Page 209 - Distributed model predictive control for plant-wide systems

Local Cost Optimization Based Distributed Predictive Control with Constraints 183

8.4.2 Stability Analysis of Entire Closed-loop System

The stability of the closed-loop system is analyzed in this subsection.

n
Theorem 8.2 Suppose Assumptions 8.1–8.3 hold, x(k )∈ ℝ x , constraints (8.10), (8.11), and
0
(8.13) are satisfied, and the following parametric condition holds
(N − 1) 1 1
− + < 0 (8.32)
2 2
Then, by application of Algorithm 8.1, the closed-loop system of (8.2) is asymptotically stable
at the origin.

Proof. By Algorithm 8.1 and Lemma 8.1, when x(k) enters Ω( ), the terminal controllers
take over to keep it in there and stabilize the system at the origin. Therefore, it remains to show
that if x(0)∈ X∖Ω( ), then by the application of Algorithm 8.1, the state of system (8.2) is
driven to the set Ω( ) in finite time.
Define the nonnegative function for S
N
∑ p
V(k)= ‖x (k + s|k)‖ P (8.33)
s=1
In what follows, we will show that, for any k ≥ 0, if x(k)∈ X∖Ω( ), then there exists a
constant ∈ (0, ∞) such that V(k) ≤ V(k − 1) − . Constraint (8.12) implies that
‖ f
p
‖
‖x (k + s|k)‖ ≤ ‖x (k + s|k)‖ + (8.34)
P
‖ ‖P N
Therefore,
N
∑ ‖ f
‖
V(k) ≤ ‖x (k + s|k)‖ +
‖ ‖P
s=1
p
Subtracting V(k − 1) from V(k) and using x (k + s|k − 1)= ̂ x(k + s|k), s = 1, 2, … , N − 1,
gives
p
V(k)− V(k − 1) ≤ −‖x (k|k − 1)‖ + ‖ f ‖
P + ‖x (k + N|k)‖
‖ ‖P
N−1
∑ ‖ f
‖
+ (‖x (k + s|k)‖ − ‖̂ x (k + s|k)‖ ) (8.35)
P
‖ ‖P
s=1
Assuming x(k)∈ X∖Ω( ) yields
p
‖x (k|k − 1)‖ > (8.36)
P
Also, by Theorem 8.1 we have
‖ f ‖
‖x (k + N|k)‖ ≤ ∕2 (8.37)
‖ ‖P