Page 242 - Distributed model predictive control for plant-wide systems

216 Distributed Model Predictive Control for Plant-Wide Systems

The existence of an > 0 such that Kx ∈ U for all x ∈Ω( ) follows from the fact that P is
positive definite, which implies that the set Ω( ) shrinks to the origin as decreases to zero.
This completes the proof.
In the optimization problem of each subsystem-based MPC, the terminal-state constraint set
for each S can then be set to be
i
√
Ω ( )={x ∈ ℝ n xi ∶ x ≤ ∕ m} (10.11)
‖ ‖
i
i
‖ i‖P i
Clearly, if x ∈Ω ( ) × ··· × Ω ( ), then the decoupled controllers will stabilize the system at
1 m
the origin, since
2
2
x ≤ , ∀i ∈ P
‖ ‖
m
‖ i‖P i
implies that
∑ 2 2
x ≤
‖ ‖
‖ i‖P i
i∈P
which in turn implies that x ∈Ω( ). Suppose at some time k , x (k ) ∈Ω ( ) for every sub-
0 i 0 i
system. Then, by Lemma 8.1, stabilization can be achieved if every C employs its decoupled
i
static feedback controller K x (k) after time instant k .
i i 0
Thus, the objective of each subsystem-based MPC law is to drive the state of each subsystem
S to the set Ω ( ). Once all subsystems have reached these sets, they switch to their decoupled
i
i
controllers for stabilization. Such switching from an MPC law to a terminal controller once the
state reaches a suitable neighborhood of the origin is referred to as the dual mode MPC. For
this reason, the DMPC algorithm we propose in this chapter is a dual mode DMPC algorithm.
In what follows, we formulate the optimization problem for each subsystem-based MPC.
Problem 10.1 Consider subsystem S .Let > 0 be as specified in Lemma 10.1. Let the
i
update time be k ≥ 1. Given x (k), x (k) and ̂ x (k + s|k), s = 1, 2, … , N, and ̂ u (k + s|k),
i
i
i
i
P
s = 0, 2, … , N − 1, find the control sequence u (k + s|k)∶{0, 1, … , N − 1} → U that
i
i
minimizes
∑ ‖ p 2
‖
J (k)= ‖x (k + N |k)‖
i
‖ j,i ‖P j
j∈P i
(10.12)
( )
N−1
∑ ∑ ‖ p ‖ 2 ‖ p ‖ 2
+ ‖x (k + s |k)‖ + ‖u (k + s |k)‖
‖ j,i ‖Q j ‖ i ‖R i
s=0 j∈P i
subject to the following constraints:
s
∑
‖ p ‖
s−l ‖u (k + l|k) − ̂ u (k + l|k)‖
i
l=0 ‖ i ‖2
(10.13)
(1 − )
≤ √ , s = 1, 2, … , N − 1
2 mm
1
s
∑
‖ p ‖
s−l ‖x (k + l|k) − ̂ x (k + l|k)‖
i
‖ i
l=1 ‖2
(10.14)

, s = 1, 2, … , N − 1
≤ √
2 mm
2