Page 97 - Distributed model predictive control for plant-wide systems

Local Cost Optimization-based Distributed Model Predictive Control 71

5.2.2.2 Predictive Model
To predict the future state of the current subsystem S , the only information that each controller
i
C , i = 1, … , n, needs is the future behavior of subsystems S controlled by the agents C ∈ P .
i j j +i
Similarly, C should broadcast the future behavior of the local variables only to the agents
i
C ∈ P . Then the states and outputs of the downstream neighbors in l-step ahead can be
j −i
predicted by
l
⎧
∑
l
⎪ ̂ x (k + l|k) = A ̂ x (k|k)+ A s−1 B u (k + l − s|k)
i ii i ii ii i
⎪
s=1
⎪
l (5.6)
⎨ ∑ s−1
+ A ̂ w (k + l − s|k − 1)
⎪ i
ii
⎪ s=1
̂ y (k + l|k)= C ̂ x (k + l|k)+ ̂ v (k + l|k − 1)
⎪
⎩ i ii i i
5.2.2.3 Optimization Problem
From above, the optimization problem for each subsystem-based local cost optimization-based
MPC (LCO-MPC) in each control cycle can be concluded as follows.
Problem 5.1 For each independent controller C , i = 1, … , m, the unconstrained LCO-
i
DMPC problem with the prediction horizon P and control horizon M, M < P,attime k solves
the following optimization problem:
P M
∑ d 2 ∑ 2
‖̂ y (k + l |k) − y (k + l|k)‖⌢ +
min J (k)= ‖ i ‖ ‖ Δu (k + l − 1 |k) ‖ (5.7)
i
i
ΔU i (k,M|k) ‖ i ‖Q ‖ ‖R i
l=1 i l=1
for i = 1, … , m, subject to constraints
l
∑
l
̂ x (k + l|k)= A ̂ x (k|k)+ A s−1 B u (k + l − s|k)
i ii i ii ii i
s=1
l
∑ s−1
+ A ̂ w (k + l − s|k − 1) (5.8)
ii i
s=1
̂ y (k + l|k)= C ̂ x (k + l|k)+ ̂ v (k + l|k − l) (5.9)
i
i
ii i
where ΔU (k, M|k) = {Δu (k|k), … , Δu (k + M − 1|k)}.
i
i
i
Each controller C is composed of three parts: an optimizer, a state predictor, and an
i
interaction predictor. At time k, based on the exchanged information, the interaction predictor
of the MPC controller C estimates the future interaction sequence over the prediction horizon
i
ˆ w (k + l − 1|k − 1), l = 1, … , P. Then, combing with the local state measurement of x (k),
i
i
Problem 5.1 can be solved by the optimizer that computes the optimal manipulated variable