Page 60 - Distributed model predictive control for plant-wide systems

34 Distributed Model Predictive Control for Plant-Wide Systems

Proof. Under Assumption 2.1, there is a positive definite matrix Q such that
T
A PA + Q = P (2.36)
c c
2
Consider the function V(k)= ‖x(k)‖ . The time difference of V(k) along the trajectories of
P
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
= x (k)(A PA − P)x(k)
c
c
(2.37)
T
≤ −x (k)Qx(k)
≤ 0
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
positive definite, which implies that the set Ω( ) shrinks to the origin as decreases to zero.
This completes the proof.
2.4.2 MPC Formulation

1. Performance Index
More specifically, the performance index of dual mode MPC at each time instant k is
defined as

N−1
∑ 2 2
2
J(k)= ‖x(k + N|k)‖ + (‖x(k + l|k)‖ + ‖u(k + l|k)‖ ) (2.38)
P Q R
l=0
T
T
T
where Q = Q > 0, R = R > 0, and P = P > 0. The matrix P is chosen to satisfy the
Lyapunov equation
T
A PA − P =−Q (2.39)
c c
T
where Q = Q + K RK > 0.
2. Predictive Model
From (2.35), the prediction of the system l-step ahead state can be deduced easily, and is
given as
l
∑ i−1
l
̂ x(k + l + 1|k)= A x(k)+ A Bu(k + l − i|k) (2.40)
i=1
3. Optimization Problem
In the optimization problem of each subsystem-based MPC, the terminal state constraint
set can be set to be x(k + N|k) ∈Ω( ). Suppose that at some time k , x(k ) ∈Ω( ). Then,
0
0
by Lemma 2.1, stabilization can be achieved if the controller employs its static feedback
controller Kx(k) after the time instant k .Thus, the objective of the MPC law is to drive the
0
state to the set Ω( ). Once all states have reached these sets, they switch to the feedback