Page 98 - Distributed model predictive control for plant-wide systems

72 Distributed Model Predictive Control for Plant-Wide Systems

∗
increments sequence ΔU (k, M|k) over the control horizon. The first element of ΔU (k, M|k),
i
i
Δu (k|k), is selected and u (k) = u (k − 1) +Δu (k|k) is applied as a control input to subsystem
i
i
i
i
S . Finally, the state predictor computes an estimate of the future state trajectory over the
i
prediction horizon by (5.8) and broadcasts it and the optimal control sequence ΔU (k, M|k)
i
over the control horizon to its output neighbors S , j ∈ P .Attime k + 1, the interaction
j −i
prediction part of each controller uses this information to estimate the interaction predictions
ˆ w (k + l − 1|k) and the whole procedure is repeated.
i
It should be noticed that in Problem 5.1, the future interaction sequences are substituted
by the estimate of the future interaction sequence ˆ w (k + l − s|k − 1) and ̂ v (k + l|k − 1) based
i
i
on the information broadcast at time k − 1 from the agents C ∈ P , because at time k,the
j +i
predictions ˆ w (k + l − s|k) and ̂ v (k + l|k) are unknown for the controller C . That is why in
i i i
Equations (5.8) and (5.9) the controller C has the current formation.
i
A detailed description of the proposed LCO-DMPC is provided in the next section where
a closed-loop solution to the LCO-DMPC problem is introduced by the formalization of the
different actions performed by each controller.
5.2.3 Closed-loop Solution
The main result of this section is the computation of an explicit solution to the LCO-DMPC
problem. For this purpose, first the expressions for the interaction prediction and the state
prediction are provided in an explicit form.
5.2.3.1 Interaction Prediction
According to Equation (5.2), let the last (P − M + 1) samples of the control action predictions
Û (k, P|k − 1) are equal to the last element of U (k − 1, M|k − 1). Then, for i = 1, … , m,the
j
j
l-ahead predictions at time k based on the information computed at time k − 1of the interaction
vectors (5.2) can be expressed as
n m
⎧
∑ ∑
⎪ ̂ w (k + l|k − 1) ≜ A ̂ x (k + l|k − 1)+ B u (k + l|k − 1)
i ij j ij j
⎪
j=1( j≠1) j=1( j≠1)
⎪ m
∑ (5.10)
⎨
̂ v (k) ≜
ij j
⎪ i C ̂ x (k + l|k − 1)
j=1( j≠1)
⎪
⎪ l = 1, 2, … , P
⎩
where
u (k − 1 + l|k − 1)= u (k − 1 + M|k − 1)
j
j
(5.11)
l > M
Let us define that
̃
A ≜ [diag {A }· · · diag {A i,i−1 } diag {A i,i+1 }· · · diag {A }]
P
i,1
P
i,m
P
P
i
̃ ̃
B ≜ [diag {B }· · · diag {B } diag {B } ··· diag {B }] (5.12)
i P i,1 P i,i−1 P i,i+1 P i,m
̃
C ≜ [diag {C }· · · diag {C i,i−1 } diag {C i,i+1 }· · · diag {C }]
i,m
i
P
P
P
i,1
P