Page 218 - Distributed model predictive control for plant-wide systems

192 Distributed Model Predictive Control for Plant-Wide Systems

T
where Q = Q + K RK, and
̂
R = block-diag{R , R , … , R }
m
2
1
Since every subsystem-based controller updates synchronously, the control sequences S ,
j
j ∈ P are unknown to subsystem S . Thus, at the time instant k, presume the control sequence
i i
of S , j ∈ P be the optimal control sequence calculated by C at time k − 1 concatenated with
i
j
j
the feedback control law, that is
[
u (k|k − 1) , u (k + 1|k − 1) ,
j j
]
… , u (k + N − 2|k − 1) , K ̂ x(k + N − 1|k − 1, j)
j j
Then, the predictive model in the MPC for S can be expressed as
i
l
∑
l
̂ x(k + l|k, i)= A x(k)+ A l−h B u (k + h − 1|k)
i i
h=1
l
∑ ∑ l−h
+ A B u (k + h − 1|k − 1)
j j
h=1
j∈P i
where, for ∀ i and j ∈ P ,
i
[
B = n ui ×Σ j < i n xj B n ui ×Σ j > i n xj ] T
i i
In addition, to enlarge the feasible region, a terminal-state constraint is included in each
subsystem-based MPC. The terminal-state constraint set should guarantee that the terminal
controllers are stabilizing inside it.
n
Lemma 9.1 Under Assumption 9.1, for any positive scalar c the set Ω(c)={x ∈ ℝ x ∶ ‖x‖ P
≤ c} is a positive invariant region of attraction for the closed-loop system x(k + 1) = A x(k).
c
Additionally, there exists a small enough positive scalar such that Kx is in the feasible input
n
set U ⊂ ℝ u , for all x ∈Ω( ).
Proof. From Assumption 9.1, for all x(k) ∈Ω(c)\{0}, the closed-loop system x(k + 1) =
A x(k) is asymptotically stable. This implies that all trajectories of the closed-loop system that
c
start inside Ω(c)will remain inside and converge to the origin with P satisfying the Lyapunov
equation.
The existence of an Ω(c) > 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 S can then be set to be
Ω( )={x ∈ ℝ |‖x‖ ≤ }
n x
P
Suppose at some time k , x(k ) ∈Ω( ) for every subsystem. Then, by Lemma 9.1,
0
0
stabilization can be achieved if every C , i ∈ P employs its static feedback controller K x (k)
i
i i
for all time k ≥ k .
0
Thus, the objective of the MPC law is to drive state of all subsystem to the set Ω( ). Once all
subsystems have reached this set, they switch to their decoupled controllers for stabilization.