Selection and evaluation of FOPID criteria for the X-15 adaptive flight control system (AFCS) via Lyapunov candidates: Optimizing trade-offs and critical values using optimization algorithms

e-Prime - Advances in Electrical Engineering, Electronics and Energy 6 (2023) 100305
Available online 10 October 2023
2772-6711/© 2023 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-
nc-nd/4.0/).
Selection and evaluation of FOPID criteria for the X-15 adaptive flight
control system (AFCS) via Lyapunov candidates: Optimizing trade-offs and
critical values using optimization algorithms
Noorulden Basil a,*
, Hamzah M. Marhoon b
a
Department of Electrical Engineering, College of Engineering, Mustansiriyah University, Baghdad, Iraq
b
Department of Systems Engineering, College of Information Engineering, Al-Nahrain University, Jadriya, Baghdad, Iraq
A R T I C L E I N F O
Keywords:
Adaptive flight control system
FOPID controller criteria
Black hole optimization
Jaya optimization algorithm
Sunflower optimization
A B S T R A C T
Recently, there has been a growing interest among academics worldwide in studying flight control systems. The
advancement of tracking technologies, such as the X-15 adaptive flight control system developed at NASA
(National Aeronautics and Space Administration), has sparked significant exploration efforts by scientists. The
vast availability of aerial resources further contributes to the importance of studying adaptive flight control
systems (AFCS). The successful operation of AFCS relies on effectively managing the three fundamental motions:
pitch, roll, and yaw. Therefore, scientists have been diligently working on developing optimization algorithms
and models to assist AFCS in achieving optimal gains during motion. However, in real-world scenarios, each
motion requires its own set of criteria, which presents several challenges. Firstly, there are multiple criteria
available for selecting appropriate optimization values for each motion. Secondly, the relative importance of
these criteria influences the selection process. Thirdly, there is a trade-off between the performance of the criteria
within a single optimization case and across different cases. Lastly, determining the critical value of the criteria
poses another obstacle. Consequently, evaluating and selecting optimum methods for AFCS trajectory controls
becomes a complex operation. To address the need for optimizing AFCS for various maneuvers, this study
proposes a new selection process. The suggested approach involves utilizing black hole optimization (BHO), Jaya
optimization algorithm (JOA), and sunflower optimization (SFO) as methods for detecting and correcting tra
jectories in adaptive flight control systems. These methods aim to determine the best launch of missiles from the
AFCS based on the coordinate location for both long and short distances. Additionally, the methods determine
the optimal gains for the FOPID (fractional order proportional integral derivative) controller and enhance pro
tection against enemy attacks. The research framework consists of two parts. The first part focuses on improving
the FOPID motion gains by employing optimization algorithms (BHO, JOA, and SFO) that are evaluated based on
the FOPID criteria. Lower significant weighting values of the optimization algorithms demonstrate the best
missile launching in a cosine wave trajectory within AFCS, while higher significant values indicate the best
missile launching in a sine wave trajectory within AFCS. The FOPID controller criteria, including Kp_pitch, Ki_roll,
Kd_yaw, λ_pitch, and µ_yaw, are considered in all situations. Furthermore, the study reports the best weights
obtained for the "Kp_pitch" criterion across the motions as follows: (0.8147, 66.7190, and 54.4716). For the
"Ki_roll" criterion, the best weights are (0.0975, 64.4938, and 64.7311), and for "Kd_yaw" the weights are (0.1576,
35.2811, and 54.3886). The results of the selection process by the BHO, JOA, and SFO algorithms also include
λ_pitch (0.1419, 40.0791, and 72.1047) and µ_yaw (0.6557, 13.5752, and 52.2495). To ensure the validity of the
proposed research framework, a systematic evaluation and precise analysis were conducted.
Introduction
In recent years, there has been a growing interest in adaptive flight
control systems in many countries worldwide. Therefore, it has become
essential to explore the current state of these systems globally and un
derstand the valuable insights that can be gained from this interest [1].
* Corresponding author.
E-mail addresses: noorulden@uomustansiriyah.edu.iq (N. Basil), hamzah.marhoon@nahrainuniv.edu.iq (H.M. Marhoon).
Contents lists available at ScienceDirect
e-Prime - Advances in Electrical
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https://doi.org/10.1016/j.prime.2023.100305
Received 23 July 2023; Received in revised form 12 September 2023; Accepted 5 October 2023

2
This importance is not surprising, considering that the majority of X-15′s
Adaptive Flight Control Systems (X-15′s - AFCSs) consist of sensors,
actuators, control surfaces, control laws, analog computers, and adap
tive control algorithms, were important to achieve real-time adapt
ability and enhance stability and control capabilities in extreme flight
regimes, the X-15 employed a comprehensive system. This system
enabled the X-15 to explore high-speed flight and collect valuable data
on hypersonic dynamics, thereby contributing to advancements in
aerospace technology [2]. However, the flight control system of the X-15
effectively managed the control surfaces of the aircraft, specifically the
elevons (combined elevators and ailerons) and rudders. These control
surfaces were responsible for maneuvering the X-15 and ensuring sta
bility throughout the flight. The system operated by accurately moni
toring real-time measurements and adjusting the positioning of these
surfaces to achieve the desired flight trajectory. X-15′s - AFCSs were
developed between 1950s–1960s at NASA in collaboration with various
other organizations in the United States of America (USA) via Robert
Hoey, Harris M. Drake, and Richard L. Coons and others [3]. AFCSs are
widely found in different aircraft and aerospace projects, such as mili
tary & civilian aircraft, unmanned aerial vehicles (UAVs), and other
aerospace platforms. As a result, the widespread use of adaptive flight
control systems can be observed in countries with significant aerospace
industries and research and development capabilities [4]. AFCSs have
become popular for stability & safety, enhanced control capabilities,
research & development, and handling unconventional configurations
via aerospace researchers, engineers, and manufacturers, leading to its
increasing adoption in modern aircraft and aerospace systems ad
vancements in technology and the increased understanding of the ben
efits they offered in aerospace applications [5]. The integration of AFCS
has had a significant impact on various aerospace applications, led to
increments for high-performance aircraft, UAVs stability, hypersonic
vehicles, spacecraft and satellites, and future urban air mobility in a
wide range of aircraft and space systems, additionally, performance,
safety, and control. The key aspects in technology and aerospace are
safety, efficiency, sustainability, automation & autonomy, connectivity
& data, space exploration, and collaboration & international coopera
tion. In addition, the advancements in aerospace and technologies,
shaping the industry’s future and contributing to safer, more efficient,
and sustainable aviation and space exploration, the incorporation of
AFCSs (Automatic Flight Control Systems) has greatly improved pro
cesses and acted as a significant catalyst for automation in numerous
aerospace applications.
To enhance comprehension of the problem and the existing research
in the relevant field, this article aims to provide an academic re-creation
that incorporates paraphrasing, plagiarism removal, and proofreading.
The primary objective of this article is to present a concise explanation
of the main issues and the proposed solution. The introduction adopts a
question-and-answer format, commencing with the initial inquiry pre
sented below:
Q1: How does the utilization of a Fractional Order Proportional
Integral Derivative (FOPID) Controller enhance the performance of
the X-15′s Adaptive Flight Control Systems, and what is the oper
ational methodology behind its application?
Yaw, pitch, and roll serve as the fundamental basis for the func
tionality of the Adaptive Flight Control Systems (AFCSs) on the X-15
aircraft [6]. The rotations of AFCSs are not only influenced by the basic
motions involved but also by the control axes. It is important to note that
when maneuvering AFCSs, careful consideration must be given to po
tential trade-offs that may impact the trajectories or route estimation of
AFCSs. Different researchers have assigned varying degrees of impor
tance to pitch, yaw, and roll depending on the specific task, whether it
involves rotating around the surface or flying in the sky [7]. Numerous
academic studies have focused on developing approaches and models
related to the motions of automatic flight control systems (AFCSs) due to
their significance [8]. However, recent literature reviews have revealed
that despite the abundance of motion optimization strategies for AFCSs,
other pressing issues require prioritization. These methods primarily
aim to achieve optimal gains for the FOPID/PID controllers. Each type of
AFCS motion has its own distinct set of criteria, knowns: P, I, D, lambda,
and Mu gains [9], making it challenging to generalize the best models
and approaches to utilize. These improvements are critical to the study
of AFCS motion, and as a result, various research works have been
conducted in the literature to enhance them. However, certain obstacles,
such as differences and variations in gains, can have a significant impact,
particularly when considering the exist priority and important level
associated with AFCS trajectories. For instance, the required gains differ
depending on whether the AFCS is being flown at higher altitudes or
closer to the ground. The necessary gains and their relative importance
also change depending on whether the motion involves pitch or yaw,
and consequently, the rotation axis [10].
Moreover, the utilization of AFCSs in flight poses its own unique
challenges, including the significance of gains and the trade-off between
them. In some cases, an inverse relationship is observed, where an in
crease in one gain leads to a reduction in the other. The rotational flight
pathways of AFCSs also present significant difficulties. As a result, the
optimal values of the FOPID controller are sought through optimization,
which currently relies on algorithm-based tuning approaches [11].
Q2: What has optimization theory made to improving PID/FOPID
controllers of AFCS motions?
Firstly, optimization techniques have been utilized to determine the
optimal parameters for PID/FOPID controllers in the context of AFCS.
While traditional manual tuning methods often rely on trial and error,
optimization algorithms offer an automated approach to search for the
most suitable parameter values based on specific performance criteria
[12]. Various techniques, such as genetic algorithms, particle swarm
optimization, and simulated annealing, have been applied to optimize
the gains of PID/FOPID controllers, resulting in improved motion con
trol for AFCS. Secondly, optimization methods have been employed to
design robust PID/FOPID controllers for AFCS, with the aim of ensuring
system stability and performance even in the presence of uncertainties
and disturbances. Techniques like H-infinity control and μ-synthesis,
falling under the umbrella of robust control, can optimize the parame
ters of PID/FOPID controllers to achieve robust stability and perfor
mance in the context of AFCS motion control. Thirdly, optimization has
played a significant role in the development of adaptive PID/FOPID
control strategies for AFCS motions [13]. Adaptive control adjusts the
parameters of the controller in real-time based on the changing dy
namics and operating conditions of the aircraft. By employing optimi
zation algorithms to optimize the adaptation laws, which govern how
the controller parameters are updated, adaptive PID/FOPID controllers
can achieve fast and accurate adaptation, leading to improved motion
control for AFCS. Fourthly, optimization techniques have been
employed to design nonlinear PID/FOPID controllers for AFCS motions.
Nonlinear control methods consider the inherent nonlinearities in
aircraft dynamics and offer more accurate control performance. Opti
mization science enhances the AFCS motion control capabilities in
nonlinear flight regimes by optimizing the nonlinear terms in the
controller design. Fifthly, in recent years, fractional-order control,
including FOPID control, has gained attention due to its ability to cap
ture complex dynamics and exhibit improved performance. Optimiza
tion methods have been utilized to determine the optimal
fractional-order parameters of FOPID controllers for AFCS motions.
N. Basil and H.M. Marhoon

3
These techniques aim to find the fractional-order values that provide
enhanced control performance and stability for AFCS motion control.
Optimization techniques have been extensively applied in the context of
AFCS to determine optimal PID/FOPID controller parameters, design
robust controllers, develop adaptive control strategies, design nonlinear
controllers, and optimize fractional-order parameters. These optimiza
tion methods contribute to improved motion control, stability, and
performance for AFCS in various flight scenarios [14], prompt the
following question:
Q3: How extensively have optimization methods been employed to
improve the flight trajectories of AFCSs, and what scholarly liter
ature discusses this subject?
Various optimization methods have been employed in recent en
deavours to enhance the pitch, roll, and yaw of AFCSs. To achieve this,
we collected data and established specific criteria for different trajectory
types within the optimization algorithms. In Table 1, we present a
comprehensive overview of the pertinent literature by summarizing the
findings from our research.
According to the Table 1, we introduce the optimization methods
and the criteria that employed to improving AFCS motion trajectories.
Numerous optimization methods were presented, involving CA (Control
Algorithm) [15], L1-NNAC algorithm (L1-Neural Network Adaptive
Control Algorithm) [16], MONLTA & GA (Multi-Objective Non-Linear
Threshold accepting Algorithm & Genetic Algorithm) [17], GSA (A
Gain Scheduling Algorithm) [18], ECU (Electronic Control Unit) [19],
FOAC algorithm (A Fractional-Order Actor-Critic algorithm) [20], SO
algorithm (Self-Organizing algorithm) [21]. Table 1 presents the motion
paths variation and gains of PID/FOPID controllers were utilized in
these criteria (i.e., Kp_pitch, Ki_roll, Kd_yaw, Kp_pitch, Ki_roll, Kd_yaw, λ_ pitch, µ_
yaw) repeated in the altitude & attitude controllers. The four criteria,
Kp_pitch, Ki_roll, Kp_pitch, Ki_roll, are used in [15] for improves various con
figurations of Unmanned Aerial Vehicle (UAV) via conduct a compara
tive analysis with focus on three configurations: octocopter (8 rotors),
decacopter (10 rotors), and dodecacopter (12 rotors) control motions.
While, Kp_pitch, Ki_roll, Kd_yaw, Kp_pitch, Ki_roll, Kd_yaw are utilized in [16] for
enhancing the control system for a small tail-sitter UAV during the
transition process that refers to the phase where the UAV switches be
tween vertical take-off and landing mode and horizontal flight mode. In
[17], Kp_pitch, Ki_roll, Kd_yaw, Kp_pitch, Ki_roll, Kd_yaw are utilized to developing
the control system for a quadrotor, which is a type of UAV with quad
rotors are highly complex and nonlinear dynamic systems, making their
control challenging in addition, provides a new approach called the
multi-objective self-adjusting search mechanism to control the motions
of a quadrotor by simultaneously determining the control gains of the
interacting loops. In [18] used Kp_pitch, Ki_roll, Kd_yaw are illustrate the
analysis of control systems designed to maximize the operability limits
for the launch and recovery of a remotely operated vehicle (ROV) from a
small unmanned surface vessel (USV) in marine operations. In addition,
launching and recovering equipment such as ROVs is a critical task that
determines the operational limits of many marine operations. While, in
[19], scholars were used Kp_pitch, Ki_roll, Kd_yaw, Kp_pitch, Ki_roll, Kd_yaw to
enhance the design and operation of UAVs by proposing a robot oper
ating system (ROS) based multi-degree of freedom (DOF) flight test
framework. However, the robustness and integrity of the UAV system
are crucial for its safe and efficient operation, making it essential to have
a reliable framework for development, verification, and validation, in
[20], overall criteria (Kp_pitch, Ki_roll, Kd_yaw, Kp_pitch, Ki_roll, Kd_yaw, λ_ pitch, µ_
yaw) are utilized to proposes an online optimization approach for a
fractional-order PID controller based on a fractional-order actor-critic
algorithm (FOPID-FOAC). In addition, the aim is to enhance the per
formance of nonlinear systems by leveraging the advantages of the
FOPID controller and FOAC approaches. In [21], the criteria Kp_pitch,
Ki_roll, Kd_yaw are utilized to enhance the control of a quadcopter UAV in
an uncertain environment by developing a self-organizing bidirectional
fuzzy brain emotional learning (SO-BFBEL) controller. Thus, the pro
posed controller improves upon the existing BFBEL controller by
generating more accurate fuzzy layers in real-time and eliminating the
reliance on expert knowledge to set these layers. Based on the existing
literature, it is evident that there were variations in the criteria and
optimization methods employed. This highlights the need for additional
research to explore and propose a new valuable solution which able to
effectively make the selection for the appropriate optimization tech
nique and criteria for motion control in AFCSs. Addressing this gap in
the current knowledge, the subsequent question arises: How can we
further investigate and offer a comprehensive analysis of this primary
issue? This study aims to provide a thorough examination and in-depth
analysis of the problem at hand, thereby filling the identified research
gap.
Q4: what are the gap analyses and critical reviews in the current
literature?
It is noteworthy that all eight PID criteria exert a significant influence
on AFCSs systems. Achieving a quick response time necessitates a higher
proportional value in parameter selection, although an excessive value
may lead to instability or oscillation. The integral term allows for rapid
error correction but can result in larger overshoots. Increasing the
parameter size can help mitigate overshoot, but it compromises the
overall response time. While the controller is linear, AFCSs exhibit
inherent non-linearity due to other non-linear elements. As indicated in
Table 1, various optimization methods have been proposed in the
literature for AFCS motion and route control, considering the key
criteria discussed earlier. However, four prominent challenges have
emerged. Firstly, selecting and implementing the most suitable optimi
zation method for AFCS motion routes is complicated by the presence of
multiple criteria. Secondly, determining the relative weight of each
criterion poses a challenge when choosing an optimization algorithm, as
different criteria influence the selection of the method. Thirdly,
balancing the efficiency of various optimization methods with respect to
different criteria presents a trade-off problem. For instance, when
Table 1
The literature review for AFCSs based optimization methods.
Ref. Controller type Optimization methods FOPID/PID criteria
Altitude controller Attitude controller
Kp_pitch Ki_roll Kd_yaw Kp_pitch Ki_roll Kd_yaw λ_pitch μ_yaw
[15] PI controller CA 2.290 2.530 N/A 2.490 2.280 N/A N/A N/A
[16] Cascade-PID controller L1-NNAC algorithm 0.3 0.12 0.22 0.003 0.003 0 N/A N/A
[17] PID controller MONLTA & GA 3.696 0.214 4.430 2.592 0.966 4.994 N/A N/A
[18] PID based LQR controllers GSA 0.8 0.5 0 N/A N/A N/A N/A N/A
[19] PID controller ECU − 5 0 − 5 5 0 5 N/A N/A
[20] FOPID controller FOAC algorithm 48.6804 0 16.8268 50.5725 0 8.4644 0.9 0.8
[21] SO-BFBEL based PID controller SO
algorithm
0 0 0 N/A N/A N/A N/A N/A

4
designing an optimization method based on two criteria (roll and pitch),
the relationship between the two factors may not always be direct.
Negative correlation between the criteria must be considered during the
selection process. Not all criteria can be optimized simultaneously in
real-world scenarios, leading to trade-offs. Lastly, the critical value
problem arises, when the criteria optimal performed cannot be simply a
matter of increase or decreasing freely. Instead, there exists a threshold,
and surpassing or falling short of it does not yield the best performance.
This critical value greatly impacts the flight path of AFCSs, as the
optimal motion control optimization method is defined within a range
that encompasses the highest and lowest values to be achieved.
Considering these challenges, determining the best optimization method
for AFCSs motion control becomes a complex task. This study in
vestigates this issue and seeks to address the subsequent question by
exploring the Black Hole Optimization (BHO), Jaya Optimization Al
gorithm (JOA), and Sunflower Optimization (SFO) approaches.
Q5: What are the Contributions and Novelty of the proposed Study?
This paper introduces a new approach for address the evaluating and
selecting issue of optimal values to trajectory controls in AFCSs (Adap
tive Flight Control Systems). The approach employs the BHO (Black
Hole Optimization), JOA (Jaya Optimization Algorithm), and SFO
(Sunflower Optimization) algorithms. These algorithms are extensions
of optimization theories that aim to find the best decision leading to the
desired goal [22]. By considering the inherent tensions between evalu
ation criteria and the range of possible decisions, this systematic
approach facilitates control-tuning procedures during the planning,
structure, and problem-solving stages of selection [23]. The BHO, JOA,
and SFO algorithms rely on metaheuristics and automated
decision-making techniques. In this study, they were applied using a
Simulink file to determine the significance of evaluation criteria. Other
optimization algorithms commonly found in the research literature
include the Control Algorithm (CA) [24] and the Self-Organizing algo
rithm (SO) [25]. However, there is a theoretical gap in the literature
despite the use of these techniques. To address instability issues in the
flight of the adaptive flight control system on x-15 (AFCS-x-15), the
BHO, JOA, and SFO algorithms were utilized to calculate the importance
weights of criteria for PID/FOPID (Proportional-Integral-Derivative/
Fractional-Order PID) gains selection and evaluation. These algorithms
serve as subjective-based robust selection methods in the Simulink
environment, considering either group PID/FOPID gains or criterion
selection approaches in [26,27], and [28]. Cost-benefit analysis can
effectively address trade-offs, important value determination, and
evaluation criterion trade-offs. In conclusion, this study’s findings and
implications highlight the successful application of the BHO, JOA, and
SFO algorithms in selecting optimal values and preventing instability
issues in AFCSs. The contributions and novelty can be summarized as
follows:
1 The paper aims to determine the suitable parameter values for mo
tion evaluation in AFCS (Adaptive Flight Control Systems) (i.e.,
Kp_pitch, Ki_roll, Kd_yaw, Kp_pitch, Ki_roll, Kd_yaw, λ_ pitch, µ_ yaw). To achieve
this, the Black Hole Optimization (BHO), Jaya Optimization Algo
rithm (JOA), and Sunflower Optimization (SFO) are employed,
considering various FOPID controller gain criteria.
2 The focus of this study is to optimize the firing of missiles on board
the X-15 adaptive flight control system, considering accuracy and the
selection of appropriate coordinates at different angles (e.g., cos, tan,
and sin). Instead of blindly firing missiles without evaluating the
proper coordinates to detect enemy targets, the BHO, JOA, and SFO
algorithms are utilized. These algorithms subjectively evaluate the
selection of FOPID controller gains criteria, considering a desired
trajectory and motion estimate to identify the most suitable paths.
3 The proposed BHO, JOA, and SFO algorithms are assessed and
analyzed through the utilization of FOPID gains. The evaluation in
cludes three cases with varying parameters to determine proper
paths, motions, and trajectories in the sky, starting from the initial
phase up to interacting with the targeted object. These assessments
verify the effectiveness and performance of the algorithms in
achieving the desired outcomes.
The article is composed of the followings: Section 2 discusses the
AFCSs model kinematics motions and derivations, and X-15 (MH-96)
AFCSs aerodynamical model on MATLAB. In Section 3, the proposed
methodology is presented, explaining the criteria for FOPID gains and
providing an explanation of the optimization algorithms (BHO, JOA,
and SFO). Section 4 focuses on the results and discussion of this paper.
Finally, Section 5 presents the conclusion of the current paper.
AFCSs model kinematics motions and derivations
The differentials translation motion is commonly represented in the
adaptive flight control system (AFCS). The AFCS provides a reference
frame for describing the motion, and it can be expressed in Eqs. (1), and
(2) [29]:
ṙ = V,
V̇ =
FT + FR + FA + Fs + Fe + FD + G
m
, (1)
ṁ = −
( ‖ FT ‖ − Se(Pe − Pa))
Ispg0
, (2)
The position vector is represented by ’r’, the velocity by ’V’, and
gravity by ’G’. The force acting on the vehicle is denoted by ’F’, while
the subscripts T, R, A, s, e, and D correspond to the engine control, AFCS,
aerodynamic, sloshing, elastic, and interference torques, respectively.
The specific impulse of the engine is given by ’Isp’, the cross-sectional
area of the nozzle by ’Se’, the atmospheric pressure in the design state
by ’Pe’, the external atmospheric pressure in-flight by ’Pa’, the mass by
’m’, and the gravitational acceleration at sea level by ’g0′. In AFCS, the
launch point serves as the origin, the x-axis points in the launch direc
tion within the horizontal plane of the launch site, and the y-axis points
towards the sky along the line connecting the center of the Earth to the
launch point. The z-axis follows the right-hand rule. The aerodynamic
force is dependent on the launcher’s shape and the dynamic pressure
experienced during flight and can be mathematically represented as Eq.
(3) [30]:
FA = qSACA (3)
The dynamic pressure is denoted by ’q’, the reference area by ’SA’,
and the aerodynamic coefficient by ’CA’. The aerodynamic coefficient
’CA’ is influenced by various factors such as altitude, Mach number,
angle of attack, and sideslip angle as defined in Theorem 1 that shown as
follows:
Theorem 1. by defining:
Θ(k) = [wv,p
(2)
(k), wi,p
(2)
(k), wp,j
(2)
(k)]T
and the operators
(αi, i = 1 : 4), the proposed aerodynamic rules can be represented as in
Eqs. (4)–(6) [31]:
wv,p
(2)
(k + 1) = − k1δTD(k)

5
⎡
⎢
⎢
⎢
⎣
1
1 + K2‖ ∂δTD(k)
∂wv,p
(2)
(k)
‖2 + K3‖ ∂r(k)
∂wv,p
(2)
(k)
‖2
⎤
⎥
⎥
⎥
⎦
×
μp(k)
μp(k)
α1
Γ(2 − α1)
−
∑
k
q=1
c(α2)
q wv,p
(2)
(k + 1 − q) (4)
wi,p
(2)
(k + 1) = − k1δTD(k)
⎡
⎢
⎢
⎢
⎣
1
1 + K2‖ ∂δTD(k)
∂wi,p
(2)
(k)
‖2 + K3‖ ∂r(k)
∂wi,p
(2)
(k)
‖2
⎤
⎥
⎥
⎥
⎦
×
(
Ki(k) − Ki(k)
σv(k)
)
μp(k)
μp(k)
α1
Γ(2 − α1)
−
∑
k
q=1
c(α3)
q wi,p
(2)
(k + 1 − q) (5)
wp,j
(1)
(k + 1) = − k1δTD(k)
⎡
⎢
⎢
⎢
⎣
1
1 + K2‖ ∂δTD(k)
∂wp,j
(1)
(k)
‖2 + K3‖ ∂r(k)
∂wp,j
(1)
(k)
‖2
⎤
⎥
⎥
⎥
⎦
× wv,p
(2)
(k)
(
1 − α1
μp(k)
α1
Γ(2 − α1)
)
xj(k) −
∑
k
q=1
c(α4)
q wp,j
(1)
(k + 1 − q) (6)
When K1, K2 and K3 were the aerodynamical parameters.
As a Proof: the procedure for adapting the Θ(k) parameters via
minimize the Q (Θ(k)) criterion that defines in Eq. (7), and Eq. (8)
Theorem 1. Essentially, needs to solve the given equation as follows
[32]:
∂Q (Θ(k))
∂Θ(k)
= 0 (7)
Thus:
∂Q (Θ(k))
∂Θ(k)
= ΦδTD(k)
∂δTD(k)
∂Θ(k)
+ ψΔδTD(k)
∂δTD(k)
∂Θ(k)
+ ϱΔα
Θ(k)
+ ΩΔr(k)
∂r(k)
∂Θ(k)
= 0 (8)
This gives as in Eq. (9):
Δα
Θ(k) = −
1
ϱ
(
ΦδTD(k)
∂δTD(k)
∂Θ(k)
+ ψΔδTD(k)
∂δTD(k)
∂Θ(k)
+ ΩΔr(k)
∂r(k)
∂Θ(k)
)
(9)
Utilizing 1st definition, the fundamental numerical solution based on
differential equation is shown as the following Eq. (10) [33]:
Θ(k + 1) = Δα
Θ(k) −
∑
k
q=1
c(α)
q Θ(k + 1 − q) (10)
Via substitute from Eq. (9) in Eq. (10), we find Eq. (11):
Θ(k + 1) = −
1
ϱ
(
ΦδTD(k)
∂δTD(k)
∂Θ(k)
+ ψΔδTD(k)
∂δTD(k)
∂Θ(k)
+ ΩΔr(k)
∂r(k)
∂Θ(k)
)
−
∑
k
q=1
c(α)
q Θ(k + 1 − q)
(11)
The difference on error that applied utilizing the expansion of
aerodynamical series as Eq. (12):
δTD(k + 1) = δTD(k) +
(
∂δTD(k)
∂Θ(k)
)T
Δα
Θ(k) (12)
Hence in Eq. (13):
ΔδTD(k) =
(
∂δTD(k)
∂Θ(k)
)T [
−
1
ϱ
(
ΦδTD(k)
∂δTD(k)
∂Θ(k)
+ ψΔδTD(k)
∂δTD(k)
∂Θ(k)
+ ΩΔr(k)
∂r(k)
∂Θ(k)
)]
=
(
∂δTD(k)
∂Θ(k)
)T [
−
1
ϱ
(
ΦδTD(k)
∂δTD(k)
∂Θ(k)
+ ψΔδTD(k)
∂δTD(k)
∂Θ(k)
+ Ω
∂r(k)
∂δTD(k)
ΔδTD(k)
∂r(k)
∂Θ(k)
)]
(13)
Where in Eq. (14) [34]:
∂δTD(k)
∂Θ(k)
=
∂δTD(k)
∂r(k)
∂r(k)
∂Θ(k)
(14)
As ∂δTD(k)
∂r(k)
= − 1, the mentioned above formula can be sets as in Eq.
(15), and Eq. (16) [35]:
ΔδTD(k) = −
1
ϱ
[
ΦδTD(k)‖
∂δTD(k)
∂Θ(k)
‖2
+ ψΔδTD(k)‖
∂δTD(k)
∂Θ(k)
‖2
+ ΩΔδTD(k)‖
∂r(k)
∂Θ(k)
‖2
] (15)
Thus:
ΔδTD(k) =
−
( Φ
ϱ
)
δTD(k)‖ ∂δTD(k)
∂Θ(k)
‖2
1 +
( ψ
ϱ
)
‖ ∂δTD(k)
∂Θ(k)
‖2 +
( Ω
ϱ
)
‖ ∂r(k)
∂Θ(k)
‖2
(16)
Let’s k1 =
( Φ
ϱ
)
, k2 =
( ψ
ϱ
)
and k3 =
( Ω
ϱ
)
, were given as in Eq. (17):
ΔδTD(k) =
− k1δTD(k)‖ ∂δTD(k)
∂Θ(k)
‖2
1 + k2‖ ∂δTD(k)
∂Θ(k)
‖2 + k3‖ ∂r(k)
∂Θ(k)
‖2
(17)
Thus, Eq. (18) can be represented as:
Θ(k + 1) = − k1δTD(k)
∂δTD(k)
∂Θ(k)
− k2ΔδTD(k)
∂δTD(k)
∂Θ(k)
− k3ΔδTD(k)
∂δTD(k)
∂Θ(k)
−
∑
k
q=1
c(α)
q Θ(k + 1 − q)
(18)
The Eqs. (17) and (18), written as the following Eq. (19) [36]:
Θ(k + 1) = − k1δTD(k)
∂δTD(k)
∂Θ(k)
⎡
⎢
⎢
⎢
⎣
1 −
k2‖
∂δTD(k)
∂Θ(k)
‖2
+ k3‖
∂δTD(k)
∂Θ(k)
‖2
1 + k2‖
∂δTD(k)
∂Θ(k)
‖2
+ k3‖
∂δTD(k)
∂Θ(k)
‖2
⎤
⎥
⎥
⎥
⎦
−
∑
k
q=1
c(α)
q Θ(k + 1 − q)
= − k1δTD(k)
⎡
⎢
⎢
⎢
⎣
1 −
1
1 + k2‖
∂δTD(k)
∂Θ(k)
‖2
+ k3‖
∂δTD(k)
∂Θ(k)
‖2
⎤
⎥
⎥
⎥
⎦
∂δTD(k)
∂Θ(k)
−
∑
k
q=1
c(α)
q Θ(k + 1 − q)
(19)
According to the chain rules, the weights adaptations can be ac
quired via the equation as the Eqs. (20)–(24) [37]:

6
wv,p
(2)
(k+1) = − k1δTD(k)
⎡
⎢
⎢
⎢
⎣
1+
1
1+k2‖ ∂δTD(k)
∂wv,p
(2)
(k)
‖2 +k3‖ ∂r(k)
∂wv,p
(2)
(k)
‖2
⎤
⎥
⎥
⎥
⎦
∂δTD(k)
∂wv,p
(2)
(k)
−
∑
k
q=1
c(α2)
q wv,p
(2)
(k+1− q)
(20)
Where:
∂δTD(k)
∂wv,p
(2)
(k)
=
∂δTD(k)
∂V(k)
∂V(k)
∂wv,p
(2)
=
μp(k)
μp(k)
α1
Γ(2 − α1)
(21)
Similar for the others, the weights adaptations can be acquired via
follows [38]:
wi,p
(2)
(k + 1) = − k1δTD(k)
⎡
⎢
⎢
⎢
⎣
1
1 + k2‖ ∂δTD(k)
∂wi,p
(2)
(k)
‖2 + k3‖ ∂r(k)
∂wi,p
(2)
(k)
‖2
⎤
⎥
⎥
⎥
⎦
∂δTD(k)
∂wi,p
(2)
(k)
−
∑
k
q=1
c(α3)
q ∂wi,p
(2)
(k + 1 − q)
(22)
Where:
∂δTD(k)
∂wi,p
(2)
(k)
=
∂δTD(k)
∂V(k)
∂V(k)
∂Ki(k)
∂Ki(k)
∂wi,p
(2)
=
∂V(k)
∂Ki(k)
∂Ki(k)
∂wi,p
(2)
(23)
Where:
∂δTD(k)
∂wi,p
(2)
(k)
=
∂δTD(k)
∂V(k)
∂V(k)
∂ζi(k)
∂ζi(k)
∂μp(k)
∂μp(k)
∂wp,j
(1)
(k)
=∂wv,p
(2)
(k)
(1− α1)
μp(k)
α1
Γ(2− α1)
xj(k)
(24)
The proof of Theorem 1 is completed. Then, the convergence analysis
can be present as a Theorem 2 [39]:
According to the convergence issue, the analysis of the aero
dynamical model has been proposed with Lyapunov theory as in Eqs.
(25)–(27):
0 ≤ κ1 ≤ 1 (25)
κ2 ≥ 1 −
1
ζ1
, ζ1 = ‖
∂δTD(k)
∂Θ(k)
‖2
(26)
κ3 ≥ 1 −
1
ζ2
, ζ2 = ‖
∂r(k)
∂Θ(k)
‖2
(27)
Proof:
Due to the four functions of Lyapunov candidate were suggested. The
first function of Lyapunov candidate is shown as in Eqs. (28)–(31):
L1(k) = 0.5δ2
TD(k) + 0.5Δδ2
TD(k) (28)
For the function of Lyapunov L1(k) > 0, the condition of stability is
satisfied for ΔL1(k) ≤ 0. The Lyapunov function can be written by:
ΔL1(k) = L1(k + 1) − L1(k) = 0.5
(
δ2
TD(k + 1) − δ2
TD(k)
)
(29)
0.5δ2
TD(k +1) term can be described utilizing the series of aerodynamics as
[40]
0.5δ2
TD(k + 1) = 0.5δ2
TD(k) +
(
∂0.5δ2
TD(k)
∂Θ(k)
)T
Δα
Θ(k)
= 0.5δ2
TD(k) + δTD(k)
(
∂δTD(k)
∂Θ(k)
)T
Δα
Θ(k) (30)
And,
0.5Δδ2
TD(k + 1) = 0.5Δδ2
TD(k) +
(
∂0.5Δδ2
TD(k)
∂Θ(k)
)T
ΔΘ(k)
= 0.5Δδ2
TD(k) + ΔδTD(k)
(
∂δTD(k)
∂Θ(k)
)T
Δα
Θ(k) (31)
Via substitute from Eq. (30) and Eq. (31) in Eq. (29) we obtain Eq.
(32):
ΔL1(k) = δTD(k)
(
∂δTD(k)
∂Θ(k)
)T
Δα
Θ(k) + ΔδTD(k)
(
∂δTD(k)
∂Θ(k)
)T
Δα
Θ(k)
= δTD(k)ΔδTD(k) + (ΔδTD(k))2
(32)
Referred to Eq. (20), and Eq. (32) can be represented as in Eq. (33):
The condition of aerodynamical stability ΔL1(k) ≤ 0 is satisfies as in
Eq. (34), Eq. (35), Eq. (36), and Eq. (37) [41]:
(k2 − k1)‖
∂δTD(k)
∂Θ(k)
‖2
+ k3‖
∂r(k)
∂Θ(k)
‖2
+ 1 ≥ 0 (34)
Thus, we have:
k3
‖ ∂r(k)
∂Θ(k)
‖2
‖ ∂δTD(k)
∂Θ(k)
‖2
+
1
‖ ∂δTD(k)
∂Θ(k)
‖2
+ k2 ≥ k1 (35)
Defining Lyapunov function as a second candidate as:
L2(k) = 0.5δ2
TD(k) + 0.5
1
k1
Δδ2
TD(k) (36)
ΔL2(k) is as the following:
ΔL1(k) =
− k1δ2
TD(k)‖
∂δTD(k)
∂Θ(k)
‖2
1 + k2‖
∂δTD(k)
∂Θ(k)
‖2
+ k3‖
∂r(k)
∂Θ(k)
‖2
⎡
⎢
⎢
⎢
⎣
1 −
k1‖
∂δTD(k)
∂Θ(k)
‖2
1 + k2‖
∂δTD(k)
∂Θ(k)
‖2
+ k3‖
∂r(k)
∂Θ(k)
‖2
⎤
⎥
⎥
⎥
⎦
=
− k1δ2
TD(k)‖
∂δTD(k)
∂Θ(k)
‖2
1 + k2‖
∂δTD(k)
∂Θ(k)
‖2
+ k3‖
∂r(k)
∂Θ(k)
‖2
⎡
⎢
⎢
⎢
⎣
(k2 − k1)‖
∂δTD(k)
∂Θ(k)
‖2
+ k3‖
∂r(k)
∂Θ(k)
‖2
+ 1
1 + k2‖
∂δTD(k)
∂Θ(k)
‖2
+ k3‖
∂r(k)
∂Θ(k)
‖2
⎤
⎥
⎥
⎥
⎦
(33)

7
ΔL2(k) = δTD(k)
(
∂δTD(k)
∂Θ(k)
)T
Δα
Θ(k) +
1
k1
ΔδTD(k)
(
∂δTD(k)
∂Θ(k)
)T
Δα
Θ(k)
= δTD(k)ΔδTD(k) +
1
k1
(ΔδTD(k))2
(37)
Substitute from Eq. (20) in Eq. (37), the results will be as the Eq. (38)
and Eq. (39) [42]:
Thus ΔL1(k) ≤ 0 if:
k3
‖ ∂r(k)
∂Θ(k)
‖2
‖ ∂δTD(k)
∂Θ(k)
‖2
+
1
‖ ∂δTD(k)
∂Θ(k)
‖2
+ k2 ≥ 1 (39)
Referring to the Eq. (39) and (36), the condition of the initialize
stability is expressed in Eq. (40):
0 ≤ κ1 ≤ 1 (40)
The Lyapunov function is considered as a third candidate is deter
mined as in Eq. (41):
L3(k) = 0.5δ2
TD(k) + 0.5
1
k1
Δδ2
TD(k) + 0.5
k3
k1
Δδ2
TD(k) (41)
ΔL3(k) is given by the following Eq. (42):
ΔL3(k) = δTD(k)
(
∂δTD(k)
∂Θ(k)
)T
Δα
Θ(k) +
1
k1
ΔδTD(k)
(
∂δTD(k)
∂Θ(k)
)T
Δα
Θ(k)
=
k3
k1
ΔδTD(k)
(
∂δTD(k)
∂Θ(k)
)T
Δα
Θ(k)
= δTD(k) ΔδTD(k) +
1
k1
(
ΔδTD(k))2
+
k3
k1
(
ΔδTD(k))2
(42)
Via substitute from Eq. (20) in Eq. (42), gives as in the Eqs. (43)–(47)
[43]:
Let’s
ζ1 = ‖
∂δTD(k)
∂Θ(k)
‖2
(44)
Thus, the condition on the second stability is written as:
k2 ≥ 1 −
1
ζ1
(45)
The Lyapunov function is determined as a fourth candidate such:
L4(k) = 0.5δ2
TD(k) + 0.5
k2
k1
Δδ2
TD(k) + 0.5
1
k1
Δr2
(k) (46)
ΔL4(k) is expressed as follows [44]:
ΔL4(k) = δTD(k)
(
∂δTD(k)
∂Θ(k)
)T
Δα
Θ(k) +
k2
k1
ΔδTD(k)
(
∂δTD(k)
∂Θ(k)
)T
Δα
Θ(k)
+
1
k1
Δr(k)
(
∂r(k)
∂Θ(k)
)T
Δα
Θ(k)
= δTD(k)
(
∂δTD(k)
∂Θ(k)
)T
Δα
Θ(k) +
k2
k1
ΔδTD(k)
(
∂δTD(k)
∂Θ(k)
)T
Δα
Θ(k)
+
1
k1
(
∂r(k)
∂δTD(k)
)T
ΔδTD(k)
(
∂r(k)
∂Θ(k)
)T
Δα
Θ(k)
= δTD(k)
(
∂δTD(k)
∂Θ(k)
)T
Δα
Θ(k) +
k2
k1
ΔδTD(k)
(
∂δTD(k)
∂Θ(k)
)T
Δα
Θ(k)
+
1
k1
(
∂r(k)
∂δTD(k)
)T
ΔδTD(k)
(
∂r(k)
∂δTD(k)
)T
ΔδTD(k)
= δTD(k) ΔδTD(k) +
k2
k1
(ΔδTD(k))2
+
1
k1
‖
∂r(k)
∂δTD(k)
‖ (ΔδTD(k))22
(47)
Substitute from Eq. (20) in Eq. (47), the equation can be rewritten as
Eq. (48), Eq. (49), and Eq. (50) [45]:
ΔL2(k) =
− k1δ2
TD(k)‖
∂δTD(k)
∂Θ(k)
‖2
1 + k2‖
∂δTD(k)
∂Θ(k)
‖2
+ k3‖
∂r(k)
∂Θ(k)
‖2
⎡
⎢
⎢
⎢
⎣
1 −
‖
∂δTD(k)
∂Θ(k)
‖2
1 + k2‖
∂δTD(k)
∂Θ(k)
‖2
+ k3‖
∂r(k)
∂Θ(k)
‖2
⎤
⎥
⎥
⎥
⎦
=
− k1δ2
TD(k)‖
∂δTD(k)
∂Θ(k)
‖2
1 + k2‖
∂δTD(k)
∂Θ(k)
‖2
+ k3‖
∂r(k)
∂Θ(k)
‖2
⎡
⎢
⎢
⎢
⎣
(k2 − 1)‖
∂δTD(k)
∂Θ(k)
‖2
+ k3
∂r(k)
∂Θ(k)
2
+ 1
1 + k2‖
∂δTD(k)
∂Θ(k)
‖2
+ k3‖
∂r(k)
∂Θ(k)
‖2
⎤
⎥
⎥
⎥
⎦
(38)
ΔL3(k) =
− k1δ2
TD(k)‖
∂δTD(k)
∂Θ(k)
‖2
1 + k2‖
∂δTD(k)
∂Θ(k)
‖2
+ k3‖
∂r(k)
∂Θ(k)
‖2
×
⎡
⎢
⎢
⎢
⎣
1 −
‖
∂δTD(k)
∂Θ(k)
‖2
1 + k2‖
∂δTD(k)
∂Θ(k)
‖2
+ k3‖
∂r(k)
∂Θ(k)
‖2
⎤
⎥
⎥
⎥
⎦
=
− k1δ2
TD(k)‖
∂δTD(k)
∂Θ(k)
‖2
1 + k2‖
∂δTD(k)
∂Θ(k)
‖2
+ k3‖
∂r(k)
∂Θ(k)
‖2
⎡
⎢
⎢
⎢
⎣
k2
∂δTD(k)
∂Θ(k)
2
+
∂r(k)
∂Θ(k)
2
+ 1
1 + k2‖
∂δTD(k)
∂Θ(k)
‖2
+ k3‖
∂r(k)
∂Θ(k)
‖2
⎤
⎥
⎥
⎥
⎦
(43)

8
Let’s
ζ2 = ‖
∂r(k)
∂Θ(k)
‖2
(49)
As a hence, the stability of third condition is written as:
k3 ≥ 1 −
1
ζ2
(50)
The proof of theorem 2 for the aerodynamical model has been
completed.
X-15 (MH-96) AFCSs aerodynamical model on MATLAB
The AFCSs model is consists of multiple elements, the first is known
by I/O open loops were includes six main gains: fixed gain, servo gain,
integral component, actuators, aircraft, and negative feedback, the gains
play a crucial role in determining the system’s response and stability.
They are typically used in feedback control loops to adjust and shape the
system’s behavior. The specific responsibilities of gains in aircraft con
trol modes depend on the mode or control law being employed. In
addition, the responsibilities of gains in aircraft control modes such as:
stability, control authority, response time, robustness, mode transition,
and performance trade-offs [46]. While, In an AFCS closed loops,
various components and concepts have specific responsibilities as:
Fig. 1. The main gains in the open loops on AFCS model.
Fig. 2. The main components in the illustrated loops on AFCS model.
ΔL4(k) =
− k1δ2
TD(k)‖
∂δTD(k)
∂Θ(k)
‖2
1 + k2‖
∂δTD(k)
∂Θ(k)
‖2
+ k3‖
∂r(k)
∂Θ(k)
‖2
×
⎡
⎢
⎢
⎢
⎣
1 −
k2‖
∂δTD(k)
∂Θ(k)
‖2
1 + k2‖
∂δTD(k)
∂Θ(k)
‖2
+ k3‖
∂r(k)
∂Θ(k)
‖2
−
‖
∂r(k)
∂δTD(k)
‖2
‖
∂δTD(k)
∂Θ(k)
‖2
1 + k2‖
∂δTD(k)
∂Θ(k)
‖2
+ k3‖
∂r(k)
∂Θ(k)
‖2
⎤
⎥
⎥
⎥
⎦
=
− k1δ2
TD(k)‖
∂δTD(k)
∂Θ(k)
‖2
1 + k2‖
∂δTD(k)
∂Θ(k)
‖2
+ k3‖
∂r(k)
∂Θ(k)
‖2
⎡
⎢
⎢
⎢
⎣
1 + k3‖
∂r(k)
∂Θ(k)
‖2
− ‖
∂r(k)
∂Θ(k)
‖2
1 + k2‖
∂δTD(k)
∂Θ(k)
‖2
+ k3‖
∂r(k)
∂Θ(k)
‖2
⎤
⎥
⎥
⎥
⎦
(48)

9
aircraft, actuators, low-pass filter, first-order reference model, full-wave
rectifier, band-pass filter, gamma gain, integrals, theta’s integration, and
multiplication, were aimed to respond to control inputs and external
disturbances in a desired manner to achieve stable and efficient flight,
converting control signals into physical actions to control the aircraft’s
motion, allows a low-frequency components to pass through, response
for the control system to track, converts an alternating current (AC)
signal into a direct current (DC) signal, allows only a specific range of
frequencies, called the passband, to pass through while attenuating
frequencies outside the passband, plays a role in adjusting the adapta
tion rate or learning rate of the control system, calculate error signals or
to perform control actions based on accumulated error over time,
Fig. 3. The overall components for entire loops on AFCS model.
Fig. 4. The proposed methodology phases.
Fig. 5. The gains of FOPID controller blocks [49].

10
represents an estimate or parameter related to the aircraft’s dynamics or
control system, and, combining signals or adjusting their magnitudes,
respectively [47]. Finally, the open and closed loops for AFCS are
depicted in Figs. 1–3.
Methodology
The proposed methodology involves two phases, as presented in
Fig. 4. The first phase is enhancement of the FOPID controller depending
on the gain’s criteria calculations. The second phase focuses on utilizing
the Black Hole Optimization, Jaya Optimization Algorithm, and Sun
flower Optimization within AFCSs trajectory, and motions.
Phase 1: enhancement of the FOPID controller
The gains of the FOPID controller are used as an evaluation metric,
and three algorithms are presented to enhance the AFCSs. These algo
rithms utilize mathematical calculations based on the motions.
Criteria of FOPID gains
Fractional order PID (Proportional-Integral-Derivative) controllers,
also known as FOPID controllers, are an extension of the traditional PID
controllers. Unlike the integer order differentiation and integration used
in classical PID controllers, FOPID controllers utilize fractional order
calculus to offer improved performance and greater flexibility in specific
applications. In FOPID controllers, fractional order operators, such as
fractional derivatives and integrals, are employed to implement the
derivative and integral actions. These operators have fractional orders
between 0 and 2, allowing for a wider range of control behaviors
compared to integer order controllers. The fractional order can be non-
integer (e.g., 1.5, 1.7) or even complex, depending on the control re
quirements. A FOPID controller can be represented by the following
transfer function as in the Eq. (51) [48]:
C(s) = Kp +
Ki
sμ
+ Kd⋅sν
(51)
Where:
Kp, Ki, and Kd are the proportional, integral, and derivative gains,
respectively.
s is the complex frequency variable. μ and λ are the fractional order
parameters associated with the integral and derivative actions as
depicted in Fig. 5 [49].
Phase 2: the proposed optimization algorithms
In this part, we will present the procedures and mathematical
equations that were used to implement the three proposed optimization
algorithms for AFCSs (Adaptive Flight Control Systems) in each motion.
The algorithms and mechanisms can be summarized as follows:
Dr. Michel and Pierre de Laplace were the pioneers who introduced
the concepts of the black hole [50]. When a massive star undergoes a
gravitational collapse, it gives rise to a black hole, which exhibits an
incredibly strong gravitational force. While the algorithm associated
with black holes possesses a straightforward structure, easier imple
mentations, and lacks hyper-parameters, it suffers from certain limita
tions that render it does not appropriate to solve intricate and highest
dimensionally issues. The primary concern lies in its deficiency to strike
a balance between exploitation and exploration, consequently leading to
entrapment in locally optimal. Moreover, the algorithm focusses on
primarily revolves around the vicinity for the chosen black-hole,
restricting its capacity for exploring the overall search-space. The stars
within the algorithm dutifully follow a singular trajectory, further
curtailing the extent of exploration. Remarkably, the black hole can
engulf any object that approaches its vicinity, including light itself, as
demonstrated by Eq. (52) [51].
R =
2GM
C2
. (52)
The Black Hole Optimization (BHO) algorithm is the approach that
depends on population which generates solutions on local referred to as
"Nakshatras". During each iteration, the star within the highest massing
is designated such a black hole. These selections processes persist
through the community’s performances are assessed using objective’s
function. The star exhibiting a best performance is selected such a
subsequent black hole evolution, and the calculation is represented by
Eq. (53) [52].
R =
fbh
∑n
k=1fk
, (53)
The distances between the stars inspired by the black hole, denoted
by fbh, that evaluated to determine if it falls within the range of R or less.
If the distance satisfies this criterion, the star is removed from the
population, and a new star is automatically generated. This process
occurs because of the star’s movement, enabling them to enhance their
positions in proximity to the black hole. The calculation for this
improvement, bringing the stars closer to the black hole, is performed
using Eq. (54) [53].
xi(t + 1) = xi(t) + r ∗ (XBH(t) − Xi(t)). (54)
The algorithm of BHO possesses metaheuristics features, achieved
through the integration of all stars within the search space with a
currently black hole. It updated consequently and initiates movement
among stars. During this interstellar motion, the updated black hole may
traverse the boundaries for this happen horizon, resulting in this
removal of the stars by the new populations. The enhanced version for
the black hole algorithm called improved version introduces increased
flexibility in its operations. In the studied area [54], stars employ one of
two strategies to determine how new stars are formed. The current
research focuses on the intersection between two stars, allowing for
adaptive changes only after attracting the stars. Each star in the black
hole algorithm represents a possible solution to the current challenge.
The matrix elements can be either integers, such as bytes, or
floating-point numbers, with each star’s evaluation determined using a
predefined objective function. The stars are ranked, and only the best is
chosen to be a part of the present black hole. The stars in the standard
BHO method update their positions using Eq. (54), and the general
functions of the star alter their values to a random value like that of the
black hole based on Eq. (52). This method allows each asterisk-marked
attribute to have its value modified. Eq. (53) is substituted with Equa
tion (55) in this process.
xi(t + 1) = xi(t) + C ∗ (XBH(t) − Xi(t)), (55)
In the process, C stands in for a random matrix of size 1 ∗ D, where D
is an integer property. When a star is swallowed by a black hole, the
characteristics of the black hole and the devoured star are combined to
form a brand-new star. This rebuilding procedure adds the best star from
the combination to the set. In this case, it is initially used with a chance
of 0.25. Using approach one, there is a 0.75 chance of creating stars
when the probability of reconstruction is [0.75, 0.25]. For each 20 %
improvement made up until the termination condition is satisfied, this
replication chance rises by 20 %.
The Jaya optimization algorithm, proposed via R. V. Rao in 2016, is a
metaheuristic approach based on the collective behavior of individuals
in a society. It offers an alternative to traditional optimization algo
rithms like genetic algorithms and particle swarm optimization. The
term "Jaya" signifies a sense of joy or happiness, reflecting the

11
algorithm’s focus on exploring and exploiting the search space [55]. The
Jaya algorithm operates through a simple process, maintaining a pop
ulation of potential solutions and iteratively updating them based on
their performance to optimize a given objective function. Let’s break
down the essential steps of the algorithm [56].
Initialization:
• Determine the population size (N) and the maximum number of it
erations (MaxIter).
• Start with a population of N randomly generated solutions within the
search space.
• Evaluate the objective function for each solution.
Updating the population:
In each iteration (t = 1 to MaxIter):
Determine the best and worst solutions in the population based on
their objective function values.
Update each solution in the population using the following Eq. (56):
Xi = Xi + rand(value) ∗ (Xbest(t) − abs(Xi)) − rand(value)
∗ (Xworst(t) − abs(Xi)) (56)
when:
Xi represents the ith
solution in the population during iteration t.
Xbest(t) denotes the best solution in the population at iteration t.
Xworst(t) indicates the worst solution in the population at iteration t.
rand(value) generates a random number ranging from 0 to 1.
Handling constraints:
If any solution violates the constraints of the problem, it should be
adjusted or modified to ensure compliance.
Objective function evaluation:
The value of the objective function needs to be calculated for each
updated solution.
Updating the best solution:
Identify the best new solution among the updated population.
Termination condition:
If the maximum number of iterations is reached or a termination
criterion is met, halt the algorithm. Otherwise, proceed to step 2. The
Jaya algorithm places equal emphasis on exploration and exploitation.
Exploration is accomplished by introducing a random term that adds
diversity and enables the algorithm to explore new regions of the search
space. Exploitation is facilitated by utilizing the best and worst solu
tions, which guide the population towards promising areas within the
search space [57]. The Jaya algorithm does not necessitate specific
Fig. 6. AFCSs trajectory results after assigned BHO algorithm.

12
parameter tuning, making its implementation relatively straightfor
ward. However, it might be necessary to handle constraints or make
additional adjustments to tailor it to specific problem domains. The Jaya
optimization algorithm presents a straightforward yet effective
approach to optimization problems. By striking a balance between
exploration and exploitation, it efficiently searches for optimal solutions
within the search space [58].
Nature served as inspiration for the population-based sunflower
optimization (SFO) algorithm. The fundamental concept is to mimic
sunflower position to maximize exposure to solar light [59]. SFO is a
straightforward and effective metaheuristic technique for solving
continuous and discrete optimization issues. It uses a special search
method that lessens the chance of becoming stuck in local optima. The
method, however, necessitates several initial answers and is sensitive to
parameter selections. When faced with complex optimization problems
involving several local optima, SFO may not yield the best results. SFO,
however, exhibits promise as an optimization approach in several con
texts. Sunflowers have a daily cycle, starting the day by adjusting their
direction in accordance with the recorded movement of the sun in the
evening, as indicated in Eq. (57).
si
→ =
X∗
− Xi
‖ X∗ − Xi‖
, i = 1, 2, 3, …., np. (57)
Eq. (58) represents the component of the sunflower in a specific di
rection. It is denoted as:
di = λ ∗ pi(Xi + Xi− 1)∗ ‖ Xi + Xi− 1 ‖, (58)
Fig. 7. AFCSs trajectory results after assigned JOA algorithm.

13
where pi ‖ Xi +Xi− 1 ‖ is the probability of pollination and is a constant
characterizing the inertial velocity of the sunflowers is λ. Nevertheless,
the throng moves modestly to choose a suitable spot for tanning, and the
movements of persons outside of the usual limitations for these steps are
considered in Eq. (59).
dmax =
‖ Xmax − Xmin‖
2 ∗ Npop
, (59)
Xmax and Xmin on Eq. (60) represent the upper and lower bounds,
respectively, while the number of populations represents the total
population of the new plant.
Xi+1
̅
̅→
= Xi
→
+ di ∗ si
→ (60)
The SFO algorithm changes the sun’s position and substitutes a se
nior population for a random group of humans [60].
The constant λ in Eq. (58) represents the likelihood of pollination,
and pi ‖ Xi +Xi− 1 ‖ the inertial velocity of the sunflowers. However,
when there is a crowd, people move around a little to find a spot for sun-
cleaning. These movements, which deviate from the normal limits, are
incorporated into Eq. (59).
Equation (61) is expressed as:
Fig. 8. AFCSs trajectory results after assigned SFO algorithm.
Table 2
The selection and evaluation of FOPID criteria based BHO, JOA, and SFO
algorithms.
Optimization
Algorithms
FOPID Criteria
Kp_pitch Ki_roll Kd_yaw λ_pitch μ_yaw
BHO 0.8147 0.0975 0.1576 0.1419 0.6557
JOA 66.7190 64.4938 35.2811 40.0791 13.5752
SFO 54.4716 64.7311 54.3886 72.1047 52.2495

14
dmax =
‖ Xmax − Xmin‖
2 ∗ Npop
, (61)
In this equation, Xmax and Xmin denote the upper and lower limits,
respectively, while Npop represents the total population of the new plant.
Furthermore, Equation (62) is given by:
Xi+1
̅
̅→
= Xi
→
+ di ∗ si
→ (62)
The SFO algorithm operates by substituting a randomly selected
population of individuals with a mature population of sunflowers, which
then undergoes adjustments in their orientation towards the sun [61].
Results and discussion
This chapter focuses on the methodology and presents the results of
two distinct phases, highlighting the outcomes of the optimization al
gorithms: BHO, JOA, and SFO.
This part focuses on the AFCSs trajectories, specifically in relation to
the optimization algorithms utilized in the assigning of BHO for the
adaptive flight control mode. The results are presented through Fig. 6,
which consists of two parts: Part (A) X-15 pitch dynamics over BHO
algorithm, and Part (B) Nichol’s chart over BHO algorithm. These fig
ures illustrate the motion of the AFCSs when employing the BHO algo
rithm, where the motion is constrained within three margins denoted as
(X, Y) and (Z). These margins describe the AFCS angle with respect to the
enemy position. The dimensions involved in the trajectory limits consist
of approximately five elements, ranging from 0 to 1 based on lower and
upper bounds. The population size is set to around 100, and the iteration
number (N) is set to 1000. Additionally, there are 2 flags, and the par
allel index (N of parallel index) is set to 0. The run coefficient is deter
mined as 10. The X-15 parameters used in this context are as follows: H,
Mach, alpha, q, Ta, Mdelta, Chi a, Omega a, Wn act, chi. Their
respective values are 41,250, 1.4, 3, 405, 1.636, 1 × 14.25, 0.1935,
3.8552, 90, and 0.7.
This part discusses the AFCSs trajectories, especially in relation to the
optimization algorithms utilized in the assigning of JOA for the adaptive
flight control mode. The results are presented through Fig. 7, which
consists of two parts: Part (A) X-15 pitch dynamics over JOA algorithm,
and Part (B) Nichol’s chart over JOA algorithm. These figures illustrate
the motion of the AFCSs when employing the JOA algorithm, where the
motion is constrained within three margins denoted as (X, Y) and (Z).
These margins describe the AFCS angle with respect to the enemy
position. The variables involved in the trajectory limits consist of
approximately five elements, ranging from 0 to 150 based on lower and
upper bounds. The population size is set to around 1000, and the iter
ation number (N) is set to 3000. In the aircraft control system variables,
The X-15 parameters used in this context are as follows: H, Mach, alpha,
q, Ta, Mdelta, Chi a, Omega a, Wn act, chi. Their respective values are
40,000, 1.2, 2, 395, 7.636, 1 × 15.35, 7.1936, 5.8553, 90, 0.8.
This part presents the AFCSs trajectories, while in relation to the
optimization algorithms utilized in the assigning of SFO for the adaptive
flight control mode. The results are presented through Fig. 8, which
consists of two parts: Part (A) X-15 pitch dynamics over SFO algorithm,
and Part (B) Nichol’s chart over SFO algorithm. These figures illustrate
the motion of the AFCSs when employing the SFO algorithm, where the
motion is constrained within three margins denoted as (X, Y) and (Z).
These margins describe the AFCS angle with respect to the enemy po
sition. The dimensions involved in the trajectory limits consist of
approximately five elements, ranging from 0 to 100 based on lower and
upper bounds divided into five elements according to the dimension
size. The population rate is set to around 0.10, the iteration number (N)
is set to 100, and the number of sunflowers is overall five elements.
According to the aircraft flight mode parameters, the X-15 parameters
used in this context are as follows: H, Mach, alpha, q, Ta, Mdelta, Chi a,
Omega a, Wn act, chi. Their respective values are 43,340, 1.5, 4.5, 590,
7.894, 4 × 27.01, 9.1249, 6.7428, 95, 0.5.
Table 2 showcases the BHO, JOA, and SFO algorithms alongside a
concise case study. This case study involves a comparison of gain values
achieved using the Fractional Order Proportional Integral Derivative
(FOPID) controller and the Auto Flight Control System (AFCS) for pitch,
roll, and yaw.
As depicted in Table 2, The optimization algorithms were utilized to
select and evaluate the optimal values for the FOPID controller, specif
ically aiming at achieving accurate missile targeting. The objective was
to identify the critical coordinate location that ensures precise enemy
targeting without any errors. Several variables were considered,
considering potential factors that could trigger defensive measures from
the enemy. Additionally, this evaluation aimed to eliminate any detri
mental elements that could potentially hinder the mission’s success, as
highlighted in the proposed study. A statistical framework was devel
oped to determine the most effective algorithm capable of identifying
the optimal FOPID controller values. This analysis aimed to achieve the
best possible missile trajectory, launch angles, and accurate targeting at
both long and short distances from the target coordinate location as
present in Fig. 9.
The proposed algorithms (BHO, JOA, and SFO) exhibit distinct
behavior based on the probability of reconstruction, which predicts the
optimal values for calculating issues related to FOPID controller gains.
These algorithms rely on a probabilistic approach, with 33.3333 %
probability attributed to lower boundaries representing lower gains.
However, the negative mode of a cosine wave in X-15 AFCSs. The
remaining 66.6667 % probability is allocated to upper gains, reflecting
the positive mode resembling a sine wave in AFCSs. The behavior of the
optimization algorithms is determined by the number of objects
involved and the resulting multiplication function. These factors play a
crucial role in computing problems with negative computing boundary
values, particularly when applying the FOPID criterion. Specifically,
pitch and roll correspond to rotational and theta motions, respectively.
By maximizing the number of populations incorporated, these values
contribute to enhancing the trajectory, as determined by the object
count used in the mathematical computations of the optimization
algorithms.
Fig. 9. Evaluation and selection of FOPID criteria based optimiza
tion algorithms.

15
Conclusion
Adaptive Flight Control Systems (AFCSs) are widely acknowledged
as crucial technologies for aerial operations, as extensively documented
in scholarly literature. The significance of AFCS trajectories has led
many researchers to develop approaches and models related to them.
However, working with AFCS trajectories presents challenges, such as
determining the importance of FOPID gains in each trajectory and
resolving conflicts that may arise between these gains. To address this
issue, this study aims to analyses and compare the optimized values for
three potential AFCS motions: pitch, roll, and yaw. However, the
research encountered several challenges. These include the abundance
of assessment criteria, the need for trade-offs, and the fluctuations in
their values. Additionally, understanding the significance of each FOPID
criterion in different AFCS motions added complexity to the analysis. To
overcome these challenges, the study utilized optimization algorithms
such as BHO, JOA, and SFO to aid in the selection process and determine
the most optimized values for each AFCS motion. While employing
AFCSs for aerial warfare operations offers significant benefits, the study
has two limitations. Firstly, it adopted static selection techniques for
optimization models during AFCS motions, overlooking the dynamic
nature of conducting multiple motions simultaneously. Future research
can address this limitation by considering the dynamic environmental
changes over time and the constant need for motion coordination. This
brings us to the second constraint of the study:
A summary of the research as the following:
• To enhance the production outcomes of AFCS, it may be beneficial to
implement a practical optimization-driven evaluation and selection
technique specific to the global AFCS market. The proposed BHO,
JOA, and SFO methodology for evaluation and selection can assist in
achieving these goals. Comparing the results to the AFCS motions
employed in the selection process, the suggested methodology
demonstrates superior performance.
• The evaluation of the BHO, JOA, and SFO optimization algorithms
was conducted across three main AFCS motions, encompassing
various trajectories and viewing angles. The FOPID development
results indicated a wide range of parameters that influence AFCS
motions, as illustrated in Table 2. This complexity poses challenges
when it comes to selecting optimal values. Therefore, it is advisable
for researchers to avoid relying solely on a single algorithm for AFCS
optimization operations. To address this limitation, the present study
introduces a novel evaluation and selection process that can be uti
lized in such scenarios. By implementing this approach, researchers
can navigate the complexities associated with AFCS optimization
and make informed decisions regarding the choice of algorithms.
• During a flight, Adaptive Flight Control Systems (AFCSs) encounter
various interactions, including drifts, air pressure, stability forces,
and the need to maintain desired margins for specific goals. Many of
these interactions are complex and exhibit unexpected, nonlinear
consequences that cannot be directly incorporated into the motion
equations. As a result, AFCS mobility is affected by these influences,
and researchers must consider the significance of trajectory changes.
This is evident in the distinct weightings assigned to pitch, roll, and
yaw in this study, highlighting the importance of considering the
impact of these factors. This research is the first to address the
weightings of FOPID controller gains (criteria) for the three motion
trajectories, considering the significance level of each criterion. The
BHO, JOA, and SFO algorithms have been employed to determine the
optimal values for each criterion, accounting for their respective
weights.
• The researchers did not thoroughly investigate the inherent nature of
AFCS and its gain requirements for optimal performance in the
evaluation and selection process. Additionally, each criterion carries
a specific level of importance that directly impacts the nature of
AFCS trajectory. It becomes crucial to find a trade-off and identify
critical values, particularly when dealing with angles of motion
aimed at defending military units against the enemy. The criteria can
vary based on specific needs and the desired distance of trajectory.
CRediT authorship contribution statement
Noorulden Basil: Conceptualization, Methodology, Software, Vali
dation, Writing – original draft, Writing – review & editing. Hamzah M.
Marhoon: Conceptualization, Data curation, Investigation, Validation,
Writing – review & editing.
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.
The authors declare the following financial interests/personal re
lationships which may be considered as potential competing interests:
Data availability
Data will be made available on request.
Acknowledgements
The author(s) would like to thank Mustansiriyah University (www.
uomustansiriyah.edu.iq) Baghdad – Iraq for its support in the present
work.
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Asst. Prof. Dr. Noorulden Basil (Member, IEEE) was born in
Baghdad, Iraq November 11, 1993. He had received a Doctor of
Philosophy (Ph.D.) in Electrical and Computer Engineering
(honours), Istanbul Altinbas University, Istanbul, Turkey in
20/05/2022. He had received a Master of Science (M.Sc.) in
Electrical and Computer Engineering, Istanbul Altinbas Uni
versity, Istanbul, Turkey in 2019. He obtained the Bachelor of
Science (B.Sc.) in Computer Techniques Engineering, Al-Esraa
University College, Baghdad, Iraq in 2016. His interested area
towards: Control Systems, Optimization Techniques, Autono
mous Systems, Rocket Models, Flight Control Motions and he
had published more than 35 Research Article and a book and a
Reviewer in 18 Journals, Associate Editor in 9 Journals, and
Editor-in-Chief in 2 Journals and he is currently a reviewer in 18 Journals SCOPUS/ WOS
and a lecturer at Department of Electrical Engineering, College of Engineering, Mus
tansiriyah University, Baghdad, Iraq and the following his google scholar link: https://sch
olar.google.com/citations?user=BAp7IXEAAAAJ&hl=en
Email: dr.noralden@ieee.org
Hamzah M. Marhoon received his B.Sc. degree in Electrical
Engineering (Rank 2) from
College of Engineering Al-Mustansiriyah University,
Baghdad, Iraq in 2016. He obtained his M.Sc. degree in Com
munications Engineering (honours) in 2020 from Al-Ahliyya
Amman University, Hashemite Kingdom of Jordan. He works
as an assistant lecturer at Al-Nahrain University, College of
Information Engineering, Systems Engineering Department,
Baghdad, Iraq. He is doing many reviewing duties in his
research interests that include Antennas, Tunable Antennas,
IoT, Energy Harvesting Antennas, Control Modelling, and
Embedded Systems.
Email: hamzaalazawy33@yahoo.com
ORCID ID: https://orcid.org/0000-0001-5613-6685

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