Page 119 - Distributed model predictive control for plant-wide systems

Local Cost Optimization-based Distributed Model Predictive Control 93

with
−1 T T −1
W =[D (I − D )− A] Q(A − A S) + (A − A S) Q[D (I − D )− A]
1 E 1 E
T
T
T
+ (A − A S) Q(A − A S) + (S − I) R(S − I) + R(S − I) + (S − I) R
Then
∗ 2 ∗T ∗ ∗ 2
‖Δu (k)‖ ≤ Δu (k)‖W‖Δu (k)= ‖W‖‖Δu (k)‖
M W M M M
‖W‖ ∗ 2 ‖W‖ ∗
≤ ‖Δu (k)‖ = J
(F) M F (F)
m
m
Here, (F) is the minimal eigenvalue of F. From the above derivations, the performance rela-
m
tionship between the communication failure free and communication failure can be expressed
as
[ ]
∗
∗
∗
J dis ≤ J + ‖W‖ J = 1 + ‖W‖ J =(1 + )J ∗ (5.73)
(F) (F)
m
m
where = ‖W‖/ (F) denotes the degrading magnitude of the performance index under the
m
local communication failure.
dis
Inspection of (5.72) shows that ‖W‖ depends on G dis and D , while G dis and D dis are
E E
affected by the communication failure matrix T and T . So in the case of all existed com-
r c
munication failures, ‖W‖ can arrive at the maximal value, at this time, T D T = 0, G dis =
r E c
0, D dis = 0, A = A , S = I − D , and
E
0
E
T
−1
W max =[D (I − D )− A] Q[A − A (I − D )] + [A − A (I − D )] T
0
E
0
E
E
1
−1
T
× Q[D (I − D )− A]+[A − A (I − D )] Q[A − A (I − D )]
E
0
0
E
E
1
T
T
− D RD − RD − D R
E E E E
Therefore, the upper bound of the performance deviation under the local communication fail-
ure is
‖W max ‖
=
max
(F)
m
Theorem 5.4 The convergent condition of the distributed linear model predictive control sys-
tem under the communication failure is | (T D T)| < 1, where D , T , and T are the same
E
r
E
r
c
as defined before.
Proof. The output prediction model of the ith agent under the communication failure at the
time instant k can be described as
m
∑ dis dis
dis
dis
̃ y = ̃ y (k)+ A Δu (k)+ G Δu (k)(i = 1, … , m) (5.74)
i,PM i,P0 ii i,M ij j,M
j=1, j≠i
The local performance index for the ith agent can be expressed as
min J dis = ‖ (k)− ̃ y dis (k)‖ 2 + ‖Δu dis (k)‖ 2 (i = 1, … , m) (5.75)
i i i,PM Q i i,M R i