Page 117 - Distributed model predictive control for plant-wide systems

Local Cost Optimization-based Distributed Model Predictive Control 91

0 or 1. For a block diagonal matrix, the elements of its main diagonal block are all 0s or all 1s.
The value 0 corresponds to the communication failure existed, and 1 for no failure.

Remark 5.4 Here the communication failure is classified into three cases as follows:

• Row failure. In this case, the communication failure happens on the receiving channels. In
this case, the agent cannot receive the information coming from other agents, equivalently
dis
the corresponding row of matrix G becomes 0 and G becomes G , G dis = T G, and the
r
corresponding element of the communication failure matrix T has changed from 1 to 0.
r
• Column failure. In this case, the communication failure happens on the transmitting chan-
nels. In this case, the agent cannot send its information to other agents, equivalently the
dis
corresponding column of matrix G becomes 0 and G becomes G , G dis = GT and the
c
corresponding element of the communication failure matrix T has changed from 1 to 0.
c
• Mixed failure. In this case, both row and column failures exist, the corresponding row and
dis
column of matrix G become 0 and G becomes G .
• G dis = T GT , and the corresponding element of communication failure matrices T and
r c r
T has changed from 1 to 0.
c
With these preliminaries a theorem is presented.
Theorem 5.3 For a distributed system, assume that the prediction horizon and the control
horizon are equal and the communication failure cannot affect the stability. Its performance
at the time instant k under the local communication failure is degrading. The degrading mag-
nitude of the performance index satisfies 0 ≤ ≤ , and the upper bound of this magnitude
max
max is
‖W max ‖
max =
(F)
m
where (F) denotes the minimal eigenvalue of F, with
m
−1
T
W max =[D (I − D )− A] Q[A − A (I − D )]
E
0
E
1
−1
T
+[A − A (I − D )] × Q[D (I − D )− A]
E
E
0
1
T
+[A − A (I − D )] Q[A − A (I − D )]
0 E 0 E
T
T
− D RD − RD − D R
E E E E
−1 T −1
F =[D (I − D )− A] Q[D (I − D )− A]+ R
1 E 1 E
⎡A 11 ⎤
A = ⎢ ⋱ ⎥
0
⎣ A mm⎦
⎢
⎥
Q = block-diag(Q , … , Q )
m
1
R = block-diag(R , … , R )
m
1
Proof. Without loss of generality, take the mixed failure as an example
D dis =−D G dis =−D T GT =−T D GT = T D T
E 1 1 r c r 1 c r E c