Model-free Control Design for Hybrid Magnetic Levitation System

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Nov 15, 2013 (4 years and 11 months ago)

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Model
-
free Control Design for Hybrid Magnetic Levitation System


Rong
-
Jong Wai

1
,

M
ember
,

IEEE
,

Jeng
-
Dao Lee

2
, and
Chiung
-
Chou Liao

3

1
,2
Department of Electrical Engineering
,
Yuan Ze University, Chung Li 320, Taiwan, R.O.C.

3
Department of Electr
onic

Eng
ineering
,
Ching Yun University, Chung Li 320, Taiwan, R.O.C.


Abstract

This study
investigates three model
-
free control
strategies including a simple proportional
-
integral
-
differential
(PID) scheme, a fuzzy
-
neural
-
network (FNN) control and a
robust contro
l

for a hybrid magnetic levitation (maglev) system.
In general
,
the
lumped dynamic model

of
a
hybrid

maglev
system can be derived by

the
transform
ing
principle
from
electrical energy to mechanical energy
. In practic
e
,
th
is

hybrid
maglev system is inherent
l
y

unstable in the direction of
levitation
,

and the relationships among airgap, current and
electromagnetic force are highly nonlinear,
therefore,

the
mathematical
model
can

not
be
established
precisely
.
In order to
cope with
the

unavailable dynamics, model
-
free control design is
always required to handle
the

system behaviors. In this study,
the

experimental comparison of PID, FNN and robust control
systems for the hybrid maglev system is reported. From
the

performance comparison,
the

robust control system y
ields
superior control performance than PID and FNN control
systems. Moreover, it not only has the learning ability similar to
FNN control, but also
the

simple control structure to
the

PID
control.


I.

I
NTRODUCTION

In recent years, magnetic levitation (
m
aglev) techniques
have been respected for eliminating friction due to
mechanical contact, decreasing maintainable cost, and
achieving high
-
precision positioning. Therefore, they are
widely used in various fields, such as high
-
speed trains
[1]

[5], magnetic

bearings [6], [7], vibration isolation
systems [8], wind tunnel levitation [9] and photolithography
steppers [10].
In general, maglev systems can be
classified

into two categories: electrodynamic suspension (EDS) and
electromagnetic suspension (EMS). EDS
systems are
commonly known as

repulsive levitation

, and
superconductivity magnets [11] or permanent magnets [12]
are always taken as the levitation source. However, the
repulsive magnetic poles of superconductivity magnets can
not be reacted on low speed

so that they are only suitable for
long
-
distance and high
-
speed train systems. Basically, the
magnetic levitation force of EDS is partially stable and

it

allows a large clearance
.

Nevertheless,
the productive process
of magnetic material
s

is
more
complex
and expensive.

On the
other hand, EMS systems are commonly known as

attractive
levitation

, and the
magnetic levitation force is inherently
unstable

so that

the control problem will become more
difficult.

Ge
neral
ly speaking
,
the
manufacturing process and
cost

of EMS are lower than EDS
,

but extra

electric power is
required to maintain a predestinate levitation height
.

T
o
merge the merits of these two kinds of levitation systems, a
hybrid maglev system
adopted in this study
is combined with
an electromagneti
c
magnet
and a permanent magnet.

The
magnetic force generated by the

additional permanent magnet
is

used to alleviate
the

power consumption for levitation
.

Because
the

EMS system has unstable and nonlinear
behaviors, it is difficult to build a precision dy
namic model.
Some researches have derived various mathematical models
for many kinds of maglev systems in numerical simulation
[13], [14], but there still exist uncertainties in practical
applications. In general,
linear
ized
-
control strategies based on
a T
aylor series

expansion of the actual nonlinear
dynamic
model and
fo
rce distribution at nominal operating point
s are
often employed. N
evertheless
, t
he tracking performance of
the

linear
ized
-
control strategy

[15]

[18]

deteriorates rapidly
with
the
increasing

of
deviations from nominal operating
point
s
.

Many approaches
introduced

to solve this problem for
ensuring consistent performance independent of operating
points have been reported in opening literature. Backstepping
methods were incorporated into [12], [
19] due to the
systematic design
procedure
. Huang
et al
. [12] addressed an
adaptive backstepping controller to achieve a desired stiffness
for a repulsive maglev suspension system. In [19], a
nonlinear model of a planer rotor disk, active magnetic
bearing
system was utilized to develop a nonlinear
backstepping controller for the full
-
order electromechanical
system. Unfortunately, some constrain conditions should be
satisfied for
the

precision positioning. Moreover, the
approach of gain scheduling [20], [21]

can
linearize

the
nonlinear

relationship of the magnetic suspension

at various
operating points with

a suitable controller designed for each
of these operating

points
. In order to
achieve

better control
performance

under the entire operation range, it nee
ds to
subdivide

the operating range into appropriate intervals. By
this way, the favorable control gains collected in the lookup
table will occupy a large memory to bring about the heavy
computation burden. In addition, Sinha and Pechev [22]
presented an
a
daptive controller
to

compensate for payload
variations and external force disturbance

using the criterion
of stable maximum descent
. Overall,

the detailed
or
part
ial
mathematical model
s

acquired by complicated modeling
process
es are usually required

to de
sign
a suitable
control law

for achieving
the

positioning demand
.
The aim of this study
is attempted to introduce model
-
free control strategies for a
hybrid maglev system and to compare their superiority or
defect via
experimental

results.


II.

H
YBRID
M
AG
LEV
S
YSTEM

The configuration of a hybrid maglev system is depicted
in Fig. 1(a),
which

consists of a
hybrid

electromagnet, a
ferrous plate, a load carrier and a gap sensor. Among these,
the hybrid electromagnet is composed of a permanent magnet
and an elec
tromagnet. It forms two flux
-
loops in the E
-
type
hybrid electromagnet, and the flux passes through a
permanent magnet, a ferrous plate, an air gap and a core in
each loop. The magnetic equivalent circuit can be represented
as Fig. 1(b). The magnetomotive f
orce (mmf) of this hybrid
electromagnet is the summation of the permanent magnet
mmf (
P
F
) and the electromagnet mmf (
i
N
m
), where
m
N

is
the coil turns and
i

is the coil current. Moreover, the tot
al
reluctance of
the

magnetic path is


)
//(
)
(
c
x
Fe
P
c
x
Fe
P
R
R
R
R
R
R
R
R
R








(1)

where
P
R
,
Fe
R
,
x
R

and
c
R

are the reluctances of the
permanent magnet, ferrous plate, air gap and core in
the
magnetic path, respectively. In addition, the flux (

)
produced against the magnetic reluctance (
R
) by this hybrid
electromagnetic mmf can be denoted as


R
i
N
F
m
P



.

(2)

The energy in this magn
etic field is


iP
i
N
F
N
i
L
W
m
P
m
t
f
)
(
2
1
2
1
2



,

(3)

where
R
P
/
1


is the permeance of
the

magnetic path;
t
L

is the inductance of the hybrid electromagnet and is defined
as


R
i
N
F
i
N
i
N
i
L
m
P
m
m
t







(4)

in which


means the flux linkage.

(
a)
)
(
t
U
Gap
Sensor
x
Ferrous Plate
Permanent
Magnet
Electromagnet
mg
)
,
(
i
x
F
Power
Amplifier
Load Carrier

(
b)
Fe
R
Fe
R
P
R
P
R
x
R
x
R
c
R
c
R
P
F
i
N
m



Fig. 1. H
ybrid

maglev system
: (a) Configuration. (b) Equivalent circuit.

Assume that there is no loss in energy transmission, the
power produced by the magnetic field can be represented via
the principle of the conserva
tion of energy as


dt
dx
F
dt
d
i
dt
dW
f




(5)

where
F

is the produced mechanical force, and
x

is the
displacement of levitation. Multiply
dt

on both sides of (5),
then


dx
F
d
i
dW
f




(6)

Since the magn
etic energy
f
W

is a function of


and
x
,
one can obtain


dx
x
W
d
W
x
dW
f
f
f









)
,
(

(7)

To compare (6) with (7), the mechanical force
)
,
(
i
x
F

can
be expressed via (3) as


i
G
x
P
i
i
N
F
N
x
x
W
i
x
F
f
m
P
m
f










)
(
2
1
)
,
(
)
,
(


(8)

where the term
f
G

is related to the total magnetomotive
force, coil turns and the
permeance

in the magnetic path.

According to the Newtonian law
, the dynamic behavior of
the hybrid maglev system can be governed by
the following
equation:


)
,
(
)
(
)
,
(
)
(
t
x
M
t
U
t
x
G
m
f
g
U
m
G
G
m
f
g
i
m
G
t
x
d
i
f
d
f











(9)

where
m

is the mass of total suspension object,
g

is the
acceleration of gravity
,
d
f

is the external disturbance force,
i
G

is the function representation of a p
ower amplifier and
U

is the

control voltage. Moreover,
m
G
G
t
x
G
i
f

)
,
(

expresses the control gain
, and
m
f
g
t
x
M
d



)
,
(
.
Due to
the nonlinear and time
-
varying characteristics of the
hybrid
maglev system
, the accurate dynamics model (
)
,
(
t
x
G

and
)
,
(
t
x
M
) are assumed to be unknown in this study.

Without
loss of generality it is assumed that
)
,
(
t
x
G

is finite and
bounded away from zero for all
x
.


III
.

C
ONTROL
S
YSTEMS
D
ESIGN

A.

PID Control System

In industrial application, a PID control system is
the

common usual due to its simple scheme. Define a

tracking
error as


x
x
e
m



(
10
)

in which
m
x

represents
the

reference levitation
displacement.
The PID control law c
an be represented as


dt
de
K
e
K
e
K
U
U
U
U
D
I
P
D
I
P








(11)

where
P
U

is a proportional controller;
I
U

is an integral
controller;
D
U

is a differential controller;
P
K
,
I
K

and
D
K

are the corresponding control gains.
Selection of the
values for
the gains in the PID control system

has a
significant effect on the control performance. In general, they
are determined

according to desirable system res
ponses, e.g.,
raising time, settling time, etc.


B.

FNN
Control System

In the FNN control system, a four
-
layer network structure
with

the input

(
i

layer)
, membership

(
j

layer)
, rule
(
k

layer)
and output
(
o

layer)
layer
s is adopted [23].

The membership
la
yer acts as the membership functions. Moreover, all the
nodes in the rule layer form a fuzzy rule base.
The signal
propagation and the basic function in each layer of the
FNN

are
introduced

in the following
paragraph
.

For every node
i

in the input layer

tr
ansmits the input
variables
)
,
,
1
(
n
i
x
i



to the next layer

directly, and
n

is
the total number of the input nodes.
Moreover, each node in
the membership layer performs a membership function.

In
this study, t
he
m
embership layer represents the inp
ut values
with
the following Gaussian

membership functions
:


2
2
)
(
)
(
)
(
j
i
j
i
i
i
j
m
x
x
n e t




,
))
(
exp(
))
(
(
i
j
i
j
j
i
x
net
x
net



(12)

where
j
i
m

and
)
,
,
1
;
,
,
1
(
i
p
j
i
n
j
n
i






are,
respectively,

the mean and
the
standard deviation of the
Gaussian function
i
n
the
j
th term of the
i
th input variable
i
x

to the node of
this
layer
,
and
i
p
n

is the total number of the
linguistic variables with respect to the input nodes.

In
addition, each node
k

in the rule layer is denoted b
y

,
which multiplies the input signals and outputs the result of
the
product.
The output of this layer is given

as





n
i
i
j
j
i
k
ji
k
x
net
w
1
))
(
(



(13)

where
)
,
,
1
(
y
k
n
k




represents the
k
th output of the rule
layer
;

k
j i
w
,
the weights between the membership layer and
the rule layer,
are

assumed to be unity;

y
n

is the total
number of rules.

Furthermore, the node
o

in the output layer
is labeled with

;

e
ach
node

)
,
,
1
(
o
o
n
o
y



computes
the overall output as the summation of all input signals
, and

o
n

is the total number of output nodes
.





y
n
k
k
o
k
o
w
y
1


(14)

where the connecting weight
o
k
w

is the output action
strength of the
o
th output associated with the
k
th rule. In this
study, the inputs of the
FNN control

system are the tracking
error (
e
x

1
) and its derivative (
s
e
x

2
), and the single
output is the c
ontrol effort for the hybrid maglev system, i.e.,
o
y
U

.

To describe the on
-
line learning algorithm of th
is

FNN
control

system

via

supervised gradient decent metho
d
, first
the energy function
E

is defined as


2
/
2
/
)
(
2
2
e
x
x
E
m




(
15
)

In t
he output layer, the error term to be propagated is given
by


o
o
o
o
y
x
x
e
e
E
y
e
e
E
y
E




















(16)

and the weight is updated by the amount


k
o
w
o
k
o
o
w
o
k
w
o
k
w
y
y
E
w
E
w






























Δ

(17)

where
w


is the learning
-
rate parameter of the connecting
weights. The wei
ghts of the output layer are updated
according to the following equation:


o
k
o
k
o
k
w
N
w
N
w
Δ
)
(
)
1
(




(18)

where
N

denotes the number of iterations. Since the weights
in the rule layer are unified, only the error term t
o be
calculated and propagated.


o
k
o
k
k
w
E








(19)

In the
membership

layer, the error term is computed as
follows:






k
k
k
j
j
net
E






(20)

The update laws of
j
i
m

and
j
i


can be denoted as


j
i
j
i
j
i
m
N
m
N
m
Δ
)
(
)
1
(




(21a)


2
)
(
)
(
2
j
i
j
i
i
j
m
j
i
m
j
i
m
x
m
E
m












(21b)


j
i
j
i
j
i
N
N



Δ
)
(
)
1
(




(22a)


3
2
)
(
)
(
2
j
i
j
i
i
j
s
j
i
s
j
i
m
x
E














(22b)

where
m


and
s


are the learning
-
rate parameters of the
mean and the standard deviation of the Gaussian function.
The exact calculation of the Jacobian of th
e actual plant,
o
y
x



in (16), cannot be determined due to the
uncertainties of the plant dynamics.
Similar

to [23], the delta
adaptation law
s
o
e
e




is adopted in this study. Moreover,
varied learning rates

derived in [23
]
, which guarantee
convergence of the
tracking

error based on the analyses of a
discrete
-
type Lyapunov function, are
also used

in this
study.


C.


Robust Control System

In order to control the
levitation displacement

of the
hybrid maglev system

more
effect
ively, a robust control
system

[24] is implemented and

the state variables are
defined as follows:


x
X

1

(
23
)


2
1
X
v
X




(
24
)

where
v

represents
the
levitation velocity

of the
hybrid
maglev system
. Rewrite (
9
), then
the
hyb
rid maglev system

can be represented in the following state space form:



































)
,
(
0
)
,
(
0
0
0
1
0
2
1
2
1
t
x
M
U
t
x
G
X
X
X
X



(
25
)

The above equation can be
expressed

as

L
D
B
AX
D
D
B
B
AX
D
B
AX
X
P
P
P
P














U
U
U
)
(
)
(


(
26
)

where
T
X
X
]
[
2
1

P
X
;







0
0
1
0
A
;
T
t
x
G
]
)
,
(
0
[

B
;
T
t
x
M
]
)
,
(
0
[


D
;
B

and
D

are the nominal
parametric

matrixes

of
B

and
D
;
B


and
D


denote
the

uncertainties introduced by
parameter variation and
external disturb
ance;
D
B
L




U

is

the lumped
uncertainty.

In the robust control system design, the desired behavior
of the
hybrid maglev
system is expressed through the use of a
reference model driven by a reference i
nput. Typically, a
linear model is used. A reference model of the following state
variable form is selected:


R
R
B
A
A
m
m
m
M
M
M
M
M
B
X
A
X
X
















1
2
1
0
1
0


(
27
)

where
T
m
m
v
x
]
[

M
X

represents the reference
levitation
displacement

and
velocity
;
R

is a reference input;
M
A

and
M
B

are given constant matrices.
0

M
A

is assumed to be a
stable matrix.

The control problem is to find a control law so that the
state
P
X

can track the reference trajec
tory
M
X

in the
presence of the uncertainties. Let the control error vector be


T
s
T
m
m
P
M
e
e
v
v
x
x
]
[
]
[






X
X
E
C

(
28
)

To make the control error vector tend to zero with time, the
robust control law
U

is assumed to take the following form

[24]
:









KR
U
U
U
U
d
f
s
P
X


(
29
)

where
P
X


s
U

is
a

state feedback controller;
KR
U
f


is
a

feedforward controller;


d
U

is an

uncertainty
controller. The control gains (

,
K

and

) are adjusted
according to dynamic adaptation laws

introduced later
. After
some straightforward manipulation, the control error equation
governing the closed
-
loop system can be obtained from (
25
)
through (
29
) as follow
:


)
(
)
(
)
(
L
B
D
B
B
X
B
A
A
E
A
E
M
P
M
C
M
C










R
K



(
30
)

If the precise model dynamics and the uncertainties in
practical applications are available, there exist ideal control
gains
*
Θ
,
*
K

and
*


in the following equat
ions such that
the control error vector tend to zero with time:


)
(
*
A
A
B
Θ
M




(
31
)


M
B
B


*
K

(
32
)


)
(
*
L
D
B





(
33
)

where

B

is the left penrose pseudo inverse of
B
, i.e.,
T
T
B
B
B
B
-1
)
(


. Since the dynamic model and the
uncertainties of the controlled system may be unknown or
perturbed, the ideal control gains shown in (
31
)

(
33
) can not
be implemented in practic
e
. Reformulate (
30
), then


)
(


E
R
E
k




P
C
M
C
X
E
B
E
A
E


(
34
)

in

which the control parameter errors

E
,
k
E

and

E

are
defined as






*

E

(
35
)


K
K
E
k


*

(
36
)







*
E

(
37
)

T
HEOREM
1
: Consider the
hybrid maglev

system represented
by (
25
), if the robust control law is designed as (
29
), in which
the adaptation laws of the control gains are designed as
(
38
)

(
40
), then the stability of the robust control system can
be guaranteed.


T
P
T
X
B
E
C
1





(
38
)


R
K
T
C
E
B
2




(
39
)


C
E
B
T
3





(
40
)

where
1

,
2


and
3


are

p
ositive tuning gain
s
.

From Theorem 1, it follows that the tracking error will
tend to zero under the level of slowl
y varying uncertainties.
However, the control gains will not necessarily converge to
their ideal values in (
31
)

(
33
); it is shown only that they are
bounded. To have parameter convergence, it is necessary to
impose the persistent excitation condition

on th
e system.
Moreover, according to the unavailable system parameters,
the nominal parameter

)
,
(
t
x
G

in the tuning algorithms is
reorganized as
))
,
(
sgn(
)
,
(
t
x
G
t
x
G

in practical applications.
Therefore, the adaptation laws of the robust con
trol system
shown in (
38
)

(
40
) can be reorganized as follows:


))
,
(
sgn(
1
t
x
G
e
T
P
s
X





(
41
)


)
)
,
(
sgn(
2
t
x
G
R
e
K
s




(
42
)


))
,
(
sgn(
3
t
x
G
e
s





(
43
)

where
)
sgn(


is a sign function
;

the term
s

)
,
(
1
t
x
G

,
)
,
(
2
t
x
G


and
)
,
(
3
t
x
G


are

absorbed by the tuning
gain
,

1

,
2


and
3


individually.

Consequently, only the
sign of
)
,
(
t
x
G

is required in the design procedure, and it
can be easily obtained from the physical characteristic of the
hybrid maglev

system.


I
V.

E
XPERIMENTAL
R
ESULT
S

The block diagram of
a

computer
-
based

control system
for the
hybrid maglev system

is
depicted

in Fig.
2
.
In the
hybrid maglev system, it divide
s into two parts:
A f
errous
frame and
a
levitati
on

table. A hybrid electromagnet is fixed
on the levitati
on

table, and the attracting levitati
on

force is
produced by
the

magnetization of
the
electromagnet
ic

coil.
A
servo control card is installed in the co
ntrol computer, which
includes multi
-
channels of D/A, A/D, PIO and encoder
interface circuits. The
moving displacement

of the levitation
table is fed back using a
gap sensor
. The control system
s

in
this study are

realized in the Pentium PC
via

“Turbo C”
la
nguage

to manipulate
the

coil current (
i
) in the
electromagnetic coil by way of voltage control (
U
)
, and
t
he
control intervals are all set at
6
ms.

x
m
x
U
D/A
Converter
A/D
Converter
Servo
Control
Card
Pentium
Memory
Digital
Oscilloscope
Control Computer
x

Fig. 2. Computer
-
based control system.

0
mm
0
mm
0
A
1.428
mm
10
A
Tracking Error
Coil Current
MSE=6.041
×
10
-3
mm
2
Table Position
Position
Command
Unloading
Loading
(
a)
(
b)
0
mm
0
mm
0
A
2.01
mm
10
A
Tracking Error
Coil Current
MSE=6.902
×
10
-3
mm
2
Table Position
Position
Command
Unloading
Loading


Fig. 3. Experimental results of PID control system at load
-
var
iation
condition: (a) 1mm
-
step command. (b) 2mm
-
step command.

Some experimental results are provided here to
demonstrate

the effectiveness of the
PID, FNN

and
robust

control

systems
.

In
the

experimentation,

t
he initial condition
of this hybrid maglev syste
m is load
ed

by two piece
s

of iron
disk

with 3.7kg weight
.
The experimenta
l results

of
the PID

control system
due to step commands are depicted in
Fig.
3.
In Fig. 3(a), a 1mm
-
step command is set,
and
the

gains of the
PID co
ntrol system are given as follows:



32

P
K
,
10

I
K
,
6
.
0

D
K

(
44
)

Then, unloads one iron disk at 6s and
re
loads it at 12s, it is
obvious that the position drift of the levitati
on

is almost 1mm
when unloading. The mean
-
square
-
error (MSE) v
alue is
2
3
mm
10
041
.
6


.
Because
the

gain
s

in (44) are selected at
1mm
-
step command, these control gain
s

may not keep the
levitati
on

height at 1mm during
the

unloading duration.
Moreover, the gain
s

of the PID control system for a
2mm
-
step command
a
re
designed as

25

P
K
,
10

I
K
,
5
.
0

D
K

(45)

In Fig. 3(b), there still have similar results as Fig. 3(a), and
the MSE value is
2
3
mm
10
902
.
6


. Consequently, the
control coefficients of the PID co
ntrol system should be
redesigned for various demands to satisfy
the

desirable
dynamic behavior.

(
a)
0
mm
0
mm
0
A
1.428
mm
10
A
Table Position
Tracking Error
Coil Current
Position
Command
Unloading
Loading
MSE=1.108
×
10
-3
mm
2
0
mm
0
mm
0
A
2.
01
mm
10
A
Table
Position
Tracking Error
Coil Current
Position
Command
Unloading
Loading
MSE=1.492
×
10
-3
mm
2
(
b)

Fig. 4. Experimental results of FNN control system at load
-
variation
condition: (a) 1mm
-
step command. (b) 2mm
-
step command.

For comparison
,

the FNN control

system in Section III
-
B

is
also
applied to control the
hybrid maglev system.
T
o show
the effectiveness of the FNN with small

rule set, the FNN has
two, six, nine and one neuron at the input,

membership, rule
and output layer, respectively. It can be regar
ded

that the
associated fuzzy sets with Gaussian function for

each input
signal are divided into N (negative), Z (zero) and P

(positive)
,
and
the

number of rules with complete rule connection is nine
.

Moreover
, some heuristics can be used to roughly initia
lize

the parameters of the FNN for practical applications
.
The
effect due to the inaccurate selection of the initialized

parameters can be retrieved by the on
-
line
learning

methodology.

Therefore, for simplicity, the means of the
Gaussian functions

are set

at
-
1, 0, 1 for the N, Z, P neurons
and the standard deviations

of the Gaussian functions are set
at one.

In addition, to test the learning ability of the FNN
control system, all the initial connecting weight between
the

output layer and the rule layer ar
e set to zero in
the

experimentation.
The responses of the

table
position
, tracking
error
and co
il current

using the FNN control

system
due to
1mm
-
step and 2mm
-
step commands
are
depicted

in Fig.
4
(a)

and (b)
, where the respective MSE values are
2
3
mm
10
108
.
1



and
2
3
mm
10
492
.
1


. From the
experimental

results,
the overshoot responses at
the

transient
state are caused by the rough initialization of the network
parameters. After this, t
he
tracking

errors reduce to zero
quickly

even under the
load variations.

Although favorable
tracking performance can be obtained,
th
is control scheme
seems to be too complex in practical applications.

In the end, the
experimentation of the robust control
system is implemented. The gains of the robust control
sy
stem
are

given as follows:


1000
1


,
65
2


,
65
3



(
46
)

The selection of the positive tuning gain
s (
3
2
1
,
,



) is

concerned with the tracking
speed
. The tracking response
converges slowly
with small tuning gains, and the tracking
speed

increases with large tuning gains. Due to
the
unavailable system
dynamics
,
they are

chosen via a
trial and
error

process to achieve the superior transient response in the
experimentation considering the requi
rement of stability, the
limitation of control effort and the possible operating
conditions.
The
table position, tracking error,
and co
i
l
current

at
1mm
-
step and 2mm
-
step commands are depicted in Fig.
5(a) and (b), where the respective MSE values are
2
4
mm
10
150
.
6



and
2
4
mm
10
061
.
8


. From
the
experimental results
,
good tracking responses can be obtained
.
When the variation of load during 6s

12s,
the

table position
can be returned to the command position quickly. Comparing
Fig. 5 with Figs
. 3 and 4, it is obvious that
the

robust control
system with simple framework yields superior control
performance than
the

PID and FNN control systems.

(
a)
0
mm
0
mm
0
A
1.428
mm
10
A
Tracking Error
Coil Current
MSE=6.150
×
10
-4
mm
2
Table Position
Position
Command
Unloading
Loading
(
b)
0
mm
0
mm
0
A
2.01
mm
10
A
Tracking Error
Coil Current
MSE=8.061
×
10
-4
mm
2
Table Position
Position
Command
Unloading
Loading

Fig. 5. Experimental results of robust control system at load
-
variation
condition: (a) 1mm
-
step com
mand. (b) 2mm
-
step command.


V.

C
ONCLUSION
S

This study has successfully implemented specific PID,
FNN and robust control systems for a hybrid maglev system
to demonstrate
the

performance comparison of these three
model
-
free control strategies. The PID con
trol system
belongs to
an

event
-
based linear control
ler
. There are larger
MSE values under the occurrence of load variations;
therefore, the control gain
s

should
be
redesigned due to
different situations. Moreover
,
the

FNN control

system

could
be
d
esigned

successfully without complex mathematical
model

and possesses smaller MSE values. However, this
scheme is more complicated than the PID control system. In
addition,
the

robust control system is presented to solve this
problem, and
it
not only has good trac
king response but also
makes this system more robust under different step
commands and load variation
s
. Note that, this study just
demonstrates three specific model
-
free control strategies, not
over all possible control methods suggested in the literature.



A
CKNOWLEDGMENTS

The authors would like to acknowledge the financial
support of the National Science Council of Taiwan, R.O.C.
through grant number NSC 93
-
2213
-
E
-
155
-
014.


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