameters to provide the optimal power flow of a hybrid energy micro grid. | [127]
PSO |  PI |  Optimization of a Maximum Power Point Tracking controller to control a micro grid composed by photovoltaic panels and wind generators. | [128]
PSO-WOA |  NA |  Finding the optimal settings for IoT network. | [129]
Table 6: Applications of GSO in control systems found in literature from 2000
to 2023. NA - Not Applicable

##  4 Attraction Phenomena Applied to Control of Dynamic Systems

###  4.1 Overview

Attraction is a fundamental phenomenon that governs a wide range of
interactions across the universe. In both physical and engineering senses,
attraction and gravitational potential embody the notion of entities drawn
towards one another, whether in the physical space or within the complex
relationships of intelligent natural entities. In the context of swarm
intelligent systems, attraction plays a significant role in coordination of
social organisms, including ants, bees, and birds, embedding the principles of
collaboration, emergence, and decentralized decision-making. On the cosmos,
gravity is the primary phenomena responsible for the attraction between
bodies, formation of stars and planets, as well as to maintain stable orbits.

Inspired in the attraction phenomena, in the last decade, techniques to
control swarms of intelligent systems have been developed, aiming to control
both the swarm dynamics and its formation [131, 132]. The proposed controllers
were designed using APF functions i.e., functions that mimic specific
potential field. Considering $G(\delta)$ as a APF, it may comprise two
components, the attraction $G_{a}(\cdot)$, and the repulsion $G_{r}(\cdot)$,
and can only converge to a single equilibrium point, occurring at the minimum
potential where $G_{a}(\cdot)=G_{r}(\cdot)$. Such approach defines controllers
laws dependent on a resultant force $f(\cdot)$ due to the potential
$G(\cdot)$: $f(\delta)=-\nabla G(\delta)$. This method was already used for
trajectory path planning, where a APF is inputted in the kinetic models [133,
134, 135, 136, 137, 138], and also as control laws for dynamic models [139,
140].

###  4.2 APF in control

Eleven APF control laws were already proposed, as presented in Table 7. All
the APF were applied in the kinematics field, in particular in robotics, in
which most of them (7 in 11) are focused on the robot swarm control, and some
(4 in 11) are applications related to aircraft trajectory planning.
Considering the attraction behavior, the majority of studies (X in 11) use the
quadratic attractor $G_{a}(\delta)=k\left\|\delta^{2}\right\|$, with
$k=\frac{1}{2}\lambda_{1}$ or a similar function only differing in the
constant parameter $k$. This formulation was most likely defined due to its
simplicity and linearization of the resultant force $f(\delta)=-\nabla
G(\delta)=-\lambda_{1}\delta$. Regarding the repulsive case, the most common
established function was
$G_{r}(\delta)=\frac{1}{2}\lambda_{2}\lambda_{3}\exp\left({-\left\|\delta^{2}\right\|/\lambda_{3}}\right)$.
Even though control dynamics are performed according to attraction and
repulsion algorithms, it is important to emphasize that the proposed
functions/ control laws are not inspired by potentials with physical or
natural significance.

APF Equation |  Type |  Application | Reference
---|---|---|---
$G(\delta)=\frac{1}{2}\lambda_{1}\left\|\delta^{2}\right\|+\frac{\lambda_{2}\lambda_{3}}{2}e^{-\frac{\left\|\delta\right\|^{2}}{\lambda_{3}}}$ |  Attraction and Repulsion |  Satellite cluster formation avoiding self collisions | [133]
$G(\delta)=\frac{1}{2}\lambda_{1}\left\|\delta\right\|^{2}$ |  Attraction |  Swarm mechanical system following a target trajectory | [134]
$G(\delta)=\frac{1}{2}\lambda_{1}\left\|\delta^{2}\right\|+\frac{\lambda_{2}\lambda_{3}}{2}e^{-\frac{\left\|\delta\right\|^{2}}{\lambda_{3}}}$ |  Attraction and Repulsion |  Swarm mechanical system formation avoiding self collisions | [134]
$G(\delta)=\frac{\lambda_{1}}{\left\|\delta\right\|^{2}}-\lambda_{2}\ln{\left(\lambda_{3}-\left\|\delta\right\|^{2}\right)}+\lambda_{4}$ |  Attraction and Repulsion |  Robotic fish leader-follower formation flocking problem | [136]
$G(\delta)=\frac{\lambda_{1}}{2}\left\|\delta_{1}\right\|^{2}+\frac{\lambda_{2}}{2}\left\|\delta_{2}\right\|^{2}+\frac{\lambda_{3}\lambda_{4}}{2}e^{-\frac{\left\|\delta_{3}\right\|^{2}}{\lambda_{3}}}$ |  Attraction and Repulsion |  Path following control of a wireless sensor network avoiding self collision | [140]
$G(\delta)=\frac{1}{2}\lambda_{1}\left\|\delta\right\|^{2}$ |  Attraction |  Quad-rotor path control towards a target | [138]
$G(\delta)=\frac{1}{2}\lambda_{1}\left(\frac{1}{\left\|\delta\right\|}-\frac{1}{\lambda_{2}}\right)^{2}\textup{, if}\left\|\delta\right\|\leq\lambda_{2}$ |  Repulsion |  Quad-rotor obstacles avoidance | [138]
$G(\delta)=\lambda_{1}\left\|\delta^{2}\right\|-\lambda_{2}\ln\left(\left\|\delta\right\|\right)$ |  Attraction and Repulsion |  Swarm formation for mobile odor source localization problem avoiding self collisions | [137]
$G(\delta)=\lambda_{1}\left\|\delta^{2}\right\|-\lambda_{2}\ln\left(\frac{1}{\lambda_{3}}\left\|\delta^{2}\right\|-1\right)$ |  Attraction |  Agent swarm formation shape control | [135]
$G(\delta)=-\frac{\lambda_{1}}{2}\ln\left(\frac{\left\|\delta^{2}\right\|}{\lambda_{2}^{2}}\right)$ |  Repulsion |  Agent swarm self collision avoidance | [135]
$G(\delta)=\frac{\lambda_{1}}{2}\left\|\delta^{2}\right\|+\frac{\lambda_{2}}{2}\left(\frac{1}{\delta}-\frac{1}{\lambda_{3}}\right)^{2}$ |  Attraction and Repulsion |  Movement control of multi-UAV system with leader following and fixed obstacle avoidance | [139]
Table 7: APFs focused on control system applications. $G$ is the potential
function and $\delta$ the distance between agents and targets / obstacles.
$\left\|\cdot\right\|$ is the euclidean norm.

The APF was used to control a satellite cluster using the repulsive potential
function
$G_{r}(\delta)=\frac{1}{2}\lambda_{2}\lambda_{3}\exp\left({-\left\|\delta^{2}\right\|/\lambda_{3}}\right)$,
where $\left\|\delta^{2}\right\|$ is the distance between two agents [133].
Self-collisions are avoided by summing the resulting potentials between all
pairs of agents. A similar approach was used to control a mobile robot swarm
to track moving target [134], as well as in a multiple fish robot system with
a leader [136]. The capability of APF control to guide a swarm along a
designated trajectory and ensure formations with specific shapes is
illustrated in Fig. 5 and Fig. 5. The attractive function
$G_{a}(\delta)=\frac{1}{2}\lambda_{1}\left\|\delta^{2}\right\|$ causes the
agents to attract each other, generating the formation region. The swarm
region is unique as the controlled systems are moved towards an equilibrium
state of minimum potential. Considering the ideal kinetic model, the
application of APF is given by $v(t+1)=v(t)-\nabla G(\delta)$, where $-\nabla
G(\delta)$ defines the acceleration established by the dynamics related to the
kinematic model [134, 136, 137].

[]
[]

Figure 5: (a) Example of swarm trajectory following the leader (green mark) by
using APF control. Adapted with permission from Ref. [137]. (b) Simulation of
swarm following a trajectory $\phi(t)$ (red line) by using APF control. The
black circles are the initial positions of the agents and the dotted lines are
the paths traveled by the agents. Three temporal snapshots of the agents’
states are depicted - magenta: $t=10~{}s$; green: $t=50~{}s$; blue:
$t=80~{}s$. Adapted with permission from Ref. [135].

In such cases, the APF method is used to generate desired trajectory, but a
dynamic controller is needed to effectively track the defined trajectories
ensuring robustness, i.e., a controller with compensating ability in response
to disturbances deviating the controlled system from desired trajectories.
Differently, the APF was designed taken
$G_{a}(\delta)=-\lambda_{2}\ln{\left(\lambda_{3}-\left\|\delta\right\|^{2}\right)}$
and $G_{r}(\delta)=\frac{\lambda_{1}}{\left\|\delta\right\|^{2}}$, where
$\left\|\delta\right\|$ is also the distance between agents [136]. In such
case, a FLC was designed for path tracking defined by the potential function
method. When the APF is inputted to the dynamic model, the gradient of the
potential function is introduced on the control input of the dynamic model, as
follows: $x(k+1)=f(x(k))+u(k);u(k+1)=u(k)-\nabla G(\delta)$ [139, 140], where
$x(k)$ and $u(k)$ are the current state and the control input, respectively.
The APF was also incorporated in the dynamic model, aiming to control mobile
collectors [140]. The proposed function is composed by three components, two
of them implementing attractiveness dynamics, and another one implementing
repulsiveness dynamics. The first attractive component is used to eliminate
the state estimation error, by defining
$\frac{\lambda_{1}}{2}\left\|\delta_{1}\right\|^{2}$, where $\delta_{1}$ is
given by $x(k)-\hat{x}(k)$, and $\hat{x}(k)$ is the estimated state. The
second component was defined as
$\frac{\lambda_{2}}{2}\left\|\delta_{2}\right\|^{2}$, with
$\delta_{2}=x(k)-x_{d}(k)$, where $x_{d}(k)$ is the reference or the desired
system state, such that it is able to move the system under control towards
the desired states. In the repulsive component, $\delta_{3}$ is the distance
between the agents, and it is used to avoid self-collisions. The strategy
based on incorporating the APF in the control law is highly promising to
tackle feedback systems, involving the fundamental concept of attraction of
system state towards the required state according to a natural rationally
found in the real physical universe.

##  5 Discussion

Most studies perform comparisons with other well-established algorithms to
demonstrate their performance. Nonetheless, given the wide range of existing
metaheuristics and optimization methods, conducting a comprehensive comparison
with all of them is hardly feasible, making it difficult to conclusively
determine which is the best algorithm. Based on the analyses here performed,
the GSA exhibited superior performances in 20 (out of 20) control problems
when compared to PSO and GA. Additionally, it outperformed DE and ABC in 6
(out of 6) control problems. These findings, depicted in Fig. 6, suggest the
superiority of GSA over PSO and GA. While other comparisons are presented,
they lack significant expression for a meaningful analysis. Indeed, the BH
demonstrated a performance superiority over PSO in 2 cases (out of 2), and MVO
outperformed DE, PSO, GA, and GWO in 4 cases (out of 4). This discrepancy
highlights the disparity in the number of studies applied to control between
GSA and other methods inspired by gravitational attraction (BH and MVO). Even
so, it is important to recall the No Free Lunch theorem [141], which state
that there is not a single algorithm that outperforms all others across all
metrics and problems: performance is inherently problem-dependent.

The GSA provides several advantages, namely its minimal need for hyper-
parameter adjustment, as it only requires three parameters: $G_{0}$, $\beta$,
and $\varepsilon$. Moreover, it typically reaches rapid convergences even
though its non-complex implementation (see Fig. 2d), and features intuitive
and easily interpretable movements of agents. GSA also facilitates the
incorporation of adaptation mechanisms, as well as the resolution of
constrained problems. However, it can exhibit low agent dispersion during the
final stages, resulting in reduced exploration capabilities. Additionally, GSA
lacks memory, and may become trapped in local minima during the latter
iterations. Other limitations include the blind evolution of the variable $G$,
and the need to predefine the maximum number of iterations, as the evolution
of $G$ is dependent of the duration of the established optimization process.

[] []
[]

Figure 6: (a) The number of studies in the control field wherein the GSA
exhibits a superior optimization capacity compared to other algorithms
documented in the literature. (b) Population distributed structure of
Distributed Multi-Layer Gravitational Search Algorithm. Reproduced with
permission from Ref. [142]. (c) Illustrative population structure of
Hierarchical Multi-Layered Gravitational Search Algorithm. Reproduced with
permission from Ref. [143].

Concerning the BH algorithm, it has been formulated using a straightforward
mechanism, easily understandable and implemented, only requiring the
computation of two equations. Notably, the BH is devoid of hyper-parameters,
ensuring performances independent of user settings. This algorithm effectively
tackles problems with constraints, and maintains a strong exploration capacity
across all iterations. When a solution is eliminated, it is promptly replaced
by a new one randomly generated within the search space, ensuring population
dispersion. Despite its simplicity, its few parameters hardly allow the
inclusion of adaptation mechanisms in its original formulation, as well as the
ability to effectively balance exploration and exploitation. It relies solely
on randomization, potentially leading to extended convergence times.
Furthermore, its formulation lacks is not supported by real physical laws,
which significantly deviates from an authentic alignment with natural inspired
rationality.

The MVO is recognized for its rapid convergence and robust exploration
capabilities, facilitated by solution mutation and crossover. It only requires
one hyper-parameter to determine its exploration ability. However, MVO poses
relevant challenges in interpretation, due to its complex nature and
hypothetical astronomical objects. Moreover, it shares the limitation of
requiring the predefinition of the maximum number of iterations. Additionally,
auxiliary mechanisms need to be incorporated into its algorithm, such as the
roulette wheel selection and sorting procedures, the latter being
computationally heavier due to the sorting of solutions in each iteration.

The novelty of GSO is indeed questionable, given that its architecture bears
resemblance to multi-layered versions of other algorithms. Additionally, its
classification as an optimization algorithm is debatable, as the proposed
concept lacks an inherent optimization mechanism and relies on other
mechanisms. Therefore, GSO can be perceived more as an advancement over
existing algorithms, rather than a novel optimization approach. Furthermore,
while GSO claims inspiration from gravitational attraction between stars and
galaxies, its attraction mechanisms are contingent upon the underlying
algorithm, which may or may not incorporate gravitational principles. This
reliance on a external algorithms limit the argumentation presented by the
authors presenting the GSO as an optimization method. Although the potential
of utilizing GSO to enhance the performance of algorithms like GSA or BH
remains unexplored, such an endeavor holds strong motivation for future
research. These algorithms demonstrate complementary foundational
inspirations, suggesting that exploring their integration could yield
promising results.

To address the mentioned limitations, researchers have been developing
enhancements and modifications, introducing new mechanisms and hybridization,
as highlighted in LABEL:table:variationsGSA and LABEL:table:BH. However,
concerning variants of algorithms applied to control, the most relevant
developments have been focused on the use of GSA. In the case of BH
algorithms, only two variants have been identified, while only the original
MVO algorithm was already applied in control applications. Therefore, it is
mandatory to highlight other variants aiming to implement more sophisticated
mechanisms, even if they remain untested in the field of system control, as
they hold potential to stimulate future excellence research in a
multidisciplinary basis. Relevant studies are detailed in
LABEL:table:noControlAlg. Included are other GSA relevant variants: (1) a
Curved Space GSA, where a dimension reduction technique is applied to the
problem search space [144]; (2) a Memory-based GSA, where the concept of
personal best is introduced [145]; (3) adaptive versions, that introduce novel
mechanisms to guide agents out of local optimum trapping [146], and a personal
gravitational constant [147]; (4) hierarchical and distributed GSA, where the
overall population is divided in subsets of populations (Fig. 6) with multi
layers of hierarchy (Fig. 6) [148, 143, 142, 149]; and (5) Multiple chaos GSA
a mechanism that uses several chaotic maps [150, 151], supported by the
performance superiority of multiple embedded chaos [152]. Regarding BH, other
variants include: (1) an improved BH, which includes a crossover mechanism,
inspired in Genetic algorithms, to generate new agents, avoiding a random
generation [153]; (2) a Chaotic inertia weight BH which improves the local
search by using chaotic maps [154]; and (3) a Multi-population BH which uses
multiple populations of agents, instead of a single one [155]. To enhance MVO,
it was already proposed: (1) Chaotic MVO [156], by introducing the chaotic
behavior in the standard MVO, as already proposed in other algorithms [157,
158, 159] ; and (2) a hybrid Sine-Cosine-MVO algorithm [160, 161].

Given the wide range of metaheuristics and their variants [162, 163], the
question arises: is modification and hybridization the path to higher-
performance algorithms? Thyamianis et al. [162] reported evidence suggesting
that hybridization and the inclusion of additional mechanisms can have
positive effects on nature-inspired algorithms. However, they highlight that
the additional algorithm complexification can also result in not relevant
improvements in exploration and exploitation. Piotrowski et al. [164] also
question the complexity inherent to innovative hybrid algorithms: as
complexity of the numerous modifications to basic algorithms increases, the
risk of discouraging their use also increases. There is supporting evidence
suggesting that simplifying algorithms can enhance transparency and
performance [164].Indeed, modifications to nature-inspired algorithms often
introduce artificial (not-natural inspired) elements into basic algorithms, in
a clear opposition to their primary foundations, which stemmed from their
assumed simplicity of interpretation and use, grounded in natural origins.
Another relevant contradiction identified in the analyzed algorithms is their
formulation without explicit use of real astrophysics laws, even though they
are labeled as nature-inspired algorithms. Notice that BH and MVO algorithms
only incorporate nature-inspired concepts without a rigorous mathematical
foundation of the related phenomena. Concerning the GSA, although it seems to
have the strongest connection to the real astrophysical nature, Gauci et al.
[165] concluded that it cannot be truly inspired by Newton’s laws of gravity,
because the square of the distance is disregarded. Indeed, there is strong
evidence suggesting that the force model formulated so far for the GSA
algorithms does not rely on the distance between agents at all [165]. Thus,
the movement mechanism in the GSA is primarily proportional to the fitness of
the solutions, as the division by the distance between agents mainly serves to
normalize it. Based on this analysis, there is evidence indicating that the
movement of agents in the GSA bears similarities to that of the PSO.

Finally, we must highlight that, the use of APF emerged as the most closely
related approach to leveraging gravitational attraction or the dynamics of
black holes in the development and design of controllers. The results obtained
through the combination of attraction and repulsion functions are intuitive
and straightforward to interpret, rendering this approach particularly
appealing. Nevertheless, the use of dynamics inspired in Newtonian or
Einsteinian gravitation, including as it occurs in black holes, in control
systems has not yet been deeply researched: they have only been an
inspiration-trigger only aiming to design new methods for optimization
problems of well-known control methods (e.g. PID and FLC controllers). To our
knowledge, no control methods were developed whose formalization is directly
and truly inspired by gravitational attraction laws, and related mathematical
Newtonian/ Einsteinian-based formulation.

Table 8: Other recent relevant variants of GSA, BH and MVO, not applied in the control field. Base method |  Designation |  Concept novelty description | Reference
---|---|---|---
GSA |  Curved Space GSA |  Feasible solutions of GSA may be contained in a manifold of lower dimensions, in such a way that, according to the Euclidean distance, the agents may seem close, although they are far apart. The proposed modification calculates the distance between agents along the manifold, instead of directly calculate the Euclidean distance, by utilizing diffusion maps as dimensionality reduction | [144]
|  Memory-based GSA |  This algorithm ensures that the best position of any agent ($pbest$) is stored as the agent’s personal best position, and thus, the new positions of the agents, are always calculated based on the previous best values, such that the path towards the best solution is not lost. In this formulation $R_{ij}(t)$ is computed by: $R_{ij}(t)=\left\|X_{i}(t),pbest_{j}(t)\right\|$ | [145]
|  Adaptive position-guided GSA (disGSA) |  A novel mechanism is proposed to guide agents out of local optimum trapping in the direction of global best solution: $v_{i}^{d}(t+1)=rand\times v_{i}^{d}(t)+c_{1}a_{i}^{d}(t)+c_{2}a_{i2}^{d}(t)$, where $a_{i2}^{d}(t)=\frac{F_{i2}^{d}(t)}{M_{i}(t)}$ and $F_{i2}^{d}(t)=\sum_{j\in Dbest,j\neq i}rand_{j}F_{ij}^{d}(t)$. $Dbest$ is a sorted set of $D_{i}$ defined as $D_{i}(t)=\frac{R_{i,best}(t)}{R_{i,worst}(t)+\varepsilon}$. | [146]
|  Self-adaptive and aggregative learning GSA |  This method proposes an adaptive mechanism wherein each agent possesses its own gravitational constant, defined as: $G_{i}(t)=\left\\{\begin{matrix}G_{i}(t)r_{i}(t)&\textup{if}~{}counter>\theta~{}\textup{and}~{}rand<p\\\ G_{i}(t)&\textup{otherwise}\end{matrix}\right.$, where $r_{i}(t)=\left|\textup{log}\left(\frac{\left|a_{i}(t)\right|}{G_{i}(t)}\right)\right|$. An aggregative mechanism was also included, with (4) being replaced by: $F_{i}^{d}(t)=\sum_{j=1}^{k}Y_{i}(j)$, with $Y_{i}(j)=\frac{\sum_{k=1}^{j}G_{k}(t)}{j}\left\\{\frac{M_{i}(t)}{R_{i,k}+\varepsilon}\sum_{k=1,k\neq j}^{j}r_{k}M_{k}(t)\left[x_{k}^{d}(t)-x_{i}^{d}(t)\right]\right\\}$ | [147]
|  Multi hierarchical layer GSA |  A hierarchical population structure categorizes individuals into layers based on specific criteria, guiding their evolution in a systematically basis. Layers, organized from top to bottom like a tree, influence individuals progressively. This hierarchical arrangement fosters interactive relationships among layers, shaping the evolution of the population. | [148, 143]
|  Distributed and hierarchical GSA |  In addition to the hierarchical layers with varying levels, the population is also distributed into several sub-populations, each one segmented into hierarchical layers. | [142, 149]
|  Multiple chaos GSA |  Multiple chaotic maps into the GSA are incorporated as follows: (1) In each iteration, a new chaotic map is selected randomly; (2) The agents undergo mutation using all the different chaotic maps under consideration. Among the solutions generated, the one with the best fitness value is preserved; (3) The probability of selecting a specific chaotic map in each iteration is dynamically adapted based on its success rate. | [150, 151]
BH |  Improved BH |  With a certain probability, when a new agent is created, the process involves crossing over two existing feasible solutions instead of randomly generating a new one. | [153]
|  Chaotic inertia weight BH |  This algorithm uses chaos theory to enrich the search behavior. A hiper-parameter named inertia weight ($w$) is introduced to control the balance between exploration and exploitation, with $w$ given by: $\text{w}(t)=(\text{w}_{max}-\text{w}_{min})\left(\frac{t_{max}-t}{t_{max}}\right)+\text{w}_{min}C(t)$, where $C(t)$ is a chaotic map. Therefore, (9) is replaced by $x_{i}(t+1)=\text{w}(t)x_{i}(t)+rand\;\left(x_{BH}-x_{i}(t)\right)$ | [154]
|  Multi-Population BH |  This algorithm has the same formulation of the base version but uses multiple populations instead of a single one. At the end, the solution is the best agent of all populations | [155]
MVO |  Chaotic MVO |  This algorithm proposes to replace $r_{4}$ in (11) by a chaotic map, to improve the local search ability of standard MVO | [156]
|  Sine Cosine MVO |  By using a sine cosine mechanism, (11) is replaced by:
$x_{i}^{j}=\left\\{\begin{matrix}\left\\{\begin{matrix}\textup{AP}+\textup{TDR}\cdot\left(\left|2r_{6}X_{j}-x_{i}^{j}\right|\text{sin}(2\pi
r_{5})\right)&r_{3}<0.5\\\
\textup{AP}-\textup{TDR}\cdot\left(\left|2r_{6}X_{j}-x_{i}^{j}\right|\text{cos}(2\pi
r_{5})\right)&r_{3}\geq 0.5\end{matrix}\right.&r_{2}<\textup{WEP}\\\
x_{i}^{j}&r_{2}\geq\textup{WEP}\end{matrix}\right.$, where
$\textup{AP}=\left(X_{j}+x_{i}^{j}\right)/2$
| [160, 161]

##  6 Conclusions

Significant scientific breakthroughs have been carried out in the field of
Control Engineering using universe-inspired algorithms. Two main categories
have been the focus of such advances: optimization algorithms applied in
control problems (GSA, BH, MVO, and GSO), where main improvements were
achieved in the scope of control parameters optimization; and the
identification of a proper control method, inspired by the attraction between
bodies, known as Artificial Potential Fields, which was mainly used to guide
agents to an equilibrium state defined by choosing appropriate attraction and
repulsion functions.

GSA algorithms has been designed according to the gravity law, and the
movement of the agents is due to gravitational forces, which allow the
information transfer between agents, as masses within the gravitational system
are affected by one another. Most results obtained by GSA were able to provide
superior results in comparison with GA and PSO. Concerning the BH algorithms,
two significant advantages were identified: (i) its structure is not complex,
and its implementation is easily performed; (ii) it does not raise parameter
tuning issues. For these reasons, it is considered a feasible option when fast
and accurate results are required within a short period of time. MVO and GSO
are the most recent proposed algorithms, and, for this reason, the number of
studies analyzing their performance and characteristics remains limited.
Applications and modifications carried out to GSA, BH, MVO, and GSO algorithms
have revealed that such concepts can be used in a wide range of control
problems, and can still evolve towards improved performances. APF represents
the closest approach to natural gravitational phenomena by introducing
artificial attraction behaviors. Therefore, future high-sophisticated control
systems inspired by black-hole attraction dynamics can be engineered if they
are further analyzed and considered.

Some difficulties arise when effective conclusions must be stated. On the one
hand, only few studies present the data in a clear way or report meaningful
comparisons; on the other hand, relevant data is lacking in most studies (such
as convergence times). Besides, the comparison between the different
approaches here analyzed is hard to achieve, due to the influence of: (i) the
diversity of methodologies ; (ii) the diversity of the objective functions;
(iii) the parametrisation of the algorithms, as different parameters can
conduct to different results; and (iv) the diversity of the applications and
related scopes. Despite all these problems, the achieved results suggest that
optimization and proper control methods inspired by gravitation and black
holes attraction perform better than other approaches, namely the PSO, while
ensuring easier implementation and interpretation. Nevertheless, these results
highlight the capability of black holes, gravitational attraction, and
universe dynamics in general, to overcome many control engineering problems,
even though they are still limited to the field of optimization and
metaheuristics. Likewise, future studies may explore realist universe-related
dynamics in order to design effective control methods. To date, no control
methods have been truly designed from scientific formulations related to real
astrophysical phenomena. This fact, together with the results achieved in this
study, provides new research directions where highly innovative concepts can
be developed, namely controller ruled by astrophysics-like laws to establish
effective bridges between black hole physics and automatic control. Attraction
may behave as a feedback mechanism of the distance between the considered
masses. Consequently, future controllers can be built upon natural feedback
interactions inspired by the gravitational attraction towards the singularity
of black holes. Supported by analogies with physics, where nothing can escape
from a black hole once the Schwarzschild radius is crossed, new concepts can
established such stable equilibrium points.. These are highly promising future
prospects that overcome the methods here discussed and analyzed, as they do
not employ control approaches based on artificial phenomena, avoiding then to
neglect the rationality and effectiveness inherent of natural systems.

## Data availability statement

No data was used for the research described in the article.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Acknowledgments

R.M.C.B. and M.P.S.S. were supported by the Portuguese Foundation for Science
and Technology (FCT) (PhD grant reference: 2023.01947.BD; project references:
UIDB/00481/2020 and UIDP/00481/2020; DOI 10.54499/UIDB/00481/2020,
https://doi.org/10.54499/UIDB/00481/2020, and DOI 10.54499/UIDP/00481/2020,
https://doi.org/10.54499/UIDP/00481/2020). D.F.M.T. and C.A.R.H were supported
by FCT through project UIDB/04106/2020 (DOI 10.54499/UIDB/04106/2020,
https://doi.org/10.54499/UIDB/04106/2020).

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