Page 251 - Distributed model predictive control for plant-wide systems

Networked Distributed Predictive Control with Inputs and Information Structure Constraints 225

≤ √
m
f
that is, x (k + N |k)∈Ω ( ). This concludes the proof.
i i
n
Theorem 10.1 Suppose Assumptions 10.1–10.3 hold, x(k )∈ ℝ x and constraints
0
f
(10.13)–(10.17) and (10.18) are satisfied at k . Then, for every i ∈ P, the control u (⋅|k)
0
i
f
and state x (⋅|k), respectively, defined by (10.24) and (10.5), are a feasible solution of
j,i
Problem 10.1 at every update k.
Proof. We will prove the theorem by induction.
p f
First, consider the case of k = 1. The state sequence x (⋅|1)= x (⋅|1) trivially satisfies
j,i j,i
the dynamic equation (10.5), the stability constraint (10.16), and the consistency constraints
(10.13)–(10.15).
Observe that
p f
̂ x (1|1)= x (1|0)= x (1|1)= x (1), i ∈ P
j,i
j,i
j,i
j,i
and that
f p
x (1 + s|1)= x (1 + s|0)
j,i j,i
s = 1, 2, … , N − 1
f
Thus, x (N |1)∈Ω ( ∕2). By the invariance of Ω( ) under the terminal controller and the
j,i i
conditions in Lemma 10.1, it follows that the terminal state and control constraints are also
satisfied. This completes the proof of the case of k = 1.
p f
Now suppose u (⋅|l)= u (⋅|l) is a feasible solution for l = 1, 2, … , k − 1. We will show that
i
i
f
u (⋅|k) is a feasible solution at update k.
i
f
As before, the consistency constraint (10.13) is trivially satisfied, and u (⋅|k) is the cor-
j,i
responding state sequence that satisfies the dynamic equation. Since there is a solution for
Problem 10.1 at updates l = 1, 2, … , k − 1, Lemmas 10.2–10.4 can be invoked. Lemma 10.4
guarantees control constraint feasibility. In view of Lemmas 10.2 and 10.3, using the triangle
inequality, we have
‖ f ‖ ‖ f ‖
‖x (k + N |k)‖ ≤ ‖x (k + N |k) − ̂ x (k + N |k)‖
j,i
j,i
j,i
‖ ‖P i ‖ ‖P i
‖ ‖
j,i
+ ‖̂ x (k + N |k)‖ (10.42)
‖ ‖P i
(1 − )
≤ √ + √ = √
2 m 2 m 2 m
for each j ∈ P , i ∈ P. This shows that the terminal-state constraint is satisfied and the proof
i
of Theorem 10.1 is complete.
10.4.2 Stability
The stability of the closed-loop system is analyzed in this subsection.