Page 99 - Distributed model predictive control for plant-wide systems

Local Cost Optimization-based Distributed Model Predictive Control 73

where the diagonal blocks diag {A }, diag {B }, and diag {C } are the zero blocks of
P
ij
ij
P
P
ij
congruent dimensions if C does not belong to the input neighborhood P . Moreover,
j
+i
define that
I
⎡ (M−1)n u i ×n u i (M−1)n u i ⎤
I
⎢ ⎥
⎢ n u i ×(M−1)n u i n u i ⎥
̃
≜ ⎢ ⎥
i
⋮ ⋮
⎢ ⎥
⎣ n u i ×(M−1)n u i I n u i ⎥
⎢
⎦
̃
̃
≜ diag{ , … , }
̃
1 m
(5.13)
̃ ̃ ̃ ̃
B ≜ B ,
i
i
[ ] T
̃ T
̃ T
A ≜ A … A
̃
1 m
[ ] T
̃ T
̃ T
̃
B ≜ B … B (5.14)
1 m
[ ] T
̃ T
̃ T
̃
C ≜ C … C m
1
By representing Equation (5.10) in a stacked form for l = 1, … , P, under Assumption 5.1,
for each controller C , i = 1, … , m, the stacked predictions of the interaction vectors at time k
i
based on the information computed at time k − 1are givenby
{
̃
̃ ̂
W (k, P|k − 1)= A X(k, P|k − 1)+ B U(k − 1, M|k − 1)
̂
i
i
i
(5.15)
V (k, P|k − 1)= C X(k, P|k − 1)
̂
̂ ̂
i
i
and the complete stacked predictions have the following forms:
{
̃
W(k, P|k − 1)= AX(k, P|k − 1)+ BU(k − 1, M|k − 1)
̃ ̂
̂
(5.16)
̂ ̂
̂
V(k, P|k − 1)= CX(k, P|k − 1)
̂
Remark 5.1 In Equation (5.15), the complete stacked vectors X(k, M|k − 1) and U (k − 1,
j
M|k − 1) are defined with the predicted state trajectories of all subsystems and with the
open-loop control sequences of all subsystem-based MPC. In the real distributed implemen-
tation of LCO-DMPC, the zero blocks on the matrices A , B , and C allow us to build the
̃
̃
̃
i i i
complete stacked vectors X(k, P|k − 1) and U(k − 1, M|k − 1). We can just keep the value of
̂
̂
nonzero blocks X (k, P|k − 1) and U (k − 1, M|k − 1) produced by the controller C , j ∈ P ,
j j j +i
̂
̂
and set the other parts of X(k, P|k − 1) and U(k − 1, M|k − 1) except X (k, P|k − 1) and
j
U (k − 1, M|k − 1),j ∈ P , as the real values.
j −i
5.2.3.2 State Prediction
Now, we will introduce how to obtain the state prediction for each controller C , i = 1, … , m.
i
By Equations (5.8), (5.9), and definitions (5.9), and imposing (5.11) and
̂ v (k + p|k − 1)= ̂ v (k + p − 1|k − 1)
i
i