Sarah M. Rashid                                                   Al-Khwarizmi Engineering Journal, 4, 2, (2008) 51-58


    

This is an open access article under the CC BY license: 

 

 
Al-Khwarizmi 

Engineering 

Journal 

                        Al-Khwarizmi Engineering Journal 

           ISSN (printed): 1818 – 1171, ISSN (online): 2312 – 0789 

Vol. 20, No. 2, June, (2024), P P. 56- 64 
 

 

Modeling and Simulation of Hydraulic Proportional Control Valves 

with Different Types of Controllers 
 

   Sarah M. Rashid*, Maher Yahya Salloom** , Nor Haniza Bakhtiar Jemily *** 

and  Ab- Samat H. **** 
*,**Department of Mechatronics Engineering, Al-khwarizmi College of Engineering, University of Baghdad, Baghdad, 

Iraq 

***Mechanical Section, University Kuala Lumpur Malaysian Spanish Institute, Kulim Hi-Tech Park, Kulim 

09000,Malaysia 

****School of Mechanical Engineering, Engineering Campus, University Sains Malaysia (USM), 14300 Nibong Tebal, 

Seberang Perai Selatan, Pulau Pinang, Malaysia  
                            Corresponding Author *Email: saramr@mtu.edu.iq 

    **Email: dmahir@kecbu.uobaghdad.edu.iq 

                                                               ***Email: norhaniza@unikl.edu.my 

                                                              ****Email: hasnida@usm.my 

 
(Received 30 October 2023; Accepted 31 March 2024; Published 1 June 2024) 

https://doi.org/10.22153/kej.2024.03.001 

 
 

Abstract 
 

Numerous applications have been developed, primarily focusing on achieving platform equilibrium. The system may 

employ electric motors or hydraulic cylinders, conventional valves, or a control mechanism that utilizes pressure sensors. 

The majority of these sources exhibit inaccuracies, demonstrate a sluggish response time, and may require periodic 

reorganization. Before implementing actual systems, it is imperative to develop a virtual system through the utilization 

of simulation techniques. This research aims to develop and evaluate a platform comprising two hydraulic cylinders and 

two proportional valves. The design of a hypothetical hydraulic system involves an analysis of the system and the 

derivation of its transfer function. Subsequently, the performance of the system is assessed by identifying the optimal 

approach for acquiring the parameters kp, ki, and kd, an examination is conducted to optimize the existing genetic 

algorithm (GA) PID controller and the particle swarm optimization (PSO) algorithm through the utilization of MATLAB 

simulation. The results indicate that the genetic algorithm (GA) PID algorithm outperforms the particle swarm 

optimization (PSO) PID controller in enhancing system performance. The chosen error standard for the self-balance 

platform control system is the Integral Time Absolute Error (ITAE) type. The system exhibits an improved rate of 

convergence. 

 

Keywords: Hydraulic; proportional; Mathematical model; Simulation; PID; Controller; Genetic Algorithm (GA); 

Particle Swarm Optimization (PSO) 

 

1. Introduction 
 

In the context of scientific and technological 

advancements, modern systems necessitate the 

attainment of optimal levels of performance and 

dependability by employing platform-stabilizing 

approaches. The stability systems of platforms 

demonstrate variety depending on the strategy 

employed for platform design and the mechanisms 

utilized for governing and maintaining the 

platform's consistent level position. Maintaining 

the level of devices and equipment in laboratories 

is crucial for performance and preventing corrosion 

and damage to parts. 

Utilizing the software tool ArchiCAD [1] begin 

the process of constructing a triangular-shaped 

balancing platform. The examined system is 

outfitted with a pair of hydraulic cylinders, 

mailto:dmahir@kecbu.uobaghdad.edu.iq
https://doi.org/10.22153/kej.2024.03.001


Sarah M. Rashid                                                  Al-Khwarizmi Engineering Journal, Vol. 20, No.2, P.P. 78- 89(2024) 

57 

facilitating the manipulation of two independent 

degrees of freedom. A mathematical model was 

created to provide an accurate description of the 

behavior shown by this system. The present model 

was produced by applying an optimization 

methodology that incorporates both a genetic 

algorithm and a particle swarm optimization 

algorithm utilized, specifically, certain multi-robot 

systems. A comparison analysis was undertaken to 

assess the efficacy of these two optimization 

methodologies [2].The hexapod parallel 

manipulator's stability was ensured through a 

specialized control system, incorporating advanced 

techniques like a telescopic boom and a gyroscopic 

self-leveling head for analysis[3]. The study 

focuses on creating a pedestal assembly for a 

medium-power transportable radar system, 

utilizing electromechanical and sliding actuators to 

adjust the azimuth and elevation settings of a 7-ton 

antenna [4].The design of an electro-hydraulic-

driven parallel stabilization platform requires 

careful consideration of factors like fluid stiffness, 

damping properties, and the coupling effect 

between the fluid and structure, which increases the 

complexity of the modal and dynamic response 

characteristics[5]. The study presents a 

multipurpose amphibious platform for  geognostic 

tasks in nearshore environments, regulated by flow 

rate and pressure, and using the Inertial 

Measurement Unit for seafloor alignment.[6]The 

study proposes a 6-DOF orthogonal hexapod with 

force feedback control for preventing vibration and 

steering of space observation systems, using 

hydraulic actuator force control.[7]The T 

shipborne-stabilized platform is crucial for 

maintaining shipborne equipment stability using 

the model predictive control approach (MPMPC), 

which predicts ship motion [8]. also, The Stewart 

platform is a mechanical system that comprises a 

mobile platform and a stationary base linked by six 

extensible legs. It functions as a motion base and 

incorporates prismatic, universal, and spherical 

joints[9]. 

 

 

2. Design Structure of the Platform  

 
The system is designed utilizing the ArchiCAD 

drafting software, incorporating the necessary 

dimensions for implementation. As depicted in 

Figure (1) and presented in Table (1). 

 

 
 

Fig .1. Structural design of stabilizer platform.

Table 1,  

Mechanical properties of the platform. 

Element                                                                    Value 

The height of the platform                                           

Dimensions of the platform                                          

The initial stroke of reference                                     

The initial stroke of actuation cylinders                      

49 cm 

50 cm 

14.5 cm 

34.5cm 

 

 

2. Methodology 

2.1 Mathematical models 
 

The system is comprised of a pair of hydraulic 

cylinders equipped with two proportional control 

valves. The purpose of this mechanism is to control 

the horizontal and vertical displacements of every 

axis. The mathematical model of the system 

encompasses the proportional control valve and 

cylinder in isolation. In situations where load 

reactive forces cannot be disregarded, the input of 

the system is represented by the valve spool 

displacement, denoted as x, while the output of the 



Sarah M. Rashid                                                  Al-Khwarizmi Engineering Journal, Vol. 20, No.2, P.P. 78- 89(2024) 

58 

system is represented by the power piston 

displacement, denoted as y. 

The linearization of a hydraulic system Figure 

(2): (a) depicts a hydraulic motor. The device can 

be described as a hydraulic power amplifier and 

actuator that is regulated by a pilot valve. The  

pilot valve exhibits a balanced configuration, 

wherein the pressure forces exerted against it are 

uniformly distributed. The pilot valve possesses the 

capability to effectively manage a very high-power 

output while requiring minimal force for 

positioning. The relationship exhibits nonlinearity 

and can be transformed into a linear form by using 

ordinary differential equations (ODEs), with Ps 

representing the pressure pump. 

Figure (2) : (b) depicts an expanded depiction 

of the valve orifice area. The valve orifice areas of 

ports 1, 2, 3, and 4 shall be referred to as respective 

designations. The variables A1, A2, A3, and A4 are 

being referred to. Furthermore, let us denote the 

flow rates through ports 1, 2, 3, and 4 as q1, q2, q3, 

and q4, respectively. It should be noted that A1 is 

equal to A3 and A2 is equal to A4 due to the 

symmetrical nature of the valve. 

Given the assumption that the displacement x is 

small,[10] 

 

 
 

Fig. 2.  (a) hydraulic system (b) enlarged diagram of valve orifice area[10]. 

 

 

𝐴 = (
𝑥̥ 

2
+ 𝑋)𝐾                                      … (1)[10] 

Where A is an area, k is a constant, xo is a 

displacement. 

In addition, we presume that the return pressure 

p0 in the return line is negligible and therefore can 

be disregarded. Then, using Figure (2) :(a) as a 

reference, the flow rates through valve orifices are 

measured as follows: 

Q=CA√
2𝑔     △𝑃

𝛾
                                     … (2)[10] 

Where (C)  is discharge and (𝛾) is specific weight. 

(g) is the acceleration of gravity. The flow rate (g) 

to the left-hand side of the e power piston is: 

Q=(
𝑥̥ 

2
+ 𝑋)𝐾√

2𝑔     △𝑃

𝛾
                         … (3)[10] 

Q=C√△ 𝑃 (
𝑥̥ 

2
+ 𝑋)                              … (4)[10] 

By applying the linearization technique  

Q= k₁ x – k2 △ 𝑝                                  … (5)[10] 

Where △ 𝑝 the pressure drops across the orifice is 

a function of the supply pressure Ps and the 

pressure difference △ 𝑝 = 𝑃1 − 𝑃2. 

Also, q= A 
𝑑𝑦

𝑑𝑡
                                        …. (6)[10] 

Where 
𝑑𝑦

𝑑𝑡
 the speed of the piston Mass  

∑𝐹 = 𝑚 �̈� 

△ 𝑃 ∗ 𝐴 − 𝐾 𝑌 − 𝐶�̇� = �̇��̈�                  … (7)[10] 

Tack Laplace transform: 

Q(s)= = k₁ X(s) – k2  △ 𝑃(s) …*5 

q= A s Y(s)…*6 

A △ 𝑃(s)=Y(s) [m s²+c s + k] …*7 

rearmament equation 3. 

△ 𝑃(s)=
𝑌(𝑠)[𝑚 𝑠²+𝑐 𝑠+ 𝑘]

𝐴
                             … (8)    

After substituting the equations, we get: 

T(s)=
𝑌(𝑠)

𝑋(𝑠)
=

𝑘₁
𝑘2  

𝐴
[𝑚 𝑠2+(𝑐−𝐴)𝑠+𝑘]

                   … (9) 

Based on parameters taken from previous research 

(k₁, k2, c, k).[11] .Additionally, it has the potential 

to be utilized within this particular system. The 

values of (A,m) are assumed to be from the real 

system. Table 2 provides a comprehensive 

overview of the information presented. 

T(s)=
0.0825

1.716 𝑠2 +56.378 𝑠+3575
                       … (10) 

 

T(s)=
0.04807

 𝑠2 +32.854 𝑠+2083.33
                           … (11) 



Sarah M. Rashid                                                  Al-Khwarizmi Engineering Journal, Vol. 20, No.2, P.P. 78- 89(2024) 

59 

Table 2, 

 parameters of hydraulic system[12][13][11] 

parameters    value                      unit                                 description  

k₁                 0.0825                       m/A                  Gain of spool valve(vertical) 

k2                8.25e-5                      m/mA                  Gain of spool valve (lateral, Longi) 

A                 1.256e-3          m²                                 Piston area of lateral axis 

m                  13                       kg                                 Load mass 

c                  400                       Ns/m                              Load damping 

k                  2500                       N/m                  Load spring stiffness 

2.2 PID controller 

 
A proportional integral derivative (PID) 

controller is an automated and precise control 

system utilized for regulating various parameters, 

such as temperature, pressure, and speed, to 

achieve desired values. Control loop feedback is 

the fundamental mechanism employed in PID 

controllers. A PID algorithm computes the error by 

determining the discrepancy between the current 

value and the target value and subsequently adjusts 

the controlling parameters accordingly. This error 

is iteratively computed until the process terminates. 

The proportional component is utilized to quantify 

the discrepancy between the desired value and the 

actual value, and it is accountable for generating 

the remedial response. The integral is utilized to 

compute the cumulative sum of historical error 

values, which is then integrated to determine the 

integral term. When a mistake is removed from the 

system, the rate of increase of this integral ceases. 

The derivative is employed to forecast the 

anticipated error levels in the future, relying on the 

current values. The controlling effect can be 

enhanced by having a system with a high rate of 

change, which is likewise dependent on 

derivatives. The name "Proportional Integral 

Derivative (PID) Controller" is derived from the 

combination of these three operations [11]. 

Output=Kₚe(t)+Kᵢ ∫ 𝑒 (𝑡)dt+Kd  
𝑑

𝑑𝑡
e(t)…(12)[11] 

Where: 

e = setpoint -Input  

𝐾𝑝 = proportional constant 

𝐾𝑖 = integral constant 

𝐾𝑑 = derivative constant 

 

2.3 GA algorithm  

 
Genetic algorithms (GAs) are grounded in 

fundamental principles such as natural selection 

and natural inheritance. Natural selection refers to 

the process through which organisms with 

advantageous adaptations to their environment are 

more likely to survive and reproduce, while those 

lacking such adaptations are more likely to be 

removed. This elimination occurs due to the 

absence of favorable hereditary traits, which are 

passed on through the interbreeding of selected 

individuals to generate offspring. In the context of 

genetic algorithms (GA), the initial population is 

determined using a fully random process in 

Georgia. The subsequent stages of GA can be 

categorized into two distinct phases: genetic 

operations, which involve crossover and mutation, 

and evolutionary processes, which encompass 

selection. The initial step in problem-solving 

involves encoding a set of points, known as genetic 

algorithms (GAs). Every configuration generates a 

collection of potential solutions, referred to as a 

chromosome or individual, within the population. 

[14]Chromosomes consist of sequences of genetic 

material known as genes. 

The design process is utilized to create an 

appropriate algorithm. The genetic algorithm (GA) 

is elected as the method of choice for optimizing 

the parameters of the PID controller, specifically 

kp, ki, and kd, through a self-tuning process. The 

design process revolves around the utilization of a 

genetic algorithm (GA) to optimize the classic 

proportional-integral-derivative (PID) controller 

for self-balance platform control [15]. 

Empirical investigations have demonstrated 

that the use of the Genetic Algorithm (GA) and the 

Proportional-Integral-Derivative (PID) controller 

yields superior outcomes in the regulation of self-

balancing robots compared to traditional PID 

control methods [14].   

  

2.4 PSO algorithm  
 

Particle swarm optimization (PSO) is a widely 

recognized and influential optimization method 

that draws inspiration from natural swarm 

behavior. The popularity of this nature-inspired 

technique has experienced a significant surge due 

to its inherent characteristics of flexibility and ease 

of implementation. Particle swarm optimization 

(PSO) has garnered significant attention across 

various domains of academic inquiry [13]. 



Sarah M. Rashid                                                  Al-Khwarizmi Engineering Journal, Vol. 20, No.2, P.P. 78- 89(2024) 

60 

3. Simulation and Experiment 

 
This section presents the simulation model of a 

self-balancing platform, the input and output 

systems and the transfer functions of the platform. 

Figure 3 presents the simulation model with a 

conventional PID controller, which enhances the 

performance of the self-balancing platform. The 

simulation model is shown in Figure 4. It uses a 

genetic algorithm (GA) to find the most important 

changes needed by choosing the best values for the 

PID controller's parameters. This makes the system 

better. while Figure 5 illustrates the simulation 

model utilizing the Particle Swarm Optimization 

(PSO) algorithm to identify crucial adjustments by 

selecting optimal values for the control parameters 

of the proportional-integral-derivative (PID) 

controller for system optimization 

 

Fig. 3. Simulation model of self-balance platform with tuning PID controller. 

 

 
 

Fig. 4. Simulation model of self-balance platform with GA-PID controller. 

 

 
 

Fig. 5. Simulation model of a self-balance platform with PSO-PID controller. 

 

 



Sarah M. Rashid                                                  Al-Khwarizmi Engineering Journal, Vol. 20, No.2, P.P. 78- 89(2024) 

61 

4. Simulation Results 
 

In this section, there are three parts. The first 

part is the simulation results with the PID 

Controller shown in subsection 4.1. The second 

part is the simulation results with the GA_PID 

controller that is shown in subsection 4.2. The third 

part is the simulation results with the PSO-PID 

controller that is shown in subsection 4.3 

 

 
Table 2,  

System parameters in each optimization method. 

 

Table 3,  

The detailed MATLAB results. 

4.1 Simulation results with PID controller 

 
This section examines the system's response 

when a PID controller is employed to enhance the 

operation and performance of a self-balancing 

platform. The results presented in Figure 6 

demonstrate the system's response, which exhibits 

a reduction in overshoot to 0.181%, a rise 

 

time of 0.0247 seconds, and a settling time of 

0.189 seconds. Consequently, the system achieves 

a steady state in a shorter duration compared to the 

initial scenario in the absence of the controller. The 

parameters of the PID controller are assigned as 

follow: Kp = 36683.9058, Kd = 406.5704, and Ki 

= 819836.3269. 

 

 
 

Fig. 6. piston stroke displacement using PID Controller. 

 

 

4.2 Simulation results with PID controller 
 

This section focuses on the application of 

genetic-PID optimization to enhance the response 

of the self-balancing platform and achieve optimal 

outcomes. The aim is to enhance the performance 

and functionality of the proportional control valve 

by optimizing the parameters of the PID controller. 

Figure 7(a) depicts the system's response obtained 

by the utilization of GA-PID  

optimization. The provided data demonstrates 

that the overshoot value is 8.7663, the settling 

durations for each instance are 0.0294 seconds, the 

rise time is 0.1316 seconds, and the system 

parameters Raise time(sec) Settling time (sec) Overshoot 

PID 0.0247 0.189 0.181% 

GA-PID 0.0294 0.136 8.7663 

PSO-PID 0.0194 0.1926 0.0046 

parameters KP KI KD 

PID 36683.9058 819836.3269 406.5704 

GA-PID 0.3112*105 8.8982*105 0.0045*105 

PSO-PID 0.349*105 8.8909*105 0.0044*105 



Sarah M. Rashid                                                  Al-Khwarizmi Engineering Journal, Vol. 20, No.2, P.P. 78- 89(2024) 

62 

achieves a steady state in a significantly shorter 

duration compared to previous states. At iteration 

100, the control parameters have the following 

values: KP = 0.3112 e5, Kd = 0.0045 e5, and Ki = 

8.8982 e5. In Figure 7 (b), the optimization tools 

utilized for GA-PID are depicted.

 
 

 
 
Fig. 7. (a) Piston stroke displacement using GA_PID controller (b) Piston stroke displacement using GA_PID 

Controller at iteration 100.

4.3 Simulation results with PSO-PID 

controller 
 

This part is dedicated to examining the 

utilization of PSO-PID optimization in order to 

improve the response of the self-balancing 

platform and attain optimal results. The objective 

is to improve the operational efficiency and 

capabilities of the proportional control valve 

through the optimization of the PID controller's 

parameters. The response of the system acquired 

by the application of PSO-PID optimization is 

illustrated in Figure 8. The data supplied indicates 

that the overshoot value is 0.0046. The settling 

durations for each instance are measured at 0.1926 

seconds, while the rise time is also 0.1926 seconds. 

Furthermore, it is observed that the system 

achieves a steady state with a considerably shorter 

duration compared to previous states. At the 100th 

iteration, the control parameters are seen to possess 

the subsequent values: KP = 0.349 x 10^5, Kd = 

0.0044 x 10^5, and Ki = 8.8909 x 10^5. 

 

 
 

Fig. 8. Piston stroke displacement using PSO-PID controller. 



Sarah M. Rashid                                                 Al-Khwarizmi Engineering Journal, Vol. 20, No.2, P.P. 56- 64(2024) 

63 

5. Conclusion 
 

The paper introduces the PSO approach as a 

means of modifying the control parameters of the 

conventional PID controller utilized in the 

construction of a self-balancing platform control 

system.  

The study proposes the utilization of a genetic 

method to modify the control parameters of the 

conventional PID controller, which was originally 

built for a self-balancing platform control system. 

The genetic controller exhibits significantly faster 

temporal response qualities in comparison to the 

PID controller, and it further furnishes us with the 

starting point for the control parameter values. The 

GA-PID controller demonstrates superior 

performance in terms of settling time, rising time, 

and overshoot. Furthermore, it should be noted that 

the error margin associated with GA-PID is 

significantly lower compared to conventional 

control methods. As a result, GA-PID has proven 

to be highly effective in achieving optimal 

outcomes for the self-balance platform. 

 

 

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 (2024) 56-64، صفحة 2، العدد20مجلة الخوارزمي الهندسية المجلد                 سارة محمد رشيد                                                      

 

64 

 

 

 

 نمذجة ومحاكاة صمامات التحكم التناسبي الهيدروليكي مع أنواع مختلفة من وحدات التحكم

 
 ،***نور هانيزا باكثير جميلي  ،**ماهر يحيى سلوم،   *محمد رشيد هسار

 ****هاسنيدا عبد الصمد
العراق  ،بغداد ،جامعة بغداد ،كلية الهندسة الخوارزمي ،قسم هندسة الميكاترونكس  *،** 

ماليزيا ،09000كوليم  ،مجمع كوليم للتكنولوجيا الفائقة ،المعهد اإلسباني الماليزي بجامعة كوااللمبور ،القسم الميكانيكي  *** 

 ماليزيا ،جزيرة بينانك ،سيبيرانك بيراي الجنوبية ،يبونك تيبال 14300 ،(USM)جامعة العلوم الماليزية  ،مجمع الهندسة ،كلية الهندسة الميكانيكية ****

saramr@mtu.edu.iq :البريد االلكتروني* 
dmahir@kecbu.uobaghdad.edu.iq :البريد االلكتروني** 

norhaniza@unikl.edu.my :البريد االلكتروني*** 
hasnida@usm.my :البريد االلكتروني**** 

 
 

 

 المستخلص

 
المنصة لضمان  ترمي هذه الدراسة إلى تصميم منصة ذاتية التوازن وتطويرها وتستخدم في العديد من التطبيقات الصناعية التي تحتاج الى استواء سطح

عن طريق صمامي سيطرة هيدروليكية تناسبية. يتضمن تصميم  بهمايتم التحكم عملها بكفاية عالية. تم تصميم منصة وتنفيذها تتكون من اسطوانتين هيدروليكية 

من  .kd و  kiو kp اناتالنظام الهيدروليكي االفتراضي تحليل النظام واشتقاقه. وبعد ذلك، تم تقييم أداء النظام من خالل تحديد النهج األمثل للحصول على البي

 الحالية وخوارزمية تحسين سرب الجسيمات (GA) للخوارزمية الجينية PID التحكم ، تم إجراء فحص لتحسين وحدةMATLAB خالل استخدام محاكاة

(PSO). تشير النتائج إلى أن خوارزمية التحكم PID للخوارزمية الجينية (GA) تتفوق في تحسين أداء النظام من خوارزمية تحسين سرب الجسيمات (PSO) 

يُظهر النظام معدالا  .(ITAE) ام التحكم في منصة التوازن الذاتي هو نوع الخطأ المطلق للوقت المتكامل.وكان معيار التحكم في األخطاء الذي تم اختياره لنظ

 .محّسناا للتقارب