Page 246 - Distributed model predictive control for plant-wide systems

220 Distributed Model Predictive Control for Plant-Wide Systems

Proof. Since Problem 8.1 has a solution at time k − 1, by construction, it has
‖ p
‖ ‖ ‖
j,i
‖̂ x (k + N − 1) |k‖ = ‖x (k + N − 1) |k − 1‖
‖ ‖P j ‖ j,i ‖P j

≤ √
2 m
Define the presumed state of S , r ∈ P , in controller C as
r
i
+j
{
̂ x (N + k − 1|k − 1) , r ∉ P i
r
̂ x (k + N − 1|k)= (10.26)
r,i
̂ x (k + N − 1|k), r ∈ P
r,i i
and substitute (10.26) and the definition (10.7) into (10.5) we have
2
‖ ‖
‖̂ x (k + N |k)‖
j,i
‖ ‖P j
2
‖ ‖
‖ ∑ ‖
= A ̂ x (k + N − 1|k) + A ̂ x (k + N − 1|k) ‖
‖
‖ dj j,i
jr r,i
‖
‖ ‖
r∈P +j
‖ ‖ P j
T
T
= ̂ x (k + N − 1|k)A P A ̂ x (k + N − 1|k)
j,i dj j dj j,i
∑
T
T
+ 2̂ x (k + N − 1|k)A P A ̂ x (k + N − 1|k)
j,i dj j jr r,i
r∈P +j
⎛ ⎞
∑ T ∑
+ ⎜ ̂ x (k + N − 1|k) A T ⎟ P A ̂ x (k + N − 1|k)
r,i jr j jr r,i
⎜ ⎟
r∈V ̆ j r∈P +j
⎝ ⎠
2
T
≤ ( max (A A )
dj
4m dj
( 1 1 ) 1
1 ∑ − − 2
T 2 T 2
+ 2 max (A A ) 2 max P r A P A P
jr r
dj j
dj
dj
r∈P +j
)) 1
( ) (
∑ ∑ − 1 − 1 − 1 − 1 2
T
T
+ max P r 2 A P A P 2 max P q 2 A P A P 2
jq q
jr r
jr j
jq j
r∈P +j q∈P +j
2
j
= (10.27)
4m
√
‖
‖
Consequently, to ensure the bound ‖̂ x (k + N |k)‖ ≤ (1 − ) ∕(2 m) holds in all con-
j,i
‖ ‖P j
trollers C , i ∈ P, a sufficient condition is that
i
2
max ≤ (1 − ) , j ∈ P (10.28)
j
j∈P
This completes the proof.