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Engineering, Technology & Applied Science Research Vol. 13, No. 2, 2023, 10432-10438 10432  
 

www.etasr.com Abdulkader: Controller Design based on Fractional Calculus for AUV Yaw Control 

 

Controller Design based on Fractional Calculus 

for AUV Yaw Control 
 

Rasheed Abdulkader 

Department of Electrical Engineering, Imam Mohammad Ibn Saud Islamic University (IMSIU), Saudi 

Arabia 

rmabdulkader@imamu.edu.sa 

(corresponding author) 
 

Received: 16 January 2023 | Revised: 8 February 2023 and 12 February 2023 | Accepted: 15 February 2023 

 

ABSTRACT 

This research presents a fractional order integral controller strategy, which improves the steering angle 

for Autonomous Underwater Vehicles (AUVs). The AUV mathematical modeling is presented. A 

Fractional Order Proportional Integral (FOPI) control scheme is implemented to ensure the yaw angle 

stability of the AUV steering under system uncertainty. The FOPI controller is validated with 

MATLAB/Simulink and is compared to the conventional Integer Order PI (IOPI) controller to track the 

yaw angle of the structure. The simulation results show that the proposed FOPI controller outperforms the 

IOPI controller and improves the AUV system steering and the overall transient response while ensuring 

the system's stability with and without external disturbances such as underwater current and different 

loading conditions.  

Keywords-Autonomous Underwater Vehicle (AUV); Nelder Mean Simplex (NMS); fractional calculus; FOPI 

I. INTRODUCTION  

Water covers more than 70% of Earth's surface and the 
oceans holds nearly 96.5% of all the Earth’s water. The 
scientific area of exploration of new resources in the ocean is in 
rapid increase. However, investigating under sea water levels 
can have some limitations with manned or human operated 
systems, while the use of Automatous Underwater Vehicles 
(AUV) can provide huge benefits in terms of reducing the risk 
of human lives, exploring deep sea levels, underwater 
surveillance, and cost saving [1]. AUVs are unmanned robots 
that have the capability to move under the water surface 
typically on a pre-defined mission, with satellite networks used 
for communication. In the modeling and design stage, an 
important aspect is the mission requirements and objectives. 
Additional details on the process of designing underwater 
vehicles can be found in [2]. Underwater vehicles can be 
categorized into two major types, Remotely Operated Vehicles 
(ROVs) or manned vehicles, which need a human to function 
and send control instructions in order to operate. The other type 
is the AUV or unnamed vehicles, which can completely 
function independently. Currently, AUV’s are rapidly used in a 
wide range of applications, which include environmental 
monitoring, underwater surveillance, scientific research, anti-
submarine warfare, oceanographic discovery, subsea structure 
inspection, oil and gas natural research exploration, etc. [1]. 

The dynamics of underwater vehicles are well-known to 
contain significantly nonlinear dynamics and are dependent on 
a variant number of the system parameters. These 
nonlinearities can generate system uncertainties, time-varying 

dynamic model and severe effect by external disturbances such 
as unpredicted under water current, waves and environmental 
disturbances [1]. Controlling the AUV system is a challenging 
problem, and high control accuracy is needed to keep the 
system safe and stable when threatened by unpredicted factors. 
In order to handle AUVs' uncertainty and disturbance and to 
enhance their tracking performance, numerous control methods 
have been applied to ensure the stability of AUV systems, 
including the LQR control [3], neural networks [4], fuzzy 
control [5], PI/PID control [6], and Sliding Mode Control 
(SMC) [7]. Research findings on non-integer controllers 
indicate better quality control than Integer Order (IO) 
controllers. Some studies showing the advantages of 
implementing this control technique to stabilize the steering 
system of AUV’s are [8-12]. 

In this paper, a control method is proposed that is 
essentially found in fractional calculus theory, a non-integer or 
fractional order control. At present, fractional order controller 
based on Nelder-Mead Simlex (NMS) algorithm has not been 
utilized for controlling and stabilizing the yaw angle of AUV’s. 
The main contribution of this research is the development of a 
Fractional Order Proportional Integral (FOPI) controller 
scheme applied to enhance the steering stability and yaw angle 
for AUV dynamics with disturbances being present. The 
performance of the AUV structure and transient response is 
improved by designing the coefficients of the FOPI controller 
using the NMS method. The proposed controller is examined 
and its effectiveness to maintain the system stable with 
uncertain conditions such as underwater currents and load 
variation is shown. 



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II. AUV MODELING 

A. AUV Dynamics (Coordinate System) 

To derive a mathematical model and the equations 
representing the AUV structure, analysis of the system 
dynamics and kinematics is demonstrated, as three subsystems 
are considered. Figure 1 shows the two coordinate systems 
describing the movement of the AUV in 6 Degrees Of Freedom 
(DOF). The O-xyz axis is the motion coordinates and is static 
to the underwater vehicle, which is denoted as the body-fixed 
refence system. The movement of the body-rigid system is 
demonstrated to the earth (E) fixed system (�, �, Ψ� [13,14]. 
Thus, a non-integer order PI with feedback control scheme 
would be implemented to the AUV structure to achieve robust 
yaw angle stability with/without disturbances. Table I indicates 
the used AUV parameter values. 

 

 

Fig. 1.  AUV 6-DOF coordinate system. 

B. AUV Kinematics 

The physical demonstration of the AUV structure and the 
equation of motion follow fixed body dynamics. Consequently, 
it is valuable to utilize the physical system components by 
diminishing the number of coefficients needed to control the 
system. Thus, it is the main incentive for the development of 
the vertical description of the equations of motion, which are 
successful for computer processing [13]. The 6-DOF AUV 
equation of movement follows fixed body dynamics.  �� �  
�� ��
� ,  �� �  
� � �
�  is the position vector and �� �  
� � �
�  is the orientation vector of the body and 
earth fixed reference system. The linear and angular velocities 
are specified as �� �  
� � �
� , �� �  
� � �
�  where � �  
�� ��
� . 

The two essential equations to model the AUV dynamical 
system are obtained from Newton's laws of motion. These 
equations can be defined as [13]: �� � ���� �     (1) ��� � ����� � ����� �  ��� � !    (2)  
where ���� is the transformation matrix, ! is the control input 
matrix, � , "��� , ���� , and  ���  symbolize mass, Coriolis 
forces, damping matrix, and the gravitational matrix 
respectively.  

By including in the AUV system non-linear equations of 
motion, the kinematics equation can be expressed as [13]: �� � �#�
"���� � ����� �  ��� � !$ 
   (3) 

TABLE I.  AUV PARAMETERS 

Parameter Name Value Unit � AUV mass 50 kg %&  Cross flow drag -131 kg/m %'  Cross flow drag -0.632 kg.m/rad� +$  Cross flow drag -94 kg.m� /rad +& Cross flow drag -3.2 kg ./�  Additional mass -0.9 kg %$�  Additional mass 1.93 kg.m /rad %&�  Additional mass -36 kg +$�  Additional mass -4.9 kg.m� /rad 01 Moment of inertia 3.45 kg.m� /rad �2 AUV speed 10 m/s 
 

C. AUV Model Decoupling  

Due to the AUV system extreme nonlinear dynamics and 
coupling, a reduced system model is considered and linearized 
for the controller design, to ensure the stability and control of 
the AUV steering system. The vehicle model can be reduced 
and decoupled to study the yaw angle steering behavior. By 
considering the following three states, the sway v, yaw velocity 
r, and the yaw rate �. The steering movement can be acquired 
from the rudders and fins. Also, by disregarding the gravity 
forces, system damping and assuming an equilibrium point, the 
AUV system model can be decoupled as follows: 
3 4 %&� 
�� � �./� 4 3��5 � � %& � � %$ �,     (4) 
01 4 +$� 
�� � �%&� #./� �� � � %$� �2�  

                      �+&� � +$ � � !$     (5) 
where �� � � . The linear AUV system can be expressed in 
state-space as:  �� � 6� � 7� , � � "� � ��   (6) 
where: 

 6 �
⎣⎢
⎢⎡

;<=#;<�
�>?� #=�/@A;B=#;<� 0D<EF#DB� ;B� /@ADBEG#DB� 00 1 0⎦⎥

⎥⎤,    (7) 

7 �  ��� � L 0�EF #DB�0 M,  � �  !$ .   (8) 
From the state-space model in (7)-(8), the transfer function 

of the yaw angle control can be computed as follows: 

 �N� �  2.��PQ RA2.�2STRUA�2.SV RWA�X.XY R         (9) 
where a fractional controller is developed and applied to the 
AUV system dynamics in a feedback formation as shown in 
Figure 3. 



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III. FRACTIONAL ORDER PI CONTROL 

The controller technique implemented in this study is based 
on fractional calculus, a non-integer or fractional order PI 
control. For AUV dynamics structure, it is crucial for the 
control scheme to track the required vehicle position and yaw 
angle in a precise manner to help the system obtain steering 
stability. By introducing fractional integral and differential 
operators, the perception of only IO operators can be stretched 
over a wider range, which results in better accuracy in system 
modeling or control. One critical advantage of a fractional 
system is the memory and hereditary annotation, which can be 
described with fractional order calculus theory. However, for 
classical IO plants the memory aspect and hereditary 
annotation are of a major concern [11]. Fractional calculus 
theory is a general notion to fractional order of differentiator 

and integrator simple operator �Z[\  defined in (10) [10]: 
�Z[\ � ⎩⎨

⎧`[ `a[b            ℜ�d� e  0,1                     ℜ�d� � 0,f �`a�#[       ℜ�d� g  0.Z\
         (10) 

where h and a  are bounds of the process and i  indicates the 
fractional component. i  is assumed real, but it can be an 
imaginary component [10].  

Fractional operators can be applied directly to systems or 
can be approximated as IO operators applied into the system. 
Further, a control system can be controlled by either an IO or 
non-integer order controller, hence the system plant can also 
possibly be derived as a fractional or classical IO system. Four 
distinct combinations can occur to implement a fractional order 
control: fractional order controller with a fractional order plant, 
fractional order plant with an IO controller, IO plant with a 
fractional order controller, and IO plant with an IO controller 
[14]. In this paper a linear fractional order controller scheme is 
developed and the controller would be implemented to an IO 
plant (AUV system).  

Describing the control options by implementing a non-
integer order controller in a graphic representation, Figure 2 
indicates that all IO PID controllers are special cases of the 
non-integer order PID controller. It can be observed in Figure 2 
that the IO PID controller only transforms at four rigid points, 
whereas any point on the graphical plane can be used with a 
fractional order PID controller, i.e. the control order can 
interchange continuously and smoothly throughout the entire 
shaded region, which makes the fractional order PID design 
more flexible more effective with system uncertainties than 
IOPID [17]. Thus, in this study, the value examined for the 
fractional order integral operator j, lies in region [0-1] shown 
in Figure 2, since the system requires a lower order integral 
value to improve the transient response and maintain stability. 

The differential equation of the fractional order k0 l control is 
provided by: 

u�t� � Kp e�t� � rstu  e�t�       (11) 
By applying the Laplace transformation to (11) and with the 

assumption of zero initial conditions, the transfer function can 
be obtained as [16]: 

 v �N� � wx � yzR{           (12) 
 

 
Fig. 2.  Graphical representation of the range values of the FOPID 
controller. 

A block diagram representation of the FOPI controller for a 
closed loop structure is shown in Figure 3, where the plant is 
the AUV system dynamics and G} �s�  is the non-integer 
controller expressed as: 

 v �N� � wx � yzR{ .      (13) 
and j is the non-integer operator. 

The added tunable component j with the FOPI control in 
comparison to the IOPI control generates more flexibility in 
controlling the structure and at the same time enhances the 
control quality implementation and helps minimizing the 
steady-state error and improves the transient response of the 
AUV system. Proper tunning of the FOPI controller is required 
to meet the system specifications. With a FOPI controller, 
reaching closed-loop zero steady-state error and diminishing 
the amplification of high-frequency disturbances can be 
achieved for the AUV system. Moreover, when using the IOPI 

control there would be a lagging phase of 90° for the system, 
while with the FOPI scheme, it delivers a reduced constant 
lagging phase to the system due to the fractional element j 
which improves the transient response of the system dynamics 
as compared to the IOPI control.  

 

 

Fig. 3.  FOPI structure applied for the AUV. 

IV. THE NELDER-MEAD SIMPLEX METHOD 

The simplex algorithm developed by Nelder and Mead in 
1965 has been widely used to solve parameter estimation and 
optimization problems. The Nelder-Mead Simplex (NMS) has 
been applied by the United States army Corps Engineers for 



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www.etasr.com Abdulkader: Controller Design based on Fractional Calculus for AUV Yaw Control 

 

hydrologic engineering center-hydrologic modeling system 
(HEC-HMS) software as a search technique for optimizing the 
hydrologic parameters [18]. In this research, the value of the 
fractional operator j  is tunned using the NMS method. The 
NMS method can solve unconstrained optimization problems 
of the form 3���  ����, � � ℝ' . There are four basic 
operational steps of the NMS algorithm: reflection, expansion, 
contraction, and shrinking as demonstrated in Figure 4 [10, 18].  

The five stages for implementing the NMS algorithm are 
[18]: 

1. Order. Order � � 1 vertices in order to  meet: 
����� � ���� � ⋯ � ���'A��  

2. Reflect. Calculate the reflection point �$ : �$ � �̅ � ���̅ 4 �'A��    (14) 
where: 

�̅ � �' ∑ ��'��� ,     (15) 
is the centroid of the � best points. 

Then, �$ � ���$ � . If �� � �$ � �' , set the point �$  and 
terminate the iteration. 

3. Expand. If �$ g ��, evaluate the expansion point ��  by: �� � �̅ � ���$ 4 �̅�     (16) 
Then compute �� � ���� � . If �� g �$  then set �'A� � ��  

and terminate the iteration. Otherwise, set �'A� � �$  and 
terminate the iteration. 

4. Contract. If �$ � �', compute a contraction between �̅ and 
the best of �'A� and �$ : 

 Outer contraction. If �' � �$ g �'A�, compute: �v � �̅ �  ���$ 4 �̅�    (17) 
then determine �v � ���v � . If �v � �$ , set �'A� � �v , and 
terminae the process, else proceed to the next stage performing 
a shrinkage. 

 Inner contraction. If �$ � �'A� , execute the inner 
contraction by computing: �vv � �̅ 4  ���̅ 4 �'A��   (18) 
Then compute �vv � ���vv �. If �vv g �'A�, set �'A� � �vv , 

and terminate the iteration. Else proceed to the next stage 
performing a shrinkage.  

5. Shrink. Calculate � new points by: �� � �� � ���� 4 ���,     (19) 
where � � 2, … , � � 1. Then compute � at those vertices. 

The scalar coefficients � , χ ,  � , and �  denote the 
coefficients, expansion, contraction, and shrink reflection, 
respectively. These cofficients should obey: � e 0, χ e 1, 0 g � g 1, and 0 g � g 1. 

 

Fig. 4.  NMS flowchart. 

V. RESULTS AND DISCUSSION 

The fractional order PI controller has been applied for AUV 
dynamics using MATLAB/Similink and the FOMCON 
toolbox. Figure 5 shows the Simulink model for the AUV 
system yaw angle response for the PI/FOPI controller, as a step 
disturbance is added to emulate an underwater current 
disturbance applied to the AUV system. The FOPI control is 
developed with the vehicle travelling at a speed of 10m/s, and 
maintains the yaw angle and steering stability for the AUV 
structure. The FOPI controller was designed using NMS with 
the minimum integral square error, as the bound of fractional 
integral inspected between the closed interval [0, 1]. Further, 
by applying the NMS technique the local minimization can be 
reached by eliminating the smallest needed vertex in a given 
function. The integral term j  of the fractional controller is 
optimized to be 0.1 with �x = 200 and ��  = 79. Hence, the 
developed fractional controller is compared to the classical 
Integer-Order PI (IOPI) controller. 

A. Simulation Results with the FOPI Controller for Different 
AUV Travelling Speeds 

Figure 6 shows the simualtion results of the AUV operating 
at different speeds with the FOPI controller. It is demostarted 
that the vehicle reaches the desired steering angle of 40

o
 in 3s 

with a 50% increase and decrease in the AUV speed. As the 
speed is increased, the overshoot slightly increases, while the 
system still maintains its regulation to reach the desired yaw 
angle response.   

B. Simulation of the Yaw Angle of the AUV with PI/FOPI 
Controllers 

Figure 7 displays the response of yaw angle with the FOPI 
and IOPI controllers being applied for a desired yaw angle of 
40

o
. The response indicates improved performance of the FOPI 

over the IOPI controller, with a faster time response, settling 
time, lower overshoot, and increased damping. Table II shows 
the perfomrance comparison of the transient response of the 
controllers. We can see that the fractional controller has better 
performance in any feature.  



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Fig. 5.  Simulink model of the yaw angle for AUV using PI/FOPI. 

 

Fig. 6.  Yaw angle for different AUV speeds with FOPI controller. 

 

Fig. 7.  Yaw angle of the AUV with FOPI/IOPI controller. 

TABLE II.  TRANSIENT RESPONSE 

Response info 
Controller 

IOPI FOPI 

Settling time 4.75 1.48 

Rise time 0.46 0.39 

Overshoot 16.1 6.8 

Undershoot 0 0 

Peak 46.17 42.95 

 

C. Simulation of the Yaw Angle of the AUV under the 
Presence of Disturbancet 

Figure 8 presents the yaw angle reposne with an external 
disturbance, such as ocean curernt waves, being applied on the 
AUV system. It is seen that the FOPI controller outperforms 
the IOPI in terms of time response, overshoot, and settling time 
to reach the desired yaw angle with the presence of external 
disturance. Table III shows the respone comparizon between 
the controllers. It can be seen that the fractional controller 
improved the transisent response in all aspects. 

 

 

Fig. 8.  Yaw angle with external disturbance applied to the AUV. 



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TABLE III.  TRANSIENT RESPONSE WITH SYSTEM 
DISTURBANCE 

Response info 
Controller 

IOPI FOPI 

Settling time 4.55 1.51 

Rise time 0.47 0.38 

Overshoot 9.3 17.1 

Undershoot 0 0 

Peak 43.66 46.97 

 

D. Simulation of the Yaw Angle with Load Variation on the 
AUV System 

Figure 10 demonstrates the response of the UAV with the 
load being increased and deceased by 50% along with the 
presence of disturbances. The FOPI controller shows its 
robustness in handling uncertain environmental disturbances 
with different loading conditions, in stabilizing the system, and 
in achieving the desired yaw angle with improved time 
response, stetting time, and minimized overshoot in 
comparison with the classical IOPI controller. 

 

 

Fig. 9.  FOPI/IOPI control of the yaw angle for AUV with varying loads. 

VI. CONCLUSION 

In this paper, a non-integer (fractional) order controller is 
implemented on the AUV dynamical system. It is demonstrated 
that the proposed FOPI control, which adds an additional 
tuning parameter, improves the system transient response, 
stability, and the yaw angle control of the AUV structure. The 
underwater vehicle yaw angle control is a very challenging 
problem and with the FOPI controller the system can preserve 
improved regulation and faster time response under system 
uncertainties and disturbances such as, underwater current 
waves and load variations. The FOPI controller was compared 
to the classical integer order controller, and the simulation 
results confirm that the fractional controller outperforms the 
IOPI controller and provides improved time response and 
reduced system overshoot and settling time to achieve the 
desired yaw angle. Regarding future work, the applied 
fractional order controller can be applied to an AUV system 
modeled as a fractional order system, which can provide more 
precision and accuracy to the system model dynamics and thus, 
improve the system overall performance. 

 

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