Page 202 - Distributed model predictive control for plant-wide systems

176 Distributed Model Predictive Control for Plant-Wide Systems

Equation (8.12) will be utilized to prove that the LCO-DMPC described by Problem 8.1 is
stabilizing, where > 0 is a design parameter whose value will be specified later to satisfy
f
f
Lemma 8.1, x (k + s|k) is a feasible state sequence, and x (k + s|k) equals to the solution of
i i
(8.4) under the initial state of x (k), the presumed state of ̂ x (k + s|k), j ∈ P and the feasible
i j +i
f
control sequence u (k + s − 1|k) is defined by
i
{
p
u (k + s − 1|k − 1) , s = 1, 2, … , N − 1
i
f
u (k + s − 1|k)= (8.17)
i f
K x (k + N − 1|k), s = N
i i
It should be noticed that the terminal constraint in each optimal control problem is Ω ( /2),
i
although Lemma 8.1 ensures that the larger Ω( ) suffices for the feasibility of the terminal
controllers. In the analysis presented in the next section, it will be shown that tightening the
terminal set in this way is required to guarantee the feasibility properties.
8.3.2 Algorithm Design for Resolving Each Subsystem-based Predictive
Control
Before stating the constraint LCO-DMPC algorithm, we make the following assumption to
facilitate the initialization phase.

Assumption 8.3 At initial time k , there exists a feasible control u (k + s|k )∈ U ,
0 i 0 0 i
s = 1, 2, … , N − 1, for each i ∈ P, such that the solution to the full system x(s + 1 + k ) =
0
p p
Ax(s + k ) + Bu(s + k |k ), denoted as x (⋅|k ), satisfies x (N + k )∈Ω ( ∕2) and results in a
0 0 0 i 0 i 0 i
bounded cost J (k ). Moreover, each subsystem has access to u (⋅|k ).
i 0 i 0
Assumption 8.3 bypasses the difficult task of actually constructing an initially feasible solu-
tion in a distributed way. In fact, finding an initially feasible solution for many optimization
problems is often a primary obstacle, whether or not such problems are used in a control set-
ting. As such, many centralized implementations of MPC also assume that an initially feasible
solution is available [69].
Under the Assumption 8.3, we can get the algorithm of the networked cooperative DMPC.

Algorithm 8.1 Constraint DMPC algorithm The dual mode DMPC law for any S is con-
i
structed as follows:

Step 1: Initialization.
• Initialize x(k ), u (k + s|k ), s = 1, 2, … , N, to satisfy
i
0
0
0
• Assumption 8.3
• At time k ,if x(k ) ∈Ω( ), then apply the terminal controller u (k) = K (x (k)), for all
0
i
i
i
0
k ≥ k ,else
0
• Compute ̂ x (k + s + 1|k + 1) according to (8.4) and transmit ̂ x (k + s + 1|k + 1) to S ,
0
j
0
i
0
0
i
j ∈ P .
−i
Step 2: Communicating at time k + 1
• Measure x (k), transmit x (k), ̂ x (k + s + 1|k),to S , j ∈ P , and receive x (k), ̂ x (k + s|k)
j
j
i
i
i
−i
j
from S , j ∈ P .
j
+i