Page 275 - Distributed model predictive control for plant-wide systems

Hot-Rolled Strip Laminar Cooling Process with Distributed Predictive Control 249

s
s
where A (k) = f(x (k)) and B (k) = g(x (k)). The (s − 1)th subsystem and the (s + 1)th subsys-
s
s
tem are the upstream and the downstream neighbor of subsystem s, respectively. Assuming
that x(k) is available, the local optimization problem for subsystem s at the sampling time
instant k becomes
( )
P M
∑ ∑ j 2 ∑ j 2
min J (k)= ‖ j ‖ + ‖Δu (k + h − 1|k)‖
‖r (k + i) − ̂ y (k + i|k)‖
s
ΔU s (k) ‖ ‖Q j R j
j∈{s,s+1} i=1 h=1
j
j
j
s.t. x (i + 1|k)= A (k) ⋅ x (i|k)+ B (k) ⋅ u (i|k)
j
j
j−1
+ D ⋅ x (i|k), j ∈{s, s + 1}
n j−1
s
u s ≤ u (k + h − 1|k) ≤ u s , h = 1, … , M
min max
s
Δu s ≤ Δu (k + h − 1|k) ≤ Δu s , h = 1, … , M
min min
j j j
x ≤ x (k + i|k) ≤ x , i = 1, … , P, j ∈{s, s + 1} (11.27)
min min
where {u s , u s max },{Δu s , Δu s max }, and {x j , x j max }(j ∈ {s, s + 1}) are boundaries of manip-
min min min
ulated variables, increment of manipulated variables, and state vectors, respectively, and
[ s s s ] T
ΔU (k)= Δu (k) Δu (k + 1)· · · Δu (k + M) (11.28)
s
Define that
[ s s s ] T
X (k)= x (k + 1) x (k + 2)· · · x (k + P)
s,n s n s n s n s
[ s s s ] T
U (k)= u (k) u (k + 1) ··· u (k + M) (11.29)
s
If sequences X (k) and U (k) are available to subsystem s, problem (11.27) can
s−1,n s−1 s + 1
∗
be recast as a quadratic program (QP). Then optimal control decision sequence ΔU (k) of
s
the subsystem s can be computed at time instant k by solving (11.27) for the current states.
∗
s
∗
The first sample of U (k)= u (k − 1)I 1×M +ΔU (k) is used to compute the optimal water flux
s
s
set-point of subsystem s according to (11.14).
We remark that model (11.12) is linearized around an operating point, which, in general, is
not an equilibrium point. When evaluating the online computational burden of the proposed
scheme, one needs to account for the resources spent in computing the linear model (11.26)
and translating (11.27) into a standard QP problem. Nevertheless, for the proposed appli-
cation, complexity of problem (11.27) is reduced greatly comparing to the nonlinear model
based MPC.
11.3.5 Iterative Algorithm
According to the neighborhood optimization, the local optimal control decision for each sub-
system can be obtained by solving problem (11.27) if the local optimal control decision of its
downstream neighbors and the future optimal states of its upstream neighbors are available,
that is
{ }
∗
ΔU (k)= arg min J (k)| ∗ ∗ (s = 1, … , N) (11.30)
s
s
U (k)(j∈N −i ,j≠i),X (k)(h∈N +i ,h≠i)
ΔU s (k) j h