Page 225 - Distributed model predictive control for plant-wide systems

Cooperative Distributed Predictive Control with Constraints 199

Theorem 9.1 Suppose Assumptions 9.1 and 9.2 hold, x(k )∈ X and Equations (9.8), (9.9),
0
f
f
and (9.10) are satisfied at k . Then, for every i ∈ P, the control u (⋅|k) and state x (⋅ | k), defined
0
i
by (9.13), (9.14) and (9.16), are feasible solutions to Problem 9.1 at every update k ≥ 1.
f
Proof. First consider the case of k = 1. The state sequence ̂ x(⋅|1, i)= x (⋅|1, i) trivially sat-
isfies the dynamic equation (9.14) and the consistency constraint (9.8).
Now, suppose is a feasible solution for t = 1, … , k − 1. Lemmas 9.2–9.4 can be invoked.
The consistency constraints (9.8) are trivially satisfied, the feasibility of control constraint and
the terminal state constraint is guaranteed, and the proof of Theorem 9.1 is completed.

9.4.2 Stability

The stability of the closed-loop system is now analyzed in this subsection.
Theorem 9.2 Suppose that Assumptions 9.1 and 9.2 hold, x(k )∈ X, and Equations (9.8),
0
(9.9), and (9.10) are satisfied at k , and the following parametric conditions hold
0
′
− (0.42 +((N − 1) + 1) ) > 0
where

( 1 1 ) 1
− − 2
= min P 2 QP 2
) 1
( 1 1
′ − − 2
= P 2 QP 2
max
Then, by application of Algorithm 9.1, the closed-loop system (9.2) is asymptotically stabilized
to the origin.

Proof. By Algorithm 9.1 and Lemma 9.1, if x(k) ∈Ω( ) for any k ≥ 0, the terminal con-
trollers take over and stabilize the system to the origin. Therefore, it remains to show that if
x(k )∈ X∖Ω( ), then by application of Algorithm 9.1, the closed-loop system (9.2) is driven
0
to the set in finite time.
m
∑
Define the nonnegative function V for all system S, V = V , and
k
k
k,i
i=1
V
k,i = ‖̂ x (k + N|k, i)‖ P
N−1 ( )
∑
+ ‖̂ x (k + l|k, i)‖ + u (k + l|k) ‖
‖
Q ‖ i ‖R i
l=0
In what follows, we will show that for any k ≥ 0, if x(k)∈ X∖Ω( ), then there exists a con-
stant ∈ (0, ∞) such that V ≤ V − .
k k − 1