Page 126 - Distributed model predictive control for plant-wide systems

100 Distributed Model Predictive Control for Plant-Wide Systems

Considering the relationship between the manipulated variable and its increment
h
∑
u (k + h|k)= u (k − 1)+ Δu (k + r|k)
i i i
r=0
using the local stacked control sequence u (k, M|k) together with (5.15) and (5.25)–(5.27), we
i
can obtain the output prediction in the form
̂
̂
Y (k + 1, p|k)= N ΔU (k, m|k)+ Z (k + 1, p|k)
i i i i
By substituting this expression into the local cost function J , the optimization Problem 5.1
i
takes the form (5.24). The positive definiteness of matrices Q and R implies the same property
i
i
for matrix H .
i
Appendix B. Proof of Theorem 5.1
Problem 5.2 states that a solution to the LCO-DMPC problem minimizes the cost function
(5.24) with respect to the control sequence Δu (k, M|k). By the minimum principle, this
i
solution has the form
1 −1
Δu (k, M|k)= H G (k + 1, P|k)
i i i
2
which is, by (5.26), equivalent to
d
ΔU (k, m|k)= K [Y (k + 1, p|k)− Z (k + 1, p|k)]
̂
i i i i
In MPC algorithms, only the first element of the optimal sequence is actually applied to the
process; thus, the control law at the time instant k can be expressed as
u (k)= u (k − 1)+ ΔU (k, m|k)
i
i
i
i
which gives the final explicit form (5.30).
In addition, considering that the local control action applied at time k − 1 is given by
u (k − 1)= U (k − 1, M|k − 1)
i
i
i
the optimal control sequence U (k, M|k) for subsystem S at time k is
i
i
′
U (k, m|k)= U (k − 1, m|k − 1)+ ΔU (k, m|k)
i 1 i i i i
By (5.30), it directly follows (5.32). Then, by (5.27), (5.32), (5.16), and (5.34)–(5.37), the
complete stacked optimal manipulated variable sequence has the form (5.33).

Appendix C. Proof of Theorem 5.2
Combining the process equation (5.3), the feedback control law (5.39), the global prediction
equation (5.23), and the controller equation (5.33), the closed-loop state-space representation