Observer-based Multivariable Control of Rolling Mills

Copyright © IFAC Automation in Mining, Mineral and Metal Processing, Cologne, Germany, 1998

Observer-based Multivariable Control of Rolling Mills I. Hoshino* and H. Kimura** *Sumitomo Light Metal, Chitose Minato Nagoya. Japan **Faculty of Engineering. University of Tokyo. Bunkyo-ku Tokyo. Japan

Abstract: The synthesis methodology developed by Kimura (1985) based on the design ~eory of output regulators established basically by Wonham (1974) has been successfully applied to the control system of rolling mills. This method has the following remarkable features. (I) The obtained controller has a physically reasonable structure. (2) The disturbances are properly estimated and dealt with. (3) The synthesis can be calculated by hand. keeping the design procedure tractable. Copyright © 1998IFAC Key Words: Observers, Model-based control, Multivariable control

2. SYNTHESIS MEmOD OF OBSERVER-

1. INTRODUCTION

BASED MULTIV ARIABLE CONTROL Any good control system must appropriately deal with disturbances in the plant environment The design procedure of the synthesis method which positively treat the external signals such as the disturbances has been developed by Kimura (Kimura, 1985) based on the design theory established by Wonham and his co-workers (Wonham. 1974). This method is called feedforward type output regulator and the designed system is the ''Mixture Type" consisting of the feedforward control for detectable external signals and the feedback control realized by using an observer for undetectable external signals (Kimura and Inoue.1978).

2.1 SyntMsis Procedure Consider a plant as follows.

~x = Ax+ Bu+ Ev

(2.1) dt y = Cx + Fv (2.2) Here, x ER". U E R' and y E R- are the state variable. the manipulating variable and the controlled variable. respectively. v ER' is the disturbance. A. B. C. E. F are matrices of the appropriate dimensions. Consider the disturbance of the step type given as v(1)=0 1<0. v(t)=v o t~O (2.3) The control objective is to achieve regulation y(t)~0 (t~oo). (2.4)

In Chapter 2. the design procedure is explained which show how this synthesis method is applied to a general system.

The synthesis procedure is composed of the following three major steps.

In Chapter 3 and Chapter 4. applications of the observer-based method to the flatness control system for a 6-high rolling mill and to the roll eccentricity control via roll balancing force are presented. respectively.

(Step 1) Improve the response characteristics by state feedback. (InuT7Ull CompelUlIIion) (Step 2) Compute the feedforward control input to counterbalance the disturbance. (Feedforward Control)

(Step 3) Implement the feed forward control by feedback control based on the estimation of the disturbance by the observer. (Feedback

Applications to the thickness control system for a tandem cold mill and the tension control system for a tandem hot mill have also been presented (See Hoshino et al .• 1988 and 1996).

Realiz4tion)

241

where

The first step is carried out according to the usual pole-assignment techniques. Based on the state feedback

u=-Kx+u',

~

of x and Vo in equation (2. 11) by respectively, results in

(2.5)

u =-KX - (V.E + \S'F)v.

the closed loop system becomes

!!.. x =(A dt

BK)x + Bu' + Ev,

(2.6)

BK B][Xo] =-[E]v .

(2.7)

B]-I

(2.8)

C If there exists

u~

0

u~

F

°

A =[U1 [C 0 V. is given by u~

=-(V.E + V1'F)v

dt

d

:1:]+[~}

For example, the periodic disturbance such as the roll eccentricity described in chapter 4, which is expressed by the sine function with an CO period, is given by

A,

(2.14)

The observer, that estimates the state variable and the disturbance. is given by

If

(2.15)

:]}

![~]=[~ :I~]+[~}

is observable, the identity observer is obtained as follows.

1~JY- CX(t-t)- Fv(t-t)}.

d[x] [A EIx] [B} J~} dt V = 0 0 v + 0 1 ~ y - ex - Fv) A

=[: :l

2.2 Controller for the System with a Tinu De"'y in the Control Variable In actual cases, there are many systems with time delays in the control variables as follows. yet) =Cx(t - 't) + Fv(t - t) (2.19) Here t denotes the detection time delay. TIle first step and the second step of the design procedure are the same as described in the previous section and the state feedback and the feedforward control are given by equation (2.11).

(2.13)

F{:]

(2.19)

dt ' Here v E R'" is the state variable of the disturbance.

Combining equations (2.1) and (2.12) yields the expanded system as follows.

[~

-4C E-B(V.E+V;F)-~Flx]+[~] . -4 F v 4 y

-v=Av.

In the third step, the state variable x and the disturbance Vo in the above equation are replaced by the estimate of an observer. The differential equation of the disturbance is given by d dt v = 0 , v(O) = Vo' (2.12)

{[C F]

v

Though the step type disturbance given by equation (2.3) is assumed in the previous paragraph, the general equation of disturbances is given by

The control input by state feedback and feedforward is given by (2.11) u =-Kx - (V.E + V;F)vo

y=[c

(2.17)

(2.10)

V;=KU1+\S ·

The existence of the inverse matrix given in equation (2.8) is a necessary and sufficient condition for the existence of structurally stable regulators which achieve the control objective (2.4) for the disturbance (2.3).

:t[:]=[~

v,

Equations (2.17) and (2.18) give the final form of the controller. Equation (2.17) shows that the controller is constructed by the state feedback to guarantee a reasonably high speed of response, and the feed forward control for the disturbance. One of the most significant features of the obtained controller is that the controller parameters are analytically derived from the model parameters. This feature makes it easy to tune the control system for a large variety of small quantity production runs.

where

V.'=KU1+V.

and

(2.18)

(2.9)

O'

x

!!..[x] = [A - BK - ~C

In the second step, compute the additional input to counterbalance the disturbance. To achieve the control objective (2.4), (x(t),u'(t» should converge to ( x o' u;) which satisfies

A-

v0' and (

From equations (2.16) and (2.17),

where u' is the additional input after the state feedback .

[

x and v are the estimates of x and

, 4) denotes the observer gain. The replacement

(2.20)

A

where {y-Cx(t-t)-Fv(t-t)} is the estimation error. The time of the state variable and the disturbance

(2.16)

242

are delayed 't second to correspond with the time of the control variable. e - t I is the time ( 't ) delay operator. From the assumption dv / dt = 0 , = e - t I V . Then

detection time delay is examined in the flatness control.

v

:t [~] =[~ ~I~]+[~} 1~]{y-ce-u x-Fv}.

The object plant is a 6-high rolling mill with an intermediate roll (IMR). A flatness sensor is on the exit side of the mill and the distance between the mill stand and the sensor produces a detection time delay. Figure 3.1 shows an example of the flatness sensor output, that is, the strip elongation strain distribution calculated from the measured strip tension distribution. The aim of flatness control systems is to get a uniform distribution of the strip elongation strain. Here, the I-unit denotes a unit of strain.

(2 .21)

Substituting equation (2.17) in equation (2.21) yields

!!.-[x] = [A - BK - I;Cedt

v

tI

-~Ce-u

E-

B( ~1: + V2'F) -~F

I; FIx] + [I;] y. (2.22) v

~

Equations (2.17) and (2.22) give the final forms of the controller.

Figure 3.2 shows the flatness control actuators of the mill. The work roll (WR) bending force and IMR bending force are manipulated during rolling. To numerically express flatness, strain components ( A) are defined as I(x) = A.J::(x) + A 2 P,(x) .. + A,P,(x)+ " (3.1) I(x) : the strip elongation strain distribution in the width direction x : the normalized position in the width direction P,(x): Lagendre orthogonal function series.

The error equation of the observer is given by d

dt

[x-x] v- v = A - I;Ce-

u

E-

I;FI~_

- x].

(2.23) v-v In the case of the flatness control described in chapter 3, the initial value of the state variable is known. So, the corresponding observer gain can be set to 0, and the relation E =0 is obtained. From the relation =0 and E = 0 , the characteristic equation (C.E.) of the system is given by C.E. =det(sl- A)det(sl + ~F). (2.24) This shows that the C.E. does not involve the time delay. so high response characteristics and sufficient stability can be expected.

[

-~Ce-u

-~F

I;

The state space equations of the plant are given as follows. d (3.2) -x=Ax+Bu dt (3.3) yet) =Cx(t - 't) + Fv(t - 't) where,

x =(M.. ,AF_f , 3. FLATNESS CONTROL OF A 6-IUGH COLD

u=(M:,M::.t v =(d.. ,daa,tu'r

y=(A,2,A,.f, A'2 : 2nd order strain component variation

MILL

A,. : 4th order strain component variation

3.1 MatMmaJieal MOtkl oflM Pllmt In this chapter. the application of the observer-based method to the flatness control system is presented. The notable difference from the application to the thickness control (Hoshino et. al., 1988) is that the

M .. : WR-bending force variation Mill. : IMR-bending force variation M: : WR-bending force reference M:' : IMR-bending force reference d.. ,d_ : undetectable disturbances tu' : detectable disturbance (rolling load variation) 't : detection time delay

~.

4 2

1-.. r. I¥aximum

.IL

I>- flatness error.

2

'-I

...

•

.•1.. ~

Intermediate roll

Strip strain distribution'

~

40

Strip width

60 80

IMR-shift

-1 00

Drive-side

Center

Operator·side

Fig. 3.2

Strip width direction

Fig. 3.1

Flatness sensor output.

243

WR·bender IMR-bender

Flatness control actuators of a 6-high mill.

10

$ >-

8

'" .,

6

~

2

'ii

'0

E

.00

200

0

Rolling speed

Detection time delay

Fig. 3.4

Fig. 3.3

600

(m/min)

Block diagram of the flatness control

Ku : parameters depending on rolling Tn

•

conditions Till. : time constants of actuators

Figure 3.3 shows the block diagram of the plant. It should be noted that the output variables have the detection time delay shown in Fig. 3.4.

3.2 lnurnal Compensation and Feedforward Control The control inputs by state feedback and feedforward are given by

AF';: Morf/

an

=(1- ~:)Mn -~: =(1- T.,.)M. -!..(J

(In+K;laP)

T.'an

-

T.'an

an

(3.4)

3.4 Disturbance Transfer Characteristics and Stability In order to obtain a balance between the actuator responses and the disturbance estimation speeds, the following conditions are set : Kn = Kill. = K T':' =T/". =T' (3 .8) The transfer function from the disturbances to the outputs is then given as

A] =e {I -(T's+IXs+K) Ke [A,. ,2

+ K' aP) (3.5) 1'2

jK22 ''LKa

-u

-u

K)4Idn ]

K~

d",. +e

-v {

}

I }[KI' llAD KnJ

1-(T's+l)

(3.9)

The fIrst tenn in the above equations represent the state feedback to improve the response characteristics. where T':' and T~ are the tuned time constants of the actuators. The second terms represent the feed forward control inputs to couterbalance the disturbance.

Mn

state variables. disturbances.

In

and

Mill. are the estimates of the data are the estimates of the

This equation shows that the transfer characteristics of the (output)/(disturbance) can be controlled by T' and K . The output variations caused by the disturbance change (aP) are regulated more quickly as T' becomes smaller, and the output variations caused by the disturbance changes (dn,d_) are also regulated more quickly as K becomes larger.

and

3.30bse"er

The characteristic equation (C.E.) of the feedback system is gi ven by C.E. = (s+ I/TnXs+ 11 T':'Xs + K.. )

To estimate the unmeasurable disturbances, it is assumed that the disturbances are of the step type. Based on this assumption, the observer is derived as follows.

The C.E. does not involve the time delay, so the tuning parameters can be set independently of the time delay.

x(s + 11 T_Xs + 11 T~Xs + K_)

a : amplitude of the upper BUR eccentricity.

b : amplitude of the lower BUR eccentricity.

4. ROLL ECCENTRICITY CONTROL OF A COLD MILL

and the lower BUR eccentricities. ro, : BUR eccentricity frequency .

4.1 Mathemalical Model of the Plant The back-up roll (BUR) balancing force is to support the BUR weight and is changeable when the strip is rolling. The mechanism of BUR balancing is shown in Fig. 4.1. The dynamics of the BUR balancing force regulator is supposed to be the first order delay as follows. d -I 1 -llP. =-llP. + -llP.'" (4.1) dt ' T. ' T., ' Here. llP, is BUR balancing force. T, IS Ume constant and llP,'" is BUR balancing force reference.

4.2 SyntMsis Procedwe The synthesis procedure is composed of three steps. The first step (Internal Compensation) is skipped because the feedforward control to completely compensate the actuator dynamics and the detection time delay is constructed.

,

4.3 Fudforward Control From equations (4 . 1) and (4.2). the feed forward control to counterbalance the roll eccentricity and completely compensate the actuator dynamics is given by

Considering the force balance and spring model of the mill gives the next equations which represent the relation among the detected strip thickness variation. the BUR balancing force and the roll eccentricity. 1 M M llh (t)=----llP.(t-t)+--llS (t-t) kH M + Q ' M +Q' • (4.2)

llP.'" =- T, k ~llS , K H dt '

-J.Kk

H

llS

(4.5)

£.

With this control (4.5). the next relation is obtained. llP, =-kHllS£ (4.6)

Z

llh.(t) -llPLC(t - t) I M = -llP,(t - t)1 M' + llS,(t - t). (4.3) where llh, : detected strip thickness variation. kH : spring constant of the mill housing. M : mill constant. M' : spring constant between BUR and work roll. Q : plasticity constant of rolled material. llS, : roll gap variation caused by the roll eccentricity • llPLC : detected load variation. t : detection time delay of strip thickness.

From equation (4.2). we can see that the achievement of the above relation guarantees complete compensation of the roll eccentricity.

4.4 Feedback Rea/ivJtion The differential equation of the roll gap variation caused by the roll eccentricity is given by

[llS'J

ro'IllS,]

d [0 dt llS~ = -(.I),. 0 llS~ . Here, llS; =a · cos(ror ·t)+b·cos(ror · t+cp) =(dllS, Idt)/ro, .

(4.7)

(4.8)

Equation (4.3) is the so-called gaugemeter equation which includes the BUR balancing force complementary . The roll gap variation caused by the roll eccentricity is given as llS,(t) = a,sin(ro,. ·t)+b · sin(ro, .t+cp), (4.4)

The state observer of which the time is shifted is given by

where

tJ, and tJ; are the estimates of llS, and llS;, respectively, and le'l and le" are observer gains. (y,(t-t)-y,(t-t») is the estimation error given

~

t)] _[0

d [tJr(t dt tJ~(t-t) -

{

_ _ roll eIoock

-(.I)r

t)]

rorltJr(t 0 tJ;(t-t)

1e'lt A) . len.f,(t-t)- y,(t-t)

(4.9)

by y,(t -t) =llh,(t)- APLC(t -t)1 M +AP,(t - t)1 M' , (4.10) y,(t-t)=llS,(t-t). (4.11) Here. y, is detectable. From equation (4.3), we can see that the estimation error equals 0 if the estimates are accurate. 1be use of the gaugemetcr equation for the construction of the estimation error leads to the feature that gain tuning with the rolled material

Ion roll

FiR. 4.1 Back-up roll balancing mechanism 245

Roll Kcentriei t J eJtiMtion A.etuetOT

,, _ _________

-[:.>- :

M£] _ [cos(O)£ t) [ M; -sin 0)£ t -

( )

(1) The obtained controller has a physically reasonable

structure. (2) The disturbances are properly estimated and dealt with. (3) The synthesis can be calculated by hand. keeping the design procedure tractable. Application results show that the new control system obtained using the observer-based method is very applicable for a large variety of small quantity production runs and is easy to tune during actual implementation.

sin(O)£ t)IM£(I- t)] (4.12) cos 0)£ t M;(I- t) (

)

A

The replacement of M£ and dI':.S£! dl in equation

AS£

(4.5) by

•

i ntqrator

are as follows .

variation is not needed . The fonnula of the trigonometric function gives the following relation.

l1P.'"

,

1__ _ _ _ _ _ _ _ ...

Block diagram of the eccentricity controller.

Fig. 4.2

A

_ J

r~~e

and

AS;. respectively. results in

=[-kK -I...k K H

0) £

H

1~£]. M'

(4.l3)

c

4.5 Structure of COlltroikr Equations (4.9) - (4.l3) give the roll eccentricity controller, and the block diagram of the controller is shown in Fig. 4.2. If the observer gain is given as k£2 =0 and also if there are no actuator response delay and detection time delay. the controller is given by AP.'" = Gc(s)· A£(s) • (4.14) where,

REFERENCES

5. CONCLUSION

Edwards. W. J., P. J . Thomas and G . C. Goodwin (1987). Roll eccentricity control for strip rolling mills. Proc. 10th IFAC World Congress, 2-14. 200-211. Hosbino. I.. T. Abe, H. Kimura and H. Kimura (1995). Observer-based roll eccentricity control via roll balancing force. Trans. SICE, 31-8. 1114-1121. (in Japanese) Hosbino. I., Y. Maekawa. T. Fujimoto. H. Kimura and H . Kimura (1988). Observer-based multivariable control of the aluminum cold tandem mill. Alltomatica. 24. 741-754. Hosbino. I., M. Kawai. M. Kokubo. T . Matsuura, H. Kimura and H. Kimura (1993). Observer-based multi variable flatness control of a cold rolling mill. Control Eng. Practice. 1-6. 917-925. Hosbino. I.. Y. Okamura and H. Kimura (1996). Observer-based multi variable tension control of aluminurn hot rolling mills. Proc. tlu 35th IEEE CDC. Kimura. H. (1985). Observer-based regulator synthesis. OsaJca University Technical Report 85-

In this paper. we first described the general design procedure of the observer-based multi variable control. As application examples of this method. the flatness control and the roll eccentricity control of rolling mills are presented. The salient features of the design method

Kimura. H. and Y. Inoue (1978). Theory and applications of multivariable control systems. System and Control.l2-11. 683-692.(in Japanese) Wonham, W. M. (1974). Linear Mrdtivariable Control - A Geo~tric Approach. Springer, New York.

-k kH s

_

El

Gc(S)A£(s)

2

K

2

•

s +kEls+O). Ah.(s)- AP/.C(s)! M + AP.(s)! M'.

=

(4.15) (4.16)

1be transfer function of the controller Gc(s) has the structure of a band pass filter. Though the fact that any roll eccentricity controller has the structure of a band pass filter without regard to the design method was indicated by Edwards and his co-workers (Edwards. Thomas and Goodwin. 1987), the compensation for the actuator response delay and the detection time delay can be naturally and easily achieved without complicating the controller structure by using the observer-based design method.

04. Japan.

246