Page 173 - Distributed model predictive control for plant-wide systems

Networked Distributed Predictive Control with Information Structure Constraints 147

The local optimization problem for each subsystem at the sampling time instant k is
[ ]
P M
∑ ∑ 2 ∑ 2
min J (k)= ‖ j j ‖ + ‖ j ‖
‖r (k + s) − ̂ y (k + s|k)‖
‖Δu (k + h − 1 |k)‖
i
ΔU i,M (k) ‖ ‖Q j ‖ ‖R j
j∈P i s=1 h=1
s.t.
min(M,s)
∑ ∑ s−h
s
̂ y (k + s|k)= C A ̂ x (k)+ C A B Δu (k + h − 1|k), s = 1, … , P (7.51)
i i i i i i ij j
h=1
j∈P i
u min ≤ u (k + h − 1|k) ≤ u max , h = 1, … , M
i
i i
Δu min ≤ Δu (k + h − 1|k) ≤ Δu max , h = 1, … , M
i i i
y min ≤ ̂ y (k + s|k) ≤ y max , s = 1, … , P
i
i i
The above optimization problem can be cast as a quadratic programming (QP) with the opti-
mized vector (derivation process can be seen in Appendix A), which is solved online at each
sampling instant and can be solved by the QP algorithm available in the MATLAB toolbox.
The resulting online QP is
1 T T
min ΔU (k) ΔU i,M (k)+ f (k)ΔU i,M (k)
i
ΔU i,M (k) 2 i,M i
(7.52)
s.t. ΔU (k) ≤ b (k)
i i,M i
T
where = > , f (k), , and b (k) are given in Appendix A.
i
i
i
i
i
The diagram of the MPC unit for each subsystem in a networked MPC scheme is shown in
Figure 7.7, including the subsystem model represented by Equation (7.48), the predictive state
observer expressed in Equation (7.49), and the QP optimizer described in Equation (7.52).
Based on the “predictive state” model representation, the control decision for each subsys-
tem can be derived by using the usual state-space techniques (e.g., state observer and moving
horizon optimization) with the assumption that the local control decisions of its neighbors are
available.
It is assumed that the connectivity of the communication network is sufficient for each sub-
system to obtain information from its neighbors. Regarding the network capacity, we consider
that it is possible for each subsystem to exchange information several times during it solves its
local optimization problem at each sampling time instant, which is an ideal information model
for network communication.
7.3.3 Networked MPC Algorithm
According to neighborhood optimization, the local optimal control decision for each subsys-
tem can be obtained by solving the local problem (7.51) if the local optimal control decision
of its neighbors is available, that is,

{ }
ΔU ∗ (k)= arg ∗ (i = 1, … , m) (7.53)
i
i,M min J (k)| ΔU (k)(j∈P i , j≠i)
ΔU i,M (k) j,M