Page 245 - Distributed model predictive control for plant-wide systems

Networked Distributed Predictive Control with Inputs and Information Structure Constraints 219

occurs synchronously only when x(k) ∈Ω( ), with Ω( ) being as defined in Lemma 10.1. In
the next section, it will be shown that the N-DPC policy drives the state x(k + s)to Ω( )ina
finite number of updates. As a result, if Ω ( ) is chosen sufficiently small, then MPC can be
i
employed for all time without switching to a terminal controller, eliminating the need of the
local controllers to access the full state. Of course, in this case, instead of asymptotic stability
at the origin, we can only drive the state toward the small set Ω( ).
The analysis in the next section shows that the N-DMPC algorithm is feasible at every update
and is stabilizing.

10.4 Analysis
10.4.1 Feasibility
The main result of this section is that, provided that an initially feasible solution is avail-
p f
able and Assumption 10.3 holds true, for any S and at any time k ≥ 1, u (⋅|k)= u (⋅|k)
i
i
i
is a feasible control solution to Problem 10.1. This feasibility result refers that, for
f
f
any S and at any update k ≥ 1, the control and state pair (u (⋅|k), x (⋅|k)), j ∈ P sat-
i i j,i i
isfy the consistency constraints (10.13)–(10.15), the control constraint (10.17), and the
terminal-state constraint (10.18). Lemma 10.2 identifies sufficient conditions that ensure
′
′
̂ x (k + N |k)∈Ω ( ∕2), where = (1 − ) . Lemma 10.2 identifies sufficient conditions
j,i i
√
‖ f
‖
that ensure ‖x (s + k|k) − ̂ x (s + k|k)‖ ≤ ∕(2 m) for all j ∈ P , i ∈ P. Lemma 10.3
j,i
i
‖ j,i ‖P j
establishes that the control constraint is satisfied. Finally, Theorem 10.1 combines the results
in Lemmas 10.2–10.4 to arrive at the conclusion that, for any j ∈ P , i ∈ P, the control and
i
f
f
state pair (u (⋅|k), x (⋅|k)) are a feasible solution to Problem 8.1 at any update k ≥ 1.
j,i i
Lemma 10.2 Suppose Assumptions 10.1–10.3 hold and x(k )∈ X. For any k ≥ 0, if Problem
0
10.1 has a solution at time k − 1 and ̂ x (k + N − 1|k − 1)∈Ω ( ∕2) for any j ∈ P ,i ∈ P,
j,i
j
i
then
̂ x (k + N − 1|k)∈Ω ( ∕2)
j
j,i
and
′
̂ x (k + N |k)∈Ω ( ∕2)
j,i j
provided that Q and P satisfy
̂
j
j
max( ) ≤ (1 − ) 2 (10.25)
j
j∈P
′
where =(1 − ) , and
( 1 ) 1
1 ∑ 1 − 2
T T − T 2
= (A A )+ 2 (A A ) 2 P 2 A P A P
j max dj dj max dj dj max r dj j jr r
r∈P ̆ j
( ( 1 )
∑ ∑ − T − 1
+ P 2 A P A P 2
max r jr j jr r
r∈P +j q∈P +j
( 1 1 )) 1
− T − 2
× max P q 2 A P A P 2
jq j
jq q