- Email: [email protected]
Development of PC Mill Shape Control System
Copyright © IFAC Automation in Mining. ~Iineral and Metal Processing. Tokyo. Japan 19H6
DEVELOPMENT OF PC MILL SHAPE CONTROL SYSTEM Y. Hayama*, J. Nishizaki*, T. Kajiwara** and M. Abe*** *System Engineering Department, Mitsubishi Heavy Industries Ltd .. japan **Hiroslzima Teclznicallnstitute, Mitsubishi Heat'Y Industries Ltd., japan ***Hiroshima Shipyard, Mitsubislzi Heavy Industries Ltd, japan
Abstract. Authors developed the shape control system of roll cross type hot rolling mill (PC mill) of the excellent shape controllability. This system has the following characteristics. (1) Us i ng quadratic programming, this system determines optimum cross angles. (2 ) Learning control is performed, using interstand strip crOlms which are rationally estimated by strip crown and steepness of the product. (3) reedback control has a function of the decoupling control of strip thickness and strip crown. Verification tests were performed using test mill and a tandem simulator which is prepared simultaneous analysis of strip thickness and strip shape, and the effectiveness of this system was conformed. Keywords. Decoupling; Feedback; Learning system; Optimization; Quadratic programming; Rolling mills.
1.
Therefore, rationalization was performed under the following assumption and the equations for estimating the strip crown and the relative difference in elongation were derived. Meaning of symbols will be explained at the end of the paper.
IfnRODUCTION
In strip rolling, control of the shape and strip crown is inportant for improvement of accuracy of products, yield and workability of rolling. Authors developed the roll cross type hot rolling mill which is called pair cross mill (PC mill), of the excellent shape controllability. In succession, authors developed the PC mill shape control s ystem that can set optimum cross angles according to the rolling condition and eliminate fuctuation of the strip shape and strip crown which occur during rolling. This s ystem consists of preset control "'hich sets roll cross angles using the mathematical model of three-dimensional hot rolling and feedback control of the roll bender as final control elements.
(1) Assumption
(a) Sticking friction is assumed for all contacting parts between the strip and the roll. (b) In the lateral direction, the strip crown is approximated by quardratic and octagonal polynomials. (1)
",':ere, 2: is a variacle :or posltlons in the lateral direction of the strip and normalized by the strip width.
Each control function was evaluated as follows: For preset control, control accuracy was confirmed by aluminum strip rolling through Nitsubishi's test mill first, then for feedback control characteristics of various control methods were evaluated using the tandem simulator which enables analysis of strip thickness and shape.
(2) Estimate quation of strip crown
Ch
= J....r kc
- EfF - 1;e C8
- 1;w(CR
This paper gi ves a general description of the mathematical model of three-dimensional hot rolling, preset and feedback control model, and their verification test results.
+
+
aC l )
1'1 (C H - HEo) +
tlATHEt~ATICAL
(2 )
(3) Estimate equation of relative difference in
elongation (3)
E
2.
Cc
tl0DEL
In hot rolling the shape and strip cro,,~ are closely related with the lateral flow of material. Therefore, such relation must be taken into consideration at the time of shape control.
(4 ) Relation between steepness and relative
On the other hand, the calculating time must be
To confirm the accuracy of the system, a rolling test of aluminum strips was performed using the test mill which is approx. 1/ 3 size of the actual machine. The experimental conditions are listed in TABLE 1. Strip thickness distribution was
difference in elongation =
reduced drastically by simplifying the rolling model to incorporate computer control in the actual roll ing .
325
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Relation between change of relative crown and elongation difference
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60 50 30 20 '0 MUSUI'e:d strip crown (pm)
Strip thickness : 6tmI, ship llridth : 3CX)Trn Re6..tc1ion rate: 30U kw rKh p.ls.s
-40 ,
(2)
Mulll pnss rotting (A! slrip 6 mm thick)
3.
Comparison of calculation results and test results of strip crown
measured using a contact type profile meter and the strip crown was defined at a point 10mm from the strip edge. FIG.l(l) and (2) show a comparison of calculation results and test results of the strip crown for single pass rolling. a in the figure means standard deviation of difference between measured values and calculated values. As is obvious from these results, measured values and calculated values of the strip crown agree within the accuracy of a = 5 to 10 ~m. Moreover, crown heredity coefficient gives good accuracy because estimated accuracy of the strip crown is unchanged after mUlti-pass rolling. FIG .2 shows the relative difference in elongation obtained from the change of relative crown 6Ch/h and steepness as expressed by £
1T2
(
:jJ wl
+ '¥D2
2
2
-
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1 Experilllento' conditions
Work roil llock· up roil Rolling strip Strip thickness Strip widlh
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aown 4~ (%)
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AI strip, r 2 mm x 8450 mm
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AI tOSOP·H14 J"mm x
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t
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;r 0
60
(5)
where, :jJ w, :jJD and :jJc are steepness of the work side strip edge, drive side strip edge and cent er of the strip respectively. As shown in the figures, the linear relationship of Eq.(3) exists between 6ch and £ for strip thickness of 2mm and 4mm. Moreover, the line does not pass through the original point and drift from the original point increases with the reduction ratio. Furthermore, calculated values are nearly parallel with measured values and shape coefficients agree closely.
~260
x 610 mm
~5BO
x 610 mm AI 10501' 2,4, 6mm
Reduction ratio
300-500 10-40%
[\oiling speed Lubrication
No
III m
1011l/min
PC MILL SIII\PE CONTROL SYSTEM
FIG.3 shows the general composition of the PC mill shape control sys tem based on the nathematical model mentioned above. The sys tem consists of preset control and feedback control as explaned below. 3.1
PRESET COtITROL
Preset control consists of the cross ang l e se tting s ection and learnin g con trol sec tion, and performs control in the following procedure. 3.1.1 CROSS AnGLE SETTIrlG SECTION
Cross angle setting section ca lculat es the optimum values of roll cross angles which sa ti sfy the following conditions. (1) Cross angles must be less than their maximum values. (2) Threading must be possible, i.e. the delivery strip steepness of each stand is less than its admissible value. Moreover, the interstand delivery strip steepness must be as small as possible. (3) Desired strip crown and steepness are realized, i.e. (a) The steepness of the product is 0% (b) !ChN - Chf l-+min.
(6)
(4) Setting calculation should be carried out in short time ( less than 0.5s) From these conditions the following formulation is obtained.
327
Development of PC Mill Shape Control System Preccd;"g v,Jlues of rolling da!a
Rolling force
Strip crown Strip steepness Desired crown Pass
e!"!)!;",
Bending force
PC mill
r'dUl'
I
• Oecoupling control 01 strip thickness and ship crown • feedback contra of shape meter
III1Io(Jf!
setlinft section
Learning control scction
Corrected value or bending lorce • Determination 01 coeffiCIents for estimating strip crown and strip
• Oclernination cl esti mated optimum strip crown
Cross
-~
Feedback cOllllol
shape Strip crown correction
• Estimation of wear crown and thermal crown • Determination of opti-
• Calculation 0/ strip
crown correction
PfI!set control
,
.
+
hi_IEi-l) , Ch maxi (7)
Chi -I )+c. ) h i-I 1-1
hi
E
·<;Emax i (8)
c1
where,
c
F3-FU:
~
r...
~
1
kc/ i - Efi Fi - /;;wi (CRi
t
aC Li ) + Cci (9)
., ,
20 l!i(
Pig.4
Q
N wiE2~min (10) Z 1 i=l From (7), (8), (9) this optimization problem becomes a problem that minimize the non-negative definite quadratic function under the linear constraints of un~wn variables Ch I, ----, Cht-; o In this preset control, the quadratic programming by Lemke is used to solve this problem. FIG.4 shows the comparison of the results by this algorithm and the result by linear programming setting objective function Q' as follows.
Q = W(Chf -
Ch~)
2
'
Comparison between linear programming and quadratic programming
To compare the effectiveness of these two types of learning control, computer simulation was performed. FIG.5 shows the control results when the fluctuation of 50 Il m roll crown was input as step signal. As is obvious from FIG.5, integral type is superior to proportional type, because the former method is able to make the steady-state deviation of relative difference in elongation convergent to
(11)
O.
From this result, it is understood that the solution not only achieved the desired crown, but also made the interstand strip steepness smaller than one obtained by linear programming. Moreover, by removing the trivial inequalities in (71 (81 for 7 stand mill 26 inequalities were reduced to 14. Consequentl,', the calculation time on actual computer got less than 0.3s, and so the prospect was obtained that this optimization algorithm is appliable in actual hot rolling. 3.1.2
,
Proportional type is a method that uses the estimated crown calculated as a = 0 in (2). On the other hand, integral type uses the estimated crowns calculated as a = 1.0.
+
Q' = :Chf - Ch":
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-------, N
(2) Objective function
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.....
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(1) Constraints
General composition of shape control system for PC mill
Pig. 3
mum roll cross angle
In order to evaluate the effect of learning-control, rolling test was carried out for aluminum strip using the test mill mentioned above. TABLE 2. shows the pass schedule at this test. (1) TEST According to pass schedule@, a total of 23 rollings were carried out by cross angle Integral type ..••• Proport; ona 1 type
BD
LEARtUNG CONTROL SECTION
--0-
)0
The di fficul t,' to perform the learning control strip shape in hot rolling is that only the strip crown and steepness of the product are measurable, and that the roll cross angles of all stands (generally ~ 2) must be determined from only these measured values. To accomplish this, authors developed t"o types of learning control method (p roportional type and integral typ"e ) "hich estir.1ates the interstand strip cro"ns Chj (j = 1, 2, "-1) so that minimi:e the follo"ing function Rand "hich using these estimated strip crowns calculates correlation values as roll cro"n.
"-1
"
R = ~ =l gj (Chj - Ch5) ]
+
gN(Ch - ch~f+ G('t- E~f (12)
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....... •.•..•........•
Integral type. /Proportional type /.
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Comparison of the effectiveness of two types of learning control
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Fig. 7
Strip Cr own Fluctuation
le arning
by
control
preset t ing o f t he prototype ma th ema t ical mode l onl~'
fo !' thL' f i t',t ,t ri p :lnJ b\' ad op t ing learning control f ro m t he second str i p, FIG,6 s ho~ s the lealTIing control effect in the second pass. Brohen lines in the figure sho~ the desired
\'~llll E'S
of t he
~tr.ip
Pass SChedule
PaH schedule
Pass schedule
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Fig. 8
Strip steepness Fluctuation
by
learning
c o ntrol
Fig. 6
Learning control effect
,\s a resu lt, th e fo lloldn g is ob ta i ned ; b'en if
roll i ll ~
force fluctuates,
stri~
crO~ll
de\'iation are controlled I,ithin :: ~I" from the desir ed va l ue respectively o~ing to learning control .
for desired s tr ip
The same accuracy is assured cro~n
change .
( :' i TEST
\cc o rding to pass schedu l e
:1, ,:] ,0 , the
roll-
ing of firs t 21 s tr ips ~as carried ou t nt learing cont r o l off and th e r o ll ing of s ucce s si\'e 21 strip I,'as at l ea rnin g con tro l on. ,\t control on, t o correspo nd to th e ac tual mac hine, only the s tr ip c ro~n and s teepn ess of the s tr ip aft e r ~- pass rolling ~ e r e used as feed hack \'a l ues . i, IC. ~ ,ho,,'s the change of di fferen ce be tl,een m0asured strip c r o~n and desi re d one . Broken l in e s and so lid l i ne s indicate the res ults at learn ing control off and on respectively. ~oreove r, fIC . S sho~s the change of s teepne ss . ~s
tlle
a resul t, the aCCllrac~ ·
fo ll o~ing
is ob t ained : bo th
of s trip crOKn and s teepn ess are
i Olp r O\'ed b\' l earning con trol, Besides, ne\'er t heless pass schedu l e ~a s change d, t he flat s tr ip I, as obtained I,hen l ea rnin g cont rol is turned on, and so t he prospect i s obtained t ha t l ea rnin g contro l is possible using the same correla t ion values even if roll ing cond i t ion is changed .
3.2
FEEDBACK CONTROL
Feedhack control consists of de coup l ing con trol of str ip th ickness and s tri p cro ~n and of shape - me ter f ee dbac k con trol .
For f eedback control, th e optimum comb inat ion must be determined for many variables suc h a s type and arrangenent of detector as ~ell as feedback s\'stem. HoI,e\'e r, it is difficult to c\'aluate such variables in the actual svs tem. l~erefore, the effect of decouplin g control ~as eva lu ated for the s tr ip thickness an d strip cro ~n bv fabricating th e tand em hot rol lin g mi ll simul~tor ~hich enab les dvnamic ca lcul a tion of s trip thi ckness an d s hape by adding d)TI dmic charac teris tic s of e l ec tri ca l, mechanica l, hydraulic and control devices to th e ma the~atical model mentioned in 2. FTG .9 ShOh'S an examp l e of th e thick ness an d s hape control tand em simu l a tor consisting of 6 s tand s . The pr eset co ntrol is omi tt ed bec au se it has been explained previouslv . The s i mul ator has f eedbac k and f ee d fo r~ ard functions as sh o~n in FIG . 9 . The final control element s listed in TABLE 3 are pro\'ided and a\'a i l abl e in e\'er\' combi nation . Us in g this simu lator, the eva l uation abou t the effe c t of decoupling con trol of str i p thicknes s and s trip cro~ n. In the hot ro ll ing mill , flu ctua tion of str ip thickne ss and s trip cro~n is caused by rolling force variation due to skid ~a~ks, etc . Gatemeter t\'pe strip thickne ss control and latera l s tiffness con trol ~h ich control s rol l bending force during rollin g a re pr ovided for s tri p thi c kne ss fluctuation and s trip cro~n res pec ti velv , ho~ever, a control process hhi ch takes into conside ration interaction i s required beca use the s trip thickne ss and s trip cro~n fluctuate simultaneously even if the roll gap and roll bending force a r e controll ed . Fundamen t al charac teri s t ics of strip thi ckne ss fluctuation ~h and st rip cro ~n flu c tuati on ~Ch are expressed bv ~h
=
~p
"'y
( 13 ) (1 .) )
329
Development of PC Mill Shape Control System
AGC : Gale-meltr type Ihitknns control X-ray AGC : X- ,Oily meMor thickness controt
RF
FF APC
AGC : Fetdlorwa,d thickness control ': Roll lap conlrol
SCC : Sutcess .... e Ci f cUll DCe : Decouphnl (onlrol eRC : ero.. n Ittdbac~ conlrol GM Tt\ltkness IAlt
" SR · Mill motor speed conlrol LP loopt' control HC lOO()tf heql:ht cont(ol CM e,o-n meter
PT
Ben()H pressure del I ctO!'
"NO LC SM
Roll bender prtssur e control
: Load ceU Shape meltr
Composition of tandem simulator of thickness and shape control system (feedback control model) Type of final control elements FIG. ll shows example of si mulation charts . Tt is understood from the figure that the degree of Type change is reduced not only in th e strip crown but Hydraulic screw -down or motor screw-down also in the strip steepness under decoupling Low inertia typ.e or conventional type control for all s tand s. (Case 1 in TABLE 4) High response type or conventional type
Fig. 9
final control element Roll gap control device Looper Bender
where, symbol n of each variable means vari ation from the normal condition, and bender mill modulus MB is the bending force required t o change the roll gap 1 mm. In the shape control systeM, the roll gap an d roll bending force are controlled to n h = 0 and nCh = 0 respectively. This is called decoupling con trol between the s trip thicknes s and strip crown. The block diagram is shown in FI G. lO. In order to confirm the effect of this control, simulation was performed for a combination of one kind of pass sc hedule, two kinds of bender control systems and th ree kinds of decoupling control systems are listed j~ TABLE 4. A triangular ,,'aye, \,hich has a temperature difference of 2 sand 20·C, was used as disturbance.
In FIG .1 2 , maximum values are plotted from the calcu l ation result on th e change of strip thickne ss, strip crown, and steepness a t each s tand under the condition of TABLE 4. Consequently, the followings are clearrecr:TAnl.l~ Bar
thick. (mm)
1'1
32,0
15 ,60
Pas! schedule
dP
S iJl1ulation conditions
F3
F2
Rolling Stri p
9,34
F5
F6
speed (mprn)
width
3,01
2.30
1000
1 250
F4
6,26 4,22
Roll gap co ntrol devi c e
Hydraulic screw.down
Bende r
High re sponse type (5 Hl./ -90o) . Conventional type ( 1 HZ/-90o)
Oecoupling cont ro l (0: ON X : OH)
Disturbance Measured value of roiling lotce
1\
D e livery strip thickness (mm) .
(mm)
FI
F2
F3
F4
F5
Case 1
0
0
0
0
0
0
Case 2 Case 3
X
X
X
X
X
X
X
X
X
0
0
0
F6
Tria ngulJ r wave o f 2 sec at 20"C
Measured ... "Iue of bending force
dF
(1 ) Decoupling co ntrol reduces strip crown
variation of the product to approx. 1/3 as compared "'ith that I,hen decoupling control is turned off. Furthe r more, steepness va ri ation of the product is reduced t o 0.5 % or belo" by adjusting the control system. (2) \\hen the high re spon se bender is used, strip
crown variation is reduced to 70 to 80% o f the conventional typ e . I~wever, it makes no great diffe renc e because absolute values diffe r on l v 2 to 3 ~m. ( 3 ) Al though roll bending dS •
Preset v~lue 01 roll gap
Fig. 10
dF· Pa'set value 01 bending force
Block diagram of decoupling control between thickness and strip crown
for~e is operated for s trip cro"n con trol, the effect of decoupling control given strip thickness fluctuation equivalent to onl v gage-meter thickness contro l.
Y. Hayama et al.
330
REFERENCES
ISI~
e~
2st
-::.0
3st
~
Sst
....
------........------.
4st
-------------
_1500
~ 6st
Omori, \akajima, Tsukamoto, Morimoto, Hino and \akaza~a (1083). Precedings of the 33rd Japanese Joint Conference for the Technology of Plasticity 419-436 Hayama, Ozono, Kaj il,ara, Okura, \ishizaki and Iwatani (198 4) Precedings of the 1984 Japanese Spri ng Conference for the Technology of Plasticity, \0. 223
~ 10
(s) - - Oecoupling conlrol turned on lor all stands
/',
Kono and Yamashita \onlinear Programming, JSTU
_,~~~// ___'~,~,~__~~______-_ _ -_-~G~a~g.~-m~.~"~r~ty~p~.~th~"~k~ne~ss~ control only 10r all s(anas
Hayama, Hashimoto, Ozono, Kawanami, Kawasaki and Hamasaki ( 19 78) Mitsubishi Technical Review Vol. 15 \0. 3
1 st I
-
'~'"
~ 2st 1---------""""-==-----""',"",;.-.......:-;-,,......,:::-,-,,-------------------
~ 3 st 1---------------..2:.::::::.....:,::,":i,--. _:::-,.-,,-------------
i
4str-----------~~----~~~------
3
5str---------------------~,'~=;~',~'-----------6 st
r---
SYMBOLS
IIO.O __________________..;,. .------::--:---~''''' =""'-','--------
Strip cro~n on delivery side (mm) Rolling force (t) Bending force per chock (t) Strip cro~n on entry side (mm) Equiva l ent mechanical cro~n by roll cro ss ing (mm) Cc: Constant in equat ion of strip cro~n (mm) kc: Lateral mill modulus (t/mm) Ef: Bender influence coefficient (mm/ t ) ~~. Roll crown correlation coefficient (-) CSmax: ~laximum equivalent mechanical crown by roll crossing (mm) so: Cross angle influence coefficient n Crown heredity coeff i cien t (-) C: Relative difference in elongation for strip on delivery side ( - ) ~o: Relative difference in elongation for strip on entry side (-) c c: Constant in equation of relative difference in e lonra tion C- ) Shape coefficient (-) Strip thickness on delivery si de (mm) Strip thickness on entry s ide (mm) H B Strip width (mm) Reduction ratio (-) Suffix for stand number ( - ) \umber of stands C-) G Cross angle (0) \j! Steepness ( - ) CR : Roll initial crown (mm) a : Learning control coefficient (-) CL: Learning control correlation va lue (mm) Chf: Desired strip crown (mm) lieight factor ( - ) \\~ h', G, g: Ch: ~e s ured str ip crown of the product (mm) 'I:' . ~leasured rclati\·e difference in e l oneation ( - ) Calcul"J\ed s tri.p crOI,n b,· expression (2), USIng Chi -l as entrv strip crown (mm) ["lculat~d relative difference in elongat"i on of t.re product b,· expression (3) using C~\_land Ch In place of Ch\-l and Ch\, respecti\·el,·. (-) ~.1i 11 modulus (t / mm ) Bender mill modulus (t / mm )
10 (s)
§
i lstr_--~------------------------------------
~ ~2str_--------------------------------------- "-~
.... r3str_------------------~,~/~-~~-------------
~ ~4SIr-------------------~==~~------------- :- .& 5 st I---------------------~'=----'":-,.-..:'~--------on ~ 6st _110 ------------------=-=.;,/=-==''''''.:.:...0-------u
10
Fig. 11
4.
Example of simulation charts (high response type roll bender)
CONCLUS ION
Authors developed the PC mill shape control system consisting of preset control, which has roll cross angles as final control elements, and feedback control elements, which has roll benders as final control elements. Rolling by Mitsubishi's test mill was carried out for aluminum strips and the following results are obtained. (1) Cross angle setting method using quadratic
programming is
appliable for actual machine.
(2) By learning control method of strip shape us-
ing the interstand estimated strip cro~n, drift from the desired strip cro~n is kept within 5 m in test mill. (3) Tandem simulator ~as prepared for simultaneous analysis of strip thickness and s trip shape, then effective control was confirmed by evaluation of decoupling control for the strip thickness and strip crown.
80 .
e
~
c
60-
0
D
~ ~
Decoupling control lor all stands
~O
-
20 -
(lIlgh resp ons e bender) () Oecoup hn g control lor all stands (Conventional lIendcr)
• Gage-meler type stflP thickness control only for all stands
~
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~
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,
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.3
15-
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on
., --.
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20 .
,,
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4
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Stand
Fig. 12
Simulation results
~
... 10
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I
/
0
/
"~ 05 . "
/
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J SIJ ri J
4
.-0
DEVELOPMENT OF PC MILL SHAPE CONTROL SYSTEM Y. Hayama*, J. Nishizaki*, T. Kajiwara** and M. Abe*** *System Engineering Department, Mitsubishi Heavy Industries Ltd .. japan **Hiroslzima Teclznicallnstitute, Mitsubishi Heat'Y Industries Ltd., japan ***Hiroshima Shipyard, Mitsubislzi Heavy Industries Ltd, japan
Abstract. Authors developed the shape control system of roll cross type hot rolling mill (PC mill) of the excellent shape controllability. This system has the following characteristics. (1) Us i ng quadratic programming, this system determines optimum cross angles. (2 ) Learning control is performed, using interstand strip crOlms which are rationally estimated by strip crown and steepness of the product. (3) reedback control has a function of the decoupling control of strip thickness and strip crown. Verification tests were performed using test mill and a tandem simulator which is prepared simultaneous analysis of strip thickness and strip shape, and the effectiveness of this system was conformed. Keywords. Decoupling; Feedback; Learning system; Optimization; Quadratic programming; Rolling mills.
1.
Therefore, rationalization was performed under the following assumption and the equations for estimating the strip crown and the relative difference in elongation were derived. Meaning of symbols will be explained at the end of the paper.
IfnRODUCTION
In strip rolling, control of the shape and strip crown is inportant for improvement of accuracy of products, yield and workability of rolling. Authors developed the roll cross type hot rolling mill which is called pair cross mill (PC mill), of the excellent shape controllability. In succession, authors developed the PC mill shape control s ystem that can set optimum cross angles according to the rolling condition and eliminate fuctuation of the strip shape and strip crown which occur during rolling. This s ystem consists of preset control "'hich sets roll cross angles using the mathematical model of three-dimensional hot rolling and feedback control of the roll bender as final control elements.
(1) Assumption
(a) Sticking friction is assumed for all contacting parts between the strip and the roll. (b) In the lateral direction, the strip crown is approximated by quardratic and octagonal polynomials. (1)
",':ere, 2: is a variacle :or posltlons in the lateral direction of the strip and normalized by the strip width.
Each control function was evaluated as follows: For preset control, control accuracy was confirmed by aluminum strip rolling through Nitsubishi's test mill first, then for feedback control characteristics of various control methods were evaluated using the tandem simulator which enables analysis of strip thickness and shape.
(2) Estimate quation of strip crown
Ch
= J....r kc
- EfF - 1;e C8
- 1;w(CR
This paper gi ves a general description of the mathematical model of three-dimensional hot rolling, preset and feedback control model, and their verification test results.
+
+
aC l )
1'1 (C H - HEo) +
tlATHEt~ATICAL
(2 )
(3) Estimate equation of relative difference in
elongation (3)
E
2.
Cc
tl0DEL
In hot rolling the shape and strip cro,,~ are closely related with the lateral flow of material. Therefore, such relation must be taken into consideration at the time of shape control.
(4 ) Relation between steepness and relative
On the other hand, the calculating time must be
To confirm the accuracy of the system, a rolling test of aluminum strips was performed using the test mill which is approx. 1/ 3 size of the actual machine. The experimental conditions are listed in TABLE 1. Strip thickness distribution was
difference in elongation =
reduced drastically by simplifying the rolling model to incorporate computer control in the actual roll ing .
325
~ sign( E)
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(4 )
Y. Hayama et al.
326 1pus rallinc
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,
g,
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300
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500 AI IOSOP-H24
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e
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12mm x B4SOmm
m
One pass roiling (A! slrip 6 mm thick) 60
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& 30 :!!
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I
Relation between change of relative crown and elongation difference
~
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,
60 50 30 20 '0 MUSUI'e:d strip crown (pm)
Strip thickness : 6tmI, ship llridth : 3CX)Trn Re6..tc1ion rate: 30U kw rKh p.ls.s
-40 ,
(2)
Mulll pnss rotting (A! slrip 6 mm thick)
3.
Comparison of calculation results and test results of strip crown
measured using a contact type profile meter and the strip crown was defined at a point 10mm from the strip edge. FIG.l(l) and (2) show a comparison of calculation results and test results of the strip crown for single pass rolling. a in the figure means standard deviation of difference between measured values and calculated values. As is obvious from these results, measured values and calculated values of the strip crown agree within the accuracy of a = 5 to 10 ~m. Moreover, crown heredity coefficient gives good accuracy because estimated accuracy of the strip crown is unchanged after mUlti-pass rolling. FIG .2 shows the relative difference in elongation obtained from the change of relative crown 6Ch/h and steepness as expressed by £
1T2
(
:jJ wl
+ '¥D2
2
2
-
'" c)
1 Experilllento' conditions
Work roil llock· up roil Rolling strip Strip thickness Strip widlh
""Irri.JI : AlIOSO - )0
aown 4~ (%)
5.4pm
00
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OIanle of relative
AI strip, r 2 mm x 8450 mm
T~BLE ~ :.:
0
0
e: = T
AI strip, r 4 mm x 8300 mm
... .,
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.. o~
Fig. 1
~~ ( %)
-I
(1)
0" pus
Bmmm
-I
0
~
OI"IISS 02 PISS 01 pus
AI tOSOP·H14 J"mm x
?0~
t
'0
;r 0
60
(5)
where, :jJ w, :jJD and :jJc are steepness of the work side strip edge, drive side strip edge and cent er of the strip respectively. As shown in the figures, the linear relationship of Eq.(3) exists between 6ch and £ for strip thickness of 2mm and 4mm. Moreover, the line does not pass through the original point and drift from the original point increases with the reduction ratio. Furthermore, calculated values are nearly parallel with measured values and shape coefficients agree closely.
~260
x 610 mm
~5BO
x 610 mm AI 10501' 2,4, 6mm
Reduction ratio
300-500 10-40%
[\oiling speed Lubrication
No
III m
1011l/min
PC MILL SIII\PE CONTROL SYSTEM
FIG.3 shows the general composition of the PC mill shape control sys tem based on the nathematical model mentioned above. The sys tem consists of preset control and feedback control as explaned below. 3.1
PRESET COtITROL
Preset control consists of the cross ang l e se tting s ection and learnin g con trol sec tion, and performs control in the following procedure. 3.1.1 CROSS AnGLE SETTIrlG SECTION
Cross angle setting section ca lculat es the optimum values of roll cross angles which sa ti sfy the following conditions. (1) Cross angles must be less than their maximum values. (2) Threading must be possible, i.e. the delivery strip steepness of each stand is less than its admissible value. Moreover, the interstand delivery strip steepness must be as small as possible. (3) Desired strip crown and steepness are realized, i.e. (a) The steepness of the product is 0% (b) !ChN - Chf l-+min.
(6)
(4) Setting calculation should be carried out in short time ( less than 0.5s) From these conditions the following formulation is obtained.
327
Development of PC Mill Shape Control System Preccd;"g v,Jlues of rolling da!a
Rolling force
Strip crown Strip steepness Desired crown Pass
e!"!)!;",
Bending force
PC mill
r'dUl'
I
• Oecoupling control 01 strip thickness and ship crown • feedback contra of shape meter
III1Io(Jf!
setlinft section
Learning control scction
Corrected value or bending lorce • Determination 01 coeffiCIents for estimating strip crown and strip
• Oclernination cl esti mated optimum strip crown
Cross
-~
Feedback cOllllol
shape Strip crown correction
• Estimation of wear crown and thermal crown • Determination of opti-
• Calculation 0/ strip
crown correction
PfI!set control
,
.
+
hi_IEi-l) , Ch maxi (7)
Chi -I )+c. ) h i-I 1-1
hi
E
·<;Emax i (8)
c1
where,
c
F3-FU:
~
r...
~
1
kc/ i - Efi Fi - /;;wi (CRi
t
aC Li ) + Cci (9)
., ,
20 l!i(
Pig.4
Q
N wiE2~min (10) Z 1 i=l From (7), (8), (9) this optimization problem becomes a problem that minimize the non-negative definite quadratic function under the linear constraints of un~wn variables Ch I, ----, Cht-; o In this preset control, the quadratic programming by Lemke is used to solve this problem. FIG.4 shows the comparison of the results by this algorithm and the result by linear programming setting objective function Q' as follows.
Q = W(Chf -
Ch~)
2
'
Comparison between linear programming and quadratic programming
To compare the effectiveness of these two types of learning control, computer simulation was performed. FIG.5 shows the control results when the fluctuation of 50 Il m roll crown was input as step signal. As is obvious from FIG.5, integral type is superior to proportional type, because the former method is able to make the steady-state deviation of relative difference in elongation convergent to
(11)
O.
From this result, it is understood that the solution not only achieved the desired crown, but also made the interstand strip steepness smaller than one obtained by linear programming. Moreover, by removing the trivial inequalities in (71 (81 for 7 stand mill 26 inequalities were reduced to 14. Consequentl,', the calculation time on actual computer got less than 0.3s, and so the prospect was obtained that this optimization algorithm is appliable in actual hot rolling. 3.1.2
,
Proportional type is a method that uses the estimated crown calculated as a = 0 in (2). On the other hand, integral type uses the estimated crowns calculated as a = 1.0.
+
Q' = :Chf - Ch":
\
Fn:2 f:lTtT~ Slnrul /10.
-------, N
(2) Objective function
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-.- --
.
.,
,
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Chmaxi
FI-F2 : CUH'iI'lll.ioll<.l1 mill re .i 11
Ilud,,,tic ''''UG,",,,.I,," o: ~.§
-0-
G 2S
Chmini f Chi - ni (C hi Cmini
.....
30
< >
(1) Constraints
General composition of shape control system for PC mill
Pig. 3
mum roll cross angle
In order to evaluate the effect of learning-control, rolling test was carried out for aluminum strip using the test mill mentioned above. TABLE 2. shows the pass schedule at this test. (1) TEST According to pass schedule@, a total of 23 rollings were carried out by cross angle Integral type ..••• Proport; ona 1 type
BD
LEARtUNG CONTROL SECTION
--0-
)0
The di fficul t,' to perform the learning control strip shape in hot rolling is that only the strip crown and steepness of the product are measurable, and that the roll cross angles of all stands (generally ~ 2) must be determined from only these measured values. To accomplish this, authors developed t"o types of learning control method (p roportional type and integral typ"e ) "hich estir.1ates the interstand strip cro"ns Chj (j = 1, 2, "-1) so that minimi:e the follo"ing function Rand "hich using these estimated strip crowns calculates correlation values as roll cro"n.
"-1
"
R = ~ =l gj (Chj - Ch5) ]
+
gN(Ch - ch~f+ G('t- E~f (12)
E
.
....... •.•..•........•
Integral type. /Proportional type /.
60
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v
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....................
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1
2
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t4~ber
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8
9
10
of coils
Comparison of the effectiveness of two types of learning control
10
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2
E.x?e:-:':ne:-.t.al Condition Pass schedule
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Pass s chedule
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Pass SChedule <,JJ
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: lenntng control ON
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Fig. 7
Strip Cr own Fluctuation
le arning
by
control
preset t ing o f t he prototype ma th ema t ical mode l onl~'
fo !' thL' f i t',t ,t ri p :lnJ b\' ad op t ing learning control f ro m t he second str i p, FIG,6 s ho~ s the lealTIing control effect in the second pass. Brohen lines in the figure sho~ the desired
\'~llll E'S
of t he
~tr.ip
Pass SChedule
PaH schedule
Pass schedule
..' 1'Ohn .
. ~
C
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2.9' Number of
~
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,
·1
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·2
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p
coi Is
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·4
COI!S
-5
Fig. 8
Strip steepness Fluctuation
by
learning
c o ntrol
Fig. 6
Learning control effect
,\s a resu lt, th e fo lloldn g is ob ta i ned ; b'en if
roll i ll ~
force fluctuates,
stri~
crO~ll
de\'iation are controlled I,ithin :: ~I" from the desir ed va l ue respectively o~ing to learning control .
for desired s tr ip
The same accuracy is assured cro~n
change .
( :' i TEST
\cc o rding to pass schedu l e
:1, ,:] ,0 , the
roll-
ing of firs t 21 s tr ips ~as carried ou t nt learing cont r o l off and th e r o ll ing of s ucce s si\'e 21 strip I,'as at l ea rnin g con tro l on. ,\t control on, t o correspo nd to th e ac tual mac hine, only the s tr ip c ro~n and s teepn ess of the s tr ip aft e r ~- pass rolling ~ e r e used as feed hack \'a l ues . i, IC. ~ ,ho,,'s the change of di fferen ce be tl,een m0asured strip c r o~n and desi re d one . Broken l in e s and so lid l i ne s indicate the res ults at learn ing control off and on respectively. ~oreove r, fIC . S sho~s the change of s teepne ss . ~s
tlle
a resul t, the aCCllrac~ ·
fo ll o~ing
is ob t ained : bo th
of s trip crOKn and s teepn ess are
i Olp r O\'ed b\' l earning con trol, Besides, ne\'er t heless pass schedu l e ~a s change d, t he flat s tr ip I, as obtained I,hen l ea rnin g cont rol is turned on, and so t he prospect i s obtained t ha t l ea rnin g contro l is possible using the same correla t ion values even if roll ing cond i t ion is changed .
3.2
FEEDBACK CONTROL
Feedhack control consists of de coup l ing con trol of str ip th ickness and s tri p cro ~n and of shape - me ter f ee dbac k con trol .
For f eedback control, th e optimum comb inat ion must be determined for many variables suc h a s type and arrangenent of detector as ~ell as feedback s\'stem. HoI,e\'e r, it is difficult to c\'aluate such variables in the actual svs tem. l~erefore, the effect of decouplin g control ~as eva lu ated for the s tr ip thickness an d strip cro ~n bv fabricating th e tand em hot rol lin g mi ll simul~tor ~hich enab les dvnamic ca lcul a tion of s trip thi ckness an d s hape by adding d)TI dmic charac teris tic s of e l ec tri ca l, mechanica l, hydraulic and control devices to th e ma the~atical model mentioned in 2. FTG .9 ShOh'S an examp l e of th e thick ness an d s hape control tand em simu l a tor consisting of 6 s tand s . The pr eset co ntrol is omi tt ed bec au se it has been explained previouslv . The s i mul ator has f eedbac k and f ee d fo r~ ard functions as sh o~n in FIG . 9 . The final control element s listed in TABLE 3 are pro\'ided and a\'a i l abl e in e\'er\' combi nation . Us in g this simu lator, the eva l uation abou t the effe c t of decoupling con trol of str i p thicknes s and s trip cro~ n. In the hot ro ll ing mill , flu ctua tion of str ip thickne ss and s trip cro~n is caused by rolling force variation due to skid ~a~ks, etc . Gatemeter t\'pe strip thickne ss control and latera l s tiffness con trol ~h ich control s rol l bending force during rollin g a re pr ovided for s tri p thi c kne ss fluctuation and s trip cro~n res pec ti velv , ho~ever, a control process hhi ch takes into conside ration interaction i s required beca use the s trip thickne ss and s trip cro~n fluctuate simultaneously even if the roll gap and roll bending force a r e controll ed . Fundamen t al charac teri s t ics of strip thi ckne ss fluctuation ~h and st rip cro ~n flu c tuati on ~Ch are expressed bv ~h
=
~p
"'y
( 13 ) (1 .) )
329
Development of PC Mill Shape Control System
AGC : Gale-meltr type Ihitknns control X-ray AGC : X- ,Oily meMor thickness controt
RF
FF APC
AGC : Fetdlorwa,d thickness control ': Roll lap conlrol
SCC : Sutcess .... e Ci f cUll DCe : Decouphnl (onlrol eRC : ero.. n Ittdbac~ conlrol GM Tt\ltkness IAlt
" SR · Mill motor speed conlrol LP loopt' control HC lOO()tf heql:ht cont(ol CM e,o-n meter
PT
Ben()H pressure del I ctO!'
"NO LC SM
Roll bender prtssur e control
: Load ceU Shape meltr
Composition of tandem simulator of thickness and shape control system (feedback control model) Type of final control elements FIG. ll shows example of si mulation charts . Tt is understood from the figure that the degree of Type change is reduced not only in th e strip crown but Hydraulic screw -down or motor screw-down also in the strip steepness under decoupling Low inertia typ.e or conventional type control for all s tand s. (Case 1 in TABLE 4) High response type or conventional type
Fig. 9
final control element Roll gap control device Looper Bender
where, symbol n of each variable means vari ation from the normal condition, and bender mill modulus MB is the bending force required t o change the roll gap 1 mm. In the shape control systeM, the roll gap an d roll bending force are controlled to n h = 0 and nCh = 0 respectively. This is called decoupling con trol between the s trip thicknes s and strip crown. The block diagram is shown in FI G. lO. In order to confirm the effect of this control, simulation was performed for a combination of one kind of pass sc hedule, two kinds of bender control systems and th ree kinds of decoupling control systems are listed j~ TABLE 4. A triangular ,,'aye, \,hich has a temperature difference of 2 sand 20·C, was used as disturbance.
In FIG .1 2 , maximum values are plotted from the calcu l ation result on th e change of strip thickne ss, strip crown, and steepness a t each s tand under the condition of TABLE 4. Consequently, the followings are clearrecr:TAnl.l~ Bar
thick. (mm)
1'1
32,0
15 ,60
Pas! schedule
dP
S iJl1ulation conditions
F3
F2
Rolling Stri p
9,34
F5
F6
speed (mprn)
width
3,01
2.30
1000
1 250
F4
6,26 4,22
Roll gap co ntrol devi c e
Hydraulic screw.down
Bende r
High re sponse type (5 Hl./ -90o) . Conventional type ( 1 HZ/-90o)
Oecoupling cont ro l (0: ON X : OH)
Disturbance Measured value of roiling lotce
1\
D e livery strip thickness (mm) .
(mm)
FI
F2
F3
F4
F5
Case 1
0
0
0
0
0
0
Case 2 Case 3
X
X
X
X
X
X
X
X
X
0
0
0
F6
Tria ngulJ r wave o f 2 sec at 20"C
Measured ... "Iue of bending force
dF
(1 ) Decoupling co ntrol reduces strip crown
variation of the product to approx. 1/3 as compared "'ith that I,hen decoupling control is turned off. Furthe r more, steepness va ri ation of the product is reduced t o 0.5 % or belo" by adjusting the control system. (2) \\hen the high re spon se bender is used, strip
crown variation is reduced to 70 to 80% o f the conventional typ e . I~wever, it makes no great diffe renc e because absolute values diffe r on l v 2 to 3 ~m. ( 3 ) Al though roll bending dS •
Preset v~lue 01 roll gap
Fig. 10
dF· Pa'set value 01 bending force
Block diagram of decoupling control between thickness and strip crown
for~e is operated for s trip cro"n con trol, the effect of decoupling control given strip thickness fluctuation equivalent to onl v gage-meter thickness contro l.
Y. Hayama et al.
330
REFERENCES
ISI~
e~
2st
-::.0
3st
~
Sst
....
------........------.
4st
-------------
_1500
~ 6st
Omori, \akajima, Tsukamoto, Morimoto, Hino and \akaza~a (1083). Precedings of the 33rd Japanese Joint Conference for the Technology of Plasticity 419-436 Hayama, Ozono, Kaj il,ara, Okura, \ishizaki and Iwatani (198 4) Precedings of the 1984 Japanese Spri ng Conference for the Technology of Plasticity, \0. 223
~ 10
(s) - - Oecoupling conlrol turned on lor all stands
/',
Kono and Yamashita \onlinear Programming, JSTU
_,~~~// ___'~,~,~__~~______-_ _ -_-~G~a~g.~-m~.~"~r~ty~p~.~th~"~k~ne~ss~ control only 10r all s(anas
Hayama, Hashimoto, Ozono, Kawanami, Kawasaki and Hamasaki ( 19 78) Mitsubishi Technical Review Vol. 15 \0. 3
1 st I
-
'~'"
~ 2st 1---------""""-==-----""',"",;.-.......:-;-,,......,:::-,-,,-------------------
~ 3 st 1---------------..2:.::::::.....:,::,":i,--. _:::-,.-,,-------------
i
4str-----------~~----~~~------
3
5str---------------------~,'~=;~',~'-----------6 st
r---
SYMBOLS
IIO.O __________________..;,. .------::--:---~''''' =""'-','--------
Strip cro~n on delivery side (mm) Rolling force (t) Bending force per chock (t) Strip cro~n on entry side (mm) Equiva l ent mechanical cro~n by roll cro ss ing (mm) Cc: Constant in equat ion of strip cro~n (mm) kc: Lateral mill modulus (t/mm) Ef: Bender influence coefficient (mm/ t ) ~~. Roll crown correlation coefficient (-) CSmax: ~laximum equivalent mechanical crown by roll crossing (mm) so: Cross angle influence coefficient n Crown heredity coeff i cien t (-) C: Relative difference in elongation for strip on delivery side ( - ) ~o: Relative difference in elongation for strip on entry side (-) c c: Constant in equation of relative difference in e lonra tion C- ) Shape coefficient (-) Strip thickness on delivery si de (mm) Strip thickness on entry s ide (mm) H B Strip width (mm) Reduction ratio (-) Suffix for stand number ( - ) \umber of stands C-) G Cross angle (0) \j! Steepness ( - ) CR : Roll initial crown (mm) a : Learning control coefficient (-) CL: Learning control correlation va lue (mm) Chf: Desired strip crown (mm) lieight factor ( - ) \\~ h', G, g: Ch: ~e s ured str ip crown of the product (mm) 'I:' . ~leasured rclati\·e difference in e l oneation ( - ) Calcul"J\ed s tri.p crOI,n b,· expression (2), USIng Chi -l as entrv strip crown (mm) ["lculat~d relative difference in elongat"i on of t.re product b,· expression (3) using C~\_land Ch In place of Ch\-l and Ch\, respecti\·el,·. (-) ~.1i 11 modulus (t / mm ) Bender mill modulus (t / mm )
10 (s)
§
i lstr_--~------------------------------------
~ ~2str_--------------------------------------- "-~
.... r3str_------------------~,~/~-~~-------------
~ ~4SIr-------------------~==~~------------- :- .& 5 st I---------------------~'=----'":-,.-..:'~--------on ~ 6st _110 ------------------=-=.;,/=-==''''''.:.:...0-------u
10
Fig. 11
4.
Example of simulation charts (high response type roll bender)
CONCLUS ION
Authors developed the PC mill shape control system consisting of preset control, which has roll cross angles as final control elements, and feedback control elements, which has roll benders as final control elements. Rolling by Mitsubishi's test mill was carried out for aluminum strips and the following results are obtained. (1) Cross angle setting method using quadratic
programming is
appliable for actual machine.
(2) By learning control method of strip shape us-
ing the interstand estimated strip cro~n, drift from the desired strip cro~n is kept within 5 m in test mill. (3) Tandem simulator ~as prepared for simultaneous analysis of strip thickness and s trip shape, then effective control was confirmed by evaluation of decoupling control for the strip thickness and strip crown.
80 .
e
~
c
60-
0
D
~ ~
Decoupling control lor all stands
~O
-
20 -
(lIlgh resp ons e bender) () Oecoup hn g control lor all stands (Conventional lIendcr)
• Gage-meler type stflP thickness control only for all stands
~
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~
~ 10 -
,
E
.3
15-
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. 0
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on
., --.
10 -
20 .
,,
...- ...-- ........ fl ~
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't")...-o
5.
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4
Stand
Stand
Fig. 12
Simulation results
~
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/
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/
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