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Rolling Mill Model and Control Problem
The basic physical representation of the process and the
relevant variables involved in the control problem are given in
Fig. 1, and summarized as: fw is the rolling force, he is the strip
thickness at entry of stand; ha is the strip thickness at output of
stand; le and la are the locations of the thickness sensors, so is the
roll gap with zero rolling force; sr is the dimensions of roll stand
with zero rolling force; VI is the stand input velocity; and vw is
the stand output velocity.
The measured variables are he, ha, vi, vw, and fwj. As shown in
Fig. 1, the roll dynamics can be viewed as a damped spring
system with an effective displacement input of so, the roll gap
with rolling force zero Vw, = 0). The roll gap under operation will
be larger depending on the entry strip thickness he and the
elasticity of the stand. The control objective is to keep ha as close
as possible at a desired value ha,<,,. Basic Equations - Nonlinear Model
The bas LC equation which establishes the relationship amon
the variables is given by the equilibrium point determined by th
characterisi ics of the material and the working curve of the stand
The characteristics of the stand can be represented by the linea
relation gil en by where K is the mean yield stress (in Newtons per square meter),
o is the mean applied tension stress (in Newtons per square
meter), p is the coefficient of friction between work roll and strip
in roll gap, R' is the deformed roll radius (in meters), andfe is the
output elastic recovery (Newtons).
In order to simulate the system a simplified version of (2) was
used, The equilibrium point is given by the intersection of two curves
of forces, namely (I) and (3).
As (3) IS quadratic it is possible to obtain a closed solution for
the intersection point ha. In fact, considering The cutput thickness sensor is located la meters from the
rolling mill. According to this distance, the dead time associated
with the ineasurements of ha is Tu = Za/vw. On the other hand, for
he there i i associated a delay related to vi given by Te = Ze/vl. The The basic control objective is to keep the thickness of the
output material ha as close as possible to a reference value
Ha, rr The control variable is so and the measurements available
are ha, he,fw, vi, and vw. The evaluation index is the integral of
the square error over the time elapsed, that is The input velocity of the strip is given by a prespecified curve
with a certain period of acceleration and deceleration (a ramp up
from 1 to 5 ds, constant, then ramp down). Any changes of the
strip input thickness he are regarded as disturbances. Here, a
certain profile with steplike changes around the operating point
he = 5 mm is chosen. In order to get an idea of how the nonlinearity of the steel
rolling mill will effect a conventional PI controller based on the
linearized plant, a short description of the results obtained by the
linear approach is given. The basic structure has two parts, a
feedback controller and a feedforward controller, as shown in
Fig. 2. This figure introduces the variables Ahe, and Aso, which