Fuzzy Logic Controller of Molten Level

8/3/2019 Fuzzy Logic Controller of Molten Level

1/14

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

ARTICLE IN PRESS

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).
http://www.elsevier.com/locate/conengprachttp://www.elsevier.com/locate/conengprac


2/14

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

ARTICLE IN PRESS

Y. Park, H. Cho / Control Engineering Practice 13 (2005) 821834822


3/14

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)

ARTICLE IN PRESS

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

Y. Park, H. Cho / Control Engineering Practice 13 (2005) 821834 823


4/14

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.

ARTICLE IN PRESS

(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.



5/14

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.

ARTICLE IN PRESS

Fig. 4. Roll gap positioning system.



6/14

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

ARTICLE IN PRESS

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.



7/14

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

ARTICLE IN PRESS

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.



8/14

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.

ARTICLE IN PRESS

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.



9/14

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



10/14

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.



11/14

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



x1(t) x2(t) x3(t) x4(t) h(t)



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



12/14

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.



13/14

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



x1(t) x2(t) x3(t) x4(t) h(t)



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



14/14

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.


Fuzzy Logic Controller of Molten Level

Documents

Transcript of Fuzzy Logic Controller of Molten Level