Page 249 - Distributed model predictive control for plant-wide systems

Networked Distributed Predictive Control with Inputs and Information Structure Constraints 223

p p
Since x (⋅) and u (⋅) satisfy constraints (10.13) and (10.14) for all times l = 1, 2, … , k − 1,
i i
the following equation can be deduced:
‖ f ‖
j,i
‖x (k + s|k) − ̂ x (k + s|k)‖
j,i
‖ ‖P j
(1 − )(1 − ) (1 − )
≤ √ + √ (10.36)
2 m 2 m

= √
2 m
Thus, (10.29) holds for all s = 1, 2, … , N − 1.
In what follows, we prove that (10.29) holds for s = N. Denote the feasible states of
S , r ∈ P , used in controller S ,as
i
r
+j
{
f
x (N + k − 1|k − 1) , r ∉ P
f
x (k + N − 1|k)= r −i
r,i f
x (k + N − 1|k), r ∈ P
r,i −i
f
Then, the discrepancy between the feasible state x (k + N |k) and the presumed state
j,i
̂ x (k + N |k) is
j,i
‖ f ‖
j,i
‖x (k + N |k) − ̂ x (k + N |k)‖
j,i
‖ ‖P j
( )
f
= ‖A dj x (k + N − 1|k) − ̂ x (k + N − |k) (10.37)
j,i
j,i
( )
f
j,j ̆ j,i P j
+A ̆ x (k + N − 1|k) − ̂ x ̆ j,i (k + N − 1|k) ‖
Now consider
‖ f ‖
j,i
j,i
‖x (k + N − 1|k) − ̂ x (k + N − 1|k)‖ ≤ √
‖ ‖P j 2 m
and the constraint
‖ p ‖
‖x (k + N − 1|k − 1) − ̂ x (k + N − 1|k − 1)‖ ≤ √ , ∀j ∈ P
j
‖ j ‖P j
2 m
Then, in view of (10.25), we have
2
‖ f ‖
‖x (k + N |k) − ̂ x (k + N |k)‖
j,i
‖ j,i ‖P j
2
2 (10.38)
≤
j
4m
2
2
≤
4m
This completes the proof of (10.29).
f
In what follows, we will prove that the feasible control u (k + s|k) and the feasible state
i
f
x (k + s|k) satisfy constraints (10.13)–(8.11).
j,i