Page 253 - Distributed model predictive control for plant-wide systems

Networked Distributed Predictive Control with Inputs and Information Structure Constraints 227

Using (10.47)–(10.49) in (10.46) then yields
( )
(N − 1) 1 1
V(k)− V(k − 1) < −1 + + + (10.50)
2 2
which, in view of (10.43), implies that V(k) − V(k − 1) < 0. Thus, for any k ≥ 0, if
x(k)∈ X∖Ω( ), there is a constant ∈ (0, ∞) such that V(k) ≤ V(k − 1) − . It then follows
′
′
that there exists a finite time k such that x(k ) ∈Ω( ). This concludes the proof.
We have now established the feasibility of the N-DMPC and the stability of the resulting
closed-loop system. That is, if an initially feasible solution could be found, subsequent feasi-
bility of the algorithm is guaranteed at every update, and the resulting closed-loop system is
asymptotically stable at the origin.
It should be noticed that, with the increasing number of the downstream neighbors each
subsystem-based controller covers, the cost consumed by communication among subsystems
will become higher and higher, and the network connectivity of the entire system will become
more and more complex. When the time consumed by communicating becomes so large that
it cannot be ignored comparing to the control period, the performance of the entire system will
be more or less negatively affected. And the increasing network connectivity will inevitably
violate the error-tolerance capability of the entire control system. This is undesired in the dis-
tributed control framework. Thus the number of the downstream neighbors of each subsystem
should not be too large when the network bandwidth is limited or not large enough.
It should also be noticed that a general mathematical formulation is adopted in the N-DMPC
algorithm and its analysis. The N-DMPC and the resulting analysis can be used for any coor-
dination policy mentioned in Section 10.2 with a redefinition of P . Thus it provides a unified
i
framework for the DMPCs, which adopts the cost function based coordination strategies. This
is a very important contribution of the chapter. In the next section, we will present the formu-
lations under other coordination strategies.

10.5 Formulations Under Other Coordination Strategies
10.5.1 Local Cost Optimization Based DMPC
In this coordination strategy, each subsystem-based MPC minimizes its own cost. Redefine
P ={i}. Note that u (⋅) and x (⋅) are both nonexistent, that is, P is an empty set, and
i j,i j,i −i
= P . Consequently the optimization problem of the stabilizing DMPC, where each
thus P̃ i +i
subsystem-based MPC takes the local cost as the performance index, can be derived in the
framework of N-DMPC as
2
‖ p ‖
J (k)= J (k)= ‖x (k + N |k)‖
i i ‖ i ‖P i
(10.51)
N−1 ( )
∑ ‖ p ‖ 2 ‖ p ‖ 2
+ ‖x (k + s|k)‖ + ‖u (k + s − 1|k)‖
‖ i ‖Q i ‖ i ‖R i
s=1
subject to the constraints
s
p s ∑ s−l p
x (k + s|k)= A x (k|k)+ A u (k + l − 1|k)
i i i i i
l=1