Page 186 - Distributed model predictive control for plant-wide systems

160 Distributed Model Predictive Control for Plant-Wide Systems

Appendix B. Proof of Lemma 7.2
By (11), (12), and imposing that

u (k + P − 1|k)= u (k + P − 2|k) = ··· = u (k + M|k)= u (k + M − 1|k)
i
i
i
i
and
⌢ ̂ ⌢ ̂
v (k + P|k − 1)= v (k + P − 1|k − 1)
i
i
⌢ ̂ ⌢ ̂
also substituting W (k, p|k − 1) and V (k, p|k − 1) with their explicit expressions (7.23), it
i
i
results the following stacked state prediction for controller C i
⌢ ̂ ⌢
X (k + 1, P|k)= S [A x (k|k)+ B U (k, M|k)
̂
i
i i
i
i
i
̂
̃
̃
+A X(k, p|k − 1)+ B U(k − 1, M|k − 1)]
i
i1
[ ] T
⌢ ′ T T
Let x (k|k − 1)= ̂ x (k |k − 1) ··· ̂ x (k|k − 1) , and by definitions (7.12), (7.18)
̂
i
i 1
i mi
(7.19), the above equation becomes
(1) (2) ′
⌢ ̂ ⌢
̂
X (k + 1, P|k)= S [A ̂x(k|k)+ A x (k|k − 1)+ B U (k, M|k)
i
i
i
i
i
i
i
(1)
̃
̃
̂
+ A X(k, P|k − 1)+ B U(k − 1, M|k − 1)]
i i
(1) (1) (2)
̂
̃
= S [A ̂x(k|k)+ B U (k, M|k)+(A i + A )X(k, P|k − 1)
̃
i
i
i
i
i
̃
+ B U(k − 1, M|k − 1)]
i
(1)
= S [A ̂x(k|k)+ B U (k, M|k)
i
i
i
i
̃
+ A X(k, P|k − 1)+ B U(k − 1, M|k − 1)]
̃ ̂
i
i
By model (7.9) and the coefficient defined in (7.24), the stacked output prediction for con-
troller C can be expressed as
i
⌢ ̂ ⌢ ̂
Y (k + 1, P|k)= C X (k + 1, P|k)+ T C X(k + 1, P|k − 1).
̃ ̂
i
i
i
i
i
This proves Lemma 7.2.
Appendix C. Proof of Lemma 7.3
Making use of stacked vectors and definitions (7.30), the cost function (7.8) to be minimized
by controller C can be expressed in the equivalent form
i
⌢ ̂ ⌢ d 2 2
i
i
i
i
J = ||Y (k + 1, P|k)− Y (k + 1, P|k)|| + ||ΔU (k, M|k)||
Q i R i
⌢ ̂
The stacked local output prediction Y (k + 1, P|k) is a function of the control action,
i
therefore, in order to express J as a function of the control sequence ΔU (k, M|k),
i i
an explicit expression for such a prediction is needed. Considering that u (k + h|k)=
i
∑ h
u (k − 1)+ r=0 Δu (k + r|k), h = 1, 2, … , M, the local stacked control sequence U (k, M|k)
i
i
i