Page 222 - Distributed model predictive control for plant-wide systems

196 Distributed Model Predictive Control for Plant-Wide Systems

ˆ
X(k+S | k–1,I)
f
x (k+s|k)
Ω(αε)
k+N Ω(ε)
Ω (αε′)
k+N

δ
x(k–1)

x(k)
Figure 9.1 Schematic of the discrepancy among feasible state sequence and presumed state sequence

Lemma 9.2 Suppose Assumptions 9.1 and 9.2 hold and x(k )∈ X, for any k ≥ 0, if
0
Problem 9.1 has a solution at every update time 0, … ,k − 1, then
f
‖x (k + l|k)− ̂ x(k + l|k, i)‖ ≤
P
where 0 < < 1 is a design parameter, for ∀i ∈ P, j ∈ P , and all l ∈ {1, … , N} . In addition,
i
f
u (k + l − 1|k),l = 1, 2, … ,N − 1 satisfies the constraint (9.8).
i
Proof. First we will prove (9.18), provided there is a solution at update time
0, 1, 2, … , k − 1. Substitute (9.13) into (9.14) and consider that
∑
x(k)= Ax(k − 1)+ B u (k − 1|k − 1)
i i
i∈P
for any l = 1, 2, … , N − 1, the feasible state is given by

f
x (k + l|k)
l
= A l+1 x(k − 1)+ A B u(k − 1|k − 1)
i
l
∑ l−h f
+ A B u (k + h − 1|k)
i i
h=1
l
∑ ∑ l−h
+ A B u (k + h − 1|k − 1)
j j
h=0
j∈P i
l
∑
= A l+1 x(k − 1)+ A l−h B u (k + h − 1|k − 1)
i i
h=0
l
∑ ∑ l−h
+ A B u (k + h − 1|k − 1)
j j
h=0
j∈P i