Page 273 - Distributed model predictive control for plant-wide systems

Hot-Rolled Strip Laminar Cooling Process with Distributed Predictive Control 247

m×m
I ∈ R ;(j = 1, 2, … , n )
m s
( )
{
s s s a
x =(x ∕x ) , s ∈ C
i,j i,j i,j 0 W , (i = 1, 2, … , m, j = 1, 2, … , n ) (11.19)
s s s s
(x )= h (x ), s ∈ C
air
i,j i,j i,j A
s
s
It should be noticed that the notations g (⋅) and u are different between the subsystems
that belong to C and the subsystems which belong to C , see (11.14) and (11.19). Since
A W
the current states of subsystems are not measurable inside the cooling section, an observer is
designed to estimate the current states of each subsystem in the next subsection.
11.3.2 Design of Extended Kalman Filter
The nonlinear model of the overall system can be expressed as
{ 0
x (k + 1) = F(x(k))x(k)+ G(x(k))u(k)+ Dx (k)
(11.20)
y(k)= Cx(k)
0
N T T
1
2
n T
2 T
1 T
where x =[(x ) (x ) ··· (x ) ] and u =[u u ··· u s ] , x is the distribution of FT in
the z-direction. y is the output vector which includes the measurements of CT on the top
and bottom surfaces. The expressions of F(x(k)), G(x(k)), and D can be deduced easily from
(11.12). Coefficient matrix C is defined as
[ ]
1×(N−1)n s m 1 1×(n s m−1)
C = (11.21)
1×(N−1)n s m 1×(n s m−1) 1
This system is uniformly observable because each volume’s temperature depends on its
neighbors. Since the order of the system is inevitably high, the famous EKF is chosen to be
more convenient to design in our case. The estimation of the strip temperature distribution can
be expressed as
̂ x(k + 1)= ̂ x(k + 1|k)+ K k+1 (y(k + 1)− Ĉ x(k + 1|k)) (11.22)
The feedback coefficient K k + 1 is deduced by the difference Riccati equation. This observer
estimates the states of overall system at each control period, and transmits them to all subsys-
tems.

11.3.3 Predictor

Since there are no manipulated variables in subsystem s ∈ C , a predictor is applied for esti-
A
mating the future states X (k) where
s
[ s s s ] T
X (k)= x (k + 1) x (k + 2) ··· x (k + P) (11.23)
s
In the predictor, the prediction model is (11.12), and the measurable disturbance FT is
assumed to be a constant during the estimation of X (k). When finishing estimating X (k),
1
s
the predictor sends the estimation of X (k) to the downstream neighbors of subsystem s.
s