Nonlinear Model of the Hydraulic Automatic Gauge Control System:
Conrollability Analysis and Observability
Abdelmajid Akil, Mourad Zegrari, Abdelwahed Touati and Nabila Rabbah
Laboratory of Structural Engineering, Intelligent Systems and Electrical Energy
ENSAM, Department of Electrical Engineering – Hassan II University Casablanca, Morocco
Keywords: rollingmill; roll gap; nonlinearmodel; elastic deformation; Automatic gauge control
Abstract: In the reversible cold rolling mill, it is critical to control the gauge or thickness of the cold-rolled steel strip
for end-user requirements. To obtain a high precision in the output thickness of the strip, the automatic
control system of the gauge is used. As one of the most important functions of the basic mill automation,
there is the hydraulic roll gap controlsystem (HRGCS) which is the inner loop of the Hydraulic Automatic
Gauge Control (HAGC).In order to improve the control performance of the strip mill, a theoretical nonlinear
mathematical model of the complex HAGC has been established in this paper for a reversible cold rolling
system.The new nonlinear model of the HAGC will be used to develop nonlinear control strategies to
address the different control problems encountered.
1 INTRODUCTION
Nowadays, steel strips of various thicknesses are
vitally important products in the iron and steel
industry and are widely used in aeronautics,
household appliances, car manufacturing, machines
and many other areas.
The accuracy of the web thickness and flatness
control is directly dependent on the performance of
the web winding system (WWS) studied and
controlled by nonlinear controllers presented in
(akil, 2017), (akil, 2018a) and(akil, 2018b), the
Automatic Gauge Control (AGC) and Automatic
Flatness Control System (AFC) (John, 2010). In
particular, the Hydraulic AGC System is one of the
key techniques of modern mills to determine the
quality of the web based on Hydraulic Roll gap
controlsystem (HRGCS).
The HRGCS is an important part of the AGC
system by pressing the position adjustment of work
rolls in order to get a precision of the thickness and
flatness of the web (Soszy´nski, 2012), (Kim, 2013)
and(Hoshino, 1997). Therefore, the good dynamic
characteristics of the HRGCS are very important for
the AGC system (Zhang, 2012a).
The HRGCS is a typical of machine, electrical,
hydraulic integrated system for complete control of
the complex system. In nowadays, more and more
scholars have devoted themselves to the study of
hydraulic gap control (Zhou, 2007) and (Zhang,
2012b).
The cold rolling process has been taken as a
research topic for many decades, and currently some
theories are able to provide a valuable and detailed
description of the roll gap (Grimble, 1978). First
Siebel and von Karman (von Khmh, 1925),(Siebel,
1925) began studies on the subject; their analysis
introduced the vertical segments concept of
homogeneous compression of the sheet during
rolling. Another fundamental supposition was the
occurrence of a neutral plane in the length of the
contact arc (Freshwater, 1996).
The models proposed by (Sims ,1945),(Bland
and Ford ,1948) and (Nascimento, 2016) have
solved analytically the problem by avoiding most
numerical integrations, unlike the Orowan model
(Orowan, 1943) which is much more complex and
requires more of calculations. However, the
simplifications of Bland and Ford led to a sacrifice
of precision (Alexander, 1971).
In order to analyze and optimize the AGC, a
precise model of the HRGCS is required first.
Numerous studies on the control of the HRGCS in
Akil, A., Zegrari, M., Touati, A. and Rabbah, N.
Nonlinear Model of the Hydraulic Automatic Gauge Control System: Conrollability Analysis and Observability.
DOI: 10.5220/0009771303070316
In Proceedings of the 1st International Conference of Computer Science and Renewable Energies (ICCSRE 2018), pages 307-316
ISBN: 978-989-758-431-2
Copyright
c
2020 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
307
modeling and simulation have been carried out (Liu,
2012) and (Sun, 2014).Sun (Sun, 2015) established a
mathematical model of the HRGCS that was used to
design a controller with a predictive control theory.
In (Sun, 2014), the mathematical model of the
HRGCS was established by adopting the mechanism
modeling method and the model was dynamically
simulated and analyze
d considering the nonlinear and saturation
characteristics. However, it is difficult to determine
the important parameters in the theoretical model
with the mechanism analysis method. Moreover, real
conditions play an important role in the decision on
parameters (Ren, 2014), (Li, 2006) and (Dyja,
1996).
In order to adjust the parameters of the change of
gap of the rolling mill, the authors carried out a
number of experiments where they studied the
rolling of the web with different correction
parameters in each stand (Petryakov, 2011); (Galkin,
2011) and (Khramshin, 2010). The parameters were
selected empirically and passed to the controller of
the automatic process control system.
Based on the nonlinear equations of the elastic
and plastic deformation of the web and work roll,
and the rolling force, a nonlinear model of the
HAGC for reversible cold rolling mill was
developed in this paper.
The rest of the article is organized as follows:
section 2 presents a description of the rolling
process, Basic theories of the cold rolling mill
introduced in Section 3, The model of the HAGC is
established in Section 4 and the Simulation Results
and Discussion in Section 5. Finally, the conclusion
is given in section 6.
2 DESCRIPTION OF THE
ROLLING PROCESS
Rolling is a manufacturing process by plastic
deformation. In this method, the metal web
undergoes a reduction in thickness by crushing
between the two work rollers rotating in two
opposite directions. While the larger diameter
backup rolls (support rollers) serve to support the
work rolls to prevent them from bending too much.
The rotary motion of the work rolls exerts a
compressive force to continually decrease the initial
thickness of the web by passing several times in the
same rolling stand for the reversible single-roll mill.
A rolling mill cage is composed of:
A pair of cylinders called "work roll"
between which the material is elongated.
The working cylinders are made of
rectified cast iron;
Another pair of cylinders called "backup
roll" (a cylinder on each side of the pair of
work rolls) to reduce the deformation of the
work rolls.
Two metal columns holding the cylinders
together (one column on each side).
Figure 1 illustrates the rolling process where a
web is engaged between the two rolls rotating in
opposite direction. The thickness
web must be
able to engage between the rollers where it is
deformed in compression to stand at a thickness
corresponding to the adjustment of the clearance
between the rolls.
Figure 1: rolling Principle
The necessary compressive force is applied to
the bearings of the backup rollers by hydraulic
cylinders or by a screw arrangement driven by an
electric motor. The thickness of the rolled web is
mainly determined by the gap between two work
rolls which is initially set by a pass line adjusting
mechanism. The actual position control is performed
by the exact and fast acting hydraulic control system
called Hydraulic automatic gauge control (HAGC).
3 BASIC THEORIES OF THE
COLD ROLLING MILL
3.1 Control System of the HAGC
The structure of the HAGC is shown in figure 2. The
system is typically composed of thickness gauge,
position sensors, position controller, servoamplifier,
electrohydraulic servo valve, and hydraulic cylinder
components.
ICCSRE 2018 - International Conference of Computer Science and Renewable Energies
308
Figure 2: Basic configuration of HAGC
The position control system of hydraulic roll gap
is constructed by the following process: the position
difference is obtained by comparing the position
reference and the actual position, and the difference
is transmitted to the position controller; take the
controller output values as servoamplifier input
values; the amplified current signal is transferred to
the degree of opening of the electrohydraulic servo
valve; Finally, the hydraulic oil passes the servo
valve by alternately driving the movement of the
actuator (Zhanget al, 2009 , Wang, 2011 , Lv, 2007
and Andrew and Rui , 2000).
3.2 Symbols
Table 1: Symbol description
Symbol Parameter
the spool valve position
Servo amplifier gain
The associated time constants.
servo valve input current
Servo valve flux
the flow coefficient of the valve port
Servo valve natural frequency
and
the inlet and outlet pressure of the servo
valve respectively
the oil density
the active area of the cylinder piston
the total leakage coefficient
The oil pocket volume of the hydraulic
cylinder
the bulk modulus of elasticity
the equivalent total mass of moving parts
of the upper roller system
the viscosity coefficient of cylinder
the elastic stiffness coefficient of load
the cylinder piston displacement
Other load force acting on the piston
the plastic stiffness coefficient of the
rolled piece
and
are the input and output thickness of the
rolled piece respectively
T The constant delay time.
the unknown force including the
coulomb friction force
the rolling load
the mill modulus
the roll gap
Unloaded roll gap
and
the magnitude and period of thickness
deviation respectively
vin the entry strip is passed through the roll
stand with the velocity
Time constant of hydraulic servo
constants specified
Nonlinear Model of the Hydraulic Automatic Gauge Control System: Conrollability Analysis and Observability
309
3.3 Basic Theoretical Equations
3.3.1 Servo Valve Torque Motor Equation
The servovalve used in this paper has a critical
center (zero overlap). Its orifices are supposed
symmetrical and matched. The internal leakage of
the servo valve are neglected. The servo valve treats
the current as the input and displacement of the
valve core
as the output.
The dynamic movement of the servovalve spool is
described by the following dynamic equation:
1
3.3.2 Servo Valve Flow Equation
The flow equation of the servo valve is a typical
nonlinear loop. The outflow equation of the
servovalve is:
2
In order to satisfy the Lipschitz condition to
guarantee the existence and uniqueness of the
solution to (2) for all initial conditions, the non-
differentiable sign function is approximated by the
continuously differentiable sigmoid function defined
as:
1
1
;0
By doing so, the system described by (1)
becomes differentiable and allows the use of the
feedback linearization approach
2
1
²
;0
lim
→
2
1
²
0
Furthermore, the use of the sigmoid function is
required to ensure that the feedback linearization
conditions on the Lie derivatives of the system
dynamics are satisfied [4]. When≫1, the sigmoid
function behaves like the sign function and the
model best approximates the real electrohydraulic
system.
The outflow equation of the servovalve became:
3
3.3.3 Hydraulic Flow Equation
The flow from servo valve into the cylinder, besides
driving the piston movement, can be used to
compensate various cylinder leaks and liquid
compressed volume, etc. The continuous flow
equation of the cylinder can be expressed as
4
4
3.3.4 Hydraulic Cylinder Load Force
Balance Equation
The output rolling force of the cylinder keeps
balance with the inertia force of the moving parts,
viscous damping force, elastic load force and other
load force. The dynamic equation can be written as
5
The
is other load force acting on the piston.
And
can be expressed by
3.3.5 Elastic Deformation of the Mill and
Spring Equation
Cold rolling is a processing method by passing the
metal strip between two rolls (work rolls), rotating in
opposite directions. Due to this rotational movement
and the compression generated by the cylinders
(deformation force P), there is a continuous
reduction of the initial thickness by plastic
deformation of the metal.
The simplified thickness model normally uses
the equation gaugemeter (BISRA) or spring equation
to derive the thickness of the strip. It is based on the
relationship between the position of the roll gap, the
gap of the work rolls and the thickness of the web.
A linear approximation of the grinding stretching
characteristic is used to estimate the thickness at the
exit of a cage as by the model "BISRA" or
"gaugemeter" (Zhanget al, 2015):
Σ
6
Where Σis the compensation including thickness
compensation, e eccentricity compensation and
thermal compensation.
Figure 3: Equation gaugmeter
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310
In the gaugemeter equation (see Figure 3)
is
the amount of the mill stretch when a roll separating
force is applied. The output thickness is therefore
equal to the position of the unloaded gap plus the
stretch.When the entry strip is passed through the
roll stand with the velocity
, the disturbance ∆
is expressed as:
∆
sin
2
Where
and
are the magnitude and period of
thickness deviation respectively (Hwang et al,
1996).
There exists time delay between the output
thickness
and the thickness detected by the
thickness gauge, and it is a pure delay loop
Where is the constant delay time.
3.3.6 Dynamic Equation of the Roll Gap
The rolling model at each stand consists of the
dynamics of the roll gap and rolling direction, which
are described below.
The roll gap S is assumed to be controlled by a
local feedback loop of a hydraulic servo and is
governed by the following equation (OZAKIet al,
2012):
,
0
7
Where
denotes the command input, and
and
are constants specified by the local feedback
loop. The local feedback adjusts the roll gap S so
that h
g
(S, h) coincides with the command input
.The function
,
is defined in terms of the
thickness h i and roll gap S i as
1
Which represents the thickness modified by the
tuning parameter k (0 < k < 1).
The dynamics of the roll gap S i is governed by
1
1
0
Σ
8
3.3.7 Rolling Force Model
The mill model generates the roll force, exit strip
thickness, entry strip velocity, and exit strip velocity
using the roll gap, roll velocity, entry strip thickness,
and the other factors. The roll force and the exit strip
thickness are derived from Bland-Ford model (Bland
et al., 1948 and Ford and Bland, 1951) and the
gauge-meter equation respectively. The roll force
can be expressed as:
,
,…
9
4 NON-LINEAR GLOBAL MODEL
OF THE HAGC
4.1 Non-linear Model of the HAGC
The mathematical model of the HAGC can be
described by the system of equations including
nonlinear dynamic equations presented previously
by posing
,
,
,
4
,
5
and
6
as state variables:
1
1
1
Σ
1
4
1
Σ
Nonlinear Model of the Hydraulic Automatic Gauge Control System: Conrollability Analysis and Observability
311
The state vector is
,
,
,
,
,
and control
.
The model of the HAGC presented in the above
equation is nonlinear because of the sigmoid and
square root functions. In this state representation,
only the state variables are expressed as a function
of time. However, hydraulic and mechanical
parameters also vary during the operation of the
HRGCS.
4.2 Controllability Analysis of HAGC
The controllable canonical form is to represent the
dynamics of the system by a differential equation
relating the output variable to the control variable
HAGC. The state variable to be controlled and its
successive time derivatives represent new HAGC
state variables. The number of successive time
derivatives is determined by the number of
successive derivations performed on the output
variable in order to obtain the control variable.
Nonlinear controllable canonical form are
constructed from non-linear state representation of
the HAGC. This direct relationship between the
input variable and the state variable to enslave
enables us to develop our control laws. Consider the
non-linear state representation of the HAGC in the
space where y (t) represents the output variable
described below:
2
Σ
0
0
0
0
0
Or in compact form,
3
,
,
,
Where
is the n-dimensional state vector,
is the control input and
is the output.
Definition 1: the Lie derivative (Slotine et Li, 1991,
p. 229): Let :
→ be a smooth scalar function,
and :
→
be a smooth vector field on
,
then the Lie derivative of h with respect to is a
scalar function defined by
Definition 2: relative Degree (Khalil, 2002, p.
510): The nonlinear system describe by (sys 1) is
said to have relative degree ,1 , in a
region
⊂ if
,
0, 1,2,…,
2 ;
,
0 for all ∈
.
Thus, by successively deriving the position y (t),
we find:
,
10
,
1
1
Σ11
,
1
Σ
12
,
Σ
1
13
,
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312
Σ
14
,
,
15
with
,
,
,
with
,
,
,
and
,
;
;
;
;
;
;
;
;
;
;
;
;
;
and
The expression
,
0 if and only if
. According to the standard design of the
electro-hydraulic servo systems explained by Merritt
(1967), the pressure across the hydraulic actuator
verifies
. Thus, the control signal
appears when the output is derived 6 times.Since the
system is of the sixteenth order (six state variables),
there is no internal dynamics. The nonlinear state
representation in canonical and controllable form
becomes:
,
,
16
With
,
,
,
,
,
and
,
∑
,
With the new selected state variables such
as
,
,
,
,
and
.
The mathematical representation of equation (16)
has the previous state variables because the system
is non-linear. For the design of the exact
linearization controller, the controllable canonical
form of equation (16) is used.
Nonlinear Model of the Hydraulic Automatic Gauge Control System: Conrollability Analysis and Observability
313
4.3 HAGC Observability
The problem of designing an observer for a
nonlinear system has been widely studied. Many
different approaches have been considered to design
observers. However, the observability property of
the system must be verified before to design an
observer.
It is well-known that the observability of a
nonlinear system can be lost under some conditions.
Now, we recall the conditions for checking that a
nonlinear system is observable (Hermann, 1977).
For the system (sys3), let us define the vector of
output derivatives, H (x), as follows:
.
.
.
.
.
.
17
and the Observability matrix, O (x), as:
.
.
.
18
where
represents the i-th Lie derivative of h
(x) in the f vector field direction. In addition, let us
suppose the following condition:
Condition 1: The system (sys3) is globally
observable, in the sense that the observability rank
condition
19
is fulfilled for all x∈R
.
Remark 1. In the following, the observability
property is studied for the case of the HRGCS. Then,
from Condition 1 and by considering
the
output of system (17).Then,
6
Thus system (17) is weakly observable.
5 SIMULATION RESULTS AND
DISCUSSION
The hydraulic system of the roll gap that we
modeled is simulated using the MATLAB
SIMULINK software and the simulation is
performed over 40 seconds. Figure 4 shows the time
performance obtained for HAGC to study that a step
is applied. The HRGCS index response shows a
significant overshoot and stabilizes after 20 s.
The hydraulic HAGC is very uncertain, so it should
really be used by control algorithms with strong
robustness.
6 CONCLUSIONS
In this paper, a dynamic nonlinear mathematical
model of the HRGCS of the reversible cold rolling
mill is developed primarily from the first principles
governing mechanical and electrical components,
associated with the theory of rolling for make the
resulting model fit for decision-making and control
analysis.
The dynamic mathematical model developed
can be used to analyze and design futurement
nonlinear order to fully control the various reversible
cold rolling mill systems thus contributing to
improving the final product quality broadband mills.
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