Page 178 - Distributed model predictive control for plant-wide systems

152 Distributed Model Predictive Control for Plant-Wide Systems

The performance index of the whole system is
c
T
c
J(k)=[R (k)− Ĝ x(k)− HΔU (k)] Q[R (k)− Ĝ x(k)− HΔU (k)]
P
P
M
M
c
c
T
+(ΔU (k)) RΔU (k)
M M
(7.62)
T
c
c
T
=(ΔU (k)) (H QH + R)ΔU (k)
M M
T
c
T
− 2[H Q(R (k)− Ĝ x(k))] ΔU (k)+ constant
P
M
The integral optimal control decision of the whole system derived from the centralized
optimization is
T
−1
T
c
ΔU (k)=(H QH + R) H Q[R (k)− Ĝ x(k)] (7.63)
M P
which is equal to the result by distributed optimization.
Under the network environment, the capacity of the communication network is assumed to
be sufficient for each subsystem to obtain information from its neighbors, so it is possible for
each subsystem to exchange information several times during it solves its local optimization
problem at the sampling time instant. Furthermore, when the convergent condition is satisfied,
the solution to the local optimization problems collectively will be the global optimal control
decision of the whole system, that is, the coordinated distributed computations solve an equiv-
alent centralized MPC problem. An illustrative example will be provided in Section 7.3.6 to
test the effectiveness of the networked MPC algorithm with neighborhood optimization.
7.3.5 Nominal Stability Analysis for Distributed Control Systems
The state-space model of the whole system can be written as
{
x (k + 1) = Ax(k)+ BΔu(k)
(7.64)
y(k)= Cx(k)
where
[ ] T
T
T
Δu(k)= Δu (k) ··· Δu (k)
1 m
[ T T ] T
y(k)= y (k) ··· y (k)
m
1
A = block-diag(A , … , A )
1 m
{
B , j ∈ ℕ i
ij
B(i, j)=
, j ∉ ℕ
i
C = block-diag(C , … , C )
m
1
The observer dynamics for the whole system can be described as
̃ x(k + 1)=(I − VC)Ã x(k) (7.65)
where
[ T T ] T
̃ x(k)= ̃ x (k) ··· ̃ x (k) ,
m
1
V = block-diag(V , … , V ).
1 m