Page 112 - Distributed model predictive control for plant-wide systems

86 Distributed Model Predictive Control for Plant-Wide Systems

5.3.2 Algorithm

Algorithm 5.2
Step 1: At sampling time instant k, each controller makes initial estimation of the input vari-
ables and announces it to the other controllers; let the iterative index l = 0,
l l l l T
Δu i.M (k)=[Δu (k), Δu (k + 1), … , Δu (k + M − 1)] (i = 1, … , m)
i
i
i
Step 2: Each controller resolves its optimal problem simultaneously to obtain its solution
Δu ∗ (k)(i = 1, … , m)
i,M
Step 3: Each controller checks if its terminal iteration condition is satisfied, that is, for the
given error accuracy (i = 1, … , m), if there exist
i
(l+1) (l)
‖Δu (k)−Δu (k)‖ ≤ i (i = 1, … , m)
i,M i,M
If all the terminal conditions satisfied, then end the iteration and go to Step 4.
Otherwise, let l = l + 1,Δu l (k)= Δu ∗ (k)(i = 1, … , m), all controllers communi-
i,M i,M
cate to exchange this information, and take the latest solution to Step 2;
Step 4: Computes the instant control law
Δu (k)=[I 00 ··· 0]Δu ∗ i,M (k)(i = 1, … , m)
i
and takes them as the controller output of each agent;
Step 5: Move horizon to the next sampling time, that is, k + 1 → k, and go to Step 1, repeat
the above process.

5.3.3 Computational Convergence for Linear Systems

Consider this DMPC of linear dynamic plants. At sampling time instant k, the output prediction
model of the ith agent can be described as
m
∑
̃ y i,PM (k)= y (k)+ A Δu i,M (k)+ A Δu j,M (k)(i = 1, … , m) (5.52)
i,P
ii
ij
j=1, j≠i
where A and A are the dynamic matrix of the ith subsystem and the step response matrix of
ii
ij
the ith subsystem stimulated by the jth subsystem, respectively. They are expressed in terms
of the matrix
a (1) ··· 0
⎡ ij ⎤
⋮ ⋱ ⋮
⎢ ⎥
⎢ ⎥
A = a (M) ··· a (1) ⎥
⎢
ij
ij
ij
⎢ ⎥
⎢ ⋮ ⋮ ⋮ ⎥
⎢ ⎥
⎣a (P) ···
ij
ij a (P − M + 1)⎦
⎡A ··· A
11 1m ⎤
⎢ ⎥
A = ⋮ ⋱ ⋮
⎢ ⎥
⎢ ⎥
A
⎣ m1 ··· A mm⎦