On The Possibility of Looperless Rolling on Hot Rolling Process

\...opyngnt ... ll'AC Automation in Mining, Mineral and Metal Processing, Tokyo, Japan, 2001

ON THE possmn.ITY OF LOOPERLESS ROLLING ON HOT ROLLING PROCESS Hideo Katori, Ryu Hir~yama Tabtsugo Ueyama, Katsuhisa Furuta

Nippon Steel Corporation, Nippon Steel Corporation, Nittetsu Elex Corporation, ToJcyo Denld University Abstract: We discuss the possibility of looperless control at finishing mill of hot rolling process in this paper. In recent years, the precision of thickness and width of hot coil is becoming more and more strict. So we must develop powerful controller which stabilizes the rolling process even if plant parameters are perturbed. It is cleared that performance of control system with the compensator designed by f1 synthesis considering parameter perturbation is better than that with the compensator designed by H.,.. Copyright 02001 IFAC Keywords: H-infinity control, Industrial control, Robust control, Steel industry 1. INTRODUCTION

Torque arm equation shows the relationship among torque of mill motor QLJ torque arm coefficient a, rolling force P, forward total tension T f and back total tension Tb with the coefficient a, b, c. If the equation holds and QL, Tb, a, b, c is known, T f can be estimated, however, the precision of the Tf is not enough to use to control because coefficient a is difficult to identify and Tb is difficult to know. Therefore, looperless rolling at finishing mill has not been thought possible. The load cells installed on loopers enable us to estimate the interstand tension of strip precisely, because they measure the weight. So interstand tension can now be used as a control input. Fmthermore, high power AC motors which utilize widely in roll drive systems and help us to improve the response of it in these days, almost give us an impression of possibility of controlling interstand mass flow of the strip by roll speed control only. We discuss the possibility of looperless control at finishing mill of hot rolling process in this paper. The object of control is limited to 2 stands finishing mill for simplicity. The following notations are used, i denotes stand's number. Pi RoUingForce [kgf] [mm] Hi Entry Thickness h; Delivery Thickness [mm] [mm] Bi Entry Width [mm] bi Delivery Width [mm] Si Roll Position [-] f; Forward Slip [kgf/mm2] (J i Unit Tension [mmlscc] VRi Roll Speed [mmlsec] Vi Entry Speed [mmlsec] Vi Delivery Speed

Hot rolling process in steelworks plays a role which rolls slabs casted at continuous casting in order to produce hot sheet coils. Some of the hot sheet coils are shipped after rolled at cold rolling process in order to make cold sheet coils and the others are shipped as they are. Drastic progress in control technology on rolling process is expected, because the requirement of the precision of thickness and width of both cold coils and hot coils is becoming more and more strict in recent years. As the precision of thickness and width of hot coil depends on the performance of control systems on finishing mill of hot rolling process, many researches and developments had been carried out to construct AGC(Automatic Gauge Control), AWC(Automatic Width Control), and sophisticated looper control systems.

Especially on looper control, there are a lot of applications where latest control theories had been applied, however, the research and development on looperless control, which will enable us to install latest sensors between stands and will give us possibility of improving the precision of thickness or width. But it has not been done enough. Not more than 10 papers can be found. Those papers had concentrated on developing a method to estimate the interstand tension of the strip by torque arm equation(l) because we did not have any means to measure the interstand tension of the strip at finishing mill of hot rolling process.

QL

= aP+bTb +cTf

(1)

119

EMi

Deformation Resistance [kgf/mm2] Plasticity [kgf/mm] Friction Coeff. [-] Roll Position Ref. [mm] Interstand Distance [mm] Young's Modulus [kgf/mm2] Time Constant of Reduction [lIsec] Roll Eccentricity [mm] Temperature of Strip [degree C] Thickness of Slab [mm] Thickness of Rough Bar [mm] Minimum Skid Distance [mm] Diameter ofNo.i's Back Up Roll[mm] Moderating Ratio [-] G02 at Motor Axis [kgfom2] Propotional Gain [-] Current to Torque Coeff. [kgfm/A] Motor Coeff. [lIsec] Speed Ref. at Neutral Point [mm/s] Speed at Neutral Point [mm/s] Motor Current [A]

V oci

~Vnci

~

Qi Ili SRef

Li Ei TA; Sdi Tdi HSlab

HRbar LSlrid

D BURi a; GD2mi TCi

KMi

Kli V nRefi

Vni

= ~v,.Refi -

~v,.i

2-2. Requirements In order to maintain the stable operation of rolling, stability of mass flow control of interstands must be assured, however, the stability of mass flow control depends on the absolute temperature and the deviation of it in length direction of hot strips because the deformation resistance varies depending on the temperature of strips. So we allow the case where a strip has unusual deviation on temperature or a slab is extracted from the furnace at low temperature in our controller design, we must aware that deformation resistance may change 0.1-- 10 times that of standard value and that we must control all the cases with one compensator. 3. STATE SPACE EQUATION ON FINISHING MILL Many researchers have made efforts to construct the mathematical models of hot rolling in finishing mill. In this section, 1) the outline of the mathematical models, 2) the concept or influential coefficients and the way to calculate them, 3) the way to construct the matrix of linear state space equation based on the influential coefficients. We assume that rolling phenomena are still quicker than those of rolling actuators and that the dynamics of rolling phenomena are negligible in comparison with those of rolling actuators. Thus we have only to consider dynamics of 1) roll position system, 2) roll drive system and 3) tension generating mechanism.

[mm/s]

Motor Torque [kgfom] Rolling Torque [kgfom] Radius of Motor Axis [mm] gravity 9.80665 [m/sec2]

2. CONSTRAINTS AND REQUIREMENTS TO THE CONTROL SYSTEMS

2-1. Constraints

3-1. Mathematical model on finishing mill

We focus on the last 2 stands (F6.F7) of 7 stands finishing mill for shown in Fig. 1. In order to carry out the control system design on the real situation, we assume the latest equipments of rolling mill and entry thickness ~, H 7) and width (B6, B 7), delivery thickness
~

Rolling force Pi, delivery thickness 11;, forward slip fj, delivery width b i, delivery speed Vi, entry speed Vi

are fundamental variables of rolling and there are fundamental equations to calculate them in equation(2)--(7), however, we omit the details of them because the dynamics of rolling phenomena can be negligible. On the other hand, the details of fundamental equations on 1) roll position system, 2) roll drive system and 3) tension generating mechanism are written in equation (8), (9) and (10) respectively. I'; =1';(Hj,hj,Bp (Jp(Jj_pJ1.j,KJ (2)

Roll Speed Con-action ___ "'Vreftj

hj =hj(I';,Mj,Sj)

(3)

J; =J;(Hphp(Jj,(Jj+l) bj =bj(Bj,(Jj+I,T;) Vj

=

Vj

(J; , VRi )

(4)

(5)

(6)

V; = V;(vphj.,Hpbj,BJ (7) Fig. 1 Rolling process: observable variables and control inputs

dS 11 1- ' =--S. +-S. +-S (8) dt TAj ' TAj , TAj ,

120

dG .

<2}RolI EccentricitY (12.D-32.0 [r.dlHC])

E (V-v )

- ' =-L

dt

L.,

i

(10)

i-I

~

. aoomm",,1 Cent8r W::-Cont.r of M. ..

!~l a;. ~

3-2. Linearization offundamental equations

As the fundamental equations are non-linear equation, we construct linear state equations based on the influential coefficients, which are calculated as inclinations of graph of a standard point shown in Fig.2. ~ means the deviation of standard value. As for rolling force Pi, AP i is rewritten as

ap;

ap;

ap;

iJh;

aCT;'

E


(3)Thickne.. Deviation For Next Sand

(D.II-32.0[rad/.ec])

ap

Fig.4 Image of disturbances

M';=-Mi+-dCT. +--dCT . • + - ' Mf aCT;+1

,+

aH;

,

3-5. Disturbances on thickness and width

+ ap; M + ap; MJ (11) aK; , aB; ,

The image of disturbances on thickness and width is in Fig.4. (1) Skid Mark ~ Tdi [kgf/mm]: Skid mark is the deviation of temperature of bars caused by skids, which sustain slabs in reheating furnace. The temperature of the portion of slab to contact with skids is lower than the other portion. Frequency of skid mark W Skid can be calculated

3-3. Roll drive system Though roll drive system has many dynamics, the fewer is the better in analysis. Thus, we tried to reduce the dynamics as long as the system is 1) a servo system on roll speed and 2) step torque disturbance is rejectable. Therefore, we decided to approximate 2nd order system shown in Fig.3. In Fig.3, dot· means derivative of time and KIi can be calculated as

K _ 4g~ li -

w..,.

and T Ci, ~ are decided depending on the response of drive system.

~Vnci

=~VIlRefi -

~VIIi

(13)

3-4.State space equations on finishing mill State space equations on finishing mill is written as (14) ..... (16). xp = Apxp + Bplwp + B p2 u p (14) zp Yp

=

Cplx p

+ Dpuwp + D

pl2 U p

=l H_

H, x21r L_H_H,

(17)

and is about 0.6"'" 1. 3 [radls]. The frequency of Skid Mark is the same at each stand, which is derived from the mass flow constant law. (2) Roll Eccentricity ~ Sdi [mm]: Roll eccentricity is caused by the difference between center of mass and the geometrical center of back up roll. If the difference exists, back up rolls move up and down and when they go around causing thickness deviation. Roll eccentricity has 1st ..... 4th order harmonic. The fundamental frequency of Roll eccentricity w Ecc[radlS] can be calculated.

() 12

a;GD:"

~

R.h••tin& Furn.ce

V TrDBURi

wEcc =--'-x21r (18)

(15)

and the fundamental frequency is about 12.0 ..... 32.0[radls]. (3) Entry thickness and width deviation ~ ~ [mm], ~ Bi [mm]: The thickness or width deviations at this rolling stand affect as disturbance on thickness or width at next stand respectively.

=Cp2 x p + Dp21WP + D p22 U p (16)

Rolling

Force Pi

3-6 . Augmented system containing dynamics of distUIbances

Tension a i Fig.2 Concept of influential coefficient

If the fundamental frequency of F6 Roll eccentricity is 14[radls], then we decide the frequency weight function Wod6(S) as

s+1 s+14

WSd6 (S)=--

121

(19)

to reduce all the effect of the Roll eccentricity L\ Sdi whose frequency is more than 14[rad/s]. If we decide the frequency weight function on specific frequency, the order of weight function becomes high (4th order), however, even slight change of the frequency of Roll Eccentricity the control system becomes less effective. Considering the dynamics of Skid Mark and Entry Thickness and Width deviation also, we construct state space equation on disturbances as follows, :.tw = A..,xw + Bwuw (20)

W= Cwxw + Dwuw

(21)

Bol

H.

(22)

Ws =diag[k,,6

w

Bt

]

(26)

4-3. Meaning of

DpllC..,j (29)

k.,](41)

(j 6

in Za

Controlled Variables : Za h, a6 ~

case

x x x

1 2 3 4 5

for simplicity of design. The reason why we do not use F7 roll speed (VDRf7) as control input is that we must operate F7 roll speed Vn7 for another use. Tablel shows examples of variable selection.

0 0

Stability of Operaton

x

0

0

0

x

x

x x

0 0 0

0 0 0

0

f-·-··-Wcj"-·-- --···-·-·-·-·-··-·-··-·---·-·--·-i w i

Tablel Variables(Ll is omitted>

~

klo6

Table2 Stability of operation on ~

We consider that only F6-F7 interstand unit tension

yp

20 (39) s+1

performance we get, however, as for Za, the way to select variables is not obvious. Therefore, we searched the indispensable variables on Za from the view of stability.

exists and that the other unit tensions are O[kgf/mm]

Zp

Tdi

As fur as y. is concerned, it goes without saying that the more variables we select, the better control

4-1. Candidates ofvariables

wl!

w: . =

WT = diag(k.lnf6 k.1nf7 kYJv.iS kY-i6](42) If we set the value of KVmef6 bigger, we get the compensator which mainly operate roll gap ~f6> Sref7 and rarely operate roll speed V mef6. On the other hand, if we set the value of Kvnreffi smaller, we get it which operate both roll position ~, ~f7 and roll speed V mef6 •

(23)

4. CONTROL SYSTEM DESIGN NOT REGARDING THE PARAMETER PUTERBATION

State Variables Disturbances Controlled Variables Observable Variables Control Inputs

'

0.5(8+ 0.1) (40) s+1 where, WS,WT, are as follows.

Ca2 = [Cp2 D p21 CJ (30) Doll =lDpII D w j(31), Dol2 =Dpl2 (32) Do21 = [D p2 PJ(33), Do22 = Dp22 (34)

Xp

s+1

W, . =

=[B~~w ] (27), Bo2 = [B~2] (28) Cal = lCpI

0.02(s+0.1) (36) w: (S) = 0.02(s + 0.1) (37) s+17 ' SJ1 s+21

(S)

W, . kHi (0.1s + 1) (38)

za = Calx a + Dall wa + D al2 ua (24) Y a = Ca2 x a + Da21 wa + D a22 u a (25)

=[~

~d6(S) ~d'(S) WB6 (S) WB7 (S)](35)

w:

8.

augmented system is written as :.ta = Aaxa + Bal wa + Ba2 ua

A.

In order to design compensator with the help ofH.. theory, we make augmented system. In addition to model of rolling process, model of distmbances Wd(S), constant gain matrices W., T. are added to construct augmented system shown in Fig.5. Wd(S) is constructed as shown in equation(35)-(40) ~(s) = diag[WSd6 (s) WSd,(s) WH6 (S) WH7 (S)

SJ6

where Xw is state of disturbance and 1lw is input of it, are matrices Aw, Bw, Cw, Dw of state equation. If we combine x" and Xw as follows,

Xo=[::]

4-2. Augmented system

i

I

S6,S7,a 6,Vn6,Vn7,Vc6,Vc7 Sd6,Sd7.H6,H7,T d6,Td7,B6,B7

u : i

h7,~,a6

L.____.____ ._~.!!!!!!Jg!'.!!.!'l~!!! i

~,H7,h7,B~7'~' a6'p6'p7 8m6,Std7,Vmef6

Fig.5 Augmented system

122

y

.......-----------------------------.-----------------------.....

As shown in Table2, we found that a 6 is indispensable variable for the stable operation. In another word, keeping unit tension stable is necessary for stable rolling.

z

5. CONTROL SYSTEM DESIGN CONSIDERING PARAMETER PERTURBATION

u

~-~~y

i

._._._.. _ .. _ .._ .. _ .. _____G.'!.I)~!!'!i.~. .!!!£.I!!!~j

5-1. Quantifying parameter perturbation Elements of Ap, Bp), B p2, Cp2, Dp21 are pertmbed as the strip's temperature changes. The amount of maximum deviation of elements of Ap,Bpl,Bp2,Cp2,Dp21 as influence coefficients

apJa~,apJa~

change 0.1

......

1 0 0 0 0 1 0 0 D12 = 0 0 0 0 0 0 0 0 0 0

10 times the

standard value is shown in percent range. (3,1) element of Ap, for example, change 514% at the maximum. (1,1) element of Ap change 0% at the maximum, which means no pertmbation. Therefore, state space equation on rolling process is rewritten as xp =(Ap+Mp~p+(Bp, +Mpl)w+Bp2Up (43)

Yp

Fig.6 Block diagram considering pertmbation

= (Cp +~Cp}tp + (Dp21 +mp21 )w

[

8,

= diag [8.

~2 I~AP ~ ][C Q>2 B2X D 12x D

Apt ~ 1, 11~Q>211_ ~ 1

hold,

I'l.'ql'l.t $1)1.1J1_ =idiag[l'l...,

I'l.c"

22X

A

D12

2

]

=[~Ap

~Cp2

~Bpl

1[1' C,

~

C. I

0

=

B.I D12 D.II

B., 0 D.I ,

C., D'I D.'I

D.Z2

1

(50)

D-K iteration is carried out by checking the J1 plot. In D-K iteration, we kept in mind that 1) D-K iteration must be at most once or twice, 2) constant gain scale matrix must be used. 6. POSSmILITY OF LOOPERLESS ROLLING

6-1 . Simulation condition In order to check the appropriateness and effectiveness of the compensator designed above

thus,

with

holds.

Furthermore, if we define 0 0

0 0

0 0

1/0 I/o, I/o, 1

B, =

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

C;=

1 0 0 1 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

]

~Dp21

5-3. D-K iteration

(46 )

I'l.t $1

0 0 0 0 0 0

In this way, we obtain augmented system shown in equation (50).

Consequently,

II~

0 0 0

0 0 0 0 0 0

hold. Thus we can construct Fig.7

(44)

8) ] (4S)

8s]

D2I

D21

Though parameter perturbation can be quantified, we cannot take all perturbation into consideration because of complexity. We focus on the main pertmbation of Ap and Cp2. Thus we define normalized perturbations 8 i (-1 ~ 8 i ~ 1) and ~ A> ~ Cp2 as follows, 81 (3,1) element of Ap 82 (3,2) element of Ap 83 (3,3) element of Ap 84 (1,1) element ofCp2 85 (2,2) element ofCp2 = diag [8 I

0 0 0 0 0 0

I/o, I/o,

000 0 0 0 0 = 0 o 0 0 o 0 000 000

o o

then we can notice

5-2. Control system design considering perturbation

I'l..., I'l. c"

0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

u :

y

,i

L________ Gen'tf.!.lized P!~ Fig.7 Augmented system regarding perturbation

123

method, simulation is carried out. Fig.8 shows the relation among variables and disturbances. a) Roll Eccentricity d Sdi Roll Eccentricity consists of 1st - 4th order harmonic wave whose fundamental frequency is about 13[radlsec] and amplitude of 1st-4th order harmonic wave are about 20[ J.Un ],

lih7 lib7

Fig.8 The relation of variables

13 .2[ J.Un ],6.8[ J.Un ], 4[ J.Un ] respectively.

mill of hot rolling process is shown. It is cleared that performance of control system with the compensator designed by JJ. synthesis considering parameter perturbation is better than that with the compensator designed by H... In order to realize the looperlessrolling, we must consider the effect of noise of measured variable and the way to reduce the order of compensator.

b) Skid Mark d Tdi Skid Mark is a sine wave whose frequency is 1[rad/sec] and whose amplitude is 20 re] c) Entry Thickness Deviation d ~ d~ is a sine wave whose frequency is 1[rad/sec] and the same phase with d T d6, and whose amplitude is 120 [J.1m] . d H7 is set to d ~

calculated in simulation. d) Entry Width Deviation d B j dB6 is a sine wave whose frequency is l[radlsec] and whose phase is delayed 7r [rad] with Il T d6 , and whose amplitude is 0.5[mm]. d B7 is set to d b6 calculated in simulation.

REFERENCES A1dyoshi Ohishi et aI (1983). Looperless rolling system of finishing mill on Hot Rolling Process Hitachi Technical Report, Vol.65, No.2, p.141p.I46, A. PACKARD and J. DOYLE (1993). The Complex Structured Singular Value, Automatica, Vo1.29, No.l,71-109 Ichiro Imai et al. (1980) , Looperless rolling system of finishing mill on Hot Rolling Process, Iron and Steel, Vo1.66, No.4, p.301, Morio Saitoh et al. (1985). Development of new tension control system of finishing mill on Hot Rolling Process, NKK Technical Report, No. 107, p.12-20 Sunao Tanimoto et al. (1993). New tension control system of finishing mill on Hot Rolling Process, lEE steel industrial report, VoI.MID-93, No.I-5, p.13-20 Yoshitaka Hayashi et al. (1986). Development of new tension control system of finishing mill on Hot Rolling Process, Iron and Steel, Vol. 72, No.2, p.57 -60

6-2. Comparison Table3 shows the performance of control systems on 2 compensator types(H.,. or J1 synthesis) applied to 3 types ofplants(N:nominal, L:defonn resistance is 10 times that of nominal, S: deformation resistance is 0.1 times). Compensators designed by H.. or J1 synthesis (though constant gain scaling) can control the rolling process properly, however, performance of control system (deviation of h7 or a 6) designed by J1 synthesis is better than that designed by H... 7. CONCLUSION The possibility of looperless rolling in finishing

Table3 Deviation of variables (peak to peak) Comoensator Plant type b,; rmml h7 [mm] b 6 [mm] ~[mm]

a 6[kgf7mm2] P 6 [t] P 7 [t] SdJJ.m] Sz [JJ. m] V06 [mm/sec]

Nominal 62 35 1 1 0.36 8 14 70 300 300

Heo Large 110

90 1 1 0.12 6.2 4.5 24 45 60

u Small 180 120 1 1 0.44 16 300 200 500 400

124

Nominal 40 22 1 1 0.16 16 18 300 400 260

Large 100 70 1 1 0.2 740 700 140 130 140

Small 45 20 1 1 0.2 260 300 500 600 200