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Control Engineering Practice 13 (2005) 821834
A fuzzy logic controller for the molten steel level control of strip
casting processes
Youngjun Parka, Hyungsuck Chob,
aRobot System Dept., Mechatronics Center, Institute of Industrial Technology, Samsung Heavy Industries Co., Ltd., 103-28, Munji-Dong,
Yusong-gu, Daejeon, 305-380, Republic of KoreabDepartment of Mechanical Engineering, Korea Advanced Institute of Science and Technology, 373-1, Kusung-dong, Yusong-gu, Daejeon, 305-701,
Republic of Korea
Received 14 September 2002; accepted 13 September 2004
Abstract
The strip casting process characterized to produce steel strips of thickness ranging 15 mm directly from molten metal has been
drawing increasing interest because it skips over some of the conventional hot rolling processes. However, since there are a number
of process parameters and their relationships are somewhat complex, realization of the process design and quality control is
accordingly considered to be difficult. In this case, construction of a multi-dimensional fuzzy logic controller by conventional
methods is extremely difficult and therefore needs much time and effort. To overcome this difficulty, a new design technique of a
fuzzy logic controller is proposed, that greatly simplifies the design procedure by defining simplified design parameters associated
with the controller. In the design procedure, the major design parameters of the controller are simplified by observing several aspects
that appear in design procedures of the fuzzy membership function and the rule base. This design technique is applied to a strip
casting process and a series of simulations is carried out for various design parameters. Based on these results it is found that the
proposed design technique can drastically simplify the design procedure and that the designed fuzzy logic controller results in
satisfactory process response.r 2004 Elsevier Ltd. All rights reserved.
Keywords: Twin roll type strip casting process; Molten steel level control; Fuzzy logic controller (FLC); Simplified design parameters
1. Introduction
Because of the growth of machine industry, the
demand for thin steel strips has greatly increased in
recent years. Consequently, production techniques for
these strips have been widely studied. Recently, the strip
casting process has drawn significant interest, becauseproduction costs can be greatly reduced by eliminating
the subsequent hot rolling processes. This process has
been characterized to produce 15 mm thick steel strips
directly from molten metal, thus eliminating the need for
conventional hot rolling processes. It is thereby possible
to significantly shorten production at an enormous
savings of energy. In addition, material properties are
improved by eliminating segregation in the strips due to
the effect of rapid cooling (Shibuya and Ozawa, 1991;
Reichelt and Kapellner, 1998).
The basic concept of the strip casting process can be
traced to the work of Bessemer in 1846; it has long sincebeen a dream of steel engineers. Bessemers idea was not
realized then, because many key technical components
such as measurement devices and computer control
technology were not available at that time. However,
thanks to the phenomenal growth of steel-making and
relevant technologies, efforts to implement strip casting
technology have recently been revived. As a result of
extensive development efforts, several countries are soon
expected to announce commercialization of full sized
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www.elsevier.com/locate/conengprac
0967-0661/$- see front matter r 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.conengprac.2004.09.006
Corresponding author. Tel:+8242 8693213; fax:+8242 8693210.
E-mail addresses: [email protected] (Y. Park),
[email protected] (H. Cho).
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strip casting systems. Realization of the process,
however, has become a difficult task because of several
problems encountered in the process, such as the
identification of quantitative relationships between
process parameters, the control of surface quality, the
thickness deviation of the strips, and the materials and
cooling methods of the roll.In order to commercially produce valuable products,
solving these inherent problems has been imperative.
There have been several studies associated with this
process: analysis of solidification and metal flow in
molten pools (Miyazawa and Szekely (1981); Saitoh,
Hojo, Yaguchi, and Kang (1989); Takuda, Hatta,
Eramura, and Kokado (1990); Hwang, Lin, Hwang, &
Hu, 1995), the analysis of mechanical characteristics
(Yukumoto and Yamane, 1995; Miyake, Yamane,
Yukumoto, & Ozawa, 1991) and parameter design for
making high-quality strips Bae, Park, and Cho (1996);
Park, Bae, Cho, Lee, and Kang (1997). However, due to
the complexity of the physical phenomenon associated
with the process and a number of process parameters
affecting strip quality, parameter design to obtain good
quality still remains a difficult problem.
In an attempt to improve product quality this paper
proposes a fuzzy logic controller which controls the level
of molten steel trapped between two roll cylinders. As
previously explained, the strip casting process has the
properties of nonlinear uncertainty and coupled process
dynamics. This is due to the fact that the level of molten
steel is completely governed by flow of molten steel
which in turn is strongly affected by roll gap dynamics.
In this situation, fuzzy control techniques may besuitable for this process characterized by nonlinear
uncertainty and ambiguity (Lee, 1990).
Lee et al. proposed an adaptive fuzzy control of
molten steel level in the strip casting process (Lee, Lee,
& Kang, 1996a; Lee, 1997). They showed that the
proposed controller achieved zero steady-state error
asymptotically, through a series of simulation studies.
However, this study had limitations in showing robust-
ness against the disturbance effect because the process
model used in the simulations did not consider practical
implications of the process such as roll gap dynamics.
Moreover, the control algorithm combining threedifferent controller parts seems to be rather complex
for real implementation. In practical applications, much
time and effort are required to construct such a
controller. This is because there are many design
parameters to be considered in the controllers design
procedure.
When the design of a fuzzy system is undertaken for
the control of multi-variable systems such as in strip
casting numerous design parameters, such as fuzzy
partitions of the universes of discourse, input/output
scaling factors and assembly of an appropriate rule base,
must be confronted. Therefore, it is well-known that the
design of fuzzy controllers is more difficult than the
design of conventional controllers Lee, Park, and Cho
(1996a, b). To overcome such difficulties, Gupta has
proposed variable decomposition method for simplify-
ing their structure (Gupta, Kiszka, & Trojan, 1986). His
research was the first attempt at changing the structure
by altering the characteristics of fuzzy logic controllers.However, there have been limitations in practical
applications, because it was necessary to change the
form of these controllers (Lee, Lee, & Jeon, 1995). Thus,
much research was undertaken that attempted to
determine the database and the rule base with proper
values, while at the same time retaining the form of a
fuzzy logic controller. From this perspective, several
research projects have been reported: The first is the
fuzzy self-organizing controller (Procyk and Mamdani,
1979). This controller was based on auto-tuning the rule
base by evaluation of control performance utilizing a
performance index table. The second concerns applying
neuro-fuzzy techniques Lee, Park, & Cho (1996b). This
technique attempted to improve control performance
through learning ability achieved by combining artificial
neural networks. Recently, being inspired by the proved
effectiveness of genetic algorithms, some research has
been reported wherein genetic algorithms were used for
an optimal design of a fuzzy logic controller (Kinzel,
Klawonn, & Kruse, 1994; Karr, 1991).
The result of these research programs seemed to
adequately construct a fuzzy logic controller that would
achieve the desired control performance. However,
because many design parameters have to be considered
in controller construction, most existing methodologieshave limitations in that learning or optimization is
performed only for the rule base or only for the center
values of the membership functions. Moreover, owing to
the different characteristics among design parameters,
attaining a complete learning or optimization algorithm,
while at the same time considering overall design
parameters, has become an extremely difficult obstacle.
To resolve this, a correlation among design parameters
from overall viewpoint of control performance must be
observed.
We introduce a concept of the simplified design
parameters for the strip casting process that are definedby carefully investigating the structure of the individual
components of fuzzy logic controllers (Park, Cha, &
Cho, 1995; Park, Lee, & Cho, 1999; Park, 2001). The
design parameters characterize major design parameters
associated with the membership functions and the rule
table. The simplified design parameters for the member-
ship functions include their number n, the spacing of
their center value p and their shape factor l: For the ruletable, they include the slope of the lines ys separating
dissimilar consequent rules, and the spacing ps between
regions possessing similar rules. In the case of a fuzzy
logic controller with two inputs and one output, these
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simplified design parameters consist of nine parameters
for the membership functions and two parameters for
the rule table. Due to this simplification, effects of such
design parameters on control performance can be easily
and systematically identified. We shall see the design
procedure explained in the above is proven to be very
systematic and effective for the design of the stripcasting process control having four-inputs-one output.
To demonstrate its usefulness a series of simulations is
performed on a molten steel level control in strip casting
processes. The simulation results show that the pro-
posed design method indeed yields a great simplicity in
design and gives satisfactory control performance with
better tracking responses than a fuzzy controller with
parameters arbitrarily chosen.
2. Strip casting processes
Due to the importance of rolls in obtaining the
desired shape and surface qualities, strip casting
processes are classified into single-r and twin-roll
methods. To obtain the desired thin strip, strips
produced by the single-roll method have two distinct
surfaces, a roll-side surface and a free surface. In this
method, it is important to control the quality of the free
surface. On the other hand, the twin-roll method is
characterized by a higher heat extraction capacity than
that of the single-roll method, since the molten pool is
cooled from both sides by the twin rolls. The quality of
the two surfaces is essentially the same. However,
control of the gap of rolls and of the rolling force isdifficult, and therefore twin-roll casters generally require
more sophisticated control systems than do single-roll
case. Twin-roll strip casting is regarded as the most
promising process for producing thin strips from the
view of formability and production capacity. In this
work, we will consider the twin-roll strip casting system.
Fig. 1 shows a schematic drawing of a pilot strip
caster plant and its construction based on a twin-roll
system similar to Bessemers (Shibuya and Ozawa,
1991). As shown in the figure, molten metal is poured
from a ladle into a tundish; it then flows through a
bottom nozzle into the wedge-shaped space between tworolls, rotating in opposite directions which are internally
cooled with the water flow. Once the liquid metal
contacts the rotating rolls, a thin solidification shell is
formed on the surface of each roll. The shells gradually
grow in thickness, finally contact each other and weld
together at a position above the roll exit, called the
solidification final point pf.
Fig. 2 shows the control concept for the strip qualities.
In the case of Fig. 2(a), the roll separating force can be
excessively high when the solidification final point pfoccurs above the roll exit. This frequently results in heat
cracking and damage to the cooling roll surface in
addition to structural abnormalities in the materials. As a
result, the surface of the strip is of poor quality, and the
process can ultimately be unstable. In the case of Fig.
2(c), the solidification final point pf is not formed at the
roll exit. Consequently, the surface of the strip is ofinferior quality because of breakout and oxidation. In the
case ofFig. 2(b), the solidification final point pfis formed
at the roll exit under adequate operating conditions, and
thus strip quality is considered to be satisfactory. From
this viewpoint, it is important to control the solidification
final point pf so as not to deviate from the desirable
position.
To maintain the process states as shown in Fig. 2(b),
regulation of the molten steel level in the wedge-shaped
space is quite important. If this level is higher than the
desirable level, the process becomes unstable, as shown
in Fig. 2(a); if the level is low, the process showsbreakout states, as shown in Fig. 2(c).
2.1. Dynamics of the molten steel level
This section develops a simple mathematical model
for molten steel level dynamics. In the development of
the mathematical model, it is assumed that the molten
steel is incompressible and identical rollers are used. The
continuity equation of liquid steel is then described as
dV
dt Qin Qout, (1)
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nozzle
molten pool
cast strip
roll
cooling water
ladle
solidification
shell
tundish
flowcontroldevice
ps
pf
: solidification final pointpf
ps pf : solidification length
y
gx
molten metal
Fig. 1. The principle of strip casting
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where Qin is the control input flow into the space betweenroll cylinders, Qout is the output flow from the roll
cylinders, and V is the volume of the molten steel stored
between the twin-roll cylinders. The volume V between
the two-roll S 2 Ry0
xgt2
R ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R2 y2ph i
dy;cylinders
is SLr, where Sis the shaded area shown in Fig. 3, and Lris the length of the roll cylinders.
The shaded area S is given by
S 2Zy
0
xgt2
R ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R2 y2q !
dy, (2)
where xgt is the roll gap, R is the radius of the rollcylinder and y(t) is the height of molten metal above the
axis of rollers.
Since
dV
dt Lr dS
dt(3)
and thus
dV
dt Lr y dxg
dt xg 2R 2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 y2
q dy
dt
!. (4)
If Arxg;y is defined as xg 2R 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R2 y2p
;Eq.(1) becomes
dy
dt 1
Arxg;yLr Qin Qout Lry
dxg
dt
. (5)
Here, the input flow Qin is derived from the orifice
opening h depending on the shape of the nozzle and thestopper Lee, (1997) and given by
Qin 2:2466p h0:01585 0:2165h, (6)where the orifice opening h is equal to the height of the
stopper being controlled by an electric servomotor. Due
to fast response of the electric servomotor, the stopper
motion dynamics is assumed to be negligible. Therefore,
the orifice opening h is derived from the control input Hiand given by
h KHi (7)where K denotes the servomotor gain.
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(a) Over cooling conditions
Normal conditions
(c) Under cooling conditions
molten pool
molten pool
solidification
shell
solidification
shell
solidification
shell
roll roll
roll roll
roll roll
pf
pf
molten pool
(b)
Fig. 2. Various molten pool states of the process. (a) Over cooling
conditions, (b) Normal conditions, (c) Under cooling conditions.
Fig. 3. Volume of molten steel filled between twin rolls.
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Also, the output flow Qout is derived from the
tangential velocity vr on the roll surface and given by
Qout Lrxgvr (8)where xg denotes the roll gap being controlled by a
hydraulic roll position servo system as shown in Fig. 4,
and Lr denotes the length of the roll.The hydraulic servo system is composed of a
hydraulic power supply, an electro-hydraulic servo
valve, a double acting cylinder, mechanical linkages,
and a roll. The objective of the control is to generate the
input current such that the position of the roll is
regulated to the desired position. The piston position of
the main cylinder is controlled as follows: once the
voltage input corresponding to the position input is
transmitted to the servo controller, the input current is
generated in proportion to the error between the voltage
input and the voltage output from the potentiometer.
Then, the valve spool position is controlled according tothe input current applied to the torque motor of the
servo valve. Depending on the spool position and the
load conditions of the piston, the rate as well as the
direction of the flows supplied to each cylinder chamber
is determined. The motion of the piston then is
controlled by these flows. At the same time, the piston
is influenced by an external disturbance force generated
from the rolling force.
Since the roll gap, xg is determined directly from the
hydraulic servo system, let us briefly examine the
dynamics of the servo system. Defining the load pressure
PL as PL=P1P2 and the load flow QL as QL=(Q1+Q2)/
2, the relationship between the load pressure PL and the
load flow QL for an ideal critical center servo valve with
a matched and symmetric orifice can be expressed as
follows:
QL kxvffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPs signxvPL
p, (9)
where xv is the servo valve spool position,
k Cdw ffiffiffi
rp
represents the sizing factor of the servo
valve and Ps is the supply pressure. When the continuity
equation is applied to the fluid flowing in each chamber,
the following expression can be derived:
QL Apdxp
dt CtPL Vt
4be
dPL
dt , (10)
where Ap represents the piston ram area, Ct is the total
leakage coefficient, Vt is the total volume of the servo
valve and the cylinder, be is the effective bulk modulus
of oil, and Xp is the piston position. The motion
equation of the piston is given by
ApPL Me xp Be _xp Fd, (11)where Me represents the equivalent mass of the piston
including the roll inertia, Be is the equivalent viscous
damping coefficient, and Fd represents the external
rolling force including the friction. The position of theroll is determined by the piston position as follows:
xg xp. (12)
2.2. Effects of the process parameters
During normal operation in the strip casting process,
the molten steel level controller regulates the height of
the molten steel at a fixed preset value in order to
guarantee good quality strips. However, since themolten steel level is influenced by its nonlinear
dynamics, various process parameter changes, and
disturbances, regulation control is extremely difficult.
The main sources of disturbances come from variation
in flow rate, roll gap, and casting speed. The variation of
flow rate results from hardware wear, while the
variation in roll gap results from roll eccentricity, rolling
force variation, friction, and oil film compression
change.
Fig. 5(a) shows the effects of variation of orifice
opening h on molten pool height y in the case of a roll
gap xg of 3 mm. Fig. 5(b) also shows the effects of rollgap xg on molten pool height y in the case of an orifice
opening h of 1 mm. As shown in the figures, the
variation of molten pool height y decreases smoothly
according to the increase in operating points in the same
process conditions.
From the above two figures, it can be seen that the
strip casting system has properties of both nonlinear and
coupled parameters, as shown in Eq. (4). In particular,
the molten pool level y is significantly affected by
process state variables such as roll gap and speed;
therefore, these nonlinear and coupled effects must be
considered in the controller design.
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Fig. 4. Roll gap positioning system.
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3. Simplified design parameters of a fuzzy logic controller
3.1. Fuzzy logic controller
Fig. 6 shows a control block diagram with a simple
structured fuzzy logic controller with Gaussian-shaped
membership functions. As shown in the figure, the
controller consists of fuzzification/defuzzification parts,
a fuzzy inference engine and a knowledge base.Basically, such a fuzzy logic controller is constructed
by utilizing the following linguistic control rules:
Ri : if x1 is X1i and x2 is X2i then u is Ui
i 1; 2; . . . ; n, 13
where x1, x2 and u are the input and output variables in
the controller, respectively. These are described as the
state error, the change of state error and the control
input in general applications of fuzzy logic controllers.
The design of the controller represented by Eq. (13)
seems relatively simple; however, the design requires the
determination of the overall design parameters of a
fuzzy logic controller, such as fuzzification method,
fuzzy inference method, defuzzification method and
knowledge base. The knowledge base includes scale
factors for input and output variables, membership
functions, and fuzzy linguistic control rules. Moreover,
the design requires much time and effort to tune thedesign parameters to improve control performances.
In Fig. 6, the relationship between the input and the
output of in the controller can be represented as a
function of these parameters as follows:
ut fx1t; x2tjC, (14)where fdjC denotes a fuzzy function constructed withdesign parameters C; such as the membership functionsand the fuzzy linguistic control rules. And ut; x1t andx2t are the control input, and the feedback systemstates at time t, respectively.
To calculate the output from the controller from Eq.
(14), complete information is needed about fuzzification,membership functions, rule base, fuzzy decision making
method and defuzzification. As mentioned in the
Introduction, the fuzzy controller output to satisfy a
certain performance requirement is not easy to obtain,
since these design parameters are interacting with each
other to affect the output. Therefore, we present a
systematic approach to designing membership functions
and fuzzy linguistic control rules.
3.2. Characterization of the parameters based on
common sense knowledge (Park, 2001)
The design of a fuzzy logic controller can be achieved
through generalization of common characteristics that
appear in its design procedure. In this context, a set of
typical parameters that can be characterized in a fuzzy
logic controller are defined. If these parameters are
denoted by ~C; then Eq. (14) can be written in asimplified form in terms of such parameters,
ut fet; _etj ~C. (15)In the case of the controller having two inputs and
one output, the characteristic parameters include nine
parameters for the fuzzy membership functions and twoparameters for the linguistic control rules (Park et al.,
1995; Park et al., 1999; Park, 2001). According to the
parameters described in Figs. 79, they are given by
~C ne; nc; no;pe;pc;po; le; lc; lo; ys;psT, (16)where the number of membership functions n, the
spacing parameter of center value p and the shape factor
of membership functions l are the characteristic
parameters (CP) for the input/output membership
functions; and subscripts e, c and o denote the error,
the change in error and the control output, respectively.
In the above l is the intersection point that two
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When rollgapxgis 3 mm
Time,t[sec]
4.08.0
12.016.0
21.00.0
5.0
10.0
Orifi
ce
openin
g,
h[mm]
400
300
200
100
0Moltenpoolheight,y
[mm]
0.0
When orifice opening h is 1mm
Time,t[sec]
4.08.0
12.016.0
21.00.0
2.5
5.0
Rollg
ap,
xg
[mm]
400
300
200
100
0
Moltenpoolheight,y[m
m]
0.0
(a)
(b)
Fig. 5. Variation of molten pool height y: (a) When roll gap xg is3 mm; (b) When orifice opening h is 1 mm.
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neighboring membership functions meet each other and
is regarded as a correlation factor between the two. On
the other hand, as shown in Fig. 10, the CPs for the
linguistic control rules are the slopes of the lines
separating dissimilar consequent rules, and the spacing
between regions possessing similar rules, ys and ps:Figs. 79 show the definitions of the characteristic
parameters for membership functions, n, p and l: Asshown in the figures, construction of membership
functions can be achieved by specifying n, p and l:
Here, the center value fi can be expressed by a powerfunction as follows:
fi signin 1
2
p jijp i n 1
2; . . . ; 0; . . . ;
n 12
,
(17)
where signi denotes the sign function.These characteristic parameters for membership
functions are extracted by observing the flowing several
common sense knowledge.
The first sense about the membership function is
that the fuzzy sets are not only arranged in a row
according to their linguistic meaning but also arranged
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ERR
CER
-1 +1
OUT
(a)
(b) (c)
real value
Control block diagram
Memebership functions Linguistic rule table
decision
making
rule base
defuzzify
linguistic value
fuzzify
Plant+
-
Error
eX1
eX2
eX3
eX4
eX5
cX5
cX4
cX3
cX2
cX1
ChangeError
H2
eX1
eX2
eX3
eX4
eX5
yref ym(t)h(t)e(t), & e(t)
H2
H2
H1
H1
H3
H3
H3
H4 H4
H4
H4
H5
H5
H5
H2H1 H3 H4
H2H1 H2 H3H4
H
cX1
cX2
cX3
cX4
cX5
H1 H2 H3 H4 H5
real value
Fig. 6. Typical fuzzy logic controller with Gaussian shape membership functions: Control block diagram; Membership functions; (c) Linguistic rule
table.
Fuzzy partition by n membership functions
0.0 1.0-1.0
Symmetry
1.0
X1 X2 XnXi
2 i n1
x1
2~ = n1n
li (x1)
Fig. 7. Gaussian membership functions for a fuzzy logic controller.
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symmetrically centered around zeros (Fig. 7). This rule
illustrates that the membership function design can be
achieved by determining only one side of the fuzzy
universe of discourse. The second sense is on the general
implementation of an FLC; fuzzy sets should include at
least three membership functions: negative large
(NL), zero (ZE), and positive large (PL), in thelinguistic sense. From this we can assume that the
universe of discourse has been segmented with an odd
number of membership functions larger than three. The
third sense is that membership functions are usually
densely distributed near zero, while they are distributed
sparsely outside of near zero (see Fig. 8). It is a general
design philosophy to improve control performance by
having fast response characteristics in the range of large
errors, and accurate response in the range of small
errors. Based on this arrangement we can see the
possibility that the center value of membership functions
can be obtained by a mathematical formula expressed
by a power function. The fourth sense is that the
intersection point, at mi l; of two neighboringmembership functions is located at the midpoint of the
two centers of the two membership functions, as shown
in Fig. 9.
Fig. 10 shows the definition of CPs for control rule
table, ys and ps. In the case of the linguistic control rule,distribution of all rules in the rule table is arranged with
the location of characteristic point Fsi as shown in the
figure. The locations of the characteristic points are
determined by the separating lines and a seed line. They
xs1i; xs2i are determined by
xs1i Lsigni no12 ps ij jps
xs2i xs1i tanysi n0 1
2; . . . ; 0; . . . ;
n0 12
,
(18)
where ys and ps are the slopes of the lines
separating dissimilar consequent rules and the
spacing between the regions possessing similar rules,
respectively. A range parameter L indicative of a
range to limit the locations of the seed points in the
phase plane can be obtained from the characteristic
parameter ys as follows:
L 1 sign tanys if 1= tanys
X11= tanys otherwise:
(, (19)
These characteristic parameters for rule base are
extracted by observing the following two several
common sense knowledge.
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1.0
Xi+1XiXi 1
i1 i i+1
Fig. 9. Correlation factor among membership functions.
Fig. 10. Definition of the characteristic parameters for rule table.
0 1
1.0
-1.0
Symmetry
1.0
X1
X2
Xn
Xi
x1
2~= n1n
j
n~
p
n
j= ~
i
2
2
n
i
i (x1)
Fig. 8. Distribution of center values of membership functions.
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The first sense of the rule base is that the entire area of
the rule table is assumed to be divided into several areas
that have the same consequences, as shown in Fig. 10.
The number of these divided-areas corresponds to the
number of output membership functions. Therefore, we
know that the design of the rule base can be achieved by
designing the guidelines for dividing the areas. Thedivided areas are separated by lines in the case of an
FLC with two dimensional rule spaces. If the controller
is designed in three dimensional rule spaces, the
divisions are accomplished by plane. The second sense
concerns the consistency of the control rules. The role of
fuzzy control rules is to establish the relationships
between input and output variables. Consider two rules,
Ri and Ri1; for the case when only one input variable ischanged: this corresponds to a row or a column in the
rule table. In this case, changes in an output variable are
affected by change in the input variable. The output
variable is then monotonically increased or decreased in
linguistic meaning according to changes in the input
variable. Generally, this consistency should be satisfied
in the implementation of a fuzzy logic controller. The
PID type FLC, widely used in process control, especially
must satisfy this consistency requirement.
4. Design of a fuzzy logic controller
In the strip casting process explained in Section 2,
molten steel level is influenced by its nonlinear
dynamics, various process parameter changes and
ARTICLE IN PRESS
Strip casting process :
Dynamics of the
- Molten steel level
- Roll gap
+-
h(t)
Molten steel level
control :
Multi-dimensional
fuzzy logic
controller
Roll gap control :
PD controller
y(t)
(t)
(t)
(t) (t)
(t)
(t)
(t)
(t)
yd
ig xg
dt
d
dt
d
( ) (t)xdg
+-
ey
ey
xg
xg
ex
Fig. 11. Control block diagram of the strip casting process.
Table 1
Simple design of the fuzzy logic controller using characteristic parameters
(a) Characteristic parameters Rule table
Membership functions
x1(t) x2(t) x3(t) x4(t) h(t)
n1 P1 l1 n2 P2 l2 n3 P3 l3 n4 P4 l4 nh Ph lh ys1 ys2 ys3 Ps7 1.0 0.5 7 1.0 0.5 7 1.0 0.5 7 1.0 0.5 7 1.0 0.5 45 0 45 1.0
(b) Center values of the designed membership functions
X1(t) f11 f12 f13 f14 f15 f16 /171.00 0.67 0.33 0.00 0.33 0.67 1.00
x2(t) f21 f22 f23 f24 f25 f26 f271.00 0.67 0.33 0.00 0.33 0.67 1.00
x3(t) f31 f32 f33 f34 f35 f36 f371.00 0.67 0.33 0.00 0.33 0.67 1.00
x4(t) f41 f42 f43 f44 f45 f46 f47
1.00
0.67
0.33 0.00 0.33 0.67 1.00
h(t) fh1 fh2 fh3 fh4 fh5 fh6 fh71.00 0.67 0.33 0.00 0.33 0.67 1.00
(c) Designed rule table (when x3(t)=2.0 mm (=xg(t)) and x4(t)=0.0)
X21 X22 X23 X24 X25 X26 X27X11 H1 H1 H2 H2 H3 H3 H4X12 H1 H2 H2 H3 H3 H4 H5X13 H2 H2 H3 H3 H4 H5 H5X14 H2 H3 H3 H4 H5 H5 H6X15 H3 H3 H4 H5 H5 H6 H6X16 H3 H4 H5 H5 H6 H6 H7X17 H4 H5 H5 H6 H6 H7 H7
Y. Park, H. Cho / Control Engineering Practice 13 (2005) 821834 829
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disturbances during normal operation. In particular, the
molten steel level is greatly influenced by the roll gap
which is controlled by the hydraulic servo actuator as
can be seen from Eq. (5) and (12). However, in the
previous study (Lee et al., 1996a,b) this parameter has
not been considered as an input variable to the
controller. In order to guarantee high-quality strips,the molten steel level controller should regulate the level
of the molten steel at a preset value with consideration
of the roll gap variation during rolling. To achieve this,
we design a multi-dimensional fuzzy logic controller by
utilizing the characteristic parameters presented in
Section 3.
In Eq. (14), the input variables to the controller, x1t;x2t; x3t and x4t at time t, denote the molten steellevel error and its change, and roll gap and its change,
respectively. As shown in Fig. 11, the error is defined by
the difference between the desired ydt and actualmolten steel level y
t
as follows:
x1t ydt yt,
x2t yt 1 yt
Dt,
x3t xgt,
x4t xgt 1 xgt
Dt. 20
Here, the control input ut in Eq. (14) is the changeof orifice opening Dht: Integrating this, the orificeopening ht expressed in Eq. (7) can be expressed byht ht 1 Dht
ht 1 ut:(21)
Upon examination of the Eqs. (9)(12), it can be
easily obserbed that the dynamics of the roll gap, xg is
governed independently from the process dynamics.
Therefore, the roll gap, x3 is independently controlled by
a PD controller as indicated in Fig. (11).
For the simulation studies, the initial molten steel
level y(0) and desired molten steel level ydt were set tobe 200 and 210 mm, respectively.
4.1. Design of an FLC by using CPs
In the first design step, we attempt to design anFLC with PD control-like action. In this case, the
characteristic parameters in Eq. (16) are given by
~C n1; n2; n3; n4; nh;p1;p2;p3;p4;ph; l1,l2; l3; l4; lh; ys1; ys2; ys3;psT. 22
where n1; n2; n3; n4 and nh are the number ofmembership functions for x1; x2; x3; x4 and h expressedin Eqs. (20) and (21), respectively. And p1; p2; p3; p4 andph are the distribution of membership functions, and l1;l2; l3; l4; and lh are the correlation factor amongmembership functions. Also, since the fuzzy controller
has four inputs, it need to represent the rule base in a
four dimensional space. Therefore, the slopes of the
hyper spaces for separating dissimilar consequent
rule have three values, ys1; ys2 and ys3: Finally ps isthe spacing between the regions for possessing similar
rules.
The values of the characteristic parameters used forthis simulation are shown in Table 1(a). As shown in the
table, the characteristic parameters for input variable x1;the number of fuzzy partitions n1; the distribution ofmembership functions p1, and the correlation factor
among membership functions l1 are set to 7, 1.0, and
0.5, respectively. Also, the characteristic parameters for
input variables, x2; x3 and x4; and output variable h areset to the same values as used for the characteristic
parameters for input variable x1. In this case, the
membership functions are uniformly distributed. The
design results for membership functions are obtained by
using Eq. (17), and are shown in Table 1(b). To form the
rule base as shown in Table 1(c), the slopes ys1; ys2 andys3 are set to 451, 01, 451, respectively, and the spacing psis set to 1.0. These values are selected in engineering
intuition.
ARTICLE IN PRESS
Control surface
Time response of the molten steel level
1
-1.0 -0.5 0.0 0.5 1.0Error
-1.0
0.0
1.0
Change
error
0.0
-5.0
-10.0
-1 .0
5.0
10.0
15.0
Variationoforificeopening,
(
h),mm
0.0 1.0 2.0 3.0 4.0 5.0
Time (t), sec
215.0
210.0
205.0
200.0Moltensteellevel,(y),mm
(a)
(b)
Fig. 12. Control performance with an FLC similar to PD control: (a)
Control surface; (b) Time response of the molten steel level.
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Fig. 12 shows the control surface and regulation
performance when the roll gap is kept constant. As
shown in the figure, the control surface is a flat plane,
identical to a conventional PD controller. In this case,
the molten steel level is settled with large overshoot. The
figures also show that the regulation performance differs
according to changes in operating point due to its
nonlinear characteristics. From this simple design result,it is found that the FLC requires a more complex
structure to compensate for the nonlinear dynamic
characteristics.
4.2. Effects of the characteristic parameters
In the general design procedure of a fuzzy logic
controller, input and output membership functions
are distributed densely near zero. For this reason,
as shown in Table 2(a), the numbers of the membership
functions for input and output variables are set
to 11, and the distributions of the membership functions
are set to 1.5. In this case, membership functions
are arranged densely near zero, as shown in
Table 2(b). The rule base is constructed in the
form of an 11 11 11 11 matrix. Table 2(c) showsthe rule base for a fixed value of x3 and x4: Fig. 13shows the resulting control surface and regu-
lation performances. As shown in the figure, the
control surface is a slightly wiggled plane. However, inspite of this complex structured FLC, regu-
lation performance shows the similar result in the
above case.
Therefore, to obtain better performance of the
controller, the effects of the characteristic parameters
are analyzed. To this end, the inverse of a time-weighted
square error is used as a performance measuring
function as follows:
J Xtftt0
tyd yt2" #1
, (23)
ARTICLE IN PRESS
Table 2
General design of the fuzzy logic controller using characteristic parameters
(a) Characteristic parameters Rule table
Membership functions
x1(t) x2(t) x3(t) x4(t) h(t)
n1 P1 l1 n2 P2 l2 n3 P3 l3 n4 P4 l4 nh Ph lh ys1 ys2 ys3 Ps11 1.5 0.5 11 1.5 0.5 11 1.5 0.5 11 1.5 0.5 11 1.0 0.5 45 0 45 1.0
(b) Center values of the designed membership functions
x1(t) f11 f12 f13 f14 f15 f16 /17 f18 f19 f110 /1111.00 0.72 0.46 0.25 0.09 0.00 0.09 0.25 0.46 0.72 1.00
x2(t) f21 f22 f23 f24 f25 f26 f27 f28 f29 f210 /2111.00 0.72 0.46 0.25 0.09 0.00 0.09 0.25 0.46 0.72 1.00
x3(t) f31 f32 f33 f34 f35 f36 f37 f38 f39 f310 /3111.00 0.72 0.46 0.25 0.09 0.00 0.09 0.25 0.46 0.72 1.00
x4(t) f41 f42 f43 f44 f45 f46 f47 f48 f49 f410 /4111.00 0.72 0.46 025 0.09 0.00 0.09 0.25 0.46 0.72 1.00
h(t) fh1 fh2 fh3 fh4 fh5 fh6 fh7 fh8 fh9 fh10 /h111.00 0.80 0.60 0.40 0.20 0.00 0.20 0.40 0.60 0.80 1.00
(c) Designed rule table (when x3(t)=2.0 mm(=xg(t)) and x4(t)=0.0)
X21 X22 X23 X24 X25 X26 X27 X28 X29 X210 X211X11 H1 H1 H2 H3 H3 H3 H3 H4 H5 H5 H6X12 H1 H2 H3 H3 H4 H4 H4 H5 H5 H6 H7X13 H2 H3 H3 H4 H4 H5 H5 H5 H6 H7 H7X14 H3 H3 H4 H5 H5 H5 H6 H6 H7 H7 H8X15 H3 H4 H4 H5 H6 H6 H6 H6 H7 H8 H9X16 H3 H4 H5 H5 H6 H6 H6 H7 H7 H8 H9X17 H3 H4 H5 H6 H6 H6 H6 H7 H8 H8 H9X18 H4 H5 H5 H6 H6 H7 H7 H7 H8 H9 H9X19 H5 H5 H6 H7 H7 H7 H8 H8 H9 H9 H10X110 H5 H6 H7 H7 H8 H8 H8 H9 H9 H10 H11X111 H6 H7 H7 H8 H9 H9 H9 H9 H10 H11 H11
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where yd and yt denote the desired reference outputand the systems actual output, respectively. Fig. 14
shows that the function J defined in Eq. (23) is indeed
affected by variations in the characteristic parameters
such as the variation of the slope and the spacing of
characteristic points. From this result, the slopes ys1 51; ys2 355 5 and ys3 40; and the spacingps
1:65 are found to maximize the J function. Other
characteristic parameters can be determined in the same
manner.
4.3. Suboptimal design results
From the analysis results of the CPs in the previous
section, we obtain an optimized set of characteristic
parameters by maximizing J in Eq. (23). The results are
listed in Table 3(a). In this case, the membership
functions are distributed as shown in Table 3(b), and
the rule base is constructed in the form of a
11
13
11
11 matrix. Table 3(c) shows the rule base
for a fixed value of x3 and x4: Here, the spacing of thecharacteristic points is chosen to 1.65. The physical
meaning of this value is that the area for sameconsequent rules is densely divided near center of rule
base, while they are distributed sparsely outside of the
rule base. It is for improving the control performance by
having fast response characteristics in the range of large
errors, and accurate response in the range of small
errors. In this sub optimal, the resulting control surface
is a slightly warped plane, as illustrated in Fig. 15(a).
As shown in Fig. 15(b), the response of the molten
steel level is settled with no overshoot and no steady
state error. Also, the figure shows that the regulation
performance is little influenced by change in operating
point.
ARTICLE IN PRESS
Control surface
Time response of molten steel level
0.2
0.205
0.21
0.215
1 101 201 301 401 5
1 4 710
13
16
19
22
25
28
31
34
37
40 S1
S9
S17
S25
S33
S41
-0 .015
-0 .01
- 0.005
0
0.005
0.01
0.015
-1.0 -0.5 0.0 0.5 1.0Error
-1.0
0.0
1.0
Change
error
0.0
-5.0
-10.0
-15.0
5.0
10.0
15.0
Variationoforificeopeni
ng(h
),mm
0.0 1.0 2.0 3.0 4.0 5.0
Time (t), sec
215.0
210.0
205.0
200.0moltensteellevel(y),mm
(a)
(b)
Fig. 13. Control performance with FLC designed heuristically:
Control surface; (b) Time response of molten steel level.
1 35
79
11
13
15
17
19
0
200 0
400 0
6000
8000
10000
12000
14000
1600 0
Control performance v.s. characteristic parametersfor rule table
Control performance v.s. characteristic parametersfor output membership functions
1.70 1.45 1.20 0.95 0.70Dispersive exponentforthe rules (ps )
45
5
65
Phase
angle
(s)
Controlperformance(J)
16000
14000
12000
10000
8000
6000
4000
0
2000
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20S1
S5
S90
200 0
400 0
6000
8000
10000
12000
14000
16000
Controlperformance(J)
16000
14000
12000
10000
8000
6000
4000
0
2000
1.701.45
1.200.95
0.70Dispersive exponentforthe outputmembershipfunctions (po)
311
19
No.
ofthe
output
membe
rshi
p
Functio
ns(n o)
(b)
(a)
Fig. 14. Searching procedure of semi-optimal characteristic para-
meters (a) Control performance v.s. Characteristic parameters for rule
table; (b) Control performance v.s. Characteristic parameters for
output membership functions.
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From the simulation results discussed in the above it
is observed that the use of the characteristic para-
meters can simplify the design procedure of the FLC.
Due to the simple, it is also found that the proposed
simplifications can be effectively used to design the
optimized FLC by minimizing the response error.However, since the characteristic parameters are highly
coupled, an effective optimization algorithm such as GA
is needed.
5. Conclusions
In this paper, a new fuzzy controller design approach
has been presented for a strip casting process whose
dynamics are characterized by nonlinearity, uncertainty
and multi-variable interaction. Since these characteris-
tics present difficulties in controller design, a parametric
approach has been utilized by defining a set of
characteristic parameters involved in the design
procedure. The parameters include the number of
fuzzy membership functions, the spacing of their
center value and their shape factor, the slopes of the
hyper spaces separating dissimilar consequent rules,and the spacing between the regions possessing similar
rules.
From a series of simulations, it is found that the use of
the characteristic parameters, indeed, can simplify the
design procedure of the FLC in the case of designing a
fuzzy controller having multi-dimensional rule space.
Furthermore, it is also found that the proposed
simplification method can be effectively used to design
the fuzzy logic controller in an optimal fashion. Turing
the characteristic parameters with a genetic algorithm
(GA) may be one solution to the optimal controller
design approach.
ARTICLE IN PRESS
Table 3
Suboptimal design of the fuzzy logic controller using characteristic parameters
(a) Characteristic parameters Rule table
Membership functions
x1(t) x2(t) x3(t) x4(t) h(t)
n1 P1 l1 n2 P2 l2 n3 P3 l3 n4 P4 l4 nh Ph lh ys1 ys2 ys3 Ps11 0.85 0.5 13 1.05 0.5 11 1.0 0.5 11 1.1 0.5 11 0.95 0.5 51 5 40 1.65
(b) Center values of the designed membership functions
x1(t) f11 f12 f13 f14 f15 f16 /17 f18 f19 f110 /1111.00 0.83 0.65 0.46 0.25 0.00 0.25 0.46 0.65 0.83 1.00
x2(t) f21 f22 f23 f24 f25 f26 f27 f28 f29 f210 /211 f212 /2131.00 0.83 0.65 0.48 0.32 0.15 0.00 0.15 032 0.48 0.65 0.3 1.00
x3(t) f31 f32 f33 f34 f35 f36 f37 f38 f39 f310 /3111.00 0.80 0.60 0.40 0.20 0.00 0.20 0.40 0.60 0.80 1.00
x4(t) f41 f42 f43 f44 f45 f46 f47 f48 f49 f410 /4111.00 0.72 0.46 0.25 0.09 0.00 0.09 0.25 0.46 0.72 1.00
h(t) fh1 fh2 fh3 fh4 fh5 fh6 fh7 fh8 fh9 fh10 /h111.00 0.81 0.62 0.42 0.22 0.00 0.22 0.42 0.62 0.81 1.00
(c) Designed rule table (when x3(t)=2.0mm(=xg(t)) andx4 (t)=0.0)
X21 X22 X23 X24 X25 X26 X27 X28 X29 X210 X211 X212 X213X11 H1 H1 H1 H1 H2 H2 H3 H3 H3 H4 H5 H6 H7X12 H1 H1 H1 H2 H2 H3 H3 H3 H4 H5 H6 H7 H8X13 H1 H1 H2 H2 H3 H3 H3 H4 H5 H6 H7 H8 H8X14 H1 H2 H2 H2 H3 H3 H4 H4 H5 H7 H8 H8 H9X15 H2 H2 H2 H3 H3 H4 H5 H5 H7 H8 H8 H9 H9X16 H2 H3 H3 H4 H4 H5 H6 H7 H8 H8 H9 H9 H10X17 H3 H3 H4 H4 H5 H7 H7 H8 H9 H9 H10 H10 H10X18 H3 H4 H4 H5 H7 H8 H8 H9 H9 H10 H10 H10 H11X19 H4 H4 H5 H6 H7 H8 H9 H9 H9 H10 H10 H11 H11X110 H4 H5 H6 H7 H8 H9 H9 H9 H10 H10 H11 H11 H11X111 H5 H6 H7 H8 H9 H9 H9 H10 H10 H11 H11 H11 H11
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References
Bae, S.C., Park, Y.J. & Cho, H.S. (1996). Determination of the process
variables for quality monitoring in direct rolling processes. 96
Proceedings of the Eleventh KACC (pp. 13641367).
Gupta, M. M., Kiszka, J. B., & Trojan, G. M. (1986). Multivariable
structure of fuzzy control systems. IEEE Transactions on Systems,
Mans, and Cybernetics, SMC-, 16(5), 638656.
Hwang, J. D., Lin, H. J., Hwang, W. S., & Hu, C. T. (1995). Numerical
simulation of metal flow and heat transfer during twin roll strip
casting. ISIJ international, 35(2), 170177.
Karr, C. (1991). Genetic algorithms for fuzzy controllers. AI Expert, 2,2733.
Kinzel, J., Klawonn, F. & Kruse, R. (1994). Modifications of genetic
algorithms for designing and optimizing fuzzy controller, First
IEEE international conference evolutionary computation, pp. 2833.
Lee, C. C. (1990). Fuzzy logic in control systems: fuzzy logic
controller. Part II IEEE Transactions on Systems, Man and
Cybernetics, 20(2), 419435.
Lee, D.M. (1997). Adaptive fuzzy control to systems with state
dependent input gain. Ph.D. dissertation, Pohang University ofScience and Technology.
Lee, P. G., Lee, K. K., & Jeon, G. J. (1995). An index of applicability
for the decomposition method of multivariable fuzzy systems.
IEEE Transactions on Fuzzy Systems,, 3(3), 364369.
Lee, D. M., Lee, J. S., & Kang, T. W. (1996a). Adaptive fuzzy control
of molten steel level in strip casting process. Control Engineering
Practice, IFAC, 4(11), 15111520.
Lee, S.Y., Park, Y.J. & Cho, H.S. (1996b). A neuro-fuzzy control of an
electro-hydraulic fin position servo system. ASME International
mechanical Engineering Conference and exposition, November
1722.
Miyake, S., Yamane, H., Yukumoto, M., & Ozawa, M. (1991). Strip
quality of highly alloyed metals by twin roll casting. ISIJ
International, 31(7), 689695.
Miyazawa, K., & Szekely, J. (1981). A mathematical model of the splatcooling process using the twin-roll technique. Metallurgical
Transactions A, 12A, 10471057.
Park, Y.J. (2001). Simplified design parameter based design of fuzzy
logic controller and their applications to process control. Disserta-
tion, Korea Advanced Institute of Science and Technology.
Park, Y.J., Bae, S.C., Cho, H.S., Lee, W.H. and Kang, T.W. (1997).
Characteristic analysis and selection of process parameters for
quality inspection in strip casting processes. Proceedings of
KSPE97, pp. 384388.
Park, Y.J., Cha, D.H. & Cho, H.S. (1995). Cho, Genetic algorithm-
based optimization of fuzzy logic controller using characteristic
parameters, Proceedings of ICEC 95: vol. 2 (pp. 831836). Perth,
Australia.
Park, Y. J., Lee, S. Y., & Cho, H. S. (1999). Genetic algorithm-based
fuzzy control of an electro-hydraulic fin position servosystem. Proceedings of FUZZ-IEEE, vol. 3. Korea: Seoul pp.
13611366.
Procyk, T. J., & Mamdani, E. H. (1979). A linguistic self-organizing
process controller. Automatica, 15, 1530.
Reichelt, W., & Kapellner, W. (1998). Near-net-shape casting of flat
products. Metallurgical Plant and Technology, 2, 1835.
Saitoh, T., Hojo, H., Yaguchi, H., & Kang, C. G. (1989). Two-
dimensional model for twin-roll continuous casting. Metallurgical
Transactions B, 20B, 381390.
Shibuya, K., & Ozawa, M. (1991). Strip casting techniques for steel.
ISIJ International, 31(7), 661668.
Takuda, H., Hatta, N., Eramura, M., & Kokado, J. I. (1990). Simple
model for thermal calculation in twin-roll strip casting process.
Steel Research, 61(7), 312317.
Yukumoto, M., & Yamane, H. (1995). Thin strip casting of Ni basealloys by twin roll process. ISIJ international, 35(6), 778783.
ARTICLE IN PRESS
Control surface
Time response of molten steel level
0.2
0.205
0.21
0.215
1 101 201 301 401 501
14710
13
16
19
22
25
28
31
34
37
40 S1
S9
S17
S25
S33
S41
- 0.015
-0 .01
- 0.005
0
0. 005
0.01
0.015
-1.0 -0.5 0.0 0.5 1.0Error
-1.0
0.0
1.0
Cha
nge
error
0.0
-5.0
-10.0
-15.0
5.0
10.0
15.0
Variationoforificeopening(h
),mm
0.0 1.0 2.0 3.0 4.0 5.0
Time (t), sec
215.0
210.0
205.0
200.0moltensteellevel(y),mm
(b)
(a)
Fig. 15. Control performance with semi-optimal designed FLC: (a)
control surface; (b) Time response of molten steel level.
Y. Park, H. Cho / Control Engineering Practice 13 (2005) 821834834