Page 102 - Distributed model predictive control for plant-wide systems

76 Distributed Model Predictive Control for Plant-Wide Systems

5.2.3.4 Solution of Problem 5.1
Then, the explicit solution to the unconstrained LCO-DMPC problem is now stated as a solu-
tion to the equivalent quadratic program. By applying minimum principle to Problem 5.2 yields
Theorem 5.1.

Theorem 5.1 (Closed-loop solution). Under Assumption 5.1, for each controller C ,
i
i = 1, … , m, the explicit form of the control law applied at time k by controller C to the
i
subsystem S is given by
i
d
̂
u (k)= u (k − 1)+ K [Y (k + 1, P|k)− Z (k + 1, P|k)] (5.30)
i
i
i
i
i
where
K ≜ K ,
i i i
[ ]
≜ I n u i n u i ×(m−1)n u i (5.31)
i
T
−1
K ≜ H N Q
i i i i
In addition, the expression of the stacked open-loop optimal control sequence at time k is
d
′
U (k, M|k)= u (k − 1)+ K [Y (k + 1, P|k)− Z (k + 1, P|k)] (5.32)
̂
i 1 i i i i i
and its complete stacked expression is
U(k, m|k)= U(k − 1, m|k − 1)
(5.33)
d
̂
+ ̂ x(k|k)+ X(k, p|k − 1)+ Y (k + 1, p|k)
where
′
′
′
≜ diag{ , … , }
1 m
(5.34)
≜ diag{ , … , }
1 m
S ≜ diag{S , … , S }
m
1
(5.35)
T ≜ diag{T , … , T }
1
m
≜ diag{ K , … , K }
1 1 m m
(5.36)
≜ − SA
̃
̃
≜ − (SA + TC)
(5.37)
′ ′
̃
≜ − S(B + B)
Proof. The proof can be found in Appendix B.
In Equation (5.32), the complete stacked vectors U(k − 1, M|k − 1) built by C are included
i
to calculate the current optimal manipulated variables. Thus, all the U (k − 1, M|k − 1) of
j
its upstream neighborhood subsystems S , j ∈ P +i are used for computing the current
j
closed-loop stacked control sequence U (k, M|k). In addition, the stacked control sequence
i