Page 116 - Distributed model predictive control for plant-wide systems

90 Distributed Model Predictive Control for Plant-Wide Systems

change? In this section, the performance deviation on single-step horizon under the commu-
nication failure is discussed. Because MPC takes a receding-horizon control policy in which
the optimization is resolved online at each sampling time with updated measurements, it is
reasonable to focus on the single-step horizon.
To indicate the communication connection among agents, define a connection matrix
E = (e ).
ij
All elements in the main diagonal of E are zeros and other elements in the nonmain diago-
nal of E are 1. 0.1 denotes the communication connection between two agents existed and 0
shows no communication connection. Under the ideal situation of a communication connec-
tion without structural disturbance e = 1(i, j = 1, … , m, i ≠ j), the output prediction model and
ij
the Nash optimal solution of the ith agent at the time instant k can, respectively, be rewritten as
m
∑
̃ y (k)= ̃ y (k)+ A Δu (k)+ e A Δu (k), (i = 1, … , m) (5.64)
i,PM i,P0 ii i,M ij ij j,M
j=1, j≠i
and [ ]
m
∑ ∗
∗
Δu (k)= D − ̃ y (k) − G Δu (k) , (i = 1, … , m) (5.65)
i,M ii i i,P0 ij j,M
j=1, j≠i
where G = EA = [G ] denotes the dot multiplication with
ij
e ··· e A A ··· A
⎡ 12 1m⎤ ⎡ 11 12 1m⎤
e 21 ··· e 2m A 21 A 22 ··· A 2m
⎢ ⎥ ⎢ ⎥
G = ⎢ ⎥ ⎢ ⎥
⎢ ⋮ ⋮ ⋮ ⎥ ⎢ ⋮ ⋮ 0 ⋮ ⎥
⎢ ⎥ ⎢ ⎥
⎣e e ··· 0 ⎦ ⎣A A ··· A ⎦
m1 m2 m1 m2 mm
e A ··· e A
⎡ 12 12 1m 1m⎤
e A 21 ··· e A 2m
⎢ ⎥
2m
21
= ⎢ ⎥
⎢ ⋮ ⋮ 0 ⋮ ⎥
⎢ ⎥
⎣e A e A ··· 0 ⎦
m1 m1 m2 m2
Then the Nash optimal solution of the whole system under convergent computation is
∗
−1
Δu (k)= (I-D ) [ (k)− ̃ y (k)] (5.66)
M E P0
with
−D e A ··· −D e A
⎡ 11 12 12 11 1m 1m⎤
−D e A 21 ··· −D e A 2m
⎢ ⎥
22 21
22 2m
D =−D G = ⎢ ⎥
1
E
⎢ ⋮ ⋮ ⋮ ⎥
⎢ ⎥
⎣−D e A −D e A ··· ⎦
mm m1 m1 mm m2 m2
In the following analysis, assume that the prediction horizon and the control horizon are equal,
and the communication failure is confined within a stable region. To analyze system perfor-
mance deviation, define a communication failure matrix T. The matrix T is a diagonal matrix
or a block diagonal matrix. For a diagonal matrix, define the elements of its main diagonal as