Page 120 - Distributed model predictive control for plant-wide systems

94 Distributed Model Predictive Control for Plant-Wide Systems

According to Nash optimality, at the sampling time instant k, the Nash optimal solution of the
ith agent can be derived as
[ ]
m
∑
dis
Δu dis (k)= D − ̃ y (k) − G Δu dis (k) (i = 1, … , m) (5.76)
i,M ii i i,P0 ij j,M
j=1, j≠i
If the algorithm is convergent, the Nash optimal solution of the whole system can be written as
dis
dis
dis
Δu (k)= D [ − ̃ y (k)] + D Δu (k) (5.77)
M 1 P0 E M
In the iteration procedure, Equation (5.77) can be expressed as
dis dis
Δu (k)| l+1 = D [ (k)− ̃ y (k)] + (T D T )Δu (k)| (l = 0, 1, …) (5.78)
1
E
r
l
c
P0
M M
At the time instant k, (k) and ̃ y (k) are known in advance; hence, D [ (k) − ̃ y (k)]isthe
P0 1 P0
constant term irrelevant to the iteration. The convergence of expression (5.78) is then equiva-
lent to that of the following:
dis dis
Δu (k)| l+1 =(T D T )Δu (k)| (l = 0, 1, …) (5.79)
E
r
c
l
M M
Therefore, the convergent condition of the distributed linear model predictive control system
under the communication failure is
| (T D T)| < 1.
r E
Remark 5.5 Under the communication failure, each agent cannot exchange information prop-
erly. In the extreme case, T D T = 0, | (T D T)| < 1 is always satisfied, which corresponds
r E c r E
to the full decentralized architecture.
5.3.6 Simulation Results
Consider the Shell heavy oil fractionator benchmark control problem as shown in Figure 5.5.
The heavy oil fractionator is characterized by three product draws and three side circulating
loops. Product specifications for the top and side draw streams are determined by economics
and operating requirements. There is no product specification for the bottom draw, but there
is an operating constraint on the temperature in the lower part of the column. The three cir-
culating loops remove heat to achieve the desired product separation. The heat exchangers in
these loops re-boil columns in other parts of the plant. Therefore, they have varying heat duty
requirements. The bottom loop has an enthalpy controller which regulates heat removal in the
loop by adjusting steam make. Its heat duty can be used as a manipulated variable to control
the column. The heat duties of the other two loops act as disturbances to the column.
Prett and Morari [97] presented a model for a heavy oil fractionator as the benchmark pro-
cess for the Shell standard control problem
y = G(s)u + G (s)d
d
T
where u = [u u u ] are manipulated variables to control the process, u represents the prod-
1
3
1
2
uct draw rate from the top of the column, u represents the product draw rate from the side of
2