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1. Introduction
       Recently, customers require increasingly better quality for hotrolled
strip products, such as automotive companies expect to gain
an advantage from thinner but still very strong types of steel sheeting
which makes their vehicles more efficient and more environmentally
compatible. In addition to the alloying elements, the
cooling section is crucial for the quality of products [1]. Hot-rolled
strip laminar cooling process (HSLC) is used to cool a strip from an
initial temperature of roughly 820–920 C down to a coiling temperature
of roughly 400–680 C, according to the steel grade and
geometry. The mechanical properties of the corresponding strip
are determined by the time–temperature-course (or cooling curve)
when strip is cooled down on the run-out table [1,2]. The precise
and highly flexible control of the cooling curve in the cooling section
is therefore extremely important.

       Most of the control methods (e.g. Smith predictor control [3],
element tracking control [4], self-learning strategy [6] and adaptive
control [5]) pursue the precision of coiling temperature and
care less about the evolution of strip temperature. In these methods,
the control problem is simplified so greatly that only the coiling
temperature is controlled by the closed-loop part of the
controller. However, it is necessary to regulate the whole evolution
procedure of strip temperature if better properties of strip are
required. This is a nonlinear, large-scale, MIMO, parameter
distributed complicated system. Therefore, the problem is how to
control the whole HSLC process online precisely with the size of
HSLC process and the computational efforts required.

       Model predictive control (MPC) is widely recognized as a practical
control technology with high performance, where a control
action sequence is obtained by solving, at each sampling instant,
a finite horizon open-loop receding optimization problem and
the first control action is applied to the process [7]. An attractive
attribute of MPC technology is its ability to systematically account
for process constraints. It has been successfully applied to many
various linear [7–12], nonlinear [13–17] systems in the process
industries and is becoming more widespread [7,10]. For large-scale
and relatively fast systems, however, the on-line implementation
of centralized MPC is impractical due to its excessive on-line computation
demand. With the development of DCS, the field-bus
technology and the communication network, centralized MPC
has been gradually replaced by decentralized or distributed MPC
in large-scale systems [21,22] and [24]. DMPC accounts for the
interactions among subsystems. Each subsystem-based MPC in
DMPC, in addition to determining the optimal current response,
also generates a prediction of future subsystem behaviour. By suitably
leveraging this prediction of future subsystem behaviour, the
various subsystem-based MPCs can be integrated and therefore the
overall system performance is improved. Thus the DMPC is a good
method to control HSLC.

     Some DMPC formulations are available in the literatures
[18–25]. Among them, the methods described in [18,19] are
proposed for a set of decoupled subsystems, and the method
described in [18] is extended in [20] recently, which handles
systems with weakly interacting subsystem dynamics. For
large-scale linear time-invariant (LTI) systems, a DMPC scheme
is proposed in [21]. In the procedure of optimization of each
subsystem-based MPC in this method, the states of other subsystems
are approximated to the prediction of previous instant.
To enhance the efficiency of DMPC solution, Li et al. developed
an iterative algorithm for DMPC based on Nash optimality for
large-scale LTI processes in [22]. The whole system will arrive
at Nash equilibrium if the convergent condition of the algorithm
is satisfied. Also, in [23], a DMPC method with guaranteed feasibility
properties is presented. This method allows the practitioner
to terminate the distributed MPC algorithm at the end
of the sampling interval, even if convergence is not attained.
However, as pointed out by the authors of [22–25], the performance
of the DMPC framework is, in most cases, different from
that of centralized MPC. In order to guarantee performance
improvement and the appropriate communication burden
among subsystems, an extended scheme based on a so called
‘‘neighbourhood optimization” is proposed in [24], in which
the optimization objective of each subsystem-based MPC considers
not only the performance of the local subsystem, but also
those of its neighbours. The HSLC process is a nonlinear,
large-scale system and each subsystem is coupled with its
neighbours by states, so it is necessary to design a new DMPC
framework to optimize HSLC process. This DMPC framework
should be suitable for nonlinear system with fast computational
speed, appropriate communication burden and good global
performance.
In this work, each local MPC of the DMPC framework proposed
is formulated based on successive on-line linearization of nonlinear
model to overcome the computational obstacle caused by nonlinear
model. The prediction model of each MPC is linearized
around the current operating point at each time instant. Neighbourhood
optimization is adopted in each local MPC to improve
the global performance of HSLC and lessen the communication
burden. Furthermore, since the strip temperature can only be measured
at a few positions due to the hard ambient conditions, EKF is
employed to estimate the transient temperature of strip in the
water cooling section.
The contents are organized as follows. Section 2 describes the
HSLC process and the control problem. Section 3 presents proposed
control strategy of HSLC, which includes the modelling of subsystems,
the designing of EKF, the functions of predictor and the
development of local MPCs based on neighbourhood optimization
for subsystems, as well as the iterative algorithm for solving the
proposed DMPC. Both simulation and experiment results are presented
in Section 4. Finally, a brief conclusion is drawn to summarize
the study and potential expansions are explained.
2. Laminar cooling of hot-rolled strip
2.1. Description
The HSLC process is illustrated in Fig. 1. Strips enter cooling section
at finishing rolling temperature (FT) of 820–920 C, and are
coiled by coiler at coiling temperature (CT) of 400–680 C after
being cooled in the water cooling section. The X-ray gauge is used
to measure the gauge of strip. Speed tachometers for measuring
coiling speed is mounted on the motors of the rollers and the
mandrel of the coiler. Two pyrometers are located at the exit of
finishing mill and before the pinch rol1 respectively. Strips are
6.30–13.20 mm in thickness and 200–1100 m in length. The
run-out table has 90 top headers and 90 bottom headers. The top
headers are of U-type for laminar cooling and the bottom headers
are of straight type for low pressure spray. These headers are divided
into 12 groups. The first nine groups are for the main cooling
section and the 1ast three groups are for the fine cooling section. In
this HSLC, the number of cooling water header groups and the
water flux of each header group are taken as control variables to
adjust the temperature distribution of the strip.
2.2. Thermodynamic model
Consider the whole HSLC process from the point of view of geometrically
distributed setting system (The limits of which are represented
by the geometrical locations of FT and CT, as well as the
strip top and bottom sides), a two dimensional mathematical model
for Cartesian coordinates is developed combining academic and
industrial research findings [26]. The model assumes that there is
no direction dependency for the heat conductivity k. There is no
heat transfer in traverse and rolling direction. The latent heat is
considered by using temperature-dependent thermal property
developed in [27] and the model is expressed as
_x ¼
k
qcp
@2x
@z2 _l 
@x
@l ð1Þ
with the boundary conditions on its top and bottom surfaces
k
@x
@z ¼ h  ðx  x1Þ ð2Þ
where the right hand side of (2) is h times (x  x1) and
h ¼ hw
x  xw
x  x1
þ r0e
x4  x4
1
x  x1
ð3Þ
and x(z, l, t) strip temperature at position (z, l);
l, z length coordinate and thickness coordinate respectively;
q density of strip steel;
cp specific heat capacity;
k heat conductivity;
r0 Stefan–Boltzmann constant (5:67  108 w=m2 K4);
Water cooling section
Finishing mill
Pyrometer
Fine cooling section
7.5m 62.41m 7.5m
5.2 m
Pinch roll
Coiler
Main cooling section
X-ray
Fig. 1. Hot-rolled strip laminar cooling process.