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and Matsumoto (1973). Matsubara et al. (1989) applied the
influence coefficientmethod to predict the case ofmutual con-
tact between upper and lower work rolls during the rolling of
foil. Gunawardene et al. (1981) used the method to solve for
the 20-high clustermill using an equivalent stack of vertically
aligned rolls, and Ogawa et al. (1991) extended the method to
model 12-high rolling mills.
Due to the inherent complexity of cluster-type rollingmills,
which havemultiple roll contacting surfaces and require both
horizontal and vertical roll displacement calculations, there
have been far fewer instances of adapting the conventional
strip crown models to them. For this reason, non-physics-
based models, derived from pattern recognition/heuristics
methods such as neural networks and fuzzy techniques,
have been applied to cluster mills in greater relative num-
bers. Although the influence coefficientmethod and transport
matrixmethod have been used to simulate cluster-type rolling
mills in some cases, these methods lead to complex models
with limited opportunity for industrial application. Another
disadvantage of some conventional strip crownmodels is that
their solution time is not sufficient for real-time mill con-
trol. The influence coefficient method, which has been the
most widely studied, requires an iterative solution to satisfy
equilibriumand compatibility conditions. Although the trans-
port matrix method was extended to cluster mills, because
of the large number of rolls and contact surfaces, it is not
fast enough for real-time control (Guo and Malik, 2005). The
large-scale finite elementmethod (FEM) is themost prohibitive
of all in terms of solution time because of the vast number
of elements required to model the narrow contact interfaces
between adjacent rolls and between the working rolls and
the strip. Moreover, convergence difficulties associated with
contact-type structural FEManalyses pose additional solution
problems.
The accuracy of the conventional methods may be
examined from a theoretical viewpoint. The pattern recogni-
tion/heuristics models may be accurate if adequately trained
with significant amounts ofmanufacturing data. The accuracy
of both the influence coefficient method and the transport
matrix method depends on a large number of discretization
nodes. As accuracy is improved by increasing the node count,
solution time also increases. Another factor adversely affect-
ing the theoretical accuracy of the transport matrix method
is its use of discrete nodal springs to represent the contact
interactions between adjacent rolls and between the working
rolls and the strip. Cook et al. (2002) highlighted the risk of
using discrete springs in lieu of continuous elastic foundations
and their particular difficulty inmodeling contact interactions
near component ends.
It iswidely recognized that the rolling operation is dynamic
in nature due to, for example, changes with respect to time
of yield stress, temperature, friction coefficient, rolling force,
rolling speed, and geometric parameters of the strip. Despite
this, static models to predict the steady-state strip thick-
ness profile, based on a “snapshot” of the input parameters,
are widely used for pass-schedule setup and flatness control
systems. In the case of pass-schedule setup, this circum-
stance prevails because the pass-schedule calculations are
used to assign nominal set-point values for thickness reduc-
tions, rolling speed, and entry/exit tensions. Flatness control
algorithms operate at command frequencies on the order of
1Hz—several orders of magnitude lower than the dynamic
response of the mill to flatness actuation. The control fre-