U.S. patent application number 11/528216 was filed with the patent office on 2007-03-29 for system for controlling a rolling mill and method of controlling a rolling mill.
This patent application is currently assigned to University of Pittsburgh - Of the Commonwealth System of Higher Education. Invention is credited to John Pittner, Marwan A. Simaan.
Application Number | 20070068210 11/528216 |
Document ID | / |
Family ID | 37906693 |
Filed Date | 2007-03-29 |
United States Patent
Application |
20070068210 |
Kind Code |
A1 |
Pittner; John ; et
al. |
March 29, 2007 |
System for controlling a rolling mill and method of controlling a
rolling mill
Abstract
The invention generally relates to a method of controlled
rolling of a metal strip moving through a rolling mill. The method
comprises: monitoring a plurality of input control parameters; and
generating a plurality of responsive operational control parameters
in the rolling mill to selectively control the rolling mill and
selectively control a plurality of output parameters of the rolling
mill. A control system for controlled rolling of a metal strip
moving through a rolling mill is also provided.
Inventors: |
Pittner; John; (Pittsburgh,
PA) ; Simaan; Marwan A.; (Pittsburgh, PA) |
Correspondence
Address: |
ECKERT SEAMANS CHERIN & MELLOTT
600 GRANT STREET
44TH FLOOR
PITTSBURGH
PA
15219
US
|
Assignee: |
University of Pittsburgh - Of the
Commonwealth System of Higher Education
|
Family ID: |
37906693 |
Appl. No.: |
11/528216 |
Filed: |
September 27, 2006 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
|
60721736 |
Sep 29, 2005 |
|
|
|
Current U.S.
Class: |
72/10.1 |
Current CPC
Class: |
B21B 38/04 20130101;
B21B 38/08 20130101; B21B 38/06 20130101; B21B 37/16 20130101; B21B
37/165 20130101; B21B 37/48 20130101; B21B 37/46 20130101; B21B
37/58 20130101; B21B 2001/221 20130101 |
Class at
Publication: |
072/010.1 |
International
Class: |
B21B 37/58 20060101
B21B037/58 |
Claims
1. A control system for controlled rolling of a metal strip moving
through a rolling mill comprising: a plurality of stands each of
which includes a plurality of work rolls and a plurality of backup
rolls associated with the plurality of work rolls; a plurality of
work roll devices that monitor speed of the work rolls and may
actuate speed of the work rolls which work roll devices are
associated with at least some of the plurality of work rolls; a
plurality of loading devices and a plurality of loading device
position monitors associated with at least some of the plurality of
stands; a plurality of load cells associated with at least some of
the plurality of stands; a plurality of tensiometers located
proximate to the metal strip moving through the work rolls; a
plurality of thickness gauges located proximate to the metal strip
moving through the work rolls, a plurality of metal strip speed
monitors located proximate to the metal strip moving through the
work rolls, wherein the work roll devices, the loading devices, the
loading device position monitors, the load cells, the tensiometers,
the thickness gauges and the metal strip speed monitors monitor a
plurality of input control parameters to the control system which
generates a plurality of responsive operational control parameters
in the rolling mill to selectively control the rolling mill and
selectively control a plurality of output parameters in the rolling
mill.
2. The system of claim 1 wherein the work rolls are rotating at a
certain speed, wherein the metal strip is moving at a certain
speed, wherein a load is applied to the metal strip, wherein
tension is applied to the metal strip between the stands, wherein
the metal strip has a certain thickness, and wherein the input
control parameters are selected from the group consisting of
monitoring a position of the loading devices, the speed of the work
rolls, the speed of the metal strip, the load applied to the metal
strip, the tension applied to the metal strip between the stands,
and the thickness of the metal strip as it exits some of the work
rolls.
3. The system of claim 1 wherein the work rolls are rotating at a
selected speed, and wherein the plurality of responsive operational
control parameters are selected from the group consisting of
adjusting a position of the loading device and adjusting the speed
of the work rolls.
4. The system of claim 1 wherein the metal strip has tension
between the stands, wherein a load is applied by the work rolls to
the metal strip, wherein the metal strip has a thickness, wherein
the metal strip is moving at a certain speed, and wherein the
output control parameters are selected from the group consisting of
adjusting the tension in the metal strip between the stands, the
load applied by the rolls to the metal strip, the thickness of the
metal strip and the speed of the metal strip.
5. The system of claim 1 wherein the responsive operational control
parameters are continuously calculated.
6. The system of claim 5 wherein a deviation in the responsive
operational control parameters is determined and the responsive
operational control parameters are adjusted and updated.
7. The system of claim 1 wherein the responsive operational control
parameters are calculated about every 2 to 5 milliseconds.
8. The system of claim 1 wherein tension is applied to the metal
strip between the stands, wherein tension in the metal strip is
partially controlled by the equation -R.sup.-1(x)B'K(x)x, and
wherein -R.sup.-1 is a matrix inverse of a control weighting
matrix, B' is a matrix transposition of a control matrix, K(x) is a
state dependent solution of an algebraic Riccati equation and x is
a state vector.
9. The system of claim 1 wherein tension is applied to the metal
strip between the stands, and wherein tension in the metal strip is
partially controlled by solving matrix equations.
10. The system of claim 1 wherein the metal strip has a certain
thickness, wherein the metal strip is moving at a certain speed,
and wherein the thickness of the metal strip is partially
controlled by taking a selected metal strip speed and dividing that
metal strip speed with another selected metal strip speed and
multiplying that product by the metal strip thickness at a selected
point which is then multiplied by a correction factor.
11. A method of controlled rolling of a metal strip moving through
a rolling mill comprising: monitoring a plurality of input control
parameters; and generating a plurality of responsive operational
control parameters in the rolling mill to selectively control the
rolling mill and selectively control a plurality of output
parameters of the rolling mill.
12. The method of claim 11 wherein the work rolls are moving at a
selected speed, wherein tension is applied to the metal strip
between a plurality of stands, and wherein the monitoring of the
input control parameters includes monitoring a position of the
loading device, the speed of the work rolls and the tension applied
to the metal strip between the stands.
13. The method of claim 12 wherein a load is applied to the metal
strip, wherein the metal strip has a certain thickness, and wherein
the monitoring of the input control parameters further comprises
monitoring the load applied to the metal strip and the thickness of
the metal strip as it exits some of the work rolls.
14. The method of claim 11 wherein the rolling mill has work rolls
moving at a selected speed, and wherein the selective control of
the rolling mill includes adjusting a position of a loading device
associated with a stand and adjusting the speed of the work
rolls.
15. The method of claim 11 wherein the metal strip has tension
between a plurality of stands, wherein a load is applied by the
work rolls to the metal strip, wherein the metal strip has a
thickness, and wherein the output parameters include adjusting the
tension in the metal strip between the stands, the load applied by
the work rolls to the metal strip and the thickness of the metal
strip.
16. The method of claim 15 wherein the metal strip is moving at a
selected speed, and wherein the output parameters further comprises
adjusting the speed of the metal strip.
17. The method of claim 11 wherein the responsive operational
control parameters are continuously calculated.
18. The method of claim 17 wherein a deviation in the responsive
operational control parameters is determined and the responsive
operational control parameters are adjusted and updated.
19. The method of claim 11 wherein the responsive operational
control parameters are calculated about every 2 to 5
milliseconds.
20. The method of claim 11 wherein tension is applied to the metal
strip between a plurality of stands, wherein tension in the metal
strip is partially controlled by the equation -R.sup.-1(x)B'K(x)x,
and wherein -R.sup.-1 is a matrix inverse of a control weighting
matrix, B' is a matrix transposition of a control matrix, K(x) is a
state dependent solution of an algebraic Riccati equation and x is
a state vector.
21. The method of claim 11 wherein tension is applied to the metal
strip between a plurality of stands, and wherein tension in the
metal strip is partially controlled by solving matrix
equations.
22. The method of claim 11 wherein the metal strip has a certain
thickness, wherein the metal strip is moving at a certain speed,
and wherein the thickness of the metal strip is partially
controlled by taking a selected metal strip speed and dividing that
metal strip speed with another selected metal strip speed and
multiplying that product by the metal strip thickness at a selected
point which is then multiplied by a correction factor.
Description
PARENT CASE TEXT
[0001] This patent application claims priority under 35 USC .sctn.
119(e)(1) to provisional patent application No. 60/721,736, filed
Sep. 29, 2005, the contents of which is hereby incorporated by
reference into this patent application in its entirety as if fully
set forth herein.
FIELD OF THE INVENTION
[0002] The invention generally relates to a system for controlling
a rolling mill and a method of controlling a rolling mill.
BACKGROUND OF THE INVENTION
[0003] The cold rolling of metal strip is a complex nonlinear
multivariable process whose optimization presents significant
challenges to developing control systems for rolling mills. In
general, the current technology relies on a structure wherein the
effects of interaction between process variables are partially
mitigated by single-input-single-output ("SISO") and
single-input-multi-output ("SIMO") control loops each operating on
a selected measured variable of the rolling mill. As such, the
overall control system is based on several separate single input
variable problems which have the objective of independent
adjustment of strip tension and thickness anywhere in the rolling
mill. While such a control system and variations of it have been
effective in producing an acceptable metal strip, other control
system design techniques for rolling mills may result in
improvements in performance and in robustness to uncertainties,
disturbances and the like found in a rolling mill.
[0004] Accordingly, a need exists in the art for an improved system
for controlling a rolling mill and a method of controlling a
rolling mill.
SUMMARY OF THE INVENTION
[0005] An object of the invention is to provide a method of
controlled rolling of a metal strip moving through a rolling
mill.
[0006] Another object of the invention is to provide a method of
controlled rolling of a metal strip moving through a rolling mill
that monitors a plurality of input control parameters and generates
a plurality of responsive operational control parameters in the
rolling mill to selectively control the rolling mill and
selectively control a plurality of output parameters of the rolling
mill.
[0007] Certain objects of the invention are achieved by providing a
control system for controlled rolling of a metal strip moving
through a rolling mill. The control system has a plurality of
stands each of which includes a plurality of work rolls and a
plurality of backup rolls associated with the plurality of work
rolls. A plurality of work roll devices are provided that monitor
speed of the work rolls and may actuate speed of the work rolls
which work roll devices are associated with at least some of the
plurality of work rolls. The control system also has a plurality of
loading devices and a plurality of loading device position monitors
associated with at least some of the plurality of stands. A
plurality of load cells are associated with at least some of the
plurality of stands, a plurality of tensiometers are located
proximate to the metal strip moving through the work rolls, a
plurality of thickness gauges are located proximate to the metal
strip moving through the work rolls and a plurality of metal strip
speed monitors are located proximate to the metal strip moving
through the work rolls. The work roll devices, the loading devices,
the loading device position monitors, the load cells, the
tensiometers, the thickness gauges and the metal strip speed
monitors monitor a plurality of input control parameters to the
control system which generates a plurality of responsive
operational control parameters in the rolling mill to selectively
control the rolling mill and selectively control a plurality of
output parameters in the rolling mill.
[0008] Other objects of the invention are achieved by providing a
method of controlled rolling of a metal strip moving through a
rolling mill. The method comprises: monitoring a plurality of input
control parameters; and generating a plurality of responsive
operational control parameters in the rolling mill to selectively
control the rolling mill and selectively control a plurality of
output parameters of the rolling mill.
BRIEF DESCRIPTION OF THE DRAWINGS
[0009] FIG. 1 is a schematic drawing of a rolling mill of the
invention;
[0010] FIG. 2A is a graph of rolling mill typical entry
disturbances at one hundred percent speed of the rolling mill;
[0011] FIG. 2B is a graph of rolling mill typical entry
disturbances at five percent speed of the rolling mill;
[0012] FIG. 3 is a schematic diagram of a system configuration;
[0013] FIG. 3A is a schematic diagram of an interstand tension trim
function;
[0014] FIG. 4 is a schematic diagram of a first stand output
thickness estimation of the metal strip;
[0015] FIG. 5 is a schematic diagram of a fifth stand output
thickness estimation of the metal strip;
[0016] FIG. 6A is a graph of response in mill exit thickness to
mill entry disturbances at one hundred percent speed of the rolling
mill without uncertainties;
[0017] FIG. 6B is a graph of response in mill exit thickness to
mill entry disturbances at five percent speed of the rolling mill
without uncertainties;
[0018] FIG. 7 is a graph of response in mill exit thickness during
deceleration from one hundred percent speed to five percent speed
with mill entry disturbance applied without uncertainties; and
[0019] FIG. 8 is a graph of response in mill exit thickness during
acceleration from five percent speed to one hundred percent speed
with mill entry disturbance applied without uncertainties.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
[0020] For purposes of the description hereinafter, the terms
"upper", "lower", "vertical", "horizontal", "axial", "top",
"bottom", "aft", "behind", and derivatives thereof shall relate to
the invention, as it is oriented in the drawing FIGS. However, it
is to be understood that the invention may assume various
alternative configurations except where expressly specified to the
contrary. It is also to be understood that the specific elements
illustrated in the FIGS. and described in the following
specification are simply exemplary embodiments of the invention.
Therefore, specific dimensions, orientations and other physical
characteristics related to the embodiments disclosed herein are not
to be considered limiting.
[0021] As used herein, the symbols listed below have the meanings
provided below.
SYMBOLS
[0022] i Subscript, stand i [0023] in Subscript, stand input
parameter [0024] out Subscript, stand output parameter [0025] S
Loading device position [0026] 0 Subscript, initial value of a
parameter, or its value at an operating point [0027] S0 Intercept
of approximation of mill stretch curve [0028] U.sub.V Work roll
speed actuator reference [0029] t Time (seconds) [0030] U.sub.S
Loading device position reference [0031] P Specific roll force
[0032] .tau..sub.V Work roll speed actuator time constant [0033] F
Total roll force [0034] M Mill modulus (used in model) [0035]
.tau..sub.s Roll gap actuator time constant [0036] M.sub.e
Estimated mill modulus (used in controller) [0037] x State vector
[0038] y Output vector [0039] R Undeformed work roll radius (when
used in model) [0040] u Control vector [0041] a(x) State-dependent
system vector [0042] R' Deformed work roll radius (when used in
model) [0043] A(x) State-dependent system matrix [0044] C(x)
State-dependent output matrix [0045] L.sub.0 Length between stands
[0046] B Control matrix L.sub.m1(5) Length from stand 1(5) to
thickness gauge [0047] J Performance index [0048] Q State weighting
matrix [0049] .tau..sub.d Interstand transport lag [0050] Q(x)
State-dependent state weighting matrix [0051] .phi..sub.n Angle at
neutral plane [0052] .phi..sub.1 Angle of contact [0053] R Control
weighting matrix (when used in calculation of performance index)
[0054] f Forward slip [0055] E Young's modulus [0056] R(x)
State-dependent control weighting matrix [0057] .nu. Poisson's
ratio [0058] H.sub.a Strip annealed thickness [0059] K Solution
(matrix) of algebraic Riccati equation [0060] h Strip thickness
[0061] W Strip width [0062] K(x) State-dependent solution (matrix)
of algebraic Riccati equation [0063] .mu. Coefficient of friction
[0064] k Yield stress (hardness) [0065] {overscore (p)} Mean value
of parameter p [0066] .DELTA.k.sub.{dot over (e)} Yield stress
offset due to strain rate [0067] [ ].sup.-1 Matrix inverse [0068]
.sigma. Tension stress [0069] [ ]' Matrix or vector transpose
[0070] .delta. Draft [0071] [ ].di-elect cons.C.sup.k Elements of a
matrix or vector have continuous partial derivatives of order k
[0072] r Reduction [0073] V.sub.in(out) Stand input (output) strip
speed [0074] V Work roll linear speed
[0075] Turning to FIG. 1, an exemplary rolling mill 10 is shown in
which aspects of the invention may be used. The displayed rolling
mill 10 is a tandem cold rolling mill, but other rolling mills and
alternate configurations of the rolling mill may also fall within
the scope of the invention. Tandem cold rolling is typically
performed after a hot rolling process in which metal slabs such as
steel, for example, are heated in a furnace and then rolled into
coils of reduced thickness suitable for further processing. After
hot rolling and prior to cold rolling, the hot rolled material
typically undergoes a pickling process wherein coiled strip is
unwound and passed through an acid bath to remove oxides formed
during hot rolling. Prior to recoiling, oil is applied to the strip
to prevent rusting, eliminate damage due to scraping of adjacent
coils and often to act as a lubricant for the first stand of a
rolling mill.
[0076] The rolling mill 10 has a coil 12 which is unwound and feeds
metal strip 14 into the rolling mill 10. The rolling mill 10
includes a plurality of stands or stations 16 through which the
metal strip 14 passes. While the exemplary rolling mill 10 shows
the use of five stands 16, the teachings of the invention are
believed applicable to rolling mills having two or more stands and
the depiction of five stands 16 in the patent application should
not be considered a limitation of the invention.
[0077] Each of the stands 16 may have a plurality of rotating rolls
or work rolls 18 and a plurality of backup rolls 20 associated with
the plurality of work rolls 18. Coupled to some or all of the
plurality of work rolls 18 is a work roll device 21 for selectively
monitoring the speed of the work rolls 18 and/or actuating the
speed of the work rolls 18. As depicted in FIG. 1, there is shown a
first set of work rolls 22 and a first set of backup rolls 24 at a
first stand 25, a second set of work rolls 26 and a second set of
backup rolls 28 at a second stand 29, a third set of work rolls 30
and a third set of backup rolls 32 at a third stand 33, a fourth
set of work rolls 34 and a fourth set of backup rolls 36 at a
fourth stand 37, a fifth set of work rolls 38 and a fifth set of
backup rolls 40 at a fifth stand 41. As the metal strip 14 passes
through each of the work rolls 18, the metal strip 14 is reduced in
thickness. The reduction in thickness is caused by high compression
stress in the roll gap which is defined as a small region between
the work rolls 18. In the roll gap, the metal is plastically
deformed and there is occasional slipping between the metal strip
14 and the work rolls 18. The energy required to achieve reduction
in the metal strip 14 thickness causes a temperature rise at the
roll gap, which is reduced considerably by the cooling effects of
air and/or use of a lubricant on the metal strip 14. Cold rolling
of metal strip 14 is typically done to reduce metal thickness,
improve the surface finish or to produce desired mechanical
properties in the metal strip 14 by cold working the metal strip 14
to make it suitable for the manufacture of various products.
[0078] As shown, the backup rolls 20 may have a plurality of
hydraulic devices, pneumatic devices, loading devices, screwdowns
or the like 42 to apply a preselected load to the backup rolls 20
against the rolls work 18. The loading devices 42 may selectively
actuate the force applied to the backup rolls 20. As depicted in
FIG. 1, there is shown a first loading device 44, a second loading
device 46, a third loading device 48, a fourth loading device 50
and a fifth loading device 52.
[0079] As shown, the stands 16 may also have a plurality of loading
device position monitors used to monitor the position of the
loading devices 42. The load applied to the work rolls 18 by some
or all of the loading devices 42 via the backup rolls 20 is an
operational control parameter that acts, along with some or all of
the work roll devices 21 to selectively control output parameters
of the rolling mill 10 such as tension in the metal strip 14
between some or all of the stands 16, the load applied by some or
all of the rolls 18 to the metal strip 14, the thickness of the
metal strip 14 as it exits some or all of the stands 16 and the
speed in which the metal strip 14 exits some or all of the stands
16.
[0080] As shown, the rolling mill 10 may have monitoring equipment
such as load cells 54 that monitor the load at some or all of the
plurality of stands 16. As depicted in FIG. 1, there is shown a
first load cell 56, a second load cell 58, a third load cell 60, a
fourth load cell 62 and a fifth load cell 64 associated with each
of the plurality of stands 16. Also, the rolling mill 10 may have a
plurality of tensiometers 66 located proximate to the metal strip
14 moving through the work rolls 18 that monitor the tension
between a pair of some or all of the plurality of stands 16. As
depicted in FIG. 1, there is shown a first tensiometer 68, a second
tensiometer 70, a third tensiometer 72 and a fourth tensiometer 74
associated between a pair of each of the plurality of stands 16.
Additionally, the rolling mill 10 may have a plurality of thickness
gauges 76 located proximate to the metal strip 14 moving through
the work rolls 18 that monitor the thickness of the metal strip 14
as it exits some or all of the plurality of stands 16. As depicted
in FIG. 1, there is shown a first thickness gauge 78 and a second
thickness gauge 80 that are respectively placed beyond the exit of
first stand 25 and a fifth stand 41. Additional thickness gauges 76
could be placed after some or all of the other stands 16 in order
to monitor thickness of the metal strip 14 throughout the rolling
mill 10. Also, the rolling mill may have a plurality of strip speed
monitors 81 located proximate to the metal strip 14 moving through
some or all of the work rolls 18 that monitor the speed of the
metal strip 14 as it moves in and out of some of the work rolls 18.
As depicted in FIG. 1, there are shown strip speed monitors 81
before the second stand 29, before the third stand 33, before the
fourth stand 37, before the fifth stand 41 and after the fifth
stand 41.
[0081] Some or all of the work roll devices 21, some or all of the
loading devices 42, some or all of the loading device position
monitors 43, some or all of the load cells 54, some or all of the
tensiometers 66, some or all of the thickness gauges 76 and some or
all of the metal strip speed monitors 81 may act as input control
parameters that selectively control the position of some or all of
the loading devices 42, the speed of some or all of the work rolls
18, the speed of the work rolls 18, the speed of the metal strip
14, the load applied to the metal strip 14 between some or all of
the rolls 18, the tension in the metal strip 14 between some or all
of the stands 16, and the thickness of the metal strip 14 as it
exits some or all of the work rolls 18. As the metal strip 14 exits
the rolling mill 10, the metal strip 14 is collected and coiled on
a rewinder 82. After cold rolling, the metal strip 14 may be
cleaned and annealed to restore ductility, which was reduced by an
increase in hardness and a decrease in formability caused by the
strain hardening of the cold rolling process.
[0082] The rolling mill 10 instrumentation generally consists of
various monitoring instruments such as work roll devices 21 that
measure work roll 18 rotational speeds and/or actuate work roll 18
rotational speed, position monitors 43 that measure the positions
of the loading devices 42, load cells 54 that measure roll force at
each stand 16, tensiometers 66 that measure interstand metal strip
14 tension force, thickness gauges 78 that measure strip thickness
at the exit of some or all of the stands 16 and strip speed
monitors that measure metal strip speed 14 as it moves in and out
of some of the work rolls 18.
[0083] A mathematical model of a cold rolling process was developed
to take account of certain input control parameters and certain
operational control parameters of a rolling mill 10. The
mathematical model is a group of expressions which relate the
rolling parameters of the rolling mill 10 to each other. The type
of model used for the invention described herein is one which
relates a plurality of input control parameters of the cold rolling
process that are significant in the development of a process
control strategy, and are capable of providing a plurality of
operational control parameters for dynamic adjustment of the
operational control parameters on a continuous basis of the rolling
mill 10 operation in a straightforward manner without being
computationally demanding. Accordingly, the relationships which
comprise the mathematical model are based on a series of algebraic
equations developed for control purposes by Bryant, C. F.,
Automation of Tandem Mills, British Iron and Steel Institute,
London 1973 as a simplification of more complex classical models,
and on empirical equations given in Roberts, W. L., Cold Rolling of
Steel, Marcel Dekker, New York, 1978.
[0084] The expressions which have been derived are given below for
specific roll force (1), forward slip (3), interstand tension
stress (6), output thickness (7), work roll actuator position
(i.e., loading device position) (8), work roll speed (9), and
interstand time delay approximations (10), with symbols as
previously defined, and with stand i understood where no subscript
is given. The expressions for input thickness, mean yield stress,
mean tension stress, mean thickness, friction coefficient, deformed
work roll radius, draft, stand output strip speed, and stand input
strip speed are given in Table 6 provided below.
[0085] The specific roll force is approximated in the zone of
plastic deformation in the roll gap area of a mill stand as
P=({overscore (k)}-{overscore (.sigma.)}) {square root over
(R'.delta.)}(1+0.4.alpha.), (1) where .alpha. = h out h i .times.
.times. n .times. exp .function. ( .mu. .times. R ' .times. .delta.
h _ ) - 1. ( 2 ) ##EQU1##
[0086] The term ({overscore (k)}-{overscore (.sigma.)}) in (1)
represents the contribution of the product hardness as reduced by
the effects of an average of the entry and exit tension stresses,
while the term {square root over (R'.delta.)} is the length of the
arc of contact of the work rolls with the product being rolled, and
the term (1.times.0.4.alpha.) is an amplification effect mostly due
to friction. The elastic recovery contribution is omitted from (1)
since it is small and therefore can be neglected.
[0087] The forward slip is a measure of the increase in the speed
of the strip exiting the roll gap area, with respect to the work
roll peripheral speed which is taken as the strip speed at the
neutral plane. The forward slip is approximated by f = ( .delta. h
out ) .times. ( .PHI. n .PHI. 1 ) 2 , ( 3 ) ##EQU2## where the
angle at the neutral plane and the contact angle respectively are
.PHI. n = 1 2 .times. h out h _ .times. .delta. R ' - 1 4 .times. h
out .times. .delta. h _ .times. .times. .mu. .times. .times. R ' +
1 4 .times. h out .times. .mu. .times. .times. R ' .times. (
.sigma. out k out - .sigma. i .times. .times. n k i .times. .times.
n ) , .times. and ( 4 ) .PHI. 1 = .delta. R ' . ( 5 ) ##EQU3##
[0088] An expression for the interstand tension stress is obtained
by applying Hooke's law to a length of strip between successive
stands, assuming some average thickness and neglecting any
stretching of the strip, as d .sigma. i , i + 1 d t .ident. .sigma.
. i , i + 1 = E .function. ( V i .times. .times. n , i + 1 - V out
, i ) L 0 , .times. .sigma. i , i + 1 .function. ( 0 ) = .sigma. 0
, i , i + 1 . ( 6 ) ##EQU4##
[0089] A linear approximation for the mill stretch characteristic
is used to estimate the thickness at the exit of a stand as h out =
S + S .times. .times. 0 + F M , ( 7 ) ##EQU5## where F=PW is the
total roll force and S0 is the intercept of the linearized
approximation.
[0090] The work roll actuator position controller (i.e. the
position controller for the loading device) and the work roll speed
controller are modeled respectively as single first order lags
based on typical mill data and experience, S . = U S .tau. S - S
.tau. S , .times. S .function. ( 0 ) = S 0 , .times. and ( 8 ) V .
= U V .tau. V - V .tau. V , .times. V .function. ( 0 ) = V 0 . ( 9
) ##EQU6##
[0091] The interstand time delay is the time taken for a small
element of strip to travel a distance L.sub.0 between successive
stands and is approximated at any instant of time as .tau. d , i ,
i + 1 = L 0 V out , i . ( 10 ) ##EQU7##
[0092] The theoretical system equations (1) through (9) and the
approximations (10) for the interstand time delays are put into the
form of a state equation (11) and an output equation (12), {dot
over (x)}=a(x)+Bu, x(0)=x.sub.0, (11) y=g(x), (12) where x.di-elect
cons.R.sup.n is a vector whose elements represent the individual
state variables, a(x).di-elect cons.R.sup.n is a state-dependent
vector, y.di-elect cons.R.sup.p is a vector whose elements
represent the individual output variables, g(x).di-elect
cons.R.sup.p is a state-dependent vector, u.di-elect cons.R.sup.m
is a vector whose elements represent the individual control
variables, and B.di-elect cons.R.sup.n.times.m is a constant
matrix.
[0093] The individual state variables, control variables, and
output variables represented by the elements of the vectors x, u,
and y respectively in (11) and (12) are as shown in Table 1.
TABLE-US-00001 TABLE 1 (a) State Vector, (b) Control Vector, and
(c) Output Vector Elements, Variable Assignments (a) x.sub.1
.sigma..sub.12 x.sub.2 .sigma..sub.23 x.sub.3 .sigma..sub.34
x.sub.4 .sigma..sub.45 x.sub.5 S.sub.1 x.sub.6 S.sub.2 x.sub.7
S.sub.3 x.sub.8 S.sub.4 x.sub.9 S.sub.5 x.sub.10 V.sub.1 x.sub.11
V.sub.2 x.sub.12 V.sub.3 x.sub.13 V.sub.4 x.sub.14 V.sub.5 (b)
u.sub.1 U.sub.S1 u.sub.2 U.sub.S2 u.sub.3 U.sub.S3 u.sub.4 U.sub.S4
u.sub.5 U.sub.S5 u.sub.6 U.sub.V1 u.sub.7 U.sub.V2 u.sub.8 U.sub.V3
u.sub.9 U.sub.V4 u.sub.10 U.sub.V5 (c) y.sub.1 h.sub.out1 y.sub.2
h.sub.out2 y.sub.3 h.sub.out3 y.sub.4 h.sub.out4 y.sub.5 h.sub.out5
y.sub.6 .sigma..sub.12 y.sub.7 .sigma..sub.23 y.sub.8
.sigma..sub.34 y.sub.9 .sigma..sub.45 y.sub.10 P.sub.1 y.sub.11
P.sub.2 y.sub.12 P.sub.3 y.sub.13 P.sub.4 y.sub.14 P.sub.5
[0094] Relationships which express P, h.sub.out, and (V.sub.in,
i+1-V.sub.out, i) as functions of the state variables are derived
in Table 7 provided below.
[0095] The resulting state space model was verified by open loop
simulations using three operating points similar to the typical
production schedules given by Bryant. Open loop simulations refer
to simulating the rolling mill 10 without feedback from the
monitored or estimated input control parameters of the rolling mill
10. The results were compared to Bryant's results provided in
Bryant, C. F., Automation of Tandem Mills, British Iron and Steel
Institute, London 1973. The results were also compared to Geddes'
results provided in Geddes, E. J. M., Tandem Cold Rolling and
Robust Multivariable Control, PhD thesis, University of Leicester
UK 1998. Geddes' results were based on reduction patterns similar
to Bryant's. Simulations were performed at mill exit speeds of
about 4000 feet per minute and at thread speeds of about 200 feet
per minute. The results showed good consistency with the results of
both Bryant and Geddes.
[0096] The pointwise linear quadratic optimal control strategy
evaluated for this invention is a pointwise application of the
state-dependent Riccati equation method which has seen several
recent successful applications in aerospace technology and other
areas for control of nonlinear dynamical systems. In the pointwise
linear quadratic method, the nonlinear plant dynamics are expressed
in the form {dot over (x)}=a(x)+b(x)u, x(0)=x.sub.0, (13) y=g(x).
(14)
[0097] By factorizing the state-dependent vectors a(x) into A(x)x,
g(x) into C(x)x, and with b(x)=B, the above becomes a form
resembling linear state space equations {dot over (x)}=A(x)x+Bu,
x(0)=x.sub.0, (15) y=C(x)x, (16) where A(x).di-elect
cons.R.sup.n.times.m is a state-dependent matrix, C(x).di-elect
cons.R.sup.p.times.n is a state-dependent matrix, and x, u, y, B
are as noted previously.
[0098] The optimal control problem is to minimize the performance
index J = 1 2 .times. .intg. 0 .infin. .times. ( x ' .times. Q
.function. ( x ) .times. x + u ' .times. R .function. ( x ) .times.
u ) .times. d t ( 17 ) ##EQU8## with respect to the control vector
u, subject to the constraint (15), where Q(x).gtoreq.0, R(x)>0,
a(x).di-elect cons.C.sup.k, Q(x).di-elect cons.C.sup.k,
R(x).di-elect cons.C.sup.k, for k.gtoreq.1.
[0099] Under the assumptions a(0)=0 and B.noteq.0, the objective
(17) is to find a control law which regulates the system to the
origin.
[0100] The method of solution is first to find a factorization of
a(x) such that (13) can be expressed in the form of (15). Then the
state-dependent algebraic Riccati equation
A'(x)K(x)+K(x)A'(x)-K(x)BR.sup.-1(x)B'K(x)+Q(x)=0 (18) is solved
pointwise for K(x), resulting in the control law used in the
control system of rolling mill 10 u=-R.sup.-1(x)B'K(x)x. (19)
[0101] As can be seen from the control law and Table 1, a plurality
of input control parameters of the rolling mill 10 are monitored
and/or actuated by using work roll devices 21, loading devices 42,
loading device position monitors 43, load cells 54, tensiometers
66, thickness gauges 76 and/or strip speed monitors 81 to
selectively control the position of the loading devices 42 of some
or all of the stands 16, the speed of some or all of the rolls 18,
the speed of the metal strip 14, the load applied to the metal
strip 14 between some or all of the rolls 18, the tension in the
metal strip 14 between some or all of the stands 16, and the
thickness of the metal strip 14 as it exits some or all of the
stands 16. Such input control parameters generate a plurality of
responsive operational control parameters in the rolling mill 10
for adjusting a position of the loading devices 42 and adjusting
the rotational speed of the work rolls 18 to selectively control
the rolling mill 10 and selectively control a plurality of output
parameters of the rolling mill 10 such as tension in the metal
strip 14 between some or all of the stands 16, the load applied by
some or all of the rolls 18 to the metal strip 14, the thickness of
the metal strip 14 as it exits some or all of the stands 16 and the
speed in which the metal strip 14 exits some or all of the stands
16. The control law is continuously calculated every 2 to 5
milliseconds in order to continuously monitor the rolling mill 10.
If a deviation in the responsive operational control parameter is
determined by the control law, the responsive operational control
parameters are adjusted and updated such as position of the loading
device 42 and adjusting the speed of the work rolls 18.
[0102] In order to ensure a solution to (18) at each point, the
method requires that the pair (A(x), B) be pointwise stabilizable
(in a linear sense) for all x in the control space, assuming the
availability of full state measurement.
[0103] Local asymptotic stability is assured if (A(x), B) is
pointwise stabilizable, if there exists a matrix C.sub.1(x) such
that Q(x)=C'.sub.1(x)C.sub.1(x), and if (A(x),C.sub.1(x)) is
pointwise detectable, assuming that A(x).di-elect cons.C.sup.k.
Global asymptotic stability must be confirmed by simulation since,
except for certain special cases, at present there is no useful
theory which assures it.
[0104] In general, the necessary condition for the optimal control
problem is not satisfied in the case of pointwise linear quadratic
optimal control. However, if each element of A(x), K(x), Q(x),
R(x), and each element of their partial derivatives A.sub.x(x),
K.sub.x(x), Q.sub.x(x), R.sub.x(x) is bounded for all x in the
control space, and under global asymptotic stability, then the
state trajectories converge to the optimal state trajectories as
the states are driven to zero. This is taken to be a near optimal
(i.e. suboptimal) condition.
[0105] The application of the pointwise linear quadratic control
technique to the tandem cold rolling process relies heavily on
physical intuition and simulation to develop and confirm a
controller design. This is mostly because no useful theory
presently exists which assures global asymptotic stability or
robustness. In addition, the process is large, is highly nonlinear
with complex interactions between variables, and has significant
time delays, which make estimations of performance and robustness
to disturbances and uncertainties difficult using analytical
methods.
[0106] As an example, an operating point using a typical production
schedule, plus the mill and strip parameters, are provided in Table
2 and in Table 3 below. TABLE-US-00002 TABLE 2 Operating Point Mill
Entry Thickness 0.140 in Exit Thickness, Stand 1 0.116 Exit
Thickness, Stand 2 0.096 Exit Thickness, Stand 3 0.079 Exit
Thickness, Stand 4 0.066 Exit Thickness, Stand 5 0.062 Tension
Stress, Mill Entry 0.0 tons/in.sup.2 Tension Stress, Stands 1, 2
5.6 Tension Stress, Stands 2, 3 5.7 Tension Stress, Stands 3, 4 5.8
Tension Stress, Stands 4, 5 6.0 Tension Stress, Mill Exit 1.8
[0107] TABLE-US-00003 TABLE 3 Mill and Strip Properties Work Roll
Radius 11.5 in Mill Moduli 10.sup.4 tons/in Distance Between Stands
170 in Strip Width 36 in Annealed Thickness/Mill Entry Thickness
1.095 Young's Modulus 30 .times. 10.sup.6 lbs/in.sup.2 Poisson's
Ratio 0.3 Long Tons 2240 lbs/ton
[0108] The initial state x.sub.0 at the operating point is an open
loop equilibrium point established by the control vector u.sub.0
whose elements are given in Table 1(b). The operating point is
shifted to the origin by introducing the variable z=x-x.sub.0.
Minimization of the performance index J is then with respect to the
vector u-u.sub.0, J = 1 2 .times. .intg. 0 .infin. .times. ( z '
.times. Qz + ( u - u 0 ) ' .times. R .function. ( u - u 0 ) )
.times. d t , ( 20 ) ##EQU9## where Q and R initially are taken as
diagonal matrices with tunable constant elements.
[0109] The most significant external disturbances are deviations in
mill entry thickness and mill entry hardness over a factor of time
as depicted in FIG. 2A at one hundred percent speed of the rolling
mill 10 and FIG. 2B at five percent speed of the rolling mill 10.
These disturbances result from the contact of hot metal slabs with
colder support skids in the reheat furnace and from roll
eccentricities in the hot rolling process.
[0110] The most significant internal disturbances resulting from
cold mill roll eccentricities, for example, are assumed to be
mitigated by an active roll eccentricity compensation scheme.
[0111] A disturbance changes the matrix A(x) by .delta.A(x) which
results in a disturbance effect .delta.A(x)x as shown in FIG.
3.
[0112] A control objective of the invention is to keep deviations
in individual stand 16 output thicknesses and interstand tensions
are as low as reasonably achievable in the presence of external and
internal disturbances applied during steady speed and during speed
changes. In addition, the stand exit thicknesses and the interstand
tensions must be independently adjustable.
[0113] In the pointwise linear quadratic technique, the algebraic
Riccati equation is solved on a pointwise basis. Disturbance
rejection is improved by adding an integrator function and a
proportional function to trim the control reference for the
position of the position actuator (i.e., the position of the
loading device) of each stand. These added trim functions produce
zero steady-state error in the control of the estimated individual
stand output thickness and reduce the effect of the interstand time
delay. In addition, a function was added to estimate the unmeasured
elements of the output vector y. Monitoring instruments to measure
the strip speed at the input of the second stand 29, the third
stand 33, the fourth stand 37 and the fifth stand 41, and at the
output of the fifth stand 41 were added to provide speed signals
for the estimation of strip thicknesses at the outputs of stand 29
through stand 41 using mass flow techniques, and for tracking of
strip thickness. Elements Q(1,1), Q(2,2), Q(3,3), and Q(4,4) of
weighting matrix Q were set during initial simulation to reduce
deviations in the interstand tension stresses. In addition, a trim
function .phi..sub.r for each interstand tension was added to
correct for slight offsets from the operating point. The control
law computed by the pointwise linear quadratic controller provides
signals to the loading device position controllers and the work
roll speed controllers for the final control of the tensions, so
that excursions in the tensions are significantly reduced which is
essential for the stability of rolling.
[0114] The system configuration is depicted in FIG. 3, where each
element of the state vector x is measurable, y.sub.m represents the
measurable elements of the output vector y, y.sub.e.di-elect
cons.R.sup.p (p=14) is a vector whose elements are the measured (or
estimated) elements of y, .phi..sub.y is an algorithm which
generates y.sub.e, V.sub.ini (i=2, 3, 4, 5) are the measured strip
speeds at the inputs of stands 29, 33, 37, 41, V.sub.out5 is the
measured strip speed at the output of stand 41, h.sub.out1m and
h.sub.out5m are the measured strip thicknesses at stand 25 and
stand 41, K.sub.1.di-elect cons.R.sup.m.times.p (m=10, p=14) and
K.sub.p.di-elect cons.R.sup.m.times.p are matrices whose elements
are zero except for elements (j,j), (j=1, 2, 3, 4, 5), which are
the gains for the integral and proportional trim functions for each
stand 16. .phi..sub.r is an algorithm which implements the
interstand tension operating point trims as shown in FIG. 3A, where
x.sub.op,i (i=1, 2, 3, 4) is an element of the vector x.sub.op
which represents the operating point for the interstand tension for
stands i,i+1, .phi..sub.i,i+1, ref is the interstand tension
reference for stands i,i+1, x.sub.i is the element of the state
vector which represents the measured interstand tension for stands
i,i+1, and K.sub.i,i+1 is a gain term for stands i,i+1. A direct
feed-through is provided for elements x.sub.op, i (i=5, . . .
,14).
[0115] The algorithm .phi..sub.y computes h.sub.out1e(y.sub.1e) as
an estimate of h.sub.out1(y.sub.1) using a British Iron and Steel
Research Association measurement h.sub.out1b (21) and as depicted
in FIG. 4, h out .times. .times. 1 .times. b = x .function. ( 5 ) +
S .times. .times. 0 + F 1 M e .times. .times. 1 , ( 21 ) ##EQU10##
where the notation h.sub.out1e(y.sub.1e) indicates that variable
h.sub.out1e is represented by element 1 of vector y.sub.e, and
similarly for other variables represented by the elements of y and
y.sub.e.
[0116] The effects on h.sub.out1b of roll eccentricity and the
uncertainty in M.sub.e1 are addressed in the sequel. The time delay
from stand 25 to the thickness gauge is approximated as a.sub.e
using V.sub.in2 and L.sub.m1.
[0117] Thickness h.sub.out2e(y.sub.2e) is computed using V.sub.in2,
V.sub.in3, and h.sub.in2e as h out .times. .times. 2 .times.
.times. e = V i .times. .times. n .times. .times. 2 V i .times.
.times. n .times. .times. 3 .times. h i .times. .times. n .times.
.times. 2 .times. e .times. k 2 .times. e , ( 22 ) ##EQU11## where
h.sub.in2e is h.sub.out1m delayed by the transport lag from the
thickness gauge to second stand 29, and k.sub.2e is a correction
factor for small errors such as changes in thickness caused by
spreading, reductions in width, or other effects, which is computed
by a separate mill adaptation system which is not a part of the
controller.
[0118] Thicknesses h.sub.out3e(y.sub.3e) and h.sub.out4e(y.sub.4e)
are determined similarly, except that tracking is from the previous
stand 16. Thickness h.sub.out5e(y.sub.5e) is obtained as depicted
in FIG. 5, where h.sub.out5b is computed as h out .times. .times. 5
.times. b = V i .times. .times. n .times. .times. 5 V i .times.
.times. n .times. .times. 5 .times. h i .times. .times. n .times.
.times. 5 .times. e .times. k 5 .times. e , ( 23 ) ##EQU12## where
h.sub.in5e is h.sub.out4e(y.sub.4e) delayed by the time delay from
the fourth stand 37 to the fifth stand 41, and the time delay from
the fifth stand 41 to the thickness gauge is approximated as be
using V.sub.out5 and L.sub.m5.
[0119] Interstand tension stresses .sigma..sub.12(y.sub.6e) through
.sigma..sub.45(y.sub.9e) are computed using strip thicknesses and
direct measurement of tension forces. Specific roll forces
P.sub.1(y.sub.10e) through P.sub.5(y.sub.14e) are computed using
strip width and direct measurement of roll forces.
[0120] Adjustments of the individual stand output thicknesses and
the individual interstand tension stresses are made simply by
changing the variables represented by the elements of the vector
y.sub.op.i (i=1, . . . , 5) and the elements of the vector
x.sub.op.i (i=1, . . . , 4), respectively (Table 1). The
independence of adjustment was confirmed by simulation which showed
that an adjustment of 2% in a stand output thickness, or an
adjustment of 5% in an interstand tension, resulted in a negligible
effect on the unadjusted interstand tensions and on the unadjusted
steady-state stand output thicknesses.
[0121] Generally speaking, roll eccentricity is an axial deviation
between the roll barrel and the roll neck caused by irregularities
in the work rolls 18, in the roll bearings, or in both, which
results in cyclic variations in the metal strip 14 thickness. In
the model, roll eccentricity modifies (7) as h out = S + S .times.
.times. 0 + PW M + e , ( 24 ) ##EQU13## where e is the roll
eccentricity.
[0122] While compensation for roll eccentricity is not part of the
pointwise linear quadratic controller, it must be considered for
consistency with data reported from operating mills which usually
includes the effects of roll eccentricity. Numerous methods of
eccentricity compensation are described in the literature and many
have been successfully implemented. A method of active compensation
that fits nicely into the framework of the pointwise linear
quadratic controller is a form of adaptive noise cancellation,
similar (but not identical) to that described in Kugi, A., et al.,
2000, "Active Compensation of Roll Eccentricity in Rolling Mills,"
IEEE Transactions on Industry Applications, Vol. 36, No. 2, pp.
625-632, which relies on the eccentricity as being always periodic
with a frequency proportional to the measured angular velocity of
the rolls, so that after compensation the stand exit thickness
h.sub.out, the measured roll force F, and the measured position S
of the loading device are nearly eccentricity free. The
eccentricity components remaining in the mill exit thickness after
compensation have been estimated by simulation and combined with
the deviations in output thickness as noted later herein.
[0123] The reduction of errors caused by modeling uncertainties and
by measurement uncertainties is significant to attaining strong
robustness. Estimates of these uncertainties and the sources for
each estimate are listed in Table 4 and in Table 5. In these
tables, the listed estimated uncertainties are percentages of the
measured values except for F, .sigma., and S (Table 5) which are
percentages of full scale values, and for purposes of comparison
with other controllers (Table 11) the estimated uncertainty for
h.sub.out1m(5m) (Table 5) is taken to be zero. Assuming stability,
the steady-state errors in stand output thicknesses resulting from
these uncertainties are attenuated since they occur inside the
closed loops of the trims (FIG. 3). An exception is the errors
caused by the measurements of V.sub.ini (i=2, 3, 4) which are used
in the computation of h.sub.outie for the second, third, fourth
stands 16, e.g. (22). These errors are small since the
uncertainties in the measurement of V.sub.ini are small.
TABLE-US-00004 TABLE 4 Modeling Uncertainties Estimated Parameter
Uncertainty Source of Estimate .mu. 20% Roberts, W. L. M 10% Teoh,
E. K., et. al. k 25% experience
[0124] TABLE-US-00005 TABLE 5 Measurement Uncertainties Estimated
Parameter Uncertainty Source of Estimate h.sub.out1m(5m) 0% n/a F
0.2% Roberts, W. L. .sigma. 0.2% Ginzburg, V. B. S 0.05% Ginzburg,
V. B. V 0.05% Ginzburg, V. B. V.sub.in, V.sub.out 0.025% George
Kelk Corp.
[0125] Transient errors in thicknesses due to uncertainties also
are small because any changes in the uncertainties are slow
compared to the responses of the trim control loops, or the errors
are small. In the case of the first stand 25, where a BISRA
measurement (21) is used, the estimate of h.sub.out1b is very
sensitive to the uncertainty in M.sub.1e. To reduce the transient
effects of this uncertainty, M.sub.1e is determined using (7) and
measurements of h.sub.out1m, F.sub.1, and S.sub.1, where the
measurements of F.sub.1 and S.sub.1 are delayed by the transport
lag from the first stand 25 to the thickness gauge. M.sub.1e is
taken to be equal to M.sub.1 since changes in M.sub.1 are slow
compared to the transport lag. TABLE-US-00006 TABLE 6 Expressions
for Various Parameters h.sub.in,i(t) = h.sub.in10, (i = 1) (6-1)
h.sub.in,j(t) = h.sub.out,i-1(t - .tau..sub.di,i-1), (i = 2, 3, 4,
5). (6-2) {overscore (k)} '2 .lamda..sub.1k.sub.in + (1 -
.lamda..sub.1)k.sub.out, (6-3) where k.sub.in(out) =
k.sub.0,in(out) + .DELTA..sub.k.sub.{dot over (e)},in(out), (6-4)
k.sub.0,in(out) = a(b + r.sub.in(out)).sup.c, (6-5) r in .function.
( out ) = H a - h in .function. ( out ) H a , ##EQU14## (6-6)
.DELTA.k e . = .gamma. .function. ( 3 + log 10 .times. V h in
.times. .times. 0 .times. h in .times. .times. 0 - h out .times.
.times. 0 R ) , ##EQU15## (6-7) and .lamda..sub.1, a, b, c, .gamma.
are constants. {overscore (.sigma.)} = .lamda..sub.2.sigma..sub.in
+ (1 - .lamda..sub.2).sigma..sub.out, (6-8) where .lamda..sub.2 is
a constant. {overscore (h)} = .lamda..sub.3h.sub.out + (1 -
.lamda..sub.3)h.sub.in, (6-9) where .lamda..sub.3 is a constant.
.mu. = h in .times. .times. 0 - h out .times. .times. 0 2 .times. R
.times. ( .5 + ( K 1 - .5 ) .times. .times. exp .function. ( - K 2
.times. V ) ) , ##EQU16## (6-10) where K.sub.1, K.sub.2 are
constants. R ' = R ( 1 + 16 .times. ( 1 - .upsilon. 2 ) .times. P 0
.pi.E ( h in .times. .times. 0 - h out .times. .times. 0 ) .
##EQU17## (6-11) .delta. = h.sub.in - h.sub.out. (6-12) V.sub.out =
V(f + 1). (6-13) V in = V out .times. .times. ( h out h in ) .
##EQU18## (6-14)
A. Relationships for h.sub.out, P as Functions of the State
Variables
[0126] Using the expressions of Table 6, .xi. and .alpha. are
computed (during each scan of the controller) at a number of
equally spaced points in a predetermined neighborhood of h.sub.out0
as .xi. = .mu. .times. R ' .times. .delta. h _ , .times. and ( 7
.times. - .times. 1 ) .alpha. = h out h i .times. .times. n .times.
exp .function. ( .xi. ) - 1. ( 7 .times. - .times. 2 )
##EQU19##
[0127] Using (1) and noting that F=PW, the total roll force is then
computed (at each point) as F=({overscore (k)}-{overscore
(.sigma.)}) {square root over (R'.delta.)}(1+0.4.alpha.)W.
(7-3)
[0128] In the neighborhood of h.sub.out0, F is approximated by a
linear fit, which is reasonable because the neighborhood is not
large. F=c.sub.1h.sub.out+c.sub.2, (7-4) where c.sub.1 and c.sub.2
are constants.
[0129] Using (7) and (7-4) h.sub.out is then h out = M .function. (
S + S 0 ) + c 2 ( M - c 1 ) , ( 7 .times. - .times. 5 ) ##EQU20##
and the specific roll force is P = M .function. ( h out - ( S + S 0
) ) W . ( 7 .times. - .times. 6 ) ##EQU21##
[0130] Thus h.sub.out and P become functions of the state variables
which are represented by the state vector elements (Table 1
(a)).
B. Relationship for (V.sub.in,i+1-V.sub.out,1) as a Function of the
State Variables
[0131] From (6-13) the strip speed at the exit of the roll bite is
V.sub.out=V(f+1), (7-7) where calculation of the forward slip f is
as given in (3) and calculation of h.sub.out is as given in (7-5).
The variables used in calculation of f (and V.sub.out) thus become
functions of the state variables.
[0132] By conservation of volume through the roll bite, V i .times.
.times. n , i + 1 = V out , i + 1 .times. h out , i + 1 h i .times.
.times. n , i + 1 , ( 7 .times. - .times. 8 ) ##EQU22## and
(V.sub.in,i+1-V.sub.out,i) becomes a function of the state
variables.
Table 7. Derivations of Relationships which Express P, h.sub.out,
and (V.sub.in,i+1-V.sub.out,i) as Functions of State Variables
[0133] Open loop simulations and closed loop simulations were
performed on the invention using Matlab and Simulink. Matlab and
Simulink are products of The MathWorks, Inc., 3 Apple Hill Drive,
Natick, Mass. 01760-2098. Open loop simulations or open loop
systems refer to simulating the rolling mill 10 without feedback
from the monitored or estimated input control parameters of the
rolling mill 10. Closed loop simulations or closed loop systems
refer to operating the rolling mill by monitoring the input control
parameters, and generating (by feedback control) operational
control parameters to consistently produce metal strip 14 with
certain output control parameters. The open loop simulations
confirmed the validity of the model by comparing the simulation
results with the results of others, as noted previously. Closed
loop simulations, performed to verify control performance and
robustness, were done with the controller coupled to the model. For
these simulations, Q was set to I.sub.14 except for Q(1,1), Q(2,2),
Q(3,3) and Q(4,4), which were set to 10.sup.8, and R was set to
I.sub.10. Other parameters were set as noted in Table 8.
TABLE-US-00007 TABLE 8 Parameter Settings Parameter Setting
K.sub.l, i 1000 Stand i (i = 1, 2, 3, 4, 5) K.sub.P, i 500 Stand i
(i = 1, 2, 3, 4, 5) K.sub.g.sub.--int, i 5 Stand i (i = 1, 5)
K.sub.i, i + l 400 Stand i (i = 1, 5)
[0134] To highlight the performance and robustness of the
controller, total compensation of eccentricity was assumed for the
simulations. An estimated eccentricity component was then added to
the mill exit thickness for comparison to other systems. See, e.g.
Table 11.
[0135] The mill entry disturbances of FIG. 2A and FIG. 2B were
applied at 100% speed of the rolling mill 10 and at 5% speed of the
rolling mill 10, with zero uncertainties. The resulting percentage
changes in mill exit thicknesses at 100% speed and at 5% speed are
displayed in FIGS. 6A and 6B. The mill then was decelerated from
100% speed to 5% speed by changing U.sub.V1, U.sub.V2, U.sub.V3,
U.sub.V4, and U.sub.V5 proportionally to the master speed reference
and similarly changing the corresponding elements of x.sub.0, with
the mill entry disturbances applied and with zero uncertainties.
The master speed reference and the response in mill exit thickness
are shown in FIG. 7. The mill next was accelerated from 5% speed to
100% speed similarly to deceleration. The results are shown in FIG.
8. Table 9 summarizes the magnitude of the maximum percent
deviations of stand exit thicknesses and interstand tension
stresses from their operating point values during steady speed and
during speed changes, assuming no uncertainties. TABLE-US-00008
TABLE 9 Maximum Percent Deviation of Stand Exit Thicknesses and
Interstand Tension Stresses, with Mill Entry Disturbances Applied,
without Uncertainties Magnitude of Maximum Percent Deviation of
Variable from Operating Point Value Decel from Accel from 100% 5%
100% to 5% 5% to 100% Variable Speed Speed Speed Speed h.sub.out1
.02% <.01% .02% .02% h.sub.out2 .01 <.01 .01 .01 h.sub.out3
.01 <.01 .01 .02 h.sub.out4 .01 <.01 .01 .02 h.sub.out5 .01
<.01 .01 .01 .sigma..sub.12 0.11% 0.2% 0.1% 0.10% .sigma..sub.23
0.05 0.1 0.05 0.05 .sigma..sub.34 0.04 0.1 0.04 0.04 .sigma..sub.45
0.14 0.5 0.10 0.10
[0136] The previous simulations were repeated except with the
uncertainties of Table 4 and Table 5 applied simultaneously, with
magnitudes and directions such that the worst deviation in mill
exit thickness was realized for each case simulated. The results
are summarized in Table 10, which shows no significant deviations
from the results of Table 9, implying good robustness to external
disturbances and to modeling and measurement uncertainties.
[0137] For a change in product which changes the operating point,
the material properties, or both, the simulations are repeated to
verify stability, performance, and robustness, and to establish the
settings of weighting matrices Q and R, and of parameters
K.sub.1.i, K.sub.P.i, K.sub.g.sub.--.sub.int.1(5), and K.sub.i.i+1.
TABLE-US-00009 TABLE 10 Maximum Percent Deviation of Stand Exit
Thicknesses and Interstand Tension Stresses, with Mill Entry
Disturbances Applied, with Uncertainties Magnitude of Maximum
Percent Deviation of Variable from Operating Point Value Decel from
Accel from 100% 5% 100% to 5% 5% to 100% Variable Speed Speed Speed
Speed h.sub.out1 .161% .02% .141% .101% h.sub.out2 .07 .051 .062
.07 h.sub.out3 .08 .051 .082 .07 h.sub.out4 .08 .051 .063 .083
h.sub.out5 .077 .051 .074 .072 .sigma..sub.12 0.18% 0.02% 0.15%
0.02% .sigma..sub.23 0.05 0.01 0.05 0.05 .sigma..sub.34 0.10 0.02
0.10 0.1 .sigma..sub.45 0.22 0.06 0.11 0.17
[0138] The previous results were compared with data from two
operating industrial controllers of Tezuka, T., et al. and
Sekiguchi, K., et al. described in Tezuka, T., et. al.,
"Application of a New Automatic Gauge Control System for the Tandem
Cold Mill," in IEEE IAS 2001 Conference Record of the 36.sup.th IAS
Annual Meeting, Vol. 2, September/October, 2001 and Sekiguchi, K.,
et. al., "The Advanced Set-Up and Control System for Dofasco's
Tandem Cold Mill," in IEEE Transactions on Industry Applications,
Vol. 32, No. 3, May/June 1996. While differences in mill
properties, in operating points, and in material properties, and an
absence of disturbance data in the case of the industrial
controllers precluded specific comparisons, some general
comparisons using mill exit thickness measurements could be made.
As noted, mill exit thickness of metal strip 14 is expected by
operators to be within 0.8% of the operating point value, which is
met by the pointwise linear quadratic controller. The two
industrial controllers used for comparison purposes generally
conformed to this guideline. Table 11 summarizes the results of the
comparisons, assuming that the mill exit thickness measurements
have zero uncertainties. For comparison purposes, a maximum
eccentricity component of 0.05% (after compensation) was assumed
for the pointwise linear quadratic controller, which was confirmed
by initial simulation of an active compensation method, using an
eccentricity of 0.0012 inches and considering changes in the roll
diameter due to mechanical wear and heating. The 0.2% of Table 11
was obtained by adding the 0.05% plus 0.08% for the fifth stand 16
maximum deviation in output thickness from Table 10, plus 0.07% for
conservatism. TABLE-US-00010 TABLE 11 Comparison of Magnitudes of
Maximum Percent Deviation of Mill Exit Thickness with Other
Controllers Magnitude of Maximum Percent Controller Deviation of
Mill Exit Thickness Pointwise Linear Quadratic .2% Industrial A of
Tezuka, T., et al. .5 Industrial B of Sekiguchi, K., et al. .7
[0139] As shown in Table I 1, the pointwise linear quadratic
control method with appropriate trimming functions provides the
potential for significant improvement over exiting controllers in
maintaining the tolerance in mill exit thickness during steady
speed and during speed change, in the presence of disturbances and
uncertainties. In addition, this method offers the following
features, and advantages over existing control strategies and over
proposed control methods: [0140] 1. The structure of the controller
allows for the use of physical intuition in the design process, and
does not require a linearized model or gain scheduling as in the
case of existing control methods and certain proposed strategies
(e.g. linear H.sup..infin.). This simplifies the design and thus
reduces the design effort and cost. [0141] 2. Design effort and
cost are further reduced since the pointwise linear quadratic MIMIO
control strategy requires no feedforward control, as in the case of
most existing industrial control schemes. [0142] 3. The
configuration of the controller is very simple as compared to
existing controllers and to proposed control strategies. This
results in a user-friendly environment for commissioning and
maintenance personnel which reduces efforts and costs in these
areas. [0143] 4. The strip thicknesses and interstand tensions are
independently adjustable by the operator which is essential during
mill operation. [0144] 5. The tight control of thickness and
interstand tensions strongly contributes to the stability of
rolling. [0145] 6. The capability of the controller to maintain
tight control of thickness and interstand tensions during speed
change is especially useful for continuous rolling applications
where the speed is changed during weld seam passage. [0146] 7. The
controller is easily configured to accommodate new applications or
revamps.
[0147] While specific embodiments of the invention have been
described in detail, it will be appreciated by those skilled in the
art that various modifications and alternatives to those details
could be developed in light of the overall teachings of the
disclosure. Accordingly, the particular arrangements disclosed are
meant to be illustrative only and not limiting as to the scope of
the invention which is to be given the full breadth of the claims
appended hereto and any and all equivalents thereto.
* * * * *