Page 172 - Distributed model predictive control for plant-wide systems

146 Distributed Model Predictive Control for Plant-Wide Systems

where
Q = block-diag(Q , … , Q )
1 m
R = block-diag(R , … , R )
1 m
Q ∈ ℝ n yi ×n yi , R ∈ ℝ n ui ×n ui
i i
The local control decision is computed by solving the local optimization problem J (k) with
i
local input/output variables and constraints in the distributed MPC based on Nash optimality
presented in [29]. The whole system will arrive at Nash equilibrium if the convergent condition
of the algorithm is satisfied. However, the Nash optimal solution to the local optimization
problem collectively is not equal to the global optimal control decision of the whole system,
that is, there exists loss in control performance.
The new performance index for each subsystem can be improved by
∑
min J (k)= J (k) (i = 1, … , m) (7.47)
j
i
ΔU i,M (k)
j∈P i
It is noted that the new performance index for the subsystem S includes cost functions not
i
only by the subsystem S but also by its neighbors, which is called neighborhood optimization.
i
Cooperation between subsystems can be incorporated in the performance index of the local
optimization problem by including cost functions of neighbor subsystems. At the time instant
k, the future predictive output of the ith subsystem can be expressed as
min(M,s)
∑ ∑ s−h
s
̂ y (k + s|k)= C A x (k)+ C A i B Δu (k + h − 1|k)
j
ij
i
i
i i
i
h=1
j∈P i
s = 1, … , P (7.48)
However, all the elements in x (k) are not directly measurable except x (k)= y (k), a pre-
i
i
i1
dictive state observer for each subsystem can be constructed to estimate them as follows:
∑
⎧ ̂ x (k + 1) = A ̂ x (k)+ B Δu (k)+ V (y (k + 1)− ̂ y (k + 1))
i i i ij j i i i
⎪
j∈P i
∑
⎪
⎨ ̂ y (k + 1)= C A ̂ x (k)+ C B Δu (k) (7.49)
i ij
i i
j
i
i
⎪ j∈P i
⎪ i = 1, … , m
⎩
where V is the observer feedback matrix.
i
With the notation ̃ x (k)= x (k)− ̂ x (k), the observer dynamics for each subsystem can be
i
i
i
described as
̃ x (k + 1)=(I − V C )A ̃ x (k) (7.50)
i i i i i
It can be seen that will ̂ x (k) converge to x (k) under a proper selection of V . A more flexible
i
i
i
error correction is naturally derived by employing the predictive state observer, and the stability
for the prediction error correction can be guaranteed by analyzing the observer dynamics.
Furthermore, the selection of the observer feedback matrix for each subsystem is independent
of that of other subsystems.