Page 146 - Distributed model predictive control for plant-wide systems

120 Distributed Model Predictive Control for Plant-Wide Systems

If the system structure is centralized control, the performance index is
2 2
min J(k) =‖w(k) − Y (6.49)
M
PM (k)‖ +‖ u (k)‖ R
Q
The optimal control law is
T
T
u = (A QA + R) − A Q[w(k) − y (k)] (6.50)
M p0
If the system structure is distributed, the input of the ith subsystem is independent from the
inputs of other subsystems. So, we can obtain the performance index based on the plant-wide
optimality:

n
2 2 ∑ 2
min J =‖w (k) − Y i,PM (k)‖ +‖ u i,M (k)‖ + ‖w (k)− Y j,PM (k)‖ (6.51)
i
j
i
Δu l+1 (k) Q i R i Q j
i,M j=1, j≠i
At time k, the predictive equation for the l + 1iteration is
m
∑
y l+1 (k) = y i,P0 (k) + A u l+1 (k) + A u l (k) (6.52)
ii
ij
i,PM i,M j,M
j=1,j≠i
y l+1 (k) = y (k) + A u l+1 (k)
j,PM j,P0 jj i,M
m
+ ∑ A u l (k) + A u l+1 (k) (6.53)
jn n,M ji i,M
n = 1
n ≠ i, n ≠ j
Without the constraints, we can obtain the explicit solution
( )
n (
∑ T ) l+1
A Q A + R Δu (k)
ji j ji i i,M
j=1
{ [ ]}
m m
∑ ∑
T
= A Q w (k) − y (k)− A Δu l (k) (6.54)
ji j j j,p0 jn n,M
j=1 n=1,n≠i
Let
T
D ≜ A QA = D + D
d nd
where D is a diagonal matrix made up of diagonal elements in the matrix D, and D is a
d nd
matrix made up of nondiagonal elements in the matrix D. From Equation (14), we can obtain
the optimal control law
−1
T
Δu l+1 (k)=(D + R) A Q[w(k)− y (k)]
M d p0
l
−1
−(D + R) D Δu (k) (6.55)
d nd M