A FULL ORDER SLIDING MODE TRACKING CONTROLLER
DESIGN FOR AN ELECTROHYDRAULIC CONTROL SYSTEM
RAFIDAH BTE NGADENGON @ NGADUNGON
UNVERSITI TEKNOLOGI MALAYSIA
PSZ 19:16 (Pind. 1/97)
UNIVERSITI TEKNOLOGI MALAYSIA
BORANG PENGESAHAN STATUS TESIS 
JUDUL: A FULL ORDER SLIDING MODE TRACKING CONTROLLER
DESIGN FOR AN ELECTROHYDRAULIC CONTROL SYSTEM
SESI PENGAJIAN:
Saya
2004/2005
RAFIDAH BTE NGADENGON @ NGADUNGON
(HURUF BESAR)
mengaku membenarkan tesis (PSM/Sarjana/Doktor Falsafah)* ini disimpan di Perpustakaan
Universiti Teknologi Malaysia dengan syarat-syarat kegunaan seperti berikut:
1.
2.
3.
4.
Tesis adalah hakmilik Universiti Teknologi Malaysia.
Perpustakaan Universiti Teknologi Malaysia dibenarkan membuat salinan untuk
tujuan pengajian sahaja.
Perpustakaan dibenarkan membuat salinan tesis ini sabagai pertukaran antara institusi
pengajian tinggi.
**Sila tandakan (9 )
9
SULIT
(Mengandungi maklumat yang berdarjah keselamatan atau
kepentingan Malaysia seperti yang termaktub di dalam
(AKTA RAHSIA RASMI 1972)
TERHAD
(Mengandungi maklumat TERHAD yang telah ditentukan
oleh organisasi/badan di mana penyelidikan dijalankan)
TIDAK TERHAD
Disahkan oleh
(TANDATANGAN PENULIS)
Alamat tetap:
Nama Penyelia:
KG. PT. KADIR,_______
83210 SENGGARANG,__
BATU PAHAT, JOHOR._
Tarikh: 4 APRIL 2005____
CATATAN:
(TANDATANGAN PENYELIA)
P.M. DR. MOHAMAD NOH B. AHMAD
Tarikh: 4 APRIL 2005___
* Potong yang tidak berkenaan.
** Jika tesis ini SULIT atau TERHAD, sila lampirkan surat daripada pihak
berkuasa/organisasi berkenaan dengan menyatakan sekali sebab dan
tempoh tesis ini perlu dikelaskan sebagai SULIT atau TERHAD.
 Tesis dimaksudkan sebagai tesis bagi Ijazah Doktor Falsafah dan
Sarjana secara penyelidikan, atau disertasi bagi pengajian secara kerja
kursus dan penyelidikan, atau Laporan Projek Sarjana Muda (PSM).
“I hereby, declare that I have read this thesis and in my
opinion this thesis is sufficient in terms of scope
and quality for the award of degree of
Master of Engineering (Electrical-Mechatronics and Automatic Control)
Signature
: ______________________
Name of Supervisor : ASSOC. PROF DR. MOHAMAD NOH AHMAD
Date
: 4 APRIL 2005
A FULL ORDER SLIDING MODE TRACKING CONTROLLER DESIGN FOR
AN ELECTROHYDRAULIC CONTROL SYSTEM
RAFIDAH BTE NGADENGON @ NGADUNGON
A project report submitted in partial fulfilment of the
requirements for a award of the degree of
Master of Engineering ( Electrical-Mechatronics and Automatic Control)
Faculty of Electrical Engineering
Universiti Teknologi Malaysia
APRIL 2005
ii
I declare that this thesis “A Full Order Sliding Mode Tracking Controller design for
an Electrohydraulic Control System” is the result of my own research except for
works that have been cited in the reference. The thesis has not been accepted any
degree and not concurrently submitted in candidature of any other degree.
Signature
: ______________________
Name of Author : RAFIDAH BTE NGADENGON@ NGADUNGON
Date
: 4 APRIL 2005
iii
To my dearest father, mother and family for their encouragement and blessing
To my beloved fiance for his support and caring … … …
iv
ACKNOWLEDGEMENT
First of all, I am greatly indebted to ALLAH SWT on His blessing to make
this project successful.
I would like to express my gratitude to honourable Associate Professor Dr.
Mohamad Noh Ahmad, my supervisor of Master’s project. During the research, he
helped me a lot especially in guiding me, tried to give me encouragement and
assistance which finally leads me to the completion of this project.
I would like also to dedicate my appreciation to my parents, my family, my
fiance and my friends who helped me directly or indirectly help me in this project.
v
ABSTRACT
Electrohydraulic control system are widely use in industry due to continuous
operation, higher speed of response with fast motion etc. However, there is a
drawback that it is difficult to control because of the highly nonlinear and
parameters uncertainties. In this project, a Full Order Sliding Mode Controller is
design to control the system. First, the mathematical model of the electrohydraulic
servo control system is developed. Then the mathematic model will be transformed
into state space representation for the purposed of designing the controller. The
system will be treated as an uncertain system with bounded uncertainties where the
bounded are assumed known. The proposed controller will be designed based on
deterministic approach, such that the overall system is practically stable and tracks
the desired trajectory in spite the uncertainties and nonlinearities present in the
system. The performance and reliability of the proposal controller will be determined
by performing extensive simulation using MATLAB/SIMULINK. Lastly, the
performance of the controller is to be compared with Independent Joint Linear
Control and advanced deterministic controller.
vi
ABSTRAK
Sistem elektrohidraulik banyak digunakan secara meluas di industri kerana
operasi yang berterusan, tindakbalas halaju yang lebih tinggi dengan gerakan yang
pantas. Bagaimanapun kekurangan utama sistem ini ialah sukar untuk dikawal kerana
kadar ketaklelurusan yang tinggi dan wujudnya ketidak pastian parameter. Dalam
projek ini, sebuah pengawal ragam gelincir tertib penuh telah direkabentuk untuk
mengawal sistem. Tahap pertama melibatkan pembangunan model matematik
bersepadu yang mewakili sistem elektrohidraulik. Kemudian, model matematik
tersebut ditukar kepada perwakilan dalam bentuk keadaan ruang bagi tujuan
rekabentuk pengawal sepertimana telah dicadangkan. Sistem akan diperlakukan
sebagai sistem tidak pasti dengan ketidak pastian sempadan dimana had maksimum
sesetengah parameter dianggap diketahui. Pengawal yang dicadangkan akan
direkabentuk berdasarkan pada kaedah deterministic, dimana keseluruhan sistem
secara praktikalnya di anggap stabil dan mengikut kehendak trajektori. Walaupun
wujudnya ketidak pastian dan ketaklelurusan dalam sistem. Perlakuan atau simulasi
dan kebolehharapan cadangan kawalan akan ditentukan dengan bantuan perisian
MATLAB/SIMULINK. Akhir sekali, keupayaan diantara pengawal ragam gelincir
tertib penuh akan dibandingkan dengan kawalan lelurus bebas lipatan dan
deterministic kawalan termaju.
vii
CONTENTS
SUBJECT
PAGE
TITLE
i
DECLARATION
ii
DEDICATION
iii
ACKNOWLEDGEMENT
iv
ABSTRACT
v
ABSTRAK
vi
CONTENTS
vii
LIST OF FIGURES
ix
LIST OF SYMBOLS
x
LIST OF ABBREVIATIONS
xii
CHAPTER 1 INTRODUCTION
1
1.1
Introduction
1
1.2
Objective
8
1.3
Scope of Project
9
1.4
Research Methodology
9
1.5
Literature Review
10
1.6
Thesis Layout
11
CHAPTER 2 MATHEMATICAL MODELLING
13
2.1
Introduction
13
2.2
Mathematical Modelling
14
2.3
System in State Space
16
viii
CHAPTER 3 CONTROLLER DESIGN
3.1 Introduction to Variable Structure Control (VSC)
22
22
with sliding mode control
3.2 Decomposition Into An Uncertain Systems
23
3.3 Problem Formulation
25
3.4 System Dynamics During Sliding Mode
27
3.5 Tracking Controller Design
28
CHAPTER 4 SIMULATION RESULTS
4.1 Introduction
32
32
4.2 Simulation Using Integrated Sliding Mode Controller 33
4.2.1 The Selection of Controller Parameters
33
4.2.2 Simulation of Full Order SMC
35
4.4.3 The Effect of the Value of Controller
36
Parameter, α
4.3 Simulation Using Independent Joint Linear
43
Control (IJC)
4.4 Simulation Using Advanced Control
CHAPTER 5 CONCLUSION & SUGGESTION
47
51
5.1
Conclusion
51
5.1
Suggestion For Future Work
52
REFERENCES
53
APPENDIX A : Matlab Program Listing
56
APPENDIX B : Simulink
60
ix
LIST OF FIGURES
FIGURE NUMBER
1.1
TITLE
Physical model of nonlinear electrohydraulic servo
PAGE
3
control system
4.1
Angular Displacement vs Time with satisfied control
38
parameter
4.2
Angular Velocity vs Time with satisfied control
39
parameter
4.3
Angular Acceleration vs Time with satisfied control
39
parameter
4.4
Control Input with satisfied control parameter
40
4.5
Sliding surface with satisfied control parameter
40
4.6
Angular Displacement vs Time with unsatisfied control
41
parameter
4.7
Angular Velocity vs Time with unsatisfied control
41
parameter
4.8
Angular Acceleration vs Time with unsatisfied control
42
parameter
4.9
Angular Displacement vs Time with IJC
45
4.10
Angular Velocity vs Time with IJC
45
4.11
Angular Acceleration vs Time with IJC
46
4.12
Angular Displacement vs Time with Advanced Control
49
4.13
Angular Velocity vs Time with Advanced Control
50
4.14
Angular Acceleration vs Time with Advanced Control
50
x
LIST OF SYMBOLS
SYMBOL
1.
DESCRIPTION
UPPERCASE
A(*,*)
N x N system matrix for the intagrated electrohydraulic control
system
B(*,*)
N x 1 input matrix for the intagrated direct drive robot arm
Δ B(*,*)
Bm
matrix representing the uncertainties in the input matrix
βe
effective bulk modulus of the system
C
C1
1 x N constant matrix of the sliding surface
Cd
the coefficient
Dm
volumetric displacement
E(*)
a continuous function related to Δ B(*,*)
G(*)
a continuous function related to Δ W(*,*)
Gnθ m3
nonlinear stiffness of the spring
various damping coefficient of the load
total leakage coefficient of the motor
Jt
total inertial of the motor and load
Kc
flow pressure coefficient
Kq
PL
flow gain which varies at different operating points
load pressure
Ps
supply pressure
QL
load flow
Td
disturbance of the system
Vt
total compressed volume
U(*)
N x 1 control input vector for a N DOF robot arm
X(*)
2N x 1 state vector for the intagrated direct drive robot arm
Xv
displacement of the spool in the servo valve
Z(*)
2N x 1 error state vector between the actual and the desired states of
the overall system
xi
(*)T
T
||(*) ||
2.
transpose of (*)
Euclidean norm of (*)
LOWERCASE
w
area gradient
t
time (s)
3.
GREEK SYMBOLS
γ
norm bound of continuous function H(*)
β
ρ
norm bound of continuous function E(*)
θ
joint displacement (rad)
θ&
joint velocity (rad/s)
θ&&
joint acceleration (rad/s2)
θd
desired joint angle (rad)
θ&d
desired joint velocity (rad/s)
θ&&d
desired joint acceleration (rad/s2)
σ
Integral sliding manifold
τ
time interval for arm to travel from a given initial position to a final
fluid mass density
desired position (seconds)
xii
LIST OF ABBREVIATIONS
IJC
Independent Joint Control
LHP
Left Half Plane
PI
Proportional-Integral
PID
Proportional-Integral-Derivative
SMC
Sliding Mode Control
VSC
Variable Structure Control
1
CHAPTER 1
INTRODUCTION
1.1
Introduction
Hydraulic servo system are widely used in industry due to their capabilities of
providing large driving force or torques, higher speed of response with fast motion and
possible speed reversals and continuous operation. Many industrial applications of
electrohydraulic servo systems are in a load condition application such as suspension
of electrohydraulic servo system, fly by wire system of aircraft, sheep steering gear
system and numerical machine tools. Electrohydraulic servo system combine together
the versatile and precision available from electrical technique of measurement and
signal processing with the superior performance which high pressure hydraulic
mechanism can provide when moving heavy loads and applying large forces. Servos
of this type are commonly used to operate the control surface of aircraft with actuators
which are very compact because they operate at high pressure.
2
The control of hydraulic system is difficult because of the nonlinear dynamics,
load sensitivity and parameter uncertainties due to fluid compressibility, the flow
pressure relationship and internal leakage. In addressing this problem, many advanced
control approaches have been proposed. The other methods employed variable
structure control is sliding mode control. Sliding Mode Control has been known as an
efficient and robust approach to control the nonlinear system with uncertainties.
Hydraulic system can always be made to responds quickly than electrical
devices of the same power rating. The electrical signal processing take place almost
instantaneously and occurs at a very low power level. There is thus rapid response
even with large distance between the source of the control signal and the actual
mechanism. And its including servo valve itself. The hydraulic system also good in
moving a large mass and its responds must inevitably be relatively slow.
Electrohydraulic system uses low power electrical signals for precisely
controlling the movements of large power pistons and motors. The interface between
the electrical equipment and the hydraulic (power) equipment is called ‘hydraulic
servo valve’. These valves that use in the system must responded quickly and
accurately. One of the examples is in aircraft controls. Many mechanism which use
other methods of control particularly if they are already employed hydraulic could
benefit from incorporating electrohydraulic technique.
The physical model of a nonlinear electrohydraulic servo motor shown in
Figure 1.1.The inertial-damping with a nonlinear torsional spring system is driven by
an hydraulic motor and the rotation motion of the motor is controlled by a servo valve.
Higher control input voltage can produce larger valve flow from the servo valve and
fast rotation motion of the motor.
3
Td
G
Motor
Jt
θm
Servo valve
Figure 1.1 : Physical model of nonlinear electrohydraulic servo control system
There are many unique feature of hydraulic control compared to other types of
control. Some of the advantages are the following [Herbett E. Merrit, 1967]:
1.
Heat generated by internal loses is a basic limitation of any machine.
Lubricants deteriorate, machine parts seize, and insulation breaks
down as temperature increase. Hydraulic components are superior to
others in this respect since the fluid carries away the heat generated
to a convenient heat exchanger. This features permits smaller and
light components.
2.
The hydraulic fluid also acts as a lubricants and makes possible long
components life.
3.
There is no phenomena in hydraulic components comparable to the
saturation and loses in magnetic materials of electrical machine.
The torque developed by an electrical is proportional to currents and
is limited by magnetic saturation. The torque developed by
hydraulic actuators (examples motor and piston) is proportional to
pressure difference and is limited only by safe stress levels.
4
Therefore hydraulic actuators developed relatively large torques for
comparatively small devices.
4.
Electrical motors are basically a simple lag device from applied
voltage to speed. Hydraulic actuators are basically a quadratic
resonance from flow to speed with a high natural frequency.
Therefore hydraulic actuators have a higher speed of response with
fast start, stop and speed reversal possible. Torque to inertia ratios
are large with resulting high acceleration capability. On the whole,
higher loop gains and bandwidths are possible with hydraulic
actuators in servo loops.
5.
Hydraulic actuators may be operated under continuous, intermittent,
reversing and stalled condition without damage. With relief valve
protection, hydraulic actuators may be used for dynamics breaking.
Larger speed range are possible with hydraulic actuators. Both linear
and rotary actuators are available and add to the flexibility of
hydraulic power elements.
6.
Hydraulic actuators have higher stiffness, that is inverse of slope of
speed-torque curves, compared to other drive deices since leakage
are low. There is a little drop in speed as loads are applied. In closed
loop system, this results in greater positional stiffness and less
position error.
7.
Open and closed loop control of hydraulic actuators is relatively
simple using valve and pumps.
8.
The transmission of power is moderately easy with hydraulic line.
Energy storage is relatively simple with accumulators.
5
Although hydraulic offers many distinct advantages, several advantages tend to
limit their use. Major disadvantages are the following [Herbett E. Merrit, 1967]:
1.
Hydraulic power is not so readily available as that if electrical power.
This is not a serious threat to mobile and airborne application but most
certainly affect stationary application.
2.
Small allowable tolerance results in high cost of hydraulic components.
3.
The hydraulic fluid imposes an upper temperature limit. Fire and
explosion hazard exists if a hydraulic system is used near a source of
ignition. However, these situation have improved with the available of
high temperature and fire resistant fluids. Hydraulic systems are messy
because it is difficult to maintain a system free from leaks and there is
always a possibility of complete loss of fluid if a break in the system
occurs.
4.
It is impossible to maintain the fluid free of dirt and contamination.
Contaminated oil can clog valve and actuators and, if the contaminant
is abrasive, cause a permanent loss in performance and failure.
Contaminated oil is the chief source of hydraulic control failure. Clean
oil and reliability are synonymous terms in hydraulic control.
5.
Basic design procedure are lacking and difficult to obtain because of
the complexity of hydraulic control analysis. For example, the current
flow through a resistors is described by a simple law – Ohm’s law. In
contrast, no single law exists which describe the hydraulic resistance of
passages to flow. For this seemingly simple problem there are almost
endless details of Reynolds number, laminar or turbulent flow, passage
geometry, friction factors and discharge coefficients to cope with. This
factor limits the degree of sophistication of hydraulic control devices.
6
6.
Hydraulics are not so flexible, linear, accurate and inexpensive as
electronics and electromechanical computation, error detection,
amplification, instrumentation and compensation. Therefore, hydraulic
devices are generally not desirable in the low power portions of control
systems.
The outstanding characteristic of hydraulic power elements have combined
with their comparative inflexibility at low power levels to make hydraulic control
attractive primarily in power portions of circuit and systems. The low power portions
of systems are usually accomplished by mechanical and electromechanical means.
All control system can be reduced to a few basic groups of elements, the
elements of each group performing a specific function in the system. The division into
group of elements can be carried out in a number of different ways, but selecting the
following four groups forms a convenient structure for the deviation of hydraulic and
electro-hydraulic system.
i)
The power source.
ii)
The control elements.
iii)
The actuators.
iv)
The data transmission elements.
The power source consists invariable of a pump or combination of pumps and
ancillary equipments, examples accumulators, relief valves, producing hydraulic
energy which is processed by the control elements to achieve the required operation of
the actuators. In system which have the supply pressure maintained at a constant level,
the hydraulic power source can be either a fixed or variable displacement pump.
The control elements control the output variable by manipulating the hydraulic
variables, pressure and flow. The input variable to the control elements are usually in
7
the form of mechanical, pneumatic, hydraulic or electrical signal. The input variable is
mostly a low power electrical or digital signal.
The actuator convert the hydraulic energy generated by the power source and
processed by the control elements into useful mechanical work. The actuator have
either a linear or rotary output, can be classified into cylinders or jacks, rotary
actuators and motors. The actuator producing linear output is referred as a cylinder or
jack.Cylinder can be either single acting or double acting. Single acting cylinders are
power driven in one direction only, while double acting cylinders are power driven in
both direction. Cylinder can be constructed as an single ended or double ended.
Double ended symmetric cylinder are frequently used for high performance servo
system, but have greater overall length and more expensive than single-ended
actuators. Single ended cylinder widely used for industrial and aerospace control
system cause of smaller size and lower cost.
Hydraulic motors are essentially hydraulic pumps in which the sense of energy
conversion has been reversed. While a pump converts mechanical energy supplied to
its drive shaft by a primary mover into hydraulic energy.The motor reconverts the
hydraulic energy provided by the pump into mechanical energy at its output shaft.
The control elements act on information received from the data transmission
elements. In a simple hydraulic control system the data transmission elements are
mechanical linkage or gears. But in complex systems data transmission can take at
many form for examples electrical, electronic, pneumatic and optical or combination
of these types of data transmission. The function of the data transmission elements is
to sense he controlled output quantity and to convert it to a signal which can be used
to either monitor the output or to act as a feedback devices in a closed loop control
8
system. The control output variable in a hydraulic operated force motion can be force,
velocity, position acceleration, pressure and flow.
1.2
Objective
The objectives of this research are as follows:
1.
To transform the integrated nonlinear dynamic model of the Electrohydraulic
control system into a set of nonlinear uncertain model comprising the
nominal values and the bounded uncertainties. These structured uncertainties
exist due to the limit of the angular positions, speeds and accelerations.
2.
To design a controller using the Full Order Sliding Mode Controll approach
and prove the stability of the system using Lyapunov approach.
3.
To simulate the Electrohdraulic control system controlled by the Full Order
Sliding Mode Controller and to compare its performance with other
conventional controllers.
1.3
Scope of Project
The scopes of work for this project are
ƒ
The electrohydraulic system considered is as described in [Rong-Fong Fung,
1997].
9
ƒ
Design a controller using Full Order Sliding Mode Controller and prove that
the system is stable using Lyapunov approach.
ƒ
A simulation study using MATLAB/Simulink as platform to prove the
effectiveness of this controller.
ƒ
The performance of the Full Order Sliding Mode Controller is to be compared
with Independent Joint Linear Control (IJC) and advanced controller in [Yeoh
Aik Seng, 1998].
1.4
Research Methodology
The research work is undertaken in the following five developmental stages:
a) Decomposition of the complete model into an uncertain model.
b) Determination of the system dynamics during Sliding Mode.
c) Design a controller using Full Order Sliding Mode Control approach.
d) Prove the stability of the Full Order SMC controlled electrohydraulic system using
Lyapunov stability approach.
e) Perform simulation of this controller in controlling electroydraulic control system.
This simulation work will be carried out on MATLAB platform with Simulink as
it user interface.
f) Compare of the performance of Integral Sliding Mode Controller with
other controllers.
10
1.5
Literature Review
Electrohydraulic servomechanism is highly nonlinear with inherit parameter
uncertainties. Various type of Sliding Mode Control based on Variable Structure
Control has been proposed by researchers to control such a system. Some of the
existing results will be briefly outlined in this section.
In [ Rong-Fong Fung, 1997] a new technique of the variable structure control
is applied to an electrohydraulic servo control system which is described by thirdorder nonlinear equation with time-varying coefficient. A two-phase variable structure
controller is designed to get the precise position control of an electrohydraulic servo
system. A reaching law method is implements to the control procedure, which make
fast response in the transient phase and good stability in the steady state of a nonlinear
hydraulic servo system.
Sliding mode control with time-varying switching gain and a time-varying
boundary level has been introduced in [L-C.Huang, 1996] to modify the traditional
sliding mode control with fixed switching gain and constant width bounded layer to
enhance the control performance of electrohydraulic position and different pressure.
Under certain condition, for a time-varying switching gain and boundary layer, the
combination of weighted position error and differential pressure can be asymptotically
tracked even when the system is subject to parameters uncertainties. One of the
important feature is to use only one input to simultaneously controls the angular
position and torque the electrohydraulic servo system in a different load condition.By
using this technique, the high frequency and large amplitude of control input are
attenuated.
11
An approach using variable structure control (VSC) with integral
compensation for an electrohydraulic position servo is presented in [Tzuen-Lih
Chern,1992]. The design involves the choice of the control function to guarantee the
existence of a sliding mode.The procedure include the determination of the switching
function and the control gain such that the system has an optimal motion with respect
to a quadratic performance index and the elimination of chattering of the control input.
[Miroslav Mihajlov, 2002] introduced a new technique of the sliding mode
control which is enhanced by fuzzy Proportional-Integral (PI) controller. The position
control problem in the presence of unmodelled dynamics, parametric uncertainties and
external disturbances was investigated. Fuzzy controller is added in the feedforward
branch of the closed loop in parallel with the Sliding Mode Controller with boundary
layer to improved the performance of the system.
1.6 Thesis Layout
This thesis contains five chapters. Chapter 2 deals with the mathematical
modelling of the Electrohydraulic control system. The formulation of the integrated
dynamic model of this electroydraulic is presented. The nonlinear differential equation
of the dynamics model of the system are derived then transform into state space
representations.
Chapter 3 presents the controller design using Full Order sliding mode control.
The Electrohydraulic control system is treated as an uncertain system. The model
comprising the nominal and bounded uncertain parts is computed, based on the
allowable range of the position, velocity and acceleration of the electrohydraulic servo
12
control system. It is shown mathematically that Full Order SMC is practically stable
using Lyapunov stability approach.
Chapter 4 shows some of the simulation results. The performance of the Full
Order sliding mode controller is evaluated by simulation study using Matlab/Simulink.
Chapter 5 conclude the work undertaken, suggestions for future are also
presented in this chapter.
13
CHAPTER 2
MATHEMATICAL MODELLING
2.1
Introduction
An important initial step in designing the controller for electrohydraulic servo
control system is to obtain a complete and accurate mathematical model. This model
mathematical is useful for computer simulation of the electrohydraulic servo system and
synthesis processes before applied into real application.
Basically, this chapter deals with the formulation of a mathematical model of the
electrohydraulic servo control system in state space form for the purpose of deriving a
control algorithm for controlling the system.
14
2.2
Mathematical Modeling
Consider the electrohydraulic servo system as depicted in Figure 1.1.
The servo valve flow can be described by the [Rong-Fong Fung, 1997]:
QL = K q X v − K c PL
(2.1)
where
QL is the load flow,
X v is the displacement of the spool in the servo valve,
K c is the flow pressure coefficient,
PL is the load pressure
K q is the flow gain
The flow gain Kq, which varies at different operating point is basically nonlinear
and can be expressed as:
K q = C d w ([ Ps − PL sgn( X v )] / ρ )
where
Cd is the coefficient ,
w is the area gradient
ρ is the fluid mass density
Ps is the supply pressure.
(2.2)
15
The continuity equation to the cylinder can be formulated as:
⋅
QL = Dm θ m + C1 PL +
Vt ⋅
PL
4β e
(2.3)
where
Dm is the volumetric displacement
⋅
θ m is the angular velocity of the motor shaft
C1
is the total leakage coefficient of the motor
Vt
is the total compressed volume
βe
is the effective bulk modulus of the system.
Substituting equation (2.1) into (2.3) gives:
⋅
K q X v = Dm θ m + K1 PL +
Vt ⋅
PL
4β e
(2.4)
where
K l = K c + C1 is the total leakage coefficient of the hydraulic system.
The torque balance equation for the motor is described as:
⋅
⋅⋅
PL D m = J t θ m + B m θ m + G (θ m + G m θ m3 ) + T d
Jt
is the total inertial of the motor and load
Bm
is the various damping coefficient of the load.
Td
G nθ
is the disturbance of the system
3
m
is the nonlinear stiffness of the spring
(2.5)
16
From equation (2.4), the supply pressure can be written mathematically as:
PL =
2.3
⋅
1 ⎛
V ⋅ ⎞
⎜⎜ K q X v − Dm θ m − t PL ⎟⎟
4βe ⎠
K1 ⎝
(2.6)
System in State Space
Differentiate both side of equation (2.5) with respect to time gives:
•
⋅⋅⋅
⋅⋅
⋅
⋅
P L Dm = J t θ + Bm θ m + G (θ m + G mθ m3 ) + T d
or
•
PL =
...
⋅⋅
⋅
⋅
1
[ J t θ + Bm θ m + G (θ m + G mθ m3 ) + T d ]
Dm
(2.7)
Substitute equation (2.6) into equation (2.5)
⋅
⋅
⎛K X
⎞
⎜ q v − Dm θ m − Vt PL ⎟ D = J θ⋅⋅ + B θ⋅ + G (θ + G θ 3 ) + T
m
t m
m m
m
m m
d
⎜⎜ K
Kl
4β e K l ⎟⎟
l
⎝
⎠
or
Dm K q X v
Kl
⋅
⋅⋅
⋅
D2 θ
D V ⋅
− m m − m t PL = J t θ m + Bm θ m + G (θ m + Gmθ m3 ) + Td
Kl
4β e K l
(2.8)
Substitute equation (2.7) into equation (2.8)
⋅
Dm K q X v
Kl
D2 ⋅
J V ⋅⋅⋅
B V ⋅⋅
GVt ⋅
3GGnVt 2 θ m
θm−
θm
− m θm − t t θm − m t θ m −
Kl
4β e K l
4β e K l
4β e K l
4β e K l
⋅
⋅⋅
⋅
Vt
−
Td = J t θ m + Bm θ m + G (θ m + Gmθ m3 ) + Td
4β e K l
(2.9)
17
Rearranging equation (2.9) and solving for acceleration of the motor shaft gives:
⋅⋅⋅
θm =−
+
4β e K l Gθ m ⎛ 4β e Dm2 + 4β e K l Bm + GVt
− ⎜⎜
J tVt
J tVt
⎝
4 β e Dm K q X v
J tVt
−
⎞⋅
⎛ 4 β K J + BmVt
⎟⎟ θ m − ⎜⎜ e l t
J tVt
⎝
⎠
⋅
⎞ ⋅⋅
⎟⎟ θ m
⎠
⋅
4β e K l GGnθ
3GGnVtθ θ m 4β e K l Td Vt Td
−
−
−
J tVt
J tVt
J tVt
J tVt
3
m
2
m
(2.10)
Define the state variable as:
X 1 = θ m = angular displacement of motor shaft
⋅
(2.11)
⋅
X 2 = θ m = X 1 = angular velocity of motor shaft
..
(2.12)
⋅
X 3 = θ m = X 2 = angular velocity of motor shaft
(2.13)
X v = K vu (t ) = displacement of the spool in the servo valve
(2.14)
The state equation can be found by rewriting equation (2.10) in terms of state
variable as follows:
⎛ 4 β e Dm2 + 4β e K l Bm + GVt
4β e K l G
X3 =−
X 1 − ⎜⎜
J tVt
J tVt
⎝
⋅
⎞
⎛ 4 β K J + BmVt
⎟⎟ X 2 − ⎜⎜ e l t
J tVt
⎝
⎠
4 β e Dm K q K v
4β K GGn 3 3GGnVt 2
4β K
+
X1 −
X 1 X 2 − e l Td −
u (t ) − e l
J tVt
J tVt
J tVt
J tVt
⎞
⎟⎟ X 3
⎠
Vt ⋅
Td
J tVt
(2.15)
Equation (2.15) can be rewritten as
•
X 3 (t ) = −∑ ai X i + bu (t ) − N ( X , t ) − d (t )
(2.16)
18
where
4β K G
a1 (t ) = e l
J tVt
(2.17)
a2 (t ) =
4 β e Dm2 + 4β e K l Bm + GVt
J tVt
(2.18)
a3 (t ) =
4β e K l J t + BmVt
J tVt
(2.19)
b( X , t ) =
N ( X , t) =
d (t ) =
4 β e Dm K q K v
J tVt
4 β e K l GGn 3
3GGnVt 2
X 1 (t ) +
X 1 (t ) X 2 (t )
J t Vt
J t Vt
4β e K l
V ⋅
Td (t ) + t Td (t )
J t Vt
J tVt
(2.20)
(2.21)
(2.22)
According to the [Q.P Ha et al, 1998], the last term in the RHS of equation (2.16)
is the disturbance of the system:
d (t ) =
4β e K l
V ⋅
Td (t ) + t Td (t )
J tVt
J t Vt
(2.23)
where
Td = 56.526 X 1 (t )
⋅
⋅
Td = 56.526 X 1 (t ) = 56.526 X 2 (t )
(2.24)
19
So equation above can be rewrite as
d (t ) =
=
4 β e K l 56.526 X 1 (t ) + Vt 56.526 X 2 (t )
J tVt
226.124 β e K l X 1 (t ) + 56.526Vt X 2 (t )
J tVt
(2.25)
Define
W (t ) = − N ( X , t ) − d (t )
⎛ 4 β e K l GGn X 13 (t ) + 3GGnVX 12 X 2 (t ) + 226.124 β e K l X 1 (t ) + 56.526Vt X 2 (t ) ⎞
⎟
= −⎜⎜
⎟
J tVt
⎝
⎠
(2.26)
Using equation (2.12), (2.13), (2.16) and (2.26), the dynamics equation for the
electrohydraulic servo can be written in state space form as:
⋅
X (t ) = AX (t ) + BU (t ) + W (t )
(2.27)
where
⎡ X 1 (t ) ⎤
X (t ) = ⎢⎢ X 2 (t )⎥⎥
⎢⎣ X 3 (t ) ⎥⎦
⎡ ⋅ ⎤
⎢ X 1⋅(t ) ⎥
⋅
X = ⎢ X 2 (t )⎥
⎢ ⋅ ⎥
⎢ X 3 (t ) ⎥
⎣⎢
⎦⎥
(2.28)
(2.29)
20
1
⎡ 0
⎢
A=⎢ 0
0
⎢⎣− a31 − a32
0 ⎤
1 ⎥⎥
− a33 ⎥⎦
⎡0⎤
B = ⎢⎢ 0 ⎥⎥
⎢⎣b31 ⎥⎦
⎡ 0 ⎤
W (t ) = ⎢⎢ 0 ⎥⎥
⎢⎣ w31 ⎥⎦
(2.30)
(2.31)
(2.32)
The last row elements of matrices A,B and W are as follows:
a31 (t ) =
4β e K l G
J tVt
(2.33)
a32 (t ) =
4β e Dm2 + 4β e K l Bm + GVt
J tVt
(2.34)
a 33 ( t ) =
b31 =
4 β e K l J t + B mV t
J tV t
4 β e Dm K q K v
J tVt
(2.35)
(2.36)
21
⎛ 4 β K GG n X 13 (t ) + 3GG nVX 12 (t ) X 2 (t ) + 226.124 β e K l X 1 (t ) + 56.526Vt X 2 (t ) ⎞
⎟
w31 = −⎜⎜ e l
⎟
J tVt
⎝
⎠
(2.37)
22
CHAPTER 3
CONTROLLER DESIGN
3.1 Introduction to Variable Structure Control with Sliding Mode Control
The conventional controller such as Proportional Integral and Derivative(PID)
and Linear Quadratic Regulator(LQR) may not able to control the system very well
because these type of controllers ignore the nonlinear term and uncertainties that exist
in the system. A controller based on sliding mode will be proposed to control the
electrohydraulic tracking servo system because SMC is has been known as an
efficient approach to control the nonlinear system with parameters uncertainties.
Sliding mode plays a dominant role in variable structure system (VSS). The
core idea of designing VSS control algorithms consists of enforcing sliding mode in
some manifold of system space. Traditionally, these manifold are constructed as the
intersection of hypersurfaces in the state space. This intersection domain is normally
called a switching manifold. Once the system reaches the switching plane, the
structure of feedback loop is adaptively altered to slide the system state along the
switching plane. The system response depends thereafter on the gradient of the
switching plane and remains insensitive to variations of system parameters and
23
external an disturbances under so-called matching condition. The order of the motion
equation in sliding mode is equal to (n-m). Where n being dimension of the state space
and m the dimension of the control input. However, during the reaching phase, before
sliding mode occurs, the system possesses no such insensitivity property. Therefore,
insensitivity cannot be ensure throughout an entire response. The robustness during
the reaching phase is normally improved by high-gain feedback control. Stability
problem inevitably limit the application of such high-gain feedback control schemes.
The concept of Full Order sliding mode concentrates on robustness during the
entire response. The order of the motion equation is equal to the dimension of the
plant model. Therefore, the variance of the system to parametric uncertainty and
external disturbances is guaranteed starting from the initial time instant.
3.2
Decomposition Into An Uncertain Systems
Consider the uncertainties of the system described by the equation:
•
X (t ) = AX (t ) + B( X , t )U (t ) + W ( X , t )
= AX (t ) + [ B + ΔB( X , t ]U (t ) + [W + ΔW ( X , t )]
where
A,B and W - nominal constant matrices
ΔB( X , t ) and ΔW ( X , t ) - matrices uncertainties
U(t) - control input
B(X,t) - the control gain
W(X,t) - nonlinear term and system disturbance
(3.1)
24
The nominal value of elements A and B can be computed respectively, as
B31 =
W31 =
B31 MAX ( X , t ) + B31 MIN ( X , t )
2
W31 MAX ( X , t ) + W31 MIN ( X , t )
2
(3.2)
(3.3)
With the uncertainties ΔB and ΔW computed as
ΔB( X , t ) = B − BMIN ( X , t )
(3.4)
ΔW ( X , t ) = W − WMIN ( X , t )
(3.5)
The nominal matrices A and B as well as the bounds on nonzero element of
the matrices can be computed from the maximum and minimum value obtained from
equation (3.2), (3.3),(3.4) and (3.5).
Substituting the nominal values gives the system and input nominal matrices.
1
0
⎡ 0
⎤
⎢
⎥
A=⎢ 0
0
1
⎥
⎢⎣− 0.3998 − 2999.4 − 169.9804⎥⎦
(3.6)
⎡ 0 ⎤
B = ⎢⎢ 0 ⎥⎥
⎢⎣0.8844⎥⎦
(3.7)
⎡ 0 ⎤
W = ⎢⎢ 0 ⎥⎥
⎢⎣− 85422⎥⎦
(3.8)
25
The uncertainties for system and input matrices can be obtained by substituting
uncertainty value
⎡ 0 ⎤
ΔB = ⎢⎢ 0 ⎥⎥
⎢⎣0.2049⎥⎦
(3.9)
0
⎡
⎤
⎢
⎥
ΔW = ⎢
0
⎥
⎢⎣− 148230⎥⎦
(3.10)
3.3
Problem Formulation
Define the state vector as
X (t) = [ X1 (t) X 2 (t) X 3 (t)]T
.
..
= [θ m (t) θ m (t) θ m (t)]T
(3.11)
and the desired state trajectory
X d (t) = [ X d1 (t) X d 2 (t) X d 3 (t)]T
(3.12)
Define the tracking error as
Z (t ) = X (t ) − X d (t )
(3.13)
26
In this research the following assumptions are made:
i- The state vector X(t) can be fully observed.
ii- There exists continuous function E and G such that
ΔB(t ) = BE ( X , t ) ;
E (t ) ≤ β
ΔW (t ) = BG ( X , t ) ;
G (t ) ≤ γ
(3.14)
iii-There exist a Lebesgue function Ω(t ) ∈ R m×n
•
X d (t ) = AX d (t ) + BΩ(t )
(3.15)
iv- The pair (A,B) is controllable.
The continuous function H(X,t) and E(X,t) exist if and only if the following rank
condition is satisfied.
rank[B]=rank[B,∆B(X,t)]
rank[B]=rank[B,∆W(X,t)]
(3.16)
The error dynamics can be obtained from equations (3.12),(3.13),(3.14) and (3.15)
•
•
•
Z (t ) = X (t ) − X d (t )
= AX (t ) + [ B + ΔB( X , t )U (t ) + [W + Δ( X , t )] − ( AX d + BΩ(t )
= AZ (t ) + [ B + BE ( X , t )]U (t ) − BΩ(t ) + [W + BG ( X , t )]
(3.17)
Define the Sliding surface as [Ahmad, 2003].
σ (t ) = CZ (t ) − ∫0t [CA + CBK ]Z ( τ)dτ
(3.18)
The structure of matrix C is as follow :
C = diag [ c1 c2 …… cn]
(3.19)
27
The matrix C is also chosen such that CB ∈ R nxn
The matrix K designed such that
λ max ( A + BK ) < 0
(3.20)
The matrix K can be computed using pole-placement technique. The condition
imposed by equation above guarantees that all desired poles are located at the half
plane to ensure the stability.
3.4
System Dynamics During Sliding Mode
Differentiating equation (3.18)
•
•
σ (t ) = C Z (t ) − [CA + CBK ]Z ( t )
(3.21)
Substituting equation (3.17) into (3.21), gives
•
σ (t ) = C[ B + BE ( X , t )]U (t ) + CW − CBΩ(t )
+ CBG ( X , t ) − CBKZ (t )
(3.22)
The equivalent control Ueq(t), can be found by equating equation (3.22) to zero
U eq (t ) = −(CB ) −1 [ I n + E ( X , t )]−1{(CW + CBG ( X , t )
(3.23)
− CBΩ(t ) − CBKZ (t )}
The system dynamics during sliding mode by substituting (3.23) into (3.17 )
•
Z (t ) = AZ (t ) + [ B + BE ( X , t )U (t ) + W + BG ( X , t ) − BΩ(t )
(3.24)
= [ A + BK ]Z (t )
Equation (3.24) shows that, the error dynamics of the system during sliding
mode are independent of the system uncertainties and insensitive to the parameter
28
variation. In fact the response of the system can be pre-determined through proper
selection of the gain K.
3.5
Tracking Controller Design
Equation (3.18) is asymptotically stable in large, if the following hitting condition is
held [Yan et al, 1997]:
•
σ T (t ) σ (t )
<0
σ (t )
(3.25)
To proof it, let the positive definite function be
V (t ) = σ (t )
(3.26)
Differentiating equation (3.26) with respect to time t
•
σ T (t ) σ (t )
V (t ) =
σ (t )
•
(3.27)
Based on Lyapunov Stability Theory, if equation (3.25) holds, then the manifold σ(t)
is asymptotically stable in large.
Theorem
The hitting condition (3.25) of the manifold given by equation (3.18) is satisfy if the
control U(t) of the system (3.17) is given by:
U (t ) = −(CB ) −1 [α 1 Z (t ) + α 2 Ω(t ) + α 3 W + α 4 ]
SGN (σ(t )) + Ω(t )
(3.28)
29
where:
α 1 > ( CB ) /(1 + β)
α 2 > ( β CB ) /(1 + β)
(3.29)
α 3 > ( C ) /(1 + β)
α 4 > (γ CB ) /(1 + β)
Proof :
Substitute (3.28) into (3.22) gives:
•
σ(t ) = −(CB )[ I n + E ( X , t )]{(CB ) −1 [α 1 Z (t ) + α 2 Ω(t ) + α 3 W + α 4 ]
SGN (σ(t ))} − CBKZ (t ) + CBE ( X , t )Ω(t ) + CW + CBG ( X , t )
(3.30)
Substituting equation (3.30) into equation (3.27) gives:
•
V(t ) = (σ T (t ) / σ (t ) {−CBZ (t ) − CB[ I n + E ( X , t )](CB ) −1 [α 1 Z (t ) SGN (σ (t ))
+ CBE ( X , t )Ω(t ) − CB[ I n + E ( X , t )](CB ) −1 α 2 Ω(t ) SGN (σ (t ))
+ CW (t ) − CB[ I n + E ( X , t )](CB ) −1 α 3 W SGN (σ (t ))
(3.31)
+ CBG ( X , t ) − CB[ I n + E ( X , t )(CB ) −1 α 4 SGN (σ (t ))}
Equation (3.31) can be broken down as:
•
•
•
•
•
V (t ) = V 1 (t ) + V 2 (t ) + V 3 (t ) + V 4 (t )
(3.32)
where:
•
V 1 (t ) = (σ T (t ) / σ(t) ){−CBKZ (t )
(CB )[ I n + E ( X , t )](CB ) −1 α 1 Z (t ) SGN (σ(t ))}
(3.33)
30
•
V 2 (t ) = (σ T (t ) / σ(t) ){CBE ( X , t )Ω(t ) − (CB )[ I n + E ( X , t )]
(3.34)
(CB ) −1 α 2 Ω(t ) SGN (σ(t ))}
•
V 3 (t ) = (σ T (t ) / σ(t) ){CW − (CB )[ I n + E ( X , t )](CB ) −1
(3.35)
α 3 W (t ) SGN (σ(t ))}
•
V 4 (t ) = (σ T (t ) / σ(t) ){CBG (t ) − (CB )[ I n + E ( X , t )]
(3.36)
(CB ) α 4 SGN (σ(t ))}
−1
Note that:
(σ T (t )
SGN (σ (t )) = 1
σ (t )
(3.38)
Then the first parts for equation (3.33) can be simplified as
−
(σ T (t )
{CBKZ (t ) ≤ ( σ T (t ) / σ(t) ) CBK Z (t )
σ (t )
= − CBK Z (t )
(3.37)
The second part of equation (3.33), can be written as follow:
−
σ T (t )
{(CB )[ I n + E ( X , t )](CB ) −1 α 1 Z (t ) SGN (σ(t ))}
σ(t)
≤ − CB [ I n + E ( X , t ) ] (CB ) −1 α 1 Z (t )
= −(1 + β)α 1 Z (t )
(3.39)
31
Combining the first term and second term can be written as follow:
•
V 1 (t ) ≤ −[(1 + β)α1 + CBK ] Z (t )
(3.40)
Similarly, equation (3.34),(3.35) and (3.36) can be simplified in the same manner.
The results are summarized as follows:
•
V 2 (t ) ≤ −[(1 + β)α 2 − β CB ] Ω(t )
(3.41)
•
V 3 (t ) ≤ −[(1 + β)α 3 − C ] W
(3.42)
•
V 4 (t ) ≤ −(1 + β)α 4 + γ CB
(3.43)
If the condition (3.29) hold, then the global hitting condition (3.24) is satisfied . Based
on the Lyapunov Theory, the system dynamics is stable.
32
CHAPTER 4
SIMULATION RESULTS
4.1
Introduction
This chapter deals with the simulations carried out on the electrohydraulic servo
control system. The performance of the system determined by the controller designed
to control the system. The controller is a nonlinear controller which provides an
effective and robust for controlling a nonlinear system with uncertainties and
disturbances that exists in system. The nonlinear equations of (2.27) is used in the
simulation to represent a real electrohydraulic servo control system without any
approximation and simplification of the highly non-linear elements. The main
purpose of these simulation is to study the performance of the proposed controller in
controlling the system. This simulation work was carried out on MATLAB platform
with Simulink as it user interface.
33
4.2
Simulation Using Full Order Sliding Mode Controller
In this section, the simulation is carried out using the controller described by equation
(3.21).
4.2.1 The Selection of Controller Parameters
The selection of the values of the sliding surface constant C and the desired
poles location will determine the shape of the plant output in response to the desired
input trajectory. The constant c n , will determine the magnitude of the ith input, U i (t ) ,
while the constants c1 , c 2 ,L c ni −1 will determine the shape of the trajectories during
the reaching phase [Ahmad, 2003].
The desired poles location can be placed anywhere on the left half plane (LHP)
of the s-plane to guarantee stability during the sliding phase. However, if the
locations of the desired closed-loop poles are placed too far on the LHP of the s-plane,
high gain K will be produced and will somehow affects the shape of the Full Order
sliding surface of equation (3.18). The values of the controller parameters α i's must
be large enough to accommodate for the constraint as stated in equations (3.29) but
not too large to avoid excessive magnitude of the control input U i (t ) .
A specific tuning rule for the controller parameters is needed to overcome any
constraints that may arise due to the physical limitations of the elements of the system.
The task of choosing the right controller parameters to get the satisfactory tracking
response can be time consuming and in some cases very exhaustive. It is important to
note however that the conditions described above can be used to develop an algorithm,
which, for each setting of the controller parameters, determines in a systematic way
whether the output tracking performance is satisfactory while at the same time
34
guarantee the control input U i (t ) stays within the stipulated limit. The algorithm can
be stated as follows [Ahmad, 2003]:
Algorithm 4.1 :
Step 1. Input data: Numerical values for C =diag[c1, c2 ... c ni ],
λ max (A + BK) < 0, and α i > o.
Step 2. Check if the sliding mode exists and whether the output tracking response is
satisfactory. If the conditions do not hold then try other combinations. If
the conditions hold, proceed to Step 3.
Step 3. Check if all of the control inputs U (t ) = [U 1 (t )
U 2 (t )
....
U m (t )]T are
within the admissible range. If the condition does not hold then increase the
value of c n , and place the desired poles closer to the origin until sliding
mode exist and the control input U(t) is within the admissible limit. If the
condition holds, then proceed to Step 4.
Step 4. Check if the output trajectories are satisfactory during the reaching phase. If
the conditions do not hold then adjust the values of c1, c2 ... c ni until
satisfactory shape of the output trajectories are achieved. If the conditions
hold, then proceed to Step 5.
Step 5. Check if the tracking errors of the output trajectories are satisfactory. If the
conditions do not hold then increase the values of α i ,for i = 1,2,3 until
satisfactory tracking errors are achieved. The values of α i should not be too
large to guarantee that the control input U(t) is within the admissible limit.
If the conditions hold, then go to Step 6.
35
Step 6. Finish.
The algorithm presented above not only guarantees that the desired tracking
response is achieved, but it also assures that the system control input U(t) is within
the permissible range of operation.
4.4.2 Simulation of Full Order SMC
The proposed control law described by equation (3.28) – (3.29) will be
applied to control electrohydraulic servo control system. The bounds of E (t ) may be
computed as follows using equation (3.16).
E (t ) = [( B T B) −1 B T ]ΔB(t )
(4.1)
where ( BT B) −1 BT is called the pseudo inverse.
Hence,
E = [0.2317]
and
β ≥ E (t )
(4.2)
Therefore,
β ≥ 0.2317
(4.3)
Similarly, the bounds of G(t ) may be computed as follows using equation (3.16)
G (t ) = [( B T B) −1 B T ]ΔW (t )
(4.4)
G (t ) = [167610] and
(4.5)
Hence,
γ ≥ G (t )
36
Therefore,
γ ≥ 167610
(4.6)
Define the gain K as:
K = [− 0.45 − 3391.4 − 180.9]
(4.7)
So that the closed-loop poles of the system are:
λ = {−0.02, − 0.03, − 10}
(4.8)
Define the matrix C as:
C (t ) = [− 0.08 0.02 1]
(4.9)
The controller parameter α may be computed using equation (3.29) as follows:
α 1 > 0.718; α 2 > 0.1664; α 3 > 0.8146; α 4 > 120350
4.4.3
(4.10)
The Effect of the Value of Controller Parameter, α
In the following simulation, the effect of the controller parameter α is studied. For
comparison purposes, two sets of the controller parameters α have been considered in
the simulation.
Set 1: ( Control Parameters Condition Are Satisfied )
α 1 = 1.4432
α 2 = 0.3361
α 3 = 1.7107
α 4 = 168000
(4.11)
37
Set 2: ( Control Parameters Condition Are Not Satisfied )
α 1 = 0.3447
α 2 = 0.0749
(4.12)
α 3 = 0.4073
α 4 = 44890
The controller parameter in Set 1, are chosen to study the performance of the
system when the controller parameters’ conditions of equations (3.29) are satisfied.
On the other hand, in Set 2 the controller parameters is selected to represent a
situation where the conditions imposed by equations (3.29) are not satisfied. A fourth
order Runge-kutta numerical integration method has been used in the simulation to
solve the nonlinear differential equation due to effectiveness in terms of accuracy and
minimum computing time.
The simulation results for Sets 1 are shown in Figure (4.1 - 4.3). Form the
simulation results using the control parameter α as in Set 1, the actual output
positions can track the desired trajectory if the controller parameter conditions are
satisfied. The good tracking performance results for angular displacement as shown in
the Figure 4.1, angular velocity as shown in Figure 4.2 and angular acceleration as
shown in Figure 4.3. From the result obtained the electrohydraulic servo system able
to track the desired trajectory if the conditions of sliding mode controller parameters
are fulfilled.
The simulation result for control input as shown in Figures (4.4) and sliding
surface as shown in Figure (4.5), it is very clear that the range of the control input is
very high and there are not exist any switching in the system, although in
theoretically control input should switch very fast.
38
Figures (4.6 – 4.8) shows the tracking performance of angular displacement,
angular velocity and angular acceleration for the set 2. From the result obtained,
controller fail to track the desired positions if the controller parameters conditions are
unsatisfied.
Displacement vs Time
0.5
0.45
Displacement (rad)
0.4
0.35
0.3
0.25
0.2
0.15
0.1
Actual
Desired
0
0.2
0.4
0.6
0.8
1
1.2
Time (sec)
1.4
1.6
1.8
2
Figure 4.1 : Angular Displacement vs Time with satisfied control parameter
39
Velocity vs Time
1.5
1
Velocity (rad/sec)
0.5
0
-0.5
-1
Actual
Desired
-1.5
0
0.2
0.4
0.6
0.8
1
1.2
Time (sec)
1.4
1.6
1.8
2
Figure 4.2 : Angular Velocity vs Time with satisfied control parameter
Accelaration vs Time
8
6
Accelaration (rad/sec 2)
4
2
0
-2
-4
Actual
Desired
-6
-8
0
0.2
0.4
0.6
0.8
1
1.2
Time (sec)
1.4
1.6
1.8
2
Figure 4.3 : Angular Acceleration vs Time with satisfied control parameter
40
Control Input vs Time
14000
12000
Control input Xv
10000
8000
6000
4000
2000
0
0
0.2
0.4
0.6
0.8
1
Time
1.2
1.4
1.6
1.8
2
Figure 4.4 : Control Input vs Time with satisfied control parameter
Sliding Surface vs Time
0
-0.5
Sliding Surface
-1
-1.5
-2
-2.5
-3
-3.5
0
0.2
0.4
0.6
0.8
1
1.2
Time (sec)
1.4
1.6
1.8
2
Figure 4.5 : Sliding Surface vs Time with satisfied control parameter
41
Displacement vs Time
0.5
0.45
Displacement (rad)
0.4
0.35
0.3
0.25
0.2
Desired
Actual
0.15
0.1
0
0.2
0.4
0.6
0.8
1
1.2
Time (sec)
1.4
1.6
1.8
2
Figure 4.6 : Angular Displacement vs Time with unsatisfied control parameter
Velocity vs Time
1.5
1
Velocity (rad/sec)
0.5
0
-0.5
-1
Desired
Actual
-1.5
0
0.2
0.4
0.6
0.8
1
1.2
Time (sec)
1.4
1.6
1.8
2
Figure 4.7 : Angular Velocity vs Time with unsatisfied control parameter
42
Accelaration vs Time
8
6
Accelaration (rad/sec 2)
4
2
0
-2
-4
-6
-8
Desired
Actual
0
0.2
0.4
0.6
0.8
1
1.2
Time (sec)
1.4
1.6
1.8
2
Figure 4.8 : Angular Acceleration vs Time with unsatisfied control parameter
43
4.3 Simulation Using Independent Joint Linear Control (IJC)
Normally, Independent Joint Linear Control method is used in most industrial
robot. The performance of the Independent Joint Linear Control is used as a
comparison to the Full Order Sliding Mode Control proposed in this thesis.
This controller is designed with the dynamics of the mechanical linkage
completely ignored. Each joint of the robot arm is treated as an independent
servomechanism problem represented by an actuator state equation as follows:
X& (t ) = Aaci X i (t ) + Baci U i (t )
(4.13)
The element of matrices Aaci and Baci are calculated as follows:
Aac1
0
1
0
⎡
⎤
⎢
⎥
=⎢
0
0
1
⎥
⎢⎣− 0.3998 − 2999.4 − 169.9804⎥⎦
Bac1
⎡ 0 ⎤
= ⎢⎢ 0 ⎥⎥
⎢⎣0.8844⎥⎦
(4.14)
The linear state feedback controller employed in each of the subsystem is described
as follows:
U i (t ) = K i Z i (t ) + Ω i (t )
where,
Ki
- 1x3 linear state feedback gain
(4.15)
44
Ω i (t ) - control component to eliminate the steady state error
Z i (t ) - state vector of each sub-system
Z i (t ) = X i (t ) − X
d i
(t )
(4.16)
To ensure stability all desired poles are located in the left half plane.
Each of the sub-system has been assigned with the following closed-loop poles:
λ i ( Ai + Bi K i ) = {−2
− 1.5
− 1} ;
i=1,2
(4.17)
The feedback gains K are obtained using the pole placement method and the results is
as follows:
K = [− 0.45 − 3391.4 − 180.9]
(4.18)
From the simulation results as shown in Figure (4.9-4.11), it is very obvious that the
Independent Joint Linear Control fails to track the desired positions of displacement,
velocity and acceleration. Therefore, the simulation result confirmed that the
Independent Joint Linear Control is not suitable for electrohydraulic servo control
system control application.
45
Displacement vs Time
0.8
0.7
0.6
Displacement (rad)
0.5
0.4
0.3
0.2
0.1
0
Desired
Actual
-0.1
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Time (sec)
1.4
1.6
1.8
2
Figure 4.9 : Angular Displacement vs Time with IJC
Velocity vs Time
1.5
1
Velocity (rad/sec)
0.5
0
-0.5
-1
-1.5
Desired
Actual
-2
-2.5
0
0.2
0.4
0.6
0.8
1
1.2
Time (sec)
1.4
1.6
1.8
2
Figure 4.10 : Angular Velocity vs Time with IJC
46
Accelaration vs Time
10
5
Accelaration (rad/sec 2)
0
-5
-10
-15
-20
Desired
Actual
-25
-30
0
0.2
0.4
0.6
0.8
1
1.2
Time (sec)
1.4
1.6
1.8
2
Figure 4.11 : Angular Acceleration vs Time with IJC
47
4.3 Simulation Using Advanced Control [Yeoh Aik Seng, 1998]
An advanced control technique as described in [Yeoh Aik Seng, 1998] also
can be used and applied to control the system. The integrated model of the
electrohydraulic servo control system can be represented in the following form.
X& (t ) = A( x, t ) X (t ) + B( x, t )U (t )
= [ AN + ΔA( x, t )] X (t ) + [ BN + ΔB( x, t )]U (t )
(4.19)
where
x(t ) - state variables
U (t ) - control signal
AN and B N - constant matrices of appropriate dimension
ΔA( x, t ) - system matrix uncertainties
ΔB( x, t ) - input matrix uncertainties
Criterion to be satisfied:
rank ( B, ΔA) = rank ( B, ΔB) = rank ( B)
(4.20)
where
ΔA( x, t ) = BH ( x, t )
and
max || H ( x, t ) ||= 9.1189
(4.21)
ΔB( x, t ) = BE ( x, t )
and
max || E ( x, t ) ||= 0.3333
(4.22)
48
The error is
(4.23)
Z (t ) = X (t ) − X d (t )
The nonlinear controller, Φ (t ) will have the following form
⎧ μ (Z , t )
Φ ( Z , t ) = ⎨−
ρ (Z , t )
⎩ || μ ( Z , t ) ||
⎧ μ (Z , t )
= ⎨−
ρ (Z , t )
ε
⎩
if
if
|| μ ( Z , t ) ||> ε
(4.24)
|| μ ( Z , t ) ||≤ ε
where
ε - allowable steady state error and it is suggested that 0.001 ≤ ε ≤ 0.1
μ ( Z , t ) = B NT PL Z (t )
(4.25)
and
ρ ( Z , t )Δ[1 − max || E ( x, t ) ||]−1[max || H ( x, t ) X (t ) || + max || E ( x, t ) KZ (t ) ||]
(4.26)
The gain vector K using pole placement method
(4.27)
K = [ -28.9778 -0.4424 -3.9118-E]
PL is the solution of the Lyapunov equation
4.96 E 3 2.50 E − 9⎤
⎡ 2.48E 5
⎢
1.25 E 2 1.24 E − 3 ⎥⎥
PL = ⎢ 4.96 E 3
⎢⎣2.50 E − 9 1.24 E − 3 2.48E − 5⎥⎦
(4.28)
49
The simulation results for Advanced Control are shown in Figure (4.12-4.14).
From the results obtained, Advanced Control can track the desired positions of
displacement and velocity but cannot track the angular acceleration. Therefore, the
simulation result shows that Advanced Control is not really suitable for
electrohydraulic servo control system control for although the performance are better
than Independent Joint Linear Control.
Displacement vs Time
0.5
0.45
Displacement (rad)
0.4
0.35
0.3
0.25
0.2
Desired
Actual
0.15
0.1
0
0.2
0.4
0.6
0.8
1
1.2
Time (sec)
1.4
1.6
1.8
2
Figure 4.12 : Angular Displacement vs Time with Advanced Control
50
Velocity vs Time
1.5
1
Velocity (rad/sec)
0.5
0
-0.5
-1
-1.5
Desired
Actual
0
0.2
0.4
0.6
0.8
1
1.2
Time (sec)
1.4
1.6
1.8
2
Figure 4.13 : Angular Velocity vs Time with Advanced Control
Acceleration vs Time
50
Acceleration (rad/sec 2)
40
30
20
10
Desired
Actual
0
-10
0
0.2
0.4
0.6
0.8
1
1.2
Time (sec)
1.4
1.6
1.8
2
Figure 4.14 : Angular Acceleration vs Time with Advanced Control
51
CHAPTER 5
CONCLUSION AND SUGGESTION
5.1
Conclusion
The modelling and control of a electrohydraulic servo system is presented in
this thesis. Electrohydraulic servo system need a robust controller to handle the
highly nonlinear. A Full Order sliding mode control was designed to control the
system. Decompose the mathematical model into a system with bounded
uncertainties was the basis for Full Order sliding mode controller formulation.
The control law is formulated based on the assumption that the system uncertainties
and non-linear ties are bounded and these bounded are known. Sufficient condition
based on Lyapunov theory are provided to guarantee asymptotic tracking of a
desired position output. The Lyapunov theory proved that the proposed controller is
stable
In this project, the simulation results shows that the Full order sliding
mode controller can give good performance to track displacement, velocity and
acceleration.Basically, the objective of this project is achieved as the proposed
controller which is designed based on deterministic approach is able to control the
52
electrohydraulic position servo control system very successfully. For comparison
purpose, the electrohydraulic servo system was also simulated Independent Joint
Linear Control and Advanced Control. From the simulation results obtained, IJC
fails to track the desired position and Advanced Control not really suitable to
control the system.
5.2
Suggestion For Future Work
Currently, the performance of the proposed controller is evaluated based on
the simulation result. The voltage for control input is very high. Although in
theoretical the deterministic controller can give good performance but not suitable
for real application. Thus, for the future development, it is suggested that the
performance of the sliding surface and the voltage for control input need to be study
and investigate to make sure this controller can be applied under real application.
For the simulation, the suitable value for C and K are obtained based on try and
error. Research must be done to solve this problem.
53
REFERENCES
1.
Ahmad M.N., Osman J.H. S., (2000). “A Controller Design for Electrohydraulic
Position Servo Control System”, Proc. TENCON, Vol.3, pp 314-318.
2.
Ahmad M.N., Osman J.H. S., and Ghani M. R. A., (2002) “Sliding Mode Control
Of A Robot Manipulator Using Proportional Integral Switching Surface”, Proc.
IASTED Int. Conf. On Intelligent System and Control (ISC2002), Tsukuba, Japan,
pp 186-191.
3.
Ahmad M.N., (2003). “Modelling And Control Of Direct Drive Robot
Manipulators”, Univerti Teknologi Malaysia , PhD Thesis.
4.
Christopher Edwards and Sarah K. Spurgeon, (1998). “Sliding Mode Control:
Theory and Applications”, London: Taylor & Francis Group Ltd
5.
F.D.Norvelle,(2000). “Electrohydraulic Control System”, Prantice Hall: New
Jersey.
54
6.
Fung,R-H, Yang R-T., (1997). “Application of VSC in Position Control Of a
Nonlinear Electrohydraulic Servo System”, Pergamon, Vol 66, No 4, pp. 365372.
7.
Fung,R-H., Wang Y-C., Yang R-T., and Huang H-H., (1997). “A variable
Structure Control with Proportional And Integral Compensation For
Electrohydraulic Position Servo Control System”, Mechatronics, 7, pp 67-81.
8.
L-C-Hwang., (1996) “Sliding Mode Controller Using Time-varing Switching
Gain And Boundary Layer For Electrohydraulic Position And Different Pressure
Control”, IEEE. Proc-Control Theory Appl. Vol. 143, No 4.
9.
Miroslav Mihajlov., (2002). “Position Control Of An Electrohydraulic Servo
System Using Sliding Mode Control Enhanced By Fuzzy PI Controller “, FACTA
UNIVERSITATIS, Series Mechanical Engineering, Vol 1, No 9, pp. 1217-123.
10.
Q. P. Ha., H. Q. Nguyen., D.C. Rye., H.F. Durrant-Whyte., (1998). “Sliding
Mode Control with Fuzzy Tuning for an Electrohydraulic Position Servo System
“,IEEE Second International Conference on Knowledge-Based Intelligent
Electronic System, Vol 2, No 8, pp. 1516-873.
11.
Tzuen-Lih Chern., (1992) “ An optimal Variable structure Control with Integral
Compensation For Electrohydraulic Position Servo Control Systems” IEEE
Transaction On Industrial Electronics, Vol 39, No 5.
55
12.
Yeoh Aik Seng., (1998). “Advanced Control of Electrohydraulic Servo Control
System”, Universiti Teknologi Malaysia , PSM Thesis.
13.
Young,K-K.D., (1988). “A variable Structure Model Following Control Design
for Robotics Application.” IEEE journal of Robotics and Automation, Vol 4, No
5, pp. 556-561.
56
APPENDIX A
MATLAB PROGRAM LISTING
57
Program to calculate the value of A
ps=13.795
be=344.875
vt=0.0001639
k1=0.000002376
dm=0.000008195
jt=0.05652
bm=8.477
kv=0.508
g=0.00113
kgmin=0.00109595
kgmax=0.00175706
gn=1
x1min=-3.14
x2min=0
x1max=3.14
x2max=170.816
a11=(4*be*k1*g)/(jt*vt)
a21=[(4*be*dm*dm)+(4*be*k1*bm)+(g*vt)]/(jt*vt)
a31=[(4*be*k1*jt)+(bm*vt)]/(jt*vt)
58
Program to calculate the value of B and ΔB
ps=13.795
be=344.875
vt=0.0001639
k1=0.000002376
dm=0.000008195
jt=0.05652
bm=8.477
kv=0.508
g=0.00113
kgmin=0.00109595
kgmax=0.00175706
gn=1
x1min=-3.14
x2min=0
x1max=3.14
x2max=170.816
Bmax=(4*be*dm*kgmax*kv)/(jt*vt)
Bmin=(4*be*dm*kgmin*kv)/(jt*vt)
B=(Bmax+Bmin)/2
delB=B-Bmin
59
Program to calculate the value of W and ΔW
ps=13.795
be=344.875
vt=0.0001639
k1=0.000002376
dm=0.000008195
jt=0.05652
bm=8.477
kv=0.508
g=0.000113
gn=1
x1min=-3.14
x2min=0
x1max=3.14
x2max=170.816
wmin1=-[(4*be*k1*g*gn*x1min*x1min*x1min)+(3*g*gn*vt*x1min*x1min*x2min)
+(226.124*be*k1*x1min)+(56.526*vt*x2min)]/(jt*vt)
wmax=-[(4*be*k1*g*gn*x1max*x1max*x1max)+(3*g*gn*vt*x1max*x1max*x2max)
+(226.124*be*k1*x1max)+(56.526*vt*x2max)]/(jt*vt)
W=(wmin1+wmax)/2
delW=W-wmin1
60
APPENDIX B
SIMULINK
61
Simulink for Full Order Sliding Mode Controller
Overall
Err Pos
x12
2WS3
Position
Err Vel
D.Vel
Ctrl
x4
DesPos
2WS4
v
x2
X U1out
U1
Pos
X
2WS1
XdSout1
Vel
Plant
Full Order SMC
Acc
Desired
Ctrl1
Scope
x3
D.Acc
2WS2
t
Clock1
2WS
Err Acc
62
Full Order SMC
2
Sout1
x6
2WS6
Scope1
Scope
In1 Out1
SGN
f(u)
Fcn
CB(-1)
f(u)
1
s
PI
Integrator
CB(-I)
C
C
PI
PI[CA+CBK]
u[1]*u[2]
PI1
168000
alpha4
-85422
||W||
f(u)
2
Xd
Xd
Z
Omega
||Omega||
1
alpha3
Product3
0.3361
alpha2
Omega
U1out
1.7107
Product2
f(u)
1
X
Z
1.4432
alpha1
Product1
Product4
63
Full Order SMC/Omega
Xd_dot
1
Xd
Xd_dot
Xd
DERIVATIVE
AXd
u[2]
AXd_row1
u[3]
AXd_row2
f(u)
AXd_row3
(BTBinv )BT
(BTBinv)BT
f(u)
Omega1
1
Omega
64
Plant for Electrohydraulic Servo System
1
X
1
s
1
x3
Integrator
0.8844
U1
Gain3
-169.9804
Gain2
-2999.4
Gain1
-0.3998
Gain
f(u)
Fcn
1
s
Integrator1
x2
x1
1
s
Integrator2
65
Simulink for Independent Joint Linear Control (IJC)
Overall
Err Pos
x12
2WS3
Position
Err Vel
Ctrl
D.Vel
DesPos
velocity
x2
X1
U1
Pos
U1
X
2WS1
Xd1
Plant
Vel
IJC
Acc
Desired
Ctrl1
Accelaration
x3
D.Acc
2WS2
t
Clock1
2WS
Err Acc
66
IJC
em
f(u)
K1Z1
K1
em
1
U1
K
Xd1
Omega
Omega 1
2
Xd1
1
X1
67
IJC/ Omega 1
1
Xd
Xd1_dot
em
Xd1
DERIVATIVE
f(u)
u[2]
A1Xd1_row1
u[3]
Omega1
em
A1Xd1_row2
f(u)
A1Xd1_row3
(BTBinv )BT
(BTBinv)BT
1
Omega
68
Simulink For Advanced Control
Overall
Err Pos
x12
2WS3
Position
Ctrl
D.Vel
DesPos
Err Vel
v
x2
X
Pos
U1out
U1
X
2WS1
Xd
Vel
Plant
Advanced
Acc
Desired
Ctrl1
v1
x3
D.Acc
2WS2
t
Clock1
2WS
Err Acc
69
Advanced
f(u)
Fcn3
f(u)
Fcn4
max
0.3333
Product
MinMax
Constant
9.1189
K
Product5
max
Constant1
K
Product1
MinMax1
f(u)
1
Fcn2
U1out
1.49992
Constant2
2
Xd
Z
-100
f(u)
1
Product2
(BTN*PL)
X
PTNPL
Fcn1
Gain
f(u)
-1
Fcn5
Divide
Product4
Gain1
Switch