Thesis Mochammad Rizky Diprasetya

MODELING, SIMULATING, AND ANALYZING QUARTER CAR PASSIVE

AND ACTIVE SUSPENSION USING MATLAB SIMULINK AND V-REP

SIMULATOR

By

Mochammad Rizky Diprasetya S

11111034

BACHELORS DEGREE in

MECHANICAL ENGINEERING MECHATRONICS CONCENTRATION FACULTY OF ENGINEERING AND INFORMATION TECHNOLOGY

SWISS GERMAN UNIVERSITY

EduTown BSD City

Tangerang 15339

Indonesia

June 2015

MODELING, SIMULATING, AND ANALYZING QUARTER CAR PASSIVE AND ACTIVE Page 2 of 125 SUSPENSION USING MATLAB SIMULINK AND V-REP SIMULATOR


STATEMENT BY THE AUTHOR

I hereby declare that this submission is my own work and to the best of my knowledge,

it contains no material previously published or written by another person, nor material

which to a substantial extent has been accepted for the award of any other degree or

diploma at any educational institution, except where due acknowledgement is made in

the thesis.


____________________________________________

Student

Date

Approved by:

Kirina Boediardjo, ST, M.Sc

____________________________________________

Thesis Advisor

Date

Yunita Umniyati, PhD

____________________________________________

Thesis Co-Advisor

Date

Dr. Ir. Gembong Baskoro, M.Sc

____________________________________________

Dean

Date



ABSTRACT

MODELING, SIMULATING, AND ANALYZING QUARTER CAR PASSIVE AND

ACTIVE SUSPENSION USING MATLAB SIMULINK AND V-REP SIMULATOR

By


Kirina Boediardjo, ST, M.Sc, Advisor

Yunita Umniyati, PhD, Co-Advisor

SWISS GERMAN UNIVERSITY

This thesis works purposes are to develop a mathematical model for quarter car passive

and active suspension, to simulate its behavior, and analyze the difference of both

suspension. The active suspension use PID controller to control the displacement of the

car in vertical direction caused by different road profile. The displacement has to be

small, these condition are achieved by manipulating the force actuator inside the

suspension. The force actuator manipulated according to the output of controller

system. V-REP simulator is used to visualize the physical simulation. The physical

model of suspension developed in V-REP. The result from V-REP compared with the

result from MATLAB Simulink.

Keywords: Quarter Car Suspension, MATLAB, Simulink, V-REP Simulator, Full Car

Suspension, PID, LQR.



Copyright 2015

by Mochammad Rizky Diprasetya S

All rights reserved



DEDICATION

I dedicate this works for my parents, my friends, and myself.



ACKNOWLEDGEMENTS

I thank All for His constant blessing and supports during this thesis development.

I would like to thank to my parents for their patience in these past four years.

I would like to thank Kirina Boediardjo, ST, M.Sc for her support throughout the

development of the thesis, time, and her guidance in building the system model

schematic during development of this thesis.

I would like to thank to Yunita Umniyati, PhD for her inputs, advice, and support for

this thesis.

Last but not least I thank all of my beloved friends who have supported me from the

very beginning until the very end of this thesis development.



TABLE OF CONTENTS

Page

STATEMENT BY THE AUTHOR ................................................................................ 2

ABSTRACT ................................................................................................................... 3

DEDICATION ............................................................................................................... 5

ACKNOWLEDGEMENTS ........................................................................................... 6

TABLE OF CONTENTS ............................................................................................... 7

LIST OF FIGURES ..................................................................................................... 10

LIST OF TABLES ....................................................................................................... 13

CHAPTER 1 - INTRODUCTION ............................................................................... 14

1.1 Background ............................................................................................................ 14

1.2 Thesis Purpose ....................................................................................................... 16

1.3 Objective ................................................................................................................ 16

1.4 Hypothesis.............................................................................................................. 16

1.5 Thesis Scope .......................................................................................................... 17

1.6 Thesis Limitations .................................................................................................. 17

1.6 Thesis Organization ............................................................................................... 17

CHAPTER 2 - LITERATURE REVIEW .................................................................... 19

2.1 Theoretical Perspectives ........................................................................................ 19

2.1.1 Suspension System.............................................................................................. 19

2.1.2 Spring in Suspension System .............................................................................. 21

2.1.3 Damper in Suspension System............................................................................ 22

2.1.4 Force Actuator in Suspension System................................................................. 23

2.1.5 Hydraulic Actuator .............................................................................................. 23

2.1.6 Electromagnetic Actuator .................................................................................... 24

2.1.7 PID Control ......................................................................................................... 25

2.1.8 Linear Quadratic Regulation (LQR) ................................................................... 27

2.1.9 MATLAB Simulink ............................................................................................ 28

2.1.10 V-REP Simulator ............................................................................................... 29

2.1 Previous Studies ..................................................................................................... 31

2.1.1 Modeling, Simulating, and Analyzing an Overhead Crane Using MATLAB

Simulink and V-REP Simulator [5] .............................................................................. 31

CHAPTER 3 RESEARCH METHODS ................................................................... 33



3.1 Simulation Model Methodology ............................................................................ 33

3.2 Mathematical Model .............................................................................................. 34

3.2.1 Mathematical Model of Quarter Car Passive Suspension................................... 35

3.2.2 Mathematical Model of Quarter Car Active Suspension .................................... 38

3.2.3 Mathematical Model of Full Car Passive Suspension ........................................ 41

3.2.4 Mathematical Model of Full Car Active Suspension .......................................... 45

3.3 Control Design ....................................................................................................... 49

3.3.1 Proposed PID Control Design ............................................................................. 50

3.3.2 Linear Quadratic Regulator................................................................................. 51

3.4 Mechanical Part ..................................................................................................... 52

3.4.1 Model Parameter ................................................................................................. 56

3.4.2 Road Disturbance Environment .......................................................................... 57

CHAPTER 4 RESULTS AND DISCUSSIONS ....................................................... 60

4.1 System Model Result ............................................................................................. 60

4.1.1 Quarter Car Suspension System.......................................................................... 61

4.1.1.1 Passive Suspension System Response Analysis .............................................. 63

4.1.1.1.1 Impulse Input ................................................................................................ 63

4.1.1.1.2 Step Input ...................................................................................................... 66

4.1.2 Full Car Suspension System ............................................................................... 71

4.1.2.1 Passive Suspension System Response Analysis .............................................. 76

4.1.2.1.1 Impulse Input ................................................................................................ 77

4.1.2.1.2 Step Input ...................................................................................................... 78

4.3 Control Analysis..................................................................................................... 79

4.3.1 PID Controller for Quarter Car Suspension ........................................................ 79

4.3.2 PID Controller for Full Car Suspension.............................................................. 81

4.3.3 LQR Controller for Quarter Car Suspension ...................................................... 83

4.3.4 LQR Controller for Full Car Suspension ............................................................ 83

4.4 Quarter Car Passive and Active Suspension Comparison ...................................... 83

4.4.1 Comparison with PID Controller for Step Input ................................................. 83

4.4.2 Comparison with PID Controller for Impulse Input ........................................... 86

4.4.3 Comparison with LQR Controller ...................................................................... 88

4.5 Full Car Passive and Active Suspension Comparison for Rear Right Suspension 90

4.5.1 Comparison with PID Controller for Step Input ................................................. 91

4.5.2 Comparison with PID Controller for Impulse Input ........................................... 92

4.6 Full Car Suspension Simulink Model and Full Car Suspension V-REP Comparison

...................................................................................................................................... 92



4.7 Comparison V-REP and MATLAB Simulink Rear Right Suspension of Car Result

with Impulse Input ....................................................................................................... 95

4.8 Comparison V-REP and MATLAB Simulink V-REP Front Right Suspension of

Car Result with Step Input ......................................................................................... 103

CHAPTER 5 CONCLUSIONS AND RECCOMENDATIONS ............................ 110

5.1 Conclusions .......................................................................................................... 110

5.2 Recommendations ................................................................................................ 111

GLOSSARY ............................................................................................................... 112

REFERENCES .......................................................................................................... 113

APPENDICES ........................................................................................................... 114

APPENDIX A - MATLAB CODE FOR LQR CONTROLLER OF QUARTER CAR

SUSPENSION ........................................................................................................... 115

APPENDIX B - MATLAB CODE FOR LQR CONTROLLER OF FULL CAR

SUSPENSION ........................................................................................................... 117

APPENDIX C STATE SPACE MATRIX OF FULL CAR PASSIVE SUSPENSION.................................................................................................................................... 120

APPENDIX D STATE SPACE MATRIX OF FULL CAR ACTIVE SUSPENSION.................................................................................................................................... 123

CURRICULUM VITAE ............................................................................................ 124



LIST OF FIGURES

Figures Page

Figure 1.1 Suspension System [1] ............................................................................... 14

Figure 1.2 Road Disturbance (left), Load Disturbance (right)..................................... 15

Figure 2.1 Spring and Damper [1] ............................................................................... 19

Figure 2.2 Controllable Damper [2]............................................................................. 20

Figure 2.3 Active Suspension System with Electromagnetic Motor [2] ...................... 21

Figure 2.4 Spring ......................................................................................................... 22

Figure 2.5 Damper ....................................................................................................... 22

Figure 2.6 Hydraulic Actuator ..................................................................................... 23

Figure 2.7 Electromagnetic actuator in suspension system ......................................... 24

Figure 2.8 Common Process Block ............................................................................. 25

Figure 2.9 PID Controller ............................................................................................ 25

Figure 2.10 System Response Based on Varieties Kp, Ki, Kd .................................... 27

Figure 2.11 Overhead Crane Equation Modelling ....................................................... 31

Figure 2.12 Overhead Crane V-REP Simulator ........................................................... 32

Figure 3.1 Free Body Diagram Quarter Car Passive Suspension ................................ 35

Figure 3.2 Free Body Diagram Quarter Car Active Suspension .................................. 38

Figure 3.3 Full Car Passive Suspension Model ........................................................... 41

Figure 3.4 Full Car Active Suspension Model ............................................................. 45

Figure 3.5 Control System Overview .......................................................................... 50

Figure 3.6 Suspension Distance ................................................................................... 50

Figure 3.7 Control System in MATLAB Simulink ...................................................... 51

Figure 3.8 Overall V-REP Model ................................................................................. 53

Figure 3.9 Double Wishbone Joint Type ...................................................................... 54

Figure 3.10 Adjusting Suspension Parameter .............................................................. 54

Figure 3.11 Revolute Joint from Wheel to Arm ........................................................... 55

Figure 3.12 Revolute Joint from Arm to Chassis ......................................................... 56

Figure 3.13 Speed Bump Model .................................................................................. 58

Figure 3.14 Step Input Road Disturbance .................................................................... 59

Figure 4.1 Quarter Car Simulink Block ....................................................................... 61



Figure 4.2 Quarter Car Passive Suspension Simulink Block ....................................... 62

Figure 4.3 Quarter Car Active Suspension Simulink Block ........................................ 62

Figure 4.4 Impulse Input Road Disturbance ................................................................ 63

Figure 4.5 Car Body Displacement Due to Impulse Input ........................................... 64

Figure 4.6 Suspension Travel Due to Impulse Input.................................................... 65

Figure 4.7 Wheel Displacement Due to Impulse Input ................................................ 66

Figure 4.8 Step Input Road Disturbance ...................................................................... 67

Figure 4.9 Suspension Travel Due to Step Input ......................................................... 68

Figure 4.10 Car Body Displacement Due to Step Input .............................................. 69

Figure 4.11 Wheel Displacement Due to Step Input.................................................... 70

Figure 4.12 Full Car Suspension System ..................................................................... 71

Figure 4.13 Pitching Block Diagram ........................................................................... 73

Figure 4.14 Bouncing Block Diagram ......................................................................... 74

Figure 4.15 Rolling Block Diagram............................................................................. 75

Figure 4.16 Theta, Gamma, Body Displacement for Each Suspension ....................... 76

Figure 4.17 Impulse Input Road Disturbance .............................................................. 77

Figure 4.18 Car Body Displacement of Front Right Suspension due to Impulse Input

...................................................................................................................................... 77

Figure 4.19 Step Input Road Disturbance .................................................................... 78

Figure 4.20 Car Body Displacement of Front Right Suspension due to Step Input .... 79

Figure 4.21 PID Controller .......................................................................................... 80

Figure 4.22 Closed-loop System of Suspension System ............................................. 80

Figure 4.23 Control System for Each Suspension ....................................................... 81

Figure 4.24 Inside Control System Block .................................................................... 82

Figure 4.25 Flowchart for LQR Controller using MATLAB Code ............................. 83

Figure 4.26 Comparison Suspension Travel with PID Controller due to Step Input ... 84

Figure 4.27 Comparison Car Body Displacement with PID Controller due to Step

Input ............................................................................................................................. 84

Figure 4.28 Comparison Wheel Displacement with PID Controller due to Step Input

...................................................................................................................................... 85

Figure 4.29 Comparison Wheel Displacement with PID Controller due to Impulse

Input ............................................................................................................................. 86

Figure 4.30 Comparison Car Body Displacement with PID Controller due to Impulse

Input ............................................................................................................................. 86

Figure 4.31 Comparison Suspension Travel with PID Controller due to Impulse Input

...................................................................................................................................... 87



Figure 4.32 Car Body Displacement LQR Controller ................................................. 88

Figure 4.33 Wheel Displacement with LQR Controller .............................................. 89

Figure 4.34 Suspension Travel with LQR Controller .................................................. 90

Figure 4.35 Comparison Car Body Displacement of Rear Right Suspension with PID

Controller due to Step Input ......................................................................................... 91

Figure 4.36 Comparison Car Body Displacement of Rear Right Suspension with PID

Controller due to Impulse Input ................................................................................... 92

Figure 4.37 Car Body Displacement V-REP Result .................................................... 93

Figure 4.38 Car Body Displacement MATLAB Simulink Result ............................... 94

Figure 4.39 V-REP Result of Rear Right Suspension with Impulse Input .................. 95

Figure 4.40 Position of Car Body before Drive through the Speed Bump .................. 96

Figure 4.41 V-REP Result of Rear Right Suspension First Peak ................................. 96

Figure 4.42 V-REP Result of Rear Right Suspension Bottom after First Peak ........... 97

Figure 4.43 First Overshoot due to Suspension ........................................................... 98

Figure 4.44 MATLAB Simulink Result Rear Right Suspension due to Impulse Input

...................................................................................................................................... 99

Figure 4.45 MATLAB Simulink Result after Goes through the speed bump ............ 100

Figure 4.46 First overshoot due to Suspension .......................................................... 101

Figure 4.47 V-REP Result of Rear Right Suspension with Step Input ...................... 103

Figure 4.48 Position of Car Body before Drive through the Step Input .................... 104

Figure 4.49 V-REP Result of Rear Right Suspension First Peak ............................... 105

Figure 4.50 V-REP Result of Rear Right Suspension Bottom after First Peak ......... 106


.................................................................................................................................... 107


the first peak ............................................................................................................... 107


after the first peak ...................................................................................................... 108



LIST OF TABLES

Table Page

Table 3.1 State Space Variable 43

Table 3.2 State Space Variable Definition 44

Table 3.3 State Space Variable 48

Table 3.4 State Space Variable Definition 48

Table 3.5 V-REP Model Parameter 56

Table 4.1 List of Parameter for Full Car Suspension System 71

Table 4.2 Result Comparison V-REP Simulator and MATLAB Simulink for Rear

Right Suspension due to Impulse Input 102

Table 4.3 Result Comparison V-REP Simulator and MATLAB Simulink for Rear

Right Suspension due to Step Input 108



CHAPTER 1 - INTRODUCTION

1.1 Background

The suspension system must upkeep the vehicle, deliver directional control using

handling manoeuvres and deliver actual isolation of person along for the ride and load

disturbance. The criteria of the suspension personally depend on the purpose of the

vehicle. For example, a normal car driver will need a quite soft ride for low to medium

speed handling and at ease drive.

Figure 1.1 Suspension System [1]

There are two main types of disturbances on a vehicle, road and load disturbances. Road

disturbances have the characteristics of large amount in low rate such as mountains

and small amount in high rate such as road bumpiness. Load disturbance formed

by accelerating, braking, and cornering. Thus, a good suspension scheme is required

with altered disturbance dismissal from these disturbances to create hard or soft

suspension hinge on the purpose of the vehicle.



Figure 1.2 Road Disturbance (left), Load Disturbance (right)

A car suspension system is the instrument that physically isolated the car body from

the wheels of the car. Suspension system attaches the wheel and the vehicle body by

springs, dampers and some links that attaches a vehicle to its wheels. The spring storing

energy affected by the body mass and aids to stabilize the body from the road

disturbance, while damper spent this energy and aids to reduce the oscillation. The

main role of vehicle suspension system is to separate a vehicle body from road

disturbance which is the upright acceleration transferred to the passenger in order to

improve comfort and well-being while driving a car, and in order to maintain constant

road wheel contact to deliver road holding.



1.2 Thesis Purpose

The purpose of this thesis are first to build the system model of the quarter car passive

and active suspension system, to simulate the behaviour of the suspension system, to

design control system of the suspension system, and to analyse the result whether it can

achieve the desired condition where the car body

1.3 Objective

The main objective of this thesis are:

To model the suspension system in mathematical model

Develop and analyze the mathematical model using MATLAB Simulink

Develop and analyze the physical model using V-REP Simulator

To control the active suspension using PID control and LQR

To analyze the different between using PID control and LQR

To analyze the different between active and passive suspension

1.4 Hypothesis

The main purpose of this thesis are to analyze quarter car passive and active suspension

system, and analyze the difference between them. Determine which suspension has

good performance to achieve comfort and safety while ride a vehicle.

These are the hypothesis:

Passive suspension system will diminish vibration only affected by fix

parameters which is spring and damper. Hence, this suspension system will not

greatly isolated the body of vehicle from road disturbance.

Active suspension system will diminish vibration affected by fix parameters

which is spring and damper, and controllable force actuator. The force actuators



will add or dissipate energy according to the control scheme. Hence, this

suspension system will greatly isolated the body of vehicle from road

disturbance.

1.5 Thesis Scope

This thesis scope is to model and simulate the behaviour of two car suspensions system,

the one without controller which is passive car suspensions and the one with the

controller that applied after analysing the passive suspensions, which is active car

suspensions. This experiment will be carried out first to determine what model is

suitable for this simulation to be simulated with MATLAB Simulink. To crosscheck

and to visualize the simulation result, a V-REP is used.

1.6 Thesis Limitations

The main coverage of this thesis is on modelling and the simulation of quarter car

passive and active suspension system. Two main software programs are used which are

MATLAB Simulink and V-REP Simulator. MATLAB Simulink is used to simulate the

suspension system model along with the controller design, while V-REP is used to

simulate the visualization of the suspension system in real life and verify the previous

simulation done in numerical simulator software, because this simulator is considered

to be able to simulate any kind of robotic mechanism and movement.

The actuators is not design in dynamic model. Because the mechanism of the actuator

cannot be developed in V-REP Simulator.

1.6 Thesis Organization

Chapter 1 Introduction

This chapter briefly explains about the purpose, scope, and limitation of the thesis

Chapter 2 Literature Review

This chapter discusses about the related work that have been previously done



Chapter 3 Research Methodology

This chapter explains the steps how the experiment carried out.

Chapter 4 Result and Discussions

This chapter discusses the result and the measurements that have been done.

Chapter 5 Conclusions and Recommendations

This chapter concludes the thesis, and provides with recommendations for further

development of the thesis



CHAPTER 2 - LITERATURE REVIEW

2.1 Theoretical Perspectives

2.1.1 Suspension System

A suspension is the system of springs, dampers, and some linkage that connects a vehicle

to its wheel. The main function is to minimize the vertical acceleration transmitted

to the passenger which directly provides road comfort. The suspension itself is divided

into three types, which are: passive, semi-active, and active. [2]

A passive suspension system contains springs and dampers where springs store the

energy and damper dissipate the energy. Its factor normally fixed, being chosen to

achieve a certain level of conciliation between road handling, load transport and ride

comfort. Most suspensions in this type can be measured as a spring in parallel with

damper located at most at each corner of the vehicle. [2]

Figure 2.1 Spring and Damper [1]



A semi-active suspension system contains springs and controllable damper. The

controllers control the level of damping based on control scheme and spontaneously

adjust the damper to the desire levels. Sensors and actuators are added to sense the road

profile for control input.

Figure 2.2 Controllable Damper [2]

An active suspension system contains springs, dampers and force actuator. This

structure has capability to response to the upright changes in the road profile. The force

actuator will add or dissipate energy from the system. The force actuator controlled by

several type of controller determine by the control scheme. In this system passive

mechanisms are amplified by actuators that supply extra forces while pulling down

or pushing up the body masses. This is for achieving the desired level of comfort in

order to diminish the vibrations due to the road differences.



Figure 2.3 Active Suspension System with Electromagnetic Motor [2]

In this thesis, mathematical model of passive and active suspension system with

hydraulic actuator will be developed using MATLAB Simulink. The simulation will be

developed using V-REP Simulator.

2.1.2 Spring in Suspension System

The main function of spring in suspension application is to store energy produced by the

mass of the body. There are several type of spring rate depends on the vehicle. For race

car, it has heavy spring and for passenger car, it has soft spring. That is because race car

need hard suspension to maximize handling on high speed, but passenger car need soft

suspension to achieve comfortable drive.



Figure 2.4 Spring

2.1.3 Damper in Suspension System

The main function of damper in suspension application is to dissipate energy caused

by the spring. In semi-active suspension, the damper will vary and controllable. Damper

also known as shock absorber. Consist of piston that have small holes around the piston

that will make the fluid flow through, and the fluid itself which is hydraulic fluid.

Figure 2.5 Damper



2.1.4 Force Actuator in Suspension System

Force actuators is only exist in active suspension system. The main function of force

actuator is to produce force that will control the system. Mostly, force actuator that used

in suspension system is hydraulic actuator, and electromagnetic actuator.

2.1.5 Hydraulic Actuator

Figure 2.6 Hydraulic Actuator

In the suspension system that use hydraulic as the force actuator, the controller will give

the output to the servo which control the valve. The valve will determine whether the

hydraulic will flow through to give a certain force using the cylinder actuator or the

valve will remain keep the hydraulic in the hydraulic pump.



2.1.6 Electromagnetic Actuator

Figure 2.7 Electromagnetic actuator in suspension system

In the suspension system that use electromagnetic as the force actuator, the controller

will give the output to the magnetic coil which produce electromagnetic. As the figure

above if the magnetic coil not activated there will be no magnetic field produced on the

magnetic coil. If the magnetic coil activated there will be magnetic field produced the

magnetic coil. The magnetic field will produced force to keep the magnetic coil align

together.



2.1.7 PID Control

One of the most common forms of closed loop control system is a PID (Proportional,

Integral, and Derivative) controller. A PID controller can be found in all areas where

control is implemented. While desired speed is inserted, the error value as the difference

between the desired speed and the actual speed is calculated by the controller as an

attempt to minimize the error in outputs by adjusting the process control inputs.

Figure 2.8 Common Process Block

Inside the controller block, the error is processed with three coefficient which are:

Proportional, Integral, and Derivative, shown in figure 2.9.

Figure 2.9 PID Controller



Mathematically, PID algorithm can be described by:

() = () + () +

()

0

Where:

u = control signal

e = error value

Kp = proportional gain

Ki = integral gain

Kd = derivative gain

Here below explained each functions for each coefficient:

Proportional Coefficient

Kp is the constant of proportionality, or simply the gain of the amplifier. The larger the

value of this coefficient, the faster the system output will respond to the error. If it is

too high, the system can no directly stop at the targeted reference input, and this

phenomenon is called overshoot.

Integral Coefficient

Ki is implemented when the steady state error still occurs in the system, so as long as

an error exists, the output of Ki will grow with time, until the reference value is reached.

The usage of integral control can also create overshoot phenomenon, especially when

the proportional coefficient is large.

Derivative Coefficient

Kd is a derivative control of the output of the process. In an ideal process, an error must

be corrected as quickly as possible without overshoot, and this can be achieved if there



is maximum gain at the beginning and at the end reducing the gain as the error

approaches zero.

In real applications, there are several possible ways of how to tune PID control in the

system, such as: manual tuning with basic knowledge from characteristic of coefficient,

Ziegler-Nichols, and auto tuning software. Here below in figure is the example of the

system output based on several different usages of PID control possibilities.

Figure 2.10 System Response Based on Varieties Kp, Ki, Kd

2.1.8 Linear Quadratic Regulation (LQR)

For comparison purpose, the LQR approach will be utilized. LQR is one of the most

popular control approaches normally been used by many researches in controlling the

active suspension system.



Consider a state variable feedback regulator for the system as

() = ()

K is the state feedback gain matrix.

The optimization procedure consists of determining the control input U, which

minimizes the performance index. The performance index J represents the performance

characteristic requirement as well as the controller input limitation. The optimal

controller of given system is defined as controller design which minimizes the

following performance index.

= 1

2 (

0

+ )

The matrix gain K is represented by:

= 1

The matrix P must satisfy the reduced-matrix Riccati equation

+ 1 + = 0

Then the feedback regulator U

() = (1)()

= ()

2.1.9 MATLAB Simulink

MATLAB (Matrix Laboratory) is a tool for mathematical computation and visualization,

and usually used in all areas of applied mathematics, in education research at

universities, and in the industry. It is also a programming language, and is one of the

easiest programming languages for writing mathematical programs. With MATLAB,

the user can analyse data and develop algorithm, and create models and application.



The MATLAB has an add-on called Simulink, which is able to deliver a collaborating

graphical situation for modelling, simulating and analysing dynamic systems. In this

thesis, Simulink will be used as the main software to simulate the mathematical

modelling.

2.1.10 V-REP Simulator

V-REP (Virtual Robot Experimentation Platform) is a 3D robot simulator developed by

Coppelia Robotics. The robot simulator has control architecture that is distributed to

each object/model via an embedded script, a plugin node, a remote API client, or a

custom solution. The language of the controller can have several option: C/C++,

Python, Java, Lua, Matlab, Octave, or Urbi.

The application of V-REP including:

1. Simulation of factory automation systems

2. Remote monitoring

3. Hardware control

4. Fast prototyping and verification

5. Safety monitoring

6. Fast algorithm development

7. Robotics related education

8. Product presentation

There are three types of joint in V-REP that can be used to simulate movements in real

life:



Revolute: This type of joint can create rotational movement in cylindrical shap,

and the joint which is already connected to the center of the cylinder can be

driven as a motor or just acting as a passive joint. The speed can be adjusted

here.

Spherical: This type of joint has spherical movement, this provides rotational

movement in three axes.

Prismatic: This type of joint has linear movement that can generate back and

forth movement.

V-REP provides several calculation modules to support the operation of the elemental

object:

A forward and inverse kinematics module is used in solving kinematics calculation for

various mechanism. The module uses the damped least squares pseudo inverse method.

The inverse kinematics module is particularly helpful when dealing with manipulators.

Dynamic or physics module that allow the dynamic simulation of rigid bodies done

with the Bullet Physics Library.

A path planning module is based on random tree algorithm.

A collision detection module that can occur during one scene and this can allow the

user to calculate minimum distance between two points or bodies. Both these modules

use the method of Oriented Bounding Boxes.

To control the simulation, V-REP uses script methodology, and there two types: main

script and child script. Main script as the main scene and child script is responsible for

scene object.



2.1 Previous Studies

2.1.1 Modeling, Simulating, and Analyzing an Overhead Crane Using MATLAB

Simulink and V-REP Simulator [5]

The main purpose of this review to modeling the mathematical equation of overhead

crane using MATLAB Simulink and mechanical simulation using V-REP Simulator.

There will be only the methodology that is going to be discussed which are modeling

and simulating methodology.

The modeling methodology which are:

Identify the problem to determine the problem and the behavior of the system.

To know what is the real life condition.

Formulate the problem searching and deriving a suitable mathematical

equation that is as similar as possible to the real life condition.

Develop a model convert the mathematical equation to simulation

The MATLAB Simulink of this review thesis shown in figure 2.11.

Figure 2.11 Overhead Crane Equation Modelling



The modeling methodology which are:

Select an appropriate design the simulation will be done in numerical

computation simulation software and mechanical simulation. The mechanical

simulation will be conducted to get a better visualization and realistic condition.

Establish the scenario for the simulation consider what input that will be given

and any certain limitation must be defined.

Perform simulation and observe the result.

The V-REP Simulator of this review thesis shown in figure 2.12.

Figure 2.12 Overhead Crane V-REP Simulator



CHAPTER 3 RESEARCH METHODS

3.1 Simulation Model Methodology

This thesis will be carried out by two main steps. Modeling part and simulation part.

Modeling Part

Identify the system At very beginning of the project, system identification is

important. To know what is the real condition of the system, the stiffness of

suspension car, the mass of body car, the mass of suspension car, the mass of

unspring part, and the damper

Formulate the mathematical equation Deriving a suitable mathematical

equation that is related to the real condition.

Develop a model After finding the right mathematical equation, the next step

is to convert the equation into simulation.

Simulation Part

Select appropriate design The simulation will be done in numerical

computation simulation software which is MATLAB, and control will be

developed into the system. After the system response is achieved, the

mechanical simulation will be conducted to get better visualization of the

system behavior and realistic condition.

Establish simulation scenario Consider what input will be given into the

system and any certain limitation must be defined.

Perform simulation and observe the result.



3.2 Mathematical Model

Before designing a system model, mathematical equation is needed to be observed. Two

basic method to obtain the equation of suspension system are use the second Newton

Law and Hookes Law. The second Newton Law states that the net force on an object

is equal to the rate of change of its linear momentum p in an inertial reference frame:

=

=

()

The second law can also be stated in terms of an objects acceleration. Since Newtons

second law is only valid for constant-mass system, mass can be taken outside. Thus,

=

=

Where F = Force (N), m = Mass (Kg), and a = Acceleration (m/s2).

And the Hookes Law is a principle of physics that states that the force F needed to

extend or compress a spring by some distance X is proportional to that distance. That

is:

=

Where F = Force (N), k = Stiffness of spring (N/m), and X = displacement of the spring

(m).

First the free body diagram is developed for the passive and active suspension system.



3.2.1 Mathematical Model of Quarter Car Passive Suspension

The free body diagram of quarter car passive suspension system is shown in figure

3.1.

Figure 3.1 Free Body Diagram Quarter Car Passive Suspension

M1 = Mass of the wheel (Kg)

M2 = Mass of the car body (Kg)

r = Road disturbance

Xw = Wheel displacement (m)

Xs = Card body displacement (m)

Ka = Stiffness of car body spring (N/m)

Kt = Stiffness of tire (N/m)

Ca = Damper (Ns/m)



From the figure 3.1 obtain mathematical equation using the second Newton Law, where

positive sign for every upward direction and negative sign for every downward

direction.

For M1

= 1

( ) + ( ) + ( ) = 1

=( ) + ( ) + ( )

1

(3.1)

For M2

= 2

( ) ( ) = 2

=

( ) ( )

2

(3.2)

From the equations 3.1 and 3.2 above, let the state variable are:

1 =

2 =

3 =

4 =

where

= Suspension Travel (m)



= Car Body Velocity (m/s)

= Car Body Acceleration (m/s2)

= Wheel Deflection (m)

= Wheel Velocity (m/s)

= Wheel Acceleration (m/s2)

Therefore in state space equation, equations 3.1 and 3.2 can be written as:

() = () + ()

Where

1 = 2 4

2 =

3 = 4

4 =

Rewrite equations 3.1 and 3.2 into matrix form

[ 1234]

=

[

0 1 0 12

2

02

0 0 0 11

1

1

1 ]

[

1234

] + [

00

10

] (3.3)

State-space matrix equation 3.3 above is developed because the matrix will be given

into the LQR function. The LQR function use state-space matrix to calculate the gain

for the LQR controller output.



3.2.2 Mathematical Model of Quarter Car Active Suspension

The free body diagram of quarter car active suspension shown below in figure 3.2.

Figure 3.2 Free Body Diagram Quarter Car Active Suspension

M1 = Mass of the wheel (Kg)

M2 = Mass of the car body (Kg)

r = Road disturbance (m)

Ua = Force Actuator (N)

Xw = Wheel displacement (m)

Xs = Card body displacement (m)

Ka = Stiffness of car body spring (N/m)

Kt = Stiffness of tire (N/m)

Ca = Damper (Ns/m)



From the figure 3.2 obtain mathematical equation, where positive sign for every upward

direction and negative sign for every downward direction.

For M1

= 1

( ) + ( ) + ( ) = 1

=( ) + ( ) + ( )

1

(3.4)

For M2

= 2

( ) ( ) = 2

=

( ) ( ) + 2

(3.5)

From the equations 3.4 and 3.5 above, let the state variable are:

1 =

2 =

3 =

4 =

where

= Suspension Travel (m)

= Car Body Velocity (m/s)



= Car Body Acceleration (m/s2)

= Wheel Deflection (m)

= Wheel Velocity (m/s)

= Wheel Acceleration (m/s2)

Therefore in state space equation, equations 3.4 and 3.5 can be written as:

() = () + () + ()

Rewrite the equations 3.4 and 3.5 into matrix form

[ 1234]

=

[

0 1 0 12

2

02

0 0 0 11

1

1

1 ]

[

1234

] +

[ 01

201

1]

+ [

00

10

] (3.6)

State-space matrix equation 3.6 above is developed because the matrix will be given

into the LQR function. The LQR function use state-space matrix to calculate the gain

for the LQR controller output.



3.2.3 Mathematical Model of Full Car Passive Suspension

The free body diagram of full car passive suspension system shown below in figure 3.3.

Figure 3.3 Full Car Passive Suspension Model

In full car model the car body is free to heave roll and pitch. The suspension system

connects the car body to the four wheel which are front-left, front-right, rear-left, and

rear-right. They are free to bounce vertically with respect to the car body.

For rolling motion of the car body

= (1 1) + (2 2) (3 3)

+ (4 4) (1 1) + (2 2)

(3 3) + (4 4)

For pitching motion of the car body

= (1 1) (2 2) + (3 3) + (4 4)

(1 1) (2 2) + (3 3) + (4

4)



For bouncing of the car body

= (1 1) (2 2) (3 3) (4 4)

(1 1) (2 2) (3 3) (4 4)

And also for each side of wheel motion (vertical direction)

1 = (1 1) + (1 1) 1 + 1

2 = (2 2) + (2 2) 2 + 2

3 = (3 3) + (3 3) 3 + 3

4 = (4 4) + (4 4) 4 + 4

For Zs1

1 = + +

1 = + +

For Zs2

2 = + +

2 = + +

For Zs3

3 = + +

3 = + +

For Zs4

4 = + +

4 = + +



where

ms = mass of the car body (Kg)

muf and mur = front and rear mass of the wheel (Kg)

Ip and Ir = pitch and roll of moment inertia (Kg m2)

Zs = car body displacement (m)

Zs1, Zs2, Zs3, and Zs4 = car body displacement for each corner (m)

Zu1, Zu2, Zu3, and Zu4 = wheel displacement for each corner (m)

a = distance from center of the car body to front wheel (m)

b = distance from center of the car body to rear wheel (m)

Cf and Cr = front and rear damping (Nm/s)

Kf and Kr = stiffness of front and rear car body spring (N/m)

Ktf and Ktr = stiffness tire (N/m)

The state variables of the system are shown in Table 3.1 and the definition of each state

variable is given in Table 3.2.

Table 3.1 State Space Variable

1 = 8 =

2 = 9 =

3 = 10 =

4 = 1 11 = 1



5 = 2 12 = 2

6 = 3 13 = 3

7 = 4 14 = 4

Table 3.2 State Space Variable Definition

Variable Definition

Roll angle

Roll rate

Pitch angle

Pitch rate

Vertical displacement

Vertical velocity

1 Vertical displacement of front right wheel

1 Vertical velocity of front right wheel

2 Vertical displacement of front left wheel

2 Vertical velocity of front left wheel

3 Vertical displacement of rear right wheel

3 Vertical velocity of rear right wheel

4 Vertical displacement of rear left wheel

4 Vertical velocity of rear left wheel



State-space matrix for Full Car Passive Suspension is shown in Appendix C.

3.2.4 Mathematical Model of Full Car Active Suspension

The free body diagram of full active suspension shown in figure 3.4

Figure 3.4 Full Car Active Suspension Model

In full car model the car body is free to heave roll and pitch. The suspension system

connects the car body to the four wheel which are front-left, front-right, rear-left, and

rear-right. They are free to bounce vertically with respect to the car body.



For rolling motion of the car body

= (1 1) + (2 2) (3 3)

+ (4 4) (1 1) + (2 2)

(3 3) + (4 4) + 1 2 + 3 4

For pitching motion of the car body

= (1 1) (2 2) + (3 3) + (4 4)

(1 1) (2 2) + (3 3)

+ (4 4) + 1 + 2 3 4

For bouncing of the car body

= (1 1) (2 2) (3 3) (4 4)

(1 1) (2 2) (3 3) (4 4)

+ 1 + 2 + 3 + 4

And also for each side of wheel motion (vertical direction)

1 = (1 1) + (1 1) 1 1 + 1

2 = (2 2) + (2 2) 2 2 + 2

3 = (3 3) + (3 3) 3 3 + 3

4 = (4 4) + (4 4) 4 4 + 4

For Zs1

1 = + +

1 = + +

For Zs2

2 = + +



2 = + +

For Zs3

3 = + +

3 = + +

For Zs4

4 = + +

4 = + +

where

ms = mass of the car body (Kg)

muf and mur = front and rear mass of the wheel (Kg)

Ip and Ir = pitch and roll of moment inertia (Kg m2)

Zs = car body displacement (m)

Zs1, Zs2, Zs3, and Zs4 = car body displacement for each corner (m)

Zu1, Zu2, Zu3, and Zu4 = wheel displacement for each corner (m)

a = distance from center of the car body to front wheel (m)

b = distance from center of the car body to rear wheel (m)

Cf and Cr = front and rear damping (Nm/s)

Kf and Kr = stiffness of front and rear car body spring (N/m)

Ktf and Ktr = stiffness tire (N/m)

U1 and U2 = front right and left force actuator

U3 and U4 = rear right and left force actuator



The state variables of the system are shown in Table 3.3 and the definition of each state

variable is given in Table 3.4

Table 3.3 State Space Variable

1 = 8 =

2 = 9 =

3 = 10 =

4 = 1 11 = 1

5 = 2 12 = 2

6 = 3 13 = 3

7 = 4 14 = 4

Table 3.4 State Space Variable Definition

Variable Definition

Roll angle

Roll rate

Pitch angle

Pitch rate

Vertical displacement

Vertical velocity

1 Vertical displacement of front right wheel



1 Vertical velocity of front right wheel

2 Vertical displacement of front left wheel

2 Vertical velocity of front left wheel

3 Vertical displacement of rear right wheel

3 Vertical velocity of rear right wheel

4 Vertical displacement of rear left wheel

4 Vertical velocity of rear left wheel

State-space matrix for Full Car Passive Suspension is shown in Appendix D.

3.3 Control Design

In optimal control, it attempts to find the suitable controller that can provide best

performance for the system. Control design is a very importance part for active

suspension system. The controller will give better compromise between ride comfort

and vehicle handling. Nowadays there a lot of various controller that provided in

suspension system.

The controller generates forces to control output parameters such as body displacement,

wheel displacement, and suspension travel. Take the suspension travel as a feedback

parameter to the controller. It will keep the distance between body and wheel at normal

position which is when there is no disturbance.



3.3.1 Proposed PID Control Design

Figure 3.5 Control System Overview

The overview of control design shown in figure 3.5, on the control design, the controller

control the suspension travel between car body and wheel. The controller maintain the

suspension travel to keep the distance between them, below in figure 3.6, for more detail

Figure 3.6 Suspension Distance

In figure 3.7 shown the control design in Simulink based on the overview control

design.



Figure 3.7 Control System in MATLAB Simulink

Suspension travel displacement reference is used as main input to be compared to actual

suspension travel displacement; the output of the suspension travel displacement

control will give displacement reference that will be fed to displacement to force

converter. The converter will give force output to the suspension system. There will be

a road disturbance given to the suspension system. It will affect the suspension travel

displacement.

3.3.2 Linear Quadratic Regulator

The LQR approach of vehicle suspension control is widely used in background of many

studies in vehicle suspension control. It has been use in a simple quarter car model,

half-car, and also in full car model. The strength of LQR approach is that in using it the

factors of the performance index can be weighted according to the designers desires or

other constraints.

In this thesis, the Q and R value will be determined by using trial and error until the

system achieve its best performance. Using function lqr() in MATLAB, the Q and R

will produce the K value. The K value will be the new value of the state space variable.

In this thesis, Q is 4 x 4 identity matrix and R is a single value.

= [

1000 0 0 00 1000 0 00 0 1000 00 0 0 1000

]

= 0.03



State space formed by A, B, C, and D matrix. Rewrite these matrix and their relationship

as:

() = () + ()

() = () + ()

Matrix A is the main suspension system matrix, and relates how the current state affects

the state change . Matrix B is the control matrix, and determines how the system input

affects the state change. Matrix C is the output matrix, and determines the relationship

between the system state and the system output. Matrix D is the feed-forward matrix,

in this thesis the system is not use feed-forward controller which means matrix D is

zero matrix.

After determine the Matrix A, B, C, and D, MATLAB will calculate the gain of matrix

K using lqr() function. Then the matrix K will produce new value of matrix A:

= ( )

Matrix B, C, and D will be the same as previous value. Then the new state space will

be constructed with the new value of matrix.

3.4 Mechanical Part

Type of suspension system that is taken as main design on this thesis is double wishbone

suspension which commonly used by every car and it is more stable then single

wishbone. The design is completely designed using V-REP Simulator.

The main frame is created by cuboid shape, the wheel is created by cylindrical shape,

the joint is created by revolute joint which can generate force in rotational direction,

and the suspension is created by prismatic joint which can generate force in translational

direction.

The overall V-REP model shown in figure 3.8



Figure 3.8 Overall V-REP Model

The model consist of four wheels, 4 suspensions and the main chassis. All designed

using cuboid, sphere and cylindrical in V-REP Simulator.



Figure 3.9 Double Wishbone Joint Type

In figure 3.9 shown the wheel is connected to the chassis by double wishbone joint type

which have two arm, lower and upper arm.

Figure 3.10 Adjusting Suspension Parameter



In figure 3.10 shown that the lower arm connected to the chassis by suspension. The

stiffness suspension (K) and damper coefficient (C) can be adjust in the joint dynamic

properties. The joint can act like a suspension which contain spring and damper.

Figure 3.11 Revolute Joint from Wheel to Arm



Figure 3.12 Revolute Joint from Arm to Chassis

The lower and upper arm connected to the wheel using revolute joint, shown in figure

3.11. The lower and upper arm also connected to the chassis using revolute joint, shown

in figure 3.12.

3.4.1 Model Parameter

Below in Table 3.5, shown the parameter of the car and the suspension system.

Table 3.5 V-REP Model Parameter

Parameter Value

Distance from center to front (m) 0.3104 m

Distance from center to rear (m) 0.3104 m

Front suspension stiffness (N/m) 36927 N/m

Rear suspension stiffness (N/m) 36927 N/m



Front damper coefficient (N/ms) 4000 N/ms

Rear damper coefficient (N/ms) 4000 N/ms

Mass of the chassis (Kg) 500 Kg

Mass of the wheel (Kg) 60 Kg

Distance from center to the right (m) 0.1527 m

Distance from center to the left (m) 0.1527 m

3.4.2 Road Disturbance Environment

There are two different type of road disturbance, impulse and step input. The impulse

input is act like speed bump. In figur3 3.13 shown the speed bump created by V-REP

Simulator using cylindrical shape.



Figure 3.13 Speed Bump Model

For the step input can be seen in figure 3.14. The car will go through the speed bump

and obtain the suspension travel, wheel displacement and car body displacement.



Figure 3.14 Step Input Road Disturbance

The car will go through the step input road disturbance and obtain the suspension travel,

wheel displacement, and car body displacement. Those data will be shown in graph in

V-REP. Below in figure 3.15 is the example of using graph in V-REP.



CHAPTER 4 RESULTS AND DISCUSSIONS

4.1 System Model Result

The complete system model that is done can be shown in figure 4.1. The system model

is divided into two parts: passive suspension, and active suspension. Adjustable

parameter are located on the outer side of block.

In passive suspension systems block in figure 4.1. Both equation 4.1 and 4.2 are

generated to get desired output which is suspension travel, wheel displacement, and car

body displacement.

=

( ) + ( ) + ( )

1 (4.1)

=( ) ( )

2 (4.2)



4.1.1 Quarter Car Suspension System

Figure 4.1 Quarter Car Simulink Block



Figure 4.2 Quarter Car Passive Suspension Simulink Block

Figure 4.3 Quarter Car Active Suspension Simulink Block



4.1.1.1 Passive Suspension System Response Analysis

4.1.1.1.1 Impulse Input

Below are the parameters that are assumed:

Mass of car body (M1) = 290 Kg

Mass of wheel (M2) = 60 Kg

Damper (Ca) = 1000 Ns/m

Stiffness of car body spring (Ka) = 16812 N/m

Stiffness of tire (Kt) = 190000 N/m

The first experiment is to give an impulse road input 0.3 m at t = 1 s, can be seen in

figure 4.4.

Figure 4.4 Impulse Input Road Disturbance



The system response shown in figure 4.5 is showing the movement of the car body due

to impulse input. It occurs oscillating movement to the car body and create

uncomfortable movement for the passenger.

Figure 4.5 Car Body Displacement Due to Impulse Input



In the figure 4.6, it can be seen the suspension travel due to impulse input. The

suspension travel calculate the travel of the car body minus by wheel displacement. The

negative value means that the wheel displacement act first due to the input, and after

that the car body move because of the suspension create force due to the displacement

of the wheel to the car body.

Figure 4.6 Suspension Travel Due to Impulse Input



In figure 4.7, it can be seen that the wheel displacement occur due to impulse input.

Figure 4.7 Wheel Displacement Due to Impulse Input

4.1.1.1.2 Step Input

The parameters are still the same with the previous experiment using impulse input.

The second experiment is to give a step road input = 0.3 m at t = 1 s, can be seen in

figure 4.8.



In figure 4.9, it can be seen that the suspension travel occur due to impulse input. The

suspension travel calculate the travel of the car body minus by wheel displacement. The

negative value means that the wheel displacement act first due to the input, and after

that the car body move because of the suspension create force due to the displacement

of the wheel to the car body.

Figure 4.9 Suspension Travel Due to Step Input



In figure 4.10, it can be seen that the car body displacement occur due to impulse input.

The car body displacement is oscilating

Figure 4.10 Car Body Displacement Due to Step Input



In figure 4.11, it can be seen that the wheel displacement occur due to impulse input.

Figure 4.11 Wheel Displacement Due to Step Input

In figure 4.9, 4.10, and 4.11 shown that the suspension is not stable because the

suspension produced oscillating output due to the road disturbance which are step input

and impulse input. The suspension need controller to make it more stable.



4.1.2 Full Car Suspension System

Figure 4.12 Full Car Suspension System

Table 4.1 List of Parameter for Full Car Suspension System

Parameter Value

Front distance from center to right and left (Tf) 0.1527 m

Rear distance from center to right and left (Tr) 0.1527 m

Front Damper Coefficient 4000 Ns/m

Rear Damper Coefficient 4000 Ns/m

Front Spring Stiffness 36927 N/m

Rear Spring Stiffness 36927 N/m



Front Tire Stiffness 190000 N/m

Rear Tire Stiffness 190000 N/m

Distance from center to front 0.3104 m

Distance from center to rear 0.3104 m

Mass of front wheel 60 Kg

Mass of rear wheel 60 Kg

Mass of the car body 500 Kg

The full car suspension consist of three type of motion, pitching, rolling and bouncing.

For each motion has its own block diagram in Simulink.



The pitching block diagram shown in figure 4.13

Figure 4.13 Pitching Block Diagram

This pitching motion give output in theta which is the pitching angle.



The bouncing motion block diagram shown in figure 4.14

Figure 4.14 Bouncing Block Diagram

This bouncing motion give output in car body displacement.



The rolling motion block diagram shown in figure 4.15

Figure 4.15 Rolling Block Diagram

This rolling motion give output in gamma which is the rolling angle.

Gamma, theta, and car body displacement are all combined together for each

suspension. Each suspension affected by those output. In figure 4.16 below, shown the

input that affected one of the suspension.



Figure 4.16 Theta, Gamma, Body Displacement for Each Suspension

4.1.2.1 Passive Suspension System Response Analysis

The output that reported only the front right suspension.

Below are the parameters that are assumed:

Mass of car body (Ms) = 1160 Kg

Mass of wheel (Mu) = 60 Kg

Damper (Ca) = 1000 Ns/m

Stiffness of car body spring (Ka) = 16812 N/m

Stiffness of tire (Kt) = 190000 N/m



4.1.2.1.1 Impulse Input

The experiment is to give an impulse road input 0.3 m to the front right suspension, can

be seen in figure 4.17

Figure 4.17 Impulse Input Road Disturbance

Figure 4.18 Car Body Displacement of Front Right Suspension due to Impulse Input



In figure 4.18 above shown that the car body displacement of front right suspension due

to impulse input. The car body displacement oscillates which makes uncomfortable

driving.

4.1.2.1.2 Step Input

The experiment is to give an step road input 0.3 m to the front right suspension, can be

seen in figure 4.19




Figure 4.20 Car Body Displacement of Front Right Suspension due to Step Input

In figure 4.20 above shown that the car body displacement of front right suspension due

to step input. The car body displacement oscillates which makes uncomfortable driving.

4.3 Control Analysis

To prevent the oscillation of the car body due to road disturbance, controller is required.

There are two different types of controllers as previously discussed in chapter 3, PID

controller and linear quadratic regulation. Comparison is done in this chapter to see

which method is more effective.

4.3.1 PID Controller for Quarter Car Suspension

The system is needed to be controlled. The system of the suspension system input is

reference distance between car body and wheel, and the output is force which will be

produced by the force generator. The PID controller for suspension system transfer

function can be formed into block diagram below in figure 4.21:



Figure 4.21 PID Controller

The PID controller control the suspension travel. The output from the PID controller is

in meter unit. In the controller there is a converter from displacement to force. The

displacement derevatived by twice convert it into acceleration. The acceleration

multiply by the mass of car and wheel and make it into force unit. The force fed into

the suspension. The closed loop system can be shown below in figure 4.22.

Figure 4.22 Closed-loop System of Suspension System

The Kp, and Ki determined by trial and error. The trial and error been done until the

suspension achieve better performance which the suspension travel has small



oscillation. The car body displacement also achieve its better performance than passive

suspension as suspension travel.

After the controller is developed, this controller is integrated with the system.

4.3.2 PID Controller for Full Car Suspension

The difference between PID controller for full car suspension and quarter car

suspension is, full car suspension have three motion as mentioned in the previous

chapter which are bouncing, rolling and pitching. Each motion affected by the

suspension of each corner of the car. The force that produced by the PID controller fed

to the bouncing, rolling and pitching system. The reference is the same which is the

suspension travel of each suspension. Each suspension produced its own controlled

force.

Figure 4.23 Control System for Each Suspension



In figure 4.23 above shown the control system of full car suspension. Inside of the

control system box can be shown in figure. Each suspension has its own PID controller

which produce force with reference of its suspension travel.

Figure 4.24 Inside Control System Block

Then the value of force applied to the bouncing, rolling and pitching system.



4.3.3 LQR Controller for Quarter Car Suspension

Figure 4.25 Flowchart for LQR Controller using MATLAB Code

In figure 4.25 shown the flowchart of the LQR Controller. LQR controller for quarter

car suspension is developed using MATLAB code. The code can be shown in Appendix

A.

4.3.4 LQR Controller for Full Car Suspension

The flowchart is the same with the flowchart LQR Controller for Quarter Car

Suspension, the flowchart can be seen in figure 4.25. The LQR controller for full car

suspension is developed using MATLAB code. The code can be shown in Appendix B.

4.4 Quarter Car Passive and Active Suspension Comparison

4.4.1 Comparison with PID Controller for Step Input

After using manual tuning for the PID Controller, the better performance achieved

when the Kp = 0.01, and Ki = 4.



Figure 4.26 Comparison Suspension Travel with PID Controller due to Step Input

Figure 4.27 Comparison Car Body Displacement with PID Controller due to Step Input



Figure 4.28 Comparison Wheel Displacement with PID Controller due to Step Input

Overall from figure 4.26, 4.27, and 4.28 shown that the PID controller makes the

suspension travel, wheel displacement and car body displacement smooth movement

than the passive suspension which produced oscillating movement due to step input.

The comfortable driving achieved with this controller.



4.4.2 Comparison with PID Controller for Impulse Input

Figure 4.29 Comparison Wheel Displacement with PID Controller due to Impulse Input

Figure 4.30 Comparison Car Body Displacement with PID Controller due to Impulse

Input



Figure 4.31 Comparison Suspension Travel with PID Controller due to Impulse Input

Overall from figure 4.29, 4.30, and 4.31 shown that the PID controller makes the

suspension travel, wheel displacement and car body displacement smooth movement

than the passive suspension which produced oscillating movement due to impulse input.

The comfortable driving achieved with this controller.

Documents

Thesis Mochammad Rizky Diprasetya