iii
AN OBSERVER DESIGN FOR ACTIVE SUSPENSION SYSTEM
ADIZUL AHMAD
This project report is submitted in partial of the fulfillment of the
requirement for the award of the Master of Engineering (Electrical - Mechatronics &
Automatic Control)”
Faculty of Electrical Engineering
Universiti Teknologi Malaysia
NOVEMBER, 2005
i
PSZ 19:16 (Pind. 1/97)
UNIVERSITI TEKNOLOGI MALAYSIA
BORANG PENGESAHAN STATUS TESISυ
JUDUL:
AN OBSERVER DESIGN FOR ACTIVE SUSPENSION SYSTEM
SESI PENGAJIAN:
Saya
2005/2006
ADIZUL BIN AHMAD
(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 sebagai bahan 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
(TANDATANGAN PENULIS)
(TANDATANGAN PENYELIA)
Alamat Tetap:
NO 34. KG MANGGOL CHENGAI, MUKIM
JABI, 06400 POKOK SENA,
KEDAH DARULAMAN
PROF.MADYA DR. YAHAYA MD
SAM
Nama Penyelia
Tarikh:
Tarikh:
11 NOVEMBER 2005
CATATAN:
*
**
υ
11 NOVEMBER 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).
ii
“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 the degree of Master of Engineering (Electrical - Mechatronics & Automatic
Control)”
Signature
: .…………………………………………….
PROF. MADYA DR. YAHAYA MD SAM
Name of Supervisor : ………………………………………………
11 NOVEMBER 2005
Date
: ……………………………………………...
iv
DECLARATION
‘I declare that this thesis entitled “An Observer Design for Active Suspension System” is
the result of my own research expect as cited in references. The thesis has not been
accepted for any degree and is not concurrently submitted in candidate of any degree.”
Signature
: __________________
Name of Candidate
: ADIZUL AHMAD
Date
: November 30, 2005
v
DEDICATION
This thesis is dedicated to my beloved parent, Ahmad Bin Razak and Halimah Binti Haji
Dris, my sisters, Noormadiad and Zuhana,to my brother Azman and not forgotten to all
my friend, thank you for the sacrifice and support.
May Allah bless all of you. Amin!
vi
ACKNOWLEDGEMENT
In the name of ALLAH SWT, the most Gracious, who has given me the strength
and ability to complete this study. All perfect praises belong to ALLAH SWT, lord of
the universe. May His blessing upon the prophet Muhammad SAW and member of his
family and companions.
I gratefully acknowledge the co-operation of Associate Professor Dr. Yahaya
Md. Sam who has provided me with the reference, guidance, encouragement and support
in completing this thesis. All the regular discussion sessions that we had throughout the
period of study have contributed to the completion of this project.
Thank you to my classmate for providing an enjoyable study environment.
Finally, I would like to thank my family for their encouragement, support and patience.
vii
PUBLICATION
The following paper, based on the work described in this thesis, has been
submitted for publishing:
1. Adizul Ahmad, Yahaya Md. Sam and Nor Maniha Abd. Ghani . “An
Observer Design for Active Suspension System” Accepted for oral
presentation
in
The
International
Conference
on
Mechatronics
Technology (ICMT05), Kuala Lumpur, Malaysia, 5-8 December 2005.
viii
ABSTRACT
The purpose of this project is to construct an active suspension control for a
quarter car model subject to excitation from a road profile using an improved sliding
mode control with an observer design. The proportional-integral sliding mode was
chosen as a control strategy, and the road profile is estimated by using an observer
design. The performance of controller will be compared with the LQR controller. There
are three parameters to be observed in this study namely, wheel deflection, the body
acceleration and the suspension travel. The performance of this controller will be
determined by performing computer simulations using the MATLAB and SIMULINK
toolbox.
ix
ABSTRAK
Cadangan untuk projek ini adalah untuk membina model sistem gantungan aktif
yang subjek keatas permukaan jalan dengan menggunakan kawalan ragam gelincir yang
di pertingkatkan keupayaan dengan penggunaan pemerhati. Kawalan ragam gelincir
berkadaran-kamiran di pilih sebangai pengawal dan permukaan jalan di perhati dengan
menggunakan pemerhati. Prestasi pengawal yang di perkenalkan di analisis prestasi
dengan pegawal jenis LQR. Tiga parameter output diperhatikan yang di namai, pantulan
tayar, halaju badan dan pergerakan gantungan. Prestasi pengawal akan di analisis dengan
menggunakan penyelakuan komputer aturcara MATLAB dengan SIMULINK yang
mudah di guna pakai.
x
TABLE OF CONTENTS
CHAPTER
TITLE
PAGE
DECLARATION
ii
DEDICATION
iii
ACKNOWLEDGEMENT
iv
PUBLICATION
v
ABSTRACT
vi
ABSTRAK
x
TABLE OF CONTENT
viii
LIST OF TABLE
xvi
LIST OF FIGURES
xvii
1
2
INTRODUCTION
1.1
Suspension System
1
1.2
Active Suspension System Control Strategies
5
1.3
Objective of Study
5
1.4
Scope of Work
6
1.5
Structure and Layout of Thesis
7
SYSTEM MODEL
2.1
Introduction
8
2.2
Car Suspension System
9
xi
3
4
5
2.2.1
Passive Suspension System
9
2.2.2
Active suspension System
12
2.3
Road Profile
16
2.4
Conclusion
17
PROPORTIONAL INTEGRAL SLIDING MODE CONTROL
3.1
Introduction
18
3.2
Proportional Integral Sliding Mode Control Design
19
3.2.1
Switching Surface design
20
3.2.2
Stability During Sliding Mode
22
3.2.3
Controller Design
24
3.3
Improve Performance
25
3.4
Conclusion
26
OBSERVER DESIGN
4.1
Introduction
27
4.2
Literature Review on Observer Design
27
4.3
Observer design
29
SIMULATION
5.1
Simulations
5.1.1
Performance of PISMC on Road Handling and Ride
Comfort
5.1.2
46
Effect of the reaching Mode condition on varying Design
Parameter, ρ
5.2
36
Effect of the Suspension Performance on Various
Road Profiles
5.1.3
34
Conclusion
49
53
xii
6
CONCLUSION AND SUGGESTIONS
6.1
Conclusion
54
6.2
Suggestion For Future Research
55
REFERENCES
56
xiii
LIST OF TABLES
TABLE
TITLE
PAGE
2.1
Parameter value for quarter car model
9
xiv
LIST OF FIGURES
FIGURE
TITLE
PAGE
1.1
The passive suspension system
2
1.2
The semi-active suspension system
3
1.3
The active suspension system
4
2.1
The passive suspension system
10
2.2
The active suspension for the quarter car model
13
2.3
Road profile represented a 10 cm bump
16
5.1
Road profile that generated to the system
38
5.2
Estimated road profile for the system using PISMC
39
5.3
Estimated road profile for the system using LQR controller
39
5.4
Wheel deflection fro active suspension system
40
5.5
Body acceleration for active suspension system
40
5.6
Suspension travel of active suspension
41
5.7
Sliding surface
41
5.8
Control input ot the active suspension
42
5.9
Wheel deflection between plant observer using LQR controller
43
5.10
Wheel deflection between plant observer using PISMC
43
5.11
Body acceleration between plant observer using LQR controller
44
5.12
Body acceleration between plant observer using PISMC
44
5.13
Suspension travel between plant observer using LQR controller
45
5.14
Suspension travel between plant observer using PISMC
45
5.15
Body acceleration for active suspension system for road profile 1
46
5.16
Suspension travel of active suspension for road profile 1
47
5.17
Wheel deflection of active suspension for road profile 1
47
xv
5.18
Body acceleration for active suspension system for road profile 2
48
5.19
Wheel deflection of active suspension for road profile 2
48
5.20
Suspension travel of active suspension for road profile 2
49
5.21
Body acceleration for varying the parameter
ρ( positive sliding gain)
5.22
Body acceleration for varying the parameter
ρ( negative sliding gain)
5.23
51
Wheel deflection for varying the parameter
ρ( positive sliding gain)
5.26
51
Suspension travel for varying the parameter
ρ( negative sliding gain)
5.25
50
Suspension travel for varying the parameter
ρ( positive sliding gain)
5.24
50
52
Wheel deflection for varying the parameter
ρ( negative sliding gain)
52
1
CHAPTER 1
INTRODUCTION
1.1
Suspension System
Comfort and road-handling performance are the characteristics that have to be
considered in order to achieve a good suspension system. Ideally, the suspension should
isolate the body from road disturbance and inertial disturbances associated with
cornering and braking or acceleration. The suspension must also be able to minimize the
vertical force transmitted to the passengers for their comfort. This objective can be
achieved by minimizing the vertical car body acceleration [1].
Generally, a suspension system for a car is categorized as passive, semi-active
and active suspension system. A passive system is a conventional suspension that
consists of the conventional spring and shock absorber as shown in figure 1.1. The
springs are assumed to have almost linear characteristics; while, most of the shock
absorbers exhibit nonlinear relationship between force and velocity. In passive system,
these elements have fixed characteristics. When the suspension system must operate
over a wide range of conditions, there should be a compromise in choosing the spring
2
stiffness and damping parameters. In other words, it is desirable that the above
parameters could be adjustable to improve the performance of the system.
Figure 1.1: The passive suspension system.
The performance of passive suspension system can be improved by using the
semi-active suspension system as shown in figure 1.2. Semi-active suspensions provide
real-time dissipation of energy, which is achieved through a mechanical device called an
active damper. The active damper is installed in parallel with a conventional spring. The
main feature of this system is the ability to adjust the damping of the suspension system,
without any use of actuator. This type of system requires some form of measurement
with a controller board in order to properly tune the damping force.
3
Figure 1.2: The semi-active suspension system
The demand of better ride comfort and controllability of road vehicles has
motivated many automotive industries to consider the use of active suspension. Active
suspension employs pneumatic or hydraulic actuators which in turn creates the desired
force in suspension system as shown in figure 1.3. The actuator is secured in parallel
with spring and shock absorber. Active suspension requires 2 accelerometers that
mounted at sprung and unsprung mass, and a unit of displacement transducer to measure
the motions of the body, suspension system and the unsprung mass. This information is
used by the online controller to command the actuator in order to provide the optimum
target force. Active suspension may consume large amounts of energy in providing the
control force. Therefore, in the of active suspension system the power consumptions of
actuator should also be considered as an important factor. In analysis of suspension
system, there are varieties of performance criteria which need to be optimized. There are
three performances criteria which should be considered carefully in designing a
4
suspension system; namely, body acceleration, suspension travel and wheel deflection.
The performance of the system can be further improved by introducing the suitable
controller into the active suspension system.
Figure 1.3: The active suspension system
1.2
Active Suspension System Control Strategies
The objective of the active suspension system is to improve the suspension
system performance by directly controlled the suspension forces to suit with the
performance characteristics. There are various linear control strategies have been
established by previous researchers in the design of the active suspension system.
Amongst them are a fuzzy reasoning [4], robust linear control [7], H∞ [8], and adaptive
5
observer [9]. The results of the study have demonstrated that active suspension system is
more effective in the vibration isolation of the car body is compared with passive
system.
The purpose of this study is to utilize the concept of proportional integral sliding
mode control in active suspension system [1, 2, 5, 6,] with observer design for road
estimator. Therefore, the estimated road disturbance will be another state for the system.
1.3
Objective of study
The objectives of this research are as follows:
I. To develop the mathematical model of quarter car active suspension system with
an observer.
II. To design an observer that act as a road profile estimator for the active
suspension system.
III. To design a Proportional Integral Sliding Mode Control (PISMC) for the quarter
car active suspension system.
IV. To study the performance of the PISMC with a disturbance observer.
Theoretical verification of the proposed controller and observer on its stability and
reachability will be accomplished by using a Lyapunov’s second method. The
performance of the active suspension system will be observed by using extensive
computer simulation that will be performed using MATLAB software and SIMULINK
Toolbox subjected to various types of road profiles and parameter value.
6
1.4
Scope of work
The scopes of work for this study are as follows:
I. Familiarization of a quarter car active suspension system with ProportionalIntegral Sliding Mode Controller
II. Mathematical derivation of an observer for a quarter car active suspension
system.
III. Design an Observer for a quarter car active suspension system using Sliding
Mode Control technique.
IV. Perform a simulation works by using a MATLAB/Simulink to observe
effectiveness of the observer.
V. Compare the performance of the proposed PISMC with Observer design with
Validate with a Linear Quadratic Regulator (LQR) technique.
1.5
Structure and Layout of Thesis
The necessary components for achieving objective of the study are given in the
succeeding chapters. The mathematical modeling of a quarter car is derived in chapter 2.
The state space representation of the dynamic model of the passive, semi active and
active suspension system are outlined. Finally, various road profiles that represent the
uncertainty in the suspension systems will be represented.
In chapter 3, the proposed controller, proportional integral sliding mode control
with disturbance observer is presented. By using a Lyapunov’s stability theory, it will be
shown that for the system with mismatched uncertain, the system is bounded stable. In
chapter 4 discussion on an observer design is performed. The literature review and
mathematical model will discuss in this chapter.
7
Chapter 5 represents the computer simulation for the proposed observer and
controller. In this chapter, the performance of road handling and ride comfort will be
compared with the LQR controller. To assure the robustness of the proposed observer,
various road profiles will be applied in the system. The chattering and boundary layer
effect of the controller will be presented by varying the parameter in the continuous
switching gain.
The summary of the result and future research based on this study will be
presented in chapter 6.
8
CHAPTER 2
SYSTEM MODEL
2.1
Introduction
Generally, the vehicle suspension models are divided into three types namely
quarter car, half car and full car models. In the quarter car model, the model is based on
the interaction between the quarter car body and the single wheel. The motion of the
quarter car model is only in the vertical direction and for the half car model; the
interactions are between the car body and the wheel and also between both ends of the
car body. The first interaction in the half car models caused the vertical motion and the
second interaction produced an angular motion. In full car model, the interactions are
between the car body and the four wheels that generated the vertical motion, between the
car body and the left and right wheels that generated an angular motion called rolling
and between the car body and the front and rear wheels that produced the pitch motion
[1].
9
2.2
Car Suspension system
In this section, the modeling of the passive car suspension systems will be
discussed. In order to fulfill the objectives of the study, the quarter car active suspension
will also be described.
2.2.1
Passive Suspension System
Figure 2.1 shows the quarter car model of the passive suspension system in
which the single wheel and axle are connected to the quarter portion of the car body
through combination of the passive spring and passive damper. Furthermore, the tyre is
modeled as a simple spring without damper. The parameters of the passive suspension of
the quarter model components as presented in Hedrick et al. (1994) and also in Alleyne
and Hedrick (1995) is shown is Table 2.1.
Table 2.1: Parameter value for the quarter car model
Mass for the car body, mb
290 kg
Mass for the car wheel, mw
59 kg
Stiffness of the car body spring, kb
16812 N/m
Stiffness of the car tyre, kw
190000 N/m
Damping of the damper, cb
1000 Ns/m
The motion equations of the passive suspension system of the quarter car model
from figure 2.1 are given by:
mb &x&b + cb ( x& b − x& w ) + k b ( xb − x w ) = 0
(2.1)
mw &x&w + cb ( x&w − x&b ) + kb ( xw − xb ) + k w ( xw − w) = 0
(2.2)
10
where
mb = mass for the car body (kg)
mw = mass for the car wheel (kg)
kb = stiffness of the car body spring (N/m)
kw = stiffness of the car tyre (N/m)
cb = damping of the damper (Ns/m)
xb = vertical displacement of the car body (m)
xw= vertical displacement of the car wheel (m)
w = an irregular excitation from the road surface (m).
In the model, it is assumed that all the suspension components are linear.
Figure 2.1: The passive suspension system
The motion equation of the passive suspension system for the quarter car model may be
represented by a vector matrix form as follows:
11
MX&& (t ) + SX& (t ) + TX (t ) = EW (t )
(2.3)
Where the state and the excitation vectors are, respectively, given by
X (t ) = [xb
xw ]
T
(2.4)
W (t ) = w(t )
(2.5)
and the matrices of M, S, T and E, respectively, are given as follows:
m
M = b
0
0
mw 
(2.6)
c
S= b
− cb
− cb 
cb 
(2.7)
k
T = b
 − kb
− kb 
(kb + k w )
(2.8)
0
E= 
k w 
(2.9)
Equation (2.3) can be rewritten in the state space form as:
x& p (t ) = Ap x p (t ) + Fp w(t )
(2.10)
where
x p (t ) = [x1 (t ) x2 (t ) x3 (t ) x4 (t )]
T
= [x&b (t ) x&w (t )
xb (t ) xw (t )]
T
(2.11)
12
 cb
− m
b

cb

Ap =
 mw
 1

 0
cb
mb
c
− b
mw
0
1
kb
mb
cb
mw
0
−
0
kb


mb

(k + k w ) 
− b
mw 

0

0

 0 
 kw 
 
Fp =  mw 
 0 
 0 
2.2.2
(2.12)
(2.13)
Active suspension system
The active suspension system has the additional advantages whereby a
negative damping can be produced and that a larger range of forces can be generated at
low velocities. Hence, it potentially will increase the suspension system performance
[7]. The quarter car active suspension is illustrated in figure 2.2. The motion equations
of the active suspension system for the quarter car model are as follows:
mb &x&b + cb ( x&b − x&w ) + kb ( xb − xw ) − f a = 0
(2.14)
mw &x&w + cb ( x&w − x&b ) + kb ( xw − xb ) + k w ( xw − w) + f a= 0
(2.15)
where fa is a force input from the actuator.
The motion equations (2.14) and (2.15) can be represented in a vector matrix
form as:
MX&& (t ) + SX& (t ) + TX (t ) = FU (t ) + EW (t )
(2.16)
Where X(t) and W(t) are as stated in equations (2.4) and (2.5) while
U(t) = fa(t)
(2.17)
13
The matrices M, S, T and E are similar as in equation (2.6) to (2.9), where
1
F= 
− 1
(2.18)
Figure 2.2: The active suspension for the quarter car model
Therefore, the state space representation of the active suspension for the quarter car
model can be written as
(2.19)
x&a (t ) = Aa xa (t ) + Bau (t ) + Da w(t )
where
xa (t ) = [x1 (t ) x2 (t ) x3 (t ) x4 (t ) x5 (t )]
T
T
= [x&b (t ) x&w (t ) xb (t ) xw (t ) w(t )]
(2.20)
14
c
k
k
 cb

b
b
− b
0 
− m
m
m
 b mb

b
b
(k + k ) k 
c
k
 c
b
b
b
w
− b
− w

Aa =  m
m
m
m
m
 w
w
w
w
w
 1
0
0
0
0 
 0
1
0
0
0 


0
0
0
0 
 0
(2.21)
 1 
 m 
 b 
− 1 
Ba =  mw 
 0 


 0 
 0 
(2.22)
0
0
 
Da = 0
 
0
1
(2.23)
Equation (2.19) can be rewritten in the form of:
 cb
−
 &x&b (t )   m
 &x& (t )  c b
 w   b
 x& (t )  =  m
 b   w
 x& w (t )  1
 w& (t )   0



 0
cb
−
mb
cb
mw
0
1
0
k
− b
mb
kb
mw
0
0
0
kb
−
mb
(k w + k b )
mw
0
0
0

 1 
0 
 m 
&
(
)
x
t
0 


 b
 b 
0 



k w  x& w (t )  1 
 

−


m w   xb (t )  +  m w u (t ) + 0 w(t )
 


0   x w (t )  0 
0 


0
1
0   w(t )  
 


0




0 
(2.24)
where mb and mw are the masses of the body and wheel. xb and xw are the displacements
of car body and wheel respectively, kb and kw are the spring coefficients. cb is the damper
coefficient and w (t) is the road disturbance. The control force, u(t) from the hydraulic
15
system is assumed as the control input to the suspension system. Equation (2.19) shows
that the disturbance input is not in phase with the system input, i.e. rank [B] ≠ rank [B,
D], therefore the system suffers from mismatched condition. Hence, the proposed
controller must be robust enough to overcome the mismatched condition so that the
disturbance would not have significant effect on the performance of the system. The
following assumptions are taken as standard:
Assumption 1.
There exists a known positive constant such that f (t ) ≤ β , where • denotes the standard
Euclidean norm.
Assumption 2.
The pair (A, B) is controllable and the input matrix B has full rank.
2.3
Road Profile
The road disturbance w (t) representing a single bump may be given by the
following equation:
(a(1 − cos 8πt )) / 2 t1 ≤ t ≤ t2
w(t ) = 
otherwise
0,

(2.25)
Where a is the height of the bump, t1 and t2 are the lower and the upper time limit
of the bump. Figure 2.3 shows the bump height for 10 cm. Road profile 1 is represented
by a single bump on the road with 10 cm bump height. The mathematical model of the
bump for such road is given by as:
16
(a (1 − cos 8πt )) / 2 2.25 ≤ t ≤ 2.5
w(t ) = 
otherwise
0,

(2.26)
Where a = 10 cm and lower time limit is 2.25 second and the upper time limit is
2.5 second. Figure 2.4 shows the road profile.
Figure 2.4: road profile represented a 10 cm bump.
2.4
Conclusion
In this chapter, the mathematical model for passive and active quarter car
suspension system have been derived. The complete model of quarter car active
suspension system will be used in the propose observer and controller. The simulation of
quarter car active suspension system will use the estimated road profile as the
disturbance and study the performance of the system when using the proposed an
observer and controller.
17
CHAPTER 3
PROPORTIONAL – INTEGRAL SLIDING MODE CONTROL DESIGN FOR
QUARTER CAR ACTIVE SUSPENSION SYSTEM
3.1
Introduction
In recent years, the different approaches to the problem of robust control design
for uncertain mismatched condition have been proposed [1,2,4,5,6,7,8,9]. In this study,
Proportional Integral Sliding Mode Control (PISMC) with disturbance observer has been
proposed to control quarter car active suspension system. The proposed PISMC with
disturbance observer may be directly applied to solve the systems with mismatched
condition whereby the conventional sliding mode control has limitation to do so. In this
chapter, the proposed control law is presented to control the quarter car active
suspension system. In the mathematical development of the proposed controller, the
stability, reachability and the sensitivity factors of the proposed controller will also be
considered and presented. It will be shown through computer simulation that the
proposed controller is robust in overcoming the suspension problem with mismatched
condition.
18
3.2
Proportional Integral Sliding Mode Control Design
In this study, the proportional integral sliding mode control design is presented to
control the quarter car active suspension system. The design is started with the selection
of the sliding surface that can ensure the stability of the system. Then, the controller
based on the PISMC technique is designed to drive the state trajectory onto the sliding
surface. It will be proven mathematically that the reachability condition of the proposed
controller is achieved if the proposed theorem is satisfied to ensure that the state
trajectory slides onto the sliding surface and remains thereafter.
Throughout the study, Z denotes the Euclidean norm when Z is a vector:
Z == Z T Z
(3.1)
and an induced norm when Z is a matrix:
Z == λ max Z T Z
(3.2)
Where λmax(Z) is the largest eigenvalue of the matrix Z.
The state space representation of the quarter car active suspension system in
equation (2.19) can be written into the following form:
x& (t ) = Ax(t ) + Bu (t ) + f ( x, t )
(3.3)
Where x(t ) = ℜ n is the state vector, u (t ) = ℜ m is the control input, and the continuous
function
f(x,t)
represents
rank[ B f (t )] ≠ rank[ B ] .
the
uncertainties
with
mismatched
condition,
i.e
The following assumptions are taken as standard throughout this
study:
i)
The state vector x(t) is fully observable.
ii)
There exists a known positive constant β such the f ( x, t ) ≤ β
19
iii)
The pair (A, B) is controllable and the input matrix B has full rank.
The first in design PISMC is to design sliding surface.
3.2.1
Switching Surface Design
In this study proportional integral sliding mode is chosen based its ability to
overcome the steady state error in the system [1,2,3]. The PI sliding surface is defined as
follows [1, 2, 3];
t
∫
σ (t ) = Cx (t ) − (CA + CBK ) x (τ ) dτ
(3.4)
0
where C ∈ ℜ m×n and K ∈ ℜ m×n are constant matrices. The matrix C is chosen such that
CB ∈ ℜ m×n is nonsingular. The matrix K is chosen such that λ (A + BK) < 0. It is well
known that if the system is fulfilled the sliding mode condition, hence σ(t) =0.
Then, the differential of Equation 3.4 gives,
σ& (t ) = Cx& (t ) − (CA + CBK ) x (t )
(3.5)
Substituting Equation (3.3) into Equation (3.5) gives,
σ& (t ) = C[ Ax (t ) + Bu (t ) + f ( x, t )] − [CA + CBK ] x (t )
= CAx (t ) + CBu (t ) + Cf ( x, t ) − CAx (t ) − CBKx (t )
= CBu (t ) + Cf ( x, t ) − CBKx (t )
(3.6)
20
Equating Equation 3.8 to zero gives the equivalent control, u eq (t ) as follows,
ueq (t ) = Kx (t ) − [CB ]
−1
Cf ( x, t )
(3.7)
Substituting Equation 3.7 into Equation 3.3 gives the equivalent dynamics equation of
the system in sliding mode as follows,
−1
x& (t ) = Ax (t ) + B[ Kx (t ) − [CB ] Cf ( x, t )] + f ( x, t )
−1
x& (t ) = Ax (t ) + BKx(t ) − B[CB ] Cf ( x, t )] + f ( x, t )
−1
x& (t ) = [ A + B ] x (t ) − [ I − B[CB ] C ] f ( x , t )
n
(3.8)
It can be observed in equation 3.8 that the dynamics of the system during sliding
mode is affected by the desired closed loop poles and the mismatched uncertainties. The
desired closed loop poles may be specified by the designer by a proper selection of the
gain matrix K. the effect of the uncertainties in mismatched condition may be minimized
by a proper tuning of the matrix C. During the sliding mode, the uncertain system with
mismatched condition is stable provided that the following theorem is satisfied [1, 2, 3].
21
3.2.2 Stability during Sliding Mode
During the sliding mode, the uncertain system with mismatched condition is
stable provided the following theorem is satisfied.
Theorem 1
The uncertain system equation (3.3) is bounded stable on the sliding surface, σ (t)
= 0 if,
~
−1
F ( x, t ) ≤ β1 , where β1 I n − B (CB ) C β
and
F ( x, t ) = {I − B (CB )
n
−1
C} f ( x , t )
(3.9)
Proof:
For simplify, let
~
A = ( A + BK )
(3.10)
~
−1
F (t ) = {I − B (CB ) C} f ( x, t )
n
(3.11)
And rewrite Equations 3.10 and 3.11 as
~
~
x& (t ) = A x (t ) + F ( x , t )
(3.12)
22
Let Lyapunov function candidate for the system is chosen as
T
V (t ) = x (t ) Px (t )
(3.13)
Taking the derivative of V(t) gives
~T
~
~T
T
T ~
V& (t ) = x (t )[ A P + PA ] x (t ) + F ( x, t ) Px (t ) + x PF ( x, t )
~T
~T
T
T
= − x (t )Qx (t ) + F ( x, t ) Px (t ) + x (t ) PF ( x, t )
~
(3.14)
~
where P is solution of AT P + PA = −Q for a given positive definite symmetric matrix Q.
It can be shown that Equation (3.14) can be reduced to
V& (t ) = −λ
min
(Q ) x (t )
2
(3.15)
+ 2 β P x (t )
1
Since λmin(Q)>0, consequently V& (t ) < 0 for all t and x ∈ B c (η ) , where B c (η ) is the
complements of the closed ball B(η), centered at x =0 with radius η = [ 2 β P ] /[λ
1
min
(Q )] .
Hence the system is boundedly stable.□
Remark. For the system with uncertainties satisfy the matching condition, i.e. rank [B|f
(t)] = rank [B], then equation (3.8) can be reduced to x (t ) = [ A + Bk ] x (t ) [10]. Thus
asymptotic stability of the system during sliding mode is assured.
Now the design of the control scheme that drives the state trajectories of the
system in equation 3.8 onto the sliding surface σ (t) = 0 and the system remains in it
there after will be presented.
23
3.2.3
Controller design
The objective in the designing the control scheme is to drives the state
trajectories of the system in Equation 3.8 onto the sliding surface σ(t) and the system
remains in it thereafter. In order to fulfill the reaching condition, the control input u(t) is
divided into two part as described in equation 3.16. A linear component which equal to
the switching control input, ueq(t), and a nonlinear component which is equal to the
switching control input, unl(t)
u (t ) = u eq (t ) + u nl (t )
(3.16)
For the uncertain system in Equation 3.8 satisfying assumptions 1 and 2, the
following control law is proposed: [1]
u (t ) = Kx (t ) − (CB )
−1
Cf ( x, t ) − (CB )
−1
ρ
σ (t )
σ (t ) + δ
(3.17)
Where δ is the boundary layer thickness that to be selected to reduce the chattering
effect and ρ is a design parameter which will be specified by designer.
Theorem 2
The hitting condition of the sliding surface (4) is satisfied if
A + BK x (t ) ≥ f (t )
Proof.
The reaching condition is evaluated as follows,
σ (t )σ& (t ) = σ (t )[Cx& (t ) − [CA + CBK ] x (t )
(3.18)
24
= σ (t )[C{ Ax (t ) + Bu (t ) + f ( x, t )} − {CA + CBK }x (t )
(3.19)
= σ (t )[CBu (t ) + Cf ( x, t ) − CBKx (t )
Substituting Equation 3.17 into Equation 3.19, gives
σ (t )σ& (t ) = σ (t )[CB[ Kx (t ) − (CB )

= σ (t )  − ρ

−1
Cf ( x, t ) −
(CB )
−1
ρ
σ (t )
σ (t ) δ
] + Cf ( x, t ) − CBKx (t )]


σ (t ) + δ 
σ (t )
(3.20)
The sliding condition is established if ρ > 0 . The theorem 2 ensures that the proposed
control law drives the system state trajectory onto the sliding surface and remains on it
thereafter.
3.3
Improvement Performance
Improvement the ride comfort is a serious problem in the design of quarter car
active suspension system. In order to improve the performance of active suspension was
discuss by Yoshimura et al (2002). It is necessary to define that
A* = A −
TH
R
(3.21)
Where A matrix from equation 3.3 and R is weighting factor and matrix H and T are
respectively given by
H = [H 11
H 12
H 13
H 14
H 15 ]
(3.22)
25
T = [T11 T12
T13
T14
T15 ]
T
(3.23)
Assuming that (A*,B) satisfies the controllability condition.
3.4
Conclusion
The mathematical development and stability theorem of proposed controller have
been outlined in this chapter. The system is classified as mismatched condition with
bounded uncertainties. The proposed controller with disturbance observer will be
simulating in the next chapter and the performances of the quarter car active suspension
compare LQR control technique.
26
CHAPTER 4
OBSERVER DESIGN
4.1
Introduction
For any practical design of active suspension system, one of the main issues is its
sensor requirement. The method can minimize the sensor requirement is using the state
observer. In recent years, some researchers have applied an optimal state observer based
on a Kalman filter to active suspension system. However, a main issue is remained in
designing suitable state observers for the active suspension application is, how to select
the best measurement signals to achieve good estimation accuracy and meanwhile to
ensure the system stable.
4.2
Literature Review on Observer Design
In design of control system, it is assumed that all state variables are available for
feedback. Unfortunately, not all state variables are available for feedback in practice.
Then the unavailable state variables need to be estimated. The estimation of
unmeasurable state variable is commonly called state observer or simply an observer. If
the state observer observes all the state variables of the system, it is called a full order
state observer. An observer that estimates fewer than n state variables, where n is the
27
dimension of the state vector is called a reduced order state vector or reduced order
observer. In this study, the state variables with the road profiles are estimated..
Application of an observer to the active suspension system has been proposed by
Rajesh Rajamani and J. Karl Hedrick(1995). In their study, the adaptive observer was
developed for a class of nonlinear system. A realistic model of the suspension system
incorporating the dynamic of the hydraulic actuator was used. The observer was used to
adapt on dry friction which is usually present in significant magnitudes in hydraulic
actuator and can also be used to adapt on spring stiffness, viscous damping and
hydraulic bulk modulus. In their study, the special adaptive observer was proposed for
identification of the sprung mass of the automobile.
Taghirad et al. (1995) proposed the optimal controller design and optimal
observer design to control the half car active suspension system. In their study, two
different performance indices for optimal controller design was proposed. The
performance index is a quantification of the both ride comfort and road handling. The
result shows the optimal observer based controller, when including passenger
acceleration in the performance index, retains both excellent ride comfort and road
handling characteristic.
In designing a state observer for the quarter car active suspension system
application, the suitable selection of measurement signal is one of the important issues
to achieve good estimation accuracy and meanwhile to ensure the system stable and also
with the minimum sensor requirements. Fan Yu et al.(1999) proposed the estimation
accuracy along with the robustness performance of a Kalman filter active vehicle
suspension system. In their study the five different combinations of the measured states
were compared.
Yoshimura et al. (2001) represent the system used a pneumatic actuator to
generate the force input signal to control the active suspension. The road profile is
estimated by using the minimum order observer based on a linear system transformed
from the exact nonlinear system. The sliding mode control techniques used as are
28
control technique. The result shows the proposed controller combination with observer
more effective than LQ control theory.
Another study from Yoshimura et al. (2002) is using VSS observer for the active
suspension system. In their study, the sliding mode control was using as control scheme.
The active control is derived as a sum of LQ control and the nonlinear switching control,
where the LQ control is obtained by taking the acceleration of the car body seriously in
the performance index. The active control force is obtained by using a pneumatic
actuator and the road profile is estimated by using proposed simplified VSS observer.
The proposed active suspension system more improved the vibration isolation of the car
body than linear active suspension system based on the LQ control.
The problem of state and input estimation for class of uncertain systems is
studying by Martin Corless and Jay Tu(1998). In their study, the disturbance is estimated
using the proposed estimator. +Thus in this study, the complete formulation an observer
design is based on the formulation presented by Martin Corless and Jay Tu. The
following section will show the mathematical model for observer design and theorem for
determine the stability.
4.3
Observer Design
In this study, the estimator is described by,
x&ˆ (t ) = Axˆ (t ) + Bu (t ) + Gwˆ + L( y − Cxˆ )
(4.1)
wˆ = γK ( y − Cxˆ )
(4.2)
and consider here uncertain systems described by
x& (t ) = Ax(t ) + Bu (t ) + Dw(t )
(4.3)
29
y (t ) = Cx(t )
(4.4)
Where x(t ) ∈ ℜ n the system is state and y (t ) ∈ ℜ p is the measured output at time t ∈ ℜ .
The system is uncertain. The matrices A, B, C, and D are known constant real matrices
of appropriate dimensions. The following assumptions are taken as standard:
Assumption 1. rank (CB) = rank B
Under full-state measurement(C=I) the above assumption is clearly satisfied.
This assumption implies that p ≥ rank B, that is the number of scalar measurement is
greater than or equal to the number of effective scalar inputs.
From the system,
•
rank(CB) = 1
•
rank B = 1
It show s the system satisfied with assumption 1
Assumption 2.
For every complex number λ with nonnegative real part,
 A − λI
rank 
 C
B
= n + rankB
0 
 A − λI
rank 
 C
B
=6
0 
Satisfied assumption 2.
Assumption 3
- w are bounded functions
30
Assumption 4
-The pair (A, B) is controllable
Assumption 5
- The pair (A, C) is observable
Assumption 6
D in equation (4.3) is a bounded function. That is, there is η > 0 so that D ≤ η .
Using the Luenberger (1971) state estimator of the form
x&ˆ (t ) = ( A − LC ) xˆ (t ) + Bu (t ) + Dw(t ) + Ly (t )
(4.5)
but the observability of (A,C) the matrix L is chose so that A-LC is asymptotically stable
with prescribed eigenvalues. Hence given Q, real symmetric positive-definite, there
exists P, real symmetric positive-definite, so that
( A − LC )T P + P ( A − LC ) = −Q
(4.6)
Let e = x − x . Then
e&(t ) = Axˆ (t ) − LCxˆ (t ) + Bu (t ) + Dwˆ (t ) + Ly (t ) − Ax(t ) − Bu (t ) − Dw(t )
= ( A − LC )e(t ) − Dm(t )
Where Ly (t ) = LCx(t ) and m(t ) = wˆ (t ) − w(t )
Let V (e) = eT Pe be a Lyapunov function candidate for equation (4.7). Compute,
(4.7)
31
V& = e&T Pe + eT Pe&
= Pe[( A − LC )e − Dm]T + eT P[( A − LC )e − Dm]
= Pe( A − LC )T eT − PeDT mT + PeT ( A − LC )e − eT PDm
= eT [( A − GC )T P + P( A − GC )]e − 2eT PDm
≤ −eT Qe + 2 e P ( D )
≤ e (−λmin (Q) e + 2ηλmax ( P))
where
( D ) ≤η
Note that V& (e) < 0 outside the ball

2ηλmax ( P)
e e ≤
λmin (Q)

Hence (4.7) is practically stable with respect to any radius greater than
2η
λmax ( P) λmax ( P )
λmin ( P) λmin ( P)
It shows the proposed estimator can be use for the system.
4.4
Conclusion
In this chapter, the mathematical model for an observer has been derived. The
estimated state will be used in the proposed controller. The simulation of quarter car
active suspension system will use the estimated state variables. The performance of the
system when using the proposed with disturbance observer will analysis in the next
chapter.
32
CHAPTER 5
SIMULATIONS
5.1
Simulations
The mathematical model for quarter car active suspension system has been
obtained in chapter 2. The mathematical model of the passive suspension is given by
equations (2.10-2.13). The parameters for the springs and the damper for the passive
suspension system are considered to be linear. In passive suspension system the model is
assumed that the road disturbance as an input into the system. By substituting the
parameter values of the suspension system as tabulated in Table 2.1 into equation (2.102.13), the passive suspension system for the quarter car model may be obtained as:
58   x&b (t )   0 
 &x&b (t )  − 3.45 3.45 − 58
 &x& (t )  16.9 − 16.9 285 − 3510  x& (t ) 3220
 w =
 w(t )
 w  + 
 x&b (t )   1
0
0
0   xb (t )   0 
 
 



1
0
0   xw (t )  0 
 x&w (t )  0
(5.1)
It can be observed from equation 5.1 that the performance of the passive suspension
system is purely affected by the road disturbance, w(t). Equations (2.19-2.24) show the
mathematical model for quarter car active suspension system. The parameters for the
33
springs and the damper for the active suspension system are considered to be linear. In
active suspension system the model is assumed that the road disturbance as an input into
the system. By substituting the parameter values of the suspension system as tabulated in
Table 2.1 into equation (2.19-2.24), the active suspension system for the quarter car
model may be obtained as:
58
0   x&b (t )   0.0345 
 &x&b (t )  − 3.45 3.45 − 58
0 
 &x& (t )  16.9 − 16.9 285 − 3510 3220  x& (t ) − 0.0169
0 
 w  
 w  

 
 x& (t )  =  1
  x (t )  + 
u (t ) + 0 w(t )
0
0
0
0
0
 b  
 b  

 
1
0
0
0   xw (t ) 
0
 x& w (t )  0

0 
 w& (t )   0

1
0
0
0
0   w(t )  
0
(5.2)
It can be observed from equation 5.2 that the system suffered from the
mismatched condition. Therefore the proposed controller must robust to overcome or
minimize the mismatched problem and also to improve the performance of the quarter
car active suspension system in term of performance ride comfort and road handling.
Based on equation 3.4 and 3.17, the calculation fro the sliding surface and
controller parameters are performed as follow:
•
Determine the matrix K in equation 3.4 and 3.17, by using the LQR
method
•
Determine the matrix L in equation 4.1 using the observer formula,
L=acker(A*',C',S)
•
Select the proper value for matrix C in equation 3.4 and 3.17 by trial and
error method
•
Choose the proper value for ρ and δ in equation 3.17
•
Choose the proper value for positive scalar in equation 4.1.
•
Perform the simulation and observer the performance of the system.
34
The performance of the active suspension systems with the proposed controller
will be investigated and compared through computer simulation. The simulations are
carried out using MATLAB and SIMULINK Toolbox software will be presented in the
following section.
5.5.1
Performance of PISMC with Disturbance Observer on Road Handling and
Ride Comfort
The performance of the ride comfort may be observed through the car body
acceleration and the performance of road handling may be observed through the wheel
deflection. The external disturbance is generated by using road profile. The simulation is
done by using parameter as follow:
•
H = [100 0.15 0.4 0.2 0.8]
•
T = [100 0.1 0.4 0.1 0.3]'
•
Matrix K has calculated as follows
K = [30.3 0 − 2652.6 − 99.3 − 6530.2]
•
Matrix C has been selected :
C = [200 500 7500 8000 5000]
•
The following value of ρ and δ has been used:
ρ = 3 and δ = 15
•
The following values for γ and K for observer design:
γ = 1.5 and K = 50
35
For comparison purposes, the performance of the PISMC with disturbance
observer is compared to the linear quadratic regulator (LQR) with disturbance
observer control approach. Assumed a quadratic performance index in the form of
J=
1t T
T
∫ ( x Qx + u Ru )dt
20
(5.3)
In the design of the LQR controller, weighting matrices Q and R are selected
as Q = diag(q1,q2,q3,q4,q5) where q1 = q2 = q3 = q4 = q5 = 1x102 and R = [0.1].
Then the optimal linear feedback control law is obtained as
u (t ) = − Kx(t )
(5.4)
Figure 5.1 shows the external disturbance that generated to the active
suspension system. Figure 5.2 and 5.3 shows the estimated road disturbance for the
system when using LQR and PISMC controller. The objective of designing an active
suspension is to increase the ride comfort and road handling; there are three parameters
to be observed in the simulation. The parameters are the wheel deflection, the body
acceleration and the suspension travel. Figure 5.4 shows the wheel deflection of both
controllers for an active suspension system. The result shows the purposed controller
perform better than LQR controller. Figure 5.5 illustrates clearly how the PISMC with
disturbance observer technique can effectively absorb the vehicle vibration in
comparison to the LQR method. The body acceleration in the PISMC design system
performs better, which guarantee better ride comfort. Figure 5.6 shows the suspension
travel of both controllers for an active suspension system. The result shows that the
active suspension utilizing the PISMC with disturbance observer technique performs
better as compared to the others. Figure 5.7 shows that the proposed controller with
disturbance observer is able to drive the state trajectory of the system onto sliding
surface and remain thereafter. The control signal from the proposed controller shows in
figure 5.8. It shows the PISMC with disturbance observer is utilized to overcome
36
external disturbance due to the road profile. Therefore, it may be conclude that the active
suspension system with PISMC with disturbance observer improves the ride comfort
while retaining road handling characteristics, as compared to the LQR method.
Figure 5.1: Road profile that generated to the system
37
Figure 5.2: Estimated road profile for the system using PISMC
Figure 5.3: Estimated road profile for the system using LQR controller
38
Figure 5.4: Wheel deflection for active suspension
Figure 5.5: Body acceleration for active suspension system
39
Figure 5.6: Suspension travel of active suspension
Figure 5.7: Sliding surface of active suspension
40
Figure 5.8: Control input of the active suspension
Figure 5.9 and figure 5.10 show the wheel deflection between plant and observer
for both controllers. It shows the estimated wheel deflection follow the wheel deflection
from plant. Figure 5.11 and figure 5.12 show the estimated body acceleration from the
observer follow the body acceleration from plant for the both controller. Figure 5.13 and
figure 5.14 shows the suspension travel between plant and the observer fro PISMC and
LQR controller. It shows the estimated suspension travel follow the suspension travel
from plant. Therefore, it may be conclude that the proposed observer work properly
which the estimated and actual output is equal.
41
Figure 5.9: Wheel deflection between plant and observer using LQR controller
Figure 5.10: Wheel deflection between plant and observer using PISMC
42
Figure 5.11: Body acceleration between plant and observer using LQR controller
Figure 5.12: Body acceleration between plant and observer using PISMC
43
Figure 5.13: Suspension travel between plant and observer using LQR controller
Figure 5.14: Suspension travel between plant and observer using PISMC
44
In the next section, the performance of PISMC with disturbance observer are
evaluated under different road, variations of the sliding gain, ρ, variations of positive
scalar for observer parameter, and the variations of parameter C.
5.1.2
Effect of the Suspension Performance on Various road Profiles
In this section, the performance of the quarter car active suspension system is
evaluated under different road profile between the proposed controller and LQR
controller. Road profile 1 (RP1) represents a disturbance 10cm bump height and the road
profile 2 (RP2) represent the double disturbance 10 cm bump height. It shows the
PISMC with disturbance observer and the proposed observer is robustness to overcome
and performance better than LQR controller. Figure (5.15-5.17) shows the performance
for road profile 1 and the figure (5.18-5.20) shows the performance for road profile 2.
Figure 5.15: Body acceleration for active suspension for road profile 1
45
Figure 5.16: Suspension travel for active suspension for road profile 1
Figure 5.17: Wheel deflection for active suspension for road profile 1
46
Figure 5.18: Body acceleration for active suspension for road profile 2
Figure 5.19: Wheel deflection for active suspension for road profile 2
47
Figure 5.20: Suspension travel for active suspension for road profile 2
5.1.3
Effect of the Suspension Performance on Various road Profiles
In this section, the effect of constant ρ (sliding gain) in the proportional integral
sliding mode control is studied. The reaching condition of the proportional integral
sliding mode controller is affected by varying the sliding gain. The reaching mode
condition is satisfied if sliding gain, ρ >0 and not satisfied if ρ < 0. The following values
of sliding gain ρ have been considered in the simulation:
Case 1:
ρ = 100000 (satisfying the reaching mode condition)
Case 2:
ρ = -100000 (not satisfying the reaching mode condition)
Figure (5.21-5.26) illustrates the effects of varying the sliding gain on the body
acceleration, suspension travel and wheel deflection. From the result, it shows that the
reaching mode condition is satisfied for the positive sliding gain.
48
Figure 5.21: Body acceleration for varying the parameter ρ (positive sliding gain)
Figure 5.22: Body acceleration for varying the parameter ρ (negative sliding gain)
49
Figure 5.23: Suspension travel for varying the parameter ρ (positive sliding gain)
Figure 5.24: Suspension travel for varying the parameter ρ (negative sliding gain)
50
Figure 5.25: Wheel deflection for varying the parameter ρ (positive sliding gain)
Figure 5.26: Wheel deflection for varying the parameter ρ (negative sliding gain)
51
5.2
Conclusion
In this chapter the performance of proposed observer and PISMC with
disturbance observer has been investigated. It has been shown that the PISMC with
disturbance observer improved the ride comfort and road handling performances of
quarter car active suspension system compared to LQR with disturbance observer
control technique. The proportional integral sliding mode control with disturbance
observer is robust to various types of disturbance. Furthermore, the chattering problem
has been overcome by using the continuous switching function and the boundary layer
that can be adjusted by varying the sliding gain in the controller. In conclusion, the
mismatched condition inherent in the system dynamics has been overcome by the
proposed PISMC with disturbance observer. The results show the proposed controller
effective in controlling the active suspension.
52
CHAPTER 6
CONCLUSION AND SUGGESTIONS
6.1
Conclusion
The proportional integral sliding mode control strategy with observer design has
been successfully implemented to the active suspension system. The results clearly show
that the proportional integral sliding mode control scheme is robust in compensating the
disturbance in the system.
In the simulation study, it was demonstrated that the PISMC with disturbance
observer strategy gives better performance than LQR controller. The capability to
absorb disturbance for PISMC is much better than LQR controller. It shows the PISMC
observer can improve the ride comfort and road handling of the system.
The estimation works properly in estimating the state and disturbance. The road
profile was estimated by using observer with good agreement between exact and
estimated value.
Using Lyapunov stability theory, it is shown that the estimation
assured that the mismatched uncertainty system is ultimately bounded stable.
53
6.2
Suggestion For Future Work
A number of future works could be considered as an extension to the present
study and they are listed as follows:
•
The constant in the matrices C are determined by trial and error method.
Therefore, it is suggested that these matrices be tuning using another
technique.
•
The positive scalar in observer design is determined by trial and error
method. Therefore, it is suggested that the positive scalar can be tuning
using another technique.
54
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