Page 111 - Distributed model predictive control for plant-wide systems

Local Cost Optimization-based Distributed Model Predictive Control 85

It is assumed that the behavior of the whole system is described by m subsystems and the
nonlinear performance function L is decomposable in the distributed system. The local perfor-
mance index for the ith controller can be expressed as
P
∑
min J = L [y (k + j|k)Δu i,M (k|k)] (i = 1, … , m) (5.47)
i
i
i
Δu i,M (k|k)
j=1
where L is the nonlinear function of y (k + j|k) and Δu (k|k). This indicates that the global
i i i,M
performance index of the whole system is
m
∑
min J = J (5.48)
i
i=1
At time instant k, the future predictive output of the ith controller can be expressed as
y (k + j|k)= f (y (k), Δu (k|k), … , Δu (k|k)) ( j = 1, … , P) (5.49)
i i i 1,M m,M
It can be seen that the global performance index can be decomposed into a number of local
performance indexes, but the output of each subsystem is still related to all the input vari-
ables due to the input coupling. Such distributed control problem with different goals can be
resolved by means of the Nash optimal concept [95]. Concretely speaking, the group of con-
N
N
N
trol decisions u (t)={u (t), … , u (t)} is called to be the Nash optimal solution if for all
1 m
u , i = 1, … , m the following relations are held:
i
N
∗
N
N
N
N
J (u , … , u , … , u ) ≤ J (u , … , u N , u , u N , … , u ) (5.50)
i 1 i m i 1 i−1 i i+1 m
If the Nash optimal solution is adopted, each controller does not change its control decision
u because it has achieved the locally optimal objective under the above condition; otherwise,
i
the local performance index J will degrade. Each controller optimizes its objective (local
i
performance index) only using its own control decision assuming that other controllers’ Nash
optimal solutions have been known, that is,
min J | N (5.51)
i u ( j≠i)
u i j
Inspecting Equation (5.51) to obtain the Nash optimal solution u of the ith subsystem, it
i
N
is necessary to know other subsystems’ Nash optimal solutions u ( j ≠ i), so that the whole
j
system could arrive at Nash optimal equilibrium in this coupling decision process. By Nash
optimal equilibrium, the global optimization problem can be decomposed into a number of
local optimization problems.
An iterative algorithm is developed on the basis of the work of Du et al. [89] to seek the Nash
optimal solution of the whole system at each sampling time. Since the mutual communication
and the information exchange are adequately taken into account, each controller resolves local
optimal problem provided that the other subsystem-based MPCs’ optimal solutions have been
known. Then each subsystem-based MPC compares the newly computed optimal solution with
that obtained in last iteration and checks if the terminal condition is satisfied. If the algorithm is
convergent, all the terminal conditions of the m agents will be satisfied, and the whole system
will arrive at Nash equilibrium at this time. This Nash-optimization process will be repeated
at the next sampling time.