An Application of the Model Based Predictive Control in a Hot Rolling Mill

Copyright © IFAC Automation in Mining. Mineral and Metal Processing. Nancy, France, 2004

ElSEVIER

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AN APPLICAnON OF THE MODEL BASED PREDICTIVE CONTROL IN A HOT ROLLING MILL I. S. Cboi, J. A. Rossiter, P. J. Fleming Automatic Control & Systems Engineering The University ofSheffield Sheffield SI 3JD, UK CopOI [email protected]

Abstract: The conventional PI control scheme has been used widely to control the mass flow and strip tension in a hot strip mill. However, the mutual interaction between the looper angle and strip tension prevents the use of large control gains. Slow and linear actions from the controller may induce severe strip quality defects and instability of the process under abnormal operation conditions. To solve these problems a control scheme based on model based predictive control (MPC) is suggested because of its systematic handing of interactions and constraints. The LQMPC algorithm, which is a relatively conventional MPC design, is implemented for the looper and tension control using the closed loop paradigm (CLP). Copyright © 2004 IFAC Keywords: looper control, tension control, model predictive control, hot rolling mill, LQMPC, closed loop paradigm, constraint handling, mutual interaction

1. INTRODUCTION

i stand

A hot strip mill produces a strip of thickness 0.812mm by rolling slabs produced in a continuous casting line. A rolled strip in a hot rolling mill can be a final product or further rolled through cold rolling mills in order to produce a thinner coil. The quality defects generated in a hot rolling mill seriously affect the quality of the following process. Therefore, quality control in a hot strip mill is very important. Moreover, recently, more and more strict demands of strip quality are being made by the market.

i+l stand

ia

i

!

I !

The main specifications of dimensional quality in hot strip mills are thickness, width, profile and flatness of the strip. The control is performed by dedicated control systems such as AGC (Automatic Gauge Control), AWC (Automatic Width Control) and AFC (Automatic Flatness Control). Also, there is another specification, strip threading, which affects the stable operation of the process by controlling the mass flow of the strip.

Fig. I Conventional PI control Where, ASR: Automatic Speed Regulator, C.C: Current Controller In this scheme there is no tension feedback control loop. The strip tension is maintained to a constant value by adjusting the torque through the current of a looper motor. The reference current in CRCC (Current Reference Calculation Controller) is computed to balance the torque of a looper motor against the load torque at the given angle which depends on the strip tension, strip weight, looper weight and so on.

The tension and looper control system is important because it affects the strip threading as well as dimensional accuracy. The conventional PI (Proportional and Integral) control scheme has been used widely because it doesn't need a tension meter; it is very difficult to install and maintain tension meters at the inter-stand due to it being a hostile environment. Fig.1 shows a conventional PI control scheme.

The mass flow is controlled by the looper height control (LHC) loop where the inter-stand strip length is changed by the angular velocity of a mill motor. The LHC performs PI (Proportional and Integral)

131

control action to get rid of the angle difference between the target and measured looper angle.

E dL L .L dt

E L.L

= - r-dt+- rev

~'o,

-v .)dt -,

E, E =-(L -L0 )+-v-, )dt L L.Lr(v. ~,.,

The main problem with this scheme is that the mutual interaction between the looper angle and strip tension can make the system unstable and so prevents the use of large control gains hence resulting in slow response times. Therefore a large tension disturbance may induce severe quality defects of a strip and instability of the process because of the slow response.

(L'-L) =E L -

where, L' = ~(x' + y') +~(Lo -x)' + y' and L = Lo -

!(v,.,., -v_Jdt i+1

P(x,y)

Up to now, many control strategies which include robust control (Asada., et al., 2003, Imanary, et aI., 1997, Heams et al., 2000), multivariable control (Asano, et al., 2000), optimal control (Okada, et aI., 1998) and so on have been proposed and applied to reduce mutual interaction and reject disturbance quickly. However, most of the controllers proposed output linear control actions even under an abnormal operational condition; due to the presence of constraints, large perturbations usually benefit from non-linear control actions during transients. Moreover, the complexity of parameter tuning makes on-line application difficult.

Lo

Fig. 2. Outline of a looper 2.2 Load Torque on Looper Motor

If the looper has an inertial J L with respect to it's pivot and is driven by a motor torque M, then by applying Newton's second law

In this paper a control scheme based on the model based predictive control (MPC) is suggested to solve these problems because it can handle the mutual interactions and constraints systematically. The tuning in MPC can be done intuitively, which is a great advantage when applied to a real process. The model is derived in section 2, the controller design is discussed in section 3 and conclusions are given in section 4.

J L fJ = M - T T

(2.2)

=- T + T, + T

(2.3)

w

6

T = whal[sin(fJ+ p)-sin(fJ-a)]

(2.4)

6

•

a=sm

2. PROCESS MODEL The Dynamic Simulator Model (POSCO, 1997) based on a hot strip mill of POSCO is used as the process. From this a linear model is constructed for use in the controller design. This section gives the underlying dynamical equations and shows how these can be used to construct a useful model. This is then followed by discussion of a linearisation and how to model disturbance effects.

Yo +/sinfJ+rlp

_I

~(xo + I cos fJ)' + (Yo + I sin fJ+ r)'

. _.

p=sm

y, +/sinfJ+rlp

~(Lo -xo -/cosfJ)' +(Yo +/sinfJ+r)'

T, = gpwhLl cos fJ

(2.5)

Tw =gwlpllpcos(fJ+fJlp)

(2.6)

First, define the following variables. T: load torque, T load torque component by strip tension, T,: load torque component by strip weight, Tw : load torque 6

:

component by looper weight,

rip :

looper roll radius,

a : strip tension, w: strip width, h: strip thickness, g : acceleration of gravity, p :strip density, L : inter-stand strip length, wlp : looper weight, M : torque of a looper motor. 2.1 Inter-stand Strip Tension Model

Fig. 3. Outline of a looper arm Inter-stand strip tension is defined by eqn.(2.1) and fig. 2. E dL a=L !{dt"+(v... io , -v_J}dt

(2.1)

132

Fig. 4. Block diagram of the linearised looper and tension model 2.3 State Space Representation of Looper-tension Model To design a linear MPC controller the non-linear model of the Dynamic Simulator is linearised about an operating point of 12.9[ N / mm'] in tension and O.3S[rad] in looper angle. The block diagram of linearised model is shown in fig. 4. By converting it to the state space form the following model is constructed. j = A;x+B;u+ F;d (2.7) y=C;X+D;u where, x = [wl ( j e W. x, Xc X, x.]' -~

0

J,

_EK_

EK L ,

Lg,

0

L

I

0

g,

A;=

_&

J,

t;'P, J,

0

_ E(1 + f) 0

0

L

0

0

0

0

0

0

K L' ' K Ga • K .a , K Ta and K TB in fig.4 represent linearised gains on an equilibrium point. For example, K L' represents loop length variation by the looper angle variation. This relationship is shown in fig.S and linearised by a Taylor expansion. K

0

= dL "" LI

de

I

56&0

I

0

S560

0

S540

0

J. t;ll',+K",K,.. 0

0 0 0

-K. 0

0

0

0

0

K_K;'.

L,

0

0

0

-K_

0

1.,

0

5520

0

1.,

0

0

0

o

0

0

0

I

I

I

- -:- -

~I

I

: :

I

I

I

I

I

(2.8)

5510

I

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-

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~

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I

- -

I

~-

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1

:

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1

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-

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/1- -,- -,. --

: :

:

1- - -1- - -, - - T - -

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_!.... __ '

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0

0

1

- -1--,- -,. - - r -

Kt;;. J.R.

K ..

I.-._

1__

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..! __ ..!. __

" I

'""'1..':-,--,.o":'.'=-.='=.3:---:.'.:•----::.'=.•--.='=.•--=' •. T=---,JO.'=-.--'OL.•-----' _1'""1

Fig.S Loop length vs. looper angle 0 E

K,.,K,., L,

K,.,K.,

0

K .. 0

2.4 Disturbances

L

F;=

0 K ..

0

2- !..e.

-K~

0

J.R.

B:=

0

_R,+K,.,

0 0 0

0 0

0

J.R.

0

1., -K,.,K~

_ K ... Kt;;.

I

__ 1__ -l __ .J. _ _

I

_ Kc.K

Be

__ 1__ .J __ .1 __ L __ L __ ' __ .J __ .i I I I r I I I I

I

0

L + BT(e)

Where, L represents the loop length at the chosen equilibrium state.

0

0

.0; =0

C' = [0 I 0 0 0 0 0 0] < 001 0 0 0 0 0

There are several disturbances which generate tension fluctuation and induce an abnormal operating condition. The main disturbance comes from the AGe actions. AGC systems are there to get higher gauge quality by rejecting thickness disturbance due to skid marks, roll eccentricity and so on. While the hydraulic screw down system makes it possible to give quicker response to the AGC, it creates a

0 0 0 0 0

133

disturbance to the tension control system because of the mass flow change caused by roll gap movement.

summary of the key steps completeness.

Some disturbances come from the mass flow unbalance by impact drop of mill motors and set-up mismatch. Normally this kind of disturbance affects thickness and width before the beginning of the looper control. However, these disturbances can affect looper and tension control performance by giving a constant initial disturbance, and indirectly by AGC action. For example, the tension deviation from set-up mismatch affects strip thickness, and the resulting AGC action provides looper and tension control system with a disturbance.

In order to design the MPC controller, the continuous plant of eqn.(2.7) is converted to the following equation by discretisation.

IS

given next for

x(k + I) = Ax(k) + Bu(k) + Fd(k) y(k) = Cx(k)+Du(k)

(3.1)

The LQMPC control law predictions are given by

= -Koptx + c U = -KOP1x

Another disturbance occurs for downstream loopers at coiling (Imanari, et al. 1997). When the lead end of a strip is coiled, a large tension between the last stand and a down coiler is often caused. It also causes tension fluctuations at the finishing mill inter-stand. AlI these disturbances severely influence the strip tension and looper angle control performance, thus affecting strip thickness and width.

Uj

j

j

j

i < ne

j

Where, K...

(3.2)

i ~ ne

represents the optimal stabilizing

feedback gain without constraints. Vector c contains control perturbations to ensure constraint satisfaction during transients. Assuming the constraints can be defined, for all i, as CUj-d~O, Cmaxxj-d max ~O (3.3) then, with a conventional 2-norm performance index, the constrained optimisation can be defined as

3. MPC CONTROLLER DESIGN

min J. = rXT"i.,QX••,., +UTloiRu•• i

Ct.j.I.O.I.l~..

i_O

s.t. constraints (3.3)

A MPC controller is designed to achieve optimal performance during disturbance rejection and reference tracking in looper and tension control. Integral action is incorporated to allow both offset free tracking and disturbance observation. A major difficultly in this application is the constraints on inputs and outputs and also the desire for good a priori stability assurances. All these specifications can be included in a, by now relatively conventional, MPC design such as the LQMPC algorithm (Scokaert et al 1998) which here is implemented to looper and tension control using the CLP (Closed Loop paradigm) (Rossiter, 2003).

(3.4)

Substituting in from (3.2) and using Lyapunov equations to sum the cost to infinity and hence define a suitable matrix P, one can easily show l that the cost function J takes the form T

T

T

J=c Sec+2c Sax; --..

---t

C --+

---t

=[CO,,··,Cn

]

C

(3.5) Where, Se= HeTdiag(Q)He +H",Tdiag(R)H", + H e/ PH<2 T Sa = HeTdiag(Q)P" +H",Tdiag(R)P".. + H<2 PP"2

3.1 LQMPC Controller Design

B

LQMPC chooses an optimal state feedback (or control law) K as that arising from an unconstrained optimal control problem (for conventional MPC this implies infinite horizons). Where this control law is not predicted to cause constraint violations, it is used, and therefore, at least for the nominal case, optimal performance should ensue.

H e-

0

o ... ···1' Bo ... [
I

=

H

'"

At times the unconstrained control will give predictions that are expected to violate constraints and hence such a law is 'infeasible'. At these times, the first ne predicted control moves are modified to optimise predicted performance subject to constraint satisfaction. It can be shown (Scokaert et al 1998) that jf ne is large enough, then this algorithm will find the optimal for the constrained infinite dimensional optimisation. For a wisely chosen K, this value of ne may not be too large but such issues are beyond the remit of this paper (Rossiter, 2003) and here we assume ne is large enough. A brief


B

-KB

0 I

[ -~
oo

I

P

···1 '

=

1

[-K

P = -K
and
For this paper the following values were used to define the A,B,C matrices from (2.7).

I

134

Typically So=O.

L=5505 [mm] , J. = 2.1

[kg m'] , f=0.0703,

: • • • • • I.·••· • ·.;>
R. =0.09141 [n], L. =0.00122 [H], (!ft. =20.0

[Nm/ A] , R, =0.0610 [n] , L, = 0.00122 [H] ,

J , =1.91 [kg m'], (!ft, =3.326 [Nm/ A], g,=IO. The weights in (3.4) were taken as Q = CC , R = I . Control law (3.2-3.4) was implemented with n, =5

j

i •

:

.

PI

:

l

L .. L

...•...... )

_ /-_

:

~._

~

, 5

__ ..

_

_.•. '

·..·:

25

Fig. 8 Control law ofa main motor These simulation results show that MPC can achieve offset free tracking under the constant disturbance owing to the integral action. Moreover, it represents better control performance compared with the conventional PI control scheme. Next (figs. 9,10) a simulation is performed to investigate the effect of increased control gains to improve control performance in a conventional PI control. The result shows that control performance is initially faster for disturbance rejection with increasing control gains but the closed-loop system is becoming unstable because of interaction.

:

.

6 5

1

. . . . ·.. t

·l

.

..........l .2L-_ _L-_---:~--:!------:;';---~

. ·········-t··

2.5

I----!-

Im.(MC)

·i. · .

j ···········r·············;········__···:············· ....... __ .. ~ _ ~ _.. _.~_ _ . .................... -.----

'.5

,~,

)

I!!

With the MPC controller designed above, simulation was performed for a two inputs and two outputs looper and tension control process. Fig. 6 and Fig.7 show the performance of the tension and looper angle control.

·1

6

'''/1''

--···--I--·~··_···········~············_~·············

: '::::'.:::::::.:l::::::::J:::::::::::::L::::: ;.

3.3 Simulation Results

.:::::::::::]t:-::::::::::L:::::::::t:::l~ ~~i_

_._-_ •• ~.

B

t

under the following scenario: The simulation was performed for the no.6 and no.7 stand in rolling mills with step disturbance of strip speed 15 [mm/sec] at I sec which is generated from 0.3 [%] changes in thickness. In the simulation, it is assumed that the strip is 1240 [mm] in width, 2.53 [mm] in target thickness, 12.9 [N / mm'] in tension reference and 20 [degree] in looper angle reference.

.... ~ .. ,

'2 10

.,.

. _-- ------- .--- .. -.. --- .

i

l

-.!

-.~

,

,

.

- . ..

Fig. 6 Tension control 0.5

···········_(······__

o _._.. -

···t..··__ ·······~-- ..

~ional P1

~ .. ••• .. , 1..·..·..·..

<]5

h

2.5

.;< ;

Fig. 9 Tension control in conventional control

:i • • • • • • jt,•• • I·•• ··.·iZI·•• ••• • •· :

\;

;,'

0.5

;

:~ . .·.·.'.\\;J;~;::r • • ••I• •·••·•. 15

2

25

<].5

:

f

f

·1

·'.5

3 ·25

Fig.7. Looper angle control

Fig.10. Looper angle control in conventional control

4. CONCLUSION Firstly, this paper derives a simple model for the looper and tension control problem in a hot strip mill

135

which can be used for investigations of different control strategies. Secondly, a LQMPC controller is designed for the looper and tension control, and its perfonnance is compared with a conventional control scheme by simulations. It is demonstrated that the designed MPC can achieve offset free tracking under the constant disturbance owing to the integral action. Moreover, it has better control performance than the conventional PI control due to its systematic handling of interaction. Future work will consider the analysis of robustness for the model uncertainty and disturbance with the designed MPC controller.

REFERENCES Imanari H., Morimatsu Y., Sekiguchi K., Ezure H., Matuoka R., Tokuda A., and Otobe H.(I997),

"Looper H-Infinity control for hot strip mill". IEEE Transactions on industry applications ", Vol.33, No.3. Asano K., Yarnamoto K., Kawase T. and Nomura N.(2000), "Hot strip mill tension-Iooper control

based on decentralization and coordination ", Control Engineering Practice 8, 337-344. Okada M., Murayarna K., Urano A., Iwasaki Y., Kawano A. and Shiomi H.(l998), "Optimal

control system for hot strip finishing mill", Control Engineering Practice 6, 1029-1034. Asada H., Kitamura A., Nishino S. and Konishi M.(2003), "Adaptive and robust Control Method

with estimation of rolling characteristics for looper angle control at hot strip mill", ISH International, VoIA3, No.3, 358-365. Schuurmans 1. and Jones T.(2002), "Control ofmass flow in a hot strip mill using model predictive control", Proceedings of the 2002 IEEE International Conference on Control Applicaions, September, Glasgow. POSCO(1997), "Development ofDynamic Simulator on Hot Finishing Mills ". POSCO Technical Report Rossiter J.A. (2003), "Model-Based Predictive Control. A Practical Approach H, CRC Press Mayne, D.Q., J.B. Rawlings, C.V. Rao and P.O.M. Scokaert (2000), Constrained model predictive control: stability and optimality, Automatica, 36, pp789-814 Scokaert, P.O.M. and J. B. Rawlings (1998), Constrained linear quadratic regulation, IEEE Trans AC, 43, 8, pp I 163- I 168. Hearns G. and Grimble MJ.(2000), "Inferential Control for rolling mills ". lEE Proc.-Control Theory Appl., Vol. 147, N06, 673-679, November.

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