Page 62 - Distributed model predictive control for plant-wide systems

36 Distributed Model Predictive Control for Plant-Wide Systems

2.4.4 Feasibility and Stability

2.4.4.1 Feasibility
The main result of this section is that, provided that an initially feasible solution is available, for
f
system (2.35) and at any time k ≥ 1, u(⋅|k) = u (⋅|k) is a feasible control solution to Problem 2.3.
f
f
This feasibility result refers that, at any update k ≥ 1, the control and state pair (u (⋅|k), x (⋅|k))
satisfy the control constraint (2.42) and the terminal state constraint (2.43).
Under Assumption 2.2, we can assume that there is a feasible solution at time k − 1, which
means the existence of the control law u(k − 1 + l − 1|k − 1) and the state x(k − 1 + l|k − 1),
l = 1, 2, … , N. At time instant k, define
{
f
u (k + l − 1|k) = u(k + l − 1|k − 1), l = 1, 2, … , N − 1
(2.44)
f
u (k + N − 1|k)= Kx(k + N − 1|k − 1)
According to (2.35) and (2.44), we have
{
f
x (k + l|k) = x(k + l − 1|k − 1), l = 1, 2, … , N − 1
(2.45)
f
f
x (k + N − 1|k)= A x(k + l − 1|k − 1)= A x (k + l − 1|k)
c
c
Since x(k + l − 1|k − 1), l = 1, 2, … , N, is a feasible solution at time k − 1, it means
x(k + N − 1|k − 1)∈Ω( ).
Considering that (A ) < 1, we have
max c
f 2
‖x (k + N − 1|k)‖
P
2
= ‖A x(k + l − 1|k − 1)‖ P
c
2
≤ ‖x(k + l − 1|k − 1)‖
P
2
≤
f
Thus x (k + N − 1|k) satisfied the terminal constraint (2.43).
In addition, since x(k + N − 1|k − 1) ∈Ω( ), Kx(k + N − 1|k − 1) should be in the set of U by
the definition of .
From above, we can conclude that if there is a feasible solution for Problem 2.3 at time
k − 1, then we can find at least one feasible solution at time k for Problem 2.3. The recursive
feasibility of the dual mode predictive control is guaranteed.
2.4.4.2 Stability

By Algorithm 2.3, 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 2.3, the state of system (2.35) is driven to the set Ω( )in
finite time.
Define the nonnegative function for
N−1
∑ 2 2
2
V(k)= ‖x(k + N|k)‖ + (‖x(k + l|k)‖ + ‖u(k + l|k)‖ ) (2.46)
P Q R
l=0
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) − .