Page 200 - Distributed model predictive control for plant-wide systems

174 Distributed Model Predictive Control for Plant-Wide Systems

It, along with Assumption 8.1, helps with the design of the terminal set. This assump-
tion quantifies how strengthening the coupling among subsystems is sufficient so that the
overall system can be stabilized by the proposed DMPC here. This assumption is not nec-
essary, and some systems that do not satisfy this assumption may also be stabilized by the
proposed DMPC, and the more relaxing condition is still remaining to be designed in the
future work.

Lemma 8.1 Under Assumptions 8.1 and 8.2, for any positive scalar c the set

Ω(c)={x ∈ ℝ n x ∶ ‖x‖ ≤ c}
P
is a positive invariant region of attraction for the closed-loop system x(k + 1) = A x(k). Addi-
c
tionally, there exists a small enough positive scalar such that Ω( ) is in the feasible input set
U ∈ ℝ n u for all x ∈Ω( ).
2
Proof. Consider the function V(k)= ‖x (k)‖ . The time difference of V(k) along the trajec-
P
tories of the closed-loop system x(k + 1) = A x(k) can be evaluated as
c
T
T
T
ΔV(k)= x (k)A PA x(k)− x (k)Px(k)
c
c
T
T
T
T
T
= x (k)(A PA − P + A PA + A PA + A PA )x(k)
d d o o o d d o
T
1 T
≤ −x (k)Qx(k)+ x (k)Qx(k)
̂
̂
2
≤ 0 (8.7)
which holds for all x(k) ∈Ω(c)\{0}. This implies that all trajectories of the closed-loop system
that starts inside Ω (c) will remain inside and converge to the origin.
The existence of an > 0 such that Kx ∈ U for all x ∈Ω( ) follows from the fact that P is
a 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} (8.8)
‖ ‖
i
i
‖ i‖P i
Clearly, if x ∈Ω ( ) ×· · · ×Ω ( ), then the decoupled controllers will stabilize the system
m
1
at 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 that at some time k , x (k ) ∈Ω ( ) for every sub-
i
0
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 .
0
i i