Page 144 - Distributed model predictive control for plant-wide systems

118 Distributed Model Predictive Control for Plant-Wide Systems

6.3.1 Formulation

Considering a complicated plant wide control of n subsystems, the j step ahead output predic-
tion of the whole system at the time k is

Y(k + j) = f(y (k) , u (k), … , u (k)) (6.41)
m
0
1
where j = 1, 2, … , P, Δu (k) is the input vector of M dimensions, Y(k) and Y (k) are predictive
i i 0
output in the future and initial predictive output at the time k, respectively, f is the mapping
function vector. The input and output all have to meet the constraints:
Δu ≤ Δu(⋅) ≤ Δu , u ≤ u(⋅) ≤ u
min max min max
Y ≤ Y(⋅) ≤ Y
min max
u(k + 1)= u(k)+Δu(k)
The performance index is
P
∑
min J = L[Y(k + j|k), Δu (k), … , Δu (k)] (6.42)
n
1
Δu 1 (k),…,Δu m (k)
j=1
This part proposes a distributed MPC algorithm, considering the coupling relationship from
other subsystems to a single subsystem. The predictive equation for the ith (i = 1, 2, … , m)
subsystem is
y (k)= f [y (k), Δu (k), … , Δu (k)] (6.43)
i i i,0 1 m
The existing distributed MPC algorithms use the characteristics of the additivity of the
global performance index (6.42) to distribute the global performance index to every subsystem
controller. For the ith subsystem, the performance index based on the Nash optimality is

P
∑
J = L [y (k + j|k), Δu i,M i (k)] (6.44)
i
i
i
j=1
Every subsystem controller contains a part of the global performance index (6.42). As a
result, the control law will not converge to the optimal solution of the centralized control algo-
rithm and there exists some deviation.
This section proposes a distributed predictive control algorithm based on plant-wide opti-
mality. For the ith subsystem, the optimal performance index is up to the global performance
index (6.42). During the process of computing the control law of the ith subsystem, we need
to know the current control law of other subsystems which have the coupling input with the
ith subsystem. Supposing that every subsystem can communicate many times during a control
period, at the time k,for the l + 1th iteration, the performance index is

l+1 l+1 l
min J = L [y i (k + j|k), Δu i (k), Δu (k), j = 1, 2, … , m, j ≠ i]
i
i
j
Δu l+1 (k)
i
P
∑ l+1 l+1 l
+ L [y j (k + j|k), Δu i (k), Δu (k), n = 1, 2, … , m, n ≠ i] (6.45)
j
n
j=1, j≠i