Optimization of a Hot Rolling Mill

Optimization of a Hot Rolling MillAuthor(s): D. C. DowsonSource: Journal of the Royal Statistical Society. Series B (Methodological), Vol. 29, No. 2(1967), pp. 300-319Published by: Wiley for the Royal Statistical SocietyStable URL: http://www.jstor.org/stable/2984592 .

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300 [No. 2.

Optimization of a Hot Rolling Mill

By D. C. DOWSON

University of Salford

[Received December 1965. Revised June 1966]

SUMMARY This paper is concerned with constructing rules for determining the times at which slabs have to be fed into successive stages of a hot rolling mill so as to optimize the working of the mill. The errors entering the system are assumed to form stationary stochastic sequences and the method adopted for determining the appropriate feedback is that of linear least squares. The analysis is carried through and results tabulated for the case of uncorrelated errors and also for one, fairly general, case of correlated errors.

1. INTRODUCTION

THE problem to be considered in this paper is the theoretical design of an on-line signalling system for optimizing the flow of slabs through a hot rolling mill. The basic problem is one of maximizing the throughput of the mill consistent with a given level of safety or, what is equivalent, minimizing the average value of a suitably chosen loss function. We shall be concerned wholly with the steady-state problem so that we can ignore any transitory effects due to the start-up of the process. For the sake of simplicity we shall consider a specific kind of mill with four stages, but the main features of the analysis apply to rolling mills generally with any number of stages.

The process of rolling will be assumed to be made up of four stages: (i) a slab is pulled from the furnace, (ii) it is then passed through a stand where it is trimmed, (iii) it is then passed through a rolling mill where it is rolled into a more manageable

thickness, and (iv) it then enters a finishing mill where the final rolling is performed. Within the finishing mill the metal strip is then wound on to coilers and we shall assume that there are sufficient of these to deal with the strip even if the finishing mill rollers are working continuously.

Suppose, to begin with, we know exactly how long each slab will take to go through the whole process, by which we mean the time until the final rolling has been completed. In this case the slabs can be fed into the process in such a way that the rollers in the finishing mill are working continuously. Since any gaps between slabs become smaller as the slabs progress through the system, the insertion times would have to be calculated so as to produce zero gap at the finishing mill rollers.

In practice, however, there are factors which cause process times to be either greater or smaller than anticipated so that collisions would occur. Needless to say, this is to be avoided at all costs and can be overcome by allowing for a gap at the finishing mill between successive slabs. For convenience we shall refer to the dis- crepancy between actual and expected process times as an error and it will become clear later that errors of this kind are statistical in nature.



1967] DOWSON - Optimization of a Hot Rolling Mill 301

The size of the gap g (measured in units of time) and also the stage times ti (i = 1, 2, 3) are random variables and we wish to influence the distributions of these as far as possible so as to minimize the average value of the loss function. There are two ways that loss occurs: (a) a loss L of throughput due to a gap at the finishing mill, and (b) losses Li (i = 1, 2, 3) due to unstabilized stage times.

The occurrence of a collision is something of a catastrophe and consequently the loss function L must take very large values for negative values of the gap g to ensure that the probability of a collision is very small. The loss function L is likely to have the form shown in Fig. 1, whereas the form of the loss functions Li is rather different

Loss

L

L

L

0 Size of gap

FIG. 1. Loss function for the gap.

and can reasonably be taken to be piecewise linear, as shown in Fig. 2. In order to be able to carry through the subsequent analysis in an explicit form, however, it is necessary to approximate the loss functions L, L1, L2 and L3 by quadratic functions Q, Q1, Q2 and Q3 as indicated in Figs 1 and 2. We denote by c, Cl, c2 and c3 the constants for which the functions

Q(g) = (g-c)2 1 (1.1) Q%(t) = (t-C,)2 (i = I, 2, 3)



302 DOWSON - Optimization of a Hot Rolling Mill [No. 2,

best approximate to L and Li (i = 1, 2, 3) in some particular sense. The criterion for optimization of the process then becomes the minimization of a quadratic control function C, which is the expected value of the loss function and which is taken to be

C = E (A(g-C)2 + C (ti-c)2}, (1.2)

where A is a positive weighting factor.t The expectation operator E is taken over the joint distribution of gap size g and stage times t1, t2 and t3. Controlling the gap size and controlling the stage times are conflicting aims and by varying A we can alter the relative importance of these two aims.

Loss

Qi Qi

ti ith stage time

FIG. 2. Loss function for the overall process time.

In a controlled system the successive gap lengths at the finishing mill should form a stationary stochastic sequence, at least in the wide sense, and in fact so should the successive stage times in any particular stage. The method that we shall adopt for the

t It is shown elsewhere that, for normally distributed random variables and linear decision rules, the minimization of the expected value of an arbitrary loss function is exactly equivalent to the minimization of some quadratic control function.




control of these stationary processes is due to Newton (1952) and Newton et al. (1957) and is essentially a generalization of the methods used by Kolmogorov and Wiener in prediction theory. More recent work in this field has been done by Whittle (1963) and we shall follow his notation throughout this paper.

The minimization of the control function C leads to the unique way in which the errors have to be fed back to make the system well coiltrolled. With this set of rules for the feedback, the variances of the finishing mill gap and stage times are then determined in terms of the variances and covariances of the errors involved. How- ever, the average values of this gap and stage times are still at our disposal and can be increased or decreased by altering the intervals between decision times by constant amounts without in any way affecting the variability of the process, so that the feedback of errors is as before. The value of the control function C, however, does depend on the average gap size and the average stage times so that these also are chosen so as to minimize the control function.

One point of difficulty can arise when constructing the decision rules, particularly if the slab lengths are very variable. When making the decisions at a particular stage the amount of information available may differ from one slab to another; for example, preceding slabs may or may not have completed their current stages. Ideally, we should construct different decision rules according to the amount of information available. Unfortunately in a problem of control, as distinct from one of prediction only, the process becomes considerably more complex because of the varied nature of the input and to avoid this difficulty we shall base our decision rules on the maximum amount of information which is alwcays available.

2. NATURE OF THE RANDOM ERRORS

The random errors entering the process can be divided into two types: (a) errors in getting a slab started in a stage including delay in operator response and

delay due to a slab failing to enter a stage at the first attempt, (b) errors occurring during processing including slippage whilst rolling and also the

error due to possible slight difference in average stage times due to variation in length of the slabs.

In the first two stages of course there is no error due to slippage. The errors of first and second type which occur when the kth slab is passed through

the ith stage are denoted by Eik and mtk respectively, for

i= 1,2,3,4 and k=...,-1,0,1,2,3,. These variables are of a random nature and can only be specified statistically before actual observation. In the course of the analysis we shall make various assumptions about the second-order properties of these errors and determine the corresponding decision rules.

3. THE BASIC MODEL OF THE PROCESS

Let Sik = time when kth slab is signalled to enter the ith process,

Uik = time when kth slab starts to emerge from the ith process,

Vik = time when kth slab completes the ith process.

We make the seemingly reasonable assumption that the average time elapsing between a slab being instructed to enter a stage and it starting to emerge from that



304 DowSON - Optinmization of a Hot Rolling Mill [No. 2,

stage is independent of the length of the slab. This average "response" time for the ith stage will be denoted by ai (i = 1,2,3,4). The time elapsing between a slab starting to emerge from a stage and completing the stage will in general depend on the length of the slab. This average "process" time for the kth slab in the ith stage will be denoted by bik and its value can be computed as soon as the kth slab has been measured, which we assume is done shortly after the kth slab has been taken from the furnace, and certainly before one wishes to take the next slab from the furnace.

The following relations now constitute our basic model:

uik= Sik+ai+Ek ((i= 1,2,3,4), (3.1)

Vik Uik +bik +17ik (k= ... ,-1 0O 1, 2,...),

so that 'Ek is the error in the response time and Xik is the error in the process time. It will be assumed that {Ek} and {k}, the sequences of errors at any particular stage, are discrete-time stationary stochastic processes whose second-order properties are known. The variances of these processes will be denoted throughout by U2 and U

for i= 1, 2, 3 and 4. It should perhaps be pointed out that there is considerable flexibility in what

constitutes the beginning and end of a stage, provided we define the ai and bik accordingly. We shall be specific on only one such point: namely, the gap at the finishing mill (or at a set of rollers generally) will be the time elapsing between the instant when the rear end of one strip leaves the rollers and the instant when the front end of the next strip emerges from the rollers.

4. THE DECISION RULES

Our decision rules have to be capable of modifying the interval between successive insertions at any stage as a result of errors which have been observed in the process. As mentioned in the introduction we shall base our decisions only on errors which we know will have been observed before the decision is to be put into operation. (The actual computing time involved in the decision making will be very small, usually of the order of milliseconds.) For the particular mill considered it was possible to make the following assertions.

When the kth slab was to be signalled from the furnace (a) the (k - I)st slab had completed the first stage and had certainly entered the second

stage; (b) the (k - 2)nd slab had certainly entered the third stage; (c) the (k - 3)rd had completed the fourth stage (i.e. it had completely left the rollers

in the finishing mill). Hence we take as decision rule:

Slk = Sl,k-1 + Illk + X Y1; E1,k-j + 0 2j E2,k-j + E 3j E3,k-j + E 4jc4,k-j 1 1 2 3

+ E 6j s'71,k-j + E 02j i2,k-

+ z 63j 3,k-j + E 74,k-j' (4.1) 1 2 3 3

where the summations are over j and extend to infinity. All errors are measured from average values so that t1k is the average time between

signalling the (k - I)st and kth slabs from the furnace. Rather than formulate our remaining decisions in terms of intervals between successive insertions at a given




stage as we shall'do eventually, it is more convenient for the moment to use the times between consecutive decisions for any slab. In a like manner we then obtain

S2k - Slk = 2k + flj9 E1,k-j + E /2 E2,k-j + E P3J3,k-1 + z 4jj E4,k- o 1 1 2

+ E li1 "l,k-j + 02j '2,k-j + X 03j 3,k-j + E 041 "4,k-j1 o 1 2 2

S3k - Slk = P-3k + E Ylj l,k-j + E Y2j E2,k-j + E Y3j E3,k-j + E Y4j E4,k-j -1 0 1 1(4.2)

+ l j "Jl,k-1 + E 02 "2,k-j + E 03J "73,k-j + E 04j "74,k-p 4 -1 0 1 2

S4k Slk = i4k + E3j'El,k-j + 82j E2,k- + f S3jE 3,k-j + z 84j E4,k-1 -1 -1 0 1

+ Xij 71,k-i + X2j 72,k-j + E X32 73,kj + - X4 t4,k-;p -1 ~~0 01

Here ,2k is the average time elapsing between signalling the kth slab to enter the first and second stages with similar interpretations for /13k and PU4k. For any set of coefficients such as oej we shall denote by xi(z) the generating function Eo jzi where the summation is over all relevant j. Then introducing the shift operator U defined by

Uxt = Xt-1, (4.3)

the decision rule for the first stage can be written

Slk = S1,k-1 + tlk + 1(U) E1 + Y2(U) E2 + x3(U) E3 + cX4(U) E4

+ 01(U) l+ 02(U) % + 03(U) 773+ 04(U) 74, (4.4)

with similar equations for the other decision rules. These can be combined in the single equation

Slk S1,k-1 ylk a 2 X3 U4 Elk 61 02 03 64 ?7ik

52k _ Slk =- 2k + 1 2 P P4 E2k +

[1 02 03

S 4

S: 2k

S3k Slk ] M 1l Y2 Y3 I4 '3k 1 02 03 I4 3k

_S4U_ _ Slk _ _y4k_ _8l 82 33 84] E4k Xi X2 X3 X4 "74k (4.5)

where the elements of the matrices are operators. We defer for the moment writing this equation in a more symbolic form.

5. CHOICE OF AVERAGE GAP AND AVERAGE STAGE TiMEs The criterion for optimizing the throughput is that we minimize

C =E A)(g-c)2 +E(ti-c)2) (S. 1)

where the expectation is taken over all the stationary random sequences {c} and {r}. Clearly the gap at the finishing mill rollers between the kth and (k - 1)st slab is

(U4,k - V4,ki-) and the stage times for the kth slab are (Si+l,k -Si,k) so that 3

C = E (4k _V,- ) S+, Sik-CO) 2 (5.2)

i=i




Substituting the relevant expressions for U4,k and V4,k-l and then expressing all the Sik in terms of the random errors we find that C is made up of two components, namely,

C = Cn+ Cr, (5.3) where

Cn = Alk + IL4k - 4,k-1 -b4,k-1 - C)2 + (I12k - C02 + (it3k - IL2k -C2)2 + (P4k - 3k -C3)2

(5.4) and (in an obvious notation)

Cr = E [A { [xi + ( - U) as] Ei + [c4 + - U) (84 + 1)] E4

3 \2 + [Oi/+3 - U)+Xi] + [04+( E- U)(X4I- US4)

4 2 4 w2 +iN [.ici++ii]; +i [(Yi _ i)Ei + (oi_ i-+)i

+ Pi [8- y')ci+ (Xi - 070] ]* (5.5)

C. is that part of C due to non-random factors and can be made zero by suitable choice of yik (i = 1, ..., 4). In fact Cn = 0 provided

L2kC =C

I-3k = Cl + C2, (5.6)

I14k = Cl+C2+C3,

H1k = c+b4,k- I

Moreover, the average size of the gap at the finishing mill is easily seen to be the value of c.

These results are intuitively obvious. The only way in which non-random factors affect decisions about a slab is that the time between pulling two successive slabs from the furnace is updated by the time which the first of the two slabs is expected to take in the finishing mill rollers.

We now investigate the main part of the problem, which is the minimization of Cr, that part of C due to random errors, and we begin by considering the case where the errors are completely uncorrelated with one another.

6. DECISION RULES FOR UNCORRELATED VARIABLES

In this section we determine the transfer functions which minimize Cr in the case when there is no correlation between any of the errors involved.

We adopt the same notation as Whittle, namely:

I y(Z) 12 = y(Z) y(Z1),

00

7(Z)]Xt = E y z', [y(Z)]+ = [y(z)]o, j==m

.s`[y(z)] = Yo




When the errors are uncorrelated we see that

=a + = [X (|+(i-Z)x8j2u+2 +4?(12Z) (84z+x1) 12 I224

3 + , l*+-Z) X, 12 vi+2 Z4(Z X-12 04)

4 + I{li,12+lyi-Pl2+l1 i-ril2}ue

4 + X{bkil2+khi-kil2+lXi-~I4il2Ik72j. (6.1)

Minimizing Cr with respect to cx8(z), fl8(z), y8(z) and 38(z) for s = 1, 2, 3, we immediately obtain as(z) = IRS(z) = y8(z) = a8(z) = 0 for s = 1, 2, 3, and so none of the errors at the first three stages are used in modifying the insertion times for the slabs. This result is true, however, only while the process is behaving normally so that there is a continuous flow of slabs. Should a slab be held up at any stage for an abnormal length of time this would naturally have to be taken into account and succeeding slabs would have to be delayed until the blockage was cleared. Since we are constructing a signalling system which the operators can over-ride in a case of emergency, we shall omit any consideration of abnormal behaviour.

To perform the minimization with respect to x4(z) ... 8z), we consider increments doX4, ..., d34 and equate coefficients of dc4, ..., d34 to zero. Remembering that 014(z) = 3 ct4j zi, etc. we obtain the equations

A[o(4+(1 -z) 84+(1 -Z)]3 = 0,

[14-(74-134)L2 = Os, (6.2) Y4 - 4) - (84 -Y4)]1 = 0,

[(1 -z'-){o4+(l -Z)(84+ 1)}+(844-Y4)]1 = 0,

which can be written

[ A 0 0 A(1-z) - i4 [ 1-z1 0 2 1 09 [0 [ 1 2 - 1 0 A 0 ?v , (6.3)

(1 _z -1) o -1 1+AJ1-_Z2 84 1 1 _Zl2

where the elements of v are polynomials in z of degrees at most 2. Taking into account the forms of (X4, 34, y4 and 64, we see that v is necessarily of

the form

Vio + vilz + V12Z2 0Vo vl V12 1

V21 Z 0 V21 0 (6.40 1 V4,, 0 0 0




Using an obvious notation, equations (6.3) and (6.4) can be written

Mp = u+V (6.5) and

v=V zl. (6.6)

The matrix M(z) can easily be factorized in the form

M(z) = R'(z-1) R(z), (6.7) where

-A O O 4(A) (1 -Z)- 0 1 0 0

R(z) = 0 -1 I 0 (6.8)

_ O I I Z)]

with inverse

= 1/VA -(1-Z) -(1 -Z) (6.9)) 0 1 0 0

{R(z)J-1 0 1 1 0 .(6.9)

_0 1 11 _

Equation (6.5) becomes

R'(z-1) R(z) p = u + v, so that

R(z) p = [{R'(zl)}> (u + v)]+, and finally

p = {R(z)}-' [{R'(z-1)>-l (u + v)]+. (6.10) This formal solution holds in the general case when there are m stages. Moreover, the matrix factorization is still easily achieved even if we wish to consider the more general control function (applicable to an m-stage process),

C = E{((g-C)2+ zWi(ti-Ci)2}, (6.11)

where wl, w2, ..., wm are arbitrary weighting factors. Returning to the four-stage process, the vector v or equivalently the matrix V

has to be chosen so that p has the required form, i.e. so that

cx4j = 0 (j =0, 1, 2),

Y4j=0 (i= 0).

81j =O (j=O).




Some of the above conditions are automatically satisfied by virtue of the form of V in equation (6.6). It is convenient at this stage to write

{R(z))-1 = Ro+zRL. (6.12)

Substituting for R, u and v in the explicit solution given by equation (6.10) we find, after some simple computation,

P=F? 0 +P Z21 (6.13)

The (4 x 4) matrix P is given by

P =ROH+R1Hfl, (6.14)

where H = (R-V+R RVT) S (6.15)

with

01 10001 ~1 0

0 0 0 1 0 0 1 0 101

T= I 0 0 , S= 0 1 0 0 , 000 O O 1 (6.16) 0 1 0 LO 0 1 OJ _1 0 0 ?_

Choosing V so that p is of the required form we find after some elementary computations that

[4(Z)] 0 0 6A 94(Z) A 0 0 2A 0 z

Y4(z) (A+ 1)(6A+ 1) 0 3A+ 4A 0 z2. (6.17)

_84(Z)_ _0 6A+2 6A 0 3

In an analogous manner we find that

04(z)] [0 0 0 6A- [I1

I04(Z)I A 10 0 2A 0 I IzI18 4(Z)] 6A2+ 1OA+2 0 0 4A 01 l Z2 (6.18)

X4(Z) 0 6A+2 6A 0 3

The above matrices differ only as regards the elements in the third row and second column. This arises from the fact that in making decisions at stages 1, 2 and 4 we have assumed the most recent E and -q errors at the finishing mill relate to the same slab, whilst in making the decision at stage 3 we have assumed that the slab is still




being processed in the finishing mill so that its E-error, but not its -j-error, has been observed. The decision rules in the case of uncorrelated variables can now be written

Slk k-1 c + b4,kl F 11 [1 1 ] ~~~~~~~~~~~E4,k1J j 74,k-1 S21 = k Slk , 1+A| 4,k2 +B (6.19) SUI I'l l+C E4,k-2 '94,k-2

[S4k Slk +

| C2 + C3 | 4.k-3 4,k-3

where

F 0 0 6A1 0 0 6A1

A= A 0 2A 0 A lo 2A 01(.0 (A+1)(6A+1) 3A+1 4A 0 l 6A2+?10A+2 0 4A (6.20)

6A+2 6A 0 [6A+2 6A 0

The decision rule (6.19) constitutes the solution to our problem in that it determines the unique way in which the slabs have to be fed into the various stages so as to minimize the expected value of the loss function. From the point of view of the operators who have to implement these rules, however, it would be more convenient to relate insertion times at any stage to the preceding insertion time at this stage. Equation (6.19) can easily be modified to this end and we find

[Slk [lk1k Li b4,k-1E-l~4,- I~~ _ 1] / ~~~~b4k ' 4 F 1 L9,- |

52k 1 52,k- + + I +A* E4,k2 +B 4 (6.21)

S4kJ S4, k-1 LCJ Lb4kJ E4,k3_3 -4,k-3

where the (4 x 3) matrices A* and B* are given by

0 0 6A- F 0 0 6A]

_ A F O 2A 4A

_ A 10 2A 4A

(A+1)(6A+1) [3A+ A-i 2A B*A 6A2+ 1OA+2 [ 0 4A 2A

6A+2 -2 0 6A+2 -2 0 (6.22)

The equation (6.21) can be written in an obvious symbolic manner, namely

Sk-Sk-1 = (c + b4,k1) 1 + A 4,k-1 + BY4,k-1, (6.23)

where the vectors s7, and 1 are of dimension 4 whereas the vectors 64,k-l and 1 4,k-1 are of dimension 3.

The above form of the decision rule has a very simple structure and the terms on the right-hand side have an obvious interpretation: (i) c + b4,k1 is just the sum of the average gap c and the expected time b4,k-1 for

rolling the (k - I)st slab in the finishing mill, and (ii) A*c4,k1++B*_14,k--1 iS the unique feedback of errors in the "response" times and

"process" times which minimizes the control function.



1967] DOWSON- Optimization of a Hot Rolling Mill 311

The relation (6.23), which is the main result in this section, can be written purely in terms of the observables {U4k}, {V4k} and {Sik} (i = 1,2, 3,4). To do this we write

64,k = U4,k-S4,k-a4, (6.24)

'94,k = V4,k U4,k b4,k,

or equivalently for vectors (of dimension 3) C4,k = U4,k - S4,k - a4,

Y4,k = V4,k - U4,k - b4,k) (6.25)

The decision rules then take the alternative form Sk - Skl - (c + b4 k-l) 1 = (A* - B*) U4,k-1 + B*v4,k-1 - A*s4,k1 - A*a4 -B*b4,k1,

(6.26) where the vectors on the left-hand side are of dimension 4 while those on the right- hand side are of dimension 3. Some care must be taken to distinguish Sk-l and S4,k-1 and in fact

FSl,k-1 S4k1 S2k1 while

Sk--1 = [ ] while S4,k-1 = S4,k-2 S3,k-1

S4,k-1 S4,k-3

Having now found how best to feed slabs into the different stages, it only remains to evaluate the variances of the gap and the overall process time as functions of the parameter A. With this information we can choose a value of A which effects a balance between controlling the gap and controlling the overall process time. Having decided on a suitable value of A, the decision rules are then completely determined.

7. VARIABILITY OF GAP AND PROCESS TIME In this section we calculate the variance V1 of the gap and the variance V2 of the

total time taken by a slab to complete the whole process. Using equation (6.19), it is easily seen that the variance of the gap is given by

Vl = )[ 1 04 + - Z) (84 + )|f4+1 04+0 -Z)X4 Zl ?I@(71 Substituting from the decision rules for OC4, 84, 04 and X4 we find after some elementary computations that

______9A2_10____ 2 (17A2+8A+ 1) C2 (7.2) = (A+ + 1)2 (6i + 1)2 6 (3A2+ 5A+ 1)2) /7

The reason that the coefficient of Cu2 contains the unit term not occurring in the coefficient of C2 is that the gap is affected by the "response" type of error E4k but not the process type of error -4k for the second slab.

The processing of a slab is effectively completed when a slab leaves the finishing mill rollers which occurs at a time V4,k. The slab undergoes cooling as soon as it starts to emerge from the furnace and this occurs at a time U1,k. The total processing time is therefore given by

t = V4,k-Ul,k

= constant + {1 + 84(U)}E4+{1 +X4(U)} 4- El- (7.3)




F'rom this it easily follows that the variance of the total processing time t is given by

4A2 18(2+4 6A j )} 2+(+A2(18A2 +6A + 1)) 2 (.4 V2 = (1+ 4(A+ 1 2 (6A + 1)12) ae + + + (3A2A + A6 + 1)2 ?1 +

a26, (7.4)

We write V1 in the form V1 = a, a2 +b, ua and V2 in the form V2 = a2 ?+b2 a2 + a and in Table 1 tabulate values of a,, bl, a2 and b2 for a series of values of A. Fig. 3 is a display of the variance of gap size and variance of total process time as functions of A for the particular case a., =eu64= ar,4 (V1 and V2 are measured in units of the common variance of the errors).

TABLE 1

Components of variances V1 and V2 for uncorrelated errors

V1 = a, a,24+bl aX,2 V2= a2a.2,4 + b2 a,, + ae2 A a, b1 a2 b2

0.0 2-00 1P00 1P00 1P00 0-5 1 37 0-51 1.24 1P11 1.0 1P20 0-32 1P51 1P31 1P5 1P13 0.22 1P73 1P49 2-0 1.09 0-16 1P89 1P64 3-0 1-05 0.10 2-13 1P88 4.0 1.03 0-06 2.28 2-05 5-0 1P02 0-05 2-39 2.18 6-0 1.02 0-03 2-47 2.28 00 1.00 0.00 3 00 3.00

Variance

6 V2

4

2

. I I ~ ~ ~ ~ ~ ~~I I I 1 _ A

0 1 2 3 4 5 6

FIG. 3. Variances of gap and process time for uncorrelated errors.




8. DECISION RULES FOR CORRELATED VARIABLES In Section 6 the decision rules were determined on the assumption that all of the

errors involved were completely uncorrelated with one another. It is possible, however, that the sequence of errors occurring at a particular stage are in fact correlated and in this section we shall determine the form of the decision rules for a particular kind of correlation function.

A reasonable model is to assume that the error at any stage is composed of two components, one of which forms an uncorrelated sequence whilst the other forms a correlated sequence possessing an exponential type of autocorrelation function. More specifically, for an E-type error we assume that

Ek = + c(2) (8.1)

where (i) {e(1)} is a sequence of uncorrelated random variables with variance C2(1), (ii) {E(2)} is a sequence of correlated random variables with covariance function

u92(2) exp (-P pj ) so that U2(2) is the variance of {E(2)},

(iii) {E(1)} and {E(2)} are mutually uncorrelated. The covariance generating function for {Ek} iS then

00

g6(Z) = E E(EkEk-j)Zj 2 = _O0

= g( 1) (z) + g( 2)z

= 02(12) + 2(2)(_pZ)(1_pZ-1 UeiJFUY4( pz)(I1- pz')

(X2(1-Vz) (1-vz'-) (8.2) (1-pz)(1pZ-lz) '

where a2 and v are constants defined by the relations

(J2(1 + V2) = ur2(1) (1 + p2) + (J2(2) (1- p2)

J2 V = aJ2(l) p, ) (8.3)

IvI<1. J

Note that the variance of an E-process is given by X2(1) + (2(2) and not by the constant or2

With these assumptions as to the nature of the errors we find, as before, that the transfer functions ax8(z), fl8(z), y8(z) and 85(z) (s = 1,2,3) which minimize C are identically zero. Thus even when the errors possess some autocorrelation it is only the errors at the finishing mill which are important; a result which was to be expected. By considering the minimization of C with respect to 04(Z), :4(Z), y4(Z) and 84(Z)

(which constitute the elements in the vector p), we obtain the minimizing equations

(Mp-u)g,(z) = v + h, (8.4)

where v is a vector whose elements are polynomials in z of degrees at most 2, 1, 0 and 0 respectively and h is a vector whose elements consist wholly of negative powers of z.




In this case the matrix V defined by equation (6.6) is of the form

- V10 Vll V12

V= V20 V21 ? (8.5) V30 ? ?

_V40 ? ?

Writing g6(z) in its canonical factorized form

g6(z) = U21 0(Z) 12, (8.6) where

+6(z) = (1 - vz)/(1 - pz), (8.7) we see that the solution of (8.4) is

p = {0I,(z)}-' {R(z)}-' [{R'(z-')}--' {fe(z) u + 0-1 (z-1) v}]+. (8.8)

Substituting for R, u, v and +6(z) we find

p=( 0 + l-Zp [Z2' (8.9)

where P=ROH+R,Hfl (8.10)

and H = [ +{R'+(v-p)R'}VT+(v-p){vRo+R'}VT2]S (8,11)

Choosing V so that p has the required form we obtain

- N4(Z)- - -1+z- - 1 -{1 +v-p} (1 -p) (V-p) 3)u- - 1 :4(Z) O l-pz O O /11 ? z

74(Z) = O +l--vz O A 2t01 0 Z2 _ 11F?]

84(Z) 0 0 2A, 3t4 ? Z3

(8.12) where A1 and ,uj are constants given by

(A )(6A+ ) [3A+ 1 +(V-p){3A+2-P}] (8.13) and

= (-A+1)(6A+1) [2A+(v-p){2pA-1 +P}]. (8.14)

The minimization of C with respect to the functions AS(z), 0b(z), 04(z) and X8(z) is performed in an exactly similar manner. As before, we find the only errors that need to be fed back are those occurring at the finishing mill, i.e.

0,(z) = k8s(z) = 0,8(z) = X,(z) = 0 (s = 1, 2, 3).




Denoting the vector with elements 04(Z), 04(Z), 04(Z) and X4(Z) by q we find that the minimizing value of q satisfies the equation

q=[0 l+ll-Pz (8.15)

where the (4 x 4) matrix P is again defined by equations (8.10) and (8.11), although this time V has the structure

V10 Vll V12

V20 V21 0 .

'v [~0~1 o (8.16) V30 V31 ?

V40 0 0 -

Choosing V so that q is of the required form we find

[04(Z) z 0 -1 v-p 3PZ2 1

04(Z) - l l -Pz 0 0 lo 2 ? (817)

04(z) 0 -vz 0 0 2tZ2 0 Z2.7 I X4(Z) I L 0 _L0 2A2 3P-2

where the constants A2 and t2 are given by

2A2= 3A2+ -A+ [3A+ 1 +(v-p) A], (8.18)

lL2 = 3A2 [A-(v-p) (A+ 1)]. (8.19)

The transfer functions given by equations (8.12) and (8.17) determine the unique way in which the insertion times Sik have to be modified as a result of errors in the system. Although we have used the same p and v for both the E and -j processes, the results in equations (8.12) and (8.17) are completely independent and still apply if we use Pl, v1 for the E-process and P2, V2 for the -q-process.

In the case where the {E} and {Xq} processes at the fourth stage have the same values of p and v, the presence of the term (1 - pz)/(l - vz) in (8.12) and (8.17) shows that instead of calculating the interval times Slk - Sl,k- S2k - Slk, etc. from all "past" errors it is easier to form a recursive relation of the form

Slk S1,k-1L llk E [4,k-1 74,k-1

(1-vU) 52k Slk =2 C j4,k-2 +D I4,k-2* (8.20)

53k Slk t i3k [ _4,k-3 [74,k-3

S4k - Slk k4k - _4,k- 4 4,k-4-




The matrices C and D are easily computed and found to be

0 O 0 3y,1-p(l-p)(v-p) -3p,wk

1k P1ki 0 C = O pA -P1 (8.21a)

Al 2p, - pAl - 2pttl ?

_ 2A, 3[u1-2pAl - 3P1k 1 and

0 0 31Z2-P(V-P) -3P12 -

D = 0 H2 -PP2 0 (8.21b) o 2HL2 -2P2 ?

_ 2A2 3I2-2pA2 3PW2 ? _

Again it will be more convenient to give the decision rules in terms of the intervals at any stage and we correspondingly obtain

[Slk 1 S1,k-l b4,k-1 'E4,k-1 -

74,k-1

(1-v U) t |S2k S2,k-k1 b|E4k-2 | 74k2 (8.22) S3k S3,k-1 b4,k-1 E4,k-3 K14,k-3

-S4k- -I4,k-1- -b4,k-1 ) L E4,k-4l- Lq4,k-4-

where the matrices C* and D* are given by

0 0 3[t,-f -3ptkl

C*-O 2 +p)A1 (2-p) -3P1k- (8.22a) Al 21i- (1 +p) Al (l-2p) tj+ pAj- - 3 -3P

_ 31k - 2(1 + p) A1 - 3Pl + 2pA1- - 3p I and

[0 0 3[Z2-4 -3PP12

D* - 0 /2 (2-P) P2-4 -3 2 (8.22b) 0 2[2 (I -2p)t2- - 32

L2A2 3iU2-2(l +p) A2 -3PP2 + 2pA2- -13P12-

with e = p(l -p) (v-p) and ? = p(v-p). The recurrence relation (8.22) for the insertion times can be written in vector

notation, namely,

(1 - VU) {Sk - Sk1- b4,k-1} = C*4,k-1 + D* j4,k-1, (8.23)

where all the vectors are of dimension four. This recurrence relation enables the insertion times Sk at each stage to be calculated from the previous insertion times Sk-l, the time b4 k-1 that the (k- l)st slab is expected to take in the rolling mill and the E and -7 errors at the finishing mill for the four preceding slabs. Had we




taken different (p, v) for the e and X processes then equation (8.23) would have to be replaced by

(1 - Vl U) (1-1V2 U) {Sk-Sk-1-b4,k-ll}

= (1-V2 U) C*e4,k1+(l -VU)D*8Y4,k-l, (8.24)

where in the matrices C* and D* we take care to use values of p and v appropriate to the e4-process and p4-process respectively.

In Section 9 we evaluate the variances of the gap and the total time of processing.

9. VARIABILITY OF GAP AND PRocEss TIME FOR CORRELATED VARIABLES In this section we shall obtain the variances of the gap and process time for the

most general case considered, which is when the sequence of E-errors at the finishing mill is autocorrelated as in (8.1) with parameters a6(1), cr6(2) and Pl, whilst the corresponding sequence of 77-errors is similarly autocorrelated with parameters cr(l), a,c(2) and P2. The variance of the gap is then given by

V = [lx4 + (1-Z) (S4 + 1)12 |_ | a2 + |Z4+(1-Z)X4-Z12 2 I Z| (9.1) Vl=q/ a4+'-Z(54 I p1z 1 0+'ZX -P2z j -

where (cr(, vl) and ((2, v2) are calculated from equations of the form (8.3) with values of the parameters appropriate to the E-sequence and n-sequence respectively. On substituting for the transfer functions from equations (8.12) and (8.17) we quickly find that

V1 = [1 + {2A1-(1 + v - p)}2 + {3- - 2A1 + (1 - pl) (Vl - p0}2] u

- {(1-2A2)2 + (32-22 + v2 - p2)2} r2, (9.2)

where A1 and p, are calculated from equations (8.13) and (8.14) with p and v replaced by Pi and vl, whilst A2 and P2 are calculated from equations (8.18) and (8.19) with p and v replaced by P2 and v2. Values of the variance V1 are tabulated for a number of cases in Table 2. For simplicity of display we have taken the parameters of the E-sequence and 71-sequence to be the same, namely,

u,(1) = a,(1) = u(1), r(2) = a,(2) = ur(2), (9.3)

Pl= P2= P ) Moreover, we have taken a2(1) + cr2(2), which is the common variance of the sequences of errors, to be unity so that the tabulated values of V1 give the variance of the gap measured in units of variance of the errors.

We consider now the process time t which is given by

t = constant+{1 + 84(U))E4 +{1 +X4(U)} 74-'El (9.4) From this equation, and equations (8.12) and (8.17) for the transfer functions, we find that the variance V2 of the process time is given by

V = 1 [ | 1 _ p% Z + 2 4 Z + H i Z2 | 2 + cr2 (9 .5 ) i=1 I - Pi z




This can be evaluated by the method of residues and we find that

V2= 1 T_ ) +2+ 2 (9.6)

The coefficients 1j, mi, ni and pi (i = 1, 2) can be obtained from the equations

I = {(1 -vp) + p(l -p2) (2A + 3,p)} (p-v), m = (3,u-2pA)(v+3p,t-2A)+v-2A, (9 7) n = (1 + p2){3p,u(2A-v)-3ytt+2pA}, p = (1+p2+p4) 3pv,

by giving (A, {t, p, v) the values (Al, j, Pl, v1) and (A2, P2 P2' V2) respectively.

TABLE 2 Variance of gap for correlated errors

P 0 2 0-4 0 6 0 8 0-2 0-4 0-6 0-8 A\

a2(1) = 0-8, a2(2) = 0-2 a2(1) = 0-6, d2(2) = 0 4 0 2-92 2.84 2-75 2-65 2-84 2-67 2-49 2-27 1 1-52 1.51 1-48 1-44 1P51 1-47 1-40 1-28 2 1P25 1P24 1P23 1P19 1P25 1-22 1-17 1P08 3 1P15 1P14 1P13 1P10 1P14 1-12 1P08 1 00 4 1P10 1.09 1 08 1P05 1.09 1P08 1P04 0 96

Ca2(1) = 0.4, 2(2) = 0'6 ar2(1) = 02, a2(2) - 08 0 2-76 2-51 222 1P86 2-68 234 1-94 1-43 1 1-49 1-42 1P29 1-09 1-48 1-35 1P14 0.85 2 1-24 1P18 1P08 0-92 1P23 1.13 0 97 0-73 3 1P14 1P09 1P01 0.86 1-13 1P05 0 90 0 68 4 1P09 1*05 097 082 1P08 1P01 087 065

TABLE 3

Variance of process time for correlated errors

p 0 2 0*4 0-6 0-8 0-2 0 4 0-6 0-8

Cr2(1) = 0-8, cr2(2) = 0-2 (a2(1) = 0-6, r2(2) = 0 4 0 3.00 3.00 3-00 3.00 3-00 3.00 3-00 3.00 1 3 90 3.98 4 07 4-11 3-97 4.12 4-25 4-26 2 4 64 4-75 4-85 4-90 4-73 4 92 5 07 5-04 3 5-12 5-24 535 540 522 5.43 558 5.53 4 5-45 558 569 5-74 556 577 592 586

0r2(1) = 0-4, a(2) = 0-6 0r2(1) = 02, r2(2) = 0.8 0 3-00 3-00 3-00 3-00 3-00 3-00 3-00 3-00 1 4-03 4-24 4-38 4-31 4 09 4-33 4.49 4-27 2 4 81 5.06 5 20 5.05 4-89 5.16 5-25 4.92 3 531 5.57 570 5'50 539 568 5-73 5-30 4 5-65 5.92 6 04 5 80 574 6&03 6&06 5 56




Values of V2 are shown in Table 3 for the same cases for which V1 was calculated (u2 is also taken to be unity).

In every case the general dependence of V, and V2 on the weighting factor A is similar to that shown in Fig. 3. For the more highly autocorrelated sequences, the errors are more predictable and consequently zhe smaller the variance of the finishing mill gap.

10. CONCLUSION Equations (6.21) and (6.22) in the case of uncorrelated errors and equations (8.20),

(8.21a) and (8.21b) in the case of correlated errors constitute the main results in this paper. These equations define the unique way in which the insertion times at the different stages have to be modified as a result of errors occurring in the stage times for preceding slabs. The result for uncorrelated errors is extremely simple and, for the particular mill considered, the insertion times for a slab are affected only by the errors from the three preceding slabs. This is particularly pleasing since preliminary evidence suggests that the assumption of uncorrelated errors is reasonably good so that the amount of computation involved in controlling the whole process is then quite small.

ACKNOWLEDGEMENTS I wish to thank Mr P. Sandeman for suggesting this problem and Professor Whittle

for many helpful discussions during the preparation of this paper.

REFERENCES NEWTON, G. C. (1952). Compensation of feedback control systems subject to saturation. J.

Franklin Inst., 254, 281-286, 391-413. NEWTON, G. C., GOULD, L. A. and KAISER, J. F. (1957). Analytical Design of Linear Feedback

Controls. New York: Wiley. WHITTLE, P. (1963). Prediction and Regulation. London: English Universities Press.



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Optimization of a Hot Rolling Mill