Page 145 - Distributed model predictive control for plant-wide systems

Cooperative Distributed Predictive Control 119

where the first item is the local performance index in some existing research; the second
item considers the impact to the global performance by computing the local optimal problem.
The y l+1 (k) and y l+1 (k) can be concluded by the following predictive equation:
i j
y l+1 (k)= f [y (k), Δu l+1 (k), Δu l+1 (k), j = 1, 2, … , m, j ≠ i] (6.46)
i i i,0 i i
l+1 l+1 l
y (k)= f [y (k), Δu (k), Δu (k), n = 1, 2, … , m, n ≠ i] (6.47)
j j j,0 i n
There is a trick in Equation (6.47) that we use other subsystems’ control law and the ith
subsystem in the last iteration to represent the unknown other subsystems’ predictive output
l+1
y (k).
j

6.3.2 Algorithm

The algorithm can be concluded as follows.
Algorithm 6.1

Step 1. At time k, each subsystem transmits the initial predictive value y (k) to other subsys-
i,0
tems and receives the initial predictive values y (k)(j = 1, 2, … , m, j ≠ i) from other
j,0
subsystems. Then it sends the estimator of the control law to other subsystems. Let
the iteration number l = 0.
Step 2. Each subsystem computes the optimal problem in parallel with last optimal control
laws from other subsystems, performance index, and the predictive equations are
(6.45)–(6.47). The constraints are the same as mentioned before. We can get the
optimal solution Δu l+1 (k)(i = 1, 2, … , m) at the iteration.
i
Step 3. Check the convergence condition of all the subsystems. If the precision
(i = 1, 2, … , m) of all the subsystems meets the condition
i
l
‖Δu l+1 (k)+Δu (k)‖ ≤
i i i
∗
we can get Δu (k)=Δu l+1 (k)(i = 1, 2, … , m) and this iteration ends, and it goes to
i i
l
Step 4; otherwise, let Δu (k)=Δu l+1 (k)(i = 1, 2, … , m), l = l + 1 and go to Step 2
i i
Step 4. Compute the control law at time k
Δu (k)=[I0 ··· 0]Δu ∗ i,M (k)i = 1, 2, … , m
i
Step 5. Let k + 1 → k and go to Step 1.
During the optimal control time, each subsystem can get a local MPC control law without
considering the coupling relationship between subsystems. The local MPC control law will be
taken as the iteration initial value.

6.3.3 The DMPC Algorithm Based on Plant-Wide Optimality
Without generality, the control horizons of all subsystems are the same as M. The whole system
predictive output model is
Y PM (k) = Y (k) + A u (k) (6.48)
M
P0