Page 219 - Distributed model predictive control for plant-wide systems

Cooperative Distributed Predictive Control with Constraints 193

Switching from MPC to a terminal controller once the state reaches a suitable neighborhood
of the origin is referred to as dual mode MPC [71, 78]. For this reason, the implementation
here is considered a dual mode distributed MPC algorithm.
In what follows, we formulate the optimization problem for each subsystem-based MPC.

Problem 9.1 Consider subsystem S .Let > 0 be as specified in Lemma 9.1. Let the update
i
time be k ≥ 1. Given x(k), and u(k + l | k − 1), l = 1, 2, … , N − 1, find the control sequence
u (k + l | k):{0,1, … , N − 1} → U that minimizes
i i
N−1 ( )
∑
‖
‖
J = ‖̂ x (N|k, i)‖ + ‖̂ x (k + l|k, i)‖ + u (k + l|k) ‖
‖
j
i
‖ i
Q
‖ ‖P ‖R i
l=0
Subject to the constraints in (9.6)
l
∑
‖ u (k + h|k) − u (k + h|k − 1) ‖
l−h‖ i i ‖2
h=0
e
≤ , l = 1, 2, … , N − 1;
m − 1
u (k + l − 1|k)∈ U , l = 0, 1, … , N − 1;
i
i
̂ x(k + N|k, i)∈Ω( )
In the constraints above,
( ) 1
⎛ ( ) T 2 ⎞
l
l
⎜
=max A B PA B ⎟ ,
l max i i
i∈P ⎜ ⎟
⎝ ⎠
l = 0, 1, … , N − 1
(√ )
T
max A A c ≤ 1 −
c
0 < 1 − < 1
The constant 0 < < 1, 0 < < 0.5, and > 0 are design parameters whose value will be
chosen in the sequel.
Equation (9.8) is referred to as the consistency constraints, which requires that each pre-
dictive manipulated variables remain close to the presumed sequence. It is a key equation in
f
proving that x (⋅ |k, i) is a feasible state sequence at each update.
Note that the terminal constraint in each optimal control problem is Ω( ), 0 ≤ < 0.5,
although Lemma 9.1 ensures that the larger terminal set Ω( ) suffices as a collective region of
attraction for the terminal controllers. In the analysis presented in the next section, it is shown
that tightening the terminal set in this way is required to guarantee the feasibility properties.
9.3.2 Constraint C-DMPC Algorithm

Before stating the C-DMPC algorithm, an assumption is made to facilitate the initialization
phase.