Page 160 - Distributed model predictive control for plant-wide systems

134 Distributed Model Predictive Control for Plant-Wide Systems

The proposed networked DMPC problem stated in Equation (7.17) now can be formulated
as a QP by the introduction of the following matrices:

S = C S ,
i i
i
N = S B
i
i i i
I
⎡ n u i ⎤
′ = ⎢ ⋮ ⎥
i
(M blocks) ⎢ ⎥
⎣ I ⎦
n u i
(7.30)
I ··· 0
⎡ n u i ⎤
= ⎢ ⋮ ⋱ ⋮ ⎥
i
(M×M blocks) ⎢ ⎥
⎣ I ··· I ⎦
n u i n u i
⌢
Q = diag {Q }
i P i
R = diag {R }
i
i
P
Lemma 7.3 (Quadratic program) Under Assumption 7.1, each controller C ,i = 1, … , m,
i
has to solve at time k the following optimization problem:
T
min [ΔU k, M|k)H ΔU (k, M|k)− G(k + 1, P)ΔU (k, M|k)] (7.31)
i
i
i
ΔU(k,M|k) i
where the positive definite matrix H has the form
i
T
H = N Q N + R i (7.32)
i
i
i
i
and
d
T
G (k + 1, P|k)= 2N Q [Y (k + 1, P|k)− Z(k + 1, P|k)] (7.33)
̂
i
i
i
i
with
(1)
′
Z (k + 1, P|k)= S [B u (k − 1)+ A ̂x(k|k)+ A X(k, P|k − 1)
̃ ̂
̂
i
i
i
i
i i i
+B U(k − 1, M|k − 1)] + T C X(k + 1, P|k − 1) (7.34)
̃ ̂
̃
i i i
Making use of these definitions:
[ ]
−1
K = K , = I n u i n u i ×Mn u i , K = H N Q i (7.35)
T
i
i
i
i
i
i
i
The proof can be found in Appendix C. Based on Lemma 7.3, the following theorem can be
deduced.
Theorem 7.1 (Closed-form solution) Under Assumption 7.1, for each controller C ,
i
i = 1, … , n, the closed form of the control law applied at time k at controller C to subsystem
i
S is given by
i
d
u (k)= u(k − 1)+ K [Y (k + 1, P|k)− Z (k + 1, P|k)] (7.36)
̂
i
i
i
i
The proof can be found in Appendix D.