Page 155 - Distributed model predictive control for plant-wide systems

Networked Distributed Predictive Control with Information Structure Constraints 129

Definition 7.1 Neighboring subsystem. Subsystem S interacts with S , and the state outputs
j
i
of subsystem S are affected by subsystem S . In this case, S is called the upstream neighboring
i j i
subsystem of S and S is called the downstream neighboring subsystem of S . S and S are
j i j i j
the neighboring subsystems or neighbors.
Neighborhood of a subsystem. The upstream (downstream)-neighborhood P (P )of
−i
+i
subsystem S is the set of the subscripts of all its upstream (downstream) neighbors:
i
P +i ={S , S |S is an upstream neighbor of S }
i
j
i
j
P ={S , S |S is a downstream neighbor of S }
−i i j j i
The neighborhood P of subsystem S is the set of all its neighbors:
i i
P = P ∪ P
i +i −i
7.2.2.1 Performance Index
For the large-scale system considered here, the global performance index (5.4) can be decom-
posed in terms of the local index J for each subsystem S , i = 1, 2, … , m [101].
i i
P M
∑ 2 ∑ 2
J (k)= ‖ i d i ‖ + ‖ Δu (k + l − 1 |k) ‖ (7.6)
‖̂ y (k + l |k) − y (k + l|k)‖
i
i
‖ ‖Q i ‖ ‖R i
l=1 l=1
The local control decision of S is computed by solving the optimization problem min J (k)
i
i
ΔU(k,M|k)
with local input/output variables and constraints in the distributed MPC based on the state (or
input) estimations of neighbors at time k − 1 [46] or Nash optimality [29].
However, since the state evolution of downstream neighbors of subsystem S is affected by
i
the control decision of subsystem S , see Equation (5.1) (1), the performance of these neighbors
i
may be destroyed by the improper control decision of S in some cases. To solve this problem,
i
the so-called neighborhood optimization [19, 54] is adopted and the performance index is
expressed as
∑
J (k)= J (k)
i
i
j∈N out
i
[ ] (7.7)
P M
∑ ∑ d 2 ∑ 2
= ‖ j j ‖ + ‖ j ‖
‖Δu (k + l − 1 |k)‖
‖̂ y (k + l |k) − y (k + l|k)‖
‖ ‖Q j ‖ ‖R j
j∈N out l=1 l=1
i
Since Δu (k + l − 1|k) (j ∈ P , j ≠ i, l = 1, … , M) is unknown and independent of the
j
−i
control decision of S , Δu (k + l − 1|k − 1) is used to approximate Δu (k + l − 1|k). Then,
j
i
j
Equation (7.7) becomes
P M
∑ ∑ 2 ∑ 2
J (k)= ‖ j d j ‖ + ‖ Δu (k + l − 1 |k) ‖
‖̂ y (k + l |k) − y (k + l|k)‖
i
i
‖ ‖Q j ‖ ‖R i
j∈P −i l=1 l=1
M
∑ ∑ 2
+ ‖ Δu (k + l − 1 |k − 1) ‖
‖ i ‖R j
j∈P −i , j≠i l=1
P M
∑ ∑ ‖ d ‖ 2 ∑ 2
= ‖̂ y (k + l |k) − y (k + l|k)‖ + ‖ Δu (k + l − 1 |k) ‖ + Constant
i
j
j
‖ ‖Q j ‖ ‖R i
j∈P −i l=1 l=1