CONTROLLER DESIGN BASED ON Q-PARAMETERIZATION
METHOD
HUSSEIN SULEIMAN MOHAMED
A project report submitted in partial fulfillment
of the requirements for the award of the degree
of Master Engineering
(Electrical-Mechatronic and Automatic Control)
Faculty of Electrical Engineering
Universiti Teknologi Malaysia
MAY 2007
iii
DEDICATIONS
“To all my beloved family: Father, Mother, Brothers and Sisters”
iv
ACKNOWLEDGEMENTS
First of all, gratefulness of thanks to our creator,” ALLAH” S.W.T for this
continuous blessing, which make this work neither the first nor the last.
This dissertation is the culmination of my entire formal education, starting with
primary school when I was six years old, and I extend my appreciation to all the teachers
and other individuals who helped me along the way.
My upmost gratitude and especial “thank you” to, Associate Professor Dr.
Mohamed Noh Bin Ahmad for giving this opportunity to work under his supervision and
for sharing his great knowledge and experience with me. Also for his guidance, help and
oversight throughout my project.
I thank and send my deep appreciations to my family “parents and brothers and
sisters for their love and support all their blessings had made it possible for me to face
all the challenges. I would like to express a special “thank you “goes to my friend Salah
Ramadan for his support and stand beside me when I was need help in all time.
I would like to express my appreciation and convey my deepest gratitude to Mr.
Abdul Rashid Husain for his guidance to complete this research. Appreciation is also
extend to all people who gave the author heartfelt corporation and shared their
knowledge and for giving some their valuable time.
v
ABSTRACT
This project presents the application of a robust controller design based on
" Q-parameterization" theory (some time referred to as "Youla parameterization") for
on a Magnetic suspension balance beam system. This controller is used in order to
achieve both robust stability and good dynamic performance against the variation of
system parameters. In the Q-parameterization method, the set of all stabilizing
controllers of magnetic suspension balance beam system (MSBB) is characterized by
a free parameter Q. This free parameter is chosen to using optimization technique to
satisfy robust stability and other design requirements. The work was carried out in
three stages. First, it starts with the derivation of the mathematical model of a
magnetic suspension balance beam system (MSBB) in state space form. Second, the
proposed Q-parameterization controller design methodology is presented. It should
be noted that the degree of the resulting controller usually equals the degree of the
plant plus the degree of the Q-parameter can be chosen to obtain a lower or higher
order controller. Finally, the performance of the Q- parameterization controller in
controlling the balance beam control system will be illustrated. For comparison
purposes, the simulation of the pole placement and integral controllers were also
carried out. Simulation results show the effectiveness of the proposed controller.
vi
ABSTRAK
Projek ini melibatkan applikasi pengawal robust berasaskan teori parameter-Q
(dikenali juga sebagai parameter Youle) keatas sebuah loji Magnetic Suspension
Balance Beam (MSBB). Pengawal ini direkabentuk untuk mencapai kestabilan robust
disamping prestasi yang baik terhadap perubahan parameter sistem. Rekabentuk
pengawal untuk sistem MSBB menggunakan kaedah parameter-Q ditentukon oleh
parameter bebas Q. Parameter bebas Q ini dipilih untuk memenuhi keperluan kestabilan
robust dan keperluan rekaan yang lain. Kerja penyelidikan ini melibatkan tiga tahap
keria. Didalam tahap pertama, model
matematik dalam format kaedaan-ruang untuk
sebuah system MSBB akan dijalankan dahulu. Pada tahap kedua, metodologi rekabentuk
pengawal parameter-Q yang dicadangkan akan dibentangkan. Perlu diingat bahawa
tertib pengawal (kebiasaannya menyamai tertib loji bersta terib parameter-Q) boleh
dipilih samada untuk mendapatkan tertib pengawal yang lebih rendah atau lebih tinggi.
Pada tahap akhir, presetasi pengawal parameter-Q yang dicadangkan didalam mengawal
sistem MSBB
akan
dibentangkan.
Untuk
tujuan
perbandingan,
penyelakuan
menggunakan pengawal perletakan kutub beserta kawalan kamiran juga akan
dibentangkan. Keputusan simulasi membuktikan keberkesanan pengawal yang
dicadangkan.
vii
TABLE OF CONTENTS
CHAPTER
I
II
TITLE
PAGE
DECLARATION
ii
DEDICATION
iii
ACKNOWLEDGEMENT
iv
ABSTRACT
v
ABSTRAK
vi
CONTENTS
vii
LIST OF TABLES
xi
LIST OF FIGURES
xii
LIST OF SYMBOLS
xvii
LIST OF ABBREVIATIONS
xix
INTRODUCTION
1
1.1
Introduction
1
1.2
Objective
3
1.3
Scope of Project
3
1.4 Research Methodology
4
1.5
5
Layout of Thesis
LITERATURE REVIEW
7
2.1
The magnetically suspended balance beam System
7
2.2
the Q-parameterization Controller
9
viii
III
MAGNETICALLY SUSPENDED BALANCE BEAM
13
SYSTEM
3.1
Introduction
13
3.2
Mathematical Model of Magnetically Suspended
14
Balance Beam system
IV
3.3 State-space Model
19
3.4
Linearized Model
21
3.5
Simulation diagram for the model MSBB system
21
3.6
Summary
23
POLE PLACEMENT CONTROLLER DESIGN
24
4.1
Introduction
24
4.2
Pole Placement Technique without Integral Control
24
4.2.1
25
Pole Placement Technique without integral
to control the MSBB System
4.3
Pole Placement Technique with Integral Control
29
4.3.1
31
Pole Placement Technique with integral to
Control the MSBB System
4.4 Computer Simulation Using MATLAB/SIMULINK
4.4.1
Pole Placement Technique without integral
34
34
to control the linear MSBB System
4.4.2
Pole Placement Technique without integral
36
to control the nonlinear MSBB System
4.4.3
Pole Placement Technique with integral to
38
Control the MSBB System
4.5
Summary
40
ix
V
The Q-PARAMETERIZATION CONTROLLER
41
DESIGN
5.1
Introduction
41
5.2
The Q-parameterization Theory
42
5.3
Controller objective
43
5.4
Controller synthesis
43
5.5
Design of the proposed controller using
Q-parameterization theory
47
5.6
Computer Simulation Using MATLAB/SIMULINK
58
5.6.1 Computer Simulation Using MATLAB/
58
SIMULINK for Linear of the MSBB System
5.6.2 Computer Simulation Using MATLAB/
60
SIMULINK for nonlinear of The MSBB
System
5.7
VI
Summary
62
SIMULATION RESULTS AND DISCUSSION
63
6.1
63
Introduction
6.2
Results for linear model of MSBB System
64
6.3
Results for nonlinear model of MSBB System
68
6.4
Comparison between Various Results
71
6.4.1
71
Simulation of linear and nonlinear models
of the MSBB System
6.4.1.1
The Q-parameterization controller
71
6.4.1.2
Pole placement state feedback
73
controller
6.4.2
Comparison between Pole Placement
74
technique and Q-parameterization controller
6.4.2.1
Simulation using linear model of
the MSBB System
74
x
6.4.2.2
Simulation using nonlinear model
76
of the MSBB System
6.4.3
Comparison between pole placement
78
technique and Q-parameterization Controller
for linear and nonlinear of the MSBB System
6.5
Results for the MSBB System by suing the pole
80
Placement with integral Control
6.6
Comparison between pole placement with integral
82
and Q-parameterization Controller for the MSBB
System
VII
6.6.1
Linear model of the MSBB System
82
6.6.2
Nonlinear model of the MSBB System
84
6.7 Summary
88
CONCLUSIONS AND SUGGESTIONS
89
7.1
Conclusions
89
7.2
Suggestions for Future work
91
REFERENCES
92
xi
LIST OF TABLES
TABLE NO
TITLE
PAGE
3.1
Balance beam parameter
14
6.1
Description for design specification and poles
63
6.2
Comparison of various cases between pole placement
76
technique and Q-parameterization controller
6.3
Comparison between pole placement technique and Q-
78
parameterization Controller
6.4
Comparison of various cases
80
6.5
Comparison between Q-parameterization controller and
86
pole placement with and without integral control
xii
LIST OF FIGURES
FIGURE NO.
1.1
3.2
TITLE
Symmetric balance beam on two magnetic bearings.
The simulation block diagram for (MSBB) equations
PAGE
2
22
4.1
Pole Placement Technique without Integral Control
25
4.2
Pole Placement Technique with Integral Control
29
4.3
SIMULINK implementation of equation (3.21)
34
4.4
SIMULINK implementation of equation (3.21)
34
4.5
SIMULINK implementation of equation (4.15) for pole
35
placement Technique without integral
4.6
Complete simulation diagram of linear model for MSBB
35
system
4.7
Complete simulation diagram of pole placement technique
36
without integral control for MSBB linear system
4.8
Complete simulation diagram of nonlinear model for the
MSBB system
37
xiii
4.9
SIMULINK implementation of equation (4.15) for pole
37
placement Technique without integral
4.10
Complete simulation diagram of pole placement technique
38
without integral control for MSBB nonlinear system
4.11
SIMULINK implementation of equation (4.35) for pole
39
placement with integral
4.12
Complete simulation diagram of pole placement technique
39
with integral control for MSBB linear system
5.1
Block diagram of one parameter –control feedback system
43
5.2
Generalized region of stability
44
5.3
Basic feedback structures.
45
5.4
Block diagram of Q-parameterization
46
5.5
Q-parameterization as modification to nominal controller
46
5.6
one parameter controller feedback system
56
5.7
Complete simulation diagram of linear model for MSBB
58
system
5.8
Complete simulation diagram of linear model for the
MSBB System with Q-parameterization Controller
59
xiv
5.9
Complete simulation diagram of linear model for MSBB
59
system with Q-parameterization Controller
5.10
Complete simulation diagram of nonlinear model for the
60
MSBB System
5.11
Complete simulation diagram of linear model for MSBB
61
system with Q-parameterization Controller
5.12
Complete simulation diagram of linear model for MSBB
61
system with Q-parameterization Controller
6.1
Step disturbance force (1-N-m) at time =0.15 seconds
65
6.2
Displacement angle for linear MSBB System with 1 N-m
65
disturbance using pole placement controller.
6.3
Input voltage for linear MSBB System with 1 N-m
66
disturbance using pole placement controller.
6.4
Displacement angle for linear MSBB System with 1 N-m
67
disturbance using Q-parameterization controller.
6.5
Input voltage for linear MSBB System with 1 N-m
67
disturbance using Q-parameterization controller.
6.6
Displacement angle for nonlinear MSBB System with
68
1 N-m disturbance using pole placement controller.
6.7
Input voltage for nonlinear MSBB System with 1 N-m
disturbance using pole placement controller.
69
xv
6.8
Displacement angle for nonlinear MSBB System with
69
1 N-m disturbance using Q-parameterization controller.
6.9
Input voltage for nonlinear MSBB System with 1 N-m
70
disturbance using Q-parameterization controller.
6.10
Comparison of displacement angle between both systems
72
of the MSBB with Q-parameterization controller
6.11
Comparison of input voltage between both systems of the
72
MSBB with Q-parameterization controller
6.12
Comparison of displacement angle between both systems
73
of the MSBB with the pole placement technique
6.13
Comparison of input voltage between both systems of the
73
MSBB with the pole placement technique
6.14
Comparison of displacement angle for linear model of the
75
MSBB System
6.15
Comparison of input voltage between for linear of the
75
MSBB System
6.16
Comparison of displacement angle for nonlinear model of
77
the MSBB System
6.17
Comparison of input voltage between for nonlinear of the
MSBB System
77
xvi
6.18
Comparison of displacement angle for linear and
79
nonlinear model of the MSBB System
6.19
Comparison of input voltage between for nonlinear of the
79
MSBB System
6.20
Displacement angle for the MSBB System with 1 N-m
81
disturbance by using pole placement with integral control
6.21
Input Voltage for the MSBB System with 1 N-m by using
81
the pole placement with integral control
6.22
Comparison of displacement angle for the MSBB System
83
6.23
Comparison of input voltage for the MSBB System
83
6.24
Comparison of displacement angle for nonlinear of the
84
MSBB System
6.25
Comparison of input voltage for nonlinear of the MSBB
85
System
6.26
Comparison of displacement angle for nonlinear of the
85
MSBB System
6.27
Comparison of input voltage for nonlinear of the MSBB
System
86
xvii
LIST OF SYMBOLS
A
-
N x N system matrix for the Magnetically Suspended Balanced Beam
B
-
N x 1 input matrix for the Magnetically Suspended Balanced Beam
C
-
1 x N output matrix for the Magnetically Suspended Balanced Beam
E
-
N x 1 disturbance matrix for the Magnetically Suspended Balanced Beam
R
-
Set point (radian)
X
-
State vectors
fd
-
Disturbance of the system (N)
θ
-
Gap displacement (radian)
θ&
-
Velocity of gap displacement (radian/second)
θ&&
-
Acceleration of gap displacement (radian/seconds2)
i'
-
Overall instantaneous current (A)
iο
-
Steady current (A)
e`
-
Control voltage (V)
Kx
-
Magnetic bearing open loop stiffness (N/m)
Ki
-
Actuator current gain (N/A)
La
-
Half bearing span (m)
J
-
Mass moment of Inertia about the pivot point (kg/m2)
gο
-
Steady gap (m)
Q
-
Parameter arbitrary stable proper transfer function
K
-
Controller
xviii
S (s )
-
Sensitivity function
N (s )
-
Coprime factorization stable transfer function
D (s )
-
Coprime factorization stable transfer function
~
N (s)
-
Coprime factorization stable transfer function
~
D(s)
-
Coprime factorization stable transfer function
X (s )
-
Coprime factorization stable transfer function
Y (s )
-
Coprime factorization stable transfer function
~
X (s)
-
Coprime factorization stable transfer function
~
Y (s)
-
Coprime factorization stable transfer function
xix
LIST OF ABBREVIATIONS
MSBB
-
Magnetically Suspended Balance Beam
CHAPTER I
INTRODUCTION
1.1
Introduction
The selection of the controllers’ plays an important role in the design of any plant
controller .This is simply because it is the heart of the system. By making a wise choice,
the controller can achieve the exact predetermined results and achieve the best robust
stability and performance.
In recent years, a new control method has surfaced in the literature. The method
which is called the Q-parameterization method (sometime referred to as the Youla
parameterization method) is a modern control design method suitable for both stable and
unstable plants. It is used mainly to design a stabilizing controller for any system where
simple conventional control can not guarantees acceptable results.
There are several robust control methods available in literature. Nonlinear
methods such as sliding mode control or linear methods such as pole placement control
as well as H 2 , LQG, and Q-parameterization control are among the controllers that has
been applied to various control fields. In this thesis, the Q-parameterization controller is
applied to control a Magnetically Suspended Balance Beam (MSBB) system.
2
The Magnetically Suspended Balance Beam (MSBB) is a balancing system that
used two magnetic coils to balance the beam as shown in Figure 1.1. These two
magnetic coils are placed at each end of the beam, one at the right hand side and one in
the left hand side. It can be easily described as a small see-saw.
Figure 1.1: Symmetric balance beam on two magnetic bearings.
In this project, the main task is to control the gap displacement angle of the
beam. If the gap displacement angle is equal to the set point, it can be concluded that the
designed controller is successful in controlling the angle and make the beam become
stable.
The whole system is needed to be modeled first by using a state space equation.
It has been found that this system is having a non linear model. From this nonlinear
model, the linearization process has to be done first. After the linearized model has been
acquired, the next task to do is to control the beam until it become stable.
3
Once the model has been acquired, the Q-parameterization controller is used to
control the beam. For comparison purposes, the pole placement control (state feedback
control) and an integral controller will also be considered. In both evaluations, the
presence of step disturbance is also included in the system. The performance of both
controllers in controlling the MSBB will be evaluated through extensive computer
simulation using MATLAB/SIMULINK
1.2
Objective
The objectives of this project are as follows:
1.
To formulate the mathematical model of a Magnetically Suspended Balance
Beam (MSBB) System in state space form.
2.
To design a controller based on the Q-parameterization method for stabilization
of a Magnetic Suspension Balance Beam (MSBB) System.
3.
To simulate the system using the designed controller.
4.
To compare the results with pole placement controller (state feedback
controller), and integral controller.
1.3
Scope of Project
The work undertaken in this project is limited to the following aspects:
4
1.
The nonlinear mathematical model of MSBB system adopted in this work is as
described in (Baloh et al, 1999), (Lee. et al, 2001) and (Hu et al, 2002).
2.
The proposed Q-parameterization controller design methodology will be
designed and applied to the system.
3.
The performance of the balance beam control system will be illustrated by
simulation using MATLAB/SIMULINK as the platform.
4.
The results for Q-parameterization controller will be compared with pole
placement controller (state feedback control) and integral controller.
1.4
Research Methodology
The research work undertaken in the following four development stages:
1.
To study a Magnetic Suspension Balance Beam (MSBB) System and derive
the state space equations.
2.
The design of controller base on pole placement technique.
3.
The design of controller based on Q-parameterization method will be explain
and applied to the system.
4.
The Performance and the results simulation for a Magnetic Suspension
Balance Beam (MSBB) System using Q-parameterization controller will
be established.
5.
Analyzed the results by comparing it with pole placement controller and
integral controller.
5
1.5
Layout of Thesis
This thesis contains eight chapters. Chapter II contains a brief literature
review, first for model of the magnetically suspended balance beam (MSBB) System
and second for design controller (Q-parameterization controller).
Chapter III contains a brief introduction of MSBB. In this chapter also, the
mathematical model, which is a nonlinear model of the MSBB is presented. The linear
mathematical model of the system is derived and then transforms into the state space
representations.
Chapter IV presents the brief introduction of pole placement technique. Then the
controller is designed using pole placement technique without integral and plus integral
control. The use of integral control is to eliminate the steady-state error. The controller is
designed base on design specification.
For this design specification, the percentage
overshoot % OS and settling time Ts that have been used are:
i.
Percent of overshoot, %OS = 10%
ii.
Settling time, Ts = 0.1 second.
Chapter V presents the brief introduction of controller design based on Qparameterization method, the controller is designed using the Q-parameterization
method to satisfy our requirement. Then, using the MATLAB programming, same as in
Chapter IV, the Q-parameterization controller is also designed base on the same design
specification.
For this s design specification, the percentage overshoot % OS and settling
time Ts that have been used are:
i.
Percent of overshoot, %OS = 10%
ii.
Settling time, Ts = 0.1 second.
6
Chapter VI will be presents both the results and discussion of pole placement
technique and the controller design based on Q-parameterization method. For the
specification design there will be two graph presented. The first one is a gap
displacement’s graph and another one is an input voltage’s graph. At the end of this
chapter, the comparison between the pole placement technique and the Qparameterization controller is done. Also the comparison between the pole placement
technique with integral and the Q-parameterization controller is done.
Chapter VII conclude the work undertaken, suggestions for future work are also
presented in this chapter.
CHAPTER II
LITERATURE REVIEW
2.1
The magnetically suspended balance beam (MSBB) System
As magnetic bearing applications become more complicated, the need for
accurate models of the controlled bearing systems becomes more important. (Baloh et al,
1999) in his initial research use an adaptive estimation to identify unknown parameters
and disturbances for a simple one dimensional magnetic bearing system.
The magnetically suspended balance beam (MSBB) is a balancing system that
used two magnetic coils to balance the beam. These two magnetic coils are placed at
each end of the beam. One at the right hand side and one in the left hand side. The use of
magnetic coils, introduce the nonlinearities which is very difficult to control with
conventional controllers such as Proportional Integral Derivative (PID) controller.
Magnetic suspension system is a nonlinear system with gap displacement and coil
current as variables. The main task was to control the displacement angle of the beam. If
the displacement angle is equal to the set point, it can be concluded that the designed
controller is successful in controlling the angle and make the beam become stable.
8
To overcome this problem various controllers have been put into trial. (Lee, et.al.
2001) presented a sliding-mode controller with integral compensation for a magnetic
suspension balance beam system. The control scheme comprises an integral controller
which is designed for achieving zero steady-state error under step disturbances, and a
sliding-mode controller which is designed for enhancing robustness under plant
uncertainties. A designing procedure was developed for determining the coefficients of
the switching plane such that the overall closed-loop system has stable eigenvalues. A
proper continuous input signal is introduced to overcome the chattering problem.
Electromagnetic bearings inherently have nonlinear properties (Lee, et.al. (2001)).
With regard to such a nonlinear system, various controllers have been developed. The
integral sliding-mode control approach has been reported by a number of authors recently
as a method of servo control. The integral sliding-mode control approach consists of two
compensators: one uses an integral compensator for achieving a zero steady-state error
under an external step disturbance force, and the other uses a sliding-mode controller.
The design problem of an integral sliding-mode controller consists of two items.
The first item was concerned with the design of a switching surface upon which the
desired dynamic behavior can be guaranteed for the nominal system. The second item
was concerned with the selection of a proper nonlinear control law to handle strong
nonlinearities of the initial states of the magnetically suspended system. The simulation
and experimental results showed that an integral sliding-mode control design achieved
accurate tracking and was fairly robust to plant parameter variations and external load
disturbances.
Finally, this paper had presented a method to control the angular displacement of a
magnetically suspended balance beam with plant parametric variations and external
disturbances. To overcome the effect of the parametric variations, and to reject the
external disturbance forces, the integral sliding-mode controller utilized. The
mathematical model of the balance beam derived, including an integral compensator, and
then designed the linear and nonlinear control components.
They showed the
9
insensitivity of the controller response under the parametric variations and the disturbance
rejection by simulations.
2. 2 The Q-parameterization Controller
The need for solving real life problems provides the power for development of
control theory. In the 1920s, with the emergence of telephone industries, feedback
control to maintain a constant control gain of an amplifier became a focus point. Black
and Bode made significant contributions in developing the classical control theories.
State-space theory appeared in the 1950. Systems were described using a set of
differential equations. The variables of the differential equations were considered to be
the states of control systems. Control design problems focus on optimization problems
by minimizing cost functions via state feedback. The LQG-control of linear system with
Gaussian disturbances and quadratic criteria was formulated with a kalman filter and
state feedback. The estimator gain of the kalman filter and the feedback control gain
were computed easily from two Riccati equations. Safonov and Athans also showed that
the LQG problem has good robustness with a 60 phase margin and infinity amplitude
margin when all states are measured. But this amazing result dose not hold for output
feedback when the states of a system are unavailable and a state estimator needs to be
constructed from outputs.
Robust control was developed when control engineers tried to recover the
robustness of control system. Loop transfer recovery technique by (Doyle., Stein., 1981),
loop shape, H∞ control, etc, are different approaches. Coprime parameterization theory
was patterned after (Desoer, C.1980) and (Vidyasagar.M.1982). The key aspect of this
approach is coprime factorization of the transfer function of a given plant over a proper,
stable, real rational function space R. This means that the transfer functions:
10
P (s ) =
N (s )
N (s ), M (s ) ∈ ℜ , and N(s),M(s) are coprime in the sense that there
M (s )
exist function X(s) and Y(s) such that
N (s )X (s ) + M (s )Y (s ) = 1
(1)
The four functions, N(s), M(s), X(s), and Y(s), form a ring1 with identity. A
complete collection of the stabilizing compensators for the given plane can be
parameterized from this function ring. All compensators are in the set of
⎧ X (s ) + M (s )Q(s )
⎫
C∈⎨
, Q(s ) ∈ ℜ⎬.
⎩ Y (s ) − N (s )Q(s )
⎭
The problems of synthesizing a feedback system are reduced to solving Eq. 1 for
X(s) and Y(s) when N(s) and M(s) are given. The advantage of the coprime
parameterization is that of designing a feedback compensator that stabilizes a given plant
embraces within a single framework, 1-D as well as n-D system, stable as well as
unstable systems.
By choosing the free parameter Q (s), the control system is guided to meet
performance criteria. Left and right coprime factorization were defined for a n × m
dimension system. The research on computation of left and right coprime factorizations
have been carried on in the last two decades. The relations between the left/right coprime
factorization of a transfer matrix in ℜ and the state space controller/observer were
addressed by (Antsaklis, P.J.1986).
Deriving any right coprime factorization in ℜ is equivalent to solving a statefeedback stabilization problem. Similarly, deriving any left coprime factorization in ℜ is
equivalent to designing a full-order full-state observer.
Normalized Coprime factorization was introduced in 1988 by (Vidyasagar.M).
This concept was used in developing H∞ optimal control theory.
Applications of Q-parameterization are seen in the processes with dead-time, and
flexible arm control. For a multi-objectives system, the control objectives are not met
11
simultaneously by just tuning the parameter Q. this limits the utilization of the Qparameterization method.
Interpolation theories play a major role in stabilization problems. Kimura, 1984
used Nevanlinna-Pick interpolation theory to solve for the robust stabilization and in
1987 he applied this theory to the optimization of robust stability margin. The further
development of the Nevanlinna-Pick interpolation was made by Doyle, Francis, and
Tannenbaum to solve for model-matching problems (Doyle, J.C., 1992).
(A. Mohamed, 2000) was presented a robust controller design for the benchmark
problem for (two masses connected by spring) using the Q-parameterization theory to
reject two disturbance classes. The First one was class of impulse disturbance and the
second was the class of sinusoidal disturbance.
The set of all stabilizing controllers for the system is characterized by two free
parameters namely q1 and q 2 . (A. Mohamed, 2000) was used the more general type of
feedback system, which is called a two-parameter-control feedback. In this
configuration, the controller transfer function K is defined by the following left coprime
~ −1
~ −1
~ −1
factorization (lcf) of K: K = d k (n~k1 n~k 2 ) = (K1 K 2 ) where, K1 = d k n~k1 , K2 = dk n~k2 .
As was stated, requirements were to achieve Stability and Robustness to variation
~ ~
~ ~
of plant parameters. It can be pre-satisfied by choice of N , D, N , D, X , Y , X , Y ∈ DS , and
this can be done by choosing the matrices F1 , and F2 , such that the eigenvalues of Aο .
~
and Aο ,lie in the domain D S , . The other requirements are functions of q1 , q 2 which
must be chosen in S D . The set S D is an infinite dimensional set and this will make the
design problem quite complicated. Thus q1 and q 2
are restricted to a subset of S D .
This subset can be chosen in many different ways, and the following are some easy ways
to choose q1 and q 2 :
1. q ∈ DS
12
2. q ( s ) =
a ( s + b)
, where p s < α s is fixed and a and b are free design parameters.
(s + ps )
The plant at its control input with a compensator having poles coinciding with
the disturbance poles. Then the method of control design explained and applied for the
augmented system.
Finally, (A. Mohamed, 2000 found the Q-parameterization theory was used to
design an optimal robust controller for the benchmark problem. The controller Qparameters were chosen using optimization, so that the design goals are satisfied. In
order to achieve rejection of the class of impulse disturbance, a fourth order, minimum
phase, stable controller is obtained. In order to achieve rejection of the class of
sinusoidal disturbance, a fifth order, minimum phase controller is obtained. The results
obtained are satisfactory, and stability and performance robustness are achieved.
CHAPTER III
MAGNETICALLY SUSPENDED BALANCE BEAM SYSTEM
3.1
Introduction
This chapter contains four sub chapters. The first sub chapters will presents the
simple figure of MSBB. In this figure the main component of MSBB are presented. The
main component of MSBB consists of two magnetic bearings, two sensors and the
balanced beam. The length of balanced beam is 28.24 cm.
Section 3.2 will represents the mathematical modeling of a MSBB system.
From the mathematical model that had been shown, it is found that, MSBB is a nonlinear
system. By using a suitable assumption, the linearized model can be derived.
Next, when the linearized model had been obtained, the MSBB can be presented
in state space equation. It is shown in the section 3.3.
Using the parameters that had been given in (Lee et al, 2001), the complete
linearized model is presented in section 3.4. Section 3.5 presents the simulation diagram
for the MSBB model, while section 3.6 summarizes the entire chapter.
14
3.2
Mathematical Model of a Magnetically Suspended Balance Beam (MSBB)
Figure 1.1 shows the geometry of the symmetric balance beam with two horseshoe
shaped magnetic bearings. They are A1 and A2. These magnetic bearings will produced
a force to make displacement angle, θ of the balanced beam equal to the set point. The
magnitude of the force can be controlled by controlling the input voltage, e1 and e2 . S1
and S2 are the sensors that have been used to detect the displacement angle of the beam
from the reference line. Table 3.1 shows each parameter of the balance beam system.
Table 3.1 Balance beam parameter
Symbol
Value
Units
Angular position
θ
-
rad
Half bearing span
La
0.1412
m
Mass moment of Inertia about the pivot point
J
0.0948
kgm2
Coil current in bearing 1
i`1
-
A
Coil current in bearing 2
i`2
-
A
Control voltage in bearing 1
e`1
-
V
Control voltage in bearing 2
e`2
-
V
Coil resistance
R
0.7
Ω
Coil inductance
L
0.728
mH
Magnetic bearing open loop stiffness
Kx
2826.32
N/m
Actuator current gain
Ki
1.074
N/A
Steady current
i0
1
A
Steady gap
g0
380
μm
The balance beam shown in Figure 1.1 can be modeled by using the second order
dynamic equation, magnetic force laws and voltage equation (Lee et al, 2001).
The torque that can be produced at the balanced beam if there e a ‘pulling action’
from either one of the magnetic bearings can be described by
15
••
(3.1)
J θ = La ( f1 − f 2 ) + f d
The force that had been produced at the right hand side of the balanced beam can
be represented mathematically as
f 1 = μ ο Ag N
(i
ο
2
+ i1′
)
2
2( g ο + L a θ )
2
(3.2)
Similarly, the force that had been produced at the left hand side of the balanced beam is
f 2 = μ0 Ag N 2
(io + i '2 ) 2
2( g 0 − Laθ ) 2
(3.3)
By using the kirchoff voltage law, the control voltage, e1 at magnetic bearing, A1is
e'1 = Ri '1 + L
di '1
dt
(3.4)
Similarly, the control voltage, e2 at magnetic bearing, A2 is
di '2
e 2 = Ri 2 + L
dt
'
'
where,
J
mass moment of inertia about the pivot point
La
half bearing span
Ls
distance from pivot point to sensor
f1
electromagnetic attractive force on bearing 1
f2
electromagnetic attractive force on bearing 2
fd
external disturbance force
µo
permeability of free space
N
number of turns in coil
Ag
pole face area
R
coil resistance
L
coil inductance
e’1
control voltage in bearing 1
(3.5)
16
e’2
control voltage in bearing 2i
i’1
instantaneous coil current in bearing 1
i’2
instantaneous coil current in bearing 2
Equations (3.1)-(3.5) can be linearized with the assumption that the (i '1 ) 2 ,(i'2 ) 2
and (θ ) 2 ≈ 0 . In (3.4.) and (3.5), the reluctance of the iron path is not considered.
μο Ag N 2 (iο 2 + 2iο i1′ + (i1′ ) 2 )
f1 =
2( g ο ) 2 + 4 g ο Laθ + 2 La 2θ 2
μο Ag N 2 (iο 2 + 2iο i2′ + (i2′ ) 2 )
and f 2 =
2( g ο ) 2 − 4 g ο Laθ + 2 La 2θ 2
(3.6)
By assuming (i '1 ) 2 ,(i'2 ) 2 and (θ ) 2 ≈ 0 equation (3.6) can be simplified as follows:
μο Ag N 2 (iο 2 + 2iο i1′ )
f1 =
2( g ο ) 2 + 4 g ο Laθ
(3.7)
and
μο Ag N 2 (iο 2 + 2iο i2′ )
f2 =
2( g ο ) 2 − 4 g ο Laθ
(3.8)
The total current at the right hand side coil A1, i1 is the total between the steady
current, iο and the instantaneous current, i1′ .so i1 = iο + i1′ and the total current at the left
hand side coil A2, i2 is the total between the steady current, iο and the instantaneous
current, i2′ .i.e. i2 = iο + i2′ .
If i1 + i2 , the answer will be zero because i2′ = −i1′ .but if i1 − i2 , the overall
instantaneous, i ′ will exist can be written as:
i ′ = i1′ − i2′
The overall forces, F that exist in MSBB system can be calculated as follows:
μ0 Ag N 2 (i0 2 + 2i0i '1 ) μ0 Ag N 2 (i0 2 + 2i0i '2 )
F = f1 − f 2 =
−
2( g 0 ) 2 + 4 g 0 Laθ
2( g 0 ) 2 − 4 g 0 Laθ
(3.9)
17
μ0 Ag N 2 (i02 + 2i0i '1 )(2( g0 )2 − 4 g0 Laθ ) − [μ0 Ag N 2 (i02 + 2i0i '2 )][(2( g0 )2 + 4 g0 Laθ )]
=
(2( g0 )2 + 4 g0 Laθ )(2( g0 )2 − 4g0 Laθ )
=
(μ0 Ag N 2i02 + 2i0i '1 μ0 Ag N 2 )(2( g0 )2 − 4 g0 Laθ )
4( g0 ) − 16 g0 La θ
4
2
2
−
(3.10)
2
[μ0 Ag N 2i02 + 2i0i '2 μ0 Ag N 2 )][(2( g0 )2 + 4 g0 Laθ )]
4( g0 )4 − 16 g02 La2θ 2
As been mentioned before θ 2 ≈ 0, Thus;
F = f1 − f 2 =
( μ0 Ag N 2i0 2 + 2i0i '1 μ0 Ag N 2 )(2( g0 ) 2 − 4 g0 Laθ )
4( g0 )4
−
[ μ0 Ag N 2i0 2 + 2i0i '2 μ0 Ag N 2 )][(2( g0 )2 + 4 g0 Laθ )]
4( g 0 ) 4
Lets assume,
A = μ0 Ag N 2i0 2
B = 2i0i '1 μ0 Ag N 2
C = 2( g 0 )2
D = 4 g0 Laθ
E = 2i0i '2 μ0 Ag N 2
=
( A + B)(C − D) [( A + E )(C + D)]
−
4( g 0 )2
4( g0 )2
=
AC − AD + BC − BD − [ AD + AC + EC + ED]
4( g0 )2
=
−2 AD + C[ B − E ] − D[ B + E ]
4( g0 )2
=−
2( μ0 Ag N 2i0 2 )(4 g0 Laθ )
4 g04
+
+
2( g0 )2 [2i0 μ0 Ag N 2 (i1 '− i2 ')]
4 g0 4
4 g 0 Laθ [2i0 μ0 Ag N 2 (i1 '+ i2 ')]
4 g04
(3.12)
18
But,
4 g 0 Laθ [2i0 μ0 Ag N 2 (i1 '+ i2 ')]
4 g04
= 0 , because i1 '+ i2 ' = 0
Thus,
F = f1 − f 2 = −
2μ0 Ag N 2i0 2 Laθ
g 03
+
i0 μ0 Ag N 2 (i ')
g02
(3.13)
= −2 K x Laθ + K i i '
where K x =
μ0 Ag N 2i0 2
g 03
and K i =
i0 μ0 Ag N 2
g02
From equation (3.1)
••
J θ = La ( f1 − f 2 ) + f d
••
θ=
(2 K x La 2θ ) K i Lai ' f d
+
+
J
J
J
(3.14)
Define an auxiliary equation:
e ' = e1 '+ em − (e2 '− em )
= 2em + e1 '− e2 '
•
= 2 K i θ + e1 '− e2 '
(3.15)
•
where em is back emf and em = K i θ
Substituting
e'1 = Ri '1 + L
di '1
di '
and e' 2 = Ri ' 2 + L 2
dt
dt
into equation (3.15),
19
di '1
di ' 2
'
− Ri 2 − L
e ' = 2 K i θ + Ri + L
dt
dt
•
'
1
•
di ' 2
e ' 2 Ki θ Ri '1 Ri ' 2 di '1
−
−
+
=
−L
L
L
L
L
dt
dt
•
e ' 2 Ki θ R(i '1 − i ' 2 ) d (i '1 − i ' 2 )
−
−
=
L
L
L
dt
•
e ' 2 Ki θ R(i ' ) d (i ' )
−
−
=
L
L
L
dt
3.3
(3.16)
State-space Model
From (3.14) and (3.16), the state space equation of the MSBB system shown in
figure 1.1 can be represented as:
x& = Ax + Bu + Ef d
y = Cx
(3.17)
where,
x = [ x1
x2
x3 ]
⎡
0
⎢
⎢
K L2
A = ⎢ −2 x a
⎢
J
⎢
⎢
0
⎣⎢
T
•
⎡
= ⎢θ θ
⎣
1
0
−2
Ki
L
⎤
i '⎥
⎦
T
⎤
0 ⎥
⎥
K i La ⎥
J ⎥
−R ⎥
⎥
L ⎦⎥
20
⎡ ⎤
⎢0⎥
⎢ ⎥
B = ⎢0⎥
⎢1⎥
⎢ ⎥
⎣L⎦
⎡0⎤
⎢1⎥
E=⎢ ⎥
⎢J ⎥
⎢0⎥
⎣ ⎦
C = [1 0 0]
u = e’
(3.18)
E is an external disturbance matrix. Rearrange equation (3.18) gives
x&1 = θ& = x 2
2
K L
K L
1
x& 2 = θ&& = − 2 x a θ + i a i ′ +
fd
J
J
J
K
R
1
x& 3 = i&′ = − 2 i θ& − i ′ + u
L
L
L
(3.19)
Equation (3.19) can be written in a vector-matrix form as:
⎡
⎡ • ⎤ ⎢
0
x
⎢ 1⎥ ⎢
⎢ • ⎥ ⎢
K x L2 a
⎢ x2 ⎥ = ⎢ − 2 J
⎢ • ⎥ ⎢
⎢ x3 ⎥ ⎢
0
⎣ ⎦ ⎢
⎣
y = [1 0
1
0
−2
⎡ x1 ⎤
0 ] ⎢⎢ x 2 ⎥⎥
⎢⎣ x3 ⎥⎦
Ki
L
⎤
⎡ ⎤
0 ⎥
⎡0⎤
⎥ ⎡ x1 ⎤ ⎢ 0 ⎥
⎢1⎥
K i La ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎥ fd
x
u
+
+
0
2
⎢
⎥
J ⎥⎢ ⎥ ⎢ ⎥
⎢J ⎥
⎢x ⎥
1
⎢0⎥
−R ⎥ ⎣ 3 ⎦ ⎢ ⎥
⎣ ⎦
⎥
L
⎣
⎦
L ⎥⎦
(3.20)
21
3.4
Linearized Model
Table 3.1 shows all the parameters for linearized model of MSBB system as
described in (Lee et al, 2001). By using the parameters tabulated in Table 3.1, the
linearized model of the MSBB system can be computed, and the results are as follows:
⎡•⎤
⎢ x1 ⎥ ⎡ 0
1
0 ⎤ ⎡ x1 ⎤ ⎡ 0 ⎤
⎡ 0 ⎤
⎢•⎥ ⎢
0
1.6 ⎥⎥ ⎢⎢ x2 ⎥⎥ + ⎢⎢ 0 ⎥⎥ u + ⎢⎢10.55⎥⎥ fd
⎢ x2 ⎥ = ⎢ −1188.8
⎢•⎥ ⎢ 0
⎢⎣ 0 ⎥⎦
−2951 −962 ⎥⎦ ⎢⎣ x3 ⎥⎦ ⎢⎣1374⎥⎦
⎢ x3 ⎥ ⎣
⎣ ⎦
⎡ x1 ⎤
y = [1 0 0] ⎢⎢ x2 ⎥⎥
⎢⎣ x3 ⎥⎦
3.5
(3.21)
Simulation diagram for the MSBB model
Magnetic suspension system is a nonlinear system with gap displacement and coil
current as variables. It is important for the control system to cope with the nonlinearity in
the plant. The nonlinear equation of the MSBB system can be represented as:
x& = f
(x ) +
g ( x )u + Ef
d
(3.22)
where f d is external disturbance force. It is expressed in the general form as:
•
x = Ax + Bu + Δ ( x , u ) + Efd
(3.23)
where Δ ( x , u, ) represents the plant uncertainties, nonlinearity of the balance beam, and
input channel uncertainty; including unmodeled coil characteristics.
22
Figure 3.2 shows the simulink simulation diagram for the nonlinear MSBB system
as described by equation (3.22).
Figure 3.2: The block simulation diagram for (MSBB) system
23
3.6
Summary
This chapter presented the development of the nonlinear mathematical model of a
MSBB System. After some suitable assumption, the linearized model is also successfully
derived. It should be noted that the parameters used in the modeling are adapted from
(Lee et al, 2001).
CHAPTER
IV
POLE PLACEMENT CONTROLLER DESIGN
4.1
Introduction
After the linearized model had been obtained, the next task to do is to design the
controller based on pole placement technique according to the design specification.
This chapter contains the introduction to pole placement technique with two
cases:
1. Pole placement technique with out integral control.
2. Pole placement technique with integral control.
The purpose of integral controller is to eliminate the steady-state error.
4.2
Pole Placement Technique without Integral Control
Figure 4.1 shows the block diagram of a pole placement technique without integral
control.
25
Figure 4.1: Pole Placement Technique without Integral Control
The pole placement technique is based on second order closed loop characteristic
equation:
s 2 + 2ξωn s + ωn 2 = 0
(4.1)
Since MSBB system is third order system, one more pole should be added to
obtain a third order closed loop characteristic equation. This one added pole must be
located at least five times far than the dominant poles. So the new closed loop
characteristic equation will be
(s
2
)
+ 2ξωn s + ωn (s + a) = 0
2
(4.2)
After obtaining equation (4.2), the next step is to find the feedback vector K.
Lastly, after obtaining the feedback vector K, the computer simulation diagram using
MATLAB/SIMULINK can be constructed and simulated.
4.2.1
Pole Placement Technique without integral to Control MSBB
¾ Design Specification: Using Percentage Overshoot and settling time:
Percent of overshoot, (%OS) = 10%
Settling time, T s = 0 .1 sec
26
The formula to calculate the percentage overshoots (Nise, N.S., 2004)
% O.S = e
− πξ
1−ξ 2
(4.3)
When the overshoot is 10%:
10 = e
− πξ
1−ξ 2
πξ
2.3 = −
1− ξ 2
π 2ξ 2
5.29 =
1−ξ 2
5.29 − 5.29ξ 2 = 9.87ξ 2
Therefore the damping ratio is:
5.29
9.87 + 5.29
ξ
=
ξ
= 0.35
= 0.6
The formula to calculate the settling time, (Nise, N.S., 2004):
Ts =
4
ξωn
When, Ts = 0.1 second,
0.1 =
4
ξωn
Therefore, the natural frequency is:
ωn =
4
(0.1)(0.6)
= 66.67 rad/sec
(4.4)
27
By using these two parameters, the close-loop characteristic equation can be
obtained using the formula below:
s 2 + 2ξωn s + ωn 2 = 0
(4.5)
Substitute ξ = 0.6 and ωn = 66.67 rad/sec into equation (4.5):
s 2 + 2(0.6)(66.67) s + (66.67) 2 = 0
s 2 + 80 s + 4445 = 0
( s + 40 + j 53.33)( s + 40 − j 53.33) = 0
(4.6)
The poles are, s = − 40 + j 53.33 and s = − 40 − j 53.33 . To implement the pole
placement technique, one more pole is required to be added to make the characteristic
equation as a 3rd order equation. One pole that had been added is s = − 240 . This pole is
chosen because they are 6 times bigger than those two dominant poles. So by adding
s = − 240 to the s-plane, the system still behaves like a second order characteristic
equation. So the new characteristic equation
(s + 40 +
j 53 .33 )(s + 40 − j 53 .33 )(s + 240 ) = 0
(4.7)
s + 240 s 2 + 23645 s + 1066800 = 0
3
After obtaining equation (4.7), the feedback vector K can be calculated. Using
equation (3.21),
1
0 ⎤ ⎡ 0 ⎤
⎡ 0
⎢
0
1.6 ⎥⎥ − ⎢⎢ 0 ⎥⎥ [ K1 K 2 K 3 ]
A − BK = ⎢ −1188.8
⎢⎣ 0
−2951 −962 ⎥⎦ ⎢⎣1374 ⎥⎦
1
0 ⎤ ⎡ 0
0
0 ⎤
⎡ 0
⎢
⎥
⎢
0
1.6 ⎥ − ⎢ 0
0
0 ⎥⎥
A − BK = ⎢ −1188.8
⎢⎣ 0
−2951 −962 ⎥⎦ ⎢⎣1374 K1 1374 K 2 1374 K 3 ⎥⎦
0
⎡
⎢
A − BK = ⎢ −1188.8
⎢⎣ −1374 K1
⎤
⎥
0
1.6
⎥
− (2951 + 1374 K 2 ) −(962 + 1374 K 3 ) ⎥⎦
1
0
(4.8)
28
Then,
0
⎡ s 0 0⎤ ⎡
⎢
⎥
⎢
sI − ( A − BK ) = ⎢0 s 0⎥ − ⎢ − 1188.8
⎢⎣0 0 s ⎥⎦ ⎢⎣− 1374 K1
⎤
⎥
⎥
− (962 + 1374 K 3 )⎥⎦
1
0
0
1.6
− (2951 + 1374 K 2 )
−1
0
⎡ s
⎤
⎢
⎥
= ⎢ 1188.8
− 1.6
s
⎥
⎢⎣1374K1 2951+ 1374K 2 s + 962 + 1374K3 ⎥⎦
(4.9)
The characteristic equation can be computed as:
[(
]
)
sI − ( A − BK ) = s s 2 + 962s + 1374K 3 s + (4721.6 + 2198.4 K 2 )
− (− 1)[(1188.8s + 1143625.6 + 1633411.2 K 3 ) + 2198.4 K 1 ]
= s 3 + (962 + 1374K 3 )s 2 + (5910.4 + 2198.4K 2 )s + 1143625.6 + 1633411.2K 3 + 2198.4K1
(4.10)
Equation (4.7) and equation (4.10) are compared to each other
962 + 1374 K 3 = 240 ⇒ K 3 = − 0 . 5
5910.4 + 2198.4 K 2 = 23645 ⇒ K 2 = 8.067
(4.11)
(4.12)
1143625.6 + 1633411.2(− 0.5255) + 2198.4 K 1 = 1066800
K1 = 312.200
(4.13)
Hence, the feedback vector K is:
K = [K 1
K2
K3 ]
K = [312.20 8.067 − 0.5]
By substituting equation (4.14) into u , where u input control voltage
(4.14)
29
u = − Kx
⎡θ ⎤
u = − [312 . 20 8 . 067 − 0 . 5 ]⎢⎢θ& ⎥⎥
⎢⎣ i ′ ⎥⎦
u = − 312 . 20 θ − 8 . 067 θ& + 0 . 5i ′
4.3
(4.15)
(4.26)
Pole Placement Technique with Integral Control
Figure 4.2 shows the block diagram of pole placement technique with integral
control. The integral control is introduced to eliminate a steady state error. A feedback
path from the output has been added to form error,(e) which is fed forward to the
controlled plant via an integrator. The integrator increases the system type and reduces
the previous finite error to zero.
•
e
∫
xN
KN
x
∫
x
y
Figure 4.2: Pole Placement Technique with Integral Control
An additional state variable, xN , has been added at the output of the leftmost
integrator. The error is the derivative of this variable. Now, from Figure 4.2
•
xN = r − Cx
Writing the state equations from Figure 4.2,
(4.16)
30
•
x = Ax + Bu
(4.17)
x N = −Cx + r
(4.18)
•
y = Cx
(4.19)
Equations (4.17) - (4.19) can be written as augmented vectors and matrices. Hence,
⎡ • ⎤
⎢ x ⎥ = ⎡ A 0⎤ ⎡ x ⎤ + ⎡ B ⎤ u + ⎡0⎤ r
⎢1 ⎥
⎢ • ⎥ ⎢⎣ −C 0 ⎥⎦ ⎢⎣ xN ⎥⎦ ⎢⎣ 0 ⎥⎦
⎣ ⎦
x
N
⎣ ⎦
⎡x⎤
y = [C 0] ⎢ ⎥
⎣ xN ⎦
(4.20)
(4.21)
But,
u = − Kx + K N xN = −[ K
⎡x⎤
− KN ] ⎢ ⎥
⎣ xN ⎦
(4.22)
Substituting equation (4.22) into (4.20) and simplifying,
⎡ • ⎤
⎢ x ⎥ = ⎡( A − BK ) BK N ⎤ ⎡ x ⎤ + ⎡0⎤ r
⎢ • ⎥ ⎢⎣ −C
0 ⎥⎦ ⎢⎣ xN ⎥⎦ ⎢⎣1⎥⎦
⎣ xN ⎦
⎡x⎤
y = [C 0] ⎢ ⎥
⎣ xN ⎦
(4.23)
(4.24)
Thus, the system type has been increased. The characteristic equation associated
with equation (4.24) can be used to design K and K N to yield the desired transient
response. Realize, now an additional pole have to place be placed. The effect on the
transient response of many closed-loop zeros in the final design must also be taken into
consideration. One possible assumption is that the closed-loop zeros will be the same as
those of the open-loop plant. These assumptions, which of course must be checked,
suggest placing higher order poles at the closed-loop zero locations.
31
4.3.1
Pole Placement Technique with integral to Control MSBB
¾ Design Specification: Using Percentage Overshoot and settling time:
Percent of overshoot, %OS = 10%
Settling time, T s = 0 .1 sec
By using these two design specification, the closed-loop characteristic equation
is similar as calculated in the previous section.
s 2 + 2ξωn s + ωn 2 = 0
s 2 + 2(0.6)(66.67) s + (66.67) 2 = 0
s 2 + 80 s + 4445 = 0
( s + 40 + j 53.33)( s + 40 − j 53.33) = 0
The poles are s = − 40 + j 53.33 and s = − 40 − j 53.33 . To implement the pole
placement technique, two more poles are required to be added to make the characteristic
equation as a 4th order equation. Two more poles that had been added are s = − 240
and s = − 250 . These two poles are chosen because they are 6 times bigger than those
two dominant poles. So by adding s = − 240 and s = − 250 to the s-plane, the system still
behaves like a second order characteristic equation. So the new characteristic equation is
( s + 40 + j 53.33) ( s + 40 − j 53.33) ( s + 240) ( s + 250) = 0
s 4 + 570s 3 + 103641s 2 + 6976036s + 266453400 = 0
(4.25)
After obtaining equation (4.25), the feedback vector K can be calculated using
equation (3.21),
32
1
0 ⎤ ⎡ 0 ⎤
⎡ 0
0
1.6 ⎥⎥ − ⎢⎢ 0 ⎥⎥ [ K1
A − BK = ⎢⎢ −1188.8
−2951 −962 ⎦⎥ ⎣⎢1374 ⎦⎥
⎣⎢ 0
K2
K3 ]
1
0 ⎤ ⎡ 0
0
0 ⎤
⎡ 0
⎢
⎥
⎢
A − BK = ⎢ −1188.8
0
1.6 ⎥ − ⎢ 0
0
0 ⎥⎥
−2951 −962 ⎦⎥ ⎣⎢1374 K1 1374 K 2 1374 K 3 ⎥⎦
⎣⎢ 0
0
1
0
⎡
⎤
⎢
⎥
A − BK = ⎢ −1188.8
0
1.6
⎥
⎢⎣ −1374 K1 −(2951 + 1374 K 2 ) −(962 + 1374 K 3 ) ⎥⎦
(4.26)
Hence,
⎡s
⎢0
sI − ( A − BK ) = ⎢
⎢0
⎢
⎣0
0
s
0
0
0
0
s
0
0⎤
0 ⎥⎥
0⎥
⎥
s⎦
0 ⎤
0
1
0
⎡
⎢ −1188.8
0
1.6
0 ⎥⎥
⎢
−
⎢ −1374 K1 −(2951 + 1374 K 2 ) −(962 + 1374 K3 ) 1374 K N ⎥
⎢
⎥
0 ⎦
0
0
⎣ −1
−1
⎡ s
⎢ 1188.8
s
=⎢
⎢1374 K1 2951 + 1374 K 2
⎢
0
⎣ 1
0
−1.6
s + 962 + 1374 K3
0
⎤
⎥
⎥
−1374 K N ⎥
⎥
s
⎦
0
0
The characteristic equation is
sI − ( A − BK ) = s[ s ( s 2 + 962s + 1374sK 3 ) + 1.6(2951s + 1374sK 2 )]
+(1)(1188.8)[ s 2 + (962 + 1374 K 3 ) s ] + 1.6(1374sK1 + 1374 K N )
= s 4 + (962 + 1374 K 3 ) s 3 + (4721.6 + 2198.4 K 2 ) s 2
+ 1188.8s 2 + (1,142, 436.8 + 1, 633, 411.2 K 3 + 2198.4 K1 ) s + 2198.4 K N
(4.27)
33
= s 4 + (962 + 1374 K3 ) s 3 + (4721.6 + 2198.4 K 2 + 1188.8) s 2
+ (1,142, 436.8 + 1, 633, 411.2 K3 + 2198.4 K1 ) s + 2198.4 K N
(4.28)
Equation (4.25) and equation (4.28) are compared to each other to gives the
required gain vector, K:
962 + 1374 K 3 = 570
K 3 = −0.285
(4.29)
4721.6 + 2198.4 K 2 + 1188.8 = 103641
K 2 = 44.46
(4.30)
1,142, 436.8 + 1, 633, 411.2(−0.285) + 2198.4 K1 = 6976036
K1 = 2865.32
(4.31)
2198.4 K N = 266, 453, 400
K N = 121, 203.33
(4.32)
Hence, the feedback vector, K = [ K1
K2
K3
KN ]
K = [ 2865.32 44.46 −0.285 121, 203.33]
(4.33)
By substituting equation (4.33) into equation (4.22);
⎡θ ⎤
⎢•⎥
u = − [ 2865.32 44.46 −0.285] ⎢θ ⎥ + [121, 203.33] xN
⎢ ⎥
⎢i '⎥
⎣ ⎦
•
= − 2865.32θ − 44.46 θ + 0.285i '+ 121, 203.33 xN
(4.34)
It had been shown, in Figure 4.2, that xN = ∫ ( r − y ) . In this case the desired output,
y is θ . Hence,
•
u = − 2865.32θ − 44.46 θ + 0.285i '+ 121, 203.33 ⎡⎣ ∫ ( r − θ ) ⎤⎦
(4.35)
34
4.4
Computer Simulation Using MATLAB/SIMULINK
4.4.1 Pole Placement Technique without integral to Control linear MSBB system
From equation (3.21) and (4.15),
•
•
θ =θ
••
θ = − 1 1 8 8 .8θ + 1 .6 i `+ 1 0 .5 5 fd
•
•
i ` = − 2 9 5 1 θ − 9 6 2 i `+ 1 3 7 4 u
u = − 335 . 500 θ − 8 . 067 θ& + 0 . 5255 i ′
Then, by using those equations, the simulation diagram using MATLAB/SIMULINK
can be established as shown in Figures (4.3)-(4.5).
Figure 4.3: SIMULINK implementation of equation (3.21)
Figure 4.4: SIMULINK implementation of equation (3.21)
35
[THETA]
K1
1
K1
[THETADOT]
VOLTAGE -U
K2
K2
[CURRENT]
K3
K3
Figure 4.5: SIMULINK implementation of equation (4.15) for pole placement
technique without integral
By combining Figure 4.3 and Figure 4.4, the complete figure to do the simulation
block diagram for MSBB system is obtained. The complete figure is shown in Figure
4.6.
Figure 4.6: Complete simulation diagram of linear model for MSBB system
36
By combining Figure 4.5 and Figure 4.6, the complete figure to perform the
computer simulation is obtained. The complete figure is shown in Figure 4.7.
Figure 4.7: Complete simulation diagram of pole placement technique without
integral control for MSBB linear system.
4.4.2
Pole Placement Technique without integral to Control nonlinear
MSBB system
From equations, (3.1) to (3.5)
••
J θ = L a ( f1 − f 2 ) + f d
f1 = μ ο A g N
2
f2 = μ 0 Ag N
2
e
e
( iο + i1′ ) 2
2 ( g ο + L aθ ) 2
( io + i '2 ) 2
2 ( g 0 − L aθ ) 2
d i '1
dt
d i '2
+ L
dt
= R i '1 + L
'
1
'
2
= R i '2
37
Then, by using these equations, the simulation diagram using MATLAB/SIMULINK
can be established as shown in Figures (4.8)-(410).
Figure 4.8: Complete simulation diagram of nonlinear model for MSBB system
[THETA]
K1
1
K1
[THETADOT]
VOLTAGE -U
K2
K2
[CURRENT]
K3
K3
Figure 4.9: SIMULINK implementation of equation (4.15) for pole placement
technique without integral
38
Figure 4.10: Complete simulation diagram of pole placement technique without
integral control for MSBB nonlinear system
4.4.3
Pole Placement Technique with integral to Control linear MSBB system:
For pole placement with integral control to control the MSBB system, the
procedure to develop the complete simulation diagram is the same. The difference is that
equation (4.15) must be changed and equation (4.35). The complete simulation diagram
for pole placement technique with integral control for the MSBB system is shown in
Figures (4.11) and (4.12).
39
Figure 4.11: SIMULINK implementation of equation (4.35) for pole placement with
integral
Figure 4.12: Complete simulation diagram of pole placement technique with
integral control for the MSBB system
40
4.5
Summary
This chapter had introduced the concept of pole placement technique without
integral and with integral controller. The simulation diagram to implement the controller
were also been developed. In this chapter, two sets of feedback vector K (without
integral and with integral controllers) had been obtained.
CHAPTER V
Q-PARAMETERIZATION CONTROLLER DESIGN
5.1
Introduction
In this chapter the design of a Q-parameterization controller will be presented.
The Q-parameterization control has several advantages, such as: a stability controller
always exists; the closed loop poles can be located in a prescribed region in the open left
half plane, thereby ensuring the satisfaction of the required transient response. The order
of the Q-parameterization controller is equal to that of the transfer function of the system
being controlled plus the degree of the free parameter Q, which is much lower than the
order of comparable robust controllers, such as those based on H ∞ , LQG and μ analysis, the degree of the Q-parameter can be chosen to obtain a lower or higher order
controller.
In the Q-parameterization method, the set of all stabilizing controllers of the
MSBB system is characterized by one free parameter Q. This free parameter Q is chosen
to achieve robust stability and good dynamic performance over a certain range of system
parameter variations. The control objective is to keep the displacement of balance beam
in nominal gap.
42
5.2
The Q-parameterization Theory
A controller synthesis method based on Q-parameterization theory was addressed
by (Desoer.et.al, 1980)
Theorem: for a given a plant, P, all controllers K, which internally stabilize P, are
in the set:
{
K ∈ (Y − NQ
)− 1 ( X
+ MQ ), Q ∈ R
}
if and only if the four function in R ( N , M , X and Y ), satisfy
NX
+ MY
P =
N
M
, and
= 1
where Q is free parameter and Q is free-to-choose parameter. By carefully choosing the
Q, the control system can be designed to meet certain criteria, for example in this
project; the beam is design to have the robust stability.
In the general case, the set of all stabilizing controllers for the system is
characterized by free parameters. Where the parameter Q is constant gain or a stable
transfer function, the controller Q-parameter can be chosen using optimization to
achieve the desired robustness and performance goals.
As mentioned earlier, the degree of the resulting controller equals the degree of
the plant plus the degree of the Q-parameter and the degree of the Q-parameter can be
chosen to obtain a lower or higher order controller.
Consider the one-parameter-control feedback system show in Figure (5.1) for
controlling any of the SISO subsystems described by Eq.(3.17) where r ∈ ℜ
is the
reference input signal, and d ∈ ℜ is the disturbance force, and u ∈ ℜ is the controller
output, y ∈ ℜ is the plant output to be regulated, and K ∈ R is the stabilizing controller
for P (s).
43
Figure 5.1 Block diagram of one parameter –control feedback system
In order to characterize the set of all stabilizing controllers K for P(s), first
~ ~
~ ~
should be to construct a doubly coprime factorization N , D, N , D, X , Y , X , Y ∈ RH ∞ for P
(s). Then the set of all stabilizing controllers for the plant P (s) is given by
{(
~
K = Y − QN
5.3
) (X + QD~ ), Q ∈ RH
−1
∞
}
~
, Y − QN ≠ 0
(5.1)
Controller objective
In this research work, the Q-parameterization controller is design to achieve
robust stability for MSBB against disturbance and variation and achieve fast and well
damped transient response.
5.4
Controller synthesis
In order to satisfy above requirement, the closed loop poles must be located at a
prescribed region in the open left half plane. This can be achieved by choosing
44
~ ~
~ ~
N , D, N , D, X , Y , X , Y , Q ∈ Ds , where Ds ⊂ RH ∞ , and Ds an infinite dimensional set
defined as shown in Figure (5.2).
Figure 5.2: Generalized region of stability
Finally, the Q-parameterization controller can brief description in some circuits,
and show how it is connect with the nominal controller which stabilizes the plant.
Now, Consider the feedback structure shown in Figure (5.3), where the input
signals are w and u and the output signals are z and y , where w represents the
exogenous inputs which may include real physical disturbances (forces), actuator or
sensor noise as well as command signals represents the actuator inputs, and z represents
the regulated variables, and y represents the sensor outputs, then P represents the openloop interconnection transfer function
⎡z ⎤
⎡ w⎤
⎢ y ⎥ = P ⎢u ⎥
⎣ ⎦
⎣ ⎦
(5.2)
45
Plant
exogenous
inputs
controlled
inputs
w
u
z
P
y
regulated
variables
measured
variables
Controller
-K
Figure 5.3: Basic feedback structures.
The objective here is to design a controller K with input y and output u : u = − Ky
in order to stabilize the system and achieve the other design requirements.
The MIMO closed-loop transfer function H zw between the regulated variables z
and the exogenous inputs w contains by definition every closed-loop transfer function
of interest and can be expressed by the following linear fractional transformation (LFT).
H zw = Pzw − Pzu K (I + Pyu K ) Pyw
−1
(5.3)
where,
H zw
=
closed-loop transfer function between the regulated variables z and the
exogenous inputs w
Pzu = open-loop transfer function between the regulated variables z and the actuator
inputs u
Pyu = open-loop transfer function between the sensor outputs y and the actuator inputs u
Pyw = open-loop transfer function between the sensor outputs y and the exogenous
inputs w
Pzw = open-loop transfer function between the regulated variables z and the exogenous
inputs w
46
Now consider the block diagram of Q-parameterization with internal
interconnection. as shown in Figure (5.4). The derivation of the Q-parameterization
starts with any controller, K nom which stabilizes the plant and which we will call the
nominal controller. Let
K nom = Y −1 X
(5.4)
be a left stable coprime factorization of the nominal controller and
~ ~
Pyu = D −1 N
(5.5)
be a left stable coprime factorization of Pyu .Then every controller of the form
(
~
K = Y − QN
) (X + QD~ ), Q Stable
−1
Figure 5.4: Block diagram of Q-parameterization
Figure 5.5: Q-parameterization as modification to nominal controller
(5.6)
47
stabilizes P and conversely every controller which stabilizes P has this form for some
stable Q. There is similar characterization of the stabilizing controllers in terms of right
coprime factorization. Since K nom stabilizes Pyu , there is right coprime factorization
Pyu = ND −1
(5.7)
with
XN + YD = I
(5.8)
The identity map, with a little calculation yields the Q-parameterization formula,
H zw = T1 + T2 QT3
(5.9)
T1 = Pzw − Pzu DXPyw
(5.10)
T2 = − Pzu D
(5.11)
with
where, T1 is simply the H zw achieved with the nominal controller K nom , T2 is the map
from v to z , T3 is the map from w to e , as can be seen from Figure (5.4) and Figure
(5.5)shows block diagram of the Q-parameterization .
The key to the Q-parameterization is that the closed-loop map from v to e be
zero, so that the Q sees no feedback. Doyle has given a very nice interpretation of the Qparameterization when the nominal controller is an estimated state feedback.
The design of the controller based on Q-parameterization method by using linear
state space equations.
5.5
Design of the Q-parameterization controller for MSBB System
The state-space linearized model of the MSBB system as described by equations
(3.17) and (3.21) are
48
x& = Ax + Bu + Ef d
y = Cx
⎡•⎤
⎢ x1 ⎥ ⎡ 0
1
0 ⎤ ⎡ x1 ⎤ ⎡ 0 ⎤
⎡ 0 ⎤
⎢•⎥ ⎢
0
1.6 ⎥ ⎢ x2 ⎥ + ⎢ 0 ⎥ u + ⎢⎢10.55⎥⎥ fd
⎢ x2 ⎥ = ⎢ −1188.8
⎥⎢ ⎥ ⎢
⎥
⎢•⎥ ⎢ 0
⎢⎣ 0 ⎥⎦
⎥
⎢
⎥
⎢
⎥⎦
2951
962
1374
x
−
−
⎦⎣ 3⎦ ⎣
⎢ x3 ⎥ ⎣
⎣ ⎦
⎡ x1 ⎤
y = [1 0 0] ⎢⎢ x2 ⎥⎥
⎢⎣ x3 ⎥⎦
By taking the Laplace transform, the system under consideration may be rewritten in
the s-domain as:
sX ( s ) = AX ( s ) + BU ( s ) + Ef d ( s )
Y ( s ) = CX ( s )
Provided that the load disturbance f d is zero, the transfer function of the MSBB
system G ( s ) can be expressed in the state-space form as:
G ( s ) = C (sI − A) B
−1
(5.12)
where G ( s ) ∈ ℜ( s ) , where ℜ( s ) is the set of all real rational transfer function.
The design procedure of the proposed controller based on Q-parameterization
theory may be explained in the following steps:
1.
and
Choose the real matrices K1 and K 2 such that the two matrices
Aο = A − BK1
(5.13)
~
Aο = A − K 2C
(5.14)
are Hurwitz (all of its eigenvalues have negative real parts). Now use pole placement
method (or any other method) to obtain the value of K1 , such that Aο = A − BK1 is
~
stable. Similarly choose K 2 such that Aο = A − K 2 C is stable.
49
Notice that equations (5.13) and (5.14) are representation of system when it is
stable. Then if desired poles P1 are calculated from the design specification like what did
before when the over shooting was equal 10% and settling time equal 0.1 second and
P2 it will be choosing in the left side of s- the plant and will be very far from dominant
poles.
¾ Design Specification: Using Percentage Overshoot and settling time:
Percent overshoot, %OS= 10%
Settling time, T s = 0 .1 sec
The formula to calculate the percentage overshoots (Nise, N.S., 2004)
% O.S = e
− πξ
1−ξ 2
When the overshoot is 10%:
10 = e
2.3 = −
− πξ
1−ξ 2
πξ
1−ξ 2
π 2ξ 2
5.29 =
1−ξ 2
5.29 − 5.29ξ 2 = 9.87ξ 2
Therefore the damping ratio is:
50
5.29
9.87 + 5.29
ξ
=
ξ
= 0.35
= 0.6
The formula to calculate the settling time in (Nise, N.S., 2004):
Ts =
4
ξωn
When, Ts = 0.1 second,
0.1 =
4
ξωn
Therefore, the natural frequency is:
ωn =
4
(0.1)(0.6)
= 66.67 rad/sec
By using these two parameters, the close-loop characteristic equation can be
obtained using the formula below:
s 2 + 2ξωn s + ωn 2 = 0
Substitute ξ = 0.6 and ωn = 66.67 rad/sec into equation (4.5):
s 2 + 2(0.6)(66.67) s + (66.67) 2 = 0
s 2 + 80 s + 4445 = 0
( s + 40 + j 53.33)( s + 40 − j 53.33) = 0
The poles are, s = − 40 + j 53.33 and s = − 40 − j 53.33 . To implement the pole
placement technique, one more pole is required to be added to make the characteristic
51
equation as a 3rd order equation. One pole that had been added is s = − 240 . This pole is
chosen because they are 6 times bigger than those two dominant poles. So by adding
s = − 240 to the s-plane, the system still behaves like a second order characteristic
equation. So the new characteristic equation, from previous calculated in pole placement
technique, we found the feedback vector of K1 equal to
K1 = [312.12 8.067 − 0.5]
(5.15)
The gains can be calculated using the MATLAB command:
K1 = place( A, B, P1 )
(5.16)
where P1 is the desired poles
P1 = [− 40 + j 53.33 − 40 − j 53.33 − 240]
It can be shown that the gains are
K1 = [312.12 8.067 − 0.5]
Then, substituting equation (5.15) into equation (5.13) can be obtained
Aο = A − BK1
1
0 ⎤ ⎡ 0 ⎤
⎡ 0
⎢
= ⎢− 1188.8
0
1.6 ⎥⎥ − ⎢⎢ 0 ⎥⎥[312.12 8.067 − 0.5]
⎢⎣ 0
− 2951 − 962⎥⎦ ⎢⎣1374⎥⎦
0
0
0 ⎤
1
0 ⎤ ⎡
⎡ 0
⎢
⎢
⎥
= ⎢− 1188.8
0
0
0 ⎥⎥
0
1.6 ⎥ − ⎢
⎢⎣ 0
− 2951 − 962⎥⎦ ⎢⎣428852.88 11084.06 − 687⎥⎦
0
1
0 ⎤
⎡
⎢
Aο = ⎢ − 1188.8
0
1.6 ⎥⎥
⎢⎣− 428852.88 − 14035.06 − 275⎥⎦
(5.17)
52
~
Then, repeat previous step for Aο to obtain Aο , by choose P2 randomly but very
far from two dominant poles. So by adding s2 = −600, s3 = −800 and s4 = −1000 to the splane, the system still behaves like a second order characteristic equation.
P2 = [-600 -800 -1000]
(5.18)
Then, use MATLAB command to obtain the gain K 2 :
′
K 2 = place( A′, C ′, P2 )
(5.19)
⎡ 0.0144 ⎤ ⎡ 14442 ⎤
K 2 = 10 * ⎢⎢ 4.9073 ⎥⎥ = ⎢⎢ 490732 ⎥⎥
⎢⎣− 0.1188⎥⎦ ⎢⎣− 1188.4⎥⎦
(5.20)
Produce
5
Then, substituting equation (5.20) into equation (5.15) can be obtained
~
Aο = A − K 2C
~
Aο
1
0 ⎤ ⎡ 1442 ⎤
⎡ 0
⎢
= ⎢− 1188.8
0
1.6 ⎥⎥ − ⎢⎢ 490732 ⎥⎥[1
⎢⎣ 0
− 2951 − 962⎥⎦ ⎢⎣− 1188.4⎥⎦
0
1
0 ⎤ ⎡ 1442
⎡ 0
⎢
⎢
⎥
= ⎢− 1188.8
0
1.6 ⎥ − ⎢ 490732 0
⎢⎣ 0
− 2951 − 962⎥⎦ ⎢⎣− 11884 0
1
0 ⎤
⎡ − 1442
⎢
= ⎢− 491920.8
0
1.6 ⎥⎥
⎢⎣ 11884
− 2951 − 962⎥⎦
0 0]
0⎤
0⎥⎥
0⎥⎦
2. To synthesize the controller, the doubly coprime factorization of system
construct the doubly coprime factorization of system should be found.
~
~
Construct the doubly coprime factorization of system by finding N , D, N and D
53
~ ~
where N , D and N , D are transfer function of the right and left coprime factors of system,
respectively:
~ ~
G (s ) = ND −1 = D −1 N
(5.21)
where is G(s ) the transfer function of the system. Such as factorization are possible if the
pairs ( A, B ) and (C, A) are stabilizable and detectable pairs, respectively.
~
~
3. Define the transfer functions X , Y , X and Y , which are obtained, such that the
following identities hold:
YD + XN = I
(5.22)
~~ ~ ~
NX + DY = I
(5.23)
~ ~
~
~
where N , D, N , D, X , Y , X and Y ∈ RH ∞ where RH ∞ is a set of stable transfer functions.
Formulas this doubly coprime factorization in terms of A, B and C are expressed as:
N = C (sI − A ο
)− 1 B
D = I − K 1 (sI − A ο ) B
~ −1
~
N = C sI − A ο
B
~ −1
~
D = I − C sI − A ο
K2
1
−
~
X = K 1 sI − A ο
K2
~ −1
Y = I + K 1 sI − A ο
B
~
−1
X = K 1 (sI − A ο ) K 2
~
−1
Y = I + C (sI − A ο ) K 2
−1
(
(
(
(
)
)
)
(5.24)
)
~
But, in this application just use Y and N to calculate Q if it was constant and
~ ~
N , D , X and Y to calculate the Q-parameterization controller
54
(
)
~ −1
~
N = C sI − Aο B
~ −1
~
D = I − C sI − Aο K 2
~ −1
X = K 1 sI − Aο K 2
~ −1
Y = I + K 1 sI − Aο B
(
(
(
)
)
(5.25)
)
~
~
The coprime factorizations N ( s ), D ( s ), X ( s ) and Y ( s ) described in form of the
transfer functions such as:
− 9.095 *10 −13 s 2 − 1.397 *10 −9 s + 2198
~
N ( s) = 3
s + 2400s 2 + 1.88 *10 6 s + 4.8 *10 8
~
D( s ) =
s 3 + 962s 2 + 5910s + 1.144 *10 6
s 3 + 2400s 2 + 1.88 *10 6 s + 4.8 *10 8
(5.27)
4.413 *10 6 s 2 + 5.056 *10 9 s + 1.338 *1011
X ( s) =
s 3 + 2400s 2 + 1.88 *10 6 s + 4.8 *10 8
Y (s) =
(5.26)
s 3 + 1758s 2 + 9.745 * 10 5 s + 1.904 *10 8
s 3 + 2400s 2 + 1.88 *10 6 s + 4.8 * 10 8
(5.28)
(5.29)
~ ~
4- Calculate the set of all stabilizing controllers for G(s) in terms of X , Y , N , D, as:
{(
~
K = Y − QN
) (X + QD~ ), Q ∈ RH
−1
∞
}
~
, Y − QN ≠ 0
(5.30)
where Q is free parameter that can be chosen to satisfy all design requirements such as
stability, robustness to variation of the MSBB System parameters, good tracking and
disturbance rejection. Then Q can be chosen at any order to fit system requirement for
example:
Q = Q0
or
; Zero order
55
Q=
as + b
s+ p
; First order
Q=
as 2 + bs + c
(s + p1 )(s + p2 )
; Second order
or
or
Q( s) =
a0 s n + a1s n −1 + a2 s n − 2 + ..... + an
(s + p1 )(s + p2 ).....(s + pn )
; Nth order proper stable transfer function.
where a0 , a1 , a3 ,......, an ∈ ℜ are free designs parameters obtained through satisfying the
system design requirements and p1 , p2 ,......, pn 〉α s ∈ ℜ are constant values chosen
according to stability requirements.
The higher order of Q , the more constraints on the system performance like
(overshoot, settling time, steady-state error, rise time, bandwidth, nose minimization…)
that can be achieved.
The main target is to make sure that the displacement angle is equal to zero even
though there is a presence of disturbance in the plant.
In this thesis, the main concern is to achieve zero steady-state error the
displacement angle and fast response. So for this case Q may be chosen in its simplest
form, i.e. constant. This is due to having only one or two constraint to be achieved and
this leads to the lowest order of controller Q .
Accordingly, a simple feedback system for the controlling system described by
equation (3.17) is shown in Figure (5.6), where only a one state feed back signal is
shown for simplicity.
56
fd
E
θ
+
C (sI − A) −1
B
K (s )
Figure 5.6: One parameter controller feedback system
To get Q that achieves the required constraint, zero steady-state error; sensitivity
function S is defined as the ratio between the system output to be regulated gap
displacement angle θ and the system load disturbance f d .
S (s ) =
θ
fd
And its Laplace transfer function is given as:
(
)
~
S ( s ) = D( s ) Y ( s ) − Q( s ) N ( s) GFd ( s )
(5.31)
(5.32)
where GFd (s ) ∈ ℜ and can be expressed in the state-space form as
GFd ( s) = C (sI − A) E
−1
(5.33)
For zero steady-state error
S ( 0) = 0
or
~
Y ( 0) − Q ( 0 ) N ( 0 ) = 0
(5.34)
Therefore, Q (0)
Y (0)
Q (0) = ~
N (0)
From this equation, Q (0) can be calculate by substituting (s) equal zero
~
into N ( s ) and Y ( s ) . Therefore, transfer function
(5.35)
57
Y (s) =
s 3 + 1758 s 2 + 9 .745 × 10 5 s + 1 .904 × 10 8
s 3 + 2400 s 2 + 1 .88 × 10 6 s + 4 .8 × 10 8
Y (0) = 0.39667
(5.36)
and,
− 9 .095 × 10 −13 s 2 − 1 .397 × 10 −9 s + 2198
~
N (s) =
s 3 + 2400 s 2 + 1 .88 × 10 6 s + 4 .8 × 10 8
~
N (0) = 4.58 × 10 −6
(5.37)
Thus, substituting equations (5.36) and (5.37) into equation (5.35), Q (0) can be
obtained as:
0.39667
Y ( 0)
=
= 8.6596 × 10 4
Q ( 0) = ~
−6
N (0) 4.58 × 10
(5.38)
After Q (0) calculated, the controller proposed can be obtaining. The transfer
function for the Q-parameterization controller is
{(
~
K = Y − QN
) (X + QD~ ), Q ∈ RH
−1
∞
}
~
, Y − QN ≠ 0
Thus, by substituting equations (5.26)-(5.29) and (5.38) into equation
(5.30), K ( s ) can be obtained as:
8.66 *10 4 s 3 + 8.772 *10 7 s 2 + 5.568 *10 9 s + 2.329 *1011
K (s) =
s 3 + 1758s 2 + 9.745 *10 5 s + 6.704 *10 −8
(5.39)
Equation (5.39) represents the robust proposed control (Q-parameterization
controller) in transfer function form.
58
5.6
5.6.1
Computer Simulation Using MATLAB/SIMULINK
Computer Simulation Using MATLAB/SIMULINK for linear model
of MSBB System
To do computer simulation, the simulation diagram for both linear and non linear
of the magnetically suspended balance beam (MSBB) system must construct first.
Figure (5.7) shows the simulation diagram for linear MSBB system. This diagram
is constructed base on equation (3.21)
Figure 5.7: Complete simulation diagram of linear model for MSBB system
Figures (5.8) and (5.9) show the complete simulation diagram of linear model for
MSBB system with the proposed control.
59
2951
1188.8
962
thetadot
1
s
i
1
s
theta
1
s
1.6
[THETA]
0
[CURRENT]
Constant
1374
theta
i
DISTURBANCE
[THETADOT]
thetadot
10.55
THETA_Q
To Workspace
Step
THETA
num(s)
den(s)
Q-parameterization
Controller
Figure 5.8: Complete simulation diagram of linear model for MSBB system with
Q-parameterization Controller
Q-para Controller
voltage
[THETA]
num(s)
-1
U
den(s)
Q-parameterization
Controller
K2
VOLTG_Q
To Workspace
disturbance
DISTURBANCE
MSBB
Step
Figure 5.9: Complete simulation diagram of linear model for MSBB system with
Q-parameterization Controller
60
5.6.2 Computer Simulation Using MATLAB/SIMULINK for nonlinear model
of the MSBB System
Figure (5.10) shows the simulation diagram for nonlinear of MSBB system as
described by equation (3.22)
1
theta.dot
theta.dot.dot
i1'
-K-
1
s
2
3
U2=e2'
1
i1'
fd
f(u)
1/L
U1=e1'
3
1/J
1
s
theta.dot
1
s
theta
2
theta
-K-
f1
[theta]
1/L
f(u)
1
s
f2
-Ki2'
Figure 5.10: Complete simulation diagram of nonlinear model for MSBB system
Figure (5.11) and (5.12) show complete simulation diagram of nonlinear model
for MSBB system with the proposed control.
61
Figure 5.11: Complete simulation diagram of linear model for MSBB system with
Q-parameterization Controller
Gap_Q_parameterization
theta dot
Gap_Q
Gap_Q_parameterization
Disturbance
theta.dot
[thetadot]
U1=e1'
theta
[theta]
U2=e2'
i1'
fd
Gain3
[theta]
8.66e004s3+8.772e007s2+5.568e009s+2.329e011
s3 +1758s2+9.745e005s+6.704e-008
Q-parameterization
Controller
-1
-1
[i]
Gain1
Magnetic Beam1
voltage_Q_parameterization
, e1' & e2'
Voltage_Q_parameterization
Voltage_Q1
Figure 5.12: Complete simulation diagram of linear model for MSBB system with
Q-parameterization Controller
62
5.7
Summary
This chapter presents and explains the step by step procedure of designing the
proposed control (Q-parameterization Controller). The simulation diagrams for both
linear and nonlinear models were also established.
CHAPTER VI
SIMULATION RESULTS AND DISCUSSION
6.1
Introduction
This chapter contains all of the simulation results regarding to two different types
of controllers mentioned in Chapter IV and Chapter V.
The first controller is a state feedback controller using pole placement technique
and the second one is the proposed control (Q-parameterization controller). For
comparison purpose, the design specification of %10 over shoot and 0.1 second settling
time for a unit step input has been fixed for both controllers. Table 6.1 shows selection
of dominant poles and the added poles used as described in the previous chapters.
Table 6.1: Description for design specification and poles
NO.
CONTROLLERS
%OS, Ts
DOMINANT
ADDED POLES
POLES
1
2
Pole placement
10%,
technique
0.1 sec
Q-parameterization
10%,
Controller
0.1 sec
s = −40 ± j 53.3
s = − 240
s = −40 ± j 53.3
s = −240, s = −600
s = −800, s = −1000
64
By using the design specification tabulated in table 6.1, the simulations were
performed with linear and nonlinear model of MSBB System.
For pole placement technique, one added pole is needed. This one pole is
determined by the designer. To preserve the system to behave like a second order closed
loop characteristic equation, the added pole must be at least 5 times far from the
dominant poles s = −40 ± j 53.3 . The pole must be added because MSBB is a third order
system. Then by using the pole placement technique, the feedback vector K for the
mentioned design specification can be calculated.
For Q-parameterization controller, four added poles are needed, these four added
poles are the poles that are determined by the designer. To preserve the system to behave
like a second order closed loop characteristic equation, the added poles must be located
at least 5 times far from the dominant poles s = −40 ± j 53.3 . These four added poles will
not affect the order of the MSBB because they were selected very far from dominant
poles.
6.2
Results for linear model of MSBB System
This section presents the simulation results of pole placement controller as well as
Q-parameterization controller when applied to the linear MSBB model. It should be
noted that throughout the simulation proccess, it is assumed that the MSBB system is
subjected to a step disturbance as depicted in Figure 6.1.
65
Step disturbance force(N)
1.5
Magnitude (N)
1
0.5
0
0
0.2
0.4
0.6
Time (sec)
0.8
1
Figure (6.1): Step disturbance force (1-N-m) at time =0.15 seconds
Figures (6.2) and (6.3) show the results for linear model of the MSBB System
with pole placement technique. It can be seen from Figure 6.2 that the pole placement
controller is unable to reject the disturbance. The input voltage applied is as depicted in
Figure 6.3.
4
x 10
-3
Gap displacement angle forlinear of the MSBB System
3.5
Gap Displacement (rad)
3
2.5
2
1.5
1
0.5
0
-0.5
0
0.2
0.4
0.6
Time (sec)
0.8
1
Figure 6.2: Displacement angle for linear MSBB System with 1 N-m disturbance
using pole placement controller.
66
Input Voltage for linear of the MSBB System
2
1
Input Voltage (volt)
0
-1
-2
-3
-4
-5
-6
0
0.2
0.4
0.6
Time (sec)
0.8
1
Figure 6.3: Input voltage for linear MSBB System with 1 N-m disturbance
using pole placement controller.
Figure (6.4) and (6.5) show the results for linear model of MSBB System with
Q-parameterization controller. It can be seen from Figure 6.4 that when time is equal to
0.15 seconds, there is a step disturbance with a magnitude of 1 Newton-meter (N-m) that
enters into the plant.
It can be seen that the Q-parameterization controller has the ability to reject the
disturbance. Figure (6.5) shows the magnitude of the input voltage between +0.3 volts to
-5.7 volts. It can be concluded that the proposed controller (Q-parameterization
controller) satisfy our requirement by rejecting the disturbance force and the steady state
error zero with no need to inject high input voltage.
67
8
x 10
-4
Gap displacement angle for linear of the MSBB System
7
Gap Displacement (rad)
6
5
4
3
2
1
0
-1
-2
0
0.2
0.4
0.6
Time (sec)
0.8
1
Figure 6.4: Displacement angle for linear MSBB System with 1 N-m disturbance
using Q-parameterization controller.
2
1
0
Input Voltage (volt)
-1
-2
-3
-4
-5
-6
-7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (sec)
Figure 6.5: Input voltage for linear MSBB System with 1 N-m disturbance
using Q-parameterization controller.
68
6.3
Results for nonlinear model of MSBB System
Figure (6.6) and (6.7) show the results for nonlinear model of the MSBB System
with pole placement technique. Figure (6.6) shows the displacement angle when time is
equal to 0.15 seconds, there is a step disturbance that enters into the plant. It can be seen
that the disturbance had not been rejected.
Figure (6.7) shows the magnitude of the input voltage between +0.04 volts to 3.15 volts and stabilizes at value -1.75 volts.
3
x 10
-3
Gap displacement angle for the MSBB System
2.5
Gap Displacement (rad)
2
1.5
1
0.5
0
-0.5
-1
0
0.2
0.4
0.6
Time (sec)
0.8
1
Figure 6.6: Displacement angle for nonlinear MSBB System with 1 N-m
disturbance using pole placement controller.
69
Input Voltage for the MSBB System
2
1
Input Voltage (volt)
0
-1
-2
-3
-4
-5
-6
0
0.2
0.4
0.6
Time (sec)
0.8
1
Figure 6.7: Input voltage for nonlinear MSBB System with 1 N-m disturbance
using pole placement controller.
Figure (6.8) and (6.9) show the results for nonlinear model of the MSBB
System with Q-parameterization controller. Figure (6.8) shows that the disturbance had
been rejected.
5
x 10
-4
Gap displacement angle for the MSBB System
Gap Displacement (rad)
4
3
2
1
0
-1
0
0.2
0.4
0.6
Time (sec)
0.8
1
Figure 6.8: Displacement angle for nonlinear MSBB System with 1 N-m
disturbance using Q-parameterization controller
70
Figure (6.9) shows the magnitude of the input voltage between +0.2 volts to -4.9
volts and stabilized at around -4.6 volts.
Input Voltage for the MSBB System
2
1
Input Voltage (volt)
0
-1
-2
-3
-4
-5
-6
-7
0
0.2
0.4
0.6
Time (sec)
0.8
1
Figure 6.9: Input voltage for nonlinear MSBB System with 1 N-m disturbance
using Q-parameterization controller.
71
6.4
Comparison between the two controllers
This section discusses the comparison between the two controllers under
investigation. The comparison was made between the proposed controller (Qparameterization controller) and the pole placement controller in both linear and
nonlinear models of the MSBB System. The purpose of this comparison is to find the
‘ultimate quest’ of this project. That is to verify that the robust controllers such as Qparameterization controller can give better results compared to conventional controller
such as the pole placement technique.
6.4.1
Simulation of linear and nonlinear models of the MSBB System
6.4.1.1 Q-parameterization controller
Figure (6.10) and (6.11) show the comparison between both systems of the
MSBB system with Q-parameterization controller. Figure (6.10) shows that the Qparameterization controller is effective in rejecting the disturbance entered to the system.
Figure (5.11) shows Input voltages involved.
72
8
x 10
-4
Comparison between Gap displacement angle for the MSBB
Q-parameterization Controller for linear system
Q-parameterization Controller for nonlinear system
7
Gap Displacement (rad)
6
5
4
3
2
1
0
-1
0
0.2
0.4
0.6
Time (sec)
0.8
1
Figure 6.10: Comparison of displacement angle between both systems of the MSBB
with Q-parameterization controller
Comparison between Input Voltage for the MSBB System
1
Q-parameterization controller for linear system
Qparameterization controller for nonlinear system
0
Input Voltage (volt)
-1
-2
-3
-4
-5
-6
-7
0
0.2
0.4
0.6
Time (sec)
0.8
1
Figure 6.11: Comparison of input voltage between both systems of the MSBB with
Q-parameterization controller
73
6.4.1.2
Pole placement state feedback controller
Figure (6.12) and (6.13) show, respectively the gap displacement and input
voltage when the pole placement technique is used. Figure (6.12) shows that the pole
placement technique did not able to reject the disturbance effectively.
x 10
-3
Copmarison between Gap displacement angle for the MSBB
Pole placement technique for linear system
4
Pole placement technique for nonlinear system
Gap Displacement (rad)
3.5
3
2.5
2
1.5
1
0.5
0
-0.5
0
0.2
0.4
0.6
Time (sec)
0.8
1
Figure 6.12: Comparison of displacement angle between both systems of the MSBB
with the pole placement technique
Comparison between Input Voltage the MSBB System
1
Pole placement technique for linear system
Pole placement technique for nonlinear system
Input Voltage (volt)
0
-1
-2
-3
-4
-5
0
0.2
0.4
0.6
Time (sec)
0.8
1
Figure 6.13: Comparison of input voltage between both systems of the MSBB with
the pole placement technique
74
6.4.2
Comparison between Pole Placement Technique and Q-parameterization
Controller
The purpose of this comparison is to find the ‘ultimate quest’ of this project.
That is to, verify that the robust controllers such as Q-parameterization controller can
give a better results compare to conventional controller. In this case, the conventional
controller is the pole placement technique.
6.4.2.1
Simulation using linear model of the MSBB System
Figures (6.14) and (6.15) show the comparison between the pole placement
technique and Q-parameterization Controller. Both controllers used same dominant
poles ( s = −40 ± j 53.3 ) but the added pole are not same. Pole placement technique has
one added pole, while Q-parameterization controller has four added poles.
As can be observed in Figure (6.14) both the controllers, whether pole placement
or Q-parameterization controller, they are both can stabilize the system.
However, only the Q-parameterization controller mange to reject the disturbance
force effectively. As can be observed in Figure (6.15), both controllers require relatively
low input signal.
75
4
x 10
-3
Gap displacement angle forlinear of the MSBB System
Pole placement technique
Q-parameterization controller
3.5
Gap Displacement (rad)
3
2.5
2
1.5
1
0.5
0
-0.5
0
0.2
0.4
0.6
Time (sec)
0.8
1
Figure 6.14: Comparison of displacement angle for linear model of the MSBB System
Coparison between Input Voltage for linear of the MSBB System
1
Pole placement technique
Q-parameterization controller
0
Input Voltage (volt)
-1
-2
-3
-4
-5
-6
0
0.2
0.4
0.6
Time (sec)
0.8
1
Figure 6.15: Comparison of input voltage between for linear of the MSBB System
76
Table 6.2 shows the comparison between two controllers with same design
specification.
Table 6.2: Comparison of various cases between pole placement technique and
Q-parameterization controller
Disturbance force (1Newton )
Q-parameterization
Pole placement
At t=0.15
controller
Technique
Peak value after disturbance
0.00061 rad
0.0035 rad
enter
Range of input voltage
Steady state error
+0.3 to -5.7 volts
+0.08 to -4.5 volts
Zero
0.00315 rad
From table 6.2, it can be seen that Q-parameterization controller give a better
results compared to pole placement technique. The peak value after disturbance enters
for Q-parameterization controller is less then 6 times from pole placement technique. Qparameterization controller also achieves steady state error zero by using relatively low
input voltage.
6.4.2.2 Simulation using nonlinear model of the MSBB System
Figures (6.16) and (6.17) show the comparison between the pole placement
technique and Q-parameterization Controller for the nonlinear model of MSBB System.
Both of the controllers having same dominant poles which are s = −40 ± j 53.3 but the
added pole are not same. Pole placement technique has one added pole, while Qparameterization controller has four added poles to obtain the proposed controller as we
explained in Chapter V.
77
As can be observed in Figure (6.16), when a 1 N-m step disturbance enters in the
system at time equal to 1.5s, both of the controllers can stabilize the system. However,
only the Q-parameterization controller manages to reject effectively. As can be observed
in Figure (6.17), the value for the input voltage is about 3.6 volts only.
3
x 10
-3
Comparison between Gap displacement angle fore the MSBB
Pole placement technique for nonlinear system
Q-parameterization Controller for nonlinear system
2.5
Gap Displacement (rad)
2
1.5
1
0.5
0
-0.5
0
0.2
0.4
0.6
Time (sec)
0.8
1
Figure 5.16: Comparison of displacement angle for nonlinear model of the MSBB
System
Comparison between Input Voltage for the MSBB System
2
Pole placement technique for nonlinear system
Qparameterization Controller for nonlinear system
1
Input Voltage (volt)
0
-1
-2
-3
-4
-5
-6
0
0.2
0.4
0.6
Time (sec)
0.8
1
Figure 6.17: Comparison of input voltage between for nonlinear of the MSBB System
78
Table 6.3 shows the comparison between two controllers with same design
specification.
Table 6.3: Comparison between pole placement technique and Q-parameterization
Controller
Disturbance force (1Newton )
Q-parameterization
Pole placement
At t=0.15
controller
Technique
Peak value after disturbance
0.000425
0.00185
enter
Range of input voltage
Steady state error
+0.2 to -4.85 volts
+0.04 to -3.15 volts
Zero
0.00185
From Table 6.3, it can be seen that Q-parameterization controller give a better
results compare to pole placement technique. The peak value after disturbance enters for
Q-parameterization controller is less then 4 times from pole placement technique. Qparameterization controller also achieves steady state error zero with no need to inject
high input voltage.
6.4.3
Comparison between pole placement technique and Q-parameterization
Controller for linear and nonlinear of the MSBB System
Figures (6.18) and (6.19) show the comparison between the pole placement
technique and Q-parameterization Controller for linear and the nonlinear model of
MSBB System.
79
5
x 10
-3
Comparison between Gap displacement angle for the MSBB
Pole placement technique for linear system
Q-parameterization Controller for linea system
Pole placement technique for nonlinear system
Q-parameterization Controller for nonlinear system
Gap Displacement (rad)
4
3
2
1
0
-1
0
0.2
0.4
0.6
Time (sec)
0.8
1
Figure 6.18: Comparison of displacement angle for linear and nonlinear model of
the MSBB System
Comparison between Input Voltage for the MSBB System
2
Pole placement technique for linear system
Q-parameterization controller for linear system
1
Pole placement technique for nonlinear system
Input Voltage (volt)
0
Q-parameterization controller for nonlinear system
-1
-2
-3
-4
-5
-6
-7
0
0.2
0.4
0.6
Time (sec)
0.8
1
Figure 6.19: Comparison of input voltage between for nonlinear of the MSBB System
These two graphs have been shown all of the results for pole placement
technique and the Q-parameterization controller with two models linear and nonlinear of
the MSBB System. The gap displacement’s graph and an input voltage graph’s for all
80
cases are shown in this chapter. Table 6.4 shows the comparison of the simulation
results under various cases.
Table 6.4: Comparison of various cases
Disturbance force
Q-
Pole placement
Q-
Pole placement
(1N-m) At t=0.15
parameterization
with linear
parameterization
with nonlinear
controller with
model
controller with
model
linear model
Peak value after
0.00061 rad
nonlinear model
0.0035 rad
0.000425 rad
0.00185 rad
disturbance enter
Range of input
+0.3 to -5.7
+0.08 to -4.5
+0.2 to -4.85
+0.04 to -3.15
voltage
volts
volts
volts
volts
Steady state error
Zero
0.00315 rad
Zero
0.00185 rad
From Table 6.4, the Q-parameterization controller give a better results compare
to pole placement technique because the peak value after disturbance enters for the Qparameterization controller is less then from pole placement technique. Both of
controllers used a reasonable input voltage to control the MSBB.
6.5
Results for the MSBB System by using the pole placement with integral
Control
Figure (6.20) and (6.21) show the results for the MSBB System by using pole
placement technique with integral control. Figure (6.20) shows that the pole placement
technique has the ability to reject the disturbance. However, it uses a very high input
voltage about 8000 volt as shown in Figure (6.21).This is really unreasonable input
voltage.
81
8
x 10
Gap displacement angle
-4
7
Gap Displacement (rad)
6
5
4
3
2
1
0
-1
-2
0
0.2
0.4
0.6
Time (sec)
0.8
1
Figure 6.20: Displacement angle for the MSBB System with 1 N-m disturbance
by using the pole placement with integral control
Input Voltage
2000
Input Voltage (volt)
0
-2000
-4000
-6000
-8000
0
0.2
0.4
0.6
Time (sec)
0.8
1
Figure 6.21: Input Voltage for the MSBB System with 1 N-m disturbance
by using the pole placement with integral control.
82
6.6
Comparison between pole placement with integral and Q-parameterization
Controller
The comparison between the pole placement technique with integral control and
the Q-parameterization controller. Both of the controllers use the same dominant poles
which are s = −40 ± j 53.3 but the added poles are not same. Pole placement technique
has two added pole, while the Q-parameterization controller have four added poles.
6.6.1
Linear model of the MSBB System
Figure (6.22) shows the comparison between the pole placement technique with
integral control and the Q-parameterization controller when a 1 N-m step disturbance
enters in the system at time equal to 0.15s, both of the controllers, both of the controllers
can reject the disturbance. Even though it can be seen that pole placement technique
with integral control can reject the disturbance, it cannot rejects the disturbance as
efficient as the Q-parameterization controller. The Q-parameterization controller gives
better damping response compared to the pole placement technique.
The graph for input voltage in figure (6.23) shows that the pole placement
technique uses a very big input voltage, about 8000 volts to control the small angle of
the balance beam which is very unpractical to be used in the real world. This is another
disadvantage of pole placement technique.
83
8
x 10
-4
Comparison between Gap displacement angle
Pole placement technique with integral control
Q-parameterization controller
7
Gap Displacement (rad)
6
5
4
3
2
1
0
-1
0
0.2
0.4
0.6
Time (sec)
0.8
1
Figure 6.22: Comparison of displacement angle for the MSBB System
Comparison between Input Voltage
2000
Pole placement technique with integral control
Q-parameterization controller
1000
0
Input Voltage (volt)
-1000
-2000
-3000
-4000
-5000
-6000
-7000
-8000
-9000
0
0.2
0.4
0.6
Time (sec)
0.8
1
Figure 6.23: Comparison of input voltage for the MSBB System
84
6.6.2
Nonlinear model of the MSBB System
Figure (6.24) shows the comparison between the pole placement technique
with integral control and the Q-parameterization controller when a 1 N-m step
disturbance enters in the system at time equal to 0.15s, both of the controllers, both of
the controllers can reject the disturbance. Even though it can be seen that pole placement
technique with integral control can reject the disturbance, it cannot rejects the
disturbance as efficient as the Q-parameterization controller. The Q-parameterization
controller gives better damping response compared to the pole placement technique.
The graph for input voltage in figure (6.25) shows that the pole placement
technique and Q-parameterization controller use similar input voltage. but still the
proposed controller give better result from the pole placement controller with integral
control.
7
x 10
-4
Gap displacement angle
Pole placement with integral cntrol for the MSBB System
Q-parameterization controller for tha MSBB System
6
Gap Displacement (rad)
5
4
3
2
1
0
-1
0
0.1
0.2
0.3
0.4
0.5
0.6
Time (sec)
0.7
0.8
0.9
1
Figure 6.24: Comparison of displacement angle for nonlinear of the MSBB System
85
Input Voltage
2
Pole placement with integral control for nonlinear system
Q-parameterization controller for nonlinear system
1
0
Input Voltage (volt)
-1
-2
-3
-4
-5
-6
-7
0
0.1
0.2
0.3
0.4
0.5
0.6
Time (sec)
0.7
0.8
0.9
1
Figure 6.25: Comparison of input voltage for nonlinear of the MSBB System
In Figures (6.26) and (6.27) show the comparison between the Q-parameterization
controller and the pole placement with and without integral control.
3
x 10
-3
Comparison berween Gap displacement angle
Pole placement technique without integral control
Pole placement technique with integral control
Q-parameterization controller
Gap Displacement (rad)
2.5
2
1.5
1
0.5
0
-0.5
0
0.2
0.4
0.6
Time (sec)
0.8
1
Figure 6.26: Comparison of displacement angle for the MSBB System
86
Comparison between Input Voltage
Pole placement technique without integral control
Pole placement technique with integral control
2000
Q-parameterization Controller
Input Voltage (volt)
0
-2000
-4000
-6000
-8000
0
0.2
0.4
0.6
Time (sec)
0.8
1
Figure 6.27: Comparison of input voltage for the MSBB System
Table 6.5 shows the results in terms of peak overshoot, input voltage and steady
state error.
Table 6.5: Comparison between Q-parameterization controller and pole placement
with and without integral control
Disturbance force
Q-parameterization
Pole placement
Pole placement
(1N-m) At t=0.15
controller
with integral
without integral
Peak value after
0.000425 rad
0.00071 rad
0.0039 rad
+0.2 to -4.85 volts
+100 to -8000
+0.05 to -1.36
volts
volts
disturbance enter
Range of input
voltage
Steady state error
Zero
Zero
0.0034
87
From Table 6.5, it can be seen that the Q-parameterization controller gives a
better results. Furthermore, the pole placement technique with integral control used a
very big input voltage, about 8000 volts to control the small angle of the balance beam
which is very unpractical to be used in the real world.
Finally, from all the comparisons presented in this the chapter, it can be concluded
that the Q-parameterization controller was excellent and suitable to control the MSBB
System.
88
6.7
Summary
This chapter has shown all of the results for pole placement technique and the
Q-parameterization controller. The gap displacement’s response and the input voltage
signal for various simulation conditions are shown in this chapter. The comparisons
between the Q-parameterization controller and pole placement technique are shown by
graphs and tables.
CHAPTER VII
CONCLUSIONS AND SUGGESTIONS
7.1
Conclusions
The mathematical model of a magnetically suspended balance beam (MSBB)
system in state space equation has been presented. Beside that, the Q-parameterization
theory and controller design methodology has also been explained. The Qparameterization controller has been successfully designed by using linear state space
equation to control system.
The performance of the balance beam control system using the Qparameterization controller is illustrated by simulation. The result is compared and
verified with pole placement technique. From the results that had been shown in Chapter
VI, it is obvious that the Q-parameterization controller provide better performance than
pole placement technique because of two reasons.
First, it is proven that the controller design based on Q-parameterization theory
can rejects the disturbance more efficiently compare to pole placement technique.
Secondly, the Q-parameterization controller utilized a reasonable input voltage to
control the MSBB System.
90
As summary, it can be concluded that the Q-parameterization controller to control
the MSBB system offers many advantages over the other control method:
¾
Superior in term of disturbance rejection ability.
¾
The set of all stabilizing controllers of MSBB system, it will parameterizes in
terms of a free parameter.
¾
Controller that satisfies all design requirements can be chosen from this set
through optimization.
¾
The dynamic response has excellent and robust performance with the proposed
controller against variation of the system parameters and steady state error is
zero, when the system is subjected to a step load disturbance.
91
7.2
Suggestions for Future work
In the future other type of controls such as fuzzy logic, Control can be tried to
control MSBB. Fuzzy Control is suggested because it is modern and intelligent
controller can uses a different kind of approach. It is not base on complex mathematical
equation because fuzzy control uses its unique method that is called rule base.
92
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