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Regulation of a Two-Stand Coldrolling Mill
Copyright © IFA C Automation in ~[inillg . Mineral and Me ta l Processing. Toho. Japan 1986
REGULATION OF A TWO-STAND COLDROLLING MILL Xie Chang-yan Deparlmenl of AlIlomaliol/, Cmlra/-So uth Institute of Mining and Metallurgy, Changsha, HUMn , PR C
Abstract . The objective of this article is one of regulation of the out put product thic~ness of a two - stand cold -rolling mill . The design of a co~pensator which regulates t he cl osed loop syste~ to a nominal opera ting point against step disturbances o ccurring as sudden changes in the input mate rial thickness and variations in the rolling frictions at the two stands is developed . A linearized mathematical model derived by Smith ( 1969 ), valid for fluctU3ti o r.s about the steady - state ope rating point is multivari able , fully inter3cting, and contains a ti ~ e delay representing the travel ti me of the me t al between st~nds . Linear opti ~al control the or y combined with digital s i reulation is us ed to der i ve the control. The control proposed provides acceptable dynamic performance and is intensitive to the steady state errors in the process ~odel . Key words . ~ultiva ri able c on trol systems; Optima l control; Cont r ollabil ity; Controllers; Stabi lity; Discrete systems ; Delay time; Delay lines .
The Deformation process The model used for deforma tion in a single stand is based upon the cold rolling theory developed by Bla nd and Ford ( 1948) . The pr ocess paramete rs (steady - state point) are given in Table 1.
"1 "1 "1 ~ h n = O. 86h o +1.1 6 1:+ 0 . 10 5h i + 0 . 034 AA 1 (3) 1'0 2 __ " n 0 . 006"h o2 - 1.1 3 1: +0. 103 h" i2 + 0 .027JA 2 (4) The incremental variables with head are normalized, i.e. the y a re ' pe r unit '. The unit of 1: , the tension deviation fro ~ steady state , is m8gepo und s/in~
Tfl. B,::::;_~E~_-,-''.:. :a::..t:..e:..:r:..:l=.'a=l..:. :_~B.:.r...::.3-=,s-=s~(:...:7..;O;.:%.;.o_C;;..o::..p~p.:.e:c.r..l.;--=:3-=0-,-,%:.-..::Z:.:i:.:n",c:....:.) Quan tity Width Entry thic~ness hi Exit thickness ho Entry tension 1:: i Exi t tension t o Entry yield stress Ki Exit yield stress ko Coefficient of fricti on M Roll force F Ne utral plane thickness h n
Dni t in . in . in . Ib/in 2 Ib/in 2 Ib/in 2 Ib/in 2
in .
In or de r to develop a dynamic mode l, it is assumed that changes in defor~ation process Occure instantane ously. For simplycity , it is assumed that yield stress a r e constant and the entry tension at stand 1 and exi t tensi on at stand 2 are main tained constant by independent regulat or s . The incremental linear re lat i on shi p between input and output quantities can then be written as the following (Smit h, 1969) . ,..
f'1
"1'"
F 1 =-1.7.3n o -5. 20L. +l. 67hi +1. 36 )Al
( 1)
A
(2)
"2
"2" F2=-1.74ho - 5 . 77 't +l. 66h i -t o . 96 M. 2
Stand 40 0 .1 50 0 . 090 1700 3000 20 , 000 55 , 500 0 . 233 2 . 59* 10 6
0. 098
Stand 2 40 0 . 090 0 . 060 3000 4550 55,500 77,000 0 .14 8 2 . 65 -1 0 6 0 . 064
The upperscripts '1' and '2' refer to stand 1 and stand 2 respectively. The subsripts ' i ' , 'n', and ' 0 ' refer t o input, neu tral , and ou t put pl anc es respectively. Other rolling rel at ionships. Consideration of mass fl o w through the r oll gap establishes the relationship vi h i = vnh n = v oh o where v is linear vel oc ity of roll.
296
Xie C ha ng-ya n
h~(t ) = O . ilz'h~(t -'I ) +2 . 776S 1_ 1.35S2 -
In linearized fon. , this relationshir becO::',es
(5)
"0 entirel y determining
j= 1, 2 (6)
with
incre~entBl
definition as before .
A second rel ationship is derived fro~ the elastic st r etch of the roll housing . Defining S as a screwdown ~ovemen t (~osi tiv e urwcrds)
h~ = l~) Si
, 1 . 6F 1
(7)
+ 2 . 2F2
(b)
satisfact o ry technique f o r the linea r ortimal con t rol f or system having state delay like Eq . ( 1 5) is currently available . So the following approxi~ate solution method is used . The desire to maintain linearity led to the adop t ion of a discrete - time framework in which a sallifle interv al of T/ l 0 =0 .1 is used . This app r oa ch is o f couse parti cularly well - suited forseeable imple mentat i on o f digital instru~entation and co mputer con trol. S t a te Equation Model
h; =
t:~]S2
where S 1 ",nd ,52
~. easured
in inches .
The delayed state Eq . ( 1 5) is transformed to th e discrete - ti ~e state equation by using a succession of a uxill a ry state va r i ables . "1 Xl (~) = h o ( t - O. l )
Control Drive Model s
X
Fo r simrliciLy , the screwdown drives are assu~ed to be ideal integrating drives , defined by the equati o n
XC,,)
s =u
x3(k)~
A
where u is the control input measured in inches/sec •.
2 (K)= h~(t - O . 2)
h~(t -0 . 3)
• "1
10 (k)=h o (t - T
whe re T=l sec . is the travel time between the tw o stands and the sample inte rv al is 0 . 1=T/ l 0 . Then we have
In ter - Stand R elationships "1
x l (K +l ) =ho(t)
Since the exit quantities for the first stand are t he entry quanti t ies for the second , we have
"1
X(r;+ l )=
1 1:0
2
=~ i
= 1:;
"2 "1 hi( t )= ho(t - I)
( 10)
x (" + 1 )=h (t - O. 1 ) 2 o "1 X (K+ l )=h (t - 0 . 2) o 3
( 11 )
( 12) where T is the transit time between stands equal t o 1 sec • • Zquations ( 11 ) and ( 12) , together with the control drive dynamics , describe the dynacnic behaviour of the system . Through a very tedi ous but simple computati on the resulting equation f o r tension deviation is
The state equation will be ( 1 6)
r, 1 ( K ) , S2 ( K, ) , "v2n(k») T is the where U( ~ ) = ~ ihfut vect o r , and " "1 " " T is the disturba 1i(K)= (h . , M l ' M 2 ) nce vect o r, an~ ~a trices ( 13 )
The eigenv alue Of this dynamic Eq . ( 13) gives a time constant of 16 m. s ., and the resulting t r ansient res ~ons e can thus be considered negl i gible w:r . t. the delay time T ( = 1 sec . ) between stands . Unde r this approximation , we obtain t =O. F r om Eq . ( 13) setting t =O we obtain
1: = 0 . 206~ ~ -
2 . 776 -1. 35 - 0 .4
A=
, B=
o
1
[ 10 *10
2
"
0 . 00 11 7hi - 0 . 0508 h~ (t - T) + 0 . 083~2 +0 . 055S +0 . 6 12S +0 . 00964M1
Simultaneo u sly solving Eqs . ( 1 ) , (7) , and 1 we get ( 14)fo r o
h
11 2
o
A
( 14 )
Regulatio n of a Two-sta nd Cold-rolling Mill
0 .711 6
0 . 558
The cont r ol l a w is
- 0 .1 8
10 *3 From Eq s . ( 1 ) , (2) , ( 7 ) , (8) and ( 14 ) we obtain t he out put s t a t e eq uation
~~, 1:. ,F 1 -F2t
Y(k)=
1
ma6~~~eS1
~~~:~~~~ ~~~::::4 ::::~ 0 . 4432
, ( X(k) ••• X(k -1~ U(k) - U(K -1 )= f K , t: 1 * ------------ - ( 19) l 1, 21 Y(k) or k -1 U(k) =K1X(k)+K *D: Y(i) ( ;V O) ( 2 0) 2 The block diagr am of sys t em is sho wn as Fig . ,.
=CX(k) +DU(k) +FW(k) ( 17)
Whe[ re [ 0 . 223 1. 85 - 0 . 54 1 c= 0 *9 - 0 . 050 , D= 0 . 085 0 . 612 0 . 206 3 - 0 . 336 *1 0 - 5 .11 6 5.907 - 0 . 037
F=[
297
0 . 359 1
Y(I()
*3
]
- 0 . 2 152 3*3 Fi g . 1
BlOCK diagram of sys t em
STATE FEEDBACK AND OUTHlT INTEGRAL FEEDBACK CONTROL DETERtnNATIC1 0 :' CCfiTROL rATRIX
We have al r eady Eqs. ( 16) and ( 17)
;;21
X(k +1 )=AX(k)+BU(k)+EW(k) Y(k)
=CX(k) + DU(k)+F~(k)
in which the r ando~ step disturbances W is unmeasu r abl e . In gener al Eqs . (16) and ( 17) are independent on each othe r.
Control matrix (K,: can be obtained by any suitable mean~ , e . g . by pole assign ment , optimal linear - quadrtic cont r ol or inverse Nyquist techniques . In this paper we use the method o f op ti mal linear - q ua dratic control. Computation of Contro l Matrix ( K1
Defining vec t or Z(k) and V(k) as the f ol l owing
We have to c omp ute c o n~ro l r:.a trtx (1 ' 1: !\ 21 so that the cost functlon
Z(k)= ( X(k) - X(k -1 ) , Y(k -1 ») T V(k)= U(k) - U( k-1 )
J =~
then Eqs . ( 16) and (17) can be Cor:;bined as Eq . ( 18) which rep r esents an augfuented system . ( 16)
r.
fI= 'J
lzTU: ) • Q. Z ( k)
C
that in Eq . ( 18) the step di sturbance is eliminated by subtraction , because it is r easonable t ~ assume strictly ste~ distur banc e s to spe~d up the system sufficiently so as to make slowly va r ying disturbances seem essentially constant within the time constant of the closed - locp system . ~o t e
Froir, Eq . ( 16) if we want [ X(k) - X(k -1 ;}-O anci Y(k -1 ) . 0 as k +OO, then ( A1 , El ) r.:ust be stabilizable . Unde r the condition of (A1 , B, ) bei ng stabili zabl e we can design a li near fe edba ck cont r o l le r. Thi s cont r olle r contains a s tate feedback cmpensator and an ou tput i nteg r al feedback cmpensator t o ensure t he stability and output regulation of system .
)J (2 1 )
The xatrices A" D1 , Q, a nd R in Eq . (2!) are c ons tant . 4hen n. O, l . e . system a c hle ving steady - state we have the cont rol law V(~)=KZ(k)
A'
A1 =
+ 'I T ( k ) • R . V ( :.
has minimal value .
whe r e [
!K2 )
(22)
which is equivalent to Eg . ( l S) . In e;q . ( 22 ) 1: =(;' 1: " 2 ) ' V(k)=U(,,) - C'(i-: -1 ) . Solving ?iccati algebraic Sq . we obtain contro l ~atrix K whic h minimizes the cost functi on I n •
C =Ar?+ " f- 1"'~ -F3 1 R- l Bf~: .
'T'_ ,= -R-1 3i~
(23) (. 24)
where ~ is a weighting ~atrix whi ch deter ~ines how ~uch weight is attatched to each o f co~rone~ts o f state , and is a se~i positive - definite sy~~etrtc . Eecause we the output Y(~) o nly, as lS re qui'red , r:.a trix ~,,=0 , i , j ~1 0 . ;:: i s <. ~ o s:i "'J . tive - definite - symmetric weighting ~atrlx whic h determines how ~uch weight is atta ched to each of co~ponents of input . re~ulate
298
Xie Chang-yan
More important is to discuss how to determine matrices Q and R. The main idea to answer this question can be explained as the follow ing. The output weighting matrix Q is modified to obtain an acceptable balance between gauge "2 A A h , tension~, and force ~1-F2 transients. Howe~er, some of the controls will now exceed their satisfaction limits, while others may be small. The control weighting matrix R is now adjusted to weight the out-of-limit controls more heavily and to reduce the penality on those controls which are within limits. Unfortunately, this affects the initial gauge-tension- force balance, and the entire procedure must be repeated until a 'best' compromise is reached.
I
-r
Q=
0 3 *10
Given a pair of weighting matrix Q and Rand solving Eq. (23) we can get a control matrix K which makes cost function minimal, but it does not guarantee the transient process of system satisfying the performance criteria. So the design procedure consists simply of a trial and error routine of computer simulation using judicious choices of Q and R until the performance criteria are achieved. For each choice, the optimal state feedback gain matrix K was obtained vir the solution of a discrete algebraic Riccati equation, and the resulting closed-loop system was simulated.
_0.!0~3__
010*10 '
0
0
0
500
0
0
0
0.01
13*13
006
005 004 003
OOZ 00 I
Limit
O~~~~~~~--
-0.0 I -9.8Z
-0.03
Fig. 2
Response of output thickness of stand 2
Digi tal Simulation and the CDmputed Results 000075
As mentioned above, the control matrix K computed from Eqs. (23) and (24) ensures the cost function J minimal, but it does not guarantee the dYnamic performance. So it is necessary to have digital simulation to get the response curves of outputs. We assumed that the step disturbance is "1 hi =10%.
-0.0Da25
Performance criteria. They are as follows.
-000050
For output variable:
-0·00075
(1) To bring 112 to 0.01 (tolerance limi t) as rapidly as p8ssible, and thzn to zero. (2) To limit 1: to 0.001 ELB/in • 2 ) is not particularly important, rather it is used as a design parameters.
-0.000100
Fig. 3
limit
Response of tension
(F -F
For control variables: (1 )
(2)
S'1 , S'2~0.0016 in/sample interval, ~~~0.003/sample interval.
Response curves.
They are shown above.
Fr om Fig. 2 and Fig. 3 we can say that the deviation of output thickness h~ and tensi on ~ are acceptable such as the following. (1) 0.01 output thickness deviation is a chieved within 1~ sa mp le intervals,
Control matrix K-!K~2)' It is as follows. k1 k2 K3 k4 k5 k6 K7 ~8 0.056 0.0517 0.0479 0.0447 0.0419 0.0393 0.0367 0.0339
K{
0.000134 -0.03 k12 0.234
0.OO~4 0.0045
-0.034
-0.0082 0.00225
0.531
-0.00256
0.0112
k 10 k11 0.0233: 0.0282
0.0132 0.014 0.0074 -0.0263: 0.0281 I
-0.0385 -0.0435 -0.0491 -0.0553 -0. 0624 -0. 0708 -0.08 -0.1041-0.0843
k13
0.989
0.0067 0.0089
k9 0 .0302
1
(2) tension deviation ~eeps within deviation limit during the entire transient process. It should be noted that short intervals (1 - 3 samples) over saturation level of inputs are acceptable.
299
Regulation of a Two-stand Cold-rolling Mill
STRUCTURE OF REGULATOR The resulting control stra tegy a , proportiona l plus integral ' feed back given in Eq . (20) requires a tapped delay line f or implementation , which gene r ates the state. By using computer simulation we can get delay line . Transforffiing Eq . (20) we get an al ternative control fODlli which can easily be implemented .
Fig. 4 The sketch of delay line Structure o f pr oposed Con trol
(25)
From Eq. (25) we Obtain
(k>O)
(26)
where
~; : K2)"
k l"
[ -0.005
- 0 . 0038
- 0 . 0032
- 0 . 0028
- 0 . 0026
- 0 . 0026
- 0 . 0028
0 . 00226
0 . 002 13
0 . 002 16
0 . 00225
0 . 00227
0 . 002
0 . 0005
- 0 . 004
- 0 . 0045
- 0 . 005
- 0 . 0056
-0. 0027
- 0 . 0071
- 0 . 0068
- 0 . 006S]
- 0 . 0233
0 . 0 252
0 . 234
-0.00819J
- 0 . 03367
0 . 0263
0 . 026 1
0 . 989
0 . 00225
- 0 . 02 16
0 . 104
- 0 . 0843
0 . 53 1
- 0 . 00256
I
[00 .0%7 ] . 000 134
- 0 . 0037 - 0 . OC633
-O.011b
The st ructure of re gu l a t or is shown in Fig . 5 .
- 0 . 03 Delay line f or Ge n erating the St ate by Corquter Sir..ul a ti on
ho
.h.,
The ske tc h of t hi s delay line is sh own in Fig. 4 . ~' -
"t
F'
Fi g . 5
o.k.j
L;"~
j " ') [l,C X'J'
k,
.s'
.
SZ '
Kz St ructure o f re gulat or
u"
Xie Chang-yan
300 CONCLUSIONS
We have presented a control system which will regulate the two-stand cold-rolling mill with acceptable performance. In this design linear optimal control theory combined with digital simulation is used to derive the gain matrix of controller. The price for this improved performance is the requirement of a ten-stage delay line which simulates the delayed thickness deviation between the stands. In addition to providing acceptable dynamic performance, the control is insensitive to steady-state errors in the process model and may offer significant advantages over more conventional controls.
ACKHOWLEDGIvENTS The author wishes t o express his thanks to Prof. H.W. Smith who gave him a lot of help when visiting the Department of Engineering at University of Toronto in Canada in 1980 to 1982. REFERENCES Bland, D.R. and Ford, H. ( 1948) . The calcul a ti on of roll force and torque in c o ld strip rolling with tension. Froc . Inst. l';ecr.. Engrs. , L:22, 1411-1 63 . Kwakernaak, H. and Sivan, R. (1972). Linea r Optimal Control Systerr:s. John Wiley and Sons, Inc •• Smith, H.W. (1969). Dynamic control of a two -stand cold rr;,ill. Autorr;ica, 2., 183 -1 90 . Smith, H. W. and Davision, E.J.(1972). Design of industrial regulators. Fro c • I EEE , !..!...2..
REGULATION OF A TWO-STAND COLDROLLING MILL Xie Chang-yan Deparlmenl of AlIlomaliol/, Cmlra/-So uth Institute of Mining and Metallurgy, Changsha, HUMn , PR C
Abstract . The objective of this article is one of regulation of the out put product thic~ness of a two - stand cold -rolling mill . The design of a co~pensator which regulates t he cl osed loop syste~ to a nominal opera ting point against step disturbances o ccurring as sudden changes in the input mate rial thickness and variations in the rolling frictions at the two stands is developed . A linearized mathematical model derived by Smith ( 1969 ), valid for fluctU3ti o r.s about the steady - state ope rating point is multivari able , fully inter3cting, and contains a ti ~ e delay representing the travel ti me of the me t al between st~nds . Linear opti ~al control the or y combined with digital s i reulation is us ed to der i ve the control. The control proposed provides acceptable dynamic performance and is intensitive to the steady state errors in the process ~odel . Key words . ~ultiva ri able c on trol systems; Optima l control; Cont r ollabil ity; Controllers; Stabi lity; Discrete systems ; Delay time; Delay lines .
The Deformation process The model used for deforma tion in a single stand is based upon the cold rolling theory developed by Bla nd and Ford ( 1948) . The pr ocess paramete rs (steady - state point) are given in Table 1.
"1 "1 "1 ~ h n = O. 86h o +1.1 6 1:+ 0 . 10 5h i + 0 . 034 AA 1 (3) 1'0 2 __ " n 0 . 006"h o2 - 1.1 3 1: +0. 103 h" i2 + 0 .027JA 2 (4) The incremental variables with head are normalized, i.e. the y a re ' pe r unit '. The unit of 1: , the tension deviation fro ~ steady state , is m8gepo und s/in~
Tfl. B,::::;_~E~_-,-''.:. :a::..t:..e:..:r:..:l=.'a=l..:. :_~B.:.r...::.3-=,s-=s~(:...:7..;O;.:%.;.o_C;;..o::..p~p.:.e:c.r..l.;--=:3-=0-,-,%:.-..::Z:.:i:.:n",c:....:.) Quan tity Width Entry thic~ness hi Exit thickness ho Entry tension 1:: i Exi t tension t o Entry yield stress Ki Exit yield stress ko Coefficient of fricti on M Roll force F Ne utral plane thickness h n
Dni t in . in . in . Ib/in 2 Ib/in 2 Ib/in 2 Ib/in 2
in .
In or de r to develop a dynamic mode l, it is assumed that changes in defor~ation process Occure instantane ously. For simplycity , it is assumed that yield stress a r e constant and the entry tension at stand 1 and exi t tensi on at stand 2 are main tained constant by independent regulat or s . The incremental linear re lat i on shi p between input and output quantities can then be written as the following (Smit h, 1969) . ,..
f'1
"1'"
F 1 =-1.7.3n o -5. 20L. +l. 67hi +1. 36 )Al
( 1)
A
(2)
"2
"2" F2=-1.74ho - 5 . 77 't +l. 66h i -t o . 96 M. 2
Stand 40 0 .1 50 0 . 090 1700 3000 20 , 000 55 , 500 0 . 233 2 . 59* 10 6
0. 098
Stand 2 40 0 . 090 0 . 060 3000 4550 55,500 77,000 0 .14 8 2 . 65 -1 0 6 0 . 064
The upperscripts '1' and '2' refer to stand 1 and stand 2 respectively. The subsripts ' i ' , 'n', and ' 0 ' refer t o input, neu tral , and ou t put pl anc es respectively. Other rolling rel at ionships. Consideration of mass fl o w through the r oll gap establishes the relationship vi h i = vnh n = v oh o where v is linear vel oc ity of roll.
296
Xie C ha ng-ya n
h~(t ) = O . ilz'h~(t -'I ) +2 . 776S 1_ 1.35S2 -
In linearized fon. , this relationshir becO::',es
(5)
"0 entirel y determining
j= 1, 2 (6)
with
incre~entBl
definition as before .
A second rel ationship is derived fro~ the elastic st r etch of the roll housing . Defining S as a screwdown ~ovemen t (~osi tiv e urwcrds)
h~ = l~) Si
, 1 . 6F 1
(7)
+ 2 . 2F2
(b)
satisfact o ry technique f o r the linea r ortimal con t rol f or system having state delay like Eq . ( 1 5) is currently available . So the following approxi~ate solution method is used . The desire to maintain linearity led to the adop t ion of a discrete - time framework in which a sallifle interv al of T/ l 0 =0 .1 is used . This app r oa ch is o f couse parti cularly well - suited forseeable imple mentat i on o f digital instru~entation and co mputer con trol. S t a te Equation Model
h; =
t:~]S2
where S 1 ",nd ,52
~. easured
in inches .
The delayed state Eq . ( 1 5) is transformed to th e discrete - ti ~e state equation by using a succession of a uxill a ry state va r i ables . "1 Xl (~) = h o ( t - O. l )
Control Drive Model s
X
Fo r simrliciLy , the screwdown drives are assu~ed to be ideal integrating drives , defined by the equati o n
XC,,)
s =u
x3(k)~
A
where u is the control input measured in inches/sec •.
2 (K)= h~(t - O . 2)
h~(t -0 . 3)
• "1
10 (k)=h o (t - T
whe re T=l sec . is the travel time between the tw o stands and the sample inte rv al is 0 . 1=T/ l 0 . Then we have
In ter - Stand R elationships "1
x l (K +l ) =ho(t)
Since the exit quantities for the first stand are t he entry quanti t ies for the second , we have
"1
X(r;+ l )=
1 1:0
2
=~ i
= 1:;
"2 "1 hi( t )= ho(t - I)
( 10)
x (" + 1 )=h (t - O. 1 ) 2 o "1 X (K+ l )=h (t - 0 . 2) o 3
( 11 )
( 12) where T is the transit time between stands equal t o 1 sec • • Zquations ( 11 ) and ( 12) , together with the control drive dynamics , describe the dynacnic behaviour of the system . Through a very tedi ous but simple computati on the resulting equation f o r tension deviation is
The state equation will be ( 1 6)
r, 1 ( K ) , S2 ( K, ) , "v2n(k») T is the where U( ~ ) = ~ ihfut vect o r , and " "1 " " T is the disturba 1i(K)= (h . , M l ' M 2 ) nce vect o r, an~ ~a trices ( 13 )
The eigenv alue Of this dynamic Eq . ( 13) gives a time constant of 16 m. s ., and the resulting t r ansient res ~ons e can thus be considered negl i gible w:r . t. the delay time T ( = 1 sec . ) between stands . Unde r this approximation , we obtain t =O. F r om Eq . ( 13) setting t =O we obtain
1: = 0 . 206~ ~ -
2 . 776 -1. 35 - 0 .4
A=
, B=
o
1
[ 10 *10
2
"
0 . 00 11 7hi - 0 . 0508 h~ (t - T) + 0 . 083~2 +0 . 055S +0 . 6 12S +0 . 00964M1
Simultaneo u sly solving Eqs . ( 1 ) , (7) , and 1 we get ( 14)fo r o
h
11 2
o
A
( 14 )
Regulatio n of a Two-sta nd Cold-rolling Mill
0 .711 6
0 . 558
The cont r ol l a w is
- 0 .1 8
10 *3 From Eq s . ( 1 ) , (2) , ( 7 ) , (8) and ( 14 ) we obtain t he out put s t a t e eq uation
~~, 1:. ,F 1 -F2t
Y(k)=
1
ma6~~~eS1
~~~:~~~~ ~~~::::4 ::::~ 0 . 4432
, ( X(k) ••• X(k -1~ U(k) - U(K -1 )= f K , t: 1 * ------------ - ( 19) l 1, 21 Y(k) or k -1 U(k) =K1X(k)+K *D: Y(i) ( ;V O) ( 2 0) 2 The block diagr am of sys t em is sho wn as Fig . ,.
=CX(k) +DU(k) +FW(k) ( 17)
Whe[ re [ 0 . 223 1. 85 - 0 . 54 1 c= 0 *9 - 0 . 050 , D= 0 . 085 0 . 612 0 . 206 3 - 0 . 336 *1 0 - 5 .11 6 5.907 - 0 . 037
F=[
297
0 . 359 1
Y(I()
*3
]
- 0 . 2 152 3*3 Fi g . 1
BlOCK diagram of sys t em
STATE FEEDBACK AND OUTHlT INTEGRAL FEEDBACK CONTROL DETERtnNATIC1 0 :' CCfiTROL rATRIX
We have al r eady Eqs. ( 16) and ( 17)
;;21
X(k +1 )=AX(k)+BU(k)+EW(k) Y(k)
=CX(k) + DU(k)+F~(k)
in which the r ando~ step disturbances W is unmeasu r abl e . In gener al Eqs . (16) and ( 17) are independent on each othe r.
Control matrix (K,: can be obtained by any suitable mean~ , e . g . by pole assign ment , optimal linear - quadrtic cont r ol or inverse Nyquist techniques . In this paper we use the method o f op ti mal linear - q ua dratic control. Computation of Contro l Matrix ( K1
Defining vec t or Z(k) and V(k) as the f ol l owing
We have to c omp ute c o n~ro l r:.a trtx (1 ' 1: !\ 21 so that the cost functlon
Z(k)= ( X(k) - X(k -1 ) , Y(k -1 ») T V(k)= U(k) - U( k-1 )
J =~
then Eqs . ( 16) and (17) can be Cor:;bined as Eq . ( 18) which rep r esents an augfuented system . ( 16)
r.
fI= 'J
lzTU: ) • Q. Z ( k)
C
that in Eq . ( 18) the step di sturbance is eliminated by subtraction , because it is r easonable t ~ assume strictly ste~ distur banc e s to spe~d up the system sufficiently so as to make slowly va r ying disturbances seem essentially constant within the time constant of the closed - locp system . ~o t e
Froir, Eq . ( 16) if we want [ X(k) - X(k -1 ;}-O anci Y(k -1 ) . 0 as k +OO, then ( A1 , El ) r.:ust be stabilizable . Unde r the condition of (A1 , B, ) bei ng stabili zabl e we can design a li near fe edba ck cont r o l le r. Thi s cont r olle r contains a s tate feedback cmpensator and an ou tput i nteg r al feedback cmpensator t o ensure t he stability and output regulation of system .
)J (2 1 )
The xatrices A" D1 , Q, a nd R in Eq . (2!) are c ons tant . 4hen n. O, l . e . system a c hle ving steady - state we have the cont rol law V(~)=KZ(k)
A'
A1 =
+ 'I T ( k ) • R . V ( :.
has minimal value .
whe r e [
!K2 )
(22)
which is equivalent to Eg . ( l S) . In e;q . ( 22 ) 1: =(;' 1: " 2 ) ' V(k)=U(,,) - C'(i-: -1 ) . Solving ?iccati algebraic Sq . we obtain contro l ~atrix K whic h minimizes the cost functi on I n •
C =Ar?+ " f- 1"'~ -F3 1 R- l Bf~: .
'T'_ ,= -R-1 3i~
(23) (. 24)
where ~ is a weighting ~atrix whi ch deter ~ines how ~uch weight is attatched to each o f co~rone~ts o f state , and is a se~i positive - definite sy~~etrtc . Eecause we the output Y(~) o nly, as lS re qui'red , r:.a trix ~,,=0 , i , j ~1 0 . ;:: i s <. ~ o s:i "'J . tive - definite - symmetric weighting ~atrlx whic h determines how ~uch weight is atta ched to each of co~ponents of input . re~ulate
298
Xie Chang-yan
More important is to discuss how to determine matrices Q and R. The main idea to answer this question can be explained as the follow ing. The output weighting matrix Q is modified to obtain an acceptable balance between gauge "2 A A h , tension~, and force ~1-F2 transients. Howe~er, some of the controls will now exceed their satisfaction limits, while others may be small. The control weighting matrix R is now adjusted to weight the out-of-limit controls more heavily and to reduce the penality on those controls which are within limits. Unfortunately, this affects the initial gauge-tension- force balance, and the entire procedure must be repeated until a 'best' compromise is reached.
I
-r
Q=
0 3 *10
Given a pair of weighting matrix Q and Rand solving Eq. (23) we can get a control matrix K which makes cost function minimal, but it does not guarantee the transient process of system satisfying the performance criteria. So the design procedure consists simply of a trial and error routine of computer simulation using judicious choices of Q and R until the performance criteria are achieved. For each choice, the optimal state feedback gain matrix K was obtained vir the solution of a discrete algebraic Riccati equation, and the resulting closed-loop system was simulated.
_0.!0~3__
010*10 '
0
0
0
500
0
0
0
0.01
13*13
006
005 004 003
OOZ 00 I
Limit
O~~~~~~~--
-0.0 I -9.8Z
-0.03
Fig. 2
Response of output thickness of stand 2
Digi tal Simulation and the CDmputed Results 000075
As mentioned above, the control matrix K computed from Eqs. (23) and (24) ensures the cost function J minimal, but it does not guarantee the dYnamic performance. So it is necessary to have digital simulation to get the response curves of outputs. We assumed that the step disturbance is "1 hi =10%.
-0.0Da25
Performance criteria. They are as follows.
-000050
For output variable:
-0·00075
(1) To bring 112 to 0.01 (tolerance limi t) as rapidly as p8ssible, and thzn to zero. (2) To limit 1: to 0.001 ELB/in • 2 ) is not particularly important, rather it is used as a design parameters.
-0.000100
Fig. 3
limit
Response of tension
(F -F
For control variables: (1 )
(2)
S'1 , S'2~0.0016 in/sample interval, ~~~0.003/sample interval.
Response curves.
They are shown above.
Fr om Fig. 2 and Fig. 3 we can say that the deviation of output thickness h~ and tensi on ~ are acceptable such as the following. (1) 0.01 output thickness deviation is a chieved within 1~ sa mp le intervals,
Control matrix K-!K~2)' It is as follows. k1 k2 K3 k4 k5 k6 K7 ~8 0.056 0.0517 0.0479 0.0447 0.0419 0.0393 0.0367 0.0339
K{
0.000134 -0.03 k12 0.234
0.OO~4 0.0045
-0.034
-0.0082 0.00225
0.531
-0.00256
0.0112
k 10 k11 0.0233: 0.0282
0.0132 0.014 0.0074 -0.0263: 0.0281 I
-0.0385 -0.0435 -0.0491 -0.0553 -0. 0624 -0. 0708 -0.08 -0.1041-0.0843
k13
0.989
0.0067 0.0089
k9 0 .0302
1
(2) tension deviation ~eeps within deviation limit during the entire transient process. It should be noted that short intervals (1 - 3 samples) over saturation level of inputs are acceptable.
299
Regulation of a Two-stand Cold-rolling Mill
STRUCTURE OF REGULATOR The resulting control stra tegy a , proportiona l plus integral ' feed back given in Eq . (20) requires a tapped delay line f or implementation , which gene r ates the state. By using computer simulation we can get delay line . Transforffiing Eq . (20) we get an al ternative control fODlli which can easily be implemented .
Fig. 4 The sketch of delay line Structure o f pr oposed Con trol
(25)
From Eq. (25) we Obtain
(k>O)
(26)
where
~; : K2)"
k l"
[ -0.005
- 0 . 0038
- 0 . 0032
- 0 . 0028
- 0 . 0026
- 0 . 0026
- 0 . 0028
0 . 00226
0 . 002 13
0 . 002 16
0 . 00225
0 . 00227
0 . 002
0 . 0005
- 0 . 004
- 0 . 0045
- 0 . 005
- 0 . 0056
-0. 0027
- 0 . 0071
- 0 . 0068
- 0 . 006S]
- 0 . 0233
0 . 0 252
0 . 234
-0.00819J
- 0 . 03367
0 . 0263
0 . 026 1
0 . 989
0 . 00225
- 0 . 02 16
0 . 104
- 0 . 0843
0 . 53 1
- 0 . 00256
I
[00 .0%7 ] . 000 134
- 0 . 0037 - 0 . OC633
-O.011b
The st ructure of re gu l a t or is shown in Fig . 5 .
- 0 . 03 Delay line f or Ge n erating the St ate by Corquter Sir..ul a ti on
ho
.h.,
The ske tc h of t hi s delay line is sh own in Fig. 4 . ~' -
"t
F'
Fi g . 5
o.k.j
L;"~
j " ') [l,C X'J'
k,
.s'
.
SZ '
Kz St ructure o f re gulat or
u"
Xie Chang-yan
300 CONCLUSIONS
We have presented a control system which will regulate the two-stand cold-rolling mill with acceptable performance. In this design linear optimal control theory combined with digital simulation is used to derive the gain matrix of controller. The price for this improved performance is the requirement of a ten-stage delay line which simulates the delayed thickness deviation between the stands. In addition to providing acceptable dynamic performance, the control is insensitive to steady-state errors in the process model and may offer significant advantages over more conventional controls.
ACKHOWLEDGIvENTS The author wishes t o express his thanks to Prof. H.W. Smith who gave him a lot of help when visiting the Department of Engineering at University of Toronto in Canada in 1980 to 1982. REFERENCES Bland, D.R. and Ford, H. ( 1948) . The calcul a ti on of roll force and torque in c o ld strip rolling with tension. Froc . Inst. l';ecr.. Engrs. , L:22, 1411-1 63 . Kwakernaak, H. and Sivan, R. (1972). Linea r Optimal Control Systerr:s. John Wiley and Sons, Inc •• Smith, H.W. (1969). Dynamic control of a two -stand cold rr;,ill. Autorr;ica, 2., 183 -1 90 . Smith, H. W. and Davision, E.J.(1972). Design of industrial regulators. Fro c • I EEE , !..!...2..